CRC Encyclopedia Of Mathematics

Kiss Surface

Kissing Number 989

Steiner TRIPLE SYSTEMS of order 3 and 9 are Kirkman
triple systems with n = and 1, Solution to Kirkman's
Schoolgirl Problem requires construction of a Kirk-
man triple system of order n = 2.

Ray-Chaudhuri and Wilson (1971) showed that there ex-
ists at least one Kirkman triple system for every NON-
NEGATIVE order n. Earlier editions of Ball and Cox-
eter (1987) gave constructions of Kirkman triple systems
with 9 < v < 99. For n = 1, there is a single unique (up
to an isomorphism) solution, while there are 7 different
systems for n = 2 (Mulder 1917, Cole 1922, Ball and
Coxeter 1987).

see also Steiner Triple System

References

Abel, R. J. R. and Furino, S. C. "Kirkman Triple Systems,"
§1.6.3 in The CRC Handbook of Combinatorial Designs
(Ed. C. J. Colbourn and J. H. Dinitz). Boca Raton, FL;
CRC Press, pp. 88-89, 1996.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 287-
289, 1987.

Cole, F. N. Bull. Amer. Math. Soc. 28, 435-437, 1922.

Kirkman, T\ R Cambridge and Dublin Math. J. 2, 191-204,
1947.

Lindner, C. C. and Rodger, C. A. Design Theory, Boca
Raton, FL: CRC Press, 1997.

Mulder, P. Kirkman- Systemen. Groningen Dissertation. Lei-
den, Netherlands, 1917.

Ray-Chaudhuri, D. K. and Wilson, R. M. "Solution of Kirk-
man's Schoolgirl Problem." Combinatorics, Proc. Sympos.
Pure Math., Univ. California, Los Angeles, Calif., 1968
19, 187-203, 1971.

Ryser, H. J. Combinatorial Mathematics. Buffalo, NY:
Math. Assoc. Amer., pp. 101-102, 1963.

Kiss Surface

The QUINTIC SURFACE given by the equation

iz 5 + iz 4 -(y 2 + * 2 ) = 0.

References

Nordstrand, T. "Surfaces." http : //www . uib . no /people/
nfytn/surf aces. htm.

Kissing Circles Problem

see Descartes Circle Theorem, Soddy Circles

Kissing Number

The number of equivalent Hyperspheres in n-D which
can touch an equivalent HYPERSPHERE without any in-
tersections, also sometimes called the NEWTON NUM-
BER, Contact Number, Coordination Number, or
LlGANCY. Newton correctly believed that the kissing
number in 3-D was 12, but the first proofs were not pro-
duced until the 19th century (Conway and Sloane 1993,
p. 21) by Bender (1874), Hoppe (1874), and Giinther
(1875). More concise proofs were published by Schutte
and van der Waerden (1953) and Leech (1956). Exact
values for lattice packings are known for n = 1 to 9 and
n = 24 (Conway and Sloane 1992, Sloane and Nebe).
Odlyzko and Sloane (1979) found the exact value for
24-D.

The following table gives the largest known kissing num-
bers in Dimension D for lattice (L) and nonlattice (NL)
packings (if a nonlattice packing with higher number ex-
ists). In nonlattice packings, the kissing number may
vary from sphere to sphere, so the largest value is given
below (Conway and Sloane 1993, p. 15). An more exten-
sive and up-to-date tabulation is maintained by Sloane
and Nebe.

D

L

NL

D

L

NL

1

2

13

>918

> 1, 130

2

6

14

> 1,422

> 1,582

3

12

15

> 2, 340

4

24

16

>4,320

5

40

17

> 5, 346

6

72

18

> 7, 398

7

126

19

> 10, 668

8

240

20

> 17,400

9

272

>306

21

> 27,720

10

> 336

> 500

22

> 49, 896

11

>438

>582

23

> 93, 150

12

> 756

> 840

24

196,560

The lattices having maximal packing numbers in 12- and
24-D have special names: the Coxeter-Todd Lattice
and LEECH Lattice, respectively. The general form of
the lower bound of n-D lattice densities given by

V>

2 n-l

where £(ra) is the RlEMANN Zeta FUNCTION, is known
as the Minkowski-Hlawka Theorem.
see also Coxeter-Todd Lattice, Hermite Con-
stants, HYPERSPHERE PACKING, LEECH LATTICE,
Minkowski-Hlawka Theorem

References

Bender, C. "Bestimmung der grossten Anzahl gleich Kugeln,
welche sich auf eine Kugel von demselben Radius, wie die
iibrigen, auflegen lassen." Archiv Math. Physik (Grunert)
56, 302-306, 1874.

Conway, J. H. and Sloane, N. J. A. "The Kissing Number
Problem" and "Bounds on Kissing Numbers." §1.2 and
Ch. 13 in Sphere Packings, Lattices, and Groups, 2nd ed.
New York: Springer- Verlag, pp. 21-24 and 337-339, 1993.

990

Kite

Klein-Beltrami Model

Edel, Y.; Rains, E. M.; Sloane, N. J. A. "On Kissing Numbers

in Dimensions 32 to 128." Electronic J. Combinatorics 5,

No. 1, R22, 1-5, 1998. http://www.combinatorics.org/

Volume_5/v5iltoc.html.
Giinther, S. "Ein stereometrisches Problem." Archiv Math.

Physik 57, 209-215, 1875.
Hoppe, R, "Bemerkung der Redaction." Archiv Math.

Physik. (Grunert) 56, 307-312, 1874.
Kuperberg, G. "Average Kissing Numbers for Sphere Pack-
ings." Preprint.
Kuperberg, G. and Schramm, O. "Average Kissing Numbers

for Non-Congruent Sphere Packings." Math. Res. Let. 1,

339-344, 1994.
Leech, J. "The Problem of Thirteen Spheres." Math. Gaz.

40, 22-23, 1956.
Odlyzko, A. M. and Sloane, N. J. A. "New Bounds on the

Number of Unit Spheres that Can Touch a Unit Sphere in

n Dimensions." J. Combin. Th. A 26, 210-214, 1979.
Schiitte, K. and van der Waerden, B. L. "Das Problem der

dreizehn Kugeln." Math. Ann. 125, 325-334, 1953.
Sloane, N. J. A. Sequence A001116/M1585 in "An On-Line

Version of the Encyclopedia of Integer Sequences."
Sloane, N. J. A. and Nebe, G. "Table of Highest Kissing

Numbers Presently Known." http://www.research.att.

com/~njas/lattices/kiss .html.
Stewart, I. The Problems of Mathematics, 2nd ed. Oxford,

England: Oxford University Press, pp. 82-84, 1987,

Kite

see Diamond, Lozenge, Parallelogram, Penrose
Tiles, Quadrilateral, Rhombus

Klarner-Rado Sequence

The thinnest sequence which contains 1, and whenever
it contains x } also contains 2x 7 3x + 2, and Qx 4- 3: 1,2,
4, 5, 8, 9, 10, 14, 15, 16, 17, . . . (Sloane's A005658).

References

Guy, R. K. "Klarner-Rado Sequences." §E36 in Unsolved

Problems in Number Theory, 2nd ed. New York: Springer-

Verlag, p. 237, 1994.
Klarner, D. A. and Rado, R. "Linear Combinations of Sets of

Consecutive Integers." Amer. Math. Monthly 80, 985-989,

1973.
Sloane, N. J. A. Sequence A005658/M0969 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Klarner's Theorem

An a x b Rectangle can be packed with 1 x n strips
Iff n\a or n\b.

see also Box- Packing Theorem, Conway Puz-
zle, de Bruijn's Theorem, Slothouber-Graatsma
Puzzle

References

Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., p. 88, 1976.

Klein's Absolute Invariant

(Cohn 1994), where q = e iirt is the Nome, \{q) is the
Elliptic Lambda Function

X(q)~k 2 (q)

Mi)

Mq)

J(q) =

4 [l-A(g) + A 2 (g)] 3
27 \*(q)[l-\(q)]*

[E 4 (q)] 3

$i(q) is a Theta FUNCTION, and the Ei(q) are
Ramanujan-Eisenstein Series. J(t) is Gamma-
Modular.

see also ELLIPTIC LAMBDA FUNCTION, j-FUNCTION,

Pi, Ramanujan-Eisenstein Series, Theta Func-
tion

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in

Analytic Number Theory and Computational Complexity.

New York: Wiley, pp. 115 and 179, 1987.
Cohn, H. Introduction to the Construction of Class Fields.

New York: Dover, p. 73, 1994.
1$ Weisstein, E. W. "j-Function." http: //www. astro.

virginia.edu/-eww6n/math/notebooks/jFunct ion. m.

Klein-Beltrami Model

The Klein-Beltrami model of HYPERBOLIC GEOMETRY
consists of an Open Disk in the Euclidean plane whose
open chords correspond to hyperbolic lines. Two lines I
and m are then considered parallel if their chords fail to
intersect and are PERPENDICULAR under the following
conditions,

1. If at least one of I and m is a diameter of the Disk,
they are hyperbolically perpendicular Iff they are
perpendicular in the Euclidean sense.

2. If neither is a diameter, I is perpendicular to m Iff
the Euclidean line extending / passes through the
pole of m (defined as the point of intersection of the
tangents to the disk at the "endpoints" of ra).

There is an isomorphism between the Poincare Hy-
perbolic Disk model and the Klein-Beltrami model.
Consider a Klein disk in Euclidean 3-space with a
Sphere of the same radius seated atop it, tangent at the
Origin. If we now project chords on the disk orthog-
onally upward onto the Sphere's lower Hemisphere,
they become arcs of CIRCLES orthogonal to the equator.
If we then stereographically project the Sphere's lower
Hemisphere back onto the plane of the Klein disk from
the north pole, the equator will map onto a disk some-
what larger than the Klein disk, and the chords of the
original Klein disk will now be arcs of CIRCLES orthog-
onal to this larger disk. That is, they will be Poincare
lines. Now we can say that two Klein lines or angles are
congruent iff their corresponding Poincare lines and an-
gles under this isomorphism are congruent in the sense
of the Poincare model.

see also HYPERBOLIC GEOMETRY, POINCARE HYPER-
BOLIC Disk

[E4q)] B - [E Q (q) 2

Klein Bottle
Klein Bottle

A closed NONORIENTABLE SURFACE of GENUS one hav-
ing no inside or outside. It can be physically realized
only in 4-D (since it must pass through itself without
the presence of a Hole). Its TOPOLOGY is equivalent
to a pair of CROSS-CAPS with coinciding boundaries. It
can be cut in half along its length to make two Mobius
Strips.

The above picture is an Immersion of the Klein bottle in
M 3 (3-space). There is also another possible IMMERSION
called the "figure-8" IMMERSION (Geometry Center).

The equation for the usual Immersion is given by the
implicit equation

(x 2 + y 2 +z 2 + 2y- l)[{x 2 + y + z 2 - 2y - l) 2 - Sz 2 }
+ 16xz(x 2 -f y 2 + z 2 - 2y - 1) = (1)

(Stewart 1991). Nordstrand gives the parametric form

x = cos u[cos(^u) (y/2 + cos v) + sin(^u) sin v cos v]

(2)

y = sinu[cos(^u)(y/2 + cosv) + sin(^u) sin t; cos v]

(3)
z = — sin(|n)(\/2 4- cosv) + cos(|u) sin v cos v. (4)

Klein Quartic 991

The image of the Cross-Cap map of a TORUS centered
at the Origin is a Klein bottle (Gray 1993, p. 249).

Any set of regions on the Klein bottle can be colored
using ss colors only (Franklin 1934, Saaty 1986).

see also Cross-Cap, Etruscan Venus Surface, Ida
Surface, Map Coloring Mobius Strip

References

Dickson, S. "Klein Bottle Graphic," http:// www .

maths our ce . com/ cgi- bin /Math Source /Applications /

Graphics/3D/0201-801.
Franklin, P. "A Six Colour Problem." J. Math. Phys. 13,

363-369, 1934.
Geometry Center. "The Klein Bottle." http://www.geom.

umn.edu/zoo/topt3rpe/klein/.
Geometry Center. "The Klein Bottle in Four-Space."

http : // www . geom . umn . edu / - banchof f / Klein4D /

Klein4D.html.
Gray, A. "The Klein Bottle." §12.4 in Modem Differential

Geometry of Curves and Surfaces. Boca Raton, FL: CRC

Press, pp. 239-240, 1993.
Nordstrand, T. "The Famed Klein Bottle." http://www.uib.

no/people/nfytn/kleintxt.htm.
Pappas, T. "The Moebius Strip & the Klein Bottle." The

Joy of Mathematics. San Carlos, CA: Wide World Publ./

Tetra, pp. 44-46, 1989.
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:

Assaults and Conquest. New York: Dover, p. 45, 1986.
Stewart, I. Game, Set and Math. New York: Viking Penguin,

1991.
Wang, P. "Renderings." http : //www . ugcs . caltech . edu/

-peterw/portf olio/renderings/.

Klein's Equation

If a REAL curve has no singularities except nodes and
Cusps, Bitangents, and Inflection Points, then

n + 2t2 4- 1 — m + 25' 2 4- k ,

where n is the order, r is the number of conjugate tan-
gents, il is the number of REAL inflections, m is the
class, 6' is the number of Real conjugate points, and
k! is the number of Real Cusps. This is also called
Klein's Theorem.

see also PLUCKER'S EQUATION

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 114, 1959.

The "figure-8" form of the Klein bottle is obtained by
rotating a figure eight about an axis while placing a twist
in it, and is given by parametric equations

Klein Four- Group

see VlERGRUPPE

x{u,v) = [a + cos(iu) sin(t;) - sin(£u) sin(2u)] cos(<z) Klein-Gordon Equation

(5)
y(u,v) — [a + cos(^u) sin(v) — sin(^u) sin(2v)] sin(w)

(6)
z(u,v) — sin(|ii) sin(v) + cos(|u) sin(2t;) (7)

for u £ [0, 2tt), v G [0, 2tt) ; and a > 2 (Gray 1993).

1 d 2 jj

c 2 dt 2

d 2 ^J

dx 2

VV.

see also Sine-Gordon Equation, Wave Equation

Klein Quartic

The 3-holed TORUS.

992

Klein's Theorem

Knights Problem

Klein's Theorem

see Klein's Equation

Kleinian Group

A finitely generated discontinuous group of linear frac-
tional transformation acting on a domain in the COM-
PLEX Plane.

References

Iyanaga, S. and Kawada, Y. (Eds,). Encyclopedic Dictionary

of Mathematics. Cambridge, MA: MIT Press, p. 425, 1980.
Kra, I. Automorphic Forms and Kleinian Groups. Reading,

MA: W. A. Benjamin, 1972.

Kloosterman's Sum

5(zx, v,n) = > ^exp

2ni(uh + vh)

(i)

where h runs through a complete set of residues RELA-
TIVELY Prime to n, and h is defined by

hh = 1 (mod n) . (2)

If (n, ri) = 1 (if n and ri are Relatively Prime), then

S(u, v,n)S(uyv\n) — S(u y vri + v'n ,nri). (3)

Kloosterman's sum essentially solves the problem intro-
duced by Ramanujan of representing sufficiently large
numbers by QUADRATIC FORMS ax\ 2 + bx 2 2 + cx 3 2 +
dx 2 , Weil improved on Kloosterman's estimate for Ra-
manujan's problem with the best possible estimate

Knapsack Problem

Given a Sum and a set of WEIGHTS, find the WEIGHTS
which were used to generate the SUM. The values of
the weights are then encrypted in the sum. The system
relies on the existence of a class of knapsack problems
which can be solved trivially (those in which the weights
are separated such that they can be "peeled off" one at
a time using a GREEDY-like algorithm), and transfor-
mations which convert the trivial problem to a difficult
one and vice versa. Modular multiplication is used as
the Trapdoor Function. The simple knapsack sys-
tem was broken by Shamir in 1982, the Graham-Shamir
system by Adleman, and the iterated knapsack by Ernie
Brickell in 1984.

References

Coppersmith, D. "Knapsack Used in Factoring." §4.6 in

Open Problems in Communication and Computation (Ed.

T. M. Cover and B. Gopinath). New York: Springer-

Verlag, pp. 117-119, 1987.
Honsberger, R. Mathematical Gems III. Washington, DC:

Math. Assoc. Amer., pp. 163-166, 1985.

Kneser-Sommerfeld Formula

Let J u be a Bessel Function of the First Kind, N u
a Neumann Function, and >, n the zeros of z~ v J„(z) in
order of ascending REAL PART. Then for < x < X < 1

and R[z] > 0,

H^^[J v {z)N v (Xz) - N u {z)J v (Xz)]

=£

Jv(jv i nX)Jv(jv,nX)

\S(u,u,n)\ < 2\fn

(4)

(Duke 1997).

see also Gaussian Sum

References

Duke, W. "Some Old Problems and New Results about Quad-
ratic Forms." Not. Amer. Math. Soc. 44, 190-196, 1997.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, p. 56, 1979.

Katz, N. M. Gauss Sums, Kloosterman Sums, and Mon-
odromy Groups. Princeton, NJ: Princeton University
Press, 1987.

Kloosterman, H. D. "On the Representation of Numbers in
the Form ax 2 +by 2 + cz 2 + eft 2 ." Acta Math. 49, 407-464,
1926.

Ramanujan, S. "On the Expression of a Number in the Form
ax 2 + by 2 + cz 2 + du 2 ." Collected Papers. New York:
Chelsea, 1962.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary

of Mathematics. Cambridge, MA: MIT Press, p. 1474,

1980.

Knights Problem

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

The problem of determining how many nonattacking
knights K(n) can be placed on an n x n CHESSBOARD.
For n = 8, the solution is 32 (illustrated above). In
general, the solutions are

rsf \ J i n n > 2

*>> = ( }(„» + !) „>1

2 even
odd,

Knights of the Round Table

Knight's Tour 993

giving the sequence 1, 4, 5, 8, 13, 18, 25, ... (Sloane's
A030978, Dudeney 1970, p. 96; Madachy 1979).

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

Kt

The minimal number of knights needed to occupy or
attack every square on an n x n CHESSBOARD is given
by l, 4, 4, 4, 5, 8, 10, ... (Sloane's A006075). The
number of such solutions are given by 1, 1, 2, 3, 8, 22,
3, ... (Sloane's A006076).

see also BISHOPS PROBLEM, CHESS, KINGS PROBLEM,

Knight's Tour, Queens Problem, Rooks Problem

References

Dudeney, H. E. "The Knight-Guards." §319 in Amusements
in Mathematics. New York: Dover, p. 95, 1970.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 38-39, 1979.

Moser, L. "King Paths on a Chessboard." Math. Gaz. 39,
54, 1955.

Sloane, N. J. A. Sequences A030978, A006076/M0884, and
A006075/M3224 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. Extended entry for M3224 in
The Encyclopedia of Integer Sequences. San Diego: Aca-
demic Press, 1995.

Vardi,T. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 196-197, 1991.

Wilf, H. S. "The Problem of Kings." Electronic J. Combi-
natorics2, 3, 1-7, 1995, http://www.combinatorics.org/
Volume_2/volume2 . html#3.

Knights of the Round Table

see Necklace

Knight's Tour

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A knight's tour of a Chessboard (or any other grid)
is a sequence of moves by a knight CHESS piece (which
may only make moves which simultaneously shift one
square along one axis and two along the other) such
that each square of the board is visited exactly once
(i.e., a HAMILTONIAN Circuit). If the final position is
a knight's move away from the first position, the tour is
called re-entrant. The first figure above shows a knight's
tour on a 6 x 6 CHESSBOARD. The second set of figures
shows six knight's tours on an 8 x 8 CHESSBOARD, all
but the first of which are re-entrant. The final tour has
the additional property that it is a SEMIMAGIC SQUARE
with row and column sums of 260 and main diagonal
sums of 348 and 168.

Lobbing and Wegener (1996) computed the number
of cycles covering the directed knight's graph for an
8x8 Chessboard. They obtained a 2 , where a =
2,849,759,680, i.e., 8,121,130,233,753,702,400. They
also computed the number of undirected tours, obtain-
ing an incorrect answer 33,439,123,484,294 (which is not
divisible by 4 as it must be), and so are currently redoing
the calculation.

The following results are given by Kraitchik (1942). The
number of possible tours on a 4& x 4fc board for k = 3,
4, . . . are 8, 0, 82, 744, 6378, 31088, 189688, 1213112,
. . . (Kraitchik 1942, p. 263). There are 14 tours on the
3x7 rectangle, two of which are symmetrical. There are
376 tours on the 3x8 rectangle, none of which is closed.
There are 16 symmetric tours on the 3x9 rectangle and
8 closed tours on the 3 x 10 rectangle. There are 58
symmetric tours on the 3 x 11 rectangle and 28 closed
tours on the 3 x 12 rectangle. There are five doubly
symmetric tours on the 6x6 square. There are 1728
tours on the 5x5 square, 8 of which are symmetric.
The longest "uncrossed" knight's tours on an nxn board
for n = 3, 4, . . . are 2, 5, 10, 17, 24, 35, ... (Sloane's
A003192).

see also Chess, Kings Problem, Knights Problem,
Magic Tour, Queens Problem, Tour

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 175—
186, 1987.

994

Knodel Numbers

Knot

Chartrand, G. "The Knight's Tour." §6.2 in Introductory
Graph Theory. New York: Dover, pp. 133-135, 1985.

Gardner, M. "Knights of the Square Table." Ch. 14 in Math-
ematical Magic Show: More Puzzles, Games, Diversions,
Illusions and Other Mathematical Sleight- of- Mind from
Scientific American. New York: Vintage, pp. 188-202,
1978.

Guy, R. K. "The n Queens Problem." §C18 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 133-135, 1994.

Kraitchik, M. "The Problem of the Knights." Ch. liin Math-
ematical Recreations. New York: W. W. Norton, pp. 257-
266, 1942.

Madachy, J. S. Madachy } s Mathematical Recreations. New
York: Dover, pp. 87-89, 1979.

Ruskey, F. "Information on the n Knight's Tour Problem."
http: //sue . esc .uvic . ca/~cos/inf /misc/Knight .html.

Sloane, N. J. A. Sequences A003192/M1369 and A006075/
M3224 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

van der Linde, A. Geschichte und Literatur des Schachspiels,
Vol 2. Berlin, pp. 101-111, 1874.

Volpicelli, P. "Soluzione completa e generale, mediante la ge-
ometria di situazione, del problema relativo alle corse del
cavallo sopra qualunque scacchiere." Atti della Reale Ac-
cad, dei Lincei 25, 87-162, 1872.

Wegener, I. and Lobbing, M. "The Number of Knight's
Tours Equals 33,439,123,484,294— Counting with Binary
Decision Diagrams." Electronic J. Combinatorics 3,
R5, 1-4, 1996. http: //www. combinatorics. org/Volume^3/
volume3 . html#R5.

Knodel Numbers

For every k > 1, let C k be the set of COMPOSITE num-
bers n > k such that if 1 < a < n, GCD(a, n) = 1
(where GCD is the Greatest Common Divisor), then
a n ~ k = 1 (mod n). C\ is the set of Carmichael Num-
bers. Makowski (1962/1963) proved that there are in-
finitely many members of Ck for k > 2.

see also Carmichael Number, L>-Number, Great-
est Common Divisor

References

Makowski, A. "Generalization of Morrow's D-Numbers." Si-
mon Stevin 36, 71, 1962/1963.

Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, p. 101, 1989.

Knot

A knot is defined as a closed, non-self-intersecting curve
embedded in 3-D. A knot is a single component Link.
Klein proved that knots cannot exist in an Even-
numbered dimensional space > 4. It has since been
shown that a knot cannot exist in any dimension > 4.
Two distinct knots cannot have the same Knot COM-
PLEMENT (Gordon and Luecke 1989), but two Links
can! (Adams 1994, p. 261). The Knot Sum of any
number of knots cannot be the Unknot unless each
knot in the sum is the UNKNOT.

Knots can be cataloged based on the minimum num-
ber of crossings present. Knots are usually further bro-
ken down into PRIME KNOTS. Knot theory was given
its first impetus when Lord Kelvin proposed a theory
that atoms were vortex loops, with different chemical

elements consisting of different knotted configurations
(Thompson 1867). P. G. Tait then cataloged possible
knots by trial and error.

Thistlethwaite has used Dowker NOTATION to enumer-
ate the number of PRIME KNOTS of up to 13 crossings,
and Alternating Knots up to 14 crossings. In this
compilation, MIRROR Images are counted as a single
knot type. The number of distinct PRIME KNOTS N(n)
for knots from n = 3 to 13 crossings are 1, 1, 2, 3, 7, 21,
49, 165, 552, 2176, 9988 (Sloane's A002863). Combining
Prime Knots gives one additional type of knot each for
knots six and seven crossings.

Let C(n) be the number of distinct PRIME Knots of
n crossings, counting Chiral versions of the same knot
separately. Then

|(2"

1) < N(n) <S e n

(Ernst and Summers 1987). Welsh has shown that the
number of knots is bounded by an exponential in n.

A pictorial enumeration of PRIME KNOTS of up to 10
crossings appears in Rolfsen (1976, Appendix C). Note,
however, that in this table, the PERKO PAIR 10i 6 i and
IO162 are actually identical, and the uppermost crossing
in IO144 should be changed (Jones 1987). The fcth knot
having n crossings in this (arbitrary) ordering of knots
is given the symbol n^. Another possible representation
for knots uses the Braid Group. A knot with n + 1
crossings is a member of the Braid Group n. There
is no general method known for deciding whether two
given knots are equivalent or interlocked. There is no
general Algorithm to determine if a tangled curve is a
knot. Haken (1961) has given an ALGORITHM, but it is
too complex to apply to even simple cases.

If a knot is Amphichiral, the "amphichirality" is A =
1, otherwise A = (Jones 1987). Arf Invariants
are designated a. Braid WORDS are denoted b (Jones
1987). Conway's Knot Notation C for knots up to 10
crossings is given by Rolfsen (1976). Hyperbolic volumes
are given (Adams, Hildebrand, and Weeks 1991; Adams
1994). The Braid Index % is given by Jones (1987). Al-
exander Polynomials A are given in Rolfsen (1976),
but with the Polynomials for 10 83 and lOose reversed
(Jones 1987). The Alexander Polynomials are nor-
malized according to Conway, and given in abbreviated
form [ai, <i2, . . . for a\ + ai{x~ + x) + . . ..

The Jones Polynomials W for knots of up to 10
crossings are given by Jones (1987), and the Jones
POLYNOMIALS V can be either computed from these, or
taken from Adams (1994) for knots of up to 9 crossings
(although most POLYNOMIALS are associated with the
wrong knot in the first printing). The JONES POLYNO-
MIALS are listed in the abbreviated form {n} ao ai ... for
t"~ Tl (ao + ait + . . .), and correspond either to the knot
depicted by Rolfsen or its Mirror Image, whichever

Knot

Knot 995

has the lower POWER of t" 1 . The HOMFLY POLY-
NOMIAL P(l,m) and Kauffman Polynomial F(a,x)
are given in Lickorish and Millett (1988) for knots of up
to 7 crossings.

M, B. Thistlethwaite has tabulated the HOMFLY
Polynomial and Kauffman Polynomial F for
Knots of up to 13 crossings.

4i

5i

6i 62 63

7 6 7 7

87 8 8

82 83 84

7 2 7 3 7 4 7 5

85

316

*16

9 6 9 7

9 3 6 9 3 7

8l7 818

820

9i 9 2 9 3 9 4 9s

9 9 9 10

9i3 9i4 9is

9i8 9i9 9 2 o

921 9 2 2 923 924 9 2 5

927 928 9 2 9 930

9 3 i 9 3 2 9 3 3 9 3 4 9 3 5

939 940

3n 812 813 814 81

996 Knot

Knot

941 §42 943 ^44 945

9 4 6 9 4 7 948 9 4 9

10i 10 2 10 3 10 4 10 5

10 6 10 7 10 8 10 9 IO10

IO11 IO12 IO13 IO14 IO15

lOie IO17 lOis IO19 IO20

IO21 10 22 10 23 IO24 IO25

10 26 10 27 IO28 10 29 IO30

IO31 IO32 IO33 IQ34 IO35

IO41 IO42 IO43 IO44 IO45

10 4 6 IO47 10 48 IO49 IO50

IO51 IO52 IO53 IO54 IO55

10 56 IO57 10 5 8 IO59 10 60

10ei 10 6 2 10 6 3 10 6 4 10 6 5

10 66 10 6 7 1068 1069 10 7

IO71 IO72 IO73 1074 IO75

10 7 6 IO77 1078 IO79 1080

10 8 i 10 82 10 8 3 10 8 4 10 85

10 3 6 IO37 IO38 1039 10 40

10 8 6 1087 10 8 8 1089 1090

Knot

Knot 997

io 9 i

io 92

1093

1094

1095

10 96

IO97

10 9 8

IO99

IO100

10i36

10i37

10i40

IO166
see also ALEXANDER POLYNOMIAL, ALEXANDER'S

Horned Sphere, Ambient Isotopy, Amphichiral,
Antoine's Necklace, Bend (Knot), Bennequin's
Conjecture, Borromean Rings, Braid Group,
Brunnian Link, Burau Representation, Chefalo
Knot, Clove Hitch, Colorable, Conway's Knot,
Crookedness, Dehn's Lemma, Dowker Notation,

FlGURE-OF-ElGHT KNOT, GRANNY KNOT, HlTCH, IN-

vertible Knot, Jones Polynomial, Kinoshita-
Terasaka Knot, Knot Polynomial, Knot Sum,
Linking Number, Loop (Knot), Markov's The-
orem, Menasco's Theorem, Milnor's Conjec-
ture, Nasty Knot, Pretzel Knot, Prime Knot,
Reidemeister Moves, Ribbon Knot, Running
Knot, Schonflies Theorem, Shortening, Signa-
ture (Knot), Skein Relationship, Slice Knot,
Slip Knot, Smith Conjecture, Solomon's Seal
Knot, Span (Link), Splitting, Square Knot,
Stevedore's Knot, Stick Number, Stopper Knot,
Tait's Knot Conjectures, Tame Knot, Tangle,
Torsion Number, Trefoil Knot, Unknot, Un-
knotting Number, Vassiliev Polynomial, White-
head Link

998

Knot

Knot Diagram

References

Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 280-286, 1994.

Adams, C; Hildebrand, M.; and Weeks, J. "Hyperbolic In-
variants of Knots and Links." Trans. Amer. Math. Soc. 1,
1-56, 1991.

Anderson, J. "The Knotting Dictionary of Kannet." http://
www . netg , se/- j an/knopar /english/ index . htm.

Ashley, C. W. The Ashley Book of Knots. New York:
McGraw-Hill, 1996.

Bogomolny, A. "Knots " http : //www. cut— the-knot. com/

do_you_know/knots .html.

Conway, J. H. "An Enumeration of Knots and Links."
In Computational Problems in Abstract Algebra (Ed.
J. Leech). Oxford, England: Pergamon Press, pp. 329-
358, 1970.

Eppstein, D. "Knot Theory." http: //www . ics . uci . edu/~
eppste in/ junkyard/knot .html.

Eppstein, D. "Knot Theory." http : //www . ics . uci . edu/
-eppste in/ junkyard/knot/.

Erdener, K.; Candy, C; and Wu, D. "Verification and Ex-
tension of Topological Knot Tables." ftp://chs.cusd.
claremont . edu/pub/knot/FinalReport . sit .hqx.

Ernst, C. and Sumner, D. W. "The Growth of the Number of
Prime Knots." Proc. Cambridge Phil. Soc. 102, 303-315,
1987.

Gordon, C. and Luecke, J. "Knots are Determined by their
Complements." J. Amer. Math. Soc. 2, 371-415, 1989.

Haken, W. "Theorie der Normalflachen." Acta Math. 105,
245-375, 1961.

Kauffman, L. Knots and Applications. River Edge, NJ:
World Scientific, 1995.

Kauffman, L. Knots and Physics. Teaneck, NJ: World Sci-
entific, 1991.

Lickorish, W. B. R. and Millett, B. R. "The New Polynomial
Invariants of Knots and Links." Math. Mag. 61, 1-23,
1988.

Livingston, C. Knot Theory. Washington, DC: Math. Assoc.
Amer., 1993.

Praslov, V. V. and Sossinsky, A. B. Knots, Links, Braids and
3- Manifolds: An Introduction to the New Invariants in
Low- Dimensional Topology. Providence, RI: Amer. Math.
Soc, 1996.

Rolfsen, D. "Table of Knots and Links." Appendix C in
Knots and Links. Wilmington, DE: Publish or Perish
Press, pp. 280-287, 1976.

"Ropers Knots Page." http://huizen.dds.nl/-erpprs/
kne/kroot .htm.

Sloane, N. J. A. Sequences A002863/M0851 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.

Stoimenow, A. "Polynomials of Knots with Up to 10 Cross-
ings." Rev. March 16, 1998. http://www.informatik.hu-
berlin . de/ -st oimeno/poly . ps.

Suber, O. "Knots on the Web." http://www.earlham.edu/
suber /knot link. htm.

Tait, P. G. "On Knots I, II, and III." Scientific Papers,
Vol. 1. Cambridge: University Press, pp. 273-347, 1898.

Thistlethwaite, M. B. "Knot Tabulations and Related Top-
ics." In Aspects of Topology in Memory of Hugh Dowker
1912-1982 (Ed. I. M. James and E. H. Kronheimer). Cam-
bridge, England: Cambridge University Press, pp. 2—76,
1985.

Thistlethwaite, M. B. ftp://chs.cusd.claremont.edu/pub/
knot/Thistlethwaite_Tables/.

Thompson, W. T. "On Vortex Atoms." Philos. Mag. 34,
15-24, 1867.

Weisstein, E. W. "Knots." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Knots.m.

Knot Complement

Two distinct knots cannot have the same Knot Com-
plement (Gordon and Luecke 1989).

References

Cipra, B. "To Have and Have Knot: When are Two Knots

Alike?" Science 241, 1291-1292, 1988.
Gordon, C. and Luecke, J. "Knots are Determined by their

Complements." J. Amer. Math. Soc. 2, 371-415, 1989.

Knot Curve

(s 2 -l) 2 =t/ J (3 + 2y).

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.

Knot Determinant

The determinant of a knot is |A( — 1)|, where A(z) is the
Alexander Polynomial.

Knot Diagram

A picture of a projection of a KNOT onto a PLANE. Usu-
ally, only double points are allowed (no more than two
points are allowed to be superposed), and the double or
crossing points must be "genuine crossings" which trans-
verse in the plane. This means that double points must
look like the below diagram on the left, and not the one
on the right.

Also, it is usually demanded that a knot diagram con-
tain the information if the crossings are overcrossings or
undercrossings so that the original knot can be recon-
structed. Here is a knot diagram of the TREFOIL KNOT,

Knot Polynomials can be computed from knot dia-
grams. Such Polynomials often (but not always) al-
low the knots corresponding to given diagrams to be
uniquely identified.

Knot Exterior

Koch Antisnowflake 999

Knot Exterior

The Complement of an open solid Torus knotted at
the Knot. The removed open solid TORUS is called a

tubular NEIGHBORHOOD.

Knot Linking

In general, it is possible to link two n-D HYPERSPHERES
in (n + 2)-D space in an infinite number of inequivalent
ways. In dimensions greater than n + 2 in the piece-
wise linear category, it is true that these spheres are
themselves unknotted. However, they may still form
nontrivial links. In this way, they are something like
higher dimensional analogs of two 1-spheres in 3-D. The
following table gives the number of nontrivial ways that
two n-D HYPERSPHERES can be linked in k-D.

D of spheres

D of space

Distinct Linkings

23

40

239

31

48

959

102

181

3

102

182

10438319

102

183

3

Two 10-D HYPERSPHERES link up in 12, 13, 14, 15, and
16-D, then unlink in 17-D, link up again in 18, 19, 20,
and 21-D. The proof of these results consists of an "easy
part" (Zeeman 1962) and a "hard part" (Ravenel 1986).
The hard part is related to the calculation of the (stable
and unstable) HOMOTOPY GROUPS of SPHERES.

References

Bing, R. H, The Geometric Topology of 3-Manifolds. Provi-
dence, RI: Amer. Math. Soc, 1983.

Ravenel, D. Complex Cobordism and Stable Homotopy
Groups of Spheres. New York: Academic Press, 1986.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 7, 1976.

Zeeman. "Isotopies and Knots in Manifolds." In Topology of
3-Manifolds and Related Topics (Ed. M. K. Fort). Engle-
wood Cliffs, NJ: Prentice- Hall, 1962.

Knot Polynomial

A knot invariant in the form of a POLYNOMIAL such
as the Alexander Polynomial, BLM/Ho Poly-
nomial, Bracket Polynomial, Conway Polynom-
ial, Jones Polynomial, Kauffman Polynomial F,
Kauffman Polynomial X, and Vassiliev Polynom-
ial.

References

Lickorish, W. B. R. and Millett, K. C. "The New Polynomial

Invariants of Knots and Links." Math. Mag. 61, 3-23,

1988.

Knot Problem

The problem of deciding if two KNOTS in 3-space are
equivalent such that one can be continuously deformed
into another.

Knot Sum

Two oriented knots (or links) can be summed by placing
them side by side and joining them by straight bars so
that orientation is preserved in the sum. This operation
is denoted #, so the knot sum of knots Ki and Ki is

written

K 1 #K 2 =K 2 #K 1 .

see also CONNECTED SUM

Knot Theory

The mathematical study of Knots. Knot theory con-
siders questions such as the following:

1. Given a tangled loop of string, is it really knotted or
can it, with enough ingenuity and/or luck, be untan-
gled without having to cut it?

2. More generally, given two tangled loops of string,
when are they deformable into each other?

3. Is there an effective algorithm (or any algorithm to
speak of) to make these determinations?

Although there has been almost explosive growth in the
number of important results proved since the discov-
ery of the Jones Polynomial, there are still many
"knotty" problems and conjectures whose answers re-
main unknown.
see also Knot, Link

Knot Vector

see B-Spline

Koch Antisnowflake

A Fractal derived from the Koch Snowflake. The
base curve and motif for the fractal are illustrated below.

A

The Area after the nth iteration is

A„-A„-i 3 a 3n ,

where A is the area of the original Equilateral Trian-
gle, so from the derivation for the KOCH SNOWFLAKE,

A = lim A n = (l- |)A= |A.

Knot Shadow

A LINK DIAGRAM which does not specify whether cross-
ings are under- or over crossings.

see also Exterior Snowflake, Flowsnake Frac-
tal, Koch Snowflake, Pentaflake, Sierpinski

Curve

1000

Koch Island

Koch Snowflake

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 66-67, 1989.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 36-
37, 1991.
# Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.

Koch Island

see Koch Snowflake

Koch Snowflake

A Fractal, also known as the Koch Island, which was
first described by Helge von Koch in 1904. It is built by
starting with an Equilateral Triangle, removing the
inner third of each side, building another EQUILATERAL
TRIANGLE at the location where the side was removed,
and then repeating the process indefinitely. The Koch
snowflake can be simply encoded as a LlNDENMAYER
System with initial string "F — F — F", String Rewrit-
ing rule "F" -> "F+F—F+F", and angle 60°. The zeroth
through third iterations of the construction are shown
above. The fractal can also be constructed using a base
curve and motif, illustrated below.

V

Let N n be the number of sides, L n be the length of a
single side, £ n be the length of the PERIMETER, and A n
the snowflake's AREA after the nth iteration. Further,
denote the Area of the initial n = Triangle A, and
the length of an initial n = side 1. Then

N n = 3 * 4 n
t n = N n L n = 3{±) n

A n = A n -! + ±N n L n 2 A = A n -! +

(1)

(2)
(3)

3<4 n (1

(§)■

, 3-4 n - x A . 3*4'

9 n --- g.gn-i

= A n - 1 + ±ar- i A.

The Capacity Dimension is then

, r In N n ln(3 - 4 n )

4a P = - hm -—— = - hm - v J

n-^oo lnl/ n n-s-oo ln(3 -T *)

ln 3 + n ln 4

(4)

lim
n-+oo nln3

ln4 _ 2 In 2

ln3 ~~ ln3 "

1.261859507....

(5)

Now compute the Area explicitly,
A = A

(6)
(7)

"-'•♦HSr^HG)'}

JT-71

k=Q

(8)
(9)

so as n — > oo,

A — Aqo —

= |A-

' +i 4M-( i+i ^y

(10)

Some beautiful TILINGS, a few examples of which are
illustrated above, can be made with iterations toward
Koch snowflakes.

In addition, two sizes of Koch snowflakes in AREA ratio
1:3 Tile the Plane, as shown above (Gosper).

Kochansky's Approximation

Another beautiful modification of the Koch snowflake
involves inscribing the constituent triangles with filled-in
triangles, possibly rotated at some angle. Some sample
results are illustrated above for 3 and 4 iterations.
see also Cesaro Fractal, Exterior Snowflake,
Gosper Island, Koch Antisnowflake, Peano-
Gosper Curve, Pentaflake, Sierpinski Sieve

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 65-66, 1989.

Dickau, R. M. "Two-Dimensional L- Systems." http://
forum. swart hmore . edu/advanced/robertd/lsy s2d.html.

Dixon, R. Mathographics. New York: Dover, pp. 175-177
and 179, 1991.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 28-29
and 32-36, 1991.

Pappas, T. "The Snowflake Curve." The Joy of Mathemat-
ics. San Carlos, CA: Wide World Publ./Tetra, pp. 78 and
160-161, 1989.

Peitgen, H.-O.; Jiirgens, H.; and Saupe, D. Chaos and Frac-
tals: New Frontiers of Science. New York: Springer-
Verlag, 1992.

Peitgen, H.-O. and Saupe, D. (Eds.). "The von Koch Snow-
flake Curve Revisited." §C2 in The Science of Fractal
Images. New York: Springer-Verlag, pp. 275-279, 1988.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 185-195, 1991.
# Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.

Kochansky's Approximation

The approximation for Pi,

40
3

^ = 3.141533.

Koebe's Constant

A Constant equal to one Quarter, 1/4.
see also Quarter

References

Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 24, 1983.

Kolmogorov-Arnold-Moser Theorem 1001
Koebe Function

The function

/(*)

(i - z y

It has a Minimum at z = -1, where

/'(*)

l + z

= 0,

(Z - 1)3

and an INFLECTION POINT at z = -2, where

References

Stewart, I. From Here to Infinity: A Guide to Today's
Mathematics. Oxford, England: Oxford University Press,
pp. 164-165, 1996.

Kollros' Theorem

For every ring containing p SPHERES, there exists a ring
of q Spheres, each touching each of the p Spheres,
where

The Hexlet is a special case with p = 3.
see also Hexlet, Sphere

References

Honsberger, R. Mathematical Gems II. Washington, DC:
Math, Assoc. Amer., p. 50, 1976.

Kolmogorov-Arnold-Moser Theorem

A theorem outlined in 1954 by Kolmogorov which was
subsequently proved in the 1960s by Arnold and Moser
(Tabor 1989, p. 105). It gives conditions under which
CHAOS is restricted in extent. Moser's 1962 proof was
valid for TWIST MAPS

0' = + 2vf(l) +9(0,1)
f = I + f(6,I).

(1)
(2)

In 1963, Arnold produced a proof for Hamiltonian sys-
tems

£T = JTo(I) + efTi(I). (3)

The original theorem required perturbations e ~ 10~ 48 ,
although this has since been significantly increased.
Arnold's proof required C°°, and Moser's original proof

1002 Kolmogorov-Arnold-Moser Theorem

Kolmogorov- Sinai Entropy

required C 333 . Subsequently, Moser's version has been
reduced to C 6 , then C 2+e , although counterexamples
are known for C 2 . Conditions for applicability of the
KAM theorem are:

1. small perturbations,

2. smooth perturbations, and

3. sufficiently irrational Winding Number.

Moser considered an integrable Hamiltonian function Ho
with a TORUS To and set of frequencies u; having an in-
commensurate frequency vector u)* (i.e., UJ-k ^ for all
INTEGERS fc). Let Ho be perturbed by some periodic
function Hi. The KAM theorem states that, if Hi is
small enough, then for almost every w* there exists an
invariant TORUS T(lo*) of the perturbed system such
that T((V*) is "close to" Tq(w*). Moreover, the TORI
T(u/) form a set of POSITIVE measures whose comple-
ment has a measure which tends to zero as \Hi\ — > 0.
A useful paraphrase of the KAM theorem is, "For suf-
ficiently small perturbation, almost all TORI (excluding
those with rational frequency vectors) are preserved."
The theorem thus explicitly excludes TORI with ratio-
nally related frequencies, that is, n — 1 conditions of the
form

w ■ k = 0. (4)

These TORI are destroyed by the perturbation. For a
system with two DEGREES OF FREEDOM, the condition
of closed orbits is

UJi

(5)

r

For a Quasiperiodic Orbit, <t is Irrational. KAM
shows that the preserved TORI satisfy the irrationality
condition

U>2

>

K{e)

s

,2.5

(6)

for all r and s, although not much is known about K(e).

The KAM theorem broke the deadlock of the small di-
visor problem in classical perturbation theory, and pro-
vides the starting point for an understanding of the ap-
pearance of Chaos. For a Hamiltonian System, the
Isoenergetic Nondegeneracy condition

d 2 H

dljdlj

^0

(7)

guarantees preservation of most invariant TORI under
small perturbations e « 1. The Arnold version states
that

£

rrikUJk

>K(e)

(8)

for all rrik £ Z. This condition is less restrictive than
Moser's, so fewer points are excluded.

see also CHAOS, HAMILTONIAN SYSTEM, QUASIPERI-
ODIC Function, Torus

References

Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, 1989.

Kolmogorov Complexity

The complexity of a pattern parameterized as the short-
est Algorithm required to reproduce it. Also known
as Algorithmic Complexity.

References

Goetz, P. "Phil's Good Enough Complexity Dictionary."
http : //www . cs . buffalo . edu/-goetz/dict . html.

Kolmogorov Constant

The exponent 5/3 in the spectrum of homogeneous tur-
bulence, A: -5 ' 3 .

References

Le Lionnais, F. Les nombres remarquables . Paris: Hermann,
p. 41, 1983.

Kolmogorov Entropy

Also known as METRIC ENTROPY. Divide Phase Space
into £>-dimensional HYPERCUBES of Content e D . Let
Pi ,...,i n De the probability that a trajectory is in Hy-
PERCUBE i at t = 0, ii at t = T, i 2 at t = 2T, etc.
Then define

K n = h K = - J2 ^o,.^ln«o in, (1)

where Kn+i — Kn is the information needed to predict
which Hypercube the trajectory will be in at (n + 1)T
given trajectories up to nT. The Kolmogorov entropy is
then defined by

N-l

K = lim lim lim -L V(tf n+1 - K n ). (2)

T-j-0 e^0+ AT-yoo I\ 1 *—*

The Kolmogorov entropy is related to Lyapunov CHAR-
ACTERISTIC Exponents by

h K

= / ]C ffi dfJ "

Jp *i>o

(3)

see also Hypercube, Lyapunov Characteristic Ex-
ponent

References

Ott, E. Chaos in Dynamical Systems. New York: Cambridge

University Press, p. 138, 1993.
Schuster, H. G. Deterministic Chaos: An Introduction } 3rd

ed. New York: Wiley, p. 112, 1995.

Kolmogorov- Sinai Entropy

see Kolmogorov Entropy, Metric Entropy

Kolmogorov-Smirnov Test

Krawtchouk Polynomial 1003

Kolmogorov-Smirnov Test

A goodness-of-fit test for any DISTRIBUTION. The test
relies on the fact that the value of the sample cumulative
density function is asymptotically normally distributed.

To apply the Kolmogorov-Smirnov test, calculate the
cumulative frequency (normalized by the sample size)
of the observations as a function of class. Then cal-
culate the cumulative frequency for a true distribu-
tion (most commonly, the Normal Distribution).
Find the greatest discrepancy between the observed and
expected cumulative frequencies, which is called the
"D-STATISTIC." Compare this against the critical D-
Statistic for that sample size. If the calculated D-
STATISTIC is greater than the critical one, then reject
the Null Hypothesis that the distribution is of the
expected form. The test is an .R-ESTIMATE.

see also Anderson-Darling Statistic, D-Statistic,
Kuiper Statistic, Normal Distribution, R-
Estimate

References

Boes, D. C; Graybill, F. A.; and Mood, A. M. Introduction to

the Theory of Statistics, 3rd ed. New York: McGraw-Hill,

1974.
Knuth, D. E. §3. 3. IB in The Art of Computer Programming,

Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA:

Addison- Wesley, pp. 45-52, 1981.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-

terling, W. T. "Kolmogorov-Smirnov Test." In Numerical

Recipes in FORTRAN: The Art of Scientific Computing,

2nd ed. Cambridge, England: Cambridge University Press,

pp. 617-620, 1992.

Konig-Egevary Theorem

A theorem on Bipartite Graphs.

see also BIPARTITE GRAPH, FROBENIUS-KONIG THEO-
REM

Konig's Theorem

If an Analytic Function has a single simple Pole at
the Radius of Convergence of its Power Series,
then the ratio of the coefficients of its Power Series
converges to that POLE.

see also Pole

References

Konig, J. "Uber eine Eigenschaft der Potenzreihen." Math.
Ann. 23, 447-449, 1884.

Konigsberg Bridge Problem

1

by Euler, and represented the beginning of GRAPH THE-
ORY.
see also EULERIAN CIRCUIT, GRAPH THEORY

References

Bogomolny, A. "Graphs." http://www.cut-the-knot.com/
do_you_know/graphs .html.

Chartrand, G. "The Konigsberg Bridge Problem: An Intro-
duction to Eulerian Graphs." §3.1 in Introductory Graph
Theory. New York: Dover, pp. 51-66, 1985.

Kraitchik, M. §8.4.1 in Mathematical Recreations. New York:
W. W. Norton, pp. 209-211, 1942.

Newman, J. "Leonhard Euler and the Konigsberg Bridges."
Sci. Amer. 189, 66-70, 1953.

Pappas, T. "Konigsberg Bridge Problem & Topology." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 124-125, 1989.

Korselt's Criterion

n Divides a n - a for all Integers a Iff n is Square-
free and (p - l)\n/p - 1 for all PRIME DIVISORS p of

n. Carmichael Numbers satisfy this Criterion.

References

Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen-

sohn, R. "Giuga's Conjecture on Primality." Amer. Math.

Monthly 103, 40-50, 1996.

Kovalevskaya Exponent

see Leading Order Analysis

Kozyrev-Grinberg Theory

A theory of Hamiltonian Circuits.

see also Grinberg Formula, Hamiltonian Circuit

Kramers Rate

The characteristic escape rate from a stable state of a

potential in the absence of signal.

see also Stochastic Resonance

References

Bulsara, A. R. and Gammaitoni, L. "Tuning in to Noise."
Phys. Today 49, 39-45, March 1996.

Krawtchouk Polynomial

Let a(x) be a Step Function with the Jump

*/\ i ™ \ x N — x

j{x)= [ x \p q

(1)

at x = 0, 1, . . . , N, where p > 0, q > 0, and p + q = 1.
Then

k { »\x)

-1/2

(pq)- n/2

The Konigsberg bridges cannot all be traversed in a sin-
gle trip without doubling back. This problem was solved

1004 Kreisel Conjecture

Kronecker Delta

for n = 0, 1, . . . , N. It has Weight Function
N\p x q N - x

w =

r(l + x)T(N + l-x)'

(3)

where T(x) is the GAMMA FUNCTION, RECURRENCE

Relation

(n + l)k ( Si(x) + pq(N - n + ljfc^jx)

= [ iC _ n _(7V-2)]^ ) (^), (4)

and squared norm

AT!

n!(7V-n)!

(P?) n -

It has the limit

/ 2 \ n/2
lim Ur- n!*£ ?) (iVp + ^2Npqs) = ff n (s),

n-+oo \JypqJ

(5)

(6)

where tf„(x) is a HERMITE Polynomial, and is related
to the HYPERGEOMETRIC FUNCTION by

k { n p) (x,N) = k£ ) (x 1 N)

- (-1)" (^)p n 2F 1 {-n, -s; -iV; 1/p)

(-i)V 1 r(jv-a? + i)

n! r(JV-a;-n + l)
X2F1 (-n, -x; N - x - n + 1; -q/p). (7)

see also ORTHOGONAL POLYNOMIALS

References

Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical

Orthogonal Polynomials of a Discrete Variable. New York:

Springer- Verlag, 1992.
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:

Amer. Math. Soc, pp. 35-37, 1975.
Zelenkov, V. "Krawtchouk Polynomial Home Page," http : //

vvv.isir.minsk.by/~zeleiikov/physmath/kr_polyn/.

Kreisel Conjecture

A Conjecture in Decidability theory which postu-
lates that, if there is a uniform bound to the lengths of
shortest proofs of instances of 5(n), then the universal
generalization is necessarily provable in PEANO ARITH-
METIC. The Conjecture was proven true by M. Baaz
in 1988 (Baaz and Pudlak 1993).

see also DECIDABLE

References

Baaz, M. and Pudlak P. "Kreisel's Conjecture for L3 X . In
Arithmetic, Proof Theory, and Computational Complex-
ity, Papers from the Conference Held in Prague, July 2-5,
1991 (Ed. P. Clote and J. Krajicek). New York: Oxford
University Press, pp. 30—60, 1993.

Dawson, J. "The Godel Incompleteness Theorem from a
Length of Proof Perspective." Amer. Math. Monthly 86,
740-747, 1979.

Kreisel, G. "On the Interpretation of Nonfinitistic Proofs, II."
J. Symbolic Logic 17, 43-58, 1952.

Kronecker Decomposition Theorem

Every Finite ABELIAN GROUP can be written as
a Direct Product of Cyclic Groups of Prime
POWER ORDERS. In fact, the number of nonisomorphic
Abelian Finite Groups a(n) of any given Order n
is given by writing n as

i

where the pi are distinct Prime Factors, then
a(n) = JJP(a0,

i

where P is the Partition Function. This gives 1, 1,
1, 2, 1, 1, 1, 3, 2, . . . (Sloane's A000688).

see also ABELIAN GROUP, FINITE GROUP, ORDER

(Group), Partition Function P

References

Sloane, N. J. A. Sequence A000688/M0064 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Kronecker Delta

The simplest interpretation of the Kronecker delta is as
the discrete version of the DELTA FUNCTION defined by

{!

= / for i 7- j
for i = j.

It has the COMPLEX GENERATING FUNCTION

r *■ I rn — n — 1 j

5mn = ^TiJ z dz >

(i)

(2)

where m and n are INTEGERS. In 3-space, the Kronecker
delta satisfies the identities

dijUjk =

€ijk € pqk = OipOjq — Oiqdj pi

(3)

(4)
(5)
(6)

where Einstein SUMMATION is implicitly assumed,
i,j = 1,2,3, and e is the PERMUTATION SYMBOL.

Technically, the Kronecker delta is a TENSOR defined by
the relationship

~k dxi dxi _ dxi dxk _ dx±

(7)

Since, by definition, the coordinates x% and Xj are inde
pendent for i ^ j,

dx't

0_^ = *"

(8)

Kronecker's Polynomial Theorem

fi dx'i dxi fc

(9)

and dj is really a mixed second Rank Tensor. It sat-
S ab jk = e abi e jki = 8iS k - 5 k 8 j h (10)

Sabjk = QajQbk ~ 9ak9bj (ll)

€«*€*' =a«i M = 2#. (12)

see aJso Delta Function, Permutation Symbol

Kronecker's Polynomial Theorem

An algebraically soluble equation of Odd Prime degree
which is irreducible in the natural Field possesses either

1. Only a single REAL ROOT, or

2. All Real Roots.

see also Abel's Irreducibility Theorem, Abel's
Lemma, Schoenemann's Theorem

References

Dorrie, H. 100 Great Problems of Elementary Mathematics:

Their History and Solutions. New York: Dover p. 127,

1965.

Kronecker Product

see Direct Product (Matrix)

Kronecker Symbol

An extension of the Jacobi Symbol (n/m) to all In-
tegers. It can be computed using the normal rules for
the Jacobi Symbol

Ud) ~ \cd) \cd) ~\c)\d)

= (!) (;) (5) G)

plus additional rules for m = — 1,

("/ ~ 1) = { ~

1 for n<
for n > 0,

and jn — 2. The definition for (n/2) is variously written
as

{0 for n even
1 for n odd, n = ±1 (mod 8)
-1 for n odd, n = ±3 (mod 8)

or

(n/2) =

for 4|n

for n = 1 (mod 8)
— 1 for n = 5 (mod 8)

undefined otherwise

(Cohn 1980). Conn's form "undefines" (n/2) for SINGLY
Even Numbers n = 4 (mod 2) and n = — 1, 3 (mod 8),
probably because no other values are needed in applica-
tions of the symbol involving the DISCRIMINANTS d of

KS Entropy 1005

Quadratic Fields, where m > and d always satisfies
d = 0,l (mod 4).

The Kronecker Symbol is a Real Character mod-
ulo n, and is, in fact, essentially the only type of REAL
primitive character (Ayoub 1963).

see also CHARACTER (NUMBER THEORY), CLASS NUM-
ber, dlrichlet l-series, jacobi symbol, legen-
dre Symbol

References

Ayoub, R. G. An Introduction to the Analytic Theory of

Numbers. Providence, RI: Amer. Math. Soc, 1963.
Cohn, H. Advanced Number Theory. New York: Dover, p. 35,

1980.

Krull Dimension

If R is a RING (commutative with 1), the height of a
Prime Ideal p is defined as the Supremum of all n so
that there is a chain po C • * ■ p n -i C p n — P where all pi
are distinct PRIME IDEALS. Then, the Krull dimension
of R is defined as the SUPREMUM of all the heights of
all its Prime Ideals.

see also Prime Ideal

References

Eisenbud, D. Commutative Algebra with a View Toward Al-
gebraic Geometry. New York: Springer- Verlag, 1995.

Macdonald, I. G. and Atiyah, M, F. Introduction to Commu-
tative Algebra. Reading, MA: Addison- Wesley, 1969.

Kruskal's Algorithm

An Algorithm for finding a Graph's spanning TREE
of minimum length.

see also KRUSKAL'S TREE THEOREM

References

Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, pp. 248-249, 1978.

Kruskal's Tree Theorem

A theorem which plays a fundamental role in computer
science because it is one of the main tools for show-
ing that certain orderings on Trees are well-founded.
These orderings play a crucial role in proving the ter-
mination of rewriting rules and the correctness of the
Knuth-Bendix equational completion procedures.

see also Kruskal's Algorithm, Natural Indepen-
dence Phenomenon, Tree

References

Gallier, J. "What's so Special about KruskaPs Theorem and
the Ordinal Gamma[0]? A Survey of Some Results in Proof
Theory." Ann. Pure and Appl. Logic 53, 199-260, 1991.

KS Entropy

see Metric Entropy

1006 Kuen Surface

Kuen Surface

A special case of Enneper's Surfaces which can be
given parametrically by

2(cos u + u sin u) sin v

1 + u 2 sin 2 v
2\/l + u 2 cos(u — tan^ 1 u) smv

1 -f- u 2 sin 2 v
2 (sin u — u cos u) sin v

1 + u 2 sin 2 v

2\/l + u 2 sin(u — tan 1 u) svnv

z = ln[tan(|u)] +

1 + u 2 sin 2 v
2 cost;

\ + u 2 sin 2 v

(1)
(2)
(3)
(4)
(5)

for v e [0,tt), u € [0,27r) (Reckziegel et al. 1986). The
Kuen surface has constant NEGATIVE GAUSSIAN CUR-
VATURE of K — -1. The Principal Curvatures are
given by

Hi

K 2 =

ucos(^v)[—2 - u 2 + « 2 cos(2u)] 4 sin(^v)

(6)

2 [2 - ti 2 + u 2 cos(2v)](l + u 2 sin 2 v) 4
-2 - k 2 + n 2 cos(2t;)] 4 [2 - u 2 + n 2 cos(2i;)] csc(^)

64u(l + u 2 sin 2 i;) 4

(7)

see a/50 Enneper's Surfaces, Rembs' Surfaces,
Sievert's Surface

References

Fischer, G. (Ed.). Plate 86 in Mathematische Mod-

elle/ Mathematical Models, Bildband/ Photograph Volume.

Braunschweig, Germany: Vieweg, p. 82, 1986.
Gray, A. "Kuen's Surface," §19.4 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC

Press, pp. 384-386, 1993.
Kuen, T. "Ueber Flachen von constantem Krummungs-

maass." Sitzungsber. d. konigl. Bayer. Akad. Wiss. Math.-

phys. Classe, Heft II, 193-206, 1884.
Nordstrand, T. "Kuen's Surface." http : //www . uib . no/

people/nf ytn/kuentxt .htm.
Reckziegel, H. "Kuen's Surface." §3.4.4.2 in Mathematical

Models from the Collections of Universities and Museums

(Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 38,

1986.

Rummer's Conjecture

Kuhn- Tucker Theorem

A theorem in nonlinear programming which states that
if a regularity condition holds and / and the functions
hj are convex, then a solution a; which satisfies the con-
ditions hj for a Vector of multipliers A is a Global
Minimum. The Kuhn- Tucker theorem is a generaliza-
tion of Lagrange Multipliers. Farkas's Lemma is
key in proving this theorem.

see also Farkas's Lemma, Lagrange Multiplier

the Kolmogorov-

Kuiper Statistic

A statistic defined to improve

Smirnov Test in the Tails.

see also Anderson-Darling Statistic

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 621, 1992.

Kulikowski's Theorem

For every Positive Integer n, there exists a Sphere
which has exactly n Lattice Points on its surface.
The Sphere is given by the equation

(x - a) 2 + (y - b) 2 + (z - V2) 2 = c 2 + 2,

where a and b are the coordinates of the center of the
so-called SCHINZEL CIRCLE

( !B -i) a +» a = i5'- 1
(z-i) 2 + y 2 = i5 2fe

9*

for n = 2k even

for n = 2k + 1 odd

and c is its RADIUS.

see also CIRCLE LATTICE POINTS, LATTICE POINT,

Schinzel's Theorem

References

Honsberger, R. "Circles, Squares, and Lattice Points."

Ch. 11 in Mathematical Gems I. Washington, DC: Math.

Assoc. Amer., pp. 117-127, 1973.
Kulikowski, T. "Sur l'existence d'une sphere passant par un

nombre donne aux coordonnees entieres." L'Enseignement

Math. Ser. 2 5, 89-90, 1959.
Schinzel, A. "Sur l'existence d'un cercle passant par un

nombre donne de points aux coordonnees entieres."

L'Enseignement Math. Ser. 2 4, 71-72, 1958.
Sierpinski, W. "Sur quelques problemes concernant les points

aux coordonnees entieres." L'Enseignement Math. Ser. 2

4, 25-31, 1958.
Sierpinski, W. "Sur un probleme de H. Steinhaus concernant

les ensembles de points sur le plan." Fund. Math. 46,

191-194, 1959.
Sierpinski, W. A Selection of Problems in the Theory of

Numbers. New York: Pergamon Press, 1964.

Kummer's Conjecture

A conjecture concerning PRIMES.

Rummer's Differential Equation

Kummer's Differential Equation

see Confluent Hypergeometric Differential
Equation

Kummer's Formulas
Kummer's first formula is

2Fi(| + m - k, -n; 2m + 1; 1)

r(2m+l)r(m+ \ +k + n)
T{m+ \ +fc)r(2m + l + n)

, (1)

where 2 F 1 (a, b\ c; z) is the HYPERGEOMETRIC FUNCTION
with m / -1/2, -1, -3/2, . . . , and T(z) is the GAMMA
FUNCTION. The identity can be written in the more
symmetrical form as

wt u u r(|b+l)r(6-a+l)
a f 1 ( 0| 6 iCi _i ) = ______ y> (2)

where a — 6 + c — 1 and 6 is a positive integer. If b is a
negative integer, the identity takes the form

2 Fi(a,6;c; -1) = 2cos(|7r6)

(Petkovsek et al 1996).
Kummer's second formula is

iFi(| +m;2m+l;z) = M 0)in (z)

r([b|)r(b-a + l)
r(|b-a+l)

(3)

m+l/2

>+£

,2p

p=l

2 4 V( m + !)(™ + 2)-(m + p)

(4)

where iFi(a; 6; z) is the Confluent Hypergeometric
Function and m ^ -1/2, -1, -3/2, —

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, pp. 42-43 and 126, 1996.

Kummer's Function

see Confluent Hypergeometric Function

Kummer Group

A Group of Linear Fractional Transformations
which transform the arguments of Kummer solutions to
the Hypergeometric Differential Equation into
each other. Define

A(z) = 1- z
B{z) = 1/z,

then the elements of the group are {J, A, B, AB, BA,
ABA = BAB}.

Kummer Surface 1007

Kummer's Quadratic Transformation

A transformation of a HYPERGEOMETRIC FUNCTION,

a Fi a,/?; 2/3;

Az

(l + z)\
- (1 + z) 2a 2 F l (a, a + \ - ftp + h * 2 )>

Kummer's Relation

An identity which relates HYPERGEOMETRIC FUNC-
TIONS,

2 Fx (2a, 26; a + b + \ ; x) = 2 F x (a, b; a + b + \ , 4z(l - x)).

Kummer's Series

see Hypergeometric Function

Kummer's Series Transformation

Let X)H=o ak = a anc * Sfclo Cfc = c be conver g ent series
such that

lim *L = A ^ 0.

fc-»oo Cfc

Then

oo

a = Ac + 2_] ( * — ^ — ) afc -

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 16, 1972.

Kummer Surface

The Kummer surfaces are a family of Quartic SUR-
FACES given by the algebraic equation

(x +y 2 + z- yTwy - Xpqrs = 0, (1)

where

A =

_ 3m - 1

3-/i 2 '
p, q, r, and s are the TETRAHEDRAL COORDINATES

p = w — z — v2 x

q — w — z + v2x
r = u> + z + v^y
s = w + z — V2y>

(2)

(3)
(4)
(5)
(6)

1008

Kummer Surface

KuratowskVs Closure-Component Problem

and w is a parameter which, in the above plots, is set to
w = 1. The above plots correspond to jx 2 = 1/3

(3z 2 + 3y 2 + Sz 2 + l) 2 = 0,

(double sphere), 2/3, 1
x 4 - 2x 2 y 2 + y 4 + 4z 2 z + 4y 2 z + 4z V + 4y V = (7)
(Roman Surface), V2, \/3

[ ( ^_l)2_ 2 ^ ][y 2_ (z+l) 2 ]=0 (g)

(four planes), 2, and 5. The case < /x 2 < 1/3 corre-
sponds to four real points.

The following table gives the number of ORDINARY
Double Points for various ranges of ^ 2 , corresponding
to the preceding illustrations.

Range

Real Nodes Complex Nodes

< y? < |

A* = 3

| < M 2 < 1

M 2 = l

1< v? < 3

fi 2 = 3

M 2 >3

16

16

12

12

The Kummer surfaces can be represented parametrically
by hyperelliptic Theta Functions. Most of the Kum-
mer surfaces admit 16 Ordinary Double Points, the
maximum possible for a Quartic Surface. A special
case of a Kummer surface is the Tetrahedroid.

Nordstrand gives the implicit equations as
x 4 +y 4 +z 4 -x 2 -y 2 -z 2 -z 2 y 2 -x 2 z 2 -y 2 z 2 + l = (9)

4, 4 , 4. / 2 . 2. 2\ . ,/ 2 2 , 22, 2 2n

x + y + z + a(# + y + z ) + b(x y +x z +y z )

+cxyz -1 = 0. (10)

see also Quartic Surface, Roman Surface, Tetra-
hedroid

References

Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV-

Mitteilungen 4, 17-20, Apr. 1995.
Endrafi, S. "Kummer Surfaces." http://www . mathematik .

uni - mainz . de / Algebraische Geometrie / docs /

Ekummer . shtral.
Fischer, G. (Ed.). Mathematical Models from the Collections

of Universities and Museums. Braunschweig, Germany:

Vieweg, pp. 14-19, 1986.
Fischer, G. (Ed.). Plates 34-37 in Mathematische Mod-

elle/ Mathematical Models, Bildband/ Photograph Volume.

Braunschweig, Germany: Vieweg, pp. 33—37, 1986.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Verlag, p. 183, 1994.

Hudson, R. Rummer's Quartic Surface. Cambridge, Eng-
land: Cambridge University Press, 1990.

Kummer, E. "Uber die Flachen vierten Grades mit sechszehn
singularen Punkten." Ges. Werke 2, 418-432.

Kummer, E. "Uber Strahlensysteme, deren Brennflachen
Flachen vierten Grades mit sechszehn singularen Punkten
sind." Ges. Werke 2, 418-432.

Nordstrand, T. "Rummer's Surface." http://www.uib.no/
people/nf ytn/kummtxt . htm.

Rummer's Test

Given a Series of Positive terms ui and a sequence of

finite POSITIVE constants a*, let

p = lim I a n — a n+1 ) .

n-+oo y U n +1 J

1. If p > 0, the series converges.

2. If p < 0, the series diverges.

3. If p — 0, the series may converge or diverge.

The test is a general case of BERTRAND's TEST, the
Root Test, Gauss's Test, and Raabe's Test. With
a n = n and a n +i = n + 1, the test becomes Ra ABE'S

Test.

see also Convergence Tests, Raabe's Test

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 285-286, 1985.

Jingcheng, T. "Rummer's Test Gives Characterizations for
Convergence or Divergence of All Series." Amer. Math.
Monthly 101, 450-452, 1994.

Samelson, H. "More on Kummer's Test." Amer. Math.
Monthly 102, 817-818, 1995.

Kummer's Theorem

2i r i(£, -x\x + n + 1;— 1) =
a Fi(a,/3;l + a -#-!) =

r(a + n + l)r(|n+l)
r(a+!n+l)r(n+l)

r(l + a-/?)r(l+lq)
r(l + a)r(l+ia-/J)'

where 2 Fi is a Hypergeometric Function and T{z)
is the Gamma Function.

Kuratowski's Closure-Component Problem

Let X be an arbitrary TOPOLOGICAL SPACE. Denote
the Closure of a Subset A of X by A~ and the com-
plement of A by A! . Then at most 14 different Sets can
be derived from A by repeated application of closure and
complementation (Berman and Jordan 1975, Fife 1991).
The problem was first proved by Kuratowski (1922) and
popularized by Kelley (1955).
see also Kuratowski Reduction Theorem

References

Anusiak, J. and Shum, K. P. "Remarks on Finite Topological
Spaces." Colloq. Math, 23, 217-223, 1971.

Kuratowski Reduction Theorem

Kurtosis

1009

Aull, C. E. "Classification of Topological Spaces." Bull de
VAcad. Pol Sci. Math. Astron. Phys. 15, 773-778, 1967.

Baron, S. Advanced Problem 5569. Amer. Math. Monthly
75, 199, 1968.

Berman, J. and Jordan, S. L. "The Kuratowski Closure-
Complement Problem." Amer. Math. Monthly 82, 841-
842, 1975.

Buchman, E. "Problem E 3144." Amer. Math. Monthly 93,
299, 1986.

Chagrov, A. V. "Kuratowski Numbers, Application of Func-
tional Analysis in Approximation Theory." Kalinin:
Kalinin Gos. Univ., pp. 186-190, 1982.

Chapman, T. A. "A Further Note on Closure and Interior
Operators." Amer. Math. Monthly 69, 524-529, 1962.

Fife, J. H. "The Kuratowski Closure- Complement Problem."
Math. Mag. 64, 180-182, 1991.

Fishburn, P. C. "Operations on Binary Relations." Discrete
Math. 21, 7-22, 1978.

Graham, R. L.; Knuth, D. E.; and Motzkin, T. S. "Comple-
ments and Transitive Closures." Discrete Math. 2, 17-29,
1972.

Hammer, P. C. "Kuratowski's Closure Theorem." Nieuw
Arch. Wish. 8, 74-80, 1960.

Herda, H. H. and Metzler, R. C. "Closure and Interior in
Finite Topological Spaces." Colloq. Math. 15, 211-216,
1966.

Kelley, J. L. General Topology. Princeton: Van Nostrand,
p. 57, 1955.

Koenen, W. "The Kuratowski Closure Problem in the To-
pology of Convexity." Amer. Math. Monthly 73, 704-708,
1966.

Kuratowski, C. "Sur l'operation A de l'analysis situs." Fund.
Math. 3, 182-199, 1922.

Langford, E. "Characterization of Kuratowski 14-Sets."
Amer. Math. Monthly 78, 362-367, 1971.

Levine, N. "On the Commutativity of the Closure and In-
terior Operators in Topological Spaces." Amer. Math.
Monthly 68, 474-477, 1961.

Moser, L. E. "Closure, Interior, and Union in Finite Topo-
logical Spaces." Colloq. Math. 38, 41-51, 1977.

Munkresj J. R. Topology: A First Course. Englewood Cliffs,
NJ: Prentice-Hall, 1975.

Peleg, D. "A Generalized Closure and Complement Phenom-
enon." Discrete Math. 50, 285-293, 1984.

Shum, K. P. "On the Boundary of Kuratowski 14-Sets in
Connected Spaces." Glas. Mat. Ser. 7/719, 293-296, 1984.

Shum, K. P. "The Amalgamation of Closure and Boundary
Functions on Semigroups and Partially Ordered Sets." In
Proceedings of the Conference on Ordered Structures and
Algebra of Computer Languages. Singapore: World Scien-
tific, pp. 232-243, 1993.

Smith, A. Advanced Problem 5996. Amer. Math. Monthly
81, 1034, 1974.

Soltan, V. P. "On Kuratowski's Problem." Bull. Acad.
Polon. Sci. Ser. Sci. Math. 28, 369-375, 1981.

Soltan, V. P. "Problems of Kuratowski Type." Mat. Issled.
65, 121-131 and 155, 1982.

Kuratowski Reduction Theorem

Every nonplanar graph is a SlJPERGRAPH of an expan-
sion of the Utility Graph UG = iC 3 ,3 or the Com-
plete GRAPH K*>. This theorem was also proven ear-
lier by Pontryagin (1927-1928), and later by Prink and
Smith (1930). Kennedy et al (1985) give a detailed his-
tory of the theorem, and there exists a generalization
known as the Robertson-Seymour Theorem.

see also Complete Graph, Planar Graph,
Robertson-Seymour Theorem, Utility Graph

References

Kennedy, J. W.; Quintas, L. V.; and Syslo, M. M. "The

Theorem on Planar Graphs." Historia Math. 12, 356-

368, 1985.
Kuratowski, C. "Sur l'operation A de l'analysis situs." Fund.

Math. 3, 182-199, 1922.
Thomassen, C. "Kuratowski's Theorem." J. Graph Th. 5,

225-241, 1981.
Thomassen, C. "A Link Between the Jordan Curve Theorem

and the Kuratowski Planarity Criterion." Amer. Math.

Monthly 97, 216-218, 1990.

Kuratowski's Theorem

see Kuratowski Reduction Theorem

Kiirschak's Tile

An attractive tiling of the SQUARE composed of two
types of triangular tiles.

References

Alexanderson, G. L. and Seydel, K. "Kiirschak's Tile." Math.

Gaz. 62, 192-196, 1978.
Honsberger, R. Mathematical Gems III. Washington, DC:

Math. Assoc. Amer., pp. 30-32, 1985.
Schoenberg, I. Mathematical Time Exposures. Washington,

DC: Math. Assoc. Amer., p. 7, 1982.
# Weisstein, E. W. "Kiirschak's Tile." http: //www. astro.

Virginia . edu/ ~evw6n/math/notebooks/KurschaksTile . m.

Kurtosis

The degree of peakedness of a distribution, also called
the Excess or Excess Coefficient. Kurtosis is de-
noted 72 (or 62) or #2 and computed by taking the fourth
MOMENT of a distribution. A distribution with a high
peak (72 > 0) is called Leptokurtic, a flat-topped
curve (72 < 0) is called Platykurtic, and the normal
distribution (72 — 0) is called MESOKURTIC. Let m de-
note the ith Moment (a:*). The Fisher Kurtosis is
defined by

72 = 62 - — o - 3 = —7 - 3,

M2 2 <T 4

and the PEARSON KURTOSIS is defined by

P 2 = OC 4 =

M4

(1)

(2)

An Estimator for the 72 Fisher Kurtosis is given by

92 = t4"> (3)

k 2

1010 Kurtosis Kurtosis

where the fcs are k- Statistics. The Standard Devi-
ation of the estimator is

-«'«£• (4)

see also Fisher Kurtosis, Mean, Pearson Kurtosis,
Skewness, Standard Deviation

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook

of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, 9th printing. New York: Dover,

p. 928, 1972.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-

terling, W. T. "Moments of a Distribution: Mean, Vari-
ance, Skewness, and So Forth." §14.1 in Numerical Recipes

in FORTRAN: The Art of Scientific Computing, 2nd

ed. Cambridge, England: Cambridge University Press,

pp. 604-609, 1992.

Lx-Norm

L-Estimate

1011

Li-Norm

A Vector Norm defined for a Vector

Xi
X2

with Complex entries by

n

ll x l|i = J^|av|.

see also L 2 -Norm, Loo-NORM, Vector Norm

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1114-1125, 1979.

L 2 -Norm

A Vector Norm defined for a Vector

xi

X 2

1,2-Space

A Hilbert Space in which a Bracket Product is
defined by

and which satisfies the following conditions

<#/>>* = <V#)e (2)

OflAi^i + A 2 V> 2 > = Ai (0|Vi> + A 2 <0^2> (3)

{Ai0i + \ 2 <p2\i>) = Ai* (4>iW + A 2 * {0 2 |V> (4)

<V#)eM>o (5)

|(^i|^a>| a <^i|^i><^l^>- (6)

The last of these is SCHWARZ's INEQUALITY.
see also BRACKET PRODUCT, HlLBERT SPACE, L 2 -
NORM, RlESZ-FlSCHER THEOREM, SCHWARZ'S IN-
EQUALITY

Loo-Norm

A Vector Norm defined for a Vector

Xl

X 2

with Complex entries by

x oo = max ja;;|

with Complex entries by

X 2

The L 2 -norm is also called the Euclidean Norm. The
L 2 -norm is defined for a function <f>(x) by

H4(*)n = m ■ 4>{x) = (t^)i 2 ) = / i<i>(x)} 2 dx.

J a

see also Li-Norm, L 2 -Space, L^-Norm, Parallelo-
gram Law, Vector Norm

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1114-1125, 1979.

see also Lx-NORM, L 2 -NORM, VECTOR NORM

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed, San Diego, CA: Academic
Press, pp. 1114-1125, 1979.

L p f -Balance Theorem

If every component L of X/O p r (X) satisfies the
"Schreler property," then

L P ,(Y)<L P ,(X)

for every p-local SUBGROUP Y of X y where L p * is the

p-LAYER.

see also p-LAYER, SUBGROUP

L-Estimate

A Robust Estimation based on linear combinations
of Order Statistics. Examples include the Median
and TUKEY'S TRIMEAN.
see also M-Estimate, ^-ESTIMATE

References

Press, W. H.; Flannery, B. R; Teukolsky, S. A.; and Vet-
terling, W. T. "Robust Estimation." §15.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 694-700, 1992.

1012

L-Function

Ladder Graph

L- Function

see Artin L-Function, Dirichlet L-Series, Euler
L-Function, Hecke L-Function

L-Polyomino

The order n > 2 L-polyomino consists of a vertical line
of n Squares with a single additional Square attached
at the bottom.

see also L-POLYOMINO, SKEW POLYOMINO, SQUARE,

Square Polyomino, Straight Polyomino

L-Series

see Dirichlet L-Series

L-System

see Lindenmayer System

L'Hospital's Cubic

see TSCHIRNHAUSEN CUBIC

L'Hospital's Rule

Let lim stand for the LIMIT lims-^, lim x _^ c - , lini a ,_ >c +,
linix^ooj or lim a; _)._ o, and suppose that lim f{x) and
lim g(x) are both ZERO or are both ±00. If

lim

/'(*)

has a finite value or if the LIMIT is ±00, then

9(x)

9'{*Y

L'Hospitars rule occasionally fails to yield useful results,
as in the case of the function lim u ^oo n(u 2 -f-l) -1 ' 2 . Re-
peatedly applying the rule in this case gives expressions
which oscillate and never converge,

lim

— lim

00 (U 2 + l)!/2 „_>«, U ( U 2 + l)-l/2

u— )-oo U it— +00 1

= hm 7-— —-7- .

U -4oo (U 2 + 1)V2

(The actual Limit is 1.)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 13, 1972.

L'Hospital, G. de L 'analyse des infiniment petits pour
I 'intelligence des lignes courbes. 1696.

L'Huilier's Theorem

Let a Spherical Triangle have sides of length a, 6,
and c, and Semiperimeter s. Then the Spherical
Excess A is given by

tan(iA)

= Jtan(fs) tan[|(s - a)} tan[§(s - b)] tan[|(s - c)].

see also Girard's Spherical Excess Formula,
Spherical Excess, Spherical Triangle

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 148, 1987.

Labelled Graph

A labelled graph G = (V, E) is a finite series of Ver-
tices V with a set of Edges E of 2-Subsets of V.
Given a Vertex set V n = {1, 2, . . . , n}, the number
of labelled graphs is given by 2 n ^ Tl " 1 ^ 2 . Two graphs G
and H with Vertices V n = {1, 2, . . . , n} are said to
be Isomorphic if there is a Permutation p of V n such
that {u,v} is in the set of EDGES E(G) Iff {p(u),p(v)}
is in the set of EDGES E(H).

see also CONNECTED GRAPH, GRACEFUL GRAPH,

Graph (Graph Theory), Harmonious Graph,
Magic Graph, Taylor's Condition, Weighted
Tree

References

Cahit, I. "Homepage for the Graph Labelling Problems

and New Results." http://193.140.42.134/-cahit/

C0RDIAL.html.
Gallian, J. A. "Graph Labelling." Elec. J. Combin. DS6,

1-43, Mar. 5, 1998. http://www.combinatorics.org/

Surveys/.

Lacunarity

Quantifies deviation from translational invariance by de-
scribing the distribution of gaps within a set at multiple
scales. The more lacunar a set, the more heterogeneous
the spatial arrangement of gaps.

Ladder

see astroid, crossed ladders problem, ladder
Graph

Ladder Graph

A GRAPH consisting of two rows of paired nodes each
connected by an Edge. Its complement is the COCK-
TAIL Party Graph.

see also Cocktail Party Graph

Lagrange Bracket

Lagrange f s Identity 1013

Lagrange Bracket

Let F and G be infinitely differ entiable functions of £,
Uj and p. Then the Lagrange bracket is defined by

Lagrange Expansion

Let y = f(x) and yo = f(xo) where f'(xo) ^ 0, then

dpv \ dav v du )

(i)

The Lagrange bracket satisfies

[F,G] = -[G,F] (2)

[[F,G],H] + [[G,^],F] + [[ff,F],G]

= ^[ G '^ + f^F ] + f^G]. (3)

If F and G are functions of x and p only, then the La-
grange bracket [F, G] collapses the PoiSSON BRACKET

(F,G).

see also LIE BRACKET, POISSON BRACKET

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary

of Mathematics. Cambridge, MA: MIT Press, p. 1004,

1980.

Lagrange-Biirmann Theorem

see Lagrange Inversion Theorem

Lagrangian Coefficient

Coefficients which appear in Lagrange Interpo-
lating Polynomials where the points are equally
spaced along the ABSCISSA.

Lagrange's Continued Fraction Theorem

The Real Roots of quadratic expressions with integral
Coefficients have periodic Continued Fractions,
as first proved by Lagrange.

Lagrangian Derivative

see Convective Derivative

Lagrange's Equation

The Partial Differential Equation

(1 + fy 2 )f,x + 2f x f y f xy + (1 + fx 2 )fyy = 0,

whose solutions are called Minimal Surfaces.
see also Minimal Surface

References

do Carmo, M. P. "Minimal Surfaces." §3.5 in Mathemati-
cal Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 41-43, 1986.

{y - yo) k

dx*- 1

x — #o

f{x) -yo

g(x) = g(x Q )

(y-yo) k
k=i

+E-

k\

{l^r [»'<*>(

X — Xo

J(x)~yo

see also Maclaurin Series, Taylor Series

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 14, 1972.

Lagrange's Four-Square Theorem

A theorem also known as BACHET'S CONJECTURE which
was stated but not proven by Diophantus. It states that
every POSITIVE Integer can be written as the Sum
of at most four SQUARES. Although the theorem was
proved by Fermat using infinite descent, the proof was
suppressed. Euler was unable to prove the theorem. The
first published proof was given by Lagrange in 1770 and
made use of the Euler Four-Square Identity.

see also EULER FOUR-SQUARE IDENTITY, FERMAT'S

Polygonal Number Theorem, Fifteen Theorem,
Vinogradov's Theorem, Waring's Problem

Lagrange's Group Theorem

Also known as Lagrange's Lemma. If A is an Ele-
ment of a Finite Group of order n, then A n = 1. This
implies that e\n where e is the smallest exponent such
that A e = 1. Stated another way, the Order of a Sub-
group divides the Order of the Group. The converse
of Lagrange's theorem is not, in general, true (Gallian
1993, 1994).

References

BirkhofF, G. and Mac Lane, S. A Brief Survey of Modern
Algebra, 2nd ed. New York: Macmillan, p. Ill, 1965.

Gallian, J. A. "On the Converse of Lagrange's Theorem."
Math. Mag. 63, 23, 1993.

Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lex-
ington, MA: D. C. Heath, 1994.

Herstein, I. N. Abstract Algebra, 2nd ed. New York: Macmil-
lan, p. 66, 1990.

Hogan, G. T. "More on the Converse of Lagrange's Theo-
rem." Math. Mag. 69, 375-376, 1996.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 86, 1993.

Lagrange's Identity

The vector identity

(AxB)-(CxD)- (A.C)(B.D)-(A-D)(B.C). (1)

1014 Lagrange's Interpolating Fundamental Lagrange Interpolating Polynomial

This identity can be generalized to n-D, Lagrange Interpolating Polynomial

(ai x • • • x a n _i) • (bi x • • • x b n _i)

ai • bi • • ai * b n _i

a n -i ■ bi • • ■ a n _i ■ b„_i
where |A| is the DETERMINANT of A, or

(2)

/ ^kbk

, fc=i

■ IS-') IS-'J

— 2_^ ( a kbj — a,jbk) . (3)

1<k<j<n

see also Vector Triple Product, Vector Quad-
ruple Product

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1093, 1979.

Lagrange's Interpolating Fundamental Poly-
nomial

Let l(x) be an nth degree POLYNOMIAL with zeros at
xi, . . . , x m . Then the fundamental POLYNOMIALS are

lu{x)

l(x)

V(x v )(x - X u ) '

They have the property

Lu\Xj = *V^,

(1)

(2)

where <5„ M is the Kronecker. Delta. Now let /i , . . . ,
f n be values. Then the expansion

L„(x) = yj v l v (x)

(3)

gives the unique Lagrange Interpolating POLY-
NOMIAL assuming the values f v at x v . Let da(x) be
an arbitrary distribution on the interval [a, b], {p n (x)}
the associated ORTHOGONAL POLYNOMIALS, and Zi(x),
. . . , l n (x) the fundamental POLYNOMIALS corresponding
to the set of zeros of p n (x). Then

J a

l i/ {x)l^{x)da{x) = XpSv

(4)

for i/, fj, = 1, 2, . . . , n, where X u are CHRISTOFFEL NUM-
BERS.

References

Szego, G. Orthogonal Polynomials, 4th ed. Providence, RI:
Amer. Math. Soc, pp. 329 and 332, 1975.

The Lagrange interpolating polynomial is the POLY-
NOMIAL of degree n — 1 which passes through the n
points yi = /(an), y 2 = f{x 2 ), ..., y n = /(z„). It
is given by

P(x) = ^P,(x),

j=i

where

p >w = n £

Written explicitly,

x — Xk

k=i

Xj Xk

-Vi-

(1)

(2)

_ (x - x 2 ){x -x 3 )---(x- x n )
■l\X) — , w . , .y\

(Xl - X 2 ){Xi - X 3 ) • * ■ (Xl - X n )

(x - xi)(x - X3) ; • • (x - x n )
(x 2 - xi)(x 2 - x 3 ) ■ ■ * (x 2 - Xn
(a? - xi)(x - x 2 ) • • • (x - x n -i)

2/2 H- • • *

yn. (3)

(x n - X 1 )(x n — X 2 ) ' ' ' (X n — X n -i)

For n = 3 points,

(x-x 2 )(x-x 3 ) ,, , (a:-xi)(x-x 3 ) .,
P(x) - -yi + r~ ry 2

[Xl - X 2 )(Xi - X 3 ) (X 2 - Xi){X 2 - X3)

(x -xi)(x - x 2 )

+ r B ^2/3

{xs - xiJ(X3 - x 2 )

_,, . 2a - £ 2 - X3 , 2x - xi - 2:3

P ( x ) = 7 \7 ^ +

(4)

(xi — x 2 )(xi — 0)3) (x2 — xi)(x 2 — Xs)

1x — xi — x 2
+ 7 72/3-

2/2

(x 3 - xi){x% - x 2 )"

(5)

Note that the function P(x) passes through the points
(%ii Vi), as can De seen f° r the case n = 3,

0/ , (xi - x 2 )(xi - x 3 ) , (xi - xi)(xi - x 3 )
P \ x ^) = "h: w„ ^T2/i + 7Z w„ ZT\V 2

(xx - x 2 )(xi - xs) (x 2 — asi)(aj 2 - x 3 )*

(xi - xi)(xi - x 2 )
+ 7 r? 72/3 = 2/1

(x 3 — xi)(x 3 — x 2 ) "

(6)

(a? 2 ~x 2 )(x 2 - x z ) (x 2 - xi)(x 2 ~x 3 ) _

P{X 2 ) = 7 T7 rj/l + 7 T7 r2/2

(Xl — X2XX1 — X3J [X2 — Xij(X2 — X3J

Lagrange Inversion Theorem

Lagrange Multiplier 1015

(X 2 ~ X!)(X 2 ~ X 2 ) _

(X3 - xi)(a:3 - X2)

(7)

_, (X 3 ~ X 2 ){xz - Xz) , (%3 ~ Xi)(x 3 ~ X 3 )

(cci - x 2 )(xi - x 3 ) (z 2 — xi)(x 2 - x 3 ) *

+ 7 r? ^2/3 = 2/3.

(l 3 - Xi)(x 3 - X 2 ) L

(8)

Generalizing to arbitrary n,

n n

P( Xj ) = J]P fc fe) = J]<W - y,-. (9)

The Lagrange interpolating polynomials can also be
written using

jt(x) = JJ(x - x fc ), (10)

fc=l

n

A x j) = Y[( X 3 ~Vk), (11)

fc=l

fc^;

p(x) = x:

?r ( a; )

fc=i

(a; - Xfc)7r'-(arfc)

yk-

(13)

Lagrange interpolating polynomials give no error esti-
mate. A more conceptually straightforward method for
calculating them is NEVILLE'S ALGORITHM.

see also AlTKEN INTERPOLATION, LEBESGUE CON-
STANTS (Lagrange Interpolation), Neville's Al-
gorithm, Newton's Divided Difference Interpo-
lation Formula

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 878-879 and 883, 1972.

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 439, 1987.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Polynomial Interpolation and Extrapolation"
and "Coefficients of the Interpolating Polynomial." §3.1
and 3.5 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 102-104 and 113-116, 1992.

Lagrange Inversion Theorem

Let z be defined as a function of w in terms of a param-
eter a by

z — w -j- a<p(z).

Then any function of z can be expressed as a POWER
Series in a which converges for sufficiently small a and
has the form

F(z) = F(w) + j<P(w)F'(w) + ^ £{[0( w )f F'M}

References

Goursat, E. Functions of a Complex Variable, Vol. 2, Pt. 1.

New York: Dover, 1959.
Moulton, F. R. An Introduction to Celestial Mechanics, 2nd

rev. ed. New York: Dover, p. 161, 1970.
Williamson, B. "Remainder in Lagrange's Series." §119 in

An Elementary Treatise on the Differential Calculus, 9th

ed. London: Longmans, pp. 158-159, 1895.

Lagrange's Lemma

see Lagrange's Four-Square Theorem

Lagrange Multiplier

Used to find the EXTREMUM of /(#i, x 2l . . . , x n ) sub-
ject to the constraint g(xi } x 2 , . . . ,x n ) = C, where
/ and g are functions with continuous first PARTIAL
Derivatives on the Open Set containing the curve
g(xi, #2, . . . , x n ) — 0, and Vg ^ at any point on the
curve (where V is the Gradient). For an Extremum
to exist,

df=p-dx 1 + p-dx 2 + ... + p-dx n = 0. (1)

OX\ OX 2 OX n

But we also have

dg = ^ dx! + -^- dx 2 + . . . + ^- dx n = 0. (2)
ax\ ox 2 dx n

Now multiply (2) by the as yet undetermined parameter
A and add to (1),

(^ + x*L) dxi + (°L + xgL) dta

V ox\ oxi J \ ox 2 ox 2 }

+ - + (& + *it)*- ft (3)

Note that the differentials are all independent, so we can
set any combination equal to 0, and the remainder must
still give zero. This requires that

OXk OXk

(4)

for all k = 1, . . . , n. The constant A is called the
Lagrange multiplier. For multiple constraints, g\ = 0,
92 = 0, ... ,

V/ = AiVsi +A 2 Vs2 + -...

see also Kuhn-Tucker Theorem

(5)

References

Arfken, G. "Lagrange Multipliers." §17.6 in Mathematical

Methods for Physicists, 3rd ed. Orlando, FL: Academic

Press, pp. 945-950, 1985.

1016 Lagrange Number (Diophantine Equation)

Laguerre Differential Equation

Lagrange Number (Diophantine Equation)

Given a Fermat Difference Equation (a quadratic
Diophantine Equation)

2 2 A
r y =4

with r a QUADRATIC SURD, assign to each solution x\y
the Lagrange number

z= \(x + yr).

The product and quotient of two Lagrange numbers are
also Lagrange numbers. Furthermore, every Lagrange
number is a Power of the smallest Lagrange number
with an integral exponent.

see also PELL EQUATION

References

Dorrie, H. 100 Great Problems of Elementary Mathematics:

Their History and Solutions. New York: Dover, pp. 94-95,

1965.

Lagrange Number (Rational
Approximation)

Hurwitz's Irrational Number Theorem gives the
best rational approximation possible for an arbitrary ir-
rational number a as

V

W

The L n are called Lagrange numbers and get steadily
larger for each "bad" set of irrational numbers which is
excluded.

n Exclude L n

1 none

2 (f)

3 V2

/221
5

Lagrange numbers are of the form

9 ™2>

where m is a MARKOV NUMBER. The Lagrange numbers
form a Spectrum called the Lagrange Spectrum.

see also Hurwitz's Irrational Number Theo-
rem, Liouville's Rational Approximation The-
orem, LlOUVILLE-ROTH CONSTANT, MARKOV NUM-
BER, Roth's Theorem, Spectrum Sequence, Thue-
Siegel-Roth Theorem

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 187-189, 1996.

Lagrange Polynomial

see Lagrange Interpolating Polynomial

Lagrange Remainder

Given a TAYLOR Series, the error after n terms is
bounded by

/ (B) (0,

Rn —

'-{x-a) n

for some £ 6 (a,x).

see also Cauchy Remainder Form, Taylor Series

Lagrange Resolvent

A quantity involving primitive cube roots of unity which
can be used to solve the CUBIC EQUATION.

References

Faucette, W. M. "A Geometric Interpretation of the Solution
of the General Quartic Polynomial." Amer. Math. Monthly
103, 51-57, 1996.

Lagrange Spectrum

A Spectrum formed by the Lagrange Numbers. The
only ones less than three are the Lagrange Numbers,
but the last gaps end at Freiman's Constant. Real
Numbers larger than Freiman's Constant are in the
Markov Spectrum.

see also Freiman's Constant, Lagrange Number
(Rational Approximation), Markov Spectrum,
Spectrum Sequence

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 187-189, 1996.

Laguerre Differential Equation

xy" + (1 - x)y + Ay = 0.

(i)

The Laguerre differential equation is a special case of the
more general "associated Laguerre differential equation"

xy" + (v + 1 — x)y 4- Ay =

(2)

with v = 0. Note that if A = 0, then the solution to the
associated Laguerre differential equation is of the form

y"(x) + P{x)y'(x) = 0,

(3)

and the solution can be found using an INTEGRATING
Factor

fi = exp

[ / P(x) dx j = exp I / dx

i/ + l -ac

= exp[(z/ + 1) lnx — x] — x u e

(4)

V = Cif^ + C a = C 1 f£ i dx + C*. (5)

The associated Laguerre differential equation has a
Regular Singular Point at and an Irregular

Laguerre Differential Equation

SINGULARITY at oo. It can be solved using a series ex-
pansion,

oo oo

x\ n(n — l)a n x n ~ 2 -f (y + 1) > na n x n ~

n=2 n=l

oo oo

—x y na n x n ~ -f A > a n aj n = (6)

n=l n=0

oo oo

> n(n — l)a n £ n ~ + (^ + 1) > na n cc n ~

oo oo

- J^ na n x n + A J^ a n z n = (7)

n=l n=0

oo oo

y (n + l)na n+ ia: n + (i/ + 1) ^ (ra + l)a n+ i# n

oo oo

- ^ nana; 71 + A ^2 anXn = ° ( 8 )

n— 1 n=0

[(n + l)ai + Xa ]

oo

+ ^{[(n + l ^ n + (^ + x )( n + ^W+i - na n + Aa n }x n

= (9)

[(n + l)ai + Aa ]

oo

+ ^[(n + l)( n + i/ + l)a n +i + (A - n)^ 71 = 0. (10)

n=l

This requires

ax =

Ctn+l —

1/ + 1

ao

n — A

(ra+l)(n + z/+l)

for n > 1. Therefore,

&n+l

n — A

(n + l)(n + i/ + l)

(11)
(12)

(13)

for n — 1, 2, . . . , so

y = ao

A(l-A) ^ 2

I/+1 2(z/+l)(i/ + 2)

A(l- A)(2-A)
2-3(z/+l)(i/ + 2)(i/ + 3)

(14)

If A is a Positive Integer, then the series terminates
and the solution is a Polynomial, known as an asso-
ciated Laguerre Polynomial (or, if v = 0, simply a
Laguerre Polynomial).

see also LAGUERRE POLYNOMIAL

Laguerre- Gauss Quadrature 1017

Laguerre- Gauss Quadrature

Also called Gauss-Laguerre Quadrature or La-
guerre QUADRATURE. A GAUSSIAN QUADRATURE
over the interval [0, oo) with WEIGHTING FUNCTION
W(x) = e~ x . The ABSCISSAS for quadrature order n
are given by the ROOTS of the LAGUERRE POLYNOMI-
ALS L n (x). The weights are

Wi =

7n-l

A n Z4(£i)L n +l(Xi) -An-l L n -i(Xi)L' n (Xi) '

(1)

where A n is the COEFFICIENT of x n in L n {x). For La-
guerre Polynomials,

A n = (-l) n n!,
where n! is a FACTORIAL, so

A n

= -(n + l).

Additionally,

7n = 1,

W t —

n + 1

£ n +l(#i)£n(#i) L Tl _i(a?i)Z/J l (^i)

(2)

(3)
(4)

(5)

(Note that the normalization used here is different than
that in Hildebrand 1956.) Using the recurrence relation

xL n (x) = nL n (x) — nLn-i(x)

= (x - n - l)L n (x) + (n + l)L„+i(a:) (6)

which implies

XiL' n (xi) = -nL n -i(a;i) = (n + l)L„+i(sci) (7)

gives

1 Xi

Wi =

Xi[L' n {xi)]* (n+l) 2 [L n+ i(x0] 2 '
The error term is

'"" 3 r/ (an) (0-

E =

(2n)!"

(8)

(9)

Beyer (1987) gives a table of Abscissas and weights up
to n = 6.

n Xi

2

0.585786

0.853553

3.41421

0.146447

3

0.415775

0.711093

2.29428

0.278518

6.28995

0.0103893

4

0.322548

0.603154

1.74576

0.357419

4.53662

0.0388879

9.39507

0.000539295

5

0.26356

0.521756

1.4134

0.398667

3.59643

0.0759424

7.08581

0.00361176

12.6408

0.00002337

1018 Laguerre's Method

Laguerre Polynomial

The Abscissas and weights can be computed analyti-
cally for small n.

n Xi

Wi

2 2-\/2 J(2 + >/2)
2 + \/2 J(2-v^)

For the associated Laguerre polynomial L„(x) with
Weighting Function w(x) = x e~ x ,

A n = (-IT

(10)

and

/•oo

y n = n\ x + n e- x dx = n\T(n + /? + 1). (11)

Jo

The weights are

_ ra!r(n + ff + l) _ n!r(n + /3 + l)a?i
x,[ir5.'(a; 4 )] a " [4! + i(*0] a '

«;» =

(12)

where T(z) is the Gamma Function, and the error term
En = n^(n + + l) f(2n) ^ (13)

(2n)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.

Boca Raton, FL: CRC Press, p. 463, 1987.
Chandrasekhar, S. Radiative Transfer. New York: Dover,

pp. 64-65, 1960.
Hildebrand, F. B. Introduction to Numerical Analysis. New

York: McGraw-Hill, pp. 325-327, 1956.

Laguerre's Method

A RoOT-finding algorithm which converges to a COM-
PLEX ROOT from any starting position.

P n (x) = (x - xi)(x - x 2 ) - ■ • (x - x n ) (1)

ln\P n (x)\ =ln|a:-ai|+ln|a-X2| + ... + ln|a;-a:n| (2)

P' n {x) = (X - X 2 ) * ' ' (X - X n ) + (X - Xi) ■ ■ ■ (X - X n ) + . . .

l( *)(-i- + ... + -^)

\X — X\ X — X n /

(3)

dln\P n (x)
dx

^ln|P w ( g )|
dx 2

1 1

+

X — X\ x — x 2

+ ...+ ■

Pn(x)

G(X)

(4)

+

(x — Xl) 2 (x — X2)

' KM '

P n (x)

P„(x) -

+ ...+

H(x).

1

(x - x„) 2

(5)

Now let a ~ x — xi and b~x — x\. Then

a

n

]G±y/{n-\){nH-G*)}'

(6)
(7)
(8)

Setting n = 2 gives HALLEY'S IRRATIONAL FORMULA.

see a/50 Halley's Irrational Formula, Halley's
Method, Newton's Method, Root

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 365-366, 1992.

Ralston, A. and Rabinowitz, P. §8.9-8.13 in A First Course
in Numerical Analysis, 2nd ed. New York: McGraw-Hill,
1978.

Laguerre Polynomial

Solutions to the Laguerre Differential Equation
with v — are called Laguerre polynomials. The
Laguerre polynomials L n (x) are illustrated above for
x € [0,1] and n = 1, 2, . . . , 5.

The Rodrigues formula for the Laguerre polynomials is

e x d n

K ; n\ dx n v '

(1)

and the GENERATING FUNCTION for Laguerre polyno-
mials is

*c

exp(-^-)
5 , z ) - i_^ = ! + (-* + l)z

+ {\x 2 ~~ 2x + l)z 2 + (-|z 3 + \x 2 -Zx + l)z 3 + . . .

A Contour Integral is given by

1 r e -xz/(i-z)

L n (x) = — — / — — dz.

27TI J (1 - ^)^ Tl+1

(2)

(3)

Laguerre Polynomial

Laguerre Polynomial 1019

The Laguerre polynomials satisfy the RECURRENCE RE-
LATIONS

(n+l)L n +i(z) = (2n+l-x)L n (x) - nL n -i(x) (4)

(Petkovsek et cd. 1996) and

xL' n (x) = nL n (x) - nL n -i(x). (5)

The first few Laguerre polynomials are

L (x) = 1

L\{x) — — x + 1

L 2 (x) = \{x 2 -±x + 2)

L 3 (x) = |(-z 3 + 9x 2 - I8x + 6).

Solutions to the associated Laguerre Differential
Equation with v ^ are called associated Laguerre
polynomials L^{x). In terms of the normal Laguerre
polynomials,

L n (x) = L° n (x). (6)

The Rodrigues formula for the associated Laguerre poly-
nomials is

and

n! dx n

= (-!)" £r(W*(*)]

oo

(7)

(n + fc)!

(n — m)!(& + m)!m!

T^*" 1 < 8 >

and the GENERATING FUNCTION is

exp(-a)
5(X ' Z)= (1 -*)*+*
l + (k + l-x)z+±[x 2 -2(k + 2)x + {k + l)(k + 2)]z 2 + . . .

(9)

The associated Laguerre polynomials are orthogonal
over [0, oo) with respect to the WEIGHTING FUNCTION

[ X e- x x k L k n (x)L k m {x) dx = {n+ . k)l S mn , (10)
Jo n -

where 5 mn is the Kronecker Delta. They also satisfy
f 00 e—x k+1 [L*(x)] 2 dx = (n + fc)! (2n + fc + 1). (11)

*/0

Recurrence Relations include

Y J L^\x) = L^\x)

l£\x) = LL a+1) (x) - l£%\x). (13)

The Derivative is given by

(a)

In terms of the Confluent Hypergeometric Func-
tion,

L *( x ) = (*±ik 1 F 1 (-6;& + l;x). (15)

n!

An interesting identity is

y w 1 i^ w " = ^(x W )-° /2 J«(2 A /5^), (16)

is r(n + a + 1)

where F(z) is the Gamma FUNCTION and J a (z) is the
Bessel Function of the First Kind (Szego 1975,
p. 102). An integral representation is

e- x x a/2 L { n a) (x) = -i f°° e- t t n +" /2 J a (2Vix~)dt (17)
n! Jo

for n - 0, 1, . . . and a > -1. The DISCRIMINANT is

n

Di a) = JJ I /"- 2n+2 ( I / + a)"" 1 (18)

(Szego 1975, p. 143). The KERNEL POLYNOMIAL is

K M (xv)= n + 1 /" w + a V 1
An l * ,2/j T(a + 1)^ n 7

x-y

(19)

(12)

where (™) is a Binomial Coefficient (Szego 1975,
p. 101).

The first few associated Laguerre polynomials are

L k Q {x) = l

L k (x) = -x + k + l

L k 2 (x) = \[x 2 - 2{k + 2)x + {k + 1)(A; + 2)]
L3W = |[-x 3 + 3(fe + 3)x 2 - 3(fc + 2)(& + 3)a;
+ (fc + l)(A: + 2)(fc + 3)].

see a/50 Sonine Polynomial

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal
Polynomials." Ch. 22 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 771-802, 1972.

1020 Laguerre Quadrature

Laman's Theorem

Arfken, G. "Laguerre Functions." §13.2 in Mathematical
Methods for Physicists, 3rd ed, Orlando, FL: Academic
Press, pp. 721-731, 1985.

Chebyshev, P. L. "Sur le developpement des fonctions a
une seule variable." Bull. Ph. -Math., Acad. Imp. Sc. St.
Petersbourg 1, 193-200, 1859.

Chebyshev, P. L. Oeuvres, Vol. 1. New York: Chelsea,
pp. 499-508, 1987.

Iyanaga, S. and Kawada, Y. (Eds.). "Laguerre Functions."
Appendix A, Table 20. VI in Encyclopedic Dictionary of
Mathematics. Cambridge, MA: MIT Press, p. 1481, 1980.

Laguerre, E. de. "Sur l'integrale J °° x~ 1 e~ x dx" Bull.
Soc. math. France 7, 72-81, 1879. Reprinted in Oeuvres,
Vol 1. New York: Chelsea, pp. 428-437, 1971.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, pp. 61-62, 1996.

Sansone, G. "Expansions in Laguerre and Hermite Series."
Ch. 4 in Orthogonal Functions, rev. English ed. New York:
Dover, pp. 295-385, 1991.

Spanier, J. and Oldham, K. B. "The Laguerre Polynomials
L n (x)." Ch. 23 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 209-216, 1987.

Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, 1975.

Laguerre Quadrature

A Gaussian QuADRATURE-like Formula for numerical
estimation of integrals. It fits exactly all POLYNOMIALS
of degree 2m — 1.

References

Chandrasekhar, S. Radiative Transfer. New York: Dover,
p. 61, 1960.

Laguerre's Repeated Fraction

The Continued Fraction

(x + 1)" - (x - l) n _ n n 2 - 1 n 2 - 2 2

(x + l) n + (x - l) n x+ 3x+ 5x + .

References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 13, 1959.

Laisant's Recurrence Formula

The Recurrence Relation

(n - l)A„+i = (n - l)A n 4- (n + l)A n -i + 4(-l) n

with A(l) = A(2) — 1 which solves the Married COU-
PLES Problem.

see also Married Couples Problem

Lakshmi Star

see Star of Lakshmi

Lai's Constant

Let P(N) denote the number of Primes of the form
n 2 + 1 for 1 < n < TV, then

P{N)~ 0.68641 li(iV),

(1)

where \i(N) is the LOGARITHMIC INTEGRAL (Shanks
1960, pp. 321-332). Let Q(N) denote the number of
PRIMES of the form n 4 + 1 for 1 < n < N, then

Q(N) ~ \si li(JV) = 0.669741i(JV) (2)

(Shanks 1961, 1962). Let R[N) denote the number of
pairs of PRIMES (n-l) 2 + l and (n+l) 2 + l for n < JV-1,
then

R(N) -0.487621i 2 (iV), (3)

where

lia(JV);

f dn

(4)

(Shanks 1960, pp. 201-203). Finally, let S(N) denote
the number of pairs of PRIMES (n — 1) 4 + 1 and (n+l) 4 + l
for n < N — 1. then

S{N)~ Alia(JV)

(5)

(Lai 1967), where A is called Lai's constant. Shanks
(1967) showed that A « 0.79220.

References

Lai, M. "Primes of the Form n 4 + 1." Math. Comput. 21,

245-247, 1967.
Shanks, D. "On the Conjecture of Hardy and Littlewood

Concerning the Number of Primes of the Form n 2 + a."

Math. Comput 14, 321-332, 1960.
Shanks, D, "On Numbers of the Form n 4 + 1." Math. Com-
put 15, 186-189, 1961.
Shanks, D. Corrigendum to "On the Conjecture of Hardy and

Littlewood Concerning the Number of Primes of the Form

n 2 + a." Math. Comput. 16, 513, 1962.
Shanks, D. "Lai's Constant and Generalization." Math.

Comput 21, 705-707, 1967.

Lam's Problem

Given an 111 x 111 MATRIX, fill 11 spaces in each row
in such a way that all columns also have 1 1 spaces filled.
Furthermore, each pair of rows must have exactly one
filled space in the same column. This problem is equiva-
lent to finding a PROJECTIVE PLANE of order 10. Using
a computer program, Lam showed that no such arrange-
ment exists.

see also PROJECTIVE PLANE

Laman's Theorem

Let a GRAPH G have exactly 2n — 3 Edges, where n is
the number of VERTICES in G. Then G is "generically"
RIGID in R 2 IFF e < 2n' - 3 for every SUBGRAPH of G
having ri Vertices and r' Edges.
see also RIGID

References

Laman, G. "On Graphs and Rigidity of Plane Skeletal Struc-
tures." J. Engineering Math. 4, 331-340, 1970.

Lambda Calculus

Lambert Azimuthal Equal-Area Projection 1021

Lambda Calculus

Developed by Alonzo Church and Stephen Kleene to
address the COMPUTABLE NUMBER problem. In the
lambda calculus, A is denned as the ABSTRACTION OP-
ERATOR. Three theorems of lambda calculus are A-
conversion, a-conversion, and 77-conversion.

see also Abstraction Operator, Computable

Number

References

Penrose, R. The Emperor's New Mind: Concerning Comput-
ers, Minds, and the Laws of Physics. Oxford, England:
Oxford University Press, pp. 66-70, 1989.

Lambda Function

-15

The lambda function defined by Jahnke and Emden
(1945) is

A x (z) = ^ = 2jinc(z),

(1)

(2)

Lambda Group

The set of linear fractional transformations w which sat-
isfy

w(t)

at + b

ct + d'

where a and d are ODD and b and c are EVEN. Also
called the Theta Subgroup. It is a Subgroup of the
Gamma Group.

see also GAMMA GROUP

Lambda Hypergeometric Function

00 / , , 2n \ 8
n-1 V /

(1)

where q is the NOME. The lambda hypergeometric func-
tions satisfy the recurrence relationships

A(* + 2) = X(t)
A (2iTl)=^

(2)
(3)

Lambert Azimuthal Equal- Area Projection

where J\{z) is a Bessel Function of the First Kind
and jinc(z) is the Jinc Function.

A two-variable lambda function defined by Gradshteyn
and Ryzhik (1979) is

M m f*±$*.

(3)

where T{z) is the GAMMA FUNCTION.
see also Airy Functions, Dirichlet Lambda Func-
tion, Elliptic Lambda Function, Jinc Function,
Lambda Hypergeometric Function, Mangoldt
Function, Mu Function, Nu Function

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1079, 1979.

Jahnke, E. and Emde, F. Tables of Functions with Formulae
and Curves, 4th ed. New York: Dover, 1945.

x — k cos0sin(A — Ao) (1)

y = k'[cos<pi sin0 — sin^i cos<^cos(A — Ao)], (2)

where

1 + sin 0i sin</> + cos^i cos0cos(A — Ao)
The inverse FORMULAS are

y sine cos <fii

-C

(j) = sin I cos c sin cfii +

A = Ao + tan

P
xsinc

*)

p cos (pi cos c — y sin (pi sm c

where

p = y/x 2 + y 2
c = 2sin -1 (i/>).

(3)

(4)

,(5)

(6)
(7)

References

Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 182-190, 1987.

1022 Lambert Conformal Conic Projection

Lambert's Transcendental Equation

Lambert Conformal Conic Projection

where

x = psin[n(A - Ao)]

y = po - pcos[n(A - A )],

/>-Fcot n (|7r+|0)

p o =Fcot n (|7T+|0 O )

cos^itan n (|7r+ \<f>i)

F =

ln(cos^i sec ^2)

ln[tan(±7r + \fa) cot(±7r + \^)] '
The inverse FORMULAS are

l/n"

(f> = 2 tan

(7)

-I-

A = A +

where

p = sgn(n)^/x 2 + (po -y) 2

tan

x

po-y

(1)
(2)

(3)
(4)

(5)
(6)

(7)
(8)

(9)
(10)

References

Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 104-110, 1987.

Lambert's Method

A RooT-finding method also called Bailey's Method
and Hutton's Method. If g(x) = x d - r, then

H 9 (x)

__ (d-l)x d + (rf+l)r ^
~ {d+l)x* + (d- l)r°

References

Scavo, T, R. and Thoo, J. B. "On the Geometry of Halley's
Method." Amer. Math. Monthly 102, 417-426, 1995.

Lambert Series

A series of the form

F{x) = Y^ a ^YZ-,

(1)

for \x\ < 1. Then

00 00 00

F{x) = Y j anY J * mn = Y J '

where

>nx

n=l m = l

bN = 2_^ a " •

n\N

Some beautiful series of this type include
}ji{n)x n

(2)

(3)

^-^ }Ji\n)x

Z_-/ 1 _ x n

E!

4>(n)x n

(1-x) 2
00 00

n~l n=l

n=l n=l

(4)
(5)
(6)
(7)
(8)

where /i(n) is the MOBIUS FUNCTION, <f>(n) is the To-
tient Function, d(n) = <r (n) is the number of di-
visors of n, ak(n) is the Divisor Function, and r(n)
is the number of representations of n in the form n =
A 4- B 2 where A and B are rational integers (Hardy
and Wright 1979).

References

Abramowitz, M. and Stegun, C. A. (Eds,), "Number The-
oretic Functions." §24.3.1 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, pp. 826-827, 1972.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 257-258, 1979.

Lambert's Transcendental Equation

An equation proposed by Lambert (1758) and studied
by Euler in 1779 (Euler 1921).

x a - X P = ( a - 0)vx a+fl .

When ex — > j3, the equation becomes

In a; = vx^ '

Lambert's W -Function

Lambert's W -Function

1023

which has the solution

x — exp

W(-Pv)

where W is Lambert's VF-Function.

References

Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; and Jeffrey,

D. J. "On Lambert's W Function." ftp://watdragon.

uwaterloo . ca/cs-archive/CS-93-03/W .ps . Z.
de Bruijn, N. G. Asymptotic Methods in Analysis. Amster-
dam, Netherlands: North-Holland, pp. 27-28, 1961.
Euler, L. "De Serie Lambertina Plurismique Eius Insignibus

Proprietatibus." Leonhardi Euleri Opera Omnia, Ser. 1.

Opera Mathematica, Bd. 6, 1921.
Lambert, J. H. "Observations variae in Mathesin Puram."

Acta Helvitica, physico-mathematico-anatomico-botanico-

medica 3, 128-168, 1758.

Lambert's ^-Function

1.5

-0.

The inverse of the function

f(W) = We

(1)

also called the Omega Function. The function is
implemented as the Mathematical (Wolfram Research,
Champaign, IL) function ProductLogfz] . W(l) is
called the Omega Constant and can be considered
a sort of "Golden Ratio" of exponentials since

exp[-W(l)] = W(l),

giving

In

W(l)

W(l).

(2)

(3)

Lambert's P^-Function has the series expansion

, (

n-1)!

Jj dj | n Jj ts Jj

+ ^ 5 -fx 6 + lfiZ^ + .... (4)

The Lagrange Inversion Theorem gives the equiv-
alent series expansion

(-n)"

w>w = E i =^* B

(5)

where n\ is a FACTORIAL. However, this series oscillates
between ever larger Positive and Negative values for
Real z <: 0.4, and so cannot be used for practical nu-
merical computation. An asymptotic FORMULA which
yields reasonably accurate results for z ^ 3 is

W(z) = Lnz — InLnz

oo oo

+ J2J2 CkmilnLnzr+^Lnzy"-" 1 - 1

k=0 m=0

T r , L 2 , L 2 (~2 + L 2 )

+

+

+

L 2 (6-9L 2 + 2L 2 2

6L1 2
L 2 (-12 + 36L 2 - 22L 2 2 + 3L 2 3 )

12Li 4
L 2 (60 - 300L 2 + 350L 2 2 - 125L 2 3 + 12L 2 4 )
6OL1 5
L

ffi

where

L\ = Lnz
L 2 = \n~Lnz

(6)

(7)
(8)

(Corless et a/.), correcting a typographical error in de
Bruijn (1961). Another expansion due to Gosper is the
Double Sum

W(x) = a + f2\j2

Si(n.fc)

;=a^Mf)-«] " (n-*+l)l
\ *(5)

(9)

where Si is a nonnegative STIRLING Number OF THE
First Kind and a is a first approximation which can be
used to select between branches. Lambert's W-function
is two-valued for — 1/e < x < 0. For W(x) > — 1, the
function is denoted Wo (a:) or simply W(x) y and this is
called the principal branch. For W(x) < — 1, the func-
tion is denoted W-i(x). The Derivative of W is

W'(x) =

1

W{x)

[1 + W(x)] exp[W(x)] x[l + W(x)]
for x y£ 0. For the principal branch when z > 0,
\nW{z) =\nz-W(z).

(10)

(11)

see also Iterated Exponential Constants, Omega

Constant

References

Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; and Jeffrey,
D. J. "On Lambert's W Function." ftp://vatdragon.
uwaterloo . ca/cs-archive/CS-93-03/W.ps . Z.

de Bruijn, N. G. Asymptotic Methods in Analysis. Amster-
dam, Netherlands: North- Holland, pp. 27-28, 1961.

1024

Lame Curve

Lame's Theorem

Lame Curve

A curve with Cartesian equation

(:)"+(!)"

first discussed in 1818 by Lame. If n is a rational, then
the curve is algebraic. However, for irrational n, the
curve is transcendental. For Even Integers n, the
curve becomes closer to a rectangle as n increases. For
Odd Integer values of n, the curve looks like the Even
case in the Positive quadrant but goes to infinity in
both the second and fourth quadrants (MacTutor Ar-
chive). The Evolute of an Ellipse,

(ax) 2/3 + (by) 2/3 = (a 2 -b 2 ) 2/ \

n Curve

astroid

| superellipse
3 witch of Agnesi

see also Astroid, Superellipse, Witch of Agnesi

References

MacTutor History of Mathematics Archive. "Lame Curves."
http : //www-groups . dcs . st-and . ac . uk/ -history/Curves
/Lame. html.

Lame's Differential Equation

/ 2 i2w2 2\ Uj *> . ( 2

[X - ){X - C )~r-^ + X(X

+ X

2 2^dz

c) Tx

-[m(m + l)x 2 - (b 2 + c 2 )p]z = 0. (1)

(Byerly 1959, p. 255). The solution is denoted E^x)
and is known as a LAME FUNCTION or an ELLIPSOIDAL
Harmonic. Whittaker and Watson (1990, pp. 554-555)
give the alternative forms

4A;

d 2 A

dX 2

d 2 A
du 2

+

d_
dX

dA
'dX

— + —
a 2 + A b 2 + A c 2

[n(n+l)A + C]A (2)

dA _ [n(n+l)A + C]A

dX

4A,

(3)
= [n{n + l)p(u)+C -\n{n+l){a 2 +b 2 + c 2 )]A (4)

d 2 A
d Zl 2

= [n(n + l)A; 2 sn 2 a + A]A,

(5)

where p is a WElERSTRAfi ELLIPTIC FUNCTION and

m

A(9) = l[(e - e q ) (6)

q=l

A A = v /(a2+A)(62- h A)( C 2+A)

(7)

A

_ C - \n(n + l)(a 2 + b 2 + c 2 ) + e^n{n + 1)

ei - e 3

(8)

References

Byerly, W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.
New York: Dover, 1959.

Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.

Lame's Differential Equation (Types)
Whittaker and Watson (1990, pp. 539-540) write Lame's
differential equation for ELLIPSOIDAL HARMONICS of the
four types as

*K

\mT]

iA <

\m d T]

^<

\mT]

4A(»)i

r FwT j

= [2m(2m + 1)0 + C]A(9)

(1)

= [(2m + l)(2ro + 2)0 + C]A(9)

(2)
= [{2m + 2)(2m + 3)6* + C]A(6)

(3)
= [(2m + 3)(2m + 4)9 + C]A(0),

(4)

where

A(0) = ^(a2 + e)(b* + 0)(c* + 9) (5)

m

A(e) = l[(e-e q ). (6)

g=l

References

Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, ^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.

Lame Function

see Ellipsoidal Harmonic

Lame's Theorem

If a is the smallest Integer for which there is a smaller
Integer b such that a and b generate a Euclidean Al-
gorithm remainder sequence with n steps, then a is the
Fibonacci Number F n+2 . Furthermore, the number
of steps in the EUCLIDEAN ALGORITHM never exceeds 5
times the number of digits in the smaller number.

see also EUCLIDEAN ALGORITHM

References

Honsberger, R, "A Theorem of Gabriel Lame." Ch. 7 in
Mathematical Gems II. Washington, DC: Math. Assoc.
Amer., pp. 54-57, 1976.

Lamina
Lamina

A 2-D planar closed surface L which has a mass M
and a surface density <j(x, y) (in units of mass per areas
squared) such that

/ o-(x,

M = I <j(x, y) dx dy.

The Center of Mass of a lamina is called its Cen-

troid.

see also Centroid (Geometric), Cross-Section,

Solid

Laminated Lattice

A Lattice which is built up of layers of n-D lattices in
(n + 1)-D space. The Vectors specifying how layers
are stacked are called Glue Vectors.

see also Glue Vector, Lattice

References

Conway, J. H. and Sloane, N. J. A. "Laminated Lattices."

Ch. 6 in Sphere Packings, Lattices, and Groups, 2nd ed.

New York: Springer- Verlag, pp. 157-180, 1993.

Lancret Equation

dsN = *

st + dsB j

where N is the NORMAL VECTOR, T is the TANGENT,
and B is the Binormal VECTOR.

Lancret 's Theorem

A Necessary and Sufficient condition for a curve to
be a Helix is that the ratio of Curvature to Torsion
be constant.

Lanczos Approximation

see Gamma Function

Lanczos a Factor

Writing a Fourier Series as

m

f(9) - 2 a ° + X] SinC ( 2~ ) ^ n cos ( n ^) + bn sin ( n #)h

n = l

where m is the last term and the sine a: terms are the

Lanczos <x factor, removes the Gibbs Phenomenon

(Acton 1990).

see also Fourier Series, Gibbs Phenomenon, Sinc

Function

References

Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., p. 228, 1990.

Landau-Kolmogorov Constants 1025

Landau Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Let F be the set of COMPLEX analytic functions / de-
fined on an open region containing the closure of the
unit disk D — {z : \z\ < 1} satisfying /(0) = and
df/dz(0) = 1. For each / in F, let 1(f) be the Supre-
MUM of all numbers r such that f(D) contains a disk of
radius r. Then

L = inf {/(/) :f€F}.

This constant is called the Landau constant, or the
Bloch-Landau Constant. Robinson (1938, unpub-
lished) and Rademacher (1943) derived the bounds

\<L<

= 0.5432588...

where T(z) is the Gamma Function, and conjectured
that the second inequality is actually an equality,

r(f)r(§)
r(i)

0.5432588.

see also Bloch Constant

References

Finch, S. "Favorite Mathematical Constants." http://wv.

mathsof t . c om/ as o 1 ve / const ant /bloch/bloch. html.
Rademacher, H. "On the Bloch-Landau Constant." Amer.

J. Math. 65, 387-390, 1943.

Landau-Kolmogorov Constants

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Let ll/H be the Supremum of |/(x)|, a real-valued func-
tion / defined on (0, oo). If / is twice differentiate and
both / and /" are bounded, Landau (1913) showed that

ll/'ll<2|

1/2 1 1 wi 1 .1/2

'11/

(1)

where the constant 2 is the best possible. Schoenberg
(1973) extended the result to the nth derivative of /
denned on (0, oo) if both / and f^ are bounded,

I/WII^ckjoii/ii 1

1-fc/nn r(n)ijfc/n

'11/

(2)

An explicit FORMULA for C(n, k) is not known, but par-
ticular cases are

C(3,2) = 24 1/3
C(4,l) = 4.288...
C(4,2) = 5.750...
C(4,3) = 3.708....

(3)

(4)
(5)
(6)
(7)

1026 Landau-Kolmogorov Constants

Let ll/H be the SUPREMUM of \f{x)\, a real-valued func-
tion / defined on ( — 00,00). If / is twice differentiable
and both / and /" are bounded, Hadamard (1914)
showed that

ii/'ii<^ii/n 1/2 ii/"ii 1/2 ,

(8)

where the constant y/2 is the best possible. Kolmogorov
(1962) determined the best constants C(n,k) for

||/ (fc) ||<C(n,ife)||/|| 1 " Vn ||/ (n) || Vn
in terms of the Favard CONSTANTS

j=0

(-1)'

2j + l

n + l

by

C(n t k) — a n - k a n 1+fe/n ,
Special cases derived by Shilov (1937) are

C(3,l)=(|)

C(3,2) = 3 1/3

125X 1 / 5

cm -(f)

(9)

(10)
(11)

(12)
(13)
(14)

(15)

(16)

(17)

(18)

For a real- valued function / defined on ( — 00, 00), define

>
[f(x)] 2 dx. (19)

-K

If / is n differentiable and both / and /^ are bounded,
Hardy et al. (1934) showed that

f(k)

ii/ w ii<n/ir" /n ii/ !

l-k/n 11 x(n)iifc/n

(20)

where the constant 1 is the best possible for all n and
< k < n.

For a real- valued function / defined on (0, 00), define
11/11 = jf°[/(x)Pdz. (21)

H:

Landau-Kolmogorov Constants

If / is twice differentiable and both / and /" are
bounded, Hardy et al (1934) showed that

l/'ll<V2||/ll 1/2 ll/ (n) l! 1/2 ,

(22)

where the constant y/2 is the best possible. This inequal-
ity was extended by Ljubic (1964) and Kupcov (1975)
to

||/ (fc) H<C'(n ) fc)||/|| 1 - fc/n ||/ (n) || fc/n (23)

where C(n, k) are given in terms of zeros of POLYNOMI-
ALS. Special cases are

C(3, 1) = C(3,2) = 3 l/2 [2(2 1/2 - 1)]~ 1/3

= 1.84420... (24)

/31/4 + 3-3/4
C(4,l) = C(4,3) = y^— ±-*

= 2.27432...

(25)

C(4,2) = W| =2.97963...

(26)

C(4,3) = (-)

(27)

C(5, 1) = C(5,4) = 2.70247...

(28)

C(5,2) = C(5,3) = 4.37800...,

(29)

where a is the least POSITIVE Root of

x 8 - 6x 4 - Sx 2 + 1 = (30)

and b is the least Positive Root of

x 4 - 2x 2 - 4x + 1 =

(31)

(Franco et al. 1985, Neta 1980). The constants C(n,l)
are given by

CM )=;»^, (32)

where c is the least Positive Root of

i*c poo

Jo Jo <^~

dxdy

= h- (33)

■ yx 2 + l)y/y In

An explicit Formula of this type is not known for k >
1.

The cases p = 1, 2, 00 are the only ones for which the
best constants have exact expressions (Kwong and Zettl
1992, Franco et al 1983).

References

Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft . com/asolve/constant/lk/lk.html.

Franco, Z. M.; Kaper, H. G.; Kwong, M. N.; and Zettl, A.
"Bounds for the Best Constants in Landau's Inequality on
the Line." Proc. Roy. Soc. Edinburgh 95 A, 257-262, 1983.

Landau-Ramanujan Constant

Landen's Formula 1027

Franco, Z. M.; Kaper, H. G.; Kwong, M.N.; and Zettl, A.
"Best Constants in Norm Inequalities for Derivatives on a
Half Line." Proc. Roy. Soc. Edinburgh 100 A, 67-84, 1985.

Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities.
Cambridge, England: Cambridge University Press, 1934.

Kolmogorov, A. "On Inequalities Between the Upper Bounds
of the Successive Derivatives of an Arbitrary Function on
an Infinite Integral." Amer. Math. Soc. Translations, Ser.
1 2, 233-243, 1962.

Kupcov, N. P. "Kolmogorov Estimates for Derivatives in
L 2 (0,oo)." Proc. Steklov Inst Math. 138, 101-125, 1975.

Kwong, M. K. and Zettl, A. Norm Inequalities for Deriva-
tives and Differences. New York: Springer- Verlag, 1992.

Landau, E. "Einige Ungleichungen fin* zweimal different zier-
bare Funktionen." Proc. London Math. Soc. Ser. 2 13,
43-49, 1913.

Landau, E. "Die Ungleichungen fur zweimal different zier bare
Funktionen." Danske Vid. Selsk. Math. Fys. Medd. 6,
1-49, 1925.

Ljubic, J. I. "On Inequalities Between the Powers of a Linear
Operator." Amer. Math. Soc. Trans. Ser. 2 40, 39-84,
1964.

Neta, B. "On Determinations of Best Possible Constants in
Integral Inequalities Involving Derivatives." Math. Corn-
put. 35, 1191-1193, 1980.

Schoenberg, I. J. "The Elementary Case of Landau's Prob-
lem of Inequalities Between Derivatives." Amer. Math.
Monthly 80, 121-158, 1973.

Landau- Ramanujan Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Let S(x) denote the number of POSITIVE INTEGERS not
exceeding x which can be expressed as a sum of two
squares, then

lim S(x) = K,

x— j-oo X

(1)

as proved by Landau (1908) and stated by Ramanujan.
The value of K (also sometimes called A) is

K =

i

n

= 0.764223653.

(2)

p a prime
= 3 (mod 4)

(Hardy 1940, Berndt 1994). Ramanujan found the ap-
proximate value K = 0.764. Flajolet and Vardi (1996)
give a beautiful FORMULA with fast convergence

k l n Id M c(2 " }

0(2")

l/(2" + l)

where

/?(,) = A[c(,,i)-c(0]

(3)

(4)

is the DlRICHLET BETA FUNCTION, and C{z,a) is the
HURWITZ ZETA FUNCTION. Landau proved the even
stronger fact

where

Hl

■"(£)

ld_
Ads

In

11 w ~2s

p prime
lP =4fc+3

= 0.581948659....

Here,

5.2441151086..

3 = 1

(6)

(7)

is the Arc Length of a Lemniscate with a — 1 (the
Lemniscate Constant to within a factor of 2 or 4),
and 7 is the EULER-MASCHERONI CONSTANT.

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 60-66, 1994.

Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/lr/lr.html.

Flajolet, P. and Vardi, I. "Zeta Function Expan-
sions of Classical Constants." Unpublished manu-
script, 1996. http://pauillac.inria.fr/algo/flajolet/
Publicat ions/landau. ps.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
pp. 61-63, 1940.

Landau, E. "Uber die Einteilung der positiven ganzen Zahlen
in vier Klassen nach der Mindeszahl der zu ihrer additiven
Zusammensetzung erforderlichen Quadrate." Arch. Math.
Phys. 13, 305-312, 1908.

Shanks, D. "The Second-Order Term in the Asymptotic Ex-
pansion of B(x). n Math. Comput. 18, 75-86, 1964.

Shanks, D. "Non- Hypotenuse Numbers." Fibonacci Quart.
13, 319-321, 1975.

Shanks, D. and Schmid, L. P. "Variations on a Theorem of
Landau. I." Math. Comput. 20, 551-569, 1966.

Shiu, P. "Counting Sums of Two Squares: The Meissel-
Lehmer Method." Math. Comput. 47, 351-360, 1986.

Landau Symbol

Let f(z) be a function / in an interval containing
z = 0. Let g(z) be another function also defined in this
interval such that g(z)/f(z) -* as z -» 0. Then g(z)
is said to be G(f(z)).

Landen's Formula

tf 4 (2z, 2t) ~ tf 4 (0, 2t) tfi (2z, 2t) '

where $i are THETA FUNCTIONS. This transformation
was used by Gauss to show that Elliptic Integrals
could be computed using the Arithmetic-Geometric

Mean.

lim

(lnx

,3/2

Kx

S(x)

Kx

= C,

(5)

1028 Landen's Transformation

Landen's Transformation

If x sin a = sin(2/3 - a) , then

a-

-if — g -»f » •

Jo V 1 - ^ 2 sin2 Jo Jl- -^5 sin 2

see a/so Elliptic Integral of the First Kind,
Gauss's Transformation

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Ascending
Landen Transformation" and "Landen's Transformation."
§16.14 and 17.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 573—574 and 597—598,
1972,

Lane-Emden Differential Equation

A second-order Ordinary Differential Equation
arising in the study of stellar interiors. It is given by

(i)

e«{*« +,r =

i ( 9p de 2 d<e\ d 2 e 2de

It has the BOUNDARY CONDITIONS

61(0) = 1

0.

(3)
(4)

-U=o

Solutions 0(£) for n = 0, 1, 2, 3, and 4 are shown above.
The cases n = 0, 1, and 5 can be solved analytically
(Chandrasekhar 1967, p. 91); the others must be ob-
tained numerically.

For n = (7 — 00), the Lane-Emden Differential
Equation is

+ 1 =

(5)

(Chandrasekhar 1967, pp. 91-92). Directly solving gives

d_
di

(<"!)

(6)

Lane-Emden Differential Equation

ci-K 3

di

«(0

-/-/-

I £3

d(.

(9)

(10)

m = »o - cir 1 - K 2 - (")

The Boundary Condition 0(0) = 1 then gives O = 1
and ci = 0, so

»i (0 = 1-1^, (12)

and 0i(£) is Parabolic.
For n = 1 (7 = 2), the differential equation becomes

LA. (p 2d l\

e dt {* di)

+ =

(13)

-£(*'£ 1+^ = 0,

di \: d i: ■ -* — (14)

which is the Spherical Bessel Differential Equa-
tion

d_ f_ 2 dR\

dr

with k = 1 and n = 0, so the solution is

(r 2 ^)+[fcV-n(n + l)]fl = (15)

6(i) = Ajo(i) + Bno(i). (16)

Applying the Boundary Condition 6(0) = 1 gives

sin£

62(0=30(0:

£ '

(17)

where jo(x) is a SPHERICAL BESSEL FUNCTION OF THE

First Kind (Chandrasekhar 1967, pp. 92).
For n = 5, make Emden's transformation

= Ax"z (18)

2 (19)

which reduces the Lane-Emden equation to

§ + (2w-l)J+u/(u-l)z + A B -V = (20)
ac^ at

(Chandrasekhar 1967, p. 90). After further manipula-
tion (not reproduced here), the equation becomes

d 2 z 1
dt 2 ~ 4 '

*(1-

4 \

z )

and then, finally,

0s(fl(l +

H 2 Y

-1/2

References

(21)

(22)

Chandrasekhar, S. ^4n Introduction to the Study of Stellar
Structure. New York: Dover, pp. 84-182, 1967.

Langford's Problem

Langford's Problem

Arrange copies of the n digits 1, . . . , n such that there
is one digit between the Is, two digits between the 2s,
etc. For example, the n — 3 solution is 312132 and
the n = 4 solution is 41312432. Solutions exist only if
n = 0,3 (mod 4). The number of solutions for n = 3,
4, 5, . . . are 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, . . .
(Sloane's A014552).

References

Gardner, M. Mathematical Magic Show: More Puzzles,

Games, Diversions, Illusions and Other Mathematical

Sleight- of- Mind from Scientific American. New York:

Vintage, pp. 70 and 77-78, 1978.
Sloane, N. J. A. Sequence A014551 in "An On-Line Version

of the Encyclopedia of Integer Sequences."

Langlands Program

A grand unified theory of mathematics which includes
the search for a generalization of ARTIN RECIPROCITY
(known as Langlands Reciprocity) to non-Abelian
Galois extensions of NUMBER FIELDS. Langlands pro-
posed in 1970 that the mathematics of algebra and anal-
ysis are intimately related. He was a co-recipient of the
1996 Wolf Prize for this formulation.

see also ARTIN RECIPROCITY, LANGLANDS RECI-
PROCITY

References

American Mathematical Society. "Langlands and Wiles
Share Wolf Prize." Not. Amer. Math. Soc. 43, 221-222,
1996.

Knapp, A. W. "Group Representations and Harmonic Anal-
ysis from Euler to Langlands." Not. Amer. Math. Soc. 43,
410-415, 1996.

Langlands Reciprocity

The conjecture that the ARTIN L-FUNCTION of any n-D
GALOIS GROUP representation is an L-Function ob-
tained from the General Linear Group GL 1 (A).

References

Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.

Langton's Ant

A Cellular Automaton. The Cohen-Kung Theo-
rem guarantees that the ant's trajectory is unbounded.
see also CELLULAR AUTOMATON, COHEN-KUNG THE-
OREM

References

Stewart, I. "The Ultimate in Anty-Particles." Sci. Amer.
271, 104-107, 1994.

Laplace-Beltrami Operator

A self-adjoint elliptic differential operator defined some-
what technically as

A = dS + Sd,

Laplace Distribution 1029

where d is the EXTERIOR DERIVATIVE and d and S are
adjoint to each other with respect to the Inner PROD-
UCT.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 628, 1980.

Laplace Distribution

Also called the Double Exponential Distribution.
It is the distribution of differences between two inde-
pendent variates with identical EXPONENTIAL DISTRI-
BUTIONS (Abramowitz and Stegun 1972, p. 930).

P{*) = Yb e

1 -|x-H/6

(1)

D(x) = f [1 + sgn(s - /*)(1 - e- |3C -" l/6 )]. (2)

The MOMENTS about the Mean fi n are related to the
Moments about by

where (™) is a BINOMIAL COEFFICIENT, so

n L//2 J / \ / • \

"» = E E (- 1 )"-'' ( 1 ) U k )b 2k ^ k r(2k + 1)

j=o k=o w v /

-{

(3)

' n\b n for n even
. for n odd,

where \x\ is the FLOOR FUNCTION and T(2fc + 1) is the
Gamma Function.

The Moments can also be computed using the Char-
acteristic Function,

m

jt* e -\*-»\/*> dXt

(5)

Using the FOURIER TRANSFORM OF THE EXPONENTIAL

Function

F[e

-27rfco| a! h

1 fco

7T k 2 + ko

gives

4>(t)--

ifit 1

e b

26 f2 + (i) 2 1 + W

(6)

(7)

1030 Laplace's Equation

The Moments are therefore

Mn = (~i) n 0(O) = (-»)

d n <t>

(8)

L dt n J t=0
The Mean, Variance, Skewness, and Kurtosis are

(i = li

a 2

= 2fe 2

7i

=

72

= 3.

(9)
(10)

(11)
(12)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
1972.

Laplace's Equation

The scalar form of Laplace's equation is the PARTIAL
Differential Equation

V 2 ip = 0.

(1)

It is a special case of the Helmholtz Differential
Equation

(2)

vV + fcV = o

with k = 0, or POISSON'S EQUATION

VV = -47rp (3)

with p = 0. The vector Laplace's equation is given by

V 2 F = 0.

(4)

A Function ip which satisfies Laplace's equation is said
to be Harmonic. A solution to Laplace's equation has
the property that the average value over a spherical sur-
face is equal to the value at the center of the Sphere
(Gauss's Harmonic Function Theorem). Solutions
have no local maxima or minima. Because Laplace's
equation is linear, the superposition of any two solutions
is also a solution.

A solution to Laplace's equation is uniquely determined
if (1) the value of the function is specified on all bound-
aries (Dirichlet Boundary Conditions) or (2) the
normal derivative of the function is specified on all
boundaries (NEUMANN BOUNDARY CONDITIONS).

Laplace's equation can be solved by Separation OF
VARIABLES in all 11 coordinate systems that the
Helmholtz Differential Equation can. In addi-
tion, separation can be achieved by introducing a mul-
tiplicative factor in two additional coordinate systems.
The separated form is

V>:

Xi (in )X 2 (u 2 )X 3 (u 3 )

R(ui,U2,Uz)

(5)

Laplace's Equation — Bipolar Coordinates

and setting

hih,2hs . w/ v D 2 /m

, 2 = 9i{Ui+liUi+2)fi{Ui)R , (6)

hi

where hi are SCALE FACTORS, gives the Laplace's equa-
tion

E l [l d ( dXj\
hi 2 Xi fi dm \ U dm )

- V— !— —— (f —

(7)

If the right side is equal to — ki 2 /F(ui, 112,11,3), where hi
is a constant and F is any function, and if

fcifahs = ShhhR 2 F,

(8)

where S is the Stackel Determinant, then the equa-
tion can be solved using the methods of the HELMHOLTZ
Differential Equation. The two systems where this
is the case are BlSPHERICAL and TOROIDAL, bringing
the total number of separable systems for Laplace's
equation to 13 (Morse and Feshbach 1953, pp. 665-666).

In 2-D Bipolar Coordinates, Laplace's equation
is separable, although the Helmholtz Differential
Equation is not.

see also Boundary Conditions, Harmonic Equa-
tion, Helmholtz Differential Equation, Partial
Differential Equation, Poisson's Equation, Sep-
aration of Variables, Stackel Determinant

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 17, 1972.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 125-126, 1953.

Laplace's Equation — Bipolar Coordinates

In 2-D Bipolar Coordinates, Laplace's Equation
is

(coshu-cos^) 2 fdF 2 dF 2 \

a? \ dv? + dv* J ~ ' ( }

which simplifies to

dF 2 OF 2

du 2 dv 2

0,

(2)

so Laplace's Equation is separable, although the
Helmholtz Differential Equation is not.

Lapiace's Equation — Bispherical Coordinates

Laplace's Equation — Bispherical
Coordinates

— cos u cot u + 3 cosh v cot u
cosh v — cos u

3 cosh 2 v cot u esc u + cosh 3 v esc 2 u

cosh v — cos u

d_

' dv

d

d<j> 2

-2 2 3 2

-f(cosu — coshv) sinhu— — h (cosh u — cos^) — — -

+ (cosh v — cos u) (cosh ?; cot u — sin u — cos u cot u)

d 2
+ (cosh 2 v — cos u) 2 — — — 0.
du 2

JLl
du

(1)

Let

F(u,v,<j>) = Vcosh u — cos v U(u)V(v) $(</>), (2)

then Laplace's Equation is partially separable, al-
though the Helmholtz Differential Equation is
not.

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 665-666, 1953.

Laplace's Equation — Toroidal Coordinates

2 r _ (cosht; - cosu) 3 d
; ~ a 2 du

+
+

+

(cosh v — cos u) 3 d ( sinhv df

a 2 sinh v dv
(coshv — costi) 2 d 2 f

( L___£A

V cosh v — cos udu)

(sinh v df\
cosh v — cos u dv J

a 2 sinh v d<f) 2
—3 cos coth 2 v + cosh v coth 2 v
cosh v — cos u

3 cos 2 ti coth v csch v — cos 3 u csch 2 v
cosh v — cos u

(1)

d*_

d<}> 2

d_

" du

d
+ (cos u — cosh v) sin u -^- + (cosh v — cos u) 2 -7— -

+ (cosh v — cos u) (cosh v coth v — sinh v — cos n coth v)

dv 2 '

d
+(cosh 2 v — cosu) 2

dv
(2)

Let

/(£, 77, 0) = ^cosh*,- cos £X(£)tf (»,)*(</>), (3)

then

cos v '

*W0 = ^( m <M>

(4)
(5)

Laplace Limit 1031

and the equation in rj becomes

i * ( r 8illh) ,^ > \__!!4_H_ (n >_i ) H = o.

sinh rj drj \ drj J sinhr 7/

(6)

Laplace's Equation is partially separable, although
the Helmholtz Differential Equation is not.

References

Arfken, G. "Toroidal Coordinates ({,77, <£)." §2.13 in AfatA-
ematical Methods for Physicists, 2nd ed. Orlando, FL:
Academic Press, pp. 112-114, 1970.

Byerly^ ; W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.
New York: Dover, p. 264, 1959.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 666, 1953.

Laplace's Integral

P n (x)

= 1 r p du

n Jo (x + yjx 2 — 1 cos u)

= — I (x + y a; 2 — 1 cost/) 71 du.
* Jo

Laplace Limit

The value e = 0.6627434193 . . . (Sloane's A033259) for
which Laplace's formula for solving Kepler's Equa-
tion begins diverging. The constant is defined as the
value e at which the function

/(*) =

xexp(v / l -r x 2 )

l + v/TT^"

equals /(A) = 1. The Continued FRACTION of e is
given by [0, 1, 1, 1, 27, 1, 1, 1, 8, 2, 154, ...] (Sloane's
A033260). The positions of the first occurrences of n in
the Continued Fraction of e are 2, 10, 35, 13, 15,
32, 101, 9, ... (Sloane's A033261). The incrementally
largest terms in the CONTINUED FRACTION are 1, 27,
154, 1601, 2135, . . . (Sloane's A033262), which occur at
positions 2, 5, 11, 19, 1801, . . . (Sloane's A033263).

see also Eccentric Anomaly, Kepler's Equation

References

Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/lpc/lpc.html.

Plouffe, S. "Laplace Limit Constant." http://lacim.uqam.
ca/piDATA/laplace.txt,

Sloane, N. J. A. Sequences A033259, A033260, A033261,
A033262, and A033263 in "An On-Line Version of the En-
cyclopedia of Integer Sequences."

1032 Laplace-Mehler Integral

Laplace-Mehler Integral

1 / 2 "

P n (cos8) = — / (cos# + i sin cos (j>) n d<j>

* Jo

y/2 f d cos[(n +§)<£]

= V2 r

v Jo

= V2 r

* Je

Vcos (j) — cos

d<t>

^2 f" Min+m
\/cos — cos <fi

References

Iyanaga, S. and Kawada, Y. (Eds.), Encyclopedic Dictionary

of Mathematics. Cambridge, MA: MIT Press, p. 1463,

1980.

Laplace Series

A function f{6, 4>) expressed as a double sum of Spher-
ical Harmonics is called a Laplace series. Taking / as
a Complex Function,

f(o^) = J2 ]T a, m y, m (M)-

(i)

1=0 m=-l

Now multiply both sides by Y™ sin 9 and integrate over
dO and d<j).

/ / f(0,4>)Y™'* sin dOd<f)
Jo Jo

oo I

O" * pl-K ATT

1=0 m=-i ^° ^°

(2)

Now use the ORTHOGONALITY of the SPHERICAL HAR-
MONICS

/ / Yr(0,4>)Yr'\ined$d4> = 6 mm ,6w, (3)

t/o Jo

so (2) becomes

/>2tt Air

Jo Jo

f{e,4>)Y™ s'md d6d<p

oo i

= 2_^ Z_^ airn&mm'&W = «/m , (4)
1 = m=-l

where 5 mn is the KRONECKER DELTA.
For a Real series, consider

oo i

= ^ ^ [C z m cos(m0) + 5, m sin(m0)]Pr (cos 0). (5)

i = m=-l

Laplace Transform

Proceed as before, using the orthogonality relationships

/ / Pr {cos 6) cosim&P™' {cos 6)
Jo Jo

x cos(rri <p) sm(d) dd d<l> = - . '-^Smm'Sii'

\ll -\- l){l — my.

(6)

p2tt /»7T

/ / Pr (cos 0)sm(m<l>)P™' (cos 0)

Jo Jo

x sin(m<t>) sin dO d<f> = - TuTZ ' ^ ™™'^'-

(2l + l)(l-m)\

(7)

So CP and 5, m are given by

(2l + l)(l-m)!

' 27r(Z + m)!

/•27T /»7T

x / / f(0,<f>)Pr cos cos(m(f>) sinO dOd<t> (8)
Jo Jo
= (2/ + 1)(I - m)\
' 2-K(l + m)\

p2ir /*7r

x / / f(0,<p)Pr cosQsin(m<}>) sinO d$d<f>. (9)
Jo Jo

Laplace-Stieltjes Transform

An integral transform which is often written as an ordi-
nary Laplace Transform involving the Delta Func-
tion.

see also LAPLACE TRANSFORM

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 1029, 1972.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, 1953.

Widder, D. V. The Laplace Transform. Princeton, NJ:
Princeton University Press, 1941.

Laplace Transform

The Laplace transform is an Integral Transform
perhaps second only to the Fourier TRANSFORM in its
utility in solving physical problems. Due to its useful
properties, the Laplace transform is particularly useful
in solving linear ORDINARY DIFFERENTIAL EQUATIONS
such as those arising in the analysis of electronic circuits.

The (one-sided) Laplace transform C (not to be confused
with the Lie Derivative) is defined by

poo

£(*) = £(/(«)) = / f(t)e~ st dt, (1)

Jo

Laplace Transform

Laplace Transform 1033

where f(t) is defined for t > 0. A two-sided Laplace
transform is sometimes also defined by

C(a) = C(f(t))= / f(t)e- at dt.

(2)

The Laplace transform existence theorem states that, if
f(t) is piecewise CONTINUOUS on every finite interval in
[0, oo) satisfying

|/(t)| < Me at (3)

for all t £ [0, oo), then £(/(£)) exists for all s > a. The
Laplace transform is also Unique, in the sense that,
given two functions F\ (t) and F? (t) with the same trans-
form so that

£[*!(*)] = C[F 2 (t)} = f(s), (4)

then Lerch'S Theorem guarantees that the integral

/'

Jo

N(t) dt =

(5)

vanishes for all a > for a NULL FUNCTION defined by
N(t) = F 1 (t)-F 2 {t). (6)

The Laplace transform is LINEAR since

C[af(t) + bg(t)] = / [af(t) + &<?(t-)]e- st dt

/»co />oo

= a / /(r> _st dt + 6 / p(t)e~ st dt
Jo Jo

= aC[f(t)] + bC[g(t)].

(7)

The inverse Laplace transform is given by the
Bromwich Integral (see also Duhamel's Convolu-
tion Principle). A table of several important Laplace
transforms follows.

/(*)

£W)\

Range

1
t

t n
t a

e at

cos(u?t)
sin(ujt)
cosh(u;t)
sinh(o;t)
e at sm(bt)
e at cos{bt)
5(t - c)
H e (t)
Jo(t)

i

s

1

T(a+1)
s a + l

1
s — a

3

s >
5 >

n€Z >
a>0
s > a
s >
s >
s > \a\
s > \a\
s > a
s > a

s >

s 2 +w 2

s 2 + u 2

3

b

(s-a) 2 +b 2

s — a

(s~a) 2 + b 2

e~ cs

e~ ra
s
1

^/s 2 +X

In the above table, Jo(t) is the zeroth order BESSEL
Function of the First Kind, S(t) is the Delta
Function, and H c (t) is the Heaviside Step Func-
tion. The Laplace transform has many important prop-
erties.

The Laplace transform of a CONVOLUTION is given by

C(f(t)*g(t)) = £(f(t))C(g{t)) (8)

C- 1 (F(s)G(s)) = C-'iFis)) * C-\G(s)). (9)

Now consider DIFFERENTIATION. Let f(t) be continu-
ously differentiable n — 1 times in [0,oo). If \f{i)\ <
Me at , then

C\f W (t)] = 8 n C(f(t))-s n - 1 f(0)

-a B - a /'(0)-...-/ (n_1) (0). (10)

This can be proved by INTEGRATION BY PARTS,
£[/'(*)] = lim / e- 3t f'(t)dt

]im[e- at f(t)]

1 + 8 f

Jo

*f(i)dt]

pa

= lim [e- sa f(a) - /(0) + s / e' st f{t) dt]

a ~*° Jo

= s£{f(t)}~f(0). (11)

Continuing for higher order derivatives then gives

C[f"(t)] = S 2 £[/(t)] - sf(0) - /'(0). (12)

This property can be used to transform differential equa-
tions into algebraic equations, a procedure known as the
Heaviside Calculus, which can then be inverse trans-
formed to obtain the solution. For example, applying
the Laplace transform to the equation

f"(t)+a 1 f'(t) + a o f(t) = (13)

gives

{s 2 C[f(t)} - sf(Q) - /'(0)} + ai{*C[/(t)] - /(0)}

+a o £[/(t)] = (14)

C[f(t)}(s 2 + a lS + ao)-sf{0)-f'(0)-aif(0) = 0, (15)
which can be rearranged to

«/(0) + [/'(0) + oi/(0)]

£[/(*)]

s 2 + ais + do

(16)

If this equation can be inverse Laplace transformed, then
the original differential equation is solved.

1034 Laplacian

Laplacian

Consider EXPONENTIATION. If C(f(t)) = F(s) for s >
a, then C(e at f(t)) = F(s - a) for s > a + a.

/»oo /»oo

F(s - a) = / f(t)e- (B - a)t dt= [f(t)e at }e- st dt

Jo Jo

= C(e at f(t)). (17)

Consider INTEGRATION. If f(t) is piecewise continuous

and \f(t)\ < Me at , then

/'

Jo

f(t)dt

:C[f(t)}.

(18)

The inverse transform is known as the Bromwich Inte-
gral, or sometimes the Fourier-Mellin Integral.

see also Bromwich Integral, Fourier-Mellin In-
tegral, Fourier Transform, Integral Trans-
form, Laplace-Stieltjes Transform, Opera-
tional Mathematics

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Laplace Trans-
forms." Ch. 29 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 1019-1030, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 824-863, 1985.

Churchill, R. V. Operational Mathematics. New York:
McGraw-Hill, 1958.

Doetsch, G. Introduction to the Theory and Application
of the Laplace Transformation. Berlin: Springer-Verlag,
1974.

Franklin, P. An Introduction to Fourier Methods and the La-
place Transformation. New York: Dover, 1958.

Jaeger, J. C. and Newstead, G. H. An Introduction to the La-
place Transformation with Engineering Applications. Lon-
don: Methuen, 1949.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 467-469, 1953.

Spiegel, M. R. Theory and Problems of Laplace Transforms.
New York: McGraw-Hill, 1965.

Widder, D. V. The Laplace Transform. Princeton, NJ:
Princeton University Press, 1941.

Laplacian

The Laplacian operator for a SCALAR function <fi is de-
fined by

V 2^ = i rj>_ / hits d \

hxhzhz idu\ \ h± dm J

dm V h 2 dm) dm V fa dm)\ [ }

+

in Vector notation, where the hi are the Scale Fac-
tors of the coordinate system. In TENSOR notation,
the Laplacian is written

i d ( ij dct>\
= ^dxl{^ g d^)>

d<f>
dx x

(2)

where g ;K is a COVARIANT DERIVATIVE and

n A _ i \xv \k ( ®9k\i , $9™ d9v>v \ /n\

The finite difference form is

V 2 ip(x,y,z) = -j^[^{x + h,y,z) + tp(x - h,y, z)
+ip{x, y + h,z) + tp(x, y-h,z)+ ip(x, y,z + h)

+i/>(x, y,z-h)- 6il>(x t y, z)]. (4)

For a pure radial function #(r),
V 2 g(r) = V • [Vg(r)]
= V-

dr r dO r sin d<j>

Using the VECTOR DERIVATIVE identity

V-(/A) = /(V-A) + (V/)-(A),
so

V 2 5 (r) = V.[V 5 (r)] = |v.f + v(|)

_ 2 dg d?g_
r dr dr 2

Therefore, for a radial POWER law,

(5)

(6)

(7)

V 2 r n = -nr n_1 + n(n - l)r n " 2 = [2n + n{n - l)]r n ~ 2

= n(n + l)r"

(8)

A Vector Laplacian can also be defined for a Vector
A by

V 2 A = V(V • A) - V x (V x A) (9)

in vector notation. In tensor notation, A is written A^,
and the identity becomes

V% - A wA iA - (g x "A„x). K

= g X K. K A^ x +g XK A fZ]XK . (10)

Similarly, a TENSOR Laplacian can be given by

V Aa.0 = A a/ 3;A' •

An identity satisfied by the Laplacian is

V 2 |xA|

a ,-A._ !A|2 2 -l(xA)A T [ 2

cA| 3

(11)

(12)

where |A| 2 is the HlLBERT-SCHMlDT NORM, x is a row
Vector, and A T is the Matrix Transpose of A.

Laplacian Determinant Expansion by Minors

Large Number 1035

To compute the LAPLACIAN of the inverse distance func-
tion 1/r, where r = |r— r'|, and integrate the LAPLACIAN
over a volume,

X-(^)

d 3 r.

(13)

This is equal to

-4*^-, (14)

where the integration is over a small Sphere of Radius
R. Now, for r > and R — > 0, the integral becomes 0.
Similarly, for r — R and R — > 0, the integral becomes
— 47r. Therefore,

-47r<5 3 (r-r'), (15)

where S is the DELTA FUNCTION.
see also Antilaplacian

Laplacian Determinant Expansion by Minors

see Determinant Expansion by Minors

Large Number

There are a wide variety of large numbers which crop
up in mathematics. Some are contrived, but some actu-
ally arise in proofs. Often, it is possible to prove exis-
tence theorems by deriving some potentially huge upper
limit which is frequently greatly reduced in subsequent
versions (e.g., GRAHAM'S NUMBER, KOLMOGOROV-

Arnold-Moser Theorem, Mertens Conjecture,
Skewes Number, Wang's Conjecture).

Large decimal numbers beginning with 10° are named
according to two mutually conflicting nomenclatures:
the American system (in which the prefix stands for n
in 10 3+3n ) and the British system (in which the pre-
fix stands for n in 10 6n ). The following table gives the
names assigned to various POWERS of 10 (Woolf 1982).

American

British

Power of 10

million

million

10 6

billion

milliard

10 9

trillion

billion

10 12

quadrillion

10 15

quintillion

trillion

10 18

sextillion

10 21

septillion

quadrillion

1Q 24

octillion

10 27

nonillion

quintillion

10 30

decillion

10 33

undecillion

sexillion

10 36

duodecillion

10 39

tredecillion

septillion

10 42

quattuordecillion

10 45

quindecillion

octillion

10 48

sexdecillion

10 51

septendecillion

nonillion

10 54

octodecillion

10 57

novemdecillion

decillion

10 60

vigintillion

10 63

undecillion

10 66

duodecillion

10 72

tredecillion

10 78

quattuordecillion

10 84

quindecillion

10 90

sexdecillion

10 96

septendecillion

10 102

octodecillion

10 108

novemdecillion

10 114

vigintillion

10 120

centillion

10 303

centillion

10 6oo

see also 10, ACKERMANN NUMBER, ARROW NOTATION,

Billion, Circle Notation, Eddington Number, G-

FUNCTION, GOBEL'S SEQUENCE, GOOGOL, GOOGOL-

plex, Graham's Number, Hundred, Hyperfacto-
rial, Jumping Champion, Law of Truly Large
Numbers, Mega, Megistron, Million, Monster
Group, Moser, ™-plex, Power Tower, Skewes
Number, Small Number, Steinhaus-Moser Nota-
tion, Strong Law of Large Numbers, Superfac-
torial, Thousand, Weak Law of Large Numbers,
Zillion

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New

York: Springer- Verlag, pp. 59-62, 1996.
Crandall, R. E. "The Challenge of Large Numbers." Sci.

Amer. 276, 74-79, Feb. 1997.
Davis, P. J. The Lore of Large Numbers. New York: Random

House, 1961.
Knuth, D. E. "Mathematics and Computer Science: Coping

with Finiteness. Advances in Our Ability to Compute Are

Bringing Us Substantially Closer to Ultimate Limitations."

Science 194, 1235-1242, 1976.
Munafo, R. "Large Numbers." http: //home.earthlink.

net/-mrob/largenum.
Spencer, J. "Large Numbers and Unprovable Theorems."

Amer. Math. Monthly 90, 669-675, 1983.

1036 Large Prime

Latin Rectangle

Woolf, H. B. (Ed. in Chief). Webster's New Collegiate Dic-
tionary. Springfield, MA: Merriam, p. 782, 1980.

Large Prime

see Gigantic Prime, Large Number, Titanic Prime

Laspeyres' Index

The statistical INDEX

where p n is the price per unit in period n and q n is the
quantity produced in period n.

see also Index

References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,

Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 65-67,

1962.

Latin Cross

cfb

An irregular Dodecahedron CROSS in the shape of a
dagger f. The six faces of a Cube can be cut along seven
EDGES and unfolded into a Latin cross (i.e., the Latin
cross is the Net of the Cube). Similarly, eight hyper-
surfaces of a HYPERCUBE can be cut along 17 SQUARES
and unfolded to form a 3-D Latin cross.

Another cross also called the Latin cross is illustrated
above. It is a Greek CROSS with flared ends, and is
also known as the crux immissa or cross patee.

see also CROSS, DISSECTION, DODECAHEDRON, GREEK

Cross

Latin Rectangle

A k x n Latin rectangle is a k x n Matrix with ele-
ments a,ij G {1, 2, . . . , n} such that entries in each row
and column are distinct. If A; — n, the special case of
a Latin Square results. A normalized Latin rectangle
has first row {1,2,..., n) and first column {1,2,..., k}.
Let L(k,n) be the number of normalized k x n Latin
rectangles, then the total number of k x n Latin rectan-
gles is

n\(n- l)\L(k t n)
(n - k)l

N(k,n) =

L(3,n), and Athreya, Pranesachar, and Singhi (1980)
found a summation FORMULA for L(4, n).

The asymptotic value of L(o(n 6/7 ),n) was found by
Godsil and McKay (1990). The numbers of k x n Latin
rectangles are given in the following table from McKay
and Rogoyski (1995). The entries L(l,n) and L(n,n)
are omitted, since

L(l,n) = 1

L{n,n) = L(n — l,n),

but £(1,1) and L(2, 1) are included for clarity. The
values of L(k y n) are given as a "wrap-around" series by
Sloane's A001009.

n

k

L(k)Ti)

1

1

1

2

1

1

3

2

1

4

2

3

4

3

4

5

2

11

5

3

46

5

4

56

6

2

53

6

3

1064

6

4

6552

6

5

9408

7

2

309

7

3

36792

7

4

1293216

7

5

11270400

7

6

16942080

8

2

2119

8

3

1673792

8

4

420906504

8

5

27206658048

8

6

335390189568

8

7

535281401856

9

2

16687

9

3

103443808

9

4

207624560256

9

5

112681643983776

9

6

12962605404381184

9

7

224382967916691456

9

8

377597570964258816

10

2

148329

10

3

8154999232

10

4

147174521059584

10

5

746988383076286464

(McKay and Rogoyski 1995), where n! is a FACTORIAL.
Kerewala (1941) found a RECURRENCE RELATION for

10 6 870735405591003709440

10 7 177144296983054185922560

10 8 4292039421591854273003520

10 9 7580721482160132811489280

References

Athreya, K. B.; Pranesachar, C. R-; and Singhi, N. M. "On
the Number of Latin Rectangles and Chromatic Polynom-
ial of L(K r ,,)." Europ. J. Combin. 1, 9-17, 1980.

Latin Square

Latitude 1037

Colbourn, C. J. and Dinitz, J. H. (Eds.) CRC Handbook

of Combinatorial Designs. Boca Raton, FL: CRC Press,

1996.
Godsil, CD. and McKay, B. D. "Asymptotic Enumeration

of Latin Rectangles." J. Combin. Th. Ser. B 48, 19-44,

1990.
Kerawla, S. M. "The Enumeration of Latin Rectangle of

Depth Three by Means of Difference Equation" [sic]. Bull.

Calcutta Math. Soc. 33, 119-127, 1941.
McKay, B. D. and Rogoyski, E. "Latin Squares of Order 10."

Electronic J. Combinatorics 2, N3, 1-4, 1995. http://

www . combinatorics . org/Volume_2 /volume 2 . html#N3,
Ryser, H. J. "Latin Rectangles." §3.3 in Combinatorial

Mathematics. Buffalo, NY: Math. Assoc, of Amer., pp. 35-

37, 1963.
Sloane, N. J. A. Sequence A001009 in "An On-Line Version

of the Encyclopedia of Integer Sequences."

Latin Square

An n x n Latin square is a LATIN RECTANGLE with
k — n. Specifically, a Latin square consists of n sets
of the numbers 1 to n arranged in such a way that no
orthogonal (row or column) contains the same two num-
bers. The numbers of Latin squares of order n = 1, 2,
... are 1, 2, 12, 576, ... (Sloane's A002860). A pair
of Latin squares is said to be orthogonal if the n 2 pairs
formed by juxtaposing the two arrays, are all distinct.

Two of the Latin squares of order 3 are

3 2 1
2 1 3
13 2

2 3 1
1 2 3

3 12

which are or
order 4 are

:thogonal. Two of the 576 Latin squares of

12 3 4

2 14 3

3 4 12

4 3 2 1

12 3 4

3 4 12

4 3 2 1
2 14 3

A normalized, or reduced, Latin square is a Latin square
with the first row and column given by {1,2, . . . ,ra}.
General FORMULAS for the number of normalized Latin
squares L(n,n) are given by Nechvatal (1981), Gessel
(1987), and Shao and Wei (1992). The total number of
Latin squares of order n can then be computed from

N{n,n) = n\(n - l)!L(n,n) = n\(n - l)\L(n - l,n).

The numbers of normalized Latin square of order n — 1,
2, . . . , are 1, 1, 1, 4, 56, 9408, . . . (Sloane's A000315).
McKay and Rogoyski (1995) give the number of normal-
ized Latin Rectangles L{k,n) for n = 1, . . . , 10, as
well as estimates for L(n,n) with n = 11, 12, . . . , 15.

n

L(n>n)

11

5.36 x 10 33

12

1.62 x 10 44

13

2.51 x 10 56

14

2.33 x 10 70

15

1.5 x 10 86

see also EULER SQUARE, KlRKMAN TRIPLE SYSTEM,

Partial Latin Square, Quasigroup

References

Colbourn, C. J. and Dinitz, J. H. CRC Handbook of Combi-
natorial Designs. Boca Raton, FL: CRC Press, 1996.

Gessel, L "Counting Latin Rectangles." Bull. Amer. Math.
Soc. 16, 79-83, 1987.

Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 33-34, 1975.

Kraitchik, M. "Latin Squares." §7.11 in Mathematical Recre-
ations. New York: W. W. Norton, p. 178, 1942.

Lindner, C. C. and Rodger, C. A. Design Theory. Boca
Raton, FL: CRC Press, 1997.

McKay, B. D. and Rogoyski, E. "Latin Squares of Order 10."
Electronic J. Combinatorics 2, N3, 1-4, 1995. http://
www . combinatorics , org/Volume_2/volume2 . html#N3.

Nechvatal, J. R. "Asymptotic Enumeration of Generalised
Latin Rectangles." Util. Math. 20, 273-292, 1981.

Ryser, H. J. "Latin Rectangles." §3.3 in Combinatorial
Mathematics. Buffalo, NY: Math. Assoc. Amer., pp. 35-
37, 1963.

Shao, J.-Y. and Wei, W.-D. "A Formula for the Number of
Latin Squares." Disc. Math. 110, 293-296, 1992.

Sloane, N. J. A. Sequences A002860/M2051 and A000315/
M3690 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Latin- Graeco Square

see EULER SQUARE

Latitude

The latitude of a point on a SPHERE is the elevation
of the point from the PLANE of the equator. The lat-
itude 8 is related to the COLATITUDE (the polar angle
in Spherical Coordinates) by 6 = <p - 90°. More
generally, the latitude of a point on an ELLIPSOID is the
Angle between a Line Perpendicular to the surface
of the Ellipsoid at the given point and the Plane of
the equator (Snyder 1987).

The equator therefore has latitude 0° , and the north and
south poles have latitude ±90°, respectively. Latitude is
also called GEOGRAPHIC LATITUDE or GEODETIC LAT-
ITUDE in order to distinguish it from several subtly dif-
ferent varieties of Auxiliary LATITUDES.

The shortest distance between any two points on a
Sphere is the so-called Great Circle distance, which
can be directly computed from the latitudes and LON-
GITUDES of the two points.

see also Auxiliary Latitude, Colatitude, Confor-
mal Latitude, Great Circle, Isometric Latitude,
Latitude, Longitude, Spherical Coordinates

References

Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, p. 13, 1987.

1038

Lattice

Lattice Point

Lattice

A lattice is a system K such that Wl 6 K, A C A,
and if A C B and B C A, then A = 5, where = here
means "is included in." Lattices offer a natural way
to formalize and study the ordering of objects using a
general concept known as the POSET (partially ordered
set). The study of lattices is called Lattice Theory.
Note that this type of lattice is an abstraction of the
regular array of points known as LATTICE POINTS.

The following inequalities hold for any lattice:

(xAy)V(xAz)<xA(yWz)

xV(yAz) < (x V y) A (x V z)

(xAy)\f(yAz)\/(zAx)<(xVy)A(y\/z)A(zV x)

(xAy)\f(xAz)<xA(yV(xA z))

(Gratzer 1971, p. 35). The first three are the distributive
inequalities, and the last is the modular identity.

see also Distributive Lattice, Integration Lat-
tice, Lattice Theory, Modular Lattice, Toric

Variety

References

Gratzer, G. Lattice Theory: First Concepts and Distributive
Lattices. San Francisco, CA: W. H. Freeman, 1971.

Lattice Algebraic System

A generalization of the concept of set unions and inter-
sections.

Lattice Animal

A distinct (including reflections and rotations) arrange-
ment of adjacent squares on a grid, also called fixed
POLYOMINOES.

see also Percolation Theory, Polyomino

Lattice Distribution

A Discrete Distribution of a random variable such
that every possible value can be represented in the form
a + bn, where a, b / and n is an INTEGER.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 927, 1972.

Lattice Graph

Lattice Groups

In the plane, there are 17 lattice groups, eight of which

are pure translation. In M 3 , there are 32 POINT Groups

and 230 Space Groups. In M 4 , there are 4783 space

lattice groups.

see also Point Groups, Space Groups, Wallpaper

Groups

Lattice Path

A path composed of connected horizontal and vertical
line segments, each passing between adjacent LATTICE
Points. A lattice path is therefore a Sequence of
points Po, Pi , ■ ■ ■ , Pn with n > such that each Pi
is a Lattice Point and P i+ i is obtained by offsetting
one unit east (or west) or one unit north (or south).

The number of paths of length a -f b from the ORI-
GIN (0,0) to a point (a, b) which are restricted to east
and north steps is given by the BINOMIAL COEFFICIENT

see also Ballot Problem, Golygon, Kings Prob-
lem, Lattice Point, p-Good Path, Random Walk

References

Dickau, R. M. "Shortest-Path Diagrams." http:// forum .
swarthmore.edu/advanced/robertd/manhattan.html.

Hilton, P. and Pederson, J. "Catalan Numbers, Their Gener-
alization, and Their Uses." Math. Intel 13, 64-75, 1991.

Lattice Point

A POINT at the intersection of two or more grid lines in a
ruled array. (The array of grid lines could be oriented to
form unit cells in the shape of a square, rectangle, hex-
agon, etc.) However, unless otherwise specified, lattice
points are generally taken to refer to points in a square
array, i.e., points with coordinates {m^n,, . . .), where m,
n, . . . are INTEGERS.

An n-D Z[w] -lattice L n lattice can be formally defined
as a free Z[u;] -Module in complex n-D space C n .

The Fraction of lattice points Visible from the ORI-
GIN, as derived in Castellanos (1988, pp. 155-156), is

N'(r)
N{r)

^r 2 + Q(r\nr)
4r 2 + 0{r)

The lattice graph with n nodes on a side is denoted L(n)
see also Triangular Graph

Therefore, this is also the probability that two randomly
picked integers will be RELATIVELY PRIME to one an-
other.

For 2 < n < 32, it is possible to select 2n lattice points
with x y y 6 [l,n] such that no three are in a straight

Lattice Reduction

Lattice Sum 1039

LINE. The number of distinct solutions (not counting
reflections and rotations) for n = 1, 2, . . . , are 1, 1, 4,
5, 11, 22, 57, 51, 156 ... (Sloane's A000769). For large
n, it is conjectured that it is only possible to select at
most (c + e)n lattice points with no three COLLINEAR,
where

c=(2tt 2 /3) 1/3 ^1.85

(Guy and Kelly 1968; Guy 1994, p. 242). The num-
ber of the n 2 lattice points x,y 6 [l,n] which can be
picked with no four CONCYCLIC is 0(n 2/3 - e) (Guy
1994, p. 241).

A special set of Polygons defined on the regular lat-
tice are the GOLYGONS. A NECESSARY and SUFFICIENT
condition that a linear transformation transforms a lat-
tice to itself is that it be Unimodular. M. Ajtai has
shown that there is no efficient ALGORITHM for find-
ing any fraction of a set of spanning vectors in a lattice
having the shortest lengths unless there is an efficient al-
gorithm for all of them (of which none is known) . This
result has potential applications to cryptography and
authentication (Cipra 1996).

see also Barnes- Wall Lattice, Blichfeldt's Theo-
rem, Browkin's Theorem, Circle Lattice Points,
Coxeter-Todd Lattice, Ehrhart Polynomial,
Gauss's Circle Problem, Golygon, Integra-
tion Lattice, Jarnick's Inequality, Lattice
Path, Lattice Sum, Leech Lattice, Minkowski
Convex Body Theorem, Modular Lattice, N-
Cluster, Nosarzewska's Inequality, Pick's The-
orem, Poset, Random Walk, Schinzel's Theorem,
Schroder Number, Visible Point, Voronoi Poly-
gon

References

Apostol, T. Introduction to Analytic Number Theory. New-
York: Springer- Verlag, 1995.

Castellanos, D. "The Ubiquitous Pi." Math. Mag. 61, 67-98,
1988.

Cipra, B. "Lattices May Put Security Codes on a Firmer
Footing." Science 273, 1047-1048, 1996.

Eppstein, D. "Lattice Theory and Geometry of Numbers."
http:// www . ics . uci . edu / - eppstein / junkyard /
lattice.html.

Guy, R. K. "Gaufi's Lattice Point Problem," "Lattice Points
with Distinct Distances," "Lattice Points, No Four on a
Circle," and "The No-Three-in-a-Line Problem." §F1, F2,
F3, and F4 in Unsolved Problems in Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 240-244, 1994.

Guy, R. K. and Kelly, P. A. "The No-Three-in-Line-
Problem." Canad. Math. Bull. 11, 527-531, 1968.

Hammer, J. Unsolved Problems Concerning Lattice Points.
London: Pitman, 1977.

Sloane, N. J. A. Sequence A000769/M3252 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Lattice Reduction

The process finding a reduced set of basis vectors
for a given LATTICE having certain special proper-
ties. Lattice reduction is implemented in Mathematica®
(Wolfram Research, Champaign, IL) using the function
LatticeReduce. Lattice reduction algorithms are used

in a number of modern number theoretical applications,
including in the discovery of a SPIGOT ALGORITHM for
Pi.

see also Integer Relation, PSLQ Algorithm

Lattice Sum

Cubic lattice sums include the following:

i,j = — oo

*<*>- £' M

i+j+k

i,j,k= — oo

{P + p + k 2 )°

b n (2s)

£'

(-i)

k 1 +...+k„

(V + ... + fc„ 2 )'

(1)

(2)
(3)

where the prime indicates that summation over (0, 0, 0)
is excluded. As shown in Borwein and Borwein (1987,
pp. 288-301), these have closed forms for even n

b 2 (2s) = -*0(8)ri(8) (4)

b 4 (2s) = -8rj(s)r](s - 1) (5)

b 8 (2s) = -16C{s)r](s - 3), for R[s] > 1 (6)

where f3(z) is the DlRICHLET BETA FUNCTION, r)(z) is
the Dirichlet Eta Function, and C,(z) is the Rie-
mann Zeta Function. The lattice sums evaluated
at s ~ 1 are called the Madelung Constants. Bor-
wein and Borwein (1986) prove that 6s (2) converges (the
closed form for bg(2s) above does not apply for s = 1),
but its value has not been computed.

For hexagonal sums, Borwein and Borwein (1987,
p. 292) give

h 2 (2s)

m,Ti= — oo

sin[(n+l)<9]sin[(m-H)<9] - sin(n0) sin[(m - 1)0]

[(n+|m) 2 +3(im) 2 ] i

(7)

where = 27r/3. This Madelung Constant is expres-
sible in closed form for s = 1 as

/i 2 (2)=7rln3\/3.

(8)

see also Benson's Formula, Madelung Constants

References

Borwein, D. and Borwein, J. M. "On Some Trigonometric and
Exponential Lattice Sums." J. Math. Anal 188, 209-218,
1994.

Borwein, D.; Borwein, J. M.; and Snail, R. "Analysis of Cer-
tain Lattice Sums." J. Math. Anal. 143, 126-137, 1989.

1040 Lattice Theory

Laurent Series

Borwein, D. and Borwein, J. M. "A Note on Alternating Se-
ries in Several Dimensions." Amer. Math, Monthly 93,
531-539, 1986.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.

Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/mdlung/mdlTing.html.

Glasser, M. L. and Zucker, I. J. "Lattice Sums." In Perspec-
tives in Theoretical Chemistry: Advances and Perspectives
5, 67-139, 1980.

Lattice Theory

Lattice theory is the study of sets of objects known as
Lattices. It is an outgrowth of the study of Boolean
Algebras, and provides a framework for unifying the
study of classes or ordered sets in mathematics. Its
study was given a great boost by a series of papers and
subsequent textbook written by Birkhoff (1967).

see also LATTICE

References

Birkhoff, G. Lattice Theory, 3rd ed. Providence, RI: Amer.

Math. Soc, 1967.
Gratzer, G. Lattice Theory: First Concepts and Distributive

Lattices. San Francisco, CA: W. H. Freeman, 1971.

Latus Rectum

Twice the Semilatus Rectum.

see also PARABOLA

Laurent Polynomial

A Laurent polynomial with COEFFICIENTS in the FIELD
F is an algebraic object that is typically expressed in the
form

Laurent Series

. . . + a- n t n + a-( n -i)£

-(n-1)

+ ...

+a-it + ao + o,it + . . . + a n t n + . . . ,

where the a* are elements of F, and only finitely many
of the a< are NONZERO. A Laurent polynomial is an al-
gebraic object in the sense that it is treated as a POLY-
NOMIAL except that the indeterminant "i" can also have
Negative Powers.

Expressed more precisely, the collection of Laurent poly-
nomials with Coefficients in a Field F form a Ring,
denoted F[M -1 ], with RING operations given by com-
ponentwise addition and multiplication according to the
relation

at n • bt n

abi

n+m

for all n and m in the INTEGERS. Formally, this is equiv-
alent to saying that F[i,t _1 ] is the GROUP RING of the
Integers and the Field F. This corresponds to ¥[t]
(the Polynomial ring in one variable for F) being the
Group Ring or Monoid ring for the Monoid of natu-
ral numbers and the FIELD F.
see also Polynomial

References

Lang, S. Undergraduate Algebra, 2nd ed.
Springer- Verlag, 1990.

New York:

Let there be two circular contours C2 and Ci, with the
radius of C\ larger than that of C2. Let zo be interior to
Ci and C2, and z be between C\ and C2. Now create a
cut line C c between C\ and C2, and integrate around the
path C = C\ + C c — Ci — C c , so that the plus and minus
contributions of C c cancel one another, as illustrated
above. From the Cauchy Integral Formula,

/W

2m J c z f — z

21TI J Cl z ' " z 27ri Jc c z z

2m Jc 2 z z 2m Jc c z z

2m J z> - z 2m J c z'-z

Now, since contributions from the cut line in opposite
directions cancel out,

/M

/(*')

zo) - (z - Zo)

f(z')
- zo) - (z - z )

dz

dz

2« J Cl i z ' ~

~2TiJ C2 (*

^JcA z '- z o)(l-j^- )
_ J_ f (VI dz

= j_ r m dz >

2« JcA*' -*>){!-■£%)

+

Wc

f(z')

cA z -*o){l->iE%)

dz'. (2)

For the first integral, \z' - z \ > \z — z \- For the second,
\z' - z \ < \z - z \. Now use the Taylor Expansion
(valid for |t| < 1)

OO

1 -t ±-~>

(3)

Laurent Series

to obtain

1

Law of Cosines

1041

/(*) =

2ni

+

JCx Z '- Zo ± J n Kz '- ZoJ

*1 E

n =

z' - Zq

dz

Jc, z - z o^ Q \ z - z oJ

+ ^ f> - * ))_n ~ 1 / (z ' " Zo)nf[z) dz

n— 1 2

(4)

where the second term has been re-indexed. Re-indexing
again,

/(*) = -L V^ _ ZQ y [ J^l— dz'

(5)

n— — cx>

Now, use the Cauchy Integral Theorem, which re-
quires that any CONTOUR INTEGRAL of a function which
encloses no Poles has value 0. But l/(z f — zo) n+1 is
never singular inside C% for n > 0, and l/(z f — z ) n+l is
never singular inside C\ for n < — 1. Similarly, there are
no POLES in the closed cut C c — C c . We can therefore
replace Ci and C2 in the above integrals by C without
altering their values, so

71 =

+ 2^ E(*-*) J c(z ,- Zo) ^ dz

n= — 00 °

= 1 f> - «,)» / ? , /(Z 'j +1 <**'

27TI f^ *■ ; ,/ c (z' - Z ) n+1

E an(z-zo)'

(6)

The only requirement on C is that it encloses 2, so we are
free to choose any contour 7 that does so. The RESIDUES
a n are therefore denned by

an - 2ni I (z' - z )^ '

(7)

see also Maclaurin Series, Residue (Complex
Analysis), Taylor Series

References

Arfken, G. "Laurent Expansion." §6.5 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 376-384, 1985.

Morse, P. M. and Feshbach, H. "Derivatives of Analytic Func-
tions, Taylor and Laurent Series." §4.3 in Methods of The-
oretical Physics, Part I. New York: McGraw-Hill, pp. 374-
398, 1953.

Law

A law is a mathematical statement which always holds
true. Whereas "laws" in physics are generally exper-
imental observations backed up by theoretical under-
pinning, laws in mathematics are generally THEOREMS
which can formally be proven true under the stated con-
ditions. However, the term is also sometimes used in the
sense of an empirical observation, e.g., Benford'S Law.
see also ABSORPTION LAW, BENFORD'S LAW, CON-
TRADICTION Law, de Morgan's Duality Law, de
Morgan's Laws, Elliptic Curve Group Law, Ex-
cluded Middle Law, Exponent Laws, Girko's Cir-
cular Law, Law of Cosines, Law of Sines, Law of
Tangents, Law of Truly Large Numbers, Mor-
rie's Law, Parallelogram Law, Plateau's Laws,
Quadratic Reciprocity Law, Strong Law of
Large Numbers, Strong Law of Small Numbers,
Sylvester's Inertia Law, Trichotomy Law, Vec-
tor Transformation Law, Weak Law of Large
Numbers, Zipf's Law

Law of Anomalous Numbers

see Benford's Law

Law of Cancellation

see Cancellation Law

Law of Cosines

Let a, 6, and c be the lengths of the legs of a Triangle
opposite ANGLES A, £, and C. Then the law of cosines
states

c —a -rb — 2abcosC. (1)

This law can be derived in a number of ways. The def-
inition of the Dot Product incorporates the law of
cosines, so that the length of the Vector from X to Y
is given by

Y| 2 = (X- Y) ■ (X- Y)

= XX-2X-Y + YY

= ixr

2|X||Y|cos<9,

(2)
(3)
(4)

1042 Law of Large Numbers

where is the Angle between X and Y.

b-a cos C

a cos C

The formula can also be derived using a little geometry
and simple algebra. Prom the above diagram,

<? = (a sin C) 2 + (6 - a cos C) 2

a 2 sin 2 c + b 2 - 2ab cos C + a 2 cos 2 C
■ a + b 2 - 2ab cos C.

(5)

The law of cosines for the sides of a Spherical Trian-
gle states that

cos a = cos b cos c + sin 6 sin c cos A (6)

cos b = cos c cos a + sin c sin a cos B (7)

cos c = cos a cos 6 + sin a sin 6 cos C (8)

(Beyer 1987). The law of cosines for the angles of a
Spherical Triangle states that

cos A = — cos B cos C + sin B sin C cos a (9)
cos B = — cos C cos ^4 + sin C sin A cos b (10)
cos C — — cos A cos i? + sin A sin £ cos c (11)

(Beyer 1987).

see also Law of Sines, Law of Tangents

References

Abramowitz, M. and Stegun, C. A- (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 79, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 148-149, 1987.

Law of Large Numbers

see Law of Truly Large Numbers, Strong Law of
Large Numbers, Weak Law of Large Numbers

Law of Sines

Law of Tangents

Let a, 6, and c be the lengths of the Legs of a Triangle
opposite Angles A, B, and C. Then the law of sines
states that

sin A sin B sin C

2#,

(1)

where R is the radius of the CiRCUMClRCLE. Other
related results include the identities

a(sin B - sin C) + 6(sin C - sin A) + c(sin A - sin B) =

(2)
a = b cos C + c cos B,

the Law of Cosines

cos A —

c 2 + b'-a 2
2bc

and the Law of Tangents

q + b _ tan[f(.A + ff)]
a-b " tan[|(i4-S)]'

(3)
(4)

(5)

The law of sines for oblique SPHERICAL TRIANGLES

states that

sin a sin b sin c

sin A sin B sin C
see also Law of Cosines, Law of Tangents

(6)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 79, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 148, 1987.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 1-3, 1967.

Law of Small Numbers

see Strong Law of Small Numbers

Law of Tangents

Let a Triangle have sides of lengths a, 6, and c and let
the Angles opposite these sides by A, B, and C. The
law of tangents states

a-b = tan[i(A-g)]

a + b ~ tan[|(A + £)]"

An analogous result for oblique SPHERICAL TRIANGLES

states that

tan[|(a-6)] _ tan[f(A-£)]
tan[|(a + 6)] ~ tanf|(A + B)] '

see also Law of Cosines, Law of Sines

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 79, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 145 and 149, 1987.

Law of Truly Large Numbers

Leading Order Analysis 1043

Law of Truly Large Numbers

With a large enough sample, any outrageous thing is
likely to happen (Diaconis and Mosteller 1989). Little-
wood (1953) considered an event which occurs one in
a million times to be "surprising." Taking this defini-
tion, close to 100,000 surprising events are "expected"
each year in the United States alone and, in the world
at large, "we can be absolutely sure that we will see
incredibly remarkable events" (Diaconis and Mosteller
1989).

see also Coincidence, Strong Law of Large Num-
bers, Strong Law of Small Numbers, Weak Law
of Large Numbers

References

Diaconis, P. and Mosteller, F. "Methods of Studying Coinci-
dences." J. Amev. Statist Assoc. 84, 853-861, 1989.

Littlewood, J. E. Littlewood's Miscellany. Cambridge, Eng-
land: Cambridge University Press, 1986.

Lax-Milgram Theorem

Let be a bounded Coercive bilinear Functional
on a HlLBERT SPACE H. Then for every bounded linear
Functional / on H, there exists a unique x f e H such
that

f(x) = <t>(x,x f )

for all x G H.

References

Debnath, L. and Mikusinski, P. Introduction to Hilbert
Spaces with Applications. San Diego, CA: Academic Press,
1990.

Zeidler, E. Applied Functional Analysis: Applications to
Mathematical Physics. New York: Springer- Verlag, 1995.

Lax Pair

A pair of linear OPERATORS L and A associated with
a given Partial Differential Equation which can
be used to solve the equation. However, it turns out
to be very difficult to find the L and A corresponding
to a given equation, so it is actually simpler to postu-
late a given L and A and determine to which PARTIAL
Differential Equation they correspond (Infeld and
Rowlands 1990).

see also PARTIAL DIFFERENTIAL EQUATION

References

Infeld, E. and Rowlands, G. "Integrable Equations in Two
Space Dimensions as Treated by the Zakharov Shabat
Methods." §7.10 in Nonlinear Waves, Solitons, and
Chaos. Cambridge, England: Cambridge University Press,
pp. 216-223, 1990.

Layer

see p- Layer

Le Cam's Identity

Let S n be the sum of n random variates Xi with a BER-
NOULLI Distribution with P(Xi — 1) — p*. Then

k=0

P(Sn = *)

e~ x X k

k\

< 2 $> 2 >

where

\ = j2 pi -

see also Bernoulli Distribution

References

Cox, D. A. "Introduction to Fermat's Last Theorem." Amer.
Math. Monthly 101, 3-14, 1994.

Leading Digit Phenomenon
see Benford's Law

Leading Order Analysis

A procedure for determining the behavior of an nth or-
der Ordinary Differential Equation at a Remov-
able SINGULARITY without actually solving the equa-
tion. Consider

d
dz

n y p (<F- l y dy \

where F is ANALYTIC in z and rational in its other ar-
guments. Proceed by making the substitution

y(z) = a(z - zo)"
with a < 1. For example, in the equation
d 2 y

dz 2

Gy + Ay,

(2)

(3)

making the substitution gives

aa(a-l){z-z ) a - 2 = 6a 2 (z-zo) 2a +Aa(az-z ) a . (4)

The most singular terms (those with the most NEGATIVE
exponents) are called the "dominant balance terms,"
and must balance exponents and COEFFICIENTS at the
Singularity. Here, the first two terms are dominant,
so

a-2 = 2a^>a = -2 (5)

6a = 6a => a = 1,

(6)

and the solution behaves as y(z) = (z — zq)~ 2 . The
behavior in the NEIGHBORHOOD of the SINGULARITY is
given by expansion in a LAURENT SERIES, in this case,

y( z ) = 5Z<*j(*-*o) j 2 .

(7)

j=o

1044 Leaf (Foliation)

Plugging this series in yields

y-4

X>(j-2)(j-3)(z-*,) j

3=0

oo oo oo

= 6^J]a,a fc (2-zo) J ' + ^ 4 +A^a i (2-zo) J '~ 2 . (8)

j=0 fe=0

j=o

This gives Recurrence Relations, in this case with
a& arbitrary, so the (z — zo) 6 term is called the resonance
or KOVALEVSKAYA EXPONENT. At the resonances, the
COEFFICIENT will always be arbitrary. If no resonance
term is present, the POLE present is not ordinary, and
the solution must be investigated using a Psi FUNCTION,

see also PSI FUNCTION

References

Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, p. 330, 1989.

Leaf (Foliation)

Let M n be an n-MANlFOLD and let F = {F a } denote
a PARTITION of M into DISJOINT path-connected SUB-
SETS. Then if F is a FOLIATION of M, each F a is called
a leaf and is not necessarily closed or compact.

see also Foliation

References

Rolfsen, D. Knots and Links. Wilmington, DE; Publish or
Perish Press, p. 284, 1976.

Leaf (Tree)

An unconnected end of a Tree.

see also Branch, Child, Fork, Root (Tree), Tree

Leakage

see Aliasing

Least Bound

see Supremum

Least Common Multiple

The least common multiple of two numbers m and n-i
is denoted LCM(ni,7i2) or [711,712] and can be obtained
by finding the PRIME factorization of each

ni =pi

■ *Pn

&1 b n

712 = Pi ' * " Pn ,

(1)

(2)

where the pis are all Prime Factors of m and 712, and
if pi does not occur in one factorization, then the corre-
sponding exponent is 0. The least common multiple is
then

Least Common Multiple

Let 77i be a common multiple of a and b so that

m = ha = kb. (4)

Write a = a 1 {a ) b) and b = &i(o, fe), where a\ and bi
are RELATIVELY Prime by definition of the GREATEST
Common Divisor (ai,6i) = 1. Then ha\ = kbi, and
from the Division Lemma (given that hai is Divisible
by b and (61, 01) =0), we have h is DIVISIBLE by 61, so

h = nbi

m = ha = nb\a = n

ab

The smallest m is given by n = 1,

ab

LCM(a,6) =

GCD(a,6)'

so

(5)
(6)

(7)

GCD(a,6) LCM(a,6) =

= ab

(8)

(a, 6) [a, 6] = ab.

(9)

The LCM is Idempotent

[a, a] = a,

(10)

Commutative

[a, b] = [6, a],

(11)

(13)

Associative

[a,b,c] = [[a,6],c] = [o,[6,c]], (12)

Distributive

[771a, 7716, mc] = m[a, 6, c],
and satisfies the ABSORPTION LAW

(a 1 [a,b]) = a. (14)

It is also true that

(ma)(m6) a&

ma, 7716 = ^7 — - = 777- — — = 777 a, 6 . (15)

(ma.mb) (a, b)

see also Greatest Common Divisor, Mangoldt
Function, Relatively Prime

References

Guy, R. K. "Density of a Sequence with L.C.M. of Each Pair
Less than a;." §E2 in Unsolved Problems in Number The-
ory, 2nd ed. New York: Springer- Verlag, pp. 200-201,
1994.

LCM(ni,7i 2 ) = [7ii,n 2 ] = \\Pi

max(aj ,bi)

(3)

Least Deficient Number

Least Deficient Number
A number for which

a(n) ~2n-l.

All Powers of 2 are least deficient numbers.

see also DEFICIENT NUMBER, QUASIPERFECT NUMBER

Least Period

The smallest n for which a point x is a PERIODIC POINT
of a function / so that f n (xo) = xq. For example, for
the FUNCTION f(x) = —x, all points x have period 2
(including x = 0). However, x = has a least period
of 1. The analogous concept exists for a PERIODIC SE-
QUENCE, but not for a PERIODIC FUNCTION. The least
period is also called the Exact Period.

Least Prime Factor

80

60

20

UL

20 40 60 80 100

For an INTEGER n > 2, let lpf(x) denote the LEAST
Prime Factor of n, i.e., the number pi in the factor-
ization

Pi

•Pk

with pi < pj for i < j. For n = 2, 3, ..., the first
few are 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, . . .
(Sloane's A020639). The above plot of the least prime
factor function can be seen to resemble a jagged terrain
of mountains, which leads to the appellation of "TWIN
Peaks" to a Pair of Integers (x,y) such that

1. x < y,

2. lpf(a;) = lpf(i/),

3. For all z, x < z < y IMPLIES lpf(z) < lpf(x).

The least multiple prime factors for SQUAREFUL integers
are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, . . . (Sloane's
A046027).

see also Alladi-Grinstead Constant, Distinct
Prime Factors, Erdos-Selfridge Function, Fac-
tor, Greatest Prime Factor, Least Common
Multiple, Mangoldt Function, Prime Factors,
Twin Peaks

References

Sloane, N. J. A. Sequence A020639 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Least Squares Fitting 1045
Least Squares Fitting

A mathematical procedure for finding the best fitting
curve to a given set of points by minimizing the sum of
the squares of the offsets ("the residuals") of the points
from the curve. The sum of the squares of the offsets
is used instead of the offset absolute values because this
allows the residuals to be treated as a continuous dif-
ferentiate quantity. However, because squares of the
offsets are used, outlying points can have a dispropor-
tionate effect on the fit, a property which may or may
not be desirable depending on the problem at hand.

U

u-

•-'\

-i

v-

v

*'\

*'\

vertical offsets perpendicular offsets

In practice, the vertical offsets from a line are almost
always minimized instead of the perpendicular offsets.
This allows uncertainties of the data points along the x-
and y-axes to be incorporated simply, and also provides
a much simpler analytic form for the fitting parameters
than would be obtained using a fit based on perpendic-
ular distances. In addition, the fitting technique can be
easily generalized from a best-fit line to a best-fit poly-
nomial when sums of vertical distances are used (which
is not the case using perpendicular distances). For a
reasonable number of noisy data points, the difference
between vertical and perpendicular fits is quite small.

The linear least squares fitting technique is the simplest
and most commonly applied form of LINEAR REGRES-
SION and provides a solution to the problem of finding
the best fitting straight line through a set of points. In
fact, if the functional relationship between the two quan-
tities being graphed is known to within additive or mul-
tiplicative constants, it is common practice to transform
the data in such a way that the resulting line is a straight
line, say by plotting T vs. VI instead of t vs. L For this
reason, standard forms for EXPONENTIAL, LOGARITH-
MIC, and POWER laws are often explicitly computed.
The formulas for linear least squares fitting were inde-
pendently derived by Gauss and Legendre.

For Nonlinear Least Squares Fitting to a number
of unknown parameters, linear least squares fitting may
be applied iteratively to a linearized form of the func-
tion until convergence is achieved. Depending on the
type of fit and initial parameters chosen, the nonlinear

1046 Least Squares Fitting

fit may have good or poor convergence properties. If un-
certainties (in the most general case, error ellipses) are
given for the points, points can be weighted differently
in order to give the high-quality points more weight.

The residuals of the best-fit line for a set of n points
using unsquared perpendicular distances di of points
(xi,yi) are given by

r ± = Yl di -

(1)

Since the perpendicular distance from a line y = a + bx
to point i is given by

di =

\yi - (a + bxj)

VTTW '
the function to be minimized is

\yi - (a + bxt)

R ^H

VT+&

(2)

(3)

Unfortunately, because the absolute value function does
not have continuous derivatives, minimizing R± is not
amenable to analytic solution. However, if the square of
the perpendicular distances

iii^fci

[yi - (a + bxi)] 2

■6 2

(4)

is minimized instead, the problem can be solved in closed
form. R 2 ± is a minimum when (suppressing the indices)

dR\
da

^^-(a + MK-lHO (5)

and

dR 2 ±
db

= TTb2X [j/ ~ (a + 6x)]( ~ a;)

+ > [y-(a + bx)]\-l)(2b) = Q (6)

E

(1 + 6 2 ) 2

The former gives

y^y — b^x

a — _ y _ ^

(7)

and the latter
(l + 6 2 )^[y-(a + &x)]x + 6^[y-(a + 6x)] 2 = 0. (8)

But

[y - (a 4- bx)] 2 = y 2 - 2(a 4- bx)y + (a + bx) 2

= y 2 - 2ay - 2bxy + a 2 + 2abx 4- b 2 x 2 , (9)

Least Squares Fitting
so (8) becomes
(l + 6 2 )(5]^-a^x-6^x 2 )

+ b (J2 y2 -2aj2y- 2b J2 xy+a2 J2 1

+2a& ^ x + 6 2 ^ x 2 ) = (10)

[(i + & 2 )(-6) + b(b 2 )] J2x 2 + [(i + b 2 )- 2b 2 } ^2 *y

+6 Y, V + h^ 1 + fe2 ) + 2ab 2 } ^2 / x-2ab^y
+6a 2 ^l = (11)

-6^x 2 + (l-6 2 )^xt/ + 6^2/ 2 +a(6 2 -l)^x
-2ab ^ y + ha2n = °- ( 12 )
Plugging (7) into (12) then gives

^^KE^E^E*
-^(E^E^E^+ME^E*) 2

= (13)

After a fair bit of algebra, the result is

£y 2 -E* 2 + ^[(E*) 2 -(E2/) 2 "

+ i ^ ^ L ^ l b - 1 =

^J2 x T,y-J2 x y

o.

(14)

So define

n _ l [Ey 2 -HZy) 2 ]-[Z* 2 -HZ*f
KEy 2 -V)-(E^ 2 -^ 2 )

2 «E x E2/-E x 3/

and the QUADRATIC FORMULA gives

b = -B ± -v/B 2 + 1,

(15)

(16)

with a found using (7). Note the rather unwieldy form of
the best-fit parameters in the formulation. In addition,
minimizing R? ± for a second- or higher-order POLYNOM-
IAL leads to polynomial equations having higher order,
so this formulation cannot be extended.

Vertical least squares fitting proceeds by finding the sum
of the squares of the vertical deviations R 2 of a set of n
data points

R 2 = ^Jh/i — /(xi,ai,a 2 , . . . ,a n )] 2

(17)

Least Squares Fitting

Least Squares Fitting 1047

from a function /. Note that this procedure does not
minimize the actual deviations from the line (which
would be measured perpendicular to the given function).
In addition, although the unsquared sum of distances
might seem a more appropriate quantity to minimize,
use of the absolute value results in discontinuous deriva-
tives which cannot be treated analytically. The square
deviations from each point are therefore summed, and
the resulting residual is then minimized to find the best
fit line. This procedure results in outlying points being
given disproportionately large weighting.

The condition for R 2 to be a minimum is that

d(R')

for i ■■

den
n. For a linear fit,

f(a, b) — a + bx,

R^ctyEE^lyi-ia + bxi)} 2

d(R 2
da

d(R 2 )

-2^[y i -(a + 6x < )] =

(18)

(19)

(20)
(21)

db =- 2 X>-( a + M]*; = 0. (22)

1=1

These lead to the equations

na + b^2x = ^2y (23)

a > x + & / x = y xy )

(24)

where the subscripts have been dropped for conciseness.
In Matrix form,

n J2*
E* I>

so

£*

I> £ :

£y

£y

(25)

(26)

The 2x2 Matrix Inverse is
1

n J2 x 2 ~ (E x )

E y E x 2 - E x E x v

^E^-E^E^

(27)

6 =

E^E^jiE^E^

yY,x 2 -xJ2 x v

E^ 2 — nx2
n>T, x y-I2 x T,y
^E^ 2 - (E x )

E xy - nxy

E^ 2 ~ nx2

(28)

(29)
(30)

(31)

(Kenney and Keeping 1962). These can be rewritten in
a simpler form by denning the sums of squares

n

— / ( x i — x) = ( / x — nx ) (32)

n

= ^2(yi - yf = (XI y2 ~ n ^ 2 ) ^ 33 )

y = X^ ~ £ )(^ -y) = yZl x y - n ^j > ( 34 )

& yy

which are also written as

2 _
<7 X — SSxx

(35)

<J y = SSyy

(36)

cov(z,y) = SSa;^.

(37)

Here, cov(x,y) is the COVARIANCE and a 2 and <ri are
variances. Note that the quantities £ xy and E x can
also be interpreted as the DOT PRODUCTS

^^ 2 =x.x (38)

E

aj/ = x ■ y.

(39)

In terms of the sums of squares, the REGRESSION CO-
EFFICIENT b is given by

b =

cov(x^y) _ ss xy

and a is given in terms of b using (24) as

a — y — bx.

(40)

(41)

The overall quality of the fit is then parameterized in
terms of a quantity known as the CORRELATION COEF-
FICIENT, defined by

(42)

which gives the proportion of ss yv which is accounted
for by the regression.

1048 Least Squares Fitting
The Standard Errors for a and b are

SE(o) = sj- + —
V n ssxi

SE(6) = -^=.

(43)
(44)

Let yi be the vertical coordinate of the best-fit line with
x-coordinate x^ so

j/i = a + foci,

(45)

then the error between the actual vertical point yi and
the fitted point is given by

a = y» -y».

(46)

Now define s 2 as an estimator for the variance in ei,

ao

Least Squares Fitting

Y^ ^ fc +«i XI cpA:+1 +- • -+ a * X * 2fe = Yl xky ( 56)

or, in Matrix form

E =

E* E<

E =
E

„*+!

IE-* £<

.fc+l

E<

a*

E x 2/

X> fc yJ

(57)

This is a Vandermonde Matrix. We can also obtain
the Matrix for a least squares fit by writing

1 Xi

1 X2

1 X n

Xi

ao

'yi"

ai

=

2/2

_afc_

.2/n.

(58)

n- :

(47)

Premultiplying both sides by the TRANSPOSE of the first
Matrix then gives

Then s can be given by

SSyy OSSxy

n-2

n- 2

(48)

(Acton 1966, pp. 32-35; Gonick and Smith 1993,
pp. 202-204).

Generalizing from a straight line (i.e., first degree poly-
nomial) to a kth degree POLYNOMIAL

y = a 4- aix + . . . + a k x ,
the residual is given by

(49)

R 2 = 22/[yi - (ao + aiXi + . . . + a k Xi k )] 2 . (50)

The Partial Derivatives (again dropping super-
scripts) are

0(R 2 )
da

d(R 2 )
dai

-2 ^[y - (a + aix + . . . + a k x k )] = (51)

-2^[2/-(a o + aix-K.. + a*x fc )]x = (52)

£j£i = - 2 y[y-(a +a 1 x + ... + a fc ^)]x /c = 0. (53)
These lead to the equations

ao

a n 4- a x > x + . . . + a^ \^ a? 7 " = ^ ^ y (54)

/] x + ai ^J x 2 + . . . + a fc ^J x fc+1 = ^J xy (55)

1

1 ..*

1 "

'1

Xl

351

x 2

Xn

1

£2

Xi

~ k
x 2

* k
Xn

,1

Xn

' 1

1

=

Xl

x[ k

X2

#2

Xl

x 2

ao
a x

afc

y2

L2/n

, (59)

E*

E^ E-

E* n
E^ +1

E^ n E* n+1 ••■ E* 2n J La»J LE^yJ

ao
ai

Ev
E^

(60)

As before, given m points (x^, yi) and fitting with POLY-
NOMIAL Coefficients a 0) . . . , a n gives

yi

2/2
2/m J

1 Xl Xl

1 X 2 X2

1 Xtt^ X m

Xl
X2*

ao
ao

(61)

In Matrix notation, the equation for a polynomial fit
is given by

y - Xa. (62)

This can be solved by premultiplying by the MATRIX
Transpose X t ,

X y = X Xa.

(63)

Least Squares Fitting — Exponential

Least Squares Fitting — Logarithmic 1049

This Matrix Equation can be solved numerically, or
can be inverted directly if it is well formed, to yield the
solution vector

(X T X)" 1 X T y.

(64)

Setting m = 1 in the above equations reproduces the
linear solution.

see also Correlation Coefficient, Interpolation,
Least Squares Fitting — Exponential, Least
Squares Fitting — Logarithmic, Least Squares
Fitting — Power Law, Moore-Penrose General-
ized Matrix Inverse, Nonlinear Least Squares
Fitting, Regression Coefficient, Spline

References

Acton, F. S. Analysis of Straight-Line Data. New York:
Dover, 1966.

Bevington, P. R. Data Reduction and Error Analysis for the
Physical Sciences. New York: McGraw-Hill, 1969.

Gonick, L. and Smith, W. The Cartoon Guide to Statistics.
New York: Harper Perennial, 1993.

Kenney, J. F. and Keeping, E. S. "Linear Regression, Simple
Correlation, and Contingency." Ch. 8 in Mathematics of
Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand,
pp. 199-237, 1951.

Kenney, J. F. and Keeping, E. S. "Linear Regression and
Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1,
3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285, 1962.

Lancaster, P. and Salkauskas, K. Curve and Surface Fitting:
An Introduction. London: Academic Press, 1986.

Lawson, C. and Hanson, R. Solving Least Squares Problems.
Englewood Cliffs, NJ: Prentice-Hall, 1974.

Nash, J. C. Compact Numerical Methods for Computers:
Linear Algebra and Function Minimisation, 2nd ed. Bris-
tol, England: Adam Hilger, pp. 21-24, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Fitting Data to a Straight Line" "Straight-
Line Data with Errors in Both Coordinates," and "General
Linear Least Squares." §15.2, 15.3, and 15*4 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 655-675, 1992.

York, D. "Least-Square Fitting of a Straight Line." Canad.
J. Phys. 44, 1079-1086, 1966.

Least Squares Fitting — Exponential

800

/

600

//

//
//

400

,7

200

V*

/

10 20 30 40 50

To fit a functional form

y = Ae B \
take the Logarithm of both sides

lny = h\A + BXnx.

(i)

(2)

The best-fit values are then

(3)

(4)

™J> 2 - (X»
6 _ nJ2 xln y-Y, x Y, ln y
n E^ 2 - (E x )

where B = b and A = exp(a).

This fit gives greater weights to small y values so, in
order to weight the points equally, it is often, better to
minimize the function

^2

^y(lny -a- bxY
Applying Least Squares Fitting gives

a22y + b 22 x y - },y^ n y

a^2xy + b2_^% 2 y = }^xylny

J2y T, x y
Y,xy Y^ x2 y

Yly ln y

Y^xylny

(5)

(6)
(?)
(8)

Solving for a and 6,

_ J E( x2 y)J2(y ln y) -J2( x y)J2( x y [n y)

b =

(9)
(10)

Y,yY.( x2 y) - {H x y)
_ T,yT,( x y ln y) - YK x y)H(y ln y)
Y,yY,( x2 y)- (E x y)

In the plot above, the short-dashed curve is the fit com-
puted from (3) and (4) and the long-dashed curve is the
fit computed from (9) and (10).
see also LEAST SQUARES FITTING, LEAST SQUARES

Fitting — Logarithmic, Least Squares Fitting —
Power Law

Least Squares Fitting — Logarithmic

10 20 30 40 50

Given a function of the form

y = a + bliiXj (1)

the Coefficients can be found from Least Squares
Fitting as

nJ2(y lnx ) ~ Z^ZX lna

nY,[(lnx)*]-[j:(\nx)) 2

T,y- b Y,( lnx )

n

see also Least Squares Fitting, Least Squares
Fitting — Exponential, Least Squares Fitting —
Power Law

6 =

(2)
(3)

1050 Least Squares Fitting — Power Law

Least Squares Fitting — Power Law

50000 ■

10 20 30 40 50

Given a function of the form

y = Ax B , (i)

Least Squares Fitting gives the Coefficients as
n Y, (In x In y) - ^(lnx)^(lny)

6 =

n£[(ln*) 2 ]-(£ln*)'
£(lny)-6£(lns)

(2)
(3)

where B = b and A = exp(a).

see a/so LEAST SQUARES FITTING, LEAST SQUARES

Fitting — Exponential, Least Squares Fitting —
Logarithmic

Least Upper Bound

see SUPREMUM

Lebesgue Constants (Fourier Series)

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Assume a function / is integrable over the interval
[-7T, 7r] and S n (fj x) is the nth partial sum of the FOUR-
IER Series of /, so that

a k -

& fc =

= - / f(t) cos(H) dt
f(t)sin(kt)dt

(1)
(2)

and

S n (f, x) - |a + < Y2^ ak cos ( kx ) + bk sin (k%)] > • (3)

If

for all x, then

Sn{f>x)< - f

n Jo

|sin[f(2n + l)fl]|
sin(^)

(4)

d9 = L n , (5)

Lebesgue Constants (Fourier Series)

and L n is the smallest possible constant for which this
holds for all continuous /. The first few values of L n are

L = 1

!,! = - + ^l1 = 1.435991124. .

3 7T

L 2 = 1.642188435...
L 3 = 1.778322862.

Some Formulas for L 7l include

2n + l Tr^fc \2n+lJ

oo (2n + l)fc

~ 7r2 2^ 2^ 4Jfe 2 - 1 2j - 1

k=i j~i

(Zygmund 1959) and integral FORMULAS include

(6)

(7)

(8)
(9)

(10)

f°° t arm [(2™ + 1)#] dx
J tanhx 7r 2 + 4a; 2

4 f°° sinh[(2n + l)x] r , ur w . -x n ,

= — / ^— - — ln{coth[5(2n + ljzjjdcc

7r 2 Jq sinhz

(Hardy 1942). For large n,

4 4

— Inn < L n < 3+ —Inn.

7T 2 7T J

(id

(12)

This result can be generalized for an r- different iable

function satisfying

d r f

dx r

< 1

(13)

for all x. In this case,

\f(x) - S n (f,x)\ < L n , r = -^ + © (J_) , (14)

where

L n ,',

{-T \TT ^ sJ ^\^ forr>lodd
if" \yr ^ 22^511 dx for r >1 even

(15)
(Kolmogorov 1935, Zygmund 1959).

Watson (1930) showed that

lim [l„ r ln(2n + 1)1 = c,

(16)

Lebesgue Constants (Lagrange Interpolation)

Lebesgue Integrable 1051

where

JfM^

In A;
4k 2 -

4 I"(i)

* 2 r(i)

Lj=o

A(2j + 2) -

2j + l

0.9894312738 . . . ,

+ —(2 In 2 + 7 ) (18)

7T

(19)

where F(z) is the GAMMA FUNCTION, X(z) is the
Dirichlet Lambda Function, and 7 is the Euler-
Mascheroni Constant.

References

Finch, S. "Favorite Mathematical Constants." http://www.

mathsoft.com/asolve/constant/lbsg/lbsg.html.
Hardy, G. H. "Note on Lebesgue's Constants in the Theory

of Fourier Series." J. London Math. Soc. 17, 4-13, 1942.
Kolmogorov, A. N. "Zur Grossenordnung des Restgliedes

Fourierscher reihen differenzierbarer Funktionen," Ann.

Math. 36, 521-526, 1935.
Watson, G. N. "The Constants of Landau and Lebesgue."

Quart. J. Math. Oxford 1, 310-318, 1930.
Zygmund, A. G. Trigonometric Series, 2nd ed., Vols. 1-2.

Cambridge, England: Cambridge University Press, 1959.

Lebesgue Constants (Lagrange
Interpolation)

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Define the nth Lebesgue constant for the LAGRANGE
Interpolating Polynomial by

A n (X) = max >

-1<Z<1^— '

jt=i

nX — Xj
Xk — Xj

Lebesgue Covering Dimension

An important DIMENSION and one of the first dimen-
sions investigated. It is defined in terms of covering sets,
( 17 ) and is therefore also called the Covering Dimension.
Another name for the Lebesgue covering dimension is

the Topological Dimension.

A Space has Lebesgue covering dimension m if for every
open Cover of that space, there is an open Cover that
refines it such that the refinement has order at most
m+1. Consider how many elements of the cover contain
a given point in a base space. If this has a maximum
over all the points in the base space, then this maximum
is called the order of the cover. If a SPACE does not have
Lebesgue covering dimension m for any m, it is said to
be infinite dimensional.

Results of this definition are:

1 . Two homeomorphic spaces have the same dimension,

2. W 1 has dimension n,

3. A TOPOLOGICAL Space can be embedded as a closed
subspace of a EUCLIDEAN SPACE Iff it is locally
compact, Hausdorff, second countable, and is finite
dimensional (in the sense of the Lebesgue Dimen-
sion), and

4. Every compact metrizable m-dimensional TOPO-
LOGICAL Space can be embedded in M 2m+1 .

see also LEBESGUE MINIMAL PROBLEM

References

Dieudonne, J. A. A History of Algebraic and Differential To-
pology. Boston, MA: Birkhauser, 1994.

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 414, 1980.
(1) Munkres, J. R. Topology: A First Course. Englewood Cliffs,

NJ: Prentice-Hall, 1975.

It is true that

A n > —r In n ■

TV 2

(2)

The efficiency of a Lagrange interpolation is related to
the rate at which A n increases. Erdos (1961) proved
that there exists a POSITIVE constant such that

A n > - In n - C

7T

for all n. Erdos (1961) further showed that

A n < -Inn + 4,

TV

(3)

(4)

so (3) cannot be improved upon.

References

Erdos, P. "Problems and Results on the Theory of Interpo-
lation, II." Acta Math. Acad. Sci. Hungary 12, 235-244,
1961.

Finch, S. "Favorite Mathematical Constants." http: //www.
mathsof t . com/ asolve/constant/lbsg/lbsg. html.

Lebesgue Dimension

see Lebesgue Covering Dimension

Lebesgue Integrable

A real-valued function / defined on the reals R. is called
Lebesgue integrable if there exists a Sequence of Step
Functions {f n } such that the following two conditions
are satisfied:

i-£~i/l/»l<°°.

2. f(x) — YT°=i f° r ever y x £ K sucn tnat

Er =1 /i/"i<--

Here, the above integral denotes the ordinary RlEMANN
Integral. Note that this definition avoids explicit use

of the Lebesgue Measure.

see also INTEGRAL, LEBESGUE INTEGRAL, RlEMANN IN-
TEGRAL, Step Function

1052 Lebesgue Integral

Lebesgue Sum

Lebesgue Integral

The Lebesgue Integral is defined in terms of upper
and lower bounds using the LEBESGUE MEASURE of a
Set. It uses a Lebesgue Sum S n = ViK E i) where ^
is the value of the function in subinterval i y and fi(Ei)
is the Lebesgue Measure of the Set E t of points for
which values are approximately 7]i, This type of integral
covers a wider class of functions than does the Riemann
Integral.

see also ^4-Integrable, Complete Functions, Inte-
gral

References

Kestelman, H. "Lebesgue Integral of a Non-Negative Func-
tion" and "Lebesgue Integrals of Functions Which Are
Sometimes Negative." Chs. 5-6 in Modern Theories of
Integration, 2nd rev. ed. New York: Dover, pp. 113-160,
1960.

Lebesgue Measurability Problem

A problem related to the Continuum Hypothesis
which was solved by Solovay (1970) using the Inacces-
sible Cardinals Axiom. It has been proven by Shelah
and Woodin (1990) that use of this AXIOM is essential
to the proof.

see also Continuum Hypothesis, Inaccessible Car-
dinals Axiom, Lebesgue Measure

References

Shelah, S. and Woodin, H. "Large Cardinals Imply that Ev-
ery Reasonable Definable Set of Reals is Lebesgue Measur-
able." Israel J. Math. 70, 381-394, 1990.

Solovay, R. M, "A Model of Set-Theory in which Every Set
of Reals is Lebesgue Measurable." Ann. Math. 92, 1-56,
1970.

Lebesgue Measure

An extension of the classical notions of length and
Area to more complicated sets. Given an open set
5 = X^( afc 'k fc ) containing Disjoint intervals,

V>l{S) = y^(frfc - a k ).

Given a Closed Set S' = [a, 6] - £ fc (a*. ,&*.),

W (S') = (&-a)-^(6 fc -a fc ).

k

A Line Segment has Lebesgue measure 1; the Can-
tor Set has Lebesgue measure 0. The MINKOWSKI
Measure of a bounded, Closed Set is the same as its
Lebesgue measure (Ko 1995),

see also Cantor Set, Measure, Riesz-Fischer The-
orem

References

Kestelman, H. "Lebesgue Measure." Ch. 3 in Modern Theo-
ries of Integration, 2nd rev. ed. New York: Dover, pp. 67-
91, 1960.

Ko, K.-L "A Polynomial- Time Computable Curve whose In-
terior has a Nonrecursive Measure." Theoret. Corn-put. Sci.
145, 241 270, 1995.

Lebesgue Minimal Problem

Find the plane Lamina of least Area A which is capable
of covering any plane figure of unit GENERAL DIAME-
TER. A Unit Circle is too small, but a Hexagon
circumscribed on the Unit Circle is too large. More
specifically, the Area is bounded by

0.8257...= |tt + |v / 3< A< |(3 - V3) = 0.8454. . .

(Pal 1920).

see also AREA, BORSUK'S CONJECTURE, DIAMETER

(General), Kakeya Needle Problem

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 99, 1987.

Coxeter, H. S. M. "Lebesgue's Minimal Problem." Eureka
21, 13, 1958.

Grunbaum, B. "Borsuk's Problem and Related Questions."
Proc. Sympos. Pure Math, Vol. 7. Providence, RI: Amer.
Math. Soc, pp. 271-284, 1963.

Kakeya, S. "Some Problems on Maxima and Minima Re-
garding Ovals." Sci. Reports Tohoku Imperial Univ., Ser.
1 (Math., Phys., Chem.) 6, 71-88, 1917.

Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 142-144, 1990.

Pal, J. Danske videnkabernes selskab, Copenhagen Math.-fys.
maddelelser 3, 1-35, 1920.

Yaglom, I. M. and Boltyanskii, V. G. Convex Figures. New
York: Holt, Rinehart, & Winston, pp. 18 and 100, 1961.

Lebesgue- Radon Integral

see Lebesgue-Stieltjes Integral

Lebesgue Singular Integrals

lUtf) = [

f(x)K n {x)dx,

where {K n (x)} is a Sequence of Continuous Func-
tions.

Lebesgue-Stieltjes Integral

Let a(x) be a monotone increasing function and define
an INTERVAL I = (xi,x 2 ). Then define the NONNEGA-
TIVE function

17(1) = a(x 2 + 0) - a(xi + 0).

The Lebesgue Integral with respect to a Measure
constructed using U(I) is called the Lebesgue-Stieltjes
integral, or sometimes the Lebesgue-Radon Inte-
gral.

References

lyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 326, 1980.

Lebesgue Sum

■ Vi /j,(Ei),

where fi(Ei) is the MEASURE of the Set Ei of points on
the #-axis for which f(x) as rji.

Leech Lattice

Leech Lattice

A 24-D Euclidean lattice. An AUTOMORPHISM of the
Leech lattice modulo a center of two leads to the Con-
way Group Co 1 . Stabilization of the 1- and 2-D sub-
lattices leads to the CONWAY GROUPS Co 2 and Co 3 ,
the Higman-Sims GROUP HS and the McLaughlin
Group McL.

The Leech lattice appears to be the densest Hyper-
SPHERE PACKING in 24-D, and results in each Hyper-
SPHERE touching 195,560 others.

see also Barnes- Wall Lattice, Conway Groups,
Coxeter-Todd Lattice, Higman-Sims Group, Hy-
persphere, Hypersphere Packing, Kissing Num-
ber, McLaughlin Group

References

Conway, J. H. and Sloane, N. J. A. "The 24-Dimensional
Leech Lattice A 2 4 ," "A Characterization of the Leech
Lattice," "The Covering Radius of the Leech Lattice,"
"Twenty-Three Constructions for the Leech Lattice," "The
Cellular of the Leech Lattice," "Lorentzian Forms for
the Leech Lattice." §4.11, Ch. 12, and Chs. 23-26 in
Sphere Packings, Lattices, and Groups, 2nd ed. New York:
Springer- Verlag, pp. 131-135, 331-336, and 478-526, 1993.

Leech, J. "Notes on Sphere Packings." Canad. J. Math. 19,
251-267, 1967.

Wilson, R. A. "Vector Stabilizers and Subgroups of Leech
Lattice Groups." J. Algebra 127, 387-408, 1989.

Lefshetz Fixed Point Formula

see Lefshetz Trace Formula

Lefshetz's Theorem

Each DOUBLE Point assigned to an irreducible curve
whose GENUS is NONNEGATIVE imposes exactly one con-
dition.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 104, 1959.

Lefshetz Trace Formula

A formula which counts the number of Fixed Points
for a topological transformation.

Leg

The leg of a Triangle is one of its sides.

see also HYPOTENUSE, TRIANGLE

Legendre Addition Theorem

see Spherical Harmonic Addition Theorem

Legendre's Chi-Function

The function defined by

Xu(z) = J2

JHi

(2A + 1)"

Legendre Differential Equation 1053

for integral v = 2, 3, .... It is related to the POLYLOG-
arithm by

X»{z) = \[U„{z)-U v {-z)]
= Li l/ (z)-2- u U u (z 2 ).

see also POLYLOGARITHM

References

Cvijovic, D. and Klinowski, J. "Closed-Form Summation of
Some Trigonometric Series." Math. Comput. 64, 205—210,
1995.

Lewin, L. Polylogarithms and Associated Functions. Amster-
dam, Netherlands: North-Holland, pp. 282-283, 1981.

Legendre's Constant

The number 1.08366 in Legendre's guess at the Prime
Number Theorem

7r(n)

Inn -1.08366
This expression is correct to leading term only.

References

Le Lionnais, F. Les nombres remarquables . Paris: Hermann,
p. 147, 1983.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 28-29, 1991.

Legendre Differential Equation

The second-order ORDINARY DIFFERENTIAL EQUATION

^\d 2 y

'dx 2
which can be rewritten

dy
dx

(l-^)l4-2^+/(i + l)y = > (1)

dx

2^dy

dx

+ i(Z + l)j/ = 0.

(2)

The above form is a special case of the associated Leg-
endre differential equation with m — 0. The Legendre
differential equation has REGULAR SINGULAR POINTS
at — 1, 1, and oo. It can be solved using a series expan-
sion,

)a n x

(3)
(4)
(5)

y = y^anx 71

n=0
oo
/ V~^ n-1

y = y na n x

n=Q

oo

y" = ^2n(n- 1)

71 =

Plugging in,

oo oo

(1 - x 2 ) ^ n ( n ~ l)anz n ~ 2 -2x^2 nanX 71 ' 1

oo

+Z(Z + l)^a n a; Tl = (6)

1054 Legendre Differential Equation

oo oo

y n(n — l)a n x n ~ — > n(n — l)a n x n

n=0 n=0

oo oo

-2x ^2 na n x n ~ x + 1(1 + 1) ^ a n x n = (7)

n=0 n=0

oo oo

y^nin - l)ana: n_2 - / ^n(n - l)a n £ n

n = 2 n=0

oo oo

-2 ]P na n x n + 1(1 + 1) JZ a^x 71 = (8)

n=0 n=0

oo oo

y ^(n + 2)(n + l)a n +2^ n — /_^ ri(n — l)a n x n

n=0 n=0

oo oo

-2 ^ na n x" + 1(1 + 1) ^ a n x n = (9)

n=0

]T{(n+l)(n + 2)a n+ 2

n=0

+[-n(n - 1) - 2n + /(/ + l)]a n } = 0, (10)
so each term must vanish and
(ra + l)(n + 2)a n+2 - n(n + 1) + /(/ + l)]a„ = (11)

a n +2 =

n(ra + l) -*(/ + !)

(n+l)(n + 2) an
[l + (n+l)](l-n)
(„ + !)(„ + 2) a "

Therefore,

/(i + 1)

a 2 = — -^ ^ a

C&4

1-2
(i-2)(f + 3)
3-4

a 2

[(Z-2)/][(/ + !)(/ + 3)]

= (_1) lTi^4 ao

(l-4)(i + 5)

(12)

(13)

(14)

5-6

-a4

, 3 [(<-4)(/-2)q[(i + l)(i + 3)(f + 5)]
(_1) 1- 2-3-4-5-6 a °'

(15)

so the Even solution is

n-l

[(I - 2n + 2) • • • (I - 2)l][{l + l)(f + 3) ■ • ■ (I + In - 1)]
(2n)!

Legendre Differential Equation

Similarly, the Odd solution is

y 2 (x) ^x + y^(-l) n

n = l

[(I _ 2n + 1) • • • (I - 3)(Z - !)][(« + 2)(J + 4)---(i + 2n) 2m+1

(2n + l)!

(17)

If/ is an Even Integer, the series yi reduces to a Poly-
nomial of degree I with only Even POWERS of x and
the series y<i diverges. If / is an Odd INTEGER, the series
t/2 reduces to a Polynomial of degree / with only Odd
Powers of x and the series y\ diverges. The general
solution for an INTEGER I is given by the LEGENDRE
Polynomials

p ( X )- C J V^ x ) ^ /even , }

where c n is chosen so that P n (l) — 1. If the variable x
is replaced by cos#, then the Legendre differential equa-
tion becomes

d 2 y cos 6 dy
dd 2 sin0 dx

+ /(J + l)y = 0,

(19)

as is derived for the associated Legendre differential
equation with m = 0.

The associated Legendre differential equation is

A.

dx

2\dy

."-■•'g

+

... H v m

„2x^ 2 y

f 1 -^- 2 '^

^ + 1)-T^-
1 — ,

y = o (20)

y = 0. (21)

The solutions to this equation are called the associated
Legendre polynomials. Writing x = cos 0, first establish
the identities

dy _ dy

1 dy

dx d(cos 9) sin dO

dy
dx

cos dy
sinOdO'

d 2 y

1 d

( 1 dy\

dx 2

s'mOdd

\sin6 d6 J

(22)
(23)

d0 sin 2 0d9 2 '

and

1 / -cosfl \ dy 1 d 2 y

S in0lsin 2 0/ Ja -- 2 "-" 2 ' l ;

(25)

(16)

Legendre Duplication Formula

Therefore,

/-, 2,d 2 y _ . 2 1 f -cosO \ dy 1 d 2 y

(1 X ] dx* ~ Sm sin 9 V sin 2 J d9 + sin 2 dO*

d 2 y cos 9 dy
Iff ~~ shi0 cZ0'

(26)

Plugging (22) into (26) and the result back into (21)
gives

d 2 y cos 9 dy
dtP ~ sin? eft?

l2 cos9 dy +

sin# d(9

/(/ + !)-

sin 2 9

y = (27)

d 2 y cos# dy

d9'<

+

sin 9 dx

+

*(* + !)-

sin 2 (9

y = 0. (28)

References

Abramowitz, M. and Stegun, C. A. (Eds,). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 332, 1972.

Legendre Duplication Formula

Gamma Functions of argument 2z can be expressed
in terms of GAMMA FUNCTIONS of smaller arguments.
From the definition of the Beta Function,

■ r(m)r(n) f m~i n \n-i • / 1
g(m,n) = v ' = / u (1-u) (in. (1

Now, let m = n = z, then

= /" u'-^l-u)*- 1

•A)

r(2z)

l du

)

(2)

and u = (1 + x)/2, so dxt = dx/2 and

r

= 2 i + 2( Z -i ) j Q ( i - x2 y~ ldx

= 2 x ~ 2z f (l-x'y-'dx.

Jo
Now, use the Beta Function identity

= 2 f x 2 *- 1
Jo

(3)

B(m,n)
to write the above as

'(l-x 2 )*- 1 ^

Legendre's Formula 1055

Solving for T(2z),
T(2z) =

T(z)T(z + \)2 2z ~ l _ T(z)T{z + i)2 2z " 1

r(|) V5F

= (27r)- 1/2 2 2l - 1/2 r(z)r(z+i),

(6)

since T(|) = V7T.

see also GAMMA FUNCTION, GAUSS MULTIPLICATION

Formula

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 256, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 561-562, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I, New York: McGraw-Hill, pp. 424-425, 1953.

Legendre's Factorization Method

A Prime Factorization Algorithm in which a se-
quence of Trial Divisors is chosen using a Quadra-
tic Sieve. By using Quadratic Residues of N, the
Quadratic Residues of the factors can also be found.

see also PRIME FACTORIZATION ALGORITHMS, QUAD-
RATIC Residue, Quadratic Sieve Factorization
Method, Trial Divisor

Legendre's Formula

Counts the number of POSITIVE INTEGERS less than or
equal to a number x which are not divisible by any of

the first a PRIMES,

<p(x,a) = [x\ -^

-E

+ ...,

PiPj

(1)

_PiPjPk _

where [x\ is the FLOOR FUNCTION. Taking a = x gives

<p{x,x) = ir(x) - n(y/x) -f 1 = [^J ~~ / u

Pi<pj<Vx

PiPj

s

pi<pj<Pk<V®

X

PiPjPk

+ ■

(2)

where ir(n) is the PRIME COUNTING FUNCTION. Leg-
endre's formula holds since one more than the number
of PRIMES in a range equals the number of INTEGERS
minus the number of composites in the interval.

Legendre's formula satisfies the Recurrence Rela-
tion

^M = 2 1 — B(|,z) = 2 1 — ^^. (5) </>(x,a) = *(*, a - 1) - (f-,a - l) .

T(2z) - 1(2+2) \ p ° '

(3)

1056 Legendre Function of the First Kind

Let mk = P1P2 • ■ -Pk, then

+

4>(m k ,k) — [m k \ - ^

m k
Pi

k

£

_ 1_

P2

PiPj

i-i-

Pk

= JJ(Pi- 1) = 0("ifc),

i=l

where <j>(n) is the Totient Function, and

<p(sm k +t,k) = s<f)(m k ) + <£(£, fc),
where < t < rrik. If £ > m k /2, then

0(t, fc) = <t>(mk) - <t>{mk —i-i, k).

(4)

(5)

(6)

Note that <t>{n,n) is not practical for computing ir(n)
for large arguments. A more efficient modification is
Meissel's Formula.

see also Lehmer's Formula, Mapes' Method, Meis-
sel's Formula, Prime Counting Function

Legendre Function of the First Kind

see Legendre Polynomial

Legendre Function of the Second Kind

A solution to the LEGENDRE DIFFERENTIAL EQUATION
which is singular at the origin. The Legendre functions
of the second kind satisfy the same RECURRENCE Re-
lation as the Legendre Functions of the First
KIND. The first few are

*-S-(£f)-'

hx z 2

Legendre-Gauss Quadrature

The associated Legendre functions of the second kind
have Derivative about of

dx

2"^ cos[§7r(i/ + /i)]r(|i/ + \n + 1)

(Abramowitz and Stegun 1972, p. 334). The logarithmic
derivative is

dlnQ^(z)

dz

= 2e,pU„ i8g „ (SW)} ILW + ^W-rtl'

[l(A + <.-l)|l[|(A-/.-l)ll

(Binney and Tremaine 1987, p. 654).

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Legendre Func-
tions." Ch. 8 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 331-339, 1972.

Arfken, G. "Legendre Functions of the Second Kind, Q n (x). n
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 701-707, 1985.

Binney, J. and Tremaine, S. "Associated Legendre Func-
tions." Appendix 5 in Galactic Dynamics. Princeton, NJ:
Princeton University Press, pp. 654-655, 1987.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 597-600, 1953.

Snow, C. Hypergeometric and Legendre Functions with
Applications to Integral Equations of Potential Theory.
Washington, DC: U. S. Government Printing Office, 1952.

Spanier, J. and Oldham, K. B. "The Legendre Functions
P v (x) and Q u (x). n Ch. 59 in An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 581-597, 1987.

Legendre-Gauss Quadrature

Also called "the" GAUSSIAN QUADRATURE or LEGEN-
DRE Quadrature. A Gaussian Quadrature over
the interval [— 1, 1] with WEIGHTING FUNCTION W(x) =
1. The ABSCISSAS for quadrature order n are given by
the roots of the LEGENDRE POLYNOMIALS P n (x), which
occur symmetrically about 0. The weights are

Wi — —-

^n + lTVi

A n

7n-l

A n P n (xi)P n+1 (xi) A n -! P n -x{Xi)Pk{XiY

(1)

where A n is the Coefficient of x n in P n (x). For Leg-
endre Polynomials,

si n

(2n)!

SO

2"(n!) 2 '

[2(n + l)]l 2 n (n!) 2
2»+ 1 [(n+l)!] 2 (2n)!
(2n + l)(2n + 2) _ 2n + 1
2(n + l) 2 ~ n + 1 *

(2)

Additionally,

2n+l

(3)

(4)

Legendre-Jacobi Elliptic Integral

Wi (n + l)P n +i(xi)Pk(xi) nPn-^x^PUxi) '

(5)

Using the RECURRENCE RELATION
(1 - x 2 )P' n {x) = nxP n (x) + nP n -i(z)

(6)

= (n+l)xP n (x) - {n+l)P n +i{x) (7)

gives

Wi

2(1 -Xj 2 )

(1 - Xi*)[Pttxi)]* (n + l) 2 [P n+1 (^)] 2 '
The error term is

(8)

Beyer (1987) gives a table of ABSCISSAS and weights up
to n = 16, and Chandrasekhar (1960) up to n = 8 for n

Even.

n Xi

Wi

2 ±0.57735 1.000000

3 0.888889
±0.774597 0.555556

4 ±0.339981 0.652145
±0.861136 0.347855

5 0.568889
±0.538469 0.478629
±0.90618 0.236927

The Abscissas and weights can be computed analyti-
cally for small n.

n Xi

Wi

2 ±f\/3 1

3 f

±1^ I

4 ± yi^SLL

3+2

A

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.

Boca Raton, FL: CRC Press, pp. 462-463, 1987.
Chandrasekhar, S. Radiative Transfer. New York: Dover,

pp. 56-62, 1960.
Hildebrand, F. B. Introduction to Numerical Analysis. New

York: McGraw-Hill, pp. 323-325, 1956.

Legendre-Jacobi Elliptic Integral

Any of the three standard forms in which an ELLIPTIC
Integral can be expressed.

see also ELLIPTIC INTEGRAL OF THE FIRST KIND, EL-
LIPTIC Integral of the Second Kind, Elliptic In-
tegral of the Third Kind

Legendre Polynomial 1057
Legendre Polynomial

The Legendre Functions of the First Kind are
solutions to the Legendre Differential Equation.
If I is an Integer, they are Polynomials. They are a
special case of the ULTRASPHERICAL FUNCTIONS with
a = 1/2. The Legendre polynomials P n (x) are illus-
trated above for x € [0, 1] and n = 1, 2, . , . , 5.

The Rodrigues FORMULA provides the Generating
Function

p '^ = iv.^ 2 -^

(i)

which yields upon expansion

Pl(x) - I V (-l)*(2'-2fc)! x i-*» (2)

P 'W- 2 , X, kl(l-k)\(l-2k)\ X ' {2)

k=Q ' V

where [r\ is the Floor Function. The Generating
Function is

g(t, x) = (1 - 2xt + t 2 )~ 1/2 = ]T P n (x)t n . (3)
Take dg/dt,

CO

-±(l-2xt + t 2 y 3/2 (-2x + 2t) = Y^nPnix)^- 1 . (4)

71 =

Multiply (4) by 2t,

CO

-t(l - 2xt + t 2 )~ 3/2 {-2x + 2t) = ^2 2nP n (x)t n (5)

72 =

and add (3) and (5),

(1 - 2xt + t 2 y 3/2 [(2xt - 2t 2 ) + (1 - 2xt + t 2 )]

CO

= ^(2n+l)P»(x)i n (6)

n=0

oo

(1 - 2xt + t 2 )" 3/2 (l - t 2 ) = J2(2n + l)P n (x)t n . (7)

1058 Legendre Polynomial

Legendre Polynomial

This expansion is useful in some physical problems, in-
cluding expanding the Heyney-Greenstein phase func-
tion and computing the charge distribution on a
Sphere. They satisfy the Recurrence Relation

(/ + l)fl+i(x) - (2/ + l)xPi(x) + ifl-i(s) = 0. (8)

The Legendre polynomials are orthogonal over (—1,1)
with Weighting Function 1 and satisfy

/_;

P n (x)P m (x)dx =

2n+ 1

(9)

where 5 mn is the Kronecker Delta.
A Complex Generating Function is

Pt(x)

= -^ [(l-2zx + z 2 )- 1/2 z- l - 1 dz>

2ivt J

and the Schlafli integral is

(-i)

Pi(x)

1_L fSL

2m J (z ■

-z 2 ) 1

dz.

2 l 2tti I (z~x) l + l
Additional integrals (Byerly 1959, p. 172) include

(10)

(11)

/'

Jo

Pm(x)dx

-{

o

(-i) (

m-l)/2_

m(ro+l)(m-l)!!

m even ^

m odd ( 12 )

/'

Jo

Pm(x)P n (x) dx =

m, n both even or odd m ^ n

( — 1 \(rn+n+l)/2 m\n\

2^+"+l(m-n)(m+n+l)(|m)!{[|(Tz-l)]!}2

m even, n odd

l

2n+l»

m = n.

(13)

An additional identity is

l-x 3

•-ftwi'-Ef^:

Pn(x)

*n \Xv)\X X u j

(14)

(Szego 1975, p. 348).

The first few Legendre polynomials are

1

Po(x
Pi(x
P 2 (x
P 3 (x
Pa{x
Ps(x
Pe(x

-1(3^-1)
= |(5x 3 -3a;)

= |(35x 4 -30a; 2 + 3)

= |(63x 5 - 70a; 3 + 15x)

= ^(231a; 6 - 315a; 4 + 105a; 2 - 5).

The first few POWERS in terms of Legendre polynomials
are

x = P 1

x 2 = |(P + 2P 2 )

z 3 = |(3Pi+2P 3 )

z 4 = ^(7Po + 20P 2 +8P 4 )

a: B = £(27i\ + 28ft + 8ft)

x * = 2§i( 33P o + 110p 2 + 72P 4 + 16P 5 ).

For Legendre polynomials and Powers up to exponent
12, see Abramowitz and Stegun (1972, p. 798).

The Legendre Polynomials can also be generated using
Gram-Schmidt Orthonormalization in the Open
Interval (-1,1) with the Weighting Function 1.

ft (a?) = 1
Pi(x) =

•1

P 2 (x)

l-(-l)
J x 2 dx

P 3 (x) =

f_ xdx

/-1 dx .
£[*T-i _ 1(1-1)

Mil

J x 3 dx

f^x 2 dx\ [ J^dx

j[* 4 ]^

/>(»'- §) a ds
j\{x*-\Ydx

(15)

(16)

x —

u* 3 \u

}ll

x 2 -\(17)

{* 2 -\)

■ 2 ~\fdx

J x 2 dx

a

+ i)*'

— x z x 3( 5 9 )

(18)

Normalizing so that P n (l) = 1 gives the expected Leg-
endre polynomials.

The "shifted" Legendre polynomials are a set of func-
tions analogous to the Legendre polynomials, but de-
fined on the interval (0, 1). They obey the ORTHOGO-
NALITY relationship

/'

Jo

Pm(x)P n (x) dx :

2n + l

(19)

The first few are

Po(x) = 1

Pi(x) = 2a;- 1

P 2 {x) = 6x 2 - 6x + 1

Pz{x) = 20a; 3 - 30a; 2 + 12a; - 1.

Legendre Polynomial

Legendre Polynomial 1059

The associated Legendre polynomials P/ m (:c) are so-
lutions to the associated Legendre Differential
Equation, where / is a Positive Integer and m = 0,
. . . , I. They can be given in terms of the unassociated
polynomials by

2'Z!

•(i-* 2 r /2 ^(* 2 -i)',(2o)

where Pj(aO are the unassociated Legendre POLYNO-
MIALS. Note that some authors (e.g., Arfken 1985,
p. 668) omit the Condon-Shortley Phase (-l) m ,
while others include it (e.g., Abramowitz and Stegun
1972, Press et al. 1992, and the LegendreP[l,m,z]
command of Mathematical®). Abramowitz and Stegun
(1972, p. 332) use the notation

P lm (x) = (-l) m PUx)

(21)

to distinguish these two cases.

Associated polynomials are sometimes called Ferrers'
FUNCTIONS (Sansone 1991, p. 246). If m = 0, they re-
duce to the unassociated Polynomials. The associated
Legendre functions are part of the Spherical HARMON-
ICS, which are the solution of LAPLACE'S EQUATION
in Spherical Coordinates. They are Orthogonal
over [-1,1] with the Weighting Function 1

L

Pr(*)P-(*) d s=^!f±^ a ,, (22)

Orthogonal over [-1,1] with respect to m with the
Weighting Function (1 - x 2 )~ 2

I

/< (X)P, i*)T—2= m{l _ m)l <

(23)

They obey the RECURRENCE RELATIONS

(I - m)J=J m (x) = x(2l - lJJTifs) -(l + m- l)P£ 2 {x)

(24)

dPi

de

i - A —

T*tEL

dfj,

= 1(1 - m + 1)(I + m + P< - ^ ) (25)
(2/ + l)/xPr = (/ + m)P l r l 1 + (I - m + l)P z +i (26)

(2/ + l)Vl-ji 2 iT = P^t 1 - PR 1 . (27)

An identity relating associated POLYNOMIALS with
Negative m to the corresponding functions with Pos-
itive m is

c> — m / -i\m(/ TTl). n

(28)

Additional identities are

p{(x) = (-l)'(2l - 1)!!(1 - z 2 )' /2 (29)

P/+i(*) = :r(2Z + l)P/(z). (30)

Written in terms of # and using the convention without
a leading factor of (-l) m (Arfken 1985, p. 669), the first
few associated Legendre polynomials are

P§{x
Pl(x

pi(x

P§(x

PZ(x
P?(x

p2(x

Pl{x
P\{x
P!(x
Pt(x
PHx

= 1

= X

= -(i-<0

1

2

2x1/2

-±(3x 2 -l)

= -33(l-s a ) 1/2
= 3(l-x 2 )

= \x{hx 2 - 3)

= §(1-5* 2 )(1-* 2 ) 1 ' 2

= \$x{l-x 2 )

= -15(l-z 2 ) 3/2

= §(35z 4 - 30x 2 + 3)

= §s(3-7s 2 )(l- a! 2 ) 1/2

= ¥(7s 2 -l)(l-s 2 )

= -105x(l-:c 2 ) 3/2

= 105(1 -x 2 ) 2

= |x(63a: 4 - 70a; 2 + 15).

Written in terms x = cos 0, the first few become

f sin0

Po(cos0) = 1
Pf x (cos 0):
Pf (cos0) = cos0 = /x
Pi (cos 0) = sin0
P 2 ~ 2 (cos0) = | sin 2 (9
P 2 ~~ ( c °s 0) = § sin cos
P 2 °(cos0) = ±(3cos 2 0-l)
Pa 1 (cos 0) = 3sin0cos0

= §sin 2
P 2 2 (cos0) = 3sin 2

^§(l-cos 2 0)
P 3 °(cos0) = § cos 0(5 cos 2 0-3)

= ±cos0(2-5sin 2 0)
Pa 1 (cos 0) = § (5 cos 2 0-1) sin

= §(sin0 + 5sin 3 0).

The derivative about the origin is

dP£{x)
dx

2^+ 1 sin[i 7 r(^ + ^]r(i I , + i M + l)

w V»r(i^-I/i+i)

(31)

1060 Legendre Polynomial of the Second Kind

Legendre Sum

(Abramowitz and Stegun 1972, p. 334), and the loga-
rithmic derivative is

dlniTO

dz

= 2tan[±7r(A + ju)L

[i(A + M)]![£(A-/i)]l

UX + n-

1)]![^(A

1)]!
(32)

(Binney and Tremaine 1987, p. 654).

see also Condon-Shortley Phase, Conical Func-
tion, Gegenbauer Polynomial, Kings Problem,
Laplace's Integral, Laplace-Mehler Integral,
Super Catalan Number, Toroidal Function,
Turan's Inequalities

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Legendre Func-
tions" and "Orthogonal Polynomials." Ch. 22 in Chs. 8
and 22 in Handbook of Mathematical Functions with For-
mulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 331-339 and 771-802, 1972.

Arfken, G, "Legendre Functions." Ch. 12 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 637-711, 1985.

Binney, J. and Tremaine, S. "Associated Legendre Func-
tions." Appendix 5 in Galactic Dynamics. Princeton, NJ:
Princeton University Press, pp. 654-655, 1987.

Byerly, W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
with Applications to Problems in Mathematical Physics.
New York: Dover, 1959.

Iyanaga, S. and Kawada, Y. (Eds.). "Legendre Function"
and "Associated Legendre Function." Appendix A, Tables
18.11 and 18. Ill in Encyclopedic Dictionary of Mathemat-
ics. Cambridge, MA: MIT Press, pp. 1462-1468, 1980.

Legendre, A, M. "Sur l'attraction des Spheroides." Mem.
Math, et Phys. presentes a VAc. r. des. sc. par divers sa-
vants 10, 1785.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 593-597, 1953.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 252, 1992.

Sansone, G. "Expansions in Series of Legendre Polynomials
and Spherical Harmonics." Ch. 3 in Orthogonal Functions,
rev. English ed. New York: Dover, pp. 169-294, 1991.

Snow, C. Hypergeometric and Legendre Functions with
Applications to Integral Equations of Potential Theory.
Washington, DC: U. S. Government Printing Office, 1952.

Spanier, J. and Oldham, K. B. "The Legendre Polynomials
P n (x) n and "The Legendre Functions P v (x) and Q u (x). n
Chs. 21 and 59 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 183-192 and 581-597, 1987,

Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, 1975.

Legendre Quadrature

see LEGENDRE-GAUSS QUADRATURE

Legendre Relation

Let E(k) and K(k) be complete Elliptic Integrals
of the First and Second Kinds, with E'(k) and
K'(k) the complementary integrals. Then

E(k)K'(k) + E ( {k)K{k) - K(k)K'(k) = |tt.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 591, 1972.

Legendre Series

Because the LEGENDRE FUNCTIONS OF THE FIRST
Kind form a Complete Orthogonal Basis, any
Function may be expanded in terms of them

f(x) — y^a n P n (x).

(1)

n—

Now, multiply both sides by P m (ar) and integrate

P m (x)f(x) dx = ^a n I P n (x)P rn (x) dx. (2)

1 „_n J -I

But

/ P n (x)P m (x)dx = —8„

J-! 2/71+1

(3)

where 5 m n is the KRONECKER DELTA, so

f 1 °° 2 2

/ 1 *"(*)/(*)** = E a »2^+i J ™ = 2mTi a "

J ~ 1 71=0

and

Am

2m

"jC

Pm{x)f(x)dx.

(4)

(5)

see also Fourier Series, Jackson's Theorem, Leg-
endre Polynomial, Maclaurin Series, Picone's
Theorem, Taylor Series

Legendre Sum

see Legendre's Formula

Legendre Polynomial of the Second Kind

see Legendre Function of the Second Kind

Legendre's Quadratic Reciprocity Law

see Quadratic Reciprocity Law

Legendre Symbol
Legendre Symbol

{0 if m\n
1 if n is a quadratic residue modulo m
— 1 if n is a quadratic nonresidue modulo m.

If m is an Odd Prime, then the Jacobi Symbol re-
duces to the Legendre symbol. The Legendre symbol
obeys (ab\p) = (a\p)(b\p).

1 ifp = ±1 (mod 12)
-1 ifp = ±5 (mod 12).

see also Jacobi Symbol, Kronecker Symbol, Quad-
ratic Reciprocity Theorem

References

Guy, R. K. "Quadratic Residues. Schur's Conjecture." §F5

in Unsolved Problems in Number Theory, 2nd ed. New

York: Springer- Verlag, pp. 244-245, 1994.
Shanks, D. Solved and Unsolved Problems in Number Theory,

4th ed. New York: Chelsea, pp. 33-34 and 40-42, 1993.

Legendre Transformation

Given a function of two variables

df = —— dx + -r— dy = u dx + v dy, (1)

ox dy

change the differentials from dx and dy to du and dy
with the transformation

g = f-ux

(2)

dg = df — udx — x du = udx -f v dy — udx — x du
= v dy — xdu. (3)

Then

du
£?£.

dy'

(4)

(5)

Lehmer's Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Lehmer (1938) showed that every POSITIVE IRRATIONAL
NUMBER x has a unique infinite continued cotangent
representation of the form

x = cot

^2(-l) k cot- 1 b h

Lehmer's Formula 1061

where the 6fcS are NONNEGATIVE and

b k > (6 fc -i) 2 +fefc-i + l.

The case for which the convergence is slowest occurs
when the inequality is replaced by equality, giving cq =
and

Cfc = (Cfc-i) 2 -\-c k -i + 1

for k > 1. The first few values are Ck are 0, 1, 3, 13, 183,
33673, . . . (Sloane's A024556), resulting in the constant

£ = cot(cot _1 - cot" 1 1 4- cot -1 3 - cot" 1 13
+ cot" 1 183 - cot" 1 33673 + cot" 1 1133904603

- cot" 1 1285739649838492213 + . . . + (-l) k c k + - - .)
= cot( ^7r + cot" 1 3 - cot" 1 13

+ cot" 1 183 - cot" 1 33673 + cot" 1 1133904603

- cot" 1 1285739649838492213 + . . . + (~l) k c k + . . .)
= 0.59263271...

(Sloane's A030125). £ is not an ALGEBRAIC NUMBER of
degree less than 4, but Lehmer's approach cannot show
whether or not £ is TRANSCENDENTAL.
see also Algebraic Number, Transcendental Num-
ber

References

Finch, S. "Favorite Mathematical Constants." http://vvw.
mathsoft.com/asolve/constant/lehmer/lehmer.html.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 29, 1983.

Lehmer, D. H. "A Cotangent Analogue of Continued Frac-
tions." Duke Math. J. 4, 323-340, 1938.

Plouffe, S. "The Lehmer Constant." http://lacim.uqam.ca/
piDATA/lehmer.txt.

Sloane, N. J. A. Sequences A024556 and A030125 in "An On-
Line Version of the Encyclopedia of Integer Sequences."

Lehmer's Formula

A Formula related to Meissel's Formula.

n(x) = kl - X^

l<i<j<a

PiPj

+ I(6 + a -2)(6-a + l)- £ n (^)

a<i<b ^ '

~ L KPtPj J J

bi

E

i=a+l j=i

where

_ / l/4\

a = tt(x ' )

b = 7T(x 1/2 )

bi = ir(y/x/pi)

C = 7T(X 1/3 ),
and 7r(n) is the PRIME COUNTING FUNCTION.

References

Riesel, H. "Lehmer's Formula." Prime Numbers and Com-
puter Methods for Factorization, 2nd ed. Boston, MA:
Birkhauser, pp. 13-14, 1994.

1062

Lehmer Method

Leibniz Harmonic Triangle

Lehmer Method

see Lehmer-Schur Method

Lehmer Number

A number generated by a generalization of a Lucas SE-
QUENCE. Let a and be Complex Numbers with

a + (3 = VR
<*0 = Q,

(1)

(2)

where Q and R are Relatively Prime Nonzero Inte-
gers and a/0 is a ROOT OF UNITY. Then the Lehmer
numbers are

U n {VR,Q) = - §-,

a — p

and the companion numbers

for n odd
for n even.

(3)

(4)

References

Lehmer, D. H. "An Extended Theory of Lucas' Functions."

Ann. Math. 31, 419-448, 1930.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.

New York: Springer- Verlag, pp. 61 and 70, 1989.
Williams, H. C. "The Primality of N = 2A3 n - 1." Canad.

Math. Bull 15, 585-589, 1972.

Lehmer's Phenomenon

0.06 0.07 0.08 0.09 0.1 0.11

The appearance of nontrivial zeros (i.e., those along the
Critical Strip with U[z] = 1/2) of the Riemann Zeta
Function £(z) very close together. An example is the
pair of zeros C(f + (7005 + t)i) given by t x w 0.0606918
and *2 ~ 0.100055, illustrated above in the plot of |C(| +
(7005 + £)z)| 2 .

see also Critical Strip, Riemann Zeta Function

References

Csordas, G.; Odlyzko, A. M.; Smith, W.; and Varga, R. S.
"A New Lehmer Pair of Zeros and a New Lower Bound for
the de Bruijn-Newman Constant." Elec. Trans. Numer.
Analysis 1, 104-111, 1993.

Csordas, C; Smith, W.; and Varga, R. S. "Lehmer Pairs of
Zeros, the de Bruijn-Newman Constant and the Riemann
Hypothesis." Constr. Approx. 10, 107-129, 1994.

Csordas, G.j Smith, W.; and Varga, R. S. "Lehmer Pairs
of Zeros and the Riemann ^-Function." In Mathematics
of Computation 1943-1993: A Half-Century of Computa-
tional Mathematics (Vancouver, BC, 1993). Proc. Sympos.
Appl. Math. 48, 553-556, 1994.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 357-358, 1991.

Lehmer's Problem

Do there exist any Composite Numbers n such that
(f)(n)\(n — 1)? No such numbers are known. In 1932,
Lehmer showed that such an n must be ODD and
Squarefree, and that the number of distinct PRIME
factors d(7) > 7. This was subsequently extended to
d(n) > 11. The best current results are n > 10 20
and d(n) > 14 (Cohen and Hagis 1980), if 30fn, then
d(n) > 26 (Wall 1980), and if 3|n then d(n) > 213 and
5.5 x 10 570 (Lieuwens 1970).

References

Cohen, G. L. and Hagis, P. Jr. "On the Number of Prime

Factors of n is <fc(n)\(n — 1)." Nieuw Arch. Wish. 28,

177-185, 1980.
Lieuwens, E. "Do There Exist Composite Numbers for which

k<t>(M) = M-l Holds?" Nieuw. Arch. Wish. 18, 165-169,

1970.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.

New York: Springer- Verlag, pp. 27-28, 1989.
Wall, D. W. "Conditions for <f>(N) to Properly Divide JV-1."

In A Collection of of Manuscripts Related to the Fibonacci

Sequence (Ed. V. E. Hoggatt and M. V. E. Bicknell-

Johnson). San Jose, CA: Fibonacci Assoc, pp. 205-208,

1980.

Lehmer-Schur Method

An Algorithm which isolates Roots in the Complex
Plane by generalizing 1-D bracketing.

References

Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., pp. 196-198, 1990.

Lehmer's Theorem

see Fermat's Little Theorem Converse

Lehmus' Theorem

see Steiner-Lehmus Theorem

Leibniz Criterion

Also known as the Alternating Series Test, Given
a Series

£(-ir +1 a„

with a n > 0, if a n is monotonic decreasing as n — ¥ oo
and

lim a n = 0,

n—*-oo

then the series CONVERGES.

Leibniz Harmonic Triangle

i

i T i
i 5 i 3 i

i S 12 ! 12 ! 1 i

5 20 30 20 5

In the Leibniz harmonic triangle, each Fraction is the
sum of numbers below it, with the initial and final en-
try on each row one over the corresponding entry in

Leibniz Identity

Lemniscate

1063

Pascal's Triangle. The Denominators in the sec-
ond diagonals are 6, 12, 20, 30, 42, 56, ... (Sloane's
A007622).

see also Catalan's Triangle, Clark's Triangle,
Euler's Triangle, Number Triangle, Pascal's
Triangle, Seidel-Entringer-Arnold Triangle

References

Sloane, N. J. A. Sequence A007622/M4096 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Leibniz Identity

Lemarie's Wavelet

A wavelet used in multiresolution representation to an-
alyze the information content of images. The WAVELET
is defined by

H(u) =

where

4 315-420?x + 126n 2 - 4u 3
^ ~ U > 315 - 420u + 126v 2 - 4v 3

-,1/2

u = sin (|o>)

_ • 2

v = sin cj

d n
dx n

(uv)

•> dv
dx

d n u fn\ d n ~ 1 u i
dx" V \l) dx 71 - 1 (

Therefore,
da;

dy 2

<Px
dy 3

dx

d 2 y (dy
'dx 2

\dxJ

d*y

dx 2

drydy
dx 3 dx

\dx)

(1)

(2)
(3)
(4)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 12, 1972.

Leibniz Integral Rule

dz

r.

Ja(z)

f(x,z) dx

l>b{z)

Ja(z)

8 J-dx + f{b{z),z)f z - /(a(z),z)g.

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 11, 1972.

Leibniz Series

The Series for the Inverse Tangent,

tan 1 x = x — ~x 3 -f |x 5 + .

(Mallat 1989).
see also WAVELET

References

Mallat, S. G. "A Theory for Multiresolution Signal Decom-
position: The Wavelet Representation." IEEE Trans. Pat-
tern Analysis Machine Intel. 11, 674-693, 1989.

Mallat, S. G. "Multiresolution Approximation and Wavelet
Orthonormal Bases of L 2 (M)." Trans. Amer. Math. Soc.
315, 69-87, 1989.

Lemma

A short Theorem used in proving a larger Theorem.
Related concepts are the Axiom, Porism, POSTULATE,
Principle, and Theorem.

see also Abel's Lemma, Archimedes' Lemma,
Barnes' Lemma, Blichfeldt's Lemma, Borel-Can-
telli Lemma, Burnside's Lemma, Danielson-Lan-
czos Lemma, Dehn's Lemma, Dilworth's Lemma,
Dirichlet's Lemma, Division Lemma, Farkas's
Lemma, Fatou's Lemma, Fundamental Lemma
of Calculus of Variations, Gauss's Lemma,
Hensel's Lemma, Ito's Lemma, Jordan's Lemma,
Lagrange's Lemma, Neyman-Pearson Lemma,
Poincare's Holomorphic Lemma, Poincare's
Lemma, Polya-Burnside Lemma, Riemann-Le-
besgue Lemma, Schur's Lemma, Schur's Repre-
sentation Lemma, Schwarz-Pick Lemma, Spijker's
Lemma, Zorn's Lemma

Lemniscate

A polar curve also called Lemniscate of Bernoulli
which is the Locus of points the product of whose dis-
tances from two points (called the Foci) is a constant.
Letting the Foci be located at (±a,0), the Cartesian
equation is

\(x-a?+y 2 ][{x + a) 2 +y 2 ] = a\ (1)

which can be rewritten

x 4 +y 4 + 2x 2 y 2 = 2a 2 (x 2 -y 2 ). (2)

1064 Lemniscate

Lemniscate

Letting a' = y/2a, the Polar Coordinates are given
by

(3)

r 2 =a 2 cos(26>).

An alternate form is

t — a sin(20).

The parametric equations for the lemniscate are

a cost

x ^

1 + sin 2 t

a sin t cos t

y =

1 + sin 2 1 '

The bipolar equation of the lemniscate is

' 1 2

TV = 2<Z ,

(4)

(5)
(6)

(7)

and in Pedal Coordinates with the Pedal Point at
the center, the equation is

2 3

pa = r .

(8)

The two-center Bipolar Coordinates equation with
origin at a FOCUS is

T\T2 — C .

(9)

Jakob Bernoulli published an article in Acta Eruditorum
in 1694 in which he called this curve the lemniscus ("a
pendant ribbon"). Jakob Bernoulli was not aware that
the curve he was describing was a special case of Cassini
OVALS which had been described by Cassini in 1680.
The general properties of the lemniscate were discovered
by G. Fagnano in 1750 (MacTutor Archive). Gauss's
and Euler's investigations of the Arc LENGTH of the
curve led to later work on Elliptic Functions.

The CURVATURE of the lemniscate is
3\/2cosi

y/3 - cos(2i)

(10)

The Arc Length is more problematic. Using the polar
form,

ds 2 = dr 2 + r 2 d6 2 (11)

dS= V + { T al) dr '

(12)

we have

2r dr = 2a 2 sin(29) d9

(13)

dr r 2

(14)

T d0 ~ a 2 sin(20)

<#v_

4 4

r r

r 4

i 2 (2(9) a 4 [l cos 2 (20)] a 4 r 4 '

(15)

ds= 4/1 +

dr =

)4 _ r 4

dr ■

v^*

: dr

dr

fv n r ds _, n r dr

= ds = 2 —dr = 2 ,

L Jo ^ J V T T ^

= r/a, so dt = dr/a, and

L = 2a f (1 - t 4 )~ 1/2 dt,
Jo

(16)

(17)

(18)

which, as shown in LEMNISCATE FUNCTION, is given
analytically by

L = V2aK^=r^a. (19)

If a = 1, then

L = 5.2441151086....

(20)

(21)

which is related to Gauss's Constant M by

M

The quantity L/2 or L/A is called the LEMNISCATE CON-
STANT and plays a role for the lemniscate analogous to
that of 7r for the Circle.

The Area of one loop of the lemniscate is

//>tt/4
r 2 dO = \a 2 \ cos(26') dd = Ja 2 [sin(2^)]^ / 7r 4 /4
J-ir/4

= ia 2 [sin(2^)] ^ 4 = ia 2 [sin(f ) - sinO] = fa 2 . (22)

see also LEMNISCATE FUNCTION

References

Ayoub, R. "The Lemniscate and Fagnano's Contributions to

Elliptic Integrals." Arch. Hist Exact Sci. 29, 131-149,

1984.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in

Analytic Number Theory and Computational Complexity.

New York: Wiley, 1987.
Gray, A. "Lemniscates of Bernoulli." §3.2 in Modern Differ-
ential Geometry of Curves and Surfaces. Boca Raton, FL:

CRC Press, pp. 39-41, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 120-124, 1972.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

p. 37, 1983.
Lee, X. "Lemniscate of Bernoulli." http://www.best .com/

- xah / SpecialPlaneCurves jdir / Lemnis cat eOf Bernoulli-

dir/lemniscateOf Bernoulli. html.
Lockwood, E. H. A Book of Curves. Cambridge, England:

Cambridge University Press, 1967.
MacTutor History of Mathematics Archive. "Lemniscate of

Bernoulli." http : // www - groups . dcs . st - and . ac . uk /

-history/Curves/Lemniscate.htinl.
Yates, R. C. "Lemniscate." A Handbook on Curves and

Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 143-

147, 1952.

Lemniscate of Bernoulli

Lemniscate Function 1065

Lemniscate of Bernoulli

see Lemniscate

Lemniscate Case

The case of the WEIERSTRAB ELLIPTIC FUNCTION with
invariants gi~\ and gz = 0.

see also Equianharmonic Case, Weierstrad Ellip-
tic Function, Pseudolemniscate Case

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Lemniscate Case
(#2 = 1, fls = 0)." §18.14 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, pp. 658-662, 1972.

Lemniscate Constant

Let

L =

1 mi)] 2

2tt

5.2441151086..

be the Arc Length of a Lemniscate with a =
1. Then the lemniscate constant is the quan-
tity L/2 (Abramowitz and Stegun 1972), or L/4 =
1.311028777... (Todd 1975, Le Lionnais 1983). Todd
(1975) cites T. Schneider (1937) as proving L to be a
Transcendental Number.

see also LEMNISCATE

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook

of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, 9th printing. New York: Dover,

1972.
Borwein, J. M. and Borwein, R B. Pi & the AGM: A Study in

Analytic Number Theory and Computational Complexity.

New York: Wiley, 1987.
Finch, S. "Favorite Mathematical Constants." http://www.

mathsof t . com/ asolve/constant/gauss/gauss .html.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

p. 37, 1983.
Todd, J. "The Lemniscate Constant." Comm. ACM 18, 14-

19 and 462, 1975.

Lemniscate Function

The lemniscate functions arise in rectifying the ARC
Length of the Lemniscate. The lemniscate functions
were first studied by Jakob Bernoulli and G. Fagnano.
A historical account is given by Ayoub (1984), and an
extensive discussion by Siegel (1969). The lemniscate
functions were the first functions defined by inversion of
an integral, which was first done by Gauss.

L = 2a J (l-t 4 )~ 1/2 dt.
Jo

(i)

Define the functions

<f>(x) = arcsinlemna; = / (1 - t 4 )~ 1/2 dt (2)

Jo

<p'(x) = arccoslemnx =/ (1 — t )~ dt, (3)

where

and write

x = sinlemn <f>
x — coslemn^'.

There is an identity connecting <j) and <f> since

(4)

(5)
(6)

(7)

sinlemn = coslemn(|'07 — 0). (8)

These functions can be written in terms of Jacobi El-
liptic Functions,

Jo

sd(u,fc)

[(i-*V)(i + *V)r 1/J <fo. (9)

Now, if k = k' = l/>/2, then

iisd(u,l/\/2)

Jo

psd(u,l/V2)

= (l-\y 4 r 1/2 dy. (10)

Jo

Let t = y/y/2 so dy = V2dt,

/»sd(u,l/\/2)/\/2

u=V2 (l~t 4 y 1/2 dt (11)

Jo

>sd(ti,l/v / 2)/v / 2

V2 Jo

and

(l-t 4 )~ 1/2 dt (12)

psd(uV2,\/y/2)/y/2

u= (l-~t 4 )~ 1/2 dt, (13)

Jo

sinlemn ^ = — ^sd I <pV2, —= J . (14)

Similarly,

u = r (i _ t 2 )- i/2 (fc' 2 + fcV)- i/2 M

J cn(u,fc)
Jcn(u,l/y/2)

= vt f 1

V 2 Jcn(u,l/

(l-t 4 )" 1/J dt

(15)

(l-t*)- L/ *dt (16)

1066 Lemniscate Function

(i-t 4 y 1/a dt,

and

We know

Jcn(uV2,l

coslemn = en I (/>y/2, -7= )
V y/2J

(17)
(18)

coslemn(^) = en ( \m\/2, — ) = 0. (19)

But it is true that

cn(K, k) = 0,

so

*'^H^ W = >

r 2 (^)_ 1

4y^ v/2

•H7

L = o W = aV5- ra( ^- I ' 2( « )

4^ 2 3 / 2 V^

(20)

(21)
(22)
(23)

By expanding (1 - i 4 )" 1/2 in a Binomial Series and
integrating term by term, the arcsinlemn function can
be written

ju ^ T dt ^ (|)nX 4 " +1 , N

n=0

where (a) n is the Rising Factorial (Berndt 1994). Ra-
manujan gave the following inversion FORMULA for <f>{x).
If

^2 ^n!(4n+ir ^°'

where

M

r 2 (i)

2tt 3 /2

(26)

is the constant obtained by letting x = 1 and = 7r/2,
and

^ = 2- 1/2 sd(^), (27)

then

, - i - 8 E

ncos(2n#)

" ^-^^-^'^^ (28)

n = l

(Berndt 1994). Ramanujan also showed that if < <
7r/2, then

1\ „,4n-l

V2 -^— ' n!(4n - 1) 7r ^

^ ^n!(4n-l)

n=0

, ^ ^ sin(2nfl)

227TU _ J ;

(29)

n=l

Lemniscate Inverse Curve

\)nV 4

taw+ *'-* ha+ i:fu

cos(2refl)

i / • /i\ , ^ ^ V^ costzntn . .

ln(sm0) + - 2> ——± '—. (30)

sin[(2n + 1)0]

1 tan -i V = ST- sin^n+ije/j

2 Z-, (2n + 1) cosh[| (2n + 1)tt] ' V ;

i cos -i^)-f- (-l) n cog[(2n + l)g]

4 eos ^)-Z. (2n + 1)cosh[1(2n+1)7rr (32)

and

v^y 2 2 "(n! ) 2 ^ 4n+3

^ (2n-

4^ Z^ (2n+l)!(4n + 3)

7T<9

_ 7T0 ^^

(-l) n sin[(2n + l)<9]

8 ^ ( 2n + 2 ) 2 cosh[i (2n + 1)tt]

(33)

(Berndt 1994).

A generalized version of the lemniscate function can be
defined by letting < 6 < k/2 and < v < 1. Write

dt

(34)

where \i is the constant obtained by setting 6 = 7r/2 and
u = 1. Then

A* = „ /a ^,^ ' ( 35 )

r(f)r(f)'

and Ramanujan showed
2

Z— • e 7rnV3 _ (— l) n

9v 2

(Berndt 1994).

see also Hyperbolic Lemniscate Function

References

Ayoub, R. "The Lemniscate and Fagnano's Contributions to

Elliptic Integrals." Arch. Hist. Exact Sci. 29, 131-149,

1984.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:

Springer- Verlag, pp. 245, and 247-255, 258-260, 1994.
Siegel, C. L. Topics in Complex Function Theory, Vol. 1.

New York: Wiley, 1969.

Lemniscate of Gerono

see Eight Curve

Lemniscate Inverse Curve

The Inverse Curve of a Lemniscate in a Circle cen-
tered at the origin and touching the LEMNISCATE where
it crosses the x-Axis produces a Rectangular Hy-
perbola.

Lemniscate (Mandelbrot Set)
Lemniscate (Mandelbrot Set)

Lemoine Point

1067

A curve on which points of a Map z n (such as the Man-
delbrot Set) diverge to a given value r max at the same
rate. A common method of obtaining lemniscates is to
define an INTEGER called the COUNT which is the largest
n such that \z n \ < r where r is usually taken as r — 2.
Successive COUNTS then define a series of lemniscates,
which are called Equipotential Curves by Peitgen
and Saupe (1988).
see also COUNT, MANDELBROT SET

References

Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal
Images. New York: Springer- Verlag, pp. 178-179, 1988.

Lemoine Axis

see Lemoine Line

Lemoine Circle

Also called the TRIPLICATE- RATIO CIRCLE. Draw lines
through the Lemoine Point K and parallel to the sides
of the triangle. The points where the parallel lines inter-
sect the sides then lie on a CIRCLE known as the Lemoine
circle. This circle has center at the MIDPOINT of OK,
where O is the ClRCUMCENTER. The circle has radius

\ \/R 2 + r 2 = Insect*;,

where R is the ClRCUMRADlUS, r is the INRADIUS, and
uj is the BROCARD Angle. The Lemoine circle divides

any side into segments proportional to the squares of the
sides

A2P2 : P2Q3 : Q3A3 = a* : ai : a 2 .

Furthermore, the chords cut from the sides by the
Lemoine circle are proportional to the squares of the
sides.

The Cosine CIRCLE is sometimes called the second
Lemoine circle.

see also Cosine Circle, Lemoine Line, Lemoine
Point, Tucker Circles

References

Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 273-275, 1929.

Lemoine Line

The Lemoine line, also called the Lemoine Axis, is the
perspectivity axis of a TRIANGLE and its TANGENTIAL
Triangle, and also the Trilinear Polar of the Cen-
TROID of the triangle vertices. It is also the POLAR of K
with regard to its ClRCUMCIRCLE, and is PERPENDICU-
LAR to the Brocard Axis.

The centers of the APOLLONIUS CIRCLES L x , L2, and
L 3 are Collinear on the LEMOINE LINE. This line is
Perpendicular to the Brocard Axis OK and is the
Radical Axis of the Circumcircle and the Brocard
Circle. It has equation

£ + £ + 2

a c

in terms of Trilinear COORDINATES (Oldknow 1996).

see also Apollonius Circles, Brocard Axis,
Centroid (Triangle), Circumcircle, Collinear,
Lemoine Circle, Lemoine Point, Polar, Radical
Axis, Tangential Triangle, Trilinear Polar

References

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle, Boston,
MA: Houghton Mifflin, p. 295, 1929.

Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.

Lemoine Point

The point of concurrence K of the SYMMEDIAN LINES,
sometimes also called the SYMMEDIAN POINT and
Grebe Point.

Let G be the Centroid of a Triangle AABC, L a >

Lb, and L c the ANGLE BISECTORS of ANGLES A, B 7
C, and Ga, Gb, and Gc the reflections of AG, BG,
and CG about La, Lb, and Lc- Then K is the point
of concurrence of the lines Ga, Gb, and Gc- It is the
perspectivity center of a Triangle and its Tangential
Triangle.

1068

Lemoine 7 s Problem

Length (Number)

In Areal Coordinates (actual Trilinear Coor-
dinates), the Lemoine point is the point for which
a 2 +/3 2 +7 2 is a minimum. A center X is the Centroid
of its own Pedal Triangle Iff it is the Lemoine point.

The Lemoine point lies on the BROCARD Axis, and its
distances from the Lemoine point K to the sides of the
Triangle are

KKi = \ai tanw,

where w is the Brocard Angle. A Brocard Line,
MEDIAN, and Lemoine point are concurrent, with A1O1,
A 2 K, and A3M meeting at a point. Similarly, AiQ r ,
A^M, and A3K meet at a point which is the ISOGONAL
Conjugate of the first (Johnson 1929, pp. 268-269).
The line joining the Midpoint of any side to the mid-
point of the Altitude on that side passes through the
Lemoine point K. The Lemoine point K is the STEINER
Point of the first Brocard Triangle.

see also Angle Bisector, Brocard Angle, Bro-
card Axis, Brocard Diameter, Centroid (Trian-
gle), COSYMMEDIAN TRIANGLES, GREBE POINT, ISO-

gonal Conjugate, Lemoine Circle, Lemoine Line,
Line at Infinity, Mittenpunkt, Pedal Triangle,
Steiner Points, Symmedian Line, Tangential Tri-
angle

References

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.
London: Hodgson, p. 86, 1913.

Honsberger, R. Episodes in Nineteenth and Twentieth Cen-
tury Euclidean Geometry. Washington, DC: Math. Assoc,
Amer., 1995.

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 217, 268-269, and 271-272,
1929.

Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.

Kimberling, C. "Symmedian Point." http : //www .

evansville . edu/-ck6/tcenters/class/sympt .html.

Mackay, J. S. "Early History of the Symmedian Point." Proc.
Edinburgh Math. Soc. 11, 92-103, 1892-1893.

Lemoine's Problem

Given the vertices of the three EQUILATERAL TRIAN-
GLES placed on the sides of a TRIANGLE T, construct
T. The solution can be given using KlEPERT'S HYPER-
BOLA.

see also Kiepert's Hyperbola

Lemon

A Surface of Revolution defined by Kepler. It con-
sists of less than half of a circular Arc rotated about
an axis passing through the endpoints of the Arc. The
equations of the upper and lower boundaries in the xz
plane are

z± =±\/R 2 ~{x + r) 2

for R > r and x e [-(R-r), R-r]. The CrOSS-Section
of a lemon is a LENS. The lemon is the inside surface of
a Spindle Torus.
see also Apple, Lens, Spindle Torus

Length (Curve)

Let 7(2) be a smooth curve in a MANIFOLD M from x
to y with 7(0) = x and 7(1) = y. Then <y'(t) € T 7(t) ,
where T x is the TANGENT SPACE of M at x. The length
of 7 with respect to the Riemannian structure is given

by

/

Jo

I7'(*)ll7<i)<*i.

see also ARC LENGTH, DISTANCE

Length Distribution Function

A function giving the distribution of the interpoint dis-
tances of a curve. It is defined by

pW = ^E if «=

see also Radius of Gyration

References

Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
pp. 204-206, 1995.

Length (Number)

The length of a number n in base b is the number of
Digits in the base-6 numeral for n, given by the formula

L(n,6) = Llog 6 (n)J+l,

where [x\ is the FLOOR FUNCTION.

The Multiplicative Persistence of an n-DiGiT is
sometimes also called its length.

see also Concatenation, Digit, Figures, Multi-
plicative Persistence

Length (Partial Order)

Lens Space 1069

Length (Partial Order)

For a Partial Order, the size of the longest Chain is
called the length.

see also Width (Partial Order)

Length (Size)

The longest dimension of a 3-D object.

see also Height, Width (Size)

LengyePs Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Let L denote the partition lattice of the Set
{1, 2, . . . , n}. The MAXIMUM element of L is

Lens

M = {{l,2,...,n}}
and the Minimum element is

m = {{l},{2},...,{n}}.

(1)

(2)

Let Z n denote that number of chains of any length in
L containing both M and m. Then Z n satisfies the
Recurrence Relation

Z n = }^s{n, k)Zk,

(3).

fc=i

where s(n,k) is a Stirling Number of the Second
Kind. Lengyel (1984) proved that the Quotient

r{n) =

Z n

(n!) 2 (21n2)-"n 1 -( ln2 )/ 3

(4)

is bounded between two constants as n — > oo, and Fla-
jolet and Salvy (1990) improved the result of Babai and
Lengyel (1992) to show that

A = lim r(n) = 1.0986858055

(5)

References

Babai, L. and Lengyel, T. "A Convergence Criterion for Re-
current Sequences with Application to the Partition Lat-
tice." Analysis 12, 109-119, 1992.

Finch, S. "Favorite Mathematical Constants." http://www,
maths oft . c om/ as o 1 ve / c onst ant /lngy/ lngy.html.

Flajolet, P. and Salvy, B. "Hierarchal Set Partitions and An-
alytic Iterates of the Exponential Function." Unpublished
manuscript, 1990.

Lengyel, T. "On a Recurrence Involving Stirling Numbers."
Europ. J. Comb. 5, 313-321, 1984.

Plouffe, S. "The Lengyel Constant." http://lacim.uqam.ca/
piDATA/lengyel.txt.

A figure composed of two equal and symmetrically
placed circular Arcs. It is also known as the Fish
Bladder (Pedoe 1995, p. xii) or Vesica Piscis. The
latter term is often used for the particular lens formed
by the intersection of two unit CIRCLES whose centers
are offset by a unit distance (Rawles 1997). In this case,
the height of the lens is given by letting d — r = R = 1
in the equation for a ClRCLE-ClRCLE INTERSECTION

iy^PJP"

(<P

+ R 2 ) 2

(1)

giving a = y/S. The Area of the Vesica Piscis is given
by plugging d = R into the Circle-Circle Intersec-
tion area equation with r = R y

A = 2iT cos

-(s)

\d^Jm 2 -d 2 , (2)

giving

A= J(4tt-3a/3) « 1.22837.

(3)

Renaissance artists frequently surrounded images of Je-
sus with the vesica piscis (Rawles 1997). An asymmetri-
cal lens is produced by a Circle-Circle Intersection
for unequal CIRCLES.

see also CIRCLE, ClRCLE-ClRCLE INTERSECTION,

Flower of Life, Lemon, Lune (Plane), Reuleaux
Triangle, Sector, Seed of Life, Segment, Venn
Diagram

References

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., 1995.

Rawles, B. Sacred Geometry Design Sourcebook: Universal
Dimensional Patterns. Nevada City, CA: Elysian Pub.,
p. 11, 1997.

Lens Space

A lens space L(p, q) is the 3-Manifold obtained by glu-
ing the boundaries of two solid TORI together such that
the meridian of the first goes to a (p, q) -curve on the
second, where a (j>, g)-curve has p meridians and q lon-
gitudes.

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, 1976.

1070 Lenstra Elliptic Curve Method

Letter-Value Display

Lenstra Elliptic Curve Method

A method of factoring Integers using Elliptic
Curves.

References

Montgomery, P. L. "Speeding up the Pollard and Elliptic

Curve Methods of Factorization." Math. Comput, 48,

243-264, 1987.

Leon Anne's Theorem

so that the integral

Pick a point O in the interior of a QUADRILATERAL
which is not a PARALLELOGRAM. Join this point to
each of the four VERTICES, then the LOCUS of points O
for which the sum of opposite TRIANGLE areas is half
the Quadrilateral Area is the line joining the Mid-
points Mi and M 2 of the DIAGONALS.

see also Diagonal (Polygon), Midpoint, Quadri-
lateral

References

Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 174-175, 1991.

Leonardo's Paradox

In the depiction of a row of identical columns parallel to
the plane of a Perspective drawing, the outer columns
should appear wider even though they are farther away.

see also Perspective, Vanishing Point, Zeeman's
Paradox

References

Dixon, R. Mathographics. New York: Dover, p. 82, 1991.

Leptokurtic

A distribution with a high peak so that the KURTOSIS
satisfies 72 > 0.

see also KURTOSIS

Lerch's Theorem

If there are two functions Fi(t) and F 2 (t) with the same
integral transform

T[F x {t)] = T[F*{t)\ = f{s), (1)

then a NULL FUNCTION can be defined by

S (t) = Fxit) - F 2 (t) (2)

/ So
Jo

(t) dt = Q

(3)

vanishes for all a > 0.
see also NULL FUNCTION

Lerch Transcendent

A generalization of the HURWITZ ZETA FUNCTION and
POLYLOGARITHM function. Many sums of reciprocal
POWERS can be expressed in terms of it. It is defined
by

^> s > a ) = E(^F'

(i)

where any term with a + k — is excluded.

The Lerch transcendent can be used to express the
Dirichlet Beta Function

00
0(8) = £(-l)*(2fe + l)-'2-°*(-l,s, 1), (2)

k=0

the integral of the FERMI-DlRAC DISTRIBUTION

k 3

f

Jo

e k-v _j_ 1

dk = eT(s + l)*(-e M , s + 1, 1), (3)

where T{z) is the Gamma FUNCTION, and to evaluate
the Dirichlet L-Series.

see also Dirichlet Beta Function, Dirichlet L-
Series, Fermi-Dirac Distribution, Hurwitz Zeta
Function, Polylogarithm

Less

A quantity a is said to be less than 6 if a is smaller than
6, written a < b. If a is less than or Equal to 6, the
relationship is written a < b. If a is MUCH LESS than
6, this is written a«6. Statements involving GREATER
than and less than symbols are called INEQUALITIES.

see also EQUAL, GREATER, INEQUALITY, MUCH
Greater, Much Less

Letter- Value Display

A method of displaying simple statistical parameters in-
cluding Hinges, Median, and upper and lower values.

References

Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, p. 33, 1977.

Leudesdorf Theorem

Levy Constant 1071

Leudesdorf Theorem

Let t(m) denote the set of the <f){m) numbers less than
and Relatively Prime to m, where <j>(n) is the To-
tient Function. Then if

-£J.

t(m)

then

f Sm = (mod m 2 ) if 2{m, Z\m

S m = (mod \m 2 ) if 2{m, 3|m

Sm = (mod |m 2 ) 2|m, {m, m not a power of 2

5 m = (mod \m 2 ) if 2|m, 3|m

, S m = (mod |m 2 ) if m = 2 a .

see also Bauer's Identical Congruence, Totient
Function

References

Hardy, G. H. and Wright, E. M. "A Theorem of Leudesdorf."

§8.7 in An Introduction to the Theory of Numbers, 5th ed.

Oxford, England: Clarendon Press, pp. 100-102, 1979.

Level Curve

A Level Set in 2-D.

Leviathan Number

The number (10 666 )!, where 666 is the Beast Number
and n! denotes a FACTORIAL. The number of trailing ze-
ros in the Leviathan number is 25 x 10 664 - 143 (Pickover
1995).

see also 666, Apocalypse Number, Apocalyptic
Number, Beast Number

References

Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 97-
102, 1995.

Levine-O'Sullivan Greedy Algorithm

For a sequence {x*}> tne Levine-O'Sullivan greedy algo-
rithm is given by

Xi= max + l)(i-Xi)

1<J<1— 1

for i > 1.

see also Greedy Algorithm, Levine-0 'Sullivan Se-
quence

References

Levine, E. and O'Sullivan, J. "An Upper Estimate for the

Reciprocal Sum of a Sum-Free Sequence." Acta Arith. 34,

9-24, 1977.

Level Set

The level set of c is the Set of points

{(zi,...,Xn) £ U : /(a;i,...,Xn) — c] £ M n ,

and is in the DOMAIN of the function. If n = 2, the level
set is a plane curve (a level curve). If n = 3, the level
set is a surface (a level surface).

References

Gray, A. "Level Surfaces in R ." §10.7 in Modern Differential

Geometry of Curves and Surfaces. Boca Raton, FL: CRC

Press, pp. 204-207, 1993.

Level Surface

A Level Set in 3-D.

Levi-Civita Density

see Permutation Symbol

Levi-Civita Symbol

see Permutation Symbol

Levi-Civita Tensor

see Permutation Tensor

Levine-O'Sullivan Sequence

The sequence generated by the LEVINE-0 'Sullivan
Greedy Algorithm: 1, 2, 4, 6, 9, 12, 15, 18, 21, 24,
28, 32, 36, 40, 45, 50, 55, 60, 65, . . . (Sloane's A014011).
The reciprocal sum of this sequence is conjectured to
bound the reciprocal sum of all A-SEQUENCES.

References

Finch, S. "Favorite Mathematical Constants." http://www.

mathsof t . com/ asolve/constant/erdos/erdos .html.
Levine, E. and O'Sullivan, J. "An Upper Estimate for the

Reciprocal Sum of a Sum-Free Sequence." Acta Arith. 34,

9-24, 1977.
Sloane, N. J. A, Sequence A014011 in "An On-Line Version

of the Encyclopedia of Integer Sequences."

Levy Constant

Let p n /q n be the nth Convergent of a REAL NUMBER
x. Then almost all Real Numbers satisfy

L= lira (q n ) 1/n = e 7r2/(l21n2) =3.27582291872....

see also KHINTCHINE'S CONSTANT, KHINTCHINE-LEVY

Constant

References

Le Lionnais, F. Les nombres remarquables, Paris: Hermann,
p. 51, 1983.

1072 Levy Distribution

Lexis Ratio

Levy Distribution

F[P N {k)] = exp(-N\kf),

where T is the FOURIER TRANSFORM of the probability
Pn(Jc) for AT-step addition of random variables. Levy
showed that G (0,2) for P(x) to be NONNEGATIVE.
The Levy distribution has infinite variance and some-
times infinite mean. The case — 1 gives a Cauchy
Distribution, while — 2 gives a Gaussian Distri-
bution.

see also Cauchy Distribution, Gaussian Distribu-
tion

Levy Flight

Random Walk trajectories which are composed of self-
similar jumps. They are described by the Levy Distri-
bution.

see also Levy Distribution

References

Shlesinger, M.; Zaslavsky, G. M.; and Frisch, U. (Eds.).
Levy Flights and Related Topics in Physics. New York:
Springer-Verlag, 1995.

Levy Function

see Brown Function

Levy Tapestry

The FRACTAL curve illustrated above, with base curve
and motif illustrated below.

Levy Fractal

see also Levy Fractal

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 45-
48, 1991.
$ Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/~eww6n/math/notebooks/Fractal.m.

rJ~~i r~ ^

\±y

A Fractal curve, also called the C-Curve (Beeler et
al. 1972, Item 135). The base curve and motif are illus-
trated below.

see also Levy Tapestry

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.

Cambridge, MA: MIT Artificial Intelligence Laboratory,

Memo AIM-239, Feb. 1972.
Dixon, R. Mathographics. New York: Dover, pp. 182-183,

1991.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-

ures. Princeton, NJ: Princeton University Press, pp. 45-

48, 1991.
Weisstein, E. W. "Fractals." http: //www. astro. Virginia.

edu/-eww6n/math/notebooks/Fractal.m.

Lew A>gram

Diagrams invented by Lewis Carroll which can be used
to determine the number of minimal MINIMAL COVERS
of n numbers with k members.

References

Macula, A. J. "Lewis Carroll and the Enumeration of Mini-
mal Covers." Math. Mag. 68, 269-274, 1995.

Lexicographic Order

An ordering of PERMUTATIONS in which they are listed
in increasing numerical order. For example, the PER-
MUTATIONS of {1,2,3} in lexicographic order are 123,
132, 213, 231, 312, and 321.

see also TRANSPOSITION ORDER

References

Ruskey, F. "Information on Combinations of a Set."
http: //sue . esc .uvic . ca/ -cos/ inf /comb/Combinations
Info.html.

Lexis Ratio

L =

where a is the VARIANCE in a set of s LEXIS TRIALS
and a B is the VARIANCE assuming BERNOULLI TRIALS.

Lexis Trials

Lie Bracket 1073

If L < 1, the trials are said to be Subnormal, and if
L > 1, the trials are said to be SUPERNORMAL.

see also Bernoulli Trial, Lexis Trials, Subnor-
mal, Supernormal

Lexis Trials

n sets of s trials each, with the probability of success p
constant in each set.

var f — J = spq + s(s — l)cr p 2 ,

where <r v 2 is the Variance of pi.

see also BERNOULLI TRIAL, LEXIS RATIO

Lg

The Logarithm to Base 2 is denoted lg, i.e.,

lgX = log 2 iE.

see also Base (Logarithm), e, Ln, Logarithm,
Napierian Logarithm, Natural Logarithm

Liar's Paradox

see Epimenides Paradox

Lichnerowicz Conditions

Second and higher derivatives of the METRIC TENSOR
g a b need not be continuous across a surface of disconti-
nuity, but g a b and g a b,c must be continuous across it.

Lichnerowicz Formula

D*Dj> = V* W + \Rj> - \F£ ty),

where D is the Dirac operator D : T(W+) -* r(W"),
V is the Covariant Derivative on Spinors, R is the
Curvature Scalar, and F£ is the self-dual part of the
curvature of L.

see also Lichnerowicz- Weitzenbock Formula

References

Donaldson, S. K. "The Seiberg-Witten Equations and 4-
Manifold Topology." Bull Amer. Math. Soc. 33, 45-70,
1996.

Lichnerowicz- Weitzenbock Formula

where D is the Dirac operator D : T(S + ) -> T(S~) y V
is the Covariant Derivative on Spinors, and R is
the Curvature Scalar.

see also LICHNEROWICZ FORMULA

References

Donaldson, S. K. "The Seiberg-Witten Equations and 4-
Manifold Topology." Bull Amer. Math. Soc. 33, 45-70,
1996,

Lichtenfels Surface

A Minimal Surface given by the parametric equation

x = R

y = R

2 = K

v^co S (K)VW|0

-V^cos(|C)^/cos(|C)
Jo

v^(!o

References

do Carmo, M. P. "The Helicoid." §3.5F in Mathematical

Models from the Collections of Universities and Museums

(Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 47,

1986.
Lichtenfels, O. von. "Notiz iiber eine transcendente Mini-

malnache." Sitzungsber. Kaiserl Akad. Wiss. Wien 94,

41-54, 1889.

Lie Algebra

A Nonassociative Algebra obeyed by objects such
as the Lie Bracket and Poisson Bracket. Elements
/, g, and h of a Lie algebra satisfy

lf,g] = -\s,f], (i)

[f + g,h} = [f,h] + [g,h], (2)

[f,[9,h}} + [g,[h,f]] + [h,[f,9]} = (3)

and

(the Jacobi Identity), and are not Associative. The

binary operation of a Lie algebra is the bracket

[f9,h] = f[g,h]+g[f,h}.

(4)

see also Jacobi Identities, Lie Algebroid, Lie
Bracket, Iwasawa's Theorem, Poisson Bracket

References

Jacobson, N. Lie Algebras. New York: Dover, 1979.

Lie Algebroid

The infinitesimal algebraic object associated with a LIE
GROUPOID. A Lie algebroid over a MANIFOLD B is a
Vector Bundle A over B with a Lie Algebra struc-
ture [ , ] (Lie Bracket) on its Space of smooth sections
together with its Anchor p.

see also Lie Algebra

References

Weinstein, A. "Gro lipoids: Unifying Internal and External
Symmetry." Not Amer. Math. Soc. 43, 744-752, 1996.

Lie Bracket

The commutation operation

[a, 6] — ab — ba

corresponding to the Lie PRODUCT.

see also LAGRANGE BRACKET, POISSON BRACKET

1074

Lie Commutator

Life

Lie Commutator

see Lie Product

Lie Derivative

C x T a

lim

<5u-»>0

T ab (x')-T lab {x
5u

Lie Group

A continuous GROUP with an infinite number of ele-
ments such that the parameters of a product element
are Analytic Functions. Lie groups are also C°°
Manifolds with the restriction that the group oper-
ation maps a C°° map of the Manifold into itself. Ex-
amples include 3 , SU(n), and the LORENTZ GROUP.

see also Compact Group, Lie Algebra, Lie
Groupoid, Lie-Type Group, Nil Geometry, Sol

Geometry

References

Arfken, G. "Infinite Groups, Lie Groups." Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, p. 251-252, 1985.

Chevalley, C. Theory of Lie Groups. Princeton, NJ: Prince-
ton University Press, 1946.

Knapp, A. W. Lie Groups Beyond an Introduction. Boston,
MA: Birkhauser, 1996.

Lipkin, H. J, Lie Groups for Pedestrians, 2nd ed. Amster-
dam, Netherlands: North- Holland, 1966.

Lie Groupoid

A GROUPOID G over B for which G and B are differ en-
tiable manifolds and a, /?, and multiplication are differ-
entiate maps. Furthermore, the derivatives of a and
are required to have maximal RANK everywhere. Here,
a and are maps from G onto R with a : (ar, 7, y) *->■ x
and : (x,7,y) »-» y.

see also Lie Algebroid, Nilpotent Lie Group,
Semisimple Lie Group, Solvable Lie Group

References

Weinstein, A. "Groupoids: Unifying Internal and External
Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996.

Lie Product

The multiplication operation corresponding to the LIE
Bracket.

Lie- Type Group

A finite analog of Lie Groups. The Lie-type groups
include the CHEVALLEY GROUPS [PSL(n,q), PSU(n y q),
PSp{2n,q), PQ € (n,q)], Twisted Chevalley Groups,
and the Tits GROUP.

see also Chevalley Groups, Finite Group, Lie
Group, Linear Group, Orthogonal Group, Sim-
ple Group, Symplectic Group, Tits Group,
Twisted Chevalley Groups, Unitary Group

References

Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat .bham.ac.uk/atlas#lie.

Liebmann's Theorem

A Sphere is Rigid.

see also Rigid

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 377, 1993.

O'Neill, B. Elementary Differential Geometry, 2nd ed. New
York: Academic Press, p. 262, 1997.

Life

The most well-known Cellular Automaton, invented
by John Conway and popularized in Martin Gardner's
Scientific American column starting in October 1970.
The game was originally played (i.e., successive genera-
tions were produced) by hand with counters, but imple-
mentation on a computer greatly increased the ease of
exploring patterns.

The Life AUTOMATON is run by placing a number of
filled cells on a 2-D grid. Each generation then switches
cells on or off depending on the state of the cells that
surround it. The rules are defined as follows. All eight
of the cells surrounding the current one are checked to
see if they are on or not. Any cells that are on are
counted, and this count is then used to determine what
will happen to the current cell.

1. Death: if the count is less than 2 or greater than 3,
the current cell is switched off.

2. Survival: if (a) the count is exactly 2, or (b) the
count is exactly 3 and the current cell is on, the
current cell is left unchanged.

3. Birth: if the current cell is off and the count is ex-
actly 3, the current cell is switched on.

Hensel gives a Java applet (http://www.mindspring.
com/~alanh/lif e/) implementing the Game of Life on
his web page.

A pattern which does not change from one generation to
the next is known as a Still Life, and is said to have pe-
riod 1. Conway originally believed that no pattern could
produce an infinite number of cells, and offered a $50
prize to anyone who could find a counterexample before
the end of 1970 (Gardner 1983, p. 216). Many coun-
terexamples were subsequently found, including Guns
and Puffer Trains.

A Life pattern which has no Father Pattern is known
as a Garden of Eden (for obvious biblical reasons). The
first such pattern was not found until 1971, and at least
3 are now known. It is not, however, known if a pattern
exists which has a Father Pattern, but no Grandfather
Pattern (Gardner 1983, p. 249).

Rather surprisingly, Gosper and J. H. Conway inde-
pendently showed that Life can be used to generate a
Universal Turing Machine (Berlekamp et al. 1982,
Gardner 1983, pp. 250-253).

Life Expectancy

Life Expectancy 1075

Similar CELLULAR AUTOMATON games with different
rules are HashLife, HexLife, and HiGHLlFE.

see also Cellular Automaton, HashLife, HexLife,
HighLife

References

"Alife online." http : //alii e . santaf e . edu/alif e/topics/
cas/ca-faq/lifefaq/lifef aq.html.

Berlekamp, E. R.; Conway, J. H.; and Guy, R. K. "What Is
Life." Ch. 25 in Winning Ways, For Your Mathematical
Plays, Vol. 2: Games in Particular. London: Academic
Press, 1982.

Callahan, P. "Patterns, Programs, and Links for Con-
way's Game of Life." http://www.cs.jhu.edu/-callahan/
lifepage.html.

Flammenkamp, A. "Game of Life." http://www.minet .
uni- j ena . de/-achim/gol . html.

"The Game of Life." Math Horizons, p. 9, Spring 1994.

Gardner, M. "The Game of Life, Parts I-III." Chs. 20-22 in
Wheels, Life, and other Mathematical Amusements. New
York: W. H. Freeman, 1983.

Hensel, A. "A Brief Illustrated Glossary of Terms in Con-
way's Game of Life." http://www.cs.jhu.edu/-callahan/
glossary.html.

Hensel, A. "PC Life Distribution." http: //www. mindspring.
com/-alanh/lifep,zip.

Hensel, A. "Conway's Game of Life." Includes a Java ap-
plet for the Game of Life, http://www.mindspring.com/
-alanh/lif e/.

Koenig, H. "Game of Life Information." http: //www.
halcyon.com/nkoenig/Lifelnfo/Lifelnfo.html.

Mcintosh, H. V. "A Zoo of Life Forms." http://www.es.
cinvestav.mx/mcintosh/life.html.

Poundstone, W. The Recursive Universe: Cosmic Complex-
ity and the Limits of Scientific Knowledge. New York:
Morrow, 1985.

Toffoli, T. and Margolus, N. Cellular Automata Machines:
A New Environment for Modeling, Cambridge, MA: MIT
Press, 1987.

Wainwright, R. T. "LifeLine." http://members.aol.com/
lifeiine/life/lif epage.htm.

Wainwright, R. T. LifeLine: A Quarterly Newsletter for En-
thusiasts of John Conway's Game of Life. Nos. 1-11,
1971-1973.

Life Expectancy

An l x table is a tabulation of numbers which is used to
calculate life expectancies.

X

Tl x

d x

/*

q x

L x

T x

6x

1000

200

1.00

0.20

0.90

2.70

2.70

1

800

100

0.80

0.12

0.75

1.80

2.25

2

700

200

0.70

0.29

0.60

1.05

1.50

3

500

300

0.50

0.60

0.35

0.45

0.90

4

200

200

0.20

1.00

0.10

0.10

0.50

5

0.00

—

0.00

0.00

—

r,

1000

2.70

x : Age category (x = 0, 1, . .., k). These values
can be in any convenient units, but must be chosen
so that no observed lifespan extends past category
k-1.

a x : Census size, defined as the number of individuals
in the study population who survive to the begin-
ning of age category x. Therefore, no = N (the
total population size) and rik = 0.

d x : — n x — n x +i; ^ i=0 A = ^o- Crude death rate,
which measures the number of individuals who die
within age category x.

l x : = Thx/no* Survivorship, which measures the pro-
portion of individuals who survive to the beginning
of age category x.

q x : = d x /n x ] qk-i = 1. Proportional death rate, or
"risk," which measures the proportion of individ-
uals surviving to the beginning of age category x
who die within that category.

L x : = (l x + Ja.+i)/2. Midpoint survivorship, which
measures the proportion of individuals surviving to
the midpoint of age category x. Note that the sim-
ple averaging formula must be replaced by a more
complicated expression if survivorship is nonlinear
within age categories. The sum ^2 i:=0 ^ x &* ves *^ e
total number of age categories lived by the entire
study population.

T x : = Ta;_i - L x -i; T = Y%=o L *- Measures the
total number of age categories left to be lived by
all individuals who survive to the beginning of age
category x.

e x : — T x /l x \ ek-i = 1/2. Life expectancy, which is
the mean number of age categories remaining until
death for individuals surviving to the beginning of
age category x.

For all x, e x+ i + 1 > e x . This means that the total
expected lifespan increases monotonically. For instance,
in the table above, the one-year-olds have an average
age at death of 2.25 + 1 = 3.25, compared to 2.70 for
newborns. In effect, the age of death of older individuals
is a distribution conditioned on the fact that they have
survived to their present age.

It is common to study survivorship as a semilog plot of
l x vs. x, known as a Survivorship Curve. A so-called
l x m x table can be used to calculate the mean generation
time of a population. Two l x m x tables are illustrated
below.

Population 1

l x

l x m x

Xt x TTb x

1.00

0.00

0.00

0.00

1

0.70

0.50

0.35

0.35

2

0.50

1.50

0.75

1.50

3

0.20

0.00

0.00

0.00

4

0.00

0.00

0.00

0.00

Ro = 1.10 Y, = 1 - 85

Y^xl x m x _ 1.85

Y,^ra x ~ Ho

ln.Ro In 1.10

= 1.68

= 0.057.

1076

Life Expectancy

Likelihood Ratio

Population 2

'a; ifi>x

Xlx'if'x

1.00

0.00

0.00

0.00

1

0.70

0.00

0.00

0.00

2

0.50

2.00

1.00

2.00

3

0.20

0.50

0.10

0.30

4

0.00

0.00

0.00

0.00

R = i.iQ ^ = 2.30

T =

/ ^ Xlx

2.30

InRo In 1.10

= 2.09

2.09

= 0.046.

x : Age category (a; = 0, 1, . . . , k). These values
can be in any convenient units, but must be
chosen so that no observed lifespan extends past
category k — 1 (as in an l x table).

l x ; = n x /no. Survivorship, which measures the
proportion of individuals who survive to the be-
ginning of age category x (as in an l x table).

m x : The average number of offspring produced by
an individual in age category x while in that
age category. y\_ m x therefore represents the
average lifetime number of offspring produced
by an individual of maximum lifespan.

l x m x : The average number of offspring produced by
an individual within age category x weighted
by the probability of surviving to the beginning
of that age category. 5^__ lxm x therefore rep-
resents the average lifetime number of offspring
produced by a member of the study population.
It is called the net reproductive rate per gener-
ation and is often denoted Rq.
xl x m x : A column weighting the offspring counted
in the previous column by their parents' age
when they were born. Therefore, the ratio
T — YK x ^ x7nx ) I XX^ m£C ) is the mean gener-
ation time of the population.

The Malthusian Parameter r measures the repro-
ductive rate per unit time and can be calculated as
r — {\nRo)/T. For an exponentially increasing popu-
lation, the population size N(t) at time t is then given

by

N(t) = N e rt .

In the above two tables, the populations have identical
reproductive rates of Rq — 1.10. However, the shift to-
ward later reproduction in population 2 increases the
generation time, thus slowing the rate of POPULATION
GROWTH. Often, a slight delay of reproduction de-
creases Population Growth more strongly than does
even a fairly large reduction in reproductive rate.
see also GOMPERTZ CURVE, LOGISTIC GROWTH
Curve, Makeham Curve, Malthusian Parameter,
Population Growth, Survivorship Curve

Lift

Given a Map / from a Space X to a Space Y and
another MAP g from a SPACE Z to a SPACE Y, a lift is a
MAP h from X to Z such that gh~f. In other words,
a lift of / is a Map h such that the diagram (shown
below) commutes.

Z

x-

V'

/

If / is the identity from Y to Y, a Manifold, and if
g is the bundle projection from the TANGENT BUNDLE
to Y, the lifts are precisely VECTOR FIELDS. If g is a
bundle projection from any Fiber Bundle to Y, then
lifts are precisely sections. If / is the identity from Y to
Y, a Manifold, and g a projection from the orientation
double cover of Y, then lifts exist IFF Y is an orientable
Manifold.

If / is a Map from a CIRCLE to Y, an n-MANlFOLD,
and g the bundle projection from the FIBER BUNDLE
of alternating n-FoRMS on Y, then lifts always exist
Iff Y is orientable. If / is a MAP from a region in
the Complex Plane to the Complex Plane (complex
analytic), and if g is the exponential MAP, lifts of / are
precisely LOGARITHMS of /.

see also LIFTING PROBLEM

Lifting Problem

Given a Map / from a Space X to a Space Y and
another MAP g from a SPACE Z to a SPACE Y, docs
there exist a MAP h from X to Z such that gh = /? If
such a map h exists, then h is called a Lift of /.

see also Extension Problem, Lift

Ligancy

see Kissing Number

Likelihood

The hypothetical Probability that an event which has
already occurred would yield a specific outcome. The
concept differs from that of a probability in that a prob-
ability refers to the occurrence of future events, while a
likelihood refers to past events with known outcomes.

see also Likelihood Ratio, Maximum Likelihood,
Negative Likelihood Ratio, Probability

Likelihood Ratio

A quantity used to test Nested Hypotheses. Let H'
be a Nested Hypothesis with n Degrees of Free-
dom within H (which has n Degrees OF Freedom),
then calculate the MAXIMUM LIKELIHOOD of a given
outcome, first given H\ then given H. Then

LR-

[likelihood H']
[likelihood H]

Limacon

Limit

1077

Comparison of this ratio to the critical value of the
Chi-Squared Distribution with n-n' Degrees of
Freedom then gives the Significance of the increase
in Likelihood.

The term likelihood ratio is also used (especially in med-
icine) to test nonnested complementary hypotheses as
follows,

LR:

[true positive rate] _ [sensitivity]
[false positive rate] 1 — [specificity]

see also NEGATIVE LIKELIHOOD RATIO, SENSITIVITY,

Specificity
Limacon

The limagon is a polar curve of the form

r — b-\- a cos

also called the LlMAgON OF PASCAL. It was first in-
vestigated by Diirer, who gave a method for drawing
it in Underweysung der Messung (1525). It was redis-
covered by Etienne Pascal, father of Blaise Pascal, and
named by Gilles-Personne Roberval in 1650 (MacTutor
Archive). The word "limagon" comes from the Latin
Umax, meaning "snail."

If b > 2a, we have a convex limacon. If 2a > b >
a, we have a dimpled limagon. If b = a, the limacon
degenerates to a Cardioid. If b < a, we have limacon
with an inner loop. If b = a/2, it is a TRISECTRIX
(but not the MACLAURIN Trisectrix) with inner loop
of Area

dinner loop — 4^ | 7T 6^j

and Area between the loops of

-^■between loops ~ 4^ l^T + OyO J

(MacTutor Archive). The limacon is an ANALLAGMATIC
Curve, and is also the Catacaustic of a Circle when
the Radiant Point is a finite (Nonzero) distance from
the Circumference, as shown by Thomas de St. Lau-
rent in 1826 (MacTutor Archive).

see also CARDIOID

References

Lawrence, J. D. A Catalog of Special Plane Curves.
York: Dover, pp. 113-117, 1972.

New

Lee, X. "Limacon of Pascal." http://www.best.com/-xah/

SpecialPlaneCurves_dir/LimaconOf Pascal _dir /limacon

OfPascal.html.
Lee, X. "Limacon Graphics Gallery." http://www.best.com

/ - xah / Spe c ial Plane Curves _dir / LimaconGGallery _dir /

limaconGGallery . html.
Lockwood, E. H. "The Limacon." Ch. 5 in A Book of Curves.

Cambridge, England: Cambridge University Press, pp. 44-

51, 1967.
MacTutor History of Mathematics Archive. "Limacon of Pas-
cal." http: //www-groups . des . st-and. ac .uk/ -history/

Curves/Limacon.html.
Yates, R. C. "Limacon of Pascal." A Handbook on Cu *ue$

and Their Properties. Ann Arbor, MI: J, W, Edwards,

pp. 148-151, 1952.

Limacon Evolute

The Catacaustic of a Circle for a Radiant Point
is the limagon evolute. It has parametric equations

_ a[4a 2 + 46 2 + 9abcost - ab cos(3*)]
X ~ 4(2a 2 +6 2 +3a&cos«)

a 2 b sin 3 t

2a 2 + 6 2 + 3a6cosV

Limagon of Pascal

see LlMAgON

Limit

A function f(z) is said to have a limit lim z _>. a f(z) = c if,
for all e > 0, there exists a 6 > such that \f(z) — c\ < e
whenever < \z — a\ < S.

A Lower Limit

lower lim S n — lim S n = h

is said to exist if, for every e > 0, \S n — h\ < e for
infinitely many values of n and if no number less than h
has this property.

An Upper Limit

upper lim S n = lim S n = k

is said to exist if, for every e > 0, \S n — k\ < e for
infinitely many values of n and if no number larger than
k has this property.

Indeterminate limit forms of types oo/oo and 0/0 can be
computed with L'Hospital's Rule. Types • oo can
be converted to the form 0z/0 by writing

f{x)g{x) -

fix)

1078 Limit Comparison Test

Lindenmayer System

Types 0°, oo°, and 1°° are treated by introducing a de-
pendent variable y — f(x)g(x), then calculating lim Iny.
The original limit then equals e limlny .

see also Central Limit Theorem, Continuous, Dis-
continuity, L'Hospital's Rule, Lower Limit, Up-
per Limit

References

Courant, R. and Robbins, H. "Limits. Infinite Geometrical
Series." §2.2.3 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 63-66, 1996.

Limit Comparison Test

Let Y^ a k and ^ bk be two Series with Positive terms
and suppose

lim — = p.

fc-^oo Ok

If p is finite and p > 0, then the two Series both Con-
verge or Diverge.

see also Convergence Tests

Limit Cycle

An attracting set to which orbits or trajectories converge
and upon which trajectories are periodic.

see also Hopf Bifurcation

Limit Point

A number x such that for all e > 0, there exists a mem-
ber of the Set y different from x such that \y — x\ < e.
The topological definition of limit point P of A is that P
is a point such that every Open Set around it intersects
A.

see also Closed Set, Open Set

References

Lauwerier, PL Fractals; Endlessly Repeated Geometric Fig-

ures. Princeton, NJ: Princeton University Press, pp. 25-

26, 1991.

Lin's Method

An Algorithm for finding Roots for Quartic Equa-
tions with Complex Roots.

References

Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., pp. 198-199, 1990.

Lindeberg Condition

A Sufficient condition on the Lindeberg-Feller
Central Limit Theorem. Given random variates Xi,
X 2 , . . ■ , let (Xi) — 0, the Variance v? of Xi be finite,
and Variance of the distribution consisting of a sum of
XiS

S n = X 1 +X 2 + ... + X n (1)

Let

be

^±(m

i**i

>e

— W >

(3)

(4)

then the Lindeberg condition is

lim A„(e) =

■,-n— +.00

t for all e > 0.
see also Feller-Levy Condition

References

Zabell, S. L. "Alan Turing and the Central Limit Theorem."
Amer. Math. Monthly 102, 483-494, 1995.

Lindeberg-Feller Central Limit Theorem

If the random variates Xi, X 2) ... satisfy the LINDE-
BERG Condition, then for all a < b,

lim P

(a<£<»)=*(6).

*(a),

]C^ 2 -

(2)

where $ is the Normal Distribution Function.

see also Central Limit Theorem, Feller-Levy
Condition, Normal Distribution Function

References

Zabell, S. L. "Alan Turing and the Central Limit Theorem."
Amer. Math. Monthly 102, 483-494, 1995.

Lindelof 's Theorem

The Surface of Revolution generated by the exter-
nal CATENARY between a fixed point a and its conjugate
on the Envelope of the Catenary through the fixed
point is equal in Area to the surface of revolution gen-
erated by its two Lindelof TANGENTS, which cross the
axis of rotation at the point a and are calculable from
the position of the points and Catenary.

see also CATENARY, ENVELOPE, SURFACE OF REVOLU-
TION

Lindemann-Weierstrafi Theorem

If cti, . . . , a n are linearly independent over Q, then e ai ,
. . . , e an are algebraically independent over Q.

see also Hermite-Lindemann Theorem

Lindenmayer System

A String Rewriting system which can be used to gen-
erate Fractals with Dimension between 1 and 2. The
term L-System is often used as an abbreviation.

see also ARROWHEAD CURVE, DRAGON CURVE EXTE-
RIOR Snowflake, Fractal, Hilbert Curve, Koch
Snowflake, Peano Curve, Peano-Gosper Curve,
Sierpinski Curve, String Rewriting

References

Dickau, R. M. "Two-dimensional L-systems." http: //forum
. swarthmore . edu/advanced/robertd/lsys2d . html.

Prusinkiewicz, P. and Hanan, J. Lindenmayer Systems, Frac-
tal, and Plants. New York: Springer- Verlag, 1989.

Line

Prusinkiewicz, P. and Lindenmayer, A. The Algorithmic

Beauty of Plants. New York: Springer- Verlag, 1990.
Stevens, R. T. Fractal Programming in C. New York: Holt,

1989.
Wagon, S. "Recursion via String Rewriting." §6.2 in Mathe-

matica in Action. New York: W. H. Freeman, pp. 190-196,

1991.

Line

Euclid defined a line as a "breadthless length," and a
straight line as a line which "lies evenly with the points
on itself" (Kline 1956, Dunham 1990). Lines are in-
trinsically 1-dimensional objects, but may be embedded
in higher dimensional SPACES. An infinite line pass-
ing through points A and B is denoted jfe. A LINE
SEGMENT terminating at these points is denoted AB.
A line is sometimes called a Straight Line or, more
archaically, a Right Line (Casey 1893), to emphasize
that it has no curves anywhere along its length.

Consider first lines in a 2-D PLANE. The line with x-
Intercept a and ^-Intercept b is given by the inter-
cept form

(i)

- + T = 1 -
a b

The line through (xi, yi) with Slope m is given by the
point-slope form

y-yi— m(x - xi).

(2)

The line with y-intercept b and slope m is given by the
slope-intercept form

y = mx + b.

(3)

The line through (#i,yi) and (£2,3/2) is given by the two
point form

3/2 - yi / x

y-y 1 = - (X-Xi).

X2 — X\

(4)

Line 1079

where (eR. Similarly, Vectors of the form

(10)

are Perpendicular to the line. Three points lie on a

line if

xi yi 1

x 2 y-z l

Xz 2/3 1

0.

(11)

The ANGLE between lines

A^ + Bxy + d =
A 2 x + B 2 y + C 2 =

is

tan# =

A 1 B 2 -A 2 B 1
A 1 A 2 +B 1 B 2 '

(12)
(13)

(14)

The line joining points with TRILINEAR COORDINATES
on ' ft : 71 and a 2 '• 02 ' 72 is the set of point a : : 7
satisfying

a 7

on ft 71 =0 (15)

Ot 2 02 72

(ft 72 -7ift)<*+ (71 «2 -onj 2 )0-\- (aift -fta2)7 = °*

(16)
Three lines CONCUR if their TRILINEAR COORDINATES

satisfy

Other forms are

ha -f m\0 + ni7 =
l 2 a + m 2 + ri27 =
ha -f mz0 + n 3 7 = 0,

in which case the point is

m 2 nz — n 2 mz : n 2 h — hnz : hmz — m 2 h,

or if the COEFFICIENTS of the lines

(17)
(18)
(19)

(20)

i(x -xi) + b(y-yi) =
ax -\- by + c =

X

y

1

Xi

2/i

1

x 2

y2

1

0.

(5)
(6)

(7) satisfy

A line in 2-D can also be represented as a VECTOR. The
Vector along the line

ax + by =

is given by

(8)
(9)

A^ + Bxy + Ci =0
A 2 x + B 2 y + C 2 =
A 3 x + B 3 y + C3 =

= 0.

1 Bi

Ci

2 B2

c 2

3 B3

c 3

(21)
(22)
(23)

(24)

Two lines Concur if their Trilinear Coordinates

satisfy

£1 mi Tii .

h rri2 ri2 — 0. (25)

h TI3 rtz

1080

Line

Line Element

The line through Pi is the direction (ai,6i,Ci) and the
line through P 2 in direction (fl2,&2,C2) intersect IFF

X2 -xi V2- 2/i z 2 - zi
a\ b\ c\

ai 62 C2

0.

The line through a point a' : f : 7' PARALLEL to

la + mf3 + 717 =

The lines

a j

a' (3' i

bn — cm cl — an am — bl

la + m/3 + wy —

= 0.

(26)

(27)
(28)

(29)
(30)

are PARALLEL if

a(mn — nm') -J- 6(n/' — /n') + c(lm - m£') = (31)

for all (a,b,c), and Perpendicular if

2abc(ll' + mm' + nn') — (mn' + mm) cos ^4

~(n/' + nl) cos 5 - (Im + Z'm) cos C = (32)

for all {a,b,c) (Sommerville 1924). The line through a
point a' : (3* : 7' PERPENDICULAR to (32) is given by

a 7

a' 0' 7'

I — m cos C m — n cos A n — I cos 5

— ncosl? — JcosC — mcosA

0. (33)

In 3-D Space, the line passing through the point
(xo,yo,z Q ) and Parallel to the Nonzero Vector

(34)

has parametric equations

' x — Xq + at

y = y +bt

£ = zq + ct.

(35)

see also Asymptote, Brocard Line, Collinear,
Concur, Critical Line, Des argues' Theorem,
Erdos-Anning Theorem, Line Segment, Ordinary
Line, Pencil, Point, Point-Line Distance — 2-D,
Point-Line Distance — 3-D, Plane, Range (Line
Segment), Ray, Solomon's Seal Lines, Steiner
Set, Steiner's Theorem, Sylvester's Line Prob-
lem

References

Casey, J. "The Right Line." Ch. 2 in A Treatise on the An-
alytical Geometry of the Point, Line, Circle, and Conic
Sections, Containing an Account of Its Most Recent Exten-
sions, with Numerous Examples, 2nd ed., rev. enl. Dublin:
Hodges, Figgis, & Co., pp. 30-95, 1893.

Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, p. 32, 1990.

Kline, M. "The Straight Line." Set Amer. 156, 105-114,
Mar. 1956.

MacTutor History of Mathematics Archive. "Straight Line."
http://www-groups.dcs.st-and.ac.uk/-history/Curves
/Straight. html.

Sommerville, D. M. Y. Analytical Conies. London: G. Bell,
p. 186, 1924.

Spanier, J. and Oldham, K. B. "The Linear Function bx +
c and Its Reciprocal." Ch. 7 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 53-62, 1987.

Line Bisector

■line segment bisector

The line bisecting a given Line Segment PiP 2 can be
constructed geometrically, as illustrated above.

References

Courant, R. and Robbins, H. "How to Bisect a Segment and
Find the Center of a Circle with the Compass Alone."
§3.4.4 in What is Mathematics?: An Elementary Approach
to Ideas and Methods, 2nd ed. Oxford, England: Oxford
University Press, pp. 145-146, 1996.

Dixon, R. Mathographics. New York: Dover, p. 22, 1991.

Line of Curvature

A curve on a surface whose tangents are always in the
direction of PRINCIPAL CURVATURE. The equation of
the lines of curvature can be written

9ii

du 2

912

922

&12

622

dudv

dv 2

= 0,

where g and b are the Coefficients of the first and

second FUNDAMENTAL FORMS.

see also Dupin's Theorem, Fundamental Forms,
Principal Curvatures

Line Element

Also known as the first FUNDAMENTAL FORM

ds = g a b dx a dx .
In the principal axis frame for 3-D,

ds 2

■■ g aa (dx a ) 2 + g h b(dx b ) 2 + g cc {dx c ) 2 .

Line Graph

Linear Algebra 1081

At Ordinary Points on a surface, the line element is
positive definite.

see also Area Element, Fundamental Forms, Vol-
ume Element

Line Graph

where

A Line Graph L(G) (also called an Interchange
Graph) of a graph G is obtained by associating a vertex
with each edge of the graph and connecting two vertices
with an edge Iff the corresponding edges of G meet
at one or both endpoints. In the three examples above,
the original graphs are the COMPLETE GRAPHS K3, K4,
and K 5 shown in gray, and their line graphs are shown
in black.

References

Saaty, T. L. and Kainen, P. C. "Line Graphs." §4-3 in The

Four-Color Problem; Assaults and Conquest. New York:

Dover, pp. 108-112, 1986.

Line at Infinity

The straight line on which all POINTS AT INFINITY lie.
The line at infinity is given in terms of TRILINEAR CO-
ORDINATES by

aa + b/3 + C7 = 0,

which follows from the fact that a REAL TRIANGLE will
have Positive Area, and therefore that

2A = aa + b(3 + ey > 0.

Instead of the three reflected segments concurring for
the Isogonal Conjugate of a point X on the Cir-
CUMCiRCLE of a Triangle, they become parallel (and
can be considered to meet at infinity). As X varies
around the ClRCUMCIRCLE, X~ l varies through a line
called the line at infinity. Every line is PERPENDICULAR
to the line at infinity.

see also POINT AT INFINITY

Line Integral

The line integral on a curve cr is defined by

/ F-ds= / Y{o-{t))-o-'{t)dt
Jcr J a,

-I

Fi dx + F 2 dy + F 3 dz,

(1)
(2)

F =

F 2
F 3

(3)

If V • F - (i.e., it is a Divergenceless Field), then
the line integral is path independent and

(a,!/,z)

Fi dx + F 2 dy + F 3 dz

(o,6,c)

Fi dx + / F 2 dy+ F s dz. (4)

) J(x,b t c) J{x,y,c)

L.

/»(x,6,c) p{x,y,c) Mx,y,

= / F±dx+ I F 2 dy +

J (a,fe,c) J (x,b,c) J (x,y,c

For z Complex, 7:2 = z(t), and t e [0,6],
f fdz= f f(z(t))z'(t)dt.

J *y J a

see also CONTOUR INTEGRAL, PATH INTEGRAL

Line Segment

(5)

A B

A closed interval corresponding to a FINITE portion of
an infinite LINE. Line segments are generally labelled
with two letters corresponding to their endpoints, say
A and B, and then written AB. The length of the line
segment is indicated with an overbar, so the length of
the line segment AB would be written AB.

Curiously, the number of points in a line segment
(Aleph-1; Hi) is equal to that in an entire 1-D SPACE
(a Line), and also to the number of points in an n-D
Space, as first recognized by Georg Cantor.
see also ALEPH-1 (Hi), COLLINEAR, CONTINUUM, LINE,

Ray

Line Space

see LlOUVILLE SPACE

Linear Algebra

The study of linear sets of equations and their trans-
formation properties. Linear algebra allows the analysis
of Rotations in space, Least Squares Fitting, so-
lution of coupled differential equations, determination
of a circle passing through three given points, as well
as many other other problems in mathematics, physics,
and engineering.

The Matrix and Determinant are extremely useful
tools of linear algebra. One central problem of linear
algebra is the solution of the matrix equation

Ax = b

for x. While this can, in theory, be solved using a Ma-
trix Inverse

x = A _1 b,

1082 Linear Approximation

Linear Extension

other techniques such as Gaussian Elimination are
numerically more robust.

see also Control Theory, Cramer's Rule, Deter-
minant, Gaussian Elimination, Linear Transfor-
mation, Matrix, Vector

References

Ay res, F. Jr. Theory and Problems of Matrices. New York:

Schaum, 1962.
Banchoff, T. and Wermer, J. Linear Algebra Through Geom-
etry, 2nd ed. New York: Springer- Verlag, 1992.
Bellman, R. E. Introduction to Matrix Analysis, 2nd ed. New

York: McGraw-Hill, 1970.
Faddeeva, V. N. Computational Methods of Linear Algebra,

New York: Dover, 1958.
Golub, G. and van Loan, C. Matrix Computations, 3rd ed,

Baltimore, MD: Johns Hopkins University Press, 1996.
Halmos, P. R. Linear Algebra Problem Book. Providence, RI:

Math. Assoc. Amer., 1995.
Lang, S. Introduction to Linear Algebra, 2nd ed. New York:

Springer- Verlag, 1997.
Marcus, M. and Mine, H. Introduction to Linear Algebra.

New York: Dover, 1988.
Marcus, M. and Mine, H. A Survey of Matrix Theory and

Matrix Inequalities. New York: Dover, 1992.
Marcus, M. Matrices and Matlab: A Tutorial. Englewood

Cliffs, NJ: Prentice-Hall, 1993.
Mirsky, L. An Introduction to Linear Algebra. New York:

Dover, 1990.
Muir, T. A Treatise on the Theory of Determinants. New

York: Dover, 1960.
Nash, J. C. Compact Numerical Methods for Computers:

Linear Algebra and Function Minimisation, 2nd ed. Bris-
tol, England: Adam Hilger, 1990.
Strang, G. Linear Algebra and its Applications, 3rd ed.

Philadelphia, PA: Saunders, 1988.
Strang, G. Introduction to Linear Algebra. Wellesley, MA:

Wellesley-Cambridge Press, 1993.
Strang, G. and Borre, K. Linear Algebra, Geodesy, & GPS.

Wellesley, MA: Wellesley-Cambridge Press, 1997.

Linear Approximation

A linear approximation to a function f(x) at a point Xq
can be computed by taking the first term in the Taylor

Series

f(x + Ax) = f{x ) + f'(x )Ax + . . . .

see also MACLAURIN SERIES, TAYLOR SERIES

Linear Code

A linear code over a FINITE FIELD with q elements F q
is a linear SUBSPACE C C F q n . The vectors forming
the Subspace are called code words. When code words
are chosen such that the distance between them is max-
imized, the code is called error-correcting since slightly
garbled vectors can be recovered by choosing the nearest
code word.

see also Code, Coding Theory, Error-Correcting
Code, Gray Code, Huffman Coding, ISBN

Linear Congruence

A linear congruence

ax = b (mod m)

is solvable Iff the Congruence

6 = (mod (a,m))

is solvable, where d = (a, m) is the GREATEST COMMON
Divisor, in which case the solutions are xo, xq + m/d,
xq + 2m/ d, . . . , xq + (d — l)m/d y where xq < m/d. If
d = 1, then there is only one solution.
see also CONGRUENCE, QUADRATIC CONGRUENCE

Linear Congruence Method

A Method for generating Random (Pseudorandom)
numbers using the linear RECURRENCE RELATION

X n +i — aX n -r c (mod ra),

where a and c must assume certain fixed values and Xo

is an initial number known as the Seed.

see also Pseudorandom Number, Random Number,

Seed

References

Pickover, C. A. "Computers, Randomness, Mind, and In-
finity." Ch. 31 in Keys to Infinity. New York: W. H.
Freeman, pp. 233-247, 1995.

Linear Equation

An algebraic equation of the form

y = ax + b

involving only a constant and a first-order (linear) term,
see also Line, Polynomial, Quadratic Equation

Linear Equation System

When solving a system of n linear equations with k > n
unknowns, use MATRIX operations to solve the system
as far as possible. Then solve for the first (k — n) com-
ponents in terms of the last n components to find the
solution space.

Linear Extension

A linear extension of a Partially Ordered Set P is
a Permutation of the elements pi, p2, • ■ ■ of P such
that i < j Implies pi < pj. For example, the linear ex-
tensions of the Partially Ordered Set ((1,2), (3,4))
are 1234, 1324, 1342, 3124, 3142, and 3412, all of which
have 1 before 2 and 3 before 4.

References

Brightwell, G. and Winkler, P. "Counting Linear Exten-
sions." Order 8, 225-242, 1991.

Preusse, G. and Ruskey, F. "Generating Linear Extensions
Fast." SI AM J. Comput. 23, 373-386, 1994.

Ruskey, F. "Information on Linear Extension." http://sue
. csc.uvic . ca/-cos/inf /pose/LinearExt .html.

Varol, Y. and Rotem, D. "An Algorithm to Generate All
Topological Sorting Arrangements." Comput. J. 24, 83—
84, 1981.

Linear Fractional Transformation

Linear Stability 1083

Linear Fractional Transformation

see Mobius Transformation

Linear Group

see General Linear Group, Lie-Type Group, Pro-
jective General Linear Group, Projective Spe-
cial Linear Group, Special Linear Group

References

Wilson, R. A. "ATLAS of Finite Group Representation."
http : / It or . mat , bham . ac . uk/atlas#lin.

Linear Group Theorem

Any linear system of point-groups on a curve with only
ordinary singularities may be cut by ADJOINT CURVES.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 122 and 251, 1959.

Linear Operator

An operator L is said to be linear if, for every pair of
functions / and g and SCALAR t,

Dantzig, G. B. "Programming of Interdependent Activities.
II. Mathematical Model." Econometrica 17, 200-211,
1949.

Dantzig, G. B. Linear Programming and Extensions. Prince-
ton, NJ: Princeton University Press, 1963.

Greenberg, H. J. "Mathematical Programming Glossary."
http: //www -math, cudenver . edu/-hgreenbe /glossary/
glossary.html.

Karloff, H. Linear Programming. Boston, MA: Birkhauser,
1991.

Karmarkar, N. "A New Polynomial- Time Algorithm for Lin-
ear Programming." Combinatorica 4, 373-395, 1984.

Pappas, T. "Projective Geometry & Linear Programming."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, pp. 216-217, 1989.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Linear Programming and the Simplex
Method." §10.8 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 423-436, 1992.

Sultan, A. Linear Programming: An Introduction with Ap-
plications. San Diego, CA: Academic Press, 1993.

Tokhomirov, V. M. "The Evolution of Methods of Convex
Optimization." Amer. Math. Monthly 103, 65-71, 1996.

Wood, M. K. and Dantzig, G. B. "Programming of Interde-
pendent Activities. I. General Discussion." Econometrica
17, 193-199, 1949.

Yudin, D. B. and Nemirovsky, A. S. Problem Complexity
and Method Efficiency in Optimization. New York: Wiley,
1983.

and

L(tf) = tLf.

see also Linear Transformation, Operator

Linear Ordinary Differential Equation

see Ordinary Differential Equation — First-
Order, Ordinary Differential Equation — Sec-
ond-Order

Linear Programming

The problem of maximizing a linear function over a
convex polyhedron, also known as OPERATIONS RE-
SEARCH, Optimization Theory, or Convex Opti-
mization THEORY. It can be solved using the SIMPLEX
METHOD (Wood and Dantzig 1949, Dantzig 1949) which
runs along Edges of the visualization solid to find the
best answer.

In 1979, L. G. Khachian found a 0(x b ) POLYNOMIAL-
time ALGORITHM. A much more efficient POLYNOMIAL-
time Algorithm was found by Karmarkar (1984). This
method goes through the middle of the solid and then
transforms and warps, and offers many advantages over
the simplex method.

see also Criss-Cross Method, Ellipsoidal Cal-
culus, Kuhn-Tucker Theorem, Lagrange Multi-
plier, Vertex Enumeration

References

Bellman, R. and Kalaba, R, Dynamic Programming and
Modern Control Theory. New York: Academic Press,
1965.

Linear Recurrence Sequence

see Recurrence Sequence

Linear Regression

The fitting of a straight LINE through a given set of
points according to some specified goodness-of-fit cri-
terion. The most common form of linear regression is

Least Squares Fitting.

see also LEAST SQUARES FITTING, MULTIPLE REGRES-
SION, Nonlinear Least Squares Fitting

References

Edwards, A. L. An Introduction to Linear Regression and
Correlation. San Francisco, CA: W. H. Freeman, 1976.

Edwards, A. L. Multiple Regression and the Analysis of Vari-
ance and Covariance. San Francisco, CA: W. H. Freeman,
1979.

Linear Space

see Vector Space

Linear Stability

Consider the general system of two first-order ORDI-
NARY Differential Equations

£ = f(x,y)
y = g(x,y)-

(i)

(2)

Let xq and y denote Fixed Points with x — y — 0, so

f(x ,y ) =
g(x ,yo) = 0.

(3)
(4)

1084 Linear Stability

Then expand about (a;o,yo) so

6x = fx(x ,yo)5x + f y (xo,yo)6y + fx V (xo,yo)6x5y + . . .

(5)
Sy = 9x(xo,yo)5x + g y (x ,yo)Sy + g xy (xo,yo)SxSy + —

(6)

To first-order, this gives

dt

8x

Sy

fx{x ,y ) f y {x 0j y )
g x (xo,yo) g y (xo>yo)

Sx
Sy

(?)

where the 2 x 2 Matrix is called the Stability Matrix.

In general, given an n-D MAP x' = T(x), let x be a
Fixed Point, so that

T(x ) = x . (8)

Expand about the fixed point,

T(x + tfx) = T(x ) + |^x 4- 0(8*f

= T(xo) + 5T, (9)

so

£T= ^<5x = A<5x. (10)

The map can be transformed into the principal axis
frame by finding the Eigenvectors and Eigenvalues
of the Matrix A

(A-Al)<fcc = 0,
so the Determinant

|A-AI| = 0.
The mapping is

"Ai ••• *

<5x • ' - : • . :

.0 •• A„.

When iterated a large number of times
<5T' inc ->

(11)

(12)

(13)

(14)

only if |5R(Ai)| < 1 for i = 1, . . . , n but — » oo if any |A»| >

1. Analysis of the Eigenvalues (and Eigenvectors)

of A therefore characterizes the type of Fixed POINT.

The condition for stability is |9ft(Ai)| < 1 for i = 1, . . . ,

n.

see also Fixed Point, Stability Matrix

References

Tabor, M. "Linear Stability Analysis." §1.4 in Chaos and In-

tegrability in Nonlinear Dynamics: An Introduction. New

York: Wiley, pp. 20-31, 1989.

Linearly Dependent Curves

Linear Transformation

An n x n MATRIX A is a linear transformation (linear
Map) Iff, for every pair of n- Vectors X and Y and
every SCALAR i,

A(X + Y) = A(X) + A(Y)

and

A(r-X) =*A(X).
Consider the 2-D transformation

pXi = di\X\ + CL12X2

px 2 = CL21X2 + a22#2-

(1)

(2)

(3)
(4)

Rescale by defining A = xi/£2 and A' = x f 1 /x' 2 , then the
above equations become

A'

g\ +

-yX + S'

(5)

where a5 — {3y ^ and a, /?, 7 and 6 are defined in
terms of the old constants. Solving for A gives

5\' -{3
-7A' + a '

(6)

so the transformation is ONE-TO-ONE. To find the
Fixed Points of the transformation, set A = A' to ob-
tain

7 A 2 + (<$-a)A-/3 = 0.

(7)

This gives two fixed points which may be distinct or
coincident. The fixed points are classified as follows.

variables

type

(5 — a) + 4/?7 > hyperbolic fixed point
(5 — a) 2 + 4/?7 < elliptic fixed point
(6 - a) 2 + 4/?7 = parabolic fixed point

see also ELLIPTIC Fixed POINT (Map), HYPERBOLIC
Fixed Point (Map), Involuntary, Linear Opera-
tor, Parabolic Fixed Point

References

Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, pp. 13-
15, 1961.

Linearly Dependent Curves

Two curves <j> and ^ satisfying

<t> + il> =

are said to be linearly dependent. Similarly, n curves
<f>i, i = 1, . . . , n are said to be linearly dependent if

£> = o.

see also Bertini's Theorem, Study's Theorem

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 32-34, 1959.

Linearly Dependent Functions

Link 1085

Linearly Dependent Functions

The n functions /i(#), /2(a), • • ■ , /n(#) are linearly de-
pendent if, "for some ci, c 2 , . . . , c n £ R. not all zero,

Ci/»(aj) =

(1)

(where EiNSTEIN SUMMATION is used) for all x in some
interval J. If the functions axe not linearly dependent,
they are said to be linearly independent. Now, if the

functions €

we can differentiate (1) up to n — 1

times. Therefore, linear dependence also requires

dfi =
dfi =

dfl

(n-l)

= 0,

(2)

(3)
(4)

where the sums are over i = 1, . . . , n. These equations
have a nontrivial solution Iff the Determinant

A
fi

h
ft

An-1) /(n-l)
7l J2

/»

f

J n

,(n'-l)

= o,

(5)

where the DETERMINANT is conventionally called the
Wronskian and is denoted W(/i,/2,- ■ -,/n). If the
WRONSKIAN ^ for any value c in the interval I, then
the only solution possible for (2) is a = (i = 1, . . . ,
n), and the functions are linearly independent. If, on
the other hand, W = for a range, the functions are
linearly dependent in the range. This is equivalent to
stating that if the vectors V[/i(c)], . . . , V[/ n (c)] defined

by

" fi(x)

V[/i(x)] =

fi(x)

r(»-l)

ur X} {*).

(6)

are linearly independent for at least one c E /, then the
functions fi are linearly independent in L

References

Sansone, G. "Linearly Independent Functions." §1.2 in Or-
thogonal Functions, rev. English ed. New York: Dover,
pp. 2-3, 1991.

Linearly Dependent Vectors

n VECTORS Xi, X 2 , . . . , X n are linearly dependent IFF
there exist SCALARS a, c 2 , . . . , c n such that

CilHi = 0,

(i)

where EiNSTEIN SUMMATION is used and i = 1, . . . , n.
If no such SCALARS exist, then the vectors are said to be

linearly independent. In order to satisfy the Criterion
for linear dependence,

#11

#12

ci

#21

+ C 2

#22

_ #nl _

_#n2.

+ •

#ln

'0'

#2n

Cn

—

_%nn _

_0_

(2)

#11

#12 *

#ln

"ci"

"0"

#21

#22 *

• #2n

C2

=

#nl

#n2 '

X nn _

Sn.

_0_

(3)

In order for this MATRIX equation to have a nontrivial
solution, the Determinant must be 0, so the Vectors
are linearly dependent if

#11 #12

#21 #22

#ln
X2n

= 0,

(4)

2.

pp pq
q.p q.q

= 0.

3. '

rhe 2 x n M

\TRIX

P

q

#nl #n2 * * * #n

and linearly independent otherwise.

Let p and q be n-D VECTORS. Then the following three
conditions are equivalent (Gray 1993).

1. p and q are linearly dependent.

has rank less than two.

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 186-187, 1993.

Linearly Independent

Two or more functions, equations, or vectors which are
not linearly dependent are said to be linearly indepen-
dent.

see also Linearly Dependent Curves, Linearly De-
pendent Functions, Linearly Dependent Vec-
tors, Maximally Linear Independent

Link

Formally, a link is one or more disjointly embedded CIR-
CLES in 3-space. More informally, a link is an assem-
bly of KNOTS with mutual entanglements. Kuperberg
(1994) has shown that a nontrivial Knot or link in R 3
has four COLLINEAR points (Eppstein). Doll and Hoste
(1991) list Polynomials for oriented links of nine or
fewer crossings. A listing of the first few simple links
follows, arranged by CROSSING NUMBER.

1086

Link

Link

Link

Linkage 1087

see also Andrews-Curtis Link, Borromean Rings,
Brunnian Link, Hopf Link, Knot, Whitehead

Link

References

Doll, H. and Hoste, J. "A Tabulation of Oriented Links."

Math. Comput. 57, 747-761, 1991.
Eppstein, D. "Colinear Points on Knots." http://www.ics.

uci.edu/-eppstein/junkyard/knot-coline2ir.html.
Kuperberg, G. "Quadrisecants of Knots and Links." J. Knot

Theory Ramifications 3, 41-50, 1994.
& Weisstein, E. W. "Knots." http: //www. astro. Virginia.

edu/ ~ eww6n/math/not ebooks /Knot s . m.

Link Diagram

A planar diagram depicting a Link (or Knot) as a se-
quence of segments with gaps representing undercross-
ings and solid lines overcrossings. In such a diagram,
only two segments should ever cross at a single point.
Link diagrams for the TREFOIL KNOT and FlGURE-OF-
Eight Knot are illustrated above.

Linkage

Sylvester, Kempe and Cayley developed the geometry
associated with the theory of linkages in the 1870s.
Kempe proved that every finite segment of an algebraic
curve can be generated by a linkage in the manner of

Watt's Curve.

1088 Linking Number

Liouville's Constant

see also Hart's Inversor, Kempe Linkage, Pan-
tograph, Peaucellier Inversor, Sarrus Linkage,
Watt's Parallelogram

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., 1989.

Linking Number

A Link invariant. Given a two-component oriented
LINK, take the sum of +1 crossings and -1 crossing
over all crossings between the two links and divide by 2.
For components a and (3,

pGan/3

where a n j3 is the set of crossings of a with (3 and e(p)
is the sign of the crossing. The linking number of a
splittable two-component link is always 0.

see also JONES POLYNOMIAL, LlNK

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 132-133, 1976.

Links Curve

The curve given by the Cartesian equation

(x 2 + V - 3x) 2 = 4x 2 (2-x).

The origin of the curve is a Tacnode.

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.

Linnik's Theorem

Let p(d, a) be the smallest PRIME in the arithmetic pro-
gression {a + kd} for k an Integer > 0. Let

p(d) = maxp(d,a)

such that 1 < a < d and (a, d) = 1. Then there exists a
do > 2 and an L > 1 such that p(d) < d L for all d > do.
L is known as LINNIK'S CONSTANT.

References

Linnik, U. V. "On the Least Prime in an Arithmetic Progres-
sion. I. The Basic Theorem." Mat. Sbornik N. S. 15 (57),
139-178, 1944.

Linnik, U. V. "On the Least Prime in an Arithmetic Pro-
gression. II. The Deuring-Heilbronn Phenomenon" Mat.
Sbornik N S. 15 (57), 347-368, 1944.

Liouville's Boundedness Theorem

A bounded ENTIRE FUNCTION in the COMPLEX PLANE
C is constant. The FUNDAMENTAL THEOREM OF AL-
GEBRA follows as a simple corollary.

see also Complex Plane, Entire Function, Funda-
mental Theorem of Algebra

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 381-382, 1953.

Liouville's Conformality Theorem

In Space, the only Conformal Transformations
are inversions, Similarity TRANSFORMATIONS, and
Congruence Transformations. Or, restated, ev-
ery ANGLE-preserving transformation is a Sphere-
preserving transformation.

see also Conformal Map

Liouville's Conic Theorem

The lengths of the TANGENTS from a point P to a CONIC
C are proportional to the Cube Roots of the Radii of
Curvature of C at the corresponding points of contact.

see also CONIC SECTION
Liouville's Constant

Linnik's Constant

The constant L in Linnik's Theorem. Heath-Brown
(1992) has shown that L < 5.5, and Schinzel, Sierpiri-
ski, and Kanold (Ribenboim 1989) have conjectured that
L = 2.

References

Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ asolve/constant/linnik/linnik. html.

Guy, Ft. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 13, 1994.

Heath- Brown, D. R. "Zero-Free Regions for Dirichlet L~
Functions and the Least Prime in an Arithmetic Progres-
sion." Proc. London Math, Soc. 64, 265-338, 1992.

Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, 1989.

L = ^l(T n!

0.110001000000000000000001.

(Sloane's A012245). Liouville's constant is a decimal
fraction with a 1 in each decimal place corresponding
to a FACTORIAL n!, and ZEROS everywhere else. This
number was among the first to be proven to be TRANS-
CENDENTAL. It nearly satisfies

190z + 21 = 0,

but with x = L, this equation gives -0.0000000059 . . ..
see also Liouville Number

Liouville's Elliptic Function Theorem

Liouville Function

1089

References

Conway, J. H. and Guy, R. K. "Liouville 's Number." In The

Book of Numbers. New York: Springer- Verlag, pp. 239-

241, 1996.
Courant, R. and Robbins, H. "Liouville's Theorem and the

Construction of Transcendental Numbers." §2.6.2 in What

is Mathematics?: An Elementary Approach to Ideas and

Methods, 2nd ed, Oxford, England: Oxford University

Press, pp. 104-107, 1996.
Liouville, J. "Sur des classes tres etendues de quantites dont

la valeur n'est ni algebrique, ni meme reductible a des irra-

tionelles algebriques." C. R. Acad. Sci. Paris 18, 883-885

and 993-995, 1844.
Liouville, J. "Sur des classes tres-etendues de quantites dont

la valeur n'est ni algebrique, ni meme reductible a des irra-

tionelles algebriques." J. Math, pures appl. 15, 133-142,

1850.
Sloane, N. J. A. Sequence A012245 in "An On-Line Version

of the Encyclopedia of Integer Sequences."

Liouville's Elliptic Function Theorem

An Elliptic Function with no Poles in a fundamen-
tal cell is a constant.

Liouville Function

-0.5

The function

A(n) = (-1)

r(n)

(i)

where r(n) is the number of not necessarily distinct
Prime Factors of n, with r(l) = 0. The first few
values of A(n) are 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1,

-1, The Liouville function is connected with the

Riemann Zeta Function by the equation

C(2*) =

cm

A(n)

n

(2)

(Lehman 1960).

The Conjecture that the Summatory Function

n

L(n) = 5^A(n) (3)

satisfies L(n) < for n > 2 is called the POLYA CON-
JECTURE and has been proved to be false. The first n
for which L(n) = are for n = 2, 4, 6, 10, 16, 26,
40, 96, 586, 906150256, ... (Sloane's A028488), and
n = 906150257 is, in fact, the first counterexample to
the POLYA Conjecture (Tanaka 1980). However, it is
unknown if L(x) changes sign infinitely often (Tanaka
1980). The first few values of L(n) are 1, 0, -1, 0, -1,
0, -1, -2, -1, 0, -1, -2, -3, -2, -1, 0, -1, -2, -3,
-4, ... (Sloane's A002819). L(n) also satisfies

i>©=^-

(4)

where [sbJ is the FLOOR FUNCTION (Lehman 1960).
Lehman (1960) also gives the formulas

L(x) = J2 ^M { [\

-l>>(Ls;H;

x/w

m\l
m=l

and

*<»> = E"(£) + X>>h/7

, (6)

where Jfc, /, and m are variables ranging over the POSI-
TIVE integers, fi(n) is the MOBIUS FUNCTION, M(x) is
Mertens Function, and v, w, and x are POSITIVE real
numbers with v < w < x.

see also Polya Conjecture, Prime Factors, Rie-
mann Zeta Function

References

Fawaz, A. Y. "The Explicit Formula for L (x)." Proc. Lon-
don Math. Soc. 1, 86-103, 1951.

Lehman, R. S. "On Liouville's Function." Math. Comput.
14, 311-320, 1960.

Sloane, N. J. A. Sequences A028488 and A002819/M0042 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."

Tanaka, M. "A Numerical Investigation on Cumulative Sum
of the Liouville Function." Tokyo J. Math. 3, 187-189,
1980.

1090

Liouville Measure

Liouville Measure

jQdpiC^i,

where pi and qi are momenta and positions of particles.
see also LlOUVILLE'S PHASE SPACE THEOREM, PHASE

Space

Liouville Number

A Liouville number is a Transcendental Number
which is very close to a Rational Number. An Ir-
rational Number j3 is a Liouville number if, for any
n, there exist an infinite number of pairs of Integers p
and q such that

0<

/*-*

1

< — .

q n

Mahler (1953) proved that n is not a Liouville number.

see also LlOUVILLE'S CONSTANT, LlOUVILLE'S RATIO-
NAL Approximation Theorem, Roth's Theorem,
Transcendental Number

References

Mahler, K. "On the Approximation of 7r." Nederl. Akad.
Wetensch, Proc. Ser. A. 56 / Indagationes Math. 15, 30-
42, 1953.

Liouville's Phase Space Theorem

States that for a nondissipative Hamiltonian System,
phase space density (the Area between phase space con-
tours) is constant. This requires that, given a small time
increment dt,

Ql = q(t + dt) = q + dH ^,P0j) dt + 0(<ft2)

opo

px = p(t + dt) =p - ™^lBhti dt + {dt%

dqo

(1)
(2)

the Jacobian be equal to one:

d(giiPi) _
d(Qo,Po)

dpo

dpi

dqo

dpi

dp

1 +

d 2 H

dt

dp

dt

— 7 dt

dq Q *

d 2 H
dqodpo

dt

+ G(dt 2 )

= l + (D(dt 2 ).

(3)

Expressed in another form, the integral of the LIOU-
VILLE Measure,

N

JJ / dpidqu

(4)

is a constant of motion. Symplectic Maps of Ham-
iltonian SYSTEMS must therefore be AREA preserving
(and have DETERMINANTS equal to 1).
see also Liouville Measure, Phase Space

References

Chavel, I. Riemannian Geometry: A Modern Introduction.
New York: Cambridge University Press, 1994.

Liouville-Roth Constant
Liouville Polynomial Identity

6(xi + X2 + £3 +X4 ) = (xi + x 2 ) -f (x± + X3)

+(x2+X3)*+(xi+X4) 4 +{x2+X4) 4 -\-(x3+X4) 4 +(x 1 -x 2 ) 4
+(xi ~ X3) 4 + (x2 - X3) 4 + (xi - Z4) 4 4- (x 2 - x 4 ) 4

+ (X 3 - X 4 ) 4 .

This is proven in Rademacher and Toeplitz (1957).
see also WARING'S PROBLEM

References

Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 55-56,
1957.

Liouville's Rational Approximation Theorem

For any Algebraic Number x of degree n > 1, a Ra-
tional approximation x = p/q must satisfy

7 n + l

for sufficiently large q. Writing r = n + 1 leads to the
definition of the LlOUVlLLE-ROTH CONSTANT of a given
number.

see also Lagrange Number (Rational Approxi-
mation), Liouville's Constant, Liouville Num-
ber, Liouville-Roth Constant, Markov Number,
Roth's Theorem, Thue-Siegel-Roth Theorem

References

Courant, R. and Robbins, H. "Liouville's Theorem and the
Construction of Transcendental Numbers." §2.6.2 in What
is Mathematics?: An Elementary Approach to Ideas and
Methods, 2nd ed. Oxford, England: Oxford University
Press, pp. 104-107, 1996.

Liouville-Roth Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Let x be a REAL Number, and let R be the Set of
Positive Real Numbers for which

q r

(i)

has (at most) finitely many solutions p/q for p and q
Integers. Then the Liouville-Roth constant (or Ir-
rationality Measure) is defined as the threshold at
which Liouville's Rational Approximation Theo-
rem kicks in and x is no longer approximable by Ra-
tional Numbers,

r(x) = inf r.
ren

(2)

Liouville Space

Lissajous Curve 1091

There are three regimes:

' r(x) = 1 z is rational

r(x) = 2 x is algebraic irrational
, t{x) > 2 cc is transcendental.

(3)

The best known upper bounds for common constants
are

r(L) = oo
r(e) = 2
r(7r) < 8.0161
r(ln2) <4.13
r(?r 2 ) < 6.3489
r(C(3)) < 13.42,

(4)
(5)
(6)
(T)
(8)
(9)

where L is Liouville's Constant, £(3) is Apery's
CONSTANT, and the lower bounds are 2 for the inequal-
ities.

see also Liouville's Rational Approximation The-
orem, Roth's Theorem, Thue-Siegel-Roth Theo-
rem

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.

Finch, S. "Favorite Mathematical Constants." http://vwv.
mathsoft.com/asolve/constant/lvlrth/lvlrth.html.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford: Clarendon Press, 1979.

Hata, M. "Improvement in the Irrationality Measures of n
and 7r 2 ." Proc. Japan. Acad. Ser. A Math. Sci. 68, 283-
286, 1992.

Hata, M. "Rational Approximations to 7r and Some Other
Numbers." Acta Arith. 63 335-349, 1993.

Hata, M. "A Note on Beuker's Integral." J. Austral Math,
Soc. 58, 143-153, 1995.

Stark, H. M. An Introduction to Number Theory. Cam-
bridge, MA: MIT Press, 1978.

Liouville Space

Also known as Line Space or "extended" Hilbert

Space, it is the Direct Product of two Hilbert

Spaces.

see also Direct Product (Set), Hilbert Space

Liouville's Sphere-Preserving Theorem

see Liouville's Conformality Theorem

Lipschitz Condition

A function f(x) satisfies the Lipschitz condition of order
a at x = if

\f(h)-f(0)\<B\hf

for all \h\ < e, where B and are independent of /i,
> 0, and a is an UPPER BOUND for all for which a
finite B exists.

see also HlLLAM'S THEOREM, LIPSCHITZ FUNCTION

Lipschitz Function

A function / such that

l/(*)-/(l/)l<C|z-»|
is called a Lipschitz function.
see also LIPSCHITZ CONDITION

References

Morgan, F. "What Is a Surface?" Amer. Math. Monthly 103,
369-376, 1996.

Lipschitz's Integral

/

e ax J (bx)dx= ,

Va 2 + b 2

where J (z) is the zeroth order BESSEL FUNCTION OF
the First Kind.

References

Bowman, F. Introduction to Bessel Functions. New York:
Dover, p. 58, 1958.

Lissajous Curve

Lissajous curves are the family of curves described by
the parametric equations

x(t) — Acos(u; x t - S v ) (1)

y(t) = Bcoa(u> y t-S y ) t (2)

sometimes also written in the form

x(t) = a sin(nt -\- c) (3)

y(t) = 6sint. (4)

They are sometimes known as BOWDITCH CURVES after
Nathaniel Bowditch, who studied them in 1815. They
were studied in more detail (independently) by Jules-
Antoine Lissajous in 1857 (MacTutor Archive). Lis-
sajous curves have applications in physics, astronomy,
and other sciences. The curves close Iff u; x /u> y is RA-
TIONAL.

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 53-54, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 178-179 and 181-183, 1972.

MacTutor History of Mathematics Archive. "Lissajous
Curves." http: // www - groups . dcs . st - and .ac.uk/
-history/Curves/Lissajous .html.

1092 Lissajous Figure

Lobachevsky's Formula

Lissajous Figure

see Lissajous Curve

List

A Data Structure consisting of an order Set of el-
ements, each of which may be a number, another list,
etc. A list is usually denoted (ai, a2, . .., a n ) or {ai,
a2, ... , a n }.
see also Queue, Stack

Lituus

An Archimedean Spiral with m
equation

-2, having polar

2/1 2

r V = a .

Lituus means a "crook," in the sense of a bishop's
crosier. The lituus curve originated with Cotes in 1722.
Maclaurin used the term lituus in his book Harmonia
Mensurarum in 1722 (MacTutor Archive). The lituus is
the locus of the point P moving such that the Area of
a circular SECTOR remains constant.

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 69-70, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 186 and 188, 1972.

Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 175, 1967.

MacTutor History of Mathematics Archive. "Lituus." http:
//www -groups . dcs . st-and . ac . uk/ -history /Curves/
Lituus.html.

Lituus Inverse Curve

The Inverse Curve of the Lituus is an Archimedean
Spiral with m = 2, which is Fermat's Spiral.
see also Archimedean Spiral, Fermat's Spiral,
Lituus

LLL Algorithm

An Integer Relation algorithm.

see also FERGUSON-FORCADE ALGORITHM, HJLS AL-
GORITHM, Integer Relation, PSLQ Algorithm,
PSOS Algorithm

References

Lenstra, A. K.; Lenstra, H. W.; and Lovasz, L, "Factoring

Polynomials with Rational Coefficients." Math. Ann. 261,

515-534, 1982.

Ln

The Logarithm to Base e, also called the Natural
Logarithm, is denoted ln, i.e.,

lnx = log e x.
see also BASE (LOGARITHM), E, LG, LOGARITHM,

Napierian Logarithm, Natural Logarithm
LoShu

8

1

6

3

5

7

4

9

2

The unique MAGIC SQUARE of order three. The Lo Shu
is an Associative Magic Square, but not a Pan-
magic Square.

see also Associative Magic Square, Magic Square,
Panmagic Square

References

Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 23-24, 1975.

Lobachevsky-Bolyai-Gauss Geometry

see Hyperbolic Geometry

Lobachevsky's Formula

P

nw

A C D B

Given a point P and a Line AB, draw the PERPENDIC-
ULAR through P and call it PC. Let PD be any other
line from P which meets CB in D. In a HYPERBOLIC
Geometry, as D moves off to infinity along CB, then
the line PD approaches the limiting line PE, which is
said to be parallel to CB at P. The angle LCPE which
PE makes with PC is then called the Angle of Par-
allelism for perpendicular distance cc, and is given by

n(x) = 2tan~ 1 (e~* a! ) )

which is called Lobachevsky's formula.
see also Angle of Parallelism, Hyperbolic Geom-
etry

References

Manning, H. P. Introductory Non-Euclidean Geometry. New
York: Dover, p. 58, 1963.

Lobatto Quadrature

Local Density 1093

Lobatto Quadrature

Also called RADAU QUADRATURE (Chandrasekhar
1960). A Gaussian Quadrature with Weighting
Function W(x) = 1 in which the endpoints of the in-
terval [—1,1] are included in a total of n ABSCISSAS,
giving r — n — 2 free abscissas. ABSCISSAS are symmet-
rical about the origin, and the general FORMULA is

/i n-l

f{x) dx = «;i/(-l) + w n f(l) + J^ WifM- (1)

The free ABSCISSAS xi for i = 2, . . . , n — 1 are the roots
of the Polynomial Pn-i(x), where P(x) is a Legen-
dre Polynomial. The weights of the free abscissas are

2n

(1 - Xi^P^ix^P^Xi)

__ 2

~ n(n-l)[P n _i(xO] 2 '

and of the endpoints are

2
n(n — 1)

The error term is given by

n(n-l) 3 2 2 - 1 [(n-2)!] 4
(2n-l)[(2n-2)!] 3

(2)

(3)

(4)
/ (2n ~ 2) (£), (5)

for £ £ ( — 1, 1). Beyer (1987) gives a table of parame-
ters up to n=ll and Chandrasekhar (1960) up to n~9
(although Chandrasekhar's //3,4 for m = 5 is incorrect).

n

Xi

Wi

3

±1

1.33333
0.333333

4

±0.447214

±1

0.833333
0.166667

5

0.711111

±0.654654

0.544444

±1

0.100000

6

±0.285232

0.554858

±0.765055

±1

0.378475
0.0666667

The Abscissas and weights can be comp
cally for small n.

n

Xi

Wi

3

±1

4
3
1
3

4

±£V5

1
6

±i

5
6

5

^

U 45

= t 7 VZ1 90

±! To

see also Chebyshev Quadrature, Radau Quadra-
ture

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook

of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, 9th printing. New York: Dover,

pp. 888-890, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.

Boca Raton, FL: CRC Press, p. 465, 1987.
Chandrasekhar, S. Radiative Transfer. New York: Dover,

pp. 63-64, 1960.
Hildebrand, F. B. Introduction to Numerical Analysis. New

York: McGraw-Hill, pp. 343-345, 1956.

Lobster

A 6-Polyiamond,

References

Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,

and Packings, 2nd ed. Princeton, NJ: Princeton University

Press, p. 92, 1994.

Local Cell

The Polyhedron resulting from letting each Sphere
in a Sphere Packing expand uniformly until it touches
its neighbors on flat faces.
see also LOCAL DENSITY

Local Degree

The degree of a Vertex of a Graph is the number of
Edges which touch the Vertex, also called the Local
Degree. The Vertex degree of a point A in a Graph,
denoted p(A), satisfies

Y /P (A i ) = 2E,

where E is the total number of EDGES. Directed graphs
have two types of degrees, known as the Indegree and
OUTDEGREE.

see also Indegree, Outdegree

Local Density

Let each Sphere in a Sphere Packing expand uni-
formly until it touches its neighbors on flat faces. Call
the resulting POLYHEDRON the LOCAL Cell. Then the
local density is given by

Ksphere

' T/

Mocal cell

When the Local Cell is a regular Dodecahedron,
then

pdo decahedron —

7T\/5 + y/5

= 0.7547.

see also LOCAL DENSITY CONJECTURE

1094 Local Density Conjecture

Log Normal Distribution

Local Density Conjecture

The Conjecture that the maximum Local Density

iS given by Pdodecahedron ■

see also Local Density

Local Extremum

A Local Minimum or Local Maximum.

see also Extremum, Global Extremum

Local Field

A FIELD which is complete with respect to a discrete
Valuation is called a local field if its Field of Residue
Classes is Finite. The Hasse Principle is one of the
chief applications of local field theory.

see also HASSE PRINCIPLE, VALUATION

References

Iyanaga, S. and Kawada, Y. (Eds.). "Local Fields." §257

in Encyclopedic Dictionary of Mathematics. Cambridge,

MA: MIT Press, pp. 811-815, 1980.

Local-Global Principle

see Hasse Principle

Local Group Theory

The study of a FINITE GROUP G using the LOCAL SUB-
GROUPS of G. Local group theory plays a critical role in
the Classification Theorem,

see also Sylow Theorems

Local Maximum

The largest value of a set, function, etc., within some
local neighborhood.

see also Global Maximum, Local Minimum, Maxi-
mum, Peano Surface

Local Minimum

The smallest value of a set, function, etc., within some
local neighborhood.

see also Global Minimum, Local Maximum, Mini-
mum

Local Ring

A Noetherian Ring R with a Jacobson radical which
has only a single maximal ideal.

References

Iyanaga, S. and Kawada, Y. (Eds.). "Local Rings." §281D
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, pp. 890-891, 1980.

Local Subgroup

A normalizer of a nontrivial SYLOW p-SUBGROUP of a
Group G.

see also Local Group Theory

Local Surface

see Patch

Locally Convex Space

see Locally Pathwise-Connected Space

Locally Finite Space

A locally finite Space is one for which every point of
a given space has a NEIGHBORHOOD that meets only
finitely many elements of the COVER.

Locally Pathwise-Connected Space

A Space X is locally pathwise-connected if for every
NEIGHBORHOOD around every point in X, there is a
smaller, Pathwise-Connected Neighborhood.

Loculus of Archimedes

see Stomachion

Locus

The set of all points (usually forming a curve or surface)
satisfying some condition. For example, the locus of
points in the plane equidistant from a given point is
a CIRCLE, and the set of points in 3-space equidistant
from a given point is a SPHERE.

Log

The symbol log as is used by physicists, engineers, and
calculator keypads to denote the BASE 10 LOGARITHM.
However, mathematicians generally use the same symbol
to mean the NATURAL LOGARITHM Ln, In a. In this
work, logx = log 10 x, and ln# = log e x is used for the
Natural Logarithm.

see also LG, LN, LOGARITHM, NATURAL LOGARITHM

Log Likelihood Procedure

A method for testing NESTED HYPOTHESES. To ap-
ply the procedure, given a specific model, calculate the
Likelihood of observing the actual data. Then com-
pare this likelihood to a nested model (i.e., one in which
fewer parameters are allowed to vary independently).

Log Normal Distribution

A Continuous Distribution in which the Loga-
rithm of a variable has a Normal Distribution. It is
a general case of Gilbrat's Distribution, to which
the log normal distribution reduces with 5 = 1 and

Log Normal Distribution

M = 0. The probability density and cumulative dis-
tribution functions are log normal distribution

P{x) = _JL_ e -(ln.-J*) a /<2S a ) (1)

£>(*) =

1

1 + erf

lnx - M

(2)

where erf(x) is the Erf function. This distribution is
normalized, since letting y = lnx gives dy = dx/x and

re = e% so

Jo

1 P[x) dx = i r c -<v-«> a /« a d = L (3)

The Mean, Variance, Skewness, and Kurtosis are
given by

M+S 2 /2

V

7i = \/e s - 1 (2 + e s )

72 = e (3 + 2e + e ) — 3.

These can be found by direct integration

i r

= -j-r

SV2^J- C

_ * /°" e -[-y+(w-M) 2 /2S 2 ] ,

= ^— r e - ( -

(4)
(5)
(6)
(7)

dx

e iy-Mf/2S* e y dy

>dy

dy

SV2n

f

J — c

-{[y-{S 2 + M)) 2 + S 2 (S 2 +2M)}/2S 2 ^

1 M+5 2 /2

M+5 2 /2

/oo
e -[«-
-oo

(5 2 +M) 2 ]/25 2

dy

(8)

and similarly for a 2 . Examples of variates which have
approximately log normal distributions include the size
of silver particles in a photographic emulsion, the sur-
vival time of bacteria in disinfectants, the weight and
blood pressure of humans, and the number of words
written in sentences by George Bernard Shaw.

see also Gilbrat's Distribution, Weibull Distri-
bution

References

Aitchison, J. and Brown, J. A. C. The Lognormal Distribu-
tion, with Special Reference to Its Use in Economics. New
York: Cambridge University Press, 1957.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 123, 1951.

Logarithm 1095

Log-Series Distribution

The terms in the series expansion of ln(l — 9) about
= are proportional to this distribution.

p ^ = -^hr) (1)

D(n) = ^P(i) ^^ ,

(2)
where $ is the LERCH TRANSCENDENT. The MEAN,

Variance, Skewness, and Kurtosis

M =

(0-l)ln(l-0)

2 0[0 + ln(l-0)]

(6>-l) 2 [ln(l-0)] 2

(3)
(4)

= 26> 2 + 30 ln(l - g) + (1 + g) ln 2 (l - g) _

71 ~ ln(l - 6)[9 + ln(l - 9)y -6[9 + \n(l - 9)] °

(5)
_ 6fl 3 + 129 2 In(l -_g)_+ 6(7 + 40) ln 2 (l - 6)

72 ~ 0[0 + ln(l - 6)] 2

+

(l + 40 + fl 2 )ln 3 (l-6>)
6[9 + ln{l-9)] 2

(6)

Log- Weibull Distribution

see Fisher-Tippett Distribution

Logarithm

The logarithm is defined to be the INVERSE FUNCTION
of taking a number to a given Power. Therefore, for

any x and 6,

x = b l ° Sb \ (1)

or equivalently,

x = log b (b x ).

(2)

Here, the Power b is known as the Base of the log-
arithm. For any BASE, the logarithm function has a
Singularity at re — 0. In the above plot, the solid
curve is the logarithm to Base e (the Natural Loga-
rithm), and the dotted curve is the logarithm to Base
10 (Log).

1096 Logarithm

Logarithmic Binomial Theorem

Logarithms are used in many areas of science and engi-
neering in which quantities vary over a large range. For
example, the decibel scale for the loudness of sound, the
Richtcr scale of earthquake magnitudes, and the astro-
nomical scale of stellar brightnesses are all logarithmic
scales.

x 2 ~y 2 = l (14)

■ = v^ 2 - 1>

so

(15)

log b (x + \/x 2 -l) = - log 6 (x - v^ 2 -l)- (16)

The logarithm can also be defined for COMPLEX argu-
ments, as shown above. If the logarithm is taken as
the forward function, the function taking the BASE to a
given POWER is then called the Antilogarithm.

For x = logiV, [x\ is called the CHARACTERISTIC and
x — [x\ is called the MANTISSA. Division and multipli-
cation identities follow from these

xy - 6 logb x b logh y = b logb x+logb y , (3)

from which it follows that

l °g& ( X V) = lo g b x + lc, g& V ( 4 )

(5)
(6)

lo g& I ~ J = !og6 x - log 6 y
log 6 x n = nlog b x.

There are a number of properties which can be used to
change from one Base to another

a _ a log a 6/log a 6 _ / a log a 6\l/log a 6 _ ^l/log a 6 /^

1

log 6 a :

(8)

log, z = log x (y log * z ) = log y z log, y (9)

^=!^ do)

^ y

a x = b x ^ loga b = b x logfc a

(11)

The logarithm BASE e is called the NATURAL LOGA-
RITHM and is denoted In a; (Ln). The logarithm BASE
10 is denoted logx (Log), (although mathematics texts
often use logx to mean lnx). The logarithm Base 2 is
denoted lgx (Lg).

An interesting property of logarithms follows from look-
ing for a number y such that

log 6 (x + y) = - log 6 (z - y)

1

x + y ■

x -y

(12)
(13)

Numbers of the form log a b are IRRATIONAL if a and b
are Integers, one of which has a Prime factor which
the other lacks. A, Baker made a major step forward
in TRANSCENDENTAL NUMBER theory by proving the
transcendence of sums of numbers of the form a ln /3 for
a and (3 Algebraic Numbers.

see also ANTILOGARITHM, COLOGARITHM, e, EXPO-
NENTIAL Function, Harmonic Logarithm, Lg, Ln,
Log, Logarithmic Number, Napierian Logarithm,
Natural Logarithm, Power

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Logarithmic

Function." §4.1 in Handbook of Mathematical Functions

with Formulas, Graphs, and Mathematical Tables, 9th

printing. New York: Dover, pp. 67-69, 1972.
Conway, J. H. and Guy, R. K. "Logarithms." The Book of

Numbers. New York: Springer- Verlag, pp. 248-252, 1996.
Beyer, W. H. "Logarithms." CRC Standard Mathematical

Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 159-

160, 1987.
Pappas, T. "Earthquakes and Logarithms." The Joy of

Mathematics. San Carlos, CA: Wide World Publ./Tetra,

pp. 20-21, 1989.
Spanier, J. and Oldham, K. B, "The Logarithmic Function

ln(;r)." Ch. 25 in An Atlas of Functions. Washington, DC:

Hemisphere, pp. 225-232, 1987.

Logarithmic Binomial Formula

see Logarithmic Binomial Theorem

Logarithmic Binomial Theorem

For all integers n and |x| < a,

\W(x + a) = J2

v(*)

KL k (a)x*

i(*)

where A„ is the Harmonic Logarithm and

is a

Roman Coefficient. For t = 0, the logarithmic bino-
mial theorem reduces to the classical BINOMIAL THEO-
REM for POSITIVE n, since X { ^l k {a) = a n ~ k for n > k,

A^_ fc (a) = for n < k y and

= (I) whenn> k > 0.

Similarly, taking t = 1 and n < gives the NEGATIVE
BINOMIAL SERIES. Roman (1992) gives expressions ob-
tained for the case t — 1 and n > which are not
obtainable from the BINOMIAL THEOREM.
see also HARMONIC LOGARITHM, ROMAN COEFFICIENT

References

Roman, S. "The Logarithmic Binomial Formula." Amer.
Math. Monthly 99, 641-648, 1992.

Logarithmic Distribution

Logarithmic Distribution

A Continuous Distribution for a variate with prob-
ability function

P (x ) = ^

{ ' 6(log6-l)-a(loga-l)
and distribution function

a(l — log a) — x(l — log a:)

D(x) =

a(l-loga) -6(1 -log b) '

The Mean is

_ a 2 (l-21oga)-& 2 (l-2log6)
M ~ 4[a(l-loga) -6(1 -log b)} '

but higher order moments are rather messy.
Logarithmic Integral

The logarithmic integral is defined by

(1)

The offset form appearing in the PRIME NUMBER THE-
OREM is defined so that Li(2) = 0:

Li(x) =

du

(2)

= li(x) - li(2) « li(x) - 1.04516 (3)

= ei(lnx), (4)

where ei(x) is the EXPONENTIAL INTEGRAL. (Note that
the NOTATION Li n (z) is also used for the Polyloga-
RITHM.) Nielsen (1965, pp. 3 and 11) showed and Ra-
manujan independently discovered (Berndt 1994) that

_ =7 + lnl na; + £^, (5)

where 7 is the Euler-Mascheroni Constant and fi
is Soldner's Constant. Another Formula due to

Ramanujan which converges more rapidly is

pX

J u

dt ^_1 1

= 7 -f In In x

lni

+*£ ( " V-'i"" E STT < 6 >

Logarithmic Spiral 1097

(Berndt 1994).

see also Polylogarithm, Prime Constellation,

Prime Number Theorem, Skewes Number

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 126-131, 1994.

Nielsen, N. Theorie des Integrallogarithms. New York:
Chelsea, 1965.

Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 151, 1991.

Logarithmic Number

A Coefficient of the Maclaurin Series of

- - 4- i 4- ±x 2 - -^-x 3 + -^-x 4 +

ln(l + x) X

160"

(Sloane's A002206 and A002207), the multiplicative in-
verse of the MERCATOR Series function ln(l + x).

see also Mercator Series

References

Sloane, N. J. A. Sequences A002206/M5066 and A002207/
M2017 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Logarithmic Spiral

A curve whose equation in POLAR COORDINATES is

given by

(i)

be

where r is the distance from the ORIGIN, is the angle
from the x-axis, and a and b are arbitrary constants.
The logarithmic spiral is also known as the GROWTH
Spiral, Equiangular Spiral, and Spira Mirabilis.
It can be expressed parametrically using

(2)

— n _ 1 _ 1

X X

VI -tan** ^ 1 + £

V^ 2 +2/ 2 r
1

which gives

1

x = r cos Q — a cos 6e

(

y = x tan 9 = r sin =

a sin Qe .

(3)
(4)

The logarithmic spiral was first studied by Descartes in
1638 and Jakob Bernoulli. Bernoulli was so fascinated

1098 Logarithmic Spiral

Logarithmic Spiral Evolute

by the spiral that he had one engraved on his tomb-
stone (although the engraver did not draw it true to
form). Torricelli worked on it independently and found
the length of the curve (MacTutor Archive).

The rate of change of RADIUS is

dr , be

Te= abe

br,

(5)

and the ANGLE between the tangent and radial line at
the point (r, 6) is

ib = tan 1 I -j- ] = tan x ( - ) =

cot b.

(6)

So, as b — >
Circle.

0, ip — > 7r/2 and the spiral approaches a

If P is any point on the spiral, then the length of the spi-
ral from P to the origin is finite. In fact, from the point
P which is at distance r from the origin measured along
a RADIUS vector, the distance from P to the POLE along
the spiral is just the Arc Length. In addition, any Ra-
dius from the origin meets the spiral at distances which
are in Geometric PROGRESSION (MacTutor Archive).

The Arc Length, Curvature, and Tangential An-
gle of the logarithmic spiral are

= f ds =f^T^ dt = ?!/l + v t

7-Vl + b 2

(7)

J ~ f - t„lt

—fcfa- <*&*>")-' m

■=|»

(s)ds = 9.

The Cesaro Equation is

bs'

(9)

(10)

On the surface of a SPHERE, the analog is a Loxo-
drome. This Spiral is related to Fibonacci Numbers
and the Golden Mean.

References

Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 184-186, 1972.
Lee, X. "Equiangular Spiral." http://www.best.com/-xah/

Special Plane Curves _ dir / Equiangular Spiral _ dir /

equiangular Spiral . html.
Lockwood, E. H. "The Equiangular Spiral." Ch. 11 in .A Book

of Curves. Cambridge, England: Cambridge University

Press, pp. 98-109, 1967.
MacTutor History of Mathematics Archive. "Equiangular

Spiral." http: //www- groups .dcs.st-and.ac .uk/ -history

/Curve s /Equiangular .html.

Logarithmic Spiral Caustic Curve

The Caustic of a Logarithmic Spiral, where the pole
is taken as the RADIANT Point, is an equal LOGARITH-
MIC Spiral.

Logarithmic Spiral Evolute

(r 2 +r e 2 ) 3/2

R =

r 2 + 2r 2 r$ 2 — wee '

Using

gives

R:

be t be ,2 be

r = ae r$ = aoe Tee = ao e

(a 2 e 2be + a 2 6 2 e 2M ) 3/2

(ae M ) 2 + 2(abe be ) 2 - (o6 M ')(o6 2 e M )

(1 + b 2 ) 3 ' 2 a 3 e 3be

(1)

(2)

2a 2 b 2

+ a 2 e 2

a 2 6 2 e

_ (l + 6 2 ) 3/ W M (l + & 2 ) 3 /Ve 36 *
a 2 b 2 e 2he + a 2 e 2be a 2 (1 + b 2 )e 2b9

= ay/l + b 2 e be

(3)

and

X

y

=

ae be cos 9
ae be sin 9

r '
X

'abe b8 cosO-ae be smO

y'\

abe be sinO + ae be cosQ

= c

be
le

b cos 9 — sin 9

b sin 9 -h cos 9

y

(4)

|r'| = ae be y/(b cos 9 - sin0) 2 + (6 sin (9 + cos<9) 2
= ae b9 ^fl + tf, (5)

and the Tangent Vector is given by
a r' 1

ae be Vl+¥

ae be cos 9
ae b9 sin9

cos 9
sin0

The coordinates of the Evolute are therefore

£ = — abe sin 9
7] = abe cos 9.

(6)

(7)
(8)

So the Evolute is another logarithmic spiral with a' =
a&, as first shown by Johann Bernoulli. However, in
some cases, the Evolute is identical to the original, as
can be demonstrated by making the substitution to the
new variable

9 = <f>- §7r±2n7r. (9)

Logarithmic Spiral Inverse Curve

Then the above equations become

= abe b *e b{ -* /2±2n ' ) cos<l>
r) = o6e i,( *~ ,r/a±2n,r) cos(<£ - tt/2 ± 2twt)
= a&e i V ( -' r/a±an,r) sin0,

(10)

(11)

which are equivalent to the form of the original equation
if

be

b(-^-K±2nir) _

1

In 6

ln6 + 6(-|7r±2Twr) =
= §7r =F 2n7r = -(2n - |)7r,

(12)
(13)

(14)

where only solutions with the minus sign in =j= exist.
Solving gives the values summarized in the following ta-
ble.

n

&n

1p — COt" 1 b n

1
2
3
4
5
6
7
8
9
10

0.2744106319. .
0.1642700512,.
0.1218322508..
0.0984064967. .
0.0832810611..
0.0725974881..
0.0645958183. .
0.0583494073. .
0.0533203211..
0.0491732529. .

74°39'18.53"
80°40'16.80"
83°03'13.53"
84°22'47.53"
85°14'21.60"
85°50 , 51.92"
86° 18' 14.64"
86°39'38.20"
86°56'52.30"
87°ll'05.45"

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 60-
64, 1991.

Logarithmic Spiral Inverse Curve

The Inverse Curve of the Logarithmic Spiral

with Inversion Center at the origin and inversion ra-
dius k is the Logarithmic Spiral

ke

Logarithmic Spiral Pedal Curve

Logic 1099

The Pedal Curve of a Logarithmic Spiral with
parametric equation

/ = e at cos t

at • *

g — e sint

(i)

(2)

for a PEDAL Point at the pole is an identical LOGA-
RITHMIC Spiral

(a sint + cos£)e at

1 + a 2
(sint — acost)e at

r == y/x 2 + y 2

VTTa2

(3)
(4)

(5)

Logarithmic Spiral Radial Curve

The Radial Curve of the Logarithmic Spiral is an-
other Logarithmic Spiral.

Logarithmically Convex Function

A function f(x) is logarithmically convex on the interval
[a, b] if / > and In /(e) is Concave on [a, 6]. If f(x)
and g(x) are logarithmically convex on the interval [a, 6],
then the functions f(x) + g(x) and f(x)g(x) are also
logarithmically convex on [a, 6].
see also CONVEX FUNCTION

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1100, 1980.

Logic

The formal mathematical study of the methods, struc-
ture, and validity of mathematical deduction and proof.
Formal logic seeks to devise a complete, consistent for-
mulation of mathematics such that propositions can be
formally stated and proved using a small number of sym-
bols with well-defined meanings. While this sounds like
an admirable pursuit in principle, in practice the study
of mathematical logic can rapidly become bogged down
in pages of dense and unilluminating mathematical sym-
bols, of which Whitehead and Russell's Principia Math-
ematica (1925) is perhaps the best (or worst) example.

A very simple form of logic is the study of "Truth Ta-
bles" and digital logic circuits in which one or more
outputs depend on a combination of circuit elements
(AND, NAND, OR, XOR, etc.; "gates") and the input

1100 Logical Paradox

values. In such a circuit, values at each point can take
on values of only TRUE (l) or FALSE (0). DE Morgan's
Duality Law is a useful principle for the analysis and
simplification of such circuits.

A generalization of this simple type of logic in which pos-
sible values are True, False, and "undecided" is called
Three- Valued LOGIC. A further generalization called
FUZZY LOGIC treats "truth" as a continuous quantity
ranging from to 1.

see also Absorption Law, Alethic, Boolean Alge-
bra, Boolean Connective, Bound, Caliban Puz-
zle, Contradiction Law, de Morgan's Duality
Law, de Morgan's Laws, Deducible, Excluded
Middle Law, Free, Fuzzy Logic, Godel's Incom-
pleteness Theorem, Khovanski's Theorem, Log-
ical Paradox, Logos, Lowenheimer-Skolem The-
orem, Metamathematics, Model Theory, Quan-
tifier, Sentence, Tarski's Theorem, Tautology,
Three- Valued Logic, Topos, Truth Table, Tur-
ing Machine, Universal Statement, Universal
Turing Machine, Venn Diagram, Wilkie's Theo-
rem

References

Adamowicz, Z. and Zbierski, P. Logic of Mathematics: A

Modern Course of Classical Logic, New York: Wiley, 1997.
Bogomolny, A. "Falsity Implies Anything." http://www.cut-

the-knot.com/do_you_know/falsity.html.
Carnap, R. Introduction to Symbolic Logic and Its Applica-
tions. New York: Dover, 1958.
Church, A. Introduction to Mathematical Logic, Vol. 1.

Princeton, NJ: Princeton University Press, 1996.
Godel, K. On Formally Undecidable Propositions of Prin-

cipia Mathematica and Related Systems. New York:

Dover, 1992.
Jeffrey, R. C. Formal Logic: Its Scope and Limits. New York:

McGraw-Hill, 1967.
Kac, M. and Ulam, S. M. Mathematics and Logic: Retrospect

and Prospects. New York: Dover, 1992.
Kleene, S. C. Introduction to Metamathematics. Princeton,

NJ: Van Nostrand, 1971.
Whitehead, A. N. and Russell, B. Principia Mathematica,

2nd ed. Cambridge, England: Cambridge University Press,

1962.

Logical Paradox

see Paradox

Logistic Distribution

P(x)

( x -m)/b

D W = l + e (m-_)/|6|'

(1)

(2)

Logistic Equation

and the Mean, Variance, Skewness, and Kurtosis

H — m

7i

72 = 5

5*

(3)
(4)
(5)
(6)

see also Logistic Equation, Logistic Growth

Curve

References

von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 250, 1993.

Logistic Equation

The logistic equation (sometimes called the VERHULST
MODEL since it was first published in 1845 by the Bel-
gian P.-F. Verhulst) is defined by

Xri-\-l — TZCnyl %n)i

(i)

where r (sometimes also denoted fi) is a POSITIVE con-
stant (the "biotic potential"). We will start xq in the
interval [0, 1]. In order to keep points in the interval, we
must find appropriate conditions on r. The maximum
value x n +i can take is found from

O/Xn

+ 1

aXji

r(l - 2x n ) = 0,

(2)

so the largest value of x n +i occurs for x n = 1/2. Plug-
ging this in, max(_ n +i) — r/4. Therefore, to keep the
Map in the desired region, we must have r £ (0, 4]. The
Jacobian is

(XX x\,

+1

|r(l - 2x n

(3)

and the Map is stable at a point Xq if J(xo) < 1. Now
we wish to find the FIXED POINTS of the MAP, which
occur when x n +i — x n - Drop the n subscript on x n

f(x) = rx(l — x) = x

(4)

c[l — r(l — x)] = x(l — r + rx) = rx [x — (l — r *)] = 0,
so the Fixed Points are x\

(i)

and x 2

1

An interesting thing happens if a value of r greater than
3 is chosen. The map becomes unstable and we get a
Pitchfork Bifurcation with two stable orbits of pe-
riod two corresponding to the two stable FIXED POINTS
of / (x). The fixed points of order two must satisfy

#7i + 2 = a?n, SO

X n +2 = rX n+ i(l — X n + l)

= r[rX n (l - X n )][l — TX n {l - X n )]

= r 2 x n (l - x n )(l - rx n + rx n 2 ) = x n . (6)

Logistic Equation

Logistic Equation 1101

Now, drop the n subscripts and rewrite

x{r 2 [1 - x(l + r) + 2rx 2 - rx 3 } - 1} = (7)
x[-r 3 x 3 + 2rV - r 2 (l + r)x + (r 2 - 1)] = (8)

-r 3 x[x - (1 - r _1 )][a: 2 - (1 + r~ 1 )x + r _1 (l + r" 1 )]

= 0. (9)

Notice that we have found the first-order Fixed POINTS
as well, since two iterations of a first-order FIXED POINT
produce a trivial second-order Fixed Point. The true
2-CYCLES are given by solutions to the quadratic part

,( 2 )

§[(1 + r" 1 ) ± y/il + r-iy-ir-iil + r- 1 )]

= l[( 1 + r ~ 1 ) ± \/l + 2r" 1 + r- 2 - 4r-! - 4r~ 2 ]

(10)

= |[(1 + r" 1 ) ± Vl - 2r- 2 - 3r" 2 ]
= i[(l + r- 1 )±r- 1 v /(^-3)(r + l)]-

These solutions are only REAL for r > 3, so this is where
the 2-Cycle begins. Now look for the onset of the 4-
CYCLE. To eliminate the 2- and 1-CYCLES, consider

f 4 (x)

P{x)

0.

(11)

This gives

1,2./ 2 3 4

1+r + (— r — r

- r )x

+ (2r 3 + r 4 + 4r 5 + r 6 + 2r 7 )x 2

+ (-r 3 - 5r 5 - 4r 6 - 5r 7 - 4r 8 - r 9 )x 3

+ (2r 5 + 6r 6 + 4r 7 + 14r 8 + 5r 9 + 3r 10 )x 4

+ (-4r 6 - r 7 - 18r 8 - 12r 9 - 12r 10 - 3r n )x 5

+ (r 6 + 10r 8 + 17r 9 + 18r 10 + ISr 11 + r 12 )x 6

+ (-2r 8 - 14r 9 - 12r 10 - 30r n - 6r 12 )x 7

+ (6r 9 + 3r 10 + 30r n + 15r 12 )x 8

+ (-r 9 - ISr 11 - 20r 12 )z 9 + (3r n + 15r 12 )aj :

-er^+r 12 * 12 .

(12)

The ROOTS of this equation are all IMAGINARY for
r < 1 4- v 7 ^, but two of them convert to REAL roots
at this value (although this is difficult to show ex-
cept by plugging in). The 4- CYCLE therefore starts at
1 + -y/6 = 3.449490 . . .. The BIFURCATIONS come faster
and faster (8, 16, 32, ...), then suddenly break off.
Beyond a certain point known as the Accumulation
Point, periodicity gives way to Chaos.

A table of the CYCLE type and value of r n at which the
cycle 2" appears is given below.

n

cycle (2 n )

r n

1

2

3

2

4

3.449490

3

8

3.544090

4

16

3.564407

5

32

3.568750

6

64

3.56969

7

128

3.56989

8

256

3.569934

9

512

3.569943

10

1024

3.5699451

11

2048

3.569945557

00

ace. pt.

3.569945672

For additional values, see Rasband (1990, p. 23). Note
that the table in Tabor (1989, p. 222) is incorrect, as
is the n = 2 entry in Lauweirer 1991. In the middle of
the complexity, a window suddenly appears with a reg-
ular period like 3 or 7 as a result of MODE LOCKING.
The period 3 BIFURCATION occurs at r = 1 + f l\f 7 i =
3.828427..., as is derived below. Following the 3-
Cycle, the Period Doublings then begin again with
CYCLES of 6, 12, . . .and 7, 14, 28, . . . , and then once
again break off to CHAOS.

A set of n + 1 equations which can be solved to give the

onset of an arbitrary n-cycle (Saha and Strogatz 1995)

is

' X2 = ra?i(l — Xi)

xs = rx 2 (l - x 2 )

x n = rx n -i[l — X n -i)
xi = rx n (l — x n )

.^n: =1 (i-2x fc ) = i.

The first n of these give f(x), f 2 (x), . . . , f n (x), and the
last uses the fact that the onset of period n occurs by a
Tangent Bifurcation, so the nth Derivative is 1.

1102 Logistic Equation

Logistic Equation

For n = 2, the solutions (xi, . . . , x n , r) are (0, 0, ±1)
and (2/3, 2/3, 3), so the desired Bifurcation occurs
at T2 = 3. Taking n = 3 gives

«*[/'(»)] = <*[/»(»)] d^fr)] d[/(x)]

dx d[f 2 (x)] d[f(x)\ dx

= d[f(z)} d[f(y)) d[f(x)}

dz dy dx
= r\l-2z)(l-2y)(l-2x). (14)

Solving the resulting CUBIC EQUATION using computer
algebra gives

Xi

2 5 1^2

63 -7 1 / 3 63-28 1 / 3

9-98 1 / 3 21 \ 9 -T^s 71/3

11/3

1 10 + ^2 /4-2 5/6 2 1 1 _x

25 -28 1/3 -44-2 1/6 7 1/3 _ 2

H « c

X2

(15)
2 2/3 10+ a/2

/ 1 2 5 / 6 \ ,

\63.28V3 + 63*7 1 / 3 y ° 9-7 2 /3 c + 2 1

. 2 5/6 2 • 2 1/3 \ _!

+ (f— -— lc

+

. 71/3 71/3

44 . 2 i/6yi/3 „25-28 1/3
9 '

(16)

^3 = o ^o-i/^ c "l ^i 1 n ^ /q c (17)

3 • 98 1 / 3 21

r = 1 + 2\Z2,

where

3 • 7^3

(18)

c = (-25 + 22\/2 + 3\/3\/ll00\/2 - 1593 ) 1/3 . (19)
Numerically,

an =0.514355...
x 2 =0.956318...
x z =0.159929...

r = 3.828427....

(20)
(21)
(22)
(23)

Saha and Strogatz (1995) give a simplified algebraic
treatment which involves solving

r 3 (l - 2a + 4/3 - 87) = 1, (24)

together with three other simultaneous equations, where

a = x\ 4- X2 + xz

(3 = x±x 2 -r X ± X3 + ^3:3

7 = xix 2 a; 3 .

(25)
(26)
(27)

Further simplifications still are provided in Bechhoeffer
(1996) and Gordon (1996), but neither of these tech-
niques generalizes easily to higher Cycles. Bechhoeffer
(1996) expresses the three additional equations as

lot = 3 + r

(28)

40 = f + 5r _1 + f r~ 2

(29)

8 7 = -i + |r-- 1 + fr- 2 + fr- 3 ,

(30)

giving

2r-7 = 0.

(31)

Gordon (1996) derives not only the value for the onset of
the 3-Cycle, but also an upper bound for the r-values
supporting stable period 3 orbits. This value is obtained
by solving the CUBIC EQUATION

s 3 - lis 2 + 37s- 108 =

for s, then

r — 1 + yfs

(32)

(33)

v^+i

1915
54

1 +

3.841499007543....

+ IV201) 1 / 3 + (W - f ^ / 201 ) 1/3

(34)

The logistic equation has CORRELATION EXPONENT
0.500±0.005 (Grassberger and Procaccia 1983), CAPAC-
ITY Dimension 0.538 (Grassberger 1981), and Infor-
mation Dimension 0.5170976 (Grassberger and Pro-
caccia 1983).

see also BIFURCATION, FEIGENBAUM CONSTANT, LO-
GISTIC Distribution, Logistic Equation — r = 4,
Logistic Growth Curve, Period Three Theorem,
Quadratic Map

References

Bechhoeffer, J. "The Birth of Period 3, Revisited." Math,
Mag. 69, 115-118, 1996.

Bogomolny, A. "Chaos Creation (There is Order in Chaos)."
http : //www . cut-the-knot . com/blue/chaos . html.

Dickau, R. M. "Bifurcation Diagram." http:// forum .
swarthmore.edu/advanced/robertd/bifurcation.html.

Gleick, J. Chaos: Making a New Science. New York: Pen-
guin Books, pp. 69-80, 1988.

Gordon, W. B. "Period Three Trajectories of the Logistic
Map." Math. Mag. 69, 118-120, 1996.

Grassberger, P. "On the Hausdorff Dimension of Fractal At-
tractors." J. Stat. Phys. 26, 173-179, 1981.

Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.

Lauwerier, H. Fractals: Endlessly Repeated Geometrical Fig-
ures. Princeton, NJ: Princeton University Press, pp. 119—
122, 1991.

May, R. M. "Simple Mathematical Models with Very Com-
plicated Dynamics." Nature 261, 459-467, 1976.

Peitgen, H.-O.; Jurgens, H.; and Saupe, D. Chaos and Frac-
tals: New Frontiers of Science. New York: Springer-
Verlag, pp. 585-653, 1992.

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, p. 23, 1990.

Logistic Equation — r = 4

Russell, D. A.; Hanson, J. D.; and Ott, E. "Dimension of

Strange Attractors." Phys. Rev. Let. 45, 1175-1178, 1980.
Saha, P. and Strogatz, S. H. "The Birth of Period Three."

Math. Mag. 68, 42-47, 1995.
Strogatz, S. H. Nonlinear Dynamics and Chaos. Reading,

MA: Addison- Wesley, 1994.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:

An Introduction. New York: Wiley, 1989.
Wagon, S. "The Dynamics of the Quadratic Map." §4.4

in Mathematica in Action. New York: W. H. Freeman,

pp. 117-140, 1991.

Logistic Equation — r = 4

With r = 4, the LOGISTIC EQUATION becomes

Now let

x n+ i = 4x n (l - x n ).

x = sin 2 (§7n/) = |[1 - cos{ny)]

y/x = sin(§7ry)

3/= -siiT^Vs)

dy
dx

It' 1 / 2

7r^/as(l — x)

(1)

(2)
(3)
(4)

(5)

7T y/l-X

Manipulating (2) gives

sin 2 (§7ry„+i)

- 4±[1 - cos(7ry n )]{l - |[1 - i(l - cos(7ry n )]}
= 2[1 - cos(7n/ = 1 - cos 2 (7T2/„)sin 2 (7ry„), (6)

so

§7n/ n +i - ±y n + S7T (7)

y n +i = ±2y n + \s. (8)

But y e [0, 1]. Taking y n e [0, 1/2], then s = and

Vn+i = 2y n .
For t/ e [1/2,1], s = 1 and

2/n+i = 2 - 2y n .
Combining

2/n

2y n for y n e [0, f ]

2-2y n fory n £ [f,l],

which can be written

y n = 1 - 2^

■fc|,

(9)

(10)

(11)

(12)

the Tent Map with (x = 1, so the Natural Invariant
in y is

p(y) = i. (13)

Logit Transformation

Transforming back to x gives
p{x)~-

1103

P(2/0*0) = Z

7T v 7 ! - X '

1 -1/2

1

(14)

(15)

7ry x(l — x)
This can also be derived from

where J(x) is the DELTA FUNCTION.
see also LOGISTIC EQUATION

Logistic Growth Curve

The Population Growth law which arises frequently
in biology and is given by the differential equation

dN
dt

r(K - N)
K '

(1)

where r is the Malthusian Parameter and K is the
so-called Carrying Capacity (i.e., the maximum sus-
tainable population). Rearranging and integrating both
sides gives

[ N dN = r f
J No K-N Kj

( N -K \ r.
N(t) = K + (N - K)e~ Tt/K .

The curve

(2)

(3)
(4)

(5)

* 1 + bq x
is sometimes also known as the logical curve.

see also Gompertz Curve, Life Expectancy, Logis-
tic Equation, Makeham Curve, Malthusian Pa-
rameter, Population Growth

Logistic Map

see Logistic Equation

Logit Transformation

The function

= /<*) = m (rh).

1104

Logos

Lommel Polynomial

This function has an inflection point at x = 1/2, where

Applying the logit transformation to values obtained by
iterating the LOGISTIC EQUATION generates a sequence
of Random Numbers having distribution

i

which is very close to a GAUSSIAN DISTRIBUTION.

References

Collins, J.; Mancilulli, M.; Hohlfeld, R.; Finch, D.; San-

dri, G.; and Shtatland, E. "A Random Number Generator

Based on the Logit Transform of the Logistic Variable."

Computers in Physics 6, 630-632, 1992.
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,

pp. 244-245, 1995.

Logos

A generalization of a HEYTING ALGEBRA which replaces
Boolean Algebra in "intuitionistic" Logic,

see also TOPOS

Lommel Differential Equation

A generalization of the BESSEL DIFFERENTIAL EQUA-
TION (Watson 1966, p. 345),

^S + ^-(^ + ^ = ^ +1

dz 2

y dy
' dz

A further generalization gives
2d 2 y dy

- +Z f z -( Z > + S)y = ±k Z ^.

The solutions are Lommel FUNCTIONS.
see also Lommel FUNCTION

References

Watson, G. N. A Treatise on the Theory of Bessel Functions,

2nd ed. Cambridge, England: Cambridge University Press,

1966.

Lommel Function

There are several functions called "Lommel functions."
One type of Lommel function is the solution to the LOM-
MEL Differential Equation with a Plus Sign,

Here, J v (z) and Y»{z) are BESSEL FUNCTIONS OF THE
First and Second Kinds (Watson 1966, p. 346). If a
minus sign precedes fc, then the solution is

s { ~l = I v (z) / z»K v (z) dz - J v (z) / z»I u (z) dz,

J z J C2

(3)

where K u (z) and h{z) are Modified Bessel FUNC-
TIONS of the First and Second Kinds.

Lommel functions of two variables are related to the
Bessel Function of the First Kind and arise in the
theory of diffraction and, in particular, Mie scattering
(Watson 1966, p. 537),

U n (w,z)=J2(- l ) m {™) Jn+2m(z) (4)

m =

V n (w,z) = J2(- i r(™) n ^ J-n-2m{z). (5)

see also LOMMEL DIFFERENTIAL EQUATION, LOMMEL

Polynomial

References

Chandrasekhar, S. Radiative Transfer. New York: Dover,

p. 369, 1960.
Watson, G. N. A Treatise on the Theory of Bessel Functions,

2nd ed. Cambridge, England: Cambridge University Press,

1966.

Lommel's Integrals

(P 2

I

= x[aJ' n (ax)J n (px) - 0J' n ((3x)J n (ax)]
xJ n (ax)dx~\x [J n (ax) -{- J n -i(oLx)J n +i(ax)],

where J n (x) is a Bessel Function of the First
Kind.

References

Bowman, F. Introduction to Bessel Functions. New York:
Dover, p. 101, 1958.

Lommel Polynomial

y = ks^, u {z),

(1) RmAz)

where

*$(*) = ¥

Y u {z) f z fX J u (z)dz
Jo

-Ju(z) I z yL Y v (z)dz
Jo

(2)

= T{u)(z/2) m 2F 3 (f(l-m),-|m;i/,-m,l-i/-m;z 2 )

[Jv I m(z)J-v + l(z)

2 sin(i/7r)

+ (-l) m J- v - m (z)J v - 1 {z]\,

Long Division

Look and Say Sequence 1105

where T(z) is a Gamma Function, J n (x) is a Bessel
Function of the First Kind, and 2 F 3 {a,b\c,d i e;z)
is a Generalized Hypergeometric Function.

see also Lommel FUNCTION

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary

of Mathematics. Cambridge, MA: MIT Press, p. 1477,

1980.

Long Division

72

726

1 1 123456.

17|123456.

17|123456.

-119

44

-119
44

-119

44

-34

-34

105

105
-102
36

7262.1

7262.11...

7 1 123456.0

17|123456.00

-119
44

-119

44

-34
105

-34
105

-102
36

-102
36

-34
20

-34
20
-17
30

Long division is an algorithm for dividing two numbers,
obtaining the Quotient one Digit at a time. The
above example shows how the division of 123456/17 is
performed to obtain the result 7262.11

see also DIVISION

Long Exact Sequence of a Pair Axiom

One of the Eilenberg-Steenrod Axioms. It states
that, for every pair (X, A), there is a natural long exact
sequence

. . . -> H n (A) ^ H n (X) -+

H n (X,A)^H n . 1 (A) ->...,

where the MAP H n (A) -> H n (X) is induced by the IN-
CLUSION Map A -> X, H n (X) -> H n (X,A) is induced
by the Inclusion Map (X, <p)' -> {X,A). The Map
H n (X,A) -► H n -i(A) is called the Boundary Map.

see also Eilenberg-Steenrod Axioms

Long Prime

see Decimal Expansion

Longitude

The azimuthal coordinate on the surface of a SPHERE
(0 in Spherical Coordinates) or on a Spheroid
(in Prolate or Oblate Spheroidal Coordinates).
Longitude is defined such that 0° = 360°. Lines of con-
stant longitude are generally called Meridians. The
other angular coordinate on the surface of a SPHERE is
called the LATITUDE.

The shortest distance between any two points on a
Sphere is the so-called Great Circle distance, which
can be directly computed from the Latitude and lon-
gitudes of two points.

see also GREAT CIRCLE, LATITUDE, MERIDIAN,

Oblate Spheroidal Coordinates, Prolate Spher-
oidal Coordinates

Look and Say Sequence

The INTEGER Sequence beginning with a single digit in
which the next term is obtained by describing the previ-
ous term. Starting with 1, the sequence would be defined
by "one 1, two Is, one 2 two Is," etc., and the result is
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,
... (Sloane's A005150).

Starting the sequence instead with the digit d for 2 <
d < 9 gives d, Id, Hid, 311d, 13211d, 111312211d,
31131122211^, 1321132132211^, ... The sequences for
d = 2 and 3 are Sloane's A006715 and A006751. The
number of DIGITS in the nth term of both the sequences
for 1 < n < 9 is asymptotic to CX n , where C is a con-
stant and

A= 1.303577269034296...

(Sloane's A014715) is CONWAY'S CONSTANT. A is given
by the largest ROOT of the POLYNOMIAL

= x 71

r 69

53

x ™ _ 2s 68 + 2x 66 + 2a; 65 + x 64 - r 63

^60 _ ^59 + 2x 58 + 5a ,57 + 3^6 „_55

63 62 61

x — x — X

*, — a, ~r ***; -r u^u ~r u*, — 2x — lOx
3a .« _ 2x 52 + 6x Sl + 6x 50 + ^49 + ^48 _ ^47

_ 7x *G _ &,« _ 8cc 44 + 1Qx 43 + 6x 42 + 8;c 41 _ ^40

- 12a; 39 + 7x 38 - 7x 37 + 7a; 36 - 3x 34 + a; 35 + 10a; 33
+ x 32 - 6a; 31 - 2x 3Q - 10a; 29 - 3a; 28 + 2x 27 + 9a; 26

o 25 , -i A 24 23 ~ 21 . n 20 Q 19 i 18

- Sx + 14a; — 8x - 7x +9x - 3x -Ax

- 10a: 17 - 7a: 16 + 12a; 15 + 7x 14 + 2a; 13 -

-4a; u -2a; 10 -5a; 9 +x 7 -7x 6

+ 7a; 5 - 4x 4 + 12a; 3 - 6x 2 + 3a; - 6.

12a; 1

In fact, the constant is even more general than this, ap-
plying to all starting sequences (i.e., even those starting
with arbitrary starting digits), with the exception of 22,
a result which follows from the COSMOLOGICAL THE-
OREM. Conway discovered that strings sometimes fac-
tor as a concatenation of two strings whose descendants

1106 Loop (Algebra)

Lorentz Tensor

never interfere with one another. A string with no non-
trivial splittings is called an "element," and other strings
are called "compounds." Every string of Is, 2s, and 3s
eventually "decays" into a compound of 92 special ele-
ments, named after the chemical elements.

see also Conway's Constant, Cosmological Theo-
rem

References

Conway, J. H. "The Weird and Wonderful Chemistry of Au-
dioactive Decay." Eureka, 5-18, 1985.

Conway, J. H. "The Weird and Wonderful Chemistry of
Audioactive Decay." §5.11 in Open Problems in Com-
munications and Computation. (Ed. T. M. Cover and
B. Gopinath). New York: Springer- Verlag, pp. 173-188,
1987.

Conway, J. H. and Guy, R. K. "The Look and Say Sequence."
In The Book of Numbers. New York: Springer- Verlag,
pp. 208-209, 1996.

Sloane, N. J. A. Sequences A005150/M4780, A006715/
M2965, and A6751/M2052 in "An On-Line Version of the
Encyclopedia of Integer Sequences."

Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 13-14, 1991.

Loop (Algebra)

A nonassociative Algebra (and Quasigroup) which
has a single binary operation.

Loop Gain

The loop gain is usually assigned a value between 0.1
and 0.5. The CLEAN Algorithm performs better for
extended structures if \x is set to the lower part of this
range. However, the time required for the CLEAN Al-
gorithm increases rapidly for small //. From Thompson
et at. (1986), the number of cycles needed for one point

source is

ln(SNR)

[cycles] = —

ln(l- 7 )'

see also CLEAN ALGORITHM

References

Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr.

Interferometry and Synthesis in Radio Astronomy. New

York: Wiley, p. 348, 1986.

Loop Space

Let Y x be the set of continuous mappings / : X — > Y.
Then the TOPOLOGICAL SPACE for Y x supplied with a
compact-open topology is called a MAPPING SPACE, and
if y = J is taken as the interval (0, 1), then Y 1 = Q(Y)
is called a loop space (or Space of Closed Paths).

see also MACHINE, MAPPING SPACE, MAY-THOMASON

Uniqueness Theorem

References

Brylinski, J.-L. Loop Spaces, Characteristic Classes and Ge-
ometric Quantization. Boston, MA: Birkhauser, 1993.

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 658, 1980.

Lorentz Group

The Lorentz group is the GROUP L of time-preserving
linear Isometries of Minkowski Space R 4 with the
pseudo-Riemannian metric

dr 2 = -dt 2 -f dx 2 + dy 2 + dz 2 .

It is also the GROUP of ISOMETRIES of 3-D HYPER-
BOLIC Space. It is time-preserving in the sense that the
unit time VECTOR (1, 0, 0, 0) is sent to another Vector
(i, x, y, z) such that t > 0.

A consequence of the definition of the Lorentz group
is that the full GROUP of time-preserving isometries of
Minkowski R 4 is the Direct Product of the group
of translations of R 4 (i.e., IR itself, with addition as the
group operation), with the Lorentz group, and that the
full isometry group of the MINKOWSKI R 4 is a group
extension of Z2 by the product L (g> R .

The Lorentz group is invariant under space rotations
and Lorentz Transformations.

see also Lorentz Tensor, Lorentz Transforma-
tion

References

Arfken, G. "Homogeneous Lorentz Group." §4.13 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 271-275, 1985.

Loop (Graph)

A degenerate edge of a graph which joins a vertex to
itself.

Loop (Knot)

A Knot or HITCH which holds its form rigidly.

References

Owen, P. Knots. Philadelphia, PA: Courage, p. 35, 1993,

Lorentz Tensor

The Tensor in the Lorentz Transformation given

by

7 -7/3 0"

L =

-7/3 7
10
1_

(1)

where beta and gamma are defined by

t-\ m

1

*y —

(3)

V1-/3 2

see also Lorentz Group, Lorentz Transformation

Lorentz Transformation

Lorentz Transformation

A 4-D transformation satisfied by all FOUR- VECTORS

= A'V\

(1)

In the theory of special relativity, the Lorentz trans-
formation replaces the GALILEAN TRANSFORMATION as
the valid transformation law between reference frames
moving with respect to one another at constant VE-
LOCITY. Let x v be the Position Four- Vector with
2° = c£, and let the relative motion be along the x l axis
with Velocity v. Then (1) becomes

Lorenz System

1107

where is called the rapidity,

x± = ict,

(13)

and

tanhO = fi = -
c

(14)

(15)
(16)

sinh 9 = /?7-

where the LORENTZ TENSOR is given by

(2)

Here,

a8 a? M M

AJ A} A^ A\

Ag A? A3 A§

Ag A? A? A|

0=1

c

7

-7/3

7 =

^T 2

'ifi o

0"

7

1

1_

(3)

(4)
(5)

see also Hyperbolic Rotation, Lorentz Group,
Lorentz Tensor

References

Fraundorf, P. "Accel-ID: Frame-Dependent Relativity at
UM-StL." http://www.umsl.edu/-fraundor/altoc.html.

Griffiths, D. J. Introduction to Electrodynamics. Englewood
Cliffs, NJ: Prentice-Hall, pp. 412-414, 1981.

Morse, P. M. and Feshbach, H. "The Lorentz Transforma-
tion, Four- Vectors, Spinors." §1.7 in Methods of Theoreti-
cal Physics, Part I. New York: McGraw-Hill, pp. 93-107,
1953.

Lorentzian Distribution

see Cauchy Distribution

Written explicitly, the transformation between x v and
x ul coordinate is

x ' = y(x° - fix 1 )

(6)

x = 7(0; — fix )

(7)

2' 2
X = X

(8)

™ 3 ' ~ 3

X = X .

(9)

The Determinant of the upper left 2x2 Matrix in
(3) is

D

(7) 2 -(-7/?) 2 =7 2 (l-/? 2 ) = «l, (10)

7

Lorentzian Function

The Lorentzian function is given by

1

L(x) = -

K

7T (x- x Q y + {\vy

Its Full Width at Half Maximum is I\ This function
gives the shape of certain types of spectral lines and is
the distribution function in the Cauchy Distribution.
The Lorentzian function has Fourier Transform

1

2 1

«( x -xoF + Gry

_ -2irikx — rVjfc|

SO

L" x =

(A" 1 )*
(A" 1 )?

(A- ! )8

(A" 1 )?

(A- 1 )!

(A" 1 )?
(A" 1 )?

7 7/3

7/? 7

10

1

(A- X )S
(A- 1 )^
(A" 1 )!
(A" 1 )!

(A- 1 )?" 1

(A- 1 )!

(A" 1 )!

(11)

see also DAMPED EXPONENTIAL COSINE INTEGRAL,

Fourier Transform — Lorentzian Function

Lorenz System

A simplified system of equations describing the 2-D flow
of fluid of uniform depth H, with an imposed tempera-
ture difference AT, under gravity <?, with buoyancy a,
thermal diffusivity k, and kinematic viscosity v. The
full equations are

A Lorentz transformation along the cci-axis can also be
written

rsi'i

X 2 '

Xl

_X4

cosh i sinh

10

10

-zsinhtf cosh#

Xl
X2
X3
X4

, (12)

9 /■t?2j.\ 9 ^ 9 it,* n

di> d

dT

-^£(vV) + ,v'(vV) + ^ (i)

dT
dt

dx dz

dTdip
dz dx

dx

1108 Lorenz System

Lotka-Volterra Equations

Here, ip is the "stream function," as usual defined such

that

dip _ dip

dx ' dx

(3)

In the early 1960s, Lorenz accidentally discovered the
chaotic behavior of this system when he found that, for
a simplified system, periodic solutions of the form

ij) = Vo sin (^r) sin [^ J (4)

^ = ^ocos(^)sin(5) (5)

grew for Rayleigh numbers larger than the critical value,
Ra > Ra c . Furthermore, vastly different results were
obtained for very small changes in the initial values, rep-
resenting one of the earliest discoveries of the so-called
Butterfly Effect.

Lorenz included the following terms in his system of
equations,

X = -011 oc convective intensity (6)

Y = Tn oc AT between descending and

ascending currents (7)

Z = T02 oc A vertical temperature profile from

linearity, (8)

and obtained the simplified equations

X = a(Y- X)
Y = -XZ + rX
Z^XY - bZ,

(9)
(10)

(11)

The Critical Points at (0, 0, 0) correspond to no
convection, and the CRITICAL POINTS at

(^/b(r-l)^b(r-l),r-l)

and

(-^/b(r-l),-y/b(r-l),r-l)

(15)

(16)

correspond to steady convection. This pair is stable only

if

__ a(a + b 4- 3)

cr-b-l

(17)

which can hold only for POSITIVE r if a > 6+1.
The Lorenz attractor has a CORRELATION EXPONENT
of 2.05 ± 0.01 and CAPACITY DIMENSION 2.06 ± 0.01
(Grassberger and Procaccia 1983). For more details,
see Lichtenberg and Lieberman (1983, p. 65) and Tabor
(1989, p. 204).

see also Butterfly Effect, Rossler Model

References

Gleick, J. Chaos: Making a New Science. New York: Pen-
guin Books, pp. 27-31, 1988.

Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.

Lichtenberg, A. and Lieberman, M. Regular and Stochastic
Motion. New York: Springer- Verlag, 1983.

Lorenz, E. N. "Deterministic Nonperiodic Flow." J. Atmos.
Sci. 20, 130-141, 1963.

Peitgen, H.-O.; Jiirgens, H.; and Saupe, D. Chaos and Frac-
tals: New Frontiers of Science. New York: Springer-
Verlag, pp. 697-708, 1992.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, 1989.

Lorraine Cross

see GAULLIST CROSS

where

Prandtl number

Ra

(12)

r = = normalized Rayleigh number (13)

Ra c

b~

1 + a 2
Lorenz took b = 8/3 and a

geometric factor.
10.

(14)

Lotka-Volterra Equations

An ecological model which assumes that a population
x increases at a rate dx = Axdt, but is destroyed at a
rate dx — —Bxydt. Population y decreases at a rate
dy = —Cydt, but increases at dy — Dxydt, giving the
coupled differential equations

dx

— = Ax- Bxy

at

dy

dt

= ~Cy + Dxy.

Critical points occur when dx/dt = dy/dt = 0, so

A - By =

-C + Dx^ 0.

The sole STATIONARY POINT is therefore located at
{x,y) = (C/D,A/B).

Low-Dimensional Topology

Loxodrome

1109

Low-Dimensional Topology

Low-dimensional topology usually deals with objects
that are 2-, 3-, or 4-dimensional in nature. Properly
speaking, low-dimensional topology should be part of
Differential Topology, but the general machin-
ery of Algebraic and Differential Topology gives
only limited information. This fact is particularly no-
ticeable in dimensions three and four, and so alternative
specialized methods have evolved.

see also ALGEBRAIC TOPOLOGY, DIFFERENTIAL TO-
POLOGY, Topology

Lowenheimer-Skolem Theorem

A fundamental result in MODEL THEORY which states
that if a countable theory has a model, then it has a
countable model. Furthermore, it has a model of every
Cardinality greater than or equal to tt (Aleph-0).
This theorem established the existence of "nonstandard"
models of arithmetic*

see also Aleph-0 (N ), Cardinality, Model Theory

References

Chang, C. C. and Keisler, H. J. Model Theory, 3rd enL ed.
New York: Elsevier, 1990.

Lower Bound

see Greatest Lower Bound

Lower Denjoy Sum

see Lower Sum

Lower Integral

The limit of a Lower Sum, when it exists, as the Mesh
Size approaches 0.

see also Lower Sum, Riemann Integral, Upper In-
tegral

Lower Limit

Let the least term h of a SEQUENCE be a term which is
smaller than all but a finite number of the terms which
are equal to h. Then h is called the lower limit of the
Sequence.

A lower limit of a SERIES

is said to exist if, for every e > 0, \S n — h\ < e for
infinitely many values of n and if no number less than h
has this property.

see also LIMIT, UPPER LIMIT

References

Bromwich, T. J. Fa and MacRobert, T. M. "Upper and Lower
Limits of a Sequence." §5.1 in An Introduction to the The-
ory of Infinite Series, 3rd ed. New York: Chelsea, p. 40
1991.

Lower Sum

For a given function f(x) over a partition of a given
interval, the lower sum is the sum of box areas f(xl)Axk
using the smallest value of the function f(xl) in each
subinterval Axk-

see also LOWER INTEGRAL, RlEMANN INTEGRAL, UP-
PER Sum

Lower- Trimmed Subsequence

The lower-trimmed subsequence of x = {x n } is the se-
quence V(x) obtained by subtracting 1 from each x n
and then removing all 0s. If x is a FRACTAL SEQUENCE,
then V(x) is a FRACTAL SEQUENCE, If z is a SIGNA-
TURE Sequence, then V(x) = x.

see also Signature Sequence, Upper-Trimmed Sub-
sequence

References

Kimberling, C. "Fractal Sequences and Interspersions." Ars
Combin. 45, 157-168, 1997.

Lowest Terms Fraction

A FRACTION p/q for which (p,q) = 1, where (p,q) de-
notes the Greatest Common Divisor.

Loxodrome

A path, also known as a Rhumb Line, which cuts a
Meridian on a given surface (usually a Sphere, in
which case the loxodrome is also called a SPHERICAL
HELIX) at any constant ANGLE but a RIGHT ANGLE.
The loxodrome is the path taken when a compass is kept
pointing in a constant direction. It is not the shortest
distance between two points.

see also Great Circle

lower lim S n = lim S n

1110 Lozenge

Lozenge

A Parallelogram whose Acute Angles are 45°,

see also DIAMOND, PARALLELOGRAM, QUADRILAT-
ERAL, Rhombus

Lozenge Method

A method for constructing MAGIC SQUARES of Odd or-
der.
see also MAGIC SQUARE

Lozi Map

A 2-D map similar to the Henon Map which is given
by the equations

X n +l = 1 - a\x n \ +Vn
2/n+l = 0X n .

see also Henon Map

References

Dickau, R. M. "Lozi Attractor." http://www.prairienet .

org/-pops/lozi,html.
Peitgen, H.-O.; Jiirgens, H.; and Saupe, D. §12.1 in Chaos

and Fractals: New Frontiers of Science. New York:

Springer- Verlag, p. 672, 1992.

LU Decomposition

A procedure for decomposing an N x N matrix A into
a product of a lower Triangular Matrix L and an
upper Triangular Matrix U,

LU = A.

(1)

Written explicitly for a 3 x 3 Matrix, the decomposition
is

hi

1

hi

h2

hi

h2

J33.

till

U12

Ul3~

U22

W23

=

^33.

All Al2 Al3

fl21 A22 A23

_fl31 A32 A33 _

(2)

hiun I11U12 hiu\z

hiU\\ I21U22 4- I22U22 hiU\z + £22^23

hlUn h\U\2 + h2U22 hlUlZ + ^32^23 + ^33^23

an fli2 «i3

«21 A22 A23
.fl31 ^32 A33 .

This gives three types of equations

i < j li\Uij + li2U2j + . . . + liiUij = Ciij

i = j h\Uij + U2U2J + - - - + UiUjj = aij

i > j hiuij + li2U 2 j + . . . + hjUjj = a^.

(3)

(4)
(5)
(6)

Lucas Correspondence Theorem

This gives N 2 equations for N 2 + N unknowns (the
decomposition is not unique), and can be solved using
Crout's Method. To solve the Matrix equation

Ax = (LU)x = L(Ux) = b,

(7)

first solve Ly = b for y. This can be done by forward
substitution

yi =

Hi

bi — 2> hj

Vi

(9)

i=i

for i — 2, . . . , N. Then solve Ux = y for x. This can
be done by back substitution

Xn =

UNN

Xi

I

(10)

(ii)

\ j = i+l /

for i = JV-1, ..., 1.

see also Cholesky Decomposition, QR Decomposi-
tion, Triangular Matrix

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "LU Decomposition and Its Applications." §2.3
in Numerical Recipes in FORTRAN: The Art of Scientific
Computing, 2nd ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 34-42, 1992.

Lucas Correspondence

The correspondence which relates the HANOI Graph to
the Isomorphic Graph of the Odd Binomial Coef-
ficients in Pascal's Triangle, where the adjacencies
are determined by adjacency (either horizontal or diag-
onal) in Pascal's Triangle. The proof that the cor-
respondence is given by the LUCAS CORRESPONDENCE
Theorem.

see also Binomial Coefficient, Hanoi Graph, Pas-
cal's Triangle

References

Poole, David G. "The Towers and Triangles of Professor

Claus (or, Pascal Knows Hanoi)." Math. Mag. 67, 323-

344, 1994.

Lucas Correspondence Theorem

Let p be Prime and

r = r m p m + . . . + np + r (0 < n < p) (1)
k = k^™ + . . . + feip + fc (0 < ki < p), (2)

then

(0-5(2)

(mod p).

(3)

This is proved in Fine (1947).

References

Fine, N. J. "Binomial Coefficients Modulo a Prime." Amer.
Math. Monthly 54, 589-592, 1947.

Lucas-Lehmer Residue

Lucas-Lehmer Residue

see Lucas-Lehmer Test

Lucas-Lehmer Test

A MERSENNE Number M p is prime IFF M p divides
Sp_2, where so = 4 and

Si = si-! 2 - 2(mod 2 P - 1)

(1)

for i > 1. The first few terms of this series are 4, 14,
194, 37634, 1416317954, ... (Sloane's A003010). The
remainder when s p -2 is divided by M p is called the
Lucas-Lehmer Residue for p. The Lucas-Lehmer
Residue is Iff M p is Prime. This test can also be
extended to arbitrary Integers.

A generalized version of the Lucas-Lehmer test lets

n
3 = 1

with qj the distinct PRIME factors, and f3j their respec-
tive Powers. If there exists a Lucas Sequence U u
such that

GCr>(U (N+1)/qj ,N) = l (3)

for j = 1, . . . , n and

U N+ i = (mod N) ,

(4)

then iV is a PRIME. The test is particularly simple for
Mersenne Numbers, yielding the conventional Lucas-
Lehmer test.

see also Lucas Sequence, Mersenne Number,
Rabin-Miller Strong Pseudoprime Test

References

Sloane, N. J. A. Sequence A003010/M3494 in "An On-Line
Version of the Encyclopedia of Integer Sequences,"

Lucas' Married Couples Problem

see Married Couples Problem

Lucas Number

The numbers produced by the V recurrence in the Lu-
CAS Sequence with (P,Q) — (1,-1) are called Lucas
numbers. They are the companions to the Fibonacci
Numbers F n and satisfy the same recurrence

L n — Ln-l + £n-2.

(1)

where Li = 1, L 2 = 3. The first few are 1, 3, 4, 7, 11,
18, 29, 47, 76, 123, . . . (Sloane's A000204).

In terms of the FIBONACCI NUMBERS,

Ln = Fn — 1 + Fn+1-

(2)

Lucas Number 1111

The analog of Binet's Formula for Lucas numbers is

-(^H 1 ^ 1 )"

Another formula is

Ln = [4> n ],

(3)

(4)

where <f> is the Golden Ratio and [x] denotes the Nint
function. Given L n ,

Ln+l —

L n (l + \/5) + 1

where [^J is the FLOOR FUNCTION,

L n — L n —\L

'n + l

5(-l)"

and

/ ^ Lk = L n L n+ i — 2.

(5)

(6)

(7)

Let p be a Prime > 3 and fc be a Positive Inte-
ger. Then L 2p k ends in a 3 (Honsberger 1985, p. 113).
Analogs of the Cesaro identities for FIBONACCI NUM-
BERS are

fc=o v 7

fc=0

(8)
(9)

where (]j) is a Binomial Coefficient.

L n \Fm (L n Divides F m ) Iff n Divides into m an Even
number of times. L n \L m IFF n divides into m an Odd
number of times. 2 n L n always ends in 2 (Honsberger
1985, p. 137).

Defining

D n

3

i

•

•

i

1

i

•

•

i

1

i -

■

i

1 •

-

-

• 1

i

.

• i

1

Ln

+1

(10)

gives

D n = D n -i + D n -2 (11)

(Honsberger 1985, pp. 113-114).

The number of ways of picking a set (including the
Empty Set) from the numbers 1, 2, ..., n without
picking two consecutive numbers (where 1 and n are
now consecutive) is L n (Honsberger 1985, p. 122).

1112 Lucas Polynomial

Lucas Pseudoprime

The only SQUARE NUMBERS in the Lucas sequence are
1 and 4, as proved by John H. E. Cohn (Alfred 1964).
The only TRIANGULAR Lucas numbers are 1, 3, and 5778
(Ming 1991). The only Lucas Cubic Number is 1. The
first few Lucas PRIMES L n occur for n = 2, 4, 5, 7, 8,
11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313,
353, . . . (Dubner and Keller 1998, Sloane's A001606).

see also Fibonacci Number

References

Alfred, Brother U. "On Square Lucas Numbers." Fib. Quart.
2, 11-12, 1964.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 94-101, 1987.

Brillhart, J.; Montgomery, P. L.; and Solverman, R. D. "Ta-
bles of Fibonacci and Lucas Factorizations." Math. Corn-
put 50, 251-260 and S1-S15, 1988.

Brown, J. L. Jr. "Unique Representation of Integers as Sums
of Distinct Lucas Numbers." Fib. Quart. 7,243-252,1969.

Dubner, H. and Keller, W. "New Fibonacci and Lucas
Primes." Math. Comput 1998.

Guy, R. K. "Fibonacci Numbers of Various Shapes." §D26 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 194-195, 1994.

Hoggatt, V. E. Jr. The Fibonacci and Lucas Numbers.
Boston, MA: Houghton Mifflin, 1969.

Honsberger, R. "A Second Look at the Fibonacci and Lucas
Numbers." Ch. 8 in Mathematical Gems HI. Washington,
DC: Math. Assoc. Arner., 1985.

Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/
lucas.Z.

Ming, L. "On Triangular Lucas Numbers." Applica-
tions of Fibonacci Numbers, Vol. 4 (Fd. G. E. Bergum,
A. N. Philippou, and A. F. Horadam). Dordrecht, Nether-
lands: Kluwer, pp. 231-240, 1991.

Sloane, N. J. A. Sequences A000692/M2341 and A001606/
M0961 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Lucas Polynomial

The w Polynomials obtained by setting p(x) = x and

q(x) — 1 in the LUCAS POLYNOMIAL SEQUENCE. The
first few are

x 2 + 2

Fi(x) = x

F 2 (x)

F 3 {x) =r3z 3 + 3;c

F 4 (x) = z 4 + 4x 2 + 2

F 5 (x) = x 5 + bx 3 + $x.

The corresponding W POLYNOMIALS are called FI-
BONACCI Polynomials. The Lucas polynomials satisfy

Lucas Polynomial Sequence

A pair of generalized POLYNOMIALS which generalize the
Lucas Sequence to Polynomials is given by

where

wk = A k ( x )[a«(x)-(-l) k b n (x)}
A(x)

w k n (x) = A k (x)[a n (x) + (-l)V(x)],

a(x) H- b(x) = p(x)
a(x)b(x) — —q(x)

a(x) - b(x) = vV(z) +4q(x) = A(x)
(Horadam 1996). Setting n = gives

w k (x) = A k (x)[l + (-!)%

giving

WS(x) =
w° (x) = 2.

p(x) q(x) Polynomial 1

Polynomial 2

(i)

(2)

(3)
(4)
(5)

(6)
(7)

(8)
(9)

When k = 1,

W^(x) = w° n (x) = w n (x) (10)

W*(x) = A 2 (x)W°(x) = A 2 (x)W n (x). (11)
Special cases are given in the following table.

Lucas L n (x)
Pell-Lucas Q n (x)
Jacobsthal-Lucas j n (x)
Fermat-Lucas f n (x)

x 1 Fibonacci F n (x)

2x 1 PellP n (a;)

1 2x Jacobsthal J n (x)

3x -2 Fermat F n (x)

2x -1 Chebyshev U n -i(x) Chebyshev 2T n (x)

see also Lucas Sequence

References

Horadam, A. F. "Extension of a Synthesis for a Class of Poly-
nomial Sequences." Fib. Quart 34, 68-74, 1996.

Lucas Pseudoprime

When P and Q are Integers such that D = P 2 - 4Q ^
0, define the Lucas Sequence {U k } by

where the L n s are LUCAS Numbers.

see also Fibonacci Polynomial, Lucas Number, Lu-
cas Polynomial Sequence

U k =

b k

a — b

for k > 0, with a and b the two ROOTS of x 2 — Px + Q =
0. Then define a Lucas pseudoprime as an Odd COM-
POSITE number n such that n{Q, the JACOBI Symbol
(D/n) = — 1, and n\U n +i.

Lucas Sequence

Lucas Sequence 1113

There are no EVEN Lucas pseudoprimes (Bruckman
1994). The first few Lucas pseudoprimes are 705, 2465,
2737, 3745, . . . (Sloane's A005845).
see also EXTRA STRONG LUCAS PSEUDOPRIME, LUCAS

Sequence, Pseudoprime, Strong Lucas Pseudo-
prime

References

Bruckman, P. S. "Lucas Pseudoprimes are Odd." Fib. Quart

32, 155-157, 1994.
Ribenboim, P. "Lucas Pseudoprimes (lpsp(P, Q))." §2.X.B

in The New Book of Prime Number Records, 3rd ed. New

York: Springer- Verlag, p. 129, 1996.
Sloane, N. J. A. Sequence A005845/M5469 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Lucas Sequence

Let P, Q be Positive Integers. The Roots of

x 2 - Px + Q =

where

D = P Z

■4Q,

The first few values are therefore

U (P,Q) =

Ux(P,Q) = l
V (P,Q) = 2
V 1 (P,Q) = P.

The sequences

U(P,Q) = {U n (P,Q):n>l}
V(P,Q) = {V n (P,Q):n>l}

(1)

(2)
(3)

(4)

a + b = P

(5)

ab={(P 2 -D) = Q

(6)

a - 6 = Vd.

(7)

Then define

n in

U n {P,Q) = ±—±-

a — o

(8)

V n {P,Q) = a n +b n .

(9)

(10)

(11)

(12)
(13)

(14)

(15)

are called Lucas sequences, where the definition is usu-
ally extended to include

U-i =

a ' 1 -b~

a — b

-1
ab

1_

Q'

For (P,Q) = (1,-1), the U n are the Fibonacci Num-
bers and V n are the LUCAS NUMBERS. For (F, Q) =
(2,-1), the Pell Numbers and Pell-Lucas numbers are
obtained. (P, Q) = (1,-2) produces the JACOBSTHAL
Numbers and Pell-Jacobsthal Numbers.

The Lucas sequences satisfy the general RECURRENCE

Relations

Um-\-n —

m+n •Lm+n

a - b

(a m -& m )(a n -f-6 n )

a n 6 n (a m " n -6 m_n )

a — b

a — b

UmVn — a n b n Um-n

(17)

m+n — a +0

= (a m + b m )(a n + b n ) - a n b n (a m - n + b m ~ n )
= V m V n -a n b n V m -n. (18)

Taking n = 1 then gives

U m (P,Q) = PU m -i{P,Q) - QU m -2{P,Q) (19)
V m (P,Q) = PV m -i{P,Q) - QV m -2{P,Q). (20)

Other identities include

u 2n = u n v n

U 2 n+1 = Un+lVn — Q

V 2n = V n 2 - 2(ab) n = V n 2 - 2Q n
V 2n +i = V n+1 V n - PQ n .

(21)
(22)
(23)
(24)

These formulas allow calculations for large n to be de-
composed into a chain in which only four quantities must
be kept track of at a time, and the number of steps
needed is ~ lgn. The chain is particularly simple if n
has many 2s in its factorization.

The Us in a Lucas sequence satisfy the CONGRUENCE

if

where

Up^-Hp-iD/p)] = ( mod P n )

GCD(2QcD,p) = 1,

P 2 - 4Q 2 = c 2 D.

(25)

(26)

(27)

(16)

This fact is used in the proof of the general LUCAS-

Lehmer Test.

see also FIBONACCI NUMBER, JACOBSTHAL NUMBER,

Lucas-Lehmer Test, Lucas Number, Lucas Poly-
nomial Sequence, Pell Number, Recurrence Se-
quence, Sylvester Cyclotomic Number

References

Dickson, L. E. "Recurring Series; Lucas' u n , v n ." Ch. 17 in
History of the Theory of Numbers, Vol. 1: Divisibility and
Primality. New York: Chelsea, pp. 393-411, 1952.

Ribenboim, P. The Little Book of Big Primes. New York:
Springer-Verlag, pp. 35-53, 1991.

1114 Lucas's Theorem

Ludwig's Inversion Formula

Lucas's Theorem

The primitive factors Q n (#, y) of x n + y n can be written
in the form

Qn{x } y) = U 2 (x,y)±nxyV 2 {x,y)

for SQUAREFREE n where U and V are HOMOGENEOUS
Polynomials with the sign chosen according to

{ + for n = 4/ + 1

for n = 4Z + 3
either for n = 4/ + 2.

Lucky Number

Write out all the ODD numbers: 1, 3, 5, 7, 9, 11, 13, 15,

17, 19, The first Odd number > 1 is 3, so strike

out every third number from the list: 1, 3, 7, 9, 13, 15,
19, .... The first ODD number greater than 3 in the list
is 7, so strike out every seventh number: 1, 3, 7, 9, 13,
15, 21, 25, 31, ....

Numbers remaining after this procedure has been car-
ried out completely are called lucky numbers. The first
few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, . . . (Sloane's
A000959). Many asymptotic properties of the Prime
Numbers are shared by the lucky numbers. The asymp-
totic density is 1/ In N, just as the Prime Number The-
orem, and the frequency of Twin PRIMES and twin
lucky numbers are similar. A version of the GOLDBACH
Conjecture also seems to hold.

It therefore appears that the Sieving process accounts
for many properties of the PRIMES.

see also GOLDBACH CONJECTURE, LUCKY NUMBER OF

Euler, Prime Number, Prime Number Theorem,
Sieve

References

Gardner, M. "Mathematical Games: Tests Show whether a
Large Number can be Divided by a Number from 2 to 12."
Sci. Amer. 207, 232, Sep. 1962.

Gardner, M. "Lucky Numbers and 2187." Math. Intell. 19,
26, 1997.

Guy, R. K. "Lucky Numbers." §C3 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 108-109, 1994.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, pp. 100-102, 1988.

Peterson, I. "MathTrek: Martin Gardner's Luck Num-
ber." http://www.sciencenevs.org/snjirc97/9_6_97/
mathland.htm.

Sloane, N. J. A. Sequence A000959/M2616 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Ulam, S. M. A Collection of Mathematical Problems. New
York: Interscience Publishers, p. 120, 1960.

Wells, D. G. The Penguin Dictionary of Curious and Inter-
esting Numbers. London: Penguin, p. 32, 1986.

Lucky Number of Euler

A number p such that the Prime-Generating Poly-
nomial

2

n — n + p

is Prime for n — 0, 1, . . . , p — 2. Such numbers are
related to the Complex Quadratic Field in which
the Ring of Integers is factorable. Specifically, the
Lucky numbers of Euler (excluding the trivial case p =
3) are those numbers p such that the QUADRATIC Field
Q(V1 - 4 p) has Class Number 1 (Rabinowitz 1913,
Le Lionnais 1983, Conway and Guy 1996).

As established by Stark (1967), there are only nine num-
bers -d such that h(-d) = 1 (the Heegner Numbers
-2, -3, -7, -11, -19, -43, -67, and -163), and of
these, only 7, 11, 19, 43, 67, and 163 are of the re-
quired form. Therefore, the only Lucky numbers of
Euler are 2, 3, 5, 11, 17, and 41 (Le Lionnais 1983,
Sloane's A014556), and there does not exist a better
Prime-Generating Polynomial of Euler's form.

see also Class Number, Heegner Number, Prime-
Generating Polynomial

References

Conway, J. H. and Guy, R. K. "The Nine Magic Discrimi-
nants." In The Book of Numbers. New York: Springer-

Verlag, pp. 224-226, 1996.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

pp. 88 and 144, 1983.
Rabinowitz, G. "Eindeutigkeit der Zerlegung in Primzahlfak-

toren in quadratischen Zahlkorpern." Proc. Fifth Internat.

Congress Math. (Cambridge) 1, 418-421, 1913.
Sloane, N. J. A. Sequence A014556 in "An On-Line Version

of the Encyclopedia of Integer Sequences."
Stark, H. M. "A Complete Determination of the Complex

Quadratic Fields of Class Number One." Michigan Math.

J. 14, 1-27, 1967.

LUCY

A nonlinear DECONVOLUTION technique used in decon-
volving images from the Hubble Space Telescope before
corrective optics were installed.

see also CLEAN Algorithm, Deconvolution, Max-
imum Entropy Method

Ludolph's Constant

see Pi

Ludwig's Inversion Formula

Expresses a function in terms of its Radon Trans-
form,

f(x,y) = K- 1 (nf)(x,y)

1 1 f°° £W)(P.°0 , A
— _ I . dp d a

77 27T J_ 00 x cos a + y sin a — p

see also Radon Transform

Lukacs Theorem

Lukacs Theorem

Let p(x) be an mth degree POLYNOMIAL which is NON-
NEGATIVE in [—1,1]- Then p(x) can be represented in
the form

/ [A(x)} 2 + (1 - x 2 )[B(x)} 2 for m even

\ (1 + x)[C(x)] 2 + (1 - x)[D(x)} 2 for m odd,

where A(x), B(x) t C(x), and D(z) are REAL POLYNO-
MIALS whose degrees do not exceed m.

References

Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 4, 1975.

Lune (Plane)

Liiroth's Theorem

1115

A figure bounded by two circular ARCS of unequal
Radii. Hippocrates of Chios Squared the above left
lune, as well as two others, in the fifth century BC. Two
more SQUARABLE lunes were found by T. Clausen in the
19th century (Dunham 1990 attributes these discoveries
to Euler in 1771). In the 20th century, N. G. Tscheba-
torew and A. W. Dorodnow proved that these are the
only five squarable lunes (Shenitzer and Steprans 1994).
The left lune above is squared as follows,

Ahalf small circle — 2^1 /K J ~~ 4 1

A h

Aiu

-•^quarter big circle
12 12

^triangle

1« 2

^half small circle ^lens — ^
-^triangle ,

so the lune and TRIANGLE have the same AREA. In the
right figure, A\ + A 2 = Aa-

References

Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1
in Journey Through Genius: The Great Theorems of
Mathematics. New York: Wiley, pp. 1-20, 1990.

Heath, T. L. A History of Greek Mathematics. New York:
Dover, p. 185, 1981.

Pappas, T. "Lunes." The Joy of Mathematics. San Carlos,
CA: Wide World Publ./Tetra, pp. 72-73, 1989.

Shenitzer, A. and Steprans, J. "The Evolution of Integra-
tion." Amer. Math. Monthly 101, 66-72, 1994.

Lune (Solid)

A geometric figure consisting of two TRIANGLES at-
tached to opposite sides of a SQUARE.
see also SQUARE, TRIANGLE

Lune (Surface)

A sliver of the surface of a Sphere of Radius r cut out
by two planes through the azimuthal axis with Dihe-
dral Angle 0. The Surface Area of the lune is

5 - 2r 2 <9,

which is just the area of the Sphere times 0/(2ir).
see also Lune (Plane), Sphere

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 130, 1987.

Lunule

see Lune (Plane)

Liiroth's Theorem

If x and y are nonconstant rational functions of a param-
eter, the curve so defined has Genus 0. Furthermore, x
and y may be expressed rationally in terms of a param-
eter which is rational in them.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 246, 1959.

For the above lune,

see also Annulus, Arc, Circle, Lune (Surface)

1116

Lusin's Theorem

Lyapunov Characteristic Exponent

Lusin's Theorem

Let f(x) be a finite and Measurable Function in
(—00,00), and let e be freely chosen. Then there is a
function g(x) such that

1. g(x) is continuous in (—00,00),

2. The Measure of {x : f(x) ^ g(x)} is < e,

3. M(\g\ ] R 1 )<M(\f\;R 1 ) y

where M(f; S) denotes the upper bound of the aggregate
of the values of f(P) as P runs through all values of S.

References

Kestelman, H. §4.4 in Modern Theories of Integration, 2nd
rev. ed. New York: Dover, pp. 30 and 109-112, 1960.

LUX Method

A method for constructing Magic Squares of Singly
Even order n > 6.

see also MAGIC SQUARE

Lyapunov Characteristic Exponent

The Lyapunov characteristic exponent [LCE] gives the
rate of exponential divergence from perturbed initial
conditions. To examine the behavior of an orbit around
a point X*(£), perturb the system and write

X{t) = X'(t) + U{t),

(1)

have initial conditions (x',2/') = (#o + <ix,j/o +dy). The
distance between trajectories at iteration k is then

dk

(x -x ,y ~yo)

(5)

and the mean exponential rate of divergence of the tra-
jectories is denned by

<ri

lim

k— s-oo k

Ml)-

(6)

For an n-dimensional phase space (Map), there are n
Lyapunov characteristic exponents o~\ > <T2 > . . . > c n .
However, because the largest exponent o~\ will dominate,
this limit is practically useful only for finding the largest
exponent. Numerically, since dk increases exponentially
with &, after a few steps the perturbed trajectory is no
longer nearby. It is therefore necessary to renormalize
frequently every t steps. Defining

Vkr =

dkr

do

one can then compute

(7)

(8)

where U(t) is the average deviation from the unper-
turbed trajectory at time t. In a CHAOTIC region, the
LCE a is independent of X*(0). It is given by the OSED-
elec Theorem, which states that

<n = lim ln|U(t)|.

t—tQQ

(2)

For an n-dimensional mapping, the Lyapunov charac-
teristic exponents are given by

lim In I A, (AT) I

(3)

for i — 1, . . . , n, where Aj is the LYAPUNOV CHARAC-
TERISTIC Number.

One Lyapunov characteristic exponent is always 0, since
there is never any divergence for a perturbed trajec-
tory in the direction of the unperturbed trajectory. The
larger the LCE, the greater the rate of exponential di-
vergence and the wider the corresponding SEPARATRIX
of the Chaotic region. For the Standard Map, an
analytic estimate of the width of the CHAOTIC zone by
Chirikov (1979) finds

SI = Be'

(4)

Since the Lyapunov characteristic exponent increases
with increasing X, some relationship likely exists con-
necting the two. Let a trajectory (expressed as a Map)
have initial conditions (xo,yo) and a nearby trajectory

Numerical computation of the second (smaller) Lya-
punov exponent may be carried by considering the evo-
lution of a 2-D surface. It will behave as

(<7l+<T 2 )t

(9)

so (72 can be extracted if a\ is known. The process may
be repeated to find smaller exponents.

For Hamiltonian Systems, the LCEs exist in additive
inverse pairs, so if a is an LCE, then so is —a. One
LCE is always 0. For a 1-D oscillator (with a 2-D phase
space), the two LCEs therefore must be a\ — ui = 0, so
the motion is QUASIPERIODIC and cannot be CHAOTIC.
For higher order HAMILTONIAN SYSTEMS, there are al-
ways at least two LCEs, but other LCEs may enter
in plus-and-minus pairs I and —L If they, too, are both
zero, the motion is integrable and not CHAOTIC. If they
are Nonzero, the Positive LCE / results in an expo-
nential separation of trajectories, which corresponds to
a CHAOTIC region. Notice that it is not possible to have
all LCEs Negative, which explains why convergence of
orbits is never observed in Hamiltonian Systems.

Now consider a dissipative system. For an arbitrary n-
D phase space, there must always be one LCE equal
to 0, since a perturbation along the path results in no
divergence. The LCEs satisfy ^\ o~i < 0. Therefore, for
a 2-D phase space of a dissipative system, cr-i = 0, cr 2 <
0. For a 3-D phase space, there are three possibilities:

1. (Integrable): ai = 0,<t 2 = 0,cr 3 < 0,

Lyapunov Characteristic Number

Lyapunov's Second Theorem 1117

2. (Integrable): ai = 0,<J2,cr 3 < 0,

3. (Chaotic): <n = 0, <r 2 > 0,0-3 < -cr 2 < 0.

see also Chaos, Hamiltonian System, Lyapunov
Characteristic Number, Osedelec Theorem

References

Chirikov, B. V. "A Universal Instability of Many-
Dimensional Oscillator Systems." Phys. Rep. 52,264-379,
1979.

Lyapunov Characteristic Number

Given a LYAPUNOV CHARACTERISTIC EXPONENT (T iy
the corresponding Lyapunov characteristic number A;
is denned as

Xi = e ai . (1)

For an n- dimensional linear MAP,

Xn+i — MX n .

(2)

The Lyapunov characteristic numbers Ai, ..., A n are
the EIGENVALUES of the Map Matrix. For an arbitrary
Map

X n +l = fl(x n ,Vn) (3)

y n +i = h{x n ,y n ), (4)

the Lyapunov numbers are the EIGENVALUES of the limit

lim [J(x n ,y n )J(x n -i,y n -i) • • • J(zi,2/i)] ,

where J(x,y) is the JACOBIAN
J(x,y) =

(5)

dfi(x,y) dfi(x,y)
dx dy

9f2(x,y) df 2 (x,y)
dx dy

(6)

If Aj — for all i, the system is not CHAOTIC. If A ^
and the Map is Area-Preserving (Hamiltonian),
the product of Eigenvalues is 1.

see also Adiabatic Invariant, Chaos, Lyapunov
Characteristic Exponent

Lyapunov Condition

If the third MOMENT exists for a DISTRIBUTION of x»
and the LEBESGUE INTEGRAL is given by

r n 3 = J]

\x\*dFi{x),

then if

lim — = 0,

n— >-co S n

the Central Limit Theorem holds,
see also Central Limit Theorem

Lyapunov Dimension

For a 2-D MAP with cr 2 > a\ ,

^Lya = 1 ,

where cr n are the Lyapunov Characteristic Expo-
nents.

see also Capacity Dimension, Kaplan- Yorke Con-
jecture

References

Frederickson, P.; Kaplan, J. L.; Yorke, E. D.; and Yorke, J. A.

"The Liapunov Dimension of Strange Attractors." J. Diff.

Eq. 49, 185-207, 1983.
Nayfeh, A. H. and Balachandran, B. Applied Nonlinear

Dynamics: Analytical, Computational, and Experimental

Methods. New York: Wiley, p. 549, 1995.

Lyapunov's First Theorem

A Necessary and Sufficient condition for all the
Eigenvalues of a Real n x n matrix A to have Neg-
ative Real Parts is that the equation

A T V + VA=-I

has as a solution where V is an n x n matrix and (x, Vx)
is a positive definite quadratic form.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1122, 1979.

Lyapunov Function

A function which is continuous, nonnegative, and has
continuous PARTIAL DERIVATIVES. The existence of a
Lyapunov function guarantees the NONLINEAR STABIL-
ITY of a Fixed Point.

References

Jordan, D. W. and Smith, P. Nonlinear Ordinary Differential
Equations. Oxford, England: Clarendon Press, p. 283,
1977.

Lyapunov's Second Theorem

If all the Eigenvalues of a Real Matrix A have Real
Parts, then to an arbitrary negative definite quadratic
form (x, Wx) with x = x(i) there corresponds a positive
definite quadratic form (x, Vx) such that if one takes

dx .
then (x, Wx) and (x, Wx) satisfy

^(x,Vx) = (x,Wx).

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1122, 1979.

1118 Lyndon Word Lyons Group

Lyndon Word

A Lyndon word is an aperiodic notation for representing
a Necklace.

see also DE BRUIJN SEQUENCE, NECKLACE

References

Ruskey, F. "Information on Necklaces, Lyndon Words, de

Bruijn Sequences." http://sue . esc .uvic . ca/~cos/inf /

neck/Necklacelnfo.html.
Sloane, N. J. A. Sequence A001037/M0116 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Lyons Group

The Sporadic Group Ly.

see also SPORADIC GROUP

References

Wilson, R. A. "ATLAS of Finite Group Representation."
http : //for .mat . bham. ac . uk/ atlas /Ly .html.

M-Estimate

M

M-Estimate

A Robust Estimation based on maximum likelihood

argument.

see also L-Estimate, ^-Estimate

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Robust Estimation." §15.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 694-700, 1992.

Mac Lane's Theorem

A theorem which treats constructions of FIELDS of
Characteristic p.

see also CHARACTERISTIC (FIELD), FIELD
Machin's Formula

±7r = 4tan" 1 (f)-tan" 1 (^).

There are a whole class of MACHlN-LlKE FORMULAS

with various numbers of terms (although only four such

formulas with only two terms). The properties of these

formulas are intimately connected with COTANGENT

identities.

see also 196-Algorithm, Gregory Number, Mach-

in-Like Formulas, Pi

Machin-Like Formulas

Machin-like formulas have the form

m cot 1 u + n cot x v = \ kit,

(i)

where u, v, and k are POSITIVE INTEGERS and m and
n are NONNEGATIVE INTEGERS. Some such FORMU-
LAS can be found by converting the Inverse Tangent
decompositions for which c n ^ in the table of Todd
(1949) to Inverse Cotangents. However, this gives
only Machin-like formulas in which the smallest term is
±1.

Maclaurin-like formulas can be derived by writing

and looking for a^ and Uk such that

}] ah cot" 1 Uk = |7r, (3)

cot z = — - In f
2%

Machin-Like Formulas 1119

Machin-like formulas exist Iff (4) has a solution in In-
tegers. This is equivalent to finding Integer values
such that

(l-i) k (u + i) m (v + i) n (5)

is Real (Borwein and Borwein 1987, p. 345). An equiv-
alent formulation is to find all integral solutions to one
of

1 + x 2 = 2y n (6)

l + x 2 =y n

for n = 3, 5, . . . .

There are only four such FORMULAS,

i7r = 4tan- 1 (i)-tan- 1 ( 5 i 5 )
i» = tan- 1 (i) + tan- l (|)
. |7r = 2tan -1 (|)-tan _1 (i)

I)

i7r = 2tan _1 (|)-htan *(£),

(7)

(8)

(9)

(10)

(11)

known as MACHIN'S FORMULA, EULER'S MACHIN-LlKE

Formula, Hermann's Formula, and Hutton's For-
mula. These follow from the identities

/5_-H\ 4 /239j-i\ _1
U-J V239-J

(12)
(13)

iK^r-** 4

(4)

Machin-like formulas with two terms can also be gener-
ated which do not have integral arc cotangent arguments
such as Euler's

|7r = 5tan- 1 (^) + 2tan- 1 (4) (16)

(Wetherfield 1996), and which involve inverse SQUARE
Roots, such as

f = » ta -(^) + U.-(i). (H)

Three-term Machin-like formulas include GAUSS'S
Machin-Like Formula

\tt = 12 cot" 1 18 + 8 cot" 1 57 - Scot" 1 239, (18)

Strassnitzky's Formula

±tt = cot" 1 2 + cot" 1 5 + cot" 1 8, (19)

1120

Machin-Like Formulas

Machin-Like Formulas

and the following,

\k = 6 cot" 1 8 + 2 cot" 1 57 + cot" 1 239 (20)

|?r = 4 cot" 1 5 - 1 cot" 1 70 + cot" 1 99 (21)

|tt = 1 cot" 1 2 + 1 cot" 1 5 + cot" 1 8 (22)

±tt = 8 cot" 1 10 - 1 cot" 1 239 - 4 cot" 1 515 (23)

\tt = 5 cot" 1 7 + 4 cot" 1 53 + 2 cot" 1 4443. (24)

The first is due to St0rmer, the second due to Ruther-
ford, and the third due to Dase.

Using trigonometric identities such as

cot" 1 a; = 2cot~ 1 (2z) - cot _1 (4a; 3 + 3z), (25)

it is possible to generate an infinite sequence of Machin-
like formulas. Systematic searches therefore most often
concentrate on formulas with particularly "nice" prop-
erties (such as "efficiency").

The efficiency of a FORMULA is the time it takes to cal-
culate 7r with the POWER series for arctangent

7r = a\ cot(6i) + ai cot(&2) + ■ ■ ■ ,

(26)

and can be roughly characterized using Lehmer's "mea-
sure" formula

The number of terms required to achieve a given preci-
sion is roughly proportional to e, so lower e- values cor-
respond to better sums. The best currently known effi-
ciency is 1.51244, which is achieved by the 6-term series

\k = 183 cot" 1 239 + 32 cot" 1 1023 - 68 cot" 1 5832
+12 cot" 1 110443 - 12 cot" 1 4841182

-100 cot" 1 6826318 (28)

discovered by C.-L. Hwang (1997). Hwang (1997) also
discovered the remarkable identities

\n - Pcot" 1 2 - Mcot" 1 3 + Lcot" 1 5 + K cot" 1 7
+(JV + K + L - 2M + 3P - 5) cot" 1 8

+ (2N + M-P + 2-L) cot" 1 18
-(2P -3-M + L + K-N) cot" 1 57 - iVcot" 1 239,

(29)

where K, L, M, N, and P are Positive Integers, and

Itt = (JV+2) cot" 1 2-N cot" 1 3-(JV+l) cot" 1 N. (30)

The following table gives the number N(n) of Machin-
like formulas of n terms in the compilation by Wether-
field and Hwang. Except for previously known identities
(which are included), the criteria for inclusion are the
following:

1. first term < 8 digits: measure < 1.8.

2. first term = 8 digits: measure < 1.9.

3. first term = 9 digits: measure < 2.0.

4. first term =10 digits: measure < 2.0.

n

N{n)

mine

1

1

2

4

1.85113

3

106

1.78661

4

39

1.58604

5

90

1.63485

6

120

1.51244

7

113

1.54408

8

18

1.65089

9

4

1.72801

10

78

1.63086

11

34

1.6305

12

188

1.67458

13

37

1.71934

14

5

1.75161

15

24

1.77957

16

51

1.81522

17

5

1.90938

18

570

1.87698

19

1

1.94899

20

11

1.95716

21

1

1.98938

Total

1500

1.51244

see also Euler's Machin-Like Formula, Gauss's
Machin-Like Formula, Gregory Number, Her-
mann's Formula, Hutton's Formula, Inverse Co-
tangent, Machin's Formula, Pi, Stormer Num-
ber, Strassnitzky's Formula

References

Arndt, J. "Arctan Formulas." http://jjj.spektracom.de/

jjf .dvi.
Arndt, J. "Big ArcTan Formula Bucket." http://jjj.

spektracom. de/f ox . dvi.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 347-

359, 1987.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in

Analytic Number Theory and Computational Complexity.

New York: Wiley, 1987.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.

61, 67-98, 1988.
Conway, J. H. and Guy, R. K. The Book of Numbers. New

York: Springer- Verlag, pp. 241-248, 1996.
Hwang, C.-L. "More Machin-Type Identities." Math. Gaz.,

120-121, March 1997.
Lehmer, D. H. "On Arccotangent Relations for 7r." j4mer.

Math. Monthly 45, 657-664, 1938.
Lewin, L. Poly logarithms and Associated Functions. New

York: North-Holland, 1981.
Lewin, L. Structural Properties of Poly logarithms. Provi-
dence, RI: Amer. Math. Soc, 1991.
Nielsen, N. Der Euler'sche Dilogarithms. Leipzig, Germany:

Halle, 1909.
St0rmer, C. "Sur l'Application de la Theorie des Nombres

Entiers Complexes a la Solution en Nombres Rationels Xi,

Machine

Maclaurin Series

1121

£2, • ■ ■ , ci, c 2) . . . , k de l'Equation. . . ." Arc/iiu /or Math-
ematik og Naturvidenskab B 19, 75-85, 1896.

Todd, J. "A Problem on Arc Tangent Relations." Amer.
Math. Monthly 56, 517-528, 1949.
^ Weisstein, E. W. "Machin-Like Formulas." http://www.
astro . Virginia . edu/-eww6n/math/notebooks/
MachinFormulas .m.

Wetherfield, M. "The Enhancement of Machin's Formula by
Todd's Process." Math. Gaz. 80, 333-344, 1996.

Wetherfield, M. "Machin Revisited." Math. Gaz., 121-123,
March 1997.

Williams, R. "Arctangent Formulas for Pi." http://www.
cacr.caltech.edu/~roy/pi.formulas. html. [Contains er-
rors].

Machine

A method for producing infinite LOOP Spaces and spec-
tra.
see also GADGET, LOOP SPACE, MAY-THOMASON

Uniqueness Theorem, Turing Machine

Mackey's Theorem

Let E and F be paired spaces with S a family of ab-
solutely convex bounded sets of F such that the sets of
S generate F and, if Bi,B2 6 5, there exists a B3 € S
such that Bs D B\ and B3 D B 2 . Then the dual space
of Es is equal to the union of the weak completions of
XB, where A > and B e S.

see also Grothendieck's Theorem

References

Iyanaga, S. and Kawada, Y. (Eds.). "Mackey's Theorem."
§407M in Encyclopedic Dictionary of Mathematics. Cam-
bridge, MA: MIT Press, p. 1274, 1980.

Maclaurin- Bezout Theorem

The Maclaurin-Bezout theorem says that two curves of
degree n intersect in n 2 points, so two Cubics intersect
in nine points. This means that n(n + 3)/2 points do
not always uniquely determine a single curve of order n.

see also Cramer-Euler Paradox

Maclaurin- Cauchy Theorem

If f(x) is POSITIVE and decreases to 0, then an EuLER
Constant

7/ = lim

71— >00

71 pn

f{x) dx
can be defined. If f(x) = 1/x, then

71— >-00 \ ^ ■* K

, fc = l

where 7 is the EULER-MASCHERONI Constant.

Maclaurin Integral Test

see Integral Test

Maclaurin Polynomial

see Maclaurin Series

Maclaurin Series

A series expansion of a function about 0,

2!

3!

+ ... + - — Ms +■

(i)

named after the Scottish mathematician Maclaurin.
Maclaurin series for common functions include

1

= 1 + x + x 2 + x 3 + x 4 + x 5 + . . .

cn(x, k 2 ) = 1 - ±x 2 + i (1 + 4k 2 )x 4 +
± x 2 + ± x * - ±x Q -

2! ^ 4! 6! ' * *

-i< x < 1 (2)

(3)

cosx

1

—00 < x < 00 (4)

cos X

x 6^ 40^ 112 ^

-1< x < 1 (5)
coshx = l + §x 2 + ^^ 4 + t~x 6 + io^2oZ 8 + ... (6)

cosh-^l + x) = v / 2^(l - \x + ^x 2 - ^x 3 + . . .)(7)

COtX — X 3 X 45 ^ 945^ 4725

X nAt^X An^tnX . . . \P )

cot 1 X ■■

„ 7T X ~~\~ n X k X ~\~ y 9 — ' ' '

(9)

iar 5 + is- 5 -i*- 7 + S*-' + ---( 10 )

J -a; 7 +

cothx = x * + |x — ^rx 4 -f gfgic 5

coth _1 (l + x) = \ ln2- §lnx + |x - ^x 2 + .

— 1 i 1 _i_ 7 3 ,

6*" ' 360^ ' 15120
3

^-z 5 + ,

CSCllX — X 6 X ^~360 X "^15120

^-x 5 + ...

csch x = In 2 — In x + \x

4" 32

i i, 2 2 , i ^2

35 "T" QfiS

dn(x, r )x = i - £fcV + ^r (4 + r )x 4 + . .

erf x = -1= (2a; - fa: 3 + ±x 5 - £x 7 + . . .)

(11)
(12)
(13)
(14)
(15)
(16)

(17)

l+X+ix 2 +ix 3 + ^X 4 + .

-oo < x < oo (18)

a/3 a(a + 1)0(0 + 1) 2

2 F 1 (a,/3 l7 ;x) = l+— x+^^- Tly

x* + . . .

(19)

ln(l + x) = x — \x + |x

^x 4 +

*m

-1< x < 1 (20)

2x+ |x 3 + fx 5 + fx 7 + ...

sec Z = 1 + |x 2 + ^a; 4

720 •*-

-1 < x < 1 (21)
^- 8 + --- (22)

i 1 _ 1 2 - 5_ 4_ 61 6 , 277 8 ,

seen x — i 2 x -t- 2 4 :c 720 * "^ 8064 ^ "^ • * ■
sech" 1 x ~ In 2 — lncc— \x — -^x — ...

(23)
(24)

sm x = x

J_ 3 1 1_ 5 _ _1_„7 1

3!*^ "■" 5!^ 7! "■" '

1122

Maclaurin Series

Maclaurin Trisectrix

-oo < x < oo (25)
sin"* x = a+^ + ^ B + ^ x' + ^x 9 + . . . (26)
sinhx = x + §x 3 + ^x 5 + skox 7 + ^sqX 9 + . . . (27)
sinh" 1 x = x - |x 3 + ^x 5 - ^x 7 + ^x 9 -... (28)
sn(x, k 2 ) = jt(l + & 2 )z 3 + |[ (1 + 14A; 2 + fc 4 )x 5 + . . . (29)
tanx = x + \x z + A^ 5 + ^x 7 + ^x 9 + . . . (30)

Maclaurin Trisectrix

3~ ' 15

3 - +i-

tan * x = x — |x + ^x 5 — ^x +.

315 B

-Kx<l (31)
tan _1 (l + x) = |tt + §x - \x 2 + ^x 3 + ±x 5 + . . . (32)

(33)
(34)

tannic — x q*^ — > T^*^ ^i^*^

tanh -1 x = x + §x 3 + |x 5 + i^x 7 + §x 9 +

The explicit forms for some of these are

oo

71 =

COS X = > -

(2n)!

:x = 2

n + lrt/rtSTl — 1

(-l) n+1 2(2-

(2n)!

l)^2n 2n-l

oo

e = > —x
^■— ' n!

^ (-i) n+1

n=l

oo

n = l

E (-l) n £ , 2n 2n
(2n)!

n=0
_ V^ (-1) 2»-l

-2^(2n-l)!

secx

tanx =

_ Y> (-l) n+1 2 2n (2 2n -l)ff 2n

n=l

(2n)!

tan

J -E

-ir

2n-l

OO

tanh" 1 * = V — ^— as 3 "- 1 ,
^ 2n - 1

(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)

where B n are BERNOULLI NUMBERS and E n are Euler

Numbers.

see also Alcuin's Sequence, Lagrange Expansion,
Legendre Series, Taylor Series

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, pp. 299-300, 1987.

A curve first studied by Colin Maclaurin in 1742. It was
studied to provide a solution to one of the GEOMETRIC
Problems of Antiquity, in particular Trisection of
an ANGLE, whence the name trisectrix. The Maclaurin
trisectrix is an ANALLAGMATIC CURVE, and the origin
is a Crunode.

The Maclaurin trisectrix has CARTESIAN equation

y =

x 2 (x + 3a)

or the parametric equations

t 2 -3

y = a

t 2 -hi

t(t 2 - 3)

t 2 + l '

(1)

(2)
(3)

The Asymptote has equation x = a, and the center
of the loop is as (2a, 0). Draw a line L with Angle
3a through the loop center. Then the angle made by
the line through the center and point of intersection of
L with the trisectrix is a. The trisectrix is sometimes
defined instead as

x(x 2 + y 2 ) = a(y 2 -3x 2 )

2 _ x 2 (3a + x)
a — x

__ 2asin(3(9)

(4)
(5)
(6)

sin(20)
Another form of the equation is the POLAR EQUATION

r = asec(~0),

(7)

where the origin is inside the loop and the crossing point
is on the Negative x-Axis.

The tangents to the curve at the origin make angles of
±60° with the x-Axis. The Area of the loop is

^.loop — 3v3a ,

(8)

and the Negative x-intercept is (-3a, 0) (MacTutor
Archive) .

Maclaurin Trisectrix Inverse Curve

Maeder's Owl Minimal Surface 1123

The Maclaurin trisectrix is the PEDAL CURVE of the
Parabola where the Pedal Point is taken as the re-
flection of the Focus in the Directrix.

see also Catalan's Trisectrix, Strophoid

References

Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 103-106, 1972.
Lee, X. "Trisectrix." http://www.best.com/-xah/Special

PlaneCurves_dir/Trisectrix_dir/trisectrix.html.
Lee, X. "Trisectrix of Maclaurin." http://www.best . com/-

xah / Special Plane Curves _ dir / TriOf

Maclaurin _ dir / triOf

Maclaurin.html.
MacTutor History of Mathematics Archive, "Trisectrix of

Maclaurin." http: //www -groups . dcs . st-and .ac.uk/

-hist ory/Curves/Tr isectrix.html.

Maclaurin Trisectrix Inverse Curve

1

The Inverse Curve of the Maclaurin Trisectrix
with Inversion Center at the Negative ^-intercept
is a Tschirnhausen Cubic.

MacMahon's Prime Number of
Measurement

see Prime Number of Measurement

MacRobert's E-Function

E(p;a r : p s : x)

T(a,+i)

Jo

r(pi -cu)r(p2 ~ a 2 )---T(p q - a q )

9 /*oo

x Y[ / A/'*- a ' 4 - 1 (l + A,,)- p '*dA M
p-q-i /.oo

1 +

i/ = 2

e -A pAp a p -l

A 9+2^q+3 * * " A p

(1 + Ai)---(1 + A 3 )a>

dXr,

where r(;z) is the GAMMA FUNCTION and other details

are discussed by Gradshteyn and Ryzhik (1980).

see also Fox's if-FuNCTiON, Meijer's G-Function

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 896-903 and 1071-1072, 1979.

Madelung Constants

The quantities obtained from cubic, hexagonal, etc.,
LATTICE SUMS, evaluated at s = 1, are called Madelung
constants. For cubic LATTICE SUMS, they are expressi-
ble in closed form for Even indices,

62(2) =

-4/3(1)77(1) =

-\\ In2 = -7rln2

(1)

64(2) =

-87,(1)^(0) =

-81n2- 1 = -41n2.

(2)

63(1) is given by BENSON'S FORMULA,

-(.,(1)

£

= 12tt

y/i 2 +j 2 +k 2

m, n=l, 3,

sech 2 (|7r V / m 2 +n 2 ), (3)

where the prime indicates that summation over (0, 0, 0)
is excluded. 63(1) is sometimes called "the" Madelung
constant, corresponds to the Madelung constant for a 3-
D NaCl crystal, and is numerically equal to —1.74756 —

For hexagonal Lattice Sum, /i2(2) is expressible in
closed form as ,

/i 2 (2) = 7rln3v / 3. (4)

see also BENSON'S FORMULA, LATTICE SUM

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.

Buhler, J. and Wagon, S. "Secrets of the Madelung Con-
stant." Mathematica in Education and Research 5, 49-55,
Spring 1996.

Crandall, R. E. and Buhler, J. P. "Elementary Function Ex-
pansions for Madelung Constants." J. Phys. Ser. A: Math,
and Gen. 20, 5497-5510, 1987.

Finch, S. "Favorite Mathematical Constants," http: //www.
mathsoft.com/asolve/constant/mdlung/mdlung.html.

Maeder's Owl Minimal Surface

A Minimal Surface which resembles a Cross-Cap. It
is given by the polar equations

x = 1
y^<Jz

(1)
(2)
(3)

1124 Maehly's Procedure

or the parametric equations

x — r cos — |r 2 cos(2#)

12 .

y = — rsin# — |r sin(20),

4 3/2 /3/i\

Z = gT" ' COS(f 0).

see also Cross-Cap, Minimal Surface

(4)
(5)
(6)

References

Maeder, R. Programming in Mathematical 3rd ed. Reading,
MA: Addison- Wesley, pp. 29-30, 1997.

Maehly's Procedure

A method for finding ROOTS which defines

Pi(x) =

so the derivative is

P'(x)

P(x)

(x — X\) ■ • ■ (x — Xj) '

(1)

p'M

(x — Xi) ■ • ' (x — Xj)

P(x)

(X — Xi) • - - (x — Xj)

J2(x- Xi )-\ (2)

One step of Newton's Method can then be written as

P(Xk)

Xk+1 = Xk

p'(x k ) - p(x k )YlLi( x k - x^- 1

(3)

Mainardi-Codazzi Equations

l-a-TS. + wrt.-rid-rf,

■x — ^- = er 22 + /(r 2 2 - r i2 ) - #ri 2 ,

ov au

(1)

(2)

where e, /, and g are coefficients of the second FUNDA-
MENTAL Form and r£- are Christoffel Symbols of
the Second Kind. Therefore,

de
dv

2l

du

~ ^ (I + g)

= \ G « (I + §)

a(in/) _ r i r 2

H — - 1 - 11 — J- 12

du
5(1*/)

<%

- r 2

— L 22

■ri,

C-

In

-F 2 J

du \^EG
d_ ( In/ \ =

-2ri

-2r'

(3)

(4)

(5)
(6)

(7)
(8)

Magic Constant

where E> F, and t? are coefficients of the first Funda-
mental Form.

References

Gray, A. "The Mainardi-Codazzi Equations." §20.4 in Mod-
ern Differential Geometry of Curves and Surfaces. Boca
Raton, FL: CRC Press, pp. 401-402, 1993.

Green, A. E. and Zerna, W. Theoretical Elasticity f 2nd ed.
New York: Dover, p. 37, 1992.

Magic Circles

A set of n magic circles is a numbering of the intersection
of the n CIRCLES such that the sum over all intersections
is the same constant for all circles. The above sets of
three and four magic circles have magic constants 14 and
39 (Madachy 1979).

see also MAGIC GRAPH, MAGIC SQUARE

References

Madachy, J, S. Madachy's Mathematical Recreations. New
York: Dover, p. 86, 1979.

Magic Constant

The number

n

M 2 (n) = -^2k=±n{n 2 + l)

to which the n numbers in any horizontal, vertical, or
main diagonal line must sum in a MAGIC SQUARE. The
first few values are 1, 5 (no such magic square), 15, 34,
65, 111, 175, 260, ... (Sloane's A006003). The magic
constant for an nth order magic square starting with an
INTEGER A and with entries in an increasing ARITH-
METIC SERIES with difference D between terms is

M 2 (n;A,D) = \n[2a + D{n - 1)]

(Hunter and Madachy 1975, Madachy 1979). In a Pan-
MAGIC SQUARE, in addition to the main diagonals, the
broken diagonals also sum to M2{n).

For a Magic Cube, the magic constant is

M 3 (n) = —2^2,k = |n(n 3 + l) = \n(l+n){n -n + 1).

The first few values are 1, 9, 42, 130, 315, 651, 1204, . . .
(Sloane's A027441).

Magic Cube

Magic Cube 1125

There is a corresponding multiplicative magic constant
for Multiplication Magic Squares.

see also MAGIC CUBE, MAGIC GEOMETRIC CON-
STANTS, Magic Hexagon, Magic Square, Multipli-
cation Magic Square, Panmagic Square

References

Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3
in Mathematical Diversions. New York: Dover, pp. 23-34,
1975.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 86, 1979.

Sloane, N. J. A. Sequences A027441 and A006003/M3849 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."

Magic Cube

An n x n x n 3-D version of the MAGIC SQUARE in
which the n 2 rows, n 2 columns, n 2 pillars (or "files"),
and four space diagonals each sum to a single number
M 3 (n) known as the MAGIC CONSTANT. If the cross-
section diagonals also sum to Mz{n), the magic cube is
called a Perfect Magic Cube; if they do not, the cube
is called a SEMIPERFECT MAGIC Cube, or sometimes an
Andrews Cube (Gardner 1988). A pandiagonal cube
is a perfect or semiperfect magic cube which is magic
not only along the main space diagonals, but also on
the broken space diagonals.

A magic cube using the numbers 1, 2, . . . , n , if it exists,
has Magic Constant

M 3 (n) = -^J2 k " l n ( n3 + 1 ) = \n{n + l){n-n+l).

For n = 1, 2, . . . , the magic constants are 1, 9, 42, 130,
315, 651, . . . (Sloane's A027441).

4

12

26

20

7

15

18

23

1

11

25

6

9

14

19

22

3

17

27

5

10

13

21

8

2

16

24

60

37

12

21

7

26

55

42

57

40

9

24

6

27

54

43

13

20

61

36

50

47

2

31

16

17

64

33

51

46

3

30

56

41

8

25

11

22

59

38

53

44

5

28

10

23

58

39

1

32

49

48

62

35

14

19

4

29

52

45

63

34

15

18

The above semiperfect magic cubes of orders three
(Hunter and Madachy 1975, p. 31; Ball and Coxeter
1987, p. 218) and four (Ball and Coxeter 1987, p, 220)
have magic constants 42 and 130, respectively. There
is a trivial semiperfect magic cube of order one, but no
semiperfect cubes of orders two or three exist. Semiper-
fect cubes of Odd order with n > 5 and DOUBLY EVEN
order can be constructed by extending the methods used

for Magic Squares.

19

497

255

285

432

78

324

162

134

360

106

396

313

219

469

55

306

212

478

64

141

3 67

97

387

303

205

451

33

148

370

128

414

442

92

342

184

5

437

233

267

14

496

226

260

433

83

349

191

336

174

420

66

243

273

31

509

473

59

309

215

102

392

138

364

109

399

129

355

466

52

318

224

116402

160

382

463

45

291

193

229

263

9

491

346

188

438

88

337

179

445

95

23B

272

2

434

466

8

266

236

69

443

181

343

371

145

415

125

208

302

36

450

199

293

43

4 57

380

154

408

118

218

316

54

472

357

135

393

107

79

429

163

321

500

18

288

254

507

25

279

245

72

422

172

330

185

347

85

439

262

232

490

12

48

462

196

290

403

113

383

157

412

122

376

150

39

453

203

297

389

103

361

139

58

476

214

312

276

242

512

30

175

333

67

417

168

326

76

426

283

249

503

21

423

69

331

169

28

506

248

278

381

159

401

115

194

292

46

464

492

10

264

230

87

437

187

345

155

377

119 405

296

198

460

42

65

419

173

335

510

32 274

244

216

310

60

474

363

137

391

101

252

282

24

502

327

165

427

73

34

452

206304

413

127

369

147

183

341

91

441

268 234

488

6

456

38

300

202

123 409

151

373

286

256

498

20

161

323

77

431

395

105

359

133

56

470

220

314

82

436

190

352

493

15

257

227

140

362

104

390

311

213

475

57

29

511

241

275

418

68

334

176

366

144

386

100

209

307

61

479

440

86

348

186

11

489

231

261

289

195

461

47

15B

384

114

404

269

239

481

3

178

340

94

448

471

53

315

217

108

394

136

358

322

164

430

80

253

287

17

499

49

467

221

319

39S

112

354

132

235

265

7

485

344

182

444

90

126

416

146

372

449

35

301

207

96

446

180

333

483

1

271

237

201

299

37 455

374

152

410

124

356

130

400

110

223

317

51

465

501

23

281251

74

428

166

328

259

225

495

13

192

350

84

434

406

120

373

156

41

459

197

295

63 477

211

305

388

98

368

142

170

332

70

424

277

247

505

27

425

75

325

167

22

504

250

284

320

222468

50

131

353

111

397

14 9

375

121

411

298

204

454

40

4

482

240

270

447

93

339

177

246

280

26

508

329

171

421

71

99

385

143

365

480

62

308

210

458

44

294

200

117

407

153

379

351

189

435

81

228

258

16

494

There are no perfect magic cubes of order four (Beeler
et al. 1972, Item 50; Gardner 1988), No perfect magic
cubes of order five are known, although it is known that
such a cube must have a central value of 63 (Beeler et
al. 1972, Item 51; Gardner 1988). No order-six per-
fect magic cubes are known, but Langman (1962) con-
structed a perfect magic cube of order seven. An order-
eight perfect magic cube was published anonymously in
1875 (Barnard 1888, Benson and Jacoby 1981, Gard-
ner 1988). The construction of such a cube is discussed
in Ball and Coxeter (1987). Rosser and Walker redis-
covered the order-eight cube in the late 1930s (but did
not publish it), and Myers independently discovered the
cube illustrated above in 1970 (Gardner 1988). Order 9
and 11 magic cubes have also been discovered, but none
of order 10 (Gardner 1988).

Semiperfect pandiagonal cubes exist for all orders 8n
and all Odd n > 8 (Ball and Coxeter 1987). A perfect
pandiagonal magic cube has been constructed by Planck
(1950), cited in Gardner (1988).

Berlekamp et al (1982, p. 783) give a magic Tesseract.
see also MAGIC CONSTANT, MAGIC GRAPH, MAGIC

Hexagon, Magic Square

References

Adler, A. and Li, S.-Y. R. "Magic Cubes and Prouhet Se-
quences." Amer. Math. Monthly 84, 618-627, 1977.

Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New
York: Dover, 1960.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 216-
224, 1987.

Barnard, F. A. P. "Theory of Magic Squares and Cubes."
Mem. Nat. Acad. Sci. 4, 209-270, 1888.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.

1126 Magic Geometric Constants

Magic Graph

Benson, W. H. and Jacoby, O. Magic Cubes: New Recre-
ations. New York: Dover, 1981.

Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning
Ways, For Your Mathematical Plays, Vol, 2: Games in
Particular. London: Academic Press, 1982.

Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time
Travel and Other Mathematical Bewilderments. New
York: W. H. Freeman, pp. 213-225, 1988.

Hendricks, J. R. "Ten Magic Tesseracts of Order Three." J.
Recr. Math. 18, 125-134, 1985-1986.

Hirayama, A. and Abe, G. Researches in Magic Squares. Os-
aka, Japan: Osaka Kyoikutosho, 1983.

Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3
in Mathematical Diversions. New York: Dover, p. 31,
1975.

Langman, H. Play Mathematics. New York: Hafner, pp. 75-
76, 1962.

Lei, A. "Magic Cube and Hypercube." http://www.cs.ust.
hk/-philipl/magic/mcube2.html.

Madachy, J. S, Madachy 's Mathematical Recreations. New
York: Dover, pp. 99-100, 1979.

Pappas, T. "A Magic Cube." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, p. 77, 1989.

Planck, C. Theory of Path Nasiks. Rugby, England: Pri-
vately Published, 1905.

Rosser, J. B. and Walker, R. J. "The Algebraic Theory of
Diabolical Squares." Duke Math. J. 5, 705-728, 1939.

Sloane, N. J. A. Sequence A027441 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Wynne, B. E. "Perfect Magic Cubes of Order 7." J. Recr.
Math. 8, 285-293, 1975-1976.

Magic Geometric Constants

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Let E be a compact connected subset of d-dimensional
Euclidean Space. Gross (1964) and Stadje (1981)
proved that there is a unique REAL NUMBER a(E) such
that for all x\ , xi , . . . , x n € E, there exists y € E with

n

n ^-^

i>M-y*) 2 = <*(£)-

= 1 \ k=l

The magic constant m(E) of E is defined by

a(E)

(1)

m{E) = —

diam(S) '

(2)

If 7 is a subinterval of the Line and D is a circular Disk
in the Plane, then

m(I) = m(D) = l
If C is a Circle, then

m{C)= - =0.6366.

7T

(5)

(6)

An expression for the magic constant of an ELLIPSE in
terms of its Semimajor and Semiminor Axes lengths
is not known. Nikolas and Yost (1988) showed that for
a Reuleaux Triangle T

0.6675276 < m(T) < 0.6675284.

(7)

Denote the Maximum value of m(E) in n-D space by
M(n). Then

M(l) = \

M (2) : m(T) < M(2) < 2 - ^ 3 < 0,7182336

M(d) :

d+1

< M(d) <

3v/3

[Y{\d)f2 d - 2 V2d

(8)
(9)

T{d-\)y/{d+V^

<

(10)

where

where T(z) is the GAMMA Function (Nikolas and Yost
1988).

An unrelated quantity characteristic of a given MAGIC
Square is also known as a Magic Constant.

References

Finch, S. "Favorite Mathematical Constants." http://www.

mathsoft.com/asolve/constant/magic/magichtnil.
Cleary, J.; Morris, S. A.; and Yost, D. "Numerical

Geometry — Numbers for Shapes." Amer. Math. Monthly

95, 260-275, 1986.
Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Prob-
lems in Geometry. New York: Springer- Verlag, 1994.
Gross, O. The Rendezvous Value of Metric Space. Princeton,

NJ: Princeton University Press, pp. 49-53, 1964.
Nikolas, P. and Yost, D. "The Average Distance Property

for Subsets of Euclidean Space." Arch. Math. (Basel) 50,

380-384, 1988.
Stadje, W. "A Property of Compact Connected Spaces."

Arch. Math. (Basel) 36, 275-280, 1981.

diam(i£) = max A
v } u,vee\

^2{u k -v k ) 2 .

(3)

These numbers are also called Dispersion Numbers
and Rendezvous Values. For any E, Gross (1964)
and Stadje (1981) proved that

I < m(E) < 1.

(4)

Magic Graph

Magic Hexagon

Magic Square 1127

A Labelled Graph with e EDGES labeled with distinct
elements {1, 2, . . . , e} so that the sum of the EDGE
labels at each Vertex is the same. Another type of
magic graph,. such as the PENTAGRAM shown above, has
labelled VERTICES which give the same sum along every
straight line segment (Madachy 1979).
see also Antimagic Graph, Labelled Graph, Magic
Circles, Magic Constant, Magic Cube, Magic
Hexagon, Magic Square

References

Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A

Comprehensive Introduction. San Diego, CA: Academic

Press, 1990.
Heinz, H. "Magic Stars." http://www.geocities.com/Cape

Canaveral/Launchpad/4057/magicstar.htm.
Madachy, J. S, Madachy '$ Mathematical Recreations. New

York: Dover, pp. 98-99, 1979.

Magic Hexagon

An arrangement of close-packed HEXAGONS containing
the numbers 1, 2, . . . , H n = 3n(n — 1) -f 1, where H n
is the nth Hex Number, such that the numbers along
each straight line add up to the same sum. In the above
magic hexagon, each line (those of lengths both 3 and
4) adds up to 38. This is the only magic hexagon of the
counting numbers for any size hexagon. It was discov-
ered by C. W. Adam, who worked on the problem from
1910 to 1957.

see also Hex Number, Hexagon, Magic Graph,
Magic Square, Talisman Hexagon

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.

Cambridge, MA: MIT Artificial Intelligence Laboratory,

Memo AIM-239, Item 49, Feb. 1972.
Gardner, M. "Permutations and Paradoxes in Combinatorial

Mathematics." Sci. Amer. 209, 112-119, Aug. 1963.
Honsberger, R. Mathematical Gems I. Washington, DC:

Math. Assoc. Amer., pp. 69-76, 1973.
Madachy, J. S. Madachy's Mathematical Recreations. New

York: Dover, pp. 100-101, 1979.

Magic Labelling

It is conjectured that every TREE with e edges whose
nodes are all trivalent or monovalent can be given a
"magic" labelling such that the INTEGERS 1, 2, . . . , e
can be assigned to the edges so that the SUM of the three
meeting at a node is constant.

see also Magic Constant, Magic Cube, Magic
Graph, Magic Hexagon, Magic Square

References

Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903-909, 1989.

Magic Number
see Magic Constant

Magic Series

n numbers form a magic series of degree p if the sum of
their kth POWERS is the MAGIC CONSTANT of degree k
for all ke [l,p].
see also Magic Constant, Magic Square

References

Kraitchik,.M. "Magic Series." §7.13.3 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 183-186, 1942.

Magic Square

8 16

3 5 7

4 9 2

16

2

3

13

5

11

10

8

9

7

6

12

4

14

15

1

17

24

1

8

15

23

5

7

14

16

4

6

13

20

22

10

12

19

21

3

11

18

25

2

9

32

29

4

1

24

21

30

31

2

3

22

23

12

9

17

20

28

25

10

11

18

19

26

27

13

16

36

33

5

8

14

15

34

35

6

7

30

39

48

1

10

19

28

38

47

7

9

18

27

29

46

6

8

17

26

35

37

5

14

16

25

34

36

45

13

15

24

33

42

44

4

21

23

32

41

43

3

12

22

31

40

49

2

11

20

64

2

3

61

60

6

7

57

9

55

54

12

13

51

50

16

17

47

46

20

21

43

42

24

40

26

27

37

36

30

31

33

32

34

35

29

28

38

39

25

41

23

22

44

45

19

18

48

49

15

14

52

53

11

10

56

8

58

59

5

4

62

63

1

A (normal) magic square consists of the distinct POSI-
TIVE Integers 1, 2, . . . , n 2 such that the sum of the
n numbers in any horizontal, vertical, or main diagonal
line is always the same MAGIC CONSTANT

M2(n) = ~^k= \n(n + 1).
fc=i

The unique normal square of order three was known
to the ancient Chinese, who called it the Lo Shu. A
version of the order 4 magic square with the numbers
15 and 14 in adjacent middle columns in the bottom
row is called DURER's MAGIC SQUARE. Magic squares
of order 3 through 8 are shown above.

The Magic Constant for an nth order magic square
starting with an INTEGER A and with entries in an in-
creasing Arithmetic Series with difference D between
terms is

M 2 (n; A, D) = \n[2a + D{n - 1)]

(Hunter and Madachy 1975). If every number in a
magic square is subtracted from n 2 + 1, another magic

1128 Magic Square

Magic Square

square is obtained called the complementary magic
square. Squares which are magic under multiplica-
tion instead of addition can be constructed and are
known as Multiplication Magic Squares. In ad-
dition, squares which are magic under both addition
and multiplication can be constructed and are known as
Addition-Multiplication Magic Squares (Hunter
and Madachy 1975).

A square that fails to be magic only because one or
both of the main diagonal sums do not equal the MAGIC
Constant is called a Semimagic Square. If all diag-
onals (including those obtained by wrapping around)
of a magic square sum to the MAGIC Constant, the
square is said to be a PANMAGIC SQUARE (also called
a Diabolical Square or Pandiagonal Square). If
replacing each number rii by its square n 2 produces an-
other magic square, the square is said to be a BlMAGlC
Square (or Doubly Magic Square). If a square is
magic for m, rii 2 , and n* 3 , it is called a TREBLY MAGIC
SQUARE. If all pairs of numbers symmetrically opposite
the center sum to n 2 -f 1, the square is said to be an
Associative Magic Square.

16

17 24

23 j5

A

A r 6 r 13 1

10 12 19

11 18 25

it) z: a a a\

y > / /

8 r 15

r 14 16 ^

7*

D iJ ZU ZZ ^

-y-/v-/

12 19 21 A y

tV

15 17 24 1 8

Kraitchik (1942) gives general techniques of construct-
ing Even and Odd squares of order n. For n Odd, a
very straightforward technique known as the Siamese
method can be used, as illustrated above (Kraitchik
1942, pp. 148-149). It begins by placing a 1 in any lo-
cation (in the center square of the top row in the above
example), then incrementally placing subsequent num-
bers in the square one unit above and to the right. The
counting is wrapped around, so that falling off the top
returns on the bottom and falling off the right returns
on the left. When a square is encountered which is al-
ready filled, the next number is instead placed below the
previous one and the method continues as before. The
method, also called de la Loubere's method, is purpor-
ted to have been first reported in the West when de la
Loubere returned to France after serving as ambassador
to Siam.

A generalization of this method uses an "ordinary vec-
tor" (x,y) which gives the offset for each noncolliding
move and a "break vector" (u,v) which gives the off-
set to introduce upon a collision. The standard Siamese
method therefore has ordinary vector (1, —1) and break
vector (0, 1). In order for this to produce a magic square,
each break move must end up on an unfilled cell. Special
classes of magic squares can be constructed by consider-
ing the absolute sums |u + v|, \(u — x) + (v — j/)|, \u — u|,
and \(u — x) — (v — y)\ = \u + y — x — v\. Call the set
of these numbers the sumdiffs (sums and differences). If
all sumdiffs are Relatively Prime to n and the square
is a magic square, then the square is also a PANMAGIC
SQUARE. This theory originated with de la Hire. The
following table gives the sumdiffs for particular choices
of ordinary and break vectors.

Ordinary Break Sumdiffs
Vector Vector

Magic Panmagic
Squares Squares

(1, -1)

(0,1)

(1,3)

2fc + l

none

(1, -1)

(0,2)

(0,2)

6fc±l

none

(2,1)

(1, -2)

(1, 2, 3, 4)

6fc±l

none

(2,1)

(1, -1)

(0, 1, 2, 3)

6fc±l

6&±1

(2,1)

(1,0)

(0, 1, 2)

2fc + l

none

(2,1)

(1,2)

(0, 1, 2, 3)

6fc±l

none

18

22

24 ^5r 6

20

/ /

r f * ^ v

10 ^11 17 23 A .

/ v/ / / X

b*

/

12

16

/

/'

y

A second method for generating magic squares of Odd
order has been discussed by J. H. Conway under the
name of the "lozenge" method. As illustrated above, in
this method, the Odd numbers are built up along diag-
onal lines in the shape of a DIAMOND in the central part
of the square. The EVEN numbers which were missed
are then added sequentially along the continuation of
the diagonal obtained by wrapping around the square
until the wrapped diagonal reaches its initial point. In
the above square, the first diagonal therefore fills in 1,
3, 5, 2, 4, the second diagonal fills in 7, 9, 6, 8, 10, and
so on.

Magic Square

Magic Square 1129

64

-V

■+

40

32.

^V

49

y\

55

26

34

15

-**

54

-^

27

35

-> fc

fv

61

12

20

37

29

52

60.

^

13

-*

36

28

-V

45

53

7"

51,

43

30

38

19

11

->*■

50

31

N?

39

10

57

16

33.

25

7*7

48

56

^

An elegant method for constructing magic squares of
Doubly Even order n = 4m is to draw xs through
each 4x4 subsquare and fill all squares in sequence.
Then replace each entry a^- on a crossed-off diagonal
by (n 2 + 1) — dij or, equivalently, reverse the order of
the crossed-out entries. Thus in the above example for
n = 8, the crossed-out numbers are originally 1, 4, ... ,
61, 64, so entry 1 is replaced with 64, 4 with 61, etc.

z
u

X

68
1

66

65

96 [ 93

4

1

32

29

60

57

67

94

95

2

3

30

31

58

59

92

I
90

89

20

17

28

25

56

53

64

61

91

1

18

19

I
26

27

I
54

55

I

62

63

16
!

14

13

24

21

49

52

80

77

88

85

15

"" ""I

22

23

■ I

50

I

51

I

78

79

I
86

87

37

40

45

48

76

73

81

84

9

12

38

39

46

47

74

75

82

83

10

11

41

J

43

44

69

72

97

100

5

8

33

36

42

71 1 70

99 1 98

vie

35 1 34

A very elegant method for constructing magic squares
of Singly Even order n = Am + 2 with m > 1 (there is
no magic square of order 2) is due to J. H. Conway, who
calls it the "LUX" method. Create an array consisting
of m + 1 rows of Ls, 1 row of Us, and m — 1 rows of
Xs, all of length n/2 = 2ro + 1. Interchange the middle
U with the L above it. Now generate the magic square
of order 2m + 1 using the Siamese method centered on
the array of letters (starting in the center square of the
top row), but fill each set of four squares surrounding
a letter sequentially according to the order prescribed
by the the letter. That order is illustrated on the left
side of the above figure, and the completed square is
illustrated to the right. The "shapes" of the letters L,
U, and X naturally suggest the filling order, hence the
name of the algorithm.

It is an unsolved problem to determine the number of
magic squares of an arbitrary order, but the number
of distinct magic squares (excluding those obtained by
rotation and reflection) of order n = 1, 2, ... are 1, 0, 1,
880, 275305224, . . . (Sloane's A006052; Madachy 1979,
p. 87). The 880 squares of order four were enumerated

by Frenicle de Bessy in the seventeenth century, and are
illustrated in Berlekamp et al. (1982, pp. 778-783). The
number of 6 x 6 squares is not known.

67

1

43

13

37

61

31

73

7

3

61

19

37

43

31

5

41

7

11

73

29

67

17

23

13

The above magic squares consist only of Primes and
were discovered by E. Dudeney (1970) and A. W. John-
son, Jr. (Dewdney 1988). Madachy (1979, pp. 93-96)
and Rivera discuss other magic squares composed of
Primes.

52

61

4

13

20

29

36

45

14

3

62

51

46

35

30

19

53

60

5

12

21

28

37

44

11

6

59

54

43

38

27

22

55

58

7

10

23

26

39

42

9

8

57

56

41

40

25

24

50

63

2

15

18

31

34

47

16

1

64

49

48

33

32

17

Benjamin Franklin constructed the above 8x8 Pan-
magic Square having Magic Constant 260. Any
half-row or half-column in this square totals 130, and
the four corners plus the middle total 260. In addition,
bent diagonals (such as 52-3-5-54-10-57-63-16) also total
260 (Madachy 1979, p. 87).

1430028159

1480023153

1480028201

1480028213

14S002B171

14B0028129

14B0023141

14B00281B9

1480023133

In addition to other special types of magic squares, a
3x3 square whose entries are consecutive Primes, illus-
trated above, has been discovered by H. Nelson (Rivera).
Variations on magic squares can also be constructed us-
ing letters (either in defining the square or as entries in
it), such as the Alphamagic Square and Templar
Magic Square.

1130 Magic Square

Magic Tour

4 9 2

3 5 7

8 16

4

14

15

1

9

7

6

12

5

11

10

8

16

2

3

13

11

24

7

20

3

4

12

25

8

16

17

5

13

21

9

10

18

1

14

22

23

6

19

2

15

6

32

3

34

35

1

7

11

27

28

8

30

19

14

16

15

23

24

18

20

22

21

17

13

25

29

10

9

26

12

36

5

33

4

2

31

22

47

16

41

10

35

4

5

23

48

17

42

11

29

30

6

24

49

18

36

12

13

31

7

25

43

19

37

38

14

32

1

26

44

20

21

39

8

33

2

27

45

46

15

40

9

34

3

28

8

58

59

5

4

62

63

1

49

15

14

52

53

11

10

56

41

23

22

44

45

19

18

48

32

34

35

29

28

38

39

25

40

26

27

37

36

30

31

33

17

47

46

20

21

43

42

24

9

55

54

12

13

51

50

16

64

2

3

61

60

6

7

57

79

60

52

Various numerological properties have also been associ-
ated with magic squares. Pivari associates the squares
illustrated above with Saturn, Jupiter, Mars, the Sun,
Venus, Mercury, and the Moon, respectively. Attractive
patterns are obtained by connecting consecutive num-
bers in each of the squares (with the exception of the
Sun magic square).

see also Addition-Multiplication Magic Square
Alphamagic Square, Antimagic Square, Asso-
ciative Magic Square, Bimagic Square, Border
Square, Durer's Magic Square, Euler Square,
Franklin Magic Square, Gnomon Magic Square,
Heterosquare, Latin Square, Magic Circles,
Magic Constant, Magic Cube, Magic Hexa-
gon, Magic Labelling, Magic Series, Magic
Tour, Multimagic Square, Multiplication Magic
Square, Panmagic Square, Semimagic Square,
Talisman Square, Templar Magic Square, Tri-
magic Square

References

Abe, G. "Unsolved Problems on Magic Squares." Disc.

Math. 127, 3-13, 1994.
Alejandre, S. "Suzanne Alejandre's Magic Squares." http://

forum. swarthmore . edu/ale jandre/magic . square .html.
Andrews, W. S. Magic Squares and Cubes, 2nd rev. ed. New

York: Dover, 1960.
Ball, W. W. R. and Coxeter, H. S. M. "Magic Squares."

Ch. 7 in Mathematical Recreations and Essays, 13th ed.

New York: Dover, 1987.
Barnard, F. A. P. "Theory of Magic Squares and Cubes."

Memoirs Natl. Acad. Sci. 4, 209-270, 1888.
Benson, W. H. and Jacoby, O. New Recreations with Magic

Squares. New York: Dover, 1976.
Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning

Ways, For Your Mathematical Plays, Vol. 2: Games in

Particular. London: Academic Press, 1982.
Dewdney, A. K. "Computer Recreations: How to Pan for

Primes in Numerical Gravel." Sci. Amer. 259, pp. 120-

123, July 1988.

Dudeney, E. Amusements in Mathematics. New York:
Dover, 1970.

Fults, J. L. Magic Squares. Chicago, IL: Open Court, 1974.

Gardner, M. "Magic Squares." Ch. 12 in The Second Sci-
entific American Book of Mathematical Puzzles & Diver-
sions: A New Selection. New York: Simon and Schuster,
1961.

Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time
Travel and Other Mathematical Bewilderments. New
York: W. H. Freeman, 1988.

Grogono, A. W. "Magic Squares by Grog." http: //www.
grogono . com/magic/.

Heinz, H. "Magic Squares." http : //www. geocities.

com/CapeCanaveral/Launchpad/4057/magicsquare.htm.

Hirayama, A. and Abe, G. Researches in Magic Squares. Os-
aka, Japan: Osaka Kyoikutosho, 1983.

Horner, J. "On the Algebra of Magic Squares, I., II. , and
III." Quart. J. Pure Appl. Math. 11, 57-65, 123-131, and
213-224, 1871.

Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3
in Mathematical Diversions. New York: Dover, pp. 23-34,
1975.

Kanada, Y. "Magic Square Page." http://www.st.rim.or.
jp/-kanada/puzzles /magic -square. html.

Kraitchik, M. "Magic Squares." Ch. 7 in Mathematical
Recreations. New York: Norton, pp. 142-192, 1942.

Lei, A. "Magic Square, Cube, Hypercube." http://www.es.
ust , hk/*philipl /magic /magic . html.

Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in
Madachy 's Mathematical Recreations. New York: Dover,
pp. 85-113, 1979.

Moran, J. The Wonders of Magic Squares. New York: Vin-
tage, 1982.

Pappas, T. "Magic Squares," "The 'Special' Magic Square,"
"The Pyramid Method for Making Magic Squares," "An-
cient Tibetan Magic Square," "Magic 'Line.'," and "A Chi-
nese Magic Square." The Joy of Mathematics. San Carlos,
CA: Wide World Publ./Tetra, pp. 82-87, 112, 133, 169,
and 179, 1989.

Pivari, F. "Nice Examples." http://www.geocities.com/
CapeCanaveral/Lab/3469/examples.html.

Pivari, F. "Simple Magic Square Checker and GIF
Maker." http : //www . geocities . com/CapeCanaveral/

Lab/3469/squaremaker.html.

Rivera, C. "Problems & Puzzles (Puzzles): Magic Squares
with Consecutive Primes." http : //www . sci . net . mx/
-crivera/ppp/puzzJ)03 .htm.

Rivera, C. "Problems & Puzzles (Puzzles): Prime-
Magical Squares." http://www.sci.net.mx/-crivera/
ppp/puzz_004 . htm.

Sloane, N. J. A. Sequence A006052/M5482 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Suzuki, M. "Magic Squares." http://www.pse. che.tohoku.
ac , jp/-msuzuki/MagicSquare . html.
$ Weisstein, E. W. "Magic Squares." http: //www. astro.
Virginia. edu/~e ww6n/math/notebooks /Magic Square s .m.

Magic Star

see Magic Graph

Magic Tour

Let a chess piece make a TOUR on an n x n Chess-
board whose squares are numbered from 1 to n 2 along
the path of the chess piece. Then the TOUR is called a
magic tour if the resulting arrangement of numbers is a
Magic SQUARE. If the first and last squares traversed
are connected by a move, the tour is said to be closed (or

Magic Tour

Majorant 1131

"re-entrant"); otherwise it is open. The MAGIC CON-
STANT for the 8 X 8 CHESSBOARD is 260.

6 ^r

2 %

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Magic Knight's Tours are not possible onnxn boards
for n Odd, and are believed to be impossible for n —
8. The "most magic" knight tour known on the 8x8
board is the SEMIMAGIC SQUARE illustrated in the above
left figure (Ball and Coxeter 1987, p. 185) having main
diagonal sums of 348 and 168. Combining two half-
knights' tours one above the other as in the above right
figure does, however, give a MAGIC SQUARE (Ball and
Coxeter 1987, p. 185).

1M

21*3

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The above illustration shows a 16 x 16 closed magic
Knight's Tour (Madachy 1979).

A magic tour for king moves is illustrated above (Cox-
eter 1987, p. 186).

see also Chessboard, Knight's Tour, Magic
Square, Semimagic Square, Tour

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 185-
187, 1987.

Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 87-89, 1979.

Mahler-Lech Theorem

Let K be a Field of Characteristic (e.g., the ra-
tional Q) and let {u n } be a SEQUENCE of elements of
K which satisfies a difference equation of the form

U n = CQU n + CiU n+ i + . . . + CkU n +k,

where the COEFFICIENTS d are fixed elements of K.
Then, for any c € K, we have either u n — c for only
finitely many values of n, or u n = c for the values of n
in some ARITHMETIC PROGRESSION.

The proof involves embedding certain fields inside the
p-ADic Numbers Q p for some Primer, and using prop-
erties of zeros of Power series over Q p (Strassman'S
Theorem).

see also Arithmetic Progression, p-adic Number,
Strassman's Theorem

Mahler's Measure

For a Polynomial P,

■f

Jo

M(P)=ex P / ln|P(e w )||£.

It is related to Jensen's Inequality.
see also Jensen's Inequality

Major Axis

see Semimajor Axis

Major Triangle Center

A Triangle Center a : /? : 7 is called a ma-
jor center if the Triangle Center Function a =
/(a, 6, c, A, B,C) is a function of ANGLE A alone, and
therefore (3 and 7 of B and C alone, respectively.

see also Regular Triangle Center, Triangle Cen-
ter

References

Kimberling, C. "Major Centers of Triangles.'
Monthly 104, 431-438, 1997.

Amer. Math.

Majorant

A function used to study Ordinary Differential
Equations.

1132

Makeham Curve

Malmsten's Differential Equation

Makeham Curve

The function defined by

V =

ks*b q

which is used in actuarial science for specifying a sim-
plified mortality law. Using s(x) as the probability that
a newborn will achieve age x, the Makeham law (1860)
uses

s(x) — exp( — Ax — m(c x — 1))

forB>0,A>-B,c>l,x> 0.

see also GOMPERTZ CURVE, LIFE EXPECTANCY, LOGIS-
TIC Growth Curve, Population Growth

References

Bowers, N. L. Jr.; Gerber, H. U.; Hickman, J. C; Jones,
D. A.; and Nesbitt, C. J. Actuarial Mathematics, Itasca,
IL: Society of Actuaries, p. 71, 1997.

Makeham, W. M. "On the Law of Mortality, and the Con-
struction of Annuity Tables." JIA 8, 1860.

Malfatti Circles

Three circles packed inside a RIGHT TRIANGLE which
are tangent to each other and to two sides of the Tri-
angle.
see also Malfatti's Right Triangle Problem

Malfatti Points

see Ajima-Malfatti Points

Draw within a given TRIANGLE three Circles, each of
which is Tangent to the other two and to two sides
of the Triangle. Denote the three Circles so con-
structed r^, Ts, and Tc. Then Fa is tangent to AB
and AC, Tb is tangent to BC and BA, and Tc is tan-
gent to AC and BC.

see also Ajima-Malfatti Points, Malfatti's Right
Triangle Problem

References

Dorrie, H. "Malfatti's Problem." §30 in 1 00 Great Problems
of Elementary Mathematics: Their History and Solutions.
New York: Dover, pp. 147-151, 1965.

Forder, H. G. Higher Course Geometry. Cambridge, Eng-
land: Cambridge University Press, pp. 244-245, 1931.

Fukagawa, H. and Pedoe, D. Japanese Temple Geometry
Problems (San Gaku). Winnipeg: The Charles Babbage
Research Centre, pp. 106-120, 1989.

Gardner, M. Fractal Music } HyperCards, and More Mathe-
matical Recreations from Scientific American Magazine.
New York: W. H. Freeman, pp. 163-165, 1992.

Goldberg, M. "On the Original Malfatti Problem." Math,
Mag. 40, 241-247, 1967.

Lob, H. and Richmond, H. W. "On the Solution of Malfatti's
Problem for a Triangle." Proc. London Math. Soc. 2,
287-304, 1930.

Woods, F. S. Higher Geometry. New York: Dover, pp. 206-
209, 1961.

Malliavin Calculus

An infinite-dimensional DIFFERENTIAL CALCULUS on

the Wiener Space. Also called Stochastic Calcu-
lus of Variations.

Malfatti's Right Triangle Problem

Find the maximum total Area of three Circles (of
possibly different sizes) which can be packed inside a
Right Triangle of any shape without overlapping. In
1803, Malfatti gave the solution as three CIRCLES (the
Malfatti Circles) tangent to each other and to two
sides of the TRIANGLE. In 1929, it was shown that the
Malfatti Circles were not always the best solution.
Then Goldberg (1967) showed that, even worse, they are
never the best solution.

see also MALFATTI'S TANGENT TRIANGLE PROBLEM

References

Eves, II. A Survey of Geometry, Vol. 2. Boston; Allyn &

Bacon, p. 245, 1965.
Goldberg, M. "On the Original Malfatti Problem." Math.

Mag. 40, 241-247, 1967.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,

pp. 145-147, 1990.

Malfatti's Tangent Triangle Problem

C

Mallow's Sequence

An Integer Sequence given by the recurrence relation

a(n) = a(a(n — 2)) -f- a(n — a(n — 2))

with a(l) = a(2) = 1. The first few values are 1, 1, 2,
3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, . . .
(Sloane's A005229).

see also Hofstadter-Conway $10,000 Sequence,
Hofstadter's Q-Sequence

References

Mallows, C. "Conway's Challenging Sequence." Amer. Math.

Monthly 98, 5-20, 1991.
Sloane, N. J. A. Sequence A005229/M0441 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Malmsten's Differential Equation

ft . r I ( A m . S \

y +-y =[Az +^)y.

References

Watson, G. N. A Treatise on the Theory of Bessel Functions,

2nd ed. Cambridge, England: Cambridge University Press,

pp. 99-100, 1966.

Maltese Cross

Mandelbrot Set

1133

Maltese Cross

Mandelbrot Set

L^J l^J

An irregular DODECAHEDRON CROSS shaped like a +
sign but whose points flange out at the end: ^. The
conventional proportions as computed on a 5 x 5 grid as
illustrated above.

see also CROSS, DISSECTION, DODECAHEDRON

References

Frederickson, G. "Maltese Crosses." Ch. 14 in Dissections:
Plane and Fancy. New York: Cambridge University Press,
pp. 157-162, 1997.

Maltese Cross Curve

The plane curve with Cartesian equation

/ 2 2\ 2.2

xy(x - y J = x -\-y

and polar equation

cos 9 sin #(cos 2 - sin 2 0)

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 71, 1989.

Malthusian Parameter

The parameter a in the exponential POPULATION
GROWTH equation

Ni(t) = N e at .

see also LIFE EXPECTANCY, POPULATION GROWTH

Mandelbar Set

A Fractal set analogous to the Mandelbrot Set or
its generalization to a higher power with the variable z
replaced by its COMPLEX CONJUGATE z* .

see also Mandelbrot Set

The set obtained by the QUADRATIC RECURRENCE

Zn+i = z n 2 + C, (1)

where points C for which the orbit z = does not tend
to infinity are in the Set. It marks the set of points
in the COMPLEX PLANE such that the corresponding
Julia Set is Connected and not Computable. The
Mandelbrot set was originally called a p MOLECULE by
Mandelbrot.

J. Hubbard and A. Douady proved that the Mandel-
brot set is Connected. Shishikura (1994) proved that
the boundary of the Mandelbrot set is a FRACTAL with
Hausdorff Dimension 2. However, it is not yet known
if the Mandelbrot set is pathwise-connected. If it is
pathwise-connected, then Hubbard and Douady's proof
implies that the Mandelbrot set is the image of a Cir-
cle and can be constructed from a Disk by collapsing
certain arcs in the interior (Douady 1986).

The Area of the set is known to lie between 1.5031 and
1.5702; it is estimated as 1.50659. . . .

Decomposing the COMPLEX coordinate z = x + iy and
zq = a-\- ib gives

y +a

x = x

y = 2xy + 6.

In practice, the limit is approximated by

lim

: lim \z n

<r n

(2)
(3)

(4)

Beautiful computer-generated plots can be created by
coloring nonmember points depending on how quickly
they diverge to r max - A common choice is to define
an Integer called the Count to be the largest n such
that \z n \ < r, where r is usually taken as r = 2, and
to color points of different COUNT different colors. The
boundary between successive COUNTS defines a series
of "Lemniscates," called Equipotential Curves by
Peitgen and Saupe (1988), \L n {C)\ = r which have dis-
tinctive shapes. The first few LEMNISCATES are

L 1 (C) = C (5)

L 2 (C) = C(C + 1) (6)

L 3 (C) -C + (C + C 2 ) 2 (7)

L 4 (C) = C + [C + (C + C 2 ) 2 ] 2 . (8)

1134

Mandelbrot Set

Mandelbrot Set

When written in CARTESIAN COORDINATES, the first
three of these are

r 2 = x 2 + y 2

r 2 = (x 2 +y 2 )[(x + l) 2 +y 2 ]

(9)
(10)

r 2 - ( x 2 + y 2 )(l -f 2x + 5x 2 + 6x 3 + 6z 4 + 4x 5 + x 6

- 3y 2 - 2x2/ 2 + Sx 2 y 2 + 8aV

+ 3a? 4 1/ 2 + 2y 4 + 4xy 4 + 3z 2 y 4 + y 6 ),

(ii)

which are a Circle, an Oval, and a PEAR CURVE. In
fact, the second LemniSCATE Li can be written in terms
of a new coordinate system with x = x — 1/2 as

[(x'-|) 2 +2/ 2 ][(x' + l) 2 +j/ 2 ] = r 2 ) (12)

which is just a Cassini Oval with a = 1/2 and b 2 =
r. The LEMNISCATES grow increasingly convoluted with
higher COUNT and approach the Mandelbrot set as the
COUNT tends to infinity.

The kidney bean-shaped portion of the Mandelbrot set
is bordered by a CARDIOID with equations

4x = 2 cos t — cos(2t)
Ay = 2sint-sin(2t).

(13)
(14)

The adjoining portion is a CIRCLE with center at ( — 1,0)
and RADIUS 1/4. One region of the Mandelbrot set con-
taining spiral shapes is known as Sea Horse Valley
because the shape resembles the tail of a sea horse.

Generalizations of the Mandelbrot set can be con-
structed by replacing z n 2 with z n k or z* & , where A; is a
Positive Integer and z* denotes the Complex Con-
jugate of z. The following figures show the Fractals
obtained for k = 2, 3, and 4 (Dickau). The plots on the
right have z replaced with z* and are sometimes called
"Mandelbar Sets."

*t +

see also Cactus Fractal, Fractal, Julia Set,
lemniscate (mandelbrot set), mandelbar set,
Quadratic Map, Randelbrot Set, Sea Horse Val-
ley

References

Alfeld, P. "The Mandelbrot Set." http://www.math.utah.
edu/-alfeld/math/mandelbrot/mandelbrotl.html.

Branner, B. "The Mandelbrot Set." In Chaos and Fractals:
The Mathematics Behind the Computer Graphics, Proc,
Sympos. Appl. Math., Vol, 39 (Ed. R. L. Devaney and
L. Keen). Providence, RI: Amer. Math. Soc, 75-105, 1989.

Dickau, R. M. "Mandelbrot (and Similar) Sets." http://
forum . swarthmore . edu / advanced / robertd /
mandelbrot .html.

Douady, A. "Julia Sets and the Mandelbrot Set." In The
Beauty of Fractals: Images of Complex Dynamical Sys-
tems (Ed. H.-O. Peitgen and D. H. Richter). Berlin:
Springer- Verlag, p. 161, 1986.

Eppstein, D. "Area of the Mandelbrot Set." http:// www .
ics. uci.edu/-eppstein/junkyard/mand-area. html.

Fisher, Y. and Hill, J. "Bounding the Area of the Mandelbrot
Set." Submitted.

Hill, J. R. "Fractals and the Grand Internet Parallel Process-
ing Project." Ch. 15 in Fractal Horizons: The Future Use
of Fractals. New York: St. Martin's Press, pp. 299-323,
1996.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 148—
151 and 179-180, 1991.

Munafo, R. "Mu-Ency — The Encyclopedia of the Mandelbrot
Set." http: //home . earthlink.net/-mrob/muency.html.

Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal
Images. New York: Springer- Verlag, pp. 178-179, 1988.

Shishikura, M. "The Boundary of the Mandelbrot Set has
Hausdorff Dimension Two." Asterisque, No. 222, 7, 389-
405, 1994.

Mandelbrot Tree

Mangoldt Function 1135

Mandelbrot Tree

, 1 b

"X

The Fractal illustrated above.

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 71-
73, 1991.
# Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.

Mangoldt Function

The function defined by

A(n)

— f lnp if n = p k for p a prime
I otherwise.

(1)

exp(A(n)) is also given by [1, 2, ... , n]/[l, 2, . . . , n- 1],
where [a, 6, c, . . .] denotes the Least COMMON Multi-
ple. The first few values of exp(A(n)) for n = 1, 2,
. . . , plotted above, are 1, 2, 3, 2, 5, 1, 7, 2, . . . (Sloane's
A014963). The Mangoldt function is related to the RlE-
mann Zeta Function C(z) by

C(s)
CM

A(n)

n 3 '

(2)

where 5ft[s] > 1.

20 40 60 80 100

The Summatory Mangoldt function, illustrated above,
is defined by

^(x) = J2Hn), (3)

nKx

where A(n) is the Mangoldt Function. This has the
explicit formula

jP(x) = x - J2 — - ln ( 2?r ) - 2 ln (! - A ( 4 )
p

where the second Sum is over all complex zeros p of the
Riemann Zeta Function ((s) and interpreted as

lim T X -.

i-*-oo ^— ' p
|9(P)I<*

(5)

Vardi (1991, p. 155) also gives the interesting formula
ln([z]!) = i,( x ) + 1>(\x) + V(b) + . . . , (6)

where [x] is the Nint function and n! is a Factorial.

Vallee Poussin's version of the Prime Number Theo-
rem states that

4>(x) = x + <D{xe- as/T ™)

(7)

for some a (Davenport 1980, Vardi 1991). The Riemann
Hypothesis is equivalent to

il>(x) =x + 0(^(\nx) 2 )

(8)

(Davenport 1980, p. 114; Vardi 1991).

see also Bombieri's Theorem, Greatest Prime Fac-
tor, Lambda Function, Least Common Multiple,
Least Prime Factor, Riemann Function

References

Davenport, H, Multiplicative Number Theory, 2nd ed. New
York: Springer- Verlag, p. 110, 1980.

Sloane, N. J. A. Sequence AO 14963 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA; Addison- Wesley, pp. 146-147, 152-153, and 249,
1991.

1136

Manifold

Map Coloring

Manifold

Rigorously, an n-D (topological) manifold is a TOPO-
LOGICAL SPACE M such that any point in M has a
Neighborhood U c M which is Homeomorphic to n-
D Euclidean Space. The Homeomorphism is called a
chart, since it lays that part of the manifold out flat, like
charts of regions of the Earth. So a preferable statement
is that any object which can be "charted" is a manifold.

The most important manifolds are DlFFERENTlABLE
MANIFOLDS. These are manifolds where overlapping
charts "relate smoothly" to each other, meaning that
the inverse of one followed by the other is an infinitely
differentiable map from EUCLIDEAN SPACE to itself.

Manifolds arise naturally in a variety of mathematical
and physical applications as "global objects." For exam-
ple, in order to precisely describe all the configurations
of a robot arm or all the possible positions and momenta
of a rocket, an object is needed to store all of these pa-
rameters. The objects that crop up are manifolds. From
the geometric perspective, manifolds represent the pro-
found idea having to do with global versus local proper-
ties.

Consider the ancient belief that the Earth was flat com-
pared to the modern evidence that it is round. The
discrepancy arises essentially from the fact that on the
small scales that we see, the Earth does look flat. We
cannot see it curve because we are too small (although
the Greeks did notice that the last part of a ship to
disappear over the horizon was the mast). We can de-
tect curvature only indirectly from our vantage point on
the Earth. The basic idea for this "problem" was codi-
fied by Poincare. The problem is that on a small scale,
the Earth is nearly flat. In general, any object which is
nearly "flat" on small scales is a manifold, and so mani-
folds constitute a generalization of objects we could live
on in which we would encounter the round/flat Earth
problem.

see also Cobordant Manifold, Compact Mani-
fold, Connected Sum Decomposition, Differ-
entiable Manifold, Flag Manifold, Grassmann
Manifold, Heegaard Splitting, Isospectral
Manifolds, Jaco-Shalen-Johannson Torus De-
composition, Kahler Manifold, Poincare Con-
jecture, Poisson Manifold, Prime Manifold,

RlEMANNIAN MANIFOLD, SET, SMOOTH MANIFOLD,

Space, Stiefel Manifold, Stratified Manifold,
submanifold, surgery, symplectic manifold,
Thurston's Geometrization Conjecture, Topo-
logical Manifold, Topological Space, White-
head Manifold, Wiedersehen Manifold

References

Conlon, L. Differentiable Manifolds:
Boston, MA: Birkhauser, 1993.

A First Course.

Mantissa

For a Real Number x, the mantissa is defined as the
POSITIVE fractional part x - [x\ = frac(x), where [x\
denotes the FLOOR FUNCTION.

see also Characteristic (Real Number), Floor
Function, Scientific Notation

Map

A way of associating unique objects to every point in a
given Set. So a map from A h-> B is an object / such
that for every a£i, there is a unique object f(a) G B.
The terms Function and Mapping are synonymous
with map.

The following table gives several common types of com-
plex maps.

Mapping

Formula

Domain

inversion

/« = I

magnification

f{z) = az

aGlR/0

magnification+rotation

f{z) = az

aeC^Q

Mobius

f(z) = e***

a, 6, c, d G C

rotation

translation

f(z) = z + a

aeC

see also 2x mod 1 Map, Arnold's Cat Map, Baker's
Map, Boundary Map, Conformal Map, Func-
tion, Gauss Map, Gingerbreadman Map, Har-
monic Map, Henon Map, Identity Map, Inclusion
Map, Kaplan-Yorke Map, Logistic Map, Mandel-
brot Set, Map Projection, Pullback Map, Quad-
ratic Map, Tangent Map, Tent Map, Transfor-
mation, Zaslavskii Map

References

Arfken, G. "Mapping." §6.6 in Mathematical Methods for

Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 384-

392, 1985.
Lee, X. "Transformation of the Plane." http://www.best.

com / - xah / Math Graphics Gallery_dir / Transform 2D

Plot_dir/transf orm2DPlot.html.

Map Coloring

Given a map with GENUS g > 0, Heawood showed in
1890 that the maximum number N u of colors necessary
to color a map (the CHROMATIC NUMBER) on an un-
bounded surface is

N u =

f(7+ V / 48^TT)j = [§(7+^/49 -24 X )J

where [x\ is the Floor Function, g is the Genus,
and x is the EULER CHARACTERISTIC. This is the Hea-
WOOD CONJECTURE. In 1968, for any orientable surface
other than the SPHERE (or equivalently, the Plane) and
any nonorientable surface other than the Klein Bot-
tle, N u was shown to be not merely a maximum, but
the actual number needed (Ringel and Youngs 1968).

When the FOUR-COLOR THEOREM was proven, the Hea-
wood FORMULA was shown to hold also for all orientable
and nonorientable surfaces with the exception of the

Map Folding

Mapes' Method 1137

Klein Bottle. For this case (which has Euler Char-
acteristic 1, and therefore can be considered to have
g = 1/2), the actual number of colors N needed is six —
one less than N u = 7 (Franklin 1934; Saaty 1986, p. 45).

surface

9

N u

N

Klein bottle

1

7

6

Mobius strip

l

6

6

plane

4

4

projective plane

i

2

6

6

sphere

4

4

torus

1

7

7

see also CHROMATIC NUMBER, FOUR-COLOR THEO-
REM, Heawood Conjecture, Six-Color Theorem,
Torus Coloring

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 237-

238, 1987.
Barnette, D. Map Coloring, Polyhedra, and the Four-Color

Problem. Washington, DC: Math. Assoc. Amer., 1983.
Franklin, P. "A Six Colour Problem." J. Math. Phys. 13,

363-369, 1934.
Franklin, P. The Four-Color Problem. New York: Scripta

Mathematica, Yeshiva College, 1941.
Ore, 0. The Four-Color Problem. New York: Academic

Press, 1967.
Ringel, G. and Youngs, J. W. T. "Solution of the Heawood

Map-Coloring Problem." Proc. Nat. Acad. Sci. USA 60,

438-445, 1968.
Saaty, T. L. and Kainen, P. C. The Four- Color Problem:

Assaults and Conquest. New York: Dover, 1986.

Map Folding

A general FORMULA giving the number of distinct ways
of folding an iV = m x n rectangular map is not known.
A distinct folding is defined as a permutation of N num-
bered cells reading from the top down. Lunnon (1971)
gives values up to n = 28.

n

1 x n

2 x n

3 x n

4 x n

5 x n

1

1

1

2

2

8

3

6

60

1368

4

16

1980

300608

5

59

19512

18698669

6

144

15552

The limiting ratio of the number of 1 x (n + 1) strips to
the number of 1 x n strips is given by

lim l lx ( n + 1 )) e [3.3868,3.98211.

n^oo [1 X 7l) L ' J

see also STAMP FOLDING

References

Gardner, M. "The Combinatorics of Paper Folding." Ch. 7 in
Wheels, Life, and Other Mathematical Amusements. New
York: W. H. Freeman, 1983.

Koehler, J. E. "Folding a Strip of Stamps." J. Combin. Th.
5, 135-152, 1968.

Lunnon, W. F. "A Map-Folding Problem." Math. Comput.
22, 193-199, 1968.

Lunnon, W. F. "Mult i- Dimensional Strip Folding." Com-
puter J. 14, 75-79, 1971.

Map Projection

A projection which maps a SPHERE (or Spheroid) onto
a Plane. No projection can be simultaneously Con-
formal and Area-Preserving.

see also AlRY PROJECTION, ALBERS EQUAL-
Area Conic Projection, Axonometry, Azimuthal
Equidistant Projection, Azimuthal Projection,
Behrmann Cylindrical Equal-Area Projection,
Bonne Projection, Cassini Projection, Chro-
matic Number, Conic Equidistant Projection,
Conic Projection, Cylindrical Equal-Area Pro-
jection, Cylindrical Equidistant Projection,
Cylindrical Projection, Eckert IV Projection,
Eckert VI Projection, Four-Color Theorem,
Gnomic Projection, Guthrie's Problem, Ham-
mer- Aitoff Equal- Area Projection, Lambert
Azimuthal Equal- Area Projection, Lambert

CONFORMAL CONIC PROJECTION, MAP COLORING,

Mercator Projection, Miller Cylindrical Pro-
jection, Mollweide Projection, Orthographic
Projection, Polyconic Projection, Pseudocylin-
drical Projection, Rectangular Projection, Si-
nusoidal Projection, Six-Color Theorem, Stere-
ographic Projection, van der Grinten Projec-
tion, Vertical Perspective Projection

References

Dana, P. H. "Map Projections." http://www.utexas.edu/

depts/grg/gcraft/notes/mapproj/mapproj .html.
Hunter College Geography. "The Map Projection Home

Page." http : //everest . hunter . cuny . edu/mp/.
Snyder, J. P. Map Projections — A Working Manual U. S.

Geological Survey Professional Paper 1395. Washington,

DC: U. S. Government Printing Office, 1987.

Mapes' Method

A method for computing the PRIME COUNTING FUNC-
TION. Define the function

r fc (x,a) = (-l)' 5o+/9l+ - + ^-i

p l 0Op 2 ^l . ..p o 0a-l

(1)

where [x\ is the FLOOR FUNCTION and the f3i are the
binary digits (0 or 1) in

A: = 2 a ~ 1 /3 a - 1 + 2 a " 2 /? a _ 2 + . . . + 2 1 /?! + 2%. (2)

The LEGENDRE SUM can then be written

<f>(x,a) = ^^ T k (x,a).

(3)

1138 Mapping (Function)

Markov Chain

The first few values of Tk{a

1) are

T (x,3)=LzJ (4)

Ti(x,3) = -

X

(5)

T 2 (x,3) = -

X

(6)

Ts(x,3) =

X

PlP2 m

(7)

T 4 (x,3) = -

X

_P3_

(8)

T s {x,3) =

X

_PiPz _

(9)

T 6 (x,3) =

X
P2PZ _

(10)

T 7 (x,3) = -

X

(11)

_P1P

2P3_

Mapes' method takes time ~ x 0,7 , which is slightly faster
than the Lehmer-Schur Method.

see also Lehmer-Schur Method, Prime Counting
Function

References

Mapes, D. C. "Fast Method for Computing the Number of
Primes Less than a Given Limit," Math. Comput. 17,
179-185, 1963.

Riesel, H. "Mapes' Method." Prime Numbers and Com-
puter Methods for Factorization, 2nd ed. Boston, MA:
Birkhauser, p. 23, 1994.

Mapping (Function)

see Map

Mapping Space

Let Y x be the set of continuous mappings / : X — > Y.
Then the TOPOLOGICAL SPACE for Y x supplied with a
compact-open topology is called a mapping space.

see also LOOP SPACE

References

Iyanaga, S. and Kawada, Y. (Eds.). "Mapping Spaces."
§204B in Encyclopedic Dictionary of Mathematics. Cam-
bridge, MA: MIT Press, p. 658, 1980.

Marginal Analysis

Let R{x) be the revenue for a production x> C(x) the
cost, and P(x) the profit. Then

P(x) = R{x) -C(a;),

and the marginal profit for the x th unit is defined by

P t (x )=R'(xo)-C'(x ),

where P'(x), R'(x), and C'(x) are the DERIVATIVES of
P(x), R(x), and C(x), respectively.

see also Derivative

Marginal Probability

Let S be partitioned into r x s disjoint sets Ei and Fj
where the general subset is denoted Ei H Fj . Then the
marginal probability of Ei is

p(E i ) = Y f P(E i nF i ;

J=l

Markoff's Formulas

Formulas obtained from differentiating NEWTON'S FOR-
WARD Difference Formula,

/'(ao+p/i) = -[A + §(2p-l)Ag

where

* =»"/<->< („>,)

+ -R n ,

+h

n+1

n + 1 / dx

,(n + l)

(0,

(£) is a Binomial Coefficient, and a < £ < a„.
Abramowitz and Stegun (1972) and Beyer (1987) give
derivatives h^f^ in terms of A k and derivatives in
terms of S k and V fc .
see also FINITE DIFFERENCE

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook

of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, 9th printing. New York: Dover,

p. 883, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.

Boca Raton, FL: CRC Press, pp. 449-450, 1987.

Markoff Number

see Markov Number

Markov Algorithm

An Algorithm which constructs allowed mathematical
statements from simple ingredients.

Markov Chain

A collection of random variables {X t }, where the index
t runs through 0, 1,

References

Kemeny, J. G. and Snell, J. L. Finite Markov Chains. New

York: Springer- Verlag, 1976.
Stewart, W. J. Introduction to the Numerical Solution of

Markov Chains. Princeton, NJ: Princeton University

Press, 1995.

Markov's Inequality

Markov's Theorem

1139

Markov's Inequality

If x takes only Nonnegative values, then

P(x > a) <

(x)

To prove the theorem, write

noo pa poo

(x) — / xf(x)dx= / xf(x)dx+ / xf(x)dx.

Jo JO J a

Since P(x) is a probability density, it must be > 0. We
have stipulated that x > 0, so

iaL

a^j^fa^ / w/^w^

Markov Number

The Markov numbers m occur in solutions to the Dio-
phantine Equation

2,2.2

x +y + z

3xyz,

and are related to LAGRANGE NUMBERS L n by

The first few solutions are (x,y, z) = (1,1,1), (1, 1, 2),
(1, 2, 5), (1, 5, 13), (2, 5, 29), .... The solutions can be
arranged in an infinite tree with two smaller branches
on each trunk. It is not known if two different regions
can have the same label. Strangely, the regions adjacent

f^ 1 Uvn olf^n,^ fTDAMA^PT MtTUT,^Q 1 O C IP OA

J a

a / f(x) dx = aP(x > a),

iV, then

Q. E. D.

Markov Matrix

see Stochastic Matrix

Markov Moves

A type I move (Conjugation) takes AB ->■ BA for A,
B e B n where B n is a Braid Group.

12/1-1 12 n-\

A type II move (Stabilization) takes A -► Ab n or A
Abn' 1 for A £ B n , and b n , A6 n , and yl&n -1 £ Bn+i-
1 2 «-l 12 w-1 i

see also BRAID GROUP, CONJUGATION, REIDEMEISTER

Moves, Stabilization

M{n) = C(lnN) + <9((ln JV) 1+€ ),

where C « 0.180717105 (Guy 1994, p. 166).
see a/so Hurwitz Equation, Hurwitz's Irrational
Number Theorem, Lagrange Number (Ratio-
nal Approximation) Liouville's Rational Ap-
proximation Theorem, Liouville-Roth Constant,
Roth's Theorem, Segre's Theorem, Thue-Siegel-
Roth Theorem

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New

York: Springer- Verlag, pp. 187-189, 1996.
Guy, R. K. "MarkofF Numbers." §D12 in Unsolved Problems

in Number Theory, 2nd ed. New York: Springer- Verlag,

pp. 166-168, 1994.

Markov Process

A random process whose future probabilities are deter-
mined by its most recent values.
see also DOOB'S THEOREM

Markov Spectrum

A Spectrum containing the Real Numbers larger
than Freiman's Constant.

see also Freiman's Constant, Spectrum Sequence

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 188-189, 1996.

Markov's Theorem

Published by A. A. Markov in 1935, Markov's theorem
states that equivalent Braids expressing the same Link
are mutually related by successive applications of two
types of Markov Moves. Markov's theorem is difficult
to apply in practice, so it is difficult to establish the
equivalence or nonequivalence of Links having different
Braid representations.
see also Braid, Link, Markov Moves

1140 Marriage Theorem

Mascheroni Construction

Marriage Theorem

If a group of men and women may date only if they have
previously been introduced, then a complete set of dates
is possible Iff every subset of men has collectively been
introduced to at least as many women, and vice versa.

References

Chartrand, G. Introductory Graph Theory.
Dover, p. 121, 1985.

New York:

Married Couples Problem

Also called the Menage Problem. In how many ways
can n married couples be seated around a circular table
in such a manner than there is always one man between
two women and none of the men is next to his own
wife? The solution (Ball and Coxeter 1987, p. 50) uses
Discordant Permutations and can be given in terms
of Laisant's Recurrence Formula

(n - l)A n+ i = (n 2 - l)A n + (n + 1)j4„_i + 4(-l) n ,

with A\ = A 2 = 1.
Touchard (1934) is

A closed form expression due to

where (£) is a Binomial Coefficient (Vardi 1991),

The first few values of A n are -1, 1, 0, 2, 13, 80,
579, . . . (Sloane's A000179), which are sometimes called
MENAGE NUMBERS. The desired solution is then 2n\A n
The numbers A n can be considered a special case of a
restricted ROOKS PROBLEM.

see also DISCORDANT PERMUTATION, LAISANT'S RE-
CURRENCE Formula, Rooks Problem

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 50, 1987.

Ddrrie, H. §8 in 100 Great Problems of Elementary Mathe-
matics: Their History and Solutions. New York: Dover,
pp. 27-33, 1965.

Halmos, P. R.; Vaughan, H. E. "The Marriage Problem."
Amer. J. Math. 72, 214-215, 1950.

Lucas, E. Theorie des Nombres. Paris, pp. 215 and 491-495,
1891.

MacMahon, P. A. Combinatory Analysis, Vol. 1. London:
Cambridge University Press, pp. 253-256, 1915.

Newman, D. J. "A Problem in Graph Theory." Amer. Math.
Monthly 65, 611, 1958.

Sloane, N. J. A. Sequence A000179/M2062 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Touchard, J. "Sur un probleme de permutations." C. R.
Acad. Sci. Paris 198, 631-633, 1934.

Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 123, 1991.

Marshall-Edgeworth Index

The statistical Index

p = X>n(gO+gn)

where p n is the price per unit in period n, q n is the
quantity produced in period n, and v n = p n Qn is the
value of the n units.

see also Index

References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
PL 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 66-67,
1962.

Martingale

A sequence of random variates such that the Condi-
tional Probability of x n+ i given xi, #2, . ••, %n is
x n - The term was first used to describe a type of wa-
gering in which the bet is doubled or halved after a loss
or win, respectively.

see also Gambler's Ruin, Saint Petersburg Para-
dox

Mascheroni Constant

see Euler-Mascheroni Constant

Mascheroni Construction

A geometric construction done with a movable COMPASS
alone. All constructions possible with a COMPASS and
Straightedge are possible with a movable Compass
alone, as was proved by Mascheroni (1797). Mascher-
oni's results are now known to have been anticipated
largely by Mohr (1672).

see also COMPASS, GEOMETRIC CONSTRUCTION, NEU-

sis Construction, Straightedge

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 96-97,
1987.

Bogomolny, A. "Geometric Constructions with the Compass
Alone." http : //www . cut— fche-knot . com/do_youJtnow/

compass.html.

Courant, R. and Robbins, H. "Constructions with Other
Tools. Mascheroni Constructions with Compass Alone."
§3.5 in What is Mathematics?: An Elementary Approach
to Ideas and Methods, 2nd ed. Oxford, England; Oxford
University Press, pp. 146-158, 1996.

Dorrie, H. "Mascheroni's Compass Problem." §33 in 100
Great Problems of Elementary Mathematics: Their His-
tory and Solutions. New York: Dover, pp. 160-164, 1965.

Gardner, M. "Mascheroni Constructions." Ch. 17 in
Mathematical Circus: More Puzzles, Games, Paradoxes
and Other Mathematical Entertainments from Scientific
American. New York: Knopf, pp. 216-231, 1979.

Mascheroni, L. Geometry of Compass. Pavia, Italy, 1797.

Mohr, G. Euclides Danicus. Amsterdam, Netherlands, 1672.

Maschke's Theorem

Masser-Gramain Constant 1141

Maschke's Theorem

If a Matrix Group is reducible, then it is completely
reducible, i.e., if the MATRIX GROUP is equivalent to the
Matrix Group in which every Matrix has the reduced
form

Df>

then it is equivalent to the MATRIX GROUP obtained by
putting Xi = 0.

see also Matrix GROUP

References

Lomont, J. S. Applications of Finite Groups. New York:
Dover, p. 49, 1987.

Mason's abc Theorem

see Mason's Theorem

Mason's Theorem

Let there be three POLYNOMIALS a(x), b(x), and c(x)
with no common factors such that

a(x) + b(x) = c(x).

Then the number of distinct ROOTS of the three POLY-
NOMIALS is one or more greater than their largest degree.
The theorem was first proved by Stothers (1981).

Mason's theorem may be viewed as a very special case
of a Wronskian estimate (Chudnovsky and Chudnovsky
1984). The corresponding Wronskian identity in the
proof by Lang (1993) is

c 3 * W{a t 6, c) = W(W(a, c), W(b, c)),

so if a, 6, and c are linearly dependent, then so are
W(cL)C) and W(b> c). More powerful Wronskian esti-
mates with applications toward diophantine approxima-
tion of solutions of linear differential equations may be
found in Chudnovsky and Chudnovsky (1984) and Os-
good (1985).

The rational function case of FERMAT's LAST THEO-
REM follows trivially from Mason's theorem (Lang 1993,
p. 195).

see also ABC CONJECTURE

References

Chudnovsky, D. V. and Chudnovsky, G. V. "The Wronskian
Formalism for Linear Differential Equations and Pade Ap-
proximations." Adv. Math. 53, 28-54, 1984.

Lang, S. "Old and New Conjectured Diophantine Inequali-
ties." Bull. Amer. Math. Soc. 23, 37-75, 1990.

Lang, S. Algebra, 3rd ed. Reading, MA: Addison- Wesley,
1993.

Osgood, C. F. "Sometimes Effective Thue-Siegel-Roth-
Schmidt-Nevanlinna Bounds, or Better." J. Number Th.
21, 347-389, 1985.

Stothers, W. W. "Polynomial Identities and Hauptmodulen."
Quart. J. Math. Oxford Ser. II 32, 349-370, 1981.

Masser-Gramain Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

If f(z) is an Entire Function such that f(n) is an
Integer for each Positive Integer n. Polya (1915)
showed that if

where

lim sup l ^ L < In 2 = 0.693 .

M r = SUp \f(x)

\Z\<T

(1)

(2)

is the SUPREMUM, then / is a POLYNOMIAL. Further-
more, In 2 is the best constant (i.e., counterexamples
exist for every smaller value).

If f(z) is an Entire Function with /(n) a Gaussian
Integer for each Gaussian Integer n, then Gelfond
(1929) proved that there exists a constant a such that

lnM r
lim sup — - — < a

(3)

implies that / is a POLYNOMIAL. Gramain (1981, 1982)
showed that the best such constant is

a= — =0.578....

2e

(4)

Maser (1980) proved the weaker result that / must be
a POLYNOMIAL if

lnM r i / . , 4c\ /cX

lim sup — — < a = | exp I -d H I , (5)

V — K30 ' \ 7T /

where

c = 7/9(1) + 0'(1) = 0.6462454398948114 . . . , (6)

7 is the Euler-Mascheroni Constant, 0(z) is the
Dirichlet Beta Function,

5 = lim [ } - Inn ,

(7)

and rk is the minimum NONNEGATIVE r for which there
exists a COMPLEX NUMBER z for which the CLOSED
Disk with center z and radius r contains at least A; dis-
tinct Gaussian Integer. Gosper gave

c = 7r{-ln[r(|)] + |7r+|ln2+|7}- (8)

Gramain and Weber (1985, 1987) have obtained

1.811447299 < <5 < 1.897327117, (9)

which implies

0.1707339 < a < 0.1860446. (10)

1142

Match Problem

Mathieu Differential Equation

Gramain (1981, 1982) conjectured that

ao =2?

which would imply

Ac

S = l+ — = 1.822825249....

7T

(ii)

(12)

References

Finch, S. "Favorite Mathematical Constants." http: //www.
mathsof t . com/ asolve/constant/masser/masser .html.

Gramain, F. "Sur le theoreme de Fukagawa-Gel'fond." In-
vent Math. 63, 495-506, 1981.

Gramain, F. "Sur le theoreme de Fukagawa-Gel'fond-Gru-
man-Masser." Seminaire Delange-Pisot-Poitou (Theorie
des Nombres), 1980-1981. Boston, MA: Birkhauser, 1982.

Gramain, F. and Weber, M. "Computing and Arithmetic
Constant Related to the Ring of Gaussian Integers." Math.
Comput 44, 241-245, 1985.

Gramain, F. and Weber, M. "Computing and Arithmetic
Constant Related to the Ring of Gaussian Integers." Math.
Comput. 48, 854, 1987.

Masser, D. W. "Sur les fonctions entieres a valeurs entieres."
C. R. Acad. Sci. Paris Ser. A-B 291, A1-A4, 1980.

Match Problem

Given n matches, find the number of topologically dis-
tinct planar arrangements T(n) which can be made. The
first few values are 1, 1, 3, 5, 10, 19, 39, ... (Sloane's
A003055).
see also Cigarettes, Matchstick Graph

References

Gardner, M. "The Problem of the Six Matches." In The

Unexpected Hanging and Other Mathematical Diversions.

Chicago, IL: Chicago University Press, pp. 79-81, 1991.
Sloane, N. J. A. Sequence A003055/M2464 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Matchstick Graph

A Planar Graph whose Edges are all unit line seg-
ments. The minimal number of Edges for matchstick
graphs of various degrees are given in the table below.
The minimal degree 1 matchstick graph is a single EDGE,
and the minimal degree 2 graph is an EQUILATERAL
Triangle.

n

v

1 1 2

2 3 3

3 12 8

4 < 42

Mathematical Induction

see Induction

Mathematics

Mathematics is a broad-ranging field of study in which
the properties and interactions of idealized objects are
examined. Whereas mathematics began merely as a cal-
culational tool for computation and tabulation of quan-
tities, it has blossomed into an extremely rich and di-
verse set of tools, terminologies, and approaches which
range from the purely abstract to the utilitarian.

Bertrand Russell once whimsically defined mathematics
as, "The subject in which we never know what we are
talking about nor whether what we are saying is true"
(Bergamini 1969).

The term "mathematics" is often shortened to "math"
in informal American speech and, consistent with the
British penchant for adding superfluous letters, "maths"
in British English.

see also Metamathematics

References

Bergamini, D. Mathematics.
p. 9, 1969.

New York: Time-Life Books,

Mathematics Prizes

Several prizes are awarded periodically for outstanding
mathematical achievement. There is no Nobel Prize
in mathematics, and the most prestigious mathematical
award is known as the FIELDS Medal. In rough order of
importance, other awards are the $100,000 Wolf Prize of
the Wolf Foundation of Israel, the Leroy P. Steele Prize
of the American Mathematical Society, followed by the
Bocher Memorial Prize, Frank Nelson Cole Prizes in Al-
gebra and Number Theory, and the Delbert Ray Fulker-
son Prize, all presented by the American Mathematical
Society.

see also FIELDS MEDAL

References

"AMS Funds and Prizes." http:// www . ams . org / ams /
prizes.html.

MacTutor History of Mathematics Archives. "The Fields
Medal." http:// www - groups . des . st - and .ac.uk/
-history/Societies/FieldsMedal.html. "Winners of the
Bocher Prize of the AMS." http:// www -groups . des . st
-and. ac . uk/ -history /Societies/ AMSBocherPrize.html.
"Winners of the Frank Nelson Cole Prize of the AMS."
http : // www - groups . des . st - and . ac . uk / - history/
Societies/AMSColePrize .html.

MacTutor History of Mathematics Archives. "Mathematical
Societies, Medals, Prizes, and Other Honours." http://
www-groups .des , st-and, ac ,uk/ -history/Societies/.

Monastyrsky, M. Modern Mathematics in the Light of the
Fields Medals. Wellesley, MA: A. K. Peters, 1997.

"Wolf Prize Recipients in Mathematics." http: //www,
aquanet.co.il/wolf /wolf 5. html.

Mathieu Differential Equation

d 2 V

dv 2

+ [6- 2^003(21;)]^ = 0.

Mathieu Function

It arises in separation of variables of LAPLACE'S EQUA-
TION in Elliptic Cylindrical Coordinates. Whit-
taker and Watson (1990) use a slightly different form to
define the Mathieu Functions.

The modified Mathieu differential equation
d 2 U

Matrix

1143

du 2

- [b-2qcosh(2u)]U =

arises in Separation of Variables of the Helmholtz
Differential Equation in Elliptic Cylindrical
Coordinates.

see also Mathieu Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 722, 1972.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 556-557, 1953.

Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.

Mathieu Function

The form given by Whittaker and Watson (1990, p. 405)
defines the Mathieu function based on the equation

dz 2

+ [a+ 16gcos(2z)]u = 0.

(1)

This equation is closely related to Hill's DIFFERENTIAL
Equation. For an Even Mathieu function,

G{r)) = X J c kco5VCO80 G{6)de,

J — 7T

where k = y/32q. For an Odd Mathieu function,

G(rj) = A / sin(fc sin r? sin 6)G(0) <ffl.
Both EVEN and Odd functions satisfy

G(v)

/7T
ik sin 77 s
e
-7T

9 G(9) dB.

(2)

(3)

(4)

Letting £ = cos 2 z transforms the Mathieu Differen-
tial Equation to

4C(l-C)~^+2(l-2C)^ + (a-16g + 32gC)^ = 0. (5)

see also Mathieu Differential Equation

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Mathieu Func-
tions." Ch. 20 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 721-746, 1972.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 562-568 and 633-
642, 1953.

Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, l^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.

Mathieu Groups

The first Simple Sporadic Groups discovered. Mn,
Mi 2, M22, M23, M24 were discovered in 1861 and 1873
by Mathieu. Probenius showed that all the Mathieu
groups are SUBGROUPS of M 2 4-

The Mathieu groups are most simply denned as Au-
tomorphism groups of STEINER SYSTEMS. Mn corre-
sponds to 5(4, 5, 11) and M23 corresponds to 5(4, 7, 23).
Mn and M23 are Transitive Permutation Groups
of 11 and 23 elements.

The Orders of the Mathieu groups are

|Afn| = 2 4 -3 2 -5-11
|Mi 2 | = 2 6 -3 3 -5-ll
|M 22 | = 2 7 '3 2 .5'7-11
|M 23 | = 2 7 .3 2 -5.7-11.23.

see also SPORADIC GROUP

References

Conway, J. H. and Sloane, N. J. A. "The Golay Codes and

the Mathieu Groups." Ch. 11 in Sphere Packings, Lattices,

and Groups, 2nd ed. New York: Springer- Verlag, pp. 299-

330, 1993.
Rotman, J. J. Ch. 9 in An Introduction to the Theory of

Groups, 4th ed. New York: Springer- Verlag, 1995.
Wilson, R. A. "ATLAS of Finite Group Representation."

http://for.mat.bham.ac.uk/atlas/Mll.html, M12.html,

M22.html, M23.html, and M24.html.

Matrix

The Transformation given by the system of equations

Xi = anXi + Gl2#2 + . . . + ainXn
x' 2 = a2lXi + a22%2 + . . . + CL2nX n

X m = a m\X\ + Q>m2X2 + . . . + Q>mnX n

is denoted by the MATRIX EQUATION

x x
x 2

x'

an Q>i2

Q>21 &22

ain

0>2n

a-n

Xx

x 2

_a m i a m 2

In concise notation, this could be written

x = Ax,

where x' and x are VECTORS and A is called an n x m
matrix. A matrix is said to be SQUARE if m = n. Spe-
cial types of Square Matrices include the Identity
Matrix I, with a^ = Sij (where 5ij is the Kronecker
Delta) and the Diagonal Matrix a^ = ciSij (where
d are a set of constants).

1144

Matrix Addition

Matrix Equality

For every linear transformation there exists one and only
one corresponding matrix. Conversely, every matrix cor-
responds to a unique linear transformation. The matrix
is an important concept in mathematics, and was first
formulated by Sylvester and Cayley.

Two matrices may be added (Matrix Addition) or
multiplied (MATRIX MULTIPLICATION) together to yield
a new matrix. Other common operations on a single ma-
trix are diagonalization, inversion (Matrix Inverse),
and transposition (MATRIX TRANSPOSE). The DETER-
MINANT det(A) or |A| of a matrix A is an very important
quantity which appears in many diverse applications.
Matrices provide a concise notation which is extremely
useful in a wide range of problems involving linear equa-
tions (e.g., Least Squares Fitting).

see also Adjacency Matrix, Adjugate Matrix,
Antisymmetric Matrix, Block Matrix, Cartan
Matrix, Circulant Matrix, Condition Number,
Cramer's Rule, Determinant, Diagonal Matrix,
Dirac Matrices, Eigenvector, Elementary Ma-
trix, Equivalent Matrix, Fourier Matrix, Gram
Matrix, Hilbert Matrix, Hypermatrix, Identity
Matrix, Incidence Matrix, Irreducible Matrix,
Kac Matrix, LU Decomposition, Markov Matrix,
Matrix Addition, Matrix Decomposition The-
orem, Matrix Inverse, Matrix Multiplication,
McCoy's Theorem, Minimal Matrix, Normal Ma-
trix, Pauli Matrices, Permutation Matrix, Posi-
tive Definite Matrix, Random Matrix, Rational
Canonical Form, Reducible Matrix, Roth's Re-
moval Rule, Shear Matrix, Skew Symmetric Ma-
trix, Smith Normal Form, Sparse Matrix, Spe-
cial Matrix, Square Matrix, Stochastic Matrix,
Submatrix, Symmetric Matrix, Tournament Ma-
trix

References

Arflcen, G. "Matrices." §4.2 in Mathematical Methods for
Physicists, 3rd ed, Orlando, FL: Academic Press, pp. 176-
191, 1985.

Matrix Decomposition Theorem

Let P be a Matrix of Eigenvectors of a given Ma-
trix A and D a MATRIX of the corresponding EIGEN-
VALUES. Then A can be written

A = PDP"

(1)

where D is a DIAGONAL MATRIX and the columns of P
are Orthogonal Vectors. If P is not a Square Ma-
trix, then it cannot have a Matrix Inverse. However,
if P is m x n (with m > n), then A can be written

A = UDV T ,

(2)

where U and V are n x n Square Matrices with Or-
thogonal columns,

u T u = V T = I.

(3)

Matrix Diagonalization

Diagonalizing a MATRIX is equivalent to finding the
Eigenvectors and Eigenvalues. The Eigenvalues
make up the entries of the diagpnalized Matrix, and
the Eigenvectors make up the new set of axes corre-
sponding to the Diagonal Matrix.

see also Diagonal Matrix, Eigenvalue, Eigenvec-
tor

References

Arfken, G. "Diagonalization of Matrices," §4.6 in Mathemati-
cal Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 217-229, 1985.

Matrix Direct Product

see Direct Product (Matrix)

Matrix Equality

Two Matrices A and B are said to be equal Iff

Matrix Addition

Denote the sum of two MATRICES A and B (of the same
dimensions) by C = A-f-B. The sum is defined by adding
entries with the same indices

over all i and j. For example,

an ai2

0,21 &22

+

fell £>12
&21 &22

an + 6n ai2 + 6i2

0>21 + &21 ^22 + &22

while

Q>ij — 0%j

for all i,j. Therefore,

1 2
3 4

1 2

3 4

1 2

3 4

2
3 4

Matrix addition is therefore both Commutative and

Associative.

see also Matrix, Matrix Multiplication

Matrix Equation

Matrix Group 1145

Matrix Equation

Nonhomogeneous matrix equations of the form

Ax = b (1)

can be solved by taking the Matrix Inverse to obtain

x = A _1 b. (2)

This equation will have a nontrivial solution IFF the
Determinant det(A) ^ 0. In general, more numeri-
cally stable techniques of solving the equation include

Gaussian Elimination, LU Decomposition, or the
Square Root Method.

For a homogeneous n x n Matrix equation

(3)

to be solved for the XiS> consider the Determinant

so

an

Q>12 '

■ • din

Xi

'0'

&21

Q>22

Q>2n

X2

=

dnl

a n 2

' ' &nn _

. *^ n -

_0_

an

ai2

■ ai

a2i

a22 •

* Q>2

dnl

Q>n2

' CL n

(4)

Now multiply by Xi y which is equivalent to multiplying
the first row (or any row) by xi,

xi

(5)

The value of the Determinant is unchanged if multi-
ples of columns are added to other columns. So add x^
times column 2, . . . , and x n times column n to the first
column to obtain

an

Ol2 '

din

an^i

ai2

•• din

CL21

022 •

' • &2n

=

CL21X1

022 *

CL2n

Onl

dn2 '

* * Q>nn

CLnlXi

CLn2 •

* * &nn

Xl

an ai2

021 &22

a2n

Cinl &n2 ' ' ' a nn

anxi + ai2^2 + . . . + ai n x n a i2

0>2lXl + a22^2 + . . . + a-2nX n 0,22

Q>ln
CL2n

o>niX\ + a n 2X2 + . . . + a nn x n a n 2 ■ • ■ a nn

(6)

But from the original Matrix, each of the entries in the
first columns is zero since

ai2
a22

ftln
G>2n

= 0.

(8)

a n 2 • * * <*>n
Therefore, if there is an X\ / which is a solution, the
Determinant is zero. This is also true for X2, • ••,
x n , so the original homogeneous system has a nontrivial
solution for all x*s only if the DETERMINANT is 0. This
approach is the basis for CRAMER'S Rule.

Given a numerical solution to a matrix equation, the
solution can be iteratively improved using the follow-
ing technique. Assume that the numerically obtained
solution to

Ax = b (9)

is Xi = x + <5xi, where <$xi is an error term. The first
solution therefore gives

A(x + ^xi)=b + (5b (10)

Afe = 8b, (11)

where Sb is found by solving (10)

Sb - Axi - b. (12)

Combining (11) and (12) then gives

oxi = A^Jb = A _1 (Axi - b) = xi - A _1 b, (13)
so the next iteration to obtain x accurately should be

X2 = Xi — <$Xi. (14)

see also Cramer's Rule, Gaussian Elimination, LU
Decomposition, Matrix, Matrix Addition, Ma-
trix Inverse, Matrix Multiplication, Normal
Equation, Square Root Method

Matrix Exponential

Given a Square Matrix A, the matrix exponential is
defined by

exp(A)

^A n , A AA AAA ,

2!

3!

CLilXi + CLi2X2 + . • • + ClinXn = 0,

(7)

where I is the Identity Matrix.

see also EXPONENTIAL FUNCTION, MATRIX

Matrix Group

A Group in which the elements are Square Matri-
ces, the group multiplication law is MATRIX MULTIPLI-
CATION, and the group inverse is simply the MATRIX
INVERSE. Every matrix group is equivalent to a unitary
matrix group (Lomont 1987, pp. 47-48).
see also MASCHKE'S THEOREM

References

Lomont, J. S. "Matrix Groups." §3.1 in Applications of Fi-
nite Groups. New York: Dover, pp. 46—52, 1987.

1146

Matrix Inverse

Matrix Multiplication

Matrix Inverse

A Matrix A has an inverse Iff the Determinant

|A| ^ 0. For a 2 x 2 MATRIX

(1)

A =

a
c

b
d

j

the inverse is

"|A|

" d
—c

-ft"

a

=

1

' d
—c

-b

a

ad -

-be

(2)

For a 3 x 3 MATRIX,

A" x =

a22 ^23
^32 ^33

^23
^33

021
031

A21

»22
«32

ai3

ai2

&33

«32

an

ai3

031

a33

ai2

an

^32

^31

0.12

«22

ai3

a23

ai3 an

Ct23 ^21

an

«21

Ol2

«22

(3)

A general n x n matrix can be inverted using methods
such as the Gauss-Jordan Elimination, Gaussian
Elimination, or LU Decomposition.

The inverse of a Product AB of Matrices A and B
can be expressed in terms of A -1 and B _1 . Let

Then
and

C = AB.

B = A" 1 AB = A- 1 C
A = ABB _1 = CB _1 .

Therefore,

C = AB = (CB- 1 )(A- 1 C) = CB-'A-'C,

so

CB-'A- 1 ^!,

where I is the Identity Matrix, and

B- 1 A _1 = C _1 = (AB)- 1 .

(4)

(5)
(6)

(7)
(8)

(9)

see also Matrix, Matrix Addition, Matrix Mul-
tiplication, Moore-Penrose Generalized Matrix
Inverse, Strassen Formulas

References

Ben-Israel, A. and Greville, T. N. E. Generalized Inverses:
Theory and Applications. New York: Wiley, 1977.

Nash, J. C. Compact Numerical Methods for Computers:
Linear Algebra and Function Minimisation, 2nd ed. Bris-
tol, England: Adam Hilger, pp. 24-26, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Is Matrix Inversion an N 3 Process?" §2.11
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 95-98, 1992.

Matrix Multiplication

The product C of two MATRICES A and B is defined by

C-ih — Oij Oj /f ,

(1)

where j is summed over for all possible values of i and
k. Therefore, in order for multiplication to be denned,

the dimensions of the MATRICES must satisfy

(n x m)(m x p) = (n x p),

(2)

where (a x b) denotes a Matrix with a rows and b
columns. Writing out the product explicitly,

Cn C12

C21 C 2 2

Cip

C2p

Cnl C n 2 * * ' C n p _

an 0,12 • ■ • Ol«

021 0,22 ' ' ' 02r\

o n i a n 2 • ' * a nTl

bn bi2

&21 &22
b-ml bm2

bi p

&2p

(3)

where

di = an&n + ai2^2i + . .

Cl2 = Onbi2 + ai2&22 + - •

ci p = anbip + ai2&2p 4- . .
C21 — a>2ibu + a 2 2&2i + • •

C22 = Q>2lbi2 + »22&22 + • •
C2p = 0,2lbi p + a22&2p 4 * •
C n l — Onlbll + a n 2&21 + ■ -

c n 2 = a n ibi2 + a n2 b22 -r

C np = Onlbip + a„2&2p +

• 4- Olmbml

• + &lm&m2
. + OlmOmp

• + 02mbml

• + 02mbm2
■ . + Q>2mbmp
. . "T OnmOml
. . + Onmbm2
• • T" OnmUmp.

Matrix Multiplication is Associative, as can be
seen by taking

[(ab)c]ij = (ab)ikCkj = (aubik)ckj.

(4)

Now, since an, fy*,, and Ckj are Scalars, use the ASSO-
CIATIVITY of Scalar Multiplication to write

(o>ubik)ckj — au(bikCkj) — au(bc)ij = [a(bc)]ij. (5)

Since this is true for all i and j, it must be true that

[(ab)c]ij = [a(bc)]ij. (6)

That is, Matrix multiplication is Associative. How-
ever, Matrix Multiplication is not Commutative
unless A and B are DIAGONAL (and have the same di-
mensions) .

Matrix Norm

Max Sequence 1147

The product of two Block Matrices is given by mul-
tiplying each block

x x
x x

[o o~\ \x #1
o o\ [x x\

[o][x]

(7)

see also Matrix, Matrix Addition, Matrix In-
verse, Strassen Formulas

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 178-179, 1985.

Matrix Norm

Given a Square Matrix A with Complex (or Real)
entries, a MATRIX NORM ||A|| is a Nonnegative num-
ber associated with A having the properties

1. ||A|| > when A ^ and ||A|| = Iff A = 0,

2. ||jfeA|| = \k\ ||A|| for any SCALAR fc,

3. ||A + B||<||A|| + ||B||,

4. ||AB||<||A||||B||.

For annxn Matrix A and annxn Unitary Matrix

u,

||AU|| = ||UA|| = ||A||.
Let Ai, . . . , A n be the Eigenvalues of A, then

<|A|<||A||.

HA" 1 !

The Maximum Absolute Column Sum Norm \\n\\i,
Spectral Norm ||A|| 2 , and Maximum Absolute
Row Sum Norm ||A||oo satisfy

(l|A|| 3 ) 2 < l|A|U IIAMoo-

For a Square Matrix, the Spectral Norm, which is
the Square Root of the maximum Eigenvalue of A* A
(where A f is the Adjoint Matrix), is often referred to
as "the" matrix norm.

see also Compatible, Hilbert-Schmidt Norm, Max-
imum Absolute Column Sum Norm, Maximum Ab-
solute Row Sum Norm, Natural Norm, Norm,
Polynomial Norm, Spectral Norm, Spectral Ra-
dius, Vector Norm

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1114-1125, 1979.

Matrix Polynomial Identity

see CAYLEY-HAMILTON THEOREM

Matrix Product

The result of a MATRIX MULTIPLICATION.
see also PRODUCT

Matrix Transpose

see Transpose

Matroid

Roughly speaking, a finite set together with a general-
ization of a concept from linear algebra that satisfies a
natural set of properties for that concept. For example,
the finite set could be the rows of a MATRIX, and the
generalizing concept could be linear dependence and in-
dependence of any subset of rows of the MATRIX. The
number of matroids with n points are 1, 1, 2, 4, 9, 26,
101, 950, . . . (Sloane's A002773).

References

Sloane, N. J. A. Sequences A002773/M1197 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.

Whitely, W. "Matroids and Rigid Structures." In Matroid
Applications, Encyclopedia of Mathematics and Its Appli-
cations (Ed. N. White), Vol. 40. New York: Cambridge
University Press, pp. 1-53, 1992.

Maurer Rose

n = 4 1 d= 120 n = 6,d= 72

A Maurer rose is a plot of a "walk" along an n- (or
2n-) leafed ROSE in steps of a fixed number d degrees,
including all cosets.
see also STARR ROSE

References

Maurer, P. "A Rose is a Rose. . . " Amer. Math. Monthly 94,
631-645, 1987.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 96-102, 1991.

Max Sequence

A sequence defined from a FINITE sequence ao, ai, . . . ,
a n by defining a n +i = max* (a* + a n -t).
see also Mex Sequence

References

Guy, R. K. "Max and Mex Sequences." §E27 in Unsolved

Problems in Number Theory, 2nd ed. New York: Springer-

Verlag, pp. 227-228, 1994.

1148

Maximal Ideal

Maximum Clique Problem

Maximal Ideal

A maximal ideal of a Ring R is an Ideal /, not equal
to Rj such that there are no IDEALS "in between" I and
R. In other words, if J is an IDEAL which contains I as
a Subset, then either J — I or J = R. For example,
nL is a maximal ideal of Z Iff n is Prime, where Z is
the Ring of Integers.

see also Ideal, Prime Ideal, Regular Local Ring,
Ring

Maximal Sum-Free Set

A maximal sum- free set is a set {ai, a2, . . . , a n } of dis-
tinct Natural Numbers such that a maximum I of
them satisfy ai- + a,i k ^ a m , for 1 < j < k < /,
1 < m < n.
see also Maximal Zero-Sum-Free Set

References

Guy, R. K. "Maximal Sum-Free Sets." §C14 in Unsolved

Problems in Number Theory, 2nd ed. New York: Springer-

Verlag, pp. 128-129, 1994.

Maximal Zero-Sum-Free Set

A set having the largest number k of distinct residue
classes modulo m so that no Subset has zero sum.

see also Maximal Sum-Free Set

References

Guy, R. K. "Maximal Zero-Sum-Free Sets." §C15 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 129-131, 1994.

For a function f(x) which is Continuous at a point aso,
a Necessary but not Sufficient condition for f(x) to
have a RELATIVE MAXIMUM at x = x is that x be
a Critical Point (i.e., f(x) is either not Differen-
tiable at xo or xo is a Stationary Point, in which
case f'(xo) = 0).

The First Derivative Test can be applied to Con-
tinuous Functions to distinguish maxima from Min-
ima. For twice different iable functions of one variable,
f(x), or of two variables, f(x,y), the Second Deriv-
ative Test can sometimes also identify the nature of
an EXTREMUM. For a function f(x), the EXTREMUM
Test succeeds under more general conditions than the
Second Derivative Test.

see also CRITICAL POINT, EXTREMUM, EXTREMUM

Test, First Derivative Test, Global Maximum,
Inflection Point, Local Maximum, Midrange,
Minimum, Order Statistic, Saddle Point (Func-
tion), Second Derivative Test, Stationary Point

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 14, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Minimization or Maximization of Functions.*'
Ch. 10 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 387-448, 1992.

Tikhomirov, V. M. Stories About Maxima and Minima.
Providence, RI: Amer. Math. Soc, 1991.

Maximally Linear Independent

A set of VECTORS is maximally linearly independent
if including any other Vector in the Vector Space
would make it LINEARLY DEPENDENT (i.e., if any other
Vector in the Space can be expressed as a linear com-
bination of elements of a maximal set — the Basis).

Maximum

The largest value of a set, function, etc. The maximum
value of a set of elements A = {ai}^ =1 is denoted max A
or maxi at, and is equal to the last element of a sorted
(i.e., ordered) version of A. For example, given the set
{3, 5, 4, 1}, the sorted version is {1, 3, 4, 5}, so the
maximum is 5. The maximum and MINIMUM are the
simplest ORDER STATISTICS.

Maximum Absolute Column Sum Norm

The Natural Norm induced by the Li-NORM is called
the maximum absolute column sum norm and is denned

by

n

||A||i =maxV]|a ij -|
i=i

for a Matrix A.

see also Li-Norm, Maximum Absolute Row Sum

Norm

Maximum Absolute Row Sum Norm

The Natural Norm induced by the Loo-Norm is called
the maximum absolute row sum norm and is defined by

/'W = o

I'M > o

fix) < o

f\x) < 0,
/"(jr)>0

/'O0<0\ //'(*) >o
f'(x) =

stationary point

A continuous FUNCTION may assume a maximum at a
single point or may have maxima at a number of points.
A Global Maximum of a Function is the largest value
in the entire RANGE of the FUNCTION, and a LOCAL
Maximum is the largest value in some local neighbor-
hood.

max > \aij\

for a Matrix A.

see also Loo-Norm, Maximum Absolute Column
Sum Norm

Maximum Clique Problem

see Party Problem

Maximum Entropy Method

Maximum Likelihood

1149

Maximum Entropy Method

A Deconvolution Algorithm (sometimes abbrevi-
ated MEM) which functions by minimizing a smooth-
ness function ("ENTROPY") in an image. Maximum en-
tropy is also called the All-Poles Model or Autore-
GRESSIVE Model. For images with more than a million
pixels, maximum entropy is faster than the CLEAN Al-
gorithm.

MEM is commonly employed in astronomical synthe-
sis imaging. In this application, the resolution depends
on the signal to NOISE ratio, which must be speci-
fied. Therefore, resolution is image dependent and varies
across the map. MEM is also biased, since the ensemble
average of the estimated noise is Nonzero. However,
this bias is much smaller than the NOISE for pixels with
a SNR ^> 1. It can yield super-resolution, which can
usually be trusted to an order of magnitude in SOLID
Angle.

Several definitions of "ENTROPY" normalized to the flux
in the image are

*-5>(e)

k

(i)

(2)

where Mk is a "default image" and Ik is the smoothed
image. Some unnormalized entropy measures (Cornwell
1982, p. 3) are given by

(3)
(4)
(5)

(6)
(7)

see also CLEAN ALGORITHM, DECONVOLUTION,
LUCY

References

Cornwell, T. J. "Can CLEAN be Improved?" VLA Scientific
Memorandum No. 141, March 1982.

Cornwell, T. and Braun, R. "Deconvolution." Ch. 8 in Syn-
thesis Imaging in Radio Astronomy: Third NRAO Sum-
mer School, 1988 (Ed. R. A. Perley, F. R. Schwab, and
A. H. Bridle). San Francisco, CA: Astronomical Society of
the Pacific, pp. 167-183, 1989.

Christiansen, W. N. and Hogbom, J. A. Radiotelescopes, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 217-218, 1985.

Narayan, R. and Nityananda, R. "Maximum Entropy
Restoration in Astronomy." Ann. Rev. Astron. Astrophys.
24, 127-170, 1986.

Press, W. H.; Flannery, B, P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Power Spectrum Estimation by the Max-
imum Entropy (All Poles) Method" and "Maximum En-
tropy Image Restoration." §13.7 and 18.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 565-569 and 809-817, 1992.

Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr.
§3.2 in Interferometry and Synthesis in Radio Astronomy.
New York: Wiley, pp. 349-352, 1986.

Maximum Likelihood

The procedure of finding the value of one or more pa-
rameters for a given statistic which makes the known
LIKELIHOOD distribution a MAXIMUM. The maximum
likelihood estimate for a parameter fj, is denoted ft.

For a Bernoulli Distribution,

' N

d_
d9

Np

e" p (i - o)

Nq

Np(l-0)-0Nq = 0, (1)

so maximum likelihood occurs for = p. If p is not
known ahead of time, the likelihood function is

/(aii,.. .,x n \p) = P(Xi =aJi,...,X n = x n \p)
= p xl (l - p) 1 '* 1 • -p Xn (l - p) 1 " 1171
= p SXi (l-p) EC1_Xi) -/ Xi (l-p) n " S "S (2)

where x = or 1, and i = 1, . . . , n.

In/ = ^Jcci lnp-h (n- } j Xj j ln(l - p) (3)

dp p 1 — p

y ^ Xi — p y^ x i — np—p y_^ %t

* E x *

n

For a Gaussian Distribution,

f(x u ...,x n \p,a) = ]l-±=e-<«-rf'**

£(*i-M) 2

(4)
(5)
(6)

<7-v/27T

(27r)-" /2

exp

2<r 2

™. _. ..12

(7)

gives

— ^nln(27r) — nlner — —

2 2cr 2

- (8)

^(ln/)_E(^-M)_
dfi a 2

(9)

M = •
n

(10)

d(\nf) _ n i E(**-A*) 2
da a a 3

(11)

1150

Maximum Likelihood

May's Theorem

gives

Maxwell Distribution

£0* - A) 2

(12)

Note that in this case, the maximum likelihood Stan-
dard Deviation is the sample Standard Deviation,

which is a BIASED ESTIMATOR for the population STAN-
DARD Deviation.

For a weighted Gaussian Distribution,

Ti y/Zn

(2*)'

-n/2

exp

2o- 2

(13)

ln/ = -inln(2 7 r)-n^lna i -X;^-# (14)

2o-i 2

9(ln/)

(15)

gives

. E^

(16)

The Variance of the Mean is then

*' =£«*(£)'■

(17)

But

(18)

The distribution of speeds of molecules in thermal equi-
librium as given by statistical mechanics. The probabil-
ity and cumulative distributions are

P(x) =

3/2 2 -ax* 12

a ' as e /

Z?(a:):

27(|,^ax 2 )

(1)
(2)

where 7(0, a;) is an incomplete GAMMA FUNCTION and

x g [0,oo). The moments are

fi-2\

(3)
(4)

(5)
(6)

and the Mean, Variance, Skewness, and Kurtosis
are

3

/i2 = -

a

^3 = 8y

M4 = f ,

a 3 7r

2 V^ 2

I/*?

EUM 2 )
l/<Ti 2

[EdM 2 )] 2 E(iM 2 )'

For a POISSON DISTRIBUTION,

/(xi,... ,# n |A)

(19)

e -A A xx g-A A x„ e -r,A A

E-

X\\

X\ . x n .

In

/ = - n \ + (In A) Y^ x i ~ ln (n^ i! )

v Jj = -ra + ^— =0
A A

c_ E^

(20)
(21)

(22)

(23)

see also Bayesian Analysis

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Least Squares as a Maximum Likelihood Es-
timator." §15.1 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 651—655, 1992.

1

3

M

—

2 V

7TCL

2

3tt

-8

ttcl

8

/ 2"

7i

~

3 V

' 3tt

72 —

(7)
(8)

(9)
(10)

see also Exponential Distribution, Gaussian Dis-
tribution, Rayleigh Distribution

References

Spiegel, M. R, Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 119, 1992.

von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 252, 1993.

May's Theorem

Simple majority vote is the only procedure which is
Anonymous, Dual, and Monotonic.

References

May, K. "A Set of Independent Necessary and Sufficient Con-
ditions for Simple Majority Decision." Econometrica 20,
680-684, 1952.

May-Thomason Uniqueness Theorem

McNugget Number 1151

May-Thomason Uniqueness Theorem

For every infinite Loop Space Machine E, there is a
natural equivalence of spectra between EX and Segal's
spectrum 1SX.

References

May, J. P. and Thomason, R. W. "The Uniqueness of Infinite

Loop Space Machines." Topology 17, 205-224, 1978.
Weibel, C. A. "The Mathematical Enterprises of Robert

Thomason." Bull Amer. Math. Soc. 34, 1-13, 1996.

Maze

A maze is a drawing of impenetrable line segments (or
curves) with "paths" between them. The goal of the
maze is to start at one given point and find a path which
reaches a second given point.

References

Gardner, M. "Mazes." Ch. 10 in The Second Scientific Amer-
ican Book of Mathematical Puzzles & Diversions: A New
Selection. New York: Simon and Schuster, pp. 112-118,
1961.

Jablan, S. "Roman Mazes." http: //members, tripod. com/
-modularity/mazes . htm.

Matthews, W. H. Mazes and Labyrinths: Their History and
Development. New York: Dover, 1970.

Pappas, T. "Mazes." The Joy of Mathematics. San Carlos,
CA: Wide World Publ./Tetra, pp. 192-194, 1989.

Phillips, A. "The Topology of Roman Mazes." Leonardo 25,
321-329, 1992.

Shepard, W. Mazes and Labyrinths: A Book of Puzzles. New
York: Dover, 1961.

Mazur's Theorem

The generalization of the SCHONFLIES THEOREM to n-
D. A smoothly embedded n-HYPERSPHERE in an (n -f
1)-Hypersphere separates the (n + 1)-Hypersphere
into two components, each Homeomorphic to (n + 1)-
BALLS. It can be proved using MORSE THEORY.

see also Ball, Hypersphere

McCay Circle

If the Vertex ^4i of a Triangle describes the Neu-
berg Circle m, its Median Point describes a circle
whose radius is 1/3 that of the Neuberg Circle. Such
a CIRCLE is known as a McKay circle, and the three
McCay circles are Concurrent at the Median Point
M. Three homologous collinear points lie on the McCay
circles.

see also Circle, Concurrent, Median Point, Neu-
berg Circles

References

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 290 and 306, 1929.

McCoy's Theorem

If two Square n x n Matrices A and B are simulta-
neously upper triangularizable by similarity transforms,
then there is an ordering a\ , . . . , a n of the EIGENVAL-
UES of A and &i, . . . , b n of the EIGENVALUES of B so
that, given any POLYNOMIAL p(x y y) in noncommuting
variables, the Eigenvalues of p(A, B) are the numbers
p(di,bi) with i — 1, ..., n. McCoy's theorem states
the converse: If every POLYNOMIAL exhibits the correct
Eigenvalues in a consistent ordering, then A and B
are simultaneously triangularizable.

References

Luchins, E. H. and McLoughlin, M. A. "In Memoriam: Olga

Taussky-Todd." Not. Amer. Math. Soc. 43, 838-847,

1996.

McLaughlin Group

The Sporadic Group McL.

References

Wilson, R. A. "ATLAS of Finite Group Representation."

http : //for . mat . bham . ac . uk/atlas/McL . html.

McMohan's Theorem

Consider a GAUSSIAN BlVARIATE DISTRIBUTION. Let

f(xi,X2) be an arbitrary FUNCTION. Then

d 2 {f)
dp n

d 2n f

dx 1 n dx2 n

see also GAUSSIAN BlVARIATE DISTRIBUTION

McNugget Number

A number which can be obtained from an order of
McDonald's® Chicken McNuggets™ (prior to consum-
ing any), which originally came in boxes of 6, 9, and
20. All integers are McNugget numbers except 1, 2, 3,
4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31,
34, 37, and 43. Since the Happy Meal™ -sized nugget
box (4 to a box) can now be purchased separately, the
modern McNugget numbers are a linear combination of
4, 6, 9, and 20. These new-fangled numbers are much
less interesting than before, with only 1, 2, 3, 5, 7, and
11 remaining as non-McNugget numbers.

The Greedy Algorithm can be used to find a Mc-
Nugget expansion of a given INTEGER.

see also Complete Sequence, Greedy Algorithm

References

Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 19-20 and 233-234, 1991.
Wilson, D. rec. puzzles newsgroup posting, March 20, 1990.

1152

Mean

Mean Curvature

Mean

A mean is HOMOGENEOUS and has the property that a
mean fi of a set of numbers x% satisfies

min(xi, . . . ,z n ) < \i < max(xi, . . . , z n ).

There are several statistical quantities called means,
e.g., Arithmetic-Geometric Mean, Geometric
Mean, Harmonic Mean, Quadratic Mean, Root-
Mean-Square. However, the quantity referred to as
"the" mean is the Arithmetic Mean, also called the
Average.

see also Arithmetic-Geometric Mean, Average,
Generalized Mean, Geometric Mean, Harmonic
Mean, Quadratic Mean, Root-Mean-Square

Mean Cluster Count Per Site

see s-CLUSTER

Mean Cluster Density

see s-Cluster

Mean Curvature

Let Ki and k 2 be the PRINCIPAL CURVATURES, then
their MEAN

H=^(k 1 + k 2 ) (1)

is called the mean curvature. Let R\ and R 2 be the radii
corresponding to the PRINCIPAL Curvatures, then the
multiplicative inverse of the mean curvature H is given
by the multiplicative inverse of the Harmonic Mean,

"=2(jir + id

-Rl + R 2
2R\R 2

In terms of the GAUSSIAN CURVATURE K,
H=\{R 1 +R 2 )K.

The mean curvature of a REGULAR SURFACE in ,
point p is formally defined as

ff(p)=§tr(5(p)),

(2)

(3)

at a

(4)

where S is the SHAPE OPERATOR and tr(5) denotes the
Trace. For a Monge Patch with z = h(x,y),

H =

_ (1 + h v )h uu — 2h u h v h uv -f- (1 + h u 2 )h v

(1+^2+^2)3/2

(5)

(Gray 1993, p. 307).

If x : U

is a Regular Patch, then the mean

curvature is given by

H

eG - 2/F + gE

2{EG-F 2 ) '

(6)

where E, F, and G are coefficients of the first FUNDA-
MENTAL Form and e, /, and g are coefficients of the
second FUNDAMENTAL FORM (Gray 1993, p. 282). It
can also be written

H =

2[|x u P|x t ,P-(x u -X„)2]3/2

Qeti^X'u-uX'uX'u j jX u j

+ ;

(7)

2[|x u |2|x t ,|2-(x u .X u )2]3/2

Gray (1993, p. 285).

The GAUSSIAN and mean curvature satisfy

H 2 > K, (8)

with equality only at UMBILIC POINTS, since

H 2 -K 2 = \(k 1 -k 2 ) 2 . (9)

If p is a point on a Regular Surface McM 3 and v p
and w p are tangent vectors to M at p, then the mean
curvature of M at p is related to the SHAPE OPERATOR
Sby

5(v p ) x w p + v p x 5(w p ) = 2H (p)v p x w p . (10)

Let Z be a nonvanishing VECTOR FIELD on M which is
everywhere PERPENDICULAR to M, and let V and W be
Vector Fields tangent to M such that V x W = Z,

then

Z-(DyZ x W + V xD w Z)

H :

2|Z|3

(11)

(Gray 1993, pp. 291-292).

Wente (1985, 1986, 1987) found a nonspherical finite
surface with constant mean curvature, consisting of a
self-intersecting three-lobed toroidal surface. A family
of such surfaces exists.

see also Gaussian Curvature, Principal Curva-
tures, Shape Operator

References

Gray, A. "The Gaussian and Mean Curvatures." §14.5 in
Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 279-285, 1993.

Isenberg, C. The Science of Soap Films and Soap Bubbles,
New York: Dover, p. 108, 1992.

Peterson, I. The Mathematical Tourist: Snapshots of Modern
Mathematics. New York: W. H. Freeman, pp. 69-70, 1988.

Wente, H. C. "A Counterexample in 3-Space to a Conjec-
ture of H. Hopf." In Workshop Bonn 1984 f Proceedings of
the 25th Mathematical Workshop Held at the Max-Planck
Institut fur Mathematik, Bonn, June 15-22, 1984 (Ed.
F. Hirzebruch, J. Schwermer, and S. Suter). New York:
Springer- Verlag, pp. 421-429, 1985.

Wente, H. C. "Counterexample to a Conjecture of H. Hopf."
Pac. J. Math. 121, 193-243, 1986.

Wente, H. C. "Immersed Tori of Constant Mean Curvature
in M ." In Variational Methods for Free Surface Inter-
faces) Proceedings of a Conference Held in Menlo Park,
CA, Sept. 7-12, 1985 (Ed. P. Concus and R. Finn). New
York: Springer- Verlag, pp. 13-24, 1987.

Mean Deviation

Measure

1153

Mean Deviation

The Mean of the Absolute Deviations,

MD

where x is the MEAN of the distribution.
see also ABSOLUTE DEVIATION

Mean Distribution

For an infinite population with Mean fx, Standard De-
viation <r 2 , SKEWNESS 71, and KurtOSIS 72, the cor-
responding quantities for the distribution of means are

: N

' Vn

72

72,. = ^ .

<?X

71,*

(1)

(2)
(3)

(4)

For a population of M (Kenney and Keeping 1962,
p. 181),

(M)

2(M)

a 2 M-N
N M-l "

(5)
(6)

References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.

Mean Run Count Per Site

see 5-Run

Mean Run Density

see 5-RuN

Mean Square Error

see Root-Mean-Square

Mean- Value Theorem

Let f(x) be Differentiable on the Open Interval
(a, b) and Continuous on the Closed Interval [a, 6].
Then there is at least one point c in (a, b) such that

f'(c) =

f(b)-f(a)

b — a

see also EXTENDED MEAN- VALUE THEOREM, GAUSS'S
Mean- Value Theorem

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1097-1098, 1993.

Measurable Function

A function / : X — > Y for which the pre-image of every
measurable set in Y is measurable in X. For a BOREL
MEASURE, all continuous functions are measurable.

Measurable Set

If F is a Sigma Algebra and A is a Subset of X, then
A is called measurable if A is a member of F. X need
not have, a priori, a topological structure. Even if it
does, there may be no connection between the open sets
in the topology and the given SlGMA ALGEBRA.

see also MEASURABLE SPACE, SlGMA ALGEBRA

Measurable Space

A Set considered together with the SlGMA ALGEBRA

on the Set.

see also Measurable Set, Measure Space, Sigma

Algebra

Measure

The terms "measure," "measurable," etc., have very pre-
cise technical definitions (usually involving SlGMA AL-
GEBRAS) which makes them a little difficult to under-
stand. However, the technical nature of the definitions
is extremely important, since it gives a firm footing to
concepts which are the basis for much of ANALYSIS (in-
cluding some of the slippery underpinnings of CALCU-
LUS).

For example, every definition of an INTEGRAL is based
on a particular measure: the RlEMANN INTEGRAL is
based on Jordan Measure, and the Lebesgue In-
tegral is based on Lebesgue Measure. The study
of measures and their application to Integration is
known as MEASURE THEORY.

A measure is formally defined as a Map m : F — ^ M (the
reals) such that m(0) = and, if A n is a COUNTABLE
Sequence in F and the A n are pairwise DISJOINT, then

[JA n \ =Y,m{A n ).

If, in addition, m(X) = 1, then m is said to be a PROB-
ABILITY Measure.

A measure m may also be defined on Sets other than
those in the SlGMA Algebra F. By adding to F all
sets to which m assigns measure zero, we again obtain
a SlGMA ALGEBRA and call this the "completion" of F
with respect to m. Thus, the completion of a SlGMA
Algebra is the smallest Sigma Algebra containing
F and all sets of measure zero.

see also Almost Everywhere, Borel Measure, Er-
godic Measure, Euler Measure, Gauss Measure,
Haar Measure, Hausdorff Measure, Helson-
Szego Measure, Integral, Jordan Measure, Leb-
esgue Measure, Liouville Measure, Mahler's

1154 Measure Algebra

Medial Triangle

Measure, Measurable Space, Measure Algebra,
Measure Space, Minkowski Measure, Natural
Measure, Probability Measure, Wiener Mea-
sure

Measure Algebra

A Boolean Sigma Algebra which possesses a Mea-
sure.

Measure Polytope

see Hypercube

Measure-Preserving Transformation

see ENDOMORPHISM

Measure Space

A measure space is a Measurable Space possessing a
NONNEGATIVE MEASURE. Examples of measure spaces
include n-D EUCLIDEAN SPACE with LEBESGUE MEA-
SURE and the unit interval with Lebesgue Measure
(i.e., probability).

see also LEBESGUE MEASURE, MEASURABLE SPACE

Measure Theory

The mathematical theory of how to perform INTEGRA-
TION in arbitrary MEASURE SPACES.

see also Cantor Set, Fractal, Integral, Mea-
surable Function, Measurable Set, Measurable
Space, Measure, Measure Space

References

Doob, J. L. Measure Theory. New York: Springer-Verlag,
1994.

Evans, L. C. and Gariepy, R. F. Measure Theory and Fi-
nite Properties of Functions. Boca Raton, FL: CRC Press,
1992.

Gordon, R. A. The Integrals of Lebesgue, Denjoy, Perron,
and Henstock. Providence, RI: Amer. Math. Soc, 1994.

Halmos, P. R. Measure Theory. New York: Springer-Verlag,
1974.

Henstock, R, The General Theory of Integration. Oxford,
England: Clarendon Press, 1991.

Kestelman, H. Modern Theories of Integration, 2nd rev. ed.
New York: Dover, 1960.

Rao, M. M. Measure Theory And Integration. New York:
Wiley, 1987.

St rook, D. W. A Concise Introduction to the Theory of In-
tegration, 2nd ed. Boston, MA: Birkhauser, 1994.

Mechanical Quadrature

see Gaussian Quadrature

Mecon

Buckminster Fuller's term for the Truncated OCTA-
HEDRON.

see also Dymaxion

Medial Axis

The boundaries of the cells of a VORONOI DIAGRAM.

Medial Deltoidal Hexecontahedron

The Dual of the Rhombidodecadodecahedron.

Medial Disdyakis Triacontahedron

The Dual of the Truncated Dodecadodecahe-

DRON.

Medial Hexagonal Hexecontahedron

The Dual of the Snub Icosidodecadodecahedron.

Medial Icosacronic Hexecontahedron

The Dual of the Icosidodecadodecahedron.

Medial Inverted Pentagonal
Hexecontahedron

The Dual of the Inverted Snub Dodecadodecahe-
dron.

Medial Pentagonal Hexecontahedron

The Dual of the Snub Dodecadodecahedron.

Medial Rhombic Triacontahedron

A ZONOHEDRON which is the DUAL of the Dodecado-

DECAHEDRON. It is also called the SMALL STELLATED

Triacontahedron.

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 125, 1989.

Medial Triambic Icosahedron

The Dual of the Ditrigonal Dodecadodecahe-
dron.

Medial Triangle

The Triangle AMiM 2 M 3 formed by joining the Mid-
points of the sides of a Triangle AA!A 2 A 3 . The
medial triangle is sometimes also called the AUXILIARY
TRIANGLE (Dixon 1991). The medial triangle has TRI-
linear Coordinates

A' = : b" 1 : c' 1
B* = a" 1 : : c" 1
C = a" 1 : b" 1 : 0.

The medial triangle AM[M' 2 M^ of the medial trian-
gle AMiM 2 M 3 of a Triangle AAxA 2 A z is similar to
AA X A 2 A 3 .

Medial Triangle Locus Theorem

Median Triangle 1155

see also ANTICOMPLEMENTARY TRIANGLE

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.

Washington, DC: Math. Assoc. Amer., pp. 18-20, 1967.
Dixon > R. Mathographics. New York: Dover, p. 56, 1991.

Medial Triangle Locus Theorem

Given an original triangle (thick line), find the Medial
Triangle (outer thin line) and its Incircle. Take the
Pedal Triangle (inner thin line) of the Medial Tri-
angle with the Incenter as the Pedal Point. Now
pick any point on the original triangle, and connect it to
the point located a half-PERlMETER away (gray lines).
Then the locus of the Midpoints of these lines (the *s
in the above diagram) is the PEDAL TRIANGLE.

References

Honsberger, R. More Mathematical Morsels. Washington,

DC: Math. Assoc. Amer., pp. 261-267, 1991.
Tsintsifas, G. "Problem 674." Crux Math., p. 256, 1982.

Median Point

see Centroid (Geometric)

Median (Statistics)

The middle value of a distribution or average of the two
middle items, denoted /ii/ 2 or x. For small samples, the
Mean is more efficient than the median and approxi-
mately 7r/2 less. It is less sensitive to outliers than the
Mean (Kenney and Keeping 1962, p. 211).

For large N samples with population median Xq,

Xq

SNf 2 (xo)'

The median is an L-ESTIMATE (Press et al. 1992).
see also Mean, Midrange, Mode

References

Kenney, J, F, and Keeping, E. S. Mathematics of Statistics,
PL 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 694, 1992.

Median (Triangle)

M *2

The CEVIAN from a TRIANGLE'S VERTEX to the MID-
POINT of the opposite side is called a median of the
TRIANGLE. The three medians of any TRIANGLE are
Concurrent, meeting in the TRIANGLE'S CENTROID
(which has TRILINEAR COORDINATES l/o : 1/6 : 1/c).
In addition, the medians of a TRIANGLE divide one an-
other in the ratio 2:1. A median also bisects the AREA
of a Triangle.

Let mi denote the length of the median of the ith side
ai. Then

mi 2 -i(2a 2 2 +2a 3 2 -ai 2 ) (1)

mi 2 + m 2 2 + m 3 = f(ai + a 2 + a 3 )

(2)

(Johnson 1929, p. 68). The AREA of a TRIANGLE can
be expressed in terms of the medians by

A = | ysm{sm - mi)(s m - m2)(sm - ms), (3)

where

s m = | (mi + m 2 + m 3 ).

(4)

see also BlMEDIAN, EXMEDIAN, EXMEDIAN POINT,

Heronian Triangle, Medial Triangle

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 7-8, 1967.

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 68 and 173-175, 1929.

Median Triangle

A Triangle whose sides are equal and Parallel to the
Medians of a given Triangle. The median triangle of

the median triangle is similar to the given TRIANGLE in
the ratio 3/4.

References

Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 282-283, 1929.

1156

Mediant

Mediant

Given a FAREY SEQUENCE with consecutive terms h/k
and h' /k', then the mediant is defined as the reduced
form of the fraction (h + h')/(k + k').
see also Farey Sequence

References

Conway, J. H. and Guy, R. K. "Farey Fractions and Ford

Circles." The Book of Numbers. New York: Springer-

Verlag, pp. 152-154, 1996.

Mega

Defined in terms of CIRCLE NOTATION by Steinhaus
(1983, pp. 28-29) as

® =

A = l2r0 = S

where Steinhaus-Moser Notation has also been
used.

see also Megistron, Moser, Steinhaus-Moser No-
tation

References

Steinhaus, H, Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, 1983.

Megistron

A very LARGE Number defined in terms of CIRCLE NO-
TATION by Steinhaus (1983) as @.

see also MEGA, Moser

References

Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 28-29, 1983.

Mehler's Bessel Function Formula

2 f°°

J (x) — — I sin(#cosh£)d£,

^ Jo

where J (x) is a zeroth order BESSEL FUNCTION OF THE
First Kind.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary

of Mathematics. Cambridge, MA: MIT Press, p. 1472,

1980.

Mehler-Dirichlet Integral

y/2 f a cos[(n+£)0]

V2 r

P n (cosa) = — /
* Jo

\/cos <j) — cos a
where P n (x) is a Legendre Polynomial,

#,

Meijer's G-Function
Mehler's Hermite Polynomial Formula

E

H n (x)H n (y) fl r

= (l + 4iiT) ' exp

2xyw — (x 2 -\-y 2 )w 2
1-w 2

where H n (x) is a HERMITE POLYNOMIAL.

References

Almqvist, G. and Zeilberger, D. "The Method of Differen-
tiating Under the Integral Sign." J. Symb. Comput. 10,
571-591, 1990.

Foata, D. "A Combinatorial Proof of the Mehler Formula."
J. Comb. Th. Ser. A 24, 250-259, 1978.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, pp. 194-195, 1996.

Rainville, E. D. Special Functions. New York: Chelsea,
p. 198, 1971.

Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 380, 1975.

Mehler Quadrature

see JaCOBI-GauSS QUADRATURE

Meijer's G- Function

/°f m ' n ( *i a i'"'» p i —

u ™ r'"i h) - 2™

nr=i r ( 6 J- z )n; = ia-^+«)

L n.

^d

bi+z)U

7

,j=n+l

rfe - z)

-x z dz,

where F(z) is the Gamma FUNCTION. The CONTOUR
jl and other details are discussed by Gradshteyn and
Ryzhik (1980, pp. 896-903 and 1068-1071). Prudnikov
et al. (1990) contains an extensive nearly 200-page list-
ing of formulas for the Meijer G-function.

see also Fox's if-FuNCTiON, G-Function, Mac-
Robert's ^-Function, Ramanujan g- and G-
Functions

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.

Luke, Y. L. The Special Functions and Their Approxima-
tions, 2 vols. New York: Academic Press, 1969.

Mathai, A. M. A Handbook of Generalized Special Functions
for Statistical and Physical Sciences. New York: Oxford
University Press, 1993.

Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A.;
Integrals and Series, Vol. 3: More Special Functions.
Newark, NJ: Gordon and Breach, 1990.

MeisseVs Formula

MEM

1157

MeissePs Formula

A modification of Legendre's Formula for the Prime
Counting Function ir(x). It starts with

Mellin Transform

w-'+E I - E

+

Ki<a

E

l<i<j<a

PiPj

PiPjPk

l<i<j<k<a u

-j-n(x) - a + P 2 {x,a) + P 3 (x,a) + ■

(i)

where |_#J is the FLOOR FUNCTION, P 2 (x,a) is the num-
ber of Integers piPj < x with a + 1 < j < j y and
Ps(x,a) is the number of Integers piPjPk < £ with
a+l<2<j<fc. Identities satisfied by the Ps include

P 2 {x,a) = J2 *(j^-{i-l)

(2)

for p a < Pi < v^ an d

niy/x/pi)

E E

i=a+l j=t

Meissel's formula is

* U7 -(J-i)

.(3)

+ §(b + c-2)(6-c+l)-

PiPj

c<t<6 N /

(4)

where

6 = tt(x 1/2 )

C = 7r(x ' ).

(5)
(6)

Taking the derivation one step further yields Lehmer's
Formula.

see also Legendre's Formula, Lehmer's Formula,
Prime Counting Function

References

Riesel, H. "Meissel's Formula." Prime Numbers and Com-
puter Methods for Factorization, 2nd ed. Boston, MA:
Birkhauser, p. 12, 1994.

(f>(z) = / t z_1 /(*)d*
Jo

J — oo

see also Strassen Formulas

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 795, 1985.

Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, pp. 254-257, 1965.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 469-471, 1953.

Melnikov- Arnold Integral

/oo
cos [|m<£(£) — At] dt,
■oo

where the function

4>(t) =4tan" 1 (e t ) - tt

describes the motion along the pendulum SEPARATRIX.
Chirikov (1979) has shown that this integral has the
approximate value

A m (X) :

— pt — s — o for A >

-|fl^+rr(m + l)sin(7rm) for A < 0.

References

Chirikov, B. V. "A Universal Instability of Many-
Dimensional Oscillator Systems." Phys. Rep. 52, 264-379,
1979.

Melodic Series

If ai, a 2 , a3, -. . is an ARTISTIC Series, then 1/ai, 1/aa,
l/a3, ... is a Melodic Series. The Recurrence Re-
lation obeyed by melodic series is

, bibi+2 . &z+2 ,

O i+ 3 = — j" + T 6 *+ 2 *

Oi+1 &i+l

see also Artistic Series

References

Duffin, R. J. "On Seeing Progressions of Constant Cross Ra-
tio." Amer. Math. Monthly 100, 38-47, 1993.

MEM

see Maximum Entropy Method

1158 Memoryless

Menger Sponge

Memoryless

A variable x is memoryless with respect to t if, for all s
with t # 0,

P(x > s + t\x>t) = P{x > s).

(1)

Equivalently,

P(*>* + t>*>*) = p {x>8) (2 )

P(x>t) V ; W

P{x > s + 1) = P(» > s)P(a; > t). (3)

The Exponential Distribution, which satisfies

-At

p(x >t) = e
P(x>s + t) = e~ Hs + t \

(4)
(5)

and therefore

P(x > s + 1) = P(cc > s)P(oj > t) = e~ Xs e~ xt

_ -\(8 + t)

(6)

is the only memoryless random distribution.
see also EXPONENTIAL DISTRIBUTION

Menage Number

see Married Couples Problem

Menage Problem

see Married Couples Problem

Menasco's Theorem

For a BRAID with M strands, R components, P positive
crossings, and N negative crossings,

(P-N<U++M-R if P>iV
\P-N<U-+M-R if P<iV,

where U± are the smallest number of positive and nega-
tive crossings which must be changed to crossings of the
opposite sign. These inequalities imply BENNEQUIN's
Conjecture. Menasco's theorem can be extended to
arbitrary knot diagrams.

see also Bennequin's Conjecture, Braid, Unknot-
ting Number

References

Cipra, B. "From Knot to Unknot." What's Happening in
the Mathematical Sciences, Vol. 2. Providence, RI: Amer.
Math. Soc, pp. 8-13, 1994.

Menasco, W. W. "The Bennequin-Milnor Unknotting Con-
jectures." C. R. Acad. Sci. Paris Sir. I Math. 318, 831-
836, 1994.

Menelaus' Theorem
C

A B

For Triangles in the Plane,

AD-BE-CF = BD-CE< AF.

(1)

For Spherical Triangles,

sin AD • sin BE • sin CF = sin BD ■ sin CE • sin AF. (2)

This can be generalized to n-gons P = [Vi,...,K
where a transversal cuts the side ViVi+i in Wi for i
. . . , n, by

1,

n

WiVi

i+l

(-1)".

Here, AB\\CD and

(3)

(4)

is the ratio of the lengths [A, B] and [C, D] with a PLUS
or MINUS Sign depending if these segments have the
same or opposite directions (Griinbaum and Shepard
1995). The case n = 3 is Pasch's Axiom.

see also CEVA'S THEOREM, HOEHN'S THEOREM,

Pasch's Axiom

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 122, 1987.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 66-67, 1967.

Griinbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the
Area Principle." Math. Mag. 68, 254-268, 1995.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., p. xxi, 1995.

Menger's n-Arc Theorem

Let G be a graph with A and B two disjoint n- tuples of
Vertices. Then either G contains n pairwise disjoint
^.S-paths, each connecting a point of A and a point of
B, or there exists a set of fewer than n VERTICES that
separate A and B.

References

Menger, K. Kurventheorie. Leipzig, Germany: Teubner,
1932.

Menger Sponge

Menu's Surface

A Fractal which is the 3-D analog of the Sierpinski
Carpet. Let N n be the number of filled boxes, L n the
length of a side of a hole, and V n the fractional VOLUME
after the nth iteration.

N n = 20 n

V n = LjN n = (§y

(1)
(2)
(3)

The Capacity Dimension is therefore

r lniV n ln(20 n )

rfcap = - lim — — = - lim ( >

ln20 _ ln(2 2 -5) _ 21n2 + ln5

ln3 ln3

2.726833028,.,,

In3

(4)

J. Mosely is leading an effort to construct a large Menger
sponge out of old business cards.

see also SIERPINSKI CARPET, TETRIX

References

Dickau, R. M. "Menger (Sierpinski) Sponge." http: //forum

. swarthmore . edu/advanced/robertd/ sponge .html.
Mosely, J. "Menger's Sponge (Depth 3)." http: //world.

std.com/-j9/sponge/.

Menn's Surface

Mercator Projection 1159
Mercator Projection

The following equations place the a;- AXIS of the projec-
tion on the equator and the y-AxiS at Longitude Ao,
where A is the LONGITUDE and cf> is the LATITUDE.

x — A — A

y = ln[tan(±7r+§0)]

J y 1 — sm <j> J
= sinh~ (tan0)
= tanh - (sin</>)
= ln(tan <fi + sec <j>).

(i)

(2)

(3)

(4)
(5)
(6)

A surface given by the parametric equations

x(Uj V ) = u
y(u,v) - v

( \ 4.2 2

z{u, v) — au + u v — v .

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 631, 1993.

Mensuration Formula

A mensuration formula is simply a formula for comput-
ing the length-related properties of an object (such as
Area, Circumradius, etc., of a Polygon) based on

other known lengths, areas, etc. Beyer (1987) gives a
collection of such formulas for various plane and solid
geometric figures.

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 121-133, 1987.

The inverse FORMULAS are

(f> = ±tt - 2tan~ 1 (e~ v ) = tan" x (sinh y) (7)

A = :r-r-Ao. (8)

Loxodromes are straight lines and Great Circles are
curved.

1160 Mercator Projection

An oblique form of the Mercator projection is illustrated
above. It has equations

tan -1 [tan cos <j} p + sin</> p sin(A — Ao)] , Q .
= cos(A-Ao) l ^

(10)

x —

y=iln

cos(A -
^|±^)=tanh- 1 ^

(^

where

_ x / cos 0i sin 02 cos Ai — sin 0i cos 02 cos A2 \
p I sin 0i cos 02 sin A2 — cos 0i sin 02 sin Ai J

(11)
(12)

(13)

_i / cos(A p -Ai) \
V tan 0i J

> p = tan

— cos0p cos0sin(A — Ao).

A = sin P sin
The inverse FORMULAS are

= sin

A = Ao + tan

cos 0p sin a; \ ( .

un P tanh y + -^ (14)

cosh y J

_! / sin 0p sin x — cos P sinh y \ f .

1 I ) • (15)

V cos x /

There is also a transverse form of the Mercator projec-
tion, illustrated above. It is given by the equations

= §ln(i±|)=tanh- 1 B

tan0

y = tan

where

cos(A — Ao)

, =sin -i (be£\

V cosh x )

A = Ao + tan - 1 (E^)

V cos D J

B = cos0sin(A — Ao)
D-y + 0o.

(16)

(17)

(18)
(19)

(20)
(21)

Mergelyan-Wesler Theorem

Mercator projection with central MERIDIAN in the cen-
ter of the zone. The zones extend from 80° S to 84° N
(Dana).

see also SPHERICAL SPIRAL

References

Dana, P. H. "Map Projections." http://www.utexas.edu/
depts/grg/gcraft/notes/mapporoj/mapproj .html.

Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U, S. Government Printing Office, pp. 38-75, 1987.

Mercator Series

The Taylor Series for the Natural Logarithm

ln(l + z)

W

+ \x*

which was found by Newton, but independently discov-
ered and first published by Mercator in 1668.

see also LOGARITHMIC NUMBER, NATURAL LOGA-
RITHM

Mercer's Theorem

see Riemann-Lebesgue Lemma

Mergelyan-Wesler Theorem

Let P ~ {1)1,1)2, • . •} be an infinite set of disjoint open
Disks D n of radius r n such that the union is almost the
unit Disk. Then

X/ n

Define

M X {P)

Tl-1

(1)

(2)

Then there is a number e(P) such that M X (P) diverges
for x < e(P) and converges for x > e(P). The above
theorem gives

1 < e{P) < 2. (3)

There exists a constant which improves the inequality,
and the best value known is

S = 1.306951....

(4)

References

Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
pp. 36-37, 1983.

Mandelbrot, B. B. Fractals. San Francisco, CA: W. H. Free-
man, p. 187, 1977.

Melzack, Z. A. "On the Solid Packing Constant for Circles."
Math. Comput. 23, 1969.

Finally, the "universal transverse Mercator projection"
is a Map Projection which maps the Sphere into 60
zones of 6° each, with each zone mapped by a transverse

Meridian

Mersenne Prime

1161

Meridian

A line of constant LONGITUDE on a SPHEROID (or
Sphere). More generally, a meridian of a Surface of
Revolution is the intersection of the surface with a
PLANE containing the axis of revolution.

see also LATITUDE, LONGITUDE, PARALLEL (SURFACE

of Revolution), Surface of Revolution

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 358, 1993.

Meromorphic

A meromorphic FUNCTION is complex analytic in all but
a discrete subset of its domain, and at those singularities
it must go to infinity like a POLYNOMIAL (i.e., have no
Essential Singularities). An equivalent definition of
a meromorphic function is a complex analytic Map to
the Riemann Sphere.
see also Essential Singularity, Riemann Sphere

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 382-383, 1953.

Mersenne Number

A number of the form

M n = 2 n - 1

(1)

for n an INTEGER is known as a Mersenne number. The
Mersenne numbers are therefore 2-REPDIGITS, and also
the numbers obtained by setting x = 1 in a Fermat
Polynomial. The first few are 1, 3, 7, 15, 31, 63, 127,
255, ... (Sloane's A000225).

The number of digits D in the Mersenne number M n is

D = Llog(2" - 1) + Ij , (2)

where [a; J is the FLOOR FUNCTION, which, for large n,
gives

D « [n log 2 + IJ a; [0.30102971 + lj = L0.301029nJ + 1.

(3)

In order for the Mersenne number M n to be Prime, n
must be PRIME. This is true since for COMPOSITE n
with factors r and s, n — vs. Therefore, 2" — 1 can be
written as 2 TS - 1, which is a BINOMIAL NUMBER and
can be factored. Since the most interest in Mersenne
numbers arises from attempts to factor them, many au-
thors prefer to define a Mersenne number as a number
of the above form

M p = 2 P - 1, (4)

but with p restricted to PRIME values.

The search for Mersenne Primes is one of the most
computationally intensive and actively pursued areas of
advanced and distributed computing.

see also CUNNINGHAM NUMBER, EBERHART'S CON-
JECTURE, Fermat Number, Lucas-Lehmer Test,
Mersenne Prime, Perfect Number, Repunit,
Riesel Number, Sierpinski Number of the Sec-
ond Kind, Sophie Germain Prime, Superperfect
Number, Wieferich Prime

References

Pappas, T. "Mersenne's Number." The Joy of Mathematics.

San Carlos, CA: Wide World PubL/Tetra, p. 211, 1989.
Shanks, D. Solved and Unsolved Problems in Number Theory,

4th ed. New York: Chelsea, pp. 14, 18-19, 22, and 29-30,

1993.
Sloane, N. J. A. Sequence A000225/M2655 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Mersenne Prime

A Mersenne Number which is Prime is called a
Mersenne prime. In order for the Mersenne number M n
defined by

M n = T - 1

for n an INTEGER to be PRIME, n must be PRIME. This
is true since for Composite n with factors r and s,
n = rs. Therefore, 2 n — 1 can be written as 2 rs — 1, which
is a BINOMIAL NUMBER and can be factored. Every
Mersenne Prime gives rise to a Perfect Number.

If n = 3 (mod 4) is a PRIME, then 2n + 1 DIVIDES M n
IFF 2n-fl is PRIME. It is also true that PRIME divisors of
2 P - 1 must have the form 2kp+ 1 where k is a POSITIVE
Integer and simultaneously of either the form 8n + l or
8n — 1 (Uspensky and Heaslet). A PRIME factor p of a
Mersenne number M q - 2 q - 1 is a WIEFERICH PRIME
Iff p 2 \2 q - 1, Therefore, MERSENNE Primes are not
Wieferich Primes. All known Mersenne numbers M p
with p Prime are Squarefree. However, Guy (1994)
believes that there are M v which are not Squarefree.

Trial Division is often used to establish the Compos-
ITENESS of a potential Mersenne prime. This test im-
mediately shows M p to be Composite for p = 11, 23,
83, 131, 179, 191, 239, and 251 (with small factors 23,
47, 167, 263, 359, 383, 479, and 503, respectively). A
much more powerful primality test for M p is the Lucas-

Lehmer Test.

It has been conjectured that there exist an infinite num-
ber of Mersenne primes, although finding them is com-
putationally very challenging. The table below gives the
index p of known Mersenne primes (Sloane's A000043)
M p , together with the number of digits, discovery years,
and discoverer. A similar table has been compiled by
C. Caldwell. Note that the region after the 35th known
Mersenne prime has not been completely searched, so
identification of "the" 36th Mersenne prime is tentative.
L. Welsh maintains an extensive bibliography and his-
tory of Mersenne numbers. G. Woltman has organized

1162

Mersenne Prime

Mertens Conjecture

a distributed search program via the Internet in which
hundreds of volunteers use their personal computers to
perform pieces of the search.

#

P

Digits

Year

Published Reference

1

2

1

Anc.

2

3

1

Anc.

3

5

2

Anc.

4

7

3

Anc.

5

13

4

1461

Reguis 1536, Cataldi 1603

6

17

6

1588

Cataldi 1603

7

19

6

1588

Cataldi 1603

8

31

10

1750

Enler 1772

9

61

19

1883

Pervouchine 1883,
Seelhoff 1886

10

89

27

1911

Powers 1911

11

107

33

1913

Powers 1914

12

127

39

1876

Lucas 1876

13

521

157

1952

Lehmer 1952-3

14

607

183

1952

Lehmer 1952-3

15

1279

386

1952

Lehmer 1952-3

16

2203

664

1952

Lehmer 1952-3

17

2281

687

1952

Lehmer 1952-3

18

3217

969

1957

Riesel 1957

19

4253

1281

1961

Hurwitz 1961

20

4423

1332

1961

Hurwitz 1961

21

9689

2917

1963

Gillies 1964

22

9941

2993

1963

Gillies 1964

23

11213

3376

1963

Gillies 1964

24

19937

6002

1971

Tuckerman 1971

25

21701

6533

1978

Noll and Nickel 1980

26

23209

6987

1979

Noll 1980

27

44497

13395

1979

Nelson and Slowinski 1979

28

86243

25962

1982

Slowinski 1982

29

110503

33265

1988

Colquitt and Welsh 1991

30

132049

39751

1983

Slowinski 1988

31

216091

65050

1985

Slowinski 1989

32

756839

227832

1992

Gage and Slowinski 1992

33

859433

258716

1994

Gage and Slowinski 1994

34

1257787

378632

1996

Slowinski and Gage

35

1398269

420921

1996

Armengaud, Woltman, et al.

36?

2976221

895832

1997

Spence

37?

3021377

909526 |

1998

Clarkson, Woltman, et al.

see also CUNNINGHAM NUMBER, FERMAT-LUCAS NUM-
BER, Fermat Number, Fermat Number (Lu-
cas), Fermat Polynomial, Lucas-Lehmer Test,
Mersenne Number, Perfect Number, Repunit,
Superperfect Number

References

Bateman, P. T.; Selfridge, J. L.; and Wagstaff, S. S. "The

New Mersenne Conjecture." Amer. Math. Monthly 96,

125-128, 1989.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 66 , 1987.
Beiler, A. H, Ch, 3 in Recreations in the Theory of Numbers:

The Queen of Mathematics Entertains. New York: Dover,

1966.
Caldwell, C. "Mersenne Primes: History, Theorems

and Lists." http : //www . utm . edu/research/primes/

mersenne . shtml.
Caldwell, C. "GIMPS Finds a Prime! 2 1398269 - 1 is Prime."

http://www.utm.edu/research/primes/note5/1398269/.

Colquitt, W. N. and Welsh, L. Jr. "A New Mersenne Prime."
Math. Comput. 56, 867-870, 1991.

Conway, J. H. and Guy, R. K. "Mersenne's Numbers." In The
Book of Numbers. New York: Springer- Ver lag, pp. 135-
137, 1996.

Gillies, D. B. "Three New Mersenne Primes and a Statistical
Theory." Math Comput. 18, 93-97, 1964.

Guy, R. K. "Mersenne Primes. Repunits. Fermat Numbers.
Primes of Shape k>2 n + 2 [sic]." §A3 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 8-13, 1994.

Haghighi, M. "Computation of Mersenne Primes Using a
Cray X~MP." Intl. J. Comput. Math. 41, 251-259, 1992.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 14-16, 1979.

Kraitchik, M. "Mersenne Numbers and Perfect Numbers."
§3.5 in Mathematical Recreations. New York: W. W. Nor-
ton, pp. 70-73, 1942.

Kravitz, S. and Berg, M. "Lucas' Test for Mersenne Numbers
6000 < p < 7000." Math. Comput 18, 148-149, 1964.

Lehmer, D. H. "On Lucas's Test for the Primality of
Mersenne's Numbers." J, London Math. Soc. 10, 162-
165, 1935.

Leyland, P. ftp : //sable . ox . ac . uk/pub/math/f actors/
mersenne.

Mersenne, M. Cogitata Physico-Mathematica. 1644,

Mersenne Organization, "GIMPS Discovers 36th Known
Mersenne Prime, 2 2976221 - 1 is Now the Largest Known
Prime." http : //www . mersenne . org/2976221 . htm.

Mersenne Organization. "GIMPS Discovers 37th Known
Mersenne Prime, 2 3021377 — 1 is Now the Largest Known
Prime." http : //www . mersenne . org/302 1377 . htm.

Noll, C. and Nickel, L. "The 25th and 26th Mersenne
Primes." Math. Comput. 35, 1387-1390, 1980.

Powers, R. E. "The Tenth Perfect Number." Amer. Math.
Monthly 18, 195-196, 1911.

Powers, R. E. "Note on a Mersenne Number." Bull. Amer.
Math. Soc. 40, 883, 1934.

Sloane, N. J. A. Sequence A000043/M0672 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Slowinski, D. "Searching for the 27th Mersenne Prime." J.
Recreat. Math. 11, 258-261, 1978-1979.

Slowinski, D. Sci. News 139, 191, 9/16/1989.

Tuckerman, B. "The 24th Mersenne Prime." Proc. Nat
Acad. Sci. USA 68, 2319-2320, 1971.

Uhler, H. S. "A Brief History of the Investigations on
Mersenne Numbers and the Latest Immense Primes."
Scripta Math. 18, 122-131, 1952.

Uspensky, J. V. and Heaslet, M. A. Elementary Number The-
ory. New York: McGraw-Hill, 1939.
# Weisstem, E. W. "Mersenne Numbers." http: //www.
astro . Virginia . edu/-eww6n/math/notebooks/
Mersenne. m.
Welsh, L. "Marin Mersenne." http://www.scruznet.com/

-luke /mersenne .htm.
Welsh, L. "Mersenne Numbers & Mersenne Primes Bibliog-
raphy." http://www.scruznet.com/-luke/biblio.htm.
Woltman, G. "The GREAT Internet Mersenne Prime
Search." http : //www . mersenne . org/prime . htm.

Mertens Conjecture

Given Mertens Function defined by

mWe^w,

(i)

Mertens Conjecture

Mertens Constant 1163

where fi(n) is the MOBIUS Function, Mertens (1897)
conjecture states that

\M{x)\ <x 1/2

(2)

for x > 1. The conjecture has important implications,
since the truth of any equality of the form

\M(x)\ < cx 1/2

(3)

for any fixed c (the form of Mertens conjecture with
c = 1) would imply the RlEMANN HYPOTHESIS. In 1885,
Stieltjes claimed that he had a proof that M(x)x~ 1 ^ 2
always stayed between two fixed bounds. However, it
seems likely that Stieltjes was mistaken.

Mertens conjecture was proved false by Odlyzko and te
Riele (1985). Their proof is indirect and does not pro-
duce a specific counterexample, but it does show that

limsupM(:c);c 1/2 > 1.06

X— J'OO

liminf M(x)x~ 1/2 < -1.009.

(4)

(5)

Odlyzko and te Riele (1985) believe that there are no
counterexamples to Mertens conjecture for x < 10 , or
even 10 30 . Pintz (1987) subsequently showed that at
least one counterexample to the conjecture occurs for
x < 10 65 , using a weighted integral average of M(x)/x
and a discrete sum involving nontrivial zeros of the RlE-
MANN Zeta Function.

It is still not known if

limsup |M(x)|a; ' — oo,

(6)

although it seems very probable (Odlyzko and te Riele

1985).

see also Mertens Function, Mobius Function, Rie-
mann Hypothesis

References

Anderson, R. J. "On the Mertens Conjecture for Cusp

Forms." Mathematika 26, 236-249, 1979.
Anderson, R. J. "Corrigendum: 'On the Mertens Conjecture

for Cusp Forms.'" Mathematika 27, 261, 1980.
Devlin, K. "The Mertens Conjecture." Irish Math. Soc. Bull.

17, 29-43, 1986.
Grupp, F. "On the Mertens Conjecture for Cusp Forms."

Mathematika 29, 213-226, 1982.
Jurkat, W. and Peyerimhoff, A. "A Constructive Approach

to Kronecker Approximation and Its Application to the

Mertens Conjecture." J. reine angew. Math. 286/287,

322-340, 1976.
Mertens, F. "Uber eine zahlentheoretische Funktion."

Sitzungsber. Akad. Wiss. Wien Ha 106, 761-830, 1897.
Odlyzko, A. M. and te Riele, H. J. J. "Disproof of the Mertens

Conjecture." J. reine angew. Math. 357, 138—160, 1985.
Pintz, J. "An Effective Disproof of the Mertens Conjecture."

Asterique 147-148, 325-333 and 346, 1987.
te Riele, H. J. J. "Some Historical and Other Notes About

the Mertens Conjecture and Its Recent Disproof." Nieuw

Arch. Wisk. 3, 237-243, 1985.

Mertens Constant

A constant related to the Twin Primes Constant
which appears in the FORMULA for the sum of inverse

Primes

V" - =ln\nx + B 1 +o(l)

(1)

p prime

which is given by

ln(l-p- 1 )+ i
V

B x =7+ Ys

p prim

Flajolet and Vardi (1996) show that

oo

Y[ C(m)^ (m)/m

0.261497. (2)

e 1 = e 7

(3)

m=2

where 7 is the Euler-Mascheroni Constant, C(n) is
the Riemann Zeta Function, and fi(n) is the Mobius
Function. The constant Bi also occurs in the Sum-
matory Function of the number of Distinct Prime
Factors,

\ w(k) = n In In n + B\n + o(n)

(4)

k = 2

(Hardy and Wright 1979, p. 355).
The related constant

£2 = 7+ Y,

lnU-p- 1 )*

p-1

ftj 1.034653 (5)

appears in the Summatory Function of the Divisor
Function cr (n) = ft(n),

y^fi(fc) ^nlnlnn + ^2 + o(n)

(6)

(Hardy and Wright 1979, p. 355).

see also BRUN'S CONSTANT, PRIME NUMBER, TWIN
Primes Constant

References

Flajolet, P. and Vardi, I. "Zeta Function Expan-
sions of Classical Constants." Unpublished manu-
script. 1996. http://pauillac.inria.fr/algo/flajolet/
Publicat ions/landau. ps.

Hardy, G. H. and Weight, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, pp. 351 and 355, 1979.

1164 Mertens Function
Mertens Function

The summary function

M(n) = V/i(A) = -?n + O(v^),

* J TV

where /i(n) is the MOBIUS FUNCTION. The first few
values are 1, 0, -1, -1, -2, -1, -2, -2, -2, -1, -2,
-2, . . . (Sloane's A002321). The first few values of n at
which M(n) = are 2, 39, 40, 58, 65, 93, 101, 145, 149,
150, ... (Sloane's A028442).

Mertens function obeys

X>(s)

(Lehman 1960). The analytic form is unsolved, although
Mertens Conjecture that

1/2

\M(x)\ < x

has been disproved.

Lehman (1960) gives an algorithm for computing M(x)
with 0(x 2 ' 3+e ) operations, while the Lagarias-Odlyzko
(1987) algorithm for computing the Prime Count-
ing Function tt(x) can be modified to give M(x) in
0(x 3/5+£ ) operations.

see also MERTENS CONJECTURE, MOBIUS FUNCTION

References

Lagarias, J. and Odlyzko, A. "Computing 7r(as): An Analytic

Method." J. Algorithms 8, 173-191, 1987.
Lehman, R. S. "On Liouville's Function." Math. Comput.

14, 311-320, 1960.
Odlyzko, A. M. and te Riele, H. J. J. "Disproof of the Mertens

Conjecture." J. reine angew. Math. 357, 138-160, 1985.
Sloane, N. J. A. Sequence A028442/M002321 in "An On-Line

Version of the Encyclopedia of Integer Sequences." 0102

Mertens Theorem

ri2<p<x (i- 1)

Um g prime

x->oo §.

In x

1,

Metabiaugmented Hexagonal Prism

References

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, p. 351, 1979.

Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, pp. 66-67,
1994.

Mertz Apodization Function

An asymmetrical Apodization Function defined by

M(x y b,d) = i

{ °

for x < —b

(X-

- b)/{2b) for -b < x < b

1

for b < x < b + 2d

lo

for x < b + 2d,

where the two-sided portion is 26 long (total) and the
one-sided portion is 6 + 2d long (Schnopper and Thomp-
son 1974, p. 508). The Apparatus Function is

M A (fc,M) =

sin[27rfc(fe + 2d)

2ixk

+i

cos[27rfc(6 + 2d)]
2ixk

sin(2&) \
47T 2 k 2 bj

where 7 is the Euler-Mascheroni Constant and
e -7 = 0.56145....

References

Schnopper, H. W. and Thompson, R. I. "Fourier Spectrom-
eters." In Methods of Experimental Physics 12A. New
York: Academic Press, pp. 491-529, 1974.

Mesh Size

When a Closed Interval [a, b] is partitioned by points
a < x\ < X2 < ... < Xn-i < b, the lengths of the
resulting intervals between the points are denoted Axi,
A#2, •-., A# n , and the value max Arc*; is called the
mesh size of the partition.

see also INTEGRAL, LOWER SUM, RlEMANN INTEGRAL,

Upper Sum

Mesokurtic

A distribution with zero KuRTOSIS (72 = 0).

see also KURTOSIS, Leptokurtic

Metabiaugmented Dodecahedron

see Johnson Solid .

Metabiaugmented Hexagonal Prism

see Johnson Solid

Metabiaugmented Truncated Dodecahedron

Metric

1165

Metabiaugmented Truncated Dodecahedron

see Johnson Solid

Metabidiminished Icosahedron

see Johnson Solid

Metabidiminished Rhombicosidodecahedron

see Johnson Solid

Metabigyrate Rhombicosidodecahedron

see Johnson Solid

Metadrome

A metadrome is a number whose HEXADECIMAL digits
are in strict ascending order. The first few are 0, 1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, . . . (Sloane's A023784).

see also HEXADECIMAL

References

Sloane, N. J. A. Sequence A023784 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Metagyrate Diminished
Rhombicosidodecahedron

see Johnson Solid

Metalogic

see Metamathematics

Metamathematics

The branch of LOGIC dealing with the study of the
combination and application of mathematical symbols,
sometimes called Metalogic. Metamathematics is the
study of Mathematics itself, and one of its primary
goals is to determine the nature of mathematical rea-
soning (Hofstadter 1989).
see also LOGIC, MATHEMATICS

References

Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra,

3rd ed. New York: Macmillan, p. 326, 1965.
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden

Braid. New York: Vintage Books, p. 23, 1989.

Method

A particular way of doing something, sometimes also
called an ALGORITHM or PROCEDURE. (According to
Petkovsek et al. (1996), "a method is a trick that has
worked at least twice.")

see also ADAMS-BASHFORTH-MOULTON METHOD,

Adams' Method, Backus-Gilbert Method, Ba-
der-Deuflhard Method, Bailey's Method, Bair-
stow's Method, Brent's Factorization Meth-
od, Brent's Method, Circle Method, Conjugate
Gradient Method, Criss-Cross Method, Crout's

Method, de la Loubere's Method, Dixon's Fac-
torization Method, Dixon's Random Squares
Factorization Method, Elliptic Curve Factor-
ization Method, Euler's Factorization Method,
Excludent Factorization Method, Exhaustion
Method, False Position Method, Fermat's Fac-
torization Method, Frobenius Method, Gill's
Method, Gosper's Method, Graeffe's Meth-
od, Greene's Method, Halley's Method, Hor-
ner's Method, Hutton's Method, Jacobi Meth-
od, Kaps-Rentrop Methods, Laguerre's Meth-
od, Lambert's Method, Legendre's Factoriza-
tion Method, Lehmer Method, Lehmer-Schur
Method, Lenstra Elliptic Curve Method, Lin's
Method, Lozenge Method, LUX Method, Mapes'
Method, Maximum Entropy Method, Milne's
Method, Muller's Method, Newton's Method,
Newton-Raphson Method, Number Field Sieve
Factorization Method, Overlapping Resonance
Method, Pollard Monte Carlo Factorization
Method, Pollard p Factorization Method, Pol-
lard p - 1 Factorization Method, Predictor-
Corrector Methods, Quadratic Sieve Factor-
ization Method, Resonance Overlap Method,
rosenbrock methods, runge-kutta method,
Schroder's Method, Secant Method, Siamese
Method, Simplex Method, Snake Oil Method,
Square Root Method, Steepest Descent Meth-
od, Tangent Hyperbolas Method, Undetermined
Coefficients Method, Williams p + 1 Factoriza-
tion Method, Wynn's Epsilon Method

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A-B. Welles-
ley, MA: A. K. Peters, p. 117, 1996.

Metric

A Nonnegative function g{x,y) describing the "DIS-
TANCE" between neighboring points for a given Set. A
metric satisfies the Triangle Inequality

g(x,y)+g(y,z) >g(x,z),

with equality Iff x — y, and is symmetric, so

9fay) = g{y,x).

A Set possessing a metric is called a METRIC Space.
When viewed as a TENSOR, the metric is called a MET-
RIC Tensor.

see also CAYLEY-KLEIN-HlLBERT METRIC, DISTANCE,

Fundamental Forms, Hyperbolic Metric, Metric
Entropy, Metric Equivalence Problem, Metric
Space, Metric Tensor, Part Metric, Riemannian

Metric, Ultrametric

References

Gray, A. "Metrics on Surfaces." Ch. 13 in Modem Differen-
tial Geometry of Curves and Surfaces. Boca Raton, FL:
CRC Press, pp. 251-265, 1993.

1166 Metric Entropy

Metric Tensor

Metric Entropy

Also known as Kolmogorov Entropy, Kolmogor-
OV- Sinai Entropy, or KS Entropy. The metric entropy
is for nonchaotic motion and > for CHAOTIC motion.

References

Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, p. 138, 1993.

Metric Equivalence Problem

1. Find a complete system of invariants, or

2. decide when two METRICS differ only by a coordinate
transformation.

The most common statement of the problem is, "Given
METRICS g and g' , does there exist a coordinate trans-
formation from one to the other?" Christoffel and Lip-
schitz (1870) showed how to decide this question for two

RlEMANNIAN METRICS.

The solution by E. Cartan requires computation of the
10th order Covariant Derivatives. The demonstra-
tion was simplified by A. Karlhede using the TETRAD
formalism so that only seventh order COVARIANT
DERIVATIVES need be computed. However, in many
common cases, the first or second-order Derivatives
are Sufficient to answer the question.

References

Karlhede, A. and Lindstrom, U. "Finding Space-Time Ge-
ometries without Using a Metric." Gen. Relativity Gravi-
tation 15, 597-610, 1983.

Metric Space

A Set S with a global distance Function (the Metric
g) which, for every two points x,y in 5, gives the DIS-
TANCE between them as a Nonnegative Real Num-
ber g(x,y). A metric space must also satisfy

1. g(x,x) = Iff x~y,

2. g(x,y) =g(y,x),

3. The Triangle Inequality g(x,y) + g(y,z) >
g(x r z).

References

Munkres, J. R. Topology: A First Course. Englewood Cliffs,

NJ: Prentice-Hall, 1975.
Rudin, W. Principles of Mathematical Analysis. New York:

McGraw-Hill, 1976.

Metric Tensor

A Tensor, also called a Riemannian Metric, which
is symmetric and POSITIVE DEFINITE. Very roughly,
the metric tensor gtj is a function which tells how to
compute the distance between any two points in a given
Space. Its components can be viewed as multiplication
factors which must be placed in front of the differen-
tial displacements dxi in a generalized PYTHAGOREAN
THEOREM

In Euclidean Space, g^ = Sij where S is the Kron-
ECKER DELTA (which is for i / j and 1 for i — j),
reproducing the usual form of the PYTHAGOREAN THE-
OREM

ds = dxi -+- dx2 + . . . . (2)

The metric tensor is defined abstractly as an Inner
Product of every Tangent Space of a Manifold
such that the INNER Product is a symmetric, non-
degenerate, bilinear form on a VECTOR SPACE. This
means that it takes two VECTORS v,w as arguments
and produces a REAL NUMBER (v, w) such that

(fcv, w) — k (v, w) = (v, kw) (3)

(v + w, x) = (v, x) + (w, x) (4)

<v,w + x) = (v,w> + <v,x> (5)

<v,w) = <w,v) (6)

<v,v)>0, (7)

with equality Iff v = 0.

In coordinate NOTATION (with respect to the basis),

(8)

(9)

(10)

g a0 =<?-{?

g a {3 = e a • e/s.

_ d C d £

a f\£&

g ^~ dx»dx» Vaf3i

where rj^ is the MINKOWSKI METRIC. This can also be
written

where

g = D L r,D,

(11)

D =°?

(12)

n T = n

(13)

d ik r fc
dx™ 9 " 9 = dx" 5i

(14)

gives

Bg li

dgu

9il d^ = ~ 9 dx^- (15)

The metric is Positive Definite, so a metric's Dis-
criminant is Positive. For a metric in 2-space,

g = 011022 - gi2 > 0.

(16)

The Orthogonality of Contravariant and Covari-
ant metrics stipulated by

9i*9 ij = Si

(17)

ds = gudxi + gi2 dx\ dx-z + 022 dx-z 4- .

(i)

for i = 1, . . . , n gives n linear equations relating the
2n quantities gij and g %3 . Therefore, if n metrics are
known, the others can be determined.

Metric Tensor

In 2-space,

Mice Problem

1167

11 _ P22

12 21

9 = 9 :

22 _ 5ll

g!2

If ^ is symmetric, then

<?a0 = <7/?a

(18)
(19)
(20)

(21)
(22)

In Euclidean Space (and all other symmetric
Spaces),

&=& = £, (23)

9oiot

1

r,aa '

(24)

The Angle <j> between two parametric curves is given
by

ri £2 _ gi2

0102 '

SO

ri • r 2

sin<£

5i 92

and

£l£2

|ri x r 2 | = #ip 2 sin <f> = y/g.
The Line Element can be written

ds — dxi dxi = gftj <% dqfj
where EINSTEIN SUMMATION has been used. But

(25)
(26)
(27)

(28)

dxi = ^ dqi + _^ dq2 + _L dqz = _L dq ., (29)

9ij

2-J dqidqj'

(30)

For Orthogonal coordinate systems, gtj = for i / j,
and the Line Element becomes (for 3-space)

ds =511 dqi + 522 d?2 2 + 533 dqs 2

= (h! d qi ) 2 + (h 2 dq 2 ) 2 + (fc 3 dq 3 ) 2 , (31)

where /i* = y/gu are called the Scale Factors.

see also Curvilinear Coordinates, Discriminant
(Metric), Lichnerowicz Conditions, Line Ele-
ment, Metric, Metric Equivalence Problem,
Minkowski Space, Scale Factor, Space

Mex

The Minimum excluded value. The mex of a Set S
of Nonnegative Integers is the least Nonnegative
Integer not in the set.

see also Mex Sequence

References

Guy, R. K. "Max and Mex Sequences." §E27 in Unsolved

Problems in Number Theory, 2nd ed. New York: Springer-

Verlag, pp. 227-228, 1994.

Mex Sequence

A sequence defined from a FINITE sequence ao, ai, . . . ,
a n by defining a n +i = mexj(ai + a n _j), where mex is
the Mex (minimum excluded value).

see also Max Sequence, Mex

References

Guy, R. K. "Max and Mex Sequences." §E27 in Unsolved

Problems in Number Theory, 2nd ed. New York: Springer-

Verlag, pp. 227-228, 1994,

Mian-Chowla Sequence

The sequence produced by starting with a\ = 1 and
applying the GREEDY ALGORITHM in the following way:
for each k > 2, let a* be the least INTEGER exceeding
afc_i for which clj + a*, are all distinct, with 1 < j < k.
This procedure generates the sequence 1, 2, 4, 8, 13,
21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290,
... (Sloane's A005282). The RECIPROCAL sum of the
sequence,

satisfies

a,i

2.1568 < S < 2.1596.

see also A-Sequence, ^-Sequence

References

Guy, R. K. "B 2 -Sequences." §E28 in Unsolved Problems

in Number Theory, 2nd ed. New York: Springer- Verlag,

pp. 228-229, 1994.
Sloane, N. J. A. Sequence A005282/M1094 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Mice Problem

n mice start at the corners of a regular n-gon of unit
side length, each heading towards its closest neighboring
mouse in a counterclockwise direction at constant speed.
The mice each trace out a Spiral, meet in the center of
the Polygon, and travel a distance

i

d n —

l-«»(£)'
The first few values for n = 2, 3, . . . , are

1,1,1,1(5 + ^), 2,

2 + V2, ■

'(¥)

,3 + VE,...,

1168

Mid-Arc Points

Midpoint

giving the numerical values 0.5, 0.666667, 1, 1.44721, 2,

2.65597, 3.41421, 4.27432, 5.23607, ....

see also APOLLONIUS PURSUIT PROBLEM, PURSUIT

Curve, Spiral, Tractrix

References

Bernhart, A. "Polygons of Pursuit." Scripta Math. 24, 23-

50, 1959.
Madachy, J. S. Madachy's Mathematical Recreations. New

York: Dover, pp. 201-204, 1979,

Mid- Arc Points

M BC

Midcircle

The mid-arc points Mab, Mac, and M B c of a TRI-
ANGLE AABC are the points on the ClRCUMClRCLE of
the triangle which lie half-way along each of the three
ARCS determined by the vertices (Johnson 1929). These
points arise in the definition of the FUHRMANN CIRCLE
and FUHRMANN TRIANGLE, and lie on the extensions
of the PERPENDICULAR BISECTORS of the triangle sides
drawn from the ClRCUMCENTER O.

Kimberling (1988, 1994) and Kimberling and Veldkamp
(1987) define the mid-arc points as the POINTS which
have TRIANGLE CENTER FUNCTIONS

ai
a 2

= [cos(§£) + cos(|C)]sec(§,4)
= [cos(f£) + cos(±C)]csc(§A).

see also FUHRMANN CIRCLE, FUHRMANN TRIANGLE

References

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 228-229, 1929.

Kimberling, C. "Problem 804." Nieuw Archief voor
Wiskunde 6, 170, 1988.

Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.

Kimberling, C. and Veldkamp, G. R. "Problem 1160 and So-
lution." Crux Math. 13, 298-299, 1987.

Hoc <D(

The midcircle of two given CIRCLES is the Circle which
would Invert the circles into each other. Dixon (1991)
gives constructions for the midcircle for four of the five
possible configurations. In the case of the two given
Circles tangent to each other, there are two midcircles.

see also INVERSION, INVERSION CIRCLE

References

Dixon, R. Mathographics. New York: Dover, pp. 66-68, 1991.

Middlespoint

see Mittenpunkt

Midpoint

M

The point on a Line Segment dividing it into two seg-
ments of equal length. The midpoint of a line segment is
easy to locate by first constructing a Lens using circular
arcs, then connecting the cusps of the Lens. The point
where the cusp-connecting line intersects the segment is
then the midpoint (Pedoe 1995, p. xii). It is more chal-
lenging to locate the midpoint using only a COMPASS,
but Pedoe (1995, pp. xviii-xix) gives one solution.

In a Right Triangle, the midpoint of the Hy-
potenuse is equidistant from the three VERTICES
(Dunham 1990).

Midpoint Ellipse

Midy's Theorem 1169

Given a Triangle AAiA 2 A 3 with Area A, locate the
midpoints Mi. Now inscribe two triangles AP1P2P3 and
AQ1Q2Q3 with Vertices Pi and Q; placed so that
~pjAl = QiMi. Then AP1P2P3 and AQ1Q2Q3 have
equal areas

V ai a2 03 /

m,2m2 77137711 77117712

^2^3

0301

aia2 .

where a; are the sides of the original triangle and mi are
the lengths of the Medians (Johnson 1929),

see also ARCHIMEDES' MIDPOINT THEOREM, BROCARD

Midpoint, Circle-Point Midpoint Theorem, Line

Segment, Median (Triangle), Midpoint Ellipse

References

Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 120-121, 1990.

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 80, 1929.

Pedoe, D, Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., 1995.

Midpoint Ellipse

The unique Ellipse tangent to the Midpoints of a Tri-
angle's LEGS. The midpoint ellipse has the maximum
Area of any Inscribed Ellipse (Chakerian 1979). Un-
der an Affine Transformation, the midpoint ellipse
can be transformed into the INCIRCLE of an EQUILAT-
ERAL Triangle.

see also Affine Transformation, Ellipse, Incir-
cle, Midpoint, Triangle

References

Central Similarities. University of Minnesota College Geom-
etry Project. Distributed by International Film Bureau,
Inc.

Chakerian, G. D. "A Distorted View of Geometry." Ch. 7
in Mathematical Plums (Ed. R. Honsberger). Washington,
DC: Math. Assoc. Amer., pp. 135-136 and 145-146, 1979,

Pedoe, D. "Thinking Geometrically," Amer. Math. Monthly
77, 711-721, 1970.

Midradius

The Radius of the Midsphere of a Polyhedron, also
called the Interradius. For a Regular Polyhedron
with Schlafli Symbol {<?,p}, the Dual Polyhedron
is {p, q}. Denote the INRADIUS r, midradius p, and ClR-
CUMRADIUS R, and let the side length be a. Then

2
r =

= a cot ( — J
W

, D 2 2,2

+ R = a -f p

(1)

(2)

For Regular Polyhedra and Uniform Polyhedra,
the Dual Polyhedron has Circumradius p 2 /r and

Inradius p 2 /R. Let 6 be the Angle subtended by the
Edge of an Archimedean Solid. Then

r=lacos(i0)cot(i0)

(3)

p=facot(i0)

(4)

R= iacsc(i0),

(5)

so

r :p: R = cos(|0) : 1 : sec(|(9)

(6)

(Cundy and Rollett 1989). Expressing the midradius in
terms of the INRADIUS r and CIRCUMRADIUS R gives

1 = \y/2\Jr 2 + ryjr 2 + a 2

(7)

for an ARCHIMEDEAN SOLID.

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 126-127, 1989.

Midrange

midrange[/(:c)] = §{max[/(a;)] +min[/(a)]}.

see also Maximum, Mean, Median (Statistics),
Minimum

Midsphere

The Sphere with respect to which the Vertices of a
POLYHEDRON are the poles of the planes of the faces
of the Dual Polyhedron (and vice versa). It touches
all Edges of a Semiregular Polyhedron or Regu-
lar Polyhedron. It is also called the Intersphere
or Reciprocating Sphere.

see also ClRCUMSPHERE, DUAL POLYHEDRON, lN-

SPHERE

Midy's Theorem

If the period of a REPEATING DECIMAL for a/p has an
EVEN number of digits, the sum of the two halves is a
string of 9s, where p is Prime and a/p is a Reduced

Fraction.

see also DECIMAL EXPANSION, REPEATING DECIMAL

References

Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 158-160,
1957.

1170

MikusinskVs Problem

Miller's Solid

Mikusiriski's Problem

Is it possible to cover completely the surface of a SPHERE
with congruent, nonoverlapping arcs of GREAT CIR-
CLES? Conway and Croft (1964) proved that it can be
covered with half-open arcs, but not with open arcs.
They also showed that the Plane can be covered with
congruent closed and half-open segments, but not with
open ones.

References

Conway, J. H. and Croft, H. T. "Covering a Sphere with

Great-Circle Arcs." Proc. Cambridge Phil Soc. 60, 787-

900, 1964.
Gardner, M. "Point Sets on the Sphere." Ch. 12 in Knotted

Doughnuts and Other Mathematical Entertainments. New

York: W. H. Freeman, pp. 145-154, 1986.

Milin Conjecture

An Inequality which Implies the correctness of the
Robertson Conjecture (Milin 1971). de Branges
(1985) proved this conjecture, which led to the proof
of the full Bieberbach Conjecture.

see also BIEBERBACH CONJECTURE, ROBERTSON CON-
JECTURE

Miller Cylindrical Projection

References

Acta

de Branges, L. "A Proof of the Bieberbach Conjecture."

Math. 154, 137-152, 1985.
Milin, I. M. Univalent Functions and Orthonormal Systems.

Providence, RI: Amer. Math. Soc, 1977.
Stewart, I. Prom Here to Infinity: A Guide to Today's

Mathematics. Oxford, England: Oxford University Press,

p. 165, 1996.

Mill

The n-roll mill curve is given by the equation

v - U) x

n-2 2 , /W\ n _4 4

y + 1 4 \ x y -••• = a '

where (™) is a Binomial Coefficient.

References

von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 86, 1993.

Miller's Algorithm

For a catastrophically unstable recurrence in one direc-
tion, any seed values for consecutive Xj and Xj+i will
converge to the desired sequence of functions in the op-
posite direction times an unknown normalization factor.

Miller- Askinuze Solid

see Elongated Square Gyrobicupola

A Map Projection given by the following transforma-
tion,

x = A — Ao

y=|ln[tan(i7r+ §</>)]
= |8inh- 1 [tan(^)].

(1)
(2)
(3)

Here x and y are the plane coordinates of a projected
point, A is the longitude of a point on the globe, Ao is
central longitude used for the projection, and <f> is the
latitude of the point on the globe. The inverse FORMU-
LAS are

(^ftan-V^ 5 )
A = Ao + x.

tan-^sinMfy)] (4)
(5)

References

Snyder, J. P. Map Projections— A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 86-89, 1987.

Miller's Primality Test

If a number fails this test, it is not a PRIME. If the
number passes, it may be a PRIME. A number passing
Miller's test is called a Strong Pseudoprime to base
a. If a number n does not pass the test, then it is called a
Witness for the Compositeness of n. If n is an Odd,
Positive Composite Number, then n passes Miller's
test for at most (n — l)/4 bases with 1 < a < -1 (Long
1995). There is no analog of CARMICHAEL NUMBERS
for Strong Pseudoprimes.

The only Composite Number less than 2.5 xlO 13 which
does not have 2, 3, 5, or 7 as a Witness is 3215031751,
Miller showed that any composite n has a Witness less
than 70(lnn) 2 if the Riemann Hypothesis is true.

see also Adleman-Pomerance-Rumely Primality
Test, Strong Pseudoprime

References

Long, C. T. Th. 4.21 in Elementary Introduction to Number

Theory, 3rd ed. Prospect Heights, IL: Waveland Press,

1995.

Miller's Solid

see Elongated Square Gyrobicupola

Milliard

Milnor's Conjecture 1171

Milliard

In British, French, and German usage, one milliard
equals 10 9 . American usage does not have a number
called the milliard, instead using the term Billion to
denote 10 9 .

see also Billion, Large Number, Million, Trillion

Millin Series

The series with sum

s'

Ei = ^-^).

where F k is a Fibonacci Number (Honsberger 1985).
see also FIBONACCI NUMBER

References

Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 135-137, 1985.

Million

The number 1,000,000 = 10 6 . While one million in
America means the same thing as one million in Britain,
the words Billion, TRILLION, etc., refer to different
numbers in the two naming systems. While Americans
may say "Thanks a million" to express gratitude, Nor-
wegians offer "Thanks a thousand" ("tusen takk").

see also Billion, Large Number, Milliard, Thou-
sand, Trillion

Mills' Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Mills (1947) proved the existence of a constant 8 =
1.3064... such that

V

(i)

is Prime for all n > 1, where [x\ is the Floor Func-
tion. It is not, however, known if is IRRATIONAL.
Mills' proof was based on the following theorem by Ho-
heisel (1930) and Ingham (1937). Let p n be the nth
Prime, then there exists a constant K such that

Pn+l ~Pn <KpJ /S (2)

for all n. This has more recently been strengthened to

*, ^ If*. 1051/1920 / oA

Pn+1 ~pn < Kpn (3)

(Mozzochi 1986). If the Riemann Hypothesis is true,
then Cramer (1937) showed that

Pn + l — Pn = 0(lnpn^/p^)

(4)

(Finch).

Hardy and Wright (1979) point out that, despite the
beauty of such FORMULAS, they do not have any prac-
tical consequences. In fact, unless the exact value of
is known, the PRIMES themselves must be known in
advance to determine 6. A generalization of Mills' theo-
rem to an arbitrary sequence of POSITIVE INTEGERS is
given as an exercise by Ellison and Ellison (1985). Con-
sequently, infinitely many values for 9 other than the
number 1.3064 . . . are possible.

References

Caldwell, C. "Mills' Theorem — A Generalization." http://
www.utm.edu/research/primes/notes/proofs/A3n.html.

Ellison, W. and Ellison, F. Prime Numbers. New York: Wi-
ley, pp. 31-32, 1985.

Finch, S. "Favorite Mathematical Constants." http: //www.
mathsoft .com/asolve/constant/mills/raills.html.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.

Mills, W. H. "A Prime-Representing Function." Bull. Amer.
Math. Soc. 53, 604, 1947.

Mozzochi, C. J. "On the Difference Between Consecutive
Primes." J. Number Th. 24, 181-187, 1986.

Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 135 and 191-193, 1989.

Ribenboim, P. The Little Book of Big Primes. New York:
Springer- Verlag, pp. 109-110, 1991.

Milne's Method

A Predictor- Corrector Method for solution of
Ordinary Differential Equations. The third-order
equations for predictor and corrector are

y n+1 = t/ n _ 3 + \h(2y' n - y n ^ + 2y' n _ 2 ) + G(h 5 )

1/n+l - yn-l + |%n-l + Wn + Vn+l) + ^(^)-

Abramowitz and Stegun (1972) also give the fifth order
equations and formulas involving higher derivatives.

see also Adams' Method, Gill's Method, Predic-
tor-Corrector Methods, Runge-Kutta Method

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 896-897, 1972.

Milnor's Conjecture

The Unknotting Number for a Torus Knot (p, q)
is (p -l)(q— l)/2. This 40-year-old CONJECTURE was
proved (Adams 1994) in Kronheimer and Mrowka (1993,
1995).

see also Torus Knot, Unknotting Number

References

Adams, C. C. The Knot Book: An Elementary Introduction

to the Mathematical Theory of Knots. New York: W. H.

Freeman, p. 113, 1994.
Kronheimer, P. B. and Mrowka, T. S. "Gauge Theory for

Embedded Surfaces. I." Topology 32, 773-826, 1993.
Kronheimer, P. B. and Mrowka, T. S. "Gauge Theory for

Embedded Surfaces. II." Topology 34, 37-97, 1995.

1172

Milnor's Theorem

Minimal Surface

Milnor's Theorem

If a Compact Manifold M has Nonnegative Ricci
Curvature, then its Fundamental Group has at
most POLYNOMIAL growth. On the other hand, if M has
Negative curvature, then its Fundamental Group
has exponential growth in the sense that n(A) grows ex-
ponentially, where n(A) is (essentially) the number of
different "words" of length A which can be made in the
Fundamental Group.

References

Chavel, I. Riemannian Geometry: A Modern Introduction.
New York: Cambridge University Press, 1994.

Minimal Cover

A minimal cover is a COVER for which removal of one
member destroys the covering property. Let fj,(n,k) be
the number of minimal covers of {1, . . . , n} with k mem-
bers. Then

1 ak (l

m — k

m!s(n,7n),

where (£) is a BINOMIAL COEFFICIENT, s{n,m) is a
Stirling Number of the Second Kind, and

a*; = min(n,2 fc — 1).

Special cases include //(n, 1) = 1 and /x(n, 2) = s(n +
1,3).

k 1 2 3 I 5 6 7~

Sloane 000392 003468 016111

1 ]

2 :

L 1

3 ]

L 6

1

4 ]

L 25

22

1

5 ]

L 90

305

65

1

6 ]

L 301

3410

2540

171

1

7 1

L 966

33621

77350

17066

420 1

see also Cover, Lew /c-gram, Stirling Number of
the Second Kind

References

Hearne, T. and Wagner, C. "Minimal Covers of Finite Sets."
Disc. Math. 5, 247-251, 1973.

Macula, A. J. "Lewis Carroll and the Enumeration of Mini-
mal Covers." Math. Mag. 68, 269-274, 1995.

Minimal Discriminant

see Frey Curve

Minimal Matrix

A Matrix with Determinant whose Determinant
becomes Nonzero when any element on or below the
diagonal is changed from to 1. An example is

1-10
0-10

111-1

10

M

There are 2 n 1 minimal Special Matrices of size n x
n.

see also Special Matrix

References

Knuth, D. E. "Problem 10470." Amer. Math. Monthly 102,
655, 1995.

Minimal Residue

The value b or 6 — m, whichever is smaller in ABSOLUTE
Value, where a = b (mod m).

see also RESIDUE (CONGRUENCE)

Minimal Set

A SET for which the dynamics can be generated by the
dynamics on any subset.

Minimal Surface

Minimal surfaces are defined as surfaces with zero Mean
Curvature, and therefore satisfy Lagrange's Equa-
tion

(1 + fy 2 )f™ + 2f X fyf X y + (1 + f X 2 )fyy = 0.

Minimal surfaces may also be characterized as surfaces
of minimal AREA for given boundary conditions. A
Plane is a trivial Minimal Surface, and the first non-
trivial examples (the CATENOID and HELICOID) were
found by Meusnier in 1776 (Meusnier 1785).

Euler proved that a minimal surface is planar Iff its
Gaussian Curvature is zero at every point so that it
is locally SADDLE-shaped. The Existence of a solution
to the general case was independently proven by Douglas
(1931) and Rado (1933), although their analysis could
not exclude the possibility of singularities. Osserman
(1970) and Gulliver (1973) showed that a minimizing
solution cannot have singularities.

The only known complete (boundaryless), embedded
(no self- intersect ions) minimal surfaces of finite topol-
ogy known for 200 years were the CATENOID, HELICOID,
and Plane. Hoffman discovered a three-ended GENUS
1 minimal embedded surface, and demonstrated the ex-
istence of an infinite number of such surfaces. A four-
ended embedded minimal surface has also been found.
L. Bers proved that any finite isolated SINGULARITY of
a single- valued parameterized minimal surface is remov-
able.

A surface can be parameterized using a ISOTHERMAL
Parameterization. Such a parameterization is mini-
mal if the coordinate functions Xk are HARMONIC, i.e.,
</>k(C) are Analytic A minimal surface can therefore
be defined by a triple of Analytic FUNCTIONS such
that <fck<i>k — 0. The REAL parameterization is then ob-
tained as

x k = 9t I <f> k (0<%- (!)

Minimal Surface

Minimum 1173

But, for an Analytic Function / and a Meromor-
PHIC function #, the triple of functions

>i(C) = /(i-s 2 )

(2)

>2(() = if(l + g 2 )

(3)

>s(C) = 2/ ff

(4)

are Analytic as long as / has a zero of order > m
at every Pole of g of order m. This gives a minimal
surface in terms of the Enneper- WElERSTRAft PARAM-
ETERIZATION

U

I

f(l-9 2 )

if(l+9 2 )

2/P

dC.

(5)

see also Bernstein Minimal Surface Theorem,
Calculus of Variations, Catalan's Surface,
Catenoid, Costa Minimal Surface, Enneper-Wei-
erstraB Parameterization, Flat Surface, Hen-
neberg's Minimal Surface, Hoffman's Minimal
Surface, Immersed Minimal Surface, Lichtenfels
Surface, Maeder's Owl Minimal Surface, Niren-
berg's Conjecture, Parameterization, Plateau's
Problem, Scherk's Minimal Surfaces, Trinoid,
Unduloid

References

Dickson, S. "Minimal Surfaces." Mathematica J. 1, 38-40,
1990.

Dierkes, U.; Hildebrandt, S.; Kuster, A.; and Wohlraub, O.
Minimal Surfaces, 2 vols. Vol. 1: Boundary Value Prob-
lems. Vol. 2: Boundary Regularity. Springer- Verlag, 1992.

do Carmo, M. P. "Minimal Surfaces." §3.5 in Mathemati-
cal Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 41-43, 1986.

Douglas, J. "Solution of the Problem of Plateau." Trans.
Amer. Math. Soc. 33, 263-321, 1931.

Fischer, G. (Ed.). Plates 93 and 96 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 89 and 96, 1986.

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 280, 1993.

Gulliver, R. "Regularity of Minimizing Surfaces of Prescribed
Mean Curvature." Ann. Math. 97, 275-305, 1973.

Hoffman, D. "The Computer-Aided Discovery of New Em-
bedded Minimal Surfaces." Math. Intell. 9, 8-21, 1987.

Hoffman, D. and Meeks, W. H. III. The Global Theory of
Properly Embedded Minimal Surfaces. Amherst, MA: Uni-
versity of Massachusetts, 1987.

Lagrange. "Essai d'une nouvelle methode pour determiner les
maxima et les minima des formules integrates indefinies."
1776.

Meusnier, J. B. "Memoire sur la courbure des surfaces."
Mem. des savans etrangers 10 (lu 1776), 477-510, 1785.

Nitsche, J. C. C. Introduction to Minimal Surfaces. Cam-
bridge, England: Cambridge University Press, 1989.

Osserman, R. A Survey of Minimal Surfaces. New York:
Van Nostrand Reinhold, 1969.

Osserman, R. "A Proof of the Regularity Everywhere of the
Classical Solution to Plateau's Problem." Ann. Math. 91,
550-569, 1970.

Rado, T. "On the Problem of Plateau." Ergeben. d. Math,
u. ihrer Grenzgebiete. Berlin: Springer- Verlag, 1933.

Minimax Approximation

A minimization of the MAXIMUM error for a fixed num-
ber of terms.

Minimax Polynomial

The approximating POLYNOMIAL which has the small-
est maximum deviation from the true function. It is
closely approximated by the Chebyshev Polynomials
of the First Kind.

Minimax Theorem

The fundamental theorem of Game Theory which
states that every FINITE, Zero-Sum, two-person GAME
has optimal Mixed Strategies. It was proved by John
von Neumann in 1928.

Formally, let X and Y be Mixed Strategies for play-
ers A and B. Let A be the PAYOFF MATRIX. Then

maxminX T AY = minmaxX T AY
x y y x

v,

where v is called the VALUE of the GAME and X and Y
are called the solutions. It also turns out that if there
is more than one optimal MIXED STRATEGY, there are
infinitely many.

see also Mixed Strategy

References

Willem, M. Minimax Theorem. Boston, MA: Birkhauser,
1996.

Minimum

The smallest value of a set, function, etc. The minimum
value of a set of elements A = {ai}^ is denoted min^l
or mini a% , and is equal to the first element of a sorted
(i.e., ordered) version of A. For example, given the set
{3, 5, 4, 1}, the sorted version is {1, 3, 4, 5}, so the
minimum is 1. The MAXIMUM and minimum are the
simplest Order Statistics.

/'W =

/■(jr)<0\ f'(x)>Q

/'(*) = o

/U)>0

f'(x)<0

stationary point

A continuous Function may assume a minimum at a
single point or may have minima at a number of points.
A Global Minimum of a Function is the smallest
value in the entire Range of the Function, while a
Local Minimum is the smallest value in some local
neighborhood.

For a function f(x) which is CONTINUOUS at a point x 0y
a Necessary but not Sufficient condition for f(x)
to have a Relative Minimum at x = x is that x be
a Critical Point (i.e., f(x) is either not Differen-
tiable at x or x is a STATIONARY Point, in which
case f f (x ) = 0).

1174 Minkowski-Bouligand Dimension

Minkowski Integral Inequality

The First Derivative Test can be applied to Con-
tinuous FUNCTIONS to distinguish minima from MAX-
IMA. For twice differentiate functions of one variable,
/(a?), or of two variables, f(x,y), the Second Deriv-
ative Test can sometimes also identify the nature of
an EXTREMUM. For a function f(x), the Extremum
Test succeeds under more general conditions than the
Second Derivative Test.

see also CRITICAL POINT, EXTREMUM, FIRST DERIVA-
TIVE Test, Global Maximum, Inflection Point,
Local Maximum, Maximum, Midrange, Order
Statistic, Saddle Point (Function), Second De-
rivative Test, Stationary Point

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 14, 1972.

Brent, R. P. Algorithms for Minimization Without Deriva-
tives. Englewood Cliffs, NJ: Prentice-Hall, 1973.

Nash, J. C. "Descent to a Minimum I— II: Variable Metric
Algorithms." Chs. 15-16 in Compact Numerical Methods
for Computers: Linear Algebra and Function Minimisa-
tion, 2nd ed. Bristol, England: Adam Hilger, pp. 186-206,
1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Minimization or Maximization of Functions."
Ch. 10 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 387-448, 1992.

Tikhomirov, V. M. Stories About Maxima and Minima.
Providence, RI: Amer. Math. Soc, 1991.

Minkowski-Bouligand Dimension

In many cases, the Hausdorff Dimension correctly
describes the correction term for a resonator with FRAC-
TAL Perimeter in Lorentz's conjecture. However, in
general, the proper dimension to use turns out to be the
Minkowski-Bouligand dimension (Schroeder 1991).

Let F(r) be the AREA traced out by a small CIRCLE with
RADIUS r following a fractal curve. Then, providing the
Limit exists,

D M = hm — -^ + 2
r->-o — In r

(Schroeder 1991). It is conjectured that for all strictly
self-similar fractals, the Minkowski-Bouligand dimen-
sion is equal to the HAUSDORFF DIMENSION D\ oth-
erwise Dm > D.
see also HAUSDORFF DIMENSION

References

Berry, M. V. "Diffractals." J. Phys. A12, 781-797, 1979.

Hunt, F. V.; Beranek, L. L.; and Maa, D. Y. "Analysis of

Sound Decay in Rectangular Rooms." J. Acoust. Soc.

Amer. 11, 80-94, 1939.
Lapidus, M. L. and Fleckinger-Pelle, J. "Tambour fractal:

vers une resolution de la conjecture de Weyl-Berry put les

valeurs propres du laplacien." Compt. Rend. Acad. Sci.

Paris Math. Ser 1 306, 171-175, 1988.
Schroeder, M. Fractals, Chaos, Power Laws: Minutes from

an Infinite Paradise. New York: W. H. Freeman, pp. 41—

45, 1991.

Minkowski Convex Body Theorem

A bounded plane convex region symmetric about a LAT-
TICE Point and with Area > 4 must contain at least
three Lattice Points in the interior. In n-D, the the-
orem can be generalized to a region with Area > 2 n ,
which must contain at least three LATTICE POINTS. The
theorem can be derived from Blichfeldt's Theorem.

see also Blichfeldt's Theorem

Minkowski Geometry

see Minkowski Space

Minkowski-Hlawka Theorem

There exist lattices in n-D having Hypersphere PACK-
ING densities satisfying

T]>

C(»)

where ((n) is the RiEMANN ZETA Function. However,
the proof of this theorem is nonconstructive and it is
still not known how to actually construct packings that
are this dense.

see also Hermite Constants, Hypersphere Pack-
ing

References

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices,
and Groups, 2nd ed. New York: Springer- Verlag, pp. 14-
16, 1993.

Minkowski Integral Inequality

If p > 1, then

/

v a.

b -|1/P

\f(x) + g(x)\ p dx\

<

f

\f(x)fdx

i/p r />*>

+

/

\g(x)\ p dx

i/p

see also MINKOWSKI SUM INEQUALITY

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 11, 1972.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1099, 1993.

Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 146-150, 1988.

Minkowski, H. Geometrie der Zahlen, Vol. 1. Leipzig, Ger-
many: pp. 115-117, 1896.

Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, p. 33, 1991.

Minkowski Measure

Minkowski Sum Inequality 1175

Minkowski Measure

The Minkowski measure of a bounded, CLOSED SET is
the same as its Lebesgue Measure.

References

Ko, K.-I. "A Polynomial-Time Computable Curve whose In-
terior has a Nonrecursive Measure." Theoret. Comput. Sci,
145, 241-270, 1995.

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 37-38
and 42, 1991.

Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal
Images. New York: Springer- Verlag, p. 283, 1988.
ft Weisstein, E. W. "Fractals." http: //www. astro. Virginia,
edu/ ~ eww6n/math/not ebooks /Fr act al . m.

Minkowski Metric

In Cartesian Coordinates,

ds 2 = dx 2 + dy 2 + dz 2

dr — —c dt + dx + dy + dz ,

and

got(3 = T) a (3

-1

01

1

1

1_

In Spherical Coordinates,

is = dr + r dO + r sin d(f>

dr = —c dt + dr + r d8 + r sin 8 d(f>

and

-1

1

r 2

r 2 sin 2

(1)
(2)

(3)

(4)
(5)

(6)

see also Lorentz Transformation, Minkowski
Space

Minkowski Sausage

A Fractal created from the base curve and motif illus-
trated below.

Minkowski Space

A 4-D space with the Minkowski Metric. Alterna-
tively, it can be considered to have a Euclidean Met-
ric, but with its Vectors defined by

(i)

where c is the speed of light. The METRIC is DIAGONAL

with

1 ,.,

go* — — , (2)

Qntcn

~x ~

~ id'

Xl

.

X

X 2

y

_#3_

z

V

Vps-

(3)

Let A be the TENSOR for a LORENTZ TRANSFORMA-
TION. Then

r^'A^s = A" 7 (4)

t^A^ = Aj

A a = n a ~A 01 - ri a ^r} l3S A' > s-

(5)
(6)

The number of segments after the nth iteration is

The NECESSARY and Sufficient conditions for a met-
ric gy, u to be equivalent to the Minkowski metric 7] a
are that the RlEMANN TENSOR vanishes everywhere
{R X pvk = 0) and that at some point g tiV has three POS-
ITIVE and One NEGATIVE EIGENVALUES.

see also LORENTZ TRANSFORMATION, MINKOWSKI

Metric

References

Thompson, A. C. Minkowski Geometry. New York: Cam-
bridge University Press, 1996.

Minkowski Sum

The sum of sets A and B in a VECTOR SPACE, equal to

{a + b : a G A y b £ B}.

Minkowski Sum Inequality

If p > 1 and afc, bk > 0, then

N„ = 8 n ,

and

(1)".

so the Capacity Dimension is

D

lnJV n

- lim

n— yoo In e n

v ln8 n
- lim - — —

twoo In 4"

In 8
ln~4

3 In 2
2 In 2

^(a fc +6 fc ) r

l/p

+ D

Equality holds Iff the sequences a\ , a 2 , ... and b± , 6 2 ,
. . . are proportional.

see also Minkowski Integral Inequality

1176

Minor

Miguel's Theorem

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 11, 1972.

Gradshteyn, L S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1092, 1979.

Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 24-26, 1988.

Minor

The reduced Determinant of a Determinant Ex-
pansion, denoted Mij, which is formed by omitting the
zth row and jth column.

see also COFACTOR, DETERMINANT, DETERMINANT

Expansion by Minors

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 169-170, 1985.

Minor Axis

see Semiminor Axis

Minor Graph

A "minor" is a sort of SUBGRAPH and is what Kura-
towski means when he says "contain." It is roughly a
small graph which can be mapped into the big one with-
out merging Vertices.

Minus

The operation of SUBTRACTION, i.e., a minus b. The
operation is denoted a -b. The Minus Sign "-" is also
used to denote a Negative number, i.e., —x.

see also Minus Sign, Negative, Plus, Plus or Mi-
nus, Times

Minus or Plus

see Plus or Minus

Miquel Circles

For a TRIANGLE AABC and three points A' , B' ', and
C', one on each of its sides, the three Miquel circles are
the circles passing through each VERTEX and its neigh-
boring side points (i.e., AC B' , BA'C\ and CB f A f ).
According to MlQUEL'S THEOREM, the Miquel circles
are CONCURRENT in a point M known as the MIQUEL
POINT. Similarly, there are n Miquel circles for n lines
taken (n — 1) at a time.

see also MlQUEL POINT, MlQUEL'S THEOREM, MlQUEL
TRIANGLE

Miquel Equation

ZA2MA3 = ^A 2 AiA 3 + ZP2P1P3,

where Z is a Directed Angle.

see also Directed Angle, Miquel's Theorem

References

Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 131-144, 1929.

Miquel Point

The point of Concurrence of the Miquel Circles.

see also MlQUEL CIRCLES, MlQUEL'S THEOREM,

Miquel Triangle

Minus Sign

The symbol "— " which is used to denote a Negative
number or SUBTRACTION.

see also Minus, Plus Sign, Sign, Subtraction

Minute

see Arc Minute

Miquel's Theorem

If a point is marked on each side of a TRIANGLE A ABC,
then the three MlQUEL CIRCLES (each through a VER-
TEX and the two marked points on the adjacent sides)

Miquel Triangle

Miter Surface

1177

are CONCURRENT at a point M called the MlQUEL
POINT. This result is a slight generalization of the so-
called Pivot Theorem.

If M lies in the interior of the triangle, then it satisfies

ZP2MP3 = 180° -ax

ZP3MP1 = 180° -a 2

IP 1 MP 2 = 180° -a 3 .

The lines from the MlQUEL POINT to the marked points
make equal angles with the respective sides. (This is a
by-product of the MlQUEL EQUATION.)

Mira Fractal

A Fractal based on the map

Given four lines Li, . . . , L4 each intersecting the other
three, the four MlQUEL CIRCLES passing through each
subset of three intersection points of the lines meet in a
point known as the 4-Miquel point M. Furthermore, the
centers of these four MlQUEL CIRCLES lie on a CIRCLE
Ca (Johnson 1929, p. 139). The lines from M to given
points on the sides make equal ANGLES with respect to
the sides.

Similarly, given n lines taken by (n — l)s yield n MlQUEL
CIRCLES like C4 passing through a point P n , and their
centers lie on a CIRCLE C n +i-
see also MlQUEL CIRCLES, MlQUEL EQUATION, MlQUEL

Triangle, Nine-Point Circle, Pedal Circle,
Pivot Theorem

References

Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 131-144, 1929.

Miquel Triangle

Given a point P and a triangle AA1A2A3, the Miquel
triangle is the triangle connecting the side points Pi,
P2, and P3 of AA1A2A3 with respect to which P is the
Miquel Point. All Miquel triangles of a given point M
are directly similar, and M is the SIMILITUDE CENTER
in every case.

F(x)

ax +

2(1 -a)x 2
1-fz 2

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, p. 136,
1991.

Mirimanoff's Congruence

If the first case of FERMAT'S Last THEOREM is false for
the Prime exponent p, then 3 P_1 = 1 (mod p 2 ).

see also Fermat's Last Theorem

Mirror Image

An image of an object obtained by reflecting it in a
mirror so that the signs of one of its coordinates are
reversed.

see Amphichiral, Chiral, Enantiomer, Handed-
ness

Mirror Plane

The Symmetry Operation (x,y,z) -> (x,y,-z), etc.,
which is equivalent to 2, where the bar denotes an Im-
proper Rotation.

Misere Form

A version of NlM-like GAMES in which the player taking
the last piece is the loser. For most IMPARTIAL GAMES,
this form is much harder to analyze, but it requires only
a trivial modification for the game of NlM.

Mitchell Index

The statistical INDEX

Pm

where p n is the price per unit in period n and q n is the
quantity produced in period n.

see also Index

References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,

Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 66-67,

1962.

Miter Surface

1178 Mittag-Leffler Function

Mobius Function

A QUARTIC SURFACE named after its resemblance to
the liturgical headdress worn by bishops and given by
the equation

4x 2 (x 2 + y 2 + z 2 ) - y 2 {l - y 2 - z 2 ) = 0.

see also QUARTIC SURFACE

References

Nordstrand, T. "Surfaces." http : //www . uib . no /people/
nfytn/surf aces. htm.

Mixed Partial Derivative

A Partial Derivative of second or greater order with
respect to two or more different variables, for example

Mittag-Leffler Function

CO

k=Q

T(jk + 1)

It is related to the Generalized Hyperbolic Func-
tions by

F^ (x) = E n (x n ).

References

Muldoon, M. E. and Ungar, A. A. "Beyond Sin and Cos."
Math. Mag. 69, 3-14, 1996.

Mittenpunkt

The Lemoine Point of the Excentral Triangle, i.e.,
the point of concurrence M of the lines from the Ex-
CENTERS Ji through the corresponding Triangle side
Midpoint Mi. It is also called the Middlespoint and
has Triangle Center Function

a = J rC — a = \ cot A.

see also Excenter, Excentral Triangle, Nagel
Point

References

Baptist, P. Die Entwicklung der Neueren Dreiecksgeometrie.

Mannheim: Wissenschaftsverlag, p. 72, 1992.
Eddy, R. H. "A Generalization of Nagel's Middlespoint."

Elem. Math. 45, 14-18, 1990.
Kimberling, C. "Central Points and Central Lines in the

Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Mittenpunkt." http: //www. evansville .

edu/-ck6/t centers/class /mitt en. html.

Jxy

d 2 f

dxdy*

If the mixed partial derivatives exist and are continuous
at a point xo, then they are equal at xo regardless of
the order in which they are taken.

see also PARTIAL DERIVATIVE

Mixed Strategy

A collection of moves together with a corresponding set
of weights which are followed probabilistically in the
playing of a Game. The Minimax Theorem of Game
THEORY states that every finite, zero-sum, two-person
game has optimal mixed strategies.
see also Game Theory, Minimax Theorem, Strat-
egy

Mixed Tensor

A Tensor having Contravariant and Covariant in-
dices.

see also Contravariant Tensor, Covariant Ten-
sor, Tensor

Mnemonic

A mental device used to aid memorization. Common
mnemonics for mathematical constants such as e and Pi
consist of sentences in which the number of letters in
each word give successive digits.

see also e, JOSEPHUS PROBLEM, Pi

References

Luria, A. R. The Mind of a Mnemonist: A Little Book
about a Vast Memory. Cambridge, MA: Harvard Univer-
sity Press, 1987.

Mobius Band

see Mobius Strip

Mobius Function

0.5

-0.5-

Mobius Group

Mobius Shorts

1179

if n has one or more repeated prime factors

if n= 1

if n is a product of k distinct primes,

so mu(n) ^ indicates that n is SQUAREFREE. The
first few values are 1, —1, -1, 0, -1, 1, -1, 0, 0, 1, -1,
0, ... (Sloane's A008683).

The Summatory Function of the Mobius function is
called Mertens Function.

see also Braun's Conjecture, Mertens Func-
tion, Mobius Inversion Formula, Mobius Peri-
odic Function, Prime Zeta Function, Riemann
Function, Squarefree

References

Abramowitz, M. and Stegun, C. A. (Eds.). "The Mobius
Function." §24.3.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, p. 826, 1972.

Deleglise, M. and Rivat, J. "Computing the Summation of
the Mobius Function." Experiment. Math. 5, 291-295,
1996.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford: Clarendon Press, p. 236,
1979.

Sloane, N. J. A. Sequence A008683 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 7-8 and 223-225,
1991.

Mobius Group

The equation

Xl 2 +X 2 2 +.

. . + X-n

2xoZoo =

represents an n-D Hypersphere § n as a quadratic hy-
persurface in an (n + 1)-D real projective space P n ,
where x a are homogeneous coordinates in F n+1 . Then
the GROUP M(n) of projective transformations which
leave § n invariant is called the Mobius group.

References

Iyanaga, S. and Kawada, Y. (Eds.). "Mobius Geometry."
§78 A in Encyclopedic Dictionary of Mathematics. Cam-
bridge, MA: MIT Press, pp. 265-266, 1980.

Mobius Inversion Formula

If0(«) = £d|n/( d )» then

/(n) = 5>(<*)fl (J) ,

d\n

where the sums are over all possible INTEGERS d that
Divide n and ft(d) is the Mobius Function. The Log-
arithm of the Cyclotomic Polynomial

is the Mobius inversion formula.

see also CYCLOTOMIC POLYNOMIAL, MOBIUS FUNC-
TION

References

Hardy, G. H. and Wright, W. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, pp. 91-93, 1979.

Schroeder, M. R. Number Theory in Science and Communi-
cation, 3rd ed. New York: Springer- Verlag, 1997.

Vardi, L Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 7-8 and 223-225,
1991.

Mobius Periodic Function

A function periodic with period 27r such that

P(0 + tt) = -p(6)
for all is said to be Mobius periodic,

Mobius Problem

Let A = {01,02,...} be a free Abelian Semigroup,
where a\ is the unit element. Then do the following
properties,

1. a < b Implies ac < be for a,b,c £ A, where A has
the linear order a\ < a<i < . . .,

2. jx{a n ) = fi(n) for all n,
imply that

0"mn — OjTn.an

for all m,n > 1? The problem is known to be true for
mn < 74 for all n < 240.

see also BRAUN'S CONJECTURE, MOBIUS FUNCTION

References

Flath, A. and Zulauf, A. "Does the Mobius Function Deter-
mine Multiplicative Arithmetic?" Amer. Math. Monthly
102, 354-256, 1995.

Mobius Shorts

A,B t C -

u - ' C

A one-sided surface reminiscent of the Mobius Strip.
see also Mobius STRIP

References

Boas, R. P. Jr. "Mobius Shorts." Math. Mag. 68, 127, 1995.

* n (x) = ]\(l-* n/d )

M (d)

1180 Mobius Strip

Mobius Strip

A one-sided surface obtained by cutting a band width-
wise, giving it a half twist, and re-attaching the two
ends. According to Madachy (1979), the B. F.Goodrich
Company patented a conveyor belt in the form of a
Mobius strip which lasts twice as long as conventional
belts.

A Mobius strip can be represented parametrically by

x = [R + scos(±0)]co&9
y = [i2 + scos(§0)]sin0

z = ssin(|0),

for s £ [-1,1] and e [0,2tt). Cutting a Mobius
strip, giving it extra twists, and reconnecting the ends
produces unexpected figures called Paradromic Rings
(Listing and Tait 1847, Ball and Coxeter 1987) which are
summarized in the table below.

half-

cuts

divs.

result

twists

1

1

2

1 band, length 2

1

1

3

1 band, length 2

1 Mobius strip, length 1

1

2

4

2 bands, length 2

1

2

5

2 bands, length 2

1 Mobius strip, length 1

1

3

6

3 bands, length 2

i

3

7

3 bands, length 2

1 Mobius strip, length 1

2

1

2

2 bands, length 1

2

2

3

3 bands, length 1

2

3

4

4 bands, length 1

A TORUS can be cut into a Mobius strip with an Even
number of half- twists, and a Klein Bottle can be cut
in half along its length to make two Mobius strips. In
addition, two strips on top of each other, each with a
half- twist, give a single strip with four twists when dis-
entangled.

There are three possible SURFACES which can be ob-
tained by sewing a Mobius strip to the edge of a DISK:
the Boy Surface, Cross-Cap, and Roman Surface.

The Mobius strip has Euler Characteristic 1, and
the Heawood Conjecture therefore shows that any
set of regions on it can be colored using six- colors only.

Mobius Transformation

see also Boy Surface, Cross-Cap, Map Coloring,
Paradromic Rings, Prismatic Ring, Roman Sur-
face

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 127-
128, 1987.

Bogomolny, A. "Mobius Strip." http://vwv.cut-the-knot.
com/do _youJcnow/moebius. html.

Gardner, M. "Mobius Bands." Ch, 9 in Mathematical
Magic Show: More Puzzles, Games, Diversions, Illusions
and Other Mathematical Sleight- of- Mind from Scientific
American. New York: Vintage, pp. 123-136, 1978.

Geometry Center. "The Klein Bottle." http://www.geom.
umn.edu/zoo/features/mobius/.

Gray, A. "The Mobius Strip." §12.3 in Modern Differential
Geometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 236-238, 1993.

Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 41-45, 1975.

Kraitchik, M. §8.4.3 in Mathematical Recreations. New York:
W. W. Norton, pp. 212-213, 1942.

Listing and Tait. Vorstudien zur Topologie, Gottinger Stu-
dien, Pt. 10, 1847.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 7, 1979.

Nordstrand, T. "Mobiusband." http://www.uib.no/people/
nf ytn/moebtxt .htm.

Pappas, T. "The Moebius Strip & the Klein Bottle," "A
Twist to the Moebius Strip," "The 'Double' Moebius
Strip." The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, p. 207, 1989.

Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 269-274, 1983.

Wagon, S. "Rotating Circles to Produce a Torus or Mobius
Strip." §7.4 in Mathematica in Action. New York: W. H.
Freeman, pp. 229-232, 1991.

Wang, P. "Renderings." http://www.ugcs . caltech.edu/
~peterw/portf olio/renderings/.

Mobius Transformation

A transformation of the form

/(*) =

az + b
cz + d'

where a, 6, c, d £ C and

ad — be ^ 0,

is a CONFORMAL TRANSFORMATION and is called a
Mobius transformation. It is linear in both w and z.

Every Mobius transformation except f(z) = z has one or
two Fixed POINTS. The Mobius transformation sends
CIRCLES and lines to CIRCLES or lines. Mobius trans-
formations preserve symmetry. The CROSS-RATIO is
invariant under a Mobius transformation. A Mobius
transformation is a composition of translations, rota-
tions, magnifications, and inversions.

To determine a particular Mobius transformation, spec-
ify the map of three points which preserve orientation.
A particular Mobius transformation is then uniquely

Mobius Triangles

Modified Bessel Differential Equation 1181

determined. To determine a general Mobius transfor-
mation, pick two symmetric points a and as- Define
= f(a)> restricting as required. Compute 0s- f{&s)
then equals 0s since the Mobius transformation pre-
serves symmetry (the Symmetry Principle). Plug in
a and as into the general Mobius transformation and
set equal to and 0s- Without loss of generality, let
c — 1 and solve for a and b in terms of 0. Plug back
into the general expression to obtain a Mobius transfor-
mation.
see also SYMMETRY PRINCIPLE

Mobius Triangles

Spherical Triangles into which a Sphere is divided
by the planes of symmetry of a UNIFORM POLYHEDRON.

see also Spherical Triangle, Uniform Polyhedron

Mock Theta Function

Ramanujan was the first to extensively study these
Theta FuNCTlON-like functions

/(?) = *-o (l + s) 2 (l + ? 2 ) 2 ..-(l + ^) 2

71 =

This is the form of the unperturbed CIRCLE Map with
the Winding Number

Q

*<*> = E 71

(1 + q 2 )(l + q 4 ) .-•(! + q 2n Y

see also ^-Series, Theta Function

References

Bellman, R. E. A Brief Introduction to Theta Functions.
New York: Holt, Rinehart, and Winston, 1961.

Mod

see Congruence

Mode

The most common value obtained in a set of observa-
tions.

see also Mean, Median (Statistics), Order Statis-
tic

Mode Locking

A phenomenon in which a system being forced at an
Irrational period undergoes rational, periodic motion
which persists for a finite range of forcing values. It may
occur for strong couplings between natural and forcing
oscillation frequencies.

The phenomenon can be exemplified in the CIRCLE MAP
when, after q iterations of the map, the new angle differs
from the initial value by a RATIONAL NUMBER

I

q'

e n

= 6> n +

For Q, not a RATIONAL NUMBER, the trajectory is
QUASIPERIODIC.

see also Chaos, Quasiperiodic Function

Model Completion

Model completion is a term employed when Existen-
tial Closure is successful. The formation of the Com-
plex NUMBERS, and the move from amne to projec-
tive geometry, are successes of this kind. The theory of
existential closure gives a theoretical basis of Hilbert's
"method of ideal elements."

References

Manders, K. L. "Interpretations and the Model Theory of

the Classical Geometries." In Models and Sets. Berlin:

Springer- Verlag, pp. 297-330, 1984.
Manders, K. L. "Domain Extension and the Philosophy of

Mathematics." J. Philos. 86, 553-562, 1989.

Model Theory

Model theory is a general theory of interpretations of
an Axiomatic Set Theory. It is the branch of Logic
studying mathematical structures by considering first-
order sentences which are true of those structures and
the sets which are definable in those structures by first-
order Formulas (Marker 1996).

Mathematical structures obeying axioms in a system
are called "models" of the system. The usual axioms
of Analysis are second order and are known to have
the Real Numbers as their unique model. Weakening
the axioms to include only the first-order ones leads to
a new type of model in what is called NONSTANDARD
Analysis.

see also KHOVANSKl'S THEOREM, NONSTANDARD

Analysis, Wilkie's Theorem

References

Doets, K. Basic Model Theory. New York: Cambridge Uni-
versity Press, 1996.

Marker, D. "Model Theory and Exponentiation." Not.
Amer. Math. Soc. 43, 753-759, 1996.

Stewart, I. "Non-Standard Analysis." In From Here to Infin-
ity: A Guide to Today's Mathematics. Oxford, England:
Oxford University Press, pp. 80-81, 1996.

Modified Bessel Differential Equation

The second-order ordinary differential equation

x ~ + x~ (x +n )y = 0.

dx 2 dx

The solutions are the Modified Bessel Functions of
the First and Second Kinds. If n = 0, the modified
Bessel differential equation becomes

2 d 2 y dy 2 n

x _| -\-x-p- - x y = 0,
dx z dx

1182 Modified Bessel Function

which can also be written

d f dy y

dx

(•2)

= m-

12 3 4 5

A function I n (x) which is one of the solutions to the
Modified Bessel Differential Equation and is
closely related to the Bessel Function of the First
Kind J n (x). The above plot shows I n (x) for n = 1, 2,
. . . , 5. In terms of J n (x) }

I n (x) = i~ n J n {ix) = e~ n7rt/2 J n (xe in/2 ). (1)

For a Real Number v, the function can be computed
using

{\z 2 T

w = (H-E fc!r( ^ +1) .

(2)

where T(z) is the GAMMA FUNCTION. An integral for-
mula is

i A')

77 Jo

e zcos cos(v9)d9

f

Jo

sin(^) i e _, cosht _, tdf) (3)

which simplifies for v an INTEGER n to

In(z) = I f

n Jo

e zcose cos(n9)d0 (4)

(Abramowitz and Stegun 1972, p. 376).

A derivative identity for expressing higher order modi-
fied Bessel functions in terms of Iq(x) is

'»(*> = r.(|)/o(«),

(5)

where T n (x) is a Chebyshev Polynomial of the
First Kind.

see also Bessel Function of the First Kind, Modi-
fied Bessel Function of the First Kind, Weber's
Formula

Modified Bessel Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Modified Bessel
Functions I and K" §9.6 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, pp. 374-377, 1972.

Arfken, G. "Modified Bessel Functions, I„(x) and K u (x)"
§11.5 in Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 610-616, 1985.

Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/cntf rc/cntf re .html.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Bessel Functions of Fractional Order, Airy-
Functions, Spherical Bessel Functions." §6.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 234-245, 1992.

Spanier, J. and Oldham, K. B. "The Hyperbolic Bessel Func-
tions Iq(x) and Ii(x)" and "The General Hyperbolic Bes-
sel Function I u (x)" Chs. 49-50 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 479-487 and 489-497,
1987.

Modified Bessel Function of the Second Kind

12 3 4 5

The function K n (x) which is one of the solutions to
the Modified Bessel Differential Equation. The
above plot shows K n (x) for n = 1, 2, . . . , 5. K n (x) is
closely related to the MODIFIED BESSEL FUNCTION OF
the First Kind I n (x) and Hankel Function H n {x),

K n (x) = ^i n+l H^(ix)

\m n+1 [J n (ix) + iN n {ix)] (2)

2 sin(mr)
(Watson 1966, p. 185). A sum formula for K n is

- i/i.A-nV^ ("-fc-l)! / 1,2

(1)
(2)

(3)

*»(*) = !(§*)-"£

k\

(-Kr

+ ( _1)«+^ ( i z) j bW
+(-l) B ^(b) n f]WA + l)+^(n + fc + l)]^-- )fc

fc=0

kl(n + k)V
(4)

where ip is the DlGAMMA FUNCTION (Abramowitz and
Stegun 1972). An integral formula is

K u {z)

cos t dt

T{v+\)(2zy r

v^ J (t 2 +2 2 )"+ 1 /2

(5)

Modified Spherical Bessel Differential Equation

Modular Angle 1183

which, for v ~ 0, simplifies to

Ko(x) = / cos(xsmht) dt
Jo

f

Jo

cos(xt) dt

(6)

Other identities are

^<4^ r /°

zx (^ 2 _ ^ n -!/2

e -J "V-l)

dx (7)

for n > —1/2 and

* \n-l/2

(8)
tt e - y^ (n-§)!
2*(n-±)!£*r!(n-r-i)! 1

/>oo

/ c - t t" + - 1/2 <ft.

(9)

The modified Bessel function of the second kind is some-
times called the BASSET Function.

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Modified Bessel
Functions I and K" §9,6 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, pp. 374-377, 1972.

Arfken, G. "Modified Bessel Functions, I u (x) and K u (x)."
§11.5 in Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 610-616, 1985.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Modified Bessel Functions of Integral Order"
and "Bessel Functions of Fractional Order, Airy Functions,
Spherical Bessel Functions." §6.6 and 6.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 229-245, 1992.

Spanier, J. and Oldham, K. B. "The Basset K u (x)." Ch. 51
in An Atlas of Functions. Washington, DC: Hemisphere,
pp. 499-507, 1987.

Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.

Modified Spherical Bessel Differential
Equation

The Spherical Bessel Differential Equation with
a Negative separation constant, given by

r 2^R + ^ _ ^2 r 2 + n{n + 1)]R =

dr 2

dR

dr

The solutions are called MODIFIED SPHERICAL BESSEL
Functions.

Modified Spherical Bessel Function

Solutions to the Modified Spherical Bessel Differ-
ential Equation, given by

i n (x) = y-/n+l/2(!C)

(i)

, N sinh(x)
io{x) =

X

(2)

k n (x) = J — K n + 1/2 {x)

(3)

ko{x) = ,

X

(4)

where I n (x) is a Modified Bessel Function of the
First Kind and K n (x) is a Modified Bessel Func-
tion of the Second Kind.

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Modified Spher-
ical Bessel Functions." §10.2 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, pp. 443-445, 1972.

Modified Struve Function

(f)

2fc

^)-(H" +1 E r(fc+l) ; (fc+l/+f)

2(tr r' 2

sinh(zcos#) sin u 8dQ,

where T(z) is the Gamma Function.

see also Anger Function, Struve Function, We-
ber Functions

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Modified Struve
Function L u (x)" §12.2 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, p. 498, 1972.

Modular Angle

Given a MODULUS k in an ELLIPTIC INTEGRAL, the
modular angle is defined by k = sin a. An ELLIPTIC
Integral is written I{(f>\m) when the Parameter is
used, I{<j), k) when the MODULUS is used, and I(<f>\a)
when the modular angle is used.

see also Amplitude, Characteristic (Elliptic In-
tegral), Elliptic Integral, Modulus (Elliptic
Integral), Nome, Parameter

References

Abramowitz , M . and Stegun, C . A. (Eds. ) . Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 590, 1972.

1184 Modular Equation

Modular Equation

The modular equation of degree n gives an algebraic
connection of the form

K'(l) K'(k)
K{1) K(k)

(1)

between the Transcendental Complete Elliptic
Integrals of the First Kind with moduli k and /.
When k and / satisfy a modular equation, a relationship
of the form

M(l,k)dy

dx

V(l-y 2 )(l-ZV) y/(l-x*)(l-k>x>)

(2)

exists, and M is called the Modular Function Mul-
tiplier. In general, if p is an Odd Prime, then the
modular equation is given by

Q p (u, v) — (v — uq)(v - ui) • - - (v - Up), (3)

where

u p = (-l)tf-»'*[\tf)]»* ee (-1)^-Wu( q »), (4)

A is a Elliptic Lambda Function, and

q = e

(5)

(Borwein and Borwein 1987, p. 126). An Elliptic In-
tegral identity gives

K\k) _ n K ji+fcj

so the modular equation of degree 2 is

2y/k

I

which can be written as

1 + k

l 2 (\ + k) 2 = 4k.

(6)

(?)

(8)

A few low order modular equations written in terms of
k and I are

i7 2 = Z 2 (l + fc) 2 - 4fc = (9)

n 7 = (A,/) 1 / 4 + (jfc'z') 1/4 -i = o (io)

n 23 = (kl) 1/4 + (k'l') 1/4 + 2 2/3 (klk'l') 1/12 -1 = 0.

(ii)

In terms of u and v,

n B (u, v) = u 4 - v 4 + 2uv(l - u 2 v 2 ) = (12)

Qs{u,v) = v 6 -u 6 + 5u 2 v 2 (v 2 -u)+4uv{u 4 v 4 - 1)

\vJ \uj V u 2 v 2 J

(13)
Q 7 (u, v) = (1 - u 8 )(l - v s ) - (1 - uv) 8 = 0, (14)

where

and

2 _

= >/jfe =

v7 =

W)

Modular Form

(15)
(16)

#3 (</*)*

Here, #; are Theta Functions.

A modular equation of degree 2^ for r > 2 can be ob-
tained by iterating the equation for 2 r ~ 1 . Modular equa-
tions for PRIME p from 3 to 23 are given in Borwein and
Borwein (1987).

Quadratic modular identities include

*3(g) , =

Cubic identities include

Mq)

\mq 9 ) ;

_ W "

oW) /

>4(g)

A seventh-order identity is

.*> V)

1/2

= 9

*W)
^2 4 (g)

V(g 3 )

-1

9^1-1

(17)

(18)

(19)
(20)

VMq)Mq 7 ) - VM<i)Mq 7 ) = \AM</)W)- (21)

From Ramanujan (1913-1914),
(1 + q)(l + ? 3 )(1 + g 5 ) • • • = 2 1/6 5 1/24 (^')- 1/12 (22)
(1 _ q){ i _ g 3 )(1 _ ^) . . . = 2 1/6 q 1/24 k- 1/12 k n/6 . (23)

see a/so Schlafli's Modular Form

References

Borwein, J, M. and Borwein, P. B. Pi & the AGM: A Study in

Analytic Number Theory and Computational Complexity.

New York: Wiley, pp. 127-132, 1987.
Hanna, M. "The Modular Equations." Proc. London Math.

Soc. 28, 46-52, 1928.
Ramanujan, S. "Modular Equations and Approximations to

7T." Quart. J. Pure. AppL Math. 45, 350-372, 1913-1914.

Modular Form

A modular form is a function in the Complex Plane
with rather spectacular and special properties resulting
from a surprising array of internal symmetries. If

'(S3) -<=+«■"(.).

then F(z) is said to be a modular form of weight 2 and
level N. If it is correctly parameterized, a modular form
is Analytic and vanishes at the cusps, so it is called

Modular Function

a CUSP FORM. It is also an eigenform under a certain
Hecke Algebra.

A remarkable connection between rational ELLIPTIC
CURVES and modular forms is given by the TANIYAMA-
Shimura Conjecture, which states that any rational
Elliptic Curve is a modular form in disguise. This
result was the one proved by Andrew Wiles in his cele-
brated proof of Fermat's Last Theorem.

see also Cusp Form, Elliptic Curve, Elliptic
Function, Fermat's Last Theorem, Hecke Al-
gebra, Modular Function, Modular Function
Multiplier, Schlafli's Modular Form, Taniyama-
Shimura Conjecture

References

Knopp, M. I. Modular Functions, 2nd ed. New York:
Chelsea, 1993.

Koblitz, N. Introduction to Elliptic Curves and Modular
Forms. New York: Springer- Verlag, 1993.

Rankin, R. A. Modular Forms and Functions. Cambridge,
England: Cambridge University Press, 1977.

Sarnack, P. Some Applications of Modular Forms. Cam-
bridge, England: Cambridge University Press, 1993.

Modular Lattice 1185

The first few multipliers in terms of I and k are

1 1 + 2'

M 2 (2,fc) =

1 + fe

M 3 (l,k) =

1-

(3)
(4)

In terms of the u and v defined for MODULAR EQUA-
TIONS,

M 3 =
M 5 =
M 7 =

v 2v 6 - u

v + 2u s 3u

v(l -uv 3 ) _ u + v 5

v — u 5 5u(l + u 3 v)

v(l — uv)[l — uv -f- (uv) 2 ]

v — u (

7u(l — uv)[l — uv + (uv) 2 } '

(5)

(6)

(7)

Modular Function

/ is a modular function of level N on the upper half H
of the Complex Plane if it is Meromorphic (even at
the CUSPS), ad - be = 1 for all a, b, c, d, and N\c.

see also Elliptic Function, Elliptic Modular
Function, Modular Form

References

Apostol, T. M. Modular Functions and Dirichlet Series in
Number Theory. New York: Springer- Verlag, 1976.

Askey, R. In Ramanujan International Symposium (Ed.
N. K Thakare). pp. 1-83.

Borwein, J. M. and Borwein, P. B. Pi and the AGM: A Study
in Analytic Number Theory and Computational Complex-
ity. New York: Wiley, 1987.

Rankin, R. A. Modular Forms and Functions. Cambridge,
England: Cambridge University Press, 1977.

Schoeneberg, B. Elliptic Modular Functions: An Introduc-
tion. Berlin: New York: Springer- Verlag, 1974.

Modular Function Multiplier

When k and / satisfy a MODULAR EQUATION, a rela-
tionship of the form

M(l,k)dy

dx

y/(l - y 2 )(l - 2V) ^(l-x*)(l-k*x*)

(1)

exists, and M is called the multiplier. The multiplier of
degree n can be given by

(2)

where 1?» is a THETA FUNCTION and K(k) is a complete
Elliptic Integral of the First Kind.

Modular Gamma Function

The GAMMA GROUP r is the set of all transformations

w of the form

,. at + b

w{t) = rfTd'

where a, 6, c, and d are INTEGERS and ad — be = 1.
T-modular functions are then defined as in Borwein and
Borwein (1987, p. 114).

see also Klein's Absolute Invariant, Lambda
Group, Theta Function

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 127-132, 1987.

Modular Group

The Group of all Mobius Transformations having
Integer coefficients and Determinant equal to 1.

Modular Lambda Function

see Elliptic Lambda Function

Modular Lattice

A Lattice which satisfies the identity

(x A y) V (x A z) = x A (y V (x A z))

is said to be modular.

see also DISTRIBUTIVE Lattice

References

Gratzer, G. Lattice Theory: First Concepts and Distributive

Lattices. San Francisco, CA: W. H. Freeman, pp. 35—36,

1971.

1186 Modular System

Modulo Multiplication Group

Modular System

A set M of all POLYNOMIALS in s variables, x\, . . . , x 8
such that if P, Pi, and P2 are members, then so are
Pi + P2 and QP, where Q is any Polynomial in xi,
. . . , x s -

see also Hilbert's Theorem, Modular System Ba-
sis

Modular System Basis

A basis of a MODULAR SYSTEM M is any set of POLY-
NOMIALS Pi, P 2 , ... of M such that every POLYNOMIAL
of M is expressible in the form

where Ri } P2,

R\B\ -\- R2B2 + . . . ,
.are POLYNOMIALS.

Modular Transformation

see Modular Equation

Modulation Theorem

The important property of FOURIER TRANSFORMS
that T[cos(27rkox)f(x)] can be expressed in terms of
T[f(x)} = F{k) as follows,

T[cos{2irk x)f(x)} = I[F(fc - ko) + F(k + k )].

see also Fourier Transform

References

Brace well, R. "Modulation Theorem." The Fourier Trans-
form and Its Applications. New York: McGraw-Hill,
p. 108, 1965.

Module

A mathematical object in which things can be added to-
gether COMMUTATIVELY by multiplying COEFFICIENTS
and in which most of the rules of manipulating VEC-
TORS hold. A module is abstractly very similar to a
Vector Space, although modules have Coefficients
in much more general algebraic objects and use RINGS
as the Coefficients instead of Fields.

The additive submodule of the INTEGERS is a set of
quantities closed under Addition and Subtraction
(although it is Sufficient to require closure under Sub-
traction). Numbers of the form na±ma for n, m € Z
form a module since,

na ± ma = (n ± m)a.

Given two INTEGERS a and b, the smallest module con-
taining a and b is GCD(a, b).

References

Foote, D. and Duramit, D. Abstract Algebra. Englewood
Cliffs, NJ: Prentice-Hall, 1990.

Modulo

see Congruence

Modulo Multiplication Group

A Finite Group M m of Residue Classes prime to m
under multiplication mod m. Mm is Abelian of Order
0(m), where 0(m) is the Totient Function. The fol-
lowing table gives the modulo multiplication groups of
small orders.

M m

Group

4>(m)

Elements

M 2

(e)

1

1

M 3

z 2

2

1, 2

M 4

z 2

2

1, 3

M 5

z±

4

1, 2, 3, 4

M 6

z 2

2

1, 5

M 7

z.

6

1, 2, 3, 4, 5, 6

M 8

z 2 ® z 2

4

1, 3, 5, 7

M 9

z*

6

1, 2, 4, 5, 7, 8

M 1Q

z±

4

1, 3, 7, 9

M ia

Zio

10

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

M 12

z 2 <g> Z 2

4

1, 5, 7, 11

M 13

Z12

12

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

M 14

Z 6

6

1, 3, 5, 9, 11, 13

M 15

Z 2 ® Z A

8

1, 2, 4, 7, 8, 11, 13, 14

M 18

Z 2 (2) Z 4

8

1, 3, 5, 7, 9, 11, 13, 15

M 17

Zie

16

1, 2, 3, ..., 16

M 18

z 6

6

1, 5, 7, 11, 13, 17

M 19

Zis

18

1, 2, 3, ..., 18

M 20

z 2 ® Z A

8

1, 3, 7, 9, 11, 13, 17, 19

M 21

z 2 ®z 6

12

1, 2, 4, 5, 7, 8, 10, 11, 13, 16, 17, 19

M 22

-^10

10

1, 3, 5, 7, 9, 13, 15, 17, 19, 21

M 23

z 22

22

1, 2,3, ...,22

M 24

Z 2 % Z 2 <8> z 2

8

1, 5, 7, 11, 13, 17, 19, 23

Mm is a CYCLIC GROUP (which occurs exactly when m
has a Primitive Root) Iff m is of one of the forms
m = 2, 4, p n , or 2p n > where p is an Odd Prime and
n> 1 (Shanks 1993, p. 92).

M, M 4 M. M.

Isomorphic modulo multiplication groups can be deter-
mined using a particular type of factorization of 4>(m) as
described by Shanks (1993, pp. 92-93). To perform this

Modulo Multiplication Group

Modulo Multiplication Group 1187

factorization (denoted <£ m ), factor m in the standard
form

(i)

Or ao „ o

m — pi L p 2 - • *p n

Now write the factorization of the Totient Function
involving each power of an ODD PRIME

<t>(Pi ai ) = (Pi ~ l)Pi

ai — 1

(2)

4>{Pi ai ) = (li" 1 ) (Q2 b2 ) ■ ■ - (q. b ') (pS*- 1 ) , (3)

where

Ibi bo ba

(4)

(q b ) denotes the explicit expansion of q b (i.e., 5 2 = 25),
and the last term is omitted if a% = 1. If p\ = 2, write

0(2'

J2 for ttl =2

) = {2(2">- 2 ) for ai >2. W

Now combine terms from the odd and even primes. For
example, consider m = 104 = 2 3 * 13. The only odd
prime factor is 13, so factoring gives 13 — 1 = 12 =
(2 2 ) (3) = 3 ■ 4. The rule for the powers of 2 gives
2 3 = 2{2 3 " 2 ) = 2 (2) = 2 • 2. Combining these two
gives 0io4 = 2-2*3-4. Other explicit values of <f> m are
given below.

^3-2
04 = 2

05=4

015-2-4

016 = 2 • 4

017 = 16

0104 = 2-2-3-4

0105 = 2 • 2 ■ 3 ■ 4.

M m and M n are isomorphic Iff m and n are identical.
More specifically, the abstract Group corresponding to
a given M m can be determined explicitly in terms of a
Direct Product of Cyclic Groups of the so-called
Characteristic Factors, whose product is denoted
$„. This representation is obtained from m as the set
of products of largest powers of each factor of m . For
example, for 0io4, the largest power of 2 is 4 = 2 2 and
the largest power of 3 is 3 = 3 1 , so the first characteristic
factor is 4x3 = 12, leaving 2-2 (i.e., only powers of two).
The largest power remaining is 2 = 2 1 , so the second
Characteristic Factor is 2, leaving 2, which is the
third and last CHARACTERISTIC FACTOR. Therefore,
$104 = 2-2-4, and the group M m is isomorphic to
Z 2 <8> Z 2 <g> Z 4 .

The following table summarizes the isomorphic modulo
multiplication groups M n for the first few n and iden-
tifies the corresponding abstract GROUP. No M m is
Isomorphic to Zs, Qs, or Z> 4 . However, every finite
Abelian Group is isomorphic to a SUBGROUP of M m
for infinitely many different values of m (Shanks 1993,
p. 96). Cycle Graphs corresponding to M n for small
n are illustrated above, and more complicated CYCLE
GRAPHS are illustrated by Shanks (1993, pp. 87-92).

Group

Isomorphic M m

(e)

M 2

z 2

M 3 , M 4 , M 6

z 4

M 5l M 10

z 2 ®z 2

Ms, M12

Ze

M 7 , M 9 , M14, Mis

z 2 ®z 4

M15, Mie,

M 20 , M 30

z 2 ®z 2 ® z 2

M 24

Zio

Mn, M 22

Z\ 2

M13, M 26

Z 2 z&

M21, M 28 ,

M 36 , M 42

Zl6

Mi 7) M34

z 2 ® z$

M 32

z 2 ®z 2 % z±

M 40 , M 48 ,

M 60

Zis

M19, M 27 ,

M 38 , M54

Z 2 o

M25, M 50

Z 2 <8> Zio

M 33 , M 44 ,

M 66

Z22

M23, M 46

Z 2 (g> z 12

M35, M39,

M 45 , M 52 , M 70 , M 78) M 90

Z28

M 29 , M 58

Z30

M 3 1, M 6 2

Z36

M 37 , M74

The number of CHARACTERISTIC FACTORS r of M m
for m — 1, 2, ... are 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1,
2, ... (Sloane's A046072). The number of QUADRA-
TIC RESIDUES in M m for m > 2 are given by <6(m)/2 r
(Shanks 1993, p. 95). The first few for m = 1, 2, . . . are
0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 5,1,6,... (Sloane's A046073).

In the table below, </>(n) is the TOTIENT FUNC-
TION (Sloane's A000010) factored into CHARACTERISTIC
Factors, A(n) is the Carmichael Function (Sloane's
A011773), and gi are the smallest generators of the
group M n (of which there is a number equal to the num-
ber of Characteristic Factors).

1188 Modulus (Complex Number)

Modulus (Elliptic Integral)

n

4>{n)

\{n)

ffi

n

<Kn)

X(n)

9i

3

2

2

2

27

18

18

2

4

2

2

3

28

2-6

6

13, 3

5

4

2

2

29

28

28

2

6

2

2

5

30

2-4

4

11, 7

7

6

6

3

31

30

30

3

8

2-2

2

7

3

32

2-8

8

31, 3

9

6

6

2

33

2-10

10

10, 2

10

4

4

3

34

16

16

3

11

10

10

2

35

2-12

12

6,2

12

2-2

2

5

7

36

2-6

6

19,5

13

12

12

2

37

36

36

2

14

6

6

3

38

18

18

3

15

2-4

4

14

2

39

2- 12

12

38, 2

16

2-4

4

15

3

40

2-2-4

4

39, 11, 3

17

16

16

3

41

40

40

6

18

6

6

5

42

2-6

6

13, 5

19

18

18

2

43

42

42

3

20

2-4

4

19

3

44

2-10

10

43, 3

21

2-6

6

20

2

45

2- 12

12

44, 2

22

10

10

7

46

22

22

5

23

22

22

5

47

46

46

5

24

2-2-2

2 5

7,

13

48

2-2-4

4

47, 7, 5

25

20

20

2

49

42

42

3

26

12

12

7

50

20

20

3

see also CHARACTERISTIC FACTOR, CYCLE GRAPH, FI-
NITE Group, Residue Class

References

Riesel, H. "The Structure of the Group Af„." Prime Numbers
and Computer Methods for Factorization, 2nd ed. Boston,
MA: Birkhauser, pp. 270-272, 1994.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 61-62 and 92, 1993.

Sloane, N. J. A. Sequences A011773, A046072, A046073, and
A000010/M0299 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
# Weisstein, E. W. "Groups." http://www. astro. Virginia.
edu/~eww6n/math/notebooks/Groups.m.

Modulus (Complex Number)

The modulus of a Complex Number z is denoted \z\.

\x + iy\ = \Jx 2 + y 2

(i)

(2)

Let ci = Ae** 1 and c 2 = Be'* 2 be two Complex Num-
bers. Then

(3)

(4)

Cl

c 2

=

Ae i<f>1
Be****

~ B {

-02) | __

A
B

| Cl | _ \Ae i4,1 \ _ A\e i<pl \
\c 2 \ \Be^\ B |e^|

A
~ B'

Cl

|ci|

c 2

\C2\'

(5)

Also,

|cic 2 | = \(Ae i4>1 )(Be i<i>2 )\ = AB\e H * 1+M \ = AB

(6)
|ci||c 2 | = \Ae 2<pl \\Be i<p2 \ = AB\e t4>1 \ \e i<p2 | - AB, (7)

CiC 2 = |ci| \c 2 \

and, by extension,

(8)

(9)

The only functions satisfying identities of the form

\f{x + iy)\ = \f(x) + f(iy)\ (10)

are f(z) = Az, f(z) = Asin(bz), and f(z) = ^4sinh(6z)

(Robinson 1957).

see also ABSOLUTE SQUARE

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook

of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, 9th printing. New York: Dover,

p. 16, 1972.
Robinson, R. M. "A Curious Mathematical Identity." Amer.

Math. Monthly 64, 83-85, 1957.

Modulus (Congruence)

see Congruence

Modulus (Elliptic Integral)

A parameter k used in ELLIPTIC INTEGRALS and ELLIP-
TIC Functions defined to be k = y/m, where m is the
Parameter. An Elliptic Integral is written I(4>>k)
when the modulus is used. It can be computed explicitly
in terms of THETA FUNCTIONS of zero argument:

^2 2 (Q|r)
tf 3 2 (0|r)'

(1)

The Real period K(k) and Imaginary period K'(k) =
K{k') = K(y/l-k 2 ) are given by

4i^(fc) = 27n?3 2 (0|r)

2iK'(k)^7TT$3 2 (0\T),

(2)
(3)

where K(k) is a complete Elliptic Integral of the
FIRST Kind and the complementary modulus is defined

by

(4)

k\

with k the modulus.

see also Amplitude, Characteristic (Elliptic In-
tegral), Elliptic Function, Elliptic Integral,
Elliptic Integral Singular Value, Modular An-
gle, Nome, Parameter, Theta Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 590, 1972.

Modulus (Quadratic Invariants)

Modulus (Quadratic Invariants)

The quantity ps - rq obtained by letting

x = pX + qY
y-rX + sY

so that

ax 2 + 2bxy + cy

A = ap 4- 2bpr + cr

B = apq + 6(ps + qr) + crs

C — aq + 26gs + cs 2

and

B 2 - AC = (ps - rq) 2 (b 2 - ac),
is called the modulus.

(1)
(2)

(3)

(4)
(5)
(6)

(7)

Modulus (Set)

The name for the Set of Integers modulo m, denoted

Z\mZ. If m is a Prime p, then the modulus is a Finite

Field F p = Z\pZ.

Moessner's Theorem

Write down the POSITIVE INTEGERS in row one, cross
out every feith number, and write the partial sums of
the remaining numbers in the row below. Now cross off
every foth number and write the partial sums of the
remaining numbers in the row below. Continue. For
every POSITIVE INTEGER k > 1, if every fcth number is
ignored in row 1, every (k - l)th number in row 2, and
every (k + 1 - i)th number in row i, then the kth row of
partial sums will be the kth POWERS l fe , 2 fc , 3 fc , . . . .

References

Conway, J. H. and Guy, R. K. "Moessner's Magic." In The

Book of Numbers. New York: Springer- Verlag, pp. 63-65,

1996.
Honsberger, R. More Mathematical Morsels. Washington,

DC: Math. Assoc. Amer, pp. 268-277, 1991.
Long, C. T. "On the Moessner Theorem on Integral Powers."

Amer. Math. Monthly 73, 846-851, 1966.
Long, C. T. "Strike it Out—Add it Up." Math. Mag. 66,

273-277, 1982.
Moessner, A. "Eine Bemerkung iiber die Potenzen der

naturlichen Zahlen." S.-B. Math.-Nat. Kl. Bayer. Akad.

Wiss. 29, 1952.
Paasche, I. "Ein neuer Beweis des moessnerischen Satzes."

S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952, 1-5, 1953.
Paasche, I. "Ein zahlentheoretische-logarithmischer 'Rechen-

stab'." Math. Naturwiss. Unterr. 6, 26-28, 1953-54.
Paasche, I. "Eine Verallgemeinerung des moessnerschen

Satzes." Compositio Math. 12, 263-270, 1956.

Mollweide Projection 1189
Mohammed Sign

A curve consisting of two mirror-reversed intersecting
crescents. This curve can be traced Unicursally.
see also Unicursal Circuit

M0ire Pattern

An interference pattern produced by overlaying similar
but slightly offset templates. M0ire patterns can also be
created be plotting series of curves on a computer screen.
Here, the interference is provided by the discretization
of the finite-sized pixels.
see also ClRCLES-AND-SQUARES FRACTAL

References

Cassin, C. Visual Illusions in Motion with M0ire Screens: 60
Designs and 3 Plastic Screens. New York: Dover, 1997.

Grafton, C. B. Optical Designs in Motion with M0ire Over-
lays. New York: Dover, 1976.

Mollweide's Formulas

Let a Triangle have side lengths a, 6, and c with op-
posite angles A, S, and C. Then

b-c sin[|(B-C)]

a cos(^A)

c -a = Bin[i(C-A)]

b cos(^B)

fl-6 = sin[|(A-g)]
c cos(^C)

see also NEWTON'S FORMULAS, TRIANGLE

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 146, 1987.

Mollweide Projection

1190

Moment

Moment-Generating Function

A Map Projection also called the Elliptical Pro-
jection or HOMOLOGRAPHIC EQUAL AREA PROJEC-
TION. The forward transformation is

_ 2\/2(A-Ao)cos(9

y = 2 1/2 sm$ i

where 6 is given by

20 + sin(2#) = 7rsm<f>.

(1)
(2)

(3)

Newton's Method can then be used to compute 0'
iteratively from

A9' =

9' + sin0' — 7rs'm(j)
1 + cos W '

where

or, better yet,

9' = \e'

* = 2*r*g)

(4)

(5)
(6)

can be used as a first guess.
The inverse FORMULAS are

i = sin

A = A +

20 + sin(2<9)

2\/2 cos (9'

where

•■»-(*)■

(7)
(8)

(9)

References

Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 249-252, 1987.

Moment

The nth moment of a distribution about zero p! n is de-
fined by

vL = <*"> , (i)

where

{Fj

f{x)P{x) discrete distribution
f(x)P(x)dx continuous distribution.

(2)

fi' 1} the Mean, is usually simply denoted fi — fi lt If the
moment is instead taken about a point a,

Hn{a) = {{x - a) n ) = J2& ~ a) n P(x). (3)

The moments are most commonly taken about the
Mean. These moments are denoted fi n and are defined

by

nM^-mD, (4)

with /ii = 0. The moments about zero and about the
Mean are related by

M2 = M2 - (/4) 2

(5)
(6)

^4 = M4 - 4// 3 ^i + 6^2 (^i) 2 - 3(^i) 4 . (7)

The second moment about the MEAN is equal to the
Variance

M2 = <r 2 , (8)

where & — ^JJii is called the STANDARD DEVIATION.
The related Characteristic Function is denned by

[d n d>l

6 (n) (0) =

dt n J t=0

i n Mn(0).

(9)

The moments may be simply computed using the
Moment-Generating Function,

& = M<">(0).

(10)

A Distribution is not uniquely specified by its mo-
ments, although it is by its CHARACTERISTIC FUNC-
TION.

see also CHARACTERISTIC FUNCTION, CHARLIER'S

Check, Cumulant-Generating Function, Fac-
torial Moment, Kurtosis, Mean, Moment-
Generating Function, Skewness, Standard De-
viation, Standardized Moment, Variance

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Moments of a Distribution: Mean, Vari-
ance, Skewness, and So Forth." §14.1 in Numerical Recipes
in FORTRAN: The Art of Scientific Computing, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 604-609, 1992.

Moment-Generating Function

Given a Random Variable x G R, if there exists an
h > such that

M{t) = (e tx )

^2 R G tx P(x) for a discrete distribution

J*_ e tx P(x) dx for a continuous distribution

(i)

-{.

for \t\ < h, then

M(t) = {e tx ) (2)

is the moment-generating function.

/oo
(1 4- tx + ±t 2 x 2 + . . ,)P(x) dx
-oo

= 1 + tmi + ^t 2 m 2 + . . . , (3)

Momental Skewness

Monge Patch 1191

where m r is the rth MOMENT about zero. The moment-
generating function satisfies

M x+y (t) = (e t(x+y) ) = (e tx e tv )

= (e**)^) = M x (t)M y (t). (4)

If M(t) is differentiable at zero, then the nth MOMENTS
about the Origin are given by M n (0)

M(t) = (e tx ) M(0) = 1 (5)

M'(t) = (xe tx ) M'(0) = (x) (6)

M"(t) = (x 2 e tx ) M"{0) = (x 2 ) (7)

M (n) (i) = (x n e tx ) M (n) (0) = <x n >. (8)
The MEAN and VARIANCE are therefore

/x=(x)=M'(0) (9)

<r 2 = (x 2 ) - <z} 2 = Af"(0) - [M'(0)] 2 , (10)

It is also true that

3 = W

,) U -\ (11)

where Mo — 1 an d f^'j 1S tne Jth moment about the origin.

It is sometimes simpler to work with the LOGARITHM of
the moment-generating function, which is also called the
Cumulant-Generating Function, and is defined by

R{t) = ln[M(i)]
M'(t)

R'(t) =

M(t)

(12)
(13)

E ,, ( ^M(^ (14)

[M{t)Y

But M (0) = (1) = 1, so

M = M'(0)=#'(0) (15)

<r 2 = M"(0) - [M'(0)] 2 = JZ"(0). (16)

see also CHARACTERISTIC FUNCTION, CUMULANT,

Cumulant-Generating Function, Moment

References

Kenney, J. F. and Keeping, E. S. "Moment-Generating and
Characteristic Functions," "Some Examples of Moment-
Generating Functions," and "Uniqueness Theorem, for
Characteristic Functions." §4.6-4.8 in Mathematics of
Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand,
pp. 72-77, 1951.

Momental Skewness

-2^1 2a 3'

where 71 is the FlSHER SKEWNESS.
see also Fisher Skewness, Skewness

Monad

A mathematical object which consists of a set of a single
element. The YlN-YANG is also known as the monad.

see also HEXAD, QUARTET, QUINTET, TETRAD, TRIAD,

Yin- Yang

Money- Changing Problem

see Coin Problem

Monge- Ampere Differential Equation

A second-order PARTIAL DIFFERENTIAL EQUATION of
the form

Hr + 2Ks + Lt + M + N(rt - s 2 ) = 0,

where H, K, L, M, and N are functions of x, y, 2, p,
and qr, and r, s, £, p, and q are defined by

_ d 2 z
r ~ dx*
d 2 z

s =

* =

p =

dxdy

dy 2

dz_

dx

dz

dy'

The solutions are given by a system of differential equa-
tions given by Iyanaga and Kawada (1980).

References

Iyanaga, S. and Kawada, Y. (Eds.). "Monge- Ampere Equa-
tions." §276 in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 879-880, 1980.

Monge's Chordal Theorem

see Radical Center

Monge's Form

A surface given by the form z = F(x, y).

see also Monge Patch

Monge Patch

A Monge patch is a Patch x : U -> R 3 of the form

x(u,v) = («,«,%«)), (1)

where U is an Open Set in R 2 and h : U -> R is
a differentiable function. The coefficients of the first
Fundamental Form are given by

E = 1 + hj
F = h u h v

(2)
(3)
(4)

1192 Mongers Problem

Monica Set

and the second FUNDAMENTAL FORM by

y/l + hu 2 +h v 2

yl + /iu + h v

9w

y/l + hj + h v 2

(5)

(6)

(7)

For a Monge patch, the GAUSSIAN CURVATURE and
Mean Curvature are

K =

H =

(1 + h v )h U u — 2h u h v h uv + (1 + h u )h v
* (l + h u 2 + h v 2 )*/*

(8)
(9)

see also Monge's Form, Patch

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 305-306, 1993.

Monge's Problem

Draw a Circle that cuts three given Circles Perpen-
dicularly. The solution is obtained by drawing the
Radical Center R of the given three Circles. If it
lies outside the three Circles, then the CIRCLE with
center R and RADIUS formed by the tangent from R to
one of the given CIRCLES intersects the given CIRCLES
perpendicularly. Otherwise, if R lies inside one of the
circles, the problem is unsolvable.

see also Circle Tangents, Radical Center

References

Dorrie, H. "Monge's Problem." §31 in 100 Great Problems

of Elementary Mathematics: Their History and Solutions.

New York: Dover, pp. 151-154, 1965.

Monge's Shuffle

A Shuffle in which Cards from the top of the deck in
the left hand are alternatively moved to the bottom and
top of the deck in the right hand. If the deck is shuffled
m times, the final position Xm and initial position xq of
a card are related by

2 m+1 a;rTl -(4p+l)[2 m - 1 + (-l) m - 1 (2 m - 2 + ... + 2 + l)]
+ (-l) m ~ 1 2x + 2 m + (-1)™" 1

for a deck of 2p cards (Kraitchik 1942).
see also Cards, Shuffle

References

Conway, J. H. and Guy, R. K. "Fractions Cycle into Deci-
mals." In The Book of Numbers. New York: Springer-
Verlag, pp. 157-163, 1996.

Kraitchik, M. "Monge's Shuffle." §12.2.14 in Mathematical
Recreations. New York: W. W. Norton, pp. 321-323, 1942.

Monge's Theorem

Draw three nonintersecting CIRCLES in the plane, and
the common tangent line for each pair of two. The points
of intersection of the three pairs of tangent lines lie on
a straight line.

References

Coxeter, H. S. M. "The Problem of Apollonius." Amer.
Math. Monthly 75, 5-15, 1968.

Graham, L. A. Problem 62 in Ingenious Mathematical Prob-
lems and Methods. New York: Dover, 1959. Ogilvy, C. S.
Excursions in Geometry. New York: Dover, pp. 115-117,
1990.

Walker, W. "Monge's Theorem in Many Dimensions." Math.
Gaz. 60, 185-188, 1976.

Monic Polynomial

A Polynomial in which the Coefficient of the high-
est Order term is 1.

see also MONOMIAL

Monica Set

The nth Monica set M n is defined as the set of COM-
POSITE NUMBERS x for which n\S(x) - S p (x), where

x = a Q + ai(K) 1 ) + . . . + a d (10 d ) = p x p 2 • • -p n , (1)

and

S{x) = ^ a i

3=0

Sp(b) = y^ 5 (P*)»

(2)
(3)

Every Monica set has an infinite number of elements.
The Monica set M n is a subset of the Suzanne Set S n .

Monkey and Coconut Problem

Monkey Saddle 1193

If x is a Smith Number, then it is a member of the
Monica set M n for all n G N. For any INTEGER k > 1,
if z is a Aj-Smith Number, then x € Mk-i-
see also Suzanne Set

References

Smith, M. "Cousins of Smith Numbers: Monica and Suzanne
Sets." Fib. Quart 34, 102-104, 1996.

Monkey and Coconut Problem

A Diophantine problem (i.e., one whose solution must
be given in terms of Integers) which seeks a solution
to the following problem. Given n men and a pile of
coconuts, each man in sequence takes (l/n)th of the
coconuts and gives the m coconuts which do not divide
equally to a monkey. When all n men have so divided,
they divide the remaining coconuts five ways, and give
the m coconuts which are left-over to the monkey. How
many coconuts N were there originally? The solution is
equivalent to solving the n+1 DIOPHANTINE EQUATIONS

iV = nA + m
(n - 1)A = nB + m
(n- 1)B = nC + m

(n-l)X = nY + m
(n- 1)Y = nZ + m,

and is given by

N = kn n+1 -m(n-l),

where k is an an arbitrary INTEGER (Gardner 1961).

For the particular case of n = 5 men and m = 1 left
over coconuts, the 6 equations can be combined into the
single Diophantine Equation

1,024JV = 15, 625F + 11,529,

where F is the number given to each man in the last
division. The smallest POSITIVE solution in this case is
N = 15,621 coconuts, corresponding to k = 1 and F —
1,023 (Gardner 1961). The following table shows how
this rather large number of coconuts is divided under
the scheme described above.

Removed Given to Monkey Left

3,124
2,499
1,999

1,599
1,279
5 x 1023

15,621
12,496
9,996
7,996
6,396
5,116

If no coconuts are left for the monkey after the final n-
way division (Williams 1926), then the original number
of coconuts is

f (1 + nfc)n n - (n - 1) n odd

1 (n - 1 + nk)n n - (n - 1) n even.

The smallest POSITIVE solution for case n = 5 and m =
1 is N = 3, 121 coconuts, corresponding to k = 1 and
1,020 coconuts in the final division (Gardner 1961), The
following table shows how these coconuts are divided.

624

1

499

1

399

1

319

1

255

1

5 x 204

Removed Given to Monkey Left
" " 3,121

2,496
1,996
1,596
1,276
1,020
0_

A different version of the problem having a solution of

79 coconuts is considered by Pappas (1989).

see also DIOPHANTINE EQUATION — LINEAR, PELL

Equation

References

Anning, N. "Monkeys and Coconuts." Math. Teacher 54,

560-562, 1951.
Bowden, J. "The Problem of the Dishonest Men, the Mon-
keys, and the Coconuts." In Special Topics in Theoretical

Arithmetic. Lancaster, PA: Lancaster Press, pp. 203-212,

1936.
Gardner, M. "The Monkey and the Coconuts." Ch. 9 in The

Second Scientific American Book of Puzzles & Diversions:

A New Selection. New York: Simon and Schuster, 1961.
Kirchner, R. B. "The Generalized Coconut Problem." Amer.

Math. Monthly 67, 516-519, 1960.
Moritz, R. E. "Solution to Problem 3,242." Amer. Math.

Monthly 35, 47-48, 1928.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number

Theory. New York: Dover, pp. 52-54, 1988.
Olds, C. D. Continued Fractions. New York: Random House,

pp. 48-50, 1963.
Pappas, T. "The Monkey and the Coconuts." The Joy of

Mathematics. San Carlos, CA: Wide World Publ./Tetra,

pp. 226-227 and 234, 1989.
Williams, B. A. "Coconuts." The Saturday Evening Post,

Oct. 9, 1926.

Monkey Saddle

1194 Monkey Saddle

A SURFACE which a monkey can straddle with both his
two legs and his tail. A simply Cartesian equation for
such a surface is

z = x(x 2 - 3y 2 ), (1)

which can also be given by the parametric equations

x(u, v) = u
y{u,v) =v
z(u,v) = u — 3uv .

(2)
(3)
(4)

The coefficients of the first and second FUNDAMENTAL
FORMS of the monkey saddle are given by

6w

\/l + 9n 4 + I8u 2 v 2 + 9v 4

(5)

6v

(6)

Vl + 9u 4 + 18u 2 v 2 + 9v 4

6u

(7)

\/l + 9m 4 + 18u 2 u 2 + 9t; 4

E = 1 + 9(w 2 - v 2 ) 2

(8)

F= -18Mv(u 2 -t> 2 )

(9)

G = 1 + 36wV,

(10)

giving RlEMANNIAN METRIC

ds 2 = [1 + (3u 2 - 3v 2 ) 2 ] du - 2[l%uv(u - v 2 )] dudv

+(l + 36uv 2 )dv 2 , (11)

Area Element

dA = \A + 9 ^ 4 + 18^ 2 ^ 2 + 9^ 4 d« A dv, (12)

and Gaussian and Mean Curvatures
36(u 2 +v 2 )

H

(1 + 9u 4 + 18u 2 v 2 + 9v 4 ) 2
27^(-^ 4 + 2^V+3t; 4 )
■" (1 + 9u 4 + 18u 2 u 2 + 9v 4 ) 3 / 2

(13)
(14)

(Gray 1993). Every point of the monkey saddle except
the origin has NEGATIVE GAUSSIAN CURVATURE.

see also Crossed Trough, Partial Derivative

References

Coxeter, H. S. M. Introduction- to Geometry, 2nd ed. New
York: Wiley, p. 365, 1969.

Gray, A. Modern. Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 213-215, 262-263,
and 288-289, 1993.

Hilbert, D. and Cohn-Vosseri, S. Geometry and the Imagina-
tion. New York: Chelsea^ p. 202, 1352.

Monogenic Function

Monochromatic Forced Triangle

Given a COMPLETE GRAPH K n which is two- colored,
the number of forced monochromatic TRIANGLES is at
least

f ±u(u - l)(w - 2) for n = 2u
< |u(u - l)(4u + 1) for n = Au + 1
[ lu(u + 1)(4« - 1) for n = 4u + 3.

The first few numbers of monochromatic forced triangles

are 0, 0, 0, 0, 0, 2, 4, 8, 12, 20, 28, 40, ... (Sloane's

A014557).

see also COMPLETE GRAPH, EXTREMAL GRAPH

References

Goodman, A. W. "On Sets of Acquaintances and Strangers

at Any Party." Amer. Math. Monthly 66, 778-783, 1959.
Sloane, N. J. A. Sequence A014553 in "An On-Line Version

of the Encyclopedia of Integer Sequences."

Monodromy

A general concept in CATEGORY THEORY involving the

globalization of local Morphisms.

see also HOLONOMY

Monodromy Group

A technically defined Group characterizing a system of
linear differential equations

yj

E

ajk(x)yk

for j = 1, . . . , n, where a jk are Complex Analytic
Functions of a; in a given Complex Domain.

see also Hilbert's 21st Problem, Riemann P-Series

References

Iyanaga, S. and Kawada, Y. (Eds.). "Monodromy Groups."
§253B in Encyclopedic Dictionary of Mathematics. Cam-
bridge, MA: MIT Press, p. 793, 1980.

Monodromy Theorem

If a Complex function / is Analytic in a Disk con-
tained in a simply connected Domain D and / can be
Analytically Continued along every polygonal arc

in £>, then / can be ANALYTICALLY CONTINUED to a

single-valued Analytic Function on all of D\

see also Analytic Continuation

Monogenic Function

lim

Z->ZQ

Z — ZQ

is the same for all paths in the COMPLEX PLANE, then
f(z) is said to be monogenic at zq. Monogenic there-
fore essentially means having a single Derivative at a
point. Functions are either monogenic or have infinitely
many DERIVATIVES (in which case they are called POLY-
GENIC); intermediate cases are not possible.

Monohedral Tiling

Monster Group 1195

see also POLYGENIC FUNCTION

References

Newman, J. R. The World of Mathematics, Vol 3. New-
York: Simon & Schuster, p. 2003, 1956.

Monotone

Another word for monotonic.

see also Monotonic Function, Monotonic Se-
quence, Monotonic Voting

Monohedral Tiling

A Tiling is which all tiles are congruent.

see also Anisohedral Tiling, Isohedral Tiling

References

Berglund, J. "Is There a fc-Anisohedral Tile for k > 5?"
Amer. Math. Monthly 100, 585-588, 1993.

Griinbaum, B. and Shephard, G. C. "The 81 Types of Isohe-
dral Tilings of the Plane." Math. Proc. Cambridge Philos.
Soc. 82, 177-196, 1977.

Monoid

A GROUP-like object which fails to be a Group because
elements need not have an inverse within the object, A
monoid S must also be ASSOCIATIVE and an IDENTITY
Element I e S such that for all a e 5, la = al = a.
A monoid is therefore a SEMIGROUP with an identity
element. A monoid must contain at least one element.

The numbers of free idempotent monoids on n letters
are 1, 2, 7, 160, 332381, ... (Sloane's A005345).

see also Binary Operator, Group, Semigroup

References

Rosenfeld, A. An Introduction to Algebraic Structures. New

York: Holden-Day, 1968.
Sloane, N. J. A. Sequence A005345/M1820 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Monomial

A POLYNOMIAL consisting of a single term.

see also BINOMIAL, MONIC POLYNOMIAL, POLYNOMIAL,

Trinomial

Monomino

The unique 1-POLYOMlNO, consisting of a single
Square.

see also Domino, Triomino

References

Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer-
ican Book of Mathematical Puzzles & Diversions. New
York: Simon and Schuster, pp. 124-140, 1959.

Monomorph

An Integer which is expressible in only one way in the
form x 2 + Dy 2 or x 2 — Dy 2 where x 2 is RELATIVELY
Prime to Dy 2 . If the Integer is expressible in more
than one way, it is called a POLYMORPH.
see also Antimorph, Idoneal Number, Polymorph

Monotone Decreasing

Always decreasing; never remaining constant or increas-
ing.

Monotone Increasing

Always increasing; never remaining constant or decreas-
ing.

Monotonic Function

A function which is either entirely nonincreasing or non-
decreasing. A function is monotonic if its first Deriv-
ative (which need not be continuous) does not change
sign.

Monotonic Sequence

A Sequence {a n } such that either (1) a i+ i > a» for
every i > 1, or (2) a*+i < a* for every i > 1.

Monotonic Voting

A term in Social Choice Theory meaning a change
favorable for X does not hurt X.

see also Anonymous, Dual Voting

Monster Group

The highest order Sporadic Group M. It has Order

2 46 - 3 20 • 5 9 -7 6 - ll 2 . 13 3 - 17- 19 -23 -29 '31*41 -47-59- 71,

and is also called the Friendly Giant Group. It was
constructed in 1982 by Robert Griess as a GROUP of
Rotations in 196,883-D space.

see also Baby Monster Group, Bimonster, Leech
Lattice

References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.;
and Wilson, R. A. Atlas of Finite Groups: Maximal Sub-
groups and Ordinary Characters for Simple Groups. Ox-
ford, England: Clarendon Press, p. viii, 1985.

Conway, J. H. and Norton, S. P. "Monstrous Moonshine."
Bull. London Math. Soc. 11, 308-339, 1979.

Conway, J. H. and Sloane, N. J. A. "The Monster Group
and its 196884-Dimensional Space" and "A Monster Lie
Algebra?" Chs. 29-30 in Sphere Packings, Lattices, and
Groups, 2nd ed. New York: Springer- Verlag, pp. 554-571,
1993.

Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat . bham.ac.uk/ atlas /M. html.

Monomorphism

An Injective Morphism.

1196 Monte Carlo Integration

Monty Hall Problem

Monte Carlo Integration

In order to integrate a function over a complicated DO-
MAIN D 7 Monte Carlo integration picks random points
over some simple DOMAIN D' which is a superset of L>,
checks whether each point is within I?, and estimates
the Area of D (Volume, n-D Content, etc.) as the
AREA of D' multiplied by the fraction of points falling
within D' '.

An estimate of the uncertainty produced by this tech-
nique is given by

/

fdV*V(f)±

(f 2 ) ~ iff

N

see also MONTE CARLO METHOD

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Simple Monte Carlo Integration" and "Adap-
tive and Recursive Monte Carlo Methods." §7.6 and 7.8
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 295-299 and 306-319, 1992.

Monte Carlo Method

Any method which solves a problem by generating suit-
able random numbers and observing that fraction of
the numbers obeying some property or properties. The
method is useful for obtaining numerical solutions to
problems which are too complicated to solve analyti-
cally. The most common application of the Monte Carlo
method is Monte Carlo Integration.

see also Monte Carlo Integration

References

Sobol, I. M. A Primer for the Monte Carlo Method, Boca
Raton, FL: CRC Press, 1994.

Montel's Theorem

Let f(z) be an analytic function of z, regular in the
half-strip S defined by a < x < b and y > 0. If f(z)
is bounded in S and tends to a limit / as y — > oo for
a certain fixed value £ of x between a and 6, then f(z)
tends to this limit / on every line x = xq in 5, and
f(z) — > I uniformly for a + $<Xo<b — 8.

see also Vitali's Convergence Theorem

References

Titchmarsh, E. C The Theory of Functions, 2nd ed. Oxford,
England: Oxford University Press, p. 170, 1960.

Monty Hall Dilemma

see Monty Hall Problem

Monty Hall Problem

The Monty Hall problem is named for its similarity to
the Let's Make a Deal television game show hosted by
Monty Hall. The problem is stated as follows. Assume
that a room is equipped with three doors. Behind two
are goats, and behind the third is a shiny new car. You
are asked to pick a door, and will win whatever is behind
it. Let's say you pick door 1. Before the door is opened,
however, someone who knows what's behind the doors
(Monty Hall) opens one of the other two doors, revealing
a goat, and asks you if you wish to change your selection
to the third door (i.e., the door which neither you picked
nor he opened). The Monty Hall problem is deciding
whether you do.

The correct answer is that you do want to switch. If
you do not switch, you have the expected 1/3 chance of
winning the car, since no matter whether you initially
picked the correct door, Monty will show you a door with
a goat. But after Monty has eliminated one of the doors
for you, you obviously do not improve your chances of
winning to better than 1/3 by sticking with your original
choice. If you now switch doors, however, there is a 2/3
chance you will win the car (counterintuitive though it
seems).

di

d 2

Winning Probability

pick
pick

stick
switch

1/3

2/3

The problem can be generalized to four doors as follows.
Let one door conceal the car, with goats behind the other
three. Pick a door d\. Then the host will open one of
the nonwinners and give you the option of switching.
Call your new choice (which could be the same as d\ if
you don't switch) d 2 . The host will then open a second
nonwinner, and you must decide for choice d?> if you
want to stick to di or switch to the remaining door.
The probabilities of winning are shown below for the
four possible strategies.

d!

d 2

Winning Probability

pick stick stick 4/8

pick switch stick 3/8

pick stick switch 6/8

pick switch switch 5/8

The above results are characteristic of the best strategy
for the n-stage Monty Hall problem: stick until the last
choice, then switch.

see also Alias' Paradox

References

Barbeau, E. "The Problem of the Car and Goats." CM J 24,

149, 1993.
Bogomolny, A. "Monty Hall Dilemma." http://www.cut-

the-knot . com/hall .html.
Dewdney, A. K. 200% of Nothing. New York: Wiley, 1993.
Donovan, D. "The WWW Tackles the Monty Hall Problem."

http : //math. rice . edu/~ddonovan/montyurl .html.
Ellis, K. M. "The Monty Hall Problem." http://www.io.

com/ -kmellis/monty. html.

Moore Graph

Morera's Theorem

1197

Gardner, M. Aha! Gotcha: Paradoxes to Puzzle and Delight,

New York: W. H. Freeman, 1982.
Gillman, L. "The Car and the Goats." Amer. Math. Monthly

99, 3, 1992.
Selvin, S. "A Problem in Probability." Amer. Stat 29, 67,

1975.
vos Savant, M. The Power of Logical Thinking. New York:

St. Martin's Press, 1996.

Moore Graph

A Graph with Diameter d and Girth 2d + 1. Moore
graphs have DIAMETER of at most 2. Every Moore graph
is both REGULAR and distance regular. Hoffman and
Singleton (1960) show that fc-regular Moore graphs with
Diameter 2 have k e {2, 3, 7, 57}.

References

Godsil, C. D. "Problems in Algebraic Combinatorics." Elec-
tronic J. Combinatorics 2, Fl, 1-20, 1995. http://www.
combinatorics . org/Volume_2/volume2 .htmltFl.

Hoffman, A. J. and Singleton, R. R. "On Moore Graphs of
Diameter Two and Three." IBM J. Res. Develop. 4, 497-
504, 1960.

Moore-Penrose Generalized Matrix Inverse

Given an m x n Matrix B, the Moore-Penrose gener-
alized Matrix Inverse is a unique nxm Matrix B +
which satisfies

(i)

(2)
(3)
(4)

It is also true that

(5)

is the shortest length Least Squares solution to the
problem

Bz = c. (6)

If the inverse of (B B) exists, then

B + = (B T B)" 1 B T , (7)

where B T is the Matrix Transpose, as can be seen
by premultiplying both sides of (7). by B to create a
Square Matrix which can then be inverted,

BB+B

= B

B+BB+

= B +

(BB + ) T

= BB +

(B + B) T

= B + B

z =

B+c

giving

B T Bz=B T c,

; = (B T B)- 1 B T c
= B + c.

(8)

(9)

see also Least Squares Fitting, Matrix Inverse

References

Ben-Israel, A. and Greville, T. N. E. Generalized Inverses:

Theory and Applications. New York: Wiley, 1977.
Lawson, C. and Hanson, R. Solving Least Squares Problems.

Englewood Cliffs, NJ: Prentice-Hall, 1974.
Penrose, R. "A Generalized Inverse for Matrices." Proc.

Cambridge Phil Soc. 51, 406-413, 1955.

Mordell Conjecture

Diophantine Equations that give rise to surfaces with
two or more holes have only finite many solutions in
GAUSSIAN Integers with no common factors. Fermat's
equation has (ra-l)(n-2)/2 HOLES, so the Mordell con-
jecture implies that for each INTEGER n > 3, the FER-
MAT Equation has at most a finite number of solutions.
This conjecture was proved by Faltings (1984).
see also Fermat Equation, Fermat's Last Theo-
rem, Safarevich Conjecture, Shimura-Taniyama
Conjecture

References

Faltings, G. "Die Vermutungen von Tate und Mordell."
Jahresber. Deutsch. Math.-Verein 86, 1-13, 1984.

Ireland, K. and Rosen, M. "The Mordell Conjecture." §20.3
in A Classical Introduction to Modern Number Theory,
2nd ed. New York: Springer- Verlag, pp. 340-342, 1990.

Mordell Integral

The integral

<f>{t

■*»-/

iritx -\-2iriux

g2irix ^

dx

which is related to the Theta Functions, Mock
Theta Functions, and Riemann Zeta Function.

Mordell- Weil Theorem

For Elliptic Curves over the Rationals, Q, the num-
ber of generators of the set of RATIONAL POINTS is al-
ways finite. This theorem was proved by Mordell in 1921
and extended by Weil in 1928 to AbeLIAN VARIETIES
over Number Fields.

References

Ireland, K. and Rosen, M. "The Mordell-Weil Theorem."
Ch. 19 in A Classical Introduction to Modern Number The-
ory, 2nd ed. New York: Springer- Verlag, pp. 319-338,
1990.

Morera's Theorem

If f(z) is continuous in a simply connected region D and

satisfies

f fdz =

for all closed Contours 7 in £>, then f(z) is Analytic

in D.

see also Cauchy Integral Theorem

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 373-374, 1985.

1198 Morgado Identity

Morgado Identity

An identity satisfied by w Generalized Fibonacci
Numbers:

+e 2 q 2n (w n U4U 5 - w n+l U 2 U 6 - w n UiU s ) 2

= (Wn+lWn + 2W n +6 + W n W n +4W n + h f , (l)

where

Morgan-Voyce Polynomial

Defining

cos(9 = \{x + 2)
cosh0= |(x + 2)

gives

sin[(n+l)0]

(11)
(12)

(13)

e = pab — qa — b
U n = Wn(0,l;p,7).

see a/50 Generalized Fibonacci Number

(2)
(3)

and

References

Morgado, J. "Note on Some Results of A. F. Horadam and A.
G. Shannon Concerning a Catalan's Identity on Fibonacci
Numbers." Portugaliae Math. 44, 243-252, 1987.

Morgan-Voyce Polynomial

Polynomials related to the Brahmagupta Polynomi-
als. They are defined by the Recurrence Relations

B n (x)
B n (x)

b n (x)

b n {x) =

The Morgan-Voyce polynomials are related to the FI-
BONACCI Polynomials F n (x) by

sine/

_ sinh[(n + l)<j>]
sinh</>

(14)

cos[f(2n + l)0]
cos(^)

(15)

cosh[§(2n+ 1)0]

i / 1 s\\

(16)

b n (x) = xB n -l{x) + 6 n -l(a0

(1)

b n (x 2 ) = F 2n+1 (x) (17)
B n (x 2 ) = -F 2n+2 {x) (18)

X

B n (x) = (X + l)J5 n _i(x) + 6n-l

(x)

(2)

for n > 1, with

(Swamy 1968).

6 (x) = B (x) = l.

(3)

B n (x) satisfies the Ordinary Differential Equa-
tion

Alternative recurrences are

x(x + 4)2/" -1- 3{x + 2)y' - n(n + 2)y = 0, (19)

B n + lBn-l — B n = —1

(4)

and & n (x) the equation

6 n+ i6„_i — b n = x.

(5)

The polynomials can be given explicitly by the sums

*W = E( n - k )

(6)
(7)

Defining the Matrix

Q =

gives the identities

Q n =

x + 2 -1
1

B n —B n -1

B n -1 —B n -2

Q- _ Q"-i =

bn —bn-1

b n -! —bn-2

(8)

(9)
(10)

x(x + 4)2/" + 2(x + l)y' - n(n + l)y = 0. (20)

These and several other identities involving derivatives
and integrals of the polynomials are given by Swamy
(1968).

see also Brahmagupta Polynomial, Fibonacci

Polynomial

References

Lahr, J. "Fibonacci and Lucas Numbers and the Morgan-
Voyce Polynomials in Ladder Networks and in Electric Line
Theory." In Fibonacci Numbers and Their Applications
(Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam).
Dordrecht, Netherlands: Reidel, 1986.

Morgan-Voyce, A. M. "Ladder Network Analysis Using Fi-
bonacci Numbers." IRE Trans. Circuit Th. CT-6, 321-
322, Sep. 1959.

Swamy, M. N. S. "Properties of the Polynomials Defined by
Morgan-Voyce." Fib. Quart. 4, 73-81, 1966.

Swamy, M. N. S. "More Fibonacci Identities." Fib. Quart.
4, 369-372, 1966.

Swamy, M. N. S. "Further Properties of Morgan-Voyce Poly-
nomials." Fib. Quart. 6, 167-175, 1968.

Morley Centers

Morley's Theorem 1199

Morley Centers

The Centroid of Morley's Triangle is called Mor-
ley's first center. It has Triangle Center Function

a = cos(lA) + 2cos(| J B)cos(|C).

The Perspective Center of Morley's Triangle
with reference TRIANGLE ABC is called Morley's sec-
ond center. The Triangle Center Function is

a = sec(|i4).

see also Centroid (Geometric), Morley's Theo-
rem, Perspective Center

References

Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.

Kimberling, C "1st and 2nd Morley Centers." http://www.
evansville.edu/~ck6/t centers/recent /morley. html.

Oakley, C O. and Baker, J. C. "The Morley Trisector The-
orem." Amer. Math. Monthly 85, 737-745, 1978.

Morley's Formula

£

= 1 +

(")'

+

m(m -f 1)

1 3

1-2

r(l-fm)

[r(i - §m)P

COS(|?7l7r),

where (£) is a Binomial Coefficient and T(z) is the
Gamma Function.

Morley's Theorem

The points of intersection of the adjacent Trisectors
of the Angles of any Triangle AABC are the Ver-
tices of an Equilateral Triangle ADEF known as
Morley's Triangle. Taylor and Marr (1914) give two
geometric proofs and one trigonometric proof.

A generalization of Morley's THEOREM was discov-
ered by Morley in 1900 but first published by Taylor
and Marr (1914). Each Angle of a Triangle AABC
has six trisectors, since each interior angle trisector has
two associated lines making angles of 120° with it. The
generalization of Morley's theorem states that these tri-
sectors intersect in 27 points (denoted Dij, Eij } Fij, for
ijj = 0, 1, 2) which lie six by six on nine lines. Further-
more, these lines are in three triples of PARALLEL lines,

(1)22^22, E12D21, FiqFqi), (^22^22) -^21^12} #01-Elo)j

and (E22F22, F 12 E 2U D 10 D i), making Angles of 60°
with one another (Taylor and Marr 1914, Johnson 1929,

p. 254).

Let L, M, and N be the other trisector-trisector inter-
sections, and let the 27 points Lij, Mij, 7V^ for i,j = 0,
1, 2 be the Isogonal Conjugates of D, E, and F.
Then these points lie 6 by 6 on 9 CONICS through
AABC. In addition, these CONICS meet 3 by 3 on the
ClRCUMCIRCLE, and the three meeting points form an
Equilateral Triangle whose sides are Parallel to
those of ADEF.

see also CONIC SECTION, MORLEY CENTERS, TRISEC-
TION

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 47-50, 1967.

Gardner, M. Martin Gardner's New Mathematical Diver-
sions from Scientific American. New York: Simon and
Schuster, pp. 198 and 206, 1966.

Honsberger, R. "Morley's Theorem." Ch. 8 in Mathematical
Gems I. Washington, DC: Math. Assoc. Amer., pp. 92-98,
1973.

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 253-256, 1929.

Kimberling, C. "Hofstadter Points." Nieuw Arch. Wiskunder
12, 109-114, 1994.

Marr, W. L. "Morley's Trisection Theorem: An Extension
and Its Relation to the Circles of Apollonius." Proc. Ed-
inburgh Math. Soc. 32, 136-150, 1914.

Oakley, C. O. and Baker, J. C. "The Morley Trisector The-
orem." Amer. Math. Monthly 85, 737-745, 1978.

Pappas, T. "Trisecting & the Equilateral Triangle." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 174, 1989.

Taylor, F. G. "The Relation of Morley's Theorem to the Hes-
sian Axis and Circumcentre." Proc. Edinburgh Math. Soc.
32, 132-135, 1914.

Taylor, F. G. and Marr, W. L. "The Six Trisectors of Each
of the Angles of a Triangle." Proc. Edinburgh Math. Soc.
32, 119-131, 1914.

1200 Morley's Triangle

Moser's Circle Problem

Morley's Triangle

An Equilateral Triangle considered by Morley's
Theorem with side lengths

8Rsm(lA)sin(lB)sm(lC),

where R is the ClRCUMRADlUS of the original Trian-
gle.

Morphism

A map between two objects in an abstract CATEGORY.

1. A general morphism is called a HOMOMORPHISM,

2. An injective morphism is called a MONOMORPHISM,

3. A surjective morphism is an EPIMORPHISM,

4. A bijective morphism is called an ISOMORPHISM (if
there is an isomorphism between two objects, then
we say they are isomorphic),

5. A surjective morphism from an object to itself is
called an Endomorphism, and

6. An Isomorphism between an object and itself is
called an AUTOMORPHISM.

see also AUTOMORPHISM, EPIMORPHISM, HOMEOMOR-
PHISM, HOMOMORPHISM, ISOMORPHISM, MONOMOR-
PHISM, Object

Morrie's Law

cos(20°)cos(40°)cos(80°) = f.

This identity was referred to by Feynman (Gleick 1992).
It is a special case of the general identity

Morse Theory

"Calculus of Variations in the large" which uses
nonlinear techniques to address problems in the CAL-
CULUS OF Variations. Morse theory applied to a
Function g on a Manifold W with g(M) = and
g(M') = 1 shows that every Cobordism can be real-
ized as a finite sequence of SURGERIES. Conversely, a
sequence of SURGERIES gives a COBORDISM.

see also Calculus of Variations, Cobordism,
Surgery

Morse-Thue Sequence

see THUE-MORSE SEQUENCE

Mortal

A nonempty finite set ofnxn MATRICES with INTE-
GER entries for which there exists some product of the
Matrices in the set which is equal to the zero Matrix.

Mortality Problem

For a given n, is the problem of determining if a set is
MORTAL solvable? n = 1 is solvable, n = 2 is unknown,
and n > 3 is unsolvable.

see also Life Expectancy

Morton- Franks- Williams Inequality

Let E be the largest and e the smallest POWER of I in
the HOMFLY POLYNOMIAL of an oriented Link, and
i be the Braid Index. Then the Morton-Franks-
Williams Inequality holds,

i>\{E-e) + l

2 k Y[ cos(2 j a) =

sin(2 fe a)

j=0

with k = 3 and a = 20° (Beyer et al 1996).

References

Anderson, E. C. "Morrie's Law and Experimental Mathemat-
ics." To appear in J. Recr. Math.

Beyer, W. A.; Louck, J. D.; Zeilberger, D. "A Generalization
of a Curiosity that Feynman Remembered All His Life."
Math. Mag. 69, 43-44, 1996.

Gleick, J. Genius: The Life and Science of Richard Feyn-
man. New York: Pantheon Books, p. 47, 1992.

Morse Inequalities

Topological lower bounds in terms of Betti Numbers
for the number of critical points form a smooth function
on a smooth MANIFOLD.

(Franks and Williams 1985, Morton 1985). The inequal-
ity is sharp for all Prime Knots up to 10 crossings with
the exceptions of 09o42, 09o49, IO132, IO150, and 10i56-
see also BRAID INDEX

References

Pranks, J. and Williams, R. F. "Braids and the Jones Poly-
nomial." Trans. Amer. Math. Soc. 303, 97-108, 1987.

Mosaic

see Tessellation

Moser

The very LARGE NUMBER consisting of the number 2
inside a MEGA-gon.

see also Mega, Megistron

Moser's Circle Problem

see Circle Cutting

Moss's Egg
Moss's Egg

Mouth

1201

An Oval whose construction is illustrated in the above

diagram.

see also EGG, Oval

References

Dixon, R. Mathographics. New York: Dover, p. 5, 1991.

Motzkin Number

^L^zrii

The Motzkin numbers enumerate various combinatorial
objects. Donaghey and Shapiro (1977) give 14 different
manifestations of these numbers. In particular, they give
the number of paths from (0, 0) to (n, 0) which never
dip below y = and are made up only of the steps (1,
0), (1, 1), and (1, -1), i.e., ->, /\ and \. The first are
1, 2, 4, 9, 21, 51, . . . (Sloane's A001006). The Motzkin
number GENERATING Function M(z) satisfies

M= l + xM + x 2 M 2

and is given by

1 - x - Vl ~ 2x - 3x 2

(1)

M(x) =

2x 2
1 + x + 2x 2 + 4x 3 + 9z 4 + 21a; 5 + . . . , (2)

or by the RECURRENCE RELATION

n-2

M n = a„_i + ^ MkM n -2-k (3)

with Mo = 1. The Motzkin number M n is also given by

•— I e <- 3 >-(!)(i)

M n =

(4)

o+b=n+2
a>0,6>0

(-1)

n + 1

22n+5

a+6=n+2
a>0,6>0

(2a-

(-3) Q /2a\ /26'

-l)(26-l)^a;^

(5)

where (£) is a Binomial Coefficient.

see also Catalan Number, King Walk, Schroder

Number

References

Barcucci, E.; Pinzani, R.; and Sprugnoli, R. "The Motzkin
Family." Pure Math. Appl. Ser. A 2, 249-279, 1991.

Donaghey, R. "Restricted Plane Tree Representations of Four
Motzkin- Catalan Equations." J. Combin. Th. Ser. B 22,
114-121, 1977.

Donaghey, R. and Shapiro, L. W. "Motzkin Numbers." J.
Combin. Th. Ser. A 23, 291-301, 1977.

Kuznetsov, A.; Pak, I.; and Postnikov, A. "Trees Associated
with the Motzkin Numbers." J. Combin. Th. Ser. A 76,
145-147, 1996.

Motzkin, T. "Relations Between Hypersurface Cross Ratios,
and a Combinatorial Formula for Partitions of a Poly-
gon, for Permanent Preponderance, and for Nonassociative
Products." Bull. Amer. Math. Soc. 54, 352-360, 1948.

Sloane, N. J. A. Sequence A001006/M1184 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Moufang Plane

A Projective Plane in which every line is a transla-
tion line is called a Moufang plane.

References

Colbourn, C. J. and Dinitz, J. H. (Eds.) CRC Handbook

of Combinatorial Designs. Boca Raton, FL: CRC Press,

p. 710, 1996.

Mousetrap

A Permutation problem invented by Cayley.

References

Guy, R. K. "Mousetrap." §E37 in Unsolved Problems in

Number Theory, 2nd ed. New York: Springer- Verlag,

pp. 237-238, 1994.

Mouth

A Principal Vertex Xi of a Simple Polygon P is
called a mouth if the diagonal [zi-ijXi+i] is an extremal
diagonal (i.e., the interior of [xi-i,Xi+i] lies in the ex-
terior of P).
see also ANTHROPOMORPHIC POLYGON, EAR, ONE-

Mouth Theorem

References

Toussaint, G. "Anthropomorphic Polygons." Amer. Math.
Monthly 122, 31-35, 1991.

1202 Moving A verage

Moving Sofa Constant

Moving Average

Given a SEQUENCE {ai}fL ly an n-moving average is a
new sequence {si}^S[ n+1 denned from the a% by taking
the AVERAGE of subsequences of n terms,

This gives

Si

i+n-l

n ^-^

j=i

see also Average, Spencer's 15-Point Moving Av-
erage

References

Kenney, J. F. and Keeping, E. S. "Moving Averages." §14.2

in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ:

Van Nostrand, pp. 221-223, 1962.

Moving Ladder Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

What is the longest ladder which can be moved around
a right-angled hallway of unit width? For a straight,
rigid ladder, the answer is 2y / 2. For a smoothly-shaped
ladder, the largest diameter is > 2(1 + \/2) (Finch).

see also Moving Sofa Constant, Piano Mover's
Problem

References

Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/sof a/sof a. html.

Moving Sofa Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

What is the sofa of greatest Area S which can be moved
around a right-angled hallway of unit width? Hammer-
sley (Croft et al. 1994) showed that

S> ~-r- =2.2074....

2 7T

(1)

Gerver (1992) found a sofa with larger Area and pro-
vided arguments indicating that it is either optimal or
close to it. The boundary of Gerver's sofa is a com-
plicated shape composed of 18 Arcs. Its Area can be
given by defining the constants A, B, 0, and 9 by solving

A(cos 9 - cos 0) - 2B sin + (9 - - 1) cos 9

— sin + cos + sin = (2)
,4(3 sin 9 + sin 0) - IB cos + 3(0 - - 1) sin 9

+3 cos 9 — sin + cos — (3)

A cos — (sin <t>+\ — \ cos + B sin 0) = (4)

{A+\TT-4>-6)-[B-\{d-4>){\ + A)-\{e-<l>f\=0.

(5)

A = 0.094426560843653 . . .
B = 1.399203727333547...
= 0.039177364790084...
B = 0.681301509382725 . . . .

(6)
(7)
(8)
(9)

(10)

Now define

r(a) =
/ l

2

for < a <

\{l + A + a-<f>)
for < a < 9

A + a -

for 9 <a< ~7v~9
B - 1(1* - a - 0)(1 + A) - 1(1* - a - 0) 2 ,
, for |* — 6 < a < |* — 0,

where

s(a) = l-r(a) (11)

(B-\{a- 0)(1 + A) for < a < 9
u{a)^l -|(a-0) 2

[A-}-|*-0-a for 6 < a < \tz

(12)

_ du _f -1(1 + 4) - \(a-(t>) for <j> < a <

if < a < \ir.

(13)

D "M = Ta=\-l

Finally, define the functions

yi(a) = l- / r(t)s'mtdt (14)

Jo

2/2 (a) = 1- / s(t)s'mtdt (15)

Jo

ys(a) = l— / s (t) sin tdt-u(a) sin a. (16)
Jo

The Area of the optimal sofa is given by

/•tt/2 —

A = 2 / yi (a)r(a) cos a da

Jo

f°
+2 / y2(a)s(a) cos a da

Jo

/•ir/4

+2 / 2/3 (a) [u (a) sin a — D u (a) cos a — s(a) cos a] da

= 2.21953166887197 . . . (17)

(Finch).

see also Piano Mover's Problem

Mrs. Perkins 7 Quilt

Miiller-Lyer Illusion 1203

References

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Prob-
lems in Geometry. New York: Springer- Verlag, 1994.

Finch, S. "Favorite Mathematical Constants." http://vvw.
mathsof t . cora/asolve/constant/sof a/sofa. html.

Gerver, J. L. "On Moving a Sofa Around a Corner." Geome-
triae Dedicata42, 267-283, 1992.

Stewart, I. Another Fine Math You've Got Me Into. . . . New
York: W. H. Freeman, 1992.

Mrs. Perkins' Quilt

The Dissection of a Square of side n into a number
S n of smaller squares. Unlike a Perfect SQUARE DIS-
SECTION, however, the smaller Squares need not be all
different sizes. In addition, only prime dissections are
considered so that patterns which can be dissected on
lower order SQUARES are not permitted. The following
table gives the smallest number of coprime dissections
of an n x n quilt (Sloane's A005670).

n

s n

1

1

2

4

3

6

4

7

5

8

6-7

9

8-9

10

10-13

11

14-17

12

18-23

13

24-29

14

30-39

15

40

16

41

15

42-100

[17,19]

see also Perfect Square Dissection

References

Conway, J. H. "Mrs. Perkins's Quilt." Proc. Cambridge Phil.
Soc. 60, 363-368, 1964.

Dudeney, H. E. Problem 173 in Amusements in Mathematics.
New York: Dover, 1917.

Dudeney, H. E. Problem 177 in 536 Puzzles & Curious Prob-
lems, New York: Scribner, 1967.

Gardner, M, "Mrs. Perkins' Quilt and Other Square- Packing
Problems." Ch. 11 in Mathematical Carnival: A New
Round-Up of Tantalizers and Puzzles from Scientific
American. New York: Vintage, 1977.

Sloane, N. J. A. Sequence A005670/M3267 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Trustrum, G. B. "Mrs. Perkins's Quilt." Proc. Cambridge
Phil. Soc. 61, 7-11, 1965.

Mu Function

p{x,(3)

li{x,(3,a)

•r

Jo

r

Jo

% ipdt

r(/3 + i)r(t + 1)

where T(z) is the Gamma Function (Gradshteyn and

Ryzhik 1980, p. 1079).

see also Lambda Function, Nu Function

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.

fi Molecule

see Mandelbrot Set

Much Greater

A strong INEQUALITY in which a is not only GREATER
than 6, but much greater (by some convention), is de-
noted a^> b. For an astronomer, "much" may mean by
a factor of 100 (or even 10), while for a mathematician,
it might mean by a factor of 10 4 (or even much more).

see also Greater, Much Less

Much Less

A strong INEQUALITY in which a is not only LESS than
6, but much less (by some convention) is denoted a <^b.

see also Less, Much Greater

Muirhead's Theorem

A Necessary and Sufficient condition that [a']
should be comparable with [a] for all POSITIVE values
of the a is that one of (a') and (a) should be majorized
by the other. If (a') -< (a), then

[«'] < [«],

with equality only when (a') and (a) are identical or
when all the a are equal. See Hardy et al. (1988) for a
definition of notation.

References

Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,

2nd ed. Cambridge, England: Cambridge University Press,

pp. 44-48, 1988.

Miiller-Lyer Illusion

r,<* + t.p

tPdt

T(l3 + l)T(a + t + 1)'

An optical ILLUSION in which the orientation of arrow-
heads makes one Line Segment look longer than an-
other. In the above figure, the Line Segments on the
left and right are of equal length in both cases.

see also ILLUSION, POGGENDORFF ILLUSION, PONZO'S

Illusion, Vertical-Horizontal Illusion

References

Fineman, M. The Nature of Visual Illusion. New York:

Dover, p. 153, 1996.
Luckiesh, M. Visual Illusions: Their Causes, Characteristics

& Applications. New York: Dover, p. 93, 1965.

1204 Muller's Method

Multifractal Measure

Muller's Method

Generalizes the Secant METHOD of root finding by us-
ing quadratic 3-point interpolation

? =

Xn %n—l
•En — 1 '&n — 2

(i)

Then define

A = qP(x n ) - q(l + q)P(x n - 1 ) + q 2 P(x n - 2 ) (2)

B = (2q + l)P{x n ) - (1 + qfP{x n ^) + q 2 P{x n - 2 )

(3)

C=(l + s)P(*n), (4)

and the next iteration is

2C

max(£ ± VS 2 - 4AC )

• (5)

This method can also be used to find COMPLEX zeros of
Analytic Functions.

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 364, 1992.

Mulliken Symbols

Symbols used to identify irreducible representations of
Groups:

A — singly degenerate state which is symmetric with
respect to ROTATION about the principal C n axis,

B = singly DEGENERATE state which is antisymmetric
with respect to ROTATION about the principal C n
axis,

E = doubly Degenerate,

T = triply Degenerate,

X g = (gerade, symmetric) the sign of the wavefunction
does not change on Inversion through the center
of the atom,

X u = (ungerade, antisymmetric) the sigi. of the wave-
function changes on INVERSION through the cen-
ter of the atom,

Xi = (on a or b) the sign of the wavefunction does not
change upon ROTATION about the center of the
atom,

X 2 = (on a or b) the sign of the wavefunction changes
upon Rotation about the center of the atom,

' = symmetric with respect to a horizontal symmetry
plane ah,

" = antisymmetric with respect to a horizontal sym-
metry plane ah-

see also GROUP THEORY

Multiamicable Numbers

Two integers n and m < n are (a, 0)- multiamicable if

and

a(m) — m = an

a(n) — n = /3m,

where <r(n) is the Divisor Function and a, are Pos-
itive integers. If a = — 1, (m,n) is an AMICABLE
Pair.

m cannot have just one distinct prime factor, and if it
has precisely two prime factors, then a = 1 and m is
Even. Small multiamicable numbers for small a,0 are
given by Cohen et al. (1995). Several of these numbers
are reproduced in the below table.

a

m

n

1

6

76455288

183102192

1

7

52920

152280

1

7

16225560

40580280

1

7

90863136

227249568

1

7

16225560

40580280

1

7

70821324288

177124806144

1

7

199615613902848

499240550375424

see also Amicable Pair, Divisor Function

References

Cohen, G. L; Gretton, S.; and Hagis, P. Jr. "Multiamicable
Numbers." Math. Comput 64, 1743-1753, 1995.

Multifactorial

A generalization of the FACTORIAL and DOUBLE FAC-
TORIAL,

n
n!
nil

= n(n- l)(n-2)..-2-l
= n(n- 2)(n- 4) • • •
= n(n — 3)(n — 6) * • • ,

etc., where the product runs through positive integers.
The FACTORIALS n\ for n = 1, 2, . . . , are 1, 2, 6, 24, 120,
720, ... (Sloane's A000142); the DOUBLE FACTORIALS
n\\ are 1, 2, 3, 8, 15, 48, 105, ... (Sloane's A006882);
the triple factorials n\\\ are 1, 2, 3, 4, 10, 18, 28, 80,
162, 280, ... (Sloane's A007661); and the quadruple
factorials n!!!! are 1, 2, 3, 4, 5, 12, 21, 32, 45, 120, ...
(Sloane's A007662).

see also Factorial, Gamma Function

References

Sloane, N. J. A. Sequences A000142/M1675, A006882/
M0876, A007661/M0596, and A007662/M0534 in "An On-
Line Version of the Encyclopedia of Integer Sequences,"

Multifractal Measure

A Measure for which the ^-Dimension D q varies with
<?■

References

Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, 1993.

Multigrade Equation

Multigrade Equation

A (fc,/)-multigrade equation is a DlOPHANTlNE EQUA-
TION of the form

for j — 1, . . . , k, where m and n are /-VECTORS. Multi-
grade identities remain valid if a constant is added to
each element of m and n (Madachy 1979), so multi-
grades can always be put in a form where the minimum
component of one of the vectors is 1.

Small-order examples are the (2, 3)-multigrade with
m = {1, 6, 8} and n = {2, 4, 9}:

3 3

Y,m\=Y^ nl i= 15

i=l i-1

3 3

E™? = E n *= 101 '

i=l i=l

the (3, 4)-multigrade with m = {1,5,8,12} and n
{2,3,10,11}:

4 4

5Z m * = X^ n * = 26

i=l i=l

4 4

^ m^ = ^ n* = 234

i=l i=l

4 4

]Tm? = ^n? = 2366,

i=i i^i

and the (4, 6)-multigrade with m = {1,5,8,12,18,19}
and n = {2, 3, 9, 13, 16, 20}:

> m] = \ n\ — 63

i=l i=l

6 6

^]m, 2 = V\?=919

z=l i-1

6 6

J^m? = ^n* = 15057

6 6

J^m- = ^n- = 260755

i=i i=i

(Madachy 1979).

A spectacular example with k — 9 and I — 10 is given
byn = {±12, ±11881, ±20231, ±20885, ±23738} and

Multimagic Series 1205

m = {±436, ±11857, ±20449, ±20667, ±23750} (Guy
1994), which has sums

i=i t=i

9 9

^ m- = J^ n- = 3100255070

i~i i=i

9 9

i=i i=i

9 9

Y^rnt = J^n- = 1390452894778220678

t=l 2=1

9 9

J^ m • = ^ n- =

i=i x=i

9 9

J^m- = Y^n* = 666573454337853049941719510

i=i i=i

9 9

i=i i=i

9 9

i=i t=i

= 33095S

9 9

^m? = ^n- =0.

i=i t=i

= 330958142560259813821203262692838598

9 9

i=l z=l

see also DlOPHANTlNE EQUATION

References

Chen, S. "Equal Sums of Like Powers: On the Integer Solu-
tion of the Diophantine System." http://www.nease.net/
~chin/eslp/

Gloden, A. Mehrgeradige Gleichungen. Groningen, Nether-
lands: Noordhoff, 1944.

Gloden, A. "Sur la multigrade Ai, A 2y A 3 , A 4 , A 5 — k B lt
B 2 , B 3 , B 4i B 5 (k = 1, 3, 5, 7)." Revista Euclides 8,
383-384, 1948.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 143, 1994.

Kraitchik, M. "Multigrade." §3.10 in Mathematical Recre-
ations. New York: W. W. Norton, p. 79, 1942.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 171-173, 1979.

Multilinear

A function, form, etc., in two or more variables is said to
be multilinear if it is linear in each variable separately.

see also BILINEAR, LINEAR OPERATOR

Multimagic Series

n numbers form a p-multimagic series if the sum of their
fcth powers is the MAGIC CONSTANT of degree k for
every k = 1, . . . , p. The following table gives the number

1206 Multimagic Square

of p-multimagic series N p of given orders n (Kraitchik

1942).

n

m

N 2

N 3

2

2

3

8

4

86

2

2

5

1,394

8

2

6

98

7

1,844

8

38,039

115

9

41

10

11

961

References

Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical
Recreations. New York: W. W. Norton, pp. 176-178, 1942.

Multimagic Square

A MAGIC Square is ^-multimagic if the square formed
by replacing each element by its kth power for k — 1, 2,
. . . , p is also magic. A 2-multimagic square is called a
BlMAGiC Square, and a 3-multimagic square is called
a Trimagic Square.

see also Bimagic Square, Magic Square, Trimagic
Square

References

Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical
Recreations. New York: W. W. Norton, pp. 176-178, 1942.

Multinomial Coefficient

The multinomial COEFFICIENTS

(X1,X 2 ,. • •) :

X! + X2 + . . .
X\\X2^ * * *

are the terms in the MULTINOMIAL SERIES expansion.
They satisfy

(xi,X2 i X3 i ...) = (Xl +X2,X%,. . .)(X\,X2)

= (#1 + X2 + #3, • • .),(#!> #2, 33) = • • •

(Beeler et al 1972, Item 44).

see also Binomial Coefficient, Multinomial Se-
ries

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Multinomial
Coefficients." §24.1.2 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 823-824, 1972,

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.

Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 113, 1992.

Multinomial Theorem

Multinomial Distribution

Let a set of random variates X\ , X2 , . . . , X n have a
probability function

P(X 1 =X U ...,X n =X n )= n ' . T\0i mi (1)

where Xi are POSITIVE INTEGERS, 0» > 0, and

n

5> = 1 (2)

^Txi = N.

(3)

Then the joint distribution of Xi, . . . , X n is a multino-
mial distribution and P(Xi = a?i, . . . , X n = x n ) is given
by the corresponding coefficient of the MULTINOMIAL
Series

( tfl+ a + ... + n )". (4)

The Mean and VARIANCE of Xi are

IH = N0i
oS =N0i(l-0i).

The COVARIANCE of Xi and X, is

cnj 2 = -NOiOj.

see also BINOMIAL DISTRIBUTION

(5)
(6)

(7)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 532, 1987.

Multinomial Series

A generalization of the BINOMIAL SERIES discovered by
Johann Bernoulli and Leibniz.

(ai +a 2 + . . . + ak) n

n 1

= £

riiln^! * • * Tiki

Til no Til,

a± L a 2 * - -a k *,

n 1 ,n 2 ,...,n A .

where n = m + ri2 + . . . + n^. The multinomial series
arises in a generalization of the Binomial Distribu-
tion called the Multinomial Distribution.

see also BINOMIAL SERIES, MULTINOMIAL DISTRIBU-
TION

Multinomial Theorem

see Multinomial Series

Multiperfect Number

Multiple Regression 1207

Multiperfect Number

A number n is ^-multiperfect (also called a fc-MULTlPLY
Perfect Number or A;-Pluperfect Number) if

a(n) = kn

for some Integer k > 2, where a(n) is the DIVISOR
Function. The value of k is called the CLASS. The spe-
cial case k = 2 corresponds to PERFECT NUMBERS P 2 ,
which are intimately connected with MERSENNE PRIMES
(Sloane's A000396), The number 120 was long known
to be 3-multiply perfect (P3) since

<t(120) = 3-120.

The following table gives the first few P n for n — 2, 3,

■ ■ ■,6.

n Sloane P n

"2 000396 6, 28, 496, 8128, . . . ,

3 005820 120, 672, 523776, 459818240, . . .

4 027687 30240, 32760, 2178540, 23569920, ...

5 046060 14182439040, 31998395520, . . .

6 046061 154345556085770649600, . . .

In 1900-1901, Lehmer proved that P3 has at least three
distinct PRIME factors, P4 has at least four, P 5 at least
six, Pq at least nine, and P-j at least 14.

As of of 1911, 251 pluperfect numbers were known (Car-
michael and Mason 1911). As of 1929, 334 pluperfect
numbers were known, many of them found by Poulet.
Franqui and Garcia (1953) found 63 additional ones (five
P5S, 29 Pes, and 29 P7S), several of which were known to
Poulet but had not been published, bringing the total to
397. Brown (1954) discovered 110 pluperfects, includ-
ing 31 discovered but not published by Poulet and 25
previously published by Franqui and Garcia (1953), for
a total of 482. Franqui and Garcia (1954) subsequently
discovered 57 additional pluperfects (3 Pes, 52 P7S, and
2 Pss), increasing the total known to 539.

An outdated database is maintained by R. Schroeppel,
who lists 2,094 multiperfects, and an up-to-date list by
J. L, Moxham (1998). It is believed that all multiperfect
numbers of index 3, 4, 5, 6, and 7 are known. The
number of known n- multiperfect numbers are 1, 37, 6,
36, 65, 245, 516, 1101, 1129, 46, 0, 0, ... .

If n is a P5 number such that 3{n, then Zn is a P4 num-
ber. If 3n is a P^k number such that 3fn, then n is a
Psk number. If n is a P3 number such that 3 (but not 5
and 9) Divides n, then 45n is a P4 number.

see also e-MULTIPERFECT NUMBER, FRIENDLY PAIR,

Hyperperfect Number, Infinary Multiperfect
Number, Mersenne Prime, Perfect Number, Uni-
tary Multiperfect Number

References

Brown, A. L. "Multiperfect Numbers. 1
103-106, 1954.

Scripta Math. 20,

Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 33-38,
1952.

Flammenkamp, A. "Multiply Perfect Numbers." http://
www.uni-bielefeld.de/-achim/mpn.html.

Franqui, B. and Garcia, M. "Some New Multiply Perfect
Numbers." Amer. Math. Monthly 60, 459-462, 1953.

Franqui, B. and Garcia, M. "57 New Multiply Perfect Num-
bers." Scripta Math. 20, 169-171, 1954.

Guy, R. K. "Almost Perfect, Quasi-Perfect, Pseudoperfect,
Harmonic, Weird, Multiperfect and Hyperperfect Num-
bers." §B2 in Unsolved Problems in Number Theory, 2nd
ed. New York: Springer- Ver lag, pp. 45-53, 1994.

Helenius, F. W. "Multiperfect Numbers (MPFNs)." http://
www.netcom.com/-fredh/mpfn.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 149-151, 1979.

Moxham, J. L. "13 New MPFN's." math-fun@cs.arizona.
edu posting, Aug 13, 1998.

Poulet, P. La Chasse aux nombres, Vol. 1. Brussels, pp. 9-27,
1929.

Schroeppel, R. "Multiperfect Numbers-Multiply Perfect
Numbers-Pluperfect Numbers-MPFNs." Rev. Dec.

13, 1995. ftp://ftp.cs.arizona.edu/xkernel/rcs/

mpfn.html.

Schroeppel, R. (moderator), mpfn mailing list. e-mail
rcsQcs.arizona.edu to subscribe.

Sloane, N. J. A. Sequences A000396/M4186 and A005820/
M5376 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Multiple Integral

A repeated integral over n > 1 variables

j...jf( Xl ,..., Xn ) dxl ... dXn

is called a multiple integral. An nth order integral cor-
responds, in general, to an n-D VOLUME (Content),
with n = 2 corresponding to an AREA. In an indefinite
multiple integral, the order in which the integrals are
carried out can be varied at will; for definite multiple
integrals, care must be taken to correctly transform the
limits if the order is changed.

see also Integral, Monte Carlo Integration

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Multidimensional Integrals." §4.6 in Numeri-
cal Recipes in FORTRAN: The Art of Scientific Comput-
ing, 2nd ed. Cambridge, England: Cambridge University
Press, pp. 155-158, 1992.

Multiple Regression

A Regression giving conditional expectation values of
a given variable in terms of two or more other variables.

see also LEAST SQUARES FITTING, MULTIVARIATE

Analysis, Nonlinear Least Squares Fitting

References

Edwards, A. L. Multiple Regression and the Analysis of Vari-
ance and Covariance. San Francisco, CA: W. H. Freeman,
1979.

1208 Multiplication

Multiplicative Inverse

Multiplication

In simple algebra, multiplication is the process of cal-
culating the result when a number a is taken b times.
The result of a multiplication is called the PRODUCT of
a and b. It is denoted a x 6, a * 6, (a)(b), or simply ab.
The symbol x is known as the MULTIPLICATION SIGN.
Normal multiplication is Associative, Commutative,
and Distributive.

More generally, multiplication can also be defined for
other mathematical objects such as Groups, Matri-
ces, Sets, and Tensors.

Karatsuba and Ofman (1962) discovered that multipli-
cation of two n digit numbers can be done with a Bit
Complexity of less than n 2 using an algorithm now
known as Karatsuba Multiplication.

see also Addition, Bit Complexity, Complex Mul-
tiplication, Division, Karatsuba Multiplication,
Matrix Multiplication, Product, Russian Multi-
plication, Subtraction, Times

References

Karatsuba, A. and Ofman, Yu. "Multiplication of Many-
Digital Numbers by Automatic Computers." Doklady
Akad. Nauk SSSR 145, 293-294, 1962. Translation in
Physics-Doklady 7, 595-596, 1963.

Multiplication Magic Square

Multiplication Table

A multiplication table is an array showing the result of
applying a BINARY OPERATOR to elements of a given
set 5.

128

1

32

4

16

64

8

256

2

A square which is magic under multiplication instead
of addition (the operation used to define a conventional
MAGIC SQUARE) is called a multiplication magic square.
Unlike (normal) MAGIC SQUARES, the n 2 entries for an
nth order multiplicative magic square are not required to
be consecutive. The above multiplication magic square
has a multiplicative magic constant of 4,096.

see also Addition-Multiplication Magic Square,
Magic Square

References

Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3

in Mathematical Diversions. New York: Dover, pp. 30-31,

1975.
Madachy, J. S. Madachy '$ Mathematical Recreations. New

York: Dover, pp. 89-91, 1979.

Multiplication Principle

If one event can occur in m ways and a second can occur
independently of the first in n ways, then the two events
can occur in mn ways.

X

1

2

3

4

5

6

7

8

9

10

1

1

2

3

4

5

6

7

8

9

10

2

2

4

6

8

10

12

14

16

18

20

3

3

6

9

12

15

18

21

24

27

30

4

4

8

12

16

20

24

28

32

36

40

5

5

10

15

20

25

30

35

40

45

50

6

6

12

18

24

30

36

42

48

54

60

7

7

14

21

28

35

42

49

56

63

70

8

8

16

24

32

40

48

56

64

72

80

9

9

18

27

36

45

54

63

72

81

90

10

10

20

30

40

50

60

70

80

90

100

Multiplication Sign

The symbol x used to denote MULTIPLICATION,
a x b denotes a times b.

i.e.,

see also Binary Operator, Truth Table

Multiplicative Character

see Character (Multiplicative)

Multiplicative Digital Root

Consider the process of taking a number, multiplying
its Digits, then multiplying the DIGITS of numbers de-
rived from it, etc., until the remaining number has only
one DIGIT. The number of multiplications required to
obtain a single DIGIT from a number n is called the
Multiplicative Persistence of n, and the Digit ob-
tained is called the multiplicative digital root of n.

For example, the sequence obtained from the starting
number 9876 is (9876, 3024, 0), so 9876 has a Mul-
tiplicative Persistence of two and a multiplicative
digital root of 0. The multiplicative digital roots of the
first few positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 0,
1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6,
9, 2, 5, 8, 2, . . . (Sloane's A031347).
see also Additive Persistence, Digitadition, Digi-
tal Root, Multiplicative Persistence

References

Sloane, N. J. A. Sequence A031347 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Multiplicative Function

A function f(m) is called multiplicative if (m, m!) =
1 (i.e., the statement that m and m' are RELATIVELY
Prime) implies

f(mm) = f{m)f{rri).
see also QUADRATIC RESIDUE, TOTIENT FUNCTION

Multiplicative Inverse

The multiplicative of a REAL or COMPLEX NUMBER z
is its Reciprocal \/z. For complex z = x + iy,

11 x . y

z

Multiplicative Perfect Number

Multivalued Function

1209

Multiplicative Perfect Number

A number n for which the PRODUCT of DIVISORS is
equal to n 2 . The first few are 1, 6, 8, 10, 14, 15, 21, 22,
... (Sloane's A007422).

see also Perfect Number

References

Sloane, N. J. A. Sequence A007422/M4068 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Multiplicative Persistence

Multiply all the digits of a number n by each other,
repeating with the product until a single Digit is ob-
tained. The number of steps required is known as the
multiplicative persistence, and the final DIGIT obtained
is called the Multiplicative Digital Root of n.

For example, the sequence obtained from the starting
number 9876 is (9876, 3024, 0), so 9876 has an mul-
tiplicative persistence of two and a MULTIPLICATIVE
Digital Root of 0. The multiplicative persistences
of the first few positive integers are 0, 0, 0, 0, 0, 0,

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2,
2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 1, 1, . . . (Sloane's
A031346). The smallest numbers having multiplicative
persistences of 1, 2, ... are 10, 25, 39, 77 679 6788
68889 267889 26888999 3778888999 277777788888899
(Sloane's A003001). There is no number < 10 50 with
multiplicative persistence > 11.

The multiplicative persistence of an n-DlGIT number is
also called its LENGTH. The maximum lengths for n =
2-, 3-, . . . , digit numbers are 4, 5, 6, 7, 7, 8, 9, 9, 10, 10,
10, . . . (Sloane's A014553; (Beeler et al. 1972, Item 56;
Gottlieb 1969-1970).

The concept of multiplicative persistence can be gener-
alized to multiplying the kth. powers of the digits of a
number and iterating until the result remains constant.
All numbers other than Re PUN ITS, which converge to

1, converge to 0. The number of iterations required for
the kth powers of a number's digits to converge to
is called its ^-multiplicative persistence. The following
table gives the n-multiplicative persistences for the first
few positive integers.

n

Sloane

n-Persistences

2

031348

0, 7, 6, 6, 3, 5, 5, 4, 5, 1, ...

3

031349

0,4,5,4,3,4,4,3,3, 1, . . .

4

031350

0,4,3,3,3,3,2,2,3, 1, . . .

5

031351

0, 4, 4, 2, 3, 3, 2, 3, 2, 1, ...

6

031352

0, 3, 3, 2, 3, 3, 3, 3, 3, 1, ...

7

031353

0,4,3,3,3,3,3,2,3, 1, . . .

8

031354

0, 3, 3, 3, 2,4, 2, 3, 2, 1, ...

9

031355

0, 3, 3, 3, 3, 2, 2, 3, 2, 1, ...

10

031356

0, 2, 2, 2, 3, 2, 3, 2, 2, 1, ...

see also 196- Algorithm, Additive Persistence,
Digitadition, Digital Root, Kaprekar Number,
Length (Number), Multiplicative Digital Root,

Narcissistic Number, Recurring Digital Invari-
ant

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.

Cambridge, MA: MIT Artificial Intelligence Laboratory,

Memo AIM-239, Feb. 1972.
Gottlieb, A. J. Problems 28-29 in "Bridge, Group Theory,

and a Jigsaw Puzzle." Techn. Rev. 72, unpaginated, Dec.

1969.
Gottlieb, A. J. Problem 29 in "Integral Solutions, Ladders,

and Pentagons." Techn. Rev. 72, unpaginated, Apr. 1970.
Sloane, N. J. A. "The Persistence of a Number." J. Recr.

Math. 6, 97-98, 1973.
Sloane, N. J. A. Sequences A014553 and A003001/M4687 in

"An On-Line Version of the Encyclopedia of Integer Se-
quences."

Multiplicative Primitive Residue Class
Group

see Modulo Multiplication Group

Multiplicity

The word multiplicity is a general term meaning "the
number of values for which a given condition holds."
The most common use of the word is as the value of the

Totient Valence Function.

see also Degenerate, Noether's Fundamental
Theorem, Totient Valence Function

Multiplier

see Modular Function Multiplier

Multiply Connected

A set which is Connected but not Simply Connected
is called multiply connected. A Space is n-MULTlPLY
Connected if it is (n — l)-connected and if every MAP
from the n-SPHERE into it extends continuously over the
(n + l)-DlSK

A theorem of Whitehead says that a SPACE is infinitely

connected Iff it is contractible.

see also CONNECTIVITY, LOCALLY PATHWISE-CON-

nected Space, Pathwise-Connected, Simply Con-
nected

Multiply Perfect Number

see Multiperfect Number

Multisection

see Series Multisection

Multivalued Function

A FUNCTION which assumes two or more distinct values
at one or more points in its Domain.
see also BRANCH CUT, BRANCH POINT

References

Morse, P. M. and Feshbach, H. "Multivalued Functions."

§4.4 in Methods of Theoretical Physics, Part I. New York:

McGraw-Hill, pp. 398-408, 1953.

1210 Multivariate Analysis

Myriad

Multivariate Analysis

The study of random distributions involving more than
one variable.

see also GAUSSIAN JOINT VARIABLE THEOREM, MUL-
TIPLE Regression, Multivariate Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 927-928, 1972.

Feinstein, A. R, Multivariable Analysis. New Haven, CT:
Yale University Press, 1996.

Hair, J. F. Jr. Multivariate Data Analysis with Readings,
4th ed. Englewood Cliffs, NJ: Prentice-Hall, 1995.

Sharma, S. Applied Multivariate Techniques. New York: Wi-
ley, 1996.

Multivariate Function

A Function of more than one variable.

see also Multivariate Analysis, Univariate Func-
tion

Multivariate Theorem

see Gaussian Joint Variable Theorem

Miintz Space

A Miintz space is a technically defined SPACE

M(A) = span{a; Ao ,a; Al ,...}
which arises in the study of function approximations.

Miintz's Theorem

Miintz's theorem is a generalization of the WeierSTRAB
Approximation Theorem, which states that any con-
tinuous function on a closed and bounded interval can
be uniformly approximated by POLYNOMIALS involv-
ing constants and any INFINITE SEQUENCE of POWERS
whose Reciprocals diverge.

In technical language, Miintz's theorem states that the
MUNTZ Space M(A) is dense in C[0, 1] Iff

oo

y-

£-< A;

Xi

see also WeierstraB Approximation Theorem

Mutant Knot

Given an original Knot K, the three knots produced
by MUTATION together with K itself are called mutant
knots. Mutant knots are often difficult to distinguish.
For instant, mutants have the same HOMFLY POLY-
NOMIALS and Hyperbolic Knot volume. Many but
not all mutants also have the same Genus (Knot).

Mutation

Consider a Knot as being formed from two TANGLES.
The following three operations are called mutations.

1. Cut the knot open along four points on each of the
four strings coming out of T2, flipping T2 over, and
gluing the strings back together.

2. Cut the knot open along four points on each of the
four strings coming out of T2, flipping T2 to the right,
and gluing the strings back together.

3. Cut the knot, rotate it by 180°, and reglue. This is
equivalent to performing (1), then (2).

Mutations applied to an alternating KNOT projection
always yield an ALTERNATING Knot. The mutation of
a Knot is always another KNOT (a opposed to a LINK).

References

Adams, C. C. The Knot Book: An Elementary Introduction

to the Mathematical Theory of Knots. New York: W. H.

Freeman, p. 49, 1994.

Mutual Energy

Let Q be a Space with Measure \i > 0, and let $(P, Q)
be a real function on the PRODUCT Space ft x ft. When

Oi, nu) = J J *(P, Q) dn{Q) dv{P)
= J*(P^)dv(P)

exists for measures ^/, v > 0, (fi,v) is called the mutual
energy, (/z, /x) is then called the ENERGY.

see also Energy

References

Iyanaga, S. and Kawada, Y. (Eds.). "General Potential."
§335. B in Encyclopedic Dictionary of Mathematics. Cam-
bridge, MA: MIT Press, p. 1038, 1980.

Mutually Exclusive

Two events E x and E 2 are mutually exclusive if E\ O
Ei = 0. n events E\, E 2l . . . , E n are mutually exclusive
tfEiDEj = for i^j.

Mutually Singular

Let M be a Sigma Algebra M, and let Ai and A 2 be
Measures on M. If there Exists a pair of disjoint Sets
A and B such that Ai is CONCENTRATED on A and A2
is CONCENTRATED on B, then Ai and A2 are said to be

mutually singular, written Ai _L A2.

see also Absolutely Continuous, Concentrated,
Sigma Algebra

References

Rudin, W. Functional Analysis. New York: McGraw-Hill,
p. 121, 1991.

Myriad

The Greek word for 10,000.

Myriagon Mystic Pentagram 1211

Myriagon

A 10,000-sided POLYGON.

Mystic Pentagram

see Pentagram

Nagel Point 1213

N

N

The Set of Natural Numbers (the Positive Inte-
gers Z + 1, 2, 3, ...; Sloane's A000027), denoted N,
also called the Whole Numbers. Like whole numbers,
there is no general agreement on whether should be
included in the list of natural numbers.

Due to lack of standard terminology, the following terms
are recommended in preference to "Counting Num-
ber," "natural number," and "WHOLE NUMBER."

Set

Name

Symbol

...,-2,-1,0,1,2,...
1,2,3,4,...
0,1,2,3,4...
-1, -2, -3, -4, ...

integers Z

positive integers Z

nonnegative integers Z*
negative integers Z~

see also C, CARDINAL NUMBER, COUNTING NUMBER,

I, Integer, Q, K, Whole Number, Z, Z +

References

Sloane, N. J. A. Sequence A000027/M0472 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

N- Cluster

A Lattice Point configuration with no three points
COLLINEAR and no four CONCYCLIC. An example is
the 6-cluster (0, 0), (132, -720), (546, -272), (960,
-720), (1155, 540), (546, 1120). Call the Radius of
the smallest CIRCLE centered at one of the points of an
N-cluster which contains all the points in the N-cluster
the Extent. Noll and Bell (1989) found 91 nonequiv-
alent prime 6-clusters of Extent less than 20937, but
found no 7-clusters.

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Verlag, p. 187, 1994.
Noll, L. C. and Bell, D. I. "n-clusters for 1 < n < 7." Math.

Comput. 53, 439-444, 1989.

n-Cube

see HYPERCUBE, POLYCUBE

n-plex

n-plex is defined as 10 n .

see also Googolplex, n-MiNEX

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 16, 1996.

n- Sphere

see Hypersphere

Nabla

see Del, Laplacian

Nagel Point

Let A! be the point at which the A-ExciRCLE meets the
side BC of a Triangle AABC, and define B' and C
similarly. Then the lines AA\ BB', and CC CONCUR
in the Nagel Point.

The Nagel point can also be constructed by letting A"
be the point halfway around the PERIMETER of AABC
starting at A, and B" and C" similarly defined. Then
the lines AA" , BB" ', and CC" concur in the Nagel point.
It is therefore sometimes known as the BISECTED PER-
IMETER Point (Bennett et al. 1988, Chen et al. 1992,
Kimberling 1994).

The Nagel point has TRIANGLE CENTER FUNCTION
b-\- c — a

n-in-a-Row

see TlC-TAC-TOE

n-minex

n-minex is defined as 10 _n .

see also n-PLEX

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 16, 1996.

n-Omino

see POLYOMINO

It is the Isotomic Conjugate Point of the Ger-
gonne Point.

see also Excenter, Excentral Triangle, Excircle,
Mittenpunkt, Trisected Perimeter Point

References

Altshiller-Court, N. College Geometry: A Second Course in

Plane Geometry for Colleges and Normal Schools, 2nd ed.

New York: Barnes and Noble, pp. 160-164, 1952.
Bennett, G.; Glenn, J.; Kimberling, C.; and Cohen, J. M.

"Problem E 3155 and Solution." Amer. Math. Monthly

95, 874, 1988.
Chen, J.; Lo, C.-H.; and Lossers, O. P. "Problem E 3397 and

Solution." Amer. Math. Monthly 99, 70-71, 1992.

1214 Naive Set Theory

Napoleon Points

Eves, H. W. A Survey of Geometry, rev, ed. Boston, MA:

Allyn and Bacon, p. 83, 1972.
Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.

London: Hodgson, p. 20, 1913.
Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 184 and 225-226, 1929.
Kimberling, C. "Central Points and Central Lines in the

Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Nagel Point." http: / /www. e vans ville.

edu/*ck6/tcenters/class/nagel.html.

Naive Set Theory

A branch of mathematics which attempts to formalize
the nature of the SET using a minimal collection of in-
dependent axioms. Unfortunately, as discovered by its
earliest proponents, naive set theory quickly runs into a
number of Paradoxes (such as Russell's Paradox),
so a less sweeping and more formal theory known as
Axiomatic Set Theory must be used.

see also Axiomatic Set Theory, Russell's Para-
dox, Set Theory

Napier's Analogies

Let a Spherical Triangle have sides a, 6, and c with
A, B, and C the corresponding opposite angles. Then

Napier's Inequality

For b > a > 0,

sm[i(A-i?)] = tan[§(a-6)]
tan[|(a + &)]

sin[i(A + B)]
cos[±(A-B)]
cos[§(A + B)]

sin[|(q-fe)]
sin[i(a + 6)]
cos[f(a-ft)]

tan(fc)
tan[§(A-B)]

cot(fC)
tan[f(A + ff)]

cos[f(a + 6)] cot(fC)

see also SPHERICAL TRIGONOMETRY

(1)

(2)
(3)
(4)

Napier's Bones

Numbered rods which can be used to perform Multi-
plication. This process is also called RABDOLOGY.

see also GENAILLE RODS

References

Gardner, M. "Napier's Bones." Ch. 7 in Knotted Dough-
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, 1986.

Pappas, T. "Napier's Bones." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, pp. 64-65, 1989.

Napier's Constant

1 In b — In a 1
b o — a a

References

Nelsen, R. B. "Napier's Inequality (Two Proofs)." College
Math. J. 24, 165, 1993.

Napierian Logarithm

N = 10 7 (1 -HT 7 ) L ,

then L is the Napierian logarithm of N. This was the
original definition of a LOGARITHM, and can be given in
terms of the modern LOGARITHM as

L(N) =

The Napierian logarithm decreases with increasing num-
bers and does not satisfy many of the fundamental prop-
erties of the modern LOGARITHM, e.g.,

Nlog(xy) ^ Nloga: + Nlogy.

Napkin Ring

see Spherical Ring

Napoleon Points

The inner Napoleon point N is the CONCURRENCE of
lines drawn between VERTICES of a given TRIANGLE

Napoleon's Problem

Nappe 1215

AABC and the opposite Vertices of the correspond-
ing inner Napoleon TRIANGLE AN A bNacN bc . The
Triangle Center Function of the inner Napoleon
point is

a = csc(A — |7r).

Napoleon Triangles

The outer Napoleon point N f is the CONCURRENCE of
lines drawn between Vertices of a given Triangle
AABC and the opposite VERTICES of the correspond-
ing outer Napoleon Triangle AN ab N ac N bc . The
Triangle Center Function of the point is

a = csc(A + |7r).

see also NAPOLEON'S THEOREM, NAPOLEON TRIAN-
GLES

References

Casey, J. Analytic Geometry, 2nd ed. Dublin: Hodges, Fig-
gis, & Co., pp. 442-444, 1893.

Kimberling, C "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.

Napoleon's Problem

Given the center of a CIRCLE, divide the CIRCLE into
four equal arcs using a COMPASS alone (a Mascheroni
Construction).

see also CIRCLE, COMPASS, MASCHERONI CONSTRUC-
TION

Napoleon's Theorem

If Equilateral Triangles are erected externally on
the sides of any Triangle, then the centers form an
Equilateral Triangle (the outer Napoleon Tri-
angle). Furthermore, the inner Napoleon Triangle
is also Equilateral and the difference between the ar-
eas of the outer and inner Napoleon triangles equals the
AREA of the original TRIANGLE.
see also NAPOLEON POINTS, NAPOLEON TRIANGLES

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.

Washington, DC: Math. Assoc. Amer., pp. 60—65, 1967.
Pappas, T. "Napoleon's Theorem." The Joy of Mathematics.

San Carlos, CA: Wide World Publ./Tetra, p. 57, 1989.
Schmidt, F. "200 Jahre franzosische Revolution — Problem

und Satz von Napoleon." Didaktik der Mathematik 19,

15-29, 1990.
Wentzel, J. E. "Converses of Napoleon's Theorem." Amer.

Math. Monthly 99, 339-351, 1992.

7^*5

The inner Napoleon triangle is the Triangle
ANabNacNbc formed by the centers of inter-
nally erected EQUILATERAL TRIANGLES AABEab,
AACEac, and ABCEbc on the sides of a given TRI-
ANGLE AABC. It is an EQUILATERAL TRIANGLE.

The outer Napoleon triangle is the TRIANGLE
AN AB N' AC N B c formed by the centers of exter-
nally erected EQUILATERAL TRIANGLES AABE AB1
AACE' AC , and ABCE BC on the sides of a given Tri-
angle AABC. It is also an Equilateral Triangle.

see also EQUILATERAL TRIANGLE, NAPOLEON POINTS,

Napoleon's Theorem

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 60-65, 1967.

Nappe

nappes

One of the two pieces of a double Cone (i.e., two CONES
placed apex to apex).

see also CONE

1216

Narcissistic Number

Nasik Square

Narcissistic Number

An n-DlGIT number which is the SUM of the nth POW-
ERS of its Digits is called an n-narcissistic number, or
sometimes an Armstrong Number or Perfect Digi-
tal Invariant (Madachy 1979). The smallest example
other than the trivial 1-Digit numbers is

153

l 3 + 5 3 +3 3 .

The series of smallest narcissistic numbers of n digits
are 0, (none), 153, 1634, 54748, 548834, ... (Sloane's
A014576). Hardy (1993) wrote, "There are just four
numbers, after unity, which are the sums of the cubes of
their digits: 153 = l 3 + 5 3 + 3 3 , 370 = 3 3 H-7 3 +0 3 , 371 =
3 3 + 7 3 + l 3 , and 407 = 4 3 +0 3 + 7 3 . These are odd facts,
very suitable for puzzle columns and likely to amuse
amateurs, but there is nothing in them which appeals
to the mathematician." The following table gives the
generalization of these "unappealing" numbers to other
Powers (Madachy 1979, p. 164).

n n-Narcissistic Numbers

1 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

2 none

3 153, 370, 371, 407

4 1634, 8208, 9474

5 54748, 92727, 93084

6 548834

7 1741725, 4210818, 9800817, 9926315

8 24678050, 24678051, 88593477

9 146511208, 472335975, 534494836, 912985153
10 4679307774

A total of 88 narcissistic numbers exist in base- 10, as
proved by D. Winter in 1985 and verified by D. Hoey.
These numbers exist for only 1, 3, 4, 5, 6, 7, 8, 9, 10,
11, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 29, 31, 32, 33,
34, 35, 37, 38, and 39 digits. It can easily be shown that
base- 10 n-narcissistic numbers can exist only for n < 60,
since

n*9 n < lO 71 " 1

for n > 60. The largest base-10 narcissistic number is
the 39-narcissistic

115132219018736992565095597973971522401

A table of the largest known narcissistic numbers in var-
ious Bases is given by Pickover (1995). A tabulation of
narcissistic numbers in various bases is given by (Corn-
ing)-

A closely related set of numbers generalize the narcissis-
tic number to n-DlGIT numbers which are the sums of
any single POWER of their DIGITS. For example, 4150
is a 4-DlGIT number which is the sum of fifth POWERS
of its DIGITS. Since the number of digits is not equal to
the power to which they are taken for such numbers, it is

not a narcissistic number. The smallest numbers which
are sums of any single positive power of their digits are
1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150,
4151, 8208, 9474, ... (Sloane's A023052), with powers
1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 4, 5, 5, 4, -4, ...
(Sloane's A046074).

The smallest numbers which are equal to the nth powers
of their digits for n = 3, 4, . . . , are 153, 1634, 4150,
548834, 1741725, ... (Sloane's A003321). Then-digit
numbers equal to the sum of nth powers of their digits
(a finite sequence) are called Armstrong Numbers or
Plus Perfect Numbers and are given by 1, 2, 3, 4, 5,
6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748,
... (Sloane's A005188).

If the sum-of-fcth-powers-of-digits operation applied it-
eratively to a number n eventually returns to n,
the smallest number in the sequence is called a k-
Recurring Digital Invariant.

see also Additive Persistence, Digital Root, Digi-

TADITION, KAPREKAR NUMBER, MULTIPLICATIVE DIG-
ITAL Root, Multiplicative Persistence, Recur-
ring Digital Invariant, Vampire Number

References

Corning, T. "Exponential Digital Invariants." http://
members.aol.com/tecl53/Edi4web/Edi.html.

Hardy, G. H. A Mathematician's Apology. New York: Cam-
bridge University Press, p. 105, 1993.

Madachy, J. S. "Narcissistic Numbers." Madachy's Mathe-
matical Recreations. New York: Dover, pp. 163—173, 1979.

Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
pp. 169-170, 1995.

Rumney, M. "Digital Invariants." Recr. Math. Mag. No. 12,
6-8, Dec. 1962.

Sloane, N. J. A. Sequences A014576, A023052, A005188/
M0488, and A003321/M5403 in "An On-Line Version of
the Encyclopedia of Integer Sequences."
# Weisstein, E. W. "Narcissistic Numbers." http: //www.
astro . Virginia . edu/~eww6n/math/notebooks/
Narcissistic .dat.

Nash Equilibrium

A set of Mixed Strategies for finite, noncooperative
GAMES of two or more players in which no player can
improve his payoff by unilaterally changing strategy.

see also Fixed Point, Game, Mixed Strategy,
Nash's Theorem

Nash's Theorem

A theorem in Game THEORY which guarantees the ex-
istence of a Nash Equilibrium for Mixed Strategies
in finite, noncooperative GAMES of two or more players.

see also Mixed Strategy, Nash Equilibrium

Nasik Square

see PANMAGIC SQUARE

Nasty Knot

Natural Logarithm 1217

Nasty Knot

An UNKNOT which can only be unknotted by first in-
creasing the number of crossings.

Natural Density

see Natural Invariant

Natural Equation

A natural equation is an equation which specifies a curve
independent of any choice of coordinates or parameter-
ization. The study of natural equations began with the
following problem: given two functions of one parame-
ter, find the Space Curve for which the functions are
the Curvature and Torsion.

Euler gave an integral solution for plane curves (which
always have Torsion r = 0). Call the Angle between
the TANGENT line to the curve and the z-AxiS <j> the
Tangential Angle, then

/■

<j> = / n(s)ds, (1)

where k is the CURVATURE. Then the equations

k = k(s)

(2)
(3)

where r is the TORSION, are solved by the curve with
parametric equations

y

-I

cos (f> ds

yds.

(4)
(5)

The equations k = k(s) and r = r(s) are called the nat-
ural (or Intrinsic) equations of the space curve. An
equation expressing a plane curve in terms of s and RA-
DIUS of Curvature R (or k) is called a Cesaro Equa-
tion, and an equation expressing a plane curve in terms
of s and <j> is called a Whewell Equation.

Among the special planar cases which can be solved in
terms of elementary functions are the CIRCLE, LOGA-
RITHMIC Spiral, Circle Involute, and Epicycloid.
Enneper showed that each of these is the projection of a
Helix on a Conic surface of revolution along the axis
of symmetry. The above cases correspond to the CYL-
INDER, Cone, Paraboloid, and Sphere.

see also CESARO EQUATION, INTRINSIC EQUATION,

Whewell Equation

References

Cesaro, E. Lezioni di Geometria Intrinseca. Napoli, Italy,
1896.

Euler, L. Comment. Acad. Petropolit. 8, 66-85, 1736.

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 111-112, 1993.

Melzak, Z. A. Companion to Concrete Mathematics, Vol. 2.
New York: Wiley, 1976.

Struik, D. J. Lectures on Classical Differential Geometry.
New York: Dover, pp. 26-28, 1988.

Natural Independence Phenomenon
A type of mathematical result which is considered by
most logicians as more natural than the Metamath-
EMATICAL incompleteness results first discovered by
Godel. Finite combinatorial examples include GOOD-
stein's Theorem, a finite form of Ramsey's Theo-
rem, and a finite form of KRUSKAL'S Tree THEOREM
(Kirby and Paris 1982; Smorynski 1980, 1982, 1983; Gal-
lier 1991).

see also GODEL'S INCOMPLETENESS THEOREM, GOOD-

stein's Theorem, Kruskal's Tree Theorem, Ram-
sey's Theorem

References

Gallier, J. "What's so Special about Kruskal's Theorem and

the Ordinal Gamma[0]? A Survey of Some Results in Proof

Theory." Ann. Pure and Appl. Logic 53, 199-260, 1991.
Kirby, L. and Paris, J. "Accessible Independence Results for

Peano Arithmetic." Bull. London Math. Soc. 14, 285-293,

1982.
Smorynski, C. "Some Rapidly Growing Functions." Math.

Jntell. 2, 149-154, 1980.
Smorynski, C. "The Varieties of Arboreal Experience."

Math. Intell 4, 182-188, 1982.
Smorynski, C. "'Big' News from Archimedes to Friedman."

Not. Amer. Math. Soc. 30, 251-256, 1983.

Natural Invariant

Let p[x) dx be the fraction of time a typical dynamical
Orbit spends in the interval [x^x + dx], and let p(x) be
normalized such that

/

p(x) dx — 1

over the entire interval of the map. Then the fraction
the time an ORBIT spends in a finite interval [a, 6], is
given by

/

J a

p(x) dx.

The natural invariant is also called the INVARIANT DEN-
SITY or Natural Density.

Natural Logarithm

The Logarithm having base e, where

e = 2.718281828...
which can be defined

lnsE5 /

dt

t

(1)

(2)

for x > 0. The natural logarithm can also be defined for
Complex Numbers as

In z = In \z\ + i arg(z),

(3)

1218 Natural Logarithm

Natural Norm

where \z\ is the MODULUS and arg(z) is the ARGUMENT.
The natural logarithm is especially useful in CALCULUS
because its Derivative is given by the simple equation

d 1

— lnx = -,
ax x

(4)

whereas logarithms in other bases have the more com-
plicated Derivative

dx b x In b

(5)

An identity for the natural logarithm of 2 discovered
using the PSLQ Algorithm is

40

6^ 16 fc

k=0

16

(8fc) 2 (8A; + I) 2

28 4

+

(8& + 2) 2 (8fc + 3) 2 (8fc + 4) 2 (8fc + 5) 2
28 4 10 2

+

(8A: + 4) 2 (8fc + 5) 2 (8A; + 5) 2 (8k + 7) 2

(15)

The Mercator Series

ln(l + x) -x~ \x 2 + \x z - ...

gives a Taylor Series for the natural logarithm.

(6)

Continued Fraction representations of logarithmic
functions include

ln(H-x)

1 +

l 2 x

(7)

2 +

~T2 —
1 x

3 +

2 x

4 +

2 2 x

5 +

3 x

6 +

3 x

7 + ...

In

(£)

2x

(S)

3-

4aT

9af

16x

7

9-...

For a Complex Number z, the natural logarithm sat-
isfies

\nz = ln[re* ( * +2n7r) ] = lnr + i(9 + 2ri7r) (9)

Py(ln^) = In r + i6 y (10)

where PV is the Principal Value.
Some special values of the natural logarithm are

lnl =

InO =

— OO

ln(-l)

= Tri

i(±<) =

±\ni.

(11)

(12)
(13)
(14)

(Bailey et al. 1995, Bailey and Plouffe).

see also e, Jensen's Formula, Lg, Logarithm

References

Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Com-
putation of Various Polylogarithmic Constants." http://
www.cecm.sfu.ca/-pborwein/PAPERS/P123.ps.

Bailey, D. and Plouffe, S. "Recognizing Numerical
Constants." http : //www . cecm . sf u . ca/organics/papers/
bailey.

Natural Measure

^i(e), sometimes denoted -Pi(e), is the probability that
element i is populated, normalized such that

£/*(€) = 1.

see also INFORMATION DIMENSION, g-DlMENSION

Natural Norm

Let llzll be a Vector Norm of z such that

||A||= max ||Az||.

Then ||A|| is a Matrix Norm which is said to be the
natural norm Induced (or Subordinate) to the Vec-
tor Norm ||z||. For any natural norm,

III

where I is the IDENTITY MATRIX. The natural matrix
norms induced by the Li-NORM, JD2-NORM, and Loo-
Norm are called the Maximum Absolute Column
Sum Norm, Spectral Norm, and Maximum Abso-
lute Row Sum Norm, respectively.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1115, 1979.

Natural Number

Nearest Integer Function 1219

Natural Number

A Positive Integer 1, 2, 3, ... (Sloane's A000027).
The set of natural numbers is denoted N or Z + . Un-
fortunately, is sometimes also included in the list of
"natural" numbers (Bourbaki 1968, Halmos 1974), and
there seems to be no general agreement about whether
to include it.

Due to lack of standard terminology, the following terms
are recommended in preference to "Counting Num-
ber," "natural number " and "Whole Number."

Set

Name

Symbol

..., "2, -1,0, 1,2,
1,2,3,4, ...
0,1,2,3,4...
-1,-2,-3,-4,...

integers Z

positive integers Z

nonnegative integers Z*

negative integers Z

see also Counting Number, Integer, N, Positive,

Z,Z",Z + ,Z*

References

Bourbaki, N. Elements of Mathematics: Theory of Sets.
Paris, France: Hermann, 1968.

Courant, R. and Robbins, H. "The Natural Numbers." Ch. 1
in What is Mathematics?: An Elementary Approach to
Ideas and Methods, 2nd ed, Oxford, England: Oxford Uni-
versity Press, pp. 1—20, 1996.

Halmos, P. R. Naive Set Theory. New York: Springer- Verlag,
1974.

Sloane, N. J. A. Sequence A000027/M0472 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Naught

The British word for "ZERO." It is often used to indicate
subscripts, so ao would be spoken as "a naught."

see also Zero

Navigation Problem

A problem in the CALCULUS OF VARIATIONS. Let a
vessel traveling at constant speed c navigate on a body
of water having surface velocity

u = u(x,y)
v = v(x,y).

The navigation problem asks for the course which travels
between two points in minimal time.

References

Sagan, H. Introduction to the Calculus of Variations. New
York: Dover, pp. 226-228, 1992.

Near-Integer

see Almost Integer

Near Noble Number

A Real Number < v < 1 whose Continued Frac-
tion is periodic, and the periodic sequence of terms
is composed of a string of Is followed by an INTEGER
n > 1,

v = [l,l,...,l,n].
p
This can be written in the form

(i)

(2)

which can be solved to give

nFp-i + Fp-2

1 + 4

(3)

where F n is a FIBONACCI NUMBER. The special case
n — 2 gives

Fp+2

Fp

-1.

(4)

see also Noble Number

References

Schroeder, M. R. Number Theory in Science and Communi-
cation: With Applications in Cryptography, Physics, Digi-
tal Information, Computing, and Self- Similarity, 2nd enl.
ed., corr. printing. Berlin: Springer- Verlag, 1990.

Schroeder, M. "Noble and Near Noble Numbers." In Frac-
tals, Chaos, Power Laws: Minutes from an Infinite Par-
adise. New York: W. H. Freeman, pp. 392-394, 1991.

Near-Pencil

An arrangement of n > 3 points such that n — 1 of them

are COLLINEAR.

see also General Position, Ordinary Line, Pencil

References

Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903-909, 1989.

Nearest Integer Function

— ^ U| Ceiling

[x] Nint (Round)

[jcJ Floor

x 2

LL!

The nearest integer function nint(x) of x, also called
Nint or the Round function, is defined such that [x] is

1220 Nearest Neighbor Problem

Necklace

the Integer closest to x. It is shown as the thin solid
curve in the above plot. Note that while [x] is used to
denote the nearest integer function in this work, [x] is
more commonly used to denote the FLOOR FUNCTION

L*J.

see also CEILING FUNCTION, FLOOR FUNCTION

Nearest Neighbor Problem

The problem of identifying the point from a set of points
which is nearest to a given point according to some mea-
sure of distance. The nearest neighborhood problem in-
volves identifying the locus of points lying nearer to the
query point than to any other point in the set.

References

Martin, E. C. "Computational Geometry." http:// www .
mathsource . com/ cgi- bin /Math Source /Enhancements /
DiscreteMath/0200-181.

Necessary

A Condition which must hold for a result to be true,
but which does not guarantee it to be true. If a CON-
DITION is both Necessary and Sufficient, then the
result is said to be true Iff the Condition holds.

see also SUFFICIENT
Necker Cube

Necklace

An Illusion in which a 2-D drawing of an array of
CUBES appear to simultaneously protrude and intrude
into the page.

References

Fineman, M. The Nature of Visual Illusion. New York:

Dover, pp. 25 and 118, 1996.
Jablan, S. "Impossible Figures." http: //members. tripod.

com/-modularity/impos .htm.
Newbold, M. "Animated Necker Cube." http: //www. sover.

net / -manx/ne cker . html .

In the technical COMBINATORIAL sense, an a-ary neck-
lace JV(n, a) of length n is a string of n characters, each
of a possible types. Rotation is ignored, in the sense that
&i&2 . . . 6 n is equivalent to &fc&fc+i • • • &1&2 * * • &k-i for any
fc, but reversal of strings is respected. Necklaces there-
fore correspond to circular collections of beads in which
the FIXED necklace may not be picked up out of the
PLANE (so that opposite orientations are not considered
equivalent).

The number of distinct Free necklaces N'(n,a) of n
beads, each of a possible colors, in which opposite ori-
entations (Mirror Images) are regarded as equivalent
(so the necklace can be picked up out of the Plane and
flipped over) can be found as follows. Find the Divi-
sors of n and label them d\ = 1, cfo, . . . , d v (n) = n
where v(n) is the number of DIVISORS of n. Then

N'(n,a)=-{

'ES^W fl " M +' w( " +11/!

for n odd
E^diK^ + iMl + a)^ 2

for n even,

where <j>(x) is the TOTIENT FUNCTION. For a = 2 and
n = p an ODD PRIME, this simplifies to

N'fa2):

2 P-1 _ !

+ 2

(P"l)/2

+ 1.

Necklace

Negative Binomial Distribution 1221

A table of the first few numbers of necklaces for a = 2
and a = 3 follows. Note that N(n, 2) is larger than
JV'(n,2) for n > 6. For n = 6, the necklace 110100
is inequivalent to its MlRROR IMAGE 0110100, account-
ing for the difference of 1 between JV(6, 2) and JV'(6, 2).
Similarly, the two necklaces 0010110 and 0101110 are
inequivalent to their reversals, accounting for the differ-
ence of 2 between N(7, 2) and JV'(7, 2).

n

N(n,2)

N'(n,2)

N'(n,3)

Sloane

000031

000029

027671

1

2

2

3

2

3

3

6

3

4

4

10

4

6

6

21

5

8

8

39

6

14

13

92

7

20

18

198

8

36

30

498

9

60

46

1219

10

108

78

3210

11

188

126

8418

12

352

224

22913

13

632

380

62415

14

1182

687

173088

15

2192

1224

481598

Ball and Coxeter (1987) consider the problem of finding

the number of distinct arrangements of n people in a

ring such that no person has the same two neighbors

two or more times. For 8 people, there are 21 such

arrangements.

see also ANTOINE'S NECKLACE, DE BRUIJN SEQUENCE,

Fixed, Free, Irreducible Polynomial, Josephus
Problem, Lyndon Word

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 49-50,
1987.

Dudeney, H. E. Problem 275 in 536 Puzzles & Curious Prob-
lems, New York: Scribner, 1967.

Gardner, M. Martin Gardner's New Mathematical Diver-
sions from Scientific American. New York: Simon and
Schuster, pp. 240-246, 1966.

Gilbert, E. N. and Riordan, J. "Symmetry Types of Periodic
Sequences." Illinois J. Math. 5, 657-665, 1961.

Riordan, J. "The Combinatorial Significance of a Theorem
of Polya." J. SIAM4 y 232-234, 1957.

Riordan, J. An Introduction to Combinatorial Analysis. New
York: Wiley, p. 162, 1980.

Ruskey, F. "Information on Necklaces, Lyndon Words, de
Bruijn Sequences." http://sue.csc.uvic.ca/-cos/inf/
neck/Necklacelnf o . html.

Sloane, N. J. A. Sequences A000029/M0563, A000031/
M0564, and A001869/M3860 in "An On-Line Version of
the Encyclopedia of Integer Sequences." http : //www .
research. att . c om/ -nj as/ sequences /e is online. html.

Sloane, N. J. A. and Plouffe, S. Extended entry for M3860 in
The Encyclopedia of Integer Sequences. San Diego: Aca-
demic Press, 1995.

Needle

see Buffon-Laplace Needle Problem, Buffon's
Needle Problem, Kakeya Needle Problem

Negation

see Not

Negative

A quantity less than ZERO (< 0), denoted with a MINUS
Sign, i.e., —x.

see also Nonnegative, Nonpositive, Nonzero, Pos-
itive, Zero

Negative Binomial Distribution

Also known as the Pascal Distribution and Polya
DISTRIBUTION. The probability of r — 1 successes and x
failures in x -f r — 1 trials, and success on the (x + r)th
trial is

X + r - 1 \ p r-l^ _ ^[(x + r-l)-(r-l)]

r- 1

r-1

p t -\i-vT

x -f r — 1
r-1

P r (l~p) X , (1)

where (™) is a BINOMIAL COEFFICIENT. Let

P

V
The CHARACTERISTIC FUNCTION is given by

<i>{t) = (Q - Pe u y r ,

and the MOMENT-GENERATING FUNCTION by

M(t) = (e tx ) = JT e" ( X + /_- X V (1 " P)'

x = ^ '

but, since ffl = { N » m ),

00/ \

^(*)=P r £( a!+ ^" 1 )[(l-P)eT

x=0 ^ '

(2)
(3)

(4)
(5)

(6)

= p r [l-(l-p)e']- p
M'(t) = p r (-r)[l - (1 - p)e t r r ~ 1 (P ~ l)e*

= p r (l-p)r[l-(l-p)e t ]- r - 1 e t (7)

M"{i) = (1 -p)rp r (l - e J +pe t )- r - 2

x (-1 - e t r + e t pr)e t (8)

M'"(t) = (1 - p)rp r (l - e* + e t P y r - 3
x [1 + e'(l -p + 3r - 3pr)
+ r 2 e 2t (l-p) 2 ]e t . (9)

1222 Negative Binomial Distribution

Neighborhood

The MOMENTS about zero p! n = M n (0) are therefore

' _ _ r ( 1 ~p) __ r Q

Mi — M — — —

P P

, r(l - p)[l - r(p - 1)] _ rq(l - rq)

1*2 =

r

P z

M3

/ _ (1 - p)r(2 - p + 3r - 3pr + r 2 - 2pr 2 + p 2

(10)

(11)
(12)

, ( — 1 + p)r( — 6 + 6p — p 2 — llr + 15pr — 4p 2 r — 6r 2

^ 4 = Za

P

12pr 2 - 6pV - r 3 + 3pr 3 - 3pV + p 3 r 3 )

+ pi • ( 13 >

(Beyer 1987, p. 487, apparently gives the Mean incor-
rectly.) The MOMENTS about the mean are

fl 2 = cr

P 2

M3

/i 4

r(2-3p + p 2 ) _ r(p-l)(p-2)
p 3 p 3

r(l - p)(6-6p + p 2 +3r-3pr)

(14)

(15)

(16)

The Mean, Variance, Skewness and Kurtosis are
then

r(l-p)

\x-

P

(17)

^ 3 r(p-l)(p-2)
7i = -3 = 3

_r(2-p)(l-p)

r(l-p)

3/2

P J

2-P

'•(l-p)v / l-P

•v/Ki-p)

72 ---3

__ — 6 + 6p — p 2 — 3r + 3pr
(p - l)r

which can also be written

fj, = nP

/^2 = nPQ

Q + P

7i -

72

1 + 6PQ

rPQ

3.

The first Cumulant is

Ki = UP,

(18)

(19)

(20)
(21)

(22)
(23)

(24)

and subsequent Cumulants are given by the recurrence
relation

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.

Boca Raton, FL: CRC Press, p. 533, 1987.
Spiegel, M. R. Theory and Problems of Probability and

Statistics. New York: McGraw-Hill, p. 118, 1992.

Negative Binomial Series

The Series which arises in the Binomial Theorem for
Negative integral n,

k=0 ^ '

00

-B-»

n + k- l\ fc _ n _ fc

* ]xa

For a = 1, the negative binomial series simplifies to

(z + l)" n = l~nx+\n(n+l)x 2 -\n{n+l){n+2) + . . . .

see also Binomial Series, Binomial Theorem

Negative Likelihood Ratio

The term Negative likelihood ratio is also used (es-
pecially in medicine) to test nonnested complementary
hypotheses as follows,

NLR:

[true negative rate] __ [specificity]

[false negative rate] 1 — [sensitivity] '
see also Likelihood Ratio, Sensitivity, Specificity
Negative Integer

Negative Pedal Curve

Given a curve C and O a fixed point called the PEDAL
Point, then for a point P on C, draw a Line Perpen-
dicular to OP. The Envelope of these Lines as P
describes the curve C is the negative pedal of C.

see also Pedal Curve

References

Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 46-49, 1972,
Lockwood, E. H. "Negative Pedals.'* Ch. 19 in A Bcok

of Curves. Cambridge, England: Cambridge University

Press, pp. 156-159, 1967.

Neighborhood

The word neighborhood is a word with many different
levels of meaning in mathematics. One of the most
general concepts of a neighborhood of a point iGl n
(also called an Epsilon-Neighborhood or infinitesi-
mal Open Set) is the set of points inside an ti-Ball
with center x and Radius e > 0.

Neile's Parabola
Neile's Parabola

Nephroid Evolute 1223

The solid curve in the above figure which is the EVO-
lute of the Parabola (dashed curve). In Cartesian
Coordinates,

y :

f(2*) 2/3 + §.

Neile's parabola is also called the Semi CUBICAL

Parabola, and was discovered by William Neile in

1657. It was the first nontrivial ALGEBRAIC Curve

to have its Arc Length computed. Wallis published

the method in 1659, giving Neile the credit (MacTutor

Archive).

see also PARABOLA EVOLUTE

References

Lee, X. "Semicubic Parabola." http://www.best .com/ -ocah/
Special Plane Curves - dir / Semicubic Parabola _ dir /
semicubicParabola.html.

MacTutor History of Mathematics Archive. "Neile's Semi-
Cubical Parabola." http://www-groups.dcs.st-and.ac.
uk/ -history/Curves /Neile s .html.

Nephroid

The 2-CUSPED Epicycloid is called a nephroid. Since
n — 2, a = 6/2, and the equation for r 2 in terms of the
parameter is given by EPICYCLOID equation

2

t 2 = — [{n 2 + 2n + 2) - 2(n + 1) cos(n0)] (1)
with n = 2,

r 2 = fa I(2 2 + 2 ■ 2 + 2) - 2(2 + 1) cos(20)]

where

tan#

3 sin 4> — sin(3</>)
3cos0 — cos(30) '

(3)

This can be written

(£) a/, = [ain(W + [oo.(i«)r.
The parametric equations are

x = a[3cos£ — cos(3t)]
y = a[3sini — sin(3t)].

The Cartesian equation is

(x 2 + 2/ 2 -4a 2 ) 3 = 108ay.

(4)

(5)
(6)

(7)

The name nephroid means "kidney shaped" and was
first used for the two-cusped EPICYCLOID by Proctor
in 1878 (MacTutor Archive). The nephroid has Arc
Length 24a and Area 127r 2 a 2 . The Catacaustic for
rays originating at the CUSP of a CARDIOID and reflected
by it is a nephroid. Huygens showed in 1678 that the
nephroid is the CATACAUSTIC of a CIRCLE when the
light source is at infinity. He published this fact in Traite
de la luminere in 1690 (MacTutor Archive).

see also ASTROID, DELTOID, FREETH'S NEPHROID

References

Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 169-173, 1972.
Lee, X. "Nephroid." http://www.best.com/-xah/Special

PlaneCurves_dir/Nephroid-dir/nephroid.html.
Lockwood, E. H. "The Nephroid." Ch. 7 in A Book of

Curves. Cambridge, England: Cambridge University

Press, pp. 62-71, 1967.
MacTutor History of Mathematics Archive. "Nephroid."

http : //www-groups . dcs . st-and . ac . uk/ -history/Curves

/Nephroid. html.
Yates, R. C. "Nephroid." A Handbook on Curves and Their

Properties. Ann Arbor, MI: J. W. Edwards, pp. 152-154,

1952.

Nephroid Evolute

/

/
/
I
1

\

\
\

\

K C

^Y

A-

J\

i
\
\

\
\

i
j
/
/
/

The Evolute of the Nephroid given by

x = | [3 cos t — cos(3t)]

y = |[3sin£-sin(3£)]

= \a[X0 - 6cos(20)] = |a 2 [5 - 3cos(2<£)], (2) is given by

x = cos t

2/= i[3sint + sin(3t)],

which is another NEPHROID.

1224 Nephroid Involute

Nephroid Involute

The Involute of the Nephroid given by

x = | [3 cost — cos(3i)]
y = |[3sini-sin(3i)]

beginning at the point where the nephroid cuts the y-
AxiS is given by

x — 4 cos t

y — 3sint + sin(3t),

another Nephroid. If the Involute is begun instead
at the Cusp, the result is Cayley's Sextic.

Neron- Sever i Group

Let V be a complete normal Variety, and write G(V)
for the group of divisors, G n (V) for the group of divisors
numerically equal to 0, and G a (V) the group of divisors
algebraically equal to 0. Then the finitely generated
Quotient Group NS(V) = G(V)/G a (V) is called the
Neron-Severi group.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 75, 1980.

Nerve

The SlMPLICIAL COMPLEX formed from a family of ob-
jects by taking sets that have nonempty intersections.

See also DELAUNAY TRIANGULATION, SlMPLICIAL COM-
PLEX

Nested Hypothesis

Let S be the set of all possibilities that satisfy HYPOTH-
ESIS H, and let S' be the set of all possibilities that
satisfy HYPOTHESIS H' . Then H f is a nested hypothe-
sis within H IFF S' C 5, where C denotes the PROPER

Subset.

see also LOG LIKELIHOOD PROCEDURE

Nested Radical

A Radical of the form

y n + v n

Netto's Conjecture

For this to equal a given INTEGER x, it must be true
that

x = y n + v n + Vn + • • * = Vn + z, (2)

so

X = 71+ X (3)

and

n = x(x — 1). (4)

Nested radicals in the computation of Pi,

and in TRIGONOMETRICAL values of COSINE and SINE
for arguments of the form 7r/2 n , e.g.,

sin(j)=i^7i
C08(f) = Iv^W5

(6)
(7)

(8)

(9)

see also SQUARE ROOT

+ >Jn+ .

(1)

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 14-20, 1994.

Net

A generalization of a Sequence used in general topol-
ogy and ANALYSIS when the spaces being dealt with
are not First-Countable. (Sequences provide an ad-
equate way of dealing with CONTINUITY for FlRST-
COUNTABLE SPACES.) Nets are used in the study of
the Riemann Integral.

see also Fiber Bundle, Fiber Space, Fibration

Net (Polyhedron)

A plane diagram in which the EDGES of a Polyhedron
are shown. All convex POLYHEDRA have nets, but not
all concave polyhedra do (the constituent POLYGONS
can overlap one another when a concave Polyhedron
is flattened out). The Great Dodecahedron and
Stella Octangula are examples of a concave poly-
hedron which have nets.

Netto's Conjecture

The probability that two elements Pi and P2 of a SYM-
METRIC Group generate the entire Group tends to 3/4
asn-y 00. This was proven by Dixon in 1967.

References

Le Lionnais, F. Les nombres remarquables . Paris: Hermann,
p. 31, 1983.

Network

Network

A Directed Graph having a Source, Sink, and a
bound on each edge.

see also Graph (Graph Theory), Sink (Directed
Graph), Smith's Network Theorem, Source

Neuberg Circles

The Locus of the Vertex Ai of a Triangle on a given
base >1.2^.3 and with a given Brocard Angle u> is a
Circle on either side of A2A3. From the center Ni, the
base A 2 A 3 subtends the ANGLE 2a;. The RADIUS of the
Circle is

r = |ai\/cot 2 a; - 3.

see also Brocard Angle

References

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 287-290, 1929.

Neumann Algebra

see von Neumann Algebra

Neumann Boundary Conditions

Partial Differential Equation Boundary Condi-
tions which give the normal derivative on a surface.

see also BOUNDARY CONDITIONS, CAUCHY BOUNDARY

Conditions

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 679, 1953.

Neumann Function

see Bessel Function of the Second Kind

Neumann Polynomial

Polynomials which obey the Recurrence Relation

On+iOc) = (n + l)-0„(a:) - ^±lo„_iOi:)
x n — 1

Neumann Series (Integral Equation) 1225

£>'*> .2/1 \

H sin (s^tt)-

Neumann Series (Bessel Function)

A series of the form

y &nJv-\-n\Z)j

(i)

where v is a REAL and J [/+n (z) is a BESSEL FUNCTION
OF THE First Kind. Special cases are

~ a*w 2 +*

(H

-J„/a+»(z), ( 2 )

z" = 2T(i„+l)X;

n=0

where T(z) is the Gamma Function, and

00 00

5>Z" + " =J> (H^^ 2 J(u + n)Mz), (3)

71 = 71 =

where

a n = 2_^ Hi ~~~ On -2m, (4)

m=0

ml

and [a: J is the FLOOR FUNCTION.
see also Kapteyn Series

References

Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.

Neumann Series (Integral Equation)

A Fredholm Integral Equation of the Second
Kind

0(x) = /(x)+ / K(x,t)4>(t)dt (l)

J a

may be solved as follows. Take

*)W = /W (2)

<MaO = /(aO + A

/ K(x 9

J a

t)f(t) dt

(3)

'/

2 (x) = /(x) + A / A-(a:,ti)/(ti)dti

The first few are

pb pb

+ A 2 / / K{x,t 1 )K{tut2)f{t 2 )dt 2 dt 1 {4)

J a J a

n

<j> n {x) = ^TXuiix), (5)

O {x)
Oi{x)

where

x*

o 2 (x) = i + 4-

X X 6

see also SCHLAFLI POLYNOMIAL

References

von Seggern, D. CRC Standard Curves and Surfaces, Boca
Raton, FL: CRC Press, p. 196, 1993.

U (x) — f(x)
r b

(6)

Ul (x) = / K{x,t)f(t 1 )dt 1 (7)

J a
pb pb

u?{x)= / K(x,t 1 )K{t u t2)f(t 2 )dt2dti (8)

J a J a

pb pb pb

u n {x)= I / K(x,t 1 )K(t 1 ,t 2 )---

J a J a, J a

x K(t n -i,t n )f(t n ) dt n -.- dt x . (9)

1226

Neusis Construction

Newton's Backward Difference Formula

The Neumann series solution is then

n

<f>(x) = lim 4> n {x) = lim > X l Ui(x). (10)

rt.—^no n — i-oo * J

where H and are the Jacobi Theta Functions and
K(u) is the complete ELLIPTIC INTEGRAL OF THE FIRST

Kind.

see also JACOBI THETA FUNCTION, THETA FUNCTION

References

Arfken, G. "Neumann Series, Separable (Degenerate) Ker-
nels." §16.3 in Mathematical Methods for Physicists, 3rd
ed. Orlando, FL: Academic Press, pp. 879-890, 1985.

Neusis Construction

A geometric construction, also called a VERGING CON-
STRUCTION, which allows the classical GEOMETRIC
CONSTRUCTION rules to be bent in order to permit slid-
ing of a marked RULER. Using a Neusis construction,
Cube Duplication and angle Trisection are soluble.
Conway and Guy (1996) give Neusis constructions for
the 7-, 9-, and 13-gons which are based on angle TRI-
SECTION.

see also Cube Duplication, Geometric Construc-
tion, Mascheroni Construction, Ruler, Trisec-
tion

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 194-200, 1996.

Neville's Algorithm

An interpolation ALGORITHM which proceeds by first
fitting a Polynomial Pk of degree through the points
(x k ,yk) for A; = 0, ..., n, i.e., P*. = yk- A second
iteration is then performed in which P12 is fit through

pairs of points, yielding P12, P23, The procedure

is repeated, generating a "pyramid" of approximations
until the final result is reached

P 2 12 P 123

o ft 3 o ft 234-
Pz D ^234

p 4 Ps4

The final result is

ft(i+l)-(t+m) -

(X - St+m)ft(i+l)-(i+m-l)
Xi Xi-\-rn

(Xj - s)P( i+1 )( H _ 2 )...(i-|- m )

+ -

Xi Xi-\-rs

Newcomb's Paradox

A paradox in DECISION THEORY. Given two boxes, Bl
which contains $1000 and B2 which contains either noth-
ing or a million dollars, you may pick either B2 or both.
However, at some time before the choice is made, an om-
niscient Being has predicted what your decision will be
and filled B2 with a million dollars if he expects you to
take it, or with nothing if he expects you to take both.

see also Alias' Paradox

References

Gardner, M. The Unexpected Hanging and Other Mathemat-
ical Diversions. Chicago, IL: Chicago University Press,
1991.

Gardner, M. "Newcomb's Paradox." Ch. 13 in Knotted
Doughnuts and Other Mathematical Entertainments. New
York: W. H. Freeman, 1986.

Nozick, R. "Reflections on Newcomb's Paradox." Ch. 14 in
Gardner, M. Knotted Doughnuts and Other Mathematical
Entertainments. New York: W. H. Freeman, 1986.

Newman- Conway Sequence

The sequence 1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, . . . (Sloane's
A004001) defined by the recurrence P(l) = P(2) = 1,

P(n) = P(P(n - 1)) + P(n - P{n - 1)).

It satisfies

and

P(2*) = 2*" 1

P(2n) < 2P(n).

see also BULIRSCH-STOER ALGORITHM

References

Bloom, D. M. "Newman-Conway Sequence." Solution to

Problem 1459. Math. Mag. 68, 400-401, 1995.
Sloane, N. J. A. Sequence A004001/M0276 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Newton's Backward Difference Formula

/ P = /o+pVo + ^p(p+l)Vg + ip(p+l)(p+2)VS + ...,

Neville Theta Function

The functions

*-(*) =

H{u)
H'(0)

M*>) =

Q(k)

Mu) =

H{u)
H(K)

& n (u)=

e(u)

(1)

(2)
(3)
(4)

for p e [0, 1], where V is the Backward Difference.
see also Newton's Forward Difference Formula

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 433, 1987.

Newton-Cotes Formulas

Newton-Cotes Formulas 1227

Newton-Cotes Formulas

The Newton-Cotes formulas are an extremely useful
and straightforward family of Numerical Integra-
tion techniques.

To integrate a function f(x) over some interval [a, 6],
divide it into n equal parts such that f n = f(x n ) and
h= (b- a)/n. Then find Polynomials which approxi-
mate the tabulated function, and integrate them to ap-
proximate the Area under the curve. To find the fitting
Polynomials, use Lagrange Interpolating Poly-
nomials. The resulting formulas are called Newton-
Cotes formulas, or Quadrature Formulas.

Newton- Cotes formulas may be "closed" if the inter-
val [xi,x n ] is included in the fit, "open" if the points
[a?2, #n-i] are used, or a variation of these two. If the for-
mula uses n points (closed or open), the Coefficients
of terms sum ton- 1.

If the function f(x) is given explicitly instead of sim-
ply being tabulated at the values, Xi, the best numer-
ical method of integration is called GAUSSIAN QUAD-
RATURE. By picking the intervals at which to sample
the function, this procedure produces more accurate ap-
proximations (but is significantly more complicated to
implement).

fa)

The 2-point closed Newton- Cotes formula is called the
Trapezoidal Rule because it approximates the area
under a curve by a TRAPEZOID with horizontal base and
sloped top (connecting the endpoints x\ and x^)- If the
first point is Xi, then the other endpoint will be located
at

#2 = xi + h,

(1)

and the Lagrange Interpolating Polynomial

through the points (xi,/i) and (x2,/2) is

Integrating over the interval (i.e., finding the area of the
trapezoid) then gives

f(x)dx= I P2(x)dx

/ f(x)dx =

J Xl t/ X\

= ^{h-h)[x 2 ]ll

+

(A + 7r*-7rA)w::

= ^(/2 - /l)(z2 + El) (a* - El)
+ (x2-Zl)(/l + ^/l-^/ 2 )

= j(/a - /i)(2zi + h) + fih + a;i(/i - / a )

= *i(/ a - /1) + lh(f 2 - h) + hh ~ xi (/a " /1)

= \Hh+h)-\h*f"{X). (3)

This is the trapezoidal rule, with the final term giving
the amount of error (which, since x\ < £ < X2> is no
worse than the maximum value of /"(£) in this range).

The 3-point rule is known as Simpson's Rule. The
Abscissas are

X2 = Xl + h
X3 — Xl + 2/i

(4)
(5)

and the LAGRANGE INTERPOLATING POLYNOMIAL is

p/ x (X-X 2 )(X~X 3 ) f

P 3 (x) = -/1

(Xl -X2)(Xx -X3)

(X-Xi)(x-Xs) (X-Xi)(x-X2) ,

{X2 - XI){X2 - Xz) 2 (X 3 ~ Xi)(x 3 ~ X 2 )

X — X(X2 + X3) + £2#3

h(2h)

h

x 2 - a(a?i + £3) + X1X3 x 2 - x(xi + gg) + ai^ .
+ /i(-fc) /2+ 2/i(/i) /3

+z[-±(2zi + 3/i)/i + {2x x + 2fc)/ 2 - \{2xx + /i)]
+ [f(a; 1 -h/i)(xi+2/i)/ 1 -xi(a;i + 2/i)/ 2 + |xi (^1 + ^/3]}.

(6)

, . x - x 2 x - x x

Xi — #2 X 2 — Xi

x — Xi — h x — Xi

= lU>-h)+(h + X ih~ X ih). (2)

Integrating and simplifying gives

/ f(x)dx = / P 3 (x)dx

= ^(/i+4/ 2 + / 3 )-M 5 / (4) (0- (7)

The 4-point closed rule is Simpson's 3/8 Rule,

f{x)dx= lh(f 1+ 3f 2 +3f 3 +f 4 )-^h 5 f^^). (8)

1228 Newton-Cotes Formulas

The 5-point closed rule is Bode'S Rule,

px

fix) dx = ±h{7fi + 32/ 2 + 12/s + 32/4 + 7/5)

-&h 7 f™{i) (9)

(Abramowitz and Stegun 1972, p. 886). Higher order
rules include the 6-point

J X\

f(x) dx = 2fg/i(19/i + 75/2 + 50/s + 50/4 + 75/5

7-point

«/ XI

8-point

f

+19/6)-if§5fc7 (6) (a (10)

fix) dx = ^h(tlh + 216/2 + 27/a + 272/ 4
+27/ 5 + 216/ 6 + 41/ 7 ) - jA-fc 9 /^). (11)

f{x) dx = T? |g 5 /i(751/i +3577/2 + 1323/ 2 +2989/ 3

+2989/5 + 1323/ 6 + 3577/ 7 + 751/ 8 ) - jg^h 9 f {s) {€),

(12)

9-point

(*Xg

PX

f{x) dx = ^71(989/1 + 5888/2 - 928/ 3

+10496/4 - 4540/s + 10496/e - 928/ 7 + 5888/ 8 + 989/ 9 )

-^*"/ (1O) (0, (13)

10-point

l*X
J Xl

/(*)«te=Si§oofc[2857(/i + /io)

+15741(/ 2 + ft) + 1080(/s + fa + 19344(/ 4 + f 7 )

+5788(/ 5 + /.)] - Mo hll f m (Z)> ( 14 )

and 11-point

/'

f(x)dx

jfc[16067(/i + /11)

+106300(/ 2 + /10) - 48525(/ 3 + / 9 ) + 272400(/ 4 + / 8 )
-260550(/ 5 + h) + 427368/ 6 ] - 32Hfi§2^ 3 / (12) (0

(15)

rules.

Closed "extended" rules use multiple copies of lower
order closed rules to build up higher order rules. By
appropriately tailoring this process, rules with particu-
larly nice properties can be constructed. For n tabulated

Newton-Cotes Formulas

points, using the TRAPEZOIDAL RULE (n — 1) times and
adding the results gives

px n J px 2 />x 3 />x n \

/ f(x)dx= / + / +...+ / \f{x)dx

J X\ \^ x l J x 2 ^ X n-\J

= \h[(fi + /a) + (h + fa) + ... + (/n-2 + /»-i)

+ (/»-!+/»)] = h(\fl+f 2 +f 3 + ... + U-2+f n -l + yn)

-± 2 nh 3 f"(0- (16)

Using a series of refinements on the extended TRAPE-
ZOIDAL Rule gives the method known as Romberg In-
tegration. A 3-point extended rule for Odd n is

px

J X\

f{x)dx = h\{\h + \h + \h) + (|/3 + |/4+|/5)

+ . . . + (3/71-4 + g/n-3 + 3/71-2)

+ (|/n-2 + |/n-l + |/n)]

= \h(h + 4/2 + 2/3 + 4/4 + 2/5 + . . . + 4/„-i + /„)

^/i 5 / C4) (0. (IT)

n-1

Applying Simpson's 3/8 Rule, then Simpson's Rule
(3-point) twice, and adding gives

pX4 pXQ n

J + +

f(x)dx

= M(|/i + |/2 + f/3 + |/4)

+ (|/4 + |/5 + 1/6) + (1/6 + |/7 + 1/8)]

= Mf/i + 1/2 + 5/3 + (| + \)U + !/ 5

+ (| + |)/6+|/7+|/8]

= M|/i + f/ 2 + 1/ 3 + M/ 4

+ |/5 + f/6+|/7+l/ 8 ). (1 8)

Taking the next Simpson's 3/8 step then gives

f

f(x)dx = fc(f/8 + §/„ + |/ 10 + |/n). (19)

Combining with the previous result gives

/"

f{x) dx = h[|/l + f /2 + |/3 + M/4 + |/5

+ |/6 + |/7 + (I + |)/8 + f /9 + |/10 + f /ll]
= Klh + |/2 + |/S + M/4 + |/5 + |/6 + |/7

+ H/8 + f/9 + |/io + |/n), (20)

where terms up to /10 have now been completely deter-
mined. Continuing gives

M|/l + 1/2 + 1/3 + M/4 + f /5 + |/6 + ■ ■ ■

+ |/n-5 + |/n-4+24/n-3+8/n-2+g/n-l+g/n)- (21)

Newton-Cotes Formulas

Now average with the 3-point result

h{\h + \h + |/3 + |/4 + |/ 5 + ifn-l + \fn) (22)

to obtain

Mi/l + I/2 + ||/4+i/4 + (/5 + /6+...+/n-5 + /n-4)
+ I/n-3 + f/,-2 + i/n-1 + £/„] + 0^). (23)

Note that all the middle terms now have unity COEFFI-
CIENTS. Similarly, combining a 4-point with the (2+4)-
point rule gives

/i (^/l+lf/2+/3+/4 + ... + /r l -3 + /n-2 + Yf/"-l+l^)

Newton-Cotes Formulas 1229

+C>(n- 3 ). (24)

Other Newton-Cotes rules occasionally encountered in-
clude Durand's Rule

/"

f(x) dx

= M|/l + I5/a + /3 + ". + /n-2 + ^/n-l + f/n) (25)

(Beyer 1987), Hardy's Rule

/

</ Xt

f{x) dx = I l 5 /i(28/-s + 162/_ 2 + 22/o + 162/ 2
+28/ 3 ) + ^sh 7 [2f^{^) - fc 2 / (8) «i)], (26)

' 030 — 3/l

J-OS^ J

1400

and Weddle's Rule

C x 6n

J X\

f(x)dx=±h(fi

+5/2 + h + 6/4 + 5/5 + U + * • • + 5/ 6 „-i + fen) (27)

(Beyer 1987).

The open Newton-Cotes rules use points outside the in-
tegration interval, yielding the 1-point

px
J xn

f(x)dx = 2hfi,

(28)

2-point

px 3 pxi+2h

/ f(x)dx= / P 2 {x)dx

J xn v x\ — h

= ±(h - fi)[* a & + (* + T h ~ T h ) w -i-*

+2h

IMA + h) + hh 3 f"(t), (29)

3-point

r f(x) dx = §M2/i - / 2 + 2/3) + |§ft 5 / (4) (e), (30)

4-point

px

J Xn

I XQ

5-point

fix)dx = f i hinf 1 +f 2 +f 3 + nf 4 )+£ i h 5 f w ii),

(31)

px
J xn

-14/ a + 26/ 3 - 14/4 + ll/s) - £U7 (6) (£)> (32)

6-point

r x 7

pX

J Xn

f{x) dx = -i 5 fc(611/i - 453/2 + 562/ 3 + 562/ 4

-453/5 + 611/ 8 ) -SEftVWfc), (33)

and 7-point

px

J xn

fix) dx = 5^/1(460/1 - 954/2 + 2196/s - 2459/ 4

+2196/5 - 954/ 6 + 460/ 7 ) - ^h 9 / (8) (0 (34)
rules.
A 2-point open extended formula is

/'

fix) dx = h[i\h + h + ■ ■ ■ + /»-l + f/n)

+ &(-/o + /a + /„-i + /„ + i)] + 11( 7 2 Q 1) fe5 / (4) (0-

(35)

Single interval extrapolative rules estimate the integral
in an interval based on the points around it. An example
of such a rule is

hfi + 0(h*f) (36)

£M3/i-/ 2 ) + 0(hV") (37)

£h(23/i " I6/2 + 5/ 3 ) + 0(h 4 f {z) ) (38)

£/i(55/i - 59/2 + 37/3 - 9/4) + 0(/i 5 / (4) ). (39)

see also Bode's Rule, Difference Equation, Du-
rand's Rule, Finite Difference, Gaussian Quad-
rature, Hardy's Rule, Lagrange Interpolating
Polynomial, Numerical Integration, Simpson's
Rule, Simpson's 3/8 Rule, Trapezoidal Rule,
Weddle's Rule

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Integration."
§25.4 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing, New-
York: Dover, pp. 885-887, 1972.

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 127, 1987.

Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 160-161, 1956.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Classical Formulas for Equally Spaced Abscis-
sas." §4.1 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 124-130, 1992.

1230 Newton's Diverging Parabolas

Newton's Identities

Newton's Diverging Parabolas

Curves with CARTESIAN equation

ay 2 = x(x 2 — 2bx + c)

with a > 0. The above equation represents the third
class of Newton's classification of CUBIC CURVES, which
Newton divided into five species depending on the
ROOTS of the cubic in x on the right-hand side of the
equation. Newton described these cases as having the
following characteristics:

1. "All the ROOTS are Real and unequal Then the
Figure is a diverging Parabola of the Form of a Bell,
with an Oval at its Vertex.

2. Two of the ROOTS are equal. A Parabola will
be formed, either Nodated by touching an Oval, or
Punctate, by having the Oval infinitely small.

3. The three ROOTS are equal. This is the Neilian
Parabola, commonly called Semi-cubical.

4. Only one REAL ROOT. If two of the ROOTS are
impossible, there will be a Pure PARABOLA of a Bell-
like Form"

(MacTutor Archive).

References

MacTutor History of Mathematics Archive. "Newton's Di-
verging Parabolas." http://www-groups.dcs.st-and.ac.
uk/"history/Curves/Newtons.html.

Newton's Divided Difference Interpolation
Formula

Let

7T n (x) = l\( X ~ Xn )i

(1)

then

/fa) = /o + / J Xk-l(x)[x ,X U . . . , Xk] + Rn, (2)

where [asi,...] is a DIVIDED DIFFERENCE, and the re-
mainder is

/ (n+1) (£)

Rn(x) = 7r n (x)[xo,--- t x ni x] = ir n (x)— — -— - (3)

(n + 1)

for xo < £ < x n ,

see also Divided Difference, Finite Difference

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook

of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, 9th printing. New York: Dover,

p. 880, 1972.
Hildebrand, F. B. Introduction to Numerical Analysis. New

York: McGraw-Hill, pp.43-44 and 62-63, 1956.

Newton's Forward Difference Formula

A Finite Difference identity giving an interpolated
value between tabulated points {f p } in terms of the first
value /o and the POWERS of the Forward DIFFERENCE

A. For a G [0, 1], the formula states

f a = fo + aA+ ±a(a-l)A 2 + ±a(a-l)(a-2)A 3 + . . . .
When written in the form

~ (a)„A B /(aO

,,„ + .,.£ sa-,

with (a) n the POCHHAMMER SYMBOL, the formula looks
suspiciously like a finite analog of a TAYLOR Series ex-
pansion. This correspondence was one of the motivating
forces for the development of UMBRAL Calculus.

The Derivative of Newton's forward difference formula
gives Markoff's Formulas.

see also Finite Difference, Markoff's Formulas,
Newton's Backward Difference Formula, New-
ton's Divided Difference Interpolation For-
mula

References

Abramowitz, M. and Stegun, C. A. (Eds,). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 880, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 432, 1987.

Newton's Formulas

Let a Triangle have side lengths a, 6, and c with op-
posite angles A, B, and C. Then

b + c = cos[\{B-C))

a sm{\A)

c + a ^ cos[\{C-A))

b sin(§B)

a + b __ cos[\{A-B)}

c " sin(fC)

see also Mollweide's Formulas, Triangle

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 146, 1987.

Newton's Identities

see also Newton's Relations

Newton's Iteration

Newton's Method

1231

Newton's Iteration

An algorithm for the SQUARE ROOT of a number r
quadratically as limn-^ x n ,

But

2 V Xn/

where Xo = 1. The first few approximants to ^Jri are
given by

l,£(l + n),

1 + 6n + n 3
4(n + l) '

3 _l „4

1 + 26n + 70n J + 28rT -f ri
8(l + n)(l + 6n + n 2 )

For V^, this gives the convergent s as 1, 3/2, 17/12,
577/408, 665857/470832, ....

see also SQUARE ROOT

Newton's Method

A RoOT-finding ALGORITHM which uses the first few
terms of the TAYLOR SERIES in the vicinity of a sus-
pected Root to zero in on the root. The Taylor Se-
ries of a function f(x) about the point x -j- e is given
by

f(x + e ) = f(x) + f'(x)e+y"(x)e 2 + .

Keeping terms only to first order,

f(x + e)*f(x) + f'{x)e.

(1)

(2)

This expression can be used to estimate the amount of
offset e needed to land closer to the root starting from
an initial guess xq. Setting f(xo + e) = and solving
(2) for e gives

f(xo)

eo

f'W

(3)

which is the first-order adjustment to the Root's posi-
tion. By letting x\ — xq + eo, calculating a new ei, and
so on, the process can be repeated until it converges to
a root.

Unfortunately, this procedure can be unstable near a
horizontal Asymptote or a Local Minimum. How-
ever, with a good initial choice of the Root's position,
the algorithm can by applied iteratively to obtain

f(Xn)

X n +1 — X n . , >

/ '(Xn)

(4)

for n = 1, 2, 3,

The error e n +i after the (n + l)st iteration is given by

En + l =£» + (x n + l — Xn)
f(Xn)

(5)

/(*„) = f(x) + f'(x)e n + i/"(x)e„ 2 + . . .

= f'(x)e n + i/"(x)e n 2 + . . .
f'(x n ) = f'(x) + f"(x)e n + ...,

/(*») _f'(x)e n + ±f"(x)e n > + ...

(6)
(7)

f'(x x ) f'(x) + f"(x)e n +...

l

2

f( x )e+y"(x)e n 2

f'(x) + f"(x)e n
and (5) becomes

, , /"(g) r 2 (R s

e n +i = e n —

e + ^e 2
en+ 2f'(x) en

f"(x) 2

2/'(x)

(9)

Therefore, when the method converges, it does so
quadratically.

A Fractal is obtained by applying Newton's method to
finding a ROOT of z n - 1 = (Mandelbrot 1983, Gleick
1988, Peitgen and Saupe 1988, Press et at 1992, Dickau
1997). Iterating for a starting point Zq gives

-Zi+l — Zi —

Zi

y.n — 1

(10)

Since this is an nth order Polynomial, there are n
ROOTS to which the algorithm can converge.

Coloring the BASIN OF ATTRACTION (the set of initial
points zq which converge to the same Root) for each
ROOT a different color then gives the above plots, cor-
responding to n = 2, 3, 4, and 5.

see also HALLEY'S IRRATIONAL FORMULA, HALLEY'S

Method, Householder's Method, Laguerre's
Method

1232

Newton Number

Newton's Theorem

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 18, 1972.

Acton, F. S. Ch. 2 in Numerical Methods That Work. Wash-
ington, DC: Math. Assoc. Amer., 1990.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 963-964, 1985.

Dickau, R. M. "Basins of Attraction for z 5 = 1 Using

Newton's Method in the Complex Plane." http: //forum,
swarthmore . edu/advanced/robertd/newtons . html.

Dickau, R. M. "Variations on Newton's Method." http://
forum . swarthmore . edu / advanced / robertd /
newnewt on . html.

Dickau, R. M. "Compilation of Iterative and List Opera-
tions." Mathematica J. 7, 14-15, 1997.

Gleick, J. Chaos: Making a New Science. New York: Pen-
guin Books, plate 6 (following pp. 114) and p. 220, 1988.

Householder, A. S. Principles of Numerical Analysis.ew
York: McGraw-Hill, pp. 135-138, 1953.

Mandelbrot, B. B. The Fractal Geometry of Nature. San
Francisco, CA: W. H. Freeman, 1983.

Ortega, J. M. and Rheinboldt, W. C. Iterative Solution of
Nonlinear Equations in Several Variables. New York:
Academic Press, 1970.

Peitgen, H.-O. and Saupe, D, The Science of Fractal Images.
New York: Springer- Verlag, 1988.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Newton-Raphson Method Using Derivatives"
and "Newton-Raphson Methods for Nonlinear Systems of
Equations." §9.4 and 9.6 in Numerical Recipes in FOR-
TRAN: The Art of Scientific Computing, 2nd ed. Cam-
bridge, England: Cambridge University Press, pp. 355-362
and 372-375, 1992.

Ralston, A. and Rabinowitz, P. §8.4 in A First Course in
Numerical Analysis, 2nd ed. New York: McGraw-Hill,
1978.

IIi(ri, . . . , r n )) Si is defined for i — 1, . . . , n. For exam-
ple, the first few values of Si are

3l = n + r 2 + r 3 + r 4 + . . .

s 2 = rir 2 + nra + rir 4 + r 2 r 3 + .

s 3 = nr 2 rs + rir 2 r 4 -f r^^r^ + .

and so on. Then

Si = (-I)"

idri

(2)
(3)
(4)

(5)

This can be seen for a second Degree Polynomial by

multiplying out,

a2X 2 + a\x + ao = a^(x — ri)(x — T2)

— a 2 [x 2 - (n +r 2 )x + rir 2 ], (6)

Si = VJ Ti — T\ 4- r 2 =

ai
a 2

(?)

2
S 2 = 22 riVj = rir2 ~

do
a 2

(8)

i*3

and for a third DEGREE POLYNOMIAL,

azx 3 + a 2 x 2 + a±x + a = a 3 {x - n)(x - r 2 )(x - r 3 )
= as[x 3 -(n+r-z+r^x 2 + (rir 2 +rir3+r 2 r3)x-rir 2 r 3 ],

(9)

Newton Number

see Kissing Number

Newton's Parallelogram

Approximates the possible values of y in terms of x if

n

^ aijX % y % = 0.

Newton-Raphson Fractal

see Newton's Method

Newton-Raphson Method

see Newton's Method

Newton's Relations

Let Si be the sum of the products of distinct ROOTS Tj
of the Polynomial equation of degree n

a n x n + a n -ix n ~ + . . . + aix + ao = 0, (1)

where the roots are taken i at a time (i.e., Si is
defined as the Elementary Symmetric Function

i=l
3

as

52

= 2_] r i r J = r i r2 + rir 3 + r 2 r 3 = — (11)

ao

(10)

S3 = > TiTjT k = r\r 2 r 3 = - — . (12)

z — ' a 3

see also ELEMENTARY SYMMETRIC FUNCTION

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 1-2, 1959.

Newton's Theorem

If each of two nonparallel transversals with nonminimal
directions meets a given curve in finite points only, then
the ratio of products of the distances from the two sets
of intersections to the intersection of the lines is inde-
pendent of the position of the latter point.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 189, 1959.

Newtonian Form

Nielsen-Ramanujan Constants 1233

Newtonian Form

see Newton's Divided Difference Interpolation
Formula

Next Prime

The next prime function NP(n) gives the smallest
Prime larger than n. The function can be given ex-
plicitly as

NP(n) =pi+ w (n),

where pi is the ith Prime and ir(n) is the Prime
Counting Function. For n = 1, 2, ... the values
are 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13, 13, 17, 17, 17, 17,
19, ... (Sloane's A007918).

see also Fortunate Prime, Prime Counting Func-
tion, Prime Number

References

Sloane, N. J. A. Sequence A007918 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Nexus Number

A Figurate Number built up of the nexus of cells less
than n steps away from a given cell. In fc-D, the (n+l)th
nexus number is given by

e(*v

JV„+i(fc)

where (^) is a BINOMIAL COEFFICIENT. The first few k-
dimensional nexus numbers are given in the table below.

k

iVn+l

name

1

unit

1

l + 2n

odd number

2

1 + 3n + 3n 2

hex number

3

1 + An + 6n 2

+ 4n 3

rhombic dodecahedral

number

see also Hex Number, Odd Number, Rhombic Do-
decahedral Number

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 53-54, 1996.

Neyman-Pearson Lemma

If there exists a critical region C of size a and a N ON-
NEGATIVE constant k such that

n: = i/(*^)

nr=i /(*!«<>)

for points in C and

IE=i/(*l»o

> k

<k

n: =1 /(*ii0o)

for points not in C, then C is a best critical region of

References

Hoel, P. G.; Port, S. C; and Stone, C. J. "Testing Hypothe-
ses." Ch. 3 in Introduction to Statistical Theory. New
York: Houghton Mifflin, pp. 56-67, 1971.

Nicholson's Formula

Let J v (z) be a Bessel Function of the First Kind,
Y u (z) a Bessel Function of the Second Kind, and
K v (z) a Modified Bessel Function of the First
Kind. Also let R[z] > 0. Then

4{z) + Y?(z)

f

Jo

K (2z sinh t) cos(2ut) dt.

see also Dixon-Ferrar Formula, Watson's For-
mula

References

Gradshteyn, I. S. and Ryzhik, I. M. Eqn. 6.664.4 in Tables

of Integrals, Series, and Products, 5th ed. San Diego, CA:

Academic Press, p. 727, 1979.
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary

of Mathematics. Cambridge, MA: MIT Press, p. 1476,

1980.

Nicomachus's Theorem

The nth Cubic Number n 3 is a sum of n consecutive
Odd Numbers, for example

1 3 = 1

2 3 = 3 + 5

3 3 = 7 + 9 + 11

4 3 = 13 + 15 + 17+19,

etc. This identity follows from

n

^[n(n-l)-l + 2i] = n 3 .

i=i

It also follows from this fact that

!>'= £' ■

, fe = l

see also Odd Number THEOREM

Nicomedes' Conchoid

see Conchoid of Nicomedes

Nielsen-Ramanujan Constants

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

N. Nielsen (1909) and Ramanujan (Berndt 1985) con-
sidered the integrals

/" (lnx) fe
' Ji *-l

f2 /i . \fc

Ctk = I — dx.

(1)

1234 Nielsen's Spiral

Nim

They found the values for k = 1 and 2. The general
constants for k > 3 were found by V. Adamchik (Finch)

a p =pK(p+l)-

p(ln2)

p-i

P +

~*t

Lip+i-»(i)(ln2)*
A;!

(2)
where C,(z) is the Riemann Zeta Function and Li n (x)
is the POLYLOGARITHM. The first few values are

ai = K(2) = ^
a 2 = k(3)

(3)

(4)

a 3 = i7r 4 + K(hi2) 2 -i(ln2) 4

-6Li 4 (i)-^ln2C(3) (5)

a 4 = |7r 2 (ln2) 3 - |(ln2) 5 - 241n2Li 4 (i)

-24Li 5 (i)-^(ln2) 2 C(3) + 24C(5). (6)

see also Polylogarithm, Riemann Zeta Function

References

Berndt, B. C. Ramanujan's Notebooks, Part I. New York:
Springer- Verlag, 1985.

Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/const ant /nielram/nielram. html.

Flajolet, P. and Salvy, B. "Euler Sums and Con-
tour Integral Representation." Submitted to Experim.
Math 1997. http://pauillac.inria.fr/algo/flajolet/
Publications/publist .html.

Nielsen's Spiral

The SPIRAL with parametric equations

x(t) — aci(i)
y(t) = asi(t),

(1)
(2)

where ci(i) is the Cosine Integral and si(t) is the Sine
Integral. The Cesaro Equation is

ps/a

(3)

see also Cornu Spiral, Cosine Integral, Sine In-
tegral

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 119, 1993.

Nil Geometry

The Geometry of the Lie Group consisting of Real
Matrices of the form

1

X

y~

1

z

1.

i.e., the Heisenberg Group.

see also Heisenberg Group, Lie Group, Thurston's
Geometrization Conjecture

Nilmanifold

Let AT be a Nilpotent, connected, Simply Con-
nected Lie Group, and let D be a discrete Subgroup
of N with compact right Quotient Space. Then N/D
is called a nilmanifold.

Nilpotent Element

An element B of a Ring is nilpotent if there exists a
Positive Integer k for which B k = 0.

see also ENGEL'S THEOREM

Nilpotent Group

A Group G for which the chain of groups

I = Z C Z x C . . . C Z n

with Zk+i/Zk (equal to the Center of G/Zk) termi-
nates finitely with G = Z u is called a nilpotent group.
see also Center (Group), Nilpotent Lie Group

Nilpotent Lie Group

A Lie Group which has a simply connected covering
group HOMEOMORPHIC to M n . The prototype is any
connected closed subgroup of upper triangular Com-
plex matrices with Is on the diagonal. The HEISEN-
BERG Group is such a group.

References

Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.

Nilpotent Matrix

A Square Matrix whose Eigenvalues are all 0. A
related definition is a Square Matrix M such that M 71
is for some Positive integral Power.

see also Eigenvalue, Square Matrix

Nim

A game, also called Tactix, which is played by the fol-
lowing rules. Given one or more piles (Nim-Heaps),
players alternate by taking all or some of the counters
in a single heap. The player taking the last counter or
stack of counters is the winner. Nim-like games are also
called Take-Away Games and Disjunctive Games.

Nim-Heap

Nine-Point Circle 1235

If optimal strategies are used, the winner can be deter-
mined from any intermediate position by its associated

Nim- Value.

see also MiSERE Form, Nim-Value, Wythoff's

Game

References

Ball, W, W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 36-38,
1987.

Bogomolny, A. "The Game of Nim." http://www.cut-the-
knot . c om/bott om_nim . html .

Bouton, C. L. "Nim, A Game with a Complete Mathematical
Theory." Ann. Math. Princeton 3, 35-39, 1901-1902.

Gardner, M. "Nim and Hackenbush." Ch. 14 in Wheels, Life,
and other Mathematical Amusements. New York: W. H.
Freeman, 1983.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, pp. 117-120, 1990.

Kraitchik, M. "Nim." §3.12.2 in Mathematical Recreations.
New York: W. W. Norton, pp. 86-88, 1942.

Nim-Heap

A pile of counters in a game of NlM.

Nim- Sum

see Nim-Value

and is the Midpoint of the line between the ClRCUM-
center C and Orthocenter H. It lies on the Euler

Line.

see also Euler Line, Nine-Point Circle, Nine-
Point Conic

References

Carr, G. S. Formulas and Theorems in Pure Mathematics,

2nd ed. New York: Chelsea, p. 624, 1970.
Dixon, R. Mathographics. New York: Dover, pp. 57-58, 1991.
Kimberling, C. "Central Points and Central Lines in the

Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Nine- Point Center." http://www.

evansville.edu/-ck6/tcenters/class/npcenter.html.

Nine-Point Circle

Nim-Value

Every position of every Impartial Game has a nim-
value, making it equivalent to a Nim-Heap. To find the
nim-value (also called the Sprague-Grundy Number),
take the Mex of the nim- values of the possible moves.
The nim-value can also be found by writing the num-
ber of counters in each heap in binary, adding without
carrying, and replacing the digits with their values mod
2. If the nim-value is 0, the position is SAFE; otherwise,
it is UNSAFE. With two heaps, safe positions are (z, x)
where x e [1,7]. With three heaps, (1, 2, 3), (1, 4, 5),
(1, 6, 7), (2, 4, 6), (2, 5, 7), and (3, 4, 7).

see also Grundy's Game, Impartial Game, Mex,
Nim, Safe, Unsafe

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 36-38,
1987.

Grundy, P. M. "Mathematics and Games." Eureka 2, 6-8,
1939.

Sprague, R. "Uber mathematische Kampfspiele." Tohoku J.
Math. 41, 438-444, 1936.

Nine-Point Center

The center F (or N) of the Nine-Point CIRCLE. It has
Triangle Center Function

The Circle, also called Euler's Circle and the
FEUERBACH CIRCLE, which passes through the feet of
the Perpendicular F a , F b , and F c dropped from the
Vertices of any Triangle AABC on the sides op-
posite them. Euler showed in 1765 that it also passes
through the MIDPOINTS Ma, Mb, M c of the sides of
AABC.

By Feuerbach's Theorem, the nine-point circle also
passes through the Midpoints M H a, Mhb, Mhc of
the segments which join the Vertices and the Ortho-
center H. These three triples of points make nine in
all, giving the circle its name. The center F of the nine-
point circle is called the Nine-Point CENTER.

The Radius of the nine-point circle is i?/2, where R is
the ClRCUMRADIUS. The center of KlEPERT'S HYPER-
BOLA lies on the nine-point circle. The nine-point circle
bisects any line from the ORTHOCENTER to a point on
the ClRCUMCIRCLE. The nine-point circle of the INCEN-
ter and Excenters of a Triangle is the Circumcir-

CLE.

The sum of the powers of the VERTICES with regard to

the nine-point circle is

\(ai* +a 2 2 + a 3 2 ).

a = cos(B - C)

— bc[a b -\- a c + (b

cos A + 2 cos B cos C
c 2 ) 2 ],

Also,

FAi + FA 2 + FA* + FH = 3JT ,

1236

Nine-Point Conic

Niven's Constant

where F is the Nine-Point Center, Ai are the Ver-
tices, H is the Orthocenter, and R is the ClRCUM-
radius. All triangles inscribed in a given Circle and
having the same Orthocenter have the same nine-
point circle.

see also Complete Quadrilateral, Eight-Point
Circle Theorem, Feuerbach's Theorem, Fontene
Theorems, Griffiths' Theorem, Nine-Point Cen-
ter, Nine-Point Conic, Orthocentric System

References

Altshiller-Court, N. College Geometry; A Second Course in
Plane Geometry for Colleges and Normal Schools, 2nd ed.,
rev. enl. New York: Barnes and Noble, pp. 93-97, 1952.

Brand, L. "The Eight-Point Circle and the Nine-Point Cir-
cle." Amer. Math. Monthly 51, 84-85, 1944.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
New York: Random House, pp. 20-22, 1967.

Dorrie, H. "The Feuerbach Circle." §28 in 100 Great Prob-
lems of Elementary Mathematics: Their History and So-
lutions. New York: Dover, pp. 142-144, 1965.

Gardner, M. Mathematical Carnival: A New Round-Up of
Tantalizers and Puzzles from Scientific American. New
York: Vintage Books, p. 59, 1977.

Guggenbuhl, L. "Karl Wilhelm Feuerbach, Mathematician."
Appendix to Circles: A Mathematical View, rev. ed.
Washington, DC: Math. Assoc. Amer., pp. 89-100, 1995.

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle, Boston,
MA: Houghton Mifflin, pp. 165 and 195-212, 1929.

Lange, J. Geschichte des Feuerbach 'schen Kreises. Berlin,
1894.

Mackay, J. S. "History of the Nine-Point Circle," Proc. Ed-
inburgh Math. Soc. 11, 19-61, 1892.

Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 119-120, 1990.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., pp. 1-4, 1995.

Nine-Point Conic

A Conic Section on which the Midpoints of the sides
of any Complete Quadrangle lie. The three diagonal
points also lie on this conic.

see also Complete Quadrangle, Conic Section,
Nine-Point Circle

Nint

see Nearest Integer Function

Nint Zeta Function

Let

Sjv^^Kn 1 ^)]-*,

(1)

where [x] denotes NlNT, the INTEGER closest to x. For
s > 3,

Sa(a) = 2C(s-l) (2)

<? 3 (s) = 3C(*-2) + 4- s C(s) (3)

S 4 (5) = 4C(a-3) + C(*-l). W

Sn(ti) is a Polynomial in 7r whose Coefficients are
Algebraic Numbers whenever n - N is Odd. The
first few values are given explicitly by

ft(4) =

23046

5 5 (6) = — + - + 412

945

170912 + 49928^
25

(5)

(6)

2 7T 4 *- 6 246013 + 353664x72 tt 7

£ fl (7) = tt + — + ^^ +

18 2520

45

2 27'

(?)

References

Borwein, J. M.; Hsu, L. C; Mabry, R.; Neu, K.; Roppert,
J.; Tyler, D. B.; and de Weger, B. M. M. "Nearest Inte-
ger Zeta-Functions." Amer. Math. Monthly 101, 579-580,
1994.

Nirenberg's Conjecture

If the GAUSS Map of a complete minimal surface omits
a Neighborhood of the Sphere, then the surface is a
Plane. This was proven by Osserman (1959). Xavier
(1981) subsequently generalized the result as follows. If
the Gauss Map of a complete Minimal Surface omits
> 7 points, then the surface is a Plane.

see also Gauss Map, Minimal Surface, Neighbor-
hood

References

do Carmo, M. P. Mathematical Models from the Collections
of Universities and Museums (Ed. G. Fischer). Braun-
schweig, Germany: Vieweg, p. 42, 1986.

Osserman, R. "Proof of a Conjecture of Nirenberg." Comm,
Pure Appl. Math. 12, 229-232, 1959.

Xavier, F. "The Gauss Map of a Complete Nonflat Minimal
Surface Cannot Omit 7 Points on the Sphere." Ann. Math.
113, 211-214, 1981.

Niven's Constant

N,B, A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Given a Positive Integer m > 1, let its Prime Fac-
torization be written

m = pi P2 P3 •• *Pk .

(i)

Define the functions h and H by h(l) = 1, H(l) = 1,
and

Then

h(m) = min(ai, a2, . . . , a*)
H(m) = max(ai , ai , . . . , a^ ) -

n

lim — y h(m) = 1

(2)
(3)

(4)

Niven Number

E: =1 Mm)-n_C(f)

Noether's Fundamental Theorem

1237

lim

\/n

C(3)'

(5)

where ((z) is the RlEMANN Zeta Function (Niven
1969). Niven (1969) also proved that

n

lim - V H(m) = C,

n-J-oo n -£— '

(6)

where

C = l +

(Sloane's A033150).

1-

C(i)

= 1.705221 .

(7)

The Continued Fraction of Niven's constant is 1, 1,
2, 2, 1, 1, 4, 1, 1, 3, 4, 4, 8, 4, 1, . . . (Sloane's A033151).
The positions at which the digits 1, 2, . . . first occur in
the Continued Fraction are 1, 3, 10, 7, 47, 41, 34,
13, 140, 252, 20, ... (Sloane's A033152). The sequence
of largest terms in the CONTINUED FRACTION is 1, 2, 4,
8, 11, 14, 29, 372, 559, ... (Sloane's A0033153), which
occur at positions 1, 3, 7, 13, 20, 35, 51, 68, 96, ...
(Sloane's A033154).

References

Finch, S. "Favorite Mathematical Constants." http://www.

mathsoft.com/asolve/constant/niven/niven.html.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

p. 41, 1983.
Niven, I. "Averages of Exponents in Factoring Integers."

Proc. Amer. Math. Soc. 22, 356-360, 1969.
Plouffe, S. "The Niven Constant." http://www.lacim.uqam.

ca/piDATA/niven.txt.

Niven Number

see Harshad Number

Nobbs Points

Given a Triangle AABC, construct the Contact
TRIANGLE ADEF. Then the Nobbs points are the
three points D', E\ and F' from which AABC and
ADEF are PERSPECTIVE, as illustrated above. The
Nobbs points are COLLINEAR and fall along the Ger-
gonne Line.

see also COLLINEAR, CONTACT TRIANGLE, EVANS

Point, Fletcher Point, Gergonne Line, Perspec-
tive Triangles

References

Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.

Noble Number

A noble number is denned as an Irrational Number
which has a CONTINUED FRACTION which becomes an
infinite sequence of Is at some point,

v = [ai,a 2 ,. . .,a n ,l].

The prototype is the GOLDEN RATIO <fi whose CONTIN-
UED Fraction is composed entirely of Is, [1], Any
noble number can written as

_ A n + <Mn-l

V ~ B n + <pB n+1 '

where A k and B k are the NUMERATOR and DENOMI-
NATOR of the fcth Convergent of [a u a 2 , . . - , a n ]. The
noble numbers are a SuBFIELD of Q(y/5).

see also NEAR NOBLE NUMBER

References

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, p. 236, 1979.

Schroeder, M. "Noble and Near Noble Numbers." In Frac-
tals, Chaos, Power Laws: Minutes from an Infinite Par-
adise. New York: W. H. Freeman, pp. 392-394, 1991.

Node (Algebraic Curve)

see Ordinary Double Point

Node (Fixed Point)

A Fixed Point for which the Stability Matrix has
both Eigenvalues of the same sign (i.e., both are Pos-
itive or both are NEGATIVE). If A x < A 2 < 0, then the
node is called STABLE; if Ai > A2 > 0, then the node is
called an UNSTABLE Node.

see also STABLE NODE, UNSTABLE NODE

Node (Graph)

Synonym for the VERTICES of a GRAPH, i.e., the points
connected by EDGES.

see also Acnode, Crunode, Tacnode

Noether's Fundamental Theorem

If two curves <j> and V of MULTIPLICITIES r» ^ and
si ^ have only ordinary points or ordinary singular
points and CUSPS in common, then every curve which
has at least Multiplicity

Ti + Si - 1

1238 Noether-Lasker Theorem

at every point (distinct or infinitely near) can be written

/ = w + i><j>' = o,

where the curves 0' and yj' have Multiplicities at least
n — 1 and Sj — 1.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 29-30, 1959.

Noether-Lasker Theorem

Let M be a finitely generated Module over a commu-
tative NOETHERIAN RING R. Then there exists a finite
set {iV;|l < i < 1} of submodules of M such that

1. n- =1 iVi = and Di^i Ni is not contained in Ni Q for
all l<i <L

2. Each quotient M/Ni is primary for some prime Pi.

3. The Pi are all distinct for 1 < i < I.

4. Uniqueness of the primary component Ni is equiva-
lent to the statement that Pi does not contain Pj for
any j ^ i.

Noether's Transformation Theorem

Any irreducible curve may be carried by a factorable
Cremona Transformation into one with none but
ordinary singular points.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 207, 1959.

Noetherian Module

A Module M is Noetherian if every submodule is
finitely generated.

see also Noetherian Ring

Noetherian Ring

An abstract commutative Ring satisfying the abstract
chain condition.

see also Local Ring, Noether-Lasker Theorem

Noise

An error which is superimposed on top of a true sig-
nal. Noise may be random or systematic. Noise can be
greatly reduced by transmitting signals digitally instead
of in analog form because each piece of information is
allowed only discrete values which are spaced farther
apart than the contribution due to noise.

Coding Theory studies how to encode information ef-
ficiently, and Error- Correcting Codes devise meth-
ods for transmitting and reconstructing information in
the presence of noise.

see also Error

Nome

References

Davenport, W. B. and Root, W. L. An Introduction to the

Theory of Random Signals and Noise. New York: IEEE

Press, 1987.
McDonough, R. N. and Whalen, A. D. Detection of Signals

in Noise, 2nd ed. Orlando, FL: Academic Press, 1995.
Pierce, J. R. Symbols, Signals and Noise: The Nature and

Process of Communication. New York: Harper & Row,

1961.
Vainshtein, L. A. and Zubakov, V. D. Extraction of Signals

from Noise. New York: Dover, 1970.
van der Ziel, A. Noise: Sources, Characterization, Measure-
ment. New York: Prentice-Hall, 1954.
van der Ziel, A. Noise in Measurement. New York: Wiley,

1976.
Wax, N. Selected Papers on Noise and Stochastic Processes.

New York: Dover, 1954.

Noise Sphere

A mapping of Random Number Triples to points in
Spherical Coordinates,

= 27rX n
(j) — 7rX n+ i
r

^Xn

+2*

The graphical result can yield unexpected structure
which indicates correlations between triples and there-
fore that the numbers are not truly RANDOM.

References

Pickover, C. A. Computers and the Imagination. New York:
St. Martin's Press, 1991.

Pickover, C. A. "Computers, Randomness, Mind, and In-
finity." Ch. 31 in Keys to Infinity. New York: W. H.
Freeman, pp. 233-247, 1995.

Richards, T. "Graphical Representation of Pseudorandom
Sequences." Computers and Graphics 13, 261-262, 1989.

Nolid

An assemblage of faces forming a Polyhedron of zero
Volume (Holden 1991, p. 124).

see also ACOPTIC POLYHEDRON

References

Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.

Nome

Given a Theta Function, the nome is defined as

q(m) = e* Ti = e _1fK(1 ' m)/Jf(m) = e -**'( m )/*< m ) m

where K(k) is the complete ELLIPTIC INTEGRAL OF THE
First Kind, and m is the Parameter.

di{z,q)=<&{z\T)
&i =&(0,q).

(2)
(3)

Nomogram

Nonassociative Product

1239

Solving the nome for the Parameter m gives

m{q)

tf3 4 (0,<7)'

(4)

where &i(z,q) is a THETA FUNCTION.

see also AMPLITUDE, CHARACTERISTIC (ELLIPTIC IN-
TEGRAL), Elliptic Integral, Modular Angle,
Modulus (Elliptic Integral), Parameter

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 591, 1972.

Nomogram

A graphical plot which can be used for solving certain
types of equations.

References

lyanaga, S. and Kawada, Y. (Eds.). "Nomograms." §282

in Encyclopedic Dictionary of Mathematics. Cambridge,

MA: MIT Press, pp. 891-893, 1980.
Menzel, D. (Ed.). Fundamental Formulas of Physics, Vol. 1.

New York: Dover, p. 141, 1960.

Nonagon

The unconstructible regular POLYGON with nine sides
and SCHLAFLI SYMBOL {9}. It is sometimes called an
Enneagon.

Although the regular nonagon is not a CONSTRUCTIBLE
POLYGON, Dixon (1991) gives several close approxi-
mations. While the Angle subtended by a side is
360° /9 = 40°, Dixon gives constructions containing an-
gles of tan _1 (5/6) « 39.8805571° and 2tan" 1 ((v / 3-
l)/2)« 40.207818°.

Madachy (1979) illustrates how to construct a nonagon
by folding and knotting a strip of paper.

see also NONAGRAM, TRIGONOMETRY VALUES — 7r/9

References

Dixon, R. Mathographics. New York: Dover, pp. 40-44, 1991.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 60-61, 1979.

Nonagonal Number

A Figurate Number of the form n(7n - 5)/2, also
called an Enneagonal Number. The first few are 1,
9, 24, 46, 75, 111, 154, 204, . . . (Sloane's A001106).

References

Sloane, N. J. A. Sequence A001106/M4604 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Nonagram

A Star Polygon composed of three Equilateral
Triangles rotated at angles 0°, 40°, and 80°. It has
been called the Star of Goliath by analogy with the
Star of David (Hexagram).

see also Hexagram, Nonagon, Trigonometry

Values — 7r/9

Nonassociative Algebra

An Algebra which does not satisfy

a(bc) = (ab)c

is called a nonassociative algebra. Bott and Milnor
(1958) proved that the only nonassociative DIVISION
ALGEBRAS are for n = 1, 2, 4, and 8. Each gives rise to
an ALGEBRA with particularly useful physical applica-
tions (which, however, is not itself necessarily nonassoc-
iative), and these four cases correspond to REAL NUM-
BERS, Complex Numbers, Quaternions, and Cay-
ley Numbers, respectively.

see also Algebra, Cayley Number, Complex Num-
ber, Division Algebra, Quaternion, Real Num-
ber

References

Bott, R. and Milnor, J. "On the Parallelizability of the
Spheres." Bull. Amer, Math. Soc. 64, 87-89, 1958.

Nonassociative Product

The number of nonassociative n-products with k ele-
ments preceding the rightmost left parameter is

n + k - l\

k-i y

F(n, k) = F(n - 1, k) + F(n - 1, k - 1)
f n + k - 2^
k

where (™) is a Binomial Coefficient. The number of
rz-products in a nonassociative algebra is

3=0

n\(n~ 1)!

1240 Nonaveraging Sequence

Noncototient

References

Niven, I. M. Mathematics of Choice: Or, How to Count
Without Counting. Washington, DC: Math. Assoc. Amcr.,
pp. 140-152, 1965.

Nonaveraging Sequence

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

An infinite sequence of POSITIVE INTEGERS

1 < ai < a2 < G&3 < . . .

is a nonaveraging sequence if it contains no three terms
which are in an ARITHMETIC PROGRESSION, so that

<H + aj ^ 2ak

for all distinct a», a/, a k . Wroblewski (1984) showed
that

oo

S(A) = sup Y^ — > 3 - 00849 *

all nonaveraging a k

sequences k—1

References

Behrend, F. "On Sets of Integers which Contain no Three
Terms in an Arithmetic Progression." Proc. Nat Acad.
Sci. USA 32, 331-332, 1946.

Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/erdos/erdos.html.

Gerver, J. L. "The Sum of the Reciprocals of a Set of Integers
with No Arithmetic Progression of k Terms." Proc. Amer.
Math. Soc. 62, 211-214, 1977.

Gerver, J. L. and Ramsey, L. "Sets of Integers with no Long
Arithmetic Progressions Generated by the Greedy Algo-
rithm." Math. Comput. 33, 1353-1360, 1979.

Guy, R. K. "Nonaveraging Sets. Nondividing Sets." §C16 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 131-132, 1994.

Wroblewski, J. "A Nonaveraging Set of Integers with a Large
Sum of Reciprocals." Math. Comput. 43, 261-262, 1984.

Noncentral Distribution

see Chi-Squared Distribution, F-Distribution,
Student's ^-Distribution

Noncommutative Group

A group whose elements do not commute. The simplest
noncommutative GROUP is the DIHEDRAL GROUP D 3
of Order six.

see also COMMUTATIVE, FINITE GROUP — D z

Nonconfbrmal Mapping

Let 7 be a path in C, w = f(z), and and <f> be the
tangents to the curves 7 and /(7) at z$ and wq. If there
is an N such that

f (n \zo) =

(1)
(2)

for all n < N (or, equivalently, if f'(z) has a zero of
order N — 1), then

f(z) = f(zo) +

f (N) (zo)

(Z - Zq)

N

+ (N+l)! {Z ~ Z0) + "- (3)

f(z) - f(zo) = (Z~ZQ

so the Argument is

,JV

f (*)(*>)

m

f {N+1) (zo)

(N + iy.

(z - z ) + ■

(4)

axg[/(z) - f(z )} = Navg(z - z Q ) + arg
/ (JV+1) (*o)

f(N)(z )

+ -

(z- Zq) + ..

(5)

(AT + 1)!
As z — > zq, arg(z-zo) ->■ and | arg[/(z) - f(zo)]\ -t <f>-

(j> = N6 + arg

/(*)(*>)

TV!

= N6 + aTg[f(N)(zo)]. (6)

see also Conformal Transformation

Nonconstructive Proof

A PROOF which indirectly shows a mathematical object
exists without providing a specific example or algorithm
for producing an example.
see also PROOF

References

Courant, R. and Robbins, H. "The Indirect Method of
Proof." §2.4.4 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 86-87, 1996.

Noncototient

A Positive value of n for which x — <f>(x) — n has no
solution, where <f>(x) is the Totient FUNCTION. The
first few are 10, 26, 34, 50, 52, . . . (Sloane's A005278).
see also Nontotient, Totient Function

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Verlag, p. 91, 1994.
Sloane, N. J. A. Sequence A005278/M4688 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Noncylindrical Ruled Surface

Nonlinear Least Squares Fitting 1241

Noncylindrical Ruled Surface

A Ruled Surface parameterization x(u,v) = h(u) +
v g(u) is called noncylindrical if g x g' is nowhere 0. A
noncylindrical ruled surface always has a parameteriza-
tion of the form

x(tt, v) — <t(u) + vS(u) 7

where \S\ = 1 and cr' * 5' = 0, where cr is called the
Striction Curve of x and S the Director Curve.

see also Distribution Parameter, Ruled Surface,
Striction Curve

References

Gray, A. "Noncylindrical Ruled Surfaces." §17.3 in Modern
Differential Geometry of Curves and Surfaces. Boca Ra-
ton, FL: CRC Press, pp. 345-349, 1993.

Nondecreasing Function

A function f(x) is said to be nondecreasing on an IN-
TERVAL / if f(b) > f(a) for all b > a, where a,b € I.
Conversely, a function f(x) is said to be nonincreasing
on an Interval i" if /(&) < f(a) for all b > a with
a,& e I.

see also Decreasing Function, Nonincreasing
Function

Carslaw, H. S. The Elements of Non- Euclidean Plane Geom-
etry and Trigonometry. London: Longmans, 1916.

Coxeter, H. S. M. Non-Euclidean Geometry, 5th ed. Toronto:
University of Toronto Press, 1965.

Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 53-60, 1990.

Iversen, B. An Invitation to Hyperbolic Geometry. Cam-
bridge, England: Cambridge University Press, 1993.

Iyanaga, S. and Kawada, Y. (Eds.). "Non-Euclidean Geom-
etry." §283 in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 893-896, 1980.

Martin, G. E. The Foundations of Geometry and the Non-
Euclidean Plane. New York: Springer- Verlag, 1975.

Pappas, T. "A Non-Euclidean World." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, pp. 90-
92, 1989.

Ramsay, A. and Richtmeyer, R. D. Introduction to Hyperbolic
Geometry. New York: Springer- Verlag, 1995.

Sommerville, D. Y. The Elements of Non-Euclidean Geome-
try. London: Bell, 1914.

Sommerville, D. Y. Bibliography of Non-Euclidean Geome-
try, 2nd ed. New York: Chelsea, 1960.

Sved, M. Journey into Geometries. Washington, DC: Math.
Assoc. Amer., 1991.

Trudeau, R. J. The Non-Euclidean Revolution. Boston, MA:
Birkhauser, 1987.

Nonillion

In the American system, 10 30 .

see also LARGE Number

Nondividing Set

A Set in which no element divides the SUM of any other.

References

Guy, R. K. "Nonaveraging Sets. Nondividing Sets." §C16 in

Unsolved Problems in Number Theory, 2nd ed. New York:

Springer- Verlag, pp. 131-132, 1994.

Nonessential Singularity

see Regular Singular Point

Non-Euclidean Geometry

In 3 dimensions, there are three classes of constant cur-
vature Geometries. All are based on the first four
of Euclid's Postulates, but each uses its own ver-
sion of the Parallel Postulate. The "flat" geom-
etry of everyday intuition is called Euclidean Ge-
ometry (or Parabolic Geometry), and the non-
Euclidean geometries are called HYPERBOLIC GEOM-
ETRY (or Lobachevsky-Bolyai-Gauss Geometry)
and Elliptic Geometry (or Riemannian Geome-
try). It was not until 1868 that Beltrami proved that
non-Euclidean geometries were as logically consistent as
Euclidean Geometry.

see also Absolute Geometry, Elliptic Geometry,
Euclid's Postulates, Euclidean Geometry, Hy-
perbolic Geometry, Parallel Postulate

References

Borsuk, K. Foundations of Geometry: Euclidean and Bolyai-
Lobachevskian Geometry. Projective Geometry. Amster-
dam, Netherlands: North- Holland, 1960.

Nonincreasing Function

A function f(x) is said to be nonincreasing on an IN-
TERVAL / if f(b) < f(a) for all b > a, where a, 6 e I.
Conversely, a function f(x) is said to be nondecreasing
on an INTERVAL I if /(&) > f(a) for all b > a with
a,b e I.
see also INCREASING FUNCTION, NONDECREASING

Function

Nonlinear Least Squares Fitting

Given a function f(x) of a variable x tabulated at m val-
ues yi = f(xi), . . . , y m — /(x m ), assume the function
is of known analytic form depending on n parameters
f(x; Ai, . . . , A n ), and consider the overdetermined set of
m equations

2/i = f(xi; Ai, A 2 , . . . , A n )
y m = f(xm\ Ai, A2,. . .,A n ).

(i)

(2)

We desire to solve these equations to obtain the values
Ai, . . . , A n which best satisfy this system of equations.
Pick an initial guess for the Aj and then define

dpi = yi- f(xi] Ai, . . . , A n ).

(3)

Now obtain a linearized estimate for the changes d\i
needed to reduce dpi to 0,

d& = j2

j=i

(4)

1242 Nonlinear Least Squares Fitting

Nonnegative Integer

for i = 1, . . . , n. This can be written in component form
as

d0i = A ij dX u (5)

where A is the m x n Matrix

r 1L\ Q f i

d*2 1x2, A 9A 2 lx 2 , A

_ ^1 lx m ,A dA n lx m ,A

(6)

In more concise MATRIX form,

d/3 = AdA,

(7)

where d/3 and dX are m- VECTORS. Applying the MA-
TRIX Transpose of A to both sides gives

Defining

A T d/3=(A T A)dA.

a = A T A
b = A T d/3

(8)

(9)
(10)

in terms of the known quantities A and d/3 then gives
the Matrix Equation

adX ■

(ii)

which can be solved for dX using standard matrix tech-
niques such as Gaussian Elimination. This offset is
then applied to A and a new d/3 is calculated. By iter-
atively applying this procedure until the elements of dX
become smaller than some prescribed limit, a solution
is obtained. Note that the procedure may not converge
very well for some functions and also that convergence is
often greatly improved by picking initial values close to
the best-fit value. The sum of square residuals is given
by R 2 = d/3* d/3 after the final iteration.

An example of a nonlinear least squares fit to a noisy

Gaussian Function

is shown above, where the thin solid curve is the initial
guess, the dotted curves are intermediate iterations, and
the heavy solid curve is the fit to which the solution con-
verges. The actual parameters are (A, #o, cr) = (1, 20, 5),
the initial guess was (0.8, 15, 4), and the converged val-
ues are (1.03105, 20.1369, 4.86022), with R 2 = 0.148461.
The Partial Derivatives used to construct the matrix
A are

d f _ -(*-x ) 2 /(2<r 2 )

dA~ €

df _ A{x - go) _(x-*o) 2 /(2<r 2 )

&Xo (T 2

df_ _ A(x~Xp) 2 -(a-so) a /(2q 2 )

da " <r*

(13)
(14)
(15)

The technique could obviously be generalized to multiple
Gaussians, to include slopes, etc., although the conver-
gence properties generally worsen as the number of free
parameters is increased.

An analogous technique can be used to solve an overde-
termined set of equations. This problem might, for ex-
ample, arise when solving for the best-fit Euler AN-
GLES corresponding to a noisy ROTATION Matrix, in
which case there are three unknown angles, but nine
correlated matrix elements. In such a case, write the
n different functions as /i(Ai, . . . , A n ) for i = 1, . . . , n,
call their actual values yi, and define

A =

0^2 \\ i

... Mi. 1

dXn IXi

3Al \\i

d *2 \Xi

dfm I

and

d/3 = y- /i(Ai,...,A„),

(16)

(17)

where Xi are the numerical values obtained after the ith
iteration. Again, set up the equations as

AdX = d/3,

(18)

and proceed exactly as before.

see also Least Squares Fitting, Linear Regres-
sion, Moore-Penrose Generalized Matrix In-
verse

Nonnegative

A quantity which is either (Zero) or POSITIVE, i.e.,

>0.

see also Negative, Nonnegative Integer, Nonpos-

itive, Nonzero, Positive, Zero

Nonnegative Integer

see If

f(A,x ,<T- 1 x) = Ae-<*-*° )2/{2 ° 2)

(12)

Nonnegative Partial Sum

Nonstandard Analysis 1243

Nonnegative Partial Sum

The number of sequences with NONNEGATIVE partial
sums which can be formed from n Is and n — Is (Bailey
1996, Buraldi 1992) is given by the Catalan Numbers.
Bailey (1996) gives the number of NONNEGATIVE partial
sums of n Is and k —Is ai, a2, . . . , a n +fc, so that

ai + a 2 + ... + ai > (1)

for all 1 < i < n + k. The closed form expression is

for n > 0,

for n > 1, and

1.}.^

(2)

(3)

-fc)(n + 2)(n + 3)---(n + fc)

fc!

. (4)

for n > k > 2. Setting k — n then recovers the Catalan

Numbers

*-{«"}-;M*> (5)

see also Catalan Number

References

Bailey, D. F. "Counting Arrangements of l's and — l's."

Math. Mag. 69, 128-131, 1996.
Buraldi, R. A. Introductory Combinatorics, 2nd ed. New

York: Elsevier, 1992.

Nonorientable Surface

A surface such as the MOBIUS Strip on which there ex-
ists a closed path such that the directrix is reversed when
moved around this path. The Euler Characteristic
of a nonorientable surface is < 0. The real PROJEC-
TIVE Plane is also a nonorientable surface, as are the
Boy Surface, Cross-Cap, and Roman Surface, all
of which are homeomorphic to the REAL PROJECTIVE
PLANE (Pinkall 1986). There is a general method for
constructing nonorientable surfaces which proceeds as
follows (Banchoff 1984, Pinkall 1986). Choose three HO-
MOGENEOUS Polynomials of Positive Even degree
and consider the MAP

f = {fi(x,y,z)J 2 (x ) y,z),f 3 (x,y,z)) :!

\ (1)

Then restricting x, y, and z to the surface of a sphere
by writing

x — cos 6 sin <j>
y = sin sin <j>

Z — COS(j>

(2)
(3)
(4)

and restricting to [0, 2tt) and <p to [0,7r/2] defines a
map of the Real Projective Plane to M 3 ,

In 3-D, there is no unbounded nonorientable surface
which does not intersect itself (Kuiper 1961, Pinkall

1986).

see also BOY SURFACE, CROSS-CAP, MOBIUS STRIP,

Orientable Surface, Projective Plane, Roman

Surface

References

Banchoff, T. "Differential Geometry and Computer Graph-
ics." In Perspectives of Mathematics: Anniversary of
Oberwolfach (Ed. W. Jager, R. Remmert, and J. Moser).
Basel, Switzerland: Birkhauser, 1984.

Gray, A. "Nonorientable Surfaces." Ch. 12 in Modern Dif-
ferential Geometry of Curves and Surfaces. Boca Raton,
FL: CRC Press, pp. 229-249, 1993.

Kuiper, N, H. "Convex Immersion of Closed Surfaces in E s ."
Comment. Math. Helv. 35, 85-92, 1961.

Pinkall, U. "Models of the Real Projective Plane." Ch. 6 in
Mathematical Models from the Collections of Universities
and Museums (Ed. G. Fischer). Braunschweig, Germany:
Vieweg, pp. 63-67, 1986.

Nonpositive

A quantity which is either (Zero) or NEGATIVE, i.e.,
<0.

see also Negative, Nonnegative, Nonzero, Posi-
tive, Zero

Nonsquarefree

see SQUAREFUL

Nonstandard Analysis

Nonstandard analysis is a branch of mathematical
LOGIC which weakens the axioms of usual Analysis to
include only the first-order ones. It also introduces Hy-
PERREAL Numbers to allow for the existence of "gen-
uine Infinitesimals," numbers which are less than 1/2,
1/3, 1/4, 1/5, . . . , but greater than 0. Abraham Robin-
son developed nonstandard analysis in the 1960s. The
theory has since been investigated for its own sake and
has been applied in areas such as BANACH SPACES, dif-
ferential equations, probability theory, microeconomic
theory, and mathematical physics (Apps).

see also Ax-KOCHEN ISOMORPHISM THEOREM, LOGIC,

Model Theory

References

Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; and Lind-
str0om, T. Nonstandard Methods in Stochastic Analysis
and Mathematical Physics. New York: Academic Press,
1986.

Anderson, R. "Nonstandard Analysis with Applications to
Economics." In Handbook of Mathematical Economics,
Vol. 4. New York: Elsevier, 1991.

Apps, P. "What is Nonstandard Analysis?" http://www.
math.wisc.edu/-apps/nonstandard.html.

Dauben, J. W. Abraham Robinson: The Creation of Non-
standard Analysis, A Personal and Mathematical Odyssey.
Princeton, NJ: Princeton University Press, 1998.

1244

Nontotient

Normal (Algebraically)

Davis, P. J. and Hersch, R. The Mathematical Experience.
Boston: Birkhauser, 1981.

Keisler, H. J. Elementary Calculus: An Infinitesimal Ap-
proach. Boston: PWS, 1986.

Lindstr0om, T. "An Invitation to Nonstandard Analysis." In
Nonstandard Analysis and Its Applications (Ed. N. Cut-
land). New York: Cambridge University Press, 1988.

Robinson, A. Non-Standard Analysis. Princeton, NJ: Prince-
ton University Press, 1996.

Stewart, I. "Non-Standard Analysis." In From Here to Infin-
ity: A Guide to Today's Mathematics. Oxford, England:
Oxford University Press, pp. 80-81, 1996.

Nontotient

A Positive Even value of n for which <j)(x) = n, where
4>(x) is the TOTIENT FUNCTION, has no solution. The
first few are 14, 26, 34, 38, 50, . . . (Sloane's A005277).

see also NONCOTOTIENT, TOTIENT FUNCTION

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Verlag, p. 91, 1994.
Sloane, N. J. A. Sequence A005277/M4927 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Nonwandering

A point x in a MANIFOLD M is said to be nonwandering
if, for every open NEIGHBORHOOD U of x, it is true that
<j)~ n U U U ^ for a Map <j> for some n > 0. In other
words, every point close to x has some iterate under <\>
which is also close to x. The set of all nonwandering
points is denoted 0(0), which is known as the nonwan-
dering set of <f).

see also Anosov Diffeomorphism, Axiom A Diffeo-

MORPHISM, SMALE HORSESHOE MAP

Nonzero

A quantity which does not equal ZERO is said to be
nonzero. A REAL nonzero number must be either POS-
ITIVE or Negative, and a Complex nonzero number
can have either Real or IMAGINARY Part nonzero.

see also Negative, Nonnegative, Nonpositive,
Positive, Zero

Nordstrand's Weird Surface

An attractive Cubic SURFACE defined by Nordstrand.
It is given by the implicit equation

25[x\y + z)+ y 3 (x + z) + z 3 (x + y)] + 5Q(x 2 y 2 + x 2 z 2
+y z 2 ) — 125(£ yz + y xz + z 2 xy) + 60xyz

—4:(xy + xz 4- yz) = 0.

References

Nordstrand, T. "Weird Cube." http://www.uib.no/people/
nf ytn/weirdtxt .htm.

Norm

Given a n-D VECTOR

xi

X 2

a Vector Norm ||x|| is a Nonnegative number sat-
isfying

1. \\x\\ > when x ^ and ||x|| = Iff x = 0,

2. ||&x|| = \k\ ||x|| for any SCALAR ft,

3. ||x + y||<||x|| + ||y||.

The most common norm is the vector L2-N0RM, defined
by

||X|| 2 = |X| = ^l 2 +Z 2 2 + ...+Zn 2 .

Given a Square Matrix A, a Matrix Norm ||A|| is
a NONNEGATIVE number associated with A having the
properties

1. ||A|| > when A ^ and ||A|| = Iff A = 0,

2. ||ftA|| = |ft| ||A|| for any SCALAR ft,

3. ||A + B||<||A|| + ||B||,

4. ||AB|| < ||A||||B||.

see also Bombieri Norm, Compatible, Euclidean
Norm, Hilbert-Schmidt Norm, Induced Norm, Li-
Norm, L2-N0RM, Loo-Norm, Matrix Norm, Maxi-
mum Absolute Column Sum Norm, Maximum Ab-
solute Row Sum Norm, Natural Norm, Nor-
malized Vector, Normed Space, Parallelogram
Law, Polynomial Norm, Spectral Norm, Subor-
dinate Norm, Vector Norm

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1114-1125, 1979.

Norm Theorem

If a PRIME number divides a norm but not the bases of
the norm, it is itself a norm.

Normal

see Normal Curve, Normal Distribution, Nor-
mal Distribution Function, Normal Equation,
Normal Form, Normal Group, Normal Magic
Square, Normal Matrix, Normal Number, Nor-
mal Plane, Normal Subgroup, Normal Vector

Normal (Algebraically)

see GALOISIAN

Normal Curvature

Normal Distribution Function

1245

Normal Curvature

Let Up be a unit Tangent Vector of a Regular Sur-
face Mel 3 , Then the normal curvature of M in the
direction u D is

re(up) = S(u p )*Up,

(1)

where S is the SHAPE Operator. Let M C R be a
Regular Surface, p g M, x be an injective Regular
Patch of M with p = x(tio,^o), and

v p = ax u (n ,fo) + &x v (uo,^o),

(2)

where v p G M p . Then the normal curvature in the
direction v p is

( . __ ea 2 + 2 fab + gb 2
K[VP) ~ Ea* + 2Fab+Gb*>

(3)

where E, F, and G are first Fundamental Forms and
e, /, and g second Fundamental Forms.

The Maximum and Minimum values of the normal cur-
vature on a Regular Surface at a point on the surface
are called the Principal Curvatures K\ and kj.

see also Curvature, Fundamental Forms, Gaus-
sian Curvature, Mean Curvature, Principal Cur-
vatures, Shape Operator, Tangent Vector

References

Euler, L. "Recherches sur la coubure des surfaces." Mem. de

I'Acad. des Sciences, Berlin 16, 119-143, 1760.
Gray, A. "Normal Curvature." §14.2 in Modern Differential

Geometry of Curves and Surfaces. Boca Raton, FL: CRC

Press, pp. 270-273 and 277, 1993.
Meusnier, J. B. "Memoire sur la courbure des surfaces."

Mem. des savans etrangers 10 (lu 1776), 477-510, 1785.

Normal Curve

see Gaussian Distribution

Normal Developable

A Ruled Surface M is a normal developable of a curve
y if M can be parameterized by x(w,v) = y(u)+vN(u),
where N is the NORMAL VECTOR.

see also Binormal Developable, Tangent Devel-
opable

References

Gray, A. "Developables." §17.6 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 352-354, 1993.

Normal Distribution

Another name for a Gaussian Distribution. Given a
normal distribution in a Variate x with Mean fi and
Variance <t 2 ,

cr\/2rr

the so-called "Standard Normal Distribution" is
given by taking \i — and a 2 — 1. An arbitrary normal
distribution can be converted to a STANDARD NORMAL
Distribution by changing variables to z = (x - m)/<j,
so dz — dx/a^ yielding

P(x) dx =

2tt

" 2 ' 2 dz.

The Fisher-Behrens Problem is the determination
of a test for the equality of MEANS for two normal dis-
tributions with different VARIANCES.

see also Fisher-Behrens Problem, Gaussian Dis-
tribution, Half-Normal Distribution, Kolmogo-
rov-Smirnov Test, Normal Distribution Func-
tion, Standard Normal Distribution, Tetra-
choric Function

Normal Distribution Function

. 5h

0.4

0.3

0.2

0.1-

0.5 1 1.5 2 2.5 3

A normalized form of the cumulative Gaussian Distri-
bution function giving the probability that a variate
assumes a value in the range [0,cc],

$(x) = Q(x)

dt.

It is related to the Probability Integral

a(x)

- -7- f <-'

/2

dt

by

$(x) = \ol{x).
Let u ~ t/y/2 so du = dt/y/2. Then

r/V2

$(x)

= — — I e u du = | erf [ —= ) .

V^ Jo \y/2J

(1)

(2)
(3)

(4)

1246

Normal Distribution Function

Normal Form

Here, ERF is a function sometimes called the error func-
tion. The probability that a normal variate assumes a
value in the range [zi,^] is therefore given by

$(xux 2 ) = -

-(3)-- (3)

(5)

Neither <&(z) nor Erf can be expressed in terms of fi-
nite additions, subtractions, multiplications, and root
extractions, and so must be either computed numeri-
cally or otherwise approximated.

Note that a function different from <&(x) is sometimes
denned as "the" normal distribution function

$'(x)

1+erf

(*)]-»

+ *(x) (6)

(Beyer 1987, p. 551), although this function is less
widely encountered than the usual $(x).

The value of a for which P(x) falls within the interval
[—a, a] with a given probability P is a related quantity
called the Confidence Interval.

For small values x <C 1, a good approximation to $(#)
is obtained from the MACLAURIN SERIES for ERF,

*(*) = ^= (2« - l^ 3 + 5,*° ~ its*' + ■ ■ ■)■ (7)

For large values x 3> 1, a good approximation is ob-
tained from the asymptotic series for ERF,

$(x)

1 e /a

2 v^

(a

# + 3x

-15x~' + 105a:" 9 + ...). (8)

The value of $(x) for intermediate x can be computed
using the CONTINUED FRACTION identity

Jo

du ~

v^

-. (9)

x +

2x +

2x +

x + ...

A simple approximation of $(x) which is good to two
decimal places is given by

( 0.1x(4.4 -x) for < x < 2.2
$i(x) w <^ 0.49 for 2.2 < z < 2.6 (10)

I 0.50 for x > 2.6.

Abramowitz and Stegun (1972) and Johnson and Kotz
(1970) give other functional approximations. An ap-
proximation due to Bagby (1995) is

-xV2

* a (a0 = i{i-&[7<r

+ 16e -x=(2-V2) +(7+ i 7rx 2 )e -

]}

1/2

The plots below show the differences between <i> and the
two approximations.

The first QuARTILE of a standard NORMAL DISTRIBU-
TION occurs when

/'

Jo

3>(z)dz

(12)

(U)

The solution is t = 0.6745 The value of t giving \

is known as the PROBABLE ERROR of a normally dis-
tributed variate.

see also CONFIDENCE INTERVAL, ERF, ERFC, FlSHER-

Behrens Problem, Gaussian Distribution, Gaus-
sian Integral, Hh Function, Normal Distribu-
tion, Probability Integral, Tetrachoric Func-
tion

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook

of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, 9th printing. New York: Dover,

pp. 931-933, 1972.
Bagby, R. J. "Calculating Normal Probabilities." Amer.

Math. Monthly 102, 46-49, 1995.
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables,

28th ed. Boca Raton, FL: CRC Press, 1987.
Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous

Univariate Distributions, Vol. 1, 2nd ed. Boston, MA:

Houghton Mifflin, 1994.

Normal Equation

Given an overdetermined Matrix Equation

Ax = b,

the normal equation is that which minimizes the sum of
the square differences between left and right sides

A T Ax = A T b.

see also Least Squares Fitting, Moore-Penrose
Generalized Matrix Inverse, Nonlinear Least
Squares Fitting

Normal Form

A way of representing objects so that, although each
may have many different names, every possible name
corresponds to exactly one object.

see also CANONICAL FORM

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 7, 1996.

Normal Function

Normal Vector

1247

Normal Function

A SQUARE INTEGRABLE function <f> is said to be normal

if

/

<f>dt = l

However, the Normal Distribution Function is also
sometimes called "the normal function."

see also Normal Distribution Function, Square

INTEGRABLE

References

Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, p. 6, 1991.

Normal Group

see Normal Subgroup

Normal Magic Square

see Magic Square

Normal Matrix

A normal matrix A is a MATRIX for which

[A.A^O,

where [a, 6] is the COMMUTATOR and * denotes the AD-
JOINT Operator.

Normal Number

An Irrational Number for which any Finite pattern
of numbers occurs with the expected limiting frequency
in the expansion in any base. It is not known if -k or e arc
normal. Tests of y/n for n = 2, 3, 5, 6, 7, 8, 10, 11, 12,
13, 14, 15 indicate that these SQUARE ROOTS may be
normal. The only numbers known to be normal are ar-
tificially constructed ones such as the Champernowne
Constant and the Copeland-Erdos Constant.
see also Champernowne Constant, Copeland-
Erdos Constant, e, Pi

Normal Order

/(n) has the normal order F(n) if f(n) is approximately
F(n) for Almost All values of n. More precisely, if

(l-e)F(n) </(n)< (1 + e)F(n)

for every positive e and Almost All values of n, then
the normal order of f(n) is F(n).

see also Almost All

References

Hardy, G. H. and Weight, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, p. 356, 1979.

Normal to a Plane

see Normal Vector

Normal Section

Let M C I 3 be a REGULAR SURFACE and u p a unit
Tangent Vector to M, and let n(u p ,N(p)) be the
Plane determined by u p and the normal to the surface
N(p). Then the normal section of M is defined as the
intersection of n(u p ,N(p)) and M.

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 271, 1993.

Normal Subgroup

Let H be a SUBGROUP of a GROUP G. Then if is a
normal subgroup of G, written H < G, if

xHx~ l = H

for every element x in H. Normal subgroups are also
known as INVARIANT SUBGROUPS.

see also GROUP, SUBGROUP

Normal Vector

The normal to a PLANE specified by

f(x, y,z)=ax + by + cz + d =

is given by

N = V/;

(1)

(2)

The normal vector at a point (xo,yo) on a surface z
/(«jy)is

~ fx(xo,yo)
N= f y (x ,yo) . (3)

-1

In the PLANE, the unit normal vector is defined by

dT

N =

#'

(4)

where T is the unit TANGENT VECTOR and <f> is the
polar angle. Given a unit TANGENT VECTOR

T = uii + u 2 y

with \Ati 2 + U2 2 = 1, the normal is
N = —U2'x. + u\y.

(5)

(6)

For a function given parametrically by (f(t),g(t)), the
normal vector relative to the point (f{t),g(t)) is there-
fore given by

Normal Plane

The Plane spanned by N and B (the Normal Vector
and Binormal Vector).

see also BINORMAL VECTOR, NORMAL VECTOR, PLANE

X(t)

y(t)

/'

V/^+ff' 2

(7)
(8)

1248

Normalized Vector

Nother

To actually place the vector normal to the curve, it must
be displaced by (/(£),#(£)).

In 3-D Space, the unit normal is

dT dT

T\T = <Js dt

~ I dt I ~ I dT

I da I I dt

ldT

k ds '

(9)

where k is the CURVATURE. Given a 3-D surface
F(x,y,z)=0,

A= F x + F y + F z
^/F x 2 + F v 2 +F z 2 '

If the surface is defined parametrically in the form

x = x(4> } ip)

z = z(<p,ip),
define the Vectors

(10)

(11)

(12)
(13)

X<f>

a =

y<t>

X<tp

b =

Then the unit normal vector is

1ST- aXb

(14)

(15)

, (16)

V|a| 2 |bP-|a-bp

Let g be the discriminant of the Metric Tensor. Then
ri x r 2

N =

V9

: tijT 3 •

(17)

see also Binormal Vector, Curvature, Frenet
Formulas, Tangent Vector

References

Gray, A. "Tangent and Normal Lines to Plane Curves." §5.5

in Modern Differential Geometry of Curves and Surfaces.

Boca Raton, FL: CRC Press, pp. 85-90, 1993.

Normalized Vector

The normalized vector of X is a VECTOR in the same
direction but with NORM (length) 1. It is denoted X
and given by

x -|x|'

where |X| is the NORM of X. It is also called a Unit
Vector.

see also Unit Vector

Normalizer

A set of elements g of a GROUP such that

g-'Hg = H,

is said to be the normalizer Ng(H) with respect to a
subset of group elements H.

see also Centralizer, Tightly Embedded

Normed Space

A Vector Space possessing a Norm.

Nosarzewska's Inequality

Given a convex PLANE region with AREA A and PERI-
METER p,

A-\ V <N < A+|p + l,

where N is the number of enclosed LATTICE POINTS
(Nosarzewska 1948). This improves on Jarnick's IN-
EQUALITY

\N-A\ <p.

see also JARNICK'S INEQUALITY, LATTICE POINT

References

Nosarzewska, M. "Evaluation de la difference entre l'aire
d'une region plane convexe et le nombre des points aux
coordonnees entieres couverts par elle." Colloq. Math. 1,
305-311, 1948.

Not

An operation in LOGIC which converts TRUE to FALSE
and False to True. NOT A is denoted \A or ->A
A ^A

F
T

T
F

see also And, Or, Truth Table, XOR

Notation

A Notation is a set of well-defined rules for represent-
ing quantities and operations with symbols.

see also Arrow Notation, Chained Arrow Nota-
tion, Circle Notation, Clebsch-Aronhold Nota-
tion, Conway's Knot Notation, Dowker Nota-
tion, Down Arrow Notation, Petrov Notation,
Scientific Notation, Steinhaus-Moser Notation

References

Cajori, F. A History of Mathematical Notations, Vols. 1-2.
New York: Dover, 1993.

Miller, J. "Earliest Uses of Various Mathematical Symbols."
http ; //members . aol . com/ j ef f 570/mathsym . html.

Miller, J. "Earliest Uses of Some of the Words of Mathemat-
ics." http : //members . aol . com/ j ef f 570/mathword. html.

Nother

see Noether's Fundamental Theorem, Noether-
Lasker Theorem, Noether's Transformation
Theorem, Noetherian Module, Noetherian Ring

Novem decillion

Null Function

1249

Novemdecillion

In the American system, 10 .

see also LARGE NUMBER

NP- Complete Problem

A problem which is both NP (solvable in nondetermin-
istic Polynomial time) and NP-Hard (can be trans-
lated into any other NP-Problem). Examples of NP-
hard problems include the Hamiltonian Cycle and
Traveling Salesman Problems.

In a landmark paper, Karp (1972) showed that 21 in-
tractable combinatorial computational problems are all
NP-complete.

see also Hamiltonian Cycle, NP-Hard Problem,
NP-Problem, P-Problem, Traveling Salesman
Problem

References

Karp, R. M. "Reductibility Among Combinatorial Problems."
In Complexity of Computer Computations, (Proc. Sympos.
IBM Thomas J. Watson Res. Center, Yorktown Heights,
N.Y., 1972). New York: Plenum, pp. 85-103, 1972.

NP-Hard Problem

A problem is NP-hard if an Algorithm for solving it
can be translated into one for solving any other NP-
PROBLEM (nondeterministic POLYNOMIAL time) prob-
lem. NP-hard therefore means "at least as hard as any
NP-PROBLEM," although it might, in fact, be harder.

see also Complexity Theory, Hitting Set, NP-
Complete Problem, NP-Problem, P-Problem,
Satisfiability Problem

NP-Problem

A problem is assigned to the NP (nondeterministic
POLYNOMIAL time) class if it is solvable in polynomial
time by a nondeterministic TURING MACHINE. (A non-
deterministic Turing Machine is a "parallel" Turing
Machine which can take many computational paths
simultaneously, with the restriction that the parallel
Turing machines cannot communicate.) A P- PROBLEM
(whose solution time is bounded by a polynomial) is al-
ways also NP. If a solution to an NP problem is known,
it can be reduced to a single P (POLYNOMIAL time) ver-
ification.

Linear Programming, long known to be NP and
thought not to be P, was shown to be P by L. Khachian
in 1979. It is not known if all apparently NP problems
are actually P.

A problem is said to be NP-HARD if an ALGORITHM
for solving it can be translated into one for solving any
other NP-problem problem. It is much easier to show
that a problem is NP than to show that it is NP-Hard.
A problem which is both NP and NP-Hard is called an
NP-Complete Problem.

See also COMPLEXITY THEORY, NP-COMPLETE PROB-
LEM, NP-Hard Problem, P-Problem, Turing Ma-
chine

References

Borwein, J. M. and Borwein, P. B. Pi and the AGM: A Study
in Analytic Number Theory and Computational Complex-
ity. New York: Wiley, 1987.

Greenlaw, R.; Hoover, H. J.; and Ruzzo, W. L. Limits to
Parallel Computation: P- Completeness Theory. Oxford,
England: Oxford University Press, 1995.

NSW Number
The numbers

S2rn+l —

(l + v / 2) 2rri+1 + (l- v^) 2 ^ 1

for positive integer m. The first few terms are 1, 7, 41,
239, 1393, ... (Sloane's A002315). The indices giving
Prime NSW numbers are 3, 5, 7, 19, 29, 47, 59, 163,
257, 421, 937, 947, 1493, 1901, . . . (Sloane's A005850).

References

Ribenboim, P. "The NSW Primes." §5.9 in The New Book
of Prime Number Records. New York: Springer- Verlag,
pp. 367-369, 1996.

Sloane, N. J. A. Sequences A002315/M4423 and A005850/
M2426 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Nu Function

, . f x dt

f°° x aJrt dt

v{x ' a) ^L fWTTTTy

where T(z) is the Gamma Function. See Gradshteyn

and Ryzhik (1980, p. 1079).

see also Lambda Function, Mu Function

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.

Null Function

A null function S°(x) satisfies

»/a

6°(x)dx =

(1)

for all o, 6, so

f

\5°{x)\dx = 0. (2)

Like a Delta Function, they satisfy

5o(x) = { J

x^0

x = 0.
see also DELTA FUNCTION, LERCH'S THEOREM

(3)

1250 Null Graph

Number

Null Graph

A GRAPH containing only VERTICES and no EDGES.

Null Hypothesis

A hypothesis which is tested for possible rejection under
the assumption that it is true (usually that observations
are the result of chance). The concept was introduced
by R. A. Fisher.

Null Tetrad

*0

1

Sij =

1

-1

-1

It can be expressed as

g a b ~ laTlb + IbTla — m a fhb — mbfh a -

see also TETRAD

References

dTnverno, R. Introducing Einstein's Relativity. Oxford, Eng-
land: Oxford University Press, pp. 248-249, 1992.

Nullspace

Also called the KERNEL. If T is a linear transformation
of R n , then Null(T) is the set of all Vectors X such
that T(X) = 0, i.e.,

Null(T) = {X : T(X) = 0}.

Nullstellansatz

see Hilbert's Nullstellansatz

Number

The word "number" is a general term which refers to a
member of a given (possibly ordered) Set. The meaning
of "number" is often clear from context (i.e., does it re-
fer to a Complex Number, Integer, Real Number,
etc.?). Wherever possible in this work, the word "num-
ber" is used to refer to quantities which are Integers,
and "CONSTANT" is reserved for nonintegral numbers
which have a fixed value. Because terms such as Real
Number, Bernoulli Number, and Irrational Num-
ber are commonly used to refer to nonintegral quanti-
ties, however, it is not possible to be entirely consistent
in nomenclature.

see also ABUNDANT NUMBER, ACKERMANN NUM-
BER, Algebraic Number, Almost Perfect Num-
ber, Amenable Number, Amicable Numbers, An-
timorphic Number, Apocalypse Number, Apoc-
alyptic Number, Armstrong Number, Arrange-
ment Number, Bell Number, Bernoulli Num-
ber, Bertelsen's Number, Betrothed Numbers,

Betti Number, Bezout Numbers, Binomial Num-
ber, Brauer Number, Brown Numbers, Car-
dinal Number, Carmichael Number, Catalan
Number, Cayley Number, Centered Cube Num-
ber, Centered Square Number, Chaitin's Num-
ber, Chern Number, Choice Number, Christof-
fel Number, Clique Number, Columbian Num-
ber, Complex Number, Computable Number,
Condition Number, Congruent Numbers, Con-
structible Number, Cotes Number, Crossing
Number (Graph), Crossing Number (Link), Cu-
bic Number, Cullen Number, Cunningham Num-
ber, Cyclic Number, Cyclomatic Number, D-
Number, de Moivre Number, Deficient Number,
Delannoy Number, Demlo Number, Diagonal
Ramsey Number, c-Perfect Number, Eban Num-
ber, Eddington Number, Edge Number, Enneag-
onal Number, Entringer Number, Erdos Num-
ber, Euclid Number, Euler's Idoneal Number,
Euler Number, Eulerian Number, Euler Zigzag
Number, Even Number, Factorial Number, Fer-
mat Number, Fibonacci Number, Figurate Num-
ber, G-NUMBER, GENOCCHI NUMBER, GlUGA NUM-
BER, Gnomic Number, Gonal Number, Graham's
Number, Gregory Number, Hailstone Number,
Hansen Number, Happy Number, Harmonic Divi-
sor Number, Harmonic Number, Harshad Num-
ber, Heegner Number, Heesch Number, Helly
Number, Heptagonal Number, Heterogeneous
Numbers, Hex Number, Hex Pyramidal Num-
ber, Hexagonal Number, Homogeneous Numbers,
Hurwitz Number, Hypercomplex Number, Hy-
perperfect Number, i, Idoneal Number, Imag-
inary Number, Independence Number, Infinary
Multiperfect Number, Infinary Perfect Num-
ber, Irrational Number, Irreducible Semiper-
fect Number, Irredundant Ramsey Number, j,
Kaprekar Number, Keith Number, Kissing Num-
ber, Knodel Numbers, Lagrange Number (Dio-
phantine Equation), Lagrange Number (Ratio-
nal Approximation), Large Number, Least Defi-
cient Number, Lehmer Number, Leviathan Num-
ber, Liouville Number, Logarithmic Number, Lu-
cas Number, Lucky Number, MacMahon's Prime
Number of Measurement, Markov Number, Mc-
Nugget Number, Menage Number, Mersenne
Number, Motzkin Number, Multiplicative Per-
fect Number, Multiply Perfect Number, Nar-
cissistic Number, Natural Number, Near No-
ble Number, Nexus Number, Niven Number, No-
ble Number, Nonagonal Number, Normal Num-
ber, NSW Number, Number Guessing, Oblong
Number, Octagonal Number, Octahedral Num-
ber, Odd Number, Ore Number, Ordinal Number,
Pentagonal Number, Pentatope Number, Per-
fect Digital Invariant, Perfect Number, Persis-
tent Number, Pluperfect Number, Plus Perfect
Number, Plutarch Numbers, Polygonal Number,

Number

Number Field Sieve Factorization Method 1251

PONTRYAGIN NUMBER, POULET NUMBER, POWER-
FUL Number, Practical Number, Primary, Prim-
itive Abundant Number, Primitive Pseudoper-
fect Number, Primitive Semiperfect Number,

PSEUDOPERFECT NUMBER, PSEUDORANDOM NUMBER,
PSEUDOSQUARE, PYRAMIDAL NUMBER, Q-NUMBER,

Quasiperfect Number, Ramsey Number, Ratio-
nal Number, Real Number, Rencontres Number,
Recurring Digital Invariant, Repfigit Number,
Rhombic Dodecahedral Number, Riesel Number,
Rotation Number, RSA Number, Sarrus Number,
Schroder Number, Schur Number, Secant Num-
ber, Segmented Number, Self-Descriptive Num-
ber, Self Number, Semiperfect Number, Sierpin-
ski Number of the First Kind, Sierpinski Num-
ber of the Second Kind, Singly Even Number,
Skewes Number, Small Number, Smith Number,
Smooth Number, Sociable Numbers, Sprague-
Grundy Number, Square Number, Square Pyra-
midal Number, Star Number, Stella Octangula
Number, Stiefel- Whitney Number, Stirling Cy-
cle Number, Stirling Set Number, Stormer Num-
ber, Sublime Number, Suitable Number, Sum-
Product Number, Super-3 Number, Super Cata-
lan Number, Superabundant Number, Superper-
fect Number, Super-Poulet number, Tangent
Number, Taxicab Number, Tetrahedral Number,
Transcendental Number, Transfinite Number,
Triangular Number, Tribonacci Number, Tri-
morphic Number, Truncated Octahedral Num-
ber, Truncated Tetrahedral Number, Twist
Number, U-Number, Ulam Number, Undulating
Number, Unhappy Number, Unitary Multiper-
fect Number, Unitary Perfect Number, Un-
touchable Number, Vampire Number, van der
Waerden Number, VR Number, Weird Number,
Whole Number, Woodall Number, Z-Number,
Zag Number, Zeisel Number, Zig Number

References

Barbeau, E. J. Power Play: A Country Walk through the

Magical World of Numbers. Providence, Rl: Amer, Math.

Soc, 1997.
Bogomolny, A. "What is a Number." http://vww.cut— the-

knot . com/do _you-know/numbers .html.
Borwein, J. and Borwein, P. A Dictionary of Real Numbers.

London; Chapman & Hall, 1990.
Conway, J. H. and Guy, R. K. The Book of Numbers. New

York: Springer- Verlag, 1996.
Dantzig, T. Number: The Language of Science, ^i/i rev. ed.

New York: Free Press, 1985.
Davis, P. J. The Lore of Large Numbers. New York: Random

House, 1961.
Prege, G. Grundlagen der Arithmetik: Eine logisch mathe-

matische Untersuchung iiber den Begriff der Zahl. New

York: Georg Olms, 1997.
Prege, G. Foundations of Arithmetic: A Logico- Mathematical

Enquiry into the Concept of Number. Evanston, IL: North-
western University Press, 1968.
Ifrah, G. From One to Zero: A Universal History of Num-
bers. New York: Viking, 1987.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

1983.

Phillips, R. Numbers: Facts, Figures & Fiction. Cambridge,
England: Cambridge University Press, 1994.

Russell, B. "Definition of Number." Introduction to Mathe-
matical Philosophy. New York: Simon and Schuster, 1971.

Smelt zer, D. Man and Number. Buchanan, NY: Emerson
Books, 1974.

Wells, D. W. The Penguin Dictionary of Curious and In-
teresting Numbers. Harmonds worth, England: Penguin
Books, 1986.

Number Axis
see Real Line

Number Field

If r is an Algebraic Number of degree n, then the
totality of all expressions that can be constructed from
r by repeated additions, subtractions, multiplications,
and divisions is called a number field (or an Algebraic
NUMBER Field) generated by r, and is denoted F[r].
Formally, a number field is a finite extension Q(c*) of

the Field Q of Rational Numbers.

The numbers of a number field which are ROOTS of a
Polynomial

z + a n -iz + . . . + ao —

with integral coefficients and leading coefficient 1 are
called the ALGEBRAIC INTEGERS of that field.

see also Algebraic Function Field, Algebraic In-
teger, Algebraic Number, Field, Finite Field, Q,
Quadratic Field

References

Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 127, 1996.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 151-152, 1993.

Number Field Sieve Factorization Method

An extremely fast factorization method developed by
Pollard which was used to factor the RSA- 130 NUMBER.
This method is the most powerful known for factoring
general numbers, and has complexity

a{exp[c(logn) 1/3 (loglogn) 2/3 ]},

reducing the exponent over the Continued Fraction
Factorization Algorithm and Quadratic Sieve
Factorization Method. There are three values of
c relevant to different flavors of the method (Pomerance
1996). For the "special" case of the algorithm applied
to numbers near a large POWER,

c=(^) 1/8 = 1.523..

for the "general" case applicable to any Odd Positive
number which is not a POWER,

c=(f) 1/3 = 1.923...,

1252 Number Group

Number Theoretic Transform

and for a version using many POLYNOMIALS (Copper-
smith 1993),

c= |(92 + 26v / 13) 1/3 = 1.902....

References

Coppersmith, D. "Modifications to the Number Field Sieve."
J. Cryptology 6, 169-180, 1993.

Coppersmith, D.; Odlyzko, A. M.; and Schroeppel, R. "Dis-
crete Logarithms in GF(p)." Algorithmics 1, 1-15, 1986.

Cowie, J.; Dodson, B.; Elkenbracht-Huizing, R. M.; Lenstra,
A. K.; Montgomery, P. L.; Zayer, J. A. "World Wide Num-
ber Field Sieve Factoring Record: On to 512 Bits." In Ad-
vances in Cryptology—ASIACRYPT '96 (Kyongju) (Ed.
K. Kim and T. Matsumoto.) New York: Springer- Verlag,
pp. 382-394, 1996.

Elkenbracht-Huizing, M. "A Multiple Polynomial General
Number Field Sieve." Algorithmic Number Theory (Tal-
ence, 1996). New York: Springer- Verlag, pp. 99-114, 1996.

Elkenbracht-Huizing, M. "An Implementation of the Number
Field Sieve." Experiment. Math. 5, 231-253, 1996.

Elkenbracht-Huizing, R.-M. "Historical Background of the
Number Field Sieve Factoring Method." Nieuw Arch.
Wisk. 14, 375-389, 1996.

Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Num-
ber Theory." In Handbook of Theoretical Computer Sci-
ence, Volume A: Algorithms and Complexity (Ed. J. van
Leeuwen). New York: Elsevier, pp. 673-715, 1990.

Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996.

Number Group

see Field

Number Guessing

By asking a small number of innocent-sounding ques-
tions about an unknown number, it is possible to re-
construct the number with absolute certainty (assum-
ing that the questions are answered correctly). Ball and
Coxeter (1987) give a number of sets of questions which
can be used.

One of the simplest algorithms uses only three questions
to determine an unknown number n:

1. Triple n and announce if the result n' — Sn is Even
or Odd.

2. If you were told that n' is EVEN, ask the person to
reveal the number n" which is half of n'. If you were
told that n' is Odd, ask the person to reveal the
number n" which is half of n + 1.

3. Ask the person to reveal the number of times k which
9 divides evenly into n'" = 3n".

The original number n is then given by 2k if n was
Even, or 2k + 1 if ri was Odd. For n = 2m even,
n = 6m, n" = 3m, n" — 9m, k — m, so 2k = 2m = n.
For n = 2m + 1 odd, n f = 6m 4- 3, n" = 3m + 2,
n" 1 = 9m 4- 6, k = m, so 2k + 1 = 2m + 1 = n.

Another method asks:

1. Multiply the number n by 5.

2. Add 6 to the product.

3. Multiply the sum by 4.

4. Add 9 to the product.

5. Multiply the sum by 5 and reveal the result n'.
The original number is then given by n = (n 1 — 165)/100,
since the above steps give ri = 5(4(5n + 6) + 9) = 100n+
165.

References

Bachet, C. G. Problemes plaisans et delectables, 2nd ed.
1624.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 5-20,
1987.

Kraitchik, M. "To Guess a Selected Number." §3.3 in Mathe-
matical Recreations. New York: W. W. Norton, pp. 58-66,
1942.

Number Pyramid

A set of numbers obeying a pattern like the following.

91 • 37 = 3367

9901 • 3367 = 33336667

999001 • 333667 = 333333666667

99990001 • 33336667 = 3333333366666667

4 2 = 16
34 2 = 1156

334'

67"

= 111556

49
4489

667' = 444889.

see also AUTOMORPHIC NUMBER

References

Heinz, H. "Miscellaneous Number Patters." http://www.

geocities . com/CapeCanaveral/Launchpad/4057/

miscnuin.htm.

Number System

see Base (Number)

Number Theoretic Transform

Simplemindedly, a number theoretic transform is a gen-
eralization of a Fast Fourier Transform obtained
by replacing e - 2ntk / N w ith an nth Primitive Root
OF UNITY. This effectively means doing a transform
over the Quotient Ring Z/pZ instead of the Com-
plex NUMBERS C. The theory is rather elegant and
uses the language of Finite FIELDS and Number The-
ory.

see also FAST FOURIER TRANSFORM, FINITE FIELD

References

Arndt, J. "Numbertheoretic Transforms (NTTs)." Ch. 4

in "Remarks on FFT Algorithms." http://www.jjj.de/

fxt/.
Cohen, H. A Course in Computational Algebraic Number

Theory. New York: Springer- Verlag, 1993.

Number Theory-

Number Theory 1253

Number Theory

A vast and fascinating field of mathematics consisting of
the study of the properties of whole numbers. PRIMES
and Prime Factorization are especially important in
number theory, as are a number of functions such as the
Divisor Function, Riemann Zeta Function, and
TOTIENT FUNCTION. Excellent introductions to num-
ber theory may be found in Ore (1988) and Beiler (1966).
The classic history on the subject (now slightly dated)
is that of Dickson (1952).

see also ARITHMETIC, CONGRUENCE, DlOPHANTINE

Equation, Divisor Function, Godel's Incom-
pleteness Theorem, Peano's Axioms, Prime
Counting Function, Prime Factorization, Prime
Number, Quadratic Reciprocity Theorem, Rie-
mann Zeta Function, Totient Function

References

Andrews, G. E. Number Theory. New York: Dover, 1994.

Andrews, G. E.; Berndt, B. C.; and Rankin, R. A. (Ed.).
Ramanujan Revisited: Proceedings of the Centenary Con-
ference, University of Illinois at Urbana- Champaign, June
1-5, 1987. Boston, MA: Academic Press, 1988.

Apostol, T. M. Introduction to Analytic Number Theory.
New York: Springer- Verlag, 1976.

Ayoub, R. G. An Introduction to the Analytic Theory of
Numbers. Providence, RI: Amer. Math. Soc, 1963.

Beiler, A. H. Recreations in the Theory of Numbers: The
Queen of Mathematics Entertains, 2nd ed. New York:
Dover, 1966.

Bellman, R. E. Analytic Number Theory: An Introduction.
Reading, MA: Benjamin/Cummings, 1980.

Berndt, B. C. Ramanujan's Notebooks, Part I. New York:
Springer- Verlag, 1985.

Berndt, B. C. Ramanujan's Notebooks, Part II. New York:
Springer- Verlag, 1988.

Berndt, B. C. Ramanujan's Notebooks, Part III. New York:
Springer- Verlag, 1997.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, 1993.

Berndt, B. C. Ramanujan's Notebooks, Part V. New York:
Springer- Verlag, 1997.

Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and
Commentary. Providence, RI: Amer. Math. Soc, 1995.

Borwein, J. M. and Borwein, P. B. Pi and the AGM: A Study
in Analytic Number Theory and Computational Complex-
ity. New York: Wiley, 1987.

Brown, K. S. "Number Theory." http://www.seanet.com/
-ksbrown/ inumber.htm.

Burr, S. A. The Unreasonable Effectiveness of Number The-
ory. Providence, RI: Amer. Math. Soc, 1992.

Burton, D. M. Elementary Number Theory, J^th ed. Boston,
MA: Allyn and Bacon, 1989.

Carmichael, R. D. The Theory of Numbers, and Diophantine
Analysis. New York: Dover, 1959.

Cohn, H. Advanced Number Theory. New York: Dover, 1980.

Courant, R. and Robbins, H. "The Theory of Numbers."
Supplement to Ch. 1 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 21-51, 1996.

Davenport, H. The Higher Arithmetic: An Introduction to
the Theory of Numbers, 6th ed. Cambridge, England:
Cambridge University Press, 1992.

Davenport, H. and Montgomery, H. L. Multiplicative Number
Theory, 2nd ed. New York: Springer- Verlag, 1980.

Dickson, L. E. History of the Theory of Numbers, 3 vols.
New York: Chelsea, 1952.

Dudley, U. Elementary Number Theory. San Francisco, CA:
W. H. Freeman, 1978.

Friedberg, R. An Adventurer's Guide to Number Theory.
New York: Dover, 1994.

Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT:
Yale University Press, 1966.

Guy, R. K, Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, 1994.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
1959.

Hasse, H. Number Theory. Berlin: Springer- Verlag, 1980.

Ireland, K. F. and Rosen, M. I. A Classical Introduction to
Modern Number Theory, 2nd ed. New York: Springer-
Verlag, 1995.

Klee, V. and Wagon, S. Old and New Unsolved Problems in
Plane Geometry and Number Theory. Washington, DC:
Math. Assoc. Amer., 1991.

Koblitz, N. A Course in Number Theory and Cryptography.
New York: Springer- Verlag, 1987.

Landau, E. Elementary Number Theory, 2nd ed. New York:
Chelsea, 1988.

Lang, S. Algebraic Number Theory, 2nd ed. New York:
Springer- Verlag, 1994.

Lenstra, H. W. and Tijdeman, R. (Eds.). Computational
Methods in Number Theory, 2 vols. Amsterdam: Mathe-
matisch Centrum, 1982.

LeVeque, W. J. Fundamentals of Number Theory. New York:
Dover, 1996.

Mitrinovic, D. S. and Sandor, J. Handbook of Number The-
ory. Dordrecht, Netherlands: Kluwer, 1995.

Niven, I. M.; Zuckerman, H. S.; and Montgomery, H. L. An
Introduction to the Theory of Numbers, 5th ed. New York:
Wiley, 1991.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, 1988.

Ore, 0. Invitation to Number Theory. Washington, DC:
Math. Assoc. Amer., 1967.

Ore, 0. Number Theory and Its History. New York: Dover,
1988.

Rose, H. E. A Course in Number Theory, 2nd ed. Oxford,
England: Clarendon Press, 1995.

Rosen, K. H. Elementary Number Theory and Its Applica-
tions, 3rd ed. Reading, MA: Addison- Wesley, 1993.

Schroeder, M. R. Number Theory in Science and Communi-
cation: With Applications in Cryptography, Physics, Dig-
ital Information, Computing, and Self- Similarity, 3rd ed.
New York: Springer- Verlag, 1997.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, 1993.

Sierpinski, W. 250 Problems in Elementary Number Theory.
New York: American Elsevier, 1970.

Uspensky, J. V. and Heaslet, M. A. Elementary Number The-
ory. New York: McGraw-Hill, 1939.

Vinogradov, I. M. Elements of Number Theory, 5th rev. ed.
New York: Dover, 1954.

Weil, A. Basic Number Theory, 3rd ed. Berlin: Springer-
Verlag, 1995.

Weil, A. Number Theory: An Approach Through History
From Hammurapi to Legendre. Boston, MA: Birkhauser,
1984.

Weyl, H. Algebraic Theory of Numbers. Princeton, NJ:
Princeton University Press, 1998.

1254 Number Triangle

NURBS Surface

Number Triangle

see Bell Triangle, Clark's Triangle, Euler's
Triangle, Leibniz Harmonic Triangle, Pascal's
Triangle, Seidel-Entringer-Arnold Triangle,
Trinomial Triangle

Number Wall

see Quotient-Difference Table

Numerator

The number p in a FRACTION pjq.

see also Denominator, Fraction, Rational Num-
ber

Numeric Function

A FUNCTION / : A -+ B such that B is a Set of num-
bers.

Numerical Derivative

While it is usually much easier to compute a DERIVA-
TIVE instead of an INTEGRAL (which is a little strange,
considering that "more" functions have integrals than
derivatives), there are still many applications where
derivatives need to be computed numerically. The sim-
plest approach simply uses the definition of the DERIV-
ATIVE

/'(*) ^ lim /(* + *>-/(')

J w h-+o h

for some small numerical value of h ^ 1.
see also NUMERICAL INTEGRATION

References

Press, W. H.; Fiannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Numerical Derivatives." §5.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 180-184, 1992.

Numerical Integration

The approximate computation of an INTEGRAL. The
numerical computation of an Integral is sometimes
called Quadrature. There are a wide range of methods
available for numerical integration. A good source for
such techniques is Press et al. (1992).

The most straightforward numerical integration tech-
nique uses the Newton-Cotes Formulas (also called
Quadrature Formulas), which approximate a func-
tion tabulated at a sequent of regularly spaced Inter-
vals by various degree Polynomials. If the endpoints
are tabulated, then the 2- and 3-point formulas are
called the Trapezoidal Rule and Simpson's Rule,
respectively. The 5-point formula is called Bode'S
RULE. A generalization of the TRAPEZOIDAL Rule is
Romberg Integration, which can yield accurate re-
sults for many fewer function evaluations.

If the functions are known analytically instead of being
tabulated at equally spaced intervals, the best numeri-
cal method of integration is called Gaussian Quadra-
ture. By picking the abscissas at which to evaluate the
function, Gaussian quadrature produces the most accu-
rate approximations possible. However, given the speed
of modern computers, the additional complication of the
Gaussian Quadrature formalism often makes it less
desirable than simply brute-force calculating twice as
many points on a regular grid (which also permits the
already computed values of the function to be re-used).
An excellent reference for Gaussian QUADRATURE is
Hildebrand (1956).

see also Double Exponential Integration, Filon's
Integration Formula, Integral, Integration,
Numerical Derivative, Quadrature

References

Davis, P. J. and Rabinowitz, P. Methods of Numerical Inte-
gration, 2nd ed. New York: Academic Press, 1984.

Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 319-323, 1956.

Milne, W. E. Numerical Calculus: Approximations, Inter-
polation, Finite Differences, Numerical Integration and
Curve Fitting. Princeton, NJ: Princeton University Press,
1949.

Press, W. H.; Fiannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, 1992.

Numerology

The study of numbers for the supposed purpose of pre-
dicting future events or seeking connections with the
occult.

see also Beast Number, Number Theory

NURBS Curve

A nonuniform rational B-SPLINE curve defined by

C(i)

where p is the order, Ni tP are the B- Spline basis func-
tions, P, are control points, and the weight Wi of Pi is
the last ordinate of the homogeneous point Pf. These
curves are closed under perspective transformations and
can represent Conic Sections exactly.

see also B-Spline, Bezier Curve, NURBS Surface

References

Piegl, L. and Tiller, W. The NURBS Book, 2nd edNew York:
Springer- Verlag, 1997.

NURBS Surface

A nonuniform rational B- Spline surface of degree (p, q)
is defined by

S(u,u) =

YZLo Z^=o N iAu)N jiq (v)w i j'Pij

J2Z.0 Di=o N i,pMN jtq (v)Wij

Nyquist Frequency

Nyquist Sampling 1255

where Ni iP and Nj iq are the B-SPLINE basis functions,
Pi j are control points, and the weight Wij of Pij is the
last ordinate of the homogeneous point P^j-
see also B-Spline, Bezier Curve, NURBS Curve

Nyquist Frequency

In order to recover all FOURIER components of a periodic
waveform, it is necessary to sample twice as fast as the
highest waveform frequency i/,

/Nyquist = 2lA

The minimum sampling frequency is called the Nyquist
frequency.

see also Fourier Series, Fourier Transform,
Nyquist Sampling, Oversampling, Sampling The-
orem

Nyquist Sampling

Sampling at the Nyquist Frequency.

Obelus

o

Obelus

The symbol 4- used to indicate DIVISION. In typography,
an obelus has a more general definition as any symbol,
such as the dagger (f), used to indicate a footnote.

see also DIVISION, SOLIDUS

Object

A mathematical structure (e.g., a GROUP, Vector
Space, or Differentiable Manifold) in a Cate-
gory.

see also MORPHISM
Oblate Spheroid

A "squashed" Spheroid for which the equatorial radius
a is greater than the polar radius c, so a > c. To first
approximation, the shape assumed by a rotating fluid
(including the Earth, which is "fluid" over astronomical
time scales) is an oblate spheroid. The oblate spheroid
can be specified parametrically by the usual Spheroid
equations (for a Spheroid with 2- Axis as the symmetry

axis J,

x = a sm v cos u
y = a sin v sin u
z = ccosv.

(1)

(2)
(3)

Oblate Spheroid 1257

as a function of the Latitude 5.

The Surface Area and Volume of an oblate spheroid
are

S = 2na 2 +n C -\n( 1 -±^)
e VI - ej

T r 4 2

V = |7ra c.

(8)
(9)

An oblate spheroid with its origin at a FOCUS has equa-
tion

q(l-e 2 )

1 + e cos 4>
Define k and expand up to POWERS of e 6 ,

k = e 2 (l - e 2 )" 1 = e 2 (l + e 2 - 2e 4 + 6e 6 + . . .)

(10)

= e 2 + e 4 - 2e 6 +
k 2 = e i + e 6 + ...
fc 3 = e 6 + . . . .

(11)
(12)
(13)

Expanding r in POWERS of ELLIPTICITY to e 6 therefore

yields

- = l-l(e 2 +e 4 -2 e 4 + 6e 6 )sin 2 <S

+ |(e 4 + e°) sin 4 6 - fe° sin° 5 + .... (14)
In terms of LEGENDRE POLYNOMIALS,

L = (i- l e 2 - lie 4 - ^3_e 6 )

V x 6 C 20 c 1680 c I

+ (-le 2 -l 2 e*-f 6 e*)P 2

+ (£e 4 + ^e 6 )P4- 5 Iie 6 J P 6 + .... (15)

The ELLIPTICITY may also be expressed in terms of the
Oblateness (also called Flattening), denoted e or /.

(16)

with a > c, u G [0,27r), and v € [0, 7r]. Its Cartesian
equation is

2,2 2

The ELLIPTICITY of an oblate spheroid is defined by

e =

7.2 - r2

(5)

and

c = o(l — e)

c 2 = a 2 (l-e) 2

(l- £ ) 2 = l-e 2 ,

= i-VT

(17)
(18)
(19)

(20)

so that

1 — e = — -

(6)

Then the radial distance from the rotation axis is given

by

/ p 2 \ ~ 1/2

r(6)=al 1+^-5-^ sin 2 5 J (7)

(1 - e) 2 = 1 - (1 - 2e + e 2 ) = 2e - e 2 (21)

n-l/2

, , 2e-c' . 2x

(22)

1258 Oblate Spheroid

Define k and expand up to POWERS of e 6

k = (2e- e)(l - ey 2 = (2e - e 2 )(l + 2e - 6e 2 + . . ,)

= 2e + 4e 4 - 12e 3 - e 2 - 2e 3 + . .
= 2e + 3e 2 - 14e 3 + . . .
fe 2 = 4e 2 + 6e 3 + . . .

A; 3 = 8e 3 + . . . .

(23)
(24)
(25)

Expanding r in POWERS of the OBLATENESS to e 3 yields

- = 1 - |(2e + 3e 2 - 14e 3 ) sin 2 5

+ f (4e 2 + 6e 3 ) sin 4 S 4- 8e 3 sin 6 S + . . . . (26)
In terms of LEGENDRE POLYNOMIALS,

^ = (l-|e-fe 2 -^e 3 ) + (-|e-ie 2 -^ 3 )P 2
+ (i^ 2 -^V4-^e 3 P 6 + .... (27)

To find the projection of an oblate spheroid onto a
Plane, set up a coordinate system such that the z-Axis
is towards the observer, and the z-axis is in the Plane
of the page. The equation for an oblate spheroid is

r(0) = a

Define

l + ^cos>*
2e-e 2

-1/2

fc =

(28)
(29)

and x = sin 0. Then

r($) = a[l + *(1 - x 2 )]~ 1/2 = a(l + * - kx 2 )~ 1/2 . (30)

Now rotate that spheroid about the z-axis by an Angle
B so that the new symmetry axes for the spheroid are
x' = x, y', and z* . The projected height of a point in
the x = Plane on the y-axis is

y = T {9) cos(9 -B)= r(0)(cos0cosB - sin0sinB)
= r(0)(\A -x 2 cosB + x sin B). (31)

To find the highest projected point,
dy __ as'm( B - 0) cos(B - 0) cos0sin0 _

(32)

d0 (1 + k cos 2 0)V2 ' — (l + fccos 2 0) 3 / 2
Simplifying,

tan(B - 0)(1 + k cos 2 (9) + A; cos sin = 0. (33)
But

tan(B - 0) =

tan B — tan 6

tanB- . sln *

VI -sin 2 e

1 + tanB tan 1 + tanB^J^

Vl-sin 2 ^

y/l - sin 2 tan B - sin6

\/l - sin 2 + tan B sin

(34)

Oblate Spheroid

Plugging (34) into (33),

V^4rf "; [l+fc(l-x 2 )]+^ V / ^^ = (35)
VI ™ a; + a? tanB

and performing a number of algebraic simplifications

(Vl-a; 2 tanJ5 - x)(l -f k - kx 2 )

+kx y / l-x 2 ( yfl-x 2 + x tan B) = (36)

[(1 + fc)\/l-a> 2 tanB - fcx 2 A/l - z 2 tanB

-x - fcz + kx 3 ] + [fca:(l - x 2 ) + fcz: 2 ^! -z 2 tanB]

(37)

(1 + k) t&nB^/l-x 2 - kx(l -x 2 )-x + kx(l - x 2 ) =

(38)
(1 + jfe) t&nB^l-x 2 = x

(l + fc) 2 tan 2 B(l-:c 2 ) = x 2

(39)

(40)

x 2 [l + (1 + fc) 2 tan 2 B] = (1 + k) 2 tan 2 B (41)

finally gives the expression for x in terms of B and &,

2 _ tan 2 B(l + fc) 2

l + (l + ifc)2tan 2 B- l ^

Combine (30) and (31) and plug in for #,

y = a-

Vl — x 2 cos B + zsinB

= a-

\/l + k - kx 2
cosB-f(l + fc)- sin2B

cos B

= a

i/(l + A;)[l + (l + fc)tan 2 B]
cos 2 B + (1 + fc) sin 2 B

(43)

cos B^/(l + fe)[l + (1 + k) tan 2 B]
Now re-express A; in terms of a and c, using e = 1 — c/a,

k =

(2-e) C = (l+g)(l-f)
(1-e) 2 (f) 2

iziil!

(f) 2

so

1 +

(44)

(45)

Plug (44) and (45) into (43) to obtain the SEMIMINOR
Axis of the projected oblate spheroid,

c = a-

cos 2 B + (f) 2 sin 2 B

cos 2 B+(2) 2 sin 2 B

a =

fy'cos 2 B+(f) 2 sin 2 B

cJcos 2 B+(~) sin 2 B = y/c 2 cos 2 B + a 2 sin 2 B
ay/{\ - e) 2 cos 2 B + sin 2 B. (46)

Oblate Spheroid

Oblate Spheroid Geodesic 1259

We wish to find the equation for a spheroid which has
been rotated about the x = x'-axis by Angle P, then
the z-axis by Angle P

1 -1
X

ri o oi

cosP sin

P"

y'

=

cos P sin P

10

z'\

_ — sin B cos B _

_ — sin P cos P _

cos P

sinP

' X

=

— sin B sin P cos B

sin B cos P

y

_ — cos P sin P —sin B cosPcosP_

_z

(47)

Now, in the original coordinates (x', y\ z') y the spheroid
is given by the equation

/2 ,2 ,2

X 11 Z

1_ £ 1 — 1

o? ^ c 2 + a 2 '

which becomes in the new coordinates,

^2

(48)

(xcosP -f ysinP)

(— xsinPsinP + zcosP + y sin P cos P) 2

+

(— x cos 5 sin P — z sin B -\- y cos 5 cos P) 2

= 1. (49)

Collecting Coefficients,

Ax 2 + Py 2 + C2 2 + Dxy + Px* + Fyz = 1, (50)
where

, cos 2 P + sin 2 P sin 2 P cos 2 P sin 2 P , ,
A= + (51)

_ __ sin 2 P + sin 2 P cos 2 P cos 2 B cos 2 P
# _ 1

_, cos 2 P sin 2 P

£> = 2 cos P sin P

1 - sin 2 P cos 2 P

= 2cosPsinPcos J P
P = 2 sin P cos P sin P
P = 2 sin P cos P cos P

\b 2 a 2 J

(52)
(53)

(54)
(55)
(56)

If we are interested in computing z, the radial distance
from the symmetry axis of the spheroid (y) correspond-
ing to a point

Cz 2 + (Ex + Fy)z + (Ax 2 + By 2 + Dxy - 1)

= Cz 2 + G(x, y)z + P(x, y) = 0, (57)

where

G(x t y) = Ex + Fy (58)

H(x, y) = Ax 2 + By 2 + Dxy - 1. (59)

z can now be computed using the quadratic equation
when (x, y) is given,

-G(x,y) ± y/GP(x,y) - 4CG(x,y)

2C

(60)

If P = 0, then we have sin P = and cos P = 1, so (51)
to (56) and (58) to (59) become

_ sin 2 B cos 2 B
„ cos 2 B sin 2 P

6 2

+ ■
a'

P =

F = 2 sin P cos P i

(61)
(62)

(63)

(64)
(65)

(66)

Va 2 6 2 7
G(x,y) = Fy = 2ysinPcosP (i - 1) (67)

P(x,y) = Ar 2 + Py 2 -1

x 2 2 /sin 2 P cos 2 P

= ^ + :

a^

+ ■

6 2

1. (68)

see also Darwin-de Sitter Spheroid, Ellipsoid,
Oblate Spheroidal Coordinates, Prolate Spher-
oid, Sphere, Spheroid

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 131, 1987.

Oblate Spheroid Geodesic

The Geodesic on an Oblate Spheroid can be com-
puted analytically for a spheroid specified parametri-
cally by

x = asinucosu
y = a sin v sin u
z = ccosv,

(1)

(2)
(3)

with a > c, although it is much more unwieldy than for
a simple Sphere. Using the first Partial Derivatives

dx . dx , A .

— —a sin v sin u -^- = a cos v cos u (4)

du
dy

dv
dy

_ —asmvcosu — - = acosvsinu (5)

du dv

dz n dz

= — = — csinv,

du dv

and second Partial Derivatives

(6)

d 2 x d 2 x

— — - = -asinvcosu — — - = — asinvcostt (7)

du 2 dv 2

1260 Oblate Spheroid Geodesic

du 2

= —asmvsmu

=

dv 2

= —asmvsmu

d 2 z

du 2 dv 2

gives the Geodesics functions as

= — ZCOSV,

(8)

(9)

P =

~ \du) + \du) + \du)
a (sin v cos u + sin v sin u)

2 . 2

a sin v

_ dx dx dy dy dz dz _

~ du dv du dv du dv

(10)

(ii)

»■(*)+ (2) + (i

= a 2 + (c 2 - a 2 ) sin 2 v = a (I - e 2 sin 2 v). (12)

Since Q = and P and i? are explicit functions of v only,
we can use the special form of the GEODESIC equation.

2 P

dv

a 2 (l — e 2 sin v)

■ dv

= C1 J\<7)

dv
sinu

Integrating gives

2 F(0|^#)-6 2 n(<i 2 -i,<t>\^f)

(13)

U = — Ci -

where

Vd 2

d =

COS(f) ':

Cl

dcosv

Vd 2 ^!

(14)

(15)
(16)

F(<j>\m) is an ELLIPTIC INTEGRAL OF THE FIRST KlND

with Parameter m, and II(0|ro, k) is an Elliptic In-
tegral of the Third Kind.

Geodesics other than Meridians of an Oblate
Spheroid undulate between two parallels with latitudes
equidistant from the equator. Using the WeierstraA
Sigma Function and WeierstraB Zeta Function,
the Geodesic on the Oblate Spheroid can be written
as

x + iy = K z!£+A ( ?b-«<*+*)\ (17)

<r(u)a(a)

x ~i v - ,. *(<* ~ u ) c -tth-C("+q)] (18)

X Zy ~ K a(u)a(a) e U * j

2 cr(uj rr + u)a{u>" — u)
cr 2 (u)a 2 (a)

(19)

Oblate Spheroidal Coordinates

(Forsyth 1960, pp. 108-109; Halphen 1886-1891).
The equation of the GEODESIC can be put in the form

v^

e 2 sin 2 v sin a

Y sin 2 v

dv, (20)

sin a sin v

where a is the smallest value of v on the curve. Fur-
thermore, the difference in longitude between points of
highest and next lowest latitude on the curve is

7T-2

Y 1 — e 2 si

! sin 2 a t K dnu
a J 1 + cc

— dn u

cot 2 a sn 2 u

du, (21)

where the Modulus of the Elliptic Function is

'- . £C0Sa (22)

v/T

e 2 sin 2 a

(Forsyth 1960, p. 446).

see also Ellipsoid Geodesic, Oblate Spheroid,
Sphere Geodesic

References

Forsyth, A. R. Calculus of Variations. New York: Dover,

1960.
Halphen, G. H. Traite des fonctions elliptiques et de leurs

applications fonctions elliptiques. Vol. 2. Paris: Gauthier-

Villars, pp. 238-243, 1886-1891.

Oblate Spheroidal Coordinates

U r? =

A system of Curvilinear Coordinates in which two
sets of coordinate surfaces are obtained by revolving
the curves of the ELLIPTIC CYLINDRICAL COORDI-
NATES about the y-AxiS which is relabeled the z-AxiS.
The third set of coordinates consists of planes passing
through this axis.

x = a cosh £ cos rj cos
y = a cosh £ cos rj sin <f>
z = asinh£sinT7,

(i)

(2)

(3)

Oblate Spheroidal Coordinates

Oblate Spheroidal Wave Function 1261

where £ G [0,oo), rj € [-7r/2,7r/2], and <j> G [0,2tt). Arf-
ken (1970) uses (u,v,<p) instead of (£, 77, 0). The SCALE
Factors are

fc,£ = aysinh 2 £ + sin 2 77

hrj = ay sinh 2 £ -+- sin 2 77
/i^ = a cosh £ cos 77.

(4)
(5)
(6)

The Laplacian is

v 3 / =

1

a 3 (sinh £ + sin 77) cosh £ cos 77

— I acosh£cos»7 —

H- -7— I a cosh £ cos

Of]

1

«g)^

(sinh 2 £ + sin 2 77) d 2 f

a 3 (sinh £ + sin 2 77) cosh f cos 77

cosh £ cos 77 d(f) 2
df

d 2 f

a sinh £ cos t)
df

di

a cosh £ cos 77 -^-r- + a sinh £ cos n-
o£ 2 or)

d 2 f
+a cosh £ cos 77 — -^

+

a 2 /

a 2 (sinh 2 £ + sin 2 77)a^ 2

a 2 (sinh £ + sin 77)

9 (ca*t°J-

H ^- cos 77—

COS 7] OTf \ or)

cosh £ d£ V <9£

1

+

o> 2 /

a 2 (cosh 2 £ + cos 2 77) d(j> 2
(7)

sin 2 77 + sinh £

d 2
(sech 2 £ tan 2 77 + sec 2 tanh 2 £) -^— r-

w d 2 d d 2

+ tanh£— + -— -tan 77- + —

a£ a£ J 77 77^

d4> 2
■ (8)

An alternate form useful for "two- center" problems is
defined by

£1 = sinh £
£[ — cosh £
£2 = cos 77

6 = &

(9)
(10)

(11)

(12)

where £1 e [1, oo], £ 2 6 [-1, 1], and £ 3 £ [0, 2n). In these
coordinates,

(Abramowitz and Stegun 1972). The Scale Factors
are

h^ — a
h$ 2 — a

ha = a £n,

and the LAPLACIAN is

d

6"

ft a -6 a
i-6 2

(16)

(17)
(18)

v 2 /

+

+ 6 2 ^i
i a

+

(i-6 J )

2^/

%

d-

«i 2 + i)(i-6 2 )^3

^}

(19)

The Helmholtz Differential Equation is separa-
ble.

see also HELMHOLTZ DIFFERENTIAL EQUATION —

Oblate Spheroidal Coordinates, Latitude, Lon-
gitude, Prolate Spheroidal Coordinates, Spher-
ical Coordinates

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Definition
of Oblate Spheroidal Coordinates." §21.2 in Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 752, 1972.

Arfken, G. "Prolate Spheroidal Coordinates (u, v, </>)." §2.11
in Mathematical Methods for Physicists, 2nd ed. Orlando,
FL: Academic Press, pp. 107-109, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 663, 1953.

Oblate Spheroidal Wave Function

The wave equation in OBLATE SPHEROIDAL COORDI-
NATES is

V 2 $ + fc 2 $

j9_

«i 2 + i)

+

%

+

6 a + 6 a

d 2 §

(fl 2 + l)(l-Z2 2 )^ 2

+c 2 Ki 3 + 6 2 )* = o, (i)

where

c = \ak.

(2)

Substitute in a trial solution

V = a£i| 2 sin£j (13)

z = aV«i 2 -l)(l-6 2 ) (14)

® = a£i£ 2 cos£ 3 (I 5 )

* = i?mn(c,6)^(c,6) . (W>). (3)

sin
The radial differential equation is

d
d$ 2

(H6 a )^-^mn(c,6)
«?2

^ 2 t 2 1
C ?2 +

1 + 6

Hmn(c,6)-0, (4)

1262

Oblateness

Obtuse Triangle

and the angular differential equation is

(i-6 2 )^r^n(c,6)

-A.

■c & +

1-6 2

i2mn(c,6) = (5)

(Abramowitz and Stegun 1972, pp. 753-755).
see also Prolate Spheroidal Wave Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Spheroidal Wave
Functions." Ch. 21 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 751-759, 1972.

Oblateness

see Flattening

Oblique Angle

An Angle which is not a Right Angle.

Oblong Number

see Pronic Number

Obstruction

Obstruction theory studies the extentability of MAPS us-
ing algebraic GADGETS. While the terminology rapidly
becomes technical and convoluted (as Iyanaga and
Kawada note, "It is extremely difficult to discuss higher
obstructions in general since they involve many com-
plexities"), the ideas associated with obstructions are
very important in modern Algebraic Topology.

see also ALGEBRAIC TOPOLOGY, CHERN CLASS,

Eilenberg-Mac Lane Space, Stiefel-Whitney
Class

References

Iyanaga, S. and Kawada, Y. (Eds.). "Obstructions." §300

in Encyclopedic Dictionary of Mathematics. Cambridge,

MA: MIT Press, pp. 948-950, 1980.

Obtuse Angle

An ANGLE greater than tt/2 RADIANS (90°).

see also Acute Angle, Obtuse Triangle, Right
Angle, Straight Angle

Obtuse Triangle

An obtuse triangle is a TRIANGLE in which one of the
Angles is an Obtuse Angle. (Obviously, only a single
Angle in a Triangle can be Obtuse or it wouldn't be
a Triangle.) A triangle must be either obtuse, Acute,
or Right.

A famous problem is to find the chance that three points
picked randomly in a PLANE are the VERTICES of an
obtuse triangle (Eisenberg and Sullivan 1996). Unfor-
tunately, the solution of the problem depends on the
procedure used to pick the "random" points (Portnoy
1994). In fact, it is impossible to pick random variables
which are uniformly distributed in the plane (Eisenberg
and Sullivan 1996). Guy (1993) gives a variety of so-
lutions to the problem. Woolhouse (1886) solved the
problem by picking uniformly distributed points in the
unit DISK, and obtained

ft = i-(£-JH-£=o.7mi5....

(1)

The problem was generalized by Hall (1982) to n-D
Ball Triangle Picking, and Buchta (1986) gave
closed form evaluations for Hall's integrals.

A 2r B

Lewis Carroll (1893) posed and gave another solution
to the problem as follows. Call the longest side of a
Triangle AB, and call the Diameter 2t\ Draw arcs
from A and B of Radius 2r. Because the longest side of
the Triangle is defined to be AB, the third Vertex
of the Triangle must lie within the region ABC A. If
the third Vertex lies within the Semicircle, the Tri-
angle is an obtuse triangle. If the Vertex lies on the
Semicircle (which will happen with probability 0), the
Triangle is a Right Triangle. Otherwise, it is an
Acute Triangle. The chance of obtaining an obtuse
triangle is then the ratio of the Area of the SEMICIRCLE
to that of ABC A. The Area of ABC A is then twice the
Area of a Sector minus the Area of the Triangle.

^whole figure — ^

Therefore,

47rr

■>/3r 2 =r 2 (|7r-V3). (2)

3tt

r 2 (|7r-\/3) 8tt-6\/3

-0.63938....

(3)

Let the Vertices of a triangle in n-D be Normal
(Gaussian) variates. The probability that a Gaussian
triangle in n-D is obtuse is

_ _ 3T(n) f 1/3 s<"- 2 >/ 2 J

3i» r" . „-!„,,,

= „ , 1 \ — 7 I sin 9 do
r*(ln)2— * J

_ 6r(n) 2 Fi(|n,n,l+ |n;-|)

~~ 3"/ 2 nr 2 (|n)

(4)

Ochoa Curve

Octagonal Number 1263

where T(n) is the GAMMA FUNCTION and 2 i ? i(a, b; c; x)
is the Hypergeometric Function. For Even n = 2k,

(Eisenberg

and Sullivan 1996). The first few

cases

are

explicitly

P 2 =

3 _
~ 4

= 0.75

(6)

ft =

= 1-

.^=0.586503...

4-7T

(7)

P 4 =

_ 15
32

= 0.46875...

(8)

P 5 =

= 1-

.^=0.379755....

(9)

8tt

see also ACUTE ANGLE, ACUTE TRIANGLE, BALL TRI-
ANGLE Picking, Obtuse Angle, Right Triangle,
Triangle

References

Buchta, C. "A Note on the Volume of a Random Polytope in
a Tetrahedron." III. J. Math. 30, 653-659, 1986.

Carroll, L. Pillow Problems & A Tangled Tale. New York:
Dover, 1976.

Eisenberg, B. and Sullivan, R. "Random Triangles n Dimen-
sions." Amer. Math. Monthly 103, 308-318, 1996.

Guy, R. K. "There are Three Times as Many Obtuse- Angled
Triangles as There are Acute- Angled Ones." Math. Mag.
66, 175-178, 1993.

Hall, G. R. "Acute Triangles in the n-Ball." J. Appl. Prob.
19, 712-715, 1982.

Portnoy, S. "A Lewis Carroll Pillow Problem: Probability on
at Obtuse Triangle." Statist. Sci. 9, 279-284, 1994.

Wells, D. G. The Penguin Book of Interesting Puzzles. Lon-
don: Penguin Books, pp. 67 and 248-249, 1992.

Woolhouse, W. S. B. Solution to Problem 1350. Mathemati-
cal Questions, with Their Solutions, from the Educational
Times, 1. London: F. Hodgson and Son, 49-51, 1886.

Ochoa Curve

The Elliptic Curve

ZY 2

2X S + 386X 2 + 256X

58195,

given in WeierstraB form as

y 2 = x 3 - 440067z + 106074110.

The complete set of solutions to this equation con-
sists of (x,y) = (-761,504), (-745, 4520), (-557,
13356), (-446, 14616), (-17, 10656), (91, 8172), (227,
4228), (247, 3528), (271, 2592), (455, 200), (499, 3276),
(523, 4356), (530, 4660), (599, 7576), (751, 14112),
(1003, 25956), (1862, 75778), (3511, 204552), (5287,
381528), (23527, 3607272), (64507, 16382772), (100102,
31670478), and (1657891, 2134685628) (Stroeker and de
Weger 1994).

References

Guy, R. K. "The Ochoa Curve." Crux Math. 16, 65-69,

1990.
Ochoa Melida, J. "La ecuacion diofantica b y 3 —bxy 1 -f b 2 y ~~

b z = z 2 ." Gaceta Math. 139-141, 1978.
Stroeker, R. J. and de Weger, B. M. M. "On Elliptic Diophan-

tine Equations that Defy Thue's Method: The Case of the

Ochoa Curve." Experiment. Math. 3, 209-220, 1994.

Octacontagon

An 80-sided POLYGON.

Octadecagon

An 18-sided POLYGON, sometimes also called an Oc-
TAKAIDECAGON.

see also POLYGON, REGULAR POLYGON, TRIGONOME-
TRY Values — 7r/18

Octagon

The regular 8-sided POLYGON. The INRADIUS r, ClR-
CUMRADIUS R, and Area A can be computed directly
from the formulas for a general regular Polygon with
side length s and n = 8 sides,

r=§*cot(!) = i(l + V2)a «

R=±scsc^ = fv / 4+2V^s (2)

A=\ns\ot(^j =2(l + v / 2)5 2 . (3)

see also OCTAHEDRON, POLYGON, REGULAR POLYGON,

Trigonometry Values— 71-/8
Octagonal Number

A Polygonal Number of the form n(3n - 2). The
first few are 1, 8, 21, 40, 65, 96, 133, 176, . . . (Sloane's
A000567). The Generating Function for the octag-
onal numbers is

x(bx + 1)

xY

= x + Sx 2 + 21x 3 + 40# 4 + .

References

Sloane, N. J. A. Sequence A000567/M4493 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

1264 Octagram

Octagram

The Star Polygon {8,3}.
Octahedral Graph

The Polyhedral Graph having the topology of the
Octahedron.

see also Cubical Graph, Dodecahedral Graph,
icosahedral graph, octahedron, tetrahedral
Graph

Octahedral Group

The Point Group of symmetries of the OCTAHEDRON,
denoted Oh- It is also the symmetry group of the CUBE,
CUBOCTAHEDRON, and TRUNCATED OCTAHEDRON. It
has symmetry operations E, 8C3, 6C4, 6C2, 3C2 = C|,
i, 6S 4 , 8S 6 , 3^, and 6<r 4 (Cotton 1990).
see also Cube, Cuboctahedron, Icosahedral
Group, Octahedron, Point Groups, Tetrahe-
dral Group, Truncated Octahedron

References

Cotton, F. A. Chemical Applications of Group Theory, 3rd

ed. New York: Wiley, p. 47-49, 1990.
Lomont, J. S. "Octahedral Group." §3.10.D in Applications

of Finite Groups. New York: Dover, p. 81, 1987.

Octahedral Number

A Figurate Number which is the sum of two consec-
utive Pyramidal Numbers,

O n = P n -i + P n = \n(2n + 1).

The first few are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670,
891, 1156, ... (Sloane's A005900). The GENERATING
FUNCTION for the octahedral numbers is

X ^ X + X )l =x + §x 2 + 19z 3 + 44z 4 + . . . .
{x - l) 4

see also Truncated Octahedral Number

Octahedron

References

Conway, J. H, and Guy, R. K. The Book of Numbers. New

York: Springer- Verlag, p. 50, 1996.
Sloane, N. J. A. Sequence A005900/M4128 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Octahedron

A Platonic Solid (P 3 ) with six Vertices, 12 Edges,
and eight equivalent EQUILATERAL TRIANGULAR faces
(8{3}), given by the SCHLAFLI SYMBOL {3,4}. It is also
Uniform Polyhedron U 5 with the Wythoff Sym-
bol 4 1 23. Its Dual Polyhedron is the Cube. Like
the Cube, it has the O h Octahedral Group of sym-
metries. The octahedron can be STELLATED to give the
Stella Octangula.

The solid bounded by the two Tetrahedra of the
Stella OCTANGULA (left figure) is an octahedron (right
figure; Ball and Coxeter 1987).

In one orientation (left figure), the VERTICES are given
by (±1,0,0), (0,±1,0), (0,0, ±1). In another orien-
tation (right figure), the vertices are (±1,±1,0) and
(0,0^^/3). In the latter, the constituent TRIANGLES
are specified by

ri = {(-i ) -i,o),(i ) -i ) o) ) (o,o ) V3)}

T 2 = {(-1,-1,0), (l,-l,0),(0,0,-v / 3)}
T 3 = {(-1,1,0), (1,1,0), (0,0, V^)}
T 4 = {(-1,1,0), (1,1,0), (0,0,-^)}
T B = {(1,-1,0), (1,1,0), (0,0, V3)}
T 6 = {(-1,-1,0), (-1,1,0), (0,0,^)}

r 7 = {(1,-1,0), (1,1,0), (o,o, -V3)}

T 8 = {(-1,-1,0), (-1,1,0), (0,0,-^3)}-

Octahedron

Octahemioctacron

1265

The face planes are ±x±y±z = 1, so a solid octahedron
is given by the equation

so the Inradius is

|*| + |y| + |z|<l.

(1)

A plane PERPENDICULAR to a C$ axis of an octahedron
cuts the solid in a regular HEXAGONAL CROSS-SECTION
(Holden 1991, pp. 22-23). Since there are four such axes,
there are four possibly HEXAGONAL CROSS-SECTIONS.
Faceted forms are the CUBOCTATRUNCATED CuBOCTA-
HEDRON and TETRAHEMIHEXAHEDRON.

Let an octahedron be length a on a side. The height
of the top Vertex from the square plane is also the

ClRCUMRADIUS

where

R= ^o? - <P ,

d= fv^a

is the diagonal length, so

0.70710a.

(2)

(3)

(4)

Now compute the INRADIUS.

£= \y/ia
b=\a

s = \a tan 30° =

2\Z3'

_ i

3*

Now use similar TRIANGLES to obtain

5

-2 =

5 a
a

Zy/2

x = 6 — b'

fa,

(5)
(6)

(7)
(8)

(9)
(10)

(11)

r=y/x* + z"= a} J$ + ± = ly/Ea:

; 0.40824a.

The Interradius is

p = \a = 0.5a.

(12)

(13)

The Area of one face is the Area of an Equilateral
Triangle

(14)

A= \V?>a

The volume is two times the volume of a square-base

pyramid,

V = 2(la 2 R) = 2(|)(a 2 )(|v / 2a) = |\/2a 3 . (15)
The Dihedral Angle is

a = cos _1 (-f ) « 70.528779°. (16)

see also OCTAHEDRAL GRAPH, OCTAHEDRAL GROUP,

Octahedron 5-Compound, Stella Octangula,
Truncated Octahedron

References

Davie, T. "The Octahedron." http://www.dcs.st-and.ac.

uk/~ad/mathrecs/polynedr a/octahedron. html.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,

1991.

Octahedron 5-Compound

Jr

A Polyhedron Compound composed of five Octahe-
dra occupying the Vertices of an Icosahedron. The
30 Vertices of the compound form an Icosidodeca-
hedron (Ball and Coxeter 1987).

see also ICOSIDODECAHEDRON, OCTAHEDRON, POLYHE-
DRON Compound

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 135 and
137, 1987.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 137-138, 1989.

Wenninger, M.J. Polyhedron Models. New York: Cambridge
University Press, p. 43, 1989.

Octahemioctacron

The Dual Polyhedron of the Octahemioctahe-

DRON.

1266 Octahemioctahedron

Octahemioctahedron

The Uniform Polyhedron £/" 3 , also called the Oc-

TATETRAHEDRON, whose DUAL POLYHEDRON is the
OCTAHEMIOCTACRON. It has WYTHOFF SYMBOL § 3 | 3.

Its faces are 8{3} +4{6}. It is a FACETED Cuboctahe-
DRON. For unit edge length, its ClRCUMRADIUS is

Odd Number

One of the eight regions of SPACE defined by the eight

possible combinations of Signs (±,±,±) for x, y, and

z.

see also QUADRANT

Octatetrahedron

see Octahemioctahedron

Octic Surface

An Algebraic Surface of degree eight. The maxi-
mum number of ORDINARY DOUBLE Points known to
exist on an octic surface is 168 (the ENDRASS OCTICS),
although the rigorous upper bound is 174.
see also ALGEBRAIC SURFACE, ENDRASS OCTIC

JR=1.

References

Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 103, 1989.

Octakaidecagon

see OCTADECAGON

Octal

The base 8 notational system for representing REAL
Numbers. The digits used are 0, 1, 2, 3, 4, 5, 6,
and 7, so that 810 (8 in base 10) is represented as 10s
(10 = l-8 1 +0-8°) in base 8.

see also Base (Number), Binary, Decimal, Hexa-
decimal, Quaternary, Ternary

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 9-10,
1991.
# Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks /Bases, m.

Octant

(+, +, +)

Octillion

In the American system, 10 27 .

see also LARGE NUMBER

Octodecillion

In the American system, 10 57 .

see also LARGE NUMBER

Octonion

see CAYLEY NUMBER

Odd Function

An odd function is a function for which f(x) = — /(— x).
An Even Function times an odd function is odd.

Odd Number

An Integer of the form N = 2n + 1, where n is an
Integer. The odd numbers are therefore . . . , —3, — 1,
1, 3, 5, 7, ... (Sloane's A005408), which are also the
Gnomic Numbers. The Generating Function for
the odd numbers is

^±^ + 3z 2 + 5, 3 + 7z 4 + ....
(x - l) 2

Since the odd numbers leave a remainder of 1 when di-
vided by two, N = 1 (mod 2) for odd N. Integers which
are not odd are called Even.

see also Even Number, Gnomic Number, Nico-
machus's Theorem, Odd Number Theorem, Odd
Prime

References

Sloane, N. J. A. Sequence A005408/M2400 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Odd Number Theorem

Onduloid

1267

Odd Number Theorem

The sum of the first n ODD NUMBERS is a SQUARE NUM-
BER:

X>fc-l) = 2j>-jS = 2

n(n-\- 1)

= n(n + 1) — n = n .
see also NlCOMACHUS'S THEOREM

Odd Order Theorem

see Feit-Thompson Theorem

Odd Prime

Any Prime Number other than 2 (which is the only
Even Prime).

see also PRIME NUMBER

Odd Sequence

A SEQUENCE of n Os and Is is called an odd sequence if
each of the n Sums Y^i=i a » a H-fc for ft = 0, 1, . . . , n — 1.

References

Guy, R. K. "Odd Sequences." §E38 in Unsolved Problems

in Number Theory, 2nd ed. New York: Springer- Verlag,

pp. 238-239, 1994.

Odds

Betting odds are written in the form r : s ("r to s") and
correspond to the probability of winning P = s/(r + s).
Therefore, given a probability P, the odds of winning
are (1/P) -1:1.
see also FRACTION, RATIO, RATIONAL NUMBER

References

Kraitchik, M. "The Horses." §6.17 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 134-135, 1942.

ODE

see Ordinary Differential Equation

Offset Rings

see Surface of Revolution

Ogive

Any cumulative frequency curve.

see also HISTOGRAM

References

Kenney, J. F. and Keeping, E. S. "Ogive Curves." §2.7 in

Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ:

Van Nostrand, pp. 29-31, 1962.

Oldknow Points

The Perspective Centers of a triangle and the TAN-
GENTIAL Triangles of its inner and outer Soddy Cir-
cles, given by

01' = /-2Ge,

where I is the Incenter and Ge is the GERGONNE
Point.

see also Gergonne Point, Incenter, Perspective
Center, Soddy Circles, Tangential Triangle

References

Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.

Omega Constant

W(l) = 0.5671432904..., (1)

where W(x) is Lambert's W-Function. It is avail-
able in Mathematical (Wolfram Research, Champaign,
IL) using the function ProductLogfl] . W(l) can be
considered a sort of "GOLDEN RATIO" for exponentials
since

exp[-W(l)] = W(l), (2)

giving

In

W(l)

= W(l).

(3)

see also GOLDEN RATIO, LAMBERT'S W-FUNCTION

References

Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; and Jeffrey,

D. J. "On Lambert's W Function." ftp://watdragon.

uwaterloo . ca/cs-archive/CS-93-03/W . ps . Z.
Plouffe, S. "The Omega Constant or W(l)." http://lacim.

uqam . ca/piDATA/omega . txt .

Omega Function

see Lambert's V^-Function

Omino

see Polyomino

Omnific Integer

The appropriate notion of INTEGER for SURREAL NUM-
BERS.

O'Nan Group

The Sporadic Group O'N.

References

Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat .bham.ac.uk/atlas/0N.html.

Onduloid

see UNDULOID

1268

One

One-Ninth Constant

One

see 1

One- Form

A linear, real-valued Function of Vectors such that
u; 1 (v) \-> M. Vectors and one-forms are Dual to each
other because VECTORS are CONTRAVARIANT ("Kets":
|^}) and one-forms are Covariant Vectors ("Bras":
(<f)\), so

o; 1 (v) = v(ctJ 1 ) = (uj 1 ,^ = {<j>\$) .

The operation of applying the one-form to a VECTOR
uj 1 (v) is called Contraction.

see also Angle Bracket, Bra, Differential k-
Form, Ket

One-Mouth Theorem

Except for convex polygons, every SIMPLE POLYGON has
at least one MOUTH.

see also Mouth, Principal Vertex, Two-Ears The-
orem

References

Toussaint, G. "Anthropomorphic Polygons." Amer. Math.
Monthly 122, 31-35, 1991.

One-Ninth Constant

N.B. A detailed on-line essay by 5. Finch was the start-
ing point for this entry.

Let Xm,n be Chebyshev Constants.
(1973) proved that

lim (A ,n) 1/n - I

n— s-oo

It was conjectured that

A = lim (A n ,n)
n— too

Carpenter et al. (1984) obtained

A = 0.1076539192...

l/n _ i
9'

Schonhage

(i)

(2)
(3)

numerically. Gonchar and Rakhmanov (1980) showed
that the limit exists and disproved the 1/9 conjecture,
showing that A is given by

exp

irK(y/T=l?)

K(c)

(4)

where K is the complete Elliptic Integral of the
First Kind, and c = 0.9089085575485414. . . is the Pa-
rameter which solves

and E is the complete ELLIPTIC INTEGRAL OF THE SEC-
OND Kind. This gives the value for A computed by
Carpenter et al (1984) A is also given by the unique
Positive Root of

where

and

oo

f(z) = J2 a i zi

j=l

d\j

(6)
(7)

(8)

(Gonchar and Rakhmanov 1980). aj may also be com-
puted by writing j as

r\TTl TTll mo

where m > and mi > 1, then

•Pk

(9)

„ mi+l -i „ m.2+1 i m mt-f-1 -j

a^|2™ +1 -3| Pl ~ 1P2 ~ l Pk - 1

Pi - 1 P2 — 1

Pk - 1

(10)

(Gonchar 1990). Yet another equation for A is due to
Magnus (1986). A is the unique solution with x £ (0, 1)
of

T£(2k + l) 2 (-x]

fc(fc + l)/2

o,

(11)

k=o

K(k) = 2E(k),

(5)

an equation which had been studied and whose root had
been computed by Halphen (1886). It has therefore been
suggested (Varga 1990) that the constant be called the
Halphen Constant. 1/A is sometimes called Varga's
Constant.

see also Chebyshev Constants, Halphen Con-
stant, Varga's Constant

References

Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft , com/ asolve/constant/onenin/ onenin.html.

Carpenter, A. J.; Ruttan, A.; and Varga, R. S. "Extended
Numerical Computations on the '1/9' Conjecture in Ra-
tional Approximation Theory." In Rational approximation
and interpolation (Tampa, Fla., 1983) (Ed. P. R. Graves-
Morris, E. B. SafF, and R. S. Varga). New York: Springer-
Verlag, pp. 383-411, 1984.

Cody, W. J.; Meinardus, G.; and Varga, R. S. "Chebyshev
Rational Approximations to e~* in [0, +oo) and Applica-
tions to Heat-Conduction Problems." J. Approx. Th. 2,
50-65, 1969.

Dunham, C. B. and Taylor, G. D. "Continuity of Best Recip-
rocal Polynomial Approximation on [0, oo)." J. Approx.
Th. 30, 71-79, 1980.

Gonchar, A. A. "Rational Approximations of Analytic Func-
tions." Amer. Math. Soc. Transl. Ser. 2 147, 25-34, 1990.

Gonchar, A. A. and Rakhmanov, E. A. "Equilibrium Distri-
butions and Degree of Rational Approximation of Analytic
Functions." Math. USSR Sbornik 62, 305-348, 1980.

One-to-One

Magnus, A. P. "On Freud's Equations for Exponential
Weights, Papers Dedicated to the Memory of Geza Freud."
J. Approx. Th. 46, 65-99, 1986.

Rahman, Q. I. and Schmeisser, G. "Rational Approxima-
tion to the Exponential Function." In Pade and Ra-
tional Approximation, (Proc. Internal. Sympos., Univ.
South Florida, Tampa, Fla., 1976) (Ed. E. B. Saff and
R. S. Varga). New York: Academic Press, pp. 189-194,
1977.

Schonhage, A. "Zur rationalen Approximierbarkeit von e~ x
uber [0,oo)." J. Approx. Th. 7, 395-398, 1973.

Varga, R. S. Scientific Computations on Mathematical Prob-
lems and Conjectures. Philadelphia, PA: SIAM, 1990.

One-to-One

Let / be a FUNCTION defined on a Set 5 and taking
values in a set T. Then / is said to be one-to-one (a.k.a.
an Injection or Embedding) if, whenever f(x) = /(y),
it must be the case that x = y. In other words, / is one-
to-one if it MAPS distinct objects to distinct objects.

If the function is a linear OPERATOR which assigns a
unique MAP to each value in a VECTOR SPACE, it is
called one-to-one. Specifically, given a Vector Space
V with X, Y e V, then a TRANSFORMATION T defined
on ¥ is one-to-one if T(X) ^ T(Y) for all X/Y.

see also Bijection, Onto

One- Way Function

Consider straight-line algorithms over a Finite Field
with q elements. Then the e-straight line complexity
C e ((f>) of a function <fi is defined as the length of the
shortest straight-line algorithm which computes a func-
tion / such that f{x) = x is satisfied for at least (1 — e)q
elements of F. A function <f> is straight-line "one way"
of range < S < 1 if <f> satisfies the properties:

1. There exists an infinite set S of finite fields such that
<f> is defined in every F G S and e is One-TO-One in
every F e S.

2. For every e such that < e < 5, C e ((£ -1 ) tends to
infinity as the cardinality q of F approaches infinity.

3. For every e such that < e < J, the "work function"
7] satisfies

; lim inf rj ;

q— J-oo

: lim inf

q — ^oo

lnCeCQ-lnCe^)
InCeO)

> 1.

It is not known if there is a one-way function with work
factor rj > (\nq) 3 .

References

Ziv, J. "In Search of a One- Way Function" §4.1 in

Open Problems in Communication and Computation (Ed.

T. M. Cover and B. Gopinath). New York: Springer-

Verlag, pp. 104-105, 1987.

Open Map 1269
Only Critical Point in Town Test

If there is only one Critical Point at an Extremum,
the Critical Point must be the Extremum for func-
tions of one variable. There are exceptions for two vari-
ables, but none of degree < 4. Such exceptions include

Sxe y

*y

z = x 2 {l + yf+y 2

f«lg^l for (*,,,) 5* (0,0)
\0 for(a:,y) = (0 ) 0)

(Wagon 1991). This latter function has discontinuous
z xy and z yx , and z yx (0, 0) = 1 and z xv (0,0) = 1.

References

Ash, A. M. and Sexton, H. "A Surface with One Local Min-
imum." Math. Mag. 58, 147-149, 1985.

Calvert, B. and Vamanamurthy, M. K. "Local and Global
Extrema for Functions of Several Variables." J. Austral.
Math. Soc. 29, 362-368, 1980.

Davies, R. Solution to Problem 1235. Math. Mag. 61, 59,
1988.

Wagon, S. "Failure of the Only-Critical-Point-in-Town Test."
§3.4 in Mathematica in Action. New York: W. H. Freeman,
pp. 87-91 and 228, 1991.

Onto

Let / be a FUNCTION defined on a SET S and taking
values in a set T. Then / is said to be onto (a.k.a. a
SURJECTION) if, for any t eT, there exists em s e S for
which t — f(s).

Let the function be an Operator which Maps points
in the DOMAIN to every point in the Range and let V
be a VECTOR Space with X,Y G V. Then a TRANS-
FORMATION T defined on ¥ is onto if there is an X € V
such that T(X) - Y for all Y.

see also Bijection, One-to-One

Open Disk

An n-D open disk of RADIUS r is the collection of points
of distance less than r from a fixed point in EUCLIDEAN
n-space.

see also CLOSED DISK, DISK

Open Interval

An Interval which does not include its Limit Points,
denoted (a, b).

see also Closed Interval, Half-Closed Interval

Open Map

A Map which sends Open Sets to Open Sets.
see also Open Mapping Theorem

1270 Open Mapping Theorem

Or

Open Mapping Theorem

There are several flavors of this theorem.

1. A continuous surjective linear mapping between Ba-
nach Spaces is an Open Map.

2. A nonconstant ANALYTIC FUNCTION on a DOMAIN
D is an Open Map.

References

Zeidler, E. Applied Functional Analysis: Applications to
Mathematical Physics. New York: Springer- Verlag, 1995.

Open Set

A Set is open if every point in the set has a NEIGHBOR-
HOOD lying in the set. An open set of RADIUS r and
center xo is the set of all points x such that |x — xo | < r,
and is denoted Z? r (xo). In 1-space, the open set is an
Open Interval. In 2-space, the open set is a Disk. In
3-space, the open set is a Ball.

More generally, given a TOPOLOGY (consisting of a Set
X and a collection of Subsets T), a Set is said to be
open if it is in T. Therefore, while it is not possible for
a set to be both finite and open in the TOPOLOGY of
the REAL Line (a single point is a Closed Set), it is
possible for a more general topological Set to be both
finite and open.

The complement of an open set is a Closed Set. It is
possible for a set to be neither open nor CLOSED, e.g.,
the interval (0, 1].

see also Ball, Closed Set, Empty Set, Open Inter-
val

O per ad

A system of parameter chain complexes used for Mul-
tiplication on differential GRADED ALGEBRAS up to
HOMOTOPY.

Operand

A mathematical object upon which an OPERATOR acts.
For example, in the expression 1x2, the MULTIPLICA-
TION OPERATOR acts upon the operands 1 and 2.

see also Operad, Operator

Operational Mathematics

The theory and applications of LAPLACE TRANSFORMS
and other INTEGRAL TRANSFORMS.

References

Churchill, R. V. Operational Mathematics, 3rd ed. New
York: McGraw-Hill, 1958.

Operations Research

A branch of mathematics which encompasses many di-
verse areas of minimization and optimization. Bron-
son (1982) describes operations research as being "con-
cerned with the efficient allocation of scarce resources."
It includes the Calculus of Variations, Control
Theory, Convex Optimization Theory, Decision

Theory, Game Theory, Linear Programming,
Markov Chains, network analysis, Optimization
Theory, queuing systems, etc. The more modern term
for operations research is OPTIMIZATION THEORY.

see also Calculus of Variations, Control Theory,
Convex Optimization Theory, Decision Theory,
Game Theory, Linear Programming, Markov
Chain, Optimization Theory, Queue

References

Bronson, R. Schaum's Outline of Theory and Problems of
Operations Research. New York: McGraw-Hill, 1982.

Hiller, F. S. and Lieberman, G. J. Introduction to Operations
Research, 5th ed. New York: McGraw-Hill, 1990.

Trick, M, "Michael Trick's Operations Research Page."
http : //mat . gsia . emu . edu

Operator

An operator A : f( n > (I) i-y /(/) assigns to every function
/ € f {n) (I) a function A(f) € /(/). It is therefore a
mapping between two FUNCTION SPACES. If the range
is on the REAL LINE or in the COMPLEX PLANE, the
mapping is usually called a FUNCTIONAL instead.

see also ABSTRACTION OPERATOR, ADJOINT OP-
ERATOR, Antilinear Operator, Biharmonic Op-
erator, Binary Operator, Casimir Operator,
Convective Operator, d'Alembertian Opera-
tor, Difference Operator, Functional Analysis,
Hecke Operator, Hermitian Operator, Identity
Operator, Laplace-Beltrami Operator, Linear
Operator, Operand, Perron-Frobenius Opera-
tor, Projection Operator, Rotation Operator,
Scattering Operator, Self-Adjoint Operator,
Spectrum (Operator), Theta Operator, Wave
Operator

References

Gohberg, L; Lancaster, P.; and Shivakuar, P. N. (Eds.), Re-
cent Developments in Operator Theory and Its Applica-
tions. Boston, MA: Birkhauser, 1996.

Hutson, V. and Pym, J. S. Applications of Functional Anal-
ysis and Operator Theory. New York: Academic Press,
1980.

Optimization Theory

see Operations Research

Or

A term in LOGIC which yields TRUE if any one of a
sequence conditions is TRUE, and FALSE if all conditions
are FALSE. A OR B is denoted A\B, A + B, or A V B.
The symbol V derives from the first letter of the Latin
word "vel" meaning "or." The Binary OR operator has
the following Truth Table.

A

B

Ay B

F

F

F

F

T

T

T

F

T

T

T

T

Orbifold

A product of ORs is called a DISJUNCTION and is de-
noted

V*-

Two BINARY numbers can have the operation OR per-
formed bitwise. This operation is sometimes denoted
A\\B.

see also AND, BINARY OPERATOR, LOGIC, NOT, PRED-
ICATE, Truth Table, Union, XOR

Orbifold

The object obtained by identifying any two points of a
Map which are equivalent under some symmetry of the
Map's Group.

Orbison's Illusion

The illusion illustrated above in which the bounding
Rectangle and inner SQUARE both appear distorted.

see also ILLUSION, MULLER-LYER ILLUSION, PONZO'S
Illusion, Vertical-Horizontal Illusion

References

Fineman, M. The Nature of Visual Illusion.
Dover, p. 153, 1996.

New York:

Orbit (Group)

Given a PERMUTATION GROUP G on a set 5, the orbit
of an element s € S is the subset of S consisting of
elements to which some element G can send s.

Orbit (Map)

The Sequence generated by repeated application of a
MAP. The MAP is said to have a closed orbit if it has a
finite number of elements.

see also Dynamical System, Sink (Map)

Orchard-Planting Problem 1271
Orchard-Planting Problem

n = 3, r = 1 n = 4, r = 1 n = 5,r = 2

n~6, r - A n = 1, r = 6 w = 8, r = l

n = 9 f r- 10 n- 10, r- 12

Also known as the Tree-Planting Problem. Plant n
trees so that there will be r straight rows with k trees in
each row. The following table gives max(r) for various
k. k = 3 is Sloane's A003035 and k = 4 is Sloane's
A006065.

n

A; = 3

k = 4

fc = 5

3

1

—

—

4

1

1

—

5

2

1

1

6

4

1

1

7

6

2

1

8

7

2

1

9

10

3

2

10

12

5

2

11

16

6

2

12

19

7

3

13

[22, 24]

>9

3

14

[26, 27]

> io

4

15

[31,32]

> 12

>6

16

37

> 15

>6

17

[40, 42]

> 15

> 7

18

[46,48]

> 18

> 9

19

[52, 54]

> 19

> io

20

[57, 60]

> 21

> 11

21

[64, 67]

22

[70, 73]

23

[77,81]

24

[85, 88]

25

[92, 96]

Sylvester showed that

r(k = 3)> |_i(n-l)(n-2)J,

where |_#J is the FLOOR FUNCTION (Ball and Coxeter
1987). Burr, Grunbaum and Sloane (1974) have shown
using cubic curves that

r(k = 3)<l+ |_£n(n-3)J,

1272 Orchard Visibility Problem

Order (Modulo)

except for n = 7, 11, 16, and 19, and conjecture that
the inequality is an equality with the exception of the
preceding cases. For n > 4,

r(k = 3)> L|[|n(n-l)-[fn]]J,

where \x] is the CEILING FUNCTION.
see also ORCHARD VISIBILITY PROBLEM

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 104-105
and 129, 1987.

Burr, S. A. "Planting Trees." In The Mathematical Gardner
(Ed. David Klarner). Boston, MA: Prindle, Weber, and
Schmidt, pp. 90-99, 1981.

Dudeney, H. E. Problem 435 in 536 Puzzles & Curious Prob-
lems. New York: Scribner, 1967.

Dudeney, H. E. The Canterbury Puzzles and Other Curi-
ous Problems, 7th ed. London: Thomas Nelson and Sons,
p. 175, 1949.

Dudeney, H. E. §213 in Amusements in Mathematics. New
York: Dover, 1970.

Gardner, M. Ch. 2 in Mathematical Carnival: A New Round-
Up of Tantalizers and Puzzles from Scientific American.
New York: Vintage Books, 1977.

Gardner, M. "Tree-Plant Problems." Ch. 22 in Time Travel
and Other Mathematical Bewilderments. New York:
W. H. Freeman, pp. 277-290, 1988.

Grunbaum, B. "New Views on Some Old Questions of Com-
binatorial Geometry." Teorie Combin. 1, 451-468, 1976.

Grunbaum, B. and Sloane, N. J. A. "The Orchard Problem."
Geom. Dedic. 2, 397-424, 1974.

Jackson, J. Rational Amusements for Winter Evenings. Lon-
don, 1821.

Macmillan, R. H. "An Old Problem." Math. Gaz. 30, 109,
1946.

Sloane, N. J. A. Sequences A006065/M0290 and A003035/
M0982 in "An On-Line Version of the Encyclopedia of In-
teger Sequences." http://www.research.att.com/-njas/
sequences/eisonline .html.

Sloane, N. J. A. and Plouffe, S. Extended entry for M0982 in
The Encyclopedia of Integer Sequences. San Diego: Aca-
demic Press, 1995.

Orchard Visibility Problem

A tree is planted at each LATTICE POINT in a circular
orchard which has CENTER at the ORIGIN and Radius
r. If the radius of trees exceeds 1/r units, one is unable
to see out of the orchard in any direction. However, if
the Radii of the trees are < 1/\A™ 2 + 1, one can see out
at certain Angles.

see also LATTICE POINT, ORCHARD-PLANTING PROB-
LEM, Visibility

References

Honsberger, R. "The Orchard Problem." Ch. 4 in Mathe-
matical Gems I. Washington, DC: Math. Assoc. Amer.,
pp. 43-52, 1973.

Order (Algebraic Curve)

The order of the POLYNOMIAL defining the curve.

Order (Algebraic Surface)

The order n of an ALGEBRAIC SURFACE is the order
of the POLYNOMIAL defining a surface, which can be
geometrically interpreted as the maximum number of
points in which a line meets the surface.

Order

Surface

3

cubic surface

4

quartic surface

5

quintic surface

6

sextic surface

7

heptic surface

8

octic surface

9

nonic surface

10

decic surface

see also ALGEBRAIC SURFACE

References

Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 8, 1986.

Order (Conjugacy Class)

The number of elements of a GROUP in a given CONJU-
gacy Class.

Order (Difference Set)

Let G be Group of Order h and D be a set of k el-
ements of G. If the set of differences di — dj contains
every NONZERO element of G exactly A times, then D
is a (h,k, A)-difference set in G of order n = k — A.

Order (Field)

The number of elements in a Finite Field.

Order (Group)

The number of elements in a GROUP G, denoted |G|.
The order of an element g of a finite group G is the
smallest POWER of n such that g n = I, where I is the
Identity Element. In general, finding the order of the
element of a group is at least as hard as factoring (Meijer
1996). However, the problem becomes significantly eas-
ier if \G\ and the factorization of |G| are known. Under
these circumstances, efficient ALGORITHMS are known
(Cohen 1993).

see also Abelian Group, Finite Group

References

Cohen, H. A Course in Computational Algebraic Number

Theory. New York: Springer- Verlag, 1993.
Meijer, A. R. "Groups, Factoring, and Cryptography." Math.

Mag. 69, 103-109, 1996.

Order (Modulo)

For any INTEGER a which is not a multiple of a Prime
p, there exists a smallest exponent h > 1 such that a =
1 (mod p) IFF h\k. In that case, h is called the order of
a modulo p.
see also CARMICHAEL FUNCTION

Order (Ordinary Differential Equation)

Ordinal Number 1273

Order (Ordinary Differential Equation)

An Ordinary Differential Equation of order n is
an equation of the form

F(x,y,y\...,y (n) ) = 0.

Order (Permutation)

see Permutation

Order (Polynomial)

The highest order Power in a one-variable POLYNOM-
IAL is known as its order (or sometimes its Degree).
For example, the POLYNOMIAL

Ordered Tree

A Rooted Tree in which the order of the subtrees
is significant. There is a ONE-TO-ONE correspondence
between ordered FORESTS with n nodes and BINARY
Trees with n nodes.

see also Binary Tree, Forest, Rooted Tree

Ordering

The number of "ARRANGEMENTS" in an ordering of n
items is given by either a COMBINATION (order is ig-
nored) or a Permutation (order is significant).

see also Arrangement, Combination, Cutting, De-
rangement, Partial Order, Permutation, Sort-
ing, Total Order

a n x + . . . + o>2X + a\x + ao
is of order n.

Order Statistic

Given a sample of n variates Xi, . . . , X n , reorder them
so that X[ < X 2 < ... < X' n . Then the ith order
statistic X^ is defined as Xl, with the special cases

m n = X (1) =min(X i )

M n = X {n) - max(X,).

3

A ROBUST Estimation technique based on linear com-
binations of order statistics is called an L-ESTIMATE.

see also Extreme Value Distribution, Hinge, Max-
imum, Minimum, Mode, Ordinal Number

References

Balakrishnan, N. and Cohen, A. C. Order Statistics and In-
ference. New York: Academic Press, 1991.

David, H. A. Order Statistics, 2nd ed. New York: Wiley,
1981.

Gibbons, J. D. and Chakraborti, S. (Eds.). Nonparametric
Statistic Inference, 3rd ed. exp. rev. New York: Marcel
Dekker, 1992.

Order (Vertex)

The number of Edges meeting at a given node in a

Graph is called the order of that Vertex.

Ordered Geometry

A Geometry constructed without reference to measure-
ment. The only primitive concepts are those of points
and intermediacy. There are 10 AXIOMS underlying or-
dered Geometry.

see also ABSOLUTE GEOMETRY, AFFINE GEOMETRY,

Geometry

Ordered Pair

A PAIR of quantities (a, b) where ordering is significant,
so (a, b) is considered distinct from (6, a) for a ^ b.

see also Pair

Ordering Axioms

The four of HlLBERT'S Axioms which concern the ar-
rangement of points.

see also Congruence Axioms, Continuity Axioms,
Hilbert's Axioms, Incidence Axioms, Parallel
Postulate

References

Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago,
IL: Open Court, 1980.

Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert's System of Ax-
ioms." §163B in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 544-545, 1980.

Ordinal Number

In informal usage, an ordinal number is an adjective
which describes the numerical position of an object, e.g.,
first, second, third, etc.

In technical mathematics, an ordinal number is one of
the numbers in Georg Cantor's extension of the Whole
Numbers. The ordinal numbers are 0, 1, 2, . . . , u>, a;+l,

lj -f 2, . . . , lj + lj, lj + lj + 1, Cantor's "smallest"

TRANSFINITE Number lj is defined to be the earliest
number greater than all WHOLE NUMBERS, and is de-
noted by Conway and Guy (1996) as lj = {0,1,... |}.
The notation of ordinal numbers can be a bit counter-
intuitive, e.g., even though 1 + w = w, lj + l> lj.

Ordinal numbers have some other rather peculiar prop-
erties. The sum of two ordinal numbers can take on two
different values, the sum of three can take on five values.
The first few terms of this sequence are 2, 5, 13, 33, 81,
193, 449, 33 2 , 33 • 81, 81 2 , 81 • 193, 192 2 , . . . (Conway
and Guy 1996, Sloane's A005348). The sum of n ordi-
nals has either 193 a 81 6 or 33 ■ 81 a possible answers for
n > 15 (Conway and Guy 1996).

r x lj is the same as lj, but lj x r is equal to lj + . . . -f lj.

r

lj 2 is larger than any number of the form lj x r, lj 3 is
larger than a; 2 , and so on.

1274 Ordinary Differential Equation

Ordinary Differential Equation

There exist ordinal numbers which cannot be con-
structed from smaller ones by finite additions, multi-
plications, and exponentiations. These ordinals obey
Cantor's Equation. The first such ordinal is

e = u; = 1 + u; + a; +u +

The next is

then follow 62 , €3 , . . . , e w , e w +i > . . . , e^ x 2 , * * * , €^2 , e^ ,

«€!»

■ ) ^€2 5

• ■ , e ee

• » € e n 1

e C£ , . . . (Conway and Guy 1996).

see also Axiom of Choice, Cantor's Equation,
Cardinal Number, Order Statistic, Power Set,
Surreal Number

References

Cantor, G. Uber unendliche, lineare Punktmannigfaltigkeit-

en f Arbeiten zur Mengenlehre aus dem Jahren 1872-1884'

Leipzig, Germany: Teubner-Archiv zur Mathematik, 1884.
Conway, J. H. and Guy, R. K. "Cantor's Ordinal Numbers."

In The Book of Numbers. New York: Springer- Verlag,

pp. 266-267 and 274, 1996.
Sloane, N. J. A. Sequence A005348/M1435 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

(1953, pp. 667-674) give canonical forms and solutions
for second-order ODEs.

While there are many general techniques for analyti-
cally solving classes of ODEs, the only practical solution
technique for complicated equations is to use numeri-
cal methods (Milne 1970). The most popular of these
is the RUNGE-KUTTA METHOD, but many others have
been developed. A vast amount of research and huge
numbers of publications have been devoted to the nu-
merical solution of differential equations, both ordinary
and PARTIAL (PDEs) as a result of their importance in
fields as diverse as physics, engineering, economics, and
electronics.

The solutions to an ODE satisfy EXISTENCE and
UNIQUENESS properties. These can be formally estab-
lished by PlCARD'S EXISTENCE THEOREM for certain
classes of ODEs. Let a system of first-order ODE be
given by

dx%

-dF = fi{xi >

• * j X ni t),

(4)

for i — 1, . . . , n and let the functions f%(xi t . . . , x n ,t),
where i = 1, . . . , n, all be defined in a DOMAIN D of
the (n + 1)-D space of the variables :n, . . . , x n , t. Let
these functions be continuous in D and have continuous
first Partial Derivatives dfi/dxj for i = 1, ..., n
and j = 1, . . . , n in D. Let (2?, , . . , a£ ) be in D. Then
there exists a solution of (4) given by

Ordinary Differential Equation

An ordinary differential equation (frequently abbrevi-
ated ODE) is an equality involving a function and its
Derivatives. An ODE of order n is an equation of the
form

F(x,y,y',...,y (n) ) = 0, (1)

where y' = dy/dx is a first DERIVATIVE with respect
to x and y (n) = d n y/dx n is an nth DERIVATIVE with
respect to x. An ODE of order n is said to be linear if
it is of the form

a n (x)y {n) + an-iOzOy^" 1 ) + . . . + ai (x)y + a Q (x)y

= Q(x). (2)

A linear ODE where Q{x) = is said to be homoge-
neous. Confusingly, an ODE of the form

Xi = X\ (t) , . . . , x n ~ x n (t)

(5)

dy
dx

'(i)

(3)

is also sometimes called "homogeneous."

Simple theories exist for first-order (INTEGRATING Fac-
tor) and second-order (STURM-LlOUVlLLE THEORY)
ordinary differential equations, and arbitrary ODEs
with linear constant COEFFICIENTS can be solved when
they are of certain factorable forms. Integral transforms
such as the LAPLACE TRANSFORM can also be used
to solve classes of linear ODEs. Morse and Feshbach

for to — S < t < to + 5 (where 5 > 0) satisfying the initial
conditions

Xi(to) = a??,..., z n (£o) = a£. (6)

Furthermore, the solution is unique, so that if

Xi = xl(t),... 7 X n - Xn(t) (7)

is a second solution of (4) for to — S < t < to + S sat-
isfying (6), then Xi(t) = acj(t) for to - S < t < to + S.
Because every nth-order ODE can be expressed as a sys-
tem of n first-order differential equations, this theorem
also applies to the single nth-order ODE.

In general, an nth-order ODE has n linearly indepen-
dent solutions. Furthermore, any linear combination of
Linearly Independent Functions solutions is also a
solution.

An exact First-Order ODEs is one of the form

where

p(x, y) dx + q(x, y) dy — 0,

dp __ dq
dy dx

(8)
(9)

Ordinary Differential Equation

An equation of the form (8) with

dp dq
dy dx

is said to be nonexact. If

Op dg
dy dx

m

(10)

(ii)

in (8), it has an x-dependent integrating factor. If

dq _ 9p
dx dy

xp- yq
in (8), it has an zy-dependent integrating factor. If

dq dp

dx dy

= /(y)

(13)

in (8), it has a ^-dependent integrating factor.

Other special first-order types include cross multiple
equations

yf(xy) dx + xg(xy) dy — 0, (14)

homogeneous equations

dy __ , (y^

? = f( y -),

dx \x J

linear equations

dy
dx

+ p{x)y = q{x),

and separable equations
dx

X(x)Y{y).

Special classes of Second-Order ODES include

(x missing) and

d 2 y ,, ,,

£ = '<■•'>

(15)

(16)

(17)

(18)

(19)

(y missing). A second-order linear homogeneous ODE
g + P(«)* + 0(^ = (20)

for which

Q'{x) + 2P{x)Q{x)

= [constant] (21)

2[Q(x)]3/2
can be transformed to one with constant coefficients.

Ordinary Differential Equation 1275

The undamped equation of Simple HARMONIC MOTION

is

d 2 y

dx 2

+ w y = 0,

which becomes

S+'2w.=o

(22)

(23)

when damped, and

^+0^+m 2 y = Acos(u>t) (24)

when both forced and damped.

Systems with Constant Coefficients are of the
form

dx

~dt

= Ax(t)+p(t).

(25)

The following are examples of important ordinary dif-
ferential equations which commonly arise in problems
of mathematical physics.

Airy Differential Equation

d 2 y

dx 2

xy = 0,

Bernoulli Differential Equation
dy

dx

+ p{x)y = q(x)y n .

Bessel Differential Equation

Chebyshev Differential Equation

/., 2sd 2 y dy 2 n

(26)

(27)

(28)

(29)

Confluent Hypergeometric Differential Equa-
tion

n d 2 y , ,_ ^dy

x d^ + { ''- x) d^ +ay=0 -

Euler Differential Equation

2 d y dy

-fax — + by = S(x).

Hermite Differential Equation

dx 2, dx

Hill's Differential Equation

d 2 r °°

-^ + O + 2 ^ 0„ cos(2nz)

= 0.

(30)

(31)

(32)

(33)

1276 Ordinary Differential Equation

Hypergeometric Differential Equation

x(x -l)^ + [(l + a + ^- 7 ]J + <*0y = 0. (34)
Jacobi Differential Equation
(l-x 2 )y" + [f3-a-(a+0+2)x]y t +n(n+a+f3+l)y = 0.
Laguerre Differential Equation

.£+(-.)*+*-.

Lane-Emden Differential Equation
Legendre Differential Equation

( 1 -^ 2 )i|- 2 ^ + ^ + 1 ^ = '

dy
J dx

Linear Constant Coefficients

d n y dy , ,

Malmsten's Differential Equation

y" + - z y' = ( Azm+ i)

y-

Riccati Differential Equation
dw

(35)

dx

= qfo(aj) + qi(x)w + g2(z)w

(36)

(37)

(38)

(39)

(40)

(41)

Ordinary Differential Equation. . .

Braun, M. Differential Equations and Their Applications,
3rd ed. New York: Springer- Verlag, 1991.

Forsyth, A. R. Theory of Differential Equations, 6 vols. New-
York: Dover, 1959.

Forsyth, A. R. A Treatise on Differential Equations. New
York: Dover, 1997.

Guterman, M. M. and Nitecki, Z. H. Differential Equations:
A First Course, 3rd ed. Philadelphia, PA: Saunders, 1992.

Ince, E. L. Ordinary Differential Equations. New York:
Dover, 1956.

Milne, W. E. Numerical Solution of Differential Equations.
New York: Dover, 1970.

Morse, P. M. and Feshbach, H. "Ordinary Differential Equa-
tions." Ch. 5 in Methods of Theoretical Physics, Part I.
New York: McGraw-Hill, pp. 492-675, 1953.

Moulton, F. R. Differential Equations. New York: Dover,
1958.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Integration of Ordinary Differential Equa-
tions." Ch. 16 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 701-744, 1992.

Simmons, G. F. Differential Equations, with Applications
and Historical Notes, 2nd ed. New York: McGraw-Hill,
1991.

Zwillinger, D. Handbook of Differential Equations, 3rd ed.
Boston, MA: Academic Press, 1997.

Ordinary Differential Equation — First-Order

Given a first-order ORDINARY DIFFERENTIAL EQUA-
TION

dy

dx

F(x,y) y

(1)

if F(x, y) can be expressed using SEPARATION OF Vari-
ables as

F(x,y) = X(x)Y(y), (2)

then the equation can be expressed as
dy

Y(y)

X(x)dx

(3)

+

"l-a-

z — a

«' +

1-0-/3' ! 1-7-V
z — b ' z — c

du

dz

+

'aot{a - b){a - c) /3/3'(6 - c)(b - a)
z — a ' z — b

7 y

(c — a)(c
z — c

-6)1

u
(z — a)(z — b)(z — c)

0. (42)

RlEMANN P-DlFFERENTIAL EQUATION

dz 2

+

see also Adams' Method, Green's Function,
Isocline, Laplace Transform, Leading Order
Analysis, Majorant, Ordinary Differential
Equation — First-Order, Ordinary Differential
Equation — Second-Order, Partial Differential
Equation, Relaxation Methods, Runge-Kutta
Method, Simple Harmonic Motion

References

Boyce, W. E. and DiPrima, R. C. Elementary Differential

Equations and Boundary Value Problems, 5th ed. New

York: Wiley, 1992.

and the equation can be solved by integrating both sides
to obtain

J Y(y) J

X(x) dx.

Any first-order ODE of the form
dy

dx

-\-p(x)y = q(x)

(4)

(5)

can be solved by finding an INTEGRATING FACTOR \i =
fi(x) such that

-Hw) = A*-r + y~r = m( x )- ( 6 )

ax ax ax

Dividing through by fxy yields

1 dy 1 dfjb _ q{x)
y dx fi dx y

(7)

Ordinary Differential Equation. . .

Ordinary Differential Equation. . . 1277

However, this condition enables us to explicitly deter-
mine the appropriate fi for arbitrary p and q. To ac-
complish this, take

ldii

(8)

in the above equation, from which we recover the origi-
nal equation (5), as required, in the form

1 d y _l n („\ q ^

-—- + p{x) = .

y dx y

(9)

But we can integrate both sides of (8) to obtain

fp(x)dx^ ^=\nfi + c (10)

H = eJ p{x)dx .
Now integrating both sides of (6) gives

/

fxy = / fiq(x) dx + c

(ii)

(12)

(with \i now a known function), which can be solved for
y to obtain

Jfiq(x)dx + c feJ p(x )dx q(x) dx + c
y = _ = r x , „ , , > V 16 )

V>

j p(x')dx f

where c is an arbitrary constant of integration.

Given an nth-order linear ODE with constant Coeffi-
cients

dx

first solve the characteristic equation obtained by writ-
ing

y = e rx (15)

and setting Q(x) — to obtain the n COMPLEX ROOTS.

n rx , n — lrx, , n „„ rx \ „ „ TX n {-\ a\

r e + a n -ir e -\- . . . + aire + aoe =u {lb)

r n + an-ir 71 ' 1 + . . . + a x r + a = 0. (17)

Factoring gives the ROOTS n 7

(r - n)(r - r 2 ) • ■ * (r - r„) = 0. (18)

For a nonrepeated REAL ROOT r, the corresponding so-
lution is

y = e rx . (19)

If a REAL ROOT r is repeated k times, the solutions are
degenerate and the linearly independent solutions are

Complex Roots always come in Complex Conjugate
pairs, r± = a ± ib. For nonrepeated COMPLEX ROOTS,
the solutions are

y = e ax cos(bx)>y = e ax sin(foc). (21)

If the COMPLEX ROOTS are repeated k times, the lin-
early independent solutions are

y = e ax cos(bx),y = e ax sin(6x), . . . ,

y = x k - l e ax cos(fcc), y = x k ~^e ax sin(6x). (22)

Linearly combining solutions of the appropriate types
with arbitrary multiplicative constants then gives the
complete solution. If initial conditions are specified, the
constants can be explicitly determined. For example,
consider the sixth-order linear ODE

(D - 1)(D - 2) 3 (D 2 + D + l)y = 0, (23)

which has the characteristic equation

(r - l)(r - 2) 3 (r 2 + r + 1) = 0. (24)

The roots are 1, 2 (three times), and (—1 ± \/3i)/2, so
the solution is

y = Ae x + Be 2x + Cxe 2x + Dx 2 e* x + Ee~ x/2 cos(±V3x)

+Fe- x sin(±VZx). (25)

If the original equation is nonhomogeneous (Q(x) # 0),
now find the particular solution y* by the method of
Variation of Parameters. The general solution is

then

y{x) = y*]cjyi(x) + y*(x),

(26)

where the solutions to the linear equations are yi(x),
y2(x), . . . , y n (x), and y*(x) is the particular solution.

see also INTEGRATING FACTOR, ORDINARY DIFFEREN-
TIAL Equation — First-Order Exact, Separation
of Variables, Variation of Parameters

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 440-445, 1985.

Ordinary Differential Equation — First-Order
Exact

Consider a first-order ODE in the slightly different form

rx rx fc — 1 rx

y == e ,y = xe ,...,y = x e .

(20)

p(x, y) dx + q(x, y) dy — 0.
Such an equation is said to be exact if

dp _ dq
dy dx'

a)

(2)

1278 Ordinary Differential Equation.

Ordinary Differential Equation. . .

This statement is equivalent to the requirement that a
Conservative Field exists, so that a scalar potential
can be denned. For an exact equation, the solution is

becomes

p(x, y) dx 4- q(x t y) dy = c,

'(a=0»3/0)

where c is a constant.

A first-order ODE (1) is said to be inexact if

dp dq
dy dx'

(3)

(4)

For a nonexact equation, the solution may be obtained
by defining an INTEGRATING FACTOR u of (6) so that
the new equation

satisfies

up(x, y) dx + fiq(x, y)dy =

A (w >) = £( M ),

or, written out explicitly,

dfj, dp du dp

(5)
(6)

(7)

This transforms the nonexact equation into an exact
one. Solving (7) for p, gives

— dx dy

P ~~ dp _ dq '
dy dx

(8)

Therefore, if a function u satisfying (8) can be found,
then writing

P(x,y) = fip

(9)

Q(x t y) = uq

(10)

in equation (5) then gives

P(x t y) dx + Q(x, y) dy = 0,

(11)

which is then an exact ODE. Special cases in which p
can be found include ^-dependent, xt/-dependent, and
y-dependent integrating factors.

Given an inexact first-order ODE, we can also look for
an Integrating Factor u(x) so that

dy

For the equation to be exact in ftp and uq, the equation
for a first-order nonexact ODE

dp dp dp dp

(13)

dp _ du dp
dy dx dx

(14)

Solving for dp/dx gives

|i = /1 ( a .)JL_2l= / ( a . iy ) /1 ( a .) ) (15)

ox q

which will be integrable if

dp dq

f( x ,y)=*!—°2-=f(x), (16)

in which case

— = f(x)dx,
A*

(17)

so that the equation is integrable

H(x) = ef f(x)dx , (18)

and the equation

[pp(x,y)]dx-\- [pq(x,y)]dy = (19)

with known p(x) is now exact and can be solved as an
exact ODE.

Given in an exact first-order ODE, look for an Inte-
grating Factor p(x,y) = g(xy). Then

dv^dg_ y

dx dx

dp _dg
dy dy

Combining these two,

dp __ y dp
dx x dy'

(20)
(21)

(22)

For the equation to be exact in pp and pq, the equation
for a first-order nonexact ODE

da dp dp , dp
P d-y + »c-y =q -dx- + »dx-

becomes

{*-!>)-{%-%)*■

(12) Therefore,

dz _d£

1 Op _ dx dy

x dy xp — yq
Define a new variable

t(x,y) = xy,

(23)

(24)

(25)

(26)

Ordinary Differential Equation,
then dt/dy = x, so

dfi _ dp dy _ ax ay
dt dy dt xp - yq

Now, if

H(t) = f(x,y)n{t). (27)

dq dp

f(*,v) = ^—? 3L = f(xv) = f(t), (28)

xy -yq

then

so that

= /(*)/*(*),

fi = eJ

(29)
(30)

and the equation

[fip(x y y)} dx + [/ig(x, y)] dy = (31)

is now exact and can be solved as an exact ODE.

Given an inexact first-order ODE, assume there exists
an integrating factor

m = /(y),

(32)

so d\ijdx = 0. For the equation to be exact in \ip and
fj,q, equation (7) becomes

OJX _ dx dy

dy p

v = f(x>y)v(y)- (33)

Now, if

then

so that

dq dp

P

= /(»),

= /(y) <*y,

(34)
(35)
(36)

/i(y)-e/ /W *,
and the equation

fi,p(x, y) dx + nq(x, y) dy = (37)

is now exact and can be solved as an exact ODE.
Given a first-order ODE of the form

yf{xy) dx + xg(xy) dy = 0, (38)

define

v = xy.

(39)

Then the solution is

I** = I cl B $ ) -?W +° *"9(v)*Hv) (40)

\xy = c for flf(u) = /(v).

Ordinary Differential Equation, . . 1279

If

dx

F{x,y) = G{v),

(41)

where

_ y

v = -,

X

(42)

then letting

2/ = xv

(43)

gives

dy _
dx

= xdv/dx + u

(44)

dv
dx

■ +u = G(v).

(45)

This can be integrated by quadratures, so

lnx = / -rr\ r- c for /(v) ^ t; (46)

J f(v)-v

ex for/ (v) = u.

(47)

References

Boyce, W. E. and DiPrima, R. C. Elementary Differential

Equations and Boundary Value Problems, J^th ed. New

York: Wiley, 1986.

Ordinary Differential Equation — Second-
Order

An ODE

y" + P(x)y' + Q(x)y = (1)

has singularities for finite x = Xo under the following
conditions: (a) If either P(x) or Q(x) diverges as x — >
xo, but (x - xo)P(x) and (x - x ) 2 Q(x) remain finite
as x -> xo, then xo is called a regular or nonessential
singular point, (b) If P(x) diverges faster than (x —
xo) -1 so that (x — xo)P(x) -t oo as x — > Xo, or Q(x)
diverges faster than (x - xo) -2 so that (x - xo) 2 <2(x) ->
oo as x -> xo, then xo is called an irregular or essential
singularity.

Singularities of equation (1) at infinity are investigated
by making the substitution x = z~ , so dx = —z~ dz y
giving

dy 2 dy

dx

dz

(2)

dx 2

—•e (-•"£) ~« , (-*S-"S)

^ -xdy 4 d 2 y

dz dz z

(3)

Then (1) becomes

z* £| + [2z 3 - z 2 P(z)] g + Q(s)y = 0. (4)

1280 Ordinary Differential Equation.

Case (a): If

a(z) =
000 =

2z - P(z)
z 2

(5)
(6)

remain finite at x — ±oo (y = 0), then the point is ordi-
nary. Case (b): If either a(z) diverges no more rapidly
than 1/z or f3(z) diverges no more rapidly than 1/z 2 ,
then the point is a regular singular point. Case (c):
Otherwise, the point is an irregular singular point.

Morse and Feshbach (1953, pp. 667-674) give the canon-
ical forms and solutions for second-order ODEs classified
by types of singular points.

For special classes of second-order linear ordinary differ-
ential equations, variable COEFFICIENTS can be trans-
formed into constant COEFFICIENTS. Given a second-
order linear ODE with variable COEFFICIENTS

g+p(-)^+ 9 (*)» = 0.

Define a function z = y(x),

dy dz dy
dx dx dz

(7)

(8)

dx 2 \dx) dz 2 dx 2 dz { }

'dz\ 2 <fy
k dx / dz 2

+

d z , .dz

dy
dz

+ q{x)y = Q (10)

cfy

dz 2

5f+^)s

\dx)

dy
dz

g(s)

. \dx) .

_d 2 y

,dy

d z> +A fz +By = °- (11)

This will have constant COEFFICIENTS if A and B are
not functions of x. But we are free to set B to an ar-
bitrary POSITIVE constant for q(x) > by defining z
as

z = B~

- 1/2 J[q(x)] l/2 dx. (12)

Then

dz
dx

= B

-1/2

[<?(*)]

1/2

i± = \B-^ [q { x) ]-^ q \ X ),

(13)
(14)

and

A=±

±B-V>[ q (x)]-^ q >(x) + B-^p(x)[ q {x)}^

B~ l q{x)
l'{ x ) + 2p(x) q (x) i/2
2[q(x)]W

(15)

Ordinary Differential Equation. . .

Equation (11) therefore becomes

d 2 y q ' (x) + 2p{x)q(x) 1/2 dy _

d^ + 2[«(x)]»/> B Tz +By -°> (16)

which has constant COEFFICIENTS provided that

^^^^^- [constant]. (17)

Eliminating constants, this gives
., _ q'(x) + 2p(x)q(x)

A =

[q(z)]*/*

= [constant]. (18)

So for an ordinary differential equation in which A' is
a constant, the solution is given by solving the second-
order linear ODE with constant COEFFICIENTS

for z, where z is defined as above.

(19)

A linear second-order homogeneous differential equation
of the general form

y"(x)+P(x)y' + Q(x)y = Q (20)

can be transformed into standard form

z"(x) + q(x)z-0 (21)

with the first-order term eliminated using the substitu-
tion

(22)

In y = In z — | / P(x) dx.

Then

V - = - - kP(*)

y z 2

.." „/2 „il #2

(23)

VJL ^ }L - = Z ^^ ~ i^W (24)

V " IV ' ~ '' *" £ ~ \P'i*) (25)

y \y

z z z*

y_

y

kP(*)

z z
z z 2

+ V - V - W

P(x) + r(x) + ^-^-ip'(x), (26)

V- + P{x)V- + Q(x)
y v

1 p2

-P(x)

+ ±P<(x) + ^--\P'(x) + P(x)

\P{?)

+Q(x)

\P'(x)-±P 2 (x) + Q(x) = 0. (27)

Ordinary Differential Equation. . .

Therefore,

z" + [Q{x)-\P'{x)-\P\x)]z

= z ,, (x)+q(x)z = 0, (28)

where

q(x) = Q(x) - \P\x) - \P\x).
If Q(x) = 0, then the differential equation becomes

y" + P(x)y' - 0,
which can be solved by multiplying by

exp

Jnw

to obtain

o.|{-p[/'iV)*']*}

ci = exp / P{x)dx

dy
dx

exp [J 1 P(x') dx']

+ c 2 .

(30)
(31)

(32)

(33)
(34)

If one solution (yi) to a second-order ODE is known,
the other (y 2 ) may be found using the REDUCTION OF
Order method. Prom the Abel's Identity

where

— = -P(x) dx,

(35)

W = 2/12/2 — 2/i2/2

(36)

J a J a

(37)

In

\m-[^

(38)

x) = W(a) exp J - / P(x') dx'

• (39)

XT — ,

1 1 _ 2 d /j/2 \

(A(\\

But

Combining (39) and (40) yields

dx

A. (vl

dx \yl

W(a)

exp[~f*P(x')dx']

vl

(41)

2/2^) = yi(x)W(a) I

*exp[-j; P(x")dx"] j ,

[yi(*')} 2

dx .
(42)

Ordinary Differential Equation. . . 1281

Disregarding VF(a), since it is simply a multiplicative
constant, and the constants a and 6, which will con-
tribute a solution which is not linearly independent of

V2{x) = yi(

„/

exp

"-/ x 'p(x")dx"l

L- i dx'. (43)

bi(z')] 2

If P(a;) = 0, this simplifies to

r dx 1

For a nonhomogeneous second-order ODE in which the
x term does not appear in the function /(x,y, 2/'),

s? = /(y.»).

let v = y', then
dv

£f . dv dy dv
dx =f{v ' y)= dydx:= V dy-

So the first-order ODE

vj- = f(y,v),

(45)

(46)

(47)

if linear, can be solved for v as a linear first-order ODE.
Once the solution is known,

dy

dx

v(y)

(48)

J&-h <49)

On the other hand, if y is missing from f(x,y,y f ),

(50)
let v ^ 2/, then 1/ = y", and the equation reduces to

|£ = **•»'>•

«' = f(x,v),

(51)

which, if linear, can be solved for v as a linear first-order
ODE. Once the solution is known,

y

/.(.

x) rfx.

(52)

see a/50 Abel's Identity, Adjoint Operator

References

Arfken, G- "A Second Solution." §8.6 in Mathematical Meth-
ods for Physicists j 3rd ed. Orlando, FL: Academic Press,
pp. 467-480, 1985.

1282 Ordinary Differential Equation, . .

Ordinary Double Point

Boyce, W. E. and DiPrima, R. C. Elementary Differential
Equations and Boundary Value Problems, J^th ed. New
York: Wiley, 1986.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 667-674, 1953.

Ordinary Differential Equation — System
with Constant Coefficients

To solve the system of differential equations

^=Ax(*) + P(*),

(1)

where A is a Matrix and x and p are Vectors, first
consider the homogeneous case with p = 0. Then the
solutions to

dX =Ax(*) (2)

are given by

dt

x(*) = e At x(t).

(3)

But, by the Matrix Decomposition THEOREM, the
Matrix Exponential can be written as

e A ' = uDu-\

where the EIGENVECTOR MATRIX

is

u = [ui ■ •• u

»]

and the Eigenvalue Matrix is

V 1 *

D =

e X2t ...
■■■

e A "<_

Now consider

e At u = uDu" 1 u =

uD

"un U21 •

• • U n l '

- c Ait

1412 U22 '

*• U n 2

e A 2 t ...

.Win U2n '

' ' Unn m

••• <

5 A„t

u lie Xlt •

•■ u nl e Xrit ~
• • Un2e Xnt

_U n

ie

A l*

• • U n

2 e A " f _

(4)

(5)

(6)

(7)

The individual solutions are then

Xi = (e A *u) -ei =u;e Ai \
so the homogeneous solution is

x = y aUie Xit ,

(8)

(9)

where the c^s are arbitrary constants.

The general procedure is therefore

1. Find the Eigenvalues of the Matrix A (A x , ...,

A n ) by solving the CHARACTERISTIC EQUATION.

2. Determine the corresponding EIGENVECTORS ui,

3. Compute

Xi = e Xit m

(10)

for i = 1, . , . , n. Then the VECTORS x» which are
Real are solutions to the homogeneous equation. If
A is a 2 x 2 matrix, the COMPLEX vectors x^* corre-
spond to Real solutions to the homogeneous equa-
tion given by 5ft(xj) and &(xj).

4. If the equation is nonhomogeneous, find the partic-
ular solution given by

x'(t)=X(t)

/-'

(t) P (t)dt,

where the Matrix X is defined by

X(t) = [xi ••• x n ].

(11)

(12)

If the equation is homogeneous so that p(t) = 0,
then look for a solution of the form

x = £e .
This leads to an equation

(A - Al)£ = 0,

(13)

(14)

SO £ is an EIGENVECTOR and A an EIGENVALUE.
5. The general solution is

x(t) = x*(t) + y^CjXj.

(15)

Ordinary Double Point

A Rational Double Point of Conic Double Point
type, known as "Ai." An ordinary Double Point is
called a Node. The above plot shows the curve x 3 —
x 2 + y 2 = 0, which has an ordinary double point at the
Origin.

Ordinary Double Point

Orientable Surface 1283

A surface in complex 3-space admits at most finitely
many ordinary double points. The maximum possi-
ble number of ordinary double points fi{d) for a sur-
face of degree d = 1, 2, . . . , are 0, 1, 4, 16, 31, 65,
93 < ^(7) < 104, 168 < fi(S) < 174, 216 < ^(8) < 246,
345 < m(10) < 360, 425 < fi(ll) < 480, 576 <
^(12) < 645 . . . (Sloane's A046001; Chmutov 1992, En-
drafi 1995). The fact that /x(5) = 31 was proved by
Beauville (1980), and //(6) = 65 was proved by Jaffe
and Ruberman (1994). For d > 3, the following inequal-
ity holds:

/i(d)< %[d(d-l)-3]

(Endrafi 1995). Examples of ALGEBRAIC SURFACES
having the maximum (known) number of ordinary dou-
ble points are given in the following table.

d fx(d) Surface

3 4 Cayley cubic

4 16 Kummer surface

5 31 dervish

6 65 Barth sextic
8 168 Endrafl octic

10 345 Barth decic

see also Algebraic Surface, Barth Decic, Barth
Sextic, Cayley Cubic, Cusp, Dervish, Endrass
Octic, Kummer Surface, Rational Double Point

References

Basset, A. B. "The Maximum Number of Double Points on

a Surface." Nature 73, 246, 1906.
Beauville, A. "Sur le nombre maximum de points dou-
bles d'une surface dans P (M5) = 31)." Joumees de

geometrie algebrique d'Angers (1979). Sijthoff & Noord-

hoflf, pp. 207-215, 1980.
Chmutov, S. V. "Examples of Projective Surfaces with Many

Singularities." J. Algebraic Geom. 1, 191-196, 1992.
Endrafi, S. "Surfaces with Many Ordinary Nodes." http://

www.mathematik.uni-mainz.de/AlgebraischeGeometrie/

docs/Eflaechen.shtml.
Endrafl, S. "Flachen mit vielen Doppelpunkten." DMV-

Mitteilungen 4, 17-20, Apr. 1995.
Endran, S. Symmetrische Fldche mit vielen gewohnlichen

Doppelpunkten. Ph.D. thesis. Erlangen, Germany, 1996.
Fischer, G. (Ed.). Mathematical Models from the Collections

of Universities and Museums. Braunschweig, Germany:

Vieweg, pp. 12-13, 1986.
Jaffe, D. B. and Ruberman, D. "A Sextic Surface Cannot

have 66 Nodes." J. Algebraic Geom. 6, 151-168, 1997.
Miyaoka, Y. "The Maximal Number of Quotient Singularities

on Surfaces with Given Numerical Invariants." Math. Ann.

268, 159-171, 1984.
Sloane, N, J. A. Sequence A046001 in "An On-Line Version

of the Encyclopedia of Integer Sequences."
Togliatti, E. G. "Sulle superficie algebriche col massimo nu-

mero di punti doppi." Rend. Sem. Mat. Torino 9, 47-59,

1950.
Varchenko, A. N. "On the Semicontinuity of Spectrum and

an Upper Bound for the Number of Singular Points on a

Projective Hypersurface." Dokl. Acad. Nauk SSSR 270,

1309-1312, 1983.
Walker, R. J. Algebraic Curves. New York: Springer- Verlag,

pp. 56-57, 1978.

Ordinary Line

Given an arrangement of n > 3 points, a Line contain-
ing just two of them is called an ordinary line. Moser
(1958) proved that at least 3n/7 lines must be ordinary
(Guy 1989, p. 903).

see also GENERAL POSITION, NEAR-PENCIL, ORDINARY

Point, Special Point, Sylvester Graph

References

Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903-909, 1989.

Ordinary Point

A Point which lies on at least one Ordinary Line.

see also Ordinary Line, Special Point, Sylvester
Graph

References

Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903-909, 1989.

Ordinate

The y- (vertical) axis of a Graph.

see also ABSCISSA, x-AxiS, y-AxiS, z-AxiS

Ore's Conjecture

Define the Harmonic Mean of the Divisors of n

H(n)~

i >

n d

where r(n) is the Tau Function (the number of Di-
visors of n). If n is a PERFECT NUMBER, H(n) is an
INTEGER. Ore conjectured that if n is ODD, then H(n)
is not an INTEGER. This implies that no Odd PERFECT
Numbers exist.

see also HARMONIC DIVISOR NUMBER, HARMONIC

Mean, Perfect Number, Tau Function

Ore Number

see Harmonic Divisor Number

Ore's Theorem

If a Graph G has n > 3 Vertices such that every pair
of the n Vertices which are not joined by an Edge has
a sum of Valences which is > n, then G is Hamilton-

ian.

see also HAMILTONIAN Graph

Orientable Surface

A Regular Surface M c R n is called orientable if
each Tangent Space M p has a Complex Structure
J p : M p —> M p such that p — > J p is a continuous func-
tion.
see also Nonorientable Surface, Regular Surface

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 230, 1993.

1284 Orientation (Plane Curve)

Orthic Triangle

Orientation (Plane Curve)

A curve has positive orientation if a region R is on the
left when traveling around the outside of i?, or on the
right when traveling around the inside of R.

Orientation-Preserving

A nonsingular linear MAP A : W 1 — > W 1 is orientation-
preserving if det(yl) > 0.
see also Orientation-Reversing, Rotation

Orientation- Reversing

A nonsingular linear MAP A : W 1 — »

reversing if det(A) < 0.

see also Orientation-Preserving

t n is orientation-

Orientation (Vectors)

Let be the ANGLE between two VECTORS. If < 6 <
7r, the VECTORS are positively oriented. If it < 6 < 2rc,
the vectors are negatively oriented.

Two vectors in the plane

xi

X 2

and

are positively oriented Iff the Determinant

D =

xi yi

X 2 V2

>o,

and are negatively oriented Iff the DETERMINANT D <
0.

Origami

The Japanese art of paper folding to make 3-dimensional
objects. Cube Duplication and Trisection of an
ANGLE can be solved using origami, although they can-
not be solved using the traditional rules for Geometric
Constructions.

see also FOLDING, GEOMETRIC CONSTRUCTION, SXOM-
ACHION, TANGRAM

References

Andersen, E. "Origami on the Web." http://www.netspace.

org/users/ema/oriweb.html.
Eppstein, D. "Origami." http://www . ics . uci . edu / -

eppstein/ junkyard/origami. html.
Geretschlager, R. "Euclidean Constructions and the Geome-
try of Origami." Math. Mag. 68, 357-371, 1995.
Gurkewitz, R. and Arnstein, B. 3-D Geometric Origami.

New York: Dover, 1996.
Kasahara, K. Origami Omnibus. Tokyo: Japan Publications,

1988.
Kasahara, K. and Takahara, T. Origami for the Connoisseur.

Tokyo: Japan Publications, 1987.
Palacios, V. Fascinating Origami: 101 Models by Alfredo

Cerceda. New York: Dover, 1997.
Pappas, T. "Mathematics &c Paperfolding." The Joy of

Mathematics. San Carlos, CA: Wide World Publ./Tetra,

pp. 48-50, 1989.
Row, T. S. Geometric Exercises in Paper Folding. New York:

Dover, 1966.
Tomoko, F. Unit Origami. Tokyo: Japan Publications, 1990.
Wu, J. "Joseph Wu's Origami Page." http://www.datt.co.

jp/Origami.

Origin

The central point (r = 0) in POLAR COORDINATES, or
the point with all zero coordinates (0, . . . , 0) in CARTE-
SIAN Coordinates. In 3-D, the a>Axis, y-Axis, and
z-AxiS meet at the origin.

see also Octant, Quadrant, z-Axis, y-Axis, z-Axis

Ornstein's Theorem

An important result in ERGODIC THEORY. It states that
any two "Bernoulli schemes" with the same MEASURE-
Theoretic Entropy are Measure-Theoretically
Isomorphic,

see also ERGODIC THEORY, ISOMORPHISM, MEASURE

Theory

Orr's Theorem

If

(1 - Z )«+^-^ 2 F 1 (2a,2/3;2 7 ;z) = ^a n z n , (1)

where 2^1(^,6; c; z) is a Hypergeometric Function,
then

i Fi(a,0; T ,z)2F 1 {-y-a+ \,i-fi + \;i + l;z)

Y, a n z n . (2)

(7+2)n/(7+l)n

Furthermore, if

(1 - zf+P-'-i/ 2 2 Fx(2a - 1, 2/3; 2 7 - 1; z) = £} a n z n ,

(3)
then

2 Fi(a,/3; 7 ;2)r(7-a+§,7-/?-f;7;2)

(7"|)n/(7)n

where T(z) is the Gamma Function.
Orthic Triangle

Given a Triangle A^4iA 2 A 3 , the Triangle
AH1H2H3 with Vertices at the feet of the Altitudes

Orthobicupola

Orthocenter 1285

(perpendiculars from a point to the sides) is called the
orthic triangle. The three lines AiHi are CONCURRENT
at the Orthocenter H of AAiA 2 j4 3 .

The centroid of the orthic triangle has TRIANGLE CEN-
TER Function

a = a cos(f? — C)

(Casey 1893, Kimberling 1994). The ORTHOCENTER of
the orthic triangle has TRIANGLE CENTER FUNCTION

a = cos(2A)cos(JB - C)

(Casey 1893, Kimberling 1994). The Symmedian
Point of the orthic triangle has Triangle Center
Function

a = tan A cos(B - C)

(Casey 1893, Kimberling 1994).

see also ALTITUDE, FAGNANO'S PROBLEM, ORTHOCEN-
TER, Pedal Triangle, Schwarz's Triangle Prob-
lem, Symmedian Point

References

Casey, J. A Treatise on the Analytical Geometry of the Point,

Line, Circle, and Conic Sections, Containing an Account

of Its Most Recent Extensions, with Numerous Examples,

2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 9,

1893.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.

Washington, DC: Math. Assoc. Amer., pp. 9 and 16-18,

1967.
Kimberling, C. "Central Points and Central Lines in the

Plane of a Triangle." Math. Mag. 67, 163-187, 1994,

The intersection H of the three Altitudes of a Trian-
gle is called the orthocenter. Its Trilinear Coordi-
nates are

cos B cos C : cos C cos A : cos A cos B.

(i)

If the Triangle is not a Right Triangle, then (1)
can be divided through by cos A cos B cos C to give

sec A : sec B : sec C.

(2)

If the triangle is Acute, the orthocenter is in the interior
of the triangle. In a Right Triangle, the orthocenter
is the Vertex of the Right Angle.

The ClRCUMCENTER O and orthocenter H are ISOGO-
nal Conjugate points. The orthocenter lies on the
Euler Line.

Orthobicupola

A BlCUPOLA in which the bases are in the same orien-
tation.

see also Pentagonal Orthobicupola, Square Or-
thobicupola, Triangular Orthobicupola

Orthobirotunda

A BlROTUNDA in which the bases are in the same orien-
tation.

Orthocenter

ai +a 2 2 + a 3 +Aiff + A 2 H + A 3 H = 12iT (3)

AxH + A 2 H + A 3 H = 2(r 4- R),

AiH +A 2 H +A 3 H =4R*-4Rr,

(4)
(5)

where r is the Inradius and R is the Circumradius
(Johnson 1929, p. 191).

Any HYPERBOLA circumscribed on a TRIANGLE and
passing through the orthocenter is RECTANGULAR, and
has its center on the NlNE-PoiNT CIRCLE (Falisse 1920,
Vandeghen 1965).

see also CENTROID (TRIANGLE), ClRCUMCENTER, EU-

ler Line, Incenter, Orthic Triangle, Orthocen-
tric Coordinates, Orthocentric Quadrilateral,
Orthocentric System, Polar Circle

References

Altshiller-Court, N. College Geometry: A Second Course in

Plane Geometry for Colleges and Normal Schools, 2nd ed.

New York: Barnes and Noble, pp. 165-172, 1952.
Carr, G. S. Formulas and Theorems in Pure Mathematics,

2nd ed. New York: Chelsea, p. 622, 1970.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.

Washington, DC: Math. Assoc. Amer., pp. 36-40, 1967.
Dixon, R. Mathographics. New York: Dover, p. 57, 1991.
Falisse, V. Cours de geometrie analytique plane. Brussels,

Belgium: Office de Publicity 1920.

1286

Orthocentric Coordinates

Orthogonal Basis

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 165-172 and 191, 1929.

Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.

Kimberling, C. "Orthocenter." http : //www . evansville .
edu/-ck6/tcenters/class/orthocn.html.

Vandeghen, A. "Some Remarks on the Isogonal and Cevian
Transforms. Alignments of Remarkable Points of a Trian-
gle." Amer. Math. Monthly 72, 1091-1094, 1965.

Orthocentric Coordinates

Coordinates denned by an Orthocentric System.

see also Trilinear Coordinates

Orthocentric Quadrilateral

If two pairs of opposite sides of a Complete Quadri-
lateral are pairs of Perpendicular lines, the Quad-
rilateral is said to be orthocentric. In such a case,
the remaining sides are also PERPENDICULAR.

Orthocentric System

A set of four points, one of which is the ORTHOCEN-
TER of the other three. In an orthocentric system, each
point is the Orthocenter of the TRIANGLE of the
other three, as illustrated above. The INCENTER and
ExCENTERS of a TRIANGLE are an orthocentric system.
The centers of the ClRCUMClRCLES of an orthocentric
system form another orthocentric system congruent to
the first. The sum of the squares of any nonadjacent
pair of connectors of an orthocentric system equals the
square of the Diameter of the ClRCUMClRCLE. Or-
thocentric systems are used to define ORTHOCENTRIC
Coordinates.

The four ClRCUMClRCLES of points in an orthocentric
system taken three at a time (illustrated above) have
equal Radius.

The four triangles of an orthocentric system have a com-
mon Nine-Point Circle, illustrated above.

see also ANGLE BISECTOR, ClRCUMClRCLE, CYCLIC

Quadrangle, Nine-Point Circle, Orthic Trian-
gle, Orthocenter, Orthocentric System, Polar
Circle

References

Altshiller-Court, N. College Geometry: A Second Course in

Plane Geometry for Colleges and Normal Schools, 2nd ed.

New York: Barnes and Noble, pp. 109-114, 1952.
Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 165-176, 1929.

Orthocupolarotunda

A CUPOLAROTUNDA in which the bases are in the same

orientation.

see also Gyrocupolarotunda, Pentagonal Or-

THOCUPOLARONTUNDA

Orthodrome

see Great Circle

Orthogonal Array

An orthogonal array OA(fc, s) is a k x s 2 Array with
entries taken from an s-set S having the property that
in any two rows, each ordered pair of symbols from S
occurs exactly once.

References

Colbourn, C. J. and Dinitz, J. H. (Eds,) CRC Handbook

of Combinatorial Designs, Boca Raton, FL: CRC Press,

p. Ill, 1996.

Orthogonal Basis

A Basis of vectors x which satisfy

XjX k = CjkSjk

where Cjk, C£ are constants (not necessarily equal to
1) and Sjk is the Kronecker Delta.

see also Basis, Orthonormal Basis

Orthogonal Circles
Orthogonal Circles

Orthogonal circles are Orthogonal Curves, i.e., they
cut one another at RIGHT Angles. Two CIRCLES with
equations

z + y + 2gx + 2/y + c =

are orthogonal if

2<?</ + 2//' = c + c'.

(1)

z 2 4- Z/ 2 + 2g'x + 2/'y + c = (2)

(3)

Orthogonal Matrix 1287

Orthogonal Group

see General Orthogonal Group, Lie-Type
Group, Orthogonal Rotation Group, Projective
General Orthogonal Group, Projective Special
Orthogonal Group, Special Orthogonal Group

References

Wilson, R. A. "ATLAS of Finite Group Representation."

http://for.mat.bham.ac.Uk/atlas#orth.

Orthogonal Group Representations

Two representations of a GROUP x% an d Xj are sa ^ to
be orthogonal if

Y,Xi{R)xi(R) = o

for i ^ j, where the sum is over all elements R of the
representation.

see also GROUP

A theorem of Euclid states that, for the orthogonal cir-
cles in the above diagram,

OPxOQ = OT 2

(4)

(Dixon 1991, p. 65).

References

Dixon, R. Mathographics. New York: Dover, pp. 65-66, 1991.

Euclid. The Thirteen Books of the Elements, 2nd ed. un-
abridged, Vol 3: Books X-XIII New York: Dover, p. 36,
1956.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., p. xxiv, 1995.

Orthogonal Curves

Two intersecting curves which are PERPENDICULAR at
their INTERSECTION are said to be orthogonal.

Orthogonal Functions

Two functions f(x) and g(x) are orthogonal on the in-
terval a < x < b if

(f(x)\9(x))= / f(x)g(x)dx = 0.

I

J a

Orthogonal Lines

Two or more Lines or Line Segments which are Per-
pendicular are said to be orthogonal.

Orthogonal Matrix

Any ROTATION can be given as a composition of rota-
tions about three axes (Euler's Rotation Theorem),
and thus can be represented by a 3 x 3 MATRIX operating
on a Vector,

(i)

We wish to place conditions on this matrix so that it
is consistent with an ORTHOGONAL TRANSFORMATION
(basically, a Rotation or Rotoinversion).

In a Rotation, a Vector must keep its original length,
so it must be true that

vr

'an

ai2

o>iz'

~X!~

x 2

—

&21

«22

023

X 2

_^3.

_a3i

0LZ2

033.

_£3_

/ /

(2)

see also ORTHOGONAL POLYNOMIALS, ORTHONORMAL

Functions

for i — 1, 2, 3, where EINSTEIN SUMMATION is being
used. Therefore, from the transformation equation,

{aijXj)(aikXk) — XiXi. (3)

This can be rearranged to

aij(xjaik)xk = aij(aikXj)x k

— — Q/%jQ/%foXjXfc == XiXi. \ )

In order for this to hold, it must be true that

aijaik — Sjk (5)

1288 Orthogonal Matrix

Orthogonal Polynomials

for j,k = 1, 2, 3, where Sij is the Kronecker Delta.
This is known as the ORTHOGONALITY CONDITION, and
it guarantees that

and

A T A = I,

(6)

(7)

where A T is the Matrix Transpose and I is the Iden-
tity Matrix. Equation (7) is the identity which gives
the orthogonal matrix its name. Orthogonal matrices
have special properties which allow them to be manip-
ulated and identified with particular ease.

Let A and B be two orthogonal matrices. By the Or-
thogonality Condition, they satisfy

and

Q'ijO'ik — Ujk j

bijbik = Sjkj

(8)

(9)

where 8n is the Kronecker Delta. Now

CijCik = (ab)ij(ab)jk = a>isb 3 jaitb t k = ai 3 aitb s jb t k

= Sstbsjbtk = hjbtk = Sjkj (10)

so the product C = AB of two orthogonal matrices is
also orthogonal.

The EIGENVALUES of an orthogonal matrix must satisfy
one of the following:

1. All Eigenvalues are 1.

2. One Eigenvalue is 1 and the other two are -1.

3. One Eigenvalue is 1 and the other two are Com-
plex Conjugates of the form e %e and e~ ld .

An orthogonal MATRIX A is classified as proper (corre-
sponding to pure Rotation) if

det(A) = 1,

(11)

where det(A) is the DETERMINANT of A, or improper
(corresponding to inversion with possible rotation; Ro-
TOINVERSION) if

det(A) = -1. (12)

see also Euler's Rotation Theorem, Orthogonal
Transformation, Orthogonality Condition, Ro-
tation, Rotation Matrix, Rotoinversion

References

Arfken, G. "Orthogonal Matrices." Mathematical Methods
for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 191-205, 1985.

Goldstein, H. "Orthogonal Transformations." §4—2 in Clas-
sical Mechanics, 2nd ed. Reading, MA: Addis on- Wesley,
132-137, 1980.

Orthogonal Polynomials

Orthogonal polynomials are classes of POLYNOMIALS
{p n (x)} over a range [a, 6] which obey an Orthogo-
nality relation

J a

W(x)pm(x)p n (x) dx = SmnCn

(1)

where w(x) is a WEIGHTING FUNCTION and 5 is the
Kronecker Delta. If c m = 1, then the Polynomials
are not only orthogonal, but orthonormal.

Orthogonal polynomials have very useful properties in
the solution of mathematical and physical problems.
Just as Fourier Series provide a convenient method of
expanding a periodic function in a series of linearly inde-
pendent terms, orthogonal polynomials provide a natu-
ral way to solve, expand, and interpret solutions to many
types of important DIFFERENTIAL EQUATIONS. Orthog-
onal polynomials are especially easy to generate using
Gram-Schmidt Orthonormalization. Abramowitz
and Stegun (1972, pp. 774-775) give a table of common
orthogonal polynomials.

Type

Interval

w(x)

c n

Chebyshev First

[-i,i]

(1-tf 2 )-

-1/2

Kind

[-i.i]

f for n -
I otherwise

Chebyshev Second

VI - x 2

h-

Kind

Hermite

( — oo, oo)

e- 2

V^2 n n!

Jacobi

(-1,1)

(l-x)°

(i + *y

h n

Laguerre
Laguerre

[0,oo)
[0,oo)

e~ x
x e

1

(Associated)

Legendre
Ultraspherical

[-1,1]
[-1,1]

1

(l_ x 2 )a -l/2

2

2n+l

f n2 1 ~ 2a r(n + 2a)
J n!(« + a)[r<«)]>

f for a ^
I for a =

In the above table, the normalization constant is the
value of

C n = W(x)\p n (x)] 2 dx

and

h n =

)a+/3+l

r(n-ha-r-l)r(n + /?+l)

2n + a + (3 + 1 nW(n + a + (3 + 1)

(2)

(3)

where T(z) is a Gamma FUNCTION.

The ROOTS of orthogonal polynomials possess many
rather surprising and useful properties. For instance,
let xi < X2 < ■ . . < x n be the Roots of the p n (x) with
Xq = a and Xn+x = b. Then each interval [aVjav+i] for
v = 0, 1, . . . , n contains exactly one ROOT of p n +i(x).
Between two Roots of p n (x) there is at least one ROOT
of pm(x) for m > n.

Orthogonal Polynomials

Let c be an arbitrary Real constant, then the Poly-
nomial

Pn+l(x) ~Cp n (x) (4)

has n + 1 distinct REAL ROOTS. If c > (c < 0), these
ROOTS lie in the interior of [a, 6], with the exception of
the greatest (least) ROOT which lies in [a, b] only for

c<

Pn+l(b)
Pn(b)

c>

Pn+i(a)
Pn(a)

(5)

The following decomposition into partial fractions holds

Pn{x) \~^ l u

Pn+l{x)

E ly

(6)

where {£„} are the ROOTS of p n +i(x) and

Pn+l(£i>)Pn(&) -Pn(&)'Pn+l(&)

1„ =

> 0. (7)

Another interesting property is obtained by letting
{p n (x)} be the orthonormal set of POLYNOMIALS asso-
ciated with the distribution da(x) on [a, 6]. Then the
CONVERGENTS i? n /5 n of the CONTINUED FRACTION

c 2

A!X + Bi A 2 x + £ 2 ^ 3 x + B z

C n

A n x + B r

■ + ..

(8)

are given by
R n = R n (x)

CO

-3/2

/ n /" Pn(x) ~Pn(Q J# ^

VC0C2 - ci 2 / — _ da(t)

J a

(9)

S n = S n (x) = VcoPn(a;),

where n = 0, 1,

. and

J a

da(x).

(10)

(11)

Furthermore, the ROOTS of the orthogonal polynomials
p n (x) associated with the distribution da(x) on the in-
terval [a, b] are REAL and distinct and are located in the
interior of the interval [a, b] .

see also CHEBYSHEV POLYNOMIAL OF THE FIRST KIND,

chebyshev polynomial of the second kind,
Gram-Schmidt Orthonormalization, Hermite
Polynomial, Jacobi Polynomial, Krawtchouk
Polynomial, Laguerre Polynomial, Legendre
Polynomial, Orthogonal Functions, Spherical

Orthogonal Rotation Group 1289

Harmonic, Ultraspherical Polynomial, Zernike
Polynomial

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal
Polynomials." Ch. 22 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. "Orthogonal Polynomials." Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 520-521, 1985.

Iyanaga, S. and Kawada, Y. (Eds.). "Systems of Orthog-
onal Functions." Appendix A, Table 20 in Encyclopedic
Dictionary of Mathematics. Cambridge, MA: MIT Press,
p. 1477, 1980.

Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical
Orthogonal Polynomials of a Discrete Variable. New York:
Springer- Verlag, 1992.

Sansone, G. Orthogonal Functions. New York: Dover, 1991.

Szego, G. Orthogonal Polynomials, l^th ed. Providence, RI:
Amer. Math. Soc, pp. 44-47 and 54-55, 1975.

Orthogonal Projection

A Projection of a figure by parallel rays. In such a pro-
jection, tangencies are preserved. Parallel lines project
to parallel lines. The ratio of lengths of parallel segments
is preserved, as is the ratio of areas.

Any Triangle can be positioned such that its shadow
under an orthogonal projection is Equilateral. Also,
the Medians of a Triangle project to the Medians
of the image Triangle. Ellipses project to Ellipses,
and any ELLIPSE can be projected to form a CIRCLE.
The center of an ELLIPSE projects to the center of the
image Ellipse, The Centroid of a Triangle projects
to the CENTROID of its image. Under an ORTHOGO-
NAL Transformation, the Midpoint Ellipse can be
transformed into a Circle Inscribed in an Equilat-
eral Triangle.

Spheroids project to Ellipses (or Circle in the De-
generate case).
see also PROJECTION

Orthogonal Rotation Group

Orthogonal rotation groups are Lie GROUPS. The or-
thogonal rotation group Os(n) is the set ofnxn REAL
Orthogonal Matrices.

The orthogonal rotation group 0% (n) is the set of n x
n Real Orthogonal Matrices (having n(n - l)/2
independent parameters) with Determinant — 1.

The orthogonal rotation group 0^(n) is the set ofnxn
Real Orthogonal Matrices, having n(n-l)/2 inde-
pendent parameters, with DETERMINANT +1. 0^(n) is

1290 Orthogonal Tensors

Orthographic Projection

HOMEOMORPHIC with 527(2). Its elements can be writ-
ten using EULER ANGLES and ROTATION MATRICES as

1 =

*1

.0

0'

1
1.

(i)

R*{4>) =

"1

_0

"

cos <j> sin <j>
— sin <f> cos <f> .

(2)

Ry(0) =

~cos# — sin0"

1
_sin# cos#

(3)

RmW =

cos^ sin^

— sin rp cos i/>

1

(4)

References

Arfken, G. "Orthogonal Group, 0^~." Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
p. 252-253, 1985.

Wilson, R. A. "ATLAS of Finite Group Representation."
http : //for . mat . bham . ac . uk/atlas#orth.

Orthogonal Tensors

Orthogonal CONTRAVARIANT and COVARIANT satisfy

9ik9 %3 = <*£,

where 5% is the KRONECKER DELTA.
see also Contravariant Tensor, Covariant Ten-
sor

Orthogonal Transformation

Any linear transformation

Xi = anXi -h CL12X2 + #13^3
x 2 = 0*21X1 + CL22X2 + 023^3

£3 = G3i#i + a32#2 -f 033^3
satisfying the ORTHOGONALITY CONDITION

Q>ijQ>ik ~ Ojfe,

where EINSTEIN SUMMATION has been used and Sij is
the KRONECKER Delta, is called an orthogonal trans-
formation.

Orthogonal transformations correspond to rigid ROTA-
TIONS (or ROTOINVERSIONS), and may be represented
using Orthogonal Matrices. If A : W 1 -» W 1 is an
orthogonal transformation, then det(A) = ±1.

see also AFFINE TRANSFORMATION, ORTHOGONAL MA-
TRIX, Orthogonality Condition, Rotation, Ro-

TOINVERSION

References

Goldstein, H. "Orthogonal Transformations." §4—2 in Clas-
sical Mechanics, 2nd ed. Reading, MA: Addison-Wesley,
132-137, 1980.

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 104, 1993.

Orthogonal Vectors

Two vectors u and v whose DOT PRODUCT is u • v =
(i.e., the vectors are Perpendicular) are said to be
orthogonal. The definition can be extended to three or
more vectors which are mutually PERPENDICULAR.

see also Dot Product, Perpendicular

Orthogonality Condition

A linear transformation

X 1 = anXi + CI12X2 + #13^3
X 2 = 021^1 + «22#2 + &23#3
X3 = a^iXi + a32#2 + O33X3,

is said to be an ORTHOGONAL TRANSFORMATION if it
satisfies the orthogonality condition

CLijdik = 8jk,

where EINSTEIN Summation has been used and 8^ is

the KRONECKER DELTA.

see also ORTHOGONAL TRANSFORMATION

References

Goldstein, H. "Orthogonal Transformations." §4-2 in Clas-
sical Mechanics, 2nd ed. Reading, MA: Addison-Wesley,
132-137, 1980.

Orthogonality Theorem

see Group Orthogonality Theorem

Orthographic Projection

A projection from infinity which preserves neither Area
nor angle.

x = cos<£sin(A — Ao)

y = cos <j>i sin <f> — sin <pi cos (f> cos(A — Ao) .

(i)

(2)

The inverse FORMULAS are

4> = sin I cos c sin <f>\ +

sin f 1

y sin c cos

M

(3)

— if a? sine \

A = Ao + tan ^— — : — ,

y p cos <pi cos c — y sin <pi sm c J

(4)

Orthologic

where

p = yjx 1 + y 2
c = sin" p.

(5)
(6)

References

Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 145-153, 1987.

Orthologic

Two Triangles A1B1C1 and A 2 B 2 C 2 are orthologic
if the perpendiculars from the VERTICES Ai, £?i, Ci
on the sides B 2 C 2 , A 2 C 2 , and A 2 B 2 pass through one
point. This point is known as the orthology center of
Triangle 1 with respect to Triangle 2.

Orthonormal Basis

A Basis of Vectors x which satisfy

and

XjXk — Ojk

X X-y — Oy ,

where 5jk is the KRONECKER Delta. An orthonormal
basis is a normalized ORTHOGONAL Basis.

see also Basis, Orthogonal Basis

Orthonormal Functions

A pair of functions <f>i and <pj are orthonormal if they
are ORTHOGONAL and each normalized. These two con-
ditions can be succinctly written as

J a

<f>i(x)<f)j(x)w(x)dx = Jij,

where w(x) is a WEIGHTING FUNCTION and Sij is the
Kronecker Delta.

see also ORTHOGONAL POLYNOMIALS

Orthonormal Vectors

Unit Vectors which are Orthogonal are said to be
orthonormal.

see also ORTHOGONAL VECTORS

Orthopole

If perpendiculars are dropped on any line from the ver-
tices of a Triangle, then the perpendiculars to the
opposite sides from their Feet are CONCURRENT at a
point called the orthopole.

References

Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, p. 247, 1929.

Osborne's Rule 1291

Orthoptic Curve

An Isoptic Curve formed from the locus of Tan-
gents meeting at Right ANGLES. The orthoptic of
a Parabola is its Directrix. The orthoptic of a cen-
tral CONIC was investigated by Monge and is a Circle
concentric with the CONIC SECTION. The orthoptic of
an Astroid is a CIRCLE.

Curve

Orthoptic

astroid
cardioid
deltoid

logarithmic spiral
parabola

quadrifolium

circle or limagon

circle

equal logarithmic spiral

directrix

References

Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 58 and 207, 1972.

Orthotomic

Given a source S and a curve 7, pick a point on 7
and find its tangent T. Then the Locus of reflections
of 5 about tangents T is the orthotomic curve (also
known as the secondary CAUSTIC). The INVOLUTE of
the orthotomic is the CAUSTIC. For a parametric curve
(f{t),g(t)) with respect to the point (a?o,2/o)j the ortho-
tomic is

2g'[f'(9 -yo)-9'(f-x )]

X = Xq —

y = yo +

f2+g*2

2f t [f , (9~yo)-9 t (f-x )}
f t2 +9' 2

see also CAUSTIC, INVOLUTE

References

Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, p. 60, 1972.

Orthotope

A Parallelotope whose edges are all mutually PER-
PENDICULAR. The orthotope is a generalization of the
Rectangle and Rectangular Parallelepiped.

see also RECTANGLE, RECTANGULAR PARALLELEPIPED

References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, pp. 122-123, 1973.

Osborne's Rule

The prescription that a TRIGONOMETRY identity can
be converted to an analogous identity for HYPERBOLIC
FUNCTIONS by expanding, exchanging trigonometric
functions with their hyperbolic counterparts, and then
flipping the sign of each term involving the product of
two Hyperbolic Sines. For example, given the iden-
tity

cos(x — y) — cos x cos y + sin x sin y,

Osborne's rule gives the corresponding identity

cosh(x — y) = cosh x cosh y — sinhassinhy.
see also Hyperbolic Functions, Trigonometry

1292 Oscillation

Osculating Sphere

Oscillation

The variation of a FUNCTION which exhibits SLOPE
changes, also called the Saltus of a function.

Oscillation Land

see Carotid-Kundalini Function

Osculating Circle

Osculating Curves

The CIRCLE which shares the same TANGENT as a curve
at a given point. The RADIUS OF CURVATURE of the
osculating circle is

p(t) =

i*(*)i'

where k is the CURVATURE, and the center is

,_, (/' 2 +g'V

7 f'9" ~ f"9'

,,_„. (/' 2 + g'V

v ~ 9 f'9"-f"9"

(i)

(2)
(3)

i.e., the centers of the osculating circles to a curve form
the EVOLUTE to that curve.

In addition, let C(ti,t 2 ,h) denote the CIRCLE passing
through three points on a curve (f(t),g(t)) with t\ <
£2 < £3. Then the osculating circle C is given by

lim C(*i,t2,*3)

(4)

(Gray 1993).

see also Curvature, Evolute, Radius of Curva-
ture, Tangent

References

Gardner, M. "The Game of Life, Parts I-III." Chs. 20-22 in
Wheels, Life, and other Mathematical Amusements. New
York: W. H. Freeman, pp. 221, 237, and 243, 1983.

Gray, A. "Osculating Circles to Plane Curves." §5.6 in Mod-
ern Differential Geometry of Curves and Surfaces. Boca
Raton, FL: CRC Press, pp. 90-95, 1993.

-1.5 -1 -0.5 0.5 1 1.5

An osculating curve to f(x) at xo is tangent at that point
and has the same CURVATURE. It therefore satisfies

y (k) (xo) = f {k) (x )

for k — 0, 1, 2. The point of tangency is called a Tac-
NODE. The simplest example of osculating curves are x 2
and x 4 , which osculate at the point xo = 0.

see also TACNODE

Osculating Interpolation

see Hermite's Interpolating Fundamental Poly-
nomial

Osculating Plane

The PLANE spanned by the three points x(t), x(£ + /ii),
and x(t + /i2) on a curve as hi, /12 — > 0. Let z be a point
on the osculating plane, then

[(z-x),x',x"] = 0,

where [A, B, C] denotes the Scalar Triple Product.
The osculating plane passes through the tangent. The
intersection of the osculating plane with the NORMAL
Plane is known as the Principal Normal Vector.
The Vectors T and N (Tangent Vector and Nor-
mal Vector) span the osculating plane.

see also NORMAL VECTOR, OSCULATING SPHERE,

Scalar Triple Product, Tangent Vector

Osculating Sphere

The center of any SPHERE which has a contact of (at
least) first-order with a curve C at a point P lies in the
normal plane to C at P. The center of any SPHERE
which has a contact of (at least) second-order with C at
point P, where the CURVATURE k > 0, lies on the polar
axis of C corresponding to P. All these SPHERES inter-
sect the Osculating Plane of C at P along a circle of
curvature at P. The osculating sphere has center

a = x + pN+ -B

T

Osedelec Theorem

Ovals of Cassini 1293

where N is the unit NORMAL VECTOR, B is the unit
Binormal Vector, p is the Radius of Curvature,
and r is the TORSION, and RADIUS

p 2 +

(*)•

and has contact of (at least) third order with C.
see also Curvature, Osculating Plane, Radius of
Curvature, Sphere, Torsion (Differential Ge-
ometry)

References

Kreyszig, E. Differential Geometry. New York: Dover,
pp. 54-55, 1991.

Osedelec Theorem

For an n-D MAP, the LYAPUNOV CHARACTERISTIC EX-
PONENTS are given by

<n = lim ln|Ai(JV)|

N— voo

for i = 1, . . . , n, where A* is the LYAPUNOV CHARAC-
TERISTIC Number.

see also Lyapunov Characteristic Exponent, Lya-
punov Characteristic Number

Ostrowski's Inequality

Let f(x) be a monotonic function integrable on [a, 6] and
let f(a),f(b) < Oand |/(a)| > |/(6)|, then if g is a Real
function integrable on [a, 6],

1/

f{x)g(x) dx

max /

— s — \J a

< \f(a)\ max / g(x)dx

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1100, 1979.

Ostrowski's Theorem

Let A = aij be a Matrix with Positive Coefficients
and A be the Positive Eigenvalue in the Frobenius
Theorem, then the n - 1 Eigenvalues A-, ^ A satisfy
the Inequality

I Aj| < A

M 2 -m 2
M 2 +m 2 '

where

M = max a^

id

m = min aij

and i,j — 1, 2, . . . , n.

see also Frobenius Theorem

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1121, 1980.

Otter's Tree Enumeration Constants

see Tree

Outdegree

The number of outward directed EDGES from a given
Vertex in a Directed Graph.

see also DIRECTED GRAPH, INDEGREE, LOCAL DEGREE

Outer Automorphism Group

A particular type of AUTOMORPHISM Group which
exists only for GROUPS. For a Group G, the
outer automorphism group is the QUOTIENT Group
Aut(G)/Inn(G), which is the Automorphism Group
of G modulo its Inner Automorphism Group.

see also AUTOMORPHISM GROUP, INNER AUTOMOR-
PHISM Group, Quotient Group

Outer Product

see Direct Product (Tensor)

Oval

An oval is a curve resembling a squashed CIRCLE but,
unlike the ELLIPSE, without a precise mathematical def-
inition. The word oval derived from the Latin word
"ovus" for egg. Unlike ellipses, ovals sometimes have,
only a single axis of reflection symmetry (instead of two).

Ovals can be constructed with a COMPASS by joining to-
gether arcs of different radii such that the centers of the
arcs lie on a line passing through the join point (Dixon
1991). Albrecht Diirer used this method to design a
Roman letter font.

see also CARTESIAN OVALS, CASSINI OVALS, EGG, EL-
LIPSE, Ovoid, Superellipse

References

Critchlow, K. Time Stands Still. London: Gordon Praser,
1979.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., 1989.

Dixon, R. Mathographics. New York: Dover, pp. 3—11, 1991.

Dixon, R. "The Drawing Out of an Egg." New Sci. y July 29,
1982.

Pedoe, D. Geometry and the Liberal Arts. London: Pere-
grine, 1976.

Oval of Descartes

see CARTESIAN OVALS

Ovals of Cassini

see Cassini Ovals

1294 Overlapping Resonance Method Ovoid

Overlapping Resonance Method

see Resonance Overlap Method

Oversampling

A signal sampled at a frequency higher than the
Nyquist Frequency is said to be oversampled /? times,
where the oversampling ratio is defined as

r* ^sampling

J^Nyquist

see also Nyquist Frequency, Nyquist Sampling

Ovoid

An egg-shaped curve. Lockwood (1967) calls the NEGA-
TIVE Pedal Curve of an Ellipse with Eccentricity
e < 1/2 an ovoid.

see also Oval

References

Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 157, 1967.

p-adic Number

p-adic Number 1295

p-adic Number

A p-adic number is an extension of the Field of Ra-
tional Numbers such that Congruences Modulo
Powers of a fixed PRIME p are related to proximity in
the so called "p-adic metric."

Any Nonzero Rational Number x can be represented
by

where p is a PRIME NUMBER, r and s are INTEGERS not
Divisible by p, and a is a unique Integer. Then define
the p-adic absolute value of x by

\x\ p =p .
Also define the p-adic value

|0|, = 0.

As an example, consider the Fraction

= 2 2 .3~ 3 -5-7-ll _1 .

It has p-adic absolute values given by

I 297 I 2 4

I MO,

I 297 I 3

I 140 I _ 1

I 297 I 5 5

I 140 1 _ 1

l297t 7 7

I 140

I 297

2 = 4

27

15 =

(2)

(3)

(4)

(5)
(6)
(7)
(8)
(9)

The p-adic absolute value satisfies the relations

1. |a;|p > for all cc,

2. \x\ p = Iff x = 0,

3. \xy\ p = \x\ p \y\ p for all x and y,

4. \x 4- y\ p < \x\ p 4- \y\ P for all x and y (the Triangle
Inequality), and

5. |x + 2/| p < max(|x| p , \y\ p ) for all x and y (the STRONG
Triangle Inequality).

In the above, relation 4 follows trivially from relation 5,
but relations 4 and 5 are relevant in the more general
Valuation Theory.

The p-adics were probably first introduced by Hensel
in 1902 in a paper which was concerned with the de-
velopment of algebraic numbers in POWER SERIES, p-
adic numbers were then generalized to VALUATIONS by
Kiirschak in 1913. In the early 1920s, Hasse formulated
the Local-Global Principle (now usually called the
Hasse Principle), which is one of the chief applica-
tions of LOCAL FIELD theory. Skolem's p-adic method,

which is used in attacking certain Diophantine Equa-
tions, is another powerful application of p-adic num-
bers. Another application is the theorem that the HAR-
MONIC Numbers H n are never Integers (except for
Hi). A similar application is the proof of the VON
Staudt- Clausen Theorem using the p-adic valuation,
although the technical details are somewhat difficult.
Yet another application is provided by the MAHLER-
Lech Theorem.

Every RATIONAL x has an "essentially" unique p-adic
expansion ( "essentially" since zero terms can always be
added at the beginning)

(10)

with m an INTEGER, a,j the INTEGERS between and
p — 1 inclusive, and where the sum is convergent with
respect to p-adic valuation. If x ^ and a m 7^ 0, then
the expansion is unique. Burger and Struppeck (1996)
show that for p a PRIME and n a POSITIVE INTEGER,

Inl^p-^'^^,

(11)

where the p-adic expansion of n is

n = a + aip + a 2 p 2 + . . . + cllP L , (12)

and

A p (n) = ao + ai + . . . + cll- (13)

For sufficiently large n,

|n!| p <p-" /(2p - 2) . (14)

The p-adic valuation on Q gives rise to the p-adic metric

d(x,y) = \x-y\ p , (15)

which in turn gives rise to the p-adic topology. It can
be shown that the rationals, together with the p-adic
metric, do not form a COMPLETE METRIC SPACE. The
completion of this space can therefore be constructed,
and the set of p-adic numbers Q is defined to be this
completed space.

Just as the REAL NUMBERS are the completion of the
Rationals Q with respect to the usual absolute valu-
ation \x — 3/|, the p-adic numbers are the completion of
Q with respect to the p-adic valuation \x — y\ p . The p-
adic numbers are useful in solving DIOPHANTINE EQUA-
TIONS. For example, the equation X 2 = 2 can easily be
shown to have no solutions in the field of 2-adic numbers
(we simply take the valuation of both sides). Because
the 2-adic numbers contain the rationals as a subset, we
can immediately see that the equation has no solutions
in the Ration als. So we have an immediate proof of
the irrationality of y/2.

1296

P-Circle

P-Polynomial

This is a common argument that is used in solving these
types of equations: in order to show that an equation
has no solutions in Q, we show that it has no solutions
in a Field Extension. For another example, consider
X 2 + l = 0. This equation has no solutions in Q because
it has no solutions in the reals R, and Q is a subset of

Now consider the converse. Suppose we have an equa-
tion that does have solutions in R and in all the Q p .
Can we conclude that the equation has a solution in Q?
Unfortunately, in general, the answer is no, but there are
classes of equations for which the answer is yes. Such
equations are said to satisfy the HASSE PRINCIPLE.
see also Ax-KOCHEN ISOMORPHISM THEOREM, DlO-

phantine Equation, Harmonic Number, Hasse
Principle, Local Field, Local-Global Principle,
Mahler-Lech Theorem, Product Formula, Val-
uation, Valuation Theory, von Staudt-Clausen
Theorem

References

Burger, E. B. and Struppeck, T. "Does .$^=0 ^r Really Con-
verge? Infinite Series and p-adic Analysis." Amer. Math.
Monthly 103, 565-577, 1996.

Cassels, J. W. S. and Scott, J. W. Local Fields. Cambridge,
England: Cambridge University Press, 1986.

Gouvea, F. Q. P-adic Numbers: An Introduction, 2nd ed.
New York: Springer- Verlag, 1997.

Koblitz, N. P-adic Numbers, P-adic Analysis, and Zeta-
Functions, 2nd ed. New York: Sp ringer- Ver lag, 1984.

Mahler, K. P-adic Numbers and Their Functions, 2nd ed.
Cambridge, England: Cambridge University Press, 1981.

P-Circle

see Spieker Circle

p-Element

see Semisimple

p-Good Path

A Lattice Path from one point to another is p-good if
it lies completely below the line

y=(p- l)x.

Hilton and Pederson (1991) show that the number of
p-good paths from (1, q — 1) to (fc, n — k) under the
condition 2 < k < n — p -f 1 < p(k — 1) is

n-q\ ST A [ n -pA

where (£) is a BINOMIAL COEFFICIENT, and

n — k
_p-l_

where \x\ is the FLOOR FUNCTION.

see also Catalan Number, Lattice Path, Schroder
Number

References

Hilton, P. and Pederson, J. "Catalan Numbers, Their Gener-
alization, and Their Uses." Math. Intel 13, 64-75, 1991.

p- Group

A Finite Group of Order p a for p a Prime is called
ap-group. Sylow proved that every GROUP of this form
has a PoWER-commutator representation on n genera-
tors defined by

n < {uk)

(i)

fc=i+i

for < (3{i y k) < p, 1 < i < n and

n
k=j+l

k)

(2)

for < /3(iJ,k) < p, 1 < i < j < n. If p is PRIME and
f(p) the number of GROUPS of order p m , then

f(p) = P J

where

lim A = £

(3)

(4)

(Higman 1960a,b).
see also Finite Group

References

Higman, G. "Enumerating p-Groups. I. Inequalities." Proc.

London Math. Soc. 10, 24-30, 1960a.
Higman, G. "Enumerating p-Groups. II. Problems Whose

Solution is PORC." Proc. London Math. Soc. 10, 566-

582, 1960b.

p'- Group

X is a p'-group if p does not divide the Order of X.

p-Layer

The p- layer of H, L p > (H) is the unique minimal NORMAL
Subgroup of H which maps onto E(H/O p >{H)).

see also ^-Theorem, IMBALANCE Theorem, Sig-
nalizer Functor Theorem

P-Polynomial

see HOMFLY Polynomial

P -Problem

Pade Approximant 1297

P-Problem

A problem is assigned to the P (POLYNOMIAL time) class
if the number of steps is bounded by a Polynomial.

see also Complexity Theory, NP-Complete Prob-
lem, NP-Hard Problem, NP-Problem

References

Borwein, J. M. and Borwein, P. B. Pi and the AGM: A Study
in Analytic Number Theory and Computational Complex-
ity. New York: Wiley, 1987.

Greenlaw, R.; Hoover, H. J.; and Ruzzo, W. L, Limits to
Parallel Computation: P- Completeness Theory. Oxford,
England: Oxford University Press, 1995.

p-Series

A shorthand name for a POWER SERIES with a NEGA-
TIVE exponent, J^^ &~ p , where p > 0.
see also Power Series, Riemann Zeta Function

p-Signature

Diagonalize a form over the rationals to

diag[p a • A,p b -B, ...],

where all the entries are INTEGERS and A, B, ... are
Relatively Prime to p. Then the p-signature of the
form (for p ^ —1, 2) is

p a +p 6 + ... + 4fc (mod 8),

where k is the number of ANTISQUARES. For p = — 1,
the p-signature is SYLVESTER'S SIGNATURE.

see also Signature (Quadratic Form)

Packing

The placement of objects so that they touch in some
specified manner, often inside a container with specified
properties.

see also Box-Packing Theorem, Circle Packing,
Groemer Packing, Hypersphere Packing, Ke-
pler Problem, Kissing Number Packing Density,
Polyhedron Packing, Space-Filling Polyhedron,
Sphere Packing

References

Eppstein, D. "Covering and Packing." http://www.ics.uci
.edu/~eppstein/ junkyard/cover, html.

Packing Density

The fraction of a volume filled by a given collection of

solids.

see also Hypersphere Packing, Packing, Sphere
Packing

Pade Approximant

Approximants derived by expanding a function as a ra-
tio of two Power Series and determining both the
Numerator and Denominator Coefficients. Pade
approximations are usually superior to TAYLOR EX-
PANSIONS when functions contain Poles, because the
use of Rational Functions allows them to be well-
represented.

The Pade approximant Rl/o corresponds to the MAC-
laurin SERIES. When it exists, the R L / M = [L/M]
Pade approximant to any Power Series

P-Symbol

A symbol employed in a formal Propositional Cal-
culus.

References

Nidditch, P. H. Propositional Calculus. New York: Free
Press of Glencoe, p. 1, 1962.

P-Value

The Probability that a variate would assume a value
greater than or equal to the observed value strictly by

Chance: P(z > ^observed)-

see also Alpha Value, Significance

A(x) = y ^CLjX J

3=0

(1)

is unique. If A{x) is a Transcendental Function,
then the terms are given by the TAYLOR SERIES about

Xq

a n = ±A (n) (x ). (2)

The Coefficients are found by setting

Qm(x)

(3)

Paasche's Index

The statistical Index

Pp

J2P0Qn'

where p n is the price per unit in period n and q n is the
quantity produced in period n.

see also INDEX

References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 65, 1962.

and equating COEFFICIENTS. Qm{x) can be multiplied
by an arbitrary constant which will rescale the other
Coefficients, so an addition constraint can be applied.
The conventional normalization is

Qm{0) = 1.

Expanding (3) gives

Pl(x) = po +pix + . . . +plx
Qm{x) = 1 + qix + . . . H- qMX M

(4)

(5)
(6)

1298 Fade Approximant

These give the set of equations

ao = po (7)

ai 4- a gi = p! (8)

a 2 + aiqi + a ^2 = P2 (9)

ai, + ai-i^i + . . . -h a q L = Pl (10)

az,+i -f ai,qi + . . . + aL-M+iQM = (11)

Ql+m + a>L+M-iqi + . . . + aLqM = 0, (12)

where a n = for n < and g^ = for j > M. Solving
these directly gives

<*£,-,

Ti + 1 O'L-m + 2

Ol+l

[L/M] = -

Q>L &L + 1
L L

j — M j = M — l

L

... Y* a * x *

3=0

&L-M + 1 &L-M + 2 •'•

&L + 1

X M X M~1

1

(13)
where sums are replaced by a zero if the lower index
exceeds the upper. Alternate forms are

[L/M] = J2 a > xJ +^" M+1 wI /M W- / 1 M w VM
i=o

L+n

Ej . L+n+1 T \A/-1

a^a; +x w (L+M)/M W L/M w (L+n)/M

j'=o

for

w L/M

"a^-M+i ~ xa^-

az, — xaL+i

"a^-M+i"

^L-M+2

Wl/M =

a £

1

and

<n

<M.

aL — xcll+i

dL+M-1 - XClL+M

(14)
(15)

Fade Approximant

The first few Pade approximates for e" are

exp /o(a;) = 1
exp /i(aO = YZ^

eXp °/ 2(x) = 2-2x + S r»

r 6

exPo/sW - 6 _ 6a . + 3a;2 _ :c 3

exp 1/0 (x) = 1 + x

exPi/xCx) =

2 + 2
2 -x

exp 1/2 (a;) =

6 + 2a:

6 - Ax + a; 2

exp 1/3 (z) =

24 + 6a;

24 - 18a; + 6a; 2 - x 3

exp 2/0 (a;) =

2 + 2x-\-x 2
2

exp 2/1 (x) =

6 + 4a; + x 2
6 -2a;

exp 2/2 (z) =

12 + 6a; + x 2
12 - 6a; + x 2

exp 2/3 (a;) =

60 + 24a; + 3a; 2

60 - 36a; + 9a; 2 - x 3

exp 3/0 (x) =

6 + 6x + 3a? 2 4- x 3

6

ex P 3 /i(^) =

24 + l&x + 16a; 2 + x 3

24- 6a:

exp 3/2 (a;) =

60 + 36a; + 9a; 2 + x 3

60 - 24a; + 3a; 2

™ . r„\ _

120 + 60a: + 12a; 2 + a; 3

120 - 60a; + 12a; 2 - x s '

Two- term identities include

Pl+iJx) P' l {x) = C iL+1)/(M+1) 2 x L+M+1
Qm+i{x) Q' m (x) Qm+i(x)Q' m (x)

Pl+i(x) P'l(x) _ C(l+i)/mC( L+1 )/( M +i)X L+m+1

(16)

Qui.') Qui*)

QMix)Q' M {x)

(17)

Pl{x) P'l{x) __ Cjt/(M + l)C(i + i)/(M + l)iB'

£+M+l

Qm+i(e) Q' M (a;)

Qm(z)Q^(z)

-P&(aQ -Pl+iC^) _ g(Lj-l)/(Af+l)

2 L+M+2

a;

0M+i(ar) Q^

Pi+i ^i-iW

Qm+iQ^

(18)
(19)

C l L/(M + l)C'(L + l)/Mg I, " t " M + C , i/MC'(t + l)/(Af + l)a' I ' +M + 1

<?«(*)««(*)

(20)

Pade Approximant

Painleve Transcendents 1299

Pl(x) P' l (x)

Qm+i(x) Q' M -l( X )

Cl/(M + 1)C(L + 1)/MX L + M — C L / M C(L + l)/(M + l)X L + M + l

Qm+i(x)Q , m __ 1 (x)

(21)

where C is the C-Determinant. Three-term identities
can be derived using the Frobenius Triangle Iden-
tities (Baker 1975, p. 32).

A five-term identity is

2

fS(L-l'

(L + l)/MJ(L-l)/M " Z>L/(M + 1)&L/(M-1)

-i) = 5.

L/M

Cross ratio identities include

(Rl/M — Rl/(M+1))(R(L+1)/M — R(L+1)/(M+1))
(Rl/M — R(L+1)/m)(Rl/(M+1) — R(L+1)/{M+1))
= Cx/(M+l)C(I,+2)/(M+l)
<?(£+l)/M<?(£ + l)/(M+2)

(22)

(23)

Pade Conjecture

If P(z) is a POWER series which is regular for \z\ < 1
except for m POLES within this CIRCLE and except for
z = +1, at which points the function is assumed contin-
uous when only points \z\ < 1 are considered, then at
least a subsequence of the [AT, N] Pade APPROXIMANTS
are uniformly bounded in the domain formed by remov-
ing the interiors of small circles with centers at these
POLES and uniformly continuous at z = +1 for \z\ < 1.

see also Pade Approximant

References

Baker, G. A. Jr. "The Pade Conjecture and Some Con-
sequences." §11. D in Advances in Theoretical Physics,
Vol. 1 (Ed. K. A. Brueckner). New York: Academic Press,
pp. 23-27, 1965.

Padovan Sequence

The Integer Sequence defined by the Recurrence
Relation

(Rl/M — R(L+l)/(M+l))(R(L+l)/M — Rl/(M+1))
(Rl/M - Rl/(M+1))(R(L+1)/M - R(L+1)/(M+1))
_ C , (L+i)/(M+l) X

Cl/(M+1)C{ L +2)/(M+\)

(Rl/M ~ fl(L+l)/(M+l))(fl(L+l)/M ~ Rl/(M+1))
(Rl/M - R(L+1)/m)(Rl/(M+1) - R(L+1)/(M+1))

'(Z,+l)/M<^(L+l)/(M+2)

(Rl/m — R(l+i)/(m-i)){Rl/{m+i) — R(l+i)/m)
(Rl/m — Rl/(m+i))(R(l+i)/{m+i) - R(l+i)/m)

_ C(L+1)/mC(L+1)/(M+1)^

'L/{M+l)^(L+2)/M

(Rl/M - R(L-1)/{M+1))(R{L+1)/M - Rl/(M+1))

(Rl/m — R(l+i)/m)(R(l-i)/(m+i) — Rl/(m+i))
_ Cl/(m+i)C(l+i)/(m+i)£

C f (L+l)/MCL/(M+2)

(24)

(25)

(26)

(27)

see also C-Determinant, Economized Rational
Approximation, Frobenius Triangle Identities

References

Baker, G. A. Jr. "The Theory and Application of The Pade
Approximant Method." In Advances in Theoretical Phys-
ics, Vol. 1 (Ed. K. A, Brueckner), New York: Academic
Press, pp. 1-58, 1965.

Baker, G. A. Jr. Essentials of Pade Approximants in The-
oretical Physics. New York: Academic Press, pp. 27-38,
1975.

Baker, G. A. Jr. and Graves-Morris, P. Pade Approximants.
New York: Cambridge University Press, 1996.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Pade Approximants." §5.12 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 194-197, 1992.

P(n) = P(n - 2) + P(n - 3)

with the initial conditions P(Q) = P(l) = P(2) = 1.
The first few terms are 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, ...
(Sloane's A000931). The ratio lim n ^oo P(n)/P(n - 1)
is called the PLASTIC CONSTANT.

see also Perrin Sequence, Plastic Constant

References

Sloane, N. J. A. Sequence A000931/M0284 in "An On-Line

Version of the Encyclopedia of Integer Sequences."
Stewart, I. "Tales of a Neglected Number." Sci. Amer. 274,

102-103, June 1996.

Painleve Property

Following the work of Fuchs in classifying first-order
Ordinary Differential Equations, Painleve stud-
ied second-order ODEs of the form

dx 2

F(y\y,x) y

where F is ANALYTIC in x and rational in y and y' .
Painleve found 50 types whose only movable SINGULAR-
ITIES are ordinary POLES. This characteristic is known
as the Painleve property. Six of the transcendents de-
fine new transcendents known as Painleve Transcen-
dents, and the remaining 44 can be integrated in terms
of classical transcendents, quadratures, or the PAINLEVE
Transcendents.

see also PAINLEVE TRANSCENDENTS

Painleve Transcendents

y =<

y = 2y + xy + a

y' 2 x + „,* + & _u^

7 + oty -j H

y xy xy z x

(i)

(2)
(3)

1300

Pair

Paley Construction

Transcendents 4-6 do not have known first integrals, but
all transcendents have first integrals for special values of
their parameters except (1). Painleve found the above
transcendents (1) to (3), and the rest were investigated
by his students. The sixth transcendent was found by
Gambier and contains the other five as limiting cases.

see also PAINLEVE PROPERTY

Pair

A Set of two numbers or objects linked in some way are
said to be a pair. The pair a and b are usually denoted
(a, b). In certain circumstances, pairs are also called
Brothers or Twins.

see also Amicable Pair, Augmented Amicable
Pair, Brown Numbers, Friendly Pair, Hexad,
Homogeneous Numbers, Impulse Pair, Irregu-
lar Pair, Lax Pair, Long Exact Sequence of a
Pair Axiom, Monad, Ordered Pair, Perko Pair,
Quadruplet, Quasiamicable Pair, Quintuplet,
Reduced Amicable Pair, Smith Brothers, Triad,
Triplet, Twin Peaks, Twin Primes, Twins, Uni-
tary Amicable Pair, Wilf-Zeilberger Pair

Pair Sum

Given an Amicable Pair (m, n), the quantity

cr(m) = cr(n) — s(m) + s(n) = m + n

is called the pair sum, where a(n) is the DIVISOR FUNC-
TION and s(n) is the RESTRICTED DIVISOR FUNCTION.

see also Amicable Pair

Paired t-Test

Given two paired sets Xi and Yi of n measured values,
the paired i-test determines if they differ from each other
in a significant way. Let

Xi = {Xi — Xi)
Yi = {Yi - Y),

then define t by

t = (X-Y).

n(n — 1)

e; =1 (*<-^) 2

This statistic has n - 1 DEGREES OF FREEDOM.

A table of Student's ^-Distribution confidence in-
terval can be used to determine the significance level at
which two distributions differ.

see also Fisher Sign Test, Hypothesis Testing,
Student's £-Distribution, Wilcoxon Signed Rank
Test

References

Goulden, C. H. Methods of Statistical Analysis, 2nd ed. New
York: Wiley, pp. 50-55, 1956.

Paley Class

The Paley class of a POSITIVE INTEGER m = (mod 4)
is defined as the set of all possible Quadruples
(&, e, <?, n) where

m = 2 e (<f + 1),

q is an Odd Prime, and

k =

if 5 =

if q n - 3 = (mod 4)
2 if q n - 1 = (mod 4)

undefined otherwise .

see also HADAMARD MATRIX, PALEY CONSTRUCTION

Paley Construction

Hadamard Matrices H n can be constructed using
Galois Field GF(p m ) when p = 4£ - 1 and m is Odd.
Pick a representation r Relatively Prime to p. Then
by coloring white [(p - 1)/2J (where [a; J is the FLOOR
Function) distinct equally spaced Residues modp (r°,
r, r 2 , . . . ; r°, r 2 , r 4 , . . . ; etc.) in addition to 0, a HAD-
AMARD Matrix is obtained if the Powers of r (mod
p) run through < |_(p - 1)/ 2 J- For example,

n = 12 = ll 1 + 1 = 2(5 + 1) = 2 2 (2 + 1)

is of this form with p~ 11 = 4x3-1 and m = 1. Since
m = 1, we are dealing with GF(ll), so pick p = 2 and
compute its RESIDUES (mod 11), which are

P^2
p 2 =4
p^S
p 4 = 16 = 5
p 5 = 10
p 6 = 20 = 9

p 7 = 18 = 7
s - 1 A =

p 8 = 14 = 3

P

12 = 1.

Picking the first L n / 2 J = 5 RESIDUES and adding
gives: 0, 1, 2, 4, 5, 8, which should then be colored
in the Matrix obtained by writing out the RESIDUES
increasing to the left and up along the border (0 through
p— 1, followed by oo), then adding horizontal and vertical
coordinates to get the residue to place in each square.

Paley's Theorem

Palindromic Number Conjecture 1301

OOOOOOOOOOOOOOOOOOOOOO OO'

10

1

2

3

4

5

6

7

8

9

oo

9

10

1

2

3

4

5

6

7

8

OO

8

9

10

1

2

3

4

5

6

7

oo

7

8

9

10

1

2

3

4

5

6

oo

6

7

8

9

10

1

2

3

4

5

oo

5

6

7

8

9

10

1

2

3

4

oo

4

5

6

7

8

9

10

1

2

3

oo

3

4

5

6

7

8

9

10

1

2

oo

2

3

4

5

6

7

8

9

10

1

oo

1

2

3

4

5

6

7

8

9

10

oo

1

2

3

4

5

6

7

8

9

10

oo

where e is any POSITIVE INTEGER such that m =
(mod 4). If m is of this form, the matrix can be
constructed with a Paley CONSTRUCTION. If m is di-
visible by 4 but not of the form (1), the PALEY CLASS is
undefined. However, HADAMARD MATRICES have been
shown to exist for all m = (mod 4) for m < 428.

see also HADAMARD MATRIX, PALEY CLASS, PALEY

Construction

Palindrome Number

see Palindromic Number

Hie can be trivially constructed from H4 ® H4. H20
cannot be built up from smaller Matrices, so use n =
20 = 19 + 1 = 2(3 2 + 1) = 2 2 (2 2 + 1). Only the first
form can be used, with p = 19 = 4x5 — 1 and m = 1.
We therefore use GF(19), and color 9 Residues plus
white. H24 can be constructed from H2 <8> H3.2.

Now consider a more complicated case. For n = 28 =
3 3 -f 1 = 2(13 + 1), the only form having p = 41 - 1 is the
first, so use the GF(3 3 ) field. Take as the modulus the
Irreducible Polynomial a; 3 + 2a: + l, written 1021. A
four-digit number can always be written using only three
digits, since 1000-1021 = 0012 and 2000-2012 = 0021.
Now look at the moduli starting with 10, where each
digit is considered separately. Then

x° = 1 x 1 = 10 x 2 = 100

x 3 = 1000 = 12 x 4 = 120 x 5 = 1200 = 212

x 6 = 2120 = 111 x 7 = 1100 = 122 x 8 = 1220 = 202

x g = 2020 = 11

= 110

x 12 = 1120 = 102 x 13 = 1020 = 2

x 15 = 200

x 16 = 2000 = 21

x 11 = 1100 = 112
x 14 = 20
x 17 = 210

x 18 = 2100 = 121 x 19 = 1210 = 222 x 20 = 2220 = 211
x 21 = 2110 = 101 x 22 = 101 = 22 x 23 = 220

= 2200 = 221 x 2

; 2210 = 201 x 26 = 2010 = 1

Taking the alternate terms gives white squares as 000,
001, 020, 021, 022, 100, 102, 110, 111, 120, 121, 202,
211, and 221.

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 107-109

and 274, 1987.
Beth, T.; Jungnickel, D.; and Lenz, H. Design Theory, 2nd

ed. rev. Cambridge, England: Cambridge University Press,

1998.
Geramita, A. V. Orthogonal Designs: Quadratic Forms and

Hadamard Matrices. New York: Marcel Dekker, 1979.
Kitis, L. "Paley's Construction of Hadamard Matrices."

http : // www . mathsource . com / cgi - bin / Math Source /

Applications /Mathematics/0205-760.

Paley's Theorem

Proved in 1933. If q is an ODD PRIME or q = 0,and n
is any Positive Integer, then there is a Hadamard
Matrix of order

m = 2 e (<T + l),

Palindromic Number

A symmetrical number which is written in some base b
as 01 a-i ... 0,2 ai. The first few are 0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, . . .
(Sloane's A002113).

The first few n for which the PRONIC Number P n is
palindromic are 1, 2, 16, 77, 538, 1621, ... (Sloane's
A028336), and the first few palindromic numbers which
are Pronic are 2, 6, 272, 6006, 289982, . . . (Sloane's
A028337). The first few numbers whose squares are
palindromic are 1, 2, 3, 11, 22, 26, . . . (Sloane's
A002778), and the first few palindromic squares are 1,
4, 9, 121, 484, 676, . . . (Sloane's A002779).

see also Demlo Number, Palindromic Number Con-
jecture, Reversal

References

de Geest, P. "Palindromic Products of Two Consecutive In-
tegers." http : //www . ping . be/-ping6758/consec . htm.

de Geest, P. "Palindromic Squares." http://www.ping.be/
-ping6758/square . htm.

Pappas, T. "Numerical Palindromes." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, p. 146,
1989.

Sloane, N. J. A. Sequences A028336, A028337, A002113/
M0484, A0027778/M0807, and A002779/M3371 in "An
On-Line Version of the Encyclopedia of Integer Sequences."

Palindromic Number Conjecture

Apply the 196-Algorithm, which consists of taking
any POSITIVE INTEGER of two digits or more, revers-
ing the digits, and adding to the original number. Now
sum the two and repeat the procedure with the sum.
Of the first 10,000 numbers, only 251 do not produce a
Palindromic Number in < 23 steps (Gardner 1979).

It was therefore conjectured that all numbers will even-
tually yield a PALINDROMIC NUMBER. However, the
conjecture has been proven false for bases which are a
Power of 2, and seems to be false for base 10 as well.
Among the first 100,000 numbers, 5,996 numbers appar-
ently never generate a PALINDROMIC NUMBER (Gruen-
berger 1984). The first few are 196, 887, 1675, 7436,
13783, 52514, 94039, 187088, 1067869, 10755470, ...
(Sloane's A006960).

It is conjectured, but not proven, that there are an infi-
nite number of palindromic Primes. With the exception

1302 Pancake Cutting

Papal Cross

of 11, palindromic Primes must have an Odd number

of digits.

see also 196-Algorithm

References

Gardner, M. Mathematical Circus: More Puzzles, Games,
Paradoxes and Other Mathematical Entertainments from
Scientific American. New York: Knopf, pp. 242-245, 1979.

Gruenberger, F. "How to Handle Numbers with Thousands
of Digits, and Why One Might Want to." Sci. Amer. 250,
19-26, Apr. 1984.

Sloane, N. J. A. Sequence A006960/M5410 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Pancake Cutting

see Circle Cutting

Pancake Theorem

The 2-D version of the HAM SANDWICH THEOREM.

Pandiagonal Square

see Panmagic Square

Pandigital

A decimal INTEGER which contains each of the digits
from to 9.

Panmagic Square

8

17

1

15

24

11

25

9

18

2

19

3

12

21

10

22

6

20

4

13

5

14

23

7

16

If all the diagonals (including those obtained by "wrap-
ping around" the edges) of a MAGIC SQUARE, as well
as the usual rows, columns, and main diagonals sum
to the MAGIC Constant, the square is said to be a
Panmagic Square (also called Diabolical Square,
Nasik Square, or Pandiagonal Square). No pan-
magic squares exist of order 3 or any order 4fc+2 for k an
Integer. The Siamese method for generating Magic
Squares produces panmagic squares for orders 6k ±1
with ordinary vector (2, 1) and break vector (1, —1).

1

15

24

8

17

23

7

16

5

14

20

4

13

22

6

12

21

10

19

3

9

18

2

11

25

The Lo SHU is not panmagic, but it is an ASSOCIATIVE
Magic Square. Order four squares can be panmagic or
Associative, but not both. Order five squares are the
smallest which can be both ASSOCIATIVE and panmagic,
and 16 distinct ASSOCIATIVE panmagic squares exist,
one of which is illustrated above (Gardner 1988).

The number of distinct panmagic squares of order 1,
2, ... are 1, 0, 0, 384, 3600, 0, . . . (Sloane's A027567,
Hunter and Madachy 1975). Panmagic squares are re-
lated to Hypercubes.

see also ASSOCIATIVE MAGIC SQUARE, Hypercube,

Franklin Magic Square, Magic Square

References

Gardner, M. The Second Scientific American Book of Math-
ematical Puzzles & Diversions: A New Selection. New

York: Simon and Schuster, pp. 135-137, 1961.
Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time

Travel and Other Mathematical Bewilderments. New

York: W. H. Freeman, pp. 213-225, 1988.
Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3

in Mathematical Diversions. New York: Dover, pp. 24-25,

1975.
Kraitchik, M. "Panmagic Squares." §7.9 in Mathematical

Recreations. New York: W. W. Norton, pp. 174-176, 1942.
Madachy, J. S. Madachy 's Mathematical Recreations. New

York: Dover, p. 87, 1979.
Rosser, J. B. and Walker, R. J. "The Algebraic Theory of

Diabolical Squares." Duke Math. J. 5, 705-728, 1939.
Sloane, N. J. A. Sequence A027567 in "An On-Line Version

of the Encyclopedia of Integer Sequences."

Pantograph

A LINKAGE invented in 1630 by Christoph Scheiner for
making a scaled copy of a given figure. The linkage
is pivoted at 0; hinges are denoted 0. By placing a
Pencil at P (or P'), a Dilated image is obtained at
P' (or P).
see also Linkage

Papal Cross

see also CROSS

Pappus's Centroid Theorem

Pappus's Hexagon Theorem 1303

Pappus's Centroid Theorem

The Surface Area of a Surface of Revolution is
given by

>-?solid of rotation

= [perimenter] x [distance traveled by centroid],

and the Volume of a Solid of Revolution is given
by

''solid of rotation

= [cross-section area] x [distance traveled by centroid].
see also CENTROID (GEOMETRIC), CROSS-SECTION,

Perimeter, Solid of Revolution, Surface Area,
Surface of Revolution, Toroid, Torus

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 132, 1987.

Pappus Chain

In the Arbelos, construct a chain of Tangent Circles
starting with the Circle Tangent to the two small
interior semicircles and the large exterior one. Then the
distance from the center of the first INSCRIBED CIRCLE
to the bottom line is twice the Circle's Radius, from
the second CIRCLE is four times the Radius, and for the
nth Circle is 2n times the Radius. The centers of the
Circles lie on an Ellipse, and the Diameter of the
nth Circle C n is (l/n)th Perpendicular distance to
the base of the Semicircle. This result was known to
Pappus, who referred to it as an ancient theorem (Hood
1961, Cadwell 1966, Gardner 1979, Bankoff 1981). The
simplest proof is via INVERSIVE GEOMETRY.

If r = AB/AC, then the radius of the nth circle in the
pappus chain is

_ (1 — r)r

2[n 2 (l-r) 2 +r]'

This equation can be derived by iteratively solving the
Quadratic Formula generated by Descartes Cir-
cle Theorem for the radius of the Soddy Circle.
This general result simplifies to r n — 1/(6 + n 2 ) for
r = 2/3 (Gardner 1979). Further special cases when
AC = 1 + AB are considered by Gaba (1940).

If B divides AC in the Golden Ratio <£, then the cir-
cles in the chain satisfy a number of other special prop-
erties (Bankoff 1955).

see also Arbelos, Coxeter's Loxodromic Sequence
of Tangent Circles, Soddy Circles, Steiner
Chain

References

Bankoff, L. "The Golden Arbelos." Scripta Math. 21, 70-76,
1955.

Bankoff, L. "Are the Twin Circles of Archimedes Really
Twins?" Math. Mag. 47, 214-218, 1974.

Bankoff, L. "How Did Pappus Do It?" In The Mathematical
Gardner (Ed. D. Klarner). Boston, MA: Prindle, Weber,
and Schmidt, pp. 112-118, 1981.

Gaba, M. G. "On a Generalization of the Arbelos." Amer.
Math. Monthly 47, 19-24, 1940.

Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Sci. Amer.
240, 18-28, Jan. 1979.

Hood, R. T. "A Chain of Circles." Math. Teacher 54, 134-
137, 1961.

Johnson, R. A. Advanced Euclidean Geometry: An Elemen-
tary Treatise on the Geometry of the Triangle and the Cir-
cle. Boston, MA: Houghton Mifflin, p. 117, 1929.

Pappus-Guldinus Theorem

see Pappus's Centroid Theorem

Pappus's Harmonic Theorem

z

A W B Y

AW, AB, and AY in the above figure are in a HAR-
MONIC Range.

see also Ceva's Theorem, Menelaus' Theorem

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 67-68, 1967.

Pappus's Hexagon Theorem

D E F

HA, B, and C are three points on one LINE, D, E, and
F are three points on another Line, and AE meets BD
at X, AF meets CD at Y , and BF meets CE at Z, then
the three points X, Y, and Z are Collinear. Pappus's
hexagon theorem is essentially its own dual according to
the Duality Principle of Projective Geometry.

1304 Pappus's Theorem

Parabola

see also Cayley-Bacharach Theorem, Desargues'
Theorem, Duality Principle, Pascal's Theorem,
Projective Geometry

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.

Washington, DC: Math. Assoc. Amer., pp. 73-74, 1967.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,

pp. 92-94, 1990.
Pappas, T. "Pappus' Theorem & the Nine Coin Puzzle," The

Joy of Mathematics. San Carlos, CA: Wide World Publ./

Tetra, p. 163, 1989.

Pappus's Theorem

There are several THEOREMS that generally are known
by the generic name "Pappus's Theorem."

see also Pappus's Centroid Theorem, Pappus
Chain, Pappus's Harmonic Theorem, Pappus's
Hexagon Theorem

Parabiaugmented Dodecahedron

see Johnson Solid

Parabiaugmented Hexagonal Prism

see Johnson Solid

Parabiaugmented Truncated Dodecahedron

see Johnson Solid

Parabidiminished Rhombicosidodecahedron

see Johnson Solid

Parabigyrate Rhombicosidodecahedron

see Johnson Solid

For a parabola opening to the right, the equation in
Cartesian Coordinates is

\J(x - p) 2 + y 2 — x+p

(1)

(x~p) 2 +y 2 = (x + p) 2 (2)

x 2 - 2px + p 2 + y 2 = x 2 + 2px + p 2 (3)

y = Apx.

(4)

If the Vertex is at (x ,yo) instead of (0, 0), the equa-
tion is

(y-yof =4p(x-x ). (5)

If the parabola opens upwards,

x — Apy

(6)

(which is the form shown in the above figure at left).
The quantity 4p is known as the Latus Rectum. In
Polar Coordinates,

2a

1 — cos 9

(7)

In Pedal Coordinates with the Pedal Point at the
FOCUS, the equation is

p 2 = ar. (8)

The parametric equations for the parabola are

a: = 2at
y = at.

(9)
(10)

Parabola

directrix

The set of all points in the PLANE equidistant from a
given LINE (the DIRECTRIX) and a given point not on
the line (the FOCUS).

The parabola was studied by Menaechmus in an attempt
to achieve CUBE DUPLICATION. Menaechmus solved the
problem by finding the intersection of the two parabolas
x 2 = y and y 2 = 2x. Euclid wrote about the parabola,
and it was given its present name by Apollonius. Pascal
considered the parabola as a projection of a CIRCLE, and
Galileo showed that projectiles falling under uniform
gravity follow parabolic paths. Gregory and Newton
considered the CATACAUSTIC properties of a parabola
which bring parallel rays of light to a focus (MacTutor
Archive).

The Curvature, Arc Length, and Tangential An-
gle are

«(*) =

2(1+^2)3/2

s(t) = ty/l + t 2 + sinlT 1 1
4>(t) — tan - t.

The Tangent Vector of the parabola is
x T (t) =

yr{t)

VTTt*
t

(11)

(12)
(13)

(14)
(15)

The plots below show the normal and tangent vectors
to a parabola.

Parabola Caustic

Parabola Inverse Curve

1305

and

see also CONIC SECTION, ELLIPSE, HYPERBOLA, QUAD-
RATIC Curve, Reflection Property, Tschirn-
hausen Cubic Pedal Curve

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 198, 1987.

Casey, J. "The Parabola." Ch. 5 in A Treatise on the An-
alytical Geometry of the Point, Line, Circle, and Conic
Sections, Containing an Account of Its Most Recent Exten-
sions, with Numerous Examples, 2nd ed., rev. enl. Dublin:
Hodges, Figgis, & Co., pp. 173-200, 1893.

Coxeter, H. S. M. "Conies." §8.4 in Introduction to Geome-
try, 2nd ed. New York: Wiley, pp. 115-119, 1969.

Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 67-72, 1972.

Lee, X. "Parabola." http://www.best.com/-xah/Special
PlaneCurves-dir/Parabola_dir/parabola.html.

Lockwood, E. H. "The Parabola." Ch. 1 in A Book of Curves.
Cambridge, England: Cambridge University Press, pp. 2-
12, 1967.

MacTutor History of Mathematics Archive. "Parabola."
http: //www-groups .dcs , st~and.ac.uk/~history/Curves
/Parabola. html.

Pappas, T. "The Parabolic Ceiling of the Capitol." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 22-23, 1989.

Parabola Caustic

The Caustic of a Parabola with rays Perpendicu-
lar to the axis of the Parabola is Tschirnhausen
Cubic.

Parabola Evolute

Given a PARABOLA

2

the parametric equation and its derivatives are

x = t x' — t

y' = 2t

y = t 2 x" =

y" = 2.

The Radius of Curvature is

x'y" — x"y'

(l + 4t 2 ) 3/2
2

The Tangent Vector is

\/l + 4t 2
so the parametric equations of the evolute are

£ = -At 3

■Zt\

(1)

(2)

(3)

(4)

(5)
(6)

-ti = t*

(7)

Uv-h) = t 2

(8)

Un-l) = (-U) 2/3

(9)

h)= (_M) a/ ' = i (2er v..

(10)

The Evolute is therefore

i = !(20 a/8 + i

(ii)

This is known as Neile's Parabola and is a Semicu-
bical Parabola. Prom a point above the evolute three
normals can be drawn to the PARABOLA, while only one
normal can be drawn to the PARABOLA from a point
below the Evolute.

see also NEILE'S PARABOLA, PARABOLA, SEMICUBICAL
PARABOLA

Parabola Inverse Curve

The Inverse Curve for a Parabola given by

x = at 2
y = 2at

(1)
(2)

with Inversion Center (xo,j/o) and Inversion Ra-
dius k is

x = x +

k(at 2 — xo)

(at 2 + x ) 2 + {2at - y ) 2

k(2at — i/o )
(at 2 +x ) 2 + (2at-y ) 2 '

(3)
(4)

O- 3

For (ao,yo ) = (a, 0) at the FOCUS, the INVERSE CURVE
is the CARDIOID

x — a +

k(t 2 - 1)
a(l + i 2 ) 2
2kt

a(l + t 2 ) 2 '

(5)
(6)

1306

Parabola Involute

Parabolic Coordinates

For (x Q ,yo) = (0, 0) at the VERTEX, the INVERSE CURVE
is the ClSSOID OF DlOCLES

k

Parabolic Coordinates

y =

Parabola Involute

a(4 + £ 2 )

2k
at(4 + t 2 )*

(?)
(8)

dr

dt

T =

Vl + 4t 2

ds 2 = \dr\ 2 ^(l + 4t 2 )dt 2

d S = -y/l + 4i 2 dt

I y/T+AP dt = 1^1 + 4^ + | sinh~ 1 (2t).

So the equation of the INVOLUTE is

n = r - sT =

WT+4t2 + ! sinh- 1 (2i)

vT+lt 2

2\/l + 4t 2

i- |sinh- 1 (2t)
-sinh _1 (2t)

(1)

(2)

(3)
(4)

(5)

1
2t

(6)

Parabola Pedal Curve

On the Directrix, the Pedal Curve of a Parabola is
a STROPHOID (top left). On the foot of the DIRECTRIX,
it is a Right Strophoid (top middle). On reflection of
the FOCUS in the DIRECTRIX, it is a MACLAURIN TRI-
SECTRIX (top right). On the Vertex, it is a ClSSOID OF
DlOCLES (bottom left). On the Focus, it is a straight
line (bottom right).

References

Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 94-97, 1972.

+>y

A system of CURVILINEAR COORDINATES in which two
sets of coordinate surfaces are obtained by revolving the
parabolas of PARABOLIC CYLINDRICAL COORDINATES
about the as-Axis, which is then relabeled the z-AxiS.
There are several notational conventions. Whereas
(u, v, 9) is used in this work, Arfken (1970) uses (£, 77, <p).

The equations for the parabolic coordinates are

x = uv cos
y — uv sin

1/2 2\

z= |(u -v ),

(1)

(2)
(3)

where u G [0,oo), v € [0,oo), and 6 € [0,2tt). To solve
for u, v, and 9, examine

x 2 + y 2 + z 2 = U V + | (U 4 - 2u\ 2 + V 4 )

= i(u 4 + 2«V+ V 4 )

_ 1

i(« a +o a

^^+7+^= \{u + v 2 )

and

y/x 2 + y 2 +z 2 + z = u 2

y/x 2 + y 2 + z 2 - z — v 2
We therefore have

u= y/y/i

x 2 + y 2 + z 2 + z
v = y y/x 2 + y 2 -\- z 2 - z

The Scale Factors are

h u = y/u 2 -h v 2
y/u 2 + v 2

h

he = uv.

(4)

(5)

(6)
(7)

(8)

(9)
(10)

(11)
(12)
(13)

Parabolic Cyclide

Parabolic Cylinder Function 1307

The Line Element is

s = (u + v )(du + dv ) + u v dO ,

and the VOLUME ELEMENT is

dV = uv(u 2 -f v 2 ) du dv d6.

(14)

(15)

The LAPLACIAN is

v 2 /

uv(u 2 + v 2 ) \_du

or

V chi / dv V 0u / J

v? + v 2

1
1A 2 + v 2

i a 2 /

udu \ du) v dv \ dv)\ u 2 v 2 dO 2
u du du 2 v dv dv

i 2 J u 2 v 2 d6 2 '

(16)

The Helmholtz Differential Equation is Separa-
ble in parabolic coordinates.

see also Confocal Paraboloidal Coordinates,
Helmholtz Differential Equation — Parabolic
Coordinates, Parabolic Cylindrical Coordi-
nates

References

Arfken, G. "Parabolic Coordinates (£, 77, </>)" §2.12 in Math-
ematical Methods for Physicists, 2nd ed. Orlando, FL:
Academic Press, pp. 109-112, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, p. 660, 1953.

Parabolic Cyclide

A Cyclide formed by inversion of a Standard Torus
when the sphere of inversion is tangent to the torus.

see also Parabolic Horn Cyclide, Parabolic Ring
Cyclide, Parabolic Spindle Cyclide

Parabolic Cylinder

A Quadratic Surface given by the equation

x J + 2rz = 0.

Parabolic Cylinder Function

These functions are sometimes called Weber FUNC-
TIONS. Whittaker and Watson (1990, p. 347) define the
parabolic cylinder functions as solutions to the WEBER
Differential Equation

^ + (n+|-I^„(,)=0. (1)

The two independent solutions are given by D„(z) and

D- n -i(ze ilr/ *)> where

Dn (z) = 2 n '™'*z- l '*W n „ +1/ i,-i,<(±z 2 )

_ 2 n/2+l/4 z -l/2

r( i )2 n/2 +1 /4^-l/ 2

(2)

^11/2+1/4,-1/4(22 )

+

r( _l )2 n/2+l/4 z -l/2

2 '_! z M n/2+1/4 , 1/4 (iA (3)

Here, W at b(z) is a WHITTAKER FUNCTION and
M a ,b( z ) = iFi(a>;b;z) are CONFLUENT Hypergeomet-
ric Functions.

Abramowitz and Stegun (1972, p. 686) define the para-
bolic cylinder functions as solutions to

y" + (ax 2 + bx + c) = 0. (4)

This can be rewritten by COMPLETING THE SQUARE,

y +

Now letting

a { X+ 2a)

b\ 2 &*_
4a

+ c

y = o.

U = X +

du — dx

2a

gives

where

g + («« a + <0y = o

d = — + c.
4a

Equation (4) has the two standard forms
y" - (\x 2 + a)y =

y" + (\x 2 -a)y = 0.

(5)

(6)
(7)

(8)
(9)

(10)
(11)

For a general a, the Even and Odd solutions to (10)

yi(x)

_ c -* 2 /4

iFiiha+ki&k*')

2 / 2 (x) = x e -^ /4 iF 1 (|a+|;f;ix 2 ),

(12)

(13)

1308 Parabolic Cylinder Function

Parabolic Cylindrical Coordinates

where iFi(o;6;z) is a Confluent Hypergeometric
Function. If y(a, x) is a solution to (10), then (11) has
solutions

y(±ia 1 xe^ 7r/ %y(±ia,-xe^ / ^). (14)

Abramowitz and Stegun (1972, p. 687) define standard
solutions to (10) as

(Watson 1966, p, 308), which is similar to the ANGER
Function. The result

/oo
Dm(x)D n (x)dx = ii m „n!v^, (25)

■oo

where Sij is the Kronecker DELTA, can also be used
to determine the Coefficients in the expansion

U(a,x) = cos[tt(! + fa)]yi -sin[7r(i + §a)]y 2 (15)

T/ , , sin[7r(| + ^a)]yi +cos[7r(^ + |ap2
F(a,x) = * * — ^ ,(16)

r(|-a)

where

Yi =

i r(|-H .

0F 2«/ 2 + 1 / 4

i r(i-|o)

•2/i

4 2?> »-*'/*. p. /1„ , 1. 1. Iji

0F 2«/ 2 + 1 /4

*2 = -7= -^77^X777-3/2

^(ia+i;!;^) (17)

V¥ 2°/ 2 + 1 /4
1 r(|-ia) _

•v/tt 2 a

/2+1/4

xe- /4 ifi(ia+|;f;ix 2 ).

In terms of Whittaker and Watscn's functions,

U(a,x) = D- a -i /2 (x)

V{a, x)

_ T(i + a)[sin(7ra)Z?_ a _ 1/2 (x) + P- a -i/ 2 (-x)]

(18)

(19)
(20)

For NONNEGATIVE INTEGER n, the solution D n reduces
to

D n (x) = 2- n/2 e-* 2/4 H n (^=\ = e-* 2/4 He„(x),

(21)
where H n (x) is a Hermite Polynomial and He n is a
modified HERMITE POLYNOMIAL.

The parabolic cylinder functions D u satisfy the RECUR-
RENCE Relations

D v +i(z) - zD v (z) + vD v -x(z) = (22)

D' v (z) 4- \zD v (z) - vD v -!(z) = 0. (23)

The parabolic cylinder function for integral n can be
defined in terms of an integral by

* Jo

sin(n0 - z sin 0) d.6 (24)

f{z) = 22 a nDn

as

i r°

a n = / — / D n

(t)f(t)dt.

(26)

(27)

For v real,

f

Jo

[D l/ {t)Ydt = n 1/2 2

i/2 n -3/2 <Mf ~ \v)-<fa{-\v)

r(-i/)

(28)

(Gradshteyn and Ryzhik 1980, p. 885, 7.711.3), where
T(z) is the Gamma Function and <p Q (z) is the Poly-
gamma Function of order 0.

see also ANGER FUNCTION, BESSEL FUNCTION, DAR-
WIN'S Expansions, Hh Function, Struve Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Parabolic Cylin-
der Function." Ch. 19 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 685-700, 1972.

Gradshteyn, 1. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.

Iyanaga, S. and Kawada, Y. (Eds.). "Parabolic Cylinder
Functions (Weber Functions)." Appendix A, Table 20.111
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, p. 1479, 1980.

Spanier, J. and Oldham, K. B. "The Parabolic Cylinder
Function D l/ (x). ii Ch. 46 in An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 445-457, 1987.

Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.

Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, ^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.

Parabolic Cylindrical Coordinates

Parabolic Fixed Point

Parabolic Point

1309

A system of Curvilinear Coordinates. There are
several different conventions for the orientation and des-
ignation of these coordinates. Arfken (1970) defines co-
ordinates (£, 77, z) such that

(i)

(2)
(3)

In this work, following Morse and Feshbach (1953), the
coordinates (u, v } z) are used instead. In this convention,
the traces of the coordinate surfaces of the xy-PLANE
are confocal PARABOLAS with a common axis. The u
curves open into the Negative x-Axis; the v curves
open into the Positive x-Axis. The u and v curves
intersect along the y-Axis.

x=\{u 2 -v 2 )

y — uv
z = z,

(4)
(5)
(6)

where u 6 [0, oo), v e [0, oo), and z £ (-00,00). The
Scale Factors are

hi = yu 2 + v 2

fi2 = yu 2 + v 2
h s = 1.

Laplace's Equation is

V 2 / =

\du 2 dv 2 )

dz 2 '

(7)

(8)
(9)

(10)

The Helmholtz Differential Equation is Separa-
ble in parabolic cylindrical coordinates.

see also Confocal Paraboloidal Coordinates,
Helmholtz Differential Equation — Parabolic
Cylindrical Coordinates, Parabolic Coordi-
nates

References

Arfken, G. "Parabolic Cylinder Coordinates (£, 77, z)." §2,8
in Mathematical Methods for Physicists, 2nd ed. Orlando,
FL: Academic Press, p. 97, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, p. 658, 1953.

Parabolic Fixed Point

A Fixed Point of a Linear Transformation for
which the rescaled variables satisfy

(6 - ocf + 407 = 0.

see also Elliptic Fixed Point (Map), Hyperbolic
Fixed Point (Map), Linear Transformation

Parabolic Geometry

see Euclidean Geometry

Parabolic Horn Cyclide

A Parabolic Cyclide formed by inversion of a Horn
TORUS when the inversion sphere is tangent to the
Torus.

see also Cyclide, Parabolic Ring Cyclide, Para-
bolic Spindle Cyclide

Parabolic Partial Differential Equation

A Partial Differential Equation of second-order,
i.e., one of the form

AU XX + 2BU X y + CUyy + DU X + EUy + F = 0, (1)

is called parabolic if the Matrix

Z~

A B
B C

(2)

satisfies det(Z) = 0. The HEAT CONDUCTION EQUA-
TION and other diffusion equations are examples. Initial-
boundary conditions are used to give

u(x, t) = g(x, t) for x £ dQ, t > (3)

u(x, 0) = v(x) for xGfi, (4)

where

u xx = f(u X iU y ,u,x,y) (5)

holds in Q.

see also Elliptic Partial Differential Equation,
Hyperbolic Partial Differential Equation, Par-
tial Differential Equation

Parabolic Point

A point p on a Regular Surface M £ M 3 is said to
be parabolic if the Gaussian CURVATURE K(p) =
but 5(p) i=- (where S is the SHAPE Operator), or
equivalently, exactly one of the PRINCIPAL CURVATURES
Ki and K2 is 0.

see also Anticlastic, Elliptic Point, Gaussian
Curvature, Hyperbolic Point, Planar Point,
Synclastic

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 280, 1993.

1310 Parabolic Ring Cy elide

Parabolic Ring Cyclide

A Parabolic Cyclide formed by inversion of a Ring
TORUS when the inversion sphere is tangent to the
Torus.

see also Cyclide, Parabolic Horn Cyclide, Para-
bolic Spindle Cyclide

Parabolic Rotation

The Map

x' = x + l (1)

y=2x + y + l, (2)

which leaves the Parabola

x ' 2 - y ' = (a- + l) 2 - (2x + y + 1) = x 2 - y (3)

invariant.

see also Parabola, Rotation

Parabolic Segment

The Arc Length of the parabolic segment shown above
is given by

> v 2 {2x+ Jlx 2 + y 2 \

. = VE*T7+f^ — *_ — y -j.

The Area contained between the curves

2

y = x

y — ax + b
can be found by eliminating y,

x — ax — 6 = 0,
so the points of intersection are

x± = \{a± <sja? +46).

(1)

(2)
(3)

(4)

(5)

Parabolic Segment

Therefore, for the Area to be NONNEGATIVE, a 2 +46 >
0, and

x± = \{a 2 ± 2ay/a 2 + 6 2 + a 2 + 46)

= !(2a 2 +46±2a>A 2 +46)
= |(a 2 + 26±a v / a 2 +46),

so the Area is

A = / [{&% -\- b) — x ]dx

t/ x_

I- 2 3 -l(a-Vo2+46)/2

(6)

(T)

Now,

x+ — X-

\ (a 2 + 2a v / a 2 + 46 + a 2 +46)

- (a 2 - 2a\/a 2 +4& + a 2 + 46)1

= | 4a V / a 2 +4&l = ay^ 2 + 46 (8)
x+ 3 — a;_ 3 = (#+ — x_)(a;+ 2 + £_#+ + z- 2 )

= V^ 2 + 46 | |(a 2 + 2a^fa 2 + 46 + a 2 + 46)

+ |[a 2 - (a 2 +46)] + f (a 2 - 2a V / ^ 2 ~+4& + a 2 +46)}

= \ s/a? + 46 (4a 2 + 46) = ^a 2 + 46 (a 2 + 6). (9)

So

A= |a 2 v / a 2 +46 + 6v / a 2 +46= |(a 2 + &) v / a 2 +46
= V / a 2 +46[(|-|)a 2 +6(l-|)]
= (ia a + |6)v/?+46
= \{a 2 + 46)x/a 2 +46= |(a 2 + 4&) 3/2 , (10)

We now wish to find the maximum AREA of an inscribed
Triangle. This Triangle will have two of its Ver-
tices at the intersections, and Area

Aa = \{x-y* - x*y- - x+y* + x*y+ + x+y~ - x-y + ),

(11)
But y* = cc* 2 , so

A 1 / 2 2

Aa = ±(x-x* — x*y- — x+x*
+ x*y* + x+y- - x-y + )
= \[~x* 2 (x+ -x-) + a;*(y + - y-)
+ (x + y- -x-y+)]. (12)

The maximum AREA will occur when
dA±

dx+

±[-2(x+-x-)x. + (y+-y-)] = 0. (13)

Parabolic Segment

But

x+ — x~ — v a 2 + 46
y+ - y_ = ay'a 2 +46,

^ = i J ^-^ = |a

2 Z+ — z_
and
A A = |[-(|a) 2 (x + - z_) + (±a)(y + - y_)

(14)
(15)

(16)

+(x+y_-x_y+)]. (17)

Working on the third term

x+y- = | (a + y^ 2 + 4& )(« 2 + 2& ~ « Va 2 + 46 )

a 3 + 2a6 - ayjcfi + 46 + a 2 yV 4- 46

+ 26 Va2+46 - a(a 2 + 46)]
= |[-2a6 + 26 v / a 2 +46]
.y + - |(a - a/^ 2 + 46)(a 2 + 26 + ayja? + 46)
= \ [a 3 + 2a6 + a 2 yV + 46 - a yja? + 46
- 26x7a 2 + 46 - a(a 2 + 46)1

(18)

- i[-2a6-26 v / a 2 +46],

(19)

x+y- - x_y+ = \{4by/a 2 +4ti) = &\/a 2 + 46 (20)

and

A* - |(-|a 2 Va 2 +46+ |a 2 Va 2 +46 + 6Va 2 +6 2 )

- |Va 2 +46 [(i - \)a 2 + 6] = I ^a 2 + 46(|a 2 + 6)

= | \/« 2 + 46 (a 2 + 46) = | (a 2 + 46) 3/2 , (21)

which gives the result known to Archimedes in the third
century BC that

A

(22)

The AREA of the parabolic segment of height h opening
upward along the y-AxiS is

A = 2 [ ^dy = §/> 3/2 .
nean of y is

/»/i ph

/ yVvdy = 2 / y s/2 dy = lh 5/2

Jo Jo

The weighted mean of y is

The CENTROID is then given by

* = ¥ = !*■

(23)

(24)

(25)

see a/50 Centroid (Geometric), Parabola, Seg-
ment

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 125, 1987.

Paraboloid
Parabolic Spindle Cyclide

1311

A Parabolic Cyclide formed by inversion of a Spin-
dle TORUS when the inversion sphere is tangent to the
Torus.

see also CYCLIDE, PARABOLIC HORN CYCLIDE, PARA-
BOLIC Ring Cyclide

Parabolic Spiral

see Fermat's Spiral

Parabolic Umbilic Catastrophe

A CATASTROPHE which can occur for four control fac-
tors and two behavior axes.

Paraboloid

The Surface of Revolution of the Parabola. It is
a Quadratic Surface which can be specified by the
Cartesian equation

z = a(x 2 +y 2 ),

(1)

or parametrically by

x(u,v) = s/u C0S1>

(2)

y{u,v) = y/u sinti

(3)

z(u,v) = u,

(4)

where u G [0, ft,], v £ [0, 27r), and h is the height.
The Volume of the paraboloid is

V

Jo

zdz = \Trh .

(5)

1312 Paraboloid Geodesic

Paradox

The weighted mean of z over the paraboloid is

{*)

-J

Jo

z 2 dz= |tt/i 3 .

The Centroid is then given by

2= { 4 = *h

(6)

(7)

(Beyer 1987).

see also Elliptic Paraboloid, Hyperbolic
Paraboloid, Parabola

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 133, 1987.

Gray, A. "The Paraboloid." §11.5 in Modern Differential
Geometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 221-222, 1993.

Paraboloid Geodesic

A Geodesic on a Paraboloid has differential param-
eters defined by

cos 2 v sin 2 v H 1
= 1 + ~A + -1 = 1+ T-

(i)

R = Q

dudv dudv dudv
sin v cos v 1

= + u cos v + u sin v = u

+

2y/u 2y/u 2y/u

(2)
(cost; — sinu). (3)

The Geodesic is then given by solving the Euler-
Lagrange Differential Equation

Q + Rv'

97 + 2v a^T + v ^ _ _o. I __

2^P + 2Qv* -\-Rv f2 du I y'p + 2Qv f + iit;' 2

= 0.

(4)
As given by Weinstock (1974), the solution simplifies to

= u(l + 4c 2 ) sin 2 {t; - 2c\n[k(2yJ u - c 2 + V4u + 1)]}.

(5)

see a/so GEODESIC

References

Weinstock, R. Calculus of Variations, with Applications to
Physics and Engineering. New York: Dover, p. 45, 1974.

Paraboloidal Coordinates

see CONFOCAL PARABOLOIDAL COORDINATES

Paracompact Space

A paracompact space is a HAUSDORFF SPACE such that
every open Cover has a Locally Finite open Refine-
ment. Paracompactness is a very common property
that TOPOLOGICAL SPACES satisfy. Paracompactness is
similar to the compactness property, but generalized for
slightly "bigger" Spaces. All MANIFOLDS (e.g, second
countable and Hausdorff) are paracompact.

see also HAUSDORFF SPACE, LOCALLY FINITE SPACE,

Manifold, Topological Space

Paracycle

see Astroid

Paradox

A statement which appears self-contradictory or con-
trary to expectations, also known as an Antinomy.
Bertrand Russell classified known logical paradoxes into
seven categories.

Ball and Coxeter (1987) give several examples of geo-
metrical paradoxes.

see also Alias' Paradox, Aristotle's Wheel Para-
dox, Arrow's Paradox, Banach-Tarski Para-
dox, Barber Paradox, Bernoulli's Paradox,
Berry Paradox, Bertrand's Paradox, Cantor's
Paradox, Coastline Paradox, Coin Paradox,
Elevator Paradox, Epimenides Paradox, Eu-
bulides Paradox, Grelling's Paradox, Haus-
dorff Paradox, Hempel's Paradox, Hetero-
logical Paradox, Leonardo's Paradox, Liar's
Paradox, Logical Paradox, Potato Paradox,
Richard's Paradox, Russell's Paradox, Saint Pe-
tersburg Paradox, Siegel's Paradox, Simpson's
Paradox, Skolem Paradox, Smarandache Para-
dox, Socrates' Paradox, Sorites Paradox, Thom-
son Lamp Paradox, Unexpected Hanging Para-
dox, Zeeman's Paradox, Zeno's Paradoxes

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 84-86,
1987.

Bunch, B. Mathematical Fallacies and Paradoxes. New York:
Dover, 1982.

Carnap, R. Introduction to Symbolic Logic and Its Applica-
tions. New York: Dover, 1958.

Curry, H. B. Foundations of Mathematical Logic. New York:
Dover, 1977.

Kasner, E. and Newman, J. R. "Paradox Lost and Paradox
Regained." In Mathematics and the Imagination. Red-
mond, WA: Tempus Books, pp. 193-222, 1989.

Northrop, E. P. Riddles in Mathematics: A Book of Para-
doxes. Princeton, NJ: Van Nostrand, 1944.

O'Beirne, T. H. Puzzles and Paradoxes. New York: Oxford
University Press, 1965.

Quine, W. V. "Paradox." Sci. Amer. 206, 84-96, Apr. 1962.

Paradromic Rings

Parallel Postulate

1313

Paradromic Rings

Rings produced by cutting a strip that has been given
m half twists and been re- attached into n equal strips
(Ball and Coxeter 1987, pp. 127-128).
see also MOBIUS Strip

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 127-
128, 1987.

Paragyrate Diminished Rhombicosidodeca-
hedron

see Johnson Solid

Parallel

The two branches of the parallel curve a distance k away
from a parametrically represented curve (f(t)>g(t)) are

Two lines in 2-dimensional Euclidean Space are said
to be parallel if they do not intersect. In 3-dimensional
Euclidean Space, parallel lines not only fail to inter-
sect, but also maintain a constant separation between
points closest to each other on the two lines. (Lines in
3-space which are not parallel but do not intersect are
called Skew Lines.)

In a Non-Euclidean Geometry, the concept of par-
allelism must be modified from its intuitive meaning.
This is accomplished by changing the so-called PARAL-
LEL POSTULATE. While this has counterintuitive re-
sults, the geometries so defined are still completely self-
consistent.

see also Antiparallel, Hyperparallel, Line, Non-
Euclidean Geometry, Parallel Curve, Parallel
Postulate Perpendicular, Skew Lines

Parallel Axiom

see Parallel Postulate

Parallel Class

A set of blocks, also called a RESOLUTION Class, that
partition the set V, where (V, B) is a balanced incom-
plete Block Design.
see also BLOCK DESIGN, RESOLVABLE

References

Abel, R. J. R. and Furino, S. C. "Resolvable and Near Re-
solvable Designs." §1.6 in The CRC Handbook of Combi-
natorial Designs (Ed. C. J. Colbourn and J. H. Dinitz).
Boca Raton, FL: CRC Press, pp. 87-94, 1996.

Parallel Curve

x = f±

y = 9T

kg'

V7' 2 + </' 2

v7' 2 + <?' 2 '

The above figure shows the curves parallel to the El-
lipse.

References

Gray, A. "Parallel Curves." §5.7 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 95-97, 1993,

Lawrence, J. D. A Catalog of Special Plane Curves, New
York: Dover, pp. 42-43, 1972.

Lee, X. "Parallel." http://www.best .com/~xah/Special
PlaneCurves_dir/Parallel_dir/par allel.html.

Yates, R. C. "Parallel Curves." A Handbook on Curves and
Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 155-
159, 1952.

Parallel Postulate

Given any straight line and a point not on it, there "ex-
ists one and only one straight line which passes" through
that point and never intersects the first line, no matter
how far they are extended. This statement is equivalent
to the fifth of Euclid's Postulates, which Euclid him-
self avoided using until proposition 29 in the Elements.
For centuries, many mathematicians believed that this
statement was not a true postulate, but rather a theorem
which could be derived from the first four of EUCLID'S
Postulates. (That part of geometry which could be
derived using only postulates 1-4 came to be known as
Absolute Geometry.)

Over the years, many purported proofs of the parallel
postulate were published. However, none were correct,
including the 28 "proofs" G. S. Kliigel analyzed in his
dissertation of 1763 (Hofstadter 1989). In 1823, Janos
Bolyai and Lobachevsky independently realized that en-
tirely self-consistent "NON-EUCLIDEAN GEOMETRIES"
could be created in which the parallel postulate did not
hold. (Gauss had also discovered but suppressed the
existence of non-Euclidean geometries.)

As stated above, the parallel postulate describes the
type of geometry now known as PARABOLIC GEOME-
TRY. If, however, the phrase "exists one and only one
straight line which passes" is replace by "exist no line
which passes," or "exist at least two lines which pass,"
the postulate describes equally valid (though less intu-
itive) types of geometries known as Elliptic and Hy-
perbolic Geometries, respectively.

The parallel postulate is equivalent to the EQUIDIS-
TANCE Postulate, Playfair's Axiom, Proclus' Ax-
iom, Triangle Postulate. There is also a single par-
allel axiom in HlLBERT'S AXIOMS which is equivalent to
Euclid's parallel postulate.

1314 Parallel (Surface of Revolution)

Parallelogram

see also Absolute Geometry, Euclid's Axioms,
Euclidean Geometry, Hilbert's Axioms, Non-
Euclidean Geometry, Playfair's Axiom, Trian-
gle Postulate

References

Dixon, R. Mathographics. New York: Dover, p. 27, 1991.

Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago,
IL: Open Court, 1980.

Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, pp. 88-92, 1989.

Iyanaga, S. and Kawada, Y. (Eds.). "Hilbert's System of Ax-
ioms." §163B in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 544-545, 1980.

Parallel (Surface of Revolution)

A parallel of a Surface of Revolution is the inter-
section of the surface with a PLANE orthogonal to the
axis of revolution.

see also Meridian, Surface of Revolution

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 358, 1993.

Parallelepiped

In 3-D, a parallelepiped is a PRISM whose faces are all
Parallelograms. The volume of a 3-D parallelepiped
is given by the Scalar Triple Product

^parallelepiped = |B • (B X C)|

= |C-(AxB)| = |B-(C x A)|.

In n-D, a parallelepiped is the POLYTOPE spanned by
n Vectors vi, ..., v n in a Vector Space over the
reals,

span(vi,...,v n ) = tivt + ... + t n v n ,

where t% € [0, 1] for i — 1, . . . , n. In the usual inter-
pretation, the Vector Space is taken as Euclidean
Space, and the Content of this parallelepiped is given

by

abs(det(vi, . ..,v n )),

where the sign of the determinant is taken to be the
"orientation" of the "oriented volume" of the parallele-
piped.

see also PRISMATOID, RECTANGULAR PARALLELE-
PIPED, ZONOHEDRON

References

Phillips, A. W. and Fisher, I. Elements of Geometry. New-
York: Amer. Book Co., 1896.

Parallelism

see Angle of Parallelism

Parallelizable

A sphere § n is parallelizable if there exist n cuts contain-
ing linearly independent tangent vectors. There exist
only three parallelizable spheres: § , § , and § (Adams
1962, Le Lionnais 1983).

see also SPHERE

References

Adams, J. F. "On the Non-Existence of Elements of Hopf

Invariant One." Bull. Amer. Math. Soc. 64, 279-282,

1958.
Adams, J. F. "On the Non-Existence of Elements of Hopf

Invariant One." Ann. Math. 72, 20-104, 1960.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

p. 49, 1983.

Parallelogram

A b B

A Quadrilateral with opposite sides parallel (and
therefore opposite angles equal). A quadrilateral with
equal sides is called a RHOMBUS, and a parallelogram
whose Angles are all Right Angles is called a Rect-
angle.

A parallelogram of base b and height h has AREA

A — bh = ab sin A = ab sin B.
The height of a parallelogram is

h = a sin A = asini?,
and the DIAGONALS are

p = y/a 2 + b 2 — 2ab cos A

q = y/a 2 + b 2 — 2ab cos B

= yja 2 + b 2 + 2a6cosA

(1)

(2)

(3)
(4)
(5)

(Beyer 1987).

The Area of the parallelogram with sides formed by the
Vectors (a, c) and (6, d) is

A = det( a c b d j =\ad-bc\.

(6)

Given a parallelogram P with area A(P) and linear
transformation T, the Area of T(P) is

A(T(P))

a b
c d

MP)-

(7)

Parallelogram Illusion

Parameter

1315

As shown by Euclid, if lines parallel to the sides are
drawn through any point on a diagonal of a parallelo-
gram, then the parallelograms not containing segments
of that diagonal are equal in AREA (and conversely), so
in the above figure, A\ = A2 (Johnson 1929).

see also Diamond, Lozenge, Parallelogram Illu-
sion, Rectangle, Rhombus, Varignon Parallelo-
gram, Wittenbauer's Parallelogram

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,

28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.
Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, p. 61, 1929.

Parallelogram Illusion

The sides a and b have the same length, appearances to
the contrary.

Parallelogram Law

Let I • I denote the NORM of a quantity. Then the quan-
tities x and y satisfy the parallelogram law if

\\x + y\\ 2 + \\x-y\\ 2 = 2\\x\\ 2 + 2\\y\\ 2 .

If the NORM is defined as |/| = y/{f\f) (the so-called
L2-NORM), then the law will always hold.

see also L2-N0RM, NORM

Parallelohedron

A special class of ZONOHEDRON. There are five par-
allelohedra with an infinity of equal and similarly sit-
uated replicas which are SPACE-FILLING POLYHEDRA:
the Cube, Elongated Dodecahedron, hexagonal
Prism, Rhombic Dodecahedron, and Truncated
Octahedron.

see also Parallelotope, Space-Filling Polyhe-
dron

References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, p. 29, 1973.

Parallelotope

Move a point IIo along a LINE for an initial point to a
final point. It traces out a LINE SEGMENT IIi. When
IIi is translated from an initial position to a final po-
sition, it traces out a PARALLELOGRAM II2. When II2
is translated, it traces out a PARALLELEPIPED II3. The
generalization of n n to n-D is then called a parallelo-
tope. Iln has 2 n vertices and

N k = 2 n ~

:)

IlfcS, where (£) is a BINOMIAL COEFFICIENT and k = 0,
1, . . . , n (Coxeter 1973). These are also the coefficients
of (2k + l) n .

see also HONEYCOMB, HYPERCUBE, ORTHOTOPE, PAR-
ALLELOHEDRON

References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, pp. 122-123, 1973.

Klee, V. and Wagon, S. Old and New Unsolved Problems in
Plane Geometry and Number Theory. Washington, DC:
Math. Assoc. Araer., 1991.

Zaks, J. "Neighborly Families of Congruent Convex Poly-
topes." Amer. Math. Monthly 94, 151-155, 1987.

Paralogic Triangles

At the points where a line cuts the sides of a TRIAN-
GLE AAiA2A^ y perpendiculars to the sides are drawn,
forming a TRIANGLE A B1B2B3 similar to the given
TRIANGLE. The two triangles are also in perspective.
One point of intersection of their ClRCUMClRCLES is the
Similitude Center, and the other is the Perspective
Center. The Circumcircles meet Orthogonally.

see also ClRCUMCIRCLE, ORTHOGONAL CIRCLES, PER-
SPECTIVE Center, Similitude Center

References

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 258-262, 1929.

Parameter

A parameter m used in Elliptic INTEGRALS defined
to be m = fc 2 , where k is the Modulus. An Elliptic
Integral is written I(<t>\m) when the parameter is used.
The complementary parameter is defined by

m,

(i)

where m is the parameter. Let q be the NOME, k the
Modulus, and m = k 2 the Parameter. Then

q(m) = e -^'(m)/*<m) (2)

where K(m) is the complete ELLIPTIC INTEGRAL OF
THE FIRST Kind. Then the inverse of q(m) is given by

m(q) =

V(*)'

(3)

1316 Parameter (Quadric)

Parodies Theorem

where #* is a Theta Function.

see also AMPLITUDE, CHARACTERISTIC (ELLIPTIC IN-
TEGRAL), Elliptic Integral, Elliptic Integral of
the First Kind, Modular Angle, Modulus (El-
liptic Integral), Nome, Parameter, Theta Func-
tion

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 590, 1972.

Parameter (Quadric)

The number 6 in the QUADRIC

y

+ ■

a 2 +6 b 2 +6 c 2 + 6

is called the parameter.
see also QUADRIC

Parameterization

The specification of a curve, surface, etc., by means of
one or more variables which are allowed to take on values
in a given specified range.

see also ISOTHERMAL PARAMETERIZATION, REGULAR

Parameterization, Surface Parameterization

Parametric Latitude

An Auxiliary Latitude also called the Reduced
Latitude and denoted r) or 0. It gives the LATITUDE
on a Sphere of Radius a for which the parallel has the
same radius as the parallel of geodetic latitude <j> and
the Ellipsoid through a given point. It is given by

7] = tan" (y 1 — e 2 tan</>).
In series form,
7j = <f> — ei sin(20) + \e± 2 sin(40) - |ei 3 sin(6</>) + . . . ,

where

ei =

vr

1 + VT"

see also Auxiliary Latitude, Ellipsoid, Latitude,
Sphere

References

Adams, O. S. "Latitude Developments Connected with
Geodesy and Cartography with Tables, Including a Table
for Lambert Equal-Area Meridional Projections." Spec.
Pub. No. 67. U. S. Coast and Geodetic Survey, 1921.

Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, p. 18, 1987.

Parametric Test

A Statistical Test in which assumptions are made
about the underlying distribution of observed data.

Pareto Distribution

The distribution

'W-(f)

a+2

References

von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 252, 1993.

Parity

The parity of a number n is the sum of the bits in Bi-
nary representation (mod 2). The parities of the first
few integers (starting with 0) are 0, 1, 1, 0, 1, 0, 0, 1, 1,
0, 0, . . . (Sloane's A010060) summarized in the following
table.

N

Binary

Parity

N

Binary

Parity

1

1

1

11

1011

1

2

10

1

12

1100

3

11

13

1101

1

4

100

1

14

1110

1

5

101

15

1111

6

110

16

10000

1

7

111

1

17

10001

8

1000

1

18

10010

9

1001

19

10011

1

10

1010

20

10100

The constant generated by the sequence of parity digits
is called the Thue-Morse Constant.

see also Binary, Thue-Morse Constant

References

Sloane, N. J. A. Sequence A010060 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Parity Constant

see Thue-Morse Constant

Parking Constant

see RENYI'S PARKING CONSTANTS

Parodi's Theorem

The EIGENVALUES A satisfying P(X) — 0, where P(X) is
the Characteristic Polynomial, lie in the unions of
the Disks

\z + bi

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1119, 1979.

Parry Circle

Parseval's Theorem 1317

Parry Circle

The Circle passing through the Isodynamic Points
and the Centroid of a Triangle (Kimberling 1998,
pp. 227-228),
see also CENTROID (TRIANGLE), ISODYNAMIC POINTS,

Parry Point

References

Kimberling, C. "Triangle Centers and Central Triangles."
Congr. Numer. 129, 1-295, 1998.

Parry Point

The intersection of the Parry Circle and the ClRCUM-
circle of a Triangle. The Trilinear Coordinates
of the Parry point are

2a 2 - b 2 - c 2 * 2b 2 - c 2 - a? ' 2c 2 - a 2 - b 2

(Kimberling 1998, pp. 227-228).
see also PARRY CIRCLE

References

Kimberling, C. "Parry Point." http://www.evansville.edu/

-ck6/t centers/recent /parry. html.
Kimberling, C. "Triangle Centers and Central Triangles."

Congr. Numer. 129, 1-295, 1998.

Parseval's Integral

The Poisson Integral with n = 0.

•+\)] 2 Jo '

Jo(z) = ._, - t lA19 / cos(z cos d)d0,
[F(n -

where Jq(z) is a Bessel Function of the First Kind
and T(x) is a GAMMA FUNCTION.

ParsevaPs Relation

Let F(u) and G(u) be the Fourier Transforms of
f(t) and g(t), respectively. Then

F

v — c

f(t)g'(t)dt

/oo [" />oo floo

/ F^e^^dv I G*{u
OO L" — oo J —oo

/oo /»oo

F{v) / G*(v')8{v' -v)dv' dv
■ oo «/ — oo

)e 2ni " * dv'

di/'

F(v)G*(v)dv.

see also Fourier Transform, Parseval's Theorem

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 425, 1985.

Parseval's Theorem

Let E{t) be a continuous function and E(t) and E u be
Fourier Transform pairs so that

/oo
E u e~ 27Til/t dv
•oo

E*(t) = J ESe^'Uv'.

(1)

(2)

Then

/OO /»oo

\E{t)\ 2 dt= / E(t)E'(t)dt
-oo J ~oo

/oo r /»oo /»oo

/ E v e-^ ivt du\ £„,*e W '&/
-oo Y.J —<x> J —oo

/oo poo /*oo
/ / E V E V ," 'e 2 *^" '-"> dv du' dt
-oo J — oo J — oo
/oo /»oo /'OO
/ / E v ESe 2 * it{v '- v) dtdvdv'
-oo J —oo J — OO
/oo /»oo
/ 5{v' -v)E u E v >*dvdv
•oo 1/-00

/OO />'

E v E v *dv= 1
-oo •/ —

dt

l^rdi/.

where <5(z - x ) is the Delta Function.

For finite FOURIER TRANSFORM pairs h k and H n ,

(3)

iV-1 JV-1

(4)

fc=0

If a function has a FOURIER SERIES given by

oo oo

f(x) = |ao + ]S a n cos(nx) -f \, &« sin(nx), (5)

n—l n=l

then Bessel'S INEQUALITY becomes an equality known
as Parseval's theorem. From (5),

oo

lf( x )] 2 — i a o 2 + a 2j[a n cos(na;) +b n sm(nx)]

oo oo

-f \ y [a n a m cos(nx) cos(ma:)

n=l 771=1

+a n b m cos(nx) sin(mx) + ambn sin(nx) cos(mx)
+b n bm sin(nx) sin(mcc)]. (6)

1318 Part Metric

Integrating

r 2 i 2 r

/ lf( x )] dx = ^ao / dx

J — 7T J — 7T

/7T °°
y [a n cos(nx) -f b n sin(nx)] dx

■^ n = l

/„. OO OO

^ ^ [ana m cos(na:) cos(mx)

■ ff n =lm=l

+a n 5 m cos(na;) sin(mx) + a m 6 n sin(nx) cos(mx)

+b n b m sin(n;c) sin(mx)] dx = \aQ 2 (2n) +

00 00

H- 2J 2Z t ™ ™ 71 "^™ + + + 6n& m 7T<5nm], ( 7 )

SO

1 r °°

i / [/W] 2 dx = \a<? + ^(a n 2 + b n 2 ). (8)

For a generalized Fourier Series with a Complete
Basis {<£i}iSi, an analogous relationship holds. For a
Complex Fourier Series,

7^: / \f(x)\ 2 dx= ^2 l a "

</ — 7T

(9)

n= — 00

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1101, 1979.

Part Metric

A Metric defined by

d(z,w) = sup

\nu(z)

u(w)

:ueH +

where H + denotes the Positive Harmonic Func-
tions on a Domain. The part metric is invariant under
Conformal Maps for any Domain.

References

Bear, H. S. "Part Metric and Hyperbolic Metric." Amer.
Math. Monthly 98, 109-123, 1991.

Partial Derivative

Partial derivatives are defined as derivatives of a func-
tion of multiple variables when all but the variable of
interest are held fixed during the differentiation.

dXm

lira

J \X\ , . . . , Xm ~~r tlj • - - , Xfij j \Xx , . . . , S m , . . . , X n J

(1)

Partial Derivative

The above partial derivative is sometimes denoted f Xm
for brevity. For a "nice" 2-D function f(x,y) (i.e., one
for which /, / x , f y , f xy , f yx exist and are continuous
in a Neighborhood (a, &)), then f xy (a,b) — f yx {a,b).
Partial derivatives involving more than one variable are
called Mixed Partial Derivatives.

For nice functions, mixed partial derivatives must be
equal regardless of the order in which the differentiation
is performed so, for example,

fxy — fyx

Jxxy = Jxyx = Jyxx'
For an EXACT DIFFERENTIAL,

(2)
(3)

*-(s).* + (g)* »>

(

dy\ _ \dxjy

(5)

f(x,y)

■{.

(6)

If the continuity requirement for MIXED PARTIALS is
dropped, it is possible to construct functions for which
MIXED PARTIALS are not equal. An example is the func-
tion

^yP- far (*,») = <>
for (x,y) = 0,

which has /ay (0,0) = —1 and /y X (0,0) = 1 (Wagon
1991). This function is depicted above and by Fischer

(1986).

Abramowitz and Stegun (1972) give Finite Differ-
ence versions for partial derivatives.

see also Ablowitz-Ramani-Segur Conjecture, De-
rivative, Mixed Partial Derivative, Monkey Sad-
dle

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 883-885, 1972.

Fischer, G. (Ed.). Plate 121 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 118, 1986.

Thomas, G. B. and Finney, R. L. §16.8 in Calculus and Ana-
lytic Geometry, 9th ed.0201531747 Reading, MA: Addison-
Wesley, 1996.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 83-85, 1991.

Partial Differential Equation

Partial Fraction Decomposition 1319

Partial Differential Equation

A partial differential equation (PDE) is an equation in-
volving functions and their PARTIAL DERIVATIVES; for
example, the WAVE EQUATION

d 2 ip d 2 ip d 2 i>

V

i_dV
2 dt 2 *

„.2 £U2 V /

In general, partial differential equations are much more
difficult to solve analytically than are ORDINARY DIF-
FERENTIAL EQUATIONS. They may sometimes be solved
using a Backlund Transformation, Characteris-
tic, Green's Function, Integral Transform, Lax
Pair, Separation of Variables, or — when all else
fails (which it frequently does) — numerical methods.

Fortunately, partial differential equations of second-
order are often amenable to analytical solution. Such
PDEs are of the form

AU XX + 2BU X y + CUyy + DU X + EUy + F = . (2)

Second-order PDEs are then classified according to the
properties of the MATRIX

z =

A B
B C

(3)

as Elliptic, Hyperbolic, or Parabolic.

If Z is a Positive Definite Matrix, i.e., det(Z) > 0,
the PDE is said to be Elliptic. Laplace's Equation
and PoiSSON's EQUATION are examples. Boundary con-
ditions are used to give the constraint u(x,y) = g(x,y)
on dO, where

ti*» + u yy = f(u xy u y ,u,x,y)

(4)

holds in Q.

If det(Z) < 0, the PDE is said to be Hyperbolic. The
WAVE EQUATION is an example of a hyperbolic par-
tial differential equation. Initial-boundary conditions
are used to give

where

u(x,0) ~ v(x) for x € 0,

u xx — jyiixyUy^u^Xfy)

(10)

(11)

holds in Q.

see also BACKLUND TRANSFORMATION, BOUNDARY

Conditions, Characteristic (Partial Differen-
tial Equation), Elliptic Partial Differential
Equation, Green's Function, Hyperbolic Par-
tial Differential Equation, Integral Trans-
form, Johnson's Equation, Lax Pair, Monge-
Ampere Differential Equation, Parabolic Par-
tial Differential Equation, Separation of Vari-
ables

References

Arfken, G. "Partial Differential Equations of Theoretical
Physics." §8.1 in Mathematical Methods for Physicists,
3rd ed. Orlando, FL: Academic Press, pp. 437-440, 1985.

Bateman, H. Partial Differential Equations of Mathematical
Physics. New York: Dover, 1944.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Partial Differential Equations." Ch. 19
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 818-880, 1992.

Sobolev, S. L. Partial Differential Equations of Mathematical
Physics. New York: Dover, 1989.

Sommerfeld, A. Partial Differential Equations in Physics.
New York: Academic Press, 1964.

Webster, A. G. Partial Differential Equations of Mathemat-
ical Physics, 2nd corr. ed. New York: Dover, 1955.

Partial Fraction Decomposition

A Rational Function P(x)/Q(x) can be rewritten
using what is known as partial fraction decomposition.
This procedure often allows integration to be performed
on each term separately by inspection. For each factor
of Q(x) the form (ax + 6) m , introduce terms

* + ^i^ + ... + A

: + b (ax + b) 2

(ax + b) ri

(1)

For each factor of the form (ax 2 -f bx + c) m , introduce
terms

u(x t y, t) = g(x, y, t) for x € dO, t > (5)

u(x, y, 0) = v (x, y) in Q, (6)

u t (z, 2/, 0) = vi (x, y) in Q, (7)

where

xy = f(u x ,u u x,y)

holds in Q.

(8)

If det(Z) = 0, the PDE is said to be parabolic. The
Heat Conduction Equation equation and other dif-
fusion equations are examples. Initial-boundary condi-
tions are used to give

A x x + Bx t A 2 x + B 2 A m x + B n

ax 2 + bx + c (ax 2 -f bx + c) 2

(ax 2 + bx + cY

Then write

P(x) _ A x

(2)

Q(x) ax-\-b

+...+ A r+ B i + ... ( 3)

ax 2 + bx + c

and solve for the AiS and BiS.

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 13-15, 1987.

u(x, t) = g(x, t) for xG^,t>0 (9)

1320 Partial Latin Square

Partition

Partial Latin Square

In a normal nxn LATIN SQUARE, the entries in each row
and column are chosen from a "global" set of n objects.
Like a Latin square, a partial Latin square has no two
rows or columns which contain the same two symbols.
However, in a partial Latin square, each cell is assigned
one of its own set of n possible "local" (and distinct)
symbols, chosen from an overall set of more than three
distinct symbols, and these symbols may vary from lo-
cation to location. For example, given the possible sym-
bols {1,2,..., 6} which must be arranged as

{1,2,3} {1,3,4} {2,5,6}
{2,3,5} {1,2,3} {4,5,6}
{4,3,6} {3,5,6} {2,3,5},

the 3x3 partial Latin square

13 2
2 4 5
6 5 3

can be constructed.

see also Dinitz Problem, Latin Square

References

Cipra, B. "Quite Easily Done." In What's Happening in the
Mathematical Sciences 2, pp. 41-46, 1994.

Partial Order

A Relation "<" is a partial order on a Set 5 if it has:

1. Reflexivity: a < a for all a € S.

2. Antisymmetry: a < b and b < a implies a = b.

3. Transitivity: a < b and b < c implies a < c.

For a partial order, the size of the longest CHAIN (An-
tichain) is called the Length (Width). A partially
ordered set is also called a POSET.
see also ANTICHAIN, CHAIN, FENCE POSET, IDEAL

(Partial Order), Length (Partial Order), Lin-
ear Extension, Partially Ordered Set, Total
Order, Width (Partial Order)

References

Ruskey, F. "Information on Linear Extension." http://sue
. esc .uvic . ca/-cos/inf /pose/LinearExt .html.

Partial Quotient

If the Simple Continued Fraction of a Real Num-
ber x is given by

x = ao 4-

1

at 4-

a2

A3 4 . . .

Partially Ordered Set

A partially ordered set (or Poset) is a Set taken to-
gether with a Partial Order on it. Formally, a par-
tially ordered set is defined as an ordered pair P =
(X, <), where X is called the Ground Set of P and
< is the Partial Order of P.

see also CIRCLE ORDER, COVER RELATION, DOMI-
NANCE, Ground Set, Hasse Diagram, Interval Or-
der, Isomorphic Posets, Partial Order, Poset
Dimension, Realizer, Relation

References

Dushnik, B. and Miller, E. W. "Partially Ordered Sets."
Amer. J. Math. 63, 600-610, 1941.

Fishburn, P. C. Interval Orders and Interval Sets: A Study
of Partially Ordered Sets. New York: Wiley, 1985.

Trotter, W. T. Combinatorics and Partially Ordered Sets:
Dimension Theory. Baltimore, MD: Johns Hopkins Uni-
versity Press, 1992.

Particularly Weil-Behaved Functions

Functions which have DERIVATIVES of all orders at all
points and which, together with their DERIVATIVES, fall
off at least as rapidly as \x\~ n as \x\ -► oo, no matter
how large n is.
see also REGULAR SEQUENCE

Partisan Game

A Game for which each player has a different set of
moves in any position. Every position in an IMPARTIAL
Game has a Nim- Value.

Partition

A partition is a way of writing an Integer n as a sum
of Positive Integers without regard to order, possibly
subject to one or more additional constraints. Particu-
lar types of partition functions include the Partition
Function P, giving the number of partitions of a num-
ber without regard to order, and PARTITION FUNCTION
Q, giving the number of ways of writing the Integer n
as a sum of POSITIVE INTEGERS without regard to order
with the constraint that all INTEGERS in each sum are
distinct.

see also AMENABLE NUMBER, DURFEE SQUARE, EL-
DER'S Theorem, Ferrers Diagram, Graphical
Partition, Partition Function P, Partition Func-
tion Q, Perfect Partition, Plane Partition, Set
Partition, Solid Partition, Stanley's Theorem

References

Andrews, G. E. The Theory of Partitions. Cambridge, Eng-
land: Cambridge University Press, 1998.

Dickson, L. E. "Partitions." Ch. 3 in History of the Theory
of Numbers, Vol. 2: Diophantine Analysis. New York:
Chelsea, pp. 101-164, 1952.

then the quantities ai are called partial quotients.

see also Continued Fraction, Convergent, Simple

Continued Fraction

Partition Function P

Partition Function P 1321

Partition Function P

P(n) gives the number of ways of writing the INTEGER
n as a sum of Positive Integers without regard to
order. For example, since 4 can be written

4 = 4
= 3 + 1

= 2 + 2

=2+1+1

= 1 + 1 + 1 + 1, (1)

so P(4) = 5. P(n) satisfies

P(n)<f[P(n+l) + P(n-l)] (2)

(Honsberger 1991). The values of P(n) for n = 1, 2,
..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (Sloane's
A000041). The following table gives the value of P(n)
for selected small n.

n

P(n)

50

204226

100

190569292

200

3972999029388

300

9253082936723602

400

6727090051741041926

500

2300165032574323995027

600

458004788008144308553622

700

60378285202834474611028659

800

5733052172321422504456911979

900

415873681190459054784114365430

1000

24061467864032622473692149727991

n for which P(n) is Prime are 2, 3, 4, 5, 6, 13, 36,
77, 132, 157, 168, 186, ... (Sloane's A046063). Num-
bers which cannot be written as a PRODUCT of P(n) are
13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (Sloane's
A046064), corresponding to numbers of nonisomorphic
AB ELIAN GROUPS which are not possible for any group
order.

When explicitly listing the partitions of a number n,
the simplest form is the so-called natural representation
which simply gives the sequence of numbers in the rep-
resentation (e.g., (2, 1, 1) for the number 4 = 2 + 1 + 1).
The multiplicity representation instead gives the number
of times each number occurs together with that number
(e.g., (2, 1), (1, 2) for 4 = 2 ■ 1 + 1 • 2). The Ferrers
DIAGRAM is a pictorial representation of a partition.

Euler invented a GENERATING FUNCTION which gives
rise to a Power Series in P{n),

oo

P(n) = ^(-l) m+1 [P(n - \m{Zm - 1))

171=1

+P{n- |m(3m+l))]. (3)

A Recurrence Relation is

71-1

P(n) = - ^ a(n - m)P{m),

(4)

where a(n) is the DIVISOR FUNCTION (Berndt 1994,
p. 108). Euler also showed that, for

oo oo

f{x) = Y[ (1 - x m ) = Ys i-l) n x n(3n+1)/2 (5)

m = l n= — oo

-, 2,5,7 12 15 , 22 , 26 , s a \

= l — x — x +x + x —x —x -\-x +x +..., (6)

where the exponents are generalized PENTAGONAL
NUMBERS 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, . . . (Sloane's
A001318) and the sign of the &th term (counting as
the 0th term) is (-l)^* 1 )/^ (with [x\ the Floor
FUNCTION), the partition numbers P[n) are given by
the Generating Function

1

W)

J2P(n)x\

(7)

MacMahon obtained the beautiful RECURRENCE RELA-
TION

P(n) - P{n - 1) - P(n - 2) + P{n - 5) + P(n - 7)

-P(n - 12) - P(n - 15) + . . . = 0, (8)

where the sum is over generalized PENTAGONAL NUM-
BERS < n and the sign of the Aith term is (-l)L( fc+1 )/ 2 J ?
as above.

In 1916-1917, Hardy and Ramanujan used the CIRCLE
Method and elliptic Modular Functions to obtain
the approximate solution

1

P(n)

4nV3

Tr<sJ2n/3

(9)

Rademacher (1937) subsequently obtained an exact se-
ries solution which yields the Hardy-Ramanujan FOR-
MULA (9) as the first term:

P(n) = J2L q (n)iP q (n),

q = l

where

K = 7T

L <i( n ) = ^2"> Pfq e-

2npwi/q

= lVu ( ^ ^ - -\

9~ V 1 L 1 J 2 /

fj. — 1

A ^ = v n - h

ip q {n)

tt\/2 I dm

sinh(^)

(10)

(11)
(12)

(13)
(14)

(15)
(16)

1322

Partition Function P

Partition Function P

[x\ is the Floor Function, and p runs through the
Integers less than and Relatively Prime to q (when
q = 1 } p = 0). The remainder after Q terms is

R(Q) < CQ- 1 ^ 2 + Dtl^smh ( ^) , (17)
V n \ Q J

where C and D are fixed constants.

With f(x) as defined above, Ramanujan also showed
that

5 ^ = f>( 5 -+ 4 )* m ( 18 )

m=0

Ramanujan also found numerous CONGRUENCES such as

P(5m + 4) = (mod 5) (19)

P(7m + 5) = (mod 7) (20)

P(llm + 6) = (mod 11) . (21)

Ramanujan's Identity gives the first of these.

Let Po(n) be the number of partitions of n containing
ODD numbers only and Po(n) be the number of parti-
tions of n without duplication, then

oo

Po{n) = P D (n)= JJ (1 + x k + x 2k + x 3k + . . .)

fc=l,3,...

oo

= JJ(l + z fc ) = 1+x + x 2 + 2x 3 + 2x 4 +3x 5 + . . . , (22)

as discovered by Euler (Honsberger 1985). The first few
values of P - Pd are 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, . . .
(Sloane's A000009).

Let Fe(ti) be the number of partitions of Even num-
bers only, and let Peo{^) {Pdo{^)) be the number of
partitions in which the parts are all Even (Odd) and
all different. The first few values of PDo(n) are 1, 1, 0,
1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, . . . (Sloane's A000700).
Some additional GENERATING FUNCTIONS are given by
Honsberger (1985, pp. 241-242)

£»■

no even

part repeated \Tl)X

= JJ(1 - o: a *- 1 )- 1 (l + a: 2 *) (23)

y -trio part occurs more than 3 times \Tl)X
n = l

= H(l + x h +x 2k +x 3h ) (24)

1-x 4

/ -*iio part divisible by &\Tl)X — II

(25)

E".

no part occurs more

than d times (n)X

fc=l t=0 fc=l

y J J every part occurs 2, 3, or 5 times V^J**'
n=l

= JJ(l + x 2fc +x 8 *+x 5fc )
fc=l

^n(i+' M )d+'")-n !:£}:£ <">

fc=i fc=i

oo
/ mio part occurs exactly once^j^

w 4fc -, 6k

(i +B » + ,» + ...)=n7id

i + x b

(i-a^xi-a 3 *) 1

(28)

Some additional interesting theorems following from
these (Honsberger 1985, pp. 64-68 and 143-146) are:

1. The number of partitions of n in which no Even part
is repeated is the same as the number of partitions of
n in which no part occurs more than three times and
also the same as the number of partitions in which
no part is divisible by four.

2. The number of partitions of n in which no part oc-
curs more often than d times is the same as the num-
ber of partitions in which no term is a multiple of
d + 1.

3. The number of partitions of n in which each part ap-
pears either 2, 3, or 5 times is the same as the number
of partitions in which each part is CONGRUENT mod
12 to either 2, 3, 6, 9, or 10.

4. The number of partitions of n in which no part ap-
pears exactly once is the same as the number of par-
titions of n in which no part is CONGRUENT to 1 or
5 mod 6.

5. The number of partitions in which the parts are all
Even and different is equal to the absolute differ-
ence of the number of partitions with ODD and EVEN
parts.

P(n, ft), also written Pk(n), is the number of ways of
writing n as a sum of ft terms, and can be computed
from the RECURRENCE RELATION

P(n, ft) = P{n - 1, ft - 1) + P(n - ft, ft)

(29)

(Ruskey). The number of partitions of n with largest
part ft is the same as P(n, ft).

The function P(n, ft) can be given explicitly for the first
few values of ft,

P(n,2) = [|nj
P(n,3) = [£n a ],

(30)
(31)

Partition Function Q

Pascal Line

1323

where [x\ is the FLOOR FUNCTION and [x] is the Nint
function (Honsberger 1985, pp. 40-45).

see also Alcuin's Sequence, Elder's Theorem, Eu-
ler's Pentagonal Number Theorem, Ferrers Di-
agram, Partition Function Q, Pentagonal Num-
ber, r k (n), Rogers-Ramanujan Identities, Stan-
ley's Theorem

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Unrestricted
Partitions." §24.2.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, p. 825, 1972.

Adler, H. "Partition Identities — From Euler to the Present."
Amer. Math. Monthly 76, 733-746, 1969.

Adler, H. "The Use of Generating Functions to Discover and
Prove Partition Identities." Two-Year College Math. J.
10, 318-329, 1979.

Andrews, G. Encyclopedia of Mathematics and Its Applica-
tions, Vol. 2: The Theory of Partitions. Cambridge, Eng-
land: Cambridge University Press, 1984.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, 1994.

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 94-96, 1996.

Honsberger, R. Mathematical Gems IIL Washington, DC:
Math. Assoc. Amer., pp. 40-45 and 64-68, 1985.

Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 237-239, 1991.

Jackson, D. and Goulden, I. Combinatorial Enumeration.
New York: Academic Press, 1983.

MacMahon, P. A. Combinatory Analysis. New York:
Chelsea, 1960.

Rademacher, H. "On the Partition Function p(n)." Proc.
London Math. Soc. 43, 241-254, 1937.

Ruskey, F. "Information of Numerical Partitions." http://
sue.csc.uvic.ca/-cos/inf /nump/NumPartition.html.

Sloane, N. J. A. Sequences A000009/M0281, A000041/
M0663, and A000700/M0217 in "An On-Line Version of
the Encyclopedia of Integer Sequences."

Partition Function Q

Q(n) gives the number of ways of writing the INTEGER n
as a sum of POSITIVE INTEGERS without regard to order
with the constraint that all INTEGERS in each sum are
distinct. The values for n = 1, 2, . . . are 1, 1, 2, 2, 3, 4,
5, 6, 8, 10, ... (Sloane's A000009). The Generating
Function for Q(n) is

= 1 + x + x 2 + 2x 3 + 2x 4 + 3x 5 + . . . .

The values of n for which Q(n) is Prime are 3, 4, 5,
7, 22, 70, 100, 495, 1247, 2072, ... (Sloane's A046065),
with no others for n < 15,000.

The number of Partitions of n with < k summands is
denoted q(n,k) or qk(n). Therefore, q n (n) = P(n) and

qk(n) = q k -i(n) + q k (n - k).
see also PARTITION FUNCTION P

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Partitions into
Distinct Parts." §24.2.2 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, pp. 823-824, 1972,

Sloane, N. J. A. Sequences A046065 and A000009/M0281 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."

Party Problem

Also known as the MAXIMUM CLIQUE PROBLEM. Find
the minimum number of guests that must be invited so
that at least m will know each other or at least n will not
know each other. The solutions are known as RAMSEY

Numbers.

see also CLIQUE, RAMSEY NUMBER

Parzen Apodization Function

An Apodization Function similar to the Bartlett
Function.

see also Apodization Function, Bartlett Func-
tion

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 547, 1992.

Pascal Distribution

see Negative Binomial Distribution

Pascal's Formula

Each subsequent row of PASCAL'S TRIANGLE is obtained
by adding the two entries diagonally above. This follows
immediately from the Binomial Coefficient identity

f n \ _ n - _ ( n ~ l)- n
yrj ~ (n-r)lrl (n-r)lrl

_ (n — l)!(n — r) (n — l)!r
(n — r)\r\ (n — r)\r\

(n - 1)! + (n-1)!

(n — r — l)!r! (n — r)\(r — 1)!

see also Binomial Coefficient, Pascal's Triangle

Pascal's Hexagrammum Mysticum

see Pascal's Theorem

Pascal's Limagon

see LlMAgON

Pascal Line

The line containing the three points of the intersection
of the three pairs of opposite sides of two TRIANGLES.

see also PASCAL'S THEOREM

1324

Pascal's Rule

Pascal's Triangle

Pascal's Rule

see Pascal's Formula

Pascal's Theorem

The dual of Brianchon's Theorem. It states that,
given a (not necessarily REGULAR, or even CONVEX)
Hexagon inscribed in a Conic Section, the three
pairs of the continuations of opposite sides meet on a
straight Line, called the Pascal Line. There are 6!
(6! means 6 FACTORIAL, where 6! = 6 • 5 • 4 • 3 • 2 • 1)
possible ways of taking all VERTICES in any order, but
among these are six equivalent CYCLIC PERMUTATIONS
and two possible orderings, so the total number of dif-
ferent hexagons (not all simple) is

6!
2-6

720
12

= 60.

There are therefore a total of 60 PASCAL LINES created
by connecting Vertices in any order. These intersect
three by three in 20 STEINER POINTS.

see also Braikenridge-Maclaurin Construction,
Brianchon's Theorem, Cayley-Bacharach Theo-
rem, Conic Section, Duality Principle, Hexagon,
Pappus's Hexagon Theorem, Pascal Line, Steiner
Points

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 73-76, 1967.

Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 105-106, 1990.

Pappas, T. "The Mystic Hexagram." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, p. 118,
1989.

Yanghui (about 500 years earlier, in fact) and the Ara-
bian poet-mathematician Omar Khayyam. It is there-
fore known as the Yanghui TRIANGLE in China. Start-
ing with n = 0, the TRIANGLE is

1

1 1

1 2 1

13 3 1

14 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

(Sloane's A007318). Pascal's FORMULA shows that
each subsequent row is obtained by adding the two en-
tries diagonally above,

(n — r)\rl

-(";K=0-

(2)

10 10 5 1

6 15 20 15 6 1

In addition, the "SHALLOW DIAGONALS" of Pascal's tri-
angle sum to Fibonacci Numbers,

y^, k v (-1)^3^ (1, 2,l-n;|(3- n), 2- §n;-j)

7r(2-3n + n 2 )
jt=i

= ^+1, (3)

where 3 F 2 {a,b,c]d,e;z) is a GENERALIZED HYPERGEO-
metric Function.

Pascal's triangle contains the Figurate NUMBERS
along its diagonals. It can be shown that

PascaPs Triangle

A Triangle of numbers arranged in staggered rows
such that

(i)

r!(n — r)\ \ r /

where (") is a BINOMIAL COEFFICIENT. The trian-
gle was studied by B. Pascal, although it had been
described centuries earlier by Chinese mathematician

and

n + 1

En-h i

n+l),(j+l)

(4)

( m 1 +1 )E^+( m 2 +1 )5: fcm " 1

(5)

Pascal's Triangle

Patch

1325

The "shallow diagonals" sum to the FIBONACCI SE-
QUENCE, i.e.,

1 = 1

1 = 1

2 = 1 + 1

3 = 2+1

5 = 1 + 3+1
8 = 3 + 4 + 1.

In addition,

^aij = 2* - 1.

(6)

(?)

j'=i

It is also true that the first number after the 1 in each
row divides all other numbers in that row IFF it is a
Prime. If P n is the number of Odd terms in the first n
rows of the Pascal triangle, then

0.812... <P n n" ln2/ln3 <1

(8)

(Harborth 1976, Le Lionnais 1983).

The Binomial Coefficient (™) mod 2 can be com-
puted using the XOR operation n XOR m, making Pas-
cal's triangle mod 2 very easy to construct. Pascal's tri-
angle is unexpectedly connected with the construction
of regular POLYGONS and with the SlERPlNSKI SIEVE.

see also Bell Triangle, Binomial Coefficient, Bi-
nomial Theorem, Brianchon's Theorem, Cata-
lan's Triangle, Clark's Triangle, Euler's Tri-
angle, Fibonacci Number, Figurate Number
Triangle, Leibniz Harmonic Triangle, Number
Triangle, Pascal's Formula, Polygon, Seidel-
Entringer- Arnold Triangle, Sierpinski Sieve,
Trinomial Triangle

References

Conway, J. H. and Guy, R. K. "Pascal's Triangle." In The
Book of Numbers. New York: Springer- Verlag, pp. 68-70,
1996.

Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 17, 1996.

Harborth, H. "Number of Odd Binomial Coefficients. " Not.
Amer. Math. Soc. 23, 4, 1976.

Le Lionnais, F. Les nombres remarquables . Paris; Hermann,
p. 31, 1983.

Pappas, T. "Pascal's Triangle, the Fibonacci Sequence &
Binomial Formula," "Chinese Triangle," and "Probability
and Pascal's Triangle." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, pp. 40-41 88, and
184-186, 1989.

Sloane, N. J. A. Sequence A007318/M0082 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Smith, D. E. A Source Book in Mathematics. New York:
Dover, p. 86, 1984.

Pascal's Wager

"God is or He is not. . . Let us weigh the gain and the
loss in choosing. . . ( God is.' If you gain, you gain all, if
you lose, you lose nothing. Wager, then, unhesitatingly,
that He is."

Pasch's Axiom

In the plane, if a line intersects one side of a TRIANGLE
and misses the three VERTICES, then it must intersect
one of the other two sides. This is a special case of the
generalized MENELAUS' THEOREM with n = 3.
see also HELLY'S THEOREM, MENELAUS' THEOREM,

Pasch's Theorem

Pasch's Theorem

A theorem stated in 1882 which cannot be derived from
Euclid's Postulates. Given points a, 6, c, and d on
a Line, if it is known that the points are ordered as
(a, 6, c) and (6, c, rf), it is also true that (a, b, d).
see also EUCLID'S POSTULATES, LINE, PASCH'S AXIOM

Pass Equivalent

Two KNOTS are pass equivalent if there exists a sequence
of pass moves taking one to the other. Every KNOT
is either pass equivalent to the UNKNOT or TREFOIL
Knot. These two knots are not pass equivalent to each
other, but theENANTiOMERS of the Trefoil Knot are
pass equivalent. A Knot has Arf Invariant if the
KNOT is pass equivalent to the Unknot and 1 if it is
pass equivalent to the TREFOIL KNOT.
see also ARF INVARIANT, KNOT, PASS MOVE, TREFOIL

Knot, Unknot

References

Adams, C. C. The Knot Book: An Elementary Introduction

to the Mathematical Theory of Knots. New York: W. H.

Freeman, pp. 223-228, 1994.

Pass Move

A change in a knot projection such that a pair of oppo-
sitely oriented strands are passed through another pair
of oppositely oriented strands.
see also PASS EQUIVALENT

Patch

A patch (also called a LOCAL SURFACE) is a differen-
tiate mapping x : U — ► R n , where U is an open subset
of R 2 . More generally, if A is any Subset of R 2 , then
a map x : A — > R n is a patch provided that x can be
extended to a differentiate map from U into W 1 , where
U is an open set containing A. Here, x(U) (or more
generally, 'x.(A)) is called the TRACE of x.

see also GAUSS MAP, INJECTIVE PATCH, MONGB

Patch, Regular Patch, Trace (Map)

References

Gray, A. "Patches in R 3 ." §10.2 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 183-184 and 192-193, 1993.

1326

Path

Peacock's Tail

Path

A path 7 is a continuous mapping 7 : [a, b] h->- C, where
7(a) is the initial point and 7(6) is the final point. It is
often written parametrically as a(t).

Path Graph

The path P n is a TREE with two nodes of valency 1, and
the other n — 2 nodes of valency 2. Path graphs P n are
always Graceful for n > 4.

see also Chain (Graph), Graceful Graph, Hamil-
tonian Path, Tree

Path Integral

Let 7 be a Path given parametrically by a(t). Let s
denote Arc LENGTH from the initial point. Then

I }{s)ds= I f{a{t))\a\t)\dt

= I f(x(t),y(t),z(t))W'(t)\dt.

see also Line Integral

References

Press, VV\ H.; Flannery, B. P.; Teukolsky, S, A.; and Vetter-
ling, W. T. "Evaluation of Functions by Path Integration."
§5.14 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 201-204, 1992.

Pathwise- Connected

A Topological Space X is pathwise-connected Iff
for every two points x,y G X, there is a CONTINUOUS
Function / from [0,1] to X such that /(0) = x and
/(I) = y. Roughly speaking, a Space X is pathwise-
connected if, for every two points in X, there is a path
connecting them. For Locally Pathwise-Connected
SPACES (which include most "interesting spaces" such as
Manifolds and CW-Complexes), being Connected
and being pathwise-connected are equivalent, although
there are connected spaces which are not pathwise con-
nected. Pathwise-connected spaces are also called 0-
connected.
see also CONNECTED SPACE, CW-COMPLEX, LOCALLY

Pathwise-Connected Space, Topological Space

Patriarchal Cross

see Gaullist Cross

Pauli Matrices

Matrices which arise in Pauli's treatment of spin in
quantum mechanics. They are defined by

sPx =

0~2 = 0~ y = P2 =

os — o~ z = P3 =

"0

l"

1

"

i

—i

1

-1

(1)

(2)
(3)

The Pauli matrices plus the 2 x 2 IDENTITY MATRIX
I form a complete set, so any 2x2 matrix A can be
expressed as

A = CqI + C\0~\ + c 2 cr 2 4- C30-3.

The associated matrices

<r+ = 2
a- =2

1

"0

0"

1

"1

0'

1

(4)

(5)
(6)
(7)

can also be defined. The Pauli spin matrices satisfy the
identities

(JiCTj = \Sij + €ijklCTk

ffiffj + CjCi — 2c i

(8)

(9)

(10)

V x p x + CTyPy + a z p z = \fp x 2 +Py 2 +Pz 2 >

see also DlRAC MATRICES, QUATERNION

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 211-212, 1985.

Goldstein, H. "The Cayley-Klein Parameters and Related
Quantities." Classical Mechanics, 2nd ed. Reading, MA:
Addison- Wesley, p. 156, 1980.

Pauli Spin Matrices

see Pauli Matrices

Payoff Matrix

A m x n Matrix which gives the possible outcome of a
two-person ZERO-SUM GAME when player A has m pos-
sible strategies and player B n strategies. The analysis of
the MATRIX in order to determine optimal strategies is
the aim of Game Theory. The so-called "augmented"
payoff matrix is defined as follows:

Pa Pi P2

11

— 1 an «i2

-1 a 2 i a 22

P n P n+ i

din 1

a 2n

Pn+2

p n+ „

-1 a mi a m2
see also GAME THEORY, ZERO-SUM GAME

Peacock's Tail

One name for the figure used by Euclid to prove the
Pythagorean Theorem.

see also BRIDE'S CHAIR, WINDMILL

Peano Arithmetic

Pear Curve

1327

Peano Arithmetic

The theory of Natural Numbers defined by the five
PEANO'S Axioms. Any universal statement which is
undecidable in Peano arithmetic is necessarily True.
Undecidable statements may be either True or False.
Paris and Harrington (1977) gave the first "natural" ex-
ample of a statement which is true for the integers but
unprovable in Peano arithmetic (Spencer 1983).

see also Kreisel Conjecture, Natural Indepen-
dence Phenomenon, Number Theory, Peano's Ax-
ioms

References

Kirby, L. and Paris, J. "Accessible Independence Results for
Peano Arithmetic." Bull. London Math. Soc. 14, 285-293,
1982.

Paris, J. and Harrington, L. "A Mathematical Incomplete-
ness in Peano Arithmetic." In Handbook of Mathematical
Logic (Ed. J. Barwise). Amsterdam, Netherlands: North-
Holland, pp. 1133-1142, 1977.

Spencer, J. "Large Numbers and Unprovable Theorems."
Amer. Math. Monthly 90, 669-675, 1983.

Peano's Axioms

1. Zero is a number.

2. If a is a number, the successor of a is a number.

3. ZERO is not the successor of a number.

4. Two numbers of which the successors are equal are
themselves equal.

5. (Induction Axiom.) If a set S of numbers contains
ZERO and also the successor of every number in 5,
then every number is in S.

Peano's axioms are the basis for the version of NUMBER
Theory known as Peano Arithmetic.

see also Induction Axiom, Peano Arithmetic
Peano Curve

A Fractal curve which can be written as a Linden-
mayer System.

see also Dragon Curve, Hilbert Curve, Linden-
mayer System, Sierpinski Curve

References

Dickau, R. M. "Two-Dimensional L-Systems." http://
forum . swarthmore . edu/advanced/robertd/lsys2d . html.

Hilbert, D. "Uber die stetige Abbildung einer Linie auf ein
Flachenstuck." Math. Ann. 38, 459-460, 1891.

Peano, G. "Sur une courbe, qui remplit une aire plane."
Math. Ann. 36, 157-160, 1890.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 207, 1991.

Peano- Gosper Curve

Jtfl

A Plane-Filling Curve originally called a Flow-
snake by R. W. Gosper and M. Gardner. Mandel-
brot (1977) subsequently coined the name Peano-Gosper
curve. The GOSPER ISLAND bounds the space that the
Peano-Gosper curve fills.

see also Dragon Curve, Exterior Snowflake,
Gosper Island, Hilbert Curve, Koch Snowflake,
Peano Curve, Sierpinski Arrowhead Curve, Sier-
pinski Curve

References

Dickau, R. M. "Two-Dimensional L-Systems." http://

forum . swarthmore . edu/advanced/robertd/lsys2d . html.
Mandelbrot, B. B. Fractals: Form, Chance, & Dimension.

San Francisco, CA: W. H. Freeman, 1977.

Peano Surface

The function

f(x,y) = (2x 2 -y)(y-

V)

which does not have a Local Maximum at (0, 0), de-
spite criteria commonly touted in the second half of the
1800s which indicated the contrary.

see also Local Maximum

References

Fischer, G. (Ed.). Plate 122 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 119, 1986.

Leitere, J. "Functions." §7.1.2 in Mathematical Models from
the Collections of Universities and Museums (Ed. G. Fis-
cher). Braunschweig, Germany: Vieweg, pp. 70-71, 1986.

Pear Curve

1328 Pear-Shaped Curve

Pearson System

The LEMNISCATE L 3 in the iteration towards the MAN-
DELBROT Set. In Cartesian Coordinates with a
constant r, the equation is given by

r 2 = (x 2 +y 2 )(l + 2x + 5x 2 +6x 3 +6x 4 +4x 5 +x 6 -3y 2
-2xy 2 + Sx 2 y 2 + Sx z y 2 + 3a; V + 2y 4 + 4xy 4
+3x 2 y 4 + y 6 ).

see also Pear-Shaped Curve
Pear-Shaped Curve

A curve given by the Cartesian equation

<2 2 3/ \

by — x (a — x).

see also Pear Curve, Teardrop Curve

References

MacTutor History of Mathematics Archive. "Pear-Shaped

Cubic." http: //www-groups .dcs . st -and. ac .uk/ -history

/Curves/Pearshaped.html.

Pearson Mode Skewness

Given a DISTRIBUTION with measured MEAN, MODE,
and Standard Deviation s, the Pearson mode skew-
ness is

mean — mode

see also Mean, Mode, Pearson Skewness, Pear-
son's Skewness Coefficients, Skewness

Pearson Skewness

Let a Distribution have third Moment /z 3 and Stan-
dard Deviation <j, then the Pearson skewness is de-
fined by

*-(?)'

see also Fisher Skewness, Pearson's Skewness Co-
efficients, Skewness

Pearson's Skewness Coefficients

Given a Distribution with measured Mean, Median,
Mode, and Standard Deviation s, Pearson's first
skewness coefficient is

3 [mean] — [mode]
s

and the second coefficient is

3 [mean] — [median]

Pearson's Correlation

see Correlation Coefficient

Pearson-Cunningham Function

see Cunningham Function

Pearson's Function

A x, 2 fc-3\_r(§x s 2 ,*fi)

v^fc^i) 2

r(V)

where T(x) is the Gamma Function.

see also CHI-SQUARED Test, GAMMA FUNCTION

Pearson Kurtosis

Let [i 4 be the fourth MOMENT of a Distribution and
a its Variance. Then the Pearson kurtosis is defined

by

see also FlSHER KURTOSIS, KURTOSIS

see also FlSHER SKEWNESS, PEARSON SKEWNESS,

Skewness

Pearson System

Generalizes the differential equation for the GAUSSIAN

Distribution

dy _ y(m - x)

dx a

(i)

to

dy _ y(m - x)

(2)

dx a + bx + ex 2

Let ci, C2 be the roots of a + bx + ex 2 . Then the possible
types of curves are

0. b = c = 0, a > 0. E.g., Normal Distribution.

1. b 2 /4ac < 0, d < x < c 2 . E.g., Beta Distribu-
tion.

II. b 2 /4ac = 0, c < 0, — ci < x < a where d =

\/-c/a.

III. b 2 /4ac = 00, c = 0, c\ < x < 00 where Ci =
—a/6. E.g., Gamma Distribution. This case is
intermediate to cases I and VI.

IV. < b 2 /4ac < 1, -00 < x < 00.

V. b 2 /4ac = 1, ci < x < 00 where Ci = — 6/2a.
Intermediate to cases IV and VI.

Pearson System

VI. 6 2 /4ac > 1, ci < x < oo where ci is the larger
root. E.g., Beta Prime Distribution.

VII. 6 2 /4ac = 0, c > 0, -oo < x < oo. E.g., Stu-
dent's ^-Distribution.

Classes IX-XII are discussed in Pearson (1916). See also
Craig (in Kenney and Keeping 1951). If a Pearson curve
possesses a MODE, it will be at x = m. Let y(x) — at
c± and C2, where these may be — oo or oo. If yx r+2 also
vanishes at ci, C2, then the rth Moment and (r + l)th
Moments exist.

f 2 ^-{ax r +bx r+l +cx r+2 )dx = f 2 y{mx r -x r+l )dx

(3)
giving
[y{ax r -t-bx r+1 + cx r+2 )} c c \

t/ ci

y[arz r * + b(r + l)x r + c(r + 2)z r ~ M ] da;

*/ Ci

y(ma; r -/ +1 )^ ( 4 )

/»c 2

- / ylarx 7 " 1 + 6(r + l)x r + c(r 4- 2)x T ' +1 ] dx

-f

y(mx r -x r+1 )dx (5)

also,

--f

*/ Ci

yx r dx,

(6)

ari/ r -i + b(r + l)v r + c(r + 2)i/ r +i = — mi/ r + i>v+i. (7)
For r = 0,

so

6 + 2ci/i = — m + j^i ,

m + b

Vi =

I -2c'

(8)

(9)

For r = 1,

a 4- 26i/i + 3cv2 = -mi/i + 1^2, (10)

a-\- (m + 2b)vi

" 2 = l-3c

Now let t = (x — v x )i<J. Then

^2 = ^2 = 1

(11)

(12)
(13)
(14)

Pearson Type III Distribution 1329

Hence 6 = — m, and a = 1 — c so

(1 - 3c)ra r _i - mra r + [c(r + 2) - l]a r +i = 0. (15)

For r = 2,

2m + (l-4c)a 3 =0. (16)

For r = 3,

3(1 - 3c) - 3ma 3 - (1 - 5c)a 4 = 0. (17)

So the Skewness and KURTOSIS are
2m

7i —ots —

4c— 1

(18)

6(rn 2 - 4c 2 + c)
72=Q4 3= (4c-l)(5c-l)'

(19)

So the parameters a, 6, and c can be written

a = 1 - 3c

(20)

" = — m — — r- T-rr

(21)

where

2(1 + 2*)'

5= 2 72 -3 7 i 2

72 + 6

(22)

(23)

References

Craig, C. C. "A New Exposition and Chart for the Pearson
System of Frequency Curves." Ann. Math. Stat. 7, 16-28,
1936.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 107, 1951.

Pearson, K. "Second Supplement to a Memoir on Skew Vari-
ation." Phil Trans. A 216, 429-457, 1916.

Pearson Type III Distribution

A skewed distribution which is similar to the BINOMIAL
Distribution when p ^ q (Abramowitz and Stegun
1972, p. 930).

y = k(t + A) A -V At ,

for t G [0, oo) where

A = 2/7

K =

A A e~

r(A») '

(i)

(2)
(3)

V(z) is the Gamma Function, and t is a standardized
variate. Another form is

^-^inrf "inr)

(4)

1330 Pearls of Sluze

For this distribution, the CHARACTERISTIC FUNCTION

0(i) = e <at (l-»/3t)" P , (5)

and the MEAN, VARIANCE, SKEWNESS, and KURTOSIS
are

Pedal Circle

sliding, and was discovered in 1864. Another LINKAGE
which performs this feat using hinged squares had been
published by Sarrus in 1853 but ignored. Coxeter (1969,

p. 428) shows that

OP x OP'

OA 2 -PA 2 .

V

= oc + p/3

2

-P/3 2

7i

2

72

_ 6

(6)
(7)

(8)
(9)

References

Abramowitz, M. and Stegun, C. A. (Eds.)- Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
1972.

Pearls of Sluze

y m ^kx n (a-x)\

The curves with integral n, p, and m were studied by
de Sluze between 1657 and 1698. The name "Pearls
of Sluze" was given to these curves by Blaise Pascal
(MacTutor Archive).

References

MacTutor History of Mathematics Archive. "Pearls of

Sluze." http: //www-groups .dcs . st - and. ac .uk/ -history

/Curves /Pearls .html.

Peaucellier Cell

see Peaucellier Inversor

Peaucellier Inversor

0<

A LINKAGE with six rods which draws the inverse of a
given curve. When a pencil is placed at P, the inverse
is drawn at P' (or vice versa). If a seventh rod (dashed)
is added (with an additional pivot), P is kept on a circle
and the locus traced out by P' is a straight line. It there-
fore converts circular motion to linear motion without

see also HART'S INVERSOR, LINKAGE

References

Bogomolny, A. "Peaucellier Linkage." http: //www, cut -the-
knot . com/pythagoras /invert .html.

Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods. Oxford, Eng-
land: Oxford University Press, p. 156, 1978.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 82-83, 1969.

Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 46-48, 1990.

Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 121-126,
1957.

Smith, D. E. A Source Book in Mathematics. New York:
Dover, p. 324, 1994.

Peaucellier's Linkage

see Peaucellier Inversor

Pedal

The pedal of a curve with respect to a point P is the
locus of the foot of the PERPENDICULAR from P to
the Tangent to the curve. When a Closed Curve
rolls on a straight line, the AREA between the line and
ROULETTE after a complete revolution by any point on
the curve is twice the AREA of the pedal (taken with
respect to the generating point) of the rolling curve.

Pedal Circle

The pedal Circle of a point P in a Triangle is the
CIRCLE through the feet of the perpendiculars from P
to the sides of the TRIANGLE (the ClRCUMCIRCLE about
the Pedal Triangle). When P is on a side of the
TRIANGLE, the line between the two perpendiculars is
called the PEDAL LINE. Given four points, no three of
which are COLLINEAR, then the four PEDAL CIRCLES of
each point for the TRIANGLE formed by the other three
have a common point through which the NINE- POINT
Circles of the four Triangles pass. The radius of the
pedal circle of a point P is

A 1 P-A 2 P-A S P

T —

2(R 2 -OP 2 )

(Johnson 1929, p. 141).

see also Miquel Point, Nine-Point Circle, Pedal

Triangle

References

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.

Pedal Coordinates

Pedal Triangle 1331

Pedal Coordinates

The pedal coordinates of a point P with respect to the
curve C and the PEDAL POINT O are the radial distant
r from O to P and the PERPENDICULAR distance p from
O to the line L tangent to C at P.

References

Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 2-3, 1972.
Yates, R. C. "Pedal Equations." A Handbook on Curves

and Their Properties. Ann Arbor, MI: J- W. Edwards,

pp. 166-169, 1952.

Pedal Line

Mark a point P on a side of a TRIANGLE and draw the
perpendiculars from the point to the two other sides.
The line between the feet of these two perpendiculars is
called the pedal line.

see also PEDAL TRIANGLE, SlMSON LINE

Pedal Point

The fixed point with respect to which a Pedal Curve
is drawn.

Pedal Curve

Given a curve C, the pedal curve of C with respect to
a fixed point O (the Pedal Point) is the locus of the
point P of intersection of the PERPENDICULAR from O
to a Tangent to C. The parametric equations for a
curve (/(*),$(*)) relative to the Pedal Point (x ,yo)
are

xof 2 +fg' 2 + (yo-g)f'g l
f' 2 +g' 2

s/' 2 +w 2 + (*o-/)/y

Pedal Triangle

f' 2 +g'2 2

Curve

Pole

Pedal

astroid

center

quadrifolium

cardioid

cusp

Cayley's sextic

central conic

focus

circle

circle

any point

limagon

circle

on circumference

cardioid

circle involute

center of circle

Archimedean spiral

cissoid of Diocles

focus

cardioid

deltoid

center

trifolium

deltoid

cusp

simple folium

deltoid

on the curve

unsymmetrical
double folium

deltoid

vertex

double folium

epicycloid

center

rose

hypocycloid

center

rose

line

any point

point

logarithmic spiral

pole

logarithmic spiral

parabola

focus

line

parabola

foot of directrix

right strophoid

parabola

on directrix

strophoid

parabola

refl. of focus by dir.

Maclaurin trisectrix

parabola

vertex

cissoid of Diocles

sinusoidal spiral

pole

sinusoidal spiral

Tschirnhausen

focus of pedal

parabola

cubic

see also Negative Pedal Curve

References

Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 46-49 and 204, 1972,
Lee, X. "Pedal." http://www.best.com/-xah/SpecialPlane

Curves_dir/Pedal_dir/pedal.html.
Lockwood, E. H. "Pedal Curves." Ch. 18 in A Book

of Curves. Cambridge, England: Cambridge University

Press, pp. 152-155, 1967.
Yates, R. C. "Pedal Curves." A Handbook on Curves and

Their Properties. Ann Arbor, Ml: J. W. Edwards, pp. 160—

165, 1952.

Given a point P, the pedal triangle of P is the TRIANGLE
whose VERTICES are the feet of the perpendiculars from
P to the side lines. The pedal triangle of a TRIANGLE
with Trilinear Coordinates a : : 7 and angles A,
B, and C has Vertices with Trilinear Coordinates

: + a cos C : 7 + a cos B
a + cos C : : 7 + cos A
a + 7 cos B : + 7 cos A : 0.

(i)

(2)
(3)

The third pedal triangle is similar to the original one.
This theorem can be generalized to: the nth pedal n-
gon of any n-gon is similar to the original one. It is also

true that

P2P3 = AiPsinai (4)

(Johnson 1929, pp. 135-136). The Area A of the pedal
triangle of a point P is proportional to the POWER of P
with respect to the ClRCUMClRCLE,

R 2 ~ OP*
AR 2

A = \ (R 2 — OP ) sin ai sin 0:2 sin 0:3

(5)
(Johnson 1929, pp. 139-141).

see also Antipedal Triangle, Fagnano's Problem,
Pedal Circle, Pedal Line, Schwarz's Triangle
Problem

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.

Washington, DC: Math. Assoc. Amer., pp. 22-26, 1967.
Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, 1929.

1332 Peg Knot

Pell Equation

Peg Knot

see Clove Hitch

Peg Solitaire

1

2 3

•

• •

4
•

5 6

• •
10 11

• •

7 8
• •

9

•

12 13
• •

% % %

17 o %

1 9# 20 #

21 22
• •

23
•

24 25

• •
29 30

• •

26 27
• •

28
•

31
•

32 33
• •

A game played on a cross-shaped board with 33 holes.
All holes but the middle one are initially filled with pegs.
The goal is to remove all pegs but one by jumping pegs
from one side of an occupied peg hole to an empty space,
removing the peg which was jumped over. Strategies
and symmetries are discussed in Beeler et al. (1972, Item
75). A triangular version called Hl-Q also exists (Beeler
et al. 1972, Item 76). Kraitchik (1942) considers a board
with one additional hole placed at the vertices of the
central right angles.

see also Hl-Q

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.

Gardner, M. "Peg Solitaire." Ch. 11 in The Unexpected
Hanging and Other Mathematical Diversions. New York:
Simon and Schuster, pp. 122-135 and 250-251, 1969.

Kraitchik, M. "Peg Solitaire." §12.19 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 297-298, 1942.

Peg Top

see Piriform

Peirce's Theorem

The only linear associative algebra in which the coor-
dinates are Real Numbers and products vanish only
if one factor is zero are the Field of Real NUMBERS,
the Field of Complex Numbers, and the algebra of
Quaternions with Real Coefficients.

see also WEIERSTRAft'S THEOREM

Pell Equation

A special case of the quadratic Diophantine Equation
having the form

(1)

x 2 - Dy 2 = 1,

where D is a nonsquare Natural Number. Dorrie
(1965) defines the equation as

Dy 2 = 4

(2)

and calls it the Fermat Difference Equation. The
general Pell equation was solved by the Indian mathe-
matician Bhaskara.

Pell equations, as well as the analogous equation with
a minus sign on the right, can be solved by finding the
Continued Fraction [ai,a 2 ,...] for \/Z). (The triv-
ial solution x = 1, y = is ignored in all subsequent
discussion.) Let p n /qn denote the nth CONVERGENT
[ai,a2, - . . , a n ], then we are looking for a convergent
which obeys the identity

p n 2 - Dq n 2 = (-iy

(3)

which turns out to always be possible since the Contin-
ued Fraction of a Quadratic Surd always becomes
periodic at some term a r +i, where a r+ i = 2ai, i.e.,

vD = [ai , Gi2 , • - • , a r , 2ai ] .

(4)

Writing n = rk gives

Prk 2 ~ Dq rk 2 = (-l) r \ (5)

for k anPosiTlVE INTEGER. If r is Odd, solutions to

x 2 -Dy 2 = ±1 (6)

can be obtained if k is chosen to be EVEN or ODD, but
if r is Even, there are no values of k which can make
the exponent Odd.

If r is Even, then (-l) r is Positive and the solution
in terms of smallest INTEGERS is x = p T and y = g r ,
where p r /q r is the rth CONVERGENT. If r is Odd, then
(— l) r is Negative, but we can take k = 2 in this case,
to obtain

P2 r 2 - Dq 2r 2 = 1, (7)

so the solution in smallest Integers is x = pir, y = <?2r-
Summarizing,

( rv \-l<P"*-) for r even ,.

W,y) - <y (p2r)P2r ) for r odd W

Given one solution (x, y) = (p, q) (which can be found
as above), a whole family of solutions can be found by
taking each side to the nth Power,

Dy 2 = (p 2 - Dq 2 ) n = 1.

(9)

Factoring gives

(x + ^/Dy){x-\fDy) = {p + ^/Dq) n {p-\fDq) n (10)
and

x + VDy = (p + ^/Dq) n
x - VDy = (p - y/Dq) n ,

(11)
(12)

Pell Equation

which gives the family of solutions

X= 2

(p + qVD) n -(p-qVD) n

y = WB •

These solutions also hold for

x 2 - Dy 2 = -1,

Pell Equation 1333

(13)
(14)

(15)

except that n can take on only Odd values.

The following table gives the smallest integer solutions
(x, y) to the Pell equation with constant D < 102 (Beiler
1966, p. 254). SQUARE D = d 2 are not included, since
they would result in an equation of the form

x — d y = x — (dy) = x — y = 1, (16)

which has no solutions (since the difference of two
Squares cannot be 1).

D

X

y

D

x

y

2

3

2

54

485

66

3

2

1

55

89

12

5

9

4

56

15

2

6

5

2

57

151

20

7

8

3

58

19603

2574

8

3

1

59

530

69

10

19

6

60

31

4

11

10

3

61

1766319049

226153980

12

7

2

62

63

8

13

649

180

63

8

1

14

15

4

65

129

16

15

4

1

66

65

8

17

33

8

67

48842

5967

18

17

4

68

33

4

19

170

39

69

7775

936

20

9

2

70

251

30

21

55

12

71

3480

413

22

197

42

72

17

2

23

24

5

73

2281249

267000

24

5

1

74

3699

430

26

51

10

75

26

3

27

26

5

76

57799

6630

28

127

24

77

351

40

29

9801

1820

78

53

6

30

11

2

79

80

9

31

1520

273

80

9

1

32

17

3

82

163

18

33

23

4

83

82

9

34

35

6

84

55

6

35

6

1

85

285769

30996

37

73

12

86

10405

1122

38

37

6

87

28

3

39

25

4

88

197

21

40

19

3

89

500001

53000

41

2049

320

90

19

2

42

13

2

91

1574

165

43

3482

531

92

1151

120

44

199

30

93

12151

1260

45

161

24

94

2143295

221064

46

24335

3588

95

39

4

47

48

7

96

49

5

48

7

1

97

62809633

6377352

50

99

14

98

99

10

51

50

7

99

10

1

52

649

90

101

201

20

53

66249

9100

102

101

10

The first few minimal values of x and y for nonsquare D
are 3, 2, 9, 5, 8, 3, 19, 10, 7, 649, . . . (Sloane's A033313)
and 2, 1, 4, 2, 3, 1, 6, 3, 2, 180, . . . (Sloane's A033317),
respectively. The values of D having x = 2, 3, . . . are
3, 2, 15, 6, 35, 12, 7, 5, 11, 30, . . . (Sloane's A033314)
and the values of D having y = 1, 2, ... are 3, 2, 7, 5,
23, 10, 47, 17, 79, 26, ... (Sloane's A033318). Values
of the incrementally largest minimal x are 3, 9, 19, 649,
9801, 24335, 66249, . . . (Sloane's A033315) which occur
at D = 2, 5, 10, 13, 29, 46, 53, 61, 109, 181, . . . (Sloane's
A033316). Values of the incrementally largest minimal

1334

Pell-Lucas Number

Pell Sequence

y are 2, 4, 6, 180, 1820, 3588, 9100, 226153980, . . .
(Sloane's A033319), which occur at D = 2, 5, 10, 13, 29,
46, 53, 61, . . . (Sloane's A033320).

see also Diophantine Equation, Diophantine
Equation — Quadratic, Lagrange Number (Dio-
phantine Equation)

References

Beiler, A. H. "The Pellian." Ch. 22 in Recreations in the The-
ory of Numbers: The Queen of Mathematics Entertains.
New York: Dover, pp. 248-268, 1966.

Degan, C. F. Canon Pellianus. Copenhagen, Denmark, 1817.

Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, 1965.

Lagarias, J. C. "On the Computational Complexity of De-
termining the Solvability or Unsolvability of the Equation
X 2 -DY 2 = -1." Trans. Amer. Math. Soc. 260, 485-508,
1980.

Smarandache, F. "Un met o do de resolucion de la ecuacion
diofantica." Gaz. Math. 1, 151-157, 1988.

Smarandache, F. " Method to Solve the Diophantine Equa-
tion ax 2 — by 2 + c = 0." In Collected Papers, Vol. 1.
Lupton, AZ: Erhus University Press, 1996.

Stillwell, J. C. Mathematics and Its History. New York:
Springer- Verlag, 1989.

Whitford, E, E. Pell Equation. New York: Columbia Uni-
versity Press, 1912.

Pell-Lucas Number
see Pell Number

Pell-Lucas Polynomial

see Pell Polynomial

Pell Number

The numbers obtained by the U n s in the LUCAS SE-
QUENCE with P = 2 and Q = -1. They and the Pell-
Lucas numbers (the V n s in the Lucas Sequence) sat-
isfy the recurrence relation

P n = 2P n _i+P n

(1)

Using Pi to denote a Pell number and Qi to denote a
Pell-Lucas number,

(2)
(3)

P 2 t m = Pm(2Qm)(2Q2m)(2Q4m) ■ ■ ■ (2Q 2 «-l m ) (4)

Q m 2 = 2P m 2 + (-iy

Qlm — 2Q„

(-1)"

(5)
(6)

The Pell numbers have Po = and Pi = 1 and are 0,
1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ... (Sloane's
A000129). The Pell-Lucas numbers have Q = 2 and
Qi = 2 and are 2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786,
6726, . . . (Sloane's A002203).

The only TRIANGULAR Pell number is 1 (McDaniel
1996).

see also Brahmagupta Polynomial, Pell Polynom-
ial

References

McDaniel, W. L. "Triangular Numbers in the Pell Sequence."
Fib. Quart. 34, 105-107, 1996.

Sloane, N. J. A. Sequences A000129/M1413 and A002203/
M0360 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Pell Polynomial

The Pell polynomials P(x) and Lucas-Pell polynomi-
als Q(x) are generated by a Lucas Polynomial Se-
quence using generator (2x, 1). This gives recursive
equations for P(x) from Po(x) — Pi(x) — 1 and

Pn+ 2 (x) = 2xP n+1 (x) + P n (x). (1)

The first few are

Pi
P 2

: 1
: 2X

4x 2

P 4 = 8x d - Ax

P 5 = 16x 4 - 12a; 2 + 1.

The Pell-Lucas numbers are defined recursively by

qo(x) = 1, qi(x) = x and

q n +2(x) = 2xq n +i(x) + q n (x),

together with

Q n (x) = 2q n (x).

(2)
(3)

The first few are

Qi =2x

Q 2 = Ax 2

Qz = Sx — 6a;

Q 4 = 16a: 4 - 16a; 2 4- 2

Q 5 = 32a; 5 - 40a; 3 + 10x.

see also LUCAS POLYNOMIAL SEQUENCE

References

Horadam, A. F. and Mahon, J. M. "Pell and Pell-Lucas Poly-
nomials." Fib. Quart. 23, 7-20, 1985.

Mahon, J. M. M. A. (Honors) thesis, The University of New
England. Armidale, Australia, 1984.

Sloane, N. J. A. Sequence A000129/M1413 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Pell Sequence

see Pell Number

Pencil

Pentacle

1335

Pencil

The set of all Lines through a point. Woods (1961),
however, uses this term as a synonym for Range.

see also Near-Pencil, Perspectivity, Range (Line
Segment), Section (Pencil), Sheaf (Geometry)

References

Woods, F. S. Higher Geometry: An Introduction to Advanced

Methods in Analytic Geometry. New York; Dover, pp. 8

and 11-12, 1961.

Penrose Stairway

An Impossible Figure (also called the Schroeder
Stairs) in which a stairway in the shape of a square
appears to circulate indefinitely while still possessing
normal steps. The Dutch artist M. C. Escher included
Penrose stairways in many of his mind-bending illustra-
tions.

see also IMPOSSIBLE FIGURE

References

Hofstadter, D, R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 15, 1989.

Jablan, S. "Impossible Figures." http: //members, tripod.
com/-modularity/impos .htm.

Pappas, T. "Optical Illusions and Computer Graphics." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 5, 1989.

Robinson, J. O. and Wilson, J. A. "The Impossible Colon-
nade and Other Variations of a Well-Known Figure." Brit.
J. Psych. 64, 363-365, 1973.

Penrose Tiles

A pair of shapes which tile the plane only aperiodically
(when the markings are constrained to match at bor-
ders). The two tiles, illustrated above, are called the
"Kite" and "Dart."

To see how the plane may be tiled aperiodically using
the kite and dart, divide the kite into acute and obtuse

tiles, shown above. Now define "deflation" and "infla-
tion" operations. The deflation operator takes an acute
Triangle to the union of two Acute Triangles and
one Obtuse, and the Obtuse Triangle goes to an
Acute and an Obtuse Triangle. These operations
are illustrated below.

When applied to a collection of tiles, the deflation op-
erator leads to a more refined collection. The operators
do not respect tile boundaries, but do respect the half
tiles defined above. There are two ways to obtain aperi-
odic TILINGS with 5-fold symmetry about a single point.
They are known an the "star" and "sun" configurations,
and are show below.

Higher order versions can then be obtained by deflation.
For example, the following are third-order deflations:

^Vv

References

Gardner, M, Chs. 1-2 in Penrose Tiles and Trapdoor

Ciphers. . . and the Return of Dr. Matrix, reissue ed. New

York: W. H. Freeman, pp. 299-300, 1989.
Hurd, L. P. "Penrose Tiles." http://www.mathsource.com/

cgi - bin / Math Source / Applications / Graphics / 2D /

0206-772.
Peterson, 1. The Mathematical Tourist: Snapshots of Modem

Mathematics. New York: W. H. Freeman, pp. 86-95, 1988.
Wagon, S. "Penrose Tiles." §4.3 in Mathematica in Action.

New York: W. H. Freeman, pp. 108-117, 1991.

Penrose Triangle

see Tribar

Penrose Tribar

see Tribar

Pentabolo

A 5-POLYABOLO.

Pentacle

see Pentagram

1336 Pentacontagon

Pentaflake

Pentacontagon

A 50-sided POLYGON.

Pentad

A group of five elements.

see also MONAD, PAIR, QUADRUPLET, QUINTUPLET,

Tetrad, Triad, Triplet, Twins
Pentadecagon

A 15-sided POLYGON, sometimes also called the Pen-
TAKAIDECAGON.

see also POLYGON, REGULAR POLYGON, TRIGONOME-
TRY Values — 7r/15

Pentaflake

A Fractal with 5-fold symmetry. As illustrated above,
five PENTAGONS can be arranged around an identical
Pentagon to form the first iteration of the pentaflake.
This cluster of six pentagons has the shape of a pentagon
with five triangular wedges removed. This construction
was first noticed by Albrecht Diirer (Dixon 1991).

For a pentagon of side length 1, the first ring of pen-
tagons has centers at Radius

where <j> is the Golden RATIO. The Inradius r and
ClRCUMRADlUS R are related by

r = Rcos(Itt) = ±(VE+1)R,
and these are related to the side length s by

s = 2^/R 2 -r 2 = §#\/l0 - 2\/5.
The height h is

h = ssin(§7r) = |s\/l0 + 2v / 5 = \y/hR,
giving a RADIUS of the second ring as

d 2 = 2(R + h) = (2 + V5)R = <f) 3 R.
Continuing, the nth pentagon ring is located at

d n = <t> n ~ .

(2)

(3)

(4)

(5)

(6)

Now, the length of the side of the first pentagon com-
pound is given by

S2

2 V / (2r + R) 2 -{h + R) 2 = R\f$ + 2\/5, (7)

so the ratio of side lengths of the original pentagon to
that of the compound is

s 2 j Ra/5 + 2v / 5

s ±R^10-2V$

1 + 0.

(8)

2r= f(l + V5)i2 = 0i2,

(1)

We can now calculate the dimension of the pentaflake
fractal. Let N n be the number of black pentagons and
L n the length of side of a pentagon after the n iteration,

N n = 6 n (9)

L n = (l + <j>)- n . (10)

The Capacity Dimension is therefore

_ In N n __ In 6 _ In2 + ln3

dcap " " n ™o hTZ^ " ln(l + <j>) " ln(l + <t>)

= 1.861715.... (11)

see also PENTAGON

References

Dixon, R. Mathographics. New York: Dover, pp. 186-188,
1991.

$ Weisstein, E. W. "Fractals." http://www. astro. Virginia,
edu/ ~eww6n/math/notebooks/Fractal . m.

Pentagon

Pentagon 1337

Pentagon

The regular convex 5-gon is called the pentagon. By
Similar Triangles in the figure on the left,

d 1 A,

1 = T = &

(1)

where d is the diagonal distance. But the dashed vertical
line connecting two nonadjacent VERTICES is the same
length as the diagonal one, so

<t> 2 - <f> - 1

(2)

(3)

(4)

2 2

This number is the GOLDEN RATIO. The coordinates of
the Vertices relative to the center of the pentagon with
unit sides, starting at the right VERTEX and moving
clockwise, are (cos(|n7r),sin(|n7r)) for n = 0, 1, . . . , 4,
or

(1, 0), (Ci, Si), (C 2 , 3 2 ), (C 2 , -S 2 ), (Ci, -Si)

where

d=cos(|) = ±(V5 + l)

c 2 = cos(^)=i(V5-l)
5i = sin(|) = |\/lO-2V5
s 2 =sin('y) = J\/l0 + 2i/5.

(5)

(6)
(7)
(8)
(9)

For a regular Polygon, the Circumradius, Inradius,
Sagitta, and Area are given by

R n = ^acsc ( - J
r n = |acot I — J

R n - r n

An = -^na cot

* atan (^)
(=)■

(10)

(11)

(12)
(13)

Plugging in n = 5 gives

# = |acsc(|7r) = ^a\/50 + 10^/5

(14)

r = |acot(f tt) = ~a\/25 + lOv^

(15)

a:= ^a\/25- IQy/E

(16)

A= |a 2 \/25 + 10\/5.

(17)

Five pentagons can be arranged around an identical pen-
tagon to form the first iteration of the "PENTAFLAKE,"
which itself has the shape of a pentagon with five trian-
gular wedges removed. For a pentagon of side length 1,
the first ring of pentagons has centers at radius </>, the
second ring at 3 , and the nth at 2n_1 .

In proposition IV. 11, Euclid showed how to inscribe a
regular pentagon in a Circle. Ptolemy also gave a
Ruler and Compass construction for the pentagon in
his epoch-making work The Almagest While Ptolemy's
construction has a Simplicity of 16, a Geometric
Construction using Carlyle Circles can be made
with GEOMETROGRAPHY symbol 2Si + S 2 + 8Ci +0C 2 +
4C 3 , which has SIMPLICITY 15 (De Temple 1991).

Pentagon

The following elegant construction for the pentagon is
due to Richmond (1893). Given a point, a Circle may
be constructed of any desired RADIUS, and a DIAM-
ETER drawn through the center. Call the center O,
and the right end of the Diameter P . The Diame-
ter Perpendicular to the original Diameter may be
constructed by finding the PERPENDICULAR BISECTOR.
Call the upper endpoint of this Perpendicular Diam-
eter B. For the pentagon, find the Midpoint of OB
and call it D. Draw DP , and Bisect AODP , calling
the intersection point with OPo Ni. Draw NiPi PAR-
ALLEL to OS, and the first two points of the pentagon
are P and Pi (Coxeter 1969).

Madachy (1979) illustrates how to construct a pentagon
by folding and knotting a strip of paper.

1338 Pentagonal Antiprism

see also Cyclic Pentagon, Decagon, Dissection,
Five Disks Problem, Home Plate, Pentaflake,
Pentagram, Polygon, Trigonometry Values —
tt/5

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 95-96,
1987.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 26-28, 1969.

De Temple, D. W. "Carlyle Circles and the Lemoine Simplic-
ity of Polygonal Constructions." Amer. Math. Monthly 98,
97-108, 1991.

Dixon, R. Mathographics. New York: Dover, p. 17, 1991.

Dudeney, H. E. Amusements in Mathematics. New York:
Dover, p. 38, 1970.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 59, 1979.

Pappas, T. "The Pentagon, the Pentagram & the Golden
Triangle." The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, pp. 188-189, 1989.

Richmond, H. W. "A Construction for a Regular Polygon of
Seventeen Sides." Quart J. Pure Appl. Math. 26, 206-
207, 1893.

Wantzel, P. L. "Recherches sur les moyens de reconnaitre si
un Probleme de Geometric peut se resoudre avec la regie
et le compas." J. Math, pures appliq. 1, 366-372, 1836.

Pentagonal Antiprism

An Antiprism and Uniform Polyhedron U77 whose
Dual Polyhedron is the Pentagonal Deltahe-
dron.

Pentagonal Cupola

Pentagonal Dipyramid
Johnson Solid J 5 . The bottom 10 Vertices are

(i + VsWs + Vs 1 \

± 4V2 >=t 2'°]'

, (l + y/E)y/b-y/Z ,3W5 \

(0,±i(l + V5),0)

and the top five VERTICES are

V$ + v$ \/5->/5

' ' \/io

10

{ WW '^(i + V5), ^ y

4>/io

' ±_ 2' yio

Pentagonal Deltahedron

A Deltahedron which is the Dual Polyhedron of
the Pentagonal Antiprism.

Pentagonal Dipyramid

The pentagonal dipyramid is one of the convex DELTA-
HEDRA, and JOHNSON Solid J13. It is also the DUAL
Polyhedron of the Pentagonal Prism. The distance
between two adjacent Vertices on the base of the Pen-
tagon is

d 12 2 = [l-cos(§7r)] 2 +sin 2 (§7r)

= [1 _^_ 1)]2+ [(i±^3

= i(B->/5), (1)

Pentagonal Gyrobicupola

and the distance between the apex and one of the base
points is

di h 2 = (0 - l) 2 + (0 - 0) 2 + (h - 0) 2 = 1 + h\ (2)

But

d\2 = di2

(3)

i(5-\/5) = l + fc 2

(4)

h a = i(3-V5),

(5)

and

3-\/5

(6)

This root is of the form ya + 6i/c, so applying SQUARE
ROOT simplification gives

h= |(\/5-l) ==<£-!,

(7)

where <j> is the Golden Mean.

see also Deltahedron, Dipyramid, Golden Mean,

ICOSAHEDRON, JOHNSON SOLID, TRIANGULAR DlPYR-
AMID

Pentagonal Gyrobicupola

see Johnson Solid

Pentagonal Gyrocupolarotunda

see Johnson Solid

Pentagonal Hexecontahedron

The Dual Polyhedron of the Snub Dodecahedron.

Pentagonal Number 1339
Pentagonal Icositetrahedron

The Dual Polyhedron of the Snub Cube.
Pentagonal Number

A Polygonal Number of the form n(3n - l)/2. The
first few are 1, 5, 12, 22, 35, 51, 70, ... (Sloane's
A000326). The Generating Function for the pen-
tagonal numbers is

^+il=x + 5x 2 + 12x 3 + 22x 4 + ....

(i — xy

Every pentagonal number is 1/3 of a Triangular

Number.

The so-called generalized pentagonal numbers are given

by n(3n - l)/2 with n = 0, ±1, ±2, . . . , the first few of
which are 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (Sloane's
A001318).

see also Euler's Pentagonal Number Theorem,
Partition Function P, Polygonal Number, Tri-
angular Number

References

Guy, R. K. "Sums of Squares." §C20 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 136-138, 1994.

Pappas, T. "Triangular, Square & Pentagonal Numbers."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, p. 214, 1989.

Sloane, N. J. A. Sequences A000326/M3818 and A001318/
M1336 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

1340 Pentagonal Orthobicupola

Pentagrammic Crossed Antiprism

Pentagonal Orthobicupola

see Johnson Solid

Pentagonal Orthobirotunda

see Johnson Solid

Pentagonal Orthocupolarontunda

see Johnson Solid

Pentagonal Prism

A Prism and Uniform Polyhedron U 76 whose Dual
Polyhedron is the Pentagonal Dipyramid.

see also Pentagrammic Prism

Pentagonal Pyramid

see Johnson Solid

Pentagonal Pyramidal Number

A Pyramidal Number of the form n 2 (n 4- l)/2. The
first few are 1, 6, 18, 40, 75, . . . (Sloane's A002411). The
Generating Function for the pentagonal pyramidal
numbers is

^+^^^ + 6^ + 18x 3 + 40^ + ....
(x - l) 4

References

Sloane, N. J. A. Sequence A002411/M4116 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Pentagonal Tiling

see Tiling

Pentagram

The Star Polygon {§}, also called the Pentacle,
Pentalpha, or Pentangle.

see also DISSECTION, HEXAGRAM, HOEHN'S THEOREM,

Pentagon, Star Figure, Star of Lakshmi

References

Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 122-125, 1990.

Pappas, T. "The Pentagon, the Pentagram & the Golden
Triangle." The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, pp. 188-189, 1989.

Schwartzman, S. The Words of Mathematics: An Etymolog-
ical Dictionary of Mathematical Terms Used in English.
Washington, DC: Math. Assoc. Amer., 1994.

Pentagrammic Antiprism

An Antiprism and Uniform Polyhedron U79 whose
Dual Polyhedron is the Pentagrammic Deltahe-
dron.

Pentagonal Rotunda

Half of an ICOSIDODECAHEDRON, denoted R5. It has 10
triangular and five pentagonal faces separating a Pen-
tagonal ceiling and a Dodecahedral floor. It is
Johnson Solid J 6 , and the only true Rotunda.

see also Icosidodecahedron, Johnson Solid, Ro-
tunda

Pentagrammic Concave Deltahedron

The Dual Polyhedron of the Pentagrammic
Crossed Antiprism.

Pentagrammic Crossed Antiprism

Pentagrammic Deltahedron

Pentomino

1341

An ANTIPRISM and UNIFORM POLYHEDRON Uso whose

Dual Polyhedron is the Pentagrammic Concave
Deltahedron.

Pentagrammic Deltahedron

The Dual Polyhedron of the Pentagrammic Anti-
prism.

Pentagrammic Dipyramid

The Dual Polyhedron of the Pentagrammic Prism.

Pentagrammic Prism

A Prism and Uniform Polyhedron U 7 8 whose Dual
Polyhedron is the Pentagrammic Dipyramid.

see also Pentagonal Prism

Pentakaidecagon

see Pentadecagon

Pentakis Dodecahedron

The Dual Polyhedron of the Truncated Icosahe-

dron.

see also Archimedean Solid, Dual Polyhedron,

Truncated Icosahedron

Pentalpha

see Pentagram

Pentangle

see Pentagram

Pentatope

The simplest regular figure in 4-D.

Pentatope Number

A Figurate Number which is given by

Ptop n = \Te n (n + 3) = ±n(n + l)(n + 2)(n + 3),

where Te n is the nth Tetrahedral Number. The
first few pentatope numbers are 1, 5, 15, 35, 70, 126,
... (Sloane's A000332). The Generating Function
for the pentatope numbers is

= x + 5z 2 + 15x 3 + 35x 4 + .

(1 - x) 5

see also FlGURATE NUMBER, TETRAHEDRAL NUMBER

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New-
York: Springer- Verlag, pp. 55-57, 1996.

Pentomino

f 1 L N P T

W

The twelve 5-POLYOMINOES illustrated above and
known by the letters of the alphabet they most
closely resemble: /, /, L, N, P, T, U, V, W, X, y, Z (Gard-
ner 1960).

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 110-
111, 1987.

Dudeney, H. E. "The Broken Chessboard." Problem 74 in
The Canterbury Puzzles and Other Curious Problems, 7th
ed. London: Thomas Nelson and Sons, pp. 119-120, 1949.

Gardner, M. "Mathematical Games: More About the Shapes
that Can Be Made with Complex Dominoes." Sci. Amer.
203, 186-198, Nov. 1960.

Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 80-86, 1975.

Lei, A. "Pentominoes." http://www.cs.ust.hk/-philipl/
omino/pento .html.

Ruskey, F. "Information on Pentomino Puzzles." http://
sue . esc , uvic . ca/~cos/inf /misc/Pent Info .html.

1342 Pepin's Test

Perfect Box

Pepin's Test

A test for the Primality of Fermat Numbers F n =
2 2 " + 1, with n > 2 and k > 2. Then the two following
conditions are equivalent:

1. F n is Prime and k/F n = -1.

2. k^ Fn - 1)/2 ~~l (modF n ).

k is usually taken as 3 as a first test.

see also Fermat Number, Pepin's Theorem

References

Ribenboim, P. The Little Book of Big Primes. New York:

Springer- Verlag, p. 62, 1991.
Shanks, D. Solved and Unsolved Problems in Number Theory,

4th ed. New York: Chelsea, pp. 119-120, 1993.

Pepin's Theorem

The Fermat Number F n is Prime Iff

3 22n_1 = -1 (modF n ).

see also Fermat Number, Pepin's Test, Selfridge-
Hurwitz Residue

Percent

The use of percentages is a way of expressing RATIOS in
terms of whole numbers. Given a Ratio or Fraction,
it is converted to a percentage by multiplying by 100
and appending a "percentage sign" %. For example,
if an investment grows from a number P = 13.00 to
a number A = 22.50, then A is 22.50/13.00 = 1.7308
times as much as P, or 173.08%, and the investment
has grown by 73.08%.

see also PERCENTAGE ERROR, PERMIL

Percentage Error

The percentage error is 100% times the RELATIVE ER-
ROR.

see also Absolute Error, Error Propagation,
Percent, Relative Error

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 14, 1972.

Percolation Theory

percolation. A SITE PERCOLATION considers the lattice
vertices as the relevant entities; a BOND PERCOLATION
considers the lattice edges as the relevant entities.

see also BOND PERCOLATION, CAYLEY TREE, CLUS-
TER, Cluster Perimeter, Lattice Animal, Perco-
lation Threshold, Polyomino, s-Cluster, s-Run,
Site Percolation

References

Deutscher, G.; Z alien, R.; and Adler, J. (Eds.). Percolation

Structures and Processes. Bristol: Adam Hilger, 1983.
Finch, S. "Favorite Mathematical Constants." http://wvv.

mathsof t . com/asolve/constant/rndprc/rndprc .html.
Grimmett, G. Percolation. New York: Springer- Verlag, 1989.
Kesten, H. Percolation Theory for Mathematicians. Boston,

MA: Birkhauser, 1982.
Stauffer, D. and Aharony, A. Introduction to Percolation

Theory, 2nd ed. London: Taylor & Francis, 1992.

Percolation Threshold

The critical fraction of lattice points which must be filled
to create a continuous path of nearest neighbors from
one side to another. The following table is from Stauffer
and Aharony (1992, p. 17).

Lattice

Site

Bond

Cubic (Body-Centered)

0.246

0.1803

Cubic (Face-Centered)

0.198

0.119

Cubic (Simple)

0.3116

0.2488

Diamond

0.43

0.388

Honeycomb

0.6962

0.65271

4-Hypercubic

0.197

0.1601

5-Hypercubic

0.141

0.1182

6-Hypercubic

0.107

0.0942

7-Hypercubic

0.089

0.0787

Square

0.592746

0.50000

Triangular

0.50000

0.34729

The square bond value is 1/2 exactly, as is the triangu-
lar site. p c = 2sin(7r/18) for the triangular bond and
p c = 1 — 2sin(7r/18) for the honeycomb bond. An exact
answer for the square site percolation threshold is not
known.

see also Percolation Theory

References

Essam, J. W.; Gaunt, D. S.; and Guttmann, A. J. "Perco-
lation Theory at the Critical Dimension." J. Phys. A 11,
1983-1990, 1978.

Finch, S. "Favorite Mathematical Constants." http://vw.
mathsoft.com/asolve/constant/rndprc/rndprc.html.

Kesten, H. Percolation Theory for Mathematicians. Boston,
MA: Birkhauser, 1982.

Stauffer, D. and Aharony, A. Introduction to Percolation
Theory, 2nd ed. London: Taylor & Francis, 1992.

Perfect Box

see Euler Brick

bond percolation site percolation

Percolation theory deals with fluid flow (or any other
similar process) in random media. If the medium is a set
of regular LATTICE POINTS, then there are two types of

Perfect Cubic

Perfect Number

1343

Perfect Cubic

A perfect cubic POLYNOMIAL can be factored into a lin-
ear and a quadratic term,

(a 3 -6 3 )- (a-6)(a 2 + a6 + 6 2 )

(a 3 +b 3 ) = (a + 6)(a 2 -a& + & 2 ).

see also CUBIC EQUATION, PERFECT SQUARE, POLY-
NOMIAL

Perfect Cuboid

see Euler Brick

Perfect Difference Set

A Set of Residues {ai,a 2 ,. .. ,a fc +i} (modn) such that
every NONZERO RESIDUE can be uniquely expressed in
the form a, - a-,-. Examples include {1, 2, 4} (mod 7)
and {1, 2, 5, 7} (mod 13). A NECESSARY condition for a
difference set to exist is that n be of the form k 2 + k + 1.
A Sufficient condition is that A; be a Prime Power.
Perfect sets can be used in the construction of PERFECT
Rulers.

see also PERFECT RULER

References

Guy, R. K. "Modular Difference Sets and Error Correcting
Codes." §C10 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 118-121, 1994.

Perfect Digital Invariant

see NARCISSISTIC NUMBER

Perfect Information

A class of GAME in which players move alternately and
each player is completely informed of previous moves.
Finite, Zero-Sum, two-player Games with perfect in-
formation (including checkers and chess) have a SADDLE
POINT, and therefore one or more optimal strategies.
However, the optimal strategy may be so difficult to
compute as to be effectively impossible to determine (as
in the game of Chess).
see also Finite Game, Game, Zero-Sum Game

Perfect Magic Cube

A perfect magic cube is a MAGIC CUBE for which the
cross-section diagonals, as well as the space diagonals,
sum to the MAGIC CONSTANT.

see also Magic Cube, Semiperfect Magic Cube

References

Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time
Travel and Other Mathematical Bewilderments. New-
York: W. H. Freeman, pp. 213-225, 1988.

Perfect Number

Perfect numbers are INTEGERS n such that

n = s(n),

(i)

where s(n) is the Restricted Divisor Function (i.e.,
the Sum of Proper Divisors of n), or equivalently

cr(n) = 2n,

(2)

where a(n) is the DIVISOR FUNCTION (i.e., the SUM of
DIVISORS of n including n itself). The first few perfect
numbers are 6, 28, 496, 8128, . . . (Sloane's A000396).
This follows from the fact that

6 = ^1,2,3

28 = ^1,2,4,7,14
496 = ^1,2, 4, 8, 16, 31, 62, 124, 248,

etc.

Perfect numbers are intimately connected with a class
of numbers known as MERSENNE PRIMES. This can be
demonstrated by considering a perfect number P of the
form P = q2 p ~ 1 where q is PRIME. Then

and using

for q prime, and

<t{P) = 2P,
a(q) = q + l

<r(2 a ) = 2 a+1 - 1

gives

a{q2 p ~ l ) = a{q)a{2^ 1 ) = (q + 1)(2 P - 1)

,p-i

= 2q2 p - 1 = q2 p

q{2 p - 1) + 2 P - 1 = q2 p
q = 2 P - 1.
Therefore, if M p = q = 2 P - 1 is PRIME, then

P = \{M P + 1)M P = 2 P " 1 (2 P - 1)

(3)
(4)

(5)

(6)

(7)
(8)

(9)

is a perfect number, as was stated in Proposition IX. 36
of Euclid's Elements (Dunham 1990), The first few per-
fect numbers are summarized in the following table.

#

P

P

1

2

6

2

3

28

3

5

496

4

7

8128

5

13

33550336

6

17

8589869056

7

19

137438691328

8

31

2305843008139952128

1344

Perfect Number

All Even perfect numbers are of this form, as was proved
by Euler in a posthumous paper. The only even perfect
number of the form x 3 + 1 is 28 (Makowski 1962).

It is not known if any ODD perfect numbers exist, al-
though numbers up to 10 300 have been checked (Brent
et al. 1991, Guy 1994) without success, improving the
result of Tuckerman (1973), who checked odd numbers
up to 10 36 . Euler showed that an ODD perfect number,
if it exists, must be of the form

- ~ 4a+1 Q\

m = p

(10)

where p is an Odd Prime Relatively Prime to Q.
In 1887, Sylvester conjectured and in 1925, Gradshtein
proved that any Odd perfect number must have at least
six different prime aliquot factors (or eight if it is not
divisible by 3; Ball and Coxeter 1987). Catalan (1888)
proved that if an Odd perfect number is not divisible
by 3, 5, or 7, it has at least 26 distinct prime aliquot
factors. Stuyvaert (1896) proved that an Odd perfect
number must be a sum of squares. All EVEN perfect
numbers end in 16, 28, 36, 56, or 76 (Lucas 1891) and,
with the exception of 6, have Digital ROOT 1.

Every perfect number of the form 2 P (2 P+1 — 1) can be
written

p/2

2 P (2 P+1 -1) = ^(2&-1) 3 . (11)

fc=i

All perfect numbers are Hexagonal Numbers and
therefore TRIANGULAR Numbers. It therefore follows
that perfect numbers are always the sum of consecutive
Positive integers starting at 1, for example,

6 = E n

n=l
7

71=1
31

496 = ^n

(12)
(13)
(14)

(Singh 1997). All EVEN perfect numbers P > 6 are of
the form

P + l + 9T n , (15)

where T n is a Triangular Number

T n = ±n(n + l)

(16)

such that n = Sj + 2 (Eaton 1995, 1996). The sum of
reciprocals of all the divisors of a perfect number is 2,
since

n+... + c + & + a = 2n (17)

■v

n

n n

- + T + --
a o

. =

1 1

- + 7T + '

a b

Perfect Number

2. (19)

2n

(18)

If s(n) > n, n is said to be an Abundant Number. If
s(n) < n, n is said to be a Deficient Number. And if
s(n) = kn for a Positive Integer k > 1, n is said to
be a Multiperfect Number of order k.

see also Abundant Number, Aliquot Sequence,
Amicable Numbers, Deficient Number, Divisor
Function, c-Perfect Number, Harmonic Number,
Hyperperfect Number, Infinary Perfect Num-
ber, Mersenne Number, Mersenne Prime, Multi-
perfect Number, Multiplicative Perfect Num-
ber, Pluperfect Number, Pseudoperfect Num-
ber, Quasiperfect Number, Semiperfect Num-
ber, Smith Number, Sociable Numbers, Sublime
Number, Superperfect Number, Unitary Per-
fect Number, Weird Number

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 66—67,
1987.

Brent, R, P.; Cohen, G. L. L.; and te Riele, H. J. J. "Improved
Techniques for Lower Bounds for Odd Perfect Numbers."
Math. Comput 57, 857-868, 1991.

Conway, J. H. and Guy, R. K. "Perfect Numbers." In The
Book of Numbers. New York: Springer- Verlag, pp. 136-
137, 1996.

Dickson, L. E. "Notes on the Theory of Numbers." Amer.
Math. Monthly 18, 109-111, 1911.

Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 3-33,
1952.

Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, p. 75, 1990.

Eaton, C. F, "Problem 1482." Math. Mag. 68, 307, 1995.

Eaton, C. F. "Perfect Number in Terms of Triangular Num-
bers." Solution to Problem 1482. Math. Mag. 69, 308-
309, 1996.

Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math-
ematical Magic Show: More Puzzles, Games, Diversions,
Illusions and Other Mathematical Sleight-of-Mind from
Scientific American. New York: Vintage, pp. 160-171,
1978.

Guy, R. K. "Perfect Numbers." §B1 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
p. 145, 1994.

Kraitchik, M. "Mersenne Numbers and Perfect Numbers."
§3.5 in Mathematical Recreations. New York: W. W. Nor-
ton, pp. 70-73, 1942.

Madachy, J. S. Madachy y s Mathematical Recreations. New
York: Dover, pp. 145 and 147-151, 1979.

Makowski, A. "Remark on Perfect Numbers." Elemente
Math. 17, 109, 1962.

Powers, R. E. "The Tenth Perfect Number." Amer. Math.
Monthly 18, 195-196, 1911.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 1-13 and 25-29, 1993.

Singh, S. FermaVs Enigma: The Epic Quest to Solve
the World's Greatest Mathematical Problem. New York:
Walker, pp. 11-13, 1997.

Sloane, N. J. A. Sequence A000396/M4186 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Tuckerman, B. "Odd Perfect Numbers: A Search Procedure,
and a New Lower Bound of 10 36 ." Not. Amer. Math. Soc.
15, 226, 1968.

Perfect Partition

Perfect Square Dissection 1345

Tuckerman, B. "A Search Procedure and Lower Bound for
Odd Perfect Numbers." Math. Comp. 27, 943-949, 1973.

Zachariou, A. and Zachariou, E. "Perfect, Semi-Perfect and
Ore Numbers." Bull. Soc. Math. Grece (New Ser.) 13,
12-22, 1972.

Perfect Partition

A Partition of n which can generate any number 1,2,

. . . , n.

see also PARTITION

References

Cohen, D. I. A. Basic Techniques of Combinatorial Theory.

New York: Wiley and Sons, p. 97, 1978.
Honsberger, R. Mathematical Gems III. Washington, DC:

Math. Assoc. Amer., pp. 140-143, 1985.

(1)

Perfect Proportion

Since

2a

2ab

a + b

' {a + b)b

it follows that

a

a+b
2

2ab
a+b

~ b '

so

a

A

H

(2)

(3)

where A and H are the ARITHMETIC MEAN and HAR-
MONIC MEAN of a and b. This relationship was purport-
edly discovered by Pythagoras.

see also ARITHMETIC MEAN, HARMONIC MEAN

Perfect Rectangle

A Rectangle which cannot be built up of Squares
all of different sizes is called an imperfect rectangle. A
Rectangle which can be built up of Squares all of
different sizes is called perfect.

order

perfect

imperfect

<9

9

2

1

10

6

11

22

12

67

9

13

213

34

14

744

104

15

2609

282

Perfect Ruler

o

i

J L

J I L

... up to some maximum distance n > k. Such a ruler
can be constructed from a PERFECT DIFFERENCE SET
by subtracting one from each element. For example, the
Perfect Difference Set {1, 2, 5, 7} gives 0, 1, 4,
6, which can be used to measure 1 — = 1, 6 — 4 = 2,
4 - 1 = 3, 4 - = 4, 6 - 1 = 5, 6 - = 6 (so we get 6
distances with only four marks),

see also PERFECT DIFFERENCE SET

References

Guy, R. K. "Modular Difference Sets and Error Correcting
Codes." §C10 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 118-121, 1994.

Perfect Set

A Set P is called perfect if P = P\ where P' is the
Derived Set of P.

see also Derived Set, Set

Perfect Square

The term perfect square is used to refer to a SQUARE

Number, a Perfect Square Dissection, or a fac-
torable quadratic polynomial of the form a 2 — b 2 =
(a- b)(a + b).

see also Perfect Square Dissection, Quadratic
Equation, Square Number, Squarefree

Perfect Square Dissection

50

35

27

8

19

15

/

17

11

2 f

29

25
4

1

9

7

18

24

16

T

33

■■

M

42

A type of RULER considered by Guy (1994) which has k
distinct marks spaced such that the distances between
marks can be used to measure all the distances 1, 2, 3, 4,

A Square which can be Dissected into a number of
smaller SQUARES with no two equal is called a PERFECT
Square Dissection (or a Squared Square). Square
dissections in which the squares need not be different
sizes are called Mrs. PERKINS' Quilts. If no subset
of the Squares forms a Rectangle, then the perfect
square is called "simple." Lusin claimed that perfect
squares were impossible to construct, but this assertion
was proved erroneous when a 55-SQUARE perfect square
was published by R. Sprague in 1939 (Wells 1991).

There is a unique simple perfect square of order 21
(the lowest possible order), discovered in 1978 by

1346 Perfect Square Dissection

Periapsis

A. J. W. Duijvestijn (Bouwkamp and Duijvestijn 1992).
It is composed of 21 squares with total side length 112,
and is illustrated above. There is a simple notation
(sometimes called Bouwkamp code) used to describe
perfect squares. In this notation, brackets are used to
group adjacent squares with flush tops, and then the
groups are sequentially placed in the highest (and left-
most) possible slots. For example, the 21-square illus-
trated above is denoted [50, 35, 27], [8, 19], [15, 17, 11],
[6, 24], [29, 25, 9, 2], [7, 18], [16], [42], [4, 37], [33].

The number of simple perfect squares of order n for
n > 21 are 1, 8, 12, 26, 160, 441, . . . (Sloane's A006983),
Duijvestijn's Table I gives a list of the 441 simple perfect
squares of order 26, the smallest with side length 212 and
the largest with side length 825. Skinner (1993) gives
the smallest possible side length (and smallest order for
each) as 110 (22), 112 (21), 120 (24), 139 (22), 140 (23),
... for simple perfect squared squares, and 175 (24),
235 (25), 288 (26), 324 (27), 325 (27), ... for compound
perfect squared squares.

There are actually three simple perfect squares having
side length 110. They are [60, 50], [23, 27], [24, 22, 14],
[7, 16], [8, 6], [12, 15], [13], [2, 28}, [26], [4, 21, 3], [18],
[17] (order 22; discovered by A. J. W. Duijvestijn); [60,
50], [27, 23], [24, 22, 14], [4, 19], [8, 6], [3, 12, 16], [9],
[2, 28], [26], [21], [1, 18], [17] (order 22; discovered by
T. H. Willcocks); and [44, 29, 37], [21, 8], [13, 32], [28,
16], [15, 19], [12,4], [3, 1], [2, 14], [5], [10, 41], [38, 7],
[31] (order 23; discovered by A. J. W. Duijvestijn).

D. Sleator has developed an efficient ALGORITHM for
finding non-simple perfect squares using what he calls
rectangle and "ell" grow sequences. This algorithm finds
a slew of compound perfect squares of orders 24-32.
Weisstein gives a partial list of known simple and com-
pound perfect squares (where the number of simple per-
fect squares is exact for orders less than 27) as well as
Mathematica® (Wolfram Research, Champaign, IL) al-
gorithms for drawing them.

Order

# Simple

# Compound

21

1

22

8

23

12

24

26

1

25

160

1

26

441

2

27

?

2

28

?

4

29

7

2

30

7

3

31

7

2

32

?

2

38

1

69

1

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 115-
116, 1987.

Beiler, A. H. Recreations in the Theory of Numbers: The
Queen of Mathematics Entertains. New York: Dover,
pp. 157-161, 1966.

Bouwkamp, C. J. and Duijvestijn, A. J. W. "Catalogue of
Simple Perfect Squared Squares of Orders 21 Through
25." Eindhoven Univ. Technology, Dept. Math, Report
92-WSK-03, Nov. 1992.

Brooks, R. L.; Smith, C. A. B.; Stone, A. H.; and Tutte,
W. T. "The Dissection of Rectangles into Squares." Duke
Math. J. 7, 312-340, 1940.

Duijvestijn, A. J. W. "A Simple Perfect Square of Lowest
Order." J. Combin. Th. Ser. B 25, 240-243, 1978.

Duijvestijn, A. J. W. "A Lowest Order Simple Perfect 2x1
Squared Rectangle." J. Combin. Th. Ser. B 26, 372-374,
1979.

Duijvestijn, A. J. W. ftp://ftp.cs.utwente.nl/pub/doc/
dvs/Tablel.

Gardner, M. "Squaring the Square." Ch. 17 in The Second
Scientific American Book of Mathematical Puzzles & Di-
versions: A New Selection. New York: Simon and Schus-
ter, 1961.

Gardner, M. Fractal Music, HyperCards, and More: Math-
ematical Recreations from Scientific American Magazine.
New York: W. H. Freeman, pp. 172-174, 1992.

Kraitchik, M. Mathematical Recreations. New York:
W. W. Norton, p. 198, 1942.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 15 and 32-33, 1979.

Mauldin, R. D. (Ed.) The Scottish Book: Math at the Scot-
tish Cafe Boston, MA: Birkhauser, 1982.

Moron, Z. "O rozkladach prostokatow na kwadraty."
Przeglad matematyczno-fizyczny 3, 152-153, 1925.

Skinner, J. D. II. Squares Squares: Who's Who & What's
What Published by the author, 1993.

Sloane, N. J. A. Sequences A006983/M4482 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.

Sprague, R. "Beispiel einer Zerlegung des Quadrats in lauter
verschiedene Quadrate." Math. Z. 45, 607-608, 1939.
# Weisstein, E. W. "Perfect Squares." http: //www. astro.
virginia.edu/-eww6n/math/notebooks/PerfectSquare.rn.

Wells, D. The Penguin Dictionary of Curious and Interesting
Geometry. London: Penguin, p. 242, 1991.

Periapsis

see also Mrs. Perkins' Quilt

The smallest radial distance of an ELLIPSE as measured
from a FOCUS. Taking v = in the equation of an

Ellipse

a(l-e 2 )
r =

1 + e cos v

gives the periapsis distance

r_ = a{\ — e).

Periapsis for an orbit around the Earth is called perigee,
and periapsis for an orbit around the Sun is called per-
ihelion.

Perigon

see also APOAPSIS, ECCENTRICITY, ELLIPSE, FOCUS

Perigon

An ANGLE of 2tt radians = 360° corresponding to the
Central Angle of an entire Circle.

Permanence of Algebraic Form 1347

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 425-427, 1953.

Spanier, J. and Oldham, K. B. "Periodic Functions." Ch. 36
in An Atlas of Functions. Washington, DC: Hemisphere,
pp. 343-349, 1987.

Perimeter

The Arc Length along the boundary of a closed 2-D
region. The perimeter of a Circle is called the Cir-
cumference.

see also Circumference, Cluster Perimeter,
Semiperimeter

Periodic Point

A point xo is said to be a periodic point of a Function
/ of period n if f n (xo) = #o, where fo(x) = x and f n (x)
is defined recursively by f n (x) = f(f n ~ 1 (x)).

see also Least Period, Periodic Function, Peri-
odic Sequence

Perimeter Polynomial

A sum over all Cluster Perimeters.

Period Doubling

A characteristic of some systems making a transition
to Chaos. Doubling is followed by quadrupling, etc.
An example of a map displaying period doubling is the
Logistic Map.

see also Chaos, LOGISTIC Map

Period Three Theorem

Li and Yorke (1975) proved that any 1-D system which
exhibits a regular Cycle of period three will also dis-
play regular CYCLES of every other length as well as
completely Chaotic Cycles.

see also Chaos, Cycle (Map)

References

Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos."
Amer. Math. Monthly 82, 985-992, 1975.

Periodic Function

i ■

A FUNCTION f(x) is said to be periodic with period p

if f(x) = f(x + np) for n = 1, 2, For example, the

Sine function sinx is periodic with period 27T (as well
as with period — 2-zr, 47r, 67r, etc.).

The Constant Function f(x) = is periodic with
any period R for all NONZERO REAL NUMBERS R, so
there is no concept analogous to the LEAST PERIOD of
a PERIODIC POINT for functions.

see also Periodic Point, Periodic Sequence

Periodic Sequence

A Sequence {a^} is said to be periodic with period p

with if it satisfies a* = ai+ nv for n = 1, 2, For

example, {1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, . . . } is a
periodic sequence with LEAST PERIOD 2.

see also Eventually Periodic, Periodic Function,
Periodic Point

Perkins' Quilt

see Mrs. Perkins' Quilt

Perko Pair

The KNOTS 10i 6 i and 10i62 illustrated above. They
are listed as separate knots in the pictorial enumeration
of Rolfsen (1976, Appendix C), but were identified as
identical by Perko (1974).

References

Proc.

Perko, K. A. Jr. "On the Classification of Knots."

Amer. Math. Soc. 45, 262-266, 1974.
Rolfsen, D. "Table of Knots and Links." Appendix C in

Knots and Links. Wilmington, DE: Publish or Perish

Press, pp. 280-287, 1976.

Permanence of Algebraic Form

All Elementary Functions can be extended to the
Complex Plane. Such definitions agree with the Real
definitions on the x-AxiS and constitute an ANALYTIC
Continuation.

see also Analytic Continuation, Elementary
Function, Permanence of Mathematical Rela-
tions Principle

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 380, 1985.

1348 Permanence of Mathematical Relations Principle

Permutation

Permanence of Mathematical Relations
Principle

The metric properties discovered for a primitive fig-
ure remain applicable, without modifications other than
changes of signs, to all correlative figures which can be
considered to arise from the first.

This principle was formulated by Poncelet, and amounts
to the statement that if an analytic identity in any finite
number of variables holds for all real values of the vari-
ables, then it also holds by Analytic Continuation
for all complex values (Bell 1945). This principle is also
called Poncelet's Continuity Principle.

see also ANALYTIC CONTINUATION, CONSERVATION OF

Number Principle, Duality Principle, Perma-
nence of Algebraic Form

References

Bell, E. T. The Development of Mathematics, 2nd ed. New
York: McGraw-Hill, p. 340, 1945.

Permanent

An analog of a DETERMINANT where all the signs in
the expansion by MINORS are taken as POSITIVE. The
permanent of a MATRIX A is the coefficient of x± * • • x n
in

n
J[ J[(ailXi 4" a i2 X 2 + . . . + CLinXn)
* = 1

(Vardi 1991). Another equation is the RYSER FORMULA

n

perm(ay) = (-1)" £ (-l) w ni>''

sC{l,...,n} i=l j£s

where the SUM is over all SUBSETS of {1, . . . , n}, and
\s is the number of elements in s (Vardi 1991).

If M is a Unitary Matrix, then
|perm(M)| < 1

(Mine 1978, p. 25; Vardi 1991).

see also Determinant, Frobenius-Konig Theorem,
Immanant, Ryser Formula, Schur Matrix

References

Borovskikh, Y. V.; Korolyuk, V. S. Random Permanents.

Philadelphia, PA: Coronet Books, 1994.
Mine, H. Permanents. Reading, MA: Addis on- Wesley, 1978.
Vardi, I. "Permanents." §6.1 in Computational Recreations

in Mathematica. Reading, MA: Addis on- Wesley, pp. 108

and 110-112, 1991.

Permil

The use of percentages is a way of expressing RATIOS in
terms of whole numbers. Given a RATIO or FRACTION,
it is converted to a permil-age by multiplying by 1000
and appending a "mil sign" %o . For example, if an
investment grows from a number P = 13.00 to a number
A = 22.50, then A is 22.50/13.00 = 1.7308 times as
much as P, or 1730.8 %o .

see also Percent

Permutation

The rearrangement of elements in a set into a One-
to-One correspondence with itself, also called an Ar-
rangement or Order. The number of ways of obtain-
ing r ordered outcomes from a permutation of n elements
is

%ifr —

(n — r)\

<:)•

(i)

where n\ is n FACTORIAL and (£) is a BINOMIAL CO-
EFFICIENT. The total number of permutations for n
elements is given by n\.

A representation of a permutation as a product of Cy-
cles is unique (up to the ordering of the cycles). An
example of a cyclic decomposition is ({1, 3, 4}, {2}), cor-
responding to the permutations (1 -> 3, 3 -> 4, 4 -> 1)
and (2 — > 2), which combine to give {4, 2, 1, 3}.

Any permutation is also a product of TRANSPOSI-
TIONS. Permutations are commonly denoted in LEX-
ICOGRAPHIC or Transposition Order. There is a
correspondence between a PERMUTATION and a pair of
Young Tableaux known as the Schensted Corre-
spondence.

The number of wrong permutations of n objects is [nl/e]
where [x] is the NlNT function. A permutation of n
ordered objects in which no object is in its natural place
is called a Derangement (or sometimes, a Complete
Permutation) and the number of such permutations is
given by the SUBFACTORIAL In.

Using

(*+»>" = E ft)*-- v

with x — y = 1 gives

-r — n \ ^

(2)

(3)

so the number of ways of choosing 0, 1, . . . , or n at a
time is 2".

The set of all permutations of a set of elements 1, . . . , n
can be obtained using the following recursive procedure

(4)

(5)

Permutation Group

Permutation Symbol 1349

Let the set of Integers 1, 2, . . . , n be permuted and
the resulting sequence be divided into increasing Runs.
As n approaches Infinity, the average length of the nth
RUN is denoted L n . The first few values are

Li =e- 1 = 1.7182818...
e -2e = 1.9524...

1.9957,

L 2

L 3 = e 3 - 3e 2 + f e :

(6)
(7)
(8)

where e is the base of the NATURAL LOGARITHM (Knuth
1973, Le Lionnais 1983).

see also Alternating Permutation, Binomial Co-
efficient, Circular Permutation, Combination,
Complete Permutation, Derangement, Discor-
dant Permutation, Eulerian Number, Linear
Extension, Permutation Matrix, Subfactorial,
Transposition

References

Bogomolny, A. "Graphs.'* http://www.cut-the-knot.com/

do_you_know/permutation.html.
Conway, J. H. and Guy, R. K. "Arrangement Numbers." In

The Book of Numbers. New York: Springer- Verlag, p. 66,

1996.
Dickau, R. M. "Permutation Diagrams." http:// forum .

swarthmore.edu/advanced/robertd/permutations.html.
Knuth, D. E. The Art of Computer Programming, Vol. 1:

Fundamental Algorithms, 2nd ed. Reading, MA: Addison-

Wesley, 1973.
Kraitchik, M. "The Linear Permutations of n Different

Things." §10.1 in Mathematical Recreations. New York:

W. W. Norton, pp. 239-240, 1942.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

pp. 41-42, 1983.
Ruskey, F. "Information on Permutations." http://sue.csc

.uvic . ca/-cos/inf /perm/Permlnf o .html.
Sloane, N. J. A. Sequence A000142/M1675 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Permutation Group

A finite GROUP of substitutions of elements for each
other. For instance, the order 4 permutation group {4,
2, 1, 3} would rearrange the elements {A, B, C, D} in
the order {£>, B, A, C}. A SUBSTITUTION GROUP of
two elements is called a TRANSPOSITION. Every SUB-
STITUTION GROUP with > 2 elements can be written as
a product of transpositions. For example,

(abc) = (ab)(ac)
{abode) = (ab)(ac)(ad)(ae).

CONJUGACY Classes of elements which are inter-
changed are called CYCLES (in the above example, the
Cycles are {{1, 3, 4}, {2}}).

see also Cayley's Group Theorem, Cycle (Permu-
tation), Group, Substitution Group, Transposi-
tion

Permutation Matrix

A Matrix p.. obtained by permuting the ith and jth
rows of the IDENTITY MATRIX with i < j. Every row
and column therefore contain precisely a single 1, and
every permutation corresponds to a unique permutation
matrix. The matrix is nonsingular, so the DETERMI-
NANT is always NONZERO. It satisfies

= 1,

where I is the IDENTITY MATRIX. Applying to another
Matrix, p i; A gives A with the ith and jth rows inter-
changed, and Ap • ■ gives A with the ith and jth columns
interchanged.

Interpreting the Is in an n x n permutation matrix as
ROOKS gives an allowable configuration of nonattacking
ROOKS on an n X n CHESSBOARD.

see also ELEMENTARY MATRIX, IDENTITY, PERMUTA-
TION, Rook Number

Permutation Pseudotensor

see Permutation Tensor

Permutation Symbol

A three-index object sometimes called the Levi-Civita
Symbol defined by

e%jk

■f : :

for i = j i j = k i OTk~i

for (i,j,k) € {(1,2, 3), (2, 3,1), (3, 1,2)}

for (i,j,k) € {(1,3, 2), (3, 2,1), (2, 1,3)}.

(1)

The permutation symbol satisfies

Sijtijk = o

(2)

Cipqtjpq = 60ij

(3)

Cijktijk — 6

(4)

CijfcCpgfc — OipOjq

OiqVjp-,

(5)

where 8%j is the KRONECKER DELTA. The symbol can be
defined as the SCALAR TRIPLE PRODUCT of unit vectors
in a right-handed coordinate system,

e ijk = x; • (xj x x fc ).

(6)

The symbol can also be interpreted as a TENSOR, in
which case it is called the PERMUTATION TENSOR.

see also Permutation Tensor

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 132-133, 1985.

1350 Permutation Tensor

Perrin Pseudoprime

Permutation Tensor

A PSEUDOTENSOR which is ANTISYMMETRIC under the
interchange of any two slots. Recalling the definition
of the Permutation Symbol in terms of a Scalar
Triple Product of the Cartesian unit vectors,

€ij k = Xi • (Xj X Xfc) = [Xi,Xj-,Xfc], (1)

the pseudotensor is a generalization to an arbitrary BA-
SIS defined by

Perpendicular Bisector

A

eap.-p = y/\g\ [<*,(), . . . , p]

a/3-/* _ [gvgi --•»/*]

6 " vi^i '

(2)
(3)

where

{1 the arguments are an even permutation
—1 the arguments are an odd permutation
two or more arguments are equal,

(4)

and g = det^^), where g a p is the METRIC TENSOR.
e(xi,...,x„) is Nonzero Iff the Vectors are Lin-
early Independent.

see also Permutation Symbol, Scalar Triple
Product

Peron Integral

see Denjoy Integral

Perpendicular

A D B

Two lines, vectors, planes, etc., are said to be perpen-
dicular if they meet at a RIGHT Angle. In R n , two
VECTORS A and B are PERPENDICULAR if their Dot
Product

A • B = 0.

In M 2 , a Line with Slope m 2 = -1/mi is Perpendic-
ular to a Line with Slope mi. Perpendicular objects
are sometimes said to be "orthogonal."

In the above figure, the Line SEGMENT AB is perpen-
dicular to the Line Segment CD. This relationship is
commonly denoted with a small Square at the vertex
where perpendicular objects meet, as shown above.

see also Orthogonal Vectors, Parallel, Perpen-
dicular Bisector, Perpendicular Foot, Right
Angle

The perpendicular bisectors of a Triangle AA1A2A3
are lines passing through the Midpoint Mi of each side
which are Perpendicular to the given side. A Trian-
gle's three perpendicular bisectors meet at a point C
known as the ClRCUMCENTER (which is also the center

of the Triangle's Circumcircle).

see also ClRCUMCENTER, MIDPOINT, PERPENDICULAR,

Perpendicular Foot
Perpendicular Foot

perpendicular
foot

The Foot of the Perpendicular is the point on the
leg opposite a given vertex of a TRIANGLE at which the
Perpendicular passing through that vertex intersects
the side. The length of the Line Segment front ver-
tex to perpendicular foot is called the ALTITUDE of the
Triangle.

see also ALTITUDE, FOOT, PERPENDICULAR, PERPEN-
DICULAR Bisector

Perrin Pseudoprime

If 77 is Prime, then p\P(p), where P(p) is a member of
the Perrin Sequence 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, . . .
(Sloane's A001608). A Perrin pseudoprime is a COM-
POSITE Number n such that n\P(n). Several "unre-
stricted" Perrin pseudoprimes are known, the smallest
of which are 271441, 904631, 16532714, 24658561, ...
(Sloane's A013998).

Adams and Shanks (1982) discovered the smallest unre-
stricted Perrin pseudoprime after unsuccessful searches
by Perrin (1899), Malo (1900), Escot (1901), and Jar-
den (1966). (Stewart's 1996 article stating no Perrin
pseudoprimes were known was in error.)

Grantham (1996) generalized the definition of Perrin
pseudoprime with parameters (r, s) to be an Odd Com-
posite Number n for which either

Perrin Sequence

Perron-Frobenius Theorem

1351

1. (A/n) = 1 and n has an S-SiGNATURE, or

2. (A/n) = -1 and n has a Q-SlGNATURE,

where (a/b) is the Jacobi Symbol. All the 55 Perrin
pseudoprimes less than 50 x 10 9 have been computed
by Kurtz et al. (1986). All have S-SlGNATURE, and
form the sequence Sloane calls "restricted" Perrin pseu-
doprimes: 27664033,46672291,102690901,... (Sloane's
A018187).

see also PERRIN SEQUENCE, PSEUDOPRIME

Refereu es

Adams, W. W. "Characterizing Pseudoprimes for Third-
Order Linear Recurrence Sequences." Math Comput. 48,
1-15, 1987.

Adams, W. and Shanks, D. "Strong Primality Tests that Are
Not Sufficient." Math. Comput 39, 255-300, 1982.

Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1:
Efficient Algorithms. Cambridge, MA: MIT Press, p. 305,
1996.

Escot, E.-B. "Solution to Item 1484." L'Intermediare des
Math. 8, 63-64, 1901.

Grantham, J. "Probenius Pseudoprimes." http://www.
dark . net/pub/grantham/pseudo/pseudo . ps

Holzbaur, C. "Perrin Pseudoprimes." http://ftp.ai.
univie . ac . at /perrin .html.

Jarden, D. Recurring Sequences. Jerusalem: Riveon Le-
matematika, 1966.

Kurtz, G. C; Shanks, D.; and Williams, H. C. "Fast Primal-
ity Tests for Numbers Less than 50 * 10 9 ." Math. Comput.
46, 691-701, 1986.

Malo, E. L'Intermediare des Math. 7, 281 and 312, 1900.

Perrin, R. "Item 1484." L'Intermediare des Math. 6, 76-77,
1899.

Ribenboim, P. The New Book of Prime Number Records, 3rd
ed. New York: Springer- Verlag, p. 135, 1996.

Sloane, N. J. A. Sequences A013998, A018187, and A001608/
M0429 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Stewart, I. "Tales of a Neglected Number," Sci. Amer. 274,
102-103, Juno 1996.

Perrin Sequence

The INTEGER SEQUENCE defined by the recurrence

where

A(n).

(7)

P(n) = P(n - 2) + P(n - 3)

(1)

with the initial conditions F(0) = 3, P(l) = 0, P(2) =
2. The first few terms are 0, 2, 3, 2, 5, 5, 7, 10, 12,
17, ... (Sloane's A001608). P(n) is the solution of a
third-order linear homogeneous Difference Equation
having characteristic equation

x 6 - x - 1 = 0,
discriminant —23, and ROOTS

(2)

Perrin (1899) investigated the sequence and noticed that
if n is PRIME, then n\P(n). The first statement of this
fact is attributed to E. Lucas in 1876 by Stewart (1996).
Perrin also searched for but did not find any Compos-
ite NUMBER n in the sequence such that n\P(n). Such
numbers are now known as PERRIN PSEUDOPRIMES.
Malo (1900), Escot (1901), and Jarden (1966) subse-
quently investigated the series and also found no PER-
RIN PSEUDOPRIMES. Adams and Shanks (1982) subse-
quently found that 271,441 is such a number.

see also PADOVAN SEQUENCE, PERRIN PSEUDOPRIME,
SIGNATURE (RECURRENCE RELATION)

References

Adams, W. and Shanks, D. "Strong Primality Tests that Are

Not Sufficient." Math. Comput. 39, 255-300, 1982.
Escot, E.-B. "Solution to Item 1484." L'Intermediare des

Math. 8, 63-64, 1901.
Jarden, D. Recurring Sequences. Jerusalem: Riveon Le-

matematika, 1966.
Malo, E. L'Intermediare des Math. 7, 281 and 312, 1900.
Perrin, R. "Item 1484." L'Intermediare des Math. 6, 76-77,

1899.
Stewart, I. "Tales of a Neglected Number." Sci. Amer. 274,

102-103, June 1996.
Sloane, N. J. A. Sequence A001608/M0429 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Perron-Frobenius Operator

An Operator which describes the time evolution of
densities in Phase Space. The Operator can be de-
fined by

Pn+l = Lpn,

where p n are the Natural Densities after the nth
iten ; on of a map /. This can be explicitly written as

Lp(y)= £

zez-Mj-)

P{x)

References

Beck, C. and Schlogl, F. "Transfer Operator Methods."
Ch. 17 in Thermodynamics of Chaotic Systems. Cam-
bridge, England: Cambridge University Press, pp. 190-
203, 1995.

Perron-Frobenius Theorem

If all elements a^- of an IRREDUCIBLE MATRIX A are
NONNEGATIVE, then R = minMA is an EIGENVALUE of
A and all the Eigenvalues of A lie on the Disk

an 1.324717957 (3)

« -0.6623589786 + 0.5622795121i (4)

7 « -0.6623589786 - 0.562279512H. (5)

The solution is then

A(n) = a n + n + 7 "

(6)

1*1 < R,

where, if A = (Ai, A 2 , . . . , A„) is a set of NONNEGATIVE
numbers (which are not all zero),

M A

inf < fj, : juAi > > Wij\ Aj, 1 < i < n >

1352

Perron's Theorem

Perspective Collineation

and R = minMA. Furthermore, if A has exactly p
Eigenvalues (p < n) on the Circle \z\ — R, then the
set of all its Eigenvalues is invariant under rotations
by 2ir/p about the Origin.

see also Wielandt's Theorem

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1121, 1979.

Perron's Theorem

If /x = (^i,/i2, ...,/in) is an arbitrary set of POSITIVE
numbers, then all Eigenvalues A of the n x n Matrix
A — dij lie on the Disk \z\ < M M , where

Mu = max

ZS'°-.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1121, 1979.

Persistence

see Additive Persistence, Multiplicative Persis-
tence, Persistent Number, Persistent Process

Persistent Number

An n-persistent number is a Positive Integer k which
contains the digits 0, 1, . . . , 9, and for which 2fc, . . . , nk
also share this property. No oo-persistent numbers exist.
However, the number k = 1234567890 is 2-persistent,
since 2k = 2469135780 but 3k = 3703703670, and
the number k = 526315789473684210 is 18-persistent.
There exists at least one ^-persistent number for each
Positive Integer k.

see also Additive Persistence, Multiplicative
Persistence

References

Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 15-18, 1991.

Persistent Process

A Fractal Process for which H > 1/2, so r > 0.

see also Antipersistent Process, Fractal Process

Perspective

r vanishing points

/ / 7

•//

/
/

/

one -point
perspective

Perspective is the art and mathematics of realistically
depicting 3-D objects in a 2-D plane. The study of the
projection of objects in a plane is called PROJECTIVE
Geometry. The principles of perspective drawing were
elucidated by the Florentine architect F. Brunelleschi
(1377-1446). These rules are summarized by Dixon
(1991):

1. The horizon appears as a line.

2. Straight lines in space appear as straight lines in the
image.

3. Sets of Parallel lines meet at a Vanishing Point.

4. Lines Parallel to the picture plane appear Paral-
lel and therefore have no Vanishing Point.

There is a graphical method for selecting vanishing
points so that a Cube or box appears to have the correct
dimensions (Dixon 1991).

see also LEONARDO'S PARADOX, PERSPECTIVE AXIS,

Perspective Center, Perspective Collineation,
Perspective Triangles, Perspectivity, Projec-
tive Geometry, Vanishing Point, Zeeman's Para-
dox

References

de Vries, V. Perspective. New York: Dover, 1968.

Dixon, R. "Perspective Drawings." Ch. 3 in Mathographics.

New York: Dover, pp. 79-88, 1991.
Parramon, J. M. Perspective — How to Draw. Barcelona,

Spain: Parramon Editions, 1984.

Perspective Axis

The line joining the three collinear points of intersection
of the extensions of corresponding sides in PERSPECTIVE

Triangles.

see also Perspective Center, Perspective Trian-
gles, Sondat's Theorem

Perspective Center

The point at which the three Lines connecting the Ver-
tices of Perspective Triangles (from a point) Con-
cur.

Perspective Collineation

A perspective collineation with center O and axis o is
a Collineation which leaves all lines through O and
points of o invariant. Every perspective collineation is a
Projective Collineation.

see also COLLINEATION, ELATION, HOMOLOGY (GEOM-
ETRY), Projective Collineation

References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 247-248, 1969.

Perspective Triangles

Petrie Polygon 1353

Perspective Triangles

Two Triangles are perspective from a line if the ex-
tensions of their three pairs of corresponding sides meet
in COLLINEAR points. The line joining these points is
called the PERSPECTIVE Axis. Two TRIANGLES are per-
spective from a point if their three pairs of correspond-
ing VERTICES are joined by lines which meet in a point
of Concurrence. This point is called the Perspec-
tive Center, Desargues' Theorem guarantees that
if two Triangles are perspective from a point, they are
perspective from a line,

see also Desargues' Theorem, Homothetic Tri-
angles, Paralogic Triangles, Perspective Axis,
Perspective Center

Perspectivity

A correspondence between two Ranges that are sec-
tions of one PENCIL by two distinct lines.
see also Pencil, Projectivity, Range (Line Seg-
ment)

Pesin Theory

A theory of linear HYPERBOLIC MAPS in which the lead-
ing constants do depend on the variable x.

Peter- Weyl Theorem

Establishes completeness for a REPRESENTATION.

References

Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not Amer. Math. Soc. 43, 537-549, 1996.

Peters Projection

A CYLINDRICAL equal-area projection that shifts the
standard parallels to 45° or 47°.

see also Cylindrical Projection

References

Dana, P. H. "Map Projections." http://www.utexas.edu/
depts/grg/gcraf t /notes /mapproj /mappro j . html.

The seven graphs obtainable from the COMPLETE
Graph K& by repeated triangle-Y exchanges are also
called Petersen graphs, where the three EDGES forming
the Triangle are replaced by three Edges and a new
Vertex that form a Y, and the reverse operation is also
permitted. A GRAPH is intrinsically linked IFF it con-
tains one of the seven Petersen graphs (Robertson et al.
1993).
see also Hoffman-Singleton Graph

References

Adams, C. C. The Knot Book: An Elementary Introduction

to the Mathematical Theory of Knots. New York: W. H.

Freeman, pp. 221-222, 1994.
Robertson, N.; Seymour, P. D.; and Thomas, R. "Linkless

Embeddings of Graphs in 3-Space." Bull. Amer. Math.

Soc. 28, 84-89, 1993.
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:

Assaults and Conquest. New York: Dover, p. 102, 1986.

Petersen-Shoute Theorem

1. If AABC and AA'B'C are two directly similar tri-
angles, while AAA' A", ABB'B", and ACC'C" are
three directly similar triangles, then AA"B"C" is
directly similar to AABC.

2. When all the points P on AB are related by a Sim-
ilarity Transformation to all the points P' on
A'B', the points dividing the segment PP' in a given
ratio are distant and collinear, or else they coincide.

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 95-100, 1967.

Petersen Graphs

"The" Petersen graph is the Graph illustrated above
possessing ten Vertices all of whose nodes have De-
gree 3 (Saaty and Kainen 1986). The Petersen graph
is the only smallest-girth graph which has no Tait col-
oring.

Petrie Polygon

{3,3} {3,4}

A skew Polygon such that every two consecutive sides
(but no three) belong to a face of a regular POLYHE-
DRON. Every finite POLYHEDRON can be orthogonally
projected onto a plane in such a way that one Petrie
polygon becomes a REGULAR POLYGON with the re-
mainder of the projection interior to it. The Petrie poly-
gon of the Polyhedron {p, q} has h sides, where

cos 2 (0=cos 2 (j)+cos 2 (V).

1354 Petrov Notation

Phasor

The Petrie polygons shown above correspond to the
Platonic Solids.

see also Platonic Solid, Regular Polygon

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays } 13th ed. New York: Dover, p. 135,
1987.

Coxeter, H. S. M. "Petrie Polygons." §2.6 in Regular Poly-
topes, 3rd ed. New York: Dover, pp. 24-2 5, 1973.

Petrov Notation

A Tensor notation which considers the Riemann Ten-
sor RxpvK as a matrix R(\^)( UK ) with indices A/j and uk.

References

Weinberg, S. Gravitation and Cosmology: Principles and

Applications of the General Theory of Relativity. New

York: Wiley, p. 142, 1972.

Pfaffian Form

A 1-FORM

oj — y di(x) dxi

such that

uj = 0.

References

Knuth, D. E. "Overlapping Pfaffians." Electronic J. Com-
binatorics 3, No, 2, R5, 1-13, 1996. http://www.
combinatorics . org/Volume^3/volume3_2 . html#R5.

Phase

The angular position of a quantity. For example, the
phase of a function cos(o;t + 0o) as a function of time is

<t>{t) =UJt + 00-

The Argument of a Complex Number is sometimes
also called the phase.

see also Argument (Complex Number), Complex
Number, Phasor, Retardance

Phase Space

For a function or object with n Degrees of Freedom,
the n-D Space which is accessible to the function or
object is called its phase space.

see also World Line

Phase Transition

see Random Graph

Phasor

The representation, beloved of engineers and physicists,
of a Complex Number in terms of a Complex expo-
nential

x + iy = \z\e l4t , (1)

where i (called j by engineers) is the IMAGINARY NUM-
BER and the Modulus and Argument (also called
Phase) are

1*1 = \A 2 + y 2

tan

(2)
(3)

Here, <j> is the counterclockwise Angle from the POSI-
TIVE Real axis. In the degenerate case when x = 0,

J7r if y <

<j> = I undefined if y =
I |tt ify>0.

It is trivially true that

J2m>i} = &

$>

(4)

(5)

Now consider a SCALAR FUNCTION ip = V>oe i0 . Then

= I(^ 2 + 2^*+V>* 2 )- (6)
Look at the time averages of each term,

(V> 2 ) = (V-oV*) = Vo 2 (e 2i *> = (7)

(W*) = (ipoe'^oe-**) = V>o 2 = IV-I 2 (8)

(r 2 ) = (ih'e-'*) = V-o 2 (e~ 2i *> = 0. (9)

Therefore,

(i) = m 2 -

Consider now two scalar functions

t(fcri+*i)

-01 = Vi.oe'

(10)

(11)
(12)

Then

[R(iM + R(V*)] a = |[(V-i + i>i') + (</> 2 + fa*)} 2

= |[Wi+V- l *) 2 + (V'2+V'2*) 2

+ 2(lpl1p2 + Tplfo* + ^1*^2 + ^1*^2*)]
(I) = l[2^lVl* + 2^2* + 2^2* + 2^i'^,]
= l[Vl(01* +^2*) + tfa(tfl* + V**)]
= |(Vl +V>2)(Vl* + ^2*) = 1 |Vl + V2| 2 .

In general,

(i)

5>

(13)

(14)

(15)

see also Affix, Argument (Complex Number),
Complex Multiplication, Complex Number,
Modulus (Complex Number), Phase

Phi Curve

Pi

1355

Phi Curve

An Adjoint Curve which bears a special relation to
the base curve.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 310, 1959.

Phi Number System

For every POSITIVE INTEGER n, there is a corresponding
finite sequence of distinct Integers &i, . .., km such
that

n = fcl + ... + #*",

where <j> is the GOLDEN MEAN.

References

Bergman, G. "A Number System with an Irrational Base."
Math. Mag. 31, 98-110, 1957.

Knuth, D. The Art of Computer Programming, Vol. 1: Fun-
damental Algorithms, 2nd ed. Reading, MA: Addison-
Wesley, 1973.

Rousseau, C. "The Phi Number System Revisited." Math.
Mag. 68, 283-284, 1995.

Phragmen-Lindelof Theorem

Let f(z) be an Analytic Function in an angular do-
main W : |argz| < a7r/2. Suppose there is a constant
M such that for each e > 0, each finite boundary point
has a Neighborhood such that \f(z)\ < M + e on the
intersection of D with this NEIGHBORHOOD, and that
for some POSITIVE number j3 > a for sufficiently large
|z|, the Inequality \f(z)\ < exp(\z\ 1/f3 ) holds. Then
\f(z)\<M in D.

References

lyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 160, 1980.

Phy Hot axis

The beautiful arrangement of leaves in some plants,
called phyllotaxis, obeys a number of subtle mathemat-
ical relationships. For instance, the florets in the head
of a sunflower form two oppositely directed spirals: 55
of them clockwise and 34 counterclockwise. Surpris-
ingly, these numbers are consecutive Fibonacci Num-
bers. The ratios of alternate Fibonacci Numbers are
given by the convergents to <£ -2 , where <j> is the Golden
RATIO, and are said to measure the fraction of a turn
between successive leaves on the stalk of a plant: 1/2
for elm and linden, 1/3 for beech and hazel, 2/5 for
oak and apple, 3/8 for poplar and rose, 5/13 for willow
and almond, etc. (Coxeter 1969, Ball and Coxeter 1987).
A similar phenomenon occurs for DAISIES, pineapples,
pinecones, cauliflowers, and so on.

Lilies, irises, and the trillium have three petals; col-
umbines, buttercups, larkspur, and wild rose have five
petals; delphiniums, bloodroot, and cosmos have eight
petals; corn marigolds have 13 petals; asters have 21
petals; and daisies have 34, 55, or 84 petals — all FI-
BONACCI Numbers.

see also DAISY, FIBONACCI NUMBER, SPIRAL

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 56-57,
1987.

Church, A. H. The Relation of Phyllotaxis to Mechanical
Laws. London: Williams and Norgate, 1904.

Church, A. H. On the Interpretation of Phenomena of Phyl-
lotaxis. Riverside, NJ: Hafher, 1968.

Conway, J. H. and Guy, R. K. "Phyllotaxis." In The Book of
Numbers. New York: Springer- Verlag, pp. 113-125, 1995.

Coxeter, H. S. M. "The Golden Section and Phyllotaxis."
Ch. 11 in Introduction to Geometry, 2nd ed. New York:
Wiley, 1969.

Coxeter, H. S. M. "The Golden Section, Phyllotaxis, and
Wythoff's Game." Scripta Mathematica 19, 135-143,
1953.

Dixon, R. Mathographics. New York: Dover, 1991.

Douady, S. and Couder, Y. "Phyllotaxis as a Self-Organized
Growth Process." In Growth Patterns in Physical Sciences
and Biology (Ed. Juan M. Garcia-Ruiz et at). Plenum
Press, 1993.

Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 20-22, 1975.

Jean, R. V. Phyllotaxis: A Systematic Study in Plant Mor-
phogenesis. New York: Cambridge University Press, 1994.

Pappas, T. "The Fibonacci Sequence & Nature." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 222-225, 1989.

Prusinkiewicz, P. and Lindenmayer, A. The Algorithmic
Beauty of Plants. New York: Springer- Verlag, 1990.

Stewart, I. "Daisy, Daisy, Give Me Your Answer, Do." Sci.
Amer. 200, 96-99, Jan. 1995.

Thompson, D. W. On Growth and Form. Cambridge, Eng-
land: Cambridge University Press, 1952.

Pi

A Real NUMBER denoted n which is defined as the
ratio of a CIRCLE'S CIRCUMFERENCE C to its DIAMETER

d=2r,

n n

(1)

2r

It is equal to

7T = 3.141592653589793238462643383279502884197. . .

(2)
(Sloane's A000796). ?r has recently (August 1997) been
computed to a world record 51,539,600,000 « 3 • 2 34
Decimal Digits by Y. Kanada. This calculation
was done using Borwein's fourth-order convergent al-
gorithm and required 29 hours on a massively parallel
1024-processor Hitachi SR2201 supercomputer. It was
checked in 37 hours using the BRENT-SALAMIN FOR-
MULA on the same machine.

The Simple Continued Fraction for 7r, which gives
the "best" approximation of a given order, is [3, 7, 15,

1356

Pi-

Pi

1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...]
(Sloane's A001203). The very large term 292 means
that the CONVERGENT

[3, 7, 15,1] = [3, 7, 16]

355
113

3.14159292...

(3)

is an extremely good approximation. The first few CON-
VERGENTS are 22/7, 333/106, 355/113, 103993/33102,
104348/33215, ... (Sloane's A002485 and A002486).
The first occurrences of n in the CONTINUED FRAC-
TION are 4, 9, 1, 30, 40, 32, 2, 44, 130, 100, . . . (Sloane's
A032523).

Gosper has computed 17,001,303 terms of 7r's CONTIN-
UED Fraction (Gosper 1977, Ball and Coxeter 1987),
although the computer on which the numbers are stored
may no longer be functional (Gosper, pers. comm.,
1998). According to Gosper, a typical Continued
FRACTION term carries only slightly more significance
than a decimal DIGIT. The sequence of increasing terms
in the CONTINUED FRACTION is 3, 7, 15, 292, 436,
20776, ... (Sloane's A033089), occurring at positions
1, 2, 3, 5, 308, 432, . . . (Sloane's A033090). In the first
26,491 terms of the CONTINUED FRACTION (counting 3
as the 0th), the only five-DlGIT terms are 20,776 (the
431st), 19,055 (15,543rd), and 19,308 (23,398th) (Beeler
et al 1972, Item 140). The first 6-DlGlT term is 528,210
(the 267,314th), and the first 8-DlGlT term is 12,996,958
(453,294th). The term having the largest known value
is the whopping 9-DlGIT 87,878,3625 (the 11,504,931st
term).

The Simple Continued Fraction for 7r does not show
any obvious patterns, but clear patterns do emerge in
the beautiful non-simple CONTINUED FRACTIONS

4 - = i +

7T

(4)

2 +

2 +

2 +

2 + .

(Brouckner), giving convergents 1, 3/2, 15/13, 105/76,
315/263, . . . (Sloane's A025547 and A007509) and

7v crops up in all sorts of unexpected places in mathe-
matics besides Circles and Spheres. For example, it
occurs in the normalization of the GAUSSIAN DISTRI-
BUTION, in the distribution of PRIMES, in the construc-
tion of numbers which are very close to INTEGERS (the
Ramanujan Constant), and in the probability that
a pin dropped on a set of Parallel lines intersects a
line (BUFFON'S Needle Problem). Pi also appears as
the average ratio of the actual length and the direct dis-
tance between source and mouth in a meandering river
(St0llum 1996, Singh 1997).

A brief history of NOTATION for pi is given by Castel-
lanos (1988). it is sometimes known as LUDOLPH'S CON-
STANT after Ludolph van Ceulen (1539-1610), a Dutch
7r calculator. The symbol it was first used by William
Jones in 1706, and subsequently adopted by Euler. In
Measurement of a Circle, Archimedes (ca. 225 BC) ob-
tained the first rigorous approximation by INSCRIBING
and Circumscribing 6 - 2 n -gons on a Circle using the
Archimedes Algorithm. Using n = 4 (a 96-gon),
Archimedes obtained

3-hif <7T<3+|

(6)

(Shanks 1993, p. 140).

The Bible contains two references (I Kings 7:23 and
Chronicles 4:2) which give a value of 3 for w. It should
be mentioned, however, that both instances refer to
a value obtained from physical measurements and, as
such, are probably well within the bounds of experi-
mental uncertainty. I Kings 7:23 states, "Also he made
a molten sea of ten Cubits from brim to brim, round
in compass, and five cubits in height thereof; and a line
thirty cubits did compass it round about." This implies
7T = C/d = 30/10 = 3. The Babylonians gave an esti-
mate of 7r as 3 + 1/8 = 3. 125. The Egyptians did better
still, obtaining 2 8 /3 4 = 3.1605 ... in the Rhind papyrus,
and 22/7 elsewhere. The Chinese geometers, however,
did best of all, rigorously deriving ir to 6 decimal places.

A method similar to Archimedes' can be used to esti-
mate 7r by starting with an n-gon and then relating the
Area of subsequent 2n-gons. Let be the Angle from
the center of one of the POLYGON'S segments,

!--

1

23

1-2

3-

4-5

1-

3-4

6-7

5-6

(5)

(Stern 1833), giving convergents 1, 2/3, 4/3, 16/15,
64/45, 128/105, . . . (Sloane's A001901 and A046126).

Then

/?=I(n-3)7r.

|nsin(2/3)

cos/? cos (f) cos (^-) COS (^) ■

(7)

(8)

(Beckmann 1989, pp. 92-94). Viete (1593) was the first
to give an exact expression for n by taking n = 4 in the
above expression, giving

cos/3 = sin/3 = — = ±\/2,
v2

(9)

Pi

Pi

1357

which leads to an Infinite Product of Continued
Square Roots,

The Surface Area and Volume of the unit Sphere
are

\\\

+ ^l + Jl + y/l- (io)

5 = 4tt

(18)
(19)

(Beckmann 1989, p. 95). However, this expression was
not rigorously proved to converge until Rudio (1892).
Another exact Formula is Machin's Formula, which
is

^4tan- 1 (i)-tan- 1 (^). (11)

There are three other Machin-Like FORMULAS, as well
as other FORMULAS with more terms. An interesting
Infinite Product formula due to Euler which relates

7r and the nth Prime p n is

(12)

n°°

1 ±2=n

sin(^7rp n )
^~ Pn

it

1+ (-l)("n-l
Pn

)/2

(13)

(Blatner 1997, p. 119), plotted below as a function of
the number of terms in the product.

3.16r

3.15

3.14

3.13

1000 2000

3000

4000 5000

The AREA and CIRCUMFERENCE of the UNIT CIRCLE
are given by

Jo

1 — x 2 dx

lim — ^ \/n 2 -

k 2

(14)
(15)

and

dx.

(16)
(17)

7r is known to be Irrational (Lambert 1761, Legendre
1794) and even TRANSCENDENTAL (Lindemann 1882).
Incidentally, Lindemann's proof of the transcendence
of 7r also proved that the GEOMETRIC PROBLEM OF
Antiquity known as Circle Squaring is impossible.
A simplified, but still difficult, version of Lindemann's
proof is given by Klein (1955).

It is also known that 7r is not a LlOUVlLLE NUMBER
(Mahler 1953). In 1974, M. Mignotte showed that

<Q~

(20)

has only a finite number of solutions in INTEGERS (Le
Lionnais 1983, p. 50). This result was subsequently
improved by Chudnovsky and Chudnovsky (1984) who
showed that

7T

>q

(21)

although it is likely that the exponent can be reduced to
2 + e, where e is an infinitesimally small number (Bor-
wein et al 1989). It is not known if it is NORMAL (Wagon
1985), although the first 30 million DIGITS are very UNI-
FORMLY DISTRIBUTED (Bailey 1988). The following dis-
tribution is found for the first n DIGITS of 7T-3. It shows
no statistically SIGNIFICANT departure from a UNIFORM
Distribution (technically, in the Chi-Squared Test,
it has a value of \s 2 =5.60 for the first 5 x 10 10 terms).

digit 1 x 10 5 1 x 10° 6 x 10 9 5 x id 1

TO -

9,999
10,137

9,908
10,025

9,971
10,026
10,029
10,025

9,978

P.902

99,959

99,758

100,026

100,229

100,230

100,359

99,548

99,800

99,985

100,106

599,963,005
600,033,260
599,999,169
600,000,243
599,957,439
600,017,176
600,016,588
600,009,044
599,987,038
600,017,038

5,000,012,647
4,999,986,263
5,000,020,237
4,999,914,405
5,000,023,598
4,999,991,499
4,999,928,368
5,000,014,860
5,000,117,637
4,999,990,486

The digits of 1 1n are also very uniformly distributed
(Xs 2 = 7.04 v shown in the following table.

1358

Pi

Pi

digit

5 x 10 10

4,999,969,955

1

5,000,113,699

2

4,999,987,893

3

5,000,040,906

4

4,999,985,863

5

4,999,977,583

6

4,999,990,916

7

4,999,985,552

8

4,999,881,183

9

5,000,066,450

It is not known if 7r + e, 7r/e, or ln7r are IRRATIONAL.
However, it is known that they cannot satisfy any POLY-
NOMIAL equation of degree < 8 with INTEGER COEFFI-
CIENTS of average size 10 9 (Bailey 1988, Borwein et al.
1989).

7v satisfies the INEQUALITY

K)'

3.14097 < jr.

(22)

Beginning with any POSITIVE INTEGER n, round up to
the nearest multiple of n — 1, then up to the nearest
multiple of n — 2, and so on, up to the nearest multiple
of 1. Let f(n) denote the result. Then the ratio

lim 77~\

n— ^00 J\n)

(23)

(Brown). David (1957) credits this result to Jabotinski
and Erdos and gives the more precise asymptotic result

f(n) = — + 0(n 4/3 ).

(24)

The first few numbers in the sequence {f{n)} are 1, 2,
4, 6, 10, 12, 18, 22, 30, 34, . . . (Sloane's A002491).

A particular case of the WALLIS FORMULA gives
(2n) 2

7T

2

n

(2n- l)(2n + l)
This formula can also be written

2-2 4-4 6-6
1-3 3-5 5-7

lim

n— +oo

_*^ =irlim f yn)i'

(25)
(26)

where (™) denotes a Binomial Coefficient and T(x)
is the GAMMA FUNCTION (Knopp 1990). Euler obtained

= V 6 ( 1 + i + i + i + -)-

(27)

which follows from the special value of the RlEMANN
Zeta Function C(2) = 7r 2 /6. Similar Formulas follow

from C( 2 ™) for a *l Positive Integers n. Gregory and
Leibniz found

7T , 1 1
4 =1 -3 + 5 + -'

(28)

which is sometimes known as GREGORY'S FORMULA.
The error after the nth term of this series in GREGORY'S
FORMULA is larger than (2n) _1 so this sum converges
so slowly that 300 terms are not sufficient to calculate
7r correctly to two decimal places! However, it can be
transformed to

*=E

3 fc -l

4*

«k + l),

(29)

where ((z) is the RlEMANN ZETA FUNCTION (Vardi
1991, pp. 157-158; Flajolet and Vardi 1996), so that
the error after k terms is « (3/4) fc . Newton used

x 2 dx

(30)

/•1/4

Tr^fx/3 + 24 / y/x-
Jo

nA ( 1 1 1 1 \

+ 24 ...

Vl2 5-2 5 28 -2 7 72 • 2 9 J

(31)

3\/3
4

(Borwein et al 1989). Using Euler's Convergence Im-
provement transformation gives

tt = 1 y. (n!) 2 2" +1 = y. n\

2 2 ^ (2n + 1)! ^ (2n + 1)!!

n=0 n=0

= -IHHK<-->)))

(33)

(Beeler e£ a/. 1972, Item 120). This corresponds to plug-
ging x = l/\/2 into the Power Series for the Hyper-
geometric Function 2 Fi(a, &;c;z),

sin^s ^y^ (2s) 2 » +1 (z!) 2
VT^~^"^ 2(2i + l)!

= 2*1(1,1; §;*>. (34)

Despite the convergence improvement, series (33) con-
verges at only one bit/term. At the cost of a SQUARE
Root, Gosper has noted that x = 1/2 gives 2 bits/term,

^ 37r -2Z^72iTT)!

and x = sin(7r/10) gives almost 3.39 bits/term,

TV _ 1 ^— a

^ + 2 ~ 2 ^ 5^

(*!) 2

575^2 2 ^ 02i+i(2i + l)!'

i=0

(35)

(36)

Pi

Pi

1359

where <j) is the GOLDEN RATIO. Gosper also obtained
* = 3+ ^( 8+ 7^( 13 +I,Tlf-3

An infinite sum due to Ramanujan is

l_Y>/2ri
7r ^ V n

2n\ 42n + 5

2l2n+4

(38)

(Borwein et al. 1989). Further sums are given in Ra-
manujan (1913-14),

i _ V^ (" 1 ) n (H23 + 21460n)(2rz - l)!!(4n - 1)!!

7T

n=0

882 2n+l 3 2n( n l)3

(39)

and

oo

1 _ /- y> (1103 + 26390n)(2n - l)!!(4n - 1)!!
7T ~ V 2_^ 99 4n + 2 32"(n!) 3

(n!) 3
V8 ^ (4n)!(1103 + 26390n)

9801

E

(n!) 4 396 4 ™

(40)

(Beeler e* al 1972, Item 139; Borwein et al 1989).
Equation (40) is derived from a modular identity of or-
der 58, although a first derivation was not presented
prior to Borwein and Borwein (1987). The above series
both give

2206\/2
7T « — — -f- = 3.14159273001 . . . (41)

9801

as the first approximation and provide, respectively,
about 6 and 8 decimal places per term. Such series exist
because of the rationality of various modular invariants.
The general form of the series is

(6n)\

&« + n W ](4^

1

V Z W)

) 3 W)] T

(42)

where t is a Quadratic Form Discriminant, j(t) is
the j'-Function,

b(t) = 0[1728 _ ,-(*)]

a(t)

f{'

#4 ft)

£ 2 (i) -

iry/i

(43)

(44)

and the Ei are Ramanujan-Eisenstein Series. A
Class Number p field involves pth degree Algebraic
INTEGERS of the constants A = a(£), JS = &(£), and
C = c(t). The fastest converging series that uses only

Integer terms corresponds to the largest Class Num-
ber 1 discriminant of d = —163 and was formulated
by the Chudnovsky brothers (1987). The 163 appearing
here is the same one appearing in the fact that e 71- ^ 163
(the Ramanujan Constant) is very nearly an Inte-
ger. The series is given by

1 _ v^ (-l) n (6n)!(13591409 + 545140134n)

7T ~ ^ (n!) 3 (3n)!(640320 3 )"+V2

71=0

163-8- 27-7- 11 -19- 127

640320 3 /2

oo

13591409

163-2-9-711-19-127

+

•)

x _i^ LIT. (45)

(3n)!(n!) 3 640320 3 " K '

(Borwein and Borwein 1993). This series gives 14 digits
accurately per term. The same equation in another form
was given by the Chudnovsky brothers (1987) and is
used by Mathematica® (Wolfram Research, Champaign,
IL) to calculate ir (Vardi 1991),

426880y/lQ005

^[s-^Mgi 5' e! 1 ' 1 !- 5 ) ~~ C 3^2(5, §, ^ ;2,2;B)]

(46)

(47)
(48)
(49)

where

A = 13591409
B = -
C =

151931373056000
30285563

1651969144908540723200 *

The best formula for Class Number 2 (largest discrim-
inant —427) is

7r ^-^ (n\

(-l) n (6w)!(A + £w)

(n!) 3 (3n)!C^+ 1 /2 '

(50)

where

A = 212175710912\/61 + 1657145277365 (51)

B = 13773980892672\/6T+ 107578229802750 (52)
C = [5280(236674 + 30303\/oT] 3 (53)

(Borwein and Borwein 1993). This series adds about 25
digits for each additional term. The fastest converging
series for CLASS NUMBER 3 corresponds to d = —907
and gives 37-38 digits per term. The fastest converging
Class Number 4 series corresponds to d — —1555 and
is

v 7 ^

E

(6w)! A + nB

(3n)!(n!) 3 C 3n '

(54)

where

1360

Pi

Pi

A = 63365028312971999585426220

+ 28337702140800842046825600^

+ 384V5(108917285511711782004674 • ■ ■

• • ■ 36212395209160385656017 + 487902908657881022 ■ • •

■ • • 5077338534541688721351255040^5 ) 1/2 (55)

B = 7849910453496627210289749000

+ 3510586678260932028965606400\/5

+ 2515968^3110(62602083237890016 • • •

• • ■ 36993322654444020882161 + 2799650273060444296 • • •

• ■ • 577206890718825190235\/5 ) 1/2 (56)

C = -214772995063512240 - 96049403338648032^5
- 1296^5(10985234579463550323713318473
+ 4912746253692362754607395912^ ) 1/2 . (57)

This gives 50 digits per term. Borwein and Borwein
(1993) have developed a general ALGORITHM for gener-
ating such series for arbitrary CLASS NUMBER. Bellard
gives the exotic formula

1

740025

3P(n)

E

20379280

(58)

where
P(n) =

-885673181n 5 + 3125347237n 4

2942969225n

+1031962795n 2 - 196882274n + 10996648. (59)

A complete listing of Ramanujan's series for 1/7T found
in his second and third notebooks is given by Berndt
(1994, pp. 352-354),

4 = y, (6n + l)(f) n 3
7T £-^

16

7T

32

4"(n!) 3

~ (42n + 5)(f) n 3

(64) n (n!) 3

__ _ (42x/5 n + 5V5 + 30n - l)(f) n 3

7T ~ 2^f (64)"(n!) 3

y/E- 1

(15n + 2)(|) B (|) B (|)„ /2\«

(60)
(61)

(n!) 3

(i)'

27 _ y^

n=0

15V3_f> (33n + 4)(i)„(i) n (f) n / 4 y
2-~> (n\) 3 \125J

E

71 =

(H"+l)(a)n(|)n(|)n / 4 \ »

(n!) 3 V125/

(62)
(63)

(64)

(65)

2tt

5y^
2ttV3

85y/85 _ ^ (133n + 8)(§) n (^) n (§)„ ( 4 N"
IS^-2^ (^!)3 UJ ( 66 )

71 =

4 ^ (-l) n (20n + 3)(|) TO (l) n (|) w

(n!) 3 2

7T\/3

£

(-l) w (28n + 3)(i) w (i)»(l)n
(n!) 3 3 n 4"+!

7T ^— '

(-l) B (260n + 23)(i)„(I)„(f)„

4

7TV5

E

(n!) 3 (18) 2Tl + 1
(-l) w (644n + 41)(i) B (|) n (§) n

(n!) 3 5"(72) 2 "+ 1

I60n+1123)(|;

(n!) 3 (882) 2Tl + 1

(68)

(69)

(70)

4 _ xp (-l) m (21460n+ 1123)(i) B (l) B (§) w

n=0

2v/3 ^(8n+l)(|) n (l) n (f),

7T _ ^

(n!) 3 9 n

J_ ^(10n + l)(i)„(i) n (f)

,72 ^

n=0

oo

277^2 ^ (n!) 3 9 2 -+ 1

n=0

~ (40n + 3)(i)„(i)„(|),

3*^3 *-> (n!)»(49)»»+i

71 =

_2_ _ ^ (280n+ 19)(i)n(i)„(f )«

71-vTi

=E

71 =

(n!) 3 (99) 2n + 1

oo

(26390n+1103)(i)„(i)„(|)

2*^2 ^ (n!)3(99) 4 »+ 2

71 =

(72)
(73)
(74)
(75)
(76)

These equations were first proved by Borwein and
Borwein (1987, pp. 177-187). Borwein and Borwein
(1987b, 1988, 1993) proved other equations of this type,
and Chudnovsky and Chudnovsky (1987) found similar
equations for other transcendental constants.

A Spigot Algorithm for n is given by Rabinowitz
and Wagon (1995). Amazingly, a closed form expression
giving a digit extraction algorithm which produces digits
of 7r (or 7r 2 ) in base-16 was recently discovered by Bailey
et at. (Bailey et al 1995, Adamchik and Wagon 1997),

y (-A i i i_\ / m-

^-^\8n + l 8n + 4 Sn + 5 8n + 6/\16/ '

n=0

(77)

which can also be written using the shorthand notation

^Elefe {**} = {4,0,0,-2,-1,-1,0,0},

(78)
where {pi} is given by the periodic sequence obtained by
appending copies of {4,0,0,-2,-1,-1,0,0} (in other
words, pi = P[(i-i) (mod s)]+i for i > 8) and [^J is the
Floor Function. This expression was discovered us-
ing the PSLQ Algorithm and is equivalent to

_ f l 16y - 16

71 ~ J y 4 -2y*-h4y-

dy.

(79)

Pi

Pi 1361

A similar formula was subsequently discovered by Fergu-
son, leading to a 2-D lattice of such formulas which can
be generated by these two formulas. A related integral
is

_ _ 22 I X \

-\-x 2

dx

(80)

(Le Lionnais 1983, p. 22). F. Bellard found the more
rapidly converging digit-extraction algorithm (in HEX-
ADECIMAL)

oo

(zD!

2 6 £ * 2 10n

n=0

+

4n + 1 4n + 3 lOn + 1

+

1

lOn + 3 lOn + 5 lOn + 7 lOn + 9

(81)

More amazingly still, S. Plouffe has devised an algo-
rithm to compute the nth Digit of tt in any base in
0(n 3 (logn) 3 ) steps.

Another identity is

7r 2 = 36Li 2 (|)-36Li 2 (i)-12Li 2 (|) + 6Li 2 (^) ) (82)
where L n is the POLYLOGARITHM. (82) is equivalent to

ie = EA {*} = [1,-3,-2,-3,1,0] (83)

and

7r 2 = 12L 2 (|) + 6(ln2) 2

(84)

(Bailey et ah 1995). Furthermore

oo

w 8 2^ 64*

k=0

and

oo

144

216

72

(6fc + l) 2 (6A; + 2) 2 (6fc + 3) 2
54

+

k=0

16

16

(6A; + 4) 2 (6& + 5) 2

16

(85)

(8& + 1) 2 (8fc + 2) 2 (8A: + 3) 2
4 4 2

+

(8fc + 4) 2 (8fc-f5) 2 (8A; + 6) 2 (8k + 7) 2

(86)

(Bailey ei a/. 1995, Bailey and Plouffe).

A slew of additional identities due to Ramanujan, Cata-
lan, and Newton are given by Castellanos (1988, pp. 86-
88), including several involving sums of FIBONACCI

Numbers.

Gasfeer quotes the result

tt= ~ \ lim aJiF 2 (|;2,3;-aj 2 )l ,

(87)

where iF 2 is a Generalized Hypergeometric Func-
tion, and transforms it to

tv= lim 4xiF 2 (|;f,|;-x 2 ).

(88)

Fascinating results due to Gosper include

2n

lim TT n ^ - : . = 4 1/7r = 1.554682275 . . . (89)
n-+oo J-J- 2tan -1 z v

n-+oo J- J- 2 tan i

and

OO /

n=l V

7T 2

12e 3 =

-0.040948222 .... (90)

Gosper also gives the curious identity

3n+l/2

jlKs + O'

3-3 1/24 v^|)I

t-5/6

= 1.012378552722912.... (91)

Another curious fact is the ALMOST Integer

e^ - tt = 19.999099979 . . . , (92)

which can also be written as

(^ + 20)* = -0.9999999992 - 0.0000388927i « -1 (93)

cos(ln(7r + 20)) w -0.9999999992. (94)

Applying Co SINE a few more times gives

C0S(7T COs(7T COs(ln(7T + 20))))

« -1 + 3.9321609261 x 10'

(95)

7r may also be computed using iterative ALGORITHMS.
A quadratically converging ALGORITHM due to Borwein

is

(96)
(97)
(98)

and

xo

= V2

7T

= 2 + \/2

yi

= 2 1/4

Xn + l

_ l

" 2

l-fl

Vr

i v^+ ys^

V-r

2/n + l

In + 1

'l/n + 1

(99)

(100)
(101)

1362 Pi

7r n decreases monotonically to 7r with

7T n — 7T < 10

(102)

for n > 2. The Brent-Salamin Formula is another
quadratically converging algorithm which can be used
to calculate 7r. A quadratically convergent algorithm
for 7r/ In 2 based on an observation by Salamin is given
by defining

f(k) = k2~ k/4

x>

-'(!)

then writing

Now iterate

9o

_ f(n)

f(2n)'

9k

V 5 (^ + ^)

to obtain

7r = 2(ln2)/(n)JJff fc .

(103)

(104)

(105)

(106)

A cubically converging ALGORITHM which converges to
the nearest multiple of -k to f is the simple iteration

fn = /n-i -f sin(/ n _i)

(107)

(Beeler ei a/. 1972). For example, applying to 23 gives
the sequence

{23, 22.1537796, 21.99186453, 21.99114858, . . .}, (108)

which converges to 7tt w 21.99114858.

A quartically converging ALGORITHM is obtained by let-
ting

yo = a/2 - 1

(109)

a = 6 - 4\/2,

(110)

defining

_l-(l-y n «)^

^+ 1 -l + (1 _ J , n 4)l/4

(111)

Qn+i = (1 + 2M-i) 4 a„ - 2 2n+3 ?/ n+ i(l + y n+ i + y n+ i 2 ).

(112)
Then

7T = lim — (113)

n— J-oo CKn

and a n converges to 1/tt quartically with

Pi

(Borwein and Borwein 1987, Bailey 1988, Borwein et aL
1989). This Algorithm rests on a Modular Equa-
tion identity of order 4.

A quintically converging ALGORITHM is obtained by let-
ting

Then let

where

50 = 5(^-2)

(115)

<*o = |.

(116)

25

(117)

" n+1 ~(*+f + l) 3 s«'

(118)

1/ = (x - l) a + 7

(119)

* = [s*(!/ + \V - 4 * 3 )1 1/5 - ( 12 °)

Finally, let

a.n+1 — s n 2 a n - 5 n [|(s n 2 - 5) + \/s n (s n 2 - 2s n + 5)],

(121)
then

< a n - i < 16 • 5 n e _7r5n (122)

7T

(Borwein et al 1989). This ALGORITHM rests on a
Modular Equation identity of order 5.

Another Algorithm is due to Woon (1995). Define
a(0) = 1 and

a(n) =

\

i +

£«(*)

fc=0

(123)

It can be proved by induction that

o(n) = esc (~) . (124)

For n = 0, the identity holds. If it holds for n < t, then

a(t + 1) =

1 +

E csc (^rr)

(125)

but

esc (^L.) = cot (^L_) - cot (^) , (126)

E csc (^r)= cot (2^)-

Therefore,

l(t+l)=C8c(^j),

(127)

(128)

Pi

Pi

1363

so the identity holds for n = t + 1 and, by induction, for
all Nonnegative n, and

2 n+1
lim — —

lim 2" +1 sinf-^-)

n+1 7T sin(^ rr )

= lim 2

Tl— KX>

2n+l

2 n + !

7T lim — -r— = 7T.

(129)

Other iterative Algorithms are the Archimedes Al-
gorithm, which was derived by Pfaff in 1800, and the
Brent-Salamin Formula. Borwein et al. (1989) dis-
cuss pth order iterative algorithms.

Kochansky's Approximation is the Root of

9x 4 - 240a: 2 + 1492.

(130)

given by

.141533.

(131)

ation involving the G
* * !*' = I (4 ±i ) 2 = 1(3 + V^) = 3.14164....

An approximation involving the GOLDEN Mean is

2

Some approximations due to Ramanujan
19\/7

TV ft

16

3V^ ' 5

1 +

('+£) -(-

2222 X 1 / 4

22 2 /
^(97+|-^) 1/4 = (97 + A)V4

_ 63 /rr+jWs\

~ 25 \ 7+15^/5 /
_ 355 / 0.0003 \
~ 113 V 3533 /

12
/130

24

/142
12

In

In

(3 + v / l3)(v / 8 + y / l0)
2

\/l0 + llv / 2 + y/l0 + 7y^
2

ln[(3 + v / 10)(v / 8 + v / 10)]

In [|(3 4- v / 5)(2 + \/2 ) (5 + 2\/l0

/190
12

V61 + 20VT0]]

(132)

(133)
(134)
(135)

(136)
(137)
(138)

(139)

(140)

(141)
(142)

(143)

/522

:ln

5 + \/29

V2

(5\/29+ll\/6)

9 + 3%/e . /5 + 3\/6

(144)

which are accurate to 3, 4, 4, 8, 8, 9, 14, 15, 15, 18, 23,
31 digits, respectively (Ramanujan 1913-1914; Hardy
1952, p. 70; Berndt 1994, pp. 48-49 and 88-89).

Castellanos (1988) gives a slew of curious formulas:

7r ft (2e + e ) '
_ / 553 \ 2
~ V 311 + 17

~ (2S5\ 2

~ 1.167.'

66 3 + 86 2

55 3

(145)
(146)

(147)

(148)

(149)

1.09999901 • 1.19999911 • 1.39999931 • 1.69999961

(150)
47 3 + 20 3

30 3 X

/ 77729 \ 1/5
V 254 /

31 + ^ r 14VVS

28 4

1700 3 + 82 3 - 10 3

-6 d

69 5

93 4 + 34 4 + 17 4 + 88 V /4
75 4 )

2125 3 + 214 3 + 30 3 + 37 2
82 5

95 +

100-

1/4

(151)

(152)
(153)

(154)
(155)
(156)

(157)

which are accurate to 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11,
12, and 13 digits, respectively. An extremely accurate
approximation due to Shanks (1982) is

>/3502

ln(2«) + 7.37 X 10"

(158)

where u is the product of four simple quartic units. A
sequence of approximations due to Plouffe includes

* « 43 7 ' 23
In 2198

Ve

/13U181/1216

689

396 In (if)

f 2143 11/4
V 22 I

22

In 5280

(159)
(160)

(161)

(162)

(163)
(164)

*(B) 1/8 +J + ^(V5+l) (165)

^ 48 / 60318 \
~ 23 Vl3387/

(166)

1364

Pi

Pi

(228 +i i,) 1/41 + 2

^ ln /28102X
124 V 1277 )

276694819753963 V 18 8
226588

In 262537412640768744

+ 2

(167)
(168)
(169)
(170)

which are accurate to 4, 5, 7, 7, 8, 9, 10, 11, 11, 11, 23,
and 30 digits, respectively.

Ramanujan (1913-14) and Olds (1963) give geomet-
ric constructions for 355/113. Gardner (1966, pp. 92-
93) gives a geometric construction for 3 + 16/113 =
3.1415929.... Dixon (1991) gives constructions for

6/5(1 + 4>)
3.141533...

= 3.141640... and ^4 + (3 - tan(30°)) =
Constructions for approximations of tt are
approximations to CIRCLE SQUARING (which is itself im-
possible).

A short mnemonic for remembering the first eight DEC-
IMAL Digits of 7r is "May I have a large container of
coffee?" giving 3.1415926 (Gardner 1959; Gardner 1966,
p. 92; Eves 1990, p. 122, Davis 1993, p. 9). A more sub-
stantial mnemonic giving 15 digits (3.14159265358979)
is "How I want a drink, alcoholic of course, after the
heavy lectures involving quantum mechanics," originally
due to Sir James Jeans (Gardner 1966, p. 92; Castellanos
1988, p. 152; Eves 1990, p. 122; Davis 1993, p. 9; Blatner
1997, p. 112). A slight extension of this adds the phrase
"All of thy geometry, Herr Planck, is fairly hard," giving
24 digits in all (3.14159265358979323846264).

An even more extensive rhyming mnemonic giving 31
digits is "Now I will a rhyme construct, By chosen
words the young instruct. Cunningly devised endeav-
our, Con it and remember ever. Widths in circle here
you see, Sketched out in strange obscurity." (Note that
the British spelling of "endeavour" is required here.)

The following stanzas are the first part of a poem written
by M. Keith based on Edgar Allen Poe's "The Raven."
The entire poem gives 740 digits; the fragment below
gives only the first 80 (Blatner 1997, p. 113). Words
with ten letters represent the digit 0, and those with 11
or more digits are taken to represent two digits.

Poe, E.: Near a Raven.

Midnights so dreary, tired and weary.

Silently pondering volumes extolling all by-now obsolete

lore.
During my rather long nap-the weirdest tap!
An ominous vibrating sound disturbing my chamber's

antedoor.
'This,' I whispered quietly, 'I ignore.'

Perfectly, the intellect remembers: the ghostly fires, a
glittering ember.

Inflamed by lightning's outbursts, windows cast penum-
bras upon this floor.

Sorrowful, as one mistreated, unhappy thoughts I heed-
ed:
That inimitable lesson in elegance — Lenore —
is delighting, exciting. . . nevermore.

An extensive collection of it mnemonics in many lan-
guages is maintained by A. P. Hatzipolakis. Other
mnemonics in various languages are given by Castellanos
(1988) and Blatner (1997, pp. 112-118).

In the following, the word "digit" refers to decimal digit
after the decimal point. J. H. Conway has shown that
there is a sequence of fewer than 40 FRACTIONS Fi, F2,
. . . with the property that if you start with 2 n and re-
peatedly multiply by the first of the Fi that gives an
integral answer, then the next Power of 2 to occur will
be the 2 n th decimal digit of n.

The first occurrence of n 0s appear at digits 32, 307,
601, 13390, 17534, .... The sequence 9999998 occurs at
decimal 762 (which is sometimes called the FEYNMAN
Point). This is the largest value of any seven digits
in the first million decimals. The first time the Beast
Number 666 appears is decimal 2440. The digits 314159
appear at least six times in the first 10 million decimal
places of 7r (Pickover 1995). In the following, "digit"
means digit of n — 3. The sequence 0123456789 oc-
curs beginning at digits 17,387,594,880, 26,852,899,245,
30,243,957,439, 34,549,153,953, 41,952,536,161, and
43,289,964,000. The sequence 9876543210 occurs
beginning at digits 21,981,157,633, 29,832,636,867,
39,232,573,648, 42,140,457,481, and 43,065,796,214.
The sequence 27182818284 (the digits of e) occur be-
ginning at digit 45,111,908,393. There are also in-
teresting patterns for 1/7T. 0123456789 occurs at
6,214,876,462, 9876543210 occurs at 15,603,388,145
and 51,507,034,812, and 999999999999 occurs at
12,479,021,132 of 1/tt,

Scanning the decimal expansion of it until all n-digit
numbers have occurred, the last 1-, 2-, ... digit num-
bers appearing are 0, 68, 483, 6716, 33394, 569540, . . .
(Sloane's A032510). These end at digits 32, 606, 8555,
99849, 1369564, 14118312, ....

see also Almost Integer, Archimedes Algorithm,
Brent-Salamin Formula, Buffon-Laplace Nee-
dle Problem, Buffon's Needle Problem, Cir-
cle, Dirichlet Beta Function, Dirichlet Eta
Function, Dirichlet Lambda Function, e, Euler-
Mascheroni Constant, Gaussian Distribution,
Maclaurin Series, Machin's Formula, Machin-
Like Formulas, Relatively Prime, Riemann Zeta
Function, Sphere, Trigonometry

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Pi

Pi

1365

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1366 Pi Heptomino

Picard's Theorem

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Pi Heptomino

Piano Mover's Problem

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Given an open subset U in n-D space and two compact
subsets Co and C\ of U, where C\ is derived from Co
by a continuous motion, is it possible to move Co to C\
while remaining entirely inside Ul

see also MOVING LADDER CONSTANT, MOVING SOFA

Constant

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Points for a Two-point Body." J. Algorithms 10, 109-119,
1989.

Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/sof a/sofa. html.

Leven, D. and Sharir, M. "An Efficient and Simple Mo-
tion Planning Algorithm for a Ladder Moving in Two-
Dimensional Space Amidst Polygonal Barriers." J. Algo-
rithms 8, 192-215, 1987.

Picard's Existence Theorem

If / is a continuous function that satisfies the LlPSCHITZ
Condition

\f(x,t)-f(y f t)\<L\x-y\

in a surrounding of (xo,to) £ ft C RxR n = {(x, t) :
| a; — aso | < 6, \t — to\ < a}, then the differential equation

dx

= /(*,*)

x(to) = Xo

has a unique solution x(t) in the interval \t — to\ < d,
where d = min(a, b/B), min denotes the MINIMUM, B =
sup \f(t,x)\, and sup denotes the Supremum.

see also ORDINARY DIFFERENTIAL EQUATION

Picard's Little Theorem

Any Entire Analytic Function whose range omits
two points must be a constant.

Picard's Theorem

An Analytic Function assumes every Complex
Number, with possibly one exception, infinitely often
in any NEIGHBORHOOD of an ESSENTIAL SINGULARITY.

see also ANALYTIC FUNCTION, ESSENTIAL SINGULAR-
ITY, NEIGHBORHOOD

A Heptomino in the shape of the Greek character 7r.

Picard Variety

Pinching Theorem 1367

Picard Variety

Let V be a Variety, and write G(V) for the set of di-
visors, Gi(V) for the set of divisors linearly equivalent
to 0, and G a {V) for the group of divisors algebraically
equal to 0. Then G a (V)/Gi(V) is called the Picard va-
riety. The Albanese Variety is dual to the Picard
variety.

see also Albanese Variety

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 75, 1980,

Pick's Formula

see Pick's Theorem

Pick's Theorem

Let A be the AREA of a simply closed POLYGON whose
Vertices are lattice points. Let B denote the number
of Lattice Points on the Edges and I the number of
points in the interior of the POLYGON. Then

A = I+\B-1.

The Formula has been generalized to 3-D and higher
dimensions using Ehrhart Polynomials.

see also Blichfeldt's Theorem, Ehrhart Poly-
nomial, Lattice Point, Minkowski Convex Body
Theorem

References

Diaz, R. and Robins, S. "Pick's Formula via the Weierstrafi
p-Function." Amer. Math. Monthly 102, 431-437, 1995.

Ewald, G. Combinatorial Convexity and Algebraic Geome-
try. New York: Springer- Verlag, 1996.

Hammer, J. Unsolved Problems Concerning Lattice Points.
London: Pitman, 1977.

Morelli, R. "Pick's Theorem and the Todd Class of a Toric
Variety." Adv. Math. 100, 183-231, 1993.

Pick, G. "Geometrisches zur Zahlentheorie." Sitzenber. Lotos
(Prague) 19, 311-319, 1899.

Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 97-98, 1983.

Picone's Theorem

Let f(x) be integrable in [—1,1], let (1 - x 2 )f(x) be of
bounded variation in [—1,1], let M' denote the least up-
per bound of \f(x)(l - x 2 )\ in [-1, 1], and let V' denote
the total variation of f(x)(l — x 2 ) in [—1, 1], Given the
function

F(z) = F(-l)+ I f(x)dx,

where P n (x) is a Legendre Polynomial, satisfy the
inequalities

/"

then the terms of its LEGENDRE SERIES

F (x) ~y^a n P n (x)

n=0

2„ = §(2ra + l) / F(x)P n (x)dx,

fo /2~ M' + V' „-3/2

[2(M' + V')n- 1

-~~ m for |a?| < J < 1
2(M' + V , )n~ 1 for \x\ < 1

for n> 1 (Sansone 1991).

see also Jackson's Theorem, Legendre Series

References

Picone, M. Appunti di Analise Superiore. Naples, Italy,,

p. 260, 1940.
Sansone, G. Orthogonal Functions, rev. English ed. New

York: Dover, pp. 203-205, 1991.

Pie Cutting

see Circle Cutting, Cylinder Cutting, Pancake
Theorem, Pizza Theorem

Piecewise Circular Curve

A curve composed exclusively of circular ARCS, e.g., the
Flower of Life, Lens, Reuleaux Triangle, Seed
of Life, and Yin- Yang.

see also Arc, Reuleaux Triangle, Yin- Yang
Flower of Life, Lens, Reuleaux Polygon,
Reuleaux Triangle, Salinon, Seed of Life, Tri-
angle Arcs, Yin- Yang

References

BanchofT, T. and Giblin, P. "On The Geometry Of Piecewise

Circular Curves." Amer. Math. Monthly 101, 403-416,

1994.

Pigeonhole Principle

see Dirichlet's Box Principle

Pillai's Conjecture

For every k > 1, there exist only finite many pairs of
Powers (p,p) with p and p' Prime and k = p' - p.

References

Ribenboim, P. "Catalan's Conjecture." Amer. Math.
Monthly 103, 529-538, 1996.

Pilot Vector

see Vector Spherical Harmonic

Pinch Point

A singular point such that every NEIGHBORHOOD of the
point intersects itself. Pinch points are also called Whit-
ney singularities or branch points.

Pinching Theorem

Let g{x) < f(x) < h(x) for all x in some open interval
containing a. If

lim g(x) = lim h(x) = L,

then limAx-^a f(x) = L.

1368

Pine Cone Number

Pisot-Vijayaraghavan Constants

Pine Cone Number
see Fibonacci Number

Piriform

A plane curve also called the Peg Top and given by the
Cartesian equation

ay 2 = b 2 x 3 (2a-x)

and the parametric curves

x — a(l + sini)

y = bcost(l + sint)

(i)

(2)
(3)

for t G [— 7r/2, 7r/2]. It was studied by G. de Longchamps
in 1886. The generalization to a Quartic 3-D surface

{x 4 -x 3 ) + y 2 +z 2 =0,

(4)

is shown below (Nordstrand).

See also BUTTERFLY CURVE, DUMBBELL CURVE, EIGHT

Curve, Heart Surface, Pear Curve

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.

Stradbroke, England: Tarquin Pub. p. 71, 1989.
Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 148-150, 1972.
Nordstrand, T. "Surfaces." http : //www , uib . no/people/

nfytn/ surf aces .htm.

Pisot-Vijayaraghavan Constants

Let 6 be a number greater than 1, A a POSITIVE number,
and

(x)=x- [x\ (1)

denote the fractional part of x. Then for a given A, the
sequence of numbers (X0 n ) for n = 1, 2, . . . is uniformly
distributed in the interval (0, 1) when does not be-
long to a A-dependent exceptional set S of MEASURE
zero (Koksma 1935). Pisot (1938) and Vijayaraghavan
(1941) independently studied the exceptional values of
#, and Salem (1943) proposed calling such values Pisot-
Vijayaraghavan numbers.

Pisot (1938) proved that if 8 is such that there exists
a A ^ such that the series y^^L sin 2 (7rAff) n con-
verges, then is an ALGEBRAIC INTEGER whose conju-
gates all (except for itself) have modulus < 1, and A is
an algebraic Integer of the Field K(0). Vijayaragha-
van (1940) proved that the set of Pisot-Vijayaraghavan
numbers has infinitely many limit points. Salem (1944)
proved that the set of Pisot-Vijayaraghavan constants is
closed. The proof of this theorem is based on the LEMMA
that for a Pisot-Vijayaraghavan constant 0, there always
exists a number A such that 1 < A < 6 and the following
inequality is satisfied,

^sin 2 (7rAO<

n=0

ir 2 (20+l) 2

(2)

The smallest Pisot-Vijayaraghavan constant is given by
the Positive Root o of

•a- 1 = 0.

(3)

This number was identified as the smallest known by
Salem (1944), and proved to be the smallest possible by
Siegel (1944). Siegel also identified the next smallest
Pisot-Vijayaraghavan constant 6\ as the root of

1 = 0,

(4)

showed that B\ and 02 are isolated in S, and showed that
the roots of each POLYNOMIAL

x n (x — x — 1) + x — 1 n = 1, 2, 3, . .

r n+l

- 1

c 2 -1

x —

-1

x-l

n = 3, 5,7, , . .

n = 3, 5, 7, .

(5)

(6)

(7)

belong to 5, where 9 = <f> (the GOLDEN Mean) is the
accumulation point of the set (in fact, the smallest; Le
Lionnais 1983, p. 40). Some small Pisot-Vijayaraghavan
constants and their POLYNOMIALS are given in the fol-
lowing table. The latter two entries are from Boyd
(1977).

k Number

Order

Polynomial

1.3247179572

3

10-1-1

1 1.3802775691

4

1-10 0-1

1.6216584885

16

1-22-32-21001
-12-22-21 -1

1.8374664495

20

1-201-101-10
10-101-101-1
1 -1

All the points in S less than <j> are known (Dufresnoy
and Pisot 1955). Each point of S is a limit point from
both sides of the set T of Salem CONSTANTS (Salem
1945).

Pistol

Place (Field) 1369

see also Salem Constants

References

Boyd, D. W. "Small Salem Numbers." Duke Math. J. 44,

315-328, 1977.
Dufresnoy, J. and Pisot, C. "Etude de certaines fonctions

meromorphes bornees sur le cercle unite, application a un

ensemble ferme d'entiers algebriques." Ann. Sci. Ecole

Norm. Sup. 72, 69-92, 1955.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

pp. 38 and 148, 1983.
Koksma, J. F. "Ein mengentheoretischer Satz iiber die

Gleichverteilung modulo Ems." Comp. Math. 2, 250-258,

1935.
Pisot, C. "La repartition modulo 1 et les nornbres alge-
briques." Annali di Pisa 7, 205-248, 1938.
Salem, R. "Sets of Uniqueness and Sets of Multiplicity."

Trans, Amer. Math. Soc. 54, 218-228, 1943.
Salem, R. "A Remarkable Class of Algebraic Numbers. Proof

of a Conjecture of Vijayaraghavan." Duke Math. J. 11,

103-108, 1944.
Salem, R. "Power Series with Integral Coefficients." Duke

Math. J. 12, 153-172, 1945.
Siegel, C. L. "Algebraic Numbers whose Conjugates Lie in

the Unit Circle." Duke Math. J. 11, 597-602, 1944.
Vijayaraghavan, T. "On the Fractional Parts of the Powers

of a Number, II." Proc. Cambridge Phil. Soc. 37, 349-357,

1941.

Pistol

A 4-POLYHEX.

References

Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, p. 147, 1978.

Pitchfork Bifurcation

Let /:lxl-4lbea one-parameter family of C 3
map satisfying

called a pitchfork bifurcation. An example of an equa-
tion displaying a pitchfork bifurcation is

x = fix — x (6)

(Guckenheimer and Holmes 1997, p. 145).

see also BIFURCATION

References

Guckenheimer, J. and Holmes, P. Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields, 3rd
ed. New York: Springer- Verlag, pp. 145 and 149-150, 1997.

Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, p. 31, 1990.

Pivot Theorem

If the Vertices A, B, and C of Triangle AABC lie
on sides QR, RP, and PQ of the TRIANGLE APQR,
then the three Circles CBP, ACQ, and BAR have a
common point. In extended form, it is MlQUEL's THE-
OREM.

see also MlQUEL's THEOREM

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.

New York: Random House, pp. 61-62, 1967.
Forder, H. G. Geometry. London: Hutchinson, p. 17, 1960.

Pivoting

The element in the diagonal of a matrix by which other
elements are divided in an algorithm such as GAUSS-
Jordan Elimination is called the pivot element. Par-
tial pivoting is the interchanging of rows and full piv-
oting is the interchanging of both rows and columns in
order to place a particularly "good" element in the di-
agonal position prior to a particular operation.

see also Gauss-Jordan Elimination

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 29-30, 1992.

(i)

(2)
(3)

(4)
(5)

Then there are intervals having a single stable fixed
point and three fixed points (two of which are stable
and one of which is unstable). This BIFURCATION is

/(-*,

M) =

-/(»

'./*)

■ar

.dx\ f

= l

i=0,x=0

■or

dx\ i

i,X

■ar

,dx.

ti=Q,x=iJ l

' d 2 f

dxdf.

>0

'Jo.o

~d z f

/^ = 0,T

<

=0

0.

Pizza Theorem

If a circular pizza is divided into 8, 12, 16, . . .slices by
making cuts at equal angles from an arbitrary point,
then the sums of the areas of alternate slices are equal.

Place (Digit)

see Digit

Place (Field)

A place v of a number FIELD k is an ISOMORPHISM class
of field maps k onto a dense subfield of a nondiscrete
locally compact Field k u .

In the function field case, let Fbea function field of al-
gebraic functions of one variable over a FIELD K. Then
by a place in F, we mean a subset p of F which is the
Ideal of nonunits of some Valuation RING O over K.

1370 Place (Game)

Planar Space

References

Chevalley, C. Introduction to the Theory of Algebraic Func-
tions of One Variable. Providence, RI: Amer. Math. Soc.,
p. 2, 1951.

Knapp, A. W, "Group Representations and Harmonic Anal-
ysis, Part II." Not Amer. Math. Soc. 43, 537-549, 1996.

Place (Game)

For n players, n — 1 games are needed to fairly determine
first place, and n — 1 -f lg(n — 1) are needed to fairly
determine first and second place.

Planar Bubble Problem

see Bubble

Planar Distance

For n points in the PLANE, there are at least

Ni = yf^\-

different DISTANCES. The minimum DISTANCE can oc-
cur only < 3n - 6 times, and the MAXIMUM DISTANCE
can occur < n times. Furthermore, no DISTANCE can
occur as often as

N 2

,3/2

1 /„ /t; ~\ Tl ' Tl

i„(i + V5rr7)<__ i

times. No set of n > 6 points in the PLANE can deter-
mine only ISOSCELES TRIANGLES.

see also Distance

References

Honsberger, R. "The Set of Distances Determined by n Points
in the Plane." Ch. 12 in Mathematical Gems II. Washing-
ton, DC: Math. Assoc. Amer., pp. 111-135, 1976.

Planar Graph

A GRAPH is planar if it can be drawn in a PLANE
without Edges crossing (i.e., it has Crossing Num-
ber 0). Only planar graphs have DUALS. If G is pla-
nar, then G has VERTEX DEGREE < 5. COMPLETE
GRAPHS are planar only for n < 4. The complete BI-
PARTITE Graph 7^(3,3) in nonplanar. More generally,
Kuratowski proved in 1930 that a graph is planar Iff it
does not contain within it any graph which can be CON-
TRACTED to the pentagonal graph K(5) or the hexago-
nal graph K(3j 3). K$ can be decomposed into a union of
two planar graphs, giving it a "Depth" of E(K$) = 2.
Simple CRITERIA for determining the depth of graphs
are not known. Beineke and Harary (1964, 1965) have
shown that if n ^ 4 (mod 6), then

E(tf n )=|_i(n + 7)J.

The Depths of the graphs K n for n = 4, 10, 22, 28, 34,
and 40 are 1, 3, 4, 5, 6, and 7 (Meyer 1970).

see also Complete Graph, Fabry Imbedding, Inte-
gral Drawing, Planar Straight Line Graph

References

Beineke, L. W. and Harary, F. "On the Thickness of the
Complete Graph." Bull. Amer. Math. Soc. 70, 618-620,
1964.

Beineke, L. W, and Harary, F. "The Thickness of the Com-
plete Graph." Canad. J. Math. 17, 850-859, 1965.

Booth, K. S. and Lueker, G. S. "Testing for the Consecu-
tive Ones Property, Interval Graphs, and Graph Planarity
using PQ-Tree Algorithms." J. Comput System Sci. 13,
335-379, 1976.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 56, 1983.

Meyer, J. "L'epaisseur des graphes completes K 3 4 et K$o"
J. Comp. Th. 9, 1970.

Planar Point

A point p on a REGULAR SURFACE M £ R 3 is said to
be planar if the GAUSSIAN CURVATURE K(p) = and
5(p) = o (where S is the Shape Operator), or equiv-
alently, both of the PRINCIPAL CURVATURES m and k 2
are 0.

see also Anticlastic, Elliptic Point, Gaussian
Curvature, Hyperbolic Point, Parabolic Point,
Synclastic

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 280, 1993.

Planar Space

Let (£1,^2) De a locally EUCLIDEAN coordinate system.
Then

ds — d^\ + d£ 2

2

(i)

Now plug in

d£i = -r — dx\ + — — dx2
OX\ 0x2

(2)

At d & A ^ d ^A

"s2 = q — dx\ + - — ax 2
dx± ox 2

(3)

to obtain

ds —

V&ri/ \dxiJ

dx\ 2

+ 2 r^6^ L+ a6 5|2i dxidX2

L9a;i dxi dxi 8x2!

(£)**(£)V-

(4)

Reading off the COEFFICIENTS from

= gudxi -\- 2gi2 dx\ dx2 + g22 (dx2) (5)

gives

ffu = (ir) + (Sr)

dxi dx

\dx 2 )

gi2 =

522 =

dJ2 d£ 2

dxi 3X2

2

+ (£)

(6)
(7)
(8)

Planar Straight Line Graph

Making a change of coordinates (x x , x 2 ) — > (a>i, x 2 ) gives

9n

36

dx\

dx[

5a?i 5^1 dx2 dx^

+ 56 5a;i | <9&<9x 2
dxi 5^1 5^2 5x' x

5xi\ 5xi 5x 2 ,

gn ^d7j +2 ^ 12 5xT5xT +P22

5X2

5x|

, _ 56 5xi 56 5x 2 56 5xi 5£ 2 5x 2
Pl2 ~ 5xi 5xi 5x 2 5x 2 5xi dx[ 5x 2 5x 2

512

5xi 5x2
dx[ 5x 2

522 = 511

5xi
5x1

, 5xi 5x 2 , / 5x 2
+ 2 ^ 12 54 5xl +522 ^5xl

(9)

(10)

(11)

Planar Straight Line Graph

A PLANAR GRAPH in which only straight line segments
are used to connect the Vertices, where the Edges
may intersect.
see also Planar Graph

Plancherel's Theorem

/OO poo

f(x)g t (x)dx = / F(s)G'(s)ds,
■ oo J — oo

where F(s) = T[f{x)] and T denotes a FOURIER
Transform. If / and g are real

f

J — c

f(x)g(-x)dx

-f

J — c

F(s)G(s)ds.

see also FOURIER TRANSFORM, PARSEVAL'S THEOREM

Planck's Radiation Function

Plane 1371

It has a Maximum at x « 0.201405, where

5x-e^(5x-l) =
; [X) ~ ^(eV- - 1)*

and inflection points at x « 0.11842 and a; « 0.283757,
where

e 1/x (l + e 1/a; ) + 6x(e l/x - l)[e 1/g (5x - 2) - 5x] _ Q
~ ( e i/x _ 1)33.9

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Planck's Radia-
tion Function." §27.2 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, p. 999, 1972.

Plane

A plane is a 2-D SURFACE spanned by two linearly in-
dependent vectors. The generalization of the plane to
higher DIMENSIONS is called a HYPERPLANE.

In intercept form, a plane passing through the points
(a, 0,0), (0,6,0) and (0,0, c) is given by

x y z

- + f + - = 1.

a b c

(1)

The equation of a plane PERPENDICULAR to the NON-
ZERO VECTOR n = (a, 6, c) through the point (xo, j/o, zq)
is

= a(x-xo) + b{y-y Q )-\-c(z-z ) = 0,

(2)
ax + by + cz + d = 0, (3)

"a"

"x — Xo"

b

2/ -2/o

_c_

. 2 — Z _

where

d = —axo — byo — czq.

(4)

The function

/(*)

x 5 (e

l/x _

1)'

1372

Plane

Plane Cutting

A plane specified in this form therefore has x-, y-, and
z-intercepts at

(5)
(6)

(7)

x =

d
a

y =

d

b

z =

d
c

and lies at

a Distance

h =

\d\

Va 2

+ 6 2 + c 2

(8)

from the ORIGIN.

The plane through Pi and parallel to (ai,6i,ci) and
(a 2 ,&2,c 2 ) is

x - xi j/ - yi 2-2i
ai bi ci

a 2 &2 C2

= 0.

(9)

The plane through points Pi and P 2 parallel to direction
(a t bjc) is

x-xi y - 2/1 2-2i

£2 - Xi J/2 - 2/1 ^2 - 2i

a b c

■0.

(10)

The three-point form is

x y 2 1

Sl 2/1 2i 1

X 2 2/2 2 2 1

XS 2/3 2 3 1

x - xi y - 2/1 2 - 2i

£2 - Xl 2/2 - 2/1 Z 2 - 21
£3 "El 2/3 - 2/1 2 3 - 2i

The Distance from a point (2:1,2/1, 21) to a plane
Ax + By + Cz + D —

rf= Agi + Byi + Czi+D

The Dihedral Angle between the planes

A1X + P12/ + C12 + D1 =0
A 2 x + B 2 y + C 2 z + D 2 =0

= 0.

(11)

(12)
(13)

(14)
(15)

COS0 =

A x Ai + BiB 2 + C1C2

v/Ai a + Bx 2 + CV^ 1 + B 2 2 + C 2

(16)

In order to specify the relative distances of n > 1 points
in the plane, 1 4- 2(n — 2) = 2n — 3 coordinates are
needed, since the first can always be placed at (0, 0)
and the second at (x,0), where it defines the e-Axis.

The remaining n — 2 points need two coordinates each.
However, the total number of distances is

nC * = $ = w£ : W = * n{n ~ 1) '

(17)

where (™) is a BINOMIAL COEFFICIENT, so the distances
between points are subject to m relationships, where

m = \n{n - 1) - (2n - 3) = \{n - 2)(n - 3). (18)

For n — 2 and n — 3, there are no relationships. How-
ever, for a Quadrilateral (with n = 4), there is one
(Weinberg 1972).

It is impossible to pick random variables which are uni-
formly distributed in the plane (Eisenberg and Sullivan
1996). In 4-D, it is possible for four planes to intersect in
exactly one point. For every set of n points in the plane,
there exists a point O in the plane having the property
such that every straight line through O has at least 1/3
of the points on each side of it (Honsberger 1985).

Every RIGID motion of the plane is one of the following
types (Singer 1995):

1. Rotation about a fixed point P.

2. Translation in the direction of a line /.

3. Reflection across a line I.

4. Glide-reflections along a line I.

Every Rigid motion of the hyperbolic plane is one of
the previous types or a

5. Horocycle rotation.

see also Argand Plane, Complex Plane, Dihedral
Angle, Elliptic Plane, Fano Plane, Hyperplane,
Moufang Plane, Nirenberg's Conjecture, Nor-
mal Section, Point-Plane Distance, Projective
Plane

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 208-209, 1987,

Eisenberg, B. and Sullivan, R. "Random Triangles n Dimen-
sions." Amer. Math, Monthly 103, 308-318, 1996.

Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 189-191, 1985.

Singer, D. A. "Isometries of the Plane." Amer. Math.
Monthly 102, 628-631, 1995.

Weinberg, S. Gravitation and Cosmology: Principles and
Applications of the General Theory of Relativity. New-
York: Wiley, p. 7, 1972.

Plane Curve

see Curve

Plane Cutting

see Circle Cutting

Plane Division

Plane Partition

1373

Plane Division

Consider n intersecting CIRCLES and ELLIPSES. The
maximal number of regions in which these divide the
Plane are

iVcircle = n — n + 2
iVeiiipse = 2n 2 - 2n + 2.

see also Arrangement, Circle, Cutting, Ellipse,
Space Division

Plane-Filling Curve

see Plane-Filling Function

Plane-Filling Function

A Space-Filling Function which maps a 1-D Inter-
val into a 2-D area. Plane-filling functions were thought
to be impossible until Hilbert discovered the HlLBERT
Curve in 1891.

Plane-filling functions are often (imprecisely) defined to
be the "limit" of an infinite sequence of specified curves
which "fill" the Plane without "HOLES," hence the
more popular term PLANE-FILLING Curve. The term
"plane-filling function" is preferable to "PLANE-FILLING
Curve" because "curve" informally connotes "Graph"
(i.e., range) of some continuous function, but the GRAPH
of a plane- filling function is a solid patch of 2-space with
no evidence of the order in which it was traced (and, for
a dense set, retraced). Actually, all that is needed to
rigorously define a plane-filling function is an arbitrar-
ily refinable correspondence between contiguous subin-
tervals of the domain and contiguous subareas of the
range.

True plane-filling functions are not One-to-One. In
fact, because they map closed intervals onto closed ar-
eas, they cannot help but overfill, revisiting at least
twice a dense subset of the filled area. Thus, every point
in the filled area has at least one inverse image.

see also Hilbert Curve, Peano Curve, Peano-
Gosper Curve, Sierpinski Curve, Space-Filling
Function, Space-Filling Polyhedron

References

Bogomolny, A. "Plane Filling Curves." http://www.cut-

the-knot . com/do_you_know/hilbert .html.
Wagon, S. "A Space- Filling Curve." §6.3 in Mathematica in

Action. New York: W. H. Freeman, pp. 196-209, 1991.

Plane Geometry

That portion of GEOMETRY dealing with figures in a
Plane, as opposed to Solid Geometry. Plane geom-
etry deals with the CIRCLE, LINE, POLYGON, etc.

see also CONSTRUCTIBLE POLYGON, GEOMETRIC CON-
STRUCTION, Geometry, Solid Geometry, Spheri-
cal Geometry

References

Altshiller-Court, N. College Geometry: A Second Course in
Plane Geometry for Colleges and Normal Schools, 2nd ed. f
rev. enl. New York: Barnes and Noble, 1952,

Casey, J. A Treatise on the Analytical Geometry of the Point,
Line, Circle, and Conic Sections, Containing an Account
of Its Most Recent Extensions with Numerous Examples,
2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., 1967.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, 1969.

Dixon, R. Mathographics. New York: Dover, 1991.

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.
London: Hodgson, 1913.

Heath, T. L. The Thirteen Books of the Elements, 2nd ed.,
Vol. 1: Books I and II. New York: Dover, 1956.

Heath, T. L. The Thirteen Books of the Elements, 2nd ed.,
Vol. 2: Books III-IX. New York: Dover, 1956.

Heath, T. L. The Thirteen Books of the Elements, 2nd ed.,
Vol. 3: Books X-XIII. New York: Dover, 1956.

Hilbert, D. The Foundations of Geometry. Chicago, IL:
Open Court, 1980.

Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina-
tion. New York: Chelsea, 1952.

Honsberger, R. Episodes in Nineteenth and Twentieth Cen-
tury Euclidean Geometry. Washington, DC: Math. Assoc.
Amer., 1995.

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.

Kimberling, C. "Triangle Centers and Central Triangles."
Congr. Numer. 129, 1-295, 1998.

Klee, V. "Some Unsolved Problems in Plane Geometry."
Math. Mag. 52, 131-145, 1979.

Klee, V. and Wagon, S. Old and New Unsolved Problems in
Plane Geometry and Number Theory, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., 1991.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., 1995.

Plane Partition

A two-dimensional array of INTEGERS nonincreasing
both left to right and top to bottom which add up to a
given number, i.e., riij > n^j + i) and nij > n^ i+ i)j. For
example, a planar partition of 2 is given by

5 4 2 11
3 2
2 2.

The Generating Function for the number PL(n) of
planar partitions of n is

S^^- ICd'— )'

= 1 + x + Sx 2 + 6a; 3 + 13z 4 + 24a? 5 + . . .

1374 Plane Symmetry Groups

Plateau's Problem

(Sloane's A000219, MacMahon 1912b, Beeler et at. 1972,
Bender and Knuth 1972). The concept of planar parti-
tions can also be generalized to cubic partitions.

see also Partition, Solid Partition

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 18 in
HAKMEM. Cambridge, MA: MIT Artificial Intelligence
Laboratory, Memo AIM-239, Feb. 1972.

Bender, E. A. and Knuth, D. E. "Enumeration of Plane Par-
titions." J. Combin. Theory Ser. A. 13, 40-54, 1972.

Knuth, D. E. "A Note on Solid Partitions." Math. Comput.
24, 955-961, 1970.

MacMahon, P. A. "Memoir on the Theory of the Partitions of
Numbers. V: Partitions in Two-Dimensional Space." Phil.
Trans. Roy. Soc. London Ser. A 211, 75-110, 1912a.

MacMahon, P. A. "Memoir on the Theory of the Partitions
of Numbers. VI: Partitions in Two-Dimensional Space, to
which is Added an Adumbration of the Theory of Parti-
tions in Three-Dimensional Space." Phil. Trans. Roy. Soc.
London Ser. A 211, 345-373, 1912b.

MacMahon, P. A. Combinatory Analysis, Vol. 2. New York:
Chelsea, 1960.

Sloane, N. J. A. Sequence A000219/M2566 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Plane Symmetry Groups

see Wallpaper Groups

Plateau Curves

A curve studied by the Belgian physicist and mathe-
matician Joseph Plateau. It has Cartesian equation

_ asin[(m -f n)t]
sin[(m — n)t]
_ 2asm(mt) sin(nt)
sin[(m — n)t]

If m = 2n, the Plateau curve degenerates to a CIRCLE
with center (1,0) and radius 2.

References

MacTutor History of Mathematics Archive. "Plateau

Curves." http: // www - groups . dcs . st - and .ac.uk/

"history/Curves/Plateau. html.

Planted Planar Tree

A planted plane tree (V,E } v,a) is defined as a vertex
set V, edges set E y ROOT v, and order relation a on V
which satisfies

1. For x,y G V if p(x) < p(y), then xay, where p(x) is
the length of the path from v to #,

2. If {r,s}, {x,y} e E, p(r) = p(x) = p(a)-l = p(y)-l
and rax, then say

(Klarner 1969, Chorneyko and Mohanty 1975). The
Catalan Numbers give the number of planar trivalent
planted trees.

see also Catalan Number, Tree

References

Chorneyko, I. Z. and Mohanty, S. G. "On the Enumeration

of Certain Sets of Planted Plane Trees." J. Combin. Th.

Ser. B 18, 209-221, 1975.
Harary, F.; Prins, G.; and Tutte, W. T. "The Number of

Plane Trees." Indag. Math. 26, 319-327, 1964.
Klarner, D. A. "A Correspondence Between Sets of Trees."

Indag. Math. 31, 292-296, 1969.

Plastic Constant

The limiting ratio of the successive terms of the PADO-
van Sequence, P = 1.32471795 . . ..

see also Padovan Sequence

References

Stewart, I. "Tales of a Neglected Number."
102-103, Jun. 1996.

Sci. Amer. 274.

Plat

A Braid in which strands are intertwined in the center
and are free in "handles" on either side of the diagram.

Plateau's Laws

Bubbles can meet only at Angles of 120° (for two
Bubbles) and 109.5° (for three Bubbles), where the
exact value of 109.5° is the TETRAHEDRAL ANGLE. This
was proved by Jean Taylor using MEASURE THEORY
to study Area minimization. The DOUBLE BUBBLE is
Area minimizing, but it is not known the triple BUBBLE
is also Area minimizing. It is also unknown if empty
chambers trapped inside can minimize Area for n > 3
Bubbles.

see also BUBBLE, CALCULUS OF VARIATIONS, DOUBLE

Bubble, Plateau's Problem

References

Morgan, F. "Mathematicians, including Undergraduates,
Look at Soap Bubbles." Amer. Math. Monthly 101, 343-
351, 1994.

Taylor, J. E. "The Structure of Singularities in Soap-Bubble-
Like and Soap-Film-Like Minimal Surfaces." Ann. Math.
103, 489-539, 1976.

Plateau's Problem

The problem in Calculus of Variations to find the
MINIMAL SURFACE of a boundary with specified con-
straints. In general, there may be one, multiple, or no
Minimal Surfaces spanning a given closed curve in
space.

see also CALCULUS OF VARIATIONS, MINIMAL SURFACE

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.

Stradbroke, England: Tarquin Pub., pp. 48-49, 1989.
Stuwe, M. Plateau's Problem and the Calculus of Variations.

Princeton, NJ: Princeton University Press, 1989.

Plato's Number

Platonic Solid

1375

Plato's Number

A number appearing in The Republic which involves 216
and 12,960,000.

References

Plato. The Republic. New York: Oxford University Press,
1994.

Wells, D. G. The Penguin Dictionary of Curious and Inter-
esting Numbers. London: Penguin, p. 144, 1986.

Platonic Solid

';■■■■■■!

A solid with equivalent faces composed of congruent reg-
ular convex POLYGONS. There are exactly five such
solids: the CUBE, DODECAHEDRON, ICOSAHEDRON,

Octahedron, and Tetrahedron, as was proved by
Euclid in the last proposition of the Elements.

The Platonic solids were known to the ancient Greeks,
and were described by Plato in his Timaeus ca. 350 BC.
In this work, Plato equated the TETRAHEDRON with the
"element" fire, the CUBE with earth, the ICOSAHEDRON
with water, the Octahedron with air, and the Dodec-
ahedron with the stuff of which the constellations and
heavens were made (Cromwell 1997).

The Platonic solids are sometimes also known as the
Regular Polyhedra of Cosmic Figures (Cromwell
1997), although the former term is sometimes used to re-
fer collectively to both the Platonic solids and Kepler-
Poinsot Solids (Coxeter 1973).

If P is a Polyhedron with congruent (convex) regular
polygonal faces, then Cromwell (1997, pp. 77-78) shows
that the following statements are equivalent.

1. The vertices of P all lie on a Sphere.

2. All the Dihedral Angles are equal.

3. All the Vertex Figures are Regular Polygons.

4. All the Solid Angles are equivalent.

5. All the vertices are surrounded by the same number
of Faces.

Let v (sometimes denoted No) be the number of VER-
TICES, e (or Ni) the number of EDGES, and / (or N2)
the number of FACES. The following table gives the
Schlafli Symbol, Wythoff Symbol, and C&R sym-
bol, the number of vertices v, edges e, and faces /, and
the POINT GROUPS for the Platonic solids (Wenninger
1989).

Solid

Schlafli

Wyth.

C&R

V

e

/

Grp

cube

{4,3}

3 | 2 4

43

8

12

6

o h

dodecahedron

{5,3}

3 | 25

5 3

20

30

12

h

icosahedron

{3,5}

5 | 23

3 5

12

30

20

h

octahedron

{3,4}

4 2 3

3 4

6

12

8

o h

tetrahedron

{3,3}

3 | 23

3 3

4

6

4

T d

Let r be the Inradius, p the Midradius, and R the
CIRCUMRADIUS. The following two tables give the ana-
lytic and numerical values of these distances for Platonic
solids with unit side length.

Solid

T

P

R

cube

dodecahedron

icosahedron
octahedron
tetrahedron

1
2

|V2

1
2

§v/3
i(v^5 + V3)
1 -^10 + 2-^5

.JV5

£ 1/25O + lW5

Solid

r

P

R

cube

0.5

0.70711

0.86603

dodecahedron

1.11352

1.30902

1.40126

icosahedron

0.75576

0.80902

0.95106

octahedron

0.40825

0.5

0.70711

tetrahedron

0.20412

0.35355

0.61237

Finally, let A be the Area of a single FACE, V be the
VOLUME of the solid, the EDGES be of unit length on
a side, and a be the Dihedral Angle. The following
table summarizes these quantities for the Platonic solids.

Solid

A

V

a

cube

dodecahedron
icosahedron
octahedron
tetrahedron

1

1

j(i5 + rVS)

£(3 + x/5)
AV2

COB-M-IVS)

cos-M-IVS)

cos-^-i)

cos-HI)

Iv^s + ioVs

The number of Edges meeting at a Vertex is 2e/v.
The Schlafli Symbol can be used to specify a Platonic
solid. For the solid whose faces are p-gons (denoted {p}),
with q touching at each VERTEX, the symbol is {p,q}.
Given p and <?, the number of VERTICES, EDGES, and
faces are given by

No-
Ni =

N 2 =

4p

4-(p-2)(?-2)

2pq
4-(p-2)(qr-2)

4g
4-(p-2)(*-2)'

Minimal Surfaces for Platonic solid frames are illus-
trated in Isenberg (1992, pp. 82-83).
see also Archimedean Solid, Catalan Solid, John-
son Solid, Kepler-Poinsot Solid, Quasiregular
Polyhedron, Uniform Polyhedron

1376 Platykurtic

Plouffe's Constant

References

Artmann, B. "Symmetry Through the Ages: Highlights from
the History of Regular Polyhedra." In In Eves' Circles
(Ed. J. M. Anthony). Washington, DC: Math. Assoc.
Amer., pp. 139-148, 1994.

Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in
Mathematical Recreations and Essays, 13th ed. New York:
Dover, pp. 131-136, 1987.

Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H. (Eds.).
Fundamentals of Mathematics, Vol. 2. Cambridge, MA:
MIT Press, p. 272, 1974.

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, pp. 128-129, 1987.

Bogomolny, A. "Regular Polyhedra." http://www.cut— the-
knot . com/do_you-know/polyhedra.html.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, pp. 1-17, 93, and 107-112, 1973.

Critchlow, K. Order in Space: A Design Source Book. New
York: Viking Press, 1970.

Cromwell, P. R. Polyhedra. New York: Cambridge University
Press, pp. 51-57, 66-70, and 77-78, 1997.

Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 78-81, 1990.

Gardner, M. "The Five Platonic Solids." Ch. 1 in The Second
Scientific American Book of Mathematical Puzzles & Di-
versions: A New Selection. New York: Simon and Schus-
ter, pp. 13-23, 1961.

Heath, T. A History of Greek Mathematics, Vol. 1. Oxford,
England: Oxford University Press, p. 162, 1921.

Isenberg, C. The Science of Soap Films and Soap Bubbles.
New York: Dover, 1992.

Kepler, J. Opera Omnia, Vol. 5. Frankfort, p. 121, 1864.

Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 129-131, 1990.

Pappas, T. "The Five Platonic Solids." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, pp. 39
and 110-111, 1989.

Rawles, B. A. "Platonic and Archimedean Solids — Faces,
Edges, Areas, Vertices, Angles, Volumes, Sphere Ratios."
http : //www . intent . com/sg/polyhedra . html.

Steinhaus, H. "Platonic Solids, Crystals, Bees' Heads, and
Soap." Ch. 8 in Mathematical Snapshots, 3rd American
ed. New York: Oxford University Press, 1960.

Waterhouse, W. "The Discovery of the Regular Solids."
Arch. Hist. Exact Sci. 9, 212-221, 1972-1973.

Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, 1971.

Platykurtic

A distribution with FlSHER KURTOSIS 72 < (and
therefore having a flattened shape).

see also Fisher Kurtosis

Play fair's Axiom

Through any point in space, there is exactly one straight
line Parallel to a given straight line. This Axiom is
equivalent to the Parallel Axiom.

see also PARALLEL AXIOM

References

Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1

in Journey Through Genius: The Great Theorems of
Mathematics. New York: Wiley, p. 54, 1990.

Plethysm

A group theoretic operation which is useful in the study
of complex atomic spectra. A plethysm takes a set of
functions of a given symmetry type {/i} and forms from
them symmetrized products of a given degree r and
other symmetry type {v}. A plethysm

satisfies the rules

A ® (BC) = {A® B)(A ®C) = A&BA&C,

A®(B±C)^A®B±A®C
(A®B)®C = A®(B®C)

(A + B)® {A} = J^ r ^( A ® iv})(B ® M),
where T^ u \ is the coefficient of {A} in {/^}{f},

(A - B) ® {A} = ^(-lyv^xiA ® {/i})(B ® {*}),

where {i>} is the partition of r conjugate to {1^}, and

(AB) <g> {A} - ^2g^ x (A <g> {fi})(B ® {i/}),

where g^ u \ is the coefficient of {A} in the inner product
{fi} o {1/} (Wybourne 1970).

References

Lit tie wood, D. E. "Polynomial Concomitants and Invariant
Matrices." J. London Math. Soc. 11, 49-55, 1936.

Wybourne, B. G. "The Plethysm of S- Functions" and
"Plethysm and Restricted Groups." Chs. 6-7 in Symme-
try Principles and Atomic Spectroscopy. New York: Wiley,
pp. 49-68, 1970.

Plot

see Graph (Function)

Plouffe's Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Define the function

P(x)^{l

1 for x <
for x > 0.

(1)

Let

sin 1 for n =

a n = sin(2 n ) - <{ 2W1 -a 2 for n = 1 (2)

2a n _i(l — 2a n 2 2 ) for n > 2,

then

pja-n) _ 1

2tt

(3)

Plouffe's Constant

Plucker's Conoid 1377

For

and

for n =

, /n nx / cos 1

6 n = cos(2 )=( 26n _ i 2_ 1 forn > 1(

■ P(bn)

£

2 n+l

0.4756260767.

(4)

(5)

Letting

c n = tan(2

f tanl
n )= J ac B -i

[ 1-C n _! =

then

E P(Cn) _ 1
2 n + 1 7r'

for n =

for n > 1, ( 6 )

(7)

Plouffe asked if the above processes could be "inverted."
He considered

a n =sin(2 n sin~ 1 §)

r | for n =

= < |V5 forra = l (8)

I 2a n -i(l - 2a n -2 2 ) for n > 2,

giving

On+1 12'

(9)

and

«..«(*- B .- 1 )-{* Ul ,. I ;:;:>»

giving

and

7 n = tan(2 n tan"

(10)
(11)

f 1 for n =

*H^ *»«>i. (12)

giving

E^ = i*»-^)

(13)

The latter is known as Plouffe 's constant (Plouffe 1997).
The positions of the Is in the Binary expansion of this
constant are 3, 6, 8, 9, 10, 13, 21, 23, ... (Sloane's
A004715).

Borwein and Girgensohn (1995) extended PloufTe's j n
to arbitrary Real x, showing that if

£ n = tan(2 n tan 1 x) = <

x

2Cn

i-€™-i a

—oo

for n —
for n > 1

and |£ n -i| t^ 1
for n > 1
and |£ n -i| = 1,
(14)

then

Z_^ 2 n + l I 1 + ^

for a? >
for a? < 0.

(15)

Borwein and Girgensohn (1995) also give much more
general recurrences and formulas.

References

Borwein, J. M. and Girgensohn, R. "Addition Theorems and
Binary Expansions." Canad. J. Math. 47, 262-273, 1995.

Finch, S. "Favorite Mathematical Constants." http://vvw.
mathsof t . com/asolve/constant/plf f /plf f .html.

Plouffe, S.. "The Computation of Certain Numbers Us-
ing a Ruler and Compass." Dec. 12, 1997. http://wwv.
research.att.com/-njas/sequences/JIS/compass.html.

Plucker Characteristics

The Class m, Order n, number of Nodes 8, number of
CUSPS k, number of STATIONARY TANGENTS (INFLEC-
TION POINTS) t, number of BlTANGENTS r, and GENUS
P*

see also ALGEBRAIC CURVE, BlTANGENT, CUSP, GENUS
(Surface), Inflection Point, Node (Algebraic
Curve), Stationary Tangent

Plucker's Conoid

A Ruled Surface sometimes also called the Cylin-
DROID. von Seggern (1993) gives the general functional
form as

0,

2 , l 2 2 2

ax + by — zx — zy

whereas Fischer (1986) and Gray (1993) give

z ~ (z 2 + y 2 )*

A polar parameterization therefore gives

x(r, 9) = rcosO
y(r,9) = r sin0
z(r,9) = 2 cos sin 9.

(i)

(2)

(3)
(4)
(5)

1378 Plucker's Equations

Plus Sign

A generalization of Plucker's conoid to n folds is given

by

x(r,0) — rcosO
2/(r, 6) = rsin#
z(r, 9) = sin(n0)

(6)
(7)
(8)

(Gray 1993). The cylindroid is the inversion of the
Cross-Cap (Pinkall 1986).

see also CROSS-CAP, RIGHT CONOID, RULED SURFACE

References

Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 4-5, 1986.

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 337-339, 1993.

Pinkall, U. Mathematical Models from the Collections of Uni-
versities and Museums (Ed. G, Fischer). Braunschweig,
Germany: Vieweg, p. 64, 1986.

von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 288, 1993.

Plucker's Equations

Relationships between the number of SINGULARITIES of
plane algebraic curves. Given a PLANE CURVE,

m = n(n — 1) — 26 — She
n = m(m — 1) — 2r — 3t
i = 3n(n - 2) - 6$ - 8«
k = 3m(m — 2) — 6r — 8t,

(i)

(2)
(3)
(4)

where m is the CLASS, n the ORDER, 5 the number of
NODES, k the number of CUSPS, i the number of STA-
TIONARY Tangents (Inflection Points), and r the
number of BlTANGENTS. Only three of these equations
are LINEARLY INDEPENDENT.

see also Algebraic Curve, Bioche's Theorem,

BlTANGENT, CUSP, GENUS (SURFACE), INFLEC-
TION Point, Klein's Equation, Node (Algebraic
Curve), Stationary Tangent

References

Boyer, C. B. A History of Mathematics. New York: Wiley,

pp. 581-582, 1968.
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New

York: Dover, pp. 99-118, 1959.

Pliicker Relations

see Plucker's Equations

Plumbing

The plumbing of a p-sphere and a g-sphere is defined
as the disjoint union of S p x S 9 and W x S g with their
common D p xD 9 , identified via the identity homeomor-
phism.

see also Hypersphere

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 180, 1976.

Pluperfect Number

see Multiply Perfect Number

Plurisubharmonic Function

An upper semicontinuous function whose restrictions to
all COMPLEX lines are subharmonic (where defined).
These functions were introduced by P. Lelong and Oka
in the early 1940s. Examples of such a function are the
logarithms of moduli of holomorphic functions.

References

Range, R. M. and Anderson, R. W. "Hans-Joachim Brem-

mermann, 1926-1996." Not. Amer. Math. Soc. 43, 972-

976, 1996.

Plus

The ADDITION of two quantities, i.e., a plus b. The
operation is denoted a 4- 6, and the symbol + is called
the Plus Sign. Floating point Addition is sometimes
denoted ®.

see also Addition, Minus, Plus or Minus, Times

Plus or Minus

The symbol ± is used to denote a quantity which should
be both added and subtracted, as in a ± b. The symbol
can be used to denote a range of uncertainty, or to de-
note a pair of quantities, such as the roots given by the
Quadratic Formula

x±

-b ± y/b 2 - 4ac
2a

When order is relevant, the symbol a =f b is also used,
so an expression of the form x ± y =f z is interpreted as
x + y — z or x — y + z. In contrast, the expression x±y±z
is interpreted to mean the set of four quantities x+y+z,
x — y + z, x + y — Zj and x — y — z.

see also Minus, Minus Sign, Plus, Plus Sign, Sign

Plus Perfect Number

see Armstrong Number

Plus Sign

The symbol "+" which is used to denote a POSITIVE
number or to indicate Addition.

see also ADDITION, MINUS SIGN, SIGN

Plutarch Numbers

Poggendorff Illusion 1379

Plutarch Numbers

In Moralia, the Greek biographer and philosopher
Plutarch states "Chrysippus says that the number of
compound propositions that can be made from only ten
simple propositions exceeds a million. (Hipparchus, to
be sure, refuted this by showing that on the affirmative
side there are 103,049 compound statements, and on the
negative side 310,952.)" These numbers are known as
the Plutarch numbers. 103,049 can be interpreted as
the number Sio of Bracketings on ten letters (Stan-
ley 1997), Habsieger et aL 1998). Similarly, Plutarch's
second number is given by (sio + sn)/2 = 310,954 (Hab-
sieger et aL 1998).

References

Biermann, K.-R. and Mau, J. "Uberpriifung einer friihen An-
wendung der Kombinatorik in der Logik." J. Symbolic
Logic 23, 129-132, 1958.

Biggs, N, L. "The Roots of Combinatorics." Historia Math-
ematica 6, 109-136, 1979.

Habsieger, L.; Kazarian, M.; and Lando, S. "On the Second
Number of Plutarch." Amer. Math. Monthly 105, 446,
1998.

Heath, T. L. A History of Greek Mathematics, Vol. 2: From
Aristarchus to Diophantus. New York: Dover, p. 256,
1981.

Kneale, W. and Kneale, M. The Development of Logic. Ox-
ford, England: Oxford University Press, p. 162, 1971.

Neugebauer, O. A History of Ancient Mathematical Astron-
omy, Vol 1. New York: Springer- Verlag, p. 338, 1975.

Plutarch. §VIII.9 in Moralia, Vol. 9. Cambridge, MA: Har-
vard University Press, p. 732, 1961.

Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam-
bridge, England: Cambridge University Press, p. 63, 1996.

Stanley, R. P. "Hipparchus, Plutarch, Schroder, and Hough/'
Amer. Math, Monthly 104, 344-350, 1997.

Pochhammer Symbol

A.k.a. Rising Factorial. For an Integer n > 0,

(a)„E^=a(a + l)...(at«-l), (1)
r(a)

where T(z) is the GAMMA FUNCTION and

(a)o = 1. (2)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 256, 1972.

Spanier, J. and Oldham, K. B. "The Pochhammer Polynomi-
als (x) n ." Ch. 18 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 149-165, 1987.

Pocklington's Criterion

Let p be an Odd Prime, k be an Integer such that
p\k and 1 < k < 2(p + 1), and

N = 2kp + l.

Then the following are equivalent

1. N is Prime.

2. GCD(a fc + l,iV) = 1.

This is a modified version of the original theorem due to
Lehmer.

References

Pocklington, H. C. "The Determination of the Prime or Com-
posite Nature of Large Numbers by Fermat's Theorem."
Proc. Cambridge Phil Soc. 18, 29-30, 1914/16.

Pocklington-Lehmer Test

see Pocklington's Theorem

Pocklington's Theorem

Let n — l = FR where F is the factored part of a number

F = p l ai ---p r a *,

(1)

where (i?, F) = 1, and R < y/n. If there exists a hi for
i = 1, . . . , r such that

bi 71 ' 1 = 1 (mod n)

GCD(bi (n - 1)/pi -l,n) = l,
then n is a Prime.

(2)
(3)

The Notation conflicts with both that for ^-Series and
that for Gaussian Coefficients, so context usually
serves to distinguish the three. Additional identities are

Poggendorff Illusion

da

(a) n = (a) n [F{a + n - 1) - F(a - 1)] (3)

(a) n +k = (a + n)k(a) n ,

(4)

where F is the DlGAMMA FUNCTION. The Pochhammer
symbol arises in series expansions of HYPERGEOMET-
ric Functions and Generalized Hypergeometric
Functions.

see also Factorial, Generalized Hypergeometric
Function, Harmonic Logarithm, Hypergeomet-
ric Function

The illusion that the two ends of a straight Line Seg-
ment passing behind an obscuring Rectangle are off-
set when, in fact, they are aligned.

see also ILLUSION, MULLER-LYER ILLUSION, PONZO'S

Illusion, Vertical-Horizontal Illusion

1380

Pohlke's Theorem

Poincare-Hopf Index Theorem

References

Burmester, E. "Beitrage zu experimentellen Bestimmung

geometrisch-optischer Tauschungen." Z. Psychologic 12,

355-394, 1896.
Day, R. H. and Dickenson, R. G. "The Components of the

Poggendorff Illusion." Brit. J. Psychology 67, 537-552,

1976.
Fineman, M. "Poggendorff's Illusion." Ch. 19 in The Nature

of Visual Illusion. New York: Dover, pp. 151-159, 1996.

Pohlke's Theorem

The principal theorem of AxONOMETRY. It states that
three segments of arbitrary length o!x\ a'y' , and a! z
which are drawn in a PLANE from a point a' under arbi-
trary ANGLES form a parallel projection of three equal
segments ace, ay, and az from the ORIGIN of three PER-
PENDICULAR coordinate axes. However, only one of the
segments or one of the ANGLES may vanish.

see also AxONOMETRY

Poincare-Birkhoff Fixed Point Theorem

For the rational curve of an unperturbed system with
Rotation Number r/s under a map T (for which ev-
ery point is a FIXED POINT of J 13 ), only an even number
of Fixed Points 2ks (k = 1, 2, . . . ) will remain under
perturbation. These FIXED Points are alternately sta-
ble (Elliptic) and unstable (Hyperbolic). Around
each elliptic fixed point there is a simultaneous appli-
cation of the Poincare-Birkhoff fixed point theorem and
the KAM Theorem, which leads to a self-similar struc-
ture on all scales.

The original formulation was: Given a CONFORM AL
One-TO-One transformation from an ANNULUS to it-
self that advances points on the outer edge positively
and on the inner edge negatively, then there are at least
two fixed points.

It was conjectured by Poincare from a consideration
of the three-body problem in celestial mechanics and
proved by Birkhoff.

Poincare Conjecture

A Simply Connected 3-Manifold is Homeomor-
phic to the 3- Sphere. The generalized Poincare con-
jecture is that a Compact ti-Manifold is Homotopy
equivalent to the n-sphere Iff it is Homeomorphic to
the n-SPHERE. This reduces to the original conjecture
for n — 3.

The n = 1 case of the generalized conjecture is trivial,
the n — 2 case is classical, n — 3 remains open, n =
4 was proved by Freedman (1982) (for which he was
awarded the 1986 FIELDS Medal), n = 5 by Zeeman
(1961), n = 6 by Stallings (1962), and n > 7 by Smale in
1961 (Smale subsequently extended this proof to include
n > 5.)

see also COMPACT MANIFOLD, HOMEOMORPHIC, HO-

motopy, Manifold, Simply Connected, Sphere,
Thurston's Geometrization Conjecture

References

Freedman, M. H. "The Topology of Four- DifTerenti able Man-
ifolds." J. Diff. Geom. 17, 357-453, 1982.

Stallings, J. "The Piecewise-Linear Structure of Euclidean
Space." Proc. Cambridge Philos. Soc. 58, 481-488, 1962.

Smale, S. "Generalized Poincare's Conjecture in Dimensions
Greater than Four." Ann. Math. 74, 391-406, 1961.

Zeeman, E. C. "The Generalised Poincare Conjecture." Bull.
Amer. Math. Soc. 67, 270, 1961.

Zeeman, E. C. "The Poincare Conjecture for n > 5." In
Topology of 3- Manifolds and Related Topics, Proceedings
of the University of Georgia Institute, 1961. Englewood
Cliffs, NJ: Prentice-Hali, pp. 198-204, 1961.

Poincare Duality

The Betti Numbers of a compact orientable n-
MANIFOLD satisfy the relation

bi — b n -i.

see also Betti Number

Poincare Formula

The Polyhedral Formula generalized to a surface of
Genus p.

V-E + F = 2-2p

where V is the number of Vertices, E is the number
of Edges, F is the number of faces, and

X = 2 - 2p
is called the Euler Characteristic.

see also EULER CHARACTERISTIC, GENUS (SURFACE),

Polyhedral Formula

References

Eppstein, D. "Fourteen Proofs of Euler's Formula: V — E +

F = 2." http://www. ics .uci . edu/ -eppstein/ junkyard/

euler.

Poincare- Fuchs-Klein Automorphic Function

H \ = k i ( az + b \
I[Z) (cz + dy T \cz + d)

where $s(z) > 0.

see also Automorphic Function

Poincare Group

see Lorentz Group

Poincare's Holomorphic Lemma
Solutions to HOLOMORPHIC differential equations are
themselves HOLOMORPHIC FUNCTIONS of time, initial
conditions, and parameters.

Poincare-Hopf Index Theorem

The index of a Vector Field with finitely many zeros

on a compact, oriented MANIFOLD is the same as the

Euler Characteristic of the Manifold.

see also Gauss-Bonnet Formula

Poincare Hyperbolic Disk

Poincare Hyperbolic Disk

A 2-D space having Hyperbolic Geometry denned

Point

1381

as the 2-BALL {x €
Metric

\x\ < 1}, with Hyperbolic

dx 2 + dy 2

(1-r 2 ) 2 ^

The Poincare disk is a model for Hyperbolic Geome-
try, and there is an isomorphism between the Poincare
disk model and the Klein-Beltrami Model.

see also ELLIPTIC PLANE, HYPERBOLIC GEOMETRY,

Hyperbolic Metric, Klein-Beltrami Model

Poincare's Lemma

Let A denote the Wedge Product and D the Exte-
rior Derivative. Then

*=£*"

&«)-(***)"-■

see also DIFFERENTIAL FORM, EXTERIOR DERIVATIVE,
POINCARE'S HOLOMORPHIC LEMMA, WEDGE PROD-
UCT

Poincare Manifold

A nonsimply connected 3-manifold also called a DODEC-
AHEDRAL SPACE.

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 245, 290, and 308, 1976.

Poincare Metric

The Metric

2 _ dx 2 + dy 2

(i - W 2 ) 2

of the Poincare Hyperbolic Disk.

see also POINCARE HYPERBOLIC DISK

Poincare Separation Theorem

Let {y fc } be a set of orthonormal vectors with k = 1,
2, . . . , K, such that the Inner Product (y fc ,y fc ) = 1.
Then set

K

-£<

x = > u k y
&-1

a)

so that for any SQUARE MATRIX A for which the product
Ax is denned, the corresponding Quadratic Form is

(x,Ax) = ^<Wy\Ay').

fe,i=i

Then if

B fc = (y fc ,Ay z )
for fc, I — 1, 2, . . . , K, it follows that

A*(Bk) < Ai(A)

(2)

(3)

(4)

AK_ j (BK)>A JV -i(A) (5)

for i = 1, 2, . . . , K and j = 0, 1, . . . , K - 1.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1120, 1979.

Poinsot Solid

see Kepler-Poinsot Solid

Poinsot's Spirals

rsinh(n#) = a.

r csch(n#) = a.

References

Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp, 192 and 194, 1972.

Point

A O-Dimensional mathematical object which can be
specified in n-D space using n coordinates. Although the
notion of a point is intuitively rather clear, the mathe-
matical machinery used to deal with points and point-
like objects can be surprisingly slippery. This difficulty
was encountered by none other than Euclid himself who,
in his Elements, gave the vague definition of a point as
"that which has no part."

The basic geometric structures of higher Dimen-
sional geometry — the LlNE, PLANE, Space, and
HYPERSPACE — are all built up of infinite numbers of
points arranged in particular ways.
see also ACCUMULATION POINT, ANTIGONAL POINTS,

Antihomologous Points, Apollonius Point,
Boundary Point, Branch Point, Brianchon
Point, Brocard Midpoint, Brocard Points,

1382

Point Estimator

Point-Line Distance — 2-D

Cantor-Dedekind Axiom, Center, Circle Lat-
tice Points, Concur, Concurrent, Congru-
ent Incircles Point, Congruent Isoscelizers
Point, Conjugate Points, Critical Point, Cru-
cial Point, Cube Point Picking, Cusp Point,
de Longchamps Point, Double Point, Eckardt
Point, Elkies Point, Elliptic Fixed Point (Dif-
ferential Equations), Elliptic Fixed Point
(Map), Elliptic Point, Equal Detour Point,
Equal Parallelians Point, Equichordal Point,
Equilibrium Point, Equiproduct Point, Equire-
ciprocal Point, Evans Point, Exeter Point, Ex-
median Point, Fagnano's Point, Far-Out Point,
Fejes Toth's Problem, Fermat Point, Feuerbach
Point, Feynman Point, Fixed Point, Fletcher
Point, Gergonne Point, Grebe Point, Griffiths
Points, Harmonic Conjugate Points, Hermit
Point, Hofstadter Point, Homologous Points,
Hyperbolic Fixed Point (Differential Equa-
tions), Hyperbolic Fixed Point (Map), Hyper-
bolic Point, Ideal Point, Imaginary Point,
Invariant Point, Inverse Points, Isodynamic
Points, Isolated Point, Isoperimetric Point, Iso-
tomic Conjugate Point, Lattice Point, Lemoine
Point, Limit Point, Malfatti Points, Median
Point, Mid-Arc Points, Midpoint, Miquel Point,
Nagel Point, Napoleon Points, Nobbs Points,
Oldknow Points, Only Critical Point in Town
Test, Ordinary Point, Parabolic Point, Parry
Point, Pedal Point, Periodic Point, Planar
Point, Point at Infinity, Point-Line Distance —
2-D, Point-Line Distance — 3-D, Point-Quadratic
Distance, Point-Plane Distance, Point-Set To-
pology, Pointwise Dimension, Policeman on
Point Duty Curve, Power Point, Radial Point,
Radiant Point, Rational Point, Rigby Points,
Saddle Point (Game), Saddle Point (Func-
tion), Salient Point, Schiffler Point, Self-
Homologous Point, Similarity Point, Singular
Point (Algebraic Curve), Singular Point (Func-
tion), Soddy Points, Special Point, Stationary
Point, Steiner Points, Sylvester's Four-Point
Problem, Symmedian Point, Symmetric Points,
Tarry Point, Torricelli Point, Trisected Per-
imeter Point, Umbilic Point, Unit Point, Van-
ishing Point, Visible Point, WeierstraB Point,
Wild Point, Yff Points

References

Casey, J. "The Point." Ch. 1 in A Treatise on the Analyt-
ical Geometry of the Point, Line, Circle, and Conic Sec-
tions, Containing an Account of Its Most Recent Exten-
sions, with Numerous Examples, 2nd ed., rev. enl. Dublin:
Hodges, Figgis, & Co., pp. 1-29, 1893.

Point Estimator

An Estimator of the actual values of population.

Point Groups

The symmetry groups possible in a crystal lattice with-
out the translation symmetry element. Although an iso-
lated object may have an arbitrary SCHONFLIES SYM-
BOL, the requirement that symmetry be present in a lat-
tice requires that only 1, 2, 3, and 6-fold symmetry axes
are possible (the Crystallography Restriction),
which restricts the number of possible point groups to
32: ft, C s , Ci, C2, C3, C4, Cq, Cih-> C3/1) C^hi Cqh,
C 2v , C 3v , C 4v , C Gv , £>2, Aj, £>4, D G (the DIHEDRAL
Groups), D 2h , D 3h , D 4h , D 6h , D 2d , D M , O, O h (the
Octahedral Group), S 4 , 5 6 , T, T h , and T d (the Tet-
rahedral Group).

see also CRYSTALLOGRAPHY RESTRICTION, DIHE-
DRAL Group, Group, Group Theory, Hermann-
Mauguin Symbol, Lattice Groups, Octahedral
Group, Schonflies Symbol, Space Groups, Tet-
rahedral group

References

Arfken, G. "Crystallographic Point and Space Groups."
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, p. 248-249, 1985.

Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 379, 1990.

Lomont, J. S. "Crystallographic Point Groups." §4.4 in Ap-
plications of Finite Groups. New York: Dover, pp. 132-
146, 1993.

Point at Infinity

P is the point on the line AB such that PA/PB — 1.
It can also be thought of as the point of intersection of
two Parallel lines.

see also Line at Infinity

References

Behnke, H.; Bachmann, F.; Fladt, K.; and Suss, W. (Eds.).
Ch, 7 in Fundamentals of Mathematics, Vol. 3: Points at
Infinity. Cambridge, MA: MIT Press, 1974.

Point-Line Distance — 2-D

Given a line ax + by + c = and a point (20,2/0), in
slope-intercept form, the equation of the line is

y = — b x -Z'

a)

so the line has Slope —a/6. Points on the line have the
vector coordinates

x

a c

' b X d

Therefore, the VECTOR

=

"

c

1

6

a

-b

a

is Parallel to the line, and the Vector

[:]

(2)

(3)

(4)

Point-Line Distance — 2-D

Point Picking 1383

is Perpendicular to it. Now, a Vector from the
point to the line is given by

x — Xo

y-yo

(5)

Projecting r onto v,

|v-r|

!proj v r| =

v-r =

\q(x - xp) + b(y-yp)\

|v| ' ' Va 2 + 6 2

\ax + by — axo — byo\

y/a 2 + b 2
\axp + byp + c\
yja? + b 2

(6)

If the line is represented by the endpoints of a VECTOR
(an, 2/1) and (2:2,2/2), then the Perpendicular Vector

is

2/2 -2/1
'(x 2 -xi)

2/2 -2/1

-(xi -xi)

(7)
(8)

where

s = |v| = yj(x 2 -xi) 2 + (2/2 ~2/i) 2 , (9)

so the distance is

d _ l^ . P i = Ka/2 - yi)( g o ~ gi) ~ ( X2 ~ x 0(^° - yi)l

(10)
The distance from a point (#1,2/1) to the line y = a + bx
can be computed using VECTOR algebra. Let L be a
VECTOR in the same direction as the line

(ii)

(12)
(13)

L =

X

'0"

X

a + bx
1

"1"
b

a

bx

vV + i

A given point on the line is

Xl

2/i

-

" "
—a

=

xi
2/i ~ a_

so the point-line distance is
r = (x • L)L - x

1 + 6 2

xi + 6(2/1 - a)

(T X1 IN)

Xl

yi - a

1 + 6 2

Xl

2/i -a

1 + 6 2

1

1 + 6 2

2/i - (a + &a?i)
1 + 6 2

6(2/1 - a) - b 2 xi
bxi + b 2 yi — ab 2 — yi + a — 6 2 yi + a6 2

6[(yi - a) - 6x1]
-[(2/1 - a) — 6a; 1]

6

-1

(14)

Therefore,

I yi - (a + 6ai)
1 + 6 2

Vl + & 2

I2/1 - (o + 6ai)

(15)

This result can also be obtained much more simply by
noting that the Perpendicular distance is just cos
times the vertical distance \yi — (a -f 6a?i)|. But the
Slope 6 is just tan#, so

sin 2 + cos 2 6 = 1 => tan 2 + 1

cos 2 0'

and

1

a/1 + tan 2 v'TTF
The Perpendicular distance is then

(16)
(17)

rf:

|yi - (a + bxi)

VT+6 2 "

(18)

the same result as before.

see also Line, Point, Point-Line Distance — 3-D

Point-Line Distance — 3-D

A line in 3-D is given by the parametric VECTOR

xo + at
2/o + bt
zo + ct

(i)

The distance between a point on the line with parameter
t and the point (2:1,2/1,^1) is therefore

r 2 = (zi-z o -a£) 2 + (2/i-yo-6£) 2 + 0zi-z o -c£) 2 . (2)

To minimize the distance, take

0(r 2 )

dt

-2a(xi — xo — at) — 26(2/1 — 2/0 — &£)

-2c(zi - z - ct) = (3)

a(x 1 -xo)+b(y 1 -y )+c(zi-zo)-t(a +6 +c ) = (4)

^ _ a(xi - go) + 6(2/1 - 2/0) + c(;gi - zp) f .

l ~ a 2 +6 2 + c 2 ' {b)

so the minimum distance is found by plugging (5) into
(2) and taking the SQUARE ROOT.

see also Line, Point, Point-Line Distance — 2-D

Point Picking

see 18-Point Problem, Ball Triangle Picking,
Cube Point Picking, Cube Triangle Picking, Dis-
crepancy Theorem, Isosceles Triangle, Obtuse
Triangle, Planar Distance, Sylvester's Four-
Point Problem

1384

Point-Plane Distance

Point-Plane Distance

Given a Plane

ax + by -f cz + d =

(i)

and a point (#0,3/0,20), the Normal to the Plane is
given by

"a"

(2)

and a VECTOR from the plane to the point is given by

x — Xq
w = y - 2/0 • (3)

. z - z o _

Projecting w onto v,

Iv-wl

|proj v w| =

|v|
\a(x - xq) + b(y - yo) + c(z - go) + d[

vV + 6 2 + c 2
|ax + 6y + cz - axp - &t/ - czp |

Va 2 + b 2 + c 2
|ax + &yo + cz + d|

vV + b 2 4- c 2

(4)

Point-Point Distance — 1-D

Given a unit Line Segment [0,1], pick two points at
random on it. Call the first point x\ and the second
point X2. Find the distribution of distances d between
points. The probability of the points being a (POSI-
TIVE) distance d apart (i.e., without regard to ordering)
is given by

P(d) =

So So $( d ~ \ X2 ~ x i\)dxidx 2

S So dxi dx2
= (1 - d)[H(l - d) - H(d - 1) + H(d) - H{-d)]
_ f 2(1 - d) for 0< d< 1
\ otherwise,

(i)

where 5 is the DlRAC DELTA FUNCTION and H is the
Heaviside Step Function. The Moments are then

A4= / d m P(d)dd = 2 f
Jo Jo

d m (l-d)dd

jm+2

<4—

[m+1 m + 2j Q

Vm+ 1 m + 2/

(m + 2)-(m+l)

(m + l)(m + 2)

(ro+l)(m + 2)

for m = 2n

(n+l)(2n+l)

(n+ i)( an+3 ) for "i = 2n + 1,

(2)

Point-Point Distance — J-D

giving Moments about

A = I (3)

(4)
(5)
(6)

The Moments can also be computed directly without
explicit knowledge of the distribution

/*2

—

6

Ms

=

1
10

/i 4

=

1

15"

Ml

, _ So So \ X2 -xi\dx x dx2

So So dxi dx2

Jo Jo
fl pi

f f

= / / \x2 — xi\ dxi dX2

Jo Jo

,1,1

— JO JO ( X2 _ ;Cl ) ^ Xl ^jj

X2-zi>0

4- Jo Jo (a?i — X2) dxi dx2

12— a=i <0

JO Jaci

(a?2 — #1) <ia?i dx2

pi pxx
+ / / (X2 — Xi) dx\ dX2

Jo Jo

= / o #2 ~~ ^1^2 G^l

7 L Jxi

4- / [xia?2 - fzs 2 ]^ 1 dxi
Jo

[C-- Xl )-C- Xl *- Xl 2 )] dx,

+ f [(si a -fxi 2 )-(0-0)] dxi

JO

— / i\ — X l + X l ) ^1 = I^ 1 ~~ 2 Xl + 3 Xl 1°
JO

■/

-(|-| + |)-(o-o + o) = |

M2 = / / (|a?2 - a?i|) 2 da;2dxi
JO Jo

(7)

Jo Jo

-//

Jo Jo

-/

Jo

-/

Jo

(x2 — x\) dx\ dX2

(x2 — 2x\X2 4- xi ) dx\ dx2

r i 3 2 . 2 ii 1

[5X2 — a?ia?2 4- #1 a^Jo^^i

(3 — xi 4- £1 ) dxi = [3X1 — 2^1 + j^ilo

3 2 ■ 3 — 6'

(8)

Point-Point Distance — 2-D

The Moments about the Mean are therefore

M2

= M2

. ,' 2 _ 1 /1\2 __ 1
"Ml - 6 ~ UJ -18

(9)

M3

= M3

-3 M2 Mi + 2(Mi) 3 = ik

(10)

M4

/
= M4

- 4^3Mi + fyxaOi) 2 ~ 3 (Mi) 4 =

1

135'

(11)

3 Mean,

Variance, Skewness, and Kurtosis are

i i
M = Mi = 3

(12)

2 1

°" = ^2 = Is

(13)

•n-5-l^

(14)

•»-£-«=-!■

(15)

The probability distribution of the distance between two
points randomly picked on a Line Segment is germane
to the problem of determining the access time of com-
puter hard drives. In fact, the average access time for a
hard drive is precisely the time required to seek across
1/3 of the tracks (Benedict 1995).
see also Point-Point Distance— 2-D, Point-Point
Distance — 3-D, Point-Quadratic Distance, Tet-
rahedron Inscribing, Triangle Inscribing in a
Circle

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 930-931, 1985.

Benedict, B. Using Norton Utilities for the Macintosh. Indi-
anapolis, IN: Que, pp. B-8-B-9, 1995.

Point-Point Distance — 2-D

Given two points in the PLANE, find the curve which
minimizes the distance between them. The Line Ele-
ment is given by

ds = ^/dx 2 +dy 2 , (1)

so the Arc Length between the points x\ and x<i is

L= 1 ds= 1 xA +

y' 2 dx,

(2)

where y f
is

= dy/dx and the quantity

we are

minimizing

/ = v / i + y' 2 -

(3)

Finding the derivatives gives

§s-°

(4)

dxdy' dx [K y }

1/2 /i

(5)

so the Euler-Lagrange Differential Equation be-

df d df

dx dy' dx I ^/^ _j_

= 0.

(6)

Point-Point Distance — 2-D

Integrating and rearranging,

v' _-

/2 2/-, , /2 X

y =c (1 + y )

n t-x 2\ 2

y (1 -c ) =c

y

VT-

The solution is therefore

y — ax + b,

1385

(7)

(8)

(9)

(10)

(11)

which is a straight Line. Now verify that the Arc
Length is indeed the straight-line distance between the
points, a and b are determined from

y\ = ax\ + b
y 2 = ax2 + b.

(12)
(13)

Writing (12) and (13) as a Matrix Equation gives

(14)

X\ 1
X2 1

r *i

r -,

-1

a

Xi 1

2/i

b

X2 1

. y2 .

1

1 -l"

— X2 X\

-1

3/i

Ci - x 2

so

Xi

6 =

yi "3/2 _ 3/2 -yi

X\ — #2 ^2

a?i2/2 — #22/i
#i — a?2

(15)

(16)
(17)

f X2 r~

L= y/l + y> 2 dy = (X 2 - X!) VI

( V2-yi \ 2

\X2 - X\)

+ a*

= (X 2 - 2!l)\/l +

= ^/(x 2 - xi) 2 + (j/2 - 2/i) 2 ,

(18)

as expected.

The shortest distance between two points on a SPHERE
is the so-called Great Circle distance.

see also Calculus of Variations, Great Cir-
cle, Point-Point Distance — 1-D, Point-Point
Distance — 3-D, Point-Quadratic Distance, Tet-
rahedron Inscribing, Triangle Inscribing in a
Circle

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 930-931, 1985.

1386 Point-Point Distance— 3-D

Point-Point Distance — 3-D

The Line Element is

Point- Quadratic Distance

ds = yjdx 2 +dy 2 + dz 2 , (1)

so the Arc Length between the points x\ and X2 is

L= ds= / y/l + y' 2 + z' 2 dx

and the quantity we are minimizing is

/= ^l + y^+z' 2 .
Finding the derivatives gives

az

and

Of = y'

9y' y/l + y'2 + 2 '2

a/ _ z'

(2)

(3)

(4)
(5)

(6)
(7)

so the Euler-Lagrange Differential Equations
become

dx

yj\ + y' 2 + z' 2 j

dx \ ^l + y' 2 +z> 2 /

=

= 0.

These give

y/\ + y 12 + z' 2

v/l + i/' 2 + z' 2

= Ci

= c 2 .

Taking the ratio,

/ c 2 ,

z = — y

11 2

y = ci

ii '2 . /C 2 \ 2 ,2

Cl

(8)
(9)

(10)
(11)

(12)
(13)

which gives

Cl

1 - Cl 2 - C 2 '

:c l a +y' 2 (c 1 2 + o a a ),

(14)

= ai 2 (15)

/2 / C 2 \ /2

C2

6i 2

Therefore, y = ai and z' = &i, so the solution is

"as"

as

2/

=

aias + ao

_z_

_ 6ias + 6o .

L = \/l + ai 2 + &i 2 (^2 - xx)

where

yi

Xl

l"

ai

y 2 .

X2

1

ao

Z\

Xl

l"

V

Z2

X 2

1

_6o_

(16)

(17)

which is the parametric representation of a straight line
with parameter x € [xi, as 2 ]. Verifying the Arc LENGTH
gives

' " (18)

(19)

(20)

see also POINT-POINT DISTANCE — 1-D, POINT-POINT

Distance — 2-D, Point-Quadratic Distance

Point Probability

The portion of the probability distribution which has a
P- Value equal to the observed P- Value.

see also TAIL PROBABILITY

Point-Quadratic Distance

Find the minimum distance between a point in the plane
(aso,yo) and a quadratic PLANE Curve

y = a + oias + a 2 as 2 . (1)

The square of the distance is

r 2 = (as -as ) 2 + (y - yo) 2

= (x - xo) 2 + (ao + aix + a 2 x - yo) . (2)

Minimizing the distance squared is the equivalent to
minimizing the distance (since r 2 and \r\ have minima
at the same point), so take

r\/ 2\

= 2(x-xo)+2(a,o+aix+a2X 2 -yo)(ai+2a.2x) —

(3)

x — xo + aoai + ai 2 + aia 2 os 2 — aiyo + 2aoa 2 as

+2aia 2 as 2 + 2a 2 2 as 3 — 2a 2 yoas = (4)

2a 2 2 as +3aia 2 as + (ai + 2aoa 2 — 2a 2 yo + l)as

+(a ai — ait/o — xo) = 0. (5)

Minimizing the distance therefore requires solution of a
Cubic Equation.

see also POINT-POINT DISTANCE — 1-D, POINT-POINT

Distance — 2-D, Point-Point Distance — 3-D

Point-Set Topology

Poisson-Charlier Polynomial 1387

Point-Set Topology

The low-level language of TOPOLOGY, which is not really
considered a separate "branch" of TOPOLOGY. Point-set
topology, also called set-theoretic topology or general
topology, is the study of the general abstract nature of
continuity or "closeness" on SPACES. Basic point-set
topological notions are ones like CONTINUITY, DIMEN-
SION, Compactness, and Connectedness. The In-
termediate Value Theorem (which states that if a
path in the real line connects two numbers, then it passes
over every point between the two) is a basic topological
result. Others are that EUCLIDEAN n-space is HOMEO-
morphic to Euclidean m-space Iff m = n, and that
REAL valued functions achieve maxima and minima on
Compact Sets.

Foundational point-set topological questions are ones
like "when can a topology on a space be derived from
a metric?" Point-set topology deals with differing no-
tions of continuity and compares them, as well as deal-
ing with their properties. Point-set topology is also the
ground-level of inquiry into the geometrical properties
of spaces and continuous functions between them, and
in that sense, it is the foundation on which the remain-
der of topology (Algebraic, Differential, and Low-
Dimensional) stands.

see also ALGEBRAIC TOPOLOGY, DIFFERENTIAL TO-
POLOGY, Low-Dimensional Topology, Topology

References

Sutherland, W. A. An Introduction to Metric & Topological
Spaces. New York: Oxford University Press, 1975.

Points Problem

see Sharing Problem

Pointwise Dimension

ZMx) = lim ln ^*»,

e-+o lne

where B e (x) is an n-D BALL of RADIUS e centered at x
and fi is the PROBABILITY MEASURE.

see also Ball, Probability Measure

References

Nayfeh, A. H. and Balachandran, B. Applied Nonlinear
Dynamics: Analytical, Computational, and Experimental
Methods. New York: Wiley, pp. 541-545, 1995.

Poisson's Bessel Function Formula

For ft[i/] > -1/2,

MZ)= (^V^WTY)1 «»(*«»*) Bin 8 " t A,

where J v (z) is a Bessel Function of the First
KIND, and T{z) is the GAMMA FUNCTION.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics, Cambridge, MA: MIT Press, p. 1472,
1980.

Poisson Bracket

Let F and G be infinitely differentiate functions of x
and p. Then the Poisson bracket is defined by

(fg) = t(^^--— — V

' ^ \ dp u dx p dp„ dx„ ) '

If F and G are functions of x and p only, then the LA-
GRANGE Bracket [F, G] collapses the Poisson bracket
(F,G).
see also LAGRANGE BRACKET, LIE BRACKET

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1004,
1980.

Poisson-Charlier Function

( . (1 + v- n) n , ,

p n (v,x) = — iFi(-n;l + u - n;x) t

\/n\x n

where (a)„ is a POCHHAMMER Symbol and iFi(a\b\z)
is a Confluent Hypergeometric Function.

see also Poisson-Charlier Polynomial

Poisson-Charlier Polynomial

Polynomials p n (x) which belong to the distribution
da(x) where a(x) is a Step Function with Jump

j(x)^e' a a x (x\)' 1
at x = 0, 1, ... for a > 0,

(i)

p n (x) = «»/*(„!)- l " E(-l) n - (") v[a ~ V (*) < 2 >

= a" /2 (n!)- 1/2 (-l)"[j(a ; )]- 1 A"i(x - n) (3)
= a-"/ 2 V^! LT n {a), (4)

where (£) is a Binomial Coefficient, L^{x) is an
associated Laguerre Polynomial, and

Af(x) = f(x + l)-f(x)
A n f(x) = A[A"-V(*)]

= f{x + n) - (fj f{x + n - 1) + . . . + (-l) n /(x).

(5)
(6)

see also Poisson-Charlier Function

References

Szego, G. Orthogonal Polynomials, J^th ed. Providence, Rl:
Amer. Math. Soc, pp. 34-35, 1975.

1388 Poisson Distribution

Poisson Distribution

Poisson Distribution

A Poisson distribution is a distribution with the follow-
ing properties:

1. The number of changes in nonover lapping intervals
are independent for all intervals.

2. The probability of exactly one change in a sufficiently
small interval h = 1/n is P = uh = v/n, where v is
the probability of one change and n is the number of
Trials.

3. The probability of two or more changes in a suffi-
ciently small interval h is essentially 0.

The probability of k changes in a given interval is then
given by the limit of the BINOMIAL DISTRIBUTION

as the number of trials becomes very large,
lim P(k) =

lim

n(n — 1) • • ■ (n — k — 1) v

r(-:)"('-i)'

.(1)|^)(0(D=^

fc!

(2)

This should be normalized so that the sum of probabil-
ities equals 1. Indeed,

E p w

k=0

e 2^-JA = ee =1 -

fc=0

as required. The ratio of probabilities is given by

P(k = i + 1)
P(k = i)

(i+l)!

i + 1

(3)

(4)

The Moment-Generating Function of this distribu-
tion is

fc=0 fc=0

= c-'e""* = e^'" 1 '

(5)

M'(t) = i/e'e"*"' -1 *

(6)

M (t) = (f e ) e + ve e K '

(7)

R(t) = In Af(i) = i/(e* - 1)

(8)

fl'(t) = i/e f

(9)

fi"(t) = i/e',

(10)

so

M = J R'(0) = i/

cr 2 =R"(0) = v.

(11)

(12)

The Moments about zero can also be computed directly

/4 = i/(l + i/)

/is = i/(l + 3z/ + i/ 2 )

p 4 = z/(l + 7i/ + 6z/ 2 + i/ 3 ),

as can the Moments about the Mean.
fix = v

^4 = f(l + 3^),

(13)
(14)
(15)

(16)
(17)
(18)
(19)

so the Mean, Variance, Skewness, and Kurtosis are

/i = V

(20)

2

a = v

(21)

_ M3 V -1/2

7i = -f = ^^

(22)

7 3= ,(l + 3,) 3
cr 4 z/

i/ + 3i/ 2 -3i/ 2 _!
= , = i/ .

(23)

The Characteristic Function is

0(t) = c M«"-D (24)

and the Cumulant-Generating Function is

tf(fc) = V ( e h - 1) = „(/i + £fc 2 + £/> 3 + ■ ■ ■ ), (25)
so

K r = I/. (26)

The Poisson distribution can also be expressed in terms
of

A = -, (27)

the rate of changes, so that

(AaO fe e- Ax

P(*)

fc!

(28)

The Moment-Generating Function of a Poisson dis-
tribution in two variables is given by

M(f) = e (l/1+1/2)(eLl) .

(29)

Poisson's Equation

Poisson Kernel

1389

If the independent variables xi, X2, - . . , xn have Poisson
distributions with parameters //i, £i2, . • - , P-n, then

*=D

has a Poisson distribution with parameter

N

m = X^'-

(30)

(31)

j=i

This can be seen since the Cumulant-Generating
Function is

K j (h)=p j (e h -1), (32)

K = 5]K j (/ l ) = (e h -l)^ / Xi=M(^-l). (33)

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 532, 1987.

Press, W. E; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Incomplete Gamma Function, Error Func-
tion, Chi-Square Probability Function, Cumulative Poisson
Function." §6.2 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 209-214, 1992.

Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 111-112, 1992.

Poisson's Equation

A second-order Partial Differential Equation
arising in physics:

V -0 = — 4:7Tp.

If p = 0, it reduces Laplace's EQUATION. It is also
related to the Helmholtz Differential Equation

see also Helmholtz Differential Equation, La-
place's Equation

References

Arfken, G. "Gauss's Law, Poisson's Equation." §1.14 in

Mathematical Methods for Physicists, 3rd ed. Orlando,

FL: Academic Press, pp. 74-78, 1985.

Poisson's Harmonic Function Formula

Let <t>(z) be a Harmonic Function. Then

«" ) = sjf

K(r,0)<t>(zo + re i9 )d0, (1)

where R = \z \ and K(r,6) is the POISSON KERNEL.
For a Circle,

1 f 2n

u(x,y) = —— I u(a cos <f>, asin</>)

27r Jo

a 2 -R 2

a 2 + R 2 - 2ar cos(0 - <p)

d<t>. (2)

For a Sphere,

a?-R 2

2 +R 2 -2aRcos6) 3 / 2

dS,

where

cos = x ■ £.

(3)
(4)

see also CIRCLE, HARMONIC FUNCTION, POISSON KER-
NEL, Sphere

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 373-374, 1953.

Poisson Integral

A.k.a. Bessel's Second Integral.

Mz)= V^' M / cos(2COS0)sin 2T1 0^,

I(™ + ^Mia) Jo

where J n (z) is a Bessel Function OF the First Kind
and T(x) is a Gamma Function. It can be derived from
Sonine's Integral. With n = 0, the integral becomes
Parseval's Integral.

see also Bessel Function of the First Kind, Par-
seval's Integral, Sonine's Integral

Poisson Integral Representation

z n f*

•?"(*) = on+i I / cos(zcos0)sin 2n+1 0d0,
1 n - Jo

where j n (z) is a SPHERICAL BESSEL FUNCTION OF THE

First Kind.

Poisson Kernel
In 2-D,

~(R + re ie )(R-re- ie )

R

= R

= R

(R - re i9 )(R - re~ ie )

R 2 - rR(e

t i$

9 \ „ 2
)~r

R 2 - rR(e ie + e~ ie ) + r 2

R 2 -\-2irRsin6 - r 2

R 2 - 2Rr cos + r 2

R 2 -r 2
R 2 -2Rr cos + r 2 '

(1)

In 3-D,

u(y) ■■

R(R 2 - a 2 )

4-7T

p 2 7T A 71

*/ /

Jo Jo

f (0, <j>) sinfl dQd<t>
(R 2 -\-a 2 -2aRcosy)s/ 2

, (2)

1390 Poisson Manifold

where a = |y| and

cos 7 = y ■

R cos 9 sin <j>

R sin 6 sin <f>

R cos <f>

The Poisson kernel for the n-BALL is

F(x,z) = -J-(D n v)(z),

I — n

(3)

(4)

where D n is the outward normal derivative at point z
on a unit n-SPHERE and

v(z) = |z-x| 2 n - |x| 2

2-n

see also Poisson's Harmonic Function Formula

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1090, 1979.

Poisson Manifold

A smooth Manifold with a Poisson Bracket defined
on its Function Space.

Poisson Sum Formula

A special case of the general result

oo oo ^oo

(1)

with z = 0, yielding

°° °° /»oo

E /(»)= £ / f(*i)e- 2 " ikx d*i- (2)

i «/ — OO

n= — oo fc= — oo

An alternate form is

n=l

n=l

(3)

Another formula called the Poisson summation formula
is

Va[i#0) + tfa) + *(2a) + ...]

= y^[|V(0) + V(/3) + V>(2/3) + ...], (4)

where

a/? = 2tt.

^(t) cos(xt) dt

(5)
(6)

Poice Move

Poisson Trials

A number s TRIALS in which the probability of success
Pi varies from trial to trial. Let x be the number of
successes, then

var(cc) = spq — scr p ,

(i)

where a p 2 is the Variance of p* and q = (1 — p). Us-
pensky has shown that

P(s,x)=0-

(5) where

>3 = [1- M*)]**"'

g(*) = ^ ^TT +

3(a -m) 3 2s(s-z)

/i(z)

ms m

= p

x(a; - 1)
1 \ (x - m) 2

('♦=)-

2m

(2)

(3)
(4)

(5)

and 6 6 (0, 1). The probability that the number of suc-
cesses is at least x is given by

Qm(x) = J2 [

(6)

Uspensky gives the true probability that there are at
least x successes in s trials as

Pms(x) = Qm(x) + A,

(7)

where
|A|<

\(e*-

l)Q m (x + 1) for Q m (x + 1) > ±

l)[l-Q m (x + l)] forQ m (x + l)<i

(8)

2(5 - m)

(9)

Poke Move

-« ►

-p

poke

unpoke
The Reidemeister Move of type II.

see also REIDEMEISTER MOVES

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 466-467, 1953.

Poker

Polar Circle 1391

Poker

Poker is a Card game played with a normal deck of
52 Cards. Sometimes, additional cards called "jokers"
are also used. In straight or draw poker, each player is
normally dealt a hand of five cards. Depending on the
variant, players then discard and redraw Cards, trying
to improve their hands. Bets are placed at each discard
step. The number of possible distinct five-card hands is

N:

52

= 2,598,960,

where (£) is a Binomial Coefficient.

There are special names for specific types of hands. A
royal flush is an ace, king, queen, jack, and 10, all of
one suit. A straight flush is five consecutive cards all of
the same suit (but not a royal flush) , where an ace may
count as either high or low. A full house is three-of-a-
kind and a pair. A flush is five cards of the same suit
(but not a royal flush or straight flush). A straight is
five consecutive cards (but not a royal flush or straight
flush), where an ace may again count as either high or
low.

The probabilities of being dealt five-card poker hands of
a given type (before discarding and with no jokers) on
the initial deal are given below (Packel 1981). As usual,
for a hand with probability P, the Odds against being
dealt it are (1/V) — 1:1.

Hand

Exact Probability

royal flush N - 649740
straight flush ^ = 2]
four of a kind jv 4 it
full house KaJ N K2j =

n , 4(V>)-36-4

flush -^^ =

straight N

10 /4\(48)(44)
fHrA^ nf a l-inr> ^ 3 ' 2!

3

L6.580

55

6

4,165

_ 1,277

"*" 649,740

__ 5

1,274

_ 88
4,165

_ 198
~~ 4,165
40)

352

tnree oi a Kinc
two pair

^)-(*) 44

13(J)i«

one pair

N

833

Hand

Probability

Odds

royal flush
straight flush
four of a kind
full house
flush
straight
three of a kind
two pair
one pair

1.54 x lO -0

1.39 x HP 5

2.40 x 10~ 4
1.44 x 10 -3
1.97 x 10~ 3
3.92 x 10" 3
0.0211
0.0475
0.423

649,739.0

72,192.3

4,164.0

693.2

507.8

253.8

46.3

20.0

1.366

1
1

1
1
1
1
1
1
1

Gadbois (1996) gives probabilities for hands if two jokers
are included, and points out that it is impossible to rank

hands in any single way which is consistent with the

relative frequency of the hands.

see also Bridge Card Game, Cards

References

Cheung, Y. L. "Why Poker is Played with Five Cards."
Math. Gaz. 73, 313-315, 1989.

Conway, J. H. and Guy, R. K. "Choice Numbers with Rep-
etitions." In The Book of Numbers. New York: Springer-
Verlag, pp. 70-71, 1996.

Gadbois, S. "Poker with Wild Cards— A Paradox?" Math.
Mag. 69, 283-285, 1996.

Jacoby, O. Oswald Jacoby on Poker. New York: Doubleday,
1981.

Packel, E. W. The Mathematics of Games and Gambling.
Washington, DC: Math. Assoc. Amer., 1981.

Polar

polar

N^

If two points A and A' are Inverse with respect to a
Circle (the Inversion Circle), then the straight line
through A' which is PERPENDICULAR to the line of the
points AA' is called the polar of A with respect to the
Circle, and A is called the Pole of the polar.

see also Apollonius' Problem, Inverse Points, In-
version Circle, Polarity, Pole, Trilinear Polar

References

Dorrie, H. 100 Great Problems of Elementary Mathematics:

Their History and Solutions. New York: Dover, p. 157,

1965.
Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 100-106, 1929.

Polar Angle

The Angle a point makes from the Origin as measured

from the cc-AxiS.

see also POLAR COORDINATES

Polar Circle

Given a TRIANGLE, the polar circle has center at the
Orthocenter H. Call Hi the Feet of the Altitude.
Then the RADIUS is

= HA ± • HH! = HA 2 ■ HH 2 = HA 2 ■ HH 2 (1)

— —AR cos c\i cos oc2 cos a3
= |(ai 2 + a 2 2 +a 3 2 )-4ie 2

(2)
(3)

where R is the ClRCUMRADlUS, a* the VERTEX angles,
and ai the corresponding side lengths.

1392

Polar Coordinates

Polar Line

A Triangle is self-conjugate with respect to its polar
circle. Also, the RADICAL Axis of any two polar circles
is the Altitude from the third Vertex. Any two po-
lar circles of an Ortho CENTRIC System are orthogonal.
The polar circles of the triangles of a COMPLETE QUAD-
RILATERAL constitute a Coaxal System conjugate to
that of the circles on the diagonals.

see also COAXAL SYSTEM, ORTHOCENTRIC SYSTEM,
Polar, Pole, Radical Axis

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.

Washington, DC: Math. Assoc. Amer., pp. 136-138, 1967.
Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 176-181, 1929.

Polar Coordinates

The polar coordinates r and are defined by

x = r cos c

y — r sin 6

(i)

(2)

A polar curve is symmetric about the x-axis if replacing
by — in its equation produces an equivalent equation,
symmetric about the y-axis if replacing by 7r — in its
equation produces an equivalent equation, and symmet-
ric about the origin if replacing r by — r in its equation
produces an equivalent equation.

In Cartesian coordinates, the Position Vector and its
derivatives are

r = y/x 2 + y 2 r (11)

f = 'r^x 2 +y 2 + r (x 2 + y 2 )~ 1/2 (xx + yy) (12)
A xx + yy

\/x 2 + y 2
_ x£ + yy

~ v^ + y 2

- \{x 2 + y 2 )- 3/2 (2)(xx + yy)(xx + yy)
_ (xy~yx)(xy - yx)
Oz 2 +y 2 ) 3 / 2

(13)

(14)

In terms of x and y,

r = y/x* + y 2 (3)

* = tan->(f). (4)

The Arc Length of a polar curve given by r = r(0) is

=Cf^)-

dO.

The Line Element is given by

ds = r dO ,

and the AREA element by

dA = rdrdO.
The Area enclosed by a polar curve r = r(9) is

.i

r 2 d0.

(5)

(6)

(7)

(8)

The Slope of a polar function r = r(0) at the point
(r, 0) is given by

r + tan 6

de

-r tan 6» + %'

(9)

The ANGLE between the tangent and radial line at the
point (r, 9) is

tp = tan'

\ de /

(10)

In polar coordinates, the Unit Vectors and their
derivatives are

(15)
(16)

(17)

r cos 6

rs'mO

dr

dr

\dr\~~
\ dr \

cos 6
sin#

e =

— sinf/

COS0

-sin 00
cos 00

-cos 00
-sin 00

06

-0r

— r sin 00 + cos0r
r cos 00 -f sin0r

rQO-rfv

f^f$d + rSO + rOO + rr + rr
= rOO + r$0 + r0(-0r) + rr + r60
= (r-r6 2 )v + (2r0 + r0)O

-(r 2 0)0.

(f-r0 2 )r+^(» 2 -
r ac

(18)
(19)
(20)

(21)

see also Cardioid, Circle, Cissoid, Conchoid,
Curvilinear Coordinates, Cylindrical Coordi-
nates, Equiangular Spiral, Lemniscate, LiMAgoN,
Rose

Polar Line

see Polar

Polarity

Pollaczek Polynomial 1393

Polarity

A projective CORRELATION of period two. In a polarity,
a is called the POLAR of A, and ^1 the POLE a.

see also CHASLES'S THEOREM, CORRELATION, POLAR,

Pole (Geometry)

Pole

A Complex function / has a pole of order m at zo if, in
the Laurent Series, a n = for n < -m and a m ^ 0.
Equivalently, / has a pole of order n at zq if n is the
smallest Positive Integer for which (z - z ) n f(z) is
differentiable at z$. If /(±oo) ^ ±oo, there is no pole
at ±co. Otherwise, the order of the pole is the greatest
Positive Coefficient in the Laurent Series.

This is equivalent to finding the smallest n such that

(z - z Q ) n
f(z)

is differentiable at 0.

see also Laurent Series, Residue (Complex Anal-
ysis)

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 396-397, 1985.

Pole (Geometry)

polar

N,

If two points A and A 1 are Inverse with respect to a
Circle (the Inversion Circle), then the straight line
through A' which is PERPENDICULAR to the line of the
points AA 1 is called the POLAR of the A with respect to
the CIRCLE, and A is called the pole of the POLAR.
see also Inverse Points, Inversion Circle, Polar,
Polarity, Trilinear Polar

References

Dorrie, H. 100 Great Problems of Elementary Mathematics:

Their History and Solutions. New York: Dover, p. 157,

1965.
Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, pp. 100-106, 1929.

Pole (Origin)

see Origin

Policeman on Point Duty Curve

see Cruciform

Polignac's Conjecture

see DE POLIGNAC'S CONJECTURE

Polish Space

The HOMEOMORPHIC image of a so-called "complete
separable" Metric Space. The continuous image of
a Polish space is called a Souslin Set.
see also DESCRIPTIVE SET THEORY, STANDARD SPACE

Pollaczek Polynomial

Let a > |6|, and write

h(0) =

a cos 6 + b

(1)

2 sin
Then define P n (x;a,b) by the GENERATING FUNCTION

oo

f(x, w) = /(cos 0, w) = y P n (x; a, b)w n

n=0

= (l _ we i0 )- 1/2+ih($) (l - we i6 r 1/2 - iH9) - (2)
The Generating Function may also be written

f(x,w) = {l-2xw + w 2 )~ 1/2

exp

771

{ax + b) 22, Um-i(x)

, (3)

where U m (x) is a CHEBYSHEV POLYNOMIAL OF THE
Second Kind. They satisfy the Recurrence Rela-
tion

nP n (x]a,b) = [(2n- 1 + 2a)x + 2b]P n -i(x; a, 6)

-(n-l)P n - 2 (x;a,b) (4)

for n = 2, 3, ... with

Po = 1 (5)

Pi = {2a + l)x + 2b. (6)

In terms of the HYPERGEOMETRIC FUNCTION
2i J i(a,6;c;x),

P n (cos9;a;b) = e inB 2 F 1 (-n, \+ih{9); 1; l-e~ 2ie ). (7)

/ P n (x;a,l

They obey the orthogonality relation

, b)P m (x\ a, b)w(x\ a, b) dx

= [n+i(a + l)]- 1 *nm, (8)

where 8 nrn is the Kronecker Delta, for n,m = 0, 1,
. . . , with the Weight Function

^(cos^;a ; 6) = e^^^^^coshtTr/i^)]}" 1 . (9)

References

Szego, G. Orthogonal Polynomials , ^th ed. Providence, RI:
Amer. Math. Soc, pp. 393-400, 1975.

1394

Pollard Monte Carlo Factorization Method

Polya Distribution

Pollard Monte Carlo Factorization Method

see Pollard p Factorization Method

Pollard p — 1 Factorization Method
A Prime Factorization Algorithm which can be
implemented in a single-step or double-step form. In
the single-step version, Primes p are found if p — 1 is a
product of small PRIMES by finding an m such that

m = c q (mod n) ,

where p — l\q 7 with q a large number and (Cj-n) — 1.
Then since p — l|g, m ~ 1 (mod p), so p\m — 1. There
is therefore a good chance that n\m — 1, in which case
GCD(ra-l,n) (where GCD is the Greatest Common
Divisor) will be a nontrivial divisor of n.

In the double-step version, a PRIMES p can be factored
if p — 1 is a product of small PRIMES and a single larger
Prime.

see also Prime Factorization Algorithms, Wil-
liams p + 1 Factorization Method

References

Bressoud, D. M. Factorization and Prime Testing. New

York: Springer- Verlag, pp. 67-69, 1989.
Pollard, J. M. "Theorems on Factorization and Primality

Testing." Proc. Cambridge Phil Soc. 76, 521-528, 1974.

Pollard p Factorization Method

A Prime Factorization Algorithm also known as
Pollard Monte Carlo Factorization Method.
Let xo — 2, then compute

xi + \ = Xi 2 — Xi + 1 (mod n).

If GCD(#2i - XijTi) > 1, then n is Composite and its
factors are found. In modified form, it becomes Brent's
Factorization Method. In practice, almost any un-
favorable Polynomial can be used for the iteration
(x 2 — 2, however, cannot). Under worst conditions, the
Algorithm can be very slow.

see also Brent's Factorization Method, Prime
Factorization Algorithms

References

Brent, R. P. "Some Integer Factorization Algorithms Using

Elliptic Curves." Austral. Comp. Sci. Comm. 8, 149-163,

1986.
Bressoud, D. M. Factorization and Prime Testing. New

York: Springer- Verlag, pp. 61-67, 1989.
Eldershaw, C. and Brent, R. P. "Factorization of Large

Integers on Some Vector and Parallel Computers."

ftp : //nimbus . anu . edu . au/pub/Brent / 156tr . dvi . Z.
Montgomery, P. L. "Speeding the Pollard and Elliptic Curve

Methods of Factorization." Math. Comput. 48, 243-264,

1987.
Pollard, J. M. "A Mcnte Carlo Method for Factorization."

Nordisk Tidskrift for Informationsbehandlung (BIT) 15,

331-334, 1975.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 83 and 102-103, 1991.

Poloidal Field

A Vector Field resembling a magnetic multipole
which has a component along the 2- Axis of a SPHERE
and continues along lines of Longitude.

see also Divergenceless Field, Toroidal Field

References

Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley,
p. 239, 1977.

Polya-Burnside Lemma

see POLYA ENUMERATION THEOREM

Polya Conjecture

Let n be a POSITIVE INTEGER and r(n) the number of
(not necessarily distinct) PRIME FACTORS of n (with
r(l) = 0). Let 0(m) be the number of Positive Inte-
gers < m with an ODD number of Prime factors, and
E(m) the number of POSITIVE INTEGERS < m with an
EVEN number of Prime factors. Polya conjectured that

m

L(m) = E(m) - 0{m) = ^ X(n)

is < 0, where X(n) is the LiOUVlLLE FUNCTION.

The conjecture was made in 1919, and disproven by
Haselgrove (1958) using a method due to Ingham (1942).
Lehman (1960) found the first explicit counterexample,
L(906,180,359) = 1, and the smallest counterexample
m = 906,150,257 was found by Tanaka (1980). The first
n for which L(n) = are n = 2, 4, 6, 10, 16, 26, 40, 96,
586, 906150256, . . . (Tanaka 1980, Sloane's A028488). It
is unknown if L(x) changes sign infinitely often (Tanaka
1980).

see also Andrica's Conjecture, Liouville Func-
tion, Prime Factors

References

Haselgrove, C. B. "A Disproof of a Conjecture of Polya."
Mathematika 5, 141-145, 1958.

Ingham, A. E. "On Two Conjectures in the Theory of Num-
bers." Amer. J. Math. 64, 313-319, 1942.

Lehman, R. S. "On Liouville's Function." Math. Comput.
14, 311-320, 1960.

Sloane, N. J. A. Sequence A028488 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Tanaka, M. "A Numerical Investigation on Cumulative Sum
of the Liouville Function" [sic]. Tokyo J. Math. 3, 187-
189, 1980.

Polya Distribution

see Negative Binomial Distribution

Polya Enumeration Theorem

Polya Enumeration Theorem

A very general theorem which allows the number of dis-
crete combinatorial objects of a given type to be enu-
merated (counted) as a function of their "order." The
most common application is in the counting of the num-
ber of Graphs of n nodes, Trees and Rooted Trees
with n branches, GROUPS of order n, etc. The theorem
is an extension of BURNSIDE'S LEMMA and is sometimes
also called the Polya-Burnside Lemma.
see also Burnside's Lemma, Graph (Graph The-
ory), Group, Rooted Tree, Tree

References

Harary, F. "The Number of Linear, Directed, Rooted, and

Connected Graphs." Trans. Amer. Math. Soc. 78, 445-

463, 1955.
Polya, G. "Kombinatorische Anzahlbestimmungen fur Grup-

pen, Graphen, und chemische Verbindungen." Acta Math.

68, 145-254, 1937.

Polya Polynomial

The POLYNOMIAL giving the number of colorings, with
m colors, of a structure defined by a PERMUTATION
Group.

see also PERMUTATION GROUP, POLYA ENUMERATION

Theorem

Polya's Random Walk Constants

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

Let p(d) be the probability that a Random Walk on
a d-D lattice returns to the origin. Polya (1921) proved
that

p(l)=p(2) = l, (1)

but

P(d) < 1

(2)

for d > 2. Watson (1939), McCrea and Whipple (1940),
Domb (1954), and Glasser and Zucker (1977) showed
that

P(3) = 1 -

«(3)

0.3405373296

(3)

where

U(3):

7T />7T /*7T

-7T J — 7T J —71

" (2tt)3

= ^(18 + 12^2- W3 -7\/6)

7T 2

x{K[(2-V^)(v / 3-v / 2)]} 2
= 3(18 + 12>/2 - W3 - 7>/6 )
00
l + 2^exp(-A; 2 7rv / 6)
fc=i

3^r(i)r(A)r(i)r(Ji)

1.5163860592...,

dx dy dz
cos x — cos v — cos z

(4)
(5)

(6)

(7)
(8)

Polya-Vinogradov Inequality 1395

where K(k) is a complete Elliptic Integral of the
First Kind and T(z) is the Gamma Function. Closed
forms for d > 3 are not known, but Montroll (1956)
showed that

p(d) = 1 - [u(d)]-\ (9)

where

u(d)

W

r r-f( d -i

y_^ j-n j-^ y fc==1

COSXk

x dx\ dx2 * • • dxd

= [Hd)] de - tdt ' (10)

and I (z) is a Modified Bessel Function of the
First Kind. Numerical values from Montroll (1956)
and Flajolet (Finch) are

d

p(d)

4

0.20

5

0.136

6

0.105

7

0.0858

8

0.0729

see also Random Walk

References

Finch, S. "Favorite Mathematical Constants." http://www.

mathsoft.com/asolve/constant/polya/polya.html.
Domb, C. "On Multiple Returns in the Random- Walk Prob-
lem." Proc. Cambridge Philos. Soc. 50, 586-591, 1954.
Glasser, M. L. and Zucker, I. J. "Extended Watson Integrals

for the Cubic Lattices." Proc. Nat. Acad. Set. U.S.A. 74,

1800-1801, 1977.
McCrea, W. H. and Whipple, F. J. W. "Random Paths in

Two and Three Dimensions." Proc. Roy. Soc. Edinburgh

60, 281-298, 1940.
Montroll, E. W. "Random Walks in Multidimensional Spaces,

Especially on Periodic Lattices." J. SIAM 4, 241-260,

1956.
Watson, G. N. "Three Triple Integrals." Quart. J. Math.,

Oxford Ser. 2 10, 266-276, 1939.

Polya- Vinogradov Inequality

Let x De a nonprincipal character (mod q). Then

M+N

^2 #)«v / 9lng,

n=M+l

where < indicates MUCH LESS than.

References

Davenport, H. "The Polya-Vinogradov Inequality." Ch. 23

in Multiplicative Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 135-138, 1980.
Polya, G. "Uber die Verteilung der quadratischen Reste
und Nichtreste." Nachr. Konigl. Gesell. Wissensch.
Gottingen, Math.-Phys. Klasse, 21-29, 1918.

1396 Polyabolo

Polygamma Function

Polyabolo

An analog of the POLYOMINO composed of n ISOSCE-
LES RIGHT Triangles joined along edges of the same
length. The number of polyaboloes composed of n trian-
gles are 1, 3, 4, 14, 30, 107, 318, 1106, 3671, . . . (Sloane's
A006074).

see also DlABOLO, HEXABOLO, PENTABOLO, TETRA-
BOLO, TRIABOLO

References

Sloane, N. J. A. Sequence A006074/M2379 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Polyconic Projection

x = cot (j) sin E

y = (<j> - <£o) + cot 0(1 - cos E) 1

where

E = (A — Ao)sin<£.
The inverse FORMULAS are

sin -1 (x tan <j>)

X = ; ; \~ Aq ,

sin0

and <p is determined from

A0 =

A(^tan<£+l)-0- \{<j> 2 +£)tan<£

4>-A _ 1

tan <j>

where <f>o = A and

A = 4> + y
B = x 2 + A 2 .

(1)
(2)

(3)
(4)

, (5)

(6)
(7)

References

Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 124-137, 1987.

Polycube

3-D generalization of the POLYOMINOES to n-D. The
number of polycubes N(n) composed of n Cubes are 1,
1, 2, 8, 29, 166, 1023, ... (Sloane's A000162, Ball and
Coxeter 1987).

see also Conway Puzzle, Cube Dissection, Diabol-
ical Cube, Slothouber-Graatsma Puzzle, Soma
Cube

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 112-
113, 1987.

Gardner, M. The Second Scientific American Book of Math-
ematical Puzzles & Diversions: A New Selection. New
York: Simon and Schuster, pp. 76-77, 1961.

Gardner, M. "Polycubes." Ch. 3 in Knotted Doughnuts and
Other Mathematical Entertainments. New York: W. H.
Freeman, 1986.

Sloane, N. J. A. Sequence A000162/M1845 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Polydisk

Let c = (ci,...,c n ) be a point in C n , then the open
polydisk is defined by

S = {z:\z j -c j \<\z° j -c j \}

for j = 1, ..., n.

see also DISK, OPEN DISK

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 100, 1980.

Polygamma Function

The polygamma function is sometimes denoted F 7n (z) i
and sometimes ip m (z). In F m (z) notation,

d m+i

= (-l) m+1 m!f% i— T

71 =

= (-l) ra+1 m!C(m+l,«),

(1)
(2)
(3)

Polygamma Function

Polygon 1397

where ((a,z) is the Hurwitz Zeta FUNCTION.
In the ipm NOTATION (the form returned by the
PolyGamma[m,z] function in Mathematica® '; Wolfram
Research, Champaign, IL),

tpm{z)

jm+1

dz m
d™ T'(z)

T ln[T(*)]

d™

dz m T(z) dz"

*w,

(4)

where T(z) is the Gamma Function and V(z) is the
DlGAMMA FUNCTION. tpm(z) is therefore related to
F m {z) by

il> m (z) = F m {z-l). (5)

The function ipo(z) is equivalent to the DlGAMMA FUNC-
TION ^f(z). Note that Morse and Feshbach (1953) adopt
a notation no longer in standard use in which Morse and
Feshbach 's iprn(z) is equal to the above tpm-i(z).

The polygamma function obeys the RECURRENCE RE-
LATION

i> n (z + 1) = ^ n {z) + (-l) n n\z- n -\ (6)

the reflection FORMULA

,Mi - z) + (-ir + VnW = (-ir^^r cot (^)> ( 7 )

and the multiplication FORMULA,

1

m — 1

^ n (mz) = (i n olnmH -r > ip n (zl ), (8)

where <5 m „ is the Kronecker Delta.

In general, special values for integral indices are given
by

giving

Vn(l) = (-l)" +1 n!C(n + l)

V'n(i) = (-l) n+1 n!(2" +1 -l)C(n + l)

V-i(l) = §* 2

^X(I) = C(2) = iTT 2

^(1) = -2<(3),

(9)
(10)

(11)

(12)
(13)

(14)

and so on.

R. Manzoni has shown that the polygamma function
can be expressed in terms of CLAUSEN FUNCTIONS for

Rational arguments and integer index. Special cases
are given by

Ml)
Ml)
M\)
Ml)
Ml)

MD = -

f^ + fVslcMH-cMfTr)

|7r 2 -f^[Cl 2 (|7r)-Cl 2 (|7r)
7r 2 +4[Cl 2 (l7r)-Cl 2 (|7r)]

7r 2 -4[Cl 2 (l7r)-Cl 2 (§^)].
-8[C1 3 (0)-Cl 3 (7r)].

4tt 3

3V3

(15)
(16)
(17)
(18)
(19)

18C1 3 (0) + 9[CU(|ir) + Cl 3 (f tt)]

lMf) =

4tt j
3\/3

18 Cls(0) + 9[Cls(f ff) + Cls(|ir)]

M\) = -2tt 3 - 32[C1 3 (0) - C1 3 (tt)]
Ml ) = 2tt 3 - 32[C1 3 (0) - Ola (it)]

§tt 4 + 8lV3[Cl4(§7r) - CMItt)]
|7r 4 -81V^[Cl 4 (f7r)-Cl 4 (|7r)]
Ml) = 8*- 4 + 384[Cl 4 (l7r) - Cl 4 (f ir)]
Ml) = 8tt 4 - 384[CL,(f tt) - Cl 4 (f;r)].

Ml)

Mi)

(20)

']

(21)
(22)
(23)
(24)
(25)
(26)
(27)

see also Clausen Function, Digamma Function,
Gamma Function, Stirling's Series

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Polygamma

Functions." §6.4 in Handbook of Mathematical Functions

with Formulas, Graphs, and Mathematical Tables, 9th

printing. New York: Dover, p. 260, 1972.
Adamchik, V. S. "Polygamma Functions of Negative Order."

Submitted to J. Symb. Comput. http: //www. wolfram.

com/~victor/articles/polyg.html.
Arfken, G. "Digamma and Polygamma Functions." §10.2 in

Mathematical Methods for Physicists, 3rd ed. Orlando,

FL: Academic Press, pp. 549-555, 1985.
Davis, H. T. Tables of the Higher Mathematical Functions.

Bloomington, IN: Principia Press, 1933.
Kolbig, V. "The Polygamma Function ^* (x) for x = 1/4 and

x - 3/4." J. Comp. Appl. Math. 75, 43-46, 1996.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 422-424, 1953.

Polygenic Function

A function which has infinitely many Derivatives at a
point. If a function is not polygenic, it is MONOGENIC.

see also Monogenic Function

References

Newman, J. R. The World of Mathematics, Vol. 3.
York: Simon & Schuster, p. 2003, 1956.

New

Polygon

A closed plane figure with n sides. If all sides and angles
are equivalent, the polygon is called regular. Regular
polygons can be CONVEX or STAR. The word derives
from the Greek poly (many) and gonu (knee).

1398 Polygon

Polygon

The AREA of a polygon with VERTICES (a?i,j/i), ...,
(xn>yn) is

-K

Xl

yi

+

X 2

2/2

+

+

Xn

Vn

X2

2/2

xs

y*

Xl

yi

(1)

which can be written

v4 = 5(0:12/2 + 3J2J/1 + - ■ • + Xn-lVn + X n y\ - 2/1^2

-2/2^3 - ... - 2/n+l^n ~ l/n^l), (2)

where the signs can be found from the following diagram.

The AREA of a polygon is denned to be POSITIVE if
the points are arranged in a counterclockwise order, and
NEGATIVE if they are in clockwise order (Beyer 1987).

The sum / of internal angles in the above diagram of a
dissected PENTAGON is

n n n

/ = ^(Q i +/30 = ^(a i + A+7i)-^7i. (3)

But

n

$> = 360° (4)

1=1

and the sum of Angles of the n Triangles is

n n

£(«i + A + 7<) = 5^(180°) = n(180°). (5)

1=1 i— 1

Therefore,

I = n(180°) - 360° = (n - 2)180°. (6)

Let n be the number of sides. The regular n-gon is then
denoted {n}.

n

W

2

digon

3

equilateral triangle (trigon)

4

square (quadrilateral, tetragon)

5

pentagon

6

hexagon

7

heptagon

8

octagon

9

nonagon (enneagon)

10

decagon

11

undecagon (hendecagon)

12

dodecagon

13

tridecagon (triskaidecagon)

14

tetradecagon (tetrakaidecagon)

15

pentadecagon (pentakaidecagon)

16

hexadecagon (hexakaidecagon)

17

heptadecagon (heptakaidecagon)

18

octadecagon (octakaidecagon)

19

enneadecagon (enneakaidecagon)

20

icosagon

30

triacontagon

40

tetracontagon

50

pent acont agon

60

hexacontagon

70

heptacontagon

80

octacontagon

90

enneacontagon

100

hectogon

10000

myriagon

n = 5
Let 5 be the side length, r be the Inradius, and R the
ClRCUMRADIUS. Then

3 = 2rtan (-) = 2Rsm(-J

(7)

r=1 > scot (l)

(8)

R = * acac {l)

(9)

A = \ns 2 cot (^)

(10)

= nr 2 tan 1 — 1

(11)

= losing).

(12)

If the number of sides is doubled, then

s 2 „ = \J2R 2 - Rs/AR? - s„ 2

(13)

A. 4rj4 "

(14)

" H 1r + v/4r 2 + s„ 2

Polygon

Polygon 1399

Furthermore, if pk and Pk are the Perimeters of the
regular polygons inscribed in and circumscribed around
a given CIRCLE and a,k and Ak their areas, then

2p n P n

Pn +Pn
P2n — yPnP2n->

and

Q>2n
A 2n

Vein A n

2d2nAn
CL2n + A n

(15)
(16)

(17)
(18)

(Beyer 1987, p. 125).

Compass and Straightedge constructions dating
back to Euclid were capable of inscribing regular poly-
gons of 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, ... ,
sides. However, this listing is not a complete enumera-
tion of "constructible" polygons. In fact, a regular n-gon
is constructible only if <j>{n) is a Power of 2, where <j>
is the TOTIENT FUNCTION (this is a NECESSARY but
not Sufficient condition). More specifically, a regular
n-gon (n > 3) can be constructed by Straightedge
and COMPASS (i.e., can have trigonometric functions of
its Angles expressed in terms of finite SQUARE ROOT
extractions) IFF

2 k p x p2 '

'•Ps

(19)

where k is in Integer > and the p, are distinct Fer-
mat Primes. Fermat Numbers are of the form

F m = 2^ +1,

(20)

where m is an INTEGER > 0. The only known PRIMES
of this form are 3, 5, 17, 257, and 65537.

The fact that this condition was Sufficient was first
proved by Gauss in 1796 when he was 19 years old, and
it relies on the property of IRREDUCIBLE POLYNOMIALS
that ROOTS composed of a finite number of SQUARE
ROOT extractions exist only if the order of the equation
is of the form 2 h . That this condition was also Neces-
sary was not explicitly proven by Gauss, and the first
proof of this fact is credited to Wantzel (1836).

Constructible values of n for n < 300 were given by
Gauss (Smith 1994), and the first few are 2, 3, 4, 5, 6,
8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60,
64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192,
. . . (Sloane's A003401). Gardner (1977) and indepen-
dently Watkins (Conway and Guy 1996) noticed that
the number of sides for constructible polygons with an
ODD number of sides is given by the first 32 rows of PAS-
CAL'S Triangle (mod 2) interpreted as Binary num-
bers, giving 1, 3, 5, 15, 17, 51, 85, 255, . . . (Sloane's
A004729, Conway and Guy 1996, p. 140).

1

1 1

1 2 1

13 3 1

14 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1

1

1 1

3

1 1

5

1111

15

10 1

17

110 11 51

10 10 10 1 85

11111111 255

10 1 257

Although constructions for the regular TRIANGLE,
Square, Pentagon, and their derivatives had been
given by Euclid, constructions based on the Fermat
Primes > 17 were unknown to the ancients. The
first explicit construction of a Heptadecagon (17-gon)
was given by Erchinger in about 1800. Richelot and
Schwendenwein found constructions for the 257-GON in
1832, and Hermes spent 10 years on the construction
of the 65537-G0N at Gottingen around 1900 (Coxeter
1969). Constructions for the EQUILATERAL TRIANGLE
and Square are trivial (top figures below). Elegant con-
structions for the Pentagon and Heptadecagon are
due to Richmond (1893) (bottom figures below).

O W, ''o N 5 F O E N 3

Pentagon 17-gon

Given a point, a Circle may be constructed of any
desired Radius, and a Diameter drawn through the
center. Call the center O, and the right end of the DI-
AMETER P . The Diameter Perpendicular to the
original DIAMETER may be constructed by finding the
Perpendicular Bisector. Call the upper endpoint
of this Perpendicular Diameter B. For the Pen-
tagon, find the MIDPOINT of OB and call it D. Draw
DPo, and Bisect lODPo, calling the intersection point
with OP Ni. Draw TViPi PARALLEL to OP, and the
first two points of the PENTAGON are Po and Pi. The
construction for the HEPTADECAGON is more compli-
cated, but can be accomplished in 17 relatively simple
steps. The construction problem has now been auto-
mated (Bishop 1978).

see also 257-gon, 65537-gon, Anthropomorphic
Polygon, Bicentric Polygon, Carnot's Poly-
gon Theorem, Chaos Game, Convex Polygon,
Cyclic Polygon, de Moivre Number, Diagonal
(Polygon), Equilateral Triangle, Euler's Poly-
gon Division Problem, Heptadecagon, Hexagon,

1400 Polygon

Hexagram, Illumination Problem, Jordan Poly-
gon, Lozenge, Octagon, Parallelogram, Pas-
cal's Theorem, Pentagon, Pentagram, Petrie
Polygon, Polygon Circumscribing Constant,
Polygon Inscribing Constant, Polygonal Knot,
Polygonal Number, Polygonal Spiral, Polygon
Triangulation, Polygram, Polyhedral Formula,
Polyhedron, Polytope, Quadrangle, Quadri-
lateral, Regular Polygon, Reuleaux Poly-
gon, Rhombus, Rotor, Simple Polygon, Simplic-
ity, Square, Star Polygon, Trapezium, Trape-
zoid, Triangle, Visibility, Voronoi Polygon,
Wallace-Bolyai-Gerwein Theorem

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.

Boca Raton, FL: CRC Press, pp. 124-125 and 196, 1987.
Bishop, W. "How to Construct a Regular Polygon." Amer.

Math. Monthly 85, 186-188, 1978.
Conway, J. H. and Guy, R. K. The Book of Numbers. New

York: Springer- Verlag, pp. 140 and 197-202, 1996.
Courant, R. and Robbins, H. "Regular Polygons." §3.2 in

What is Mathematics ?: An Elementary Approach to Ideas

and Methods, 2nd ed. Oxford, England: Oxford University

Press, pp. 122-125, 1996.
Coxeter, H. S.M. Introduction to Geometry, 2nd ed. New

York: Wiley, 1969.
De Temple, D. W. "Carlyle Circles and the Lemoine Simplic-
ity of Polygonal Constructions." Amer. Math. Monthly 98,

97-108, 1991.
Gardner, M. Mathematical Carnival: A New Round-Up of

Tantalizers and Puzzles from Scientific American. New

York: Vintage Books, p. 207, 1977.
Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae.

Leipzig, Germany, 1801. Translated by A. A Clarke. New

Haven, CT: Yale University Press, 1965.
The Math Forum. "Naming Polygons and Polyhe-

dra." http: //forum, swarthmore . edu/dr .math/f aq/f aq.

polygon. names .html.
Rawles, B. Sacred Geometry Design Sourcebook: Universal

Dimensional Patterns. Nevada City, CA: Elysian Pub.,

p. 238, 1997.
Richmond, H. W. "A Construction for a Regular Polygon of

Seventeen Sides." Quart. J. Pure Appl. Math. 26, 206-

207, 1893.
Sloane, N. J. A. Sequences A004729 and A003401/M0505 in

"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Smith, D. E. A Source Book in Mathematics. New York:

Dover, p. 350, 1994.
Tietze, H. Ch. 9 in Famous Problems of Mathematics. New

York: Graylock Press, 1965.
Wantzel, P. L. "Recherches sur les moyens de reconnaitre si

un Probleme de Geometrie peut se resoudre avec la regie

et le compas." J. Math, pures appliq. 1, 366-372, 1836.

Polygon Circumscribing Constant
Polygon Circumscribing Constant

If a Triangle is Circumscribed about a Circle, an-
other Circle around the Triangle, a Square outside
the Circle, another Circle outside the Square, and
so on. Prom POLYGONS, the CIRCUMRADIUS and Inra-
DIUS for an n-gon are

R=1 * scsc (l)

r= Is cot (I)

where s is the side length. Therefore,
1

R _

r cos

(5)

(=)•

(1)

(2)
(3)

and an infinitely nested set of circumscribed polygons
and circles has

K =

^"final circle
^initial circle

= sec (|) sec (|) sec (I)-. (4)

Kasner and Newman (1989) and Haber (1964) state that
K = 12, but this is incorrect. Write

oo
oo

In K = — y. ln(cos x) .

n = 3

(5)

(6)

Define

yo{x) = -ln(cosa;) = \x 2 + ^x 4 + ±x 6 + ^x 8 + . . . .

ow define

yi(x) = \ax 2 y

(8)

ith

Vi(i) = W>(f)

(9)

Hf) 2 =ln2,

(10)

>

a = 2(-) ln2,

(11)

Polygon Circumscribing Constant

and

91n2 2

!/2W = ~^r x •

But 2/2 (#) > yi(^) for x € (0,7r/3), so

oo oo

(12)

(13)

n—1 n=l

2

n=3 n—3 n = Z

/ oo 2 \

(14)

= 91n2 I y - 7 I =2.4637

K < e 24637 = 11.75.

If the next term is included,

As before,

2/2 (z) = a (i x2 + H* 4 )-

W(f) = l»(f)
972 In 2

2/2(2;)

7r 2 (54 + 7r 2 )'
972 In 2 t a x 4

(15)

(16)

(17)
(18)

(19)

, v 9721n2 -A [l /tt\ 2 ! ,„■.

lnK< ^(54 + ^)^ 2W + 12VnJ

972 In 2 Jl r 51 tt 2 f 111

= 7 r 2 (54 + \2[ C(2) "i] + ^[ C(4) " 1 "2?J}

1 /V

972 In 2

7r 2 (54 + 7r 2 )

9(87T 6 -457r 2 -5400)ln2
80(tt 2 + 54)

4 / + 12 I 90 2 4

2.255,

and

if < e 2255 = 9.535.

(20)

(21)

The process can be automated using computer algebra,
and the first few bounds are 11.7485, 9.53528, 8.98034,
8.8016, 8.73832, 8.71483, 8.70585, 8.70235, 8.70097, and
8.70042. In order to obtain this accuracy by direct mul-
tiplication of the terms, more than 10,000 terms are
needed. The limit is

Polygon Fractal 1401

Bouwkamp (1965) produced the following INFINITE
Product formulas

K

6 exp < y^

U=i

2/br

[A(2fc)-l]2 2fc [C(2fc)-l-2- 2fe ]
k

(23)

}■

(24)

where ((x) is the Riemann ZETA FUNCTION and X(x) is
the Dirichlet Lambda Function. Bouwkamp (1965)
also produced the formula with accelerated convergence

k = ^N/eVu - i* 2 + £0(1 - h 2 + as* 4 )

/ 2 \ / *r 2 \

B, (25)

X esc

v / 6 + 2 v / 3,

esc

y/l - lyfl,

where

**n

2n 2 + 24n 4

h) sec il)

(26)

(cited in Pickover 1995).

see also Polygon Inscribing Constant

References

Bouwkamp, C. "An Infinite Product." Indag. Math. 27,

40-46, 1965.
Finch, S. "Favorite Mathematical Constants." http://www.

mathsof t . c om/ as ol ve / c ons t ant /infprd/ infprd.html.
Haber, H. "Das Mathematische Kabinett." Bild der Wis-

senschaft 2, 73, Apr. 1964.
Kasner, E. and Newman, J. R. Mathematics and the Imag-
ination. Redmond, WA: Microsoft Press, pp. 311-312,

1989.
Pappas, T. "Infinity Sc Limits." The Joy of Mathematics.

San Carlos, CA: Wide World Publ./Tetra, p. 180, 1989.
Pickover, C, A. "Infinitely Exploding Circles." Ch. 18 in

Keys to Infinity. New York: W. H. Freeman, pp. 147-151,

1995.
Pinkham, R. S. "Mathematics and Modern Technology."

Amer. Math. Monthly 103, 539-545, 1996,
Plouffe, S. "Product(cos(Pi/n),n=3..infinity)." http://

lac im.uqam.ca/piDATA/pr oductcos.txt.

Polygon Construction

see Geometric Construction, Geometrography,
Polygon, Simplicity

Polygon Division Problem

see Euler's Polygon Division Problem

Polygon Fractal

see Chaos Game

K = 8.700036625 ....

(22)

1402 Polygon Inscribing Constant

Polygonal Number

Polygon Inscribing Constant

If a Triangle is inscribed in a Circle, another Cir-
cle inside the TRIANGLE, a SQUARE inside the CIRCLE,
another CIRCLE inside the SQUARE, and so on,

Polygonal Number

K' =

^initial circle

= cos (|) cos (J) cos (J)

Numerically,

K'

1

K 8.7000366252..

= 0.1149420448.

where K is the POLYGON CIRCUMSCRIBING CONSTANT.
Kasner and Newman's (1989) assertion that K = 1/12
is incorrect.

Let a convex POLYGON be inscribed in a CIRCLE and
divided into TRIANGLES from diagonals from one Ver-
tex. The sum of the Radii of the CIRCLES inscribed in
these Triangles is the same independent of the Ver-
tex chosen (Johnson 1929, p. 193).

see also POLYGON CIRCUMSCRIBING CONSTANT

References

Finch, S. "Favorite Mathematical Constants." http://wvw.

maths oft . c om/ as o 1 ve / c ons t ant /infprd/ infprd.html.
Johnson, R. A. Modern Geometry: An Elementary Treatise

on the Geometry of the Triangle and the Circle. Boston,

MA: Houghton Mifflin, 1929.
Kasner, E. and Newman, J. R. Mathematics and the Imag-
ination, Redmond, WA: Microsoft Press, pp. 311-312,

1989.
Pappas, T. "Infinity & Limits." The Joy of Mathematics.

San Carlos, CA: Wide World Publ./Tetra, p. 180, 1989.
Plouffe, S. "Product(cos(Pi/n),n=3.. infinity)." http://

lac im.uqam.ca/piDATA/pr oductcos.txt.

Polygon Triangulation

see Euler's Polygon Division Problem

Polygonal Knot

A Knot equivalent to a Polygon in M 3 , also called
a Tame Knot. For a polygonal knot K, there exists
a Plane such that the orthogonal projection 7r on it
satisfies the following conditions:

1. The image 7r(K) has no multiple points other than
a FINITE number of double points.

2. The projections of the vertices of K are not double
points of n(K).

Such a projection 7t(K) is called a regular knot projec-
tion.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 735, 1980.

A type of FlGURATE NUMBER which is a generalization
of Triangular, Square, etc., numbers to an arbitrary
n-gonal number. The above diagrams graphically illus-
trate the process by which the polygonal numbers are
built up. Starting with the nth Triangular Number
T„, then

n + T«_i=T„. (1)

Now note that

n + 2T„-i

gives the nth SQUARE Number,

n + 3T n _! = |n(3n - 1) = P„,

(2)

(3)

gives the nth PENTAGONAL NUMBER, and so on. The
general polygonal number can be written in the form

Pr = fr[(r-l)n-2(r-2)] = |r[(n-2)r-(n-4)], (4)

where p™ is the rth n-gonal number. For example, tak-
ing n = 3 in (4) gives a Triangular Number, n = 4
gives a Square Number, etc.

Fermat proposed that every number is expressible as at
most k fc-gonal numbers (Fermat's POLYGONAL NUM-
BER Theorem). Fermat claimed to have a proof of this
result, although this proof has never been found. Ja-
cobi, Lagrange (1772), and Euler all proved the square
case, and Gauss proved the triangular case in 1796. In
1813, Cauchy proved the proposition in its entirety.

An arbitrary number N can be checked to see if it is a
n-gonal number as follows. Note the identity

8(n - 2)p r n + (n - 4) 2 = 4r(n - 2)[(r - l)n - 2(r - 2)]
+ (n - 4) 2 = 4r(r - l)n 2 + r[-8(r - 1) - 8(r - 2)}n
+16r(r-2) + (n 2 -8n + 16)
= (4r 2 - 4r + l)n 2 + (-16r 2 + 24r - 8)n

+(16r 2 - 32r + 16)
= (2r - 1)V - 8(2r 2 - 3r + l)n + 16(r 2 - 2r + 1)

= (2rn-4r-n + 4) 2 , (5)

so 8(n - 2)N + (n - 4) 2 = S 2 must be a PERFECT
SQUARE. Therefore, if it is not, the number cannot be
n-gonal. If it is a Perfect Square, then solving

5 = 2rn — 4r — n + 4

(6)

Polygonal Spiral

Polyhedral Formula 1403

for the rank r gives

S + n-4
2(n-2) '

(7)

An n-gonal number is equal to the sum of the (n — 1)-
gonal number of the same RANK and the TRIANGULAR
Number of the previous Rank.

see also CENTERED POLYGONAL NUMBER, DECAGONAL

Number, Fermat's Polygonal Number Theorem,
Figurate Number, Heptagonal Number, Hexag-
onal Number, Nonagonal Number, Octagonal
Number, Pentagonal Number, Pyramidal Num-
ber, Square Number, Triangular Number

References

Beiler, A. H. "Ball Games." Ch. 18 in Recreations in the The-
ory of Numbers: The Queen of Mathematics Entertains.
New York: Dover, pp. 184-199, 1966.

Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 3-33,
1952.

Guy, K. "Every Number is Expressible as a Sum of How
Many Polygonal Numbers?" Amer. Math. Monthly 101,
169-172, 1994.

Pappas, T. "Triangular, Square & Pentagonal Numbers."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, p. 214, 1989.

Sloane, N. J. A. Sequences A000217/M2535 in "An On-Line
Version of the Encyclopedia of Integer Sequences,"

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.

Polygonal Spiral

The length of the polygonal spiral is found by noting
that the ratio of INRADIUS to ClRCUMRADIUS of a regu-
lar Polygon of n sides is

r
R

cot (5)

^c (5)

©

(1)

The total length of the spiral for an n-gon with side
length 5 is therefore

oo

L -»'l>'(;)->[i-«(i)]

(2)

Consider the solid region obtained by filling in subse-
quent triangles which the spiral encloses. The AREA of
this region, illustrated above for n-gons of side length s,

A =

**-*(l)

(3)

References

Sandefur, J. T. "Using Self- Similarity to Find Length, Area,

and Dimension." Amer. Math. Monthly 103, 107-120,

1996.

Polygram

A self-intersecting Star Figure such as the Penta-
gram or Hexagram.

n

symbol

polygram

5

{5/2}

pentagram

6

{6/2}

hexagram

7

{7/2}

heptagram

8

{8/3}

octagram

{8/4}

star of Lakshmi

10

{10/3}

decagram

Polyhedral Formula

A formula relating the number of Vertices, Faces, and
Edges of a Polyhedron (or Polygon). It was discov-
ered independently by Euler and Descartes, so it is also
known as the Descartes-Euler Polyhedral For-
mula. The polyhedron need not be CONVEX, but the
Formula does not hold for Stellated Polyhedra.

V + F - E = 2,

(1)

1404 Polyhedral Graph

Polyhedron

where V = No is the number of VERTICES, E — N\ is
the number of EDGES, and F ~ N 2 is the number of
FACES. For a proof, see Courant and Robbins (1978,
pp. 239-240). The FORMULA can be generalized to n-D
POLYTOPES.

n x : No = 2 (2)

n 2 : No - Ni = (3)

U 3 :No-N 1 +N 2 =2 (4)

n 4 : No - Ni + N 2 - iV 3 = (5)

n n : No - JVi + iV 2 - . . . + (-lr-'iVn-i = 1 - (-1) 71 .

(6)

For a proof of this, see Coxeter (1973, pp. 166-171).
see also Dehn Invariant, Descartes Total Angu-
lar Defect

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 128, 1987.

Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, Oxford, Eng-
land: Oxford University Press, 1978.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, 1973.

Polyhedral Graph

The graphs corresponding to the skeletons of PLATONIC
Solids. They are special cases of Schlegel Graphs.

see also CUBICAL GRAPH, DODECAHEDRAL GRAPH,

icosahedral graph, octahedral graph, schle-
gel Graph, Tetrahedral Graph

Polyhedron

A 3-D solid which consists of a collection of POLYGONS,
usually joined at their EDGES. The word derives from
the Greek poly (many) plus the Indo-European hedron
(seat). A polyhedron is the 3-D version of the more
general POLYTOPE, which can be defined on arbitrary
dimensions.

A Convex Polyhedron can be defined as the set of
solutions to a system of linear inequalities

mx < b,

where m is a real s x 3 Matrix and b is a real s- Vector.
An example is illustrated above. The more simple Do-
decahedron is given by a system with s = 12. In gen-
eral, given the Matrices, the Vertices (and Faces)
can be found using VERTEX ENUMERATION.

A polyhedron is said to be regular if its FACES and
Vertex Figures are Regular (not necessarily Con-
vex) polygons (Coxeter 1973, p. 16). Using this defi-
nition, there are a total of nine REGULAR Polyhedra,
five being the CONVEX PLATONIC SOLIDS and four be-
ing the Concave (stellated) Kepler-Poinsot Solids.
However, the term "regular polyhedra" is sometimes
also used to refer exclusively to the Platonic Solids
(Cromwell 1997, p. 53). The Dual Polyhedra of the
PLATONIC Solids are not new polyhedra, but are them-
selves Platonic Solids.

A Convex polyhedron is called Semiregular if its
FACES have a similar arrangement of nonintersecting
regular plane CONVEX polygons of two or more dif-
ferent types about each Vertex (Holden 1991, p. 41).
These solids are more commonly called the ARCHIMED-
EAN Solids, and there are 13 of them. The DUAL
Polyhedra of the Archimedean Solids are 13 new
(and beautiful) solids, sometimes called the CATALAN
Solids.

A QUASIREGULAR POLYHEDRON is the solid region inte-
rior to two Dual Regular Polyhedra (Coxeter 1973,
pp. 17-20). There are only two CONVEX QUASIREGU-
lar Polyhedra: the Cuboctahedron and Icosido-
DECAHEDRON. There are also infinite families of PRISMS
and Antiprisms.

There exist exactly 92 CONVEX POLYHEDRA with REG-
ULAR POLYGONAL faces (and not necessary equivalent
vertices). They are known as the JOHNSON Solids.
Polyhedra with identical VERTICES related by a sym-
metry operation are known as UNIFORM POLYHEDRA.
There are 75 such polyhedra in which only two faces
may meet at an EDGE, and 76 in which any Even num-
ber of faces may meet. Of these, 37 were discovered
by Badoureau in 1881 and 12 by Coxeter and Miller
ca. 1930.

Polyhedra can be superposed on each other (with the
sides allowed to pass through each other) to yield ad-
ditional Polyhedron Compounds. Those made from
Regular Polyhedra have symmetries which are espe-
cially aesthetically pleasing. The graphs corresponding
to polyhedra skeletons are called SCHLEGEL GRAPHS.

Behnke et al. (1974) have determined the symmetry
groups of all polyhedra symmetric with respect to their
VERTICES.

Polyhedron

Polyhedron Compound 1405

see also ACOPTIC POLYHEDRON, APEIROGON, ARCHI-
MEDEAN Solid, Canonical Polyhedron, Catalan
Solid, Cube, Dice, Digon, Dodecahedron, Dual
Polyhedron, Echidnahedron, Flexible Poly-
hedron, Hexahedron, Hyperbolic Polyhedron,
icosahedron, isohedron, johnson solid, kepler-
Poinsot Solid, Nolid, Octahedron, Petrie Poly-
gon, Platonic Solid, Polyhedron Coloring,
Polyhedron Compound, Prismatoid, Quadricorn,
Quasiregular Polyhedron, Rigidity Theorem,
Semiregular Polyhedron, Skeleton, Tetrahe-
dron, Uniform Polyhedron, Zonohedron

References

Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in
Mathematical Recreations and Essays, 13th ed. New York:
Dover, pp. 130-161, 1987.

Behnke, H.; Bachman, F.; Fladt, K.; and Kunle, H, (Eds.).
Fundamentals of Mathematics, Vol. 2. Cambridge, MA:
MIT Press, 1974.

Bulatov, V. "Polyhedra Collection." http: //www. physics.
orst . edu/~bulatov/polyhedra/.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, 1973.

Critchlow, K. Order in Space: A Design Source Book. New
York: Viking Press, 1970.

Cromwell, P. R. Polyhedra. New York: Cambridge University
Press, 1997.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., 1989.

Davie, T. "Books and Articles about Polyhedra and
Polytopes." http : //www . dcs . st-andrews . ac.uk/-ad/

mathrecs/polyhedra/polyhedrabooks.html.

Davie, T. "The Regular (Platonic) and Semi-Regular (Ar-
chimedean) Solids." http : //www . dcs . st-andrews . ac . uk/
"ad/mathrecs/polyhedra/polyhedrat opic.html.

Eppstein, D. "Geometric Models." http://www.ics.uci.
edu/-eppstein/ junkyard/model. html.

Eppstein, D. "Polyhedra and Polytopes." http://www.ics.
uci . edu/-eppstein/ junkyard/polytope .html.

Hart, G. W. "Virtual Polyhedra." http://www.li.net/
-george/virtual -polyhedra/ vp. html.

Hilton, P. and Pedersen, J. Build Your Own Polyhedra.
Reading, MA: Addison- Wesley, 1994.

Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.

Lyusternik, L. A. Convex Figures and Polyhedra. New York:
Dover, 1963.

Malkevitch, J. "Milestones in the History of Polyhedra." In
Shaping Space: A Polyhedral Approach (Ed. M. Senechal
and G. Fleck). Boston, MA: Birkhauser, pp. 80-92, 1988.

Miyazaki, K. An Adventure in Multidimensional Space: The
Art and Geometry of Polygons, Polyhedra, and Polytopes.
New York: Wiley, 1983.

Paeth, A. W. "Exact Dihedral Metrics for Common Poly-
hedra." In Graphic Gems II (Ed. J. Arvo). New York:
Academic Press, 1991.

Pappas, T. "Crystals-Nature's Polyhedra." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 38-39, 1989.

Pugh, A. Polyhedra: A Visual Approach. Berkeley: Univer-
sity of California Press, 1976.

Schaaf, W. L. "Regular Polygons and Polyhedra." Ch. 3, §4
in A Bibliography of Recreational Mathematics. Washing-
ton, DC: National Council of Teachers of Math., pp. 57-60,
1978.

Virtual Image. "Polytopia I" and "Polytopia II" CD-
ROMs, http : //ourworld. CompuServe . com/homepages/
vir_image/html/polytopiai .html and polytopiaii.html.

Polyhedron Coloring

Define a valid "coloring" to occur when no two faces
with a common Edge share the same color. Given two
colors, there is a single way to color an OCTAHEDRON.
Given three colors, there is one way to color a Cube and
144 ways to color an ICOSAHEDRON. Given four-colors,
there are two distinct ways to color a TETRAHEDRON
and 4 ways to color a DODECAHEDRON. Given five col-
ors, there are four ways to color an ICOSAHEDRON.

see also Coloring, Polyhedron

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, 238-242,
1987.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 82-83, 1989.

Polyhedron Compound

Solid

Vertices

Symbol

cube-octahedron

both

dodec.+icos.

both

two cubes

three cubes

four cubes

five cubes

dodecahedron

2{5,3}[5{4,3}]

five octahedra

icosidodeca.

[5{3,4}]2{3,5}

five tetrahedra

dodecahedron

{5,3}[5{3,3}]2{3,5}

two dodecahedra

both

great dodecahedron-

small stellated dodec.

great icosahedron-

both

great stellated dodec.

stella octangula

cube

{4,3}[2{3,3}]{3,4}

ten tetrahedra

dodecahedron

2{5,3}[10{3,3}]2{3,5}

The above table gives some common polyhedron com-
pounds. In Coxeter's NOTATION, d distinct VERTICES
of {ra, n} taken c times are denoted

c{m,n}[d{p,q}],

or faces of {s, t} e times

{d{p,q}]e{s,t},

or both

c{m,n}[d{p,q}}e{s y t}.

The five TETRAHEDRA can be arranged in a laevo or

dextro configuration.

see also CUBE-OCTAHEDRON COMPOUND, DODECA-

hedron-icosahedron compound, octahedron 5-
Compound, Stella Octangula, Tetrahedron 5-
Compound

1406 Polyhedron Dissection

Polyking

Polyhedron Dissection

A Dissection of one or more polyhedra into other

shapes.

see also CUBE DISSECTION, DIABOLICAL CUBE, POLY-

cube, Soma Cube, Wallace-Bolyai-Gerwein The-
orem

References

Bulatov, V.v "Compounds of Uniform Polyhedra." http://
www . physics . orst . edu/ ~bulatov/polyhedra/unif orm_
compounds/.

Coffin, S. T. The Puzzling World of Polyhedral Dissections.
New York: Oxford University Press, 1990.

Polyhedron Dual

see Dual Polyhedron

Polyhedron Hinging

see Rigidity Theorem

Polyhedron Packing

see Kelvin's Conjecture, Space-Filling Polyhe-
dron

Polyhex

3

<% ^b

An analog of the POLYOMINOES and POLYIAMONDS in
which collections of regular hexagons are arranged with
adjacent sides. They are also called Hexes and HEXAS.
The number of polyhexes of n hexagons are 1, 1, 2,
7, 22, 82, 333, 1448, 6572, 30490, 143552, 683101, ...
(Sloane's A014558). For the 4-hexes (tetrahexes), the
possible arrangements are known as the Bee, Bar, PIS-
TOL, Propeller, Worm, Arch, and Wave.

References

Gardner, M. "Polyhexes and Polyaboloes." Ch. 11 in Mathe-
matical Magic Show: More Puzzles, Games, Diversions,
Illusions and Other Mathematical Sleight- of- Mind from
Scientific American. New York: Vintage, pp. 146-159,
1978.

Gardner, M. "Tiling with Polyominoes, Polyiamonds, and
Polyhexes." Ch. 14 in Time Travel and Other Mathemat-
ical Bewilderments. New York: W. H. Freeman, pp. 175—
187, 1988.

Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, pp. 92-93, 1994.

Sloane, N. J. A. Sequence A014558 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, pp. 342-343, 1993.

Polyiamond

i A

2 £7

3 £A

4 A7V ^^

5 ** A^£Z£ &

6 A7W\ A W \7 A yr Aa A7^ /0^

_ A7 A . A AA AA
£57 £57 £A £A/ W

A generalization of the POLYOMINOES using a collec-
tion of equal-sized Equilateral Triangles (instead of
Squares) arranged with coincident sides. Polyiamonds
are sometimes simply known as IAMONDS.

The number of two-sided (i.e., can be picked up and
nipped, so MIRROR IMAGE pieces are considered iden-
tical) polyiamonds made up of n triangles are 1, 1, 1,
3, 4, 12, 24, 66, 160, 448, ... (Sloane's A000577). The
number of one-sided polyiamonds composed of n trian-
gles are 1, 1, 1, 4, 6, 19, 43, 121, . . . (Sloane's A006534).
No Holes are possible with fewer than seven triangles.

The top row of 6-polyiamonds in the above figure are
known as the Bar, Crook, Crown, Sphinx, Snake,
and Yacht. The bottom row of 6-polyiamonds are
known as the CHEVRON, SIGNPOST, LOBSTER, HOOK,

Hexagon, and Butterfly.

see also Polyabolo, Polyhex, Polyomino

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.

Cambridge, MA: MIT Artificial Intelligence Laboratory,

Memo AIM-239, Feb. 1972.
Gardner, M. "Mathematical Games." Sci. Amer., Dec. 1964.
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,

and Packings, 2nd ed. Princeton, NJ: Princeton University

Press, pp. 90-92, 1994.
Sloane, N. J. A. Sequences A000577/M2374 and A006534/

M3287 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
von Seggern, D. CRC Standard Curves and Surfaces. Boca

Raton, FL: CRC Press, pp. 342-343, 1993.

Polyking

see Polyplet

Polylogarithm

Poly logarithm

The function

Li n (z) = ^2?~,

(1)

Also known as JONQUIERE'S Function. (Note that the
Notation Li(z) is also used for the Logarithmic In-
tegral.) The polylogarithm arises in Feynman Dia-
gram integrals, and the special case n = 2 is called the
DlLOGARlTHM. The polylogarithm of NEGATIVE INTE-
GER order arises in sums of the form

k = l

where {") is an EULERIAN NUMBER.

The polylogarithm satisfies the fundamental identities

-ln(l-2~ n ) = Lii(2- n )
Li a (-l) = -(l-2 1 - s )C(s),

(3)
(4)

where ((s) is the RlEMANN Zeta FUNCTION. The de-
rivative is therefore given by

ds

Li.(-1) = -2 1 " s CWln2 - (1 - 2 1 - s K'(s), (5)

or in the special case 5 = 0, by
d

.ds

Li.(-l)

= ln2 + C'(0) = ln2- \ ln(27r)

(6)

This latter fact provides a remarkable proof of the Wal-
lis Formula.

The polylogarithm identities lead to remarkable expres-
sions. Ramanujan gave the polylogarithm identities

Li 2 (i)-|Li 2 (|) = ^ 2 -i(ln3)

(7)

= -i7r 2 +ln21n3-i(ln2) 2 -|(ln3) 2 (8)

Li 2 (-i) + iLi 2 (|)

— i J 1

--18^

Li2(|) + |Li 2 (i) = ^7r 2 + 21n21n3-2(ln2) 2 -f(ln3) 2

(9)
Li 2 (-|) - |Li 2 (|) = -^tt 2 + i(ln3) 2 (10)

Li 2 (-|) + Li 2 (i) = -i(lnf) 2 (11)

(Berndt 1994), and Bailey et al. show that

7T 2 = 36Li 2 (i) - 36Li 2 (i) - 12Li 2 (|) + 6Li 2 (i) (12)

Polynomial 1407

fC(3)-7r 2 ln2

= 36Li 3 (i) - 18Li 3 (i) _ 4Li,(i) + Li,(i) (14)

2(ln2) 3 -7C(3)

= -24Li 3 (i) + 18Li 3 (i) + 4Li 3 (|) - Li 3 (£) (15)

10(ln2) 3 - 2tt 2 In 2 = -48Li 3 (|) + 54 Lis (|)

+12Li 3 (i)-3Li 3 (i), (16)

and

Li m (£) Li m (i) 2Lim(i) 4Li m (i) 5(-ln2)J|

fim-1 Qm—1 <2rn — l

+

7r 2 (-ln2)^

V(-m2f

54(m-2)! 486(m-4)!

9 9m!

403C(5)(-ln2) m - 5
1296(m-5)!

= 0. (17)

No general ALGORITHM is know for the integration of
polylogarithms of functions.

see also Dilogarithm, Eulerian Number, Leg-
endre's Chi-Function, Logarithmic Integral,

Nielsen-Ramanujan Constants

References

Bailey, D.; Borwein, P.; and Plouffe, S. "On the Rapid Com-
putation of Various Polylogarithmic Constants." http://
www. cecm. sf u. ca/-pborwein/PAPERS/P123 .ps.

Berndt, B. C. Ramanujan' s Notebooks, Part IV. New York:
Springer- Verlag, pp. 323-326, 1994.

Lewin, L. Polylogarithms and Associated Functions. New
York: North-Holland, 1981.

Lewin, L. Structural Properties of Polylogarithms. Provi-
dence, RI: Amer. Math. Soc, 1991.

Nielsen, N. Der Euler'sche Dilogarithms. Leipzig, Germany:
Halle, 1909.

Polymorph

An INTEGER which is expressible in more than one way
in the form x 2 +Dy 2 or x 2 -Dy 2 where x 2 is RELATIVELY
Prime to Dy 2 . If the INTEGER is expressible in only one
way, it is called a MONOMORPH,
see also Antimorph, Idoneal Number, Monomorph

Polynomial

A Polynomial is a mathematical expression involving
a series of Powers in one or more variables multiplied
by Coefficients. A Polynomial in one variable with
constant COEFFICIENTS is given by

a n x n + . . . + a>2X + dix + ao-

(1)

12Li 2 (|) =?r 2 -6(ln2) 2

(13)

The highest Power in a one-variable POLYNOMIAL is
called its Order. A Polynomial in two variables with
constant COEFFICIENTS is given by

a nm x n y rn + a 22 x 2 y 2 + a 21 x 2 y + ai 2 xy 2

-\-aiixy + awx + aoiy + o o- (2)

1408 Polynomial

Polynomial

Exchanging the COEFFICIENTS of a one-variable POLY-
NOMIAL end-to-end produces a Polynomial

clqx + a\X

+ . . . + a n -ix 4- o n

(3)

whose Roots are Reciprocals 1/xi of the original
Roots x if

The following table gives special names given to poly-
nomials of low orders.

Order

Polynomial Name

1

linear equation

2

quadratic equation

3

cubic equation

4

quartic equation

5

quintic equation

6

sextic equation

Polynomials of fourth degree may be computed using
three multiplications and five additions if a few quanti-
ties are calculated first (Press et ah 1989):

2 3 4

do + Q>\X + 0> 2 X + Q>3$ 4" Q>4%

[(Ax + Bf + Ax + C\ [(Ax + B) 2 + D] + E, (4)

where

- M 1/4

= as- A 2
~ 4A*

-QR 2

D^3B' + 8B 3 + aiA - 2a2B

C=2±-2B-&B 2

A 2

A 2
D

E = a -B 4 -B 2 (C + D)- CD.

(5)
(6)
(7)
(8)
(9)

Similarly, a POLYNOMIAL of fifth degree may be com-
puted with four multiplications and five additions, and
a Polynomial of sixth degree may be computed with
four multiplications and seven additions.

Polynomials of orders 1 to 4 are solvable using only
algebraic functions and finite square root extraction.
A first-order equation is trivially solvable. A second-
order equation is soluble using the QUADRATIC EQUA-
TION. A third-order equation is solvable using the CU-
BIC Equation. A fourth-order equation is solvable us-
ing the Quartic Equation. It was proved by Abel
using GROUP THEORY that higher order equations can-
not be solved by finite root extraction.

However, the general Quintic Equation may be given
in terms of the THETA FUNCTIONS, or HYPERGEOMET-
RIC FUNCTIONS in one variable. Hermite and Kronecker
proved that higher order POLYNOMIALS are not soluble
in the same manner. Klein showed that the work of
Hermite was implicit in the GROUP properties of the
ICOSAHEDRON. Klein's method of solving the quintic

in terms of Hypergeometric Functions in one vari-
able can be extended to the sextic, but for higher order
Polynomials, either Hypergeometric Functions in
several variables or "Siegel functions" must be used. In
the 1880s, Poincare created functions which give the so-
lution to the nth order POLYNOMIAL equation in finite
form. These functions turned out to be "natural" gen-
eralizations of the Elliptic Functions.

Given an nth degree POLYNOMIAL, the ROOTS can be
found by finding the Eigenvalues of the Matrix

-a /a n
1

-ai/a n

1

-a 2 /a n

1

J

(10)

This method can be computationally expensive, but is
fairly robust at finding close and multiple roots.

Polynomial identities involving sums and differences of
like POWERS include

x 2 -y 2 = (x-y)(x + y) (11)

x 3 - y 3 = (x - y){x 2 + xy + y 2 ) (12)

x 3 + y 3 = (x + y){x 2 -xy + y 2 ) (13)

x 4 -y 4 = (x-y)(x + y)(x 2 +y 2 ) (14)

x 4 + V = (x 2 + 2xy + 2y 2 )(x 2 - 2xy + 2y 2 ) (15)

x 5 - y 5 = ( x - y)(x 4 + x 3 y + x 2 y 2 + xy 3 + y 4 ) (16)
x 5 + y 5 = (x + y){x 4 - x 3 y + x 2 y - xy 3 + y 4 ) (17)
x 6 - y 6 = (x - y)(x + y)(x 2 + xy + y 2 )(x 2 - xy + y 2 )

(18)

6.6 / 2 , 2w 4 2 2 , 4\

x -\-y — (x + y ){x - x y + y ).

Further identities include

(19)

x 4 + x 2 y 2 + y 4 = (x 2 + xy + y 2 )(x 2 - xy + y 2 ) (20)
= (xix 2 + Dy 1 y 2 f - D(xiy 2 + x 2 yi) 2 (21)
= (xix 2 ± Dy!y 2 ) 2 + D(xiy 2 =f x 2 yi) 2 . (22)

(xt 2 - D yi 2 )(x 2 * - Dy2 2 )

{xi 2 + D yi 2 )(x 2 2 + Dy 2 2 )

The identity

(x+y-f z) 7 -(x 7 +y 7 +z 7 ) = 7{x+y){x+z){y+z)

x[(X 2 + Y 2 + Z 2 + XY+XZ+YZ) 2 + XYZ{X+Y+Z)}

(23)

was used by Lame in his proof that Fermat'S LAST
Theorem was true for n = 7.

Polynomial Bar Norm

Polynomial Norm 1409

see also APPELL POLYNOMIAL, BERNSTEIN POLY-
NOMIAL, Bessel Polynomial, Bezout's Theo-
rem, Binomial, Bombieri Inner Product, Bom-
bieri Norm, Chebyshev Polynomial of the
First Kind, Chebyshev Polynomial of the Sec-
ond Kind, Christoffel-Darboux Formula, Chris-
toffel Number, Complex Number, Cyclotomic
Polynomial, Descartes' Sign Rule, Discrimi-
nant (Polynomial), Durfee Polynomial, Ehr-
hart Polynomial, Euler Four-Square Identity,
Fibonacci Identity, Fundamental Theorem of
Algebra, Fundamental Theorem of Symmetric
Functions, Gauss-Jacobi Mechanical Quadra-
ture, Gegenbauer Polynomial, Gram-Schmidt
Orthonormalization, Greatest Lower Bound,
Hermite Polynomial, Hilbert Polynomial, Irre-
ducible Polynomial, Isobaric Polynomial, Iso-
graph, Jensen Polynomial, Kernel Polynomial,
Krawtchouk Polynomial, Laguerre Polynomial,
Least Upper Bound, Legendre Polynomial, Liou-
ville Polynomial Identity, Lommel Polynom-
ial, Lukacs Theorem, Monomial, Orthogonal
Polynomials, Perimeter Polynomial, Poisson-
Charlier Polynomial, Pollaczek Polynomial,
Polynomial Bar Norm, Quarter Squares Rule,
Ramanujan 6-10-8 Identity, Root, Runge-Walsh
Theorem, Schlafli Polynomial, Separation The-
orem, Stieltjes-Wigert Polynomial, Trinomial,
Trinomial Identity, WeierstraB's Polynomial
Theorem, Zernike Polynomial

References

Barbeau, E. J. Polynomials. New York: Springer- Verlag,
1989.

Bini, D. and Pan, V. Y. Polynomial and Matrix Compu-
tations, Vol. 1: Fundamental Algorithms. Boston, MA:
Birkhauser, 1994.

Borwein, P. and Erdelyi, T. Polynomials and Polynomial In-
equalities. New York: Springer- Verlag, 1995.

Cockle, J. "Notes on the Higher Algebra." Quart. J. Pure
Applied Math. 4, 49-57, 1861.

Cockle, J. "Notes on the Higher Algebra (Continued)."
Quart. J. Pure Applied Math. 5, 1-17, 1862.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in C: The Art of Scientific
Computing. Cambridge, England: Cambridge University
Press, 1989.

Project Mathematics! Polynomials. Videotape (27 min-
utes). California Institute of Technology. Available from
the Math. Assoc. Amer.

Polynomial Bar Norm

For p — Y^, a o z ^ define

ipiii

Jo

P (e «)|g

\Ph = £ k

I

|P(e^)| 2

2tt

|P|2= /5>:

where the \\P\\i norms are functions on the UNIT CIRCLE
and the \P\i norms refer to the COEFFICIENTS ao, . . . ,

a n .

see also Bombieri Norm, Norm, Unit Circle

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 151, 1989.

Polynomial Bracket Norm

see Bombieri Norm

Polynomial Curve

A curve obtained by fitting Polynomials to each ordi-
nate of an ordered sequence of points. The above plots
show Polynomial curves where the order of the fitting
Polynomial varies from p — 3 to p — 1, where p is the
number of points.

Polynomial curves have several undesirable features, in-
cluding a nonintuitive variation of fitting curve with
varying COEFFICIENTS, and numerical instability for
high orders. Splines such as the BEZIER Curve are
therefore used more commonly.

see also Bezier Curve, Polynomial, Spline

Polynomial Factor

A Factor of a Polynomial P(x) of degree n is a Poly-
nomial Q(x) of degree less than n which can be multi-
plied by another POLYNOMIAL R(x) of degree less than
n to yield P(x), i.e., a Polynomial Q(x) such that

P(x) = Q(x)R(x).

For example, since

x 2 - 1 = (x + l)(x- 1),

both x - 1 and x + 1 are Factors of x 2 - 1. The Coef-
ficients of factor Polynomials are often required to
be Real Numbers or Integers but could, in general,
be Complex Numbers.

see also Factor, Factorization, Prime Factoriza-
tion

Polynomial Norm

see Bombieri Norm, Matrix
Bar Norm, Vector Norm

Norm, Polynomial

|P||oo = max M=1 |P(z)|

|P|oo = maxj|aj|,

1410 Polynomial Remainder Theorem

Polynomial Root

Polynomial Remainder Theorem

If the Coefficients of the Polynomial

d n x n + d n -\x n ~ + . . . + d —

(i)

are specified to be INTEGERS, then integral ROOTS must
have a NUMERATOR which is a factor of d and a DE-
NOMINATOR which is a factor of d n (with either sign
possible). This follows since a Polynomial of Order
n with k integral ROOTS can be expressed as

(aix + bi)(aix + 6 2 ) * • • (a k x + bk)(c n -kX n ~ + . . . + Co)

= 0, (2)

where the ROOTS are X\ — — &i/ai, X2 = —bijai, . . . ,
and Xk — —bk/a>k- Factoring out the a»s,

a ^--- ak ( x - b i){ x ' b i)''i x - b i)

x(c n _ fc z n - fe + ... + c o ) = 0. (3)

Now, multiplying through,

aid2 * * • akCn-kX n + . . . + &i&2 • * • bkCo — 0, (4)

where we have not bothered with the other terms. Since
the first and last COEFFICIENTS are d n and do, all the in-
tegral roots of (1) are of the form [factors of do]/[factors

Of d n ].

Polynomial Ring

The Ring R[x] of Polynomials in a variable x.

see also Polynomial, Ring

Polynomial Root

If the Coefficients of the Polynomial

d n x n + dn-ix 71 ' 1 + ... 4- do = (1)

are specified to be INTEGERS, then integral roots must
have a NUMERATOR which is a factor of do and a DE-
NOMINATOR which is a factor of d n (with either sign
possible). This is known as the POLYNOMIAL REMAIN-
DER Theorem.

Let the ROOTS of the polynomial

P(x) = a n x n + dn-ix 71 ' 1 + . . . + cnx + a Q (2)

be denoted n, r 2 , . . . , r n . Then Newton's Relations
are

ECLn-l
Ti —
a n

EO>n-2
nrj = — — -

nr 2 • ■ -r fc = (-1) .

(3)
(4)
(5)

These can be derived by writing

(x-a)(x-b) =

Similarly,

s-"-G-i)-»-'

■" .V^...-.

x
aVP

(?+h)

+ 1 = 0.

(6)
(7)
(8)

(9)
(10)

Any POLYNOMIAL can be numerically factored, al-
though different ALGORITHMS have different strengths
and weaknesses.

If there are no NEGATIVE ROOTS of a POLYNOMIAL (as
can be determined by Descartes' SIGN Rule), then
the Greatest Lower Bound is 0. Otherwise, write
out the Coefficients, let n = — 1, and compute the
next line. Now, if any COEFFICIENTS are 0, set them to
minus the sign of the next higher COEFFICIENT, starting
with the second highest order COEFFICIENT. If all the
signs alternate, n is the greatest lower bound. If not,
then subtract 1 from n, and compute another line. For
example, consider the POLYNOMIAL

y = 2x 4 + 2z 3

7x + x - 7.

(11)

Performing the above Algorithm then gives

2

2

-7

1

-7

-1

2

-7

8

-15

—

2

-1

-7

8

-15

-2

2

-2

-3

7

-21

-3

2

-4

5

-14

35

so the greatest lower bound is —3.

If there are no Positive Roots of a Polynomial (as
can be determined by Descartes' Sign Rule), the
Least Upper Bound is 0. Otherwise, write out the
Coefficients of the Polynomials, including zeros as

necessary. Let n = 1. On the line below, write the
highest order COEFFICIENT. Starting with the second-
highest Coefficient, add n- times the number just writ-
ten to the original second COEFFICIENT, and write it be-
low the second COEFFICIENT. Continue through order
zero. If all the COEFFICIENTS are NONNEGATIVE, the
least upper bound is n. If not, add one to x and repeat
the process again. For example, take the POLYNOMIAL

• 7x 2 + x - 7.

Performing the above ALGORITHM gives

(12)

Polynomial Series

Polyomino 1411

2

-1

-7

1

-7

1

2

1

-6

-5

-12

2

2

3

-1

-1

-9

3

2

5

8

25

68

so the Least Upper Bound is 3.
see also Bairstow's Method, Descartes'
Sign Rule, Jenkins-Traub Method, Laguerre's
Method, Lehmer-Schur Method, Maehly's Pro-
cedure, Muller's Method, Root, Zassenhaus-
Berlekamp Algorithm

Polynomial Series

see Multinomial Series

Polyomino

A generalization of the DOMINO. An n-omino is defined
as a collection of n squares of equal size arranged with
coincident sides. Free polyominoes can be picked up
and flipped, so mirror image pieces are considered iden-
tical, whereas Fixed polyominoes are distinct if they
have different chirality or orientation. Fixed polyomi-
noes are also called LATTICE ANIMALS.

Redelmeier (1981) computed the number of Free and
Fixed polyominoes for n < 24, and Mertens (1990) gives
a simple computer program. The sequence giving the
number of Free polyominoes of each order (Sloane's
A000105, Ball and Coxeter 1987) is shown in the second
column below, and that for FIXED polyominoes in the
third column (Sloane's A014559).

n

Free

Fixed

Pos. Holes

1

1

1

2

1

2

3

2

6

4

5

19

5

12

63

6

35

216

7

108

760

1

8

369

2725

6

9

1285

9910

37

10

4655

39446

384

11

17073

135268

12

63600

505861

13

238591

1903890

14

901971

7204874

15

3426576

27394666

16

13079255

104592937

17

50107909

400795844

18

192622052

1540820542

19

742624232

5940738676

20

2870671950

22964779660

21

11123060678

88983512783

22

43191857688

345532572678

23

168047007728

1344372335524

24

654999700403

5239988770268

The best currently known bounds on the number of n-
polyominoes are

3.72 n < P(n) < 4.65 n

(Eden 1961, Klarner 1967, Klarner and Rivest 1973, Ball
and Coxeter 1987), For n = 4, the quartominoes are
called Straight, L, T, Square, and Skew. For n = 5,
the pentominoes are called /, i", £, N, P, T, U, V } W,
X, y, and Z (Golomb 1995).

1 □

2 B

EP

3

$

a %

see also Domino, Hexomino, Monomino, Pen-
tomino, Polyabolo, Polycube, Polyhex, Polyia-
mond, Polyking, Polyplet, Tetromino, Triomino

References

Atkin, A. O. L. and Birch, B. J, (Eds.). Computers in Num-
ber Theory: Proc. Sci. Research Council Atlas Symposium
No. 2 Held at Oxford from 18-23 Aug., 1969. New York:
Academic Press, 1971.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 109-
113, 1987.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 77 in
HAKMEM. Cambridge, MA: MIT Artificial Intelligence
Laboratory, Memo AIM-239, pp. 48-50, Feb. 1972.

Eden, M. "A Two-Dimensional Growth Process." Proc.
Fourth Berkeley Symposium Math. Statistics and Probabil-
ity, Held at the Statistical Laboratory, University of Cal-
ifornia, June 30- July 30, 1960. Berkeley, CA: University
of California Press, pp. 223-239, 1961.

Finch, S. "Favorite Mathematical Constants." http://vwv.
mathsof t . com/asolve/constant/rndprc/rndprc .html.

Gardner, M. "Polyominoes and Fault-Free Rectangles."
Ch. 13 in Martin Gardner's New Mathematical Diversions
from Scientific American. New York: Simon and Schuster,
1966.

Gardner, M. "Polyominoes and Rectification." Ch. 13 in
Mathematical Magic Show: More Puzzles, Games, Diver-
sions, Illusions and Other Mathematical Sleight-of-Mind
from Scientific American. New York: Vintage, pp. 172-
187, 1978.

Golomb, S. W. "Checker Boards and Polyominoes." Amer.
Math. Monthly 61, 675-682, 1954.

Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, rev. enl. 2nd ed. Princeton, NJ: Princeton
University Press, 1995.

Klarner, D. A. "Cell Growth Problems." Can. J. Math. 19,
851-863, 1967.

Klarner, D. A. and Riverst, R. "A Procedure for Improving
the Upper Bound for the Number of n-ominoes." Can. J.
Math. 25, 585-602, 1973.

Lei, A. "Bigger Polyominoes." http://www . cs . ust . hk /
-philipl/omino/bigpolyo .html.

Lei, A. "Polyominoes." http://www.cs.ust.hk/-philipl/
omino/omino . html.

1412 Polyomino Tiling

Polytope

Lunnon, W. F. "Counting Polyominoes." In Computers in
Number Theory (Ed. A. O. L. Atkin and B. J. Brich). Lon-
don: Academic Press, pp. 347-372, 1971.

Martin, G- Polyominoes: A Guide to Puzzles and Problems
in Tiling. Washington, DC: Math. Assoc. Amer., 1991.

Mertens, S. "Lattice Animals — A Fast Enumeration Algo-
rithm and New Perimeter Polynomials." J. Stat. Phys.
58, 1095-1108, 1990.

Read, R. C. "Contributions to the Cell Growth Problem."
Canad. J. Math. 14, 1-20, 1962.

Redelmeier, D. H. "Counting Polyominoes: Yet Another At-
tack." Discrete Math. 36, 191-203, 1981.

Ruskey, F. "Information on Polyominoes." http://sue.csc
.uvic.ca/-cos/inf /misc/PolyominoInfo.html.

Sloane, N. J. A. Sequences A014559 and A000105/M1425 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."

von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, pp. 342-343, 1993.

Polyomino Tiling

A Tiling of the Plane by specified types of Polyomi-
noes. Interestingly, the FIBONACCI Number F n+1 gives
the number of ways for 2 x 1 dominoes to cover a 2 x n
checkerboard.

see also FIBONACCI NUMBER

References

Gardner, M. "Tiling with Polyominoes, Polyiamonds, and
Polyhexes." Ch. 14 in Time Travel and Other Mathemat-
ical Bewilderments. New York: W. H. Freeman, 1988.

Polyplet

□

A POLYOMINO-Iike object made by attaching squares
joined either at sides or corners. Because neighboring
squares can be in relation to one another as Kings may
move on a CHESSBOARD, polyplets are sometimes also
called POLYKINGS. The number of n-polyplets (with
holes allowed) are 1, 2, 5, 22, 94, 524, 3031, . . . (Sloane's
A030222). The number of n-polyplets having bilateral
symmetry are 1, 2, 4, 10, 22, 57, 131, ... (Sloane's
A030234). The number of n-polyplets not having bilat-
eral symmetry are 0, 0, 1, 12, 72, 467, 2900, . . . (Sloane's
A030235). The number of fixed n-polyplets are 1, 4, 20,
110, 638, 3832, ... (Sloane's A030232). The number
of one-sided n-polyplets are 1, 2, 6, 34, 166, 991, ...
(Sloane's A030233).

see also POLYIAMOND, POLYOMINO

References

Sloane, N. J. A. Sequences A030222, A030232, A030233,
A030234, and A030235 in "An On-Line Version of the En-
cyclopedia of Integer Sequences."

Polytope

A convex polytope may be defined as the Convex Hull
of a finite set of points (which are always bounded), or as
the intersection of a finite set of half-spaces. Explicitly,
a d-dimensional polytope may be specified as the set of
solutions to a system of linear inequalities

mx < b,

where m is a real sxd Matrix and b is a real s- Vector.
The positions of the vertices given by the above equa-
tions may be found using a process called Vertex Enu-
meration.

A regular polytope is a generalization of the Platonic
Solids to an arbitrary Dimension. The Necessary
condition for the figure with SCHLAFLI SYMBOL {p, q, r}
to be a finite polytope is

G)

< sin ( — | sin
V

-(;)

Sufficiency can be established by consideration of the
six figures satisfying this condition. The table below
enumerates the six regular polytopes in 4-D (Coxeter
1969, p. 414).

Name

Schlafli

Symbol

No

iVi

N 2

N s

regular simplex

{3,3,3}

5

10

10

5

hyper cube

{4,3,3}

16

32

24

8

16-cell

{3,3,4}

8

24

32

16

24-cell

{3,4,3}

24

96

96

24

120-cell

{5,3,3}

600

1200

720

120

600-cell

{3,3,5}

120

720

1200

600

Here, iV is the number of VERTICES, Ni the number of
EDGES, N 2 the number of Faces, and N$ the number
of cells. These quantities satisfy the identity

No - m + N 2 - N s = 0,

which is a version of the Polyhedral Formula.

For n-D with n > 5, there are only three regular poly-
topes, the Measure Polytope, Cross Polytope,
and regular Simplex (which are analogs of the Cube,
Octahedron, and Tetrahedron).

see also 16-Cell, 24-Cell, 120-Cell, 600-Cell,
Cross Polytope, Edge (Polytope), Face, Facet,
Hypercube, Incidence Matrix, Measure Poly-
tope, Ridge, Simplex, Tesseract, Vertex (Poly-
hedron)

References

Coxeter, H. S. M. "Regular and Semi- Regular Polytopes I."

Math. Z. 46, 380-407, 1940.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New

York: Wiley, 1969.
Eppstein, D. "Polyhedra and Polytopes." http://www.ics.

uci . edu/~eppstein/junkyard/polytope.html.

Poncelet's Closure Theorem

Pontryagin Maximum Principle 1413

Poncelet's Closure Theorem

If an rc-sided PONCELET TRANSVERSE constructed for
two given CONIC SECTIONS is closed for one point of
origin, it is closed for any position of the point of origin.
Specifically, given one Ellipse inside another, if there
exists one Circuminscribed (simultaneously inscribed
in the outer and circumscribed on the inner) n-gon, then
any point on the boundary of the outer ELLIPSE is the
Vertex of some Circuminscribed n-gon.

References

Dorrie, H. 100 Great Problems of Elementary Mathematics:

Their History and Solutions. New York: Dover, p. 193,

1965.

Poncelet's Continuity Principle

see Permanence of Mathematical Relations
Principle

Poncelet-Steiner Theorem

All Euclidean GEOMETRIC CONSTRUCTIONS can be car-
ried out with a Straightedge alone if, in addition,
one is given the RADIUS of a single CIRCLE and its cen-
ter. The theorem was suggested by Poncelet in 1822
and proved by Steiner in 1833. A construction using
Straightedge alone is called a Steiner Construc-
tion.

see also GEOMETRIC CONSTRUCTION, STEINER CON-
STRUCTION

References

Dorrie, H. "Steiner 's Straight-Edge Problem." §34 in 100
Great Problems of Elementary Mathematics: Their His-
tory and Solutions. New York: Dover, pp. 165-170, 1965.

Steiner, J. Geometric Constructions with a Ruler, Given a
Fixed Circle with Its Center. New York: Scripta Mathe-
matica, 1950.

Poncelet's Theorem

see Poncelet's Closure Theorem

Poncelet Transform

see Poncelet Transverse

Poncelet Transverse

Let a Circle C\ lie inside another CIRCLE C2. Prom
any point on C2, draw a tangent to C\ and extend it
to C2. Prom the point, draw another tangent, etc. For
n tangents, the result is called an n- sided PONCELET

Transform.

References

Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 192,
1965.

Pong Hau K'i

A Chinese Tic-TAC-TOE-like game.

see also Tic-Tac-Toe

References

Evans, R. "Pong Hau K'i." Games and Puzzles 53, 19, 1976.
Straffin, P. D. Jr. "Position Graphs for Pong Hau K'i and
Mu Torere." Math. Mag. 68, 382-386, 1995.

Pons Asinorum

An elementary theorem in geometry whose name means
"ass's bridge." The theorem states that the ANGLES
at the base of an Isosceles Triangle (defined as a
Triangle with two legs of equal length) are equal.

see also Isosceles Triangle, Pythagorean Theo-
rem

References

Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, p. 38, 1990.

Pontryagin Class

The ith Pontryagin class of a Vector Bundle is (-1)*
times the ith. Chern Class of the complexification of
the Vector Bundle. It is also in the 4zth cohomology
group of the base SPACE involved.

see also Chern Class, Stiefel- Whitney Class

Pontryagin Duality

Let G be a locally compact Abelian Group. Let G*
be the group of all homeomorphisms G —> R/Z, in the
compact open topology. Then G* is also a locally com-
pact Abelian Group, where the asterisk defines a con-
travariant equivalence of the category of locally com-
pact Abelian groups with itself. The natural mapping
G -^ (G*)*, sending g to G y where G(f) = /(#), is
an isomorphism and a HOMEOMORPHISM. Under this
equivalence, compact groups are sent to discrete groups
and vice versa.

see also Abelian Group, Homeomorphism

Pontryagin Maximum Principle
A result is Control Theory. Define

H(il>, x, u) = (i/>, f(x, u)) = Y2 ^f a (x, u).

Then in order for a control u(t) and a trajectory x(t)
to be optimal, it is NECESSARY that there exist NON-
ZERO absolutely continuous vector function ip(i) =
(V'o(i), ^1 (£)>■■• iipn{t)) corresponding to the functions
u{i) and x(t) such that

1. The function H(i/;(t),x(t), u) attains its maximum at
the point u — u{t) almost everywhere in the interval
t <t< t u

H{ip(t),x(t),u(t)) ~ max if OO), #(£)>-«)•

u£C7

1414 Pontryagin Number

2. At the terminal time ti, the relations ^o(^i) < and
H(ip(ti),x(ti),u(ti)) = are satisfied.

References

Iyanaga, S. and Kawada, Y. (Eds.). "Pontrjagin's Maximum
Principle." §88C in Encyclopedic Dictionary of Mathemat-
ics. Cambridge, MA: MIT Press, p. 295-296, 1980.

Pontryagin Number

The Pontryagin number is denned in terms of the PON-
TRYAGIN Class of a Manifold as follows. For any
collection of PONTRYAGIN CLASSES such that their cup
product has the same Dimension as the Manifold,
this cup product can be evaluated on the Manifold's
Fundamental Class. The resulting number is called
the Pontryagin number for that combination of Pontrya-
gin classes. The most important aspect of Pontryagin
numbers is that they are COBORDISM invariant. To-
gether, Pontryagin and Stiefel- Whitney Numbers
determine an oriented manifold's oriented Cobordism
class.
see also Chern Number, Stiefel- Whitney Number

Ponzo's Illusion

The upper HORIZONTAL line segment in the above figure
appears to be longer than the lower line segment despite
the fact that both are the same length.
see also Illusion, Muller-Lyer Illusion, Poggen-
dorff Illusion, Vertical-Horizontal Illusion

References

Fineman, M. The Nature of Visual Illusion. New York:
Dover, p. 153, 1996.

Pop

An action which removes a single element from the top
of a Queue or Stack, turning the List (ai, <Z2, . . . , a n )
into (a2, . . . , a n ) and yielding the element ai.
see also PUSH, STACK

Population Comparison

Let X\ and X2 be the number of successes in variates
taken from two populations. Define

xi
Pi = —

Til

P2

X2_

n 2 '

(1)

(2)

The Estimator of the difference is then pi — jb- Doing
a z-Transform,

(gi ~Pt) - (gi -vg2J

(3)

Population Growth

where

<7 Pi-P2 — v a Pi'
The Standard Error is

T- 2

Til

SEc

SEx x 2 — \ h

11 m ri2

Pl(l-Pl) P 2 (l-P2)

n 2

2 _ (m - l)si 2 + (ng - l)s2 2

Spool — .

ni + n 2 — 2

(4)

(5)
(6)
(7)

see also ^-TRANSFORM

Population Growth

The differential equation describing exponential growth

is

dN _ N

~dt ~ ^r'
This can be integrated directly

[ N dN = f f
J No N Jo

dt

T

Exponentiating,

(i)

(2)
(3)

(4)

N(t) = N e t/T .

Defining N(t = 1) = N e a gives r = 1/a in (4), so

N(t) = N e at . (5)

The quantity a in this equation is sometimes known as
the Malthusian Parameter.

Consider a more complicated growth law

dN

dt

where a > 1 is a constant. This can also be integrated

directly

dN ( 1\ J± ff7 .

-w = { a -i) dt (7)

lnAT = at-lnr + C (8)

N(t)

Ce°

t

(9)

Note that this expression blows up at t = 0. We are
given the INITIAL CONDITION that N(t = 1) = N e a ,
so C = N .

N(t) = N -

(10)

The t in the DENOMINATOR of (10) greatly suppresses
the growth in the long run compared to the simple
growth law.

Porism

Positive Definite Matrix 1415

The Logistic Growth Curve, defined by

dN _ r(K - N)
dt N

(11)

is another growth law which frequently arises in biology.
It has a rather complicated solution for N(t).
see also GOMPERTZ CURVE, LIFE EXPECTANCY, LOGIS-
TIC Growth Curve, Lotka-Volterra Equations,
Makeham Curve, Malthusian Parameter, Sur-
vivorship Curve

Porism

An archaic type of mathematical proposition whose pur-
pose is not entirely known.

see also Axiom, Lemma, Postulate, Principle,
Steiner's Porism, Theorem

Porter's Constant

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

The constant appearing in FORMULAS for the efficiency
of the Euclidean Algorithm,

C =

61n2

24 11

31n2 + 4 7 -^C / (2)-2J --

Poset Dimension

The Dimension of a Poset P = (X, <) is the size of the
smallest REALIZER of P. Equivalently, it is the smallest
Integer d such that P is Isomorphic to a Dominance
order in R .
see also DIMENSION, DOMINANCE, ISOMORPHIC

Posets, Realizer

References

Dushnik, B. and Miller, E. W. "Partially Ordered Sets."
Amer. J. Math. 63, 600-610, 1941,

Trotter, W. T. Combinatorics and Partially Ordered Sets:
Dimension Theory. Baltimore, MD: Johns Hopkins Uni-
versity Press, 1992.

Position Four- Vector

The CONTRAVARIANT FOUR- VECTOR arising in special
and general relativity,

r*°i

~ct~

x k
x 2

=

X

y

U 3 J

_ z m

= 1.4670780794.

where c is the speed of light and t is time. Multiplication
of two four- vectors gives the spacetime interval

J = flM ^V = (x°) 2 - (x 1 ) 2 - (* 2 ) 2 - (* 3 ) 2
= (<*)'- (a: 1 ) 2 -(a a ) a -(* s ) a

where 7 is the Euler-Mascheroni Constant and £(z)
is the Riemann Zeta Function.

see also Euclidean Algorithm

References

Finch, S. "Favorite Mathematical Constants." http://www.

mathsoft.com/asolve/constant/porter/porter.html.
Porter, J. W. "On a Theorem of Heilbronn." Mathematika

22, 20-28, 1975.

Posa's Theorem

Let G be a Simple Graph with n Vertices.

1. If, for every k in 1 < k < (n — l)/2, the number of
Vertices of Valency not exceeding k is less than
fc, and

2. If, for n Odd, the number of Vertices with Va-
lency not exceeding (n - l)/2 is less than or equal
to (n-l)/2,

then G contains a HAMILTONIAN CIRCUIT.

see also Hamiltonian Circuit

Poset

see Partially Ordered Set

see also Four- Vector, Lorentz Transformation,
Quaternion

Position Vector

see Radius Vector

Positive

A quantity x > 0, which may be written with an explicit

Plus Sign for emphasis, -fac.

see also Negative, Nonnegative, Plus Sign, Zero

Positive Definite Function

A Positive definite Function / on a Group G is a
Function for which the Matrix {/(aua?j -1 )} is always
Positive Semidefinite Hermitian.

References

Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.

Positive Definite Matrix

A Matrix A is positive definite if

(Av) • v >

(1)

for all Vectors v ^ 0. All Eigenvalues of a posi-
tive definite matrix are POSITIVE (or, equivalently, the
Determinants associated with all upper-left Subma-

TRICES are POSITIVE).

1416 Positive Definite Quadratic Form

Postage Stamp Problem

The Determinant of a positive definite matrix is Pos-
itive, but the converse is not necessarily true (i.e., a
matrix with a Positive Determinant is not necessar-
ily positive definite).

A Real Symmetric Matrix A is positive definite Iff
there exists a REAL nonsingular MATRIX M such that

positive definite if all the principal minors in the top-
left corner of A are POSITIVE, in other words

A=MM T .

A 2 x 2 Symmetric Matrix

a b

b c

is positive definite if

av\ 2 -f 2bv\V2 + CV2 2 >

(2)

(3)

(4)

for all v = (^1,^2) 7^ 0.

A Hermitian Matrix A is positive definite if

1. an > for all i,

2. audij > \a,ij\ 2 for i ^ j,

3. The element of largest modulus must lie on the lead-
ing diagonal,

4. |A| > 0.

see also Determinant, Eigenvalue, Hermitian Ma-
trix, Matrix, Positive Semidefinite Matrix

References

Gradshteyn, I. S. and Ryzhik, L M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1106, 1979.

Positive Definite Quadratic Form

A Quadratic Form Q(x) is said to be positive definite
if Q(x) > for x / 0. A Real Quadratic Form in n
variables is positive definite IFF its canonical form is

Q(z)=Zi 2 +Z 2 2 + ...+Z n \

A Binary Quadratic Form

F(x, y) - anx 2 + 2a 12 xy + a 22 y 2

(1)

(2)

of two Real variables is positive definite if it is > for
any (x,y) ^ (0,0), therefore if an > and the DISCRIM-
INANT a = an(X22 — ai2 2 > 0. A Binary Quadratic
Form is positive definite if there exist Nonzero x and
y such that

(ax 2 + 2bxy + cy 2 ) 2 < ||oc - b'

(3)

(Le Lionnais 1983).

A Quadratic Form (x, Ax) is positive definite Iff
every EIGENVALUE of A is POSITIVE. A QUADRATIC

Form Q = (x, Ax) with A a Hermitian Matrix is

an

ai2

<221

Q>22

an

ai2

a\z

a 2 i

G&22

fl23

a$\

0-S2

«33

an >
>0

>0.

(4)
(5)

(6)

see also Indefinite Quadratic Form, Positive
Semidefinite Quadratic Form

References

Gradshteyn, I. S. and Ryzhik, L M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1106, 1979.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 38, 1983.

Positive Definite Tensor

A Tensor g whose discriminant satisfies

9 = 911922 — gi2 > 0.

Positive Integer

Positive Semidefinite Matrix

A MATRIX A is positive semidefinite if

(Av) ■ v >

for all v ^ 0.

see also Positive Definite Matrix

Positive Semidefinite Quadratic Form

A Quadratic Form Q(x) is positive semidefinite if it
is never < 0, but is for some x ^ 0. The QUADRATIC
FORM, written in the form (x, Ax), is positive semidefi-
nite Iff every Eigenvalue of A is Nonnegative.

see also Indefinite Quadratic Form, Positive Def-
inite Quadratic Form

References

Gradshteyn, I. S. and Ryzhik, I. M, Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1106, 1979.

Postage Stamp Problem

Consider a Set A k = {ai, a 2 , . . . , a*} of Integer de-
nomination postage stamps with 1 — ax < a^ < . . . <
ak. Suppose they are to be used on an envelope with
room for no more than h stamps. The postage stamp
problem then consists of determining the smallest INTE-
GER N(h,A k ) which cannot be represented by a linear
combination $^»=i XiCii w ith Xi > and y\ Xj < h.

Posterior Distribution

Poulet Number

1417

Exact solutions exist for arbitrary Ak for fc = 2 and 3.
The k — 2 solution is

n(h, A 2 ) = (h + 3- a 2 )a 2 - 2

for h > a-i — 2. The general problem consists of finding

n(h,k) = maxn(/i, iifc).

-Afc

It is known that

n(h,2) = [|(/i 2 +6/i + l)j,

(Stohr 1955, Guy 1994), where [^J is the FLOOR FUNC-
TION, the first few values of which are 2, 4, 7, 10, 14, 18,
23, 28, 34, 40, . . . (Sloane's A014616).
see also HARMONIOUS GRAPH, STAMP FOLDING

References

Guy, R. K. "The Postage Stamp Problem." §C12 in Unsolved

Problems in Number Theory, 2nd ed. New York: Springer-

Verlag, pp. 123-127, 1994.
Sloane, N. J. A. Sequence A014616 in "An On-Line Version

of the Encyclopedia of Integer Sequences."
Stohr, A. "Geloste und ungeloste Fragen iiber Basen der

naturlichen Zahlenreihe I, II." J. reine angew. Math. 194,

111-140, 1955.

Posterior Distribution

see BAYESIAN ANALYSIS

Postnikov System

An iterated Fibration of Eilenberg-Mac Lane

Spaces. Every Topological Space has this Homo-

topy type.

see also Eilenberg-Mac Lane Space, Fibration,

Homotopy

Postulate

A statement, also known as an Axiom, which is taken
to be true without PROOF. Postulates are the basic
structure from which LEMMAS and THEOREMS are de-
rived. The whole of Euclidean Geometry, for ex-
ample, is based on five postulates known as Euclid's
Postulates.

see also ARCHIMEDES' POSTULATE, AXIOM, BER-

trand's Postulate, Conjecture, Equidistance
Postulate, Euclid's Fifth Postulate, Euclid's
Postulates, Lemma, Parallel Postulate, Porism,
Proof, Theorem, Triangle Postulate

Potato Paradox

You buy 100 pounds of potatoes and are told that they
are 99% water. After leaving them outside, you discover
that they are now 98% water. The weight of the dehy-
drated potatoes is then a surprising 50 pounds!

References

Paulos, J. A. A Mathematician Reads the Newspaper. New-
York: BasicBooks, p. 81, 1995.

Potential Function

The term used in physics and engineering for a HAR-
MONIC FUNCTION. Potential functions are extremely
useful, for example, in electromagnetism, where they re-
duce the study of a 3-component VECTOR Field to a
1-component SCALAR FUNCTION.
see also Harmonic Function, Laplace's Equation,
Scalar Potential, Vector Potential

Potential Theory

The study of HARMONIC FUNCTIONS (also called PO-
TENTIAL Functions).
see also HARMONIC FUNCTION, SCALAR POTENTIAL,

Vector Potential

References

Kellogg, O. D. Foundations of Potential Theory. New York:

Dover, 1953.
MacMillan, W. D. The Theory of the Potential New York:

Dover, 1958.

Pothenot Problem

see Snellius-Pothenot Problem

Poulet Number

A Fermat Pseudoprime to base 2, denoted psp(2),

i.e., a Composite Odd Integer such that

2 n ~ 1 = 1 (modn).

The first few Poulet numbers are 341, 561, 645, 1105,
1387, . . . (Sloane's A001567). Pomerance et al. (1980)
computed all 21,853 Poulet numbers less than 25 x 10 9 .

Pomerance has shown that the number of Poulet num-
bers less than x for sufficiently large x satisfy

exp[(lna;) 5/14 ] < P 2 (x) < xexp (■

In x In In In x >
2 In In # j

(Guy 1994).

A Poulet number all of whose Divisors d satisfy d\2 d -2
is called a Super-Poulet NUMBER. There are an in-
finite number of Poulet numbers which are not Super-
Poulet Numbers. Shanks (1993) calls any integer sat-
isfying 2 Tl ~ 1 = 1 (mod n) (i.e., not limited to ODD com-
posite numbers) a Fermatian.
see also Fermat Pseudoprime, Pseudoprime, Su-
per-Poulet Number

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Verlag, pp. 28-29, 1994.
Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The

Pseudoprimes to 25-10 9 ." Math. Comput. 35, 1003-1026,

1980. Available electronically from ftp://sable.ox.ac.

uk/pub/math/primes/ps2 . Z.
Shanks, D. Solved and Unsolved Problems in Number Theory,

4th ed. New York: Chelsea, pp. 115-117, 1993.
Sloane, N. J. A. Sequence A001567/M5441 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

1418 Power

Power

The exponent to which a given quantity is raised is
known as its POWER. The expression x a is therefore
known as "as to the ath Power." The rules for com-
bining quantities containing powers are called the Ex-
ponent Laws.

Special names given to various powers are listed in the
following table.

Power Name

1/2
1/3
2
3

square root
cube root
squared
cubed

The Sum of pth Powers of the first n Positive Inte-
gers is given by FAULHABER's FORMULA,

P + l
k

a

p+i-fc^

where 5k P is the Kronecker Delta, (£) is a Binomial
Coefficient, and B k is a Bernoulli Number.

Let s n be the largest Integer that is not the Sum of
distinct nth powers of Positive Integers (Guy 1994).
The first few values for n = 2, 3, . . . are 128, 12758,
5134240, 67898771, ... (Sloane's A001661).

Catalan's Conjecture states that 8 and 9 (2 3 and
3 2 ) are the only consecutive Powers (excluding and
1), i.e., the only solution to Catalan's Diophantine
Problem. This Conjecture has not yet been proved
or refuted, although R. Tijdeman has proved that there
can be only a finite number of exceptions should the
Conjecture not hold. It is also known that 8 and 9 are
the only consecutive Cubic and Square Numbers (in
either order). Hyyro and Makowski proved that there do
not exist three consecutive Powers (Ribenboim 1996).

Very few numbers of the form n p dz 1 are PRIME (where
composite powers p = kb need not be considered, since
n { kb) ± 1 = (n k ) b ± 1). The only PRIME NUMBERS of
the form n p - 1 for n < 100 and PRIME 2 < p < 10
correspond to n = 2, i.e., 2 2 - 1 = 3, 2 3 - 1 = 7,

Power (Circle)

n p + 1 for n < 100 and Prime 2 < p < 10 correspond
to p = 2 with n = 1, 2, 4, 6, 10, 14, 16, 20, 24, 26, . . .

(Sloane's A005574).

There are no nontrivial solutions to the equation

l n +2 n + ... + m n = (m + l) n

for m < 10 2 ' 000 ' 000 (Guy 1994, p. 153).

see also Apocalyptic Number, Biquadratic Num-
ber, Catalan's Conjecture, Catalan's Diophan-
tine Problem, Cube Root, Cubed, Cubic Num-
ber, Exponent, Exponent Laws, Faulhaber's For-
mula, Figurate Number, Moessner's Theorem,
Narcissistic Number, Power Rule, Square Num-
ber, Square Root, Squared, Sum, Waring's Prob-
lem

References

Barbeau, E. J, Power Play: A Country Walk through the
Magical World of Numbers. Washington, DC: Math. As-
soc. Amer., 1997.

Beyer, W. H. "Laws of Exponents." CRC Standard Math-
ematical Tables, 28th ed. Boca Raton, FL: CRC Press,
p. 158, 1987.

Guy, R. K. "Diophantine Equations." Ch. D in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 137, 139-198, and 153-154, 1994.

Ribenboim, P. "Catalan's Conjecture." Amer. Math.
Monthly 103, 529-538, 1996.

Sloane, N. J. A. Sequences A001661/M5393 and A005574/
M1010 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Spanier, J. and Oldham, K. B. "The Integer Powers (bx-\-c) n
and z n " and "The Noninteger Powers as"." Ch. 11 and 13
in An Atlas of Functions. Washington, DC: Hemisphere,
pp. 83-90 and 99-106, 1987.

Power Center

see Radical Center

Power (Circle)

The Power of the two points P and Q with respect to
a CIRCLE is defined by

p = OP x PQ.

Let R be the RADIUS of a CIRCLE and d be the distance
between a point P and the circle's center. Then the
POWER of the point P relative to the circle is

p=d 2

R z

•1 = 31,

The only PRIME NUMBERS of the form

Power Curve

Power Series 1419

If P is outside the Circle, its Power is Positive and
equal to the square of the length of the segment from P
to the tangent to the Circle through P. If P is inside
the Circle, then the Power is Negative and equal to
the product of the Diameters through P.

The Locus of points having Power k with regard to a
fixed Circle of Radius r is a Concentric Circle of
Radius \A" 2 + k. The Chordal Theorem states that
the LOCUS of points having equal POWER with respect
to two given nonconcentric CIRCLES is a line called the
Radical Line (or Chordal; Dorrie 1965).

see also Chordal Theorem, Coaxal Circles, In-
verse Points, Inversion Circle, Inversion Ra-
dius, Inversive Distance, Radical Line

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 27-31, 1967.

Dixon, R. Mathographics. New York: Dover, p. 68, 1991.

Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 153,
1965.

Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle, Boston,
MA: Houghton Mifflin, pp. 28-34, 1929.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., pp. xxii-xxiv, 1995.

Power Curve

The cnrve with TRILINEAR COORDINATES a* : 6* : c* for
a given POWER t.

see also POWER

References

Kimberling, C. "Major Centers of Triangles." Amer. Math.
Monthly 104, 431-438, 1997.

Power Line

see Radical Axis

Power Rule

The Derivative of the Power x n is given by

dx

(* n ) = ■■

see also Chain Rule, Derivative, Exponent Laws,
Product Rule

References

Anton, H. Calculus with Analytic Geometry, 2nd ed. New
York: Wiley, p. 131, 1984.

Power Series

A power series in a variable z is an infinite SUM of the
form

y^cgz*,

(1)

where n > and a* are INTEGERS, REAL NUMBERS,
Complex Numbers, or any other quantities of a given

type.

A Conjecture of Polya is that if a Function has a
POWER series with INTEGER COEFFICIENTS and RA-
DIUS of Convergence 1, then either the Function is
Rational or the Unit Circle is a natural boundary.

A generalized POWER sum a(h) for h — 0, 1, . . . is given

by

a(h) = J2Mh)ai h ,

(2)

with distinct NONZERO ROOTS a*, COEFFICIENTS Ai(h)
which are POLYNOMIALS of degree n* - 1 for POSITIVE
Integers m, and i G [l,m]. The generalized POWER
sum has order

Z^'

(3)

Power Point

Triangle centers with Triangle Center Functions
of the form a = a n are called nth POWER points. The
0th power point is the Incenter, with TRIANGLE Cen~
ter Function a = 1.

see also Incenter, Triangle Center Function

References

Groenman, J. T. and Eddy, R. H. "Problem 858 and Solu-
tion." Crux Math. 10, 306-307, 1984.

Kimberling, C. "Problem 865." Crux Math. 10, 325-327,
1984.

Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.

For any power series, one of the following is true:

1. The series converges only for x — 0.

2. The series converges absolutely for all x.

3. The series converges absolutely for all x in some finite
open interval (—R,R) and diverges if x < — R or
x > R. At the points x = R and x = — R y the series
may converge absolutely, converge conditionally, or
diverge.

To determine the interval of convergence, apply the Ra-
tio Test for Absolute Convergence and solve for
x. A Power series may be differentiated or integrated
within the interval of convergence. Convergent power
series may be multiplied and divided (if there is no di-
vision by zero).

£*"

(4)

Converges if p > 1 and Diverges if < p < 1.

1420

Power Set

Powerful Number

see also Binomial Series, Convergence Tests,
Laurent Series, Maclaurin Series, Multino-
mial Series, p-Series, Polynomial, Power Set,
Quotient-Difference Algorithm, Recurrence
Sequence, Series, Series Reversion, Taylor Se-
ries

References

Arfken, G. "Power Series." §5.7 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 313—
321, 1985.

Myerson, G. and van der Poorten, A. J. "Some Problems
Concerning Recurrence Sequences." Amer. Math. Monthly
102, 698-705, 1995.

Polya, G. Mathematics and Plausible Reasoning, Vol. 2: Pat-
terns of Plausible Inference. Princeton, NJ: Princeton Uni-
versity Press, p. 46, 1954.

Power Set

Given a Set S, the Power Set of S is the Set of all
Subsets of S. The order of a Power set of a Set
of order n is 2 n . Power sets are larger than the Sets
associated with them.

see also Set, Subset

Power Spectrum

For a given signal, the power spectrum gives a plot of the
portion of a signal's power (energy per unit time) falling
within given frequency bins. The most common way
of generating a power spectrum is by using a FOURIER
Transform, but other techniques such as the Maxi-
mum Entropy Method can also be used.

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Power Spectra Estimation Using the FFT"
and "Power Spectrum Estimation by the Maximum En-
tropy (All Poles) Method." §13.4 and 13.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 542-551 and 565-569, 1992.

Power (Statistics)

The probability of getting a positive result for a given
test which should produce a positive result.

see also Predictive Value, Sensitivity, Speci-
ficity, Statistical Test

Power Tower

see also ACKERMANN FUNCTION, FERMAT NUMBER,

Mills' Constant

References

Knuth, D. E. "Mathematics and Computer Science: Coping
with Finiteness. Advances in our Ability to Compute are
Bringing us Substantially Closer to Ultimate Limitations."
Science 194 1235-1242, 1976.

Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 11 and 226-229, 1991.

Power (Triangle)

The total Power of a Triangle is defined by

P = \{a± 2 + a 2 2 + a 3 2 ),

(1)

where a* are the side lengths, and the "partial power"
is defined by

Then

_ 1 / 2 , 2 2\

pi = 2<a2 +a 3 - ai ).

Pi — a 2 a% cosai
P = Pl + p 2 + p 3

(2)

(3)
(4)

P 2 + Pi 2 + v 2 + V 2 = ai 4 + a 2 4 + a 3 4 (5)

A = 2 VPlPS + P3Pl + P3Pl

pi = A ± H 2 • AiA 3

aipi

a\a 2 a?> — 4AR

(6)
(7)
(8)

cosai
pi tan ai = p 2 tan a 2 — Ps tan a 3 , (9)

where A is the Area of the TRIANGLE and Hi are the

Feet of the Altitudes. Finally, if a side of the Trian-
gle and the value of any partial power are given, then
the LOCUS of the third VERTEX is a CIRCLE or straight
line.

see also ALTITUDE, FOOT, TRIANGLE

References

Johnson, R. A, Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 260-261, 1929.

Powerfree

see BlQUADRATEFREE, CUBEFREE, PRIME NUMBER,
SQUAREFREE

a tt k = a a ,
fc
where | is Knuth's (1976) ARROW NOTATION.

at fc n = a-\ k ^ [at* («-!)]•

The infinite power tower x tt oo = x x converges Iff
e~ e <x < e 1/e (0.0659 < x < 1.4446).

Powerful Number

An Integer m such that if p\m, then p 2 \m, is called a
powerful number. The first few are 1, 4, 8, 9, 16, 25, 27,
32, 36, 49, ... (Sloane's A001694). Powerful numbers
are always of the form a b for a, b > 1.

Not every NATURAL NUMBER is the sum of two powerful
numbers, but Heath-Brown (1988) has shown that every
sufficiently large NATURAL Number is the sum of at
most three powerful numbers. There are infinitely many
pairs of consecutive powerful numbers, but Erdos has

Practical Number

Pratt-Kasapi Theorem 1421

conjectured that there do not exist three consecutive
powerful numbers. The CONJECTURE that there are no
powerful number triples implies that there are infinitely
many Wieferich primes (Granville 1986, Vardi 1991).

A separate usage of the term powerful number is for
numbers which are the sums of the positive powers of
their digits. The first few are 1, 2, 3, 4, 5, 6, .7, 8, 9, 24,
43, 63, 89, . . . (Sloane's A007532).

References

Granville, A. "Powerful Numbers and Fermat's Last Theo-
rem." C. R. Math. Rep. Acad. Sci. Canada 8, 215-218,
1986.

Guy, R. K. "Powerful Numbers." §B16 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 67-73, 1994.

Heath-Brown, D. R. "Ternary Quadratic Forms and Sums of
Three Square-Full Numbers." In Seminaire de Theorie des
Nombres, Paris 1986-87 (Ed. C. Goldstein). Boston, MA:
Birkhauser, pp. 137-163, 1988.

Ribenboim, P. "Catalan's Conjecture." Amer. Math.

Monthly 103, 529-538, 1996.

Sloane, N. J. A. Sequences A001694/M3325 and A007532/
M0487 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 59-62, 1991.

Practical Number

A number n is practical if for all k < n, k is the sum
of distinct proper divisors of n. Defined in 1948 by
A. K. Srinivasen. All even Perfect Numbers are prac-
tical. The number

: 2 n_1 (2"

1)

is practical for all n = 2, 3, The first few practical

numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32,
36, 40, 42, 48, 54, 56, . . . (Sloane's A005153). G. Melfi
has computed twins, triplets, and 5-tuples of practical
numbers. The first few 5-tuples are 12, 18, 30, 198, 306,
462, 1482, 2550, 4422, ....

References

Melfi, G. "On Two Conjectures About Practical Numbers."

J. Number Th. 56, 205-210, 1996.
Melfi, G. "Practical Numbers." http://www.c3m.unipi.it/

gauss-pages/melf i/publicJvtinl/pratica.html.
Sloane, N. J. A. Sequence A005153/M0991 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Pratt Certificate

A primality certificate based on Fermat's Little The-
orem Converse. Although the general idea had been
well-established for some time, Pratt became the first to
prove that the certificate tree was of polynomial size and
could also be verified in polynomial time. He was also
the first to observe that the tree implies that Primes
are in the complexity class NP.

To generate a Pratt certificate, assume that n is a POS-
ITIVE Integer and {pi} is the set of Prime Factors
of n — 1. Suppose there exists an INTEGER x (called a

"Witness") such that x 71 ' 1 = 1 (mod n) but x e ^ 1
(mod n) whenever e is one of (n — l)/pi. Then FER-
MAT'S Little Theorem Converse states that n is
Prime (Wagon 1991, pp. 278-279).

By applying FERMAT'S LITTLE THEOREM CONVERSE
to n and recursively to each purported factor of n — 1, a
certificate for a given PRIME NUMBER can be generated.
Stated another way, the Pratt certificate gives a proof
that a number a is a Primitive Root of the multiplica-
tive GROUP (mod p) which, along with the fact that a
has order p — 1, proves that p is a PRIME.

7919

2

37 ■

107-

2
53

2
13 ■

The figure above gives a certificate for the primality of
n = 7919. The numbers to the right of the dashes are
Witnesses to the numbers to left. The set {pi} for
n - 1 = 7918 is given by {2,37,107}. Since 7 7918 =
1 (mod 7919) but 7 7918 ' 2 , 7 7918/37 , 7 7918/107 ^ 1 (mod
7919), 7 is a WITNESS for 7919. The PRIME divisors of
7918 = 7919 - 1 are 2, 37, and 107. 2 is a so-called
"self- Witness" (i.e., it is recognized as a Prime with-
out further ado), and the remainder of the witnesses are
shown as a nested tree. Together, they certify that 7919
is indeed Prime. Because it requires the Factoriza-
tion of n -^ 1, the Method of Pratt certificates is best
applied to small numbers (or those numbers n known to
have easily factorable n — 1).

A Pratt certificate is quicker to generate for small
numbers than are other types of primality certificates.
The Mathematica® (Wolfram Research, Champaign, IL)
task ProvablePrimefn] therefore generates an Atkin-
Goldwasser-Kilian-Morain Certificate only for
numbers above a certain limit (10 10 by default), and
a Pratt certificate for smaller numbers.

see also ATKIN-GOLDWASSER-KlLIAN-MORAIN CER-
TIFICATE, Fermat's Little Theorem Converse,
Primality Certificate, Witness

References

Pratt, V. "Every Prime Has a Succinct Certificate." SIAM
J. Comput. 4, 214-220, 1975.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 278-285, 1991.

Wilf, H. §4.10 in Algorithms and Complexity. Englewood
Cliffs, NJ: Prentice-Hall, 1986.

Pratt-Kasapi Theorem

see Hoehn's Theorem

1422

Precedes

Primality Certificate

Precedes

The relationship x precedes y is written x < y. The
relation x precedes or is equal to y is written x X y.

see also Succeeds

Precession

see Curve of Constant Precession

Precisely Unless

If A is true precisely unless B\ then B implies not- A and
not-f? implies A. J. H. Conway has suggested the term
"UNLESSS" for this state of affairs, by analogy with Iff.

see also IFF, UNLESS

Predicate

A function whose value is either True or False.

see also AND, FALSE, OR, PREDICATE CALCULUS,

True, XOR

Predicate Calculus

The branch of formal LOGIC dealing with representing
the logical connections between statements as well as
the statements themselves.

see also Godel's Incompleteness Theorem, Logic,
Predicate

Predictor-Corrector Methods

A general method of integrating ORDINARY DIFFEREN-
TIAL EQUATIONS. It proceeds by extrapolating a poly-
nomial fit to the derivative from the previous points to
the new point (the predictor step), then using this to
interpolate the derivative (the corrector step). Press
et at. (1992) opine that predictor-corrector methods
have been largely supplanted by the Bulirsch-Stoer
and Runge-Kutta Methods, but predictor-corrector
schemes are still in common use.

see also Adams' Method, Gill's Method, Milne's
Method, Runge-Kutta Method

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 896-897, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 493-494, 1985.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Multistep, Multivalue, and Predictor-
Corrector Methods." §16.7 in Numerical Recipes in FOR-
TRAN: The Art of Scientific Computing, 2nd ed. Cam-
bridge, England: Cambridge University Press, pp. 740-
744, 1992.

Pretzel Curve

see Knot Curve

Predictability

Predictability at a time r in the future is defined by

R(x{t),x(t + T))
H(x(t)) '

and linear predictability by

L(x(t),x(t + r))
H(x(t)) '

where R and L are the Redundancy and Linear Re-
dundancy, and H is the Entropy.

Pretzel Knot

A Knot obtained from a Tangle which can be repre-
sented by a FINITE sequence of INTEGERS.

see also TANGLE

References

Adams, C. C. The Knot Book: An Elementary Introduction

to the Mathematical Theory of Knots. New York: W. H.

Freeman, p. 48, 1994.

Prediction Paradox

see Unexpected Hanging Paradox

Predictive Value

The Positive predictive value is the probability that a
test gives a true result for a true statistic. The negative
predictive value is the probability that a test gives a
false result for a false statistic.

see also Power (Statistics), Sensitivity, Speci-
ficity, Statistical Test

Primality Certificate

A short set of data that proves the primality of a num-
ber. A certificate can, in general, be checked much
more quickly than the time required to generate the
certificate. Varieties of primality certificates include
the Pratt Certificate and Atkin-Goldwasser-
Kilian-Morain Certificate.

see also Atkin-Goldwasser-Kilian-Morain Cer-
tificate, Compositeness Certificate, Pratt Cer-
tificate

References

Wagon, S. "Prime Certificates." §8.7 in Mathematica in Ac-
tion. New York: W. H. Freeman, pp. 277-285, 1991.

Primality Test

Prime Arithmetic Progression 1423

Primality Test

A test to determine whether or not a given number is
Prime. The Rabin-Miller Strong Pseudoprime
Test is a particularly efficient Algorithm used by
Mathematical version 2.2 (Wolfram Research, Cham-
paign, IL). Like many such algorithms, it is a proba-
bilistic test using PSEUDOPRIMES, and can potentially
(although with very small probability) falsely identify
a Composite Number as Prime (although not vice
versa). Unlike PRIME FACTORIZATION, primality test-
ing is believed to be a P-Problem (Wagon 1991). In
order to guarantee primality, an almost certainly slower
algorithm capable of generating a PRIMALITY CERTIFI-
CATE must be used.

see also Adleman-Pomerance-Rumely Primality
Test, Fermat's Little Theorem Converse, Fer-
mat's Primality Test, Fermat's Theorem, Lucas-
Lehmer Test, Miller's Primality Test, Pepin's
Test, Pocklington's Theorem, Proth's Theorem,
Pseudoprime, Rabin-Miller Strong Pseudoprime
Test, Ward's Primality Test, Wilson's Theorem

References

Beauchemin, P.; Brassard, G,; Crepeau, C; Goutier, C; and
Pomerance, C. "The Generation of Random Numbers that
are Probably Prime." J. Crypt 1, 53-64, 1988.

Brillhart, J.; Lehmer, D. H.; Selfridge, J.; WagstafT, S. S. Jr.;
and Tuckerman, B. Factorizations of b n ± 1, 6 = 2,
3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence,
Rl; Amer. Math. Soc, pp. lviii-lxv, 1988.

Cohen, H. and Lenstra, A. K. "Primality Testing and Jacobi
Sums." Math. Comput. 42, 297-330, 1984.

Knuth, D. E. The Art of Computer Programming, Vol. 2:
Seminumerical Algorithms, 2nd ed. Reading, MA:
Addison-Wesley, 1981.

Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, 1994.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 15-17, 1991.

Primary

Each factor pi ai in an Integer's Prime Decomposi-
tion is called a primary.

Primary Representation

Let 7r be a unitary REPRESENTATION of a GROUP G on
a separable HlLBERT SPACE, and let R(n) be the small-
est weakly closed algebra of bounded linear operators
containing all n(g) for g e G. Then tt is primary if the
center of R(tt) consists of only scalar operations.

References

Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.

Prime

A symbol used to distinguish one quantity x ("x
prime") from another related x. Primes are most com-
monly used to denote transformed coordinates, conju-
gate points, and DERIVATIVES.
see also PRIME ALGEBRAIC NUMBER, PRIME NUMBER

Prime Algebraic Number

An irreducible ALGEBRAIC INTEGER which has the
property that, if it divides the product of two algebraic
Integers, then it Divides at least one of the factors.
1 and -1 are the only INTEGERS which Divide every
Integer. They are therefore called the Prime Units.

see also ALGEBRAIC INTEGER, PRIME UNIT

Prime Arithmetic Progression

Let the number of PRIMES of the form mk + n less than
x be denoted 7r m( „(a:). Then

lim

Li(a:)

1
0(a)'

where Li(as) is the LOGARITHMIC INTEGRAL and <j>(x) is
the TOTIENT FUNCTION.

Let P be an increasing arithmetic progression of n
Primes with minimal difference d > 0. If a PRIME
p < n does not divide d, then the elements of P must
assume all residues modulo p, specifically, some element
of P must be divisible by p. Whereas P contains only
primes, this element must be equal to p.

If d < n# (where n# is the PRIMORIAL of n), then some
prime p < n does not divide d, and that prime p is in P.
Thus, in order to determine if P has d < n#, we need
only check a finite number of possible P (those with d <
n# and containing prime p < n) to see if they contain
only primes. If not, then d > n#. If d = n#, then
the elements of P cannot be made to cover all residues
of any prime p. The PRIME PATTERNS CONJECTURE
then asserts that there are infinitely many arithmetic
progressions of primes with difference d.

A computation shows that the smallest possible common
difference for a set of n or more Primes in arithmetic
progression for n = 1, 2, 3, . . . is 0, 1, 2, 6, 6, 30, 150,
210, 210, 210, 2310, 2310, 30030, 510510, ... (Sloane's
A033188, Ribenboim 1989, Dubner and Nelson 1997,
Wilson). The values up to n = 13 are rigorous, while the
remainder are lower bounds which assume the validity
of the Prime Patterns Conjecture and are simply
given by p n -7#, where pi is the ith PRIME. The smallest
first terms of arithmetic progressions of n primes with
minimal differences are 2, 2, 3, 5, 5, 7, 7, 199, 199,
199, 60858179, 147692845283, 14933623, . . . (Sloane's
A033189; Wilson).

Smaller first terms are possible for nonminimal n-term
progressions. Examples include the 8-term progression
11 + 1210230& for k = 0, 1, ..., 7, the 12-term pro-
gression 23143 + 30030& for k = 0, 1, . . . , 11 (Golubev
1969, Guy 1994), and the 13-term arithmetic progres-
sion 766439 + 510510A; for k = 0, 1, . . . , 12 (Guy 1994).

The largest known set of primes in Arithmetic SE-
QUENCE is 22,

11, 410, 337, 850, 553 + 4, 609, 098, 694, 200A;

1424 Prime Arithmetic Progression

Prime Array

for fc = 0, 1, . . . , 21 (Pritchard et al. 1995, UTS School
of Mathematical Sciences).

The largest known sequence of consecutive Primes in
Arithmetic Progression (i.e., all the numbers be-
tween the first and last term in the progression, except
for the members themselves, are composite) is ten, given
by

100, 996, 972, 469, 714, 247, 637, 786, 655, 587, 969,

840, 329, 509, 324, 689, 190, 041, 803, 603, 417, 758,

904, 341, 703, 348, 882, 159, 067, 229, 719 + 210k

for k = 0, 1, . . . , 9, discovered by Harvey Dubner, Tony
Forbes, Manfred Toplic, et al. on March 2, 1998. This
beats the record of nine set on January 15, 1998 by the
same investigators,

99, 679, 432, 066, 701, 086, 484, 490, 653, 695, 853,

561, 638, 982, 364, 080, 991, 618, 395, 774, 048, 585,

529, 071, 475, 461, 114, 799, 677, 694, 651 + 210A;

for k = 0, 1, . . . , 8 (two sequences of nine are now
known), the progression of eight consecutive primes
given by

43, 804, 034, 644, 029, 893, 325, 717, 710, 709, 965,

599, 930, 101, 479, 007, 432, 825, 862, 362, 446, 333,

961, 919, 524, 977, 985, 103, 251, 510, 661 + 210A:

for k = 0, 1, . . . , 7, discovered by Harvey Dubner, Tony
Forbes, et al. on November 7, 1997 (several are now
known), and the progression of seven given by

1, 089, 533, 431, 247, 059, 310, 875, 780, 378, 922, 957, 732,
908, 036, 492, 993, 138, 195, 385, 213, 105, 561, 742, 150,
447, 308, 967, 213, 141, 717, 486, 151 + 210A;,

for k — 0, 1, ..., 6, discovered by H. Dubner and
H. K. Nelson on Aug. 29, 1995 (Peterson 1995, Dubner
and Nelson 1997). The smallest sequence of six consec-
utive Primes in arithmetic progression is

121,174,811 + 30*;

for k — 0, 1, . . . , 5 (Lander and Parkin 1967, Dubner and
Nelson 1997). According to Dubner et al, a trillion-fold
increase in computer speed is needed before the search
for a sequence of 11 consecutive primes is practical, so
they expect the ten-primes record to stand for a long
time to come.

It is conjectured that there are arbitrarily long sequences
of Primes in Arithmetic Progression (Guy 1994).

see also ARITHMETIC PROGRESSION, CUNNINGHAM

Chain, Dirichlet's Theorem, Linnik's Theorem,

Prime Constellation, Prime-Generating Poly-
nomial, Prime Number Theorem, Prime Patterns
Conjecture, Prime Quadruplet

References

Abel, U. and Siebert, H. "Sequences with Large Numbers of
Prime Values." Amer. Math. Monthly 100, 167-169, 1993.

Caldwell, C K. "Cunningham Chain." http://www.utm.
edu/re sear ch/primes/glossary/CuiminghamChain. html.

Courant, R. and Robbins, H. "Primes in Arithmetical Pro-
gressions." §1.2b in Supplement to Ch. 1 in What is Math-
ematics?: An Elementary Approach to Ideas and Methods,
2nd ed. Oxford, England: Oxford University Press, pp. 26-
27, 1996.

Davenport, H. "Primes in Arithmetic Progression" and
"Primes in Arithmetic Progression: The General Modu-
lus." Chs. 1 and 4 in Multiplicative Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 1-11 and 27-34, 1980.

Dubner, H. and Nelson, H. "Seven Consecutive Primes in
Arithmetic Progression." Math. Comput. 66, 1743-1749,
1997.

Forbes, T. "Searching for 9 Consecutive Primes in Arith-
metic Progression." http : //www . ltkz . demon . co . uk/ar2/
9primes.htm.

Forman, R. "Sequences with Many Primes." Amer. Math.
Monthly 99, 548-557, 1992.

Golubev, V. A. "Faktorisation der Zahlen der Form x 3 ±4x 2 +
3a: ± 1." Anz. Osterreich. Akad. Wiss. Math.-Naturwiss.
KL 184-191, 1969.

Guy, R. K. "Arithmetic Progressions of Primes" and "Con-
secutive Primes in A. P." §A5 and A6 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 15-17 and 18, 1994.

Lander, L. J. and Parkin, T. R. "Consecutive Primes in
Arithmetic Progression." Math. Comput. 21, 489, 1967.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 154-155, 1979.

Nelson, H. L. "There Is a Better Sequence." J. Recr. Math.
8, 39-43, 1975.

Peterson, I. "Progressing to a Set of Consecutive Primes."
Sci. News 148, 167, Sep. 9, 1995.

Pritchard, P. A.; Moran, A.; and Thyssen, A. "Twenty- Two
Primes in Arithmetic Progression." Math. Comput. 64,
1337-1339, 1995.

Ramare, O. and Rumely, R. "Primes in Arithmetic Progres-
sions." Math. Comput. 65, 397-425, 1996.

Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, p. 224, 1989.

Shanks, D. "Primes in Some Arithmetic Progressions and a
General Divisibility Theorem." §104 in Solved and Un-
solved Problems in Number Theory, 4th ed. New York:
Chelsea, pp. 104-109, 1993.

Sloane, N. J. A. Sequences A033188 and A033189 in "An On-
Line Version of the Encyclopedia of Integer Sequences."

Weintraub, S. "Consecutive Primes in Arithmetic Progres-
sion." J. Recr. Math. 25, 169-171, 1993.

Zimmerman, P. http://www.loria.fr/-zimmerma/records/
8primes . announce.

Prime Array

Find the m x n ARRAY of single digits which contains
the maximum possible number of PRIMES, where allow-
able Primes may lie along any horizontal, vertical, or
diagonal line. For m = n = 2, 11 Primes are maximal
and are contained in the two distinct arrays

.4(2,2)

1 3

4 7

1 3

7 9

Prime Array

Prime Constellation 1425

giving the Primes (3, 7, 13, 17, 31, 37, 41, 43, 47, 71, 73)
and (3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97), respectively.
For the 3x2 array, 18 PRIMES are maximal and are
contained in the arrays

4(3,2) =

"l 1 3

"17 2

"l 7 2"

9 7 4

)

3 5 9

j

4 3 9

'l 7 5"

1 7 9"

1 7 9"

4 3 9

)

3 2 5

j

4 3 2

"l 7 9"
4 3 4

)

"3 1 6*
4 7 9

;

"3 7 6~
4 19

The best 3 x 3, 4 x 4, and 5x5 prime arrays known were
found by C. Rivera and J. Ayala in 1998. They are

'1 1 3"

A(3,3) =

which contains 30 PRIMES,

1
6
7
3

which contains 63 Primes, and
■1 1

A(4,4) =

9 9

A(5,5):= 8 9

3 3

L3 2

which contains 116 Primes. S. C. Root found the a 6 x 6
array containing 187 primes:

■3 1 7

3l

3

7

1

9

4(6,6):

9 9 5 6

118 1

1 3

3 4
3 7

6 3

31
9
2
3

9 19 9
9 3 7 9J

In 1998, M. Oswald found five new 6x6 arrays with 187
primes:

ri

3
9
9
9

L9 1

r3 1

9 9

118 1
13 6 3

3 4
3 7

r3 1 7

9 9 5
118

13 6

3 4 9

9 9 9

9 1
9 9

91

4

3

3

7

3J

3

9

2

3

9

9

3

9

5

3

9 8
L9 1

1 9

2 3

9 9
3 3

Rivera and Ayala conjecture that the 30-prime solution
for A(3,3) is maximal and unique. The following in-
tervals have been completely searched without finding
another 30-prime or better 3x3 array: [1, 67 x 10 6 ],
[100 x 10 6 , 133 x 10 6 ], [200 x 10 6 , 228 x 10 6 ], [300 x 10 6 ,
325 x 10 6 ], and [400 x lO 6 , 418 x 10 6 ].

Heuristic arguments by Rivera and Ayala suggest that
the maximum possible number of primes in 4 x 4, 5 X
5, and 6 x 6 arrays are 58-63, 112-121, and 205-218,
respectively.

see also Array, Prime Arithmetic Progression,
Prime Constellation, Prime String

References

Dewdney, A. K. "Computer Recreations: How to Pan for

Primes in Numerical Gravel." Sci. Amer. 259, 120-123,

July 1988.
Lee, G. "Winners and Losers." Dragon User. May 1984.
Lee, G. "Gordon's Paradoxically Perplexing Primesearch

Puzzle." http : //www. geocities . com/MotorCity/7983/

primesearch.html.
Rivera, C. "Problems & Puzzles (Puzzles): The Gor-
don Lee Puzzle," http://www.sci.net.mx/-crivera/ppp/

puzz_001.htm.
$ Weisstein, E. W. "Prime Arrays." http: //www. astro.

Virginia. edu/-eww6n/math/notebooks/PrimeArray.m.

Prime Circle

A prime circle of order 2m is a CIRCULAR PERMUTA-
TION of the numbers from 1 to 2m with adjacent PAIRS
summing to a PRIME. The number of prime circles for
m= 1, 2, ..., are 1, 1, 1, 2, 48, . . . .

References

Filz, A. "Problem 1046." J. Recr. Math. 14, 64, 1982.
Filz, A. "Problem 1046." J. Recr. Math. 15, 71, 1983.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 105-106, 1994.

Prime Cluster

see PRIME CONSTELLATION

Prime Constellation

A prime constellation, also called a Prime /c-Tuple or
Prime fc-TUPLET, is a sequence of k consecutive num-
bers such that the difference between the first and last
is, in some sense, the least possible. More precisely,
a prime A;-tuplet is a sequence of consecutive PRIMES
(Pi) P2, • ••, Pk) with pk - Pi — s(k), where s(k) is
the smallest number s for which there exist k integers
h < &2 < • • . < bk, bk — 61 = s and, for every PRIME g,
not all the residues modulo q are represented by 61, 62,
. . . , bk (Forbes). For each &, this definition excludes a
finite number of clusters at the beginning of the prime
number sequence. For example, (97, 101, 103, 107, 109)
satisfies the conditions of the definition of a prime 5-
tuplet, but (3, 5, 7, 11, 13) does not because all three
residues modulo 3 are represented (Forbes).

A prime double with s(2) = 2 is of the form (p, p + 2)
and is called a pair of Twin Primes. Prime doubles of

1426

Prime Constellation

Prime Constellation

the form (p, p + 6) are called SEXY PRIMES. A prime
triplet has s(3) = 6. However, the constellation (p, p+2,
p + 4) cannot exist, since both p + 2 and p + 4 cannot
be Prime. However, there are several types of prime
triplets which can exist: (p, p + 2, p + 6), (p, p + 4,
P + 6), (p, p + 6, p + 12). A Prime Quadruplet is
a constellation of four successive Primes with minimal
distance s(4) = 8, and is of the form (p, p + 2, p + 6,
p + 8). The sequence s(n) therefore begins 2, 6, 8, and
continues 12, 16, 20, 26, 30, ... (Sloane's A008407).
Another quadruplet constellation is (p, p + 6, p + 12,
p+18).

The first First Hardy-Littlewood Conjecture
states that the number of constellations < x are asymp-
totically given by

r dx'

= 1.320323632 / -

J 2 (Inz') 2

(1)

P>3

= 1.320323632

AX

J2

dx f

/2 ( lniC/ ) 2

p>3

= 2.640647264

r dx 1

J \ (In i'

J'x(p,P + 2 ) p + 6)~-]T

) 2

P 2 (P~3)
(p-l)»

(2)

(3)

r dx'

k On*')

= 2.858248596

7 2 (lnx') 3

/" x dx'
= 2.858248596 / -^-^
J 2 (In a:') 3

p>5

= 4.151180864 / -^
7 2 (Ins'

y 2 (lnx'

(4)

(5)

) 4
(6)

p>5

= 8.302361728

r

(7)

These numbers are sometimes called the Hardy-
Littlewood Constants. (1) is sometimes called the
extended Twin Prime Conjecture, and

Cp,p+2 — 2n2,

(8)

where n 2 is the Twin Primes Constant. Riesel (1994)
remarks that the Hardy-Littlewood Constants can
be computed to arbitrary accuracy without needing the
infinite sequence of primes.

The integrals above have the analytic forms

£

dx' __ T . / \ ^ n

___ _ U(x) + _ _ _

(9)

dx' _ x x(l + lnx) 1 1

2 (W)* ~ 5 [X > (lnx) 2 + HT2 + (b72)2

[' dx' _ 1 f 2[2 + ln2 + (ln2) 2 ]

7 2 (lnx')4-6\ Ll W+ (M2)5

n[2 + lnn+(lnn) 2 ] )
(Inn) 3 J'

(10)

(11)

where Li(z) is the Logarithmic Integral.

The following table gives the number of prime constel-
lations < 10 8 , and the second table gives the values pre-
dicted by the Hardy-Littlewood formulas.

Count

10 5

10 6

10 7

10 8

(P.P + 2)

1224

8169

58980

440312

(PiP + 4)

1216

8144

58622

440258

(p,P + 6)

2447

16386

117207

879908

(p,p + 2,p + 6)

259

1393

8543

55600

(p,p + 4,p

+ 6)

248

1444

8677

55556

(p,p + 2,p

+ 6,p + 8)

38

166

899

4768

(p,p + 6,p

+ 12,p+18)

75

325

1695

9330

Hardy-Littlewood

10 5

10 6

10 7

10 8

(p.P + 2)

1249

8248

58754

440368

(P.P + 4)

1249

8248

58754

440368

(p,P + 6)

2497

16496

117508

880736

(p,p + 2,p + 6)

279

1446

8591

55491

(p,p + 4,p + 6)

279

1446

8591

55491

(p,p + 2,p + 6,p + 8)

53

184

863

4735

(p>p + 6,p+ 12, p+ 18)

Consider prime constellations in which each term is of
the form n 2 + 1. Hardy and Littlewood showed that the
number of prime constellations of this form < x is given

by

P{x) ^CVspnx)- 1 , (12)

where

n

p>2
p prime

1-

(_l)(P"l)/2

P- 1

1.3727...

(13)

(Le Lionnais 1983).

Forbes gives a list of the "top ten" prime fc-tuples for
2 < k < 17. The largest known 14-constellations are
(11319107721272355839 + 0, 2, 8, 14, 18, 20, 24, 30,
32, 38, 42, 44, 48, 50), (10756418345074847279 + 0,
2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50),

Prime Counting Function

Prime Counting Function 1427

(6808488664768715759 + 0, 2, 8, 14, 18, 20, 24, 30,
32, 38, 42, 44, 48, 50), (6120794469172998449 + 0,
2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50),
(5009128141636113611 + 0, 2, 6, 8, 12, 18, 20, 26, 30,
32, 36, 42, 48, 50).

The largest known prime 15-constellations are
(84244343639633356306067 + 0, 2, 6, 12, 14, 20, 24, 26,
30, 36, 42, 44, 50, 54, 56), (8985208997951457604337+0,
2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56),
(3594585413466972694697 + 0, 2, 6, 12, 14, 20, 26, 30,
32, 36, 42, 44, 50, 54, 56), (3514383375461541232577+0,
2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56),
(3493864509985912609487 + 0, 2, 6, 12, 14, 20, 24, 26,
30, 36, 42, 44, 50, 54, 56).

The largest known prime 16- constellations are
(3259125690557440336637+0, 2, 6, 12, 14, 20, 26, 30, 32,
36, 42, 44, 50, 54, 56, 60), (1522014304823128379267+0,
2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60),
(47710850533373130107 + 0, 2, 6, 12, 14, 20, 26, 30, 32,
36, 42, 44, 50, 54, 56, 60), (13, 17, 19, 23, 29, 31, 37, 41,
43, 47, 53, 59, 61, 67, 71, 73).

The largest known prime 17-constellations are
(3259125690557440336631 + 0, 6, 8, 12, 18, 20, 26, 32,
36, 38, 42, 48, 50, 56, 60, 62, 66), (17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83) (13, 17, 19, 23,
29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79).

see also Composite Runs, Prime Arithmetic Pro-
gression, /c-Tuple Conjecture, Prime /c-Tuples
Conjecture, Prime Quadruplet, Sexy Primes,
Twin Primes

References

Forbes, T. "Prime fc-tuplets." http://www.ltkz.demon.co.

uk/ktuplot s . htm.
Guy, R. K. "Patterns of Primes." §A9 in Unsolved Problems

in Number Theory, 2nd ed. New York: Springer- Verlag,

pp. 23-25, 1994.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,

p. 38, 1983.
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, pp. 60-74,

1994.
Sloane, N. J. A. Sequence A008407 in "An On-Line Version

of the Encyclopedia of Integer Sequences."

Prime Counting Function

2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, . . . (Sloane's A000720).
The following table gives the values of 7r(n) for powers
of 10 (Sloane's A006880; Hardy and Wright 1979, p. 4;
Shanks 1993, pp. 242-243; Ribenboim 1996, p. 237).
Deleglise and Rivat (1996) have computed 7r(10 20 ).

tt(10 31

) = 168

tt(10 4

) = 1,229

tt(10 5 ,

) = 9,592

tt(10 6.

) = 78,498

tt(10 7

) = 664, 579

tt(10 8

) = 5,761,455

tt(io 9 ;

) = 50, 847, 534

tt(io 10 ;

) =455,052,511

7r(10 n

) = 4,118,054,813

t(io 12 ,

) = 37,607,912,018

tt(io 13 ;

) =346,065,536,839

tt(10 14

1 = 3,204,941,750,802

tt(10 15 ,

( = 29,844,570,422,669

tt(io 16 ;

) = 279,238,341,033,925

tt(io 17 ;

= 2,623,557,157,654,233

tt(io 18 ;

= 24, 739, 954, 287, 740, 860

tt(io 19 ;

= 234, 057, 667, 276, 344, 607,

7r(10 9 ) is incorrectly given as 50,847,478 in Hardy and
Wright (1979). The prime counting function can be
expressed by Legendre's Formula, Lehmer's For-
mula, Mapes' Method, or Meissel's Formula. A
brief history of attempts to calculate 7r(n) is given by
Berndt (1994). The following table is taken from Riesel
(1994).

Method

Time

Storage

Legendre

G(x)

G{x^ 2 )

Meissel

0(x/(lnx) s )

0{x^ 2 /]nx)

Lehmer

G(x/{\nx) 4 )

0(x 1/s /\nx)

Mapes'

O(x - 7 )

O(x - 7 )

Lagarias- Miller- O dly zko

e>(x 2 / 3+e )

0{x 1 ^*)

Lagarias-Odlyzko 1

0(z 3 / 5+e )

0{x*)

Lagarias-Odlyzko 2

G(x^ 2+e )

0(x 1/4 +<)

A modified version of the prime counting function is
given by

7T (P) ■

Jtt(p)
\t(p)-§

for p composite
for p prime

mp) = ^2

t*(x)f(x 1/n )

50 100 150 200

The function 7r(n) giving the number of PRIMES less
than n (Shanks 1993, p. 15). The first few values are 0, 1,

where /x(n) is the MOBIUS FUNCTION and f(x) is the
Riemann-Mangoldt Function.

The notation 7r a ,b is also used to denote the number of
PRIMES of the form ak + b (Shanks 1993, pp. 21-22).

1428 Prime Counting Function

Prime Cut

Groups of Equinumerous values of n a ,b include (^3,1,

^3,2), (tT4,1i ^4,3), (7T5,ij 7T 5i 2, 7^5,3, ^5,4), (^6,H ^O.s))
(tT7,1i 7T7,2, 7T7,3, ^7,4, ^7,5, 7I"7,6), (^8,1) ^8,3, ^8,5, ^8,7),

(tt9,1i ^9,2 , 7i"9,4, ^9,5, ^9,7, ^9,8), and so on. The values
of 7T ni fc for small n are given in the following table for
the first few powers of ten (Shanks 1993).

n

7T3,l( n )

7T3,2(n)

7r 4 ,i(n)

7T4,3(n)

10 1

1

2

1

2

10 2

11

13

11

13

10 3

80

87

80

87

10 4

611

617

609

619

10 5

4784

4807

4783

4808

10 6

39231

39266

39175

39322

10 7

332194

332384

332180

332398

n

^5,i(n)

7T 5 ,2(n)

7T5,3(n)

7r 5 ,4(n)

10 1

2

1

10 2

5

7

7

5

10 3

40

47

42

38

10 4

306

309

310

303

10 5

2387

2412

2402

2390

10 6

19617

19622

19665

19593

10 7

166104

166212

166230

166032

n

7T6,l(n)

^6,5 (™)

10 1

1

1

10 2

11

12

10 3

80

86

10 4

611

616

10 5

4784

4806

10 6

39231

39265

n

7I"7,1

7T7,2

71*7,3

7I"7,4

7T7,5

7T7,6

10 1

1

1

1

10 2

3

4

5

3

5

4

10 3

28

27

30

26

29

27

10 4

203

203

209

202

211

200

10 5

1593

1584

1613

1601

1604

1596

10 6

13063

13065

13105

13069

13105

13090

n

7T 8 ,l(^)

7I"8,3(n)

7T 8 ,5(n)

1*8,7 (n)

10 1

1

1

1

10 2

5

7

6

6

10 3

37

44

43

43

10 4

295

311

314

308

10 5

2384

2409

2399

2399

10 6

19552

19653

19623

19669

10 7

165976

166161

166204

166237

Note that since 7Ts,i(n), irs^n), irs^in), and 778,7(71) are
Equinumerous,

7T4,l(n) = 7T8,l(n) +7T 8 ,5
7T4,3(n) = 7r 8) 3(7l) + 7T8,7

are also equinumerous.

Erdos proved that there exist at least one Prime of the
form 4k -f- 1 and at least one Prime of the form 4k + 3
between n and 2n for all n > 6.

The smallest x such that x > nir(x) for n = 2, 3, . . .
are 2, 27, 96, 330, 1008, . . . (Sloane's A038625), and the
corresponding tt(x) are 1, 9, 24, 24, 66, 168, . . . (Sloane's
A038626). The number of solutions of x > nir{x) for
n = 2, 3, . . . are 4, 3, 3, 6, 7, 6, . , . (Sloane's A038627).

see also BERTELSEN'S NUMBER, EQUINUMEROUS,

Prime Arithmetic Progression, Prime Num-
ber Theorem, Riemann Weighted Prime-Power
Counting Function

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 134-135, 1994.

Brent, R. P. "Irregularities in the Distribution of Primes and
Twin Primes." Math. Corn-put. 29, 43-56, 1975.

Deleglise, M. and Rivat, J. "Computing -k(x): The Meissel,
Lehmer, Lagarias, Miller, Odlyzko Method." Math. Corn-
put. 65, 235-245, 1996.

Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/hrdyltl/hrdyltl.html.

Forbes, T. "Prime fc-tuplets." http://www.ltkz.demon.co.
uk/ktuplets.htm.

Guiasu, S. "Is There Any Regularity in the Distribution of
Prime Numbers at the Beginning of the Sequence of Posi-
tive Integers?" Math. Mag. 68, 110-121, 1995.

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.

Lagarias, J.; Miller, V. S.; and Odlyzko, A. "Computing n(x):
The Meissel-Lehmer Method." Math. Comput. 44, 537-
560, 1985.

Lagarias, J. and Odlyzko, A. "Computing 7r(x): An Analytic
Method." J. Algorithms 8, 173-191, 1987.

Mapes, D. C. "Fast Method for Computing the Number of
Primes Less than a Given Limit." Math. Comput. 17,
179-185, 1963.

Meissel, E. D. F. "Uber die Bestimmung der Primzahlmenge
innerhalb gegebener Grenzen." Math. Ann. 2, 636-642,
1870.

Ribenboim, P. The New Book of Prime Number Records, 3rd
ed. New York: Springer- Verlag, 1996.

Riesel, H. "The Number of Primes Below x." Prime Numbers
and Computer Methods for Factorization, 2nd ed. Boston,
MA: Birkhauser, pp. 10-12, 1994.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, 1993.

Sloane, N. J. A. Sequences A038625, A038626, A038627,
A000720/M2056, and A006880/M3608 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 74-76, 1991.

Wolf, M. "Unexpected Regularities in the Distribution of
Prime Numbers." http://www.ift.uni.wroc.pl/-mwolf.

Prime Cut

Find two numbers such that x 2 = y 2 (mod n). If you
know the GREATEST COMMON DIVISOR of n and x - y,
there exists a high probability of determining a PRIME
factor. Taking small numbers x which additionally give
small PRIMES x 2 = p (mod n) further increases the
chances of finding a Prime factor.

Prime Decomposition

Prime Factorization Algorithms 1429

Prime Decomposition

Given an Integer n, the prime decomposition is written

n = Pl ai P 2 a2 *-'Pn an ,

where pi are the n Prime factors, each of order o^. Each
factor pi ai is called a Primary.

see also PRIMARY, PRIME FACTORIZATION ALGO-
RITHMS, Prime Number

Prime Difference Function

100 200 300 400

dn =Pn + l — Pn-

500

The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2,
4, 6, 6, . . . (Sloane's A001223). Rankin has shown that

d n >

c In n In In n In In In In n
(In In Inn) 2

for infinitely many n and for some constant c (Guy
1994).

An integer n is called a JUMPING CHAMPION if n is the
most frequently occurring difference between consecu-
tive primes n < N for some N (Odlyzko et al. ).

see also Andrica's Conjecture, Good Prime, Jump-
ing Champion, Polya Conjecture, Prime Gaps,
Shanks' Conjecture, Twin Peaks

References

Bombieri, E. and Davenport, H. "Small Differences Between

Prime Numbers." Proc. Roy. Soc. A 293, 1-18, 1966.
Erdos, P.; and Straus, E. G. "Remarks on the Differences

Between Consecutive Primes." Elem. Math. 35, 115-118,

1980.
Guy, R. K. "Gaps between Primes. Twin Primes" and "In-
creasing and Decreasing Gaps." §A8 and All in Unsolved

Problems in Number Theory, 2nd ed. New York: Springer-

Verlag, pp. 19-23 and 26-27, 1994.
Odlyzko, A.; Rubinstein, M.; and Wolf, M. "Jumping

Champions." http://www.research.att.com/-amo/doc/

recent .html.
Riesel, H. "Difference Between Consecutive Primes." Prime

Numbers and Computer Methods for Factorization, 2nd

ed. Boston, MA: Birkhauser, p. 9, 1994.
Sloane, N. J. A. Sequence A001223/M0296 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Prime Diophantine Equations

k + 2 is Prime Iff the 14 Diophantine Equations in
26 variables

wz + h + j — q —

(gk-\-2g + k + l)(h + j) + h - z =

(1)
(2)

16(fc + l) 3 (fc + 2)(n + l) 2 + 1 - f = (3)

2-n + p + q + z — gr =
e 3 (e + 2)(a+l) 2 + l-o 2 =0
(a 2 - l)y 2 + 1 - x 2 =
16r 2 y 4 (a 2 -l) + l-u 2 =

(4)
(5)
(6)
(7)

n+l+v-y^0 (8)

(a 2 -l)J 2 + l-m 2 = (9)

ai-rk + l-l-i = Q (10)

{[a + u{u - a)] 2 - l}(n + 4dy) 2 + 1 - (x + cu) 2 =

(11)
p + l(a-n-l) + b(2an + 2a - n 2 - 2n - 2) - m =

(12)
q + y(a-p-\) + s(2ap + 2a - p - 2p - 2) - x =

(13)
z+pl(a-p) + t(2ap-p 2 - 1) -pm = (14)

have a POSITIVE integral solution.

References

Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, p. 39, 1994.

Prime Factorization

see Factorization, Prime Decomposition, Prime
Factorization Algorithms, Prime Factors

Prime Factorization Algorithms

Many ALGORITHMS have been devised for determining
the PRIME factors of a given number. They vary quite a
bit in sophistication and complexity. It is very difficult
to build a general-purpose algorithm for this computa-
tionally "hard" problem, so any additional information
which is known about the number in question or its fac-
tors can often be used to save a large amount of time.

The simplest method of finding factors is so-called "DI-
RECT Search Factorization" (a.k.a. Trial Divi-
sion). In this method, all possible factors are system-
atically tested using trial division to see if they actually
DIVIDE the given number. It is practical only for very
small numbers.

see also Brent's Factorization Method, Con-
tinued Fraction Factorization Algorithm, Di-
rect Search Factorization, Dixon's Factor-
ization Method, Elliptic Curve Factorization
Method, Euler's Factorization Method, Ex-
cludent Factorization Method, Fermat's Fac-
torization Method, Legendre's Factorization

1430

Prime Factors

Prime Gaps

Method, Lenstra Elliptic Curve Method, Num-
ber Field Sieve Factorization Method, Pollard
p - 1 Factorization Method, Pollard p Factor-
ization Algorithm, Quadratic Sieve Factoriza-
tion Method, Trial Division, Williams p + l Fac-
torization Method

References

Bressoud, D. M. Factorization and Prime Testing. New
York: Springer- Verlag, 1989.

Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.;
and Tuckerman, B. Factorizations of b n ± 1, b = 2,
3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence,
RI: Amer. Math. Soc, liv-lviii, 1988.

Dickson, L. E. "Methods of Factoring." Ch. 14 in History of
the Theory of Numbers, Vol 1: Divisibility and Primality.
New York: Chelsea, pp. 357-374, 1952.

Herman, P. "The Factoring Page!" http://www.pslc.ucla.
edu/-a540pau/f actoring.

Lenstra, A. K. and Lenstra, H. W. Jr. "Algorithms in Num-
ber Theory." In Handbook of Theoretical Computer Sci-
ence, Volume A: Algorithms and Complexity (Ed. J. van
Leeuwen). New York: Elsevier, pp. 673-715, 1990.

Odlyzko, A. M. "The Complexity of Computing Discrete Log-
arithms and Factoring Integers." §4.5 in Open Problems in
Communication and Computation (Ed. T. M. Cover and
B. Gopinath). New York: Springer- Verlag, pp. 113-116,
1987.

Odlyzko, A. M. "The Future of Integer Factorization." Cryp-
toBytes: The Technical Newsletter of RSA Laboratories 1,
No. 2, 5-12, 1995.

Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996.

Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, 1994.

Williams, H. C. and Shallit, J. O. "Factoring Integers Be-
fore Computers." In Mathematics of Computation 1943-
1993, Fifty Years of Computational Mathematics (Ed.
W. Gautschi). Providence, RI: Amer. Math. Soc, pp. 481-
531, 1994.

Prime Factors

20 40 60 80 100 200 400 600 800 1000

The number of DISTINCT Prime FACTORS of a number
n is denoted td(n). The first few values for n = 1, 2,
... are 0, 1, 1, 1, 1, 2, 1,1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2,
1, 2, . . . (Sloane's A001221; top figure). The number of
not necessarily distinct prime factors of a number n is
denoted r{n). The first few values for n = 1, 2, . . . are
0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, . . .
(Sloane's A001222; bottom figure).

see also Distinct Prime Factors, Divisor Func-
tion, Greatest Prime Factor, Least Prime Fac-
tor, Liouville Function, Polya Conjecture,
Prime Factorization Algorithms

References

Sloane, N. J. A. Sequences A001222/M0094 and A001221/
M0056 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Prime Field

A Galois Field GF(p) where p is Prime.

Prime Gaps

Letting

d n = Pn+l — Pn

be the PRIME DIFFERENCE FUNCTION, Rankin has
showed that

d n >

c In n In In n In In In In n

(In In Inn) 2

for infinitely many n are for some constant c (Guy 1994).

Let p(d) be the smallest PRIME following d or more con-
secutive Composite Numbers. The largest known is

p(804) = 90, 874, 329, 412, 297.

The largest known prime gap is of length 4247, occur-
ring following 10 314 - 1929 (Baugh and O'Hara 1992),
although this gap is almost certainly not maximal (i.e.,
there probably exists a smaller number having a gap of
the same length following it).

Let c(n) be the smallest starting INTEGER c(n) for a
run of n consecutive COMPOSITE Numbers, also called
a Composite Run. No general method other than ex-
haustive searching is known for determining the first oc-
currence for a maximal gap, although arbitrarily large
gaps exist (Nicely 1998). Cramer (1937) and Shanks
(1964) conjectured that a maximal gap of length n ap-
pears at approximately exp( v / n). Wolf conjectures that
the maximal gap of length n appears approximately at

7r(n)[21n7r(n) - Inn + ln(2C 2 )] '

where 7r(n) is the Prime Counting Function and C 2
is the Twin Primes Constant.

The first few c(n) for n = 1, 2, . . . are 4, 8, 8, 24,
24, 90, 90, 114, ... (Sloane's A030296). The following
table gives the same sequence omitting degenerate runs
which are part of a run with greater n, and is a complete
list of smallest maximal runs up to 10 15 . c(n) in this
table is given by Sloane's A008950, and n by Sloane's
A008996. The ending integers for the run corresponding
to c(n) are given by Sloane's A008995. Young and Potler
(1989) determined the first occurrences of prime gaps up
to 72,635,119,999,997, with all first occurrences found

Prime Gaps

Prime- Generating Polynomial 1431

between 1 and 673. Nicely (1998) extended the list of
maximal prime gaps to a length of 915, denoting gap
lengths by the difference of bounding PRIMES, c(n) — 1.

n

c(n)

n

c(n)

1

4

319

2,300,942,550

3

8

335

3,842,610,774

5

24

353

4,302,407,360

7

90

381

10,726,904,660

13

114

383

20,678,048,298

17

524

393

22,367,084,960

19

888

455

25,056,082,088

21

1,130

463

42,652,618,344

33

1,328

467

127,976,334,672

35

9,552

473

182,226,896,240

43

15,684

485

241,160,024,144

51

19,610

489

297,501,075,800

71

31,398

499

303,371,455,242

85

155,922

513

304,599,508,538

95

360,654

515

416,608,695,822

111

370,262

531

461,690,510,012

113

492,114

533

614,487,453,424

117

1,349,534

539

738,832,927,928

131

1,357,202

581

1,346,294,310,750

147

2,010,734

587

1,408,695,493,610

153

4,652,354

601

1,968,188,556,461

179

17,051,708

651

2,614,941,710,599

209

20,831,324

673

7,177,162,611,713

219

47,326,694

715

13,828,048,559,701

221

122,164,748

765

19,581,334,192,423

233

189,695,660

777

42,842,283,925,352

247

191,912,784

803

90,874,329,411,493

249

387,096,134

805

171,231,342,420,521

281

436,273,010

905

218,209,405,436,543

287

1,294,268,492

915

1,189,459,969,825,483

291

1,453,168,142

see also Jumping Champion, Prime Constellation,
Prime Difference Function, Shanks' Conjecture

References

Baugh, D. and O'Hara, F. "Large Prime Gaps." J. Recr.
Math. 24, 186-187, 1992.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 133-134, 1994.

Bombieri, E. and Davenport, H. "Small Differences Between
Prime Numbers." Proc. Roy. Soc. A 293, 1-18, 1966.

Brent, R. P. "The First Occurrence of Large Gaps Between
Successive Primes." Math. Comput. 27, 959-963, 1973.

Brent, R. P. "The Distribution of Small Gaps Between Suc-
cessive Primes." Math. Comput. 28, 315-324, 1974.

Brent, R. P. "The First Occurrence of Certain Large Prime
Gaps." Math. Comput. 35, 1435-1436, 1980.

Cramer, H. "On the Order of Magnitude of the Difference
Between Consecutive Prime Numbers." Acta Arith. 2,
23-46, 1937.

Guy, R. K. "Gaps between Primes. Twin Primes" and "In-
creasing and Decreasing Gaps." §A8 and All in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 19-23 and 26-27, 1994.

Lander, L. J. and Parkin, T. R. "On First Appearance of
Prime Differences." Math. Comput. 21, 483-488, 1967.

Nicely, T. R. "New Maximal Prime Gaps and First Occur-
rences." http : //www . lynchburg . edu/public/academic/

math/nicely/gaps/gaps. htm. To Appear in Math. Com-
put.

Shanks, D. "On Maximal Gaps Between Successive Primes."
Math. Comput. 18, 646-651, 1964.

Sloane, N. J. A. Sequences A008950, A008995, A008996, and
A030296 in "An On-Line Version of the Encyclopedia of
Integer Sequences."

Wolf, M. "First Occurrence of a Given Gap Between Consec-
utive Primes." http://www.ift.uni.wroc.pl/-mwolf.

Young, J. and Potler, A. "First Occurrence Prime Gaps."
Math. Comput. 52, 221-224, 1989.

Prime- Generating Polynomial

Legendre showed that there is no Rational algebraic
function which always gives PRIMES. In 1752, Goldbach
showed that no POLYNOMIAL with INTEGER COEFFI-
CIENTS can give a PRIME for all integral values. How-
ever, there exists a POLYNOMIAL in 10 variables with
Integer Coefficients such that the set of Primes
equals the set of POSITIVE values of this POLYNOMIAL
obtained as the variables run through all NONNEGATIVE
Integers, although it is really a set of Diophantine
Equations in disguise (Ribenboim 1991).

P(n)

Range

#

Reference

36n 2 -810n + 2753

[0, 44]

45

Fung and Ruby

47n 2 -1701n+ 10181

[0, 42]

43

Fung and Ruby

n 2 - n + 41

[0, 39]

40

Euler

2n 2 + 29

[0, 28]

29

Legendre

n 2 -fn+ 17

[0, 15]

16

Legendre

2n 2 + 11

[0, 10]

11

n 3 + n 2 + 17

[0, 10]

11

The above table gives some low-order polynomials which
generate only PRIMES for the first few NONNEGATIVE
values (Mollin and Williams 1990). The best-known of
these formulas is that due to Euler (Euler 1772, Ball
and Coxeter 1987). Le Lionnais (1983) has christened
numbers p such that the Euler-like polynomial

■ n + p

(1)

is PRIME for p = 0, 1, . . . , p - 2 as LUCKY NUMBERS
OF Euler (where the case p — 41 corresponds to Eu-
ler's formula). Rabinovitch (1913) showed that for a
Prime p > 0, Euler's polynomial represents a Prime
for n 6 [0,p — 2] (excluding the trivial case p — 3) IFF
the Field Q(^l - 4p) has Class Number h = 1 (Rabi-
nowitz 1913, Le Lionnais 1983, Conway and Guy 1996).
As established by Stark (1967), there are only nine num-
bers -d such that h(-d) = 1 (the Heegner Numbers
-2, -3, -7, -11, -19, -43, -67, and -163), and of
these, only 7, 11, 19, 43, 67, and 163 are of the re-
quired form. Therefore, the only LUCKY NUMBERS OF
Euler are 2, 3, 5, 11, 17, and 41 (Le Lionnais 1983,
Sloane's A014556), and there does not exist a better
prime-generating polynomial of Euler's form.

Euler also considered quadratics of the form

2x +p

(2)

1432 Prime-Generating Polynomial

Prime Number

and showed this gives Primes for x G [0,p- 1] for Prime
p > Iff Q(V-2p) has Class Number 2, which per-
mits only p = 3, 5, 11, and 29. Baker (1971) and Stark
(1971) showed that there are so such Fields for p > 29.
Similar results have been found for Polynomials of the
form

px + px + n

(3)

(Hendy 1974).

see also Class Number, Heegner Number, Lucky
Number of Euler, Prime Arithmetic Progres-
sion, Prime Diophantine Equations, Schinzel's
Hypothesis

References

Abel, U. and Siebert, H. "Sequences with Large Numbers of
Prime Values." Am. Math. Monthly 100, 167-169, 1993.

Baker, A. "Linear Forms in the Logarithms of Algebraic
Numbers." Mathematika 13, 204-216, 1966.

Baker, A. "Imaginary Quadratic Fields with Class Number
Two." Ann. Math. 94, 139-152, 1971.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 60, 1987.

Boston, N. and Greenwood, M. L. "Quadratics Representing
Primes." Amer. Math. Monthly 102, 595-599, 1995.

Conway, J. H. and Guy, R. K. "The Nine Magic Discrimi-
nants." In The Book of Numbers. New York: Springer-
Verlag, pp. 224-226, 1996.

Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 26, 1996.

Euler, L. Nouveaux Memoires de VAcademie royale des Sci-
ences. Berlin, p. 36, 1772.

Forman, R. "Sequences with Many Primes." Amer. Math.
Monthly 99, 548-557, 1992.

Garrison, B. "Polynomials with Large Numbers of Prime Val-
ues." Amer. Math. Monthly 97, 316-317, 1990.

Hendy, M. D. "Prime Quadratics Associated with Complex
Quadratic Fields of Class Number 2." Proc. Amer. Math.
Soc. 43, 253-260, 1974.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
pp. 88 and 144, 1983.

Mollin, R. A. and Williams, H. C. "Class Number Problems
for Real Quadratic Fields." Number Theory and Cryptol-
ogy; LMS Lecture Notes Series 154, 1990.

Rabinowitz, G. "Eindeutigkeit der Zerlegung in Primzahlfak-
toren in quadratischen Zahlkorpern." Proc. Fifth Internat.
Congress Math. (Cambridge) 1, 418-421, 1913.

Ribenboim, P. The Little Book of Big Primes. New York:
Springer- Verlag, 1991.

Sloane, N. J. A. Sequence A014556 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Stark, H. M. "A Complete Determination of the Complex
Quadratic Fields of Class Number One." Michigan Math.
J. 14, 1-27, 1967.

Stark, H. M. "An Explanation of Some Exotic Continued
Fractions Found by Brillhart." In Computers in Num-
ber Theory, Proc. Science Research Council Atlas Sympo-
sium No. 2 held at Oxford, from 18-23 August, 1969 (Ed.
A. O. L. Atkin and B. J. Birch). London: Academic Press,
1971.

Stark, H. M. "A Transcendence Theorem for Class Number
Problems." Ann. Math. 94, 153-173, 1971.

Prime Group

When the Order h of a finite Group is a Prime num-
ber, there is only one possible GROUP of ORDER h. Fur-
thermore, the Group is Cyclic.

see also p-GROUP

Prime Ideal

An IDEAL / such that if ah 6 /, then either a £ / or

bel.

see also Dedekind Ring, Ideal, Krull Dimension,
Maximal Ideal, Stickelberger Relation, Stone
Space

Prime Knot

A KNOT other than the UNKNOT which cannot be ex-
pressed as a sum of two other KNOTS, neither of which
is unknotted. A KNOT which is not prime is called a
Composite KNOT. It is often possible to combine two
prime knots to create two different Composite Knots,
depending on the orientation of the two.

There is no known FORMULA for giving the number of
distinct prime knots as a functions of number of cross-
ings. For the first few n crossings, the numbers of prime
knots are 0, 0, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988,
... (Sloane's A002863).

see also COMPOSITE KNOT, KNOT

References

Sloane, N. J. A. Sequences A002863/M0851 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Sloane, N. J. A. and Plouffe, S. Extended entry in The Ency-
clopedia of Integer Sequences. San Diego: Academic Press,
1995.

Prime k- Tuple

see Prime Constellation

Prime k- Tuples Conjecture

see also fc-T/UPLE CONJECTURE

Prime fc-Tuplet

see Prime Constellation

Prime Manifold

An n-MANlFOLD which cannot be "nontrivially" decom-
posed into other n- Manifolds.

Prime Number

A prime number is a POSITIVE INTEGER p which has
no DIVISORS other than 1 and p itself. Although the
number 1 used to be considered a prime, it requires spe-
cial treatment in so many definitions and applications
involving primes greater than or equal to 2 that it is
usually placed into a class of its own. Since 2 is the
only EVEN prime, it is also somewhat special, so the set
of all primes excluding 2 is called the "Odd Primes."
The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,

Prime Number

Prime Number

1433

31, 37, . . . (Sloane's A000040, Hardy and Wright 1979,
p. 3). Positive Integers other than 1 which are not
prime are called COMPOSITE.

The function which gives the number of primes less than
a number n is denoted 7r(n) and is called the PRIME
COUNTING Function. The theorem giving an asymp-
totic form for n(n) is called the Prime Number The-
orem.

Prime numbers can be generated by sieving processes
(such as the Eratosthenes Sieve), and Lucky Num-
bers, which are also generated by sieving, appear to
share some interesting asymptotic properties with the
primes.

Many Prime Factorization Algorithms have been
devised for determining the prime factors of a given IN-
TEGER. They vary quite a bit in sophistication and com-
plexity. It is very difficult to build a general-purpose
algorithm for this computationally "hard" problem, so
any additional information which is known about the
number in question or its factors can often be used to
save a large amount of time. The simplest method of
finding factors is so-called "Direct Search FACTOR-
IZATION" (a.k.a. Trial Division). In this method, all
possible factors are systematically tested using trial di-
vision to see if they actually Divide the given number.
It is practical only for very small numbers.

Because of their importance in encryption algorithms
such as RSA ENCRYPTION, prime numbers can be
important commercial commodities. In fact, Roger
Schlafly has obtained U.S. Patent 5,373,560 (12/13/94)
on the following two primes (expressed in hexadecimal
notation):

98A3DF52AEAE9799325CB258D767EBD1F4630E9B
9E2 1732A4AFB 1624BA6DF9 1 1466 AD8D A960586F4
A0D5E3C36AF099660BDDC1577E54A9F402334433
ACB14BCB

and

93E8965DAFD9DFECFD00B466B68F90EA68AF5DC9
FED915278D1B3A137471E65596C37FED0C7829FF
8F8331F81A2700438ECDCC09447DC397C685F397
294F722BCC484AEDF28BED25AAAB35D35A65DB1F
D62C9D7BA55844FEB1F9401E671340933EE43C54
E4DC459400D7AD61248B83A2624835B31FFF2D95
95A5B90B276E44F9.

The Fundamental Theorem of Arithmetic states
that any POSITIVE INTEGER can be represented in ex-
actly one way as a PRODUCT of primes. EUCLID'S SEC-
OND THEOREM demonstrated that there are an infinite

number of primes. However, it is not known if there are
an infinite number of primes of the form x 2 + 1, whether
there are an INFINITE number of Twin Primes, or if a
prime can always be found between n and (n + 1) .

Prime numbers satisfy many strange and wonderful
properties. For example, there exists a CONSTANT
w 1.3064 known as MILLS' CONSTANT such that

(1)

where [x\ is the FLOOR FUNCTION, is prime for all n >
1. However, it is not known if is IRRATIONAL. There
also exists a CONSTANT w as 1.9287800 such that

(2)

(Ribenboim 1996, p. 186) is prime for every n > 1.

Explicit Formulas exist for the nth prime both as a
function of n and in terms of the primes 2, . . . , p n -i
(Hardy and Wright 1979, pp. 5-6; Guy 1994, pp. 36-
41). Let

m =

,C?-i)i + i

(3)

for integral j > 1, and define F(l) — 1, where [x\ is
again the FLOOR FUNCTION. Then

Pn = 1 + £

m=l
2 n

= ! + £

E7^w

l/n

1 + 7r(m)

l/n

(4)
(5)

where ?r(m) is the PRIME COUNTING FUNCTION. It is
also true that

Pn-fl = 1 +Pn + F{Pn + 1)

P

+F( Pn + l)F(p n + 2) + Y[F(p n +j) (6)
i=i

(Ribenboim 1996, pp. 180-182). Note that the number
of terms in the summation to obtain the nth prime is
2 n , so these formulas turn out not to be practical in
the study of primes. An interesting INFINITE PRODUCT
formula due to Euler which relates tt and the nth PRIME
Pn is

(7)

(8)

n;

sin^TTpn)

T-rcx) L ■ (-l)(P~-i)/2 l

1434 Prime Number

Prime Number

(Blatner 1997). Conway (Guy 1983, Conway and Guy
1996, p. 147) gives an algorithm for generating primes
based on 14 fractions, but it is actually just a concealed
version of a SIEVE.

Some curious identities satisfied by primes p are

p-i

£

V

(p-2)(p-l)(p+l)

(9)

(p-l)(p-2)

Yj [( k P~) 1/3 \ =U 3 P- 5)(P - 2)(P - 1) (10)

fc = l

(Doster 1993),

n

p 2 + l = 5
p 2 -l 2

(11)

(Le Lionnais 1983, p. 46),

oo

£■>** = £ E r

fc=a * -

and

CO

^(-l)* _1 e - *" In Jb

p prime fc = l

-x*

(12)

fc = l

CO CO

ln2 E^3T+ E ^E^TT (13)

fe=l P an fc=l

p an
odd prime

(Berndt 1994, p. 114).

It has been proven that the set of prime numbers is
a DlOPHANTlNE Set (Ribenboim 1991, pp. 106-107).
Ramanujan also showed that

^M^^-f^x 1 /", (14)

dx zlnx ^-^ n

n-l

where ir(x) is the PRIME COUNTING FUNCTION and
fi(n) is the MOBIUS FUNCTION (Berndt 1994, p. 117).
B. M. Bredihin proved that

f(x,y) = x 2 +y 2 + l

(15)

takes prime values for infinitely many integral pairs
(x 7 y) (Honsberger 1976, p. 30). In addition, the func-
tion

f(x,y) = \{y - 1) [\B 2 (x,y) - 1| - (B 2 (x,j/) - 1)J +2,

(16)
where

B(x,y) = x(y + l) - (y\ + l) t (17)

y\ is the FACTORIAL, and \x\ is the FLOOR FUNCTION,
generates only prime numbers for POSITIVE integral ar-
guments. It not only generates every prime number, but
generates Odd primes exactly once each, with all other
values being 2 (Honsberger 1976, p. 33). For example,

/(l, 2) = 3

(18)

/(5,4) = 5

(19)

/(103,6) = 7,

(20)

with no new primes generated for x,y < 1000.
For n an INTEGER > 2, n is prime Iff

n-l

(-1)* (modn)

for k = 0, 1, . . . , n - 1 (Deutsch 1996).

(21)

Cheng (1979) showed that for x sufficiently large, there
always exist at least two prime factors between (x — x a )
and x for a > 0.477... (Le Lionnais 1983, p. 26). Let
f(n) be the number of decompositions of n into two or
more consecutive primes. Then

X

lim - V/(n) = ln2

x-i-oo X « ^

(22)

(Moser 1963, Le Lionnais 1983, p. 30). Euler showed
that the sum of the inverses of primes is infinite

Z_-/ T>

- = oo
V

(23)

(Hardy and Wright 1979, p. 17), although it diverges
very slowly. The sum exceeds 1, 2, 3, ... after 3, 59,
361139, . . . (Sloane's A046024) primes, and its asymp-
totic equation is

E- =lnlna: + Bi+o(l), (24)

v

p prime

where £?i is Mertens Constant (Hardy and Wright
1979, p. 351). Dirichlet showed the even stronger result

that

^2 z = °° ( 25 )

prime p=b (mod a)
(a,b) = l

V

(Davenport 1980, p. 34),

Despite the fact that ^ 1/p diverges, Brun showed that

V - = B < oo,

(26)

p+2 prime

Prime Number

Prime Number 1435

where B is BRUN'S CONSTANT. The function defined by

taken over the primes converges for n > 1 and is a gen-
eralization of the RlEMANN ZETA FUNCTION known as
the Prime Zeta Function.

The probability that the largest prime factor of a RAN-
DOM NUMBER x is less than y/x is In 2 (Beeler et al.
1972, Item 29). The probability that two INTEGERS
picked at random are RELATIVELY Prime is [C(2)] _1 =
6/7r 2 , where C(z) is the Riemann Zeta Function (Ce-
saro and Sylvester 1883). Given three INTEGERS chosen
at random, the probability that no common factor will
divide them all is

K(3)r

1.202" 1 = 0.832...

(28)

where £(3) is Apery's Constant. In general, the prob-
ability that n random numbers lack a pth Power com-
mon divisor is [({np)]~ l (Beeler et al. 1972, Item 53).

Large primes include the large MERSENNE PRIMES,
Ferrier's Prime, and 391581(2 216193 -1) (Cipral989).
The largest known prime as of 1998, is the Mersenne
Prime 2 3021377 - 1.

Primes consisting of consecutive Digits (counting as
coming after 9) include 2, 3, 5, 7, 23, 67, 89, 4567, 78901,
... (Sloane's A006510).

see also Adleman-Pomerance-Rumely Primal-
ity Test, Almost Prime, Andrica's Conjec-
ture, Bertrand's Postulate, Brocard's Conjec-
ture, Brun's Constant, Carmichael's Conjec-
ture, Carmichael Function, Carmichael Num-
ber, Chebyshev Function, Chebyshev-Sylvester
Constant, Chen's Theorem, Chinese Hypothesis,
Composite Number, Composite Runs, Copeland-
Erdos Constant, Cramer Conjecture, Cunning-
ham Chain, Cyclotomic Polynomial, de Polig-
nac's Conjecture, Dirichlet's Theorem, Divi-
sor, Erdos-Kac Theorem, Euclid's Theorems,
Feit-Thompson Conjecture, Fermat Number,
Fermat Quotient, Ferrier's Prime, Fortunate
Prime, Fundamental Theorem of Arithmetic,
Gigantic Prime, Giuga's Conjecture, Goldbach
Conjecture, Good Prime, Grimm's Conjecture,
Hardy-Ramanujan Theorem, Irregular Prime,
Rummer's Conjecture, Lehmer's Problem, Lin-
nik's Theorem, Long Prime, Mersenne Number,
Mertens Function, Miller's Primality Test, Mi-
rimanoff's Congruence, Mobius Function, Palin-
dromic Number, Pepin's Test, Pillai's Conjec-
ture, Poulet Number, Primary, Prime Array,
Prime Circle, Prime Factorization Algorithms,
Prime Number of Measurement, Prime Number
Theorem, Prime Power Symbol, Prime String,

Prime Triangle, Prime Zeta Function, Primi-
tive Prime Factor, Primorial, Probable Prime,
Pseudoprime, Regular Prime, Riemann Function,
Rotkiewicz Theorem, Schnirelmann's Theorem,
Selfridge's Conjecture, Semiprime, Shah- Wilson
Constant, Sierpinski's Composite Number The-
orem, Sierpinski's Prime Sequence Theorem,
Smooth Number, Soldner's Constant, Sophie
Germain Prime, Titanic Prime, Totient Func-
tion, Totient Valence Function, Twin Primes,
Twin Primes Constant, Vinogradov's Theorem,
von Mangoldt Function, Waring's Conjecture,
Wieferich Prime, Wilson Prime, Wilson Quo-
tient, Wilson's Theorem, Witness, Wolsten-
holme's Theorem, Zsigmondy Theorem

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.

Berndt, B. C. "Ramanujan's Theory of Prime Numbers."
Ch. 24 in Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, 1994.

Blatner, D. The Joy of Pi. New York: Walker, p. 110, 1997.

Caldwell, C "Largest Primes." http://www.utm.edu/
research/primes/largest. html.

Cheng, J. R. "On the Distribution of Almost Primes in an
Interval II." Sci. Sinica 22, 253-275, 1979.

Cipra, B. A. "Math Team Vaults Over Prime Record." Sci-
ence 245, 815, 1989.

Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 130, 1996.

Courant, R. and Robbins, H. "The Prime Numbers." §1 in
Supplement to Ch. 1 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 21-31, 1996.

Davenport, H. Multiplicative Number Theory, 2nd ed. New
York: Springer- Verlag, 1980.

Deutsch, E. "Problem 1494." Math. Mag. 69, 143, 1996.

Dickson, L. E. "Factor Tables, Lists of Primes." Ch. 13 in
History of the Theory of Numbers, Vol. 1 : Divisibility and
Primality. New York: Chelsea, pp. 347-356, 1952.

Doster, D. Problem 10346. Amer. Math. Monthly 100, 951,
1993.

Giblin, P. J. Primes and Programming: Computers and
Number Theory. New York: Cambridge University Press,
1994.

Guy, R. K. "Conway's Prime Producing Machine." Math.
Mag. 56, 26-33, 1983.

Guy, R. K. "Prime Numbers," "Formulas for Primes," and
"Products Taken Over Primes." Ch. A, §A17, and §B48 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 3-43, 36-41 and 102-103, 1994.

Hardy, G. H. Ch. 2 in Ramanujan: Twelve Lectures on Sub-
jects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, 1978.

Hardy, G. H. and Wright, E. M. "Prime Numbers" and "The
Sequence of Primes." §1.2 and 1.4 in An Introduction to
the Theory of Numbers, 5th ed. Oxford, England: Claren-
don Press, pp. 1-4, 1979.

Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., p. 30, 1976.

Kraitchik, M. "Prime Numbers." §3.9 in Mathematical
Recreations. New York: W. W. Norton, pp. 78-79, 1942.

Le Lionnais, F. Les nombres remarquables, Paris: Hermann,
pp. 26, 30, and 46, 1983.

Moser, L. "Notes on Number Theory III. On the Sum of
Consecutive Primes." Can. Math. Bull 6, 159-161, 1963.

1436

Prime Number of Measurement

Pappas, T. "Prime Numbers*" The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, pp. 100-101, 1989,

Ribenboim, P. The Little Book of Big Primes. New York:
Springer- Verlag, 1991.

Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, 1996.

Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, 1994.

Schinzel, A. and Sierpiriski, W. "Sur certains hypotheses con-
cernant les nombres premiers." Acta Arith. 4, 185-208,
1958.

Schinzel, A. and Sierpiriski, W. Erratum to "Sur certains
hypotheses concernant les nombres premiers." Acta Arith.
5, 259, 1959.

Sloane, N. J. A. Sequences A046024, A000040/M0652, and
A006510/M0679 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."

Wagon, S. "Primes Numbers." Ch. 1 in Mathematica in Ac-
tion. New York: W. H. Freeman, pp. 11-37, 1991.

Zaiger, D. "The First 50 Million Prime Numbers." Math.
Intel 0, 221-224, 1977.

Prime Number of Measurement

The set of numbers generated by excluding the SUMS of
two or more consecutive earlier members is called the
prime numbers of measurement, or sometimes the SEG-
MENTED Numbers. The first few terms are 1, 2, 4, 5,
8, 10, 14, 15, 16, 21, ... (Sloane's A002048). Excluding
two and three terms gives the sequence 1, 2, 4, 5, 8, 10,
14, 15, 16, 19, 20, 21, . . . (Sloane's A005242).

References

Guy, R. K. "MacMahon's Prime Numbers of Measurement."
§E30 in Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 230-231, 1994.

Sloane, N. J. A. Sequence A002048/M0972 in "An
On-Line Version of the Encyclopedia of Integer Se-
quences." 0052420971

Prime Number Theorem

1000

The theorem giving an asymptotic form for the PRIME
Counting Function 7r(n) for number of Primes less
than some Integer n. Legendre (1808) suggested that,
for large n,

" {n) ~ Alnn + B > (1)

with A = 1 and B = —1.08366 (where B is sometimes
called Legendre's Constant), a formula which is cor-
rect in the leading term only (Wagon 1991, pp. 28-29).
In 1791, Gauss became the first to suggest instead

7r(n)

n
Inn*

Prime Number Theorem

Gauss later refined his estimate to

7r(n) ~ Li(n),

(3)

where Li(n) is the LOGARITHMIC INTEGRAL. This func-
tion has n/ In n as the leading term and has been shown
to be a better estimate than n/lnn alone. The state-
ment (3) is often known as "the" prime number theorem
and was proved independently by Hadamard and Vallee
Poussin in 1896. A plot of 7r(n) (lower curve) and Li(n)
is shown above for n < 1000.

For small n, it has been checked and always found that
7v(n) < Li(n). However, Skewes proved that the first

10 34

crossing of 7r(n) — Li(n) = occurs before 10 10 (the
SKEWES NUMBER). The upper bound for the crossing
has subsequently been reduced to 10 371 . Littlewood
(1914) proved that the Inequality reverses infinitely
often for sufficiently large n (Ball and Coxeter 1987).
Lehman (1966) proved that at least 10 500 reversals oc-
cur for numbers with 1166 or 1167 DECIMAL DIGITS.

Chebyshev (Rubinstein and Sarnak 1994) put limits on

the Ratio

7 . 7r(n) ^ 9

8

<

<

8'

and showed that if the Limit

lim

7r(n)

(4)

(5)

existed, then it would be 1. This is, in fact, the prime
number theorem.

Hadamard and Vallee Poussin proved the prime number
theorem by showing that the RiEMANN Zeta Function
C(z) has no zeros of the form 1 + it (Smith 1994, p. 128).
In particular, Vallee Poussin showed that

,( X ) = U( X ) + 0(^- x e-^)

(6)

for some constant a. A simplified proof was found by
Selberg and Erdos (1949) (Ball and Coxeter 1987, p. 63).

Riemann estimated the PRIME COUNTING FUNCTION

with

7r(n)~Li(n)-lLi(n x/2 ), (7)

which is a better approximation than Li(n) for n < 10 7 .
Riemann (1859) also suggested the RiEMANN FUNCTION

p(n)

(8)

where p is the MOBIUS FUNCTION (Wagon 1991, p. 29).
An even better approximation for small n (by a factor
of 10 for n < 10 9 ) is the Gram Series.

(2)

Prime Number Theorem

Prime Quadratic Effect 1437

The prime number theorem is equivalent to

lim^M = i,

i-4oo X

(9)

where ij)(x) is the Summatory Mangoldt Function.

The RiEMANN Hypothesis is equivalent to the asser-
tion that

|Li(x) -7r(a;)| < cy/xlnx (10)

for some value of c (Ingham 1932, Ball and Coxeter
1987). Some limits obtained without assuming the RiE-
MANN Hypothesis are

7r(x) = U(x) + 0[xe' (lnx)1/2/15 } (11)

*(x) - Li(x) + O[a . c -0-009(lnx)3/V(lnlnx)VB ]> (l2)

Lehman, R. S. "On the Difference n(x) - li(a;)." Acia AritA.
11, 397-410, 1966.

Littlewood, J. E. "Sur les distribution des nombres premiers."
C. R. Acad. Sci. Paris 158, 1869-1872, 1914.

Nagell, T. "The Prime Number Theorem." Ch. 8 in Intro-
duction to Number Theory. New York: Wiley, 1951.

Riemann, G. F, B. "Uber die Anzahl der Primzahlen unter
einer gegebenen Grosse." Monatsber. Konigl. Preuss.
Akad. Wiss. Berlin, 671, 1859.

Rubinstein, M. and Sarnak, P. "Chebyshev's Bias." Experi-
mental Math. 3, 173-197, 1994.

Selberg, A. and Erdos, P. "An Elementary Proof of the Prime
Number Theorem." Ann. Math. 50, 305-313, 1949.

Shanks, D. "The Prime Number Theorem." §1.6 in Solved
and Unsolved Problems in Number Theory, J^th ed. New
York: Chelsea, pp. 15-17, 1993.

Smith, D. E. A Source Book in Mathematics. New York:
Dover, 1994.

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 25-35, 1991.

Ramanujan showed that for sufficiently large x,

2/ . ex (x\

(13)

The largest known PRIME for which the inequality fails is
38,358,837,677 (Berndt 1994, pp. 112-113). The related
inequality

Li2 ^<£ Li (?) (14)

is true for x > 2418 (Berndt 1994, p. 114).

see also Bertrand's Postulate, Dirichlet's The-
orem, Gram Series, Prime Counting Function,
Riemann's Formula, Riemann Function, Rie-
mann-Mangoldt Function, Riemann Weighted
Prime-Power Counting Function, Skewes Num-
ber

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 62-64,
1987.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, 1994.

Courant, R. and Robbins, H. "The Prime Number Theorem."
§1.2c in Supplement to Ch. 1 in What is Mathematics?: An
Elementary Approach to Ideas and Methods, 2nd ed. Ox-
ford, England: Oxford University Press, pp. 27-30, 1996.

Davenport, H. "Prime Number Theorem." Ch. 18 in Mul-
tiplicative Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 111-114, 1980.

de la Vallee Poussin, C.-J. "Recherches analytiques la theorie
des nombres premiers." Ann. Soc. scient. Bruxelles 20,
183-256, 1896.

Hadamard, J. "Sur la distribution des zeros de la fonction
f(s) et ses consequences arithmetiques (')." Bull. Soc.
math. France 24, 199-220, 1896.

Hardy, G. H. and Wright, E. M. "Statement of the Prime
Number Theorem." §1.8 in An Introduction to the Theory
of Numbers, 5th ed. Oxford, England: Clarendon Press,
pp. 9-10, 1979.

Ingham, A. E. The Distribution of Prime Numbers. London:
Cambridge University Press, p. 83, 1932.

Legendre, A. M. Essai sur la Theorie des Nombres. Paris:
Duprat, 1808.

Prime Pairs

see Twin Primes

Prime Patterns Conjecture

see fc-TUPLE CONJECTURE

Prime Polynomial

see Prime-Generating Polynomial

Prime Power Conjecture

An Abelian planar DIFFERENCE Set of order n exists
only for n a Prime POWER. Gordon (1994) has verified
it to be true for n < 2, 000, 000.

see also Difference Set

References

Gordon, D. M. "The Prime Power Conjecture is True
for n < 2,000,000." Electronic J. Combinatorics 1,
R6, 1-7, 1994. http://www.combinatorics.org/Volume_l/
volumel.html#R6.

Prime Power Symbol

The symbol p e \\n means, for p a Prime, that p e \n } but
V

» e+1 \n.

Prime Quadratic Effect

Let TT m ,n(x) denote the number of PRIMES < x which
are congruent to n modulo m. Then one might expect
that

A(x) = 7T4,3(ic) — 7T4,l( a; ) ~ l 71 "^ ) >

(Berndt 1994). Although this is true for small numbers,
Hardy and Littlewood showed that A(x) changes sign
infinitely often. (The first number for which it is false is
26861.) The effect was first noted by Chebyshev in 1853,
and is sometimes called the CHEBYSHEV PHENOMENON.
It was subsequently studied by Shanks (1959), Hudson
(1980), and Bays and Hudson (1977, 1978, 1979). The

1438 Prime Quadruplet

Prime Ring

effect was also noted by Ramanujan, who incorrectly
claimed that lim x _j.oo A(x) = oo (Berndt 1994).

References

Bays, C. and Hudson, R. H. "The Mean Behavior of Primes
in Arithmetic Progressions." J. Reine Angew. Math. 296,
80-99, 1977.

Bays, C. and Hudson, R. H. "On the Fluctuations of Little-
wood for Primes of the Form 4n ± 1." Math. Comput. 32,
281-286, 1978.

Bays, C. and Hudson, R. H. "Numerical and Graphical De-
scription of All Axis Crossing Regions for the Moduli 4 and
8 which Occur Before 10 12 ." Internat. J. Math. Math. Sci.
2, 111-119, 1979.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 135-136, 1994.

Hudson, R. H. "A Common Principle Underlies Riemann's
Formula, the Chebyshev Phenomenon, and Other Subtle
Effects in Comparative Prime Number Theory. I." J. Reine
Angew. Math. 313, 133-150, 1980.

Shanks, D. "Quadratic Residues and the Distribution of
Primes." Math. Comput. 13, 272-284, 1959.

Prime Quadruplet

A Prime Constellation of four successive Primes
with minimal distance (p,p + 2,p + 6,p + 8). The quad-
ruplet (2, 3, 5, 7) has smaller minimal distance, but it
is an exceptional special case. With the exception of
(5, 7, 11, 13), a prime quadruple must be of the form
(30n + 11, 30n + 13, 30n + 17, 30n + 19). The first few
values of n which give prime quadruples are n = 0, 3, 6,
27, 49, 62, 69, 108, 115, . . . (Sloane's A014561), and the
first few values of p are 5 (the exceptional case), 11, 101,
191, 821, 1481, 1871, 2081, 3251, 3461, .... The asymp-
totic FORMULA for the frequency of prime quadruples is
analogous to that for other PRIME CONSTELLATIONS,

P*(p,P + 2,p + 6,p + 8)~^ J|

p>5

27 T7 p 3 (p-4)

2 11 (p-l)« J2

f x dx

4.151180864

i

dx
(lnx) 4 '

where c = 4.15118... is the Hardy-Littlewood con-
stant for prime quadruplets. Roonguthai found the large
prime quadruplets with

p = 10 99 + 349781731
p= 10 199 + 21156403891
10 299 + 140159459341

V
P
P

10 399 + 34993836001
10 499 + 883750143961

p = 10 599 + 1394283756151
p = 10 699 + 547634621251

References

Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. New York: Oxford University
Press, 1979.

Forbes, T. "Prime fc-tuplets." http://www.ltkz.demon.co.
uk/ktuplets .htm.

Rademacher, H. Lectures on Elementary Number Theory,
New York: Blaisdell, 1964.

Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, pp. 61-62,
1994.

Roonguthai, W, "Large Prime Quadruplets." http://www.
mathsoft.com/asolve/constant/hrdyltl/roonguth.html.

Sloane, N. J. A. Sequence A014561 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Prime Representation

Let a ^ 6, A, and B denote Positive Integers satis-
fying

(a,6) = l (A,B) = 1,

(i.e., both pairs are Relatively Prime), and suppose
every PRIME p = B (mod A) with (p, 2ab) = 1 is expres-
sible if the form ax 2 — by 2 for some Integers x and y.
Then every PRIME q such that q = —B (mod A) and
(g, 2ab) = 1 is expressible in the form bX 2 — aY 2 for
some Integers X and Y (Halter-Koch 1993, Williams
1991).

Prime Form

Representation

4n + l

x 2 +y 2

871 + 1,871 + 3

x 2 + 2y 2

Sn±l

x 2 - 2y 2

6n + l

x 2 + 3y 2

12n + l

x 2 - 3y 2

20n + l,20n + 9

x 2 + by 2

10n+ l,10n + 9

x 2 - by 2

14n + l,14ra + 9,14n + 25

x 2 + 7y 2

28n+l,28n + 9,28n + 25

2 n 2

x - ly

30n + l,30n + 49

x 2 + 15y 2

60n+ l,60n + 49

x 2 - Iby 2

30n-7,30n + 17

5x 2 + 3y 2

60n-7,60n + 17

5x 2 - 3y 2

2471+1,2471 + 7

x 2 + 6y 2

24n + 1,24ti + 19

x 2 - 6y 2

24n + 5,24n + 11

2z 2 + 3y 2

247i + 5,24n- 1

2x 2 - 3y 2

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 70-73, 1994.

Halter-Koch, F. "A Theorem of Ramanujan Concerning Bi-
nary Quadratic Forms." J. Number. Theory 44, 209-213,
1993.

Williams, K. S. "On an Assertion of Ramanujan Concerning
Binary Quadratic Forms." J. Number Th. 38, 118-133,
1991.

(Roonguthai).

see also PRIME ARITHMETIC PROGRESSION, PRIME

Constellation, Prime A>Tuples Conjecture,
Sexy Primes, Twin Primes

Prime Ring

A RING for which the product of any pair of IDEALS is
zero only if one of the two IDEALS is zero. All SIMPLE
Rings are prime.
see also Ideal, Ring, Semiprime Ring, Simple Ring

Prime Sequence

Prime Sequence

see Prime Arithmetic Progression, Prime Ar-
ray, Prime-Generating Polynomial, Sierpinski's
Prime Sequence Theorem

Prime Spiral

^;^>v

:^-

A-v^.V-X- 1 /''^

The numbers arranged in a SPIRAL

5 4 3

6 12

7 8 9

with Primes indicated in black, as first drawn by
S. Ulam. Unexpected patterns of diagonal lines are ap-
parent in such a plot, as illustrated in the above 199 x 199
grid.

References

Dewdney, A. K. "Computer Recreations: How to Pan for

Primes in Numerical Gravel." Sci. Amer. 259, 120-123,

July 1988.
Lane, C. "Prime Spiral." http://www.best.com/-cdl/Prime

SpiralApplet . html.
$ Weisstein, E. W. "Prime Spiral." http: //www. astro.

Virginia . edu/ -eww6n/math/notebooks/PrimeSpiral . m.

Prime String

Call a number n a prime string from the left if n and all
numbers obtained by successively removing the right-
most DIGIT are PRIME. There are 83 left prime strings
in base 10. The first few are 2, 3, 5, 7, 23, 29, 31, 37,
53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373,
379, 593, 599, . . . (Sloane's A024770), the largest being
73,939,133. Similarly, call a number n a prime string
from the right if n and all numbers obtained by suc-
cessively removing the left-most DIGIT are PRIME. The
first few are 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67,
73, 83, 97, 103, 107, 113, 137, 167, 173, ... (Sloane's
A033664). A large right prime string is 933,739,397.

see also Prime Array, Prime Number

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.

Cambridge, MA: MIT Artificial Intelligence Laboratory,

Memo AIM-239, Feb. 1972.
Rivera, C. "Problems & Puzzles (Puzzles): Prime Strings."

http://www.sci.net.mx/-crivera/ppp/puzz_002.htm.
Sloane, N. J. A. Sequence A024770 in "An On-Line Version

of the Encyclopedia of Integer Sequences. "033664

Prime Sum

Let

Prime Triangle 1439

£(n) = ^ Pi

be the sum of the first n PRIMES. The first few terms
are 2, 5, 10, 17, 28, 41, 58, 77, . . . (Sloane's A007504).
Bach and Shallit (1996) show that

E(n)

n

2 log n '

and provide a general technique for estimating such

sums.

see also Primorial

References

Bach, E. and Shallit, J. §2.7 in Algorithmic Number Theory,
Vol 1: Efficient Algorithms. Cambridge, MA: MIT Press,
1996.

Sloane, N. J. A. Sequence A007504/M1370 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Prime Theta Function

The prime theta function is defined as

9(n) = y^lnpi,

where pi is the ith PRIME. As shown by Bach and Shallit
(1996),

0(n) ~ n.

References

Bach, E. and Shallit, J. Algorithmic Number Theory, Vol 1:

Efficient Algorithms. Cambridge, MA: MIT Press, pp. 206

and 233, 1996.

Prime Triangle

1 2

1 2 3

12 3 4

14 3 2 5

14 3 2 5 6

This triangle has rows beginning with 1 and ending with

n, with the SUM of each two consecutive entries being a

Prime.

see also Pascal's Triangle

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Verlag, p. 106, 1994.
Kenney, M. J. "Student Math Notes." NCTM News Bulletin.

Nov. 1986.

1440 Prime Unit

Primitive Recursive Function

Prime Unit

1 and —1 are the only INTEGERS which divide every
INTEGER. They are therefore called the prime units.

see also Integer, Prime Number, Unit

Prime Zeta Function

The prime zeta function

p M=Zi'

(1)

where the sum is taken over Primes is a generalization
of the Riemann Zeta Function

«»> s £f'

(2)

where the sum is over all integers. The prime zeta func-
tion can be expressed in terms of the Riemann Zeta
Function by

lnC(n) = -5>(l-P _n ) = EEV

p>2 p>2 k=l

P{hn)
k

k=l p>2

(3)

=Eii;»-'-=i;

k=i
Inverting then gives

p w = Enr ln C(fcn)< (4)

fc=i

where fi(k) is the MOBIUS FUNCTION. The values for
the first few integers starting with two are

P(2) w 0.452247
i>(3)« 0.174763
P(4) « 0.0769931
P(5) « 0.035755.

(5)
(6)
(7)
(8)

see also Mobius Function, Riemann Zeta Func-
tion, Zeta Function

References

Hardy, G. H. and Weight, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, pp. 355-356, 1979.

Primequad

see Prime Quadruplet

Primitive Abundant Number
An Abundant Number for which all Proper Di-
visors are DEFICIENT is called a primitive abundant
number (Guy 1994, p. 46). The first few Odd primi-
tive abundant numbers are 945, 1575, 2205, 3465, , , .
(Sloane's A006038).

see also Abundant Number, Deficient Number,
Highly Abundant Number, Superabundant Num-
ber, Weird Number

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Verlag, p. 46, 1994.
Sloane, N. J. A. Sequence A006038/M5486 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Primitive Function

see Integral

Primitive Irreducible Polynomial

An Irreducible Polynomial which generates all ele-
ments of an extension field from a base field. For any
Prime or Prime Power q and any Positive Integer
n, there exists a primitive irreducible POLYNOMIAL of
degree n over GF(g).

see also Galois Field, Irreducible Polynomial

Primitive Polynomial Modulo 2

A special type of POLYNOMIAL of which a subclass has
Coefficients of only or l. Such Polynomials define
a Recurrence Relation which can be used to obtain
a new RANDOM bit from the n preceding ones.

Primitive Prime Factor

If n > 1 is the smallest Integer such that p\a n - b n (or
a n + 6 n ), then p is a primitive prime factor.

Primitive Pseudoperfect Number

see Primitive Semiperfect Number

Primitive Recursive Function

For-loops (which have a fixed iteration limit) are a spe-
cial case of while-loops. A function which can be imple-
mented using only for-loops is called primitive recursive.
(In contrast, a COMPUTABLE FUNCTION can be coded
using a combination of for- and while-loops, or while-
loops only.)

The ACKERMANN FUNCTION is the simplest example of
a well-defined Total Function which is Computable
but not primitive recursive, providing a counterexample
to the belief in the early 1900s that every COMPUTABLE
Function was also primitive recursive (Dotzel 1991).
see also Ackermann Function, Computable Func-
tion, Total Function

References

Dotzel, G. "A Function to End All Functions." Algorithm:
Recreational Programming 2, 16—17, 1991.

Primitive Root

Primitive Semiperfect Number 1441

Primitive Root

A number g is a primitive root of m if

g ^ 1 (mod m)

for 1 < k < m and

1 (mod m) .

(i)

(2)

Only m = 2, 4, p a , and 2p a have primitive roots (where
p > 2 and a is an INTEGER). For composite m, there
may be more than one primitive root (both 3 and 7
are primitive roots mod 10), but for prime p, there is
only one primitive root. It is the INTEGER g satisfying
1 < 9 < V — 1 sucn that 5 (mod p) has Order p — 1.

The primitive root of m can also be defined as a cyclic
generator of the multiplicative group (mod m) when m
is a prime Power or twice a PRIME POWER. Let p be
any Odd Prime k > 1, and let

p-X

-EA

j=i

Then

" { (:

— 1 (mod p) for p — X\k
(mod p) for p — lf/c.

(3)

(4)

For numbers m with primitive roots, all y satisfying
(p> y) = 1 are representable as

2/ = p* (mod m) ,

(5)

where £ = 0, 1, . . . , <p(m) — 1, £ is known as the index, and
y is an INTEGER. Kearnes showed that for any POSITIVE
Integer ra, there exist infinitely many Primes p such
that

m < g p < p — m. (6)

Call the least primitive root g p . Burgess (1962) proved
that

g p < Cp 1 '^ (7)

for C and e POSITIVE constants and p sufficiently large.

The table below gives the primitive roots (for prime
m = p; Sloane's A001918) and least primitive roots (for
composite m) for the first few INTEGERS

m g

771

9

777

9

2 1

53

2

134

7

3 2

54

5

137

3

4 3

58

3

139

2

5 2

59

2

142

7

6 5

61

2

146

5

7 3

62

3

149

2

9 2

67

2

151

6

10 3

71

7

157

5

11 2

73

5

158

3

13 2

74

5

162

5

14 3

79

3

163

2

17 3

81

2

166

5

18 5

82

7

167

5

19 2

83

2

169

2

22 7

86

3

173

2

23 5

89

3

178

3

25 2

94

5

179

2

26 7

97

5

181

2

27 2

98

3

191

19

29 2

101

2

193

5

31 3

103

5

194

5

34 3

106

3

197

2

37 2

107

2

199

3

38 3

109

6

202

3

41 6

113

3

206

5

43 3

118

11

211

2

46 5

121

2

214

5

47 5

122

7

218

11

49 3

125

2

223

3

50 3

127

3

226

3

131

2

227

2

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Primitive
Roots." §24.3.4 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, p. 827, 1972.

Guy, R. K. "Primitive Roots." §F9 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 248-249, 1994.

Sloane, N. J. A. Sequence A001918/M0242 in "An On-Line
Version of the Encyclopedia of Integer Sequences."

Primitive Root of Unity

A number r is an nth ROOT OF UNITY if r n = 1 and
a primitive nth root of unity if, in addition, n is the
smallest INTEGER of k = 1, . . . , n for which r k = 1.

see also ROOT OF UNITY

Primitive Semiperfect Number

A Semiperfect Number for which none of its Proper
Divisors are pseudoperfect (Guy 1994, p. 46). The first
few are 6, 20, 28, 88, 104, 272 .. . (Sloane's A006036).
Primitive pseudoperfect numbers are also called IRRE-
DUCIBLE Semiperfect Numbers. There are infinitely
many primitive pseudoperfect numbers which are not
Harmonic Divisor Numbers, and infinitely many
Odd primitive semiperfect numbers.

1442 Primitive Sequence

Principal Curvatures

see also Harmonic Divisor Number, Semiperfect
Number

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Yerlag, p. 46, 1994.
Sloane, N. J. A. Sequence A006036/M4133 in "An On-Line

Version of the Encyclopedia of Integer Sequences."

Primitive Sequence

A Sequence in which no term Divides any other.

References

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 202, 1994.

Primorial

For a Prime p,

primorial (pi) = p»# = JJpj,

where pi is the ith Prime. The first few values for p;#,
are 2, 6, 30, 210, 2310, 30030, 510510, ... (Sloane's
A002110).

p# - 1 is Prime for Primes p = 3, 5, 11, 41, 89, 317,
337, 991, 1873, 2053, 2377, 4093, 4297, ... (Sloane's
A014563; Guy 1994), or p n for n = 2, 3, 5, 13, 24, 66,
68, 167, 287, 310, 352, 564, 590, .... p# + 1 is known
to be Prime for the Primes p = 2, 3, 5, 7, 11, 31, 379,
1019, 1021, 2657, 3229, 4547, 4787, 11549, . . . (Sloane's
A005234; Guy 1994, Mudge 1997), or p n for n = 1, 2, 3,
4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, .... Both
forms have been tested to p = 25000 (Caldwell 1995). It
is not known if there are an infinite number of Primes
for which p# + 1 is PRIME or COMPOSITE (Ribenboim
1989).

see also FACTORIAL, FORTUNATE PRIME, PRIME

Sum Smarandache Near-to-Primorial Function,
Twin Peaks

References

Borning, A. "Some Results for kl + 1 and 2 ■ 3 * 5 ■ p + 1."

Math. Comput. 26, 567-570, 1972.
Buhler, J. R; Crandall, R. E.; and Penk, M. A. "Primes of

the form AT! + 1 and -3 • 5 ■ p + 1." Math. Comput. 38,

639-643, 1982.
Caldwell, C. "On The Primality of n!±l and 2-3-5 •• -p±l."

Math. Comput. 64, 889-890, 1995.
Dubner, H. "Factorial and Primorial Primes." J. Rec. Math.

19, 197-203, 1987.
Dubner, H. "A New Primorial Prime," J. Rec. Math, 21,

276, 1989.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.

New York: Springer- Verlag, pp. 7-8, 1994,
Leyland, P. ftp : // sable . ox . ac . uk / pub / math / factors /

primorial-. Z and primorial+.Z.
Mudge, M. "Not Numerology but Numeralogy!" Personal

Computer World, 279-280, 1997.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.

New York: Springer- Verlag, p. 4, 1989.

Sloane, N. J. A. Sequences A014563, A002110/M1691, and
A005234/M0669 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."

Temper, M. "On the Primality of k\ + 1 and -3 ■ 5 ■ • -p + 1."
Math. Comput. 34, 303-304, 1980.

Prince Rupert's Cube

The largest Cube which can be made to pass through
a given Cube. (In other words, the Cube having
a side length equal to the side length of the largest
Hole of a Square Cross-Section which can be cut
through a unit CUBE without splitting it into two
pieces.) The Prince Rupert's cube has side length
3\/2/4 = 1.06065 . . ., and any CUBE this size or smaller
can be made to pass through the original Cube.
see also CUBE, SQUARE

References

Cundy, H. and Rollett, A. "Prince Rupert's Cubes." §3.15.2

in Mathematical Models, 3rd ed. Stradbroke, England:

Tarquin Pub., pp. 157-158, 1989.
Schrek, D. J. E. "Prince Rupert's Problem and Its Extension

by Pieter Nieuwland." Scripta Math. 16, 73-80 and 261-

267, 1950.

Principal

The original amount borrowed or lent on which INTER-
EST is then paid or given.
see also Interest

Principal Curvatures

The Maximum and Minimum of the Normal Curva-
ture k\ and k 2 at a given point on a surface are called
the principal curvatures. The principal curvatures mea-
sure the Maximum and Minimum bending of a Reg-
ular Surface at each point. The Gaussian Curva-
ture K and Mean Curvature H are related to ki and
«2 by

K = K\K2

H= K«i + k 2 ).
This can be written as a Quadratic Equation
k - 2H k + K = 0,

which has solutions

k x = H + \/H 2 -K

K>2

= H- y/H 2 - K .

(1)
(2)

(3)

(4)
(5)

see also GAUSSIAN CURVATURE, MEAN CURVATURE,

Normal Curvature, Normal Section, Principal
Direction, Principal Radius of Curvature, Ro-
drigues's Curvature Formula

References

Geometry Center. "Principal Curvatures." http:// www .

geom . uirm . edu / zoo / dif f geom / surf space / concepts /

curvatures/prin-curv . html.
Gray, A. "Normal Curvature." §14.2 in Modern Differential

Geometry of Curves and Surfaces. Boca Raton, FL: CRC

Press, pp. 270-273, 277, and 283, 1993.

Principal Curve

Principal Vector 1443

Principal Curve

A curve a on a REGULAR SURFACE M is a principal
curve IFF the velocity ex! always points in a PRINCIPAL
Direction, i.e.,

S(ot , ) = K i oc\

where S is the Shape Operator and m is a Princi-
pal Curvature. If a Surface of Revolution gener-
ated by a plane curve is a REGULAR SURFACE, then the
Meridians and Parallels are principal curves.

References

Gray, A. "Principal Curves" and "The Differential Equation
for the Principal Curves." §18.1 and 21.1 in Modern Dif-
ferential Geometry of Curves and Surfaces. Boca Raton,
FL: CRC Press, pp. 410-413, 1993.

Principal Direction

The directions in which the Principal Curvatures oc-
cur.

see also Principal Direction

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 270, 1993.

Principal Ideal

An Ideal / of a Ring R is called principal if there is an
element a of R such that

I = a R= {ar : r G R}.

In other words, the Ideal is generated by the element
a. For example, the Ideals nZ of the Ring of Inte-
gers Z are all principal, and in fact all IDEALS of Z are
principal.
see also Ideal, Ring

Principal Normal Vector

see Normal Vector

Principal Quintic Form

A general Quintic EQUATION

the ROOTS and the sums of the SQUARES of the ROOTS
vanish, so

a&x 5 + a±x A + a$x z + a 2 x 2 + a\X -f ao =

can be reduced to one of the form

y + b 2 y 2 + hy + b = 0,

(1)

(2)

called the principal quintic form.

Newton's Relations for the Roots yj in terms of
the bjS is a linear system in the bj, and solving for the
bjS expresses them in terms of the POWER sums s n (yj).
These Power sums can be expressed in terms of the
ajSj so the bjS can be expressed in terms of the cljS. For
a quintic to have no quartic or cubic term, the sums of

si(2/j) = o
82(yj) = 0.

(3)
(4)

Assume that the Roots yj of the new quintic are related
to the ROOTS Xj of the original quintic by

yj = xj 2 + axj + /?.

(5)

Substituting this into (1) then yields two equations for
a and j3 which can be multiplied out, simplified by us-
ing Newton's Relations for the Power sums in the
Xj, and finally solved. Therefore, a and j3 can be ex-
pressed using Radicals in terms of the Coefficients
clj. Again by substitution into (4), we can calculate
53 (yj), S4 (yj) and 55 (yj) in terms of a and /3 and the
Xj. By the previous solution for a and and again by
using Newton's Relations for the Power sums in
the Xj, we can ultimately express these POWER sums in
terms of the aj.

see also Bring Quintic Form, Newton's Relations,
Quintic Equation

Principal Radius of Curvature

Given a 2-D SURFACE, there are two "principal" RADII

OF CURVATURE. The larger is denoted Ri, and the

smaller R2. These are PERPENDICULAR to each other,

and both PERPENDICULAR to the tangent Plane of the

surface.

see also GAUSSIAN CURVATURE, MEAN CURVATURE,

Radius of Curvature
Principal Value

see CAUCHY PRINCIPAL VALUE

Principal Vector

A tangent vector v p = vix u +v 2 X-v is a principal vector
IFF

det

V2 2 — V\V2 Vl 21

E
e

-0,

F G

f 9 J

where e, /, and g are coefficients of the first FUNDAMEN-
TAL Form and E t F, G of the second Fundamental
Form.
see also Fundamental Forms, Principal Curve

References

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 410, 1993.

1444 Principal Vertex

Prismatic Ring

Principal Vertex

A Vertex xi of a Simple Polygon P is a princi-
pal Vertex if the diagonal [xi-±,Xi+i] intersects the
boundary of P only at Xi-i and Xi+i.

see also Ear, Mouth

References

Meisters, G. H. "Polygons Have Ears." Amer. Math. Monthly

82, 648-751, 1975.
Meisters, G. H. "Principal Vertices, Exposed Points, and

Ears." Amer. Math. Monthly 87, 284-285, 1980.
Toussaint, G. "Anthropomorphic Polygons." Amer. Math.

Monthly 98, 31-35, 1991.

Principle

A loose term for a true statement which may be a POS-
TULATE, Theorem, etc.

see also AREA PRINCIPLE, ARGUMENT PRINCIPLE, AX-
IOM, Cavalieri's Principle, Conjecture, Conti-
nuity Principle, Counting Generalized Princi-
ple, Dirichlet's Box Principle, Duality Prin-
ciple, Duhamel's Convolution Principle, Eu-
clid's Principle, Fubini Principle, Hasse Prin-
ciple, Inclusion-Exclusion Principle, Indiffer-
ence Principle, Induction Principle, Insuffi-
cient Reason Principle, Lemma, Local-Global
Principle, Multiplication Principle, Perma-
nence of Mathematical Relations Principle,
Poncelet's Continuity Principle, Pontryagin
Maximum Principle, Porism, Postulate, Schwarz
Reflection Principle, Superposition Princi-
ple, Symmetry Principle, Theorem, Thomson's
Principle, Triangle Transformation Principle,
Well-Ordering Principle

Pringsheim's Theorem

Let C U {I) be the set of real ANALYTIC FUNCTIONS on i".
Then C w (7) is a Subalgebra of C°°(i"). A Necessary
and Sufficient condition for a function / e C°°(I) to
belong to C"(I) is that

|/ (n) (x)| <k n n\

for n = 0, 1, . . . for a suitable constant k.
see also ANALYTIC FUNCTION, SUBALGEBRA

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 207, 1980.

Printer's Errors

Typesetting "errors" in which exponents or multiplica-
tion signs are omitted but the resulting expression is
equivalent to the original one. Examples include

3 4 425 = 34425
31 2 325 = 312325

25
31

9*25

where a whole number followed by a fraction is inter-
preted as addition (e.g., l| = 1 + \ = f ).
see also ANOMALOUS CANCELLATION

References

Dudeney, H. E. Amusements in Mathematics. New York:

Dover, 1970.
Madachy, J. S. Madachy's Mathematical Recreations. New

York: Dover, pp. 174-175, 1979.

Prior Distribution

see Bayesian Analysis

Prism

© © ©

A Polyhedron with two congruent Polygonal faces
and all remaining faces PARALLELOGRAMS. The 3-
prism is simply the CUBE. The simple prisms and an-
tiprisms include: decagonal antiprism, decagonal prism,
hexagonal antiprism, hexagonal prism, octagonal anti-
prism, octagonal prism, pentagonal antiprism, pentago-
nal prism, square antiprism, and triangular prism. The
DUAL Polyhedron of a simple (Archimedean) prism is
a BlPYRAMID.

The triangular prism, square prism (cube), and hexag-
onal prism are all SPACE-FILLING POLYHEDRA.

see also Antiprism, Augmented Hexagonal Prism,
Augmented Pentagonal Prism, Augmented Tri-
angular Prism, Biaugmented Pentagonal Prism,
blaugmented triangular prism, cube, metabi-
augmented Hexagonal Prism, Parabiaugmented
Hexagonal Prism, Prismatoid, Prismoid, Trape-
zohedron, triaugmented hexagonal prism, tri-
augmented Triangular Prism

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,

28th ed. Boca Raton, FL: CRC Press, p. 127, 1987.
Cromwell, P. R. Polyhedra. New York: Cambridge University

Press, pp. 85-86, 1997.
^ Weisstein, E. W. "Prisms and Antiprisms." http://www.

astro. Virginia. edu/-eww6n/math/notebooks/Prism.m.

Prismatic Ring

A Mobius Strip with finite width.
see also Mobius Strip

References

Gardner, M. "Twisted Prismatic Rings." Ch. 5 in Fractal
Music, HyperCards, and More Mathematical Recreations
from Scientific American Magazine. New York: W. H.
Freeman, 1992.

Prismatoid

Probability 1445

Prismatoid

A Polyhedron having two Polygons in Parallel
planes as bases and Triangular or Trapezoidal lat-
eral faces with one side lying in one base and the oppo-
site Vertex or side lying in the other base. Examples
include the Cube, Pyramidal Frustum, Rectangu-
lar Parallelepiped, Prism, and Pyramid. Let A x
be the Area of the lower base, A2 the Area of the
upper base, M the Area of the midsection, and h the
Altitude. Then

V= f/i(Ai+4M + A 2 ).

see also GENERAL PRISMATOID, PRISMOID

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 128 and 132, 1987.

Prismoid

A Prismatoid having planar sides and the same num-
ber of vertices in both of its parallel planes. The faces
of a prismoid are therefore either TRAPEZOIDS or PAR-
ALLELOGRAMS. Ball and Coxeter (1987) use the term
to describe an ANTIPRISM.

see also Antiprism, Prism, Prismatoid

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 130,
1987.

Prisoner's Dilemma

A problem in Game Theory first discussed by
A. Tucker. Suppose each of two prisoners A and £?,
who are not allowed to communicate with each other,
is offered to be set free if he implicates the other. If
neither implicates the other, both will receive the usual
sentence. However, if the prisoners implicate each other,
then both are presumed guilty and granted harsh sen-
tences.

A Dilemma arises in deciding the best course of action
in the absence of knowledge of the other prisoner's deci-
sion. Each prisoner's best strategy would appear to be
to turn the other in (since if A makes the worst-case as-
sumption that B will turn him in, then B will walk free
and A will be stuck in jail if he remains silent). How-
ever, if the prisoners turn each other in, they obtain the
worst possible outcome for both.

see also Dilemma, Tit-for-Tat
References

Axelrod, R. The Evolution of Cooperation New York: Basic-
Books, 1985.

Goetz, P. "Phil's Good Enough Complexity Dictionary."
http://wvw. cs .buffalo.edu/~goetz/dict .html.

Probability

Probability is the branch of mathematics which studies
the possible outcomes of given events together with their
relative likelihoods and distributions. In common usage,
the word "probability" is used to mean the chance that
a particular event (or set of events) will occur expressed
on a linear scale from (impossibility) to 1 (certainty),
also expressed as a Percentage between and 100%.
The analysis of events governed by probability is called
Statistics.

There are several competing interpretations of the ac-
tual "meaning" of probabilities. Frequentists view prob-
ability simply as a measure of the frequency of out-
comes (the more conventional interpretation), while
B AYES IAN S treat probability more subjectively as a sta-
tistical procedure which endeavors to estimate parame-
ters of an underlying distribution based on the observed
distribution.

A properly normalized function which assigns a proba-
bility "density" to each possible outcome within some
interval is called a PROBABILITY FUNCTION, and its cu-
mulative value (integral for a continuous distribution or
sum for a discrete distribution) is called a DISTRIBUTION
Function.

Probabilities are defined to obey certain assumptions,
called the Probability Axioms. Let a Sample Space
contain the Union (U) of all possible events Ei, so

S =

U*>

(i)

and let E and F denote subsets of 5. Further, let F' =
not-F be the complement of i 7 *, so that

F U F' = S.
Then the set E can be written as

(2)

E = EnS = En(FUF , ) = (EnF)U(EnF , ) i (3)
where D denotes the intersection. Then
P(E) = P{E HF) + P(E n F') - P[(E n F) n (E n F')]

= P(E r\F) + P(E n F') - P[(F n F') n{En E)]
= P(E n F) + p{e n f') - p(0 n e)
= p(E n F) + P(E n f') - P(0)

= P(Er\F) + P(EC\F f ), (4)

where is the Empty Set.

Prizes

see Mathematics Prizes

1446 Probability Axioms

Let P(E\F) denote the CONDITIONAL PROBABILITY of
E given that F has already occurred, then

P(E) = P(E\F)P(F) + P(E\F')P(F') (5)

- P(E\F)P(F) + P(E|P')[1 - P(F)] (6)
P(AnB) = P(A)P(B|i4) (7)

- P(B)P(A|B) (8)
P(A'nB)=P(A')P(5|A') (9)

P(jE7 n F)

P{E\F)

(10)

P(F)
A very important result states that

P(E U F) = P(E) + P(F) - P(£ n F), (11)

which can be generalized to

p ( I) Ai J = E p (^> - E' p ^ u A *)
v»=i / » »j

+ 53" p(i4i n ^ n At) - . . . + i-iy-'P I f] A J .

ijk

(12)

see also Bayes' Formula, Conditional Probabil-
ity, Distribution, Distribution Function, Like-
lihood, Probability Axioms, Probability Func-
tion, Probability Inequality, Statistics

Probability Axioms

Given an event Eina Sample Space 5 which is either
finite with N elements or countably infinite with N — oo
elements, then we can write

/ N \

U* •

and a quantity P{Ei), called the Probability of event
Ei, is defined such that

1. < P(Ei) < 1.

2. P(S) = 1.

3. Additivity: P(E X UE 2 ) = P(E X ) + P(E 2 ), where E x
and E 2 are mutually exclusive.

4. Countable additivity: P (UJUE*) = ^=1 P ( Ei ) for
n = 1, 2, ..., iV where Pi, E 2 , ...are mutually
exclusive (i.e., Ei C\ E 2 = 0).

see ateo SAMPLE SPACE, UNION

Probability Density Function

see Probability Function

Probability Distribution Function

see Probability Function

Probability Function

Probability Function

The probability density function P(x) (also called the
Probability Density Function) of a continuous dis-
tribution is defined as the derivative of the (cumulative)
Distribution Function D(x) 7

D'{x) - [P(aO£oo = P(*) ~ P(-o°) = P(*)> (1)

D{x) = P(X<x)= / P(y)dy. (2)

J — 00

A probability density function satisfies

P(xeB) = J P(x)dx (3)

J B

and is constrained by the normalization condition,

/oo
P(x)dx = 1. (4)

-00

Special cases are

P(a < x < b) = / P(x)dx

f

J a

(5)

pa+da

P{a<x<a + da)= I P(x) dx « P(a) da (6)

P(x = a)= I P{x) dx = 0. (7)

J a

If it = u(x, y) and u = u(x, y), then

P«,t;(u,w) = Px t y(aJ,y)

d(x,?/)

£(u,v)

(8)

Given the MOMENTS of a distribution (/x, <r, and the
Gamma Statistics 7,.), the asymptotic probability
function is given by

P{x) = Z{x)

-\h^\x)\ + [^Z^(x) + ±tSz«\x)\
-{^ l3 Z^{x) + ^ lll2 Z^(x) + ^7. 3 Z (9) (x)l
+[£- 74 zW(x) + ( T ^72 2 + ^^z)Z^\x)
+ 1 ^l7i 2 7^ (1 ° ) (x) + 5ik7l 4 ^ (ia) (*)] + • • ■ , 0)

where

Z(*) :

-(!D- M ) 2 /2er 2

o-\/2ir
is the Normal Distribution, and

^ = ^

(10)

(11)

Probability Inequality

Problem

1447

for r > 1 (with re r CuMULANTS and u the STANDARD
Deviation; Abramowitz and Stegun 1972, p. 935).

see also CONTINUOUS DISTRIBUTION, CORNISH-FlSHER

Asymptotic Expansion, Discrete Distribution,
Distribution Function, Joint Distribution Func-
tion

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Probability
Functions." Ch. 26 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 925-964, 1972.

Probability Inequality

If B D A (B is a superset of A), then P(A) < P(B).

Probability Integral

i - ^

0.5 /

^T ' -2 ^1 2 4

-/. 5 ■

RetProbabilitylntegral z] Im[ProbabilityIntegral z] | p^babiHtyjntegral z|

Probability Space

A triple (S, §,P), where (5,S) is a measurable space

and P is a MEASURE on S with P(S) = 1.

see a/50 Measurable Space, Measure, Probability,

Probability Measure, Random Variable, State

Space

Probable Error

The first Quartile of a standard Normal Distribu-
tion occurs when

/'

${Z) dz :

The solution is t = 0.6745 — The value of t giving
1/4 is known as the probable error of a NORMALLY Dis-
tributed variate. However, the number S correspond-
ing to the 50% Confidence Interval,

P(S)

r\*\
= 1-2/

Jo

4>(t) dt-±

2>

is sometimes also called the probable error.

see also Significance

Probable Prime

A number satisfying Fermat'S LITTLE THEOREM (or
some other primality test) for some nontrivial base. A
probable prime which is shown to be COMPOSITE is
called a PSEUDOPRIME (otherwise, of course, it is a

Prime).

see also Prime Number, Pseudoprime

a{x) = v^J- e

= 2§(x)

(1)

(2)
(3)
(4)

where $(x) is the NORMAL DISTRIBUTION FUNCTION
and ERF is the error function.

see also Erf, Normal Distribution Function

Probability Measure

Consider a Probability Space (5,S,P) where (£,§)
is a Measurable Space and P is a Measure on S
with P(S) — 1. Then the Measure P is said to be
a probability measure. Equivalently, P is said to be
normalized.

see also Measurable Space, Measure, Probability,
Probability Space, State Space

Problem

An exercise whose solution is desired.

see also ALHAZEN'S BILLIARD PROBLEM, ALHAZEN'S

Problem, Andre's Problem, Apollonius' Prob-
lem, Apollonius Pursuit Problem, Archimedes'
Cattle Problem, Archimedes' Problem, Ballot
Problem, Basler Problem, Bertrand's Prob-
lem, Billiard Table Problem, Birthday Prob-
lem, Bishops Problem, Bolza Problem, Book
Stacking Problem, Boundary Value Problem,
Bovinum Problema, Brachistochrone Problem,
Brahmagupta's Problem, Brocard's Problem,
Buffon-Laplace Needle Problem, Buffon's Nee-
dle Problem, Burnside Problem, Busemann-
Petty Problem, Cannonball Problem, Castil-
lon's Problem, Catalan's Diophantine Prob-
lem, Catalan's Problem, Cattle Problem of
Archimedes, Cauchy Problem, Checker-Jumping
Problem, Closed Curve Problem, Coin Prob-
lem, Collatz Problem, Condom Problem, Con-
gruum Problem, Constant Problem, Coupon
Collector's Problem, Crossed Ladders Prob-
lem, Cube Dovetailing Problem, Decision Prob-
lem, Dedekind's Problem, Delian Problem, de

1448

Problem

Problem

Mere's Problem, Diagonals Problem, Dido's
Problem, Dilemma, Dinitz Problem, Dirichlet
Divisor Problem, Disk Covering Problem, Equi-
chordal Problem, Extension Problem, Fag-
nano's Problem, Fejes Toth's Problem, Fer-
mat's Problem, Fermat's Sigma Problem, Fisher-
Behrens Problem, Five Disks Problem, Four
Coins Problem, Four Travelers Problem, Fuss's
Problem, Gauss's Circle Problem, Gauss's Class
Number Problem, Glove Problem, Guthrie's
Problem, Haberdasher's Problem, Hadwiger
Problem, Halting Problem, Hansen's Problem,
Heesch's Problem, Heilbronn Triangle Problem,
Hilbert's Problems, Illumination Problem, Inde-
terminate Problems, Initial Value Problem, In-
ternal Bisectors Problem, Isoperimetric Prob-
lem, Isovolume Problem, Jeep Problem, Josephus
Problem, Kakeya Needle Problem, Kakutani's
Problem, Katona's Problem, Kepler Problem,
Kings Problem, Kirkman's Schoolgirl Prob-
lem, Kissing Circles Problem, Knapsack Prob-
lem, Knot Problem, Konigsberg Bridge Prob-
lem, Kuratowski's Closure-Component Prob-
lem, Lam's Problem, Langford's Problem, Lebes-
gue Measurability Problem, Lebesgue Minimal
Problem, Lehmer's Problem, Lemoine's Prob-
lem, Lifting Problem, Lucas' Married Couples
Problem, Malfatti's Right Triangle Problem,
Malfatti's Tangent Triangle Problem, Mar-
ried Couples Problem, Match Problem, Max-
imum Clique Problem, Menage Problem, Met-
ric Equivalence Problem, Mice Problem, Mi-
kusinski's Problem, Mobius Problem, Money-
Changing Problem, Monkey and Coconut Prob-
lem, Monty Hall Problem, Mortality Prob-
lem, Moser's Circle Problem, Napoleon's Prob-
lem, Navigation Problem, Nearest Neighbor
Problem, NP-Complete Problem, NP-Problem,
Orchard-Planting Problem, Orchard Visibil-
ity Problem, P-Problem, Party Problem, Pi-
ano Mover's Problem, Planar Bubble Problem,
Plateau's Problem, Points Problem, Postage
Stamp Problem, Pothenot Problem, Prouhet's
Problem, Queens Problem, Railroad Track
Problem, Riemann's Moduli Problem, Satisfi-
ability Problem, Schoolgirl Problem, Schur's
Problem, Schwarz's Triangle Problem, Shar-
ing Problem, Shephard's Problem, Sinclair's
Soap Film Problem, Small World Problem,
Snellius-Pothenot Problem, Steenrod's Real-
ization Problem, Steiner's Problem, Steiner's
Segment Problem, Surveying Problems, Syl-
vester's Four-Point Problem, Sylvester's Line
Problem, Sylvester's Triangle Problem, Syra-
cuse Problem, Syzygies Problem, Tarry-Escott
Problem, Tautochrone Problem, Thomson Prob-
lem, Three Jug Problem, Traveling Salesman
Problem, Trawler Problem, Ulam's Problem,

Utility Problem, Vibration Problem, Wallis's
Problem, Waring's Problem

References

Artino, R. A.; Gaglione, A. M.; and Shell, N. The Contest
Problem Book IV: Annual High School Mathematics Ex-
aminations 1973-1982. Washington, DC: Math. Assoc.
Amer., 1982,

Alexanderson, G. L.; Klosinski, L.; and Larson, L. The
William Lowell Putnam Mathematical Competition, Prob-
lems and Solutions: 1965-1984- Washington, DC: Math.
Assoc. Amer., 1986.

Barbeau, E. J.; Moser, W. O.; and Lamkin, M. S. Five Hun-
dred Mathematical Challenges. Washington, DC: Math,
Assoc. Amer., 1995.

Brown, K. S. "Most Wanted List of Elementary Un-
solved Problems." http : //www . seanet . com/-ksbrown/
mwlist.htm.

Chung, F. and Graham, R. Erdos on Graphs: His Legacy of
Unsolved Problems. New York: A. K. Peters, 1998.

Cover, T. M. and Gopinath, B. (Eds.). Open Problems in
Communication and Computation. New York: Springer-
Verlag, 1987.

Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, 1965.

Dudeney, H. E. Amusements in Mathematics. New York:
Dover, 1917.

Dudeney, H. E. The Canterbury Puzzles and Other Curious
Problems, 7th ed. London: Thomas Nelson and Sons, 1949.

Dudeney, H. E. 536 Puzzles & Curious Problems. New York:
Scribner, 1967.

Eppstein, D. "Open Problems." http://www.ics.uci.edu/-
eppstein/ junkyard/ open. html.

Erdos, P. "Some Combinatorial Problems in Geometry." In
Geometry and Differential Geometry (Ed. R. Artzy and
I. Vaisman). New York: Springer- Verlag, pp. 46-53, 1980.

Fenchel, W. (Ed.). "Problems." In Proc. Colloquium on
Convexity, 1965. K0benhavns Univ. Mat. Inst., pp. 308-
325, 1967.

Finch, S. "Unsolved Mathematical Problems." http: //www.
mathsof t . com/asolve/.

Gleason, A. M.; Greenwood, R. E.; and Kelly, L. M. The
William Lowell Putnam Mathematical Competition, Prob-
lems and Solutions: 1938-1964- Washington, DC: Math.
Assoc. Amer., 1980.

Graham, L. A. Ingenious Mathematical Problems and Meth-
ods. New York: Dover, 1959.

Graham, L. A. The Surprise Attack in Mathematical Prob-
lems. New York: Dover, 1968.

Greitzer, S. L. International Mathematical Olympiads, 1959-
1977. Providence, RI: Amer. Math. Soc, 1978.

Gruber, P. M. and Schneider, R. "Problems in Geometric
Convexity." In Contributions to Geometry (Ed. J. T61ke
and J. M. Wills.) Boston, MA: Birkhauser, pp. 255-278,
1979.

Guy, R. K. (Ed.). "Problems." In The Geometry of Metric
and Linear Spaces. New York: Springer- Verlag, pp. 233-
244, 1974.

Halmos, P. R. Problems for Mathematicians Young and Old.
Washington, DC: Math. Assoc. Amer., 1991.

Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., 1973.

Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., 1976.

Honsberger, R. Mathematical Morsels. Washington, DC:
Math. Assoc. Amer., 1979.

Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., 1985.

Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., 1991.

Problem

Product

1449

Honsberger, R. From Erdos to Kiev. Washington, DC: Math.
Assoc. Amer., 1995.

Honsberger, R. In Polya's Footsteps: Miscellaneous Prob-
lems and Essays. Washington, DC: Math. Assoc. Amer.,
1997.

Honsberger, R. (Ed.). Mathematical Plums. Washington,
DC: Math. Assoc. Amer., 1979.

Kimberling, C. "Unsolved Problems and Rewards." http://
www . evansville . edu/-ck6/integer /unsolved .html.

Klamkin, M. S. International Mathematical Olympiads,
1978-1985 and Forty Supplementary Problems. Washing-
ton, DC: Math. Assoc. Amer., 1986.

Klamkin, M. S. U.S.A. Mathematical Olympiads, 1972-1986.
Washington, DC: Math. Assoc. Amer,, 1988.

Kordemsky, B. A. The Moscow Puzzles: 359 Mathematical
Recreations. New York: Dover, 1992.

Kurschak, J. and Hajos, G. Hungarian Problem Book, Based
on the Eotvos Competitions, Vol. 1: 1894~1905. New
York: Random House, 1963.

Kurschak, J. and Hajos, G. Hungarian Problem Book, Based
on the Eotvos Competitions, Vol. 2: 1906-1928. New
York: Random House, 1963.

Larson, L. C. Problem- Solving Through Problems. New York:
Springer- Verlag, 1983.

Mott-Smith, G. Mathematical Puzzles for Beginners and En-
thusiasts. New York: Dover, 1954.

Ogilvy, C. S. Tomorrow's Math: Unsolved Problems for the
Amateur. New York: Oxford University Press, 1962.

Ogilvy, C. S. "Some Unsolved Problems of Modern Geom-
etry." Ch. 11 in Excursions in Geometry. New York:
Dover, pp. 143-153, 1990.

Posamentier, A. S. and Salkind, C. T. Challenging Problems
in Algebra. New York: Dover, 1997.

Posamentier, A. S. and Salkind, C. T. Challenging Problems
in Geometry. New York: Dover, 1997.

Rabinowitz, S. (Ed.). Index to Mathematical Problems 1980-
1984. Westford, MA: MathPro Press, 1992.

Salkind, C- T. The Contest Problem Book I: Problems from
the Annual High School Contests 1950-1960. New York:
Random House, 1961.

Salkind, C. T. The Contest Problem Book II: Problems from
the Annual High School Contests 1961-1965. Washington,
DC: Math. Assoc. Amer., 1966.

Salkind, C. T. and Earl, J. M. The Contest Problem Book
III: Annual High School Contests 1966-1972. Washington,
DC: Math. Assoc. Amer., 1973.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, 1993.

Shkliarskii, D. O.; Chentzov, N. N.; and Yaglom, I. M. The
U.S.S.R. Olympiad Problem Book: Selected Problems and
Theorems of Elementary Mathematics. New York: Dover,
1993.

Sierpiriski, W. A Selection of Problems in the Theory of
Numbers. New York: Pergamon Press, 1964. Sierpinski,
W. Problems in Elementary Number Theory. New York:
Elsevier, 1980.

Smarandache, F. Only Problems, Not Solutions!, 4^ n e< ^-
Phoenix, AZ: Xiquan, 1993.

Steinhaus, H. One Hundred Problems in Elementary Mathe-
matics. New York: Dover, 1979.

Tietze, H. Famous Problems of Mathematics. New York:
Graylock Press, 1965.

Trigg, C. W, Mathematical Quickies: 270 Stimulating Prob-
lems with Solutions. New York: Dover, 1985.

Ulam, S. M. A Collection of Mathematical Problems. New-
York: Interscience Publishers, 1960.

van Mill, J. and Reed, G. M. (Eds.). Open Problems in To-
pology. New York: Elsevier, 1990.

Procedure

A specific prescription for carrying out a task or solving
a problem. Also called an ALGORITHM, METHOD, or
Technique

see also BISECTION PROCEDURE, MAEHLY'S PROCE-
DURE

Proclus' Axiom

If a line intersects one of two parallel lines, it must in-
tersect the other also. This AXIOM is equivalent to the
Parallel Axiom.

References

Dunham, W, "Hippocrates' Quadrature of the Lune." Ch. 1

in Journey Through Genius: The Great Theorems of
Mathematics. New York: Wiley, p. 54, 1990.

Procrustian Stretch

see Hyperbolic Rotation

Product

The term "product" refers to the result of one or more
Multiplications. For example, the mathematical
statement axb = c would be read "a Times b Equals
c," where c is the product.

The product symbol is denned by

n
Y[fi = fl-h---fn.

Useful product identities include

/ oo \ oo

J\ fi = exp I ^ ln /*

For < a; < 1, then the products FIi^i( 1 + a *) and

nnii^ ~~ ai ) conver S e an d diverge as Yii=i aim

see also CROSS PRODUCT, DOT PRODUCT, INNER

Product, Matrix Product, Multiplication, Non-
associative Product, Outer Product, Sum, Ten-
sor Product, Times, Vector Triple Product

References

Guy, R. K. "Products Taken over Primes." §B87 in Unsolved

Problems in Number Theory, 2nd ed. New York: Springer-

Verlag, pp. 102-103, 1994.

1450 Product Formula

Projection Operator

Product Formula

Let a be a Nonzero Rational Number a =
±pi 0£1 P2 Q:2 • * •pL otL , where pi, . . . , pl are distinct
Primes, a t e Z and oti ^ 0. Then

H J] |a| p = Pl ai P 2 a2 ---p^ aL

xpr ai P2- a2 ---PL L = i-

References

Burger, E. B. and Struppeck, T. "Does Yl7=o ^ Reall y Con-
verge? Infinite Series and p-adic Analysis." Amer. Math.
Monthly 103, 565-577, 1996.

Product- Moment Coefficient of Correlation

see Correlation Coefficient

Product Neighborhood

see Tubular Neighborhood

Product Rule

The Derivative identity

li [f(x)9(x)] = hm

— lim

h-+0

= lim

f(x + h)g(x + h) - f{x + h)g(x)

+

/(a; + /i)p(a;)-/(x)p(a:)

/(a; + h)

g{x + h) - g(x)

+g(*)

f( x + h)-f(x)

= f(x)g(x)+g(x)f'(x).

see also CHAIN RULE, EXPONENT LAWS, QUOTIENT

Rule

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 11, 1972.

Product Space

A Cartesian product equipped with a "product topol-
ogy" is called a product space (or product topological
space, or direct product).

References

Iyanaga, S. and Kawada, Y. (Eds.). "Product Spaces."
§408L Encyclopedic Dictionary of Mathematics. Cam-
bridge, MA: MIT Press, pp. 1281-1282, 1980.

Program

A precise sequence of instructions designed to accom-
plish a given task. The implementation of an Algo-
rithm on a computer using a programming language is
an example of a program.

see also Algorithm
Projection

r^>~7

A projection is the transformation of Points and Lines
in one Plane onto another Plane by connecting corre-
sponding points on the two planes with PARALLEL lines.
This can be visualized as shining a (point) light source
(located at infinity) through a translucent sheet of paper
and making an image of whatever is drawn on it on a
second sheet of paper. The branch of geometry dealing
with the properties and invariants of geometric figures
under projection is called PROJECTIVE GEOMETRY.

The projection of a Vector a onto a Vector u is given
by

proj " a= ^F u -

and the length of this projection is

|pro Ju a| = -j-p

General projections are considered by Foley and Van-
Dam (1983).

see also Map Projection, Point-Plane Distance,
Projective Geometry, Reflection

References

Casey, J. "Theory of Projections." Ch. 11 in A Treatise on
the Analytical Geometry of the Point, Line, Circle, and
Conic Sections, Containing an Account of Its Most Recent
Extensions, with Numerous Examples, 2nd ed., rev. enl.
Dublin: Hodges, Figgis, & Co., pp. 349-367, 1893.

Foley, J. D. and VanDam, A. Fundamentals of Interactive
Computer Graphics, 2nd ed. Reading, MA: Addison-
Wesley, 1990.

Projection Operator

p=\4>i{x)){<t>i{t)\

p5^Cj-|0j(t)> =a\Mx))

Projective Collineation

£|*(x)><*(*)| = 1.

i

see also Bra, Ket

Projective Collineation

A COLLINEATION which transforms every 1-D form pro-
jectively. Any COLLINEATION which transforms one
range into a project ively related range is a projective
collineation. Every PERSPECTIVE COLLINEATION is a
projective collineation.

see also COLLINEATION, ELATION, HOMOLOGY (GEOM-
ETRY), Perspective Collineation

References

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York; Wiley, pp. 247-248, 1969.

Projective General Linear Group

The projective general linear group PGL n (q) is the
GROUP obtained from the GENERAL Linear GROUP
GL n (q) on factoring the scalar Matrices contained in
that group.

see also General Linear Group, Projective Gen-
eral Orthogonal Group, Projective General
Unitary Group

References

Conway, J. H,; Curtis, R. T.; Norton, S. P.; Parker, R. A.;
and Wilson, R. A. "The Groups GL n (q) } SL n (q), PGL n (q),
and PSL n (q) = L n (q)" §2.1 in Atlas of Finite Groups:
Maximal Subgroups and Ordinary Characters for Simple
Groups. Oxford, England: Clarendon Press, p. x, 1985.

Projective General Orthogonal Group

The projective general orthogonal group PGO n (q) is
the GROUP obtained from the GENERAL ORTHOGONAL
GROUP GO n (q) on factoring the scalar Matrices con-
tained in that group.

see also General Orthogonal Group, Projective
General Linear Group, Projective General Uni-
tary Group

References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker,
R. A.; and Wilson, R. A. "The Groups GO n (q), SO n (q),
PGO n (q), and PSO n (q), and O n (q)." §2.4 in Atlas of
Finite Groups: Maximal Subgroups and Ordinary Char-
acters for Simple Groups. Oxford, England: Clarendon
Press, pp. xi-xii, 1985.

Projective General Unitary Group

The projective general unitary group PGU n (q) is the
Group obtained from the GENERAL UNITARY GROUP
GU n (q) on factoring the scalar MATRICES contained in
that group.

see also GENERAL UNITARY GROUP, PROJECTIVE GEN-
ERAL Linear Group, Projective General Or-
thogonal Group, Projective General Unitary
Group

Projective Geometry 1451

References

Conway, J. H.; Curtis, R. T\; Norton, S. R; Parker,
R. A.; and Wilson, R. A. "The Groups GU n {q), SU n (q) y
PGU n (q) y and PSU n (q) = U n {q). n §2.2 in Atlas of Finite
Groups: Maximal Subgroups and Ordinary Characters for
Simple Groups. Oxford, England: Clarendon Press, p. x,
1985.

Projective Geometry

The branch of geometry dealing with the properties and
invariants of geometric figures under Projection. The
most amazing result arising in projective geometry is
the Duality Principle, which states that a duality
exists between theorems such as PASCAL'S THEOREM
and BRIANCHON'S THEOREM which allows one to be in-
stantly transformed into the other. More generally, all
the propositions in projective geometry occur in dual
pairs, which have the property that, starting from ei-
ther proposition of a pair, the other can be immediately
inferred by interchanging the parts played by the words
"Point" and "Line."

The Axioms of projective geometry are:

1. If A and B are distinct points on a PLANE, there is
at least one LINE containing both A and B.

2. If A and B are distinct points on a PLANE, there is
not more than one LINE containing both A and B.

3. Any two LINES on a PLANE have at least one point
of the Plane in common.

4. There is at least one LINE on a PLANE.

5. Every LINE contains at least three points of the
Plane.

6. All the points of the Plane do not belong to the

same Line

(Veblin and Young 1910-18, Kasner and Newman 1989).

see also Collineation, Desargues' Theorem, Fun-
damental Theorem of Projective Geometry, In-
volution (Line), Pencil, Perspectivity, Projec-
tivity, Range (Line Segment), Section (Pencil)

References

Birkhoff, G. and Mac Lane, S. "Projective Geometry." §9.14
in A Survey of Modern Algebra, 3rd ed. New York:
Macmillan, pp. 275-279, 1965.

Casey, J. "Theory of Projections." Ch. 11 in A Treatise on
the Analytical Geometry of the Point, Line, Circle, and
Conic Sections, Containing an Account of Its Most Recent
Extensions, with Numerous Examples, 2nd ed., rev. enl.
Dublin: Hodges, Figgis, & Co., pp. 349-367, 1893.

Coxeter, H. S. M. Projective Geometry, 2nd ed. New York:
Springer- Verlag, 1987.

Kadison, L. and Kromann, M. T. Projective Geometry and
Modern Algebra. Boston, MA: Birkhauser, 1996.

Kasner, E. and Newman, J. R. Mathematics and the Imag-
ination. Redmond, WA: Microsoft Press, pp. 150-151,
1989.

Ogilvy, C. S. "Projective Geometry." Ch. 7 in Excursions in
Geometry. New York: Dover, pp. 86-110, 1990.

Pappas, T. "Art Sz Projective Geometry." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, pp. 66-
67, 1989.

1452 Projective Plane

Projective Special Linear Group

Pedoe, D. and Sneddon, I. A. An Introduction to Projective
Geometry. New York: Pergamon, 1963.

Seidenberg, A. Lectures in Projective Geometry. Princeton,
NJ: Van Nostrand, 1962.

Struik, D. Lectures on Projected Geometry. Reading, MA:
Addison- Wesley, 1998.

Veblen, O. and Young, J. W. Projective Geometry, 2 vols.
Boston, MA: Ginn, 1910-18.

Whitehead, A. N. The Axioms of Projective Geometry, New-
York: Hafner, 1960.

Projective Plane

A projective plane is derived from a usual Plane by
addition of a Line at Infinity. A projective plane of
order n is a set of n 2 -f- n -f 1 POINTS with the properties
that:

1. Any two POINTS determine a LINE,

2. Any two Lines determine a Point,

3. Every POINT has n + 1 LINES on it, and

4. Every LINE contains n + 1 POINTS.

(Note that some of these properties are redundant.) A
projective plane is therefore a Symmetric (n 2 + n + 1,
n + 1, 1) Block Design. An Affine Plane of order
n exists Iff a projective plane of order n exists.

A finite projective plane exists when the order n is a
Power of a Prime, i.e., n = p a for a > 1. It is conjec-
tured that these are the only possible projective planes,
but proving this remains one of the most important un-
solved problems in COMBINATORICS. The first few or-
ders which are not of this form are 6, 10, 12, 14, 15,

It has been proven analytically that there are no pro-
jective planes of order 6. By answering Lam's PROB-
LEM in the negative using massive computer calculations
on top of some mathematics, it has been proved that
there are no finite projective planes of order 10 (Lam
1991). The status of the order 12 projective plane re-
mains open. The remarkable Bruck-Ryser-Chowla
THEOREM says that if a projective plane of order n ex-
ists, and n = 1 or 2 (mod 4), then n is the sum of two
Squares. This rules out n = 6.

The projective plane of order 2, also known as the FANO
Plane, is denoted PG(2, 2). It has Incidence Matrix

1110
10 110
10 11
10 10 10
10 10 1
110 1
Lo o i o i i o,

Every row and column contains 3 is, and any pair of
rows/columns has a single 1 in common.

The projective plane has Euler Characteristic 1,
and the HEAWOOD CONJECTURE therefore shows that

any set of regions on it can be colored using six colors
only (Saaty 1986).

see also Affine Plane, Bruck-Ryser-Chowla The-
orem, Fano Plane, Lam's Problem, Map Col-
oring, Moufang Plane, Projective Plane PK 2 ,
Real Projective Plane

References

Ball, W, W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 281-
287, 1987.

Lam, C. W. H. "The Search for a Finite Projective Plane of
Order 10." Amer. Math. Monthly 98, 305-318, 1991.

Lindner, C. C. and Rodger, C. A. Design Theory. Boca
Raton, FL: CRC Press, 1997.

Pinkall, U. "Models of the Real Projective Plane." Ch. 6 in
Mathematical Models from the Collections of Universities
and Museums (Ed. G. Fischer). Braunschweig, Germany:
Vieweg, pp. 63-67, 1986.

Saaty, T. L. and Kainen, P. C. The Four-Color Problem:
Assaults and Conquest. New York: Dover, p. 45, 1986.

Projective Plane PK 2

The 2-D Space consisting of the set of Triples

{(a, 6, c) : a, 6, c 6 K, not all zero},

where triples which are Scalar multiples of each other
are identified.

Projective Space

A Space which is invariant under the Group G of
all general Linear homogeneous transformation in the
Space concerned, but not under all the transformations
of any Group containing G as a SUBGROUP.

A projective space is the space of 1-D VECTOR SUB-
SPACES of a given VECTOR SPACE. For REAL VECTOR

Spaces, the Notation EP n or P n denotes the Real
projective space of dimension n (i.e., the SPACE of 1-
D Vector Subspaces of M n+1 ) and CP n denotes the
Complex projective space of Complex dimension n
(i.e., the space of 1-D Complex Vector Subspaces
of C n+1 ). P n can also be viewed as the set consisting of
W 1 together with its POINTS AT INFINITY.

Projective Special Linear Group

The projective special linear group PSL n (q) is the
Group obtained from the Special Linear Group
SL n (q) on factoring by the Scalar Matrices contained
in that Group. It is Simple for n > 2 except for

PSL 2 (2) = S 3
PSL 2 (3) = A 4 ,

and is therefore also denoted L n (Q).

see also Projective Special Orthogonal Group,
Projective Special Unitary Group, Special Lin-
ear Group

Projective Special Orthogonal Group

Prolate Cycloid 1453

References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.;
and Wilson, R. A. "The Groups GL n (q) 7 SL n (q), PGL n {q),
and PSL n (q) = 2/„(q)." §2.1 in Atlas of Finite Groups:
Maximal Subgroups and Ordinary Characters for Simple
Groups. Oxford, England: Clarendon Press, p. x, 1985.

Projective Special Orthogonal Group

The projective special orthogonal group PSO n (q) is
the GROUP obtained from the SPECIAL ORTHOGONAL
GROUP SO n {q) on factoring by the SCALAR MATRICES
contained in that GROUP. In general, this GROUP is not
Simple.

see also PROJECTIVE SPECIAL LINEAR GROUP, PRO-
JECTIVE Special Unitary Group, Special Orthog-
onal Group

References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker,
R. A.; and Wilson, R. A. "The Groups GO n {q), SO n (q),
PGO n (q), and PSO n (q), and O n (q)" §2.4 in Atlas of
Finite Groups: Maximal Subgroups and Ordinary Char-
acters for Simple Groups. Oxford, England: Clarendon
Press, pp. xi-xii, 1985.

Projective Special Unitary Group

The projective special unitary group PSU n (q) is the
GROUP obtained from the SPECIAL UNITARY GROUP
SU n (q) on factoring by the SCALAR MATRICES con-
tained in that Group. PSU n (q) is Simple except for

PSU 2 (2) = S z
PSU 2 (3) = A 4
PSUz(2) = 3 2 :Qs,

so it is given the simpler name U n (q), with U2(q) =
L2(q).

see also Projective Special Linear Group, Pro-
jective Special Orthogonal Group, Special Uni-
tary Group

References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker,
R A.; and Wilson, R. A. "The Groups GU n (q), SU n (q),
PGUn(q), and PSU n (q) = U n (q). n §2.2 in Atlas of Finite
Groups: Maximal Subgroups and Ordinary Characters for
Simple Groups. Oxford, England: Clarendon Press, p. x,
1985.

Projective Symplectic Group

The projective symplectic group PSp n (q) is the GROUP
obtained from the Symplectic Group Sp n (q) on fac-
toring by the SCALAR MATRICES contained in that
Group. PSp 2m (q) is Simple except for

PSp 2 (2) = S 3
PSp 2 (3) — A 4
PSp 4 (2) = S 6 ,

so it is given the simpler name 52m (<z), with S2{q) =
L 2 (q).

References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.;
and Wilson, R. A. "The Groups Sp n (q) and PSp n (q) =
S n (q)>" §2.3 in Atlas of Finite Groups: Maximal Sub-
groups and Ordinary Characters for Simple Groups. Ox-
ford, England: Clarendon Press, pp. x-xi, 1985.

Projectivity

The product of any number of PERSPECTIVITIES.

see also INVOLUTION (TRANSFORMATION), PERSPEC-
TIVITY

Prolate Cycloid

M

M

li

The path traced out by a fixed point at a RADIUS 6 > a,
where a is the RADIUS of a rolling CIRCLE, also some-
times called an Extended Cycloid. The prolate cy-
cloid contains loops, and has parametric equations

x — a(j> — b sin <j>

y = a — b cos <f).

The Arc Length from = is

s = 2(a + b)E(u),

where

sin(|</>) = snu

2 _ 4a6

see also CURTATE CYCLOID, CYCLOID

References

Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 46-50, 1991.

(i)

(2)

(3)

(4)
(5)

1454 Prolate Cycloid Evolute

Prolate Cycloid Evolute

X

\

/

/

The Evolute of the Prolate Cycloid is given by

_ a[— 260 + 2a0cos0 — 2a sin + &sin(20)]

2(acos0 — b)

y =

a(a — 6cos0) 2
b(a cos <j> — b)

Prolate Spheroid

A Spheroid which is "pointy" instead of "squashed,"
i.e., one for which the polar radius c is greater than the
equatorial radius a, so c > a. A prolate spheroid has
Cartesian equations

2,2 2

a 2 c 2 ~

(1)

The ELLIPTICITY of the prolate spheroid is defined by

^ (2)

a 2 \/c 2 — a 2

so that

Then

(i + T ^*?s)

-1/2

a 1 +

The Surface Area and Volume are

S = 27ra + 2tt — sin e
e

V - §7ra 2 c.

(3)
(4)

(5)
(6)

see a/so Darwin-de Sitter Spheroid, Ellipsoid,
Oblate Spheroid, Prolate Spheroidal Coordi-
nates, Sphere, Spheroid

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 131, 1987.

Prolate Spheroidal Coordinates
Prolate Spheroidal Coordinates

A system of CURVILINEAR COORDINATES in which two
sets of coordinate surfaces are obtained by revolving
the curves of the ELLIPTIC CYLINDRICAL COORDI-
NATES about the a:- Axis, which is relabeled the 2- Axis.
The third set of coordinates consists of planes passing
through this axis.

x = a sinh £ sin 77 cos <j)
y = a sinh £ sin 77 sin <j)
z = a cosh £ cos 77,

(1)
(2)
(3)

where £ e [0,oo), 77 6 [0,tt), and e [0,2tt). Arf-
ken (1970) uses (uyV^ip) instead of (£,77, z). The SCALE
Factors are

/i£ = ay sinh 2 £ + sin 2 77
^•77 = ay sinh 2 £ + sin 2 77
/i^ = a sinh £ sin 77.

(4)
(5)
(6)

The Laplacian is

v 2 /= x

a 2 (sinh 2 £ 4- sin 2 77)
1 d

sinh£

j 1 d ( .

a 2 (sin 77 + sinh £)

df
.sin 77 —

sin 77 or] \ arj

(esc 2 77 + csch 2 i)

d<t> 2

+ cot 77— - +

Of)

d 2 ^,d d 2

(7)

(8)

An alternate form useful for "two-center" problems is
defined by

£1 = cosh £
£2 = cos 77

6 = 0,

O)
(10)
(ii)

Prolate Spheroidal Wave Function

where £1 G [l,oo], 6 G [-1,1], and 6 € [0, 2tt)
(Abramowitz and Stegun 1972). In these coordinates,

z = a£i£ 2

« = aV(6 2 -l)(l"6 2 ) cos 6
y = av / (^i 2 -l)(l-6 2 ) sin^s.
In terms of the distances from the two FOCI,

6

2a

7*1 -7*2

2a

2a = ri2-
The Scale Factors are

h^ = a

'*'-

£ 2

6 2

-1

^ 2 -

> 2
- £2

1 -

t 2
S2

/i ?2 = a

^3 = a>/(*i a -l)(l-6 a ),
and the Laplacian is

(12)
(13)
(14)

(15)

(16)
(17)

(18)

(19)
(20)

V/=-j

a 2 Ui 2 -

<9

6 2 56

+ 6 2 - 6 2 56

+

(6 2 -i)f-

(1 " 6) 36
1

(d 2 - i)(i - 6 2 ) 0&

^}

(21)

The Helmholtz Differential Equation is separable
in prolate spheroidal coordinates.

see also HELMHOLTZ DIFFERENTIAL EQUATION —

Prolate Spheroidal Coordinates, Latitude, Lon-
gitude, Oblate Spheroidal Coordinates, Spheri-
cal Coordinates

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Definition
of Prolate Spheroidal Coordinates." §21.2 in Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 752, 1972.

Arfken, G. "Prolate Spheroidal Coordinates {u, v } </>)» §2.10
in Mathematical Methods for Physicists, 2nd ed. Orlando,
FL: Academic Press, pp. 103-107, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 661, 1953.

Prolate Spheroidal Wave Function

The Wave Equation in Prolate Spheroidal Coor-
dinates is

V^ + A: $

d

(6 2 - 1)

d§

d

(1 -^

c 2 c 2

(6 2 -i)(i-*2 2 )

+ c 2 « 1 2 ~6 2 )* = 0, (1)

Prolate Spheroidal Wave Function 1455

where

c = lafc.

(2)

Substitute in a trial solution

k cos.

$ - Rmn(c,tl)Sm n (c,Z 2 ) sin {m<t>) (3)

d

(& -l)-JTRrnn(c^l)

'dfc

2 & 2 , ™

■ Sh' + , 2

The radial differential equation is

Rmn{c,Zl) = Q. (4)

d&

(6 3 -l)^-Smn(c,6)

a^2

- I A mn — c £2 +

6 2 -i

and the angular differential equation is

i2mn(c,f 2 )=0, (5)

#2

(1 "6 2 )^^ 5 'mn(c,C2)
a?2

A mn — C ^2 +

1-6-

i2mn(c,6)=0- (6)

Note that these are identical (except for a sign change).
The prolate angular function of the first kind is given

by

an 1

EZi3... d r(c)P^ +r (v) forn-modd
E" 0,2,... 4-(c)P^+r(»j) for n - m even,

(7)

where P^iv) is an associated Legendre Polynomial.
The prolate angular function of the second kind is given

by

( £ d T {c)QZ+Ari) forn-modd

c(2) _ J r=...,-l,l,3,...

mn I £ dr(c)Q™ +r (»y) for n-m even,

I, r=...,-2,0,2,...

(8)
where Q™(r}) is an associated LEGENDRE Function OF
the Second Kind and the Coefficients d r satisfy
the Recurrence Relation

OLkdk+2 + (/3k - ^mn)d k + Jkdk-2 = 0, (9)

with

_ (2m + k + 2)(2m + fc + l)c 2
Qfc " (2m + 2fc + 3)(2m + 2fc + 5)
£ fc = (m + k)(m + k + l)

2(m + fc)(m + k + 1) - 2m 2 - 1 2

7fc

(2m + 2fc - l)(2m + 2A; + 3)

k(k - l)c 2

(2m + 2k - 3) (2m + 2fc - 1) "

(10)

C a (11)
(12)

1456

Pronic Number

Proper Cover

Various normalization schemes are used for the ds
(Abramowitz and Stegun 1972, p. 758). Meixner and
Schafke (1954) use

/

js mn (c,n)] dv= 2n + 1(n _ m y.

(13)

Stratton et al. (1956) use

(n + m)\ _ \ Yj7=i 3 (r+ r 2 ! " l)! ^r for n - m odd
W^V- ~ \ Er=oV ir± ^ i dr fom-meven.

(14)
Flammer (1957) uses

dmn\C

{prn+l

(0) for n — m odd
(0) for n — m even.

(15)

see also Oblate Spheroidal Wave Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Spheroidal Wave
Functions." Ch. 21 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 751-759, 1972.

Flammer, C. Spheroidal Wave Functions. Stanford, CA:
Stanford University Press, 1957.

Meixner, J. and Schafke, F. W. Mathieusche Funktionen und
Sphdroidfunktionen. Berlin: Springer- Verlag, 1954.

Stratton, J. A.; Morse, P. M.; Chu, L. J.; Little, J. D. C;
and Corbato, F. J. Spheroidal Wave Functions. New York:
Wiley, 1956.

Pronic Number

A Figurate Number of the form P n = 2T n = n(n + l),
where T n is the nth TRIANGULAR NUMBER. The first
few are 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, . . . (Sloane's
A002378). The GENERATING FUNCTION of the pronic
numbers is

2X = 2x + 6x 2 + 12z 3 + 20z 4 + . .

Proof

A rigorous mathematical argument which unequivocally
demonstrates the truth of a given PROPOSITION. A
mathematical statement which has been proven is called
a Theorem.

There is some debate among mathematicians as to just
what constitutes a proof. The FOUR-COLOR THEOREM
is an example of this debate, since its "proof" relies on
an exhaustive computer testing of many individual cases
which cannot be verified "by hand." While many mathe-
maticians regard computer-assisted proofs as valid, some
purists do not.

see also Paradox, Proposition, Theorem

References

Gamier, R. and Taylor, J. 100% Mathematical Proof. New

York: Wiley, 1996.
Solow, D. How to Read and Do Proofs: An Introduction to

Mathematical Thought Process. New York: Wiley, 1982.

Proofreading Mistakes

If proofreader A finds a mistakes and proofreader B
finds b mistakes, c of which were also found by A, how
many mistakes were missed by both A and Bl Assume
there are a total of m mistakes, so proofreader A finds a
Fraction a/m of all mistakes, and also a Fraction c/b
of the mistakes found by B. Assuming these fractions
are the same, then solving for m gives

m

ab

c

The number of mistakes missed by both is therefore ap-
proximately

N = m — a — b -\- c ~

(a — c)(b — c)

(l-x)

References

Polya, G. "Probabilities in Proofreading."
Monthly, 83, 42, 1976.

^4mer. Math.

The first few n for which P n are PALINDROMIC are 1, 2,
16, 77, 538, 1621, . . . (Sloane's A028336), and the first
few Palindromic NUMBERS which are pronic are 2, 6,
272, 6006, 289982, ... (Sloane's A028337).

References

De Geest, P. "Palindromic Products of Two Consecutive In-
tegers." http: //www. ping.be/-ping6758/consec .htm.

Sloane, N. J. A. Sequences A028336, A028337, and A002378/
M1581 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."

Propeller

A 4-POLYHEX.

References

Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, p. 147, 1978.

Proper Cover

see COVER

Proper Divisor

Pseudocrosscap 1457

Proper Divisor

A Divisor of a number n excluding n itself.

see also ALIQUANT DIVISOR, ALIQUOT DIVISOR, DIVI-
SOR

Proper Fraction

A Fraction p/q < 1.

see also FRACTION, REDUCED FRACTION

Proper Integral

An INTEGRAL which has neither limit INFINITE and from
which the Integrand does not approach INFINITY at
any point in the range of integration.
see also IMPROPER INTEGRAL, INTEGRAL

Proper fc-CoIoring

see fc-COLORING

Proper Subset

A Subset which is not the entire Set. For example,
consider a Set {1, 2, 3, 4, 5}. Then {1, 2, 4} and {1}
are proper subsets, while {1, 2, 6} and {1, 2, 3, 4, 5}
are not.

see also SET, SUBSET

Proper Superset

A Superset which is not the entire Set.

see also Set, Superset

Proportional

If a is proportional to b y then a/b is a constant. The
relationship is written a oc 6, which implies

Proth's Theorem

For JV = h • 2 n + 1 with Odd h and 2 n > h, if there
exists an Integer a such that

a = cb,

for some constant c.

Proposition

A statement which is to be proved.

Propositional Calculus

The formal basis of LOGIC dealing with the notion and
usage of words such as "Not," "Or," "And," and "Im-
plies." Many systems of propositional calculus have
been devised which attempt to achieve consistency, com-
pleteness, and independence of Axioms.

see also LOGIC, P-Symbol

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.

Stradbroke, England: Tarquin Pub., pp. 254-255, 1989.
Nidditch, P. H. Propositional Calculus. New York: Free

Press of Glencoe, 1962.

Prosthaphaeresis Formulas

Trigonometry formulas which convert a product of
functions into a sum or difference.

,(N-l)/2 _

-1 (rnodiV),

then N is Prime.

Protractor

A ruled Semicircle used for measuring and drawing
Angles.

Prouhet's Problem

A generalization of the Tarry-Escott Problem to
three or more sets of INTEGERS.
see also TARRY-ESCOTT PROBLEM

References

Wright, E. M. "Prouhet's 1851 Solution of the Tarry-Escott
Problem of 1910." Amer. Math. Monthly 102, 199-210,
1959.

Priifer Ring

A metric space Z in which the closure of a congruence
class B(j, m) is the corresponding congruence class {x e
%\x = j (mod m)}.

References

Fried, M. D. and Jarden, M. Field Arithmetic. New York:

Springer- Verlag, pp. 7-11, 1986.
Postnikov, A. G. Introduction to Analytic Number Theory.

Providence, RI: Amer. Math. Soc, 1988.

Prussian Hat

A device used in the Cornwell smoothness stabilized
modification of the CLEAN Algorithm.
see also CLEAN Algorithm

Pseudoanalytic Function

A pseudoanalytic function is a function defined using
generalized CAUCHY-RlEMANN EQUATIONS. Pseudo-
analytic functions come as close as possible to having
Complex derivatives and are nonsingular a quasiregu-

lar" functions.

see also Analytic Function, Semianalytic, Suban-

ALYTIC

Pseudocrosscap

1458 Pseudocylindrical Projection

Pseudorandom Number

A surface constructed by placing a family of figure-eight
curves into M 3 such that the first and last curves reduce
to points. The surface has parametric equations

x(u, v) = (1 — u )sini;
y(uj v) = (1 - u 2 ) sin(2v)
z(u,v) = u.

References

Gray, A. Modem Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 247-248, 1993.

Pseudocylindrical Projection

A projection in which latitude lines are parallel but
meridians are curves.

see also CYLINDRICAL PROJECTION, ECKERT IV PRO-
JECTION, Eckert VI Projection, Mollweide Pro-
jection, Robinson Projection, Sinusoidal Pro-
jection

References

Dana, P. H. "Map Projections." http://www.utexas.edu/
depts/grg/gcraft/notes/mapproj/mapproj .html.

Pseudogroup

An algebraic structure whose elements consist of se-
lected HOMEOMORPHISMS between open subsets of a
SPACE, with the composition of two transformations de-
fined on the largest possible domain. The "germs" of the
elements of a pseudogroup form a GROUPOID (Weinstein
1996).
see also GROUP, GROUPOID, INVERSE SEMIGROUP

References

Weinstein, A. "Groupoids: Unifying Internal and External
Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996.

Pseudolemniscate Case

The case of the WeierstraB Elliptic Function with

invariants 52 = — 1 and g s = 0.

see also Equianharmonic Case, Lemniscate Case

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Pseudo-
Lemniscate Case (g 2 — — 1, £3 = 0)-" §18.15 in Hand-
book of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, 9th printing. New York: Dover,
pp. 662-663, 1972.

Pseudoperfect Number

see Semiperfect Number

Pseudoprime

A pseudoprime is a COMPOSITE number which passes a
test or sequence of tests which fail for most COMPOSITE
numbers. Unfortunately, some authors drop the "COM-
POSITE" requirement, calling any number which passes
the specified tests a pseudoprime even if it is PRIME.
Pomerance, Selfridge, and WagstafF (1980) restrict their
use of "pseudoprime" to Odd COMPOSITE numbers.
"Pseudoprime" used without qualification means FER-
MAT PSEUDOPRIME.

Carmichael Numbers are Odd Composite numbers
which are pseudoprimes to every base; they are some-
times called Absolute Pseudoprimes. The follow-
ing table gives the number of FERMAT PSEUDOPRIMES
psp, Euler Pseudoprimes epsp, and Strong Pseu-
doprimes spsp to the base 2, as well as Carmichael
Numbers CN which are less the first few powers of 10
(Guy 1994).

10 3

10 4

10 5

10 6

10 7

10 8

10 9

10 10

psp(2)

3

22

78

245

750

2057

5597

14884

epsp(2)

1

12

36

114

375

1071

2939

7706

spsp(2)

5

16

46

162

488

1282

3291

CN

1

7

16

43

105

255

646

1547

see also CARMICHAEL NUMBER, ELLIPTIC PSEUDO-
PRIME, Euler Pseudoprime, Euler-Jacobi Pseu-
doprime, Extra Strong Lucas Pseudoprime,
Fermat Pseudoprime, Fibonacci Pseudoprime,
Frobenius Pseudoprime, Lucas Pseudoprime,
Perrin Pseudoprime, Probable Prime, Somer-
Lucas Pseudoprime, Strong Elliptic Pseudo-
prime, Strong Frobenius Pseudoprime, Strong
Lucas Pseudoprime, Strong Pseudoprime

References

Grantham, J. "Frobenius Pseudoprimes." http://wvv,

clark.net/pub/granthsun/pseudo/pseudo.ps
Grantham, J. "Pseudoprimes /Probable Primes." http://

www . dark . net/pub/grantham/pseudo.
Guy, R. K. "Pseudoprimes. Euler Pseudoprimes. Strong

Pseudoprimes." §A12 in Unsolved Problems in Number

Theory, 2nd ed. New York: Springer- Verlag, pp. 27-30,

1994.
Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. "The

Pseudoprimes to 25 -10 9 ." Math. Comput. 35, 1003-1026,

1980. Available electronically from ftp://sable.ox.ac.

uk/pub/math/primes/ps2 . Z.

Pseudorandom Number

A slightly archaic term for a computer-generated RAN-
DOM Number. The prefix pseudo- is used to distinguish
this type of number from a "truly" RANDOM NUMBER
generated by a random physical process such as radioac-
tive decay.
see also RANDOM NUMBER

References

Luby, M. Pseudorandomness and Cryptographic Applica-
tions. Princeton, NJ: Princeton University Press, 1996.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of

Pseudorhombicuboctahedron

Pseudovector 1459

Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 266, 1992.

Pseudorhombicuboctahedron

see Elongated Square Gyrobicupola

Pseudoscalar

A Scalar which reverses sign under inversion is called
a pseudoscalar. The SCALAR TRIPLE PRODUCT

A ■ (B x C)

is a pseudoscalar. Given a transformation Matrix A,

S' = det |A|S,

where det is the Determinant.

see also Pseudotensor, Pseudovector, Scalar

References

Arfken, G. "Pseudotensors, Dual Tensors." §3.4 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 128-137, 1985.

Pseudosmarandache Function

The pseudosmarandache function Z(n) is the smallest
integer such that

Z{n)

£>=|Z(n)[Z(n) + l]
fc=i

is divisible by n. The values for n = 1, 2, . . . are 1, 3,
2, 7, 4, 3, 6, 15, 8, 4, . . . (Sloane's A011772).

see also Smarandache Function

References

Ashbacher, C. "Problem 514." Pentagon 57, 36, 1997.

Kashihara, K. "Comments and Topics on Smarandache No-
tions and Problems." Vail: Erhus University Press, 1996.

Sloane, N. J. A. Sequence A011772 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Pseudosphere

Half the Surface of Revolution generated by a
Tractrix about its Asymptote to form a Trac-
TROID. The Cartesian parametric equations are

for u > 0.

It has constant NEGATIVE CURVATURE, and so is called
a pseudosphere by analogy with the Sphere, which has
constant POSITIVE curvature. An equation for the Geo-

DESICS is

cosh u + (v + c) = k . (4)

see also Funnel, Gabriel's Horn, Tractrix

References

Fischer, G. (Ed.). Plate 82 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 77, 1986.

Geometry Center. "The Pseudosphere." http://www.geom.
umn.edu/zoo/diffgeom/pseudosphere/.

Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 383-384, 1993.

Pseudosquare

Given an Odd Prime p> a Square Number n satisfies
(n/p) = or 1 for all p < n, where (n/p) is the LEG-
endre Symbol. A number n > 2 which satisfies this
relationship but is not a SQUARE NUMBER is called a
pseudosquare. The only pseudoprimes less than 10 8 are
3 and 6.

see also SQUARE Number

Pseudotensor

A TENSOR-like object which reverses sign under inver-
sion. Given a transformation Matrix A,

Aij = det | A\aikajiAki,

where det is the Determinant. A pseudotensor is
sometimes also called a TENSOR DENSITY.

see also Pseudoscalar, Pseudovector, Scalar,
Tensor Density

References

Arfken, G. "Pseudotensors, Dual Tensors." §3.4 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 128-137, 1985.

Pseudovector

A typical VECTOR is transformed to its NEGATIVE un-
der inversion. A VECTOR which is invariant under in-
version is called a pseudovector, also called an AXIAL
VECTOR in older literature (Morse and Feshbach 1953).
The Cross Product

A xB

(1)

x = sech u cos v
y — sech u sin v
z — u — tanh u

(i)

(2)
(3)

is a pseudovector, whereas the Vector Triple Prod-
uct

A x (B x C) (2)

is a Vector.

[pseudovector] x [pseudovector] = [pseudovector] (3)

1460

Psi Function

Public-Key Cryptography

[vector] x [pseudovector] = [vector]. (4)

Given a transformation MATRIX A,

Ci =det\A\a ij C j . (5)

see also PSEUDOSCALAR, TENSOR, VECTOR

References

Arfken, G. "Pseudotensors, Dual Tensors." §3.4 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 128-137, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 46-47, 1953.

Psi Function

*

(z, SjV) = > 7

for \z\ < 1 and v ^ 0,-1, ... (Gradshteyn and Ryzhik
1980, pp. 1075-1076).

see also Hurwitz Zeta Function, Ramanujan Psi
Sum, Theta Function

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.

PSLQ Algorithm

An Algorithm which finds Integer Relations be-
tween real numbers xi, . . . , x n such that

a±xi 4- a 2 x 2 + . . . + a n x n = 0,

with not all a» = 0. This algorithm terminates after
a number of iterations bounded by a polynomial in n
and uses a numerically stable matrix reduction proce-
dure (Ferguson and Bailey 1992), thus improving upon
the Ferguson-Forcade Algorithm. It is based on
a partial sum of squares scheme (like the PSOS ALGO-
RITHM) implemented using LQ decomposition. A much
simplified version of the algorithm was developed by Fer-
guson et al. and extended to complex numbers.

see also Ferguson-Forcade Algorithm, Integer
Relation, LLL Algorithm, PSOS Algorithm

References

Bailey, D. H.; Borwein, J. M.; and Girgensohn, R. "Experi-
mental Evaluation of Euler Sums." Exper. Math. 3, 17-30,
1994.

Bailey, D. and Plouffe, S. "Recognizing Numerical
Constants." http://www.cecm.sfu.ca/organics/papers/
bailey.

Ferguson, H. R. P. and Bailey, D. H. "A Polynomial Time,
Numerically Stable Integer Relation Algorithm." RNR
Techn. Rept. RNR-91-032, Jul. 14, 1992.

Ferguson, H. R. P.; Bailey, D. H.; and Arno, S. "Analysis of
PSLQ, An Integer Relation Finding Algorithm." Unpub-
lished manuscript.

PSOS Algorithm

An Integer-Relation algorithm which is based on a
partial sum of squares approach, from which the algo-
rithm takes its name.

see also FERGUSON-FORCADE ALGORITHM, HJLS AL-
GORITHM, Integer Relation, LLL Algorithm,
PSLQ Algorithm

References

Bailey, D. H. and Ferguson, H. R. P. "Numerical Results
on Relations Between Numerical Constants Using a New
Algorithm." Math. Comput. 53, 649-656, 1989.

Ptolemy Inequality

For a Quadrilateral which is not Cyclic, Ptol-
emy's Theorem becomes an Inequality:

ABxCD + BC xDA> ACx BD.
see also Ptolemy's Theorem, Quadrilateral
Ptolemy's Theorem

If a Quadrilateral is inscribed in a circle (i.e., for
a cyclic quadrilateral), the sum of the products of the
two pairs of opposite sides equals the product of the
diagonals

ABxCD + BC xDA = AC x BD.

This fact can be used to derive the TRIGONOMETRY ad-
dition formulas.

see also Fuhrmann's Theorem, Ptolemy Inequal-
ity

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 42-43, 1967.

Public-Key Cryptography

A type of Cryptography in which the encoding key
is revealed without compromising the encoded message.
The two best-known methods are the Knapsack PROB-
LEM and RSA Encryption.
see also Knapsack Problem, RSA Encryption

References

Dime, W. and Hellman, M. "New Directions in Cryptogra-
phy." IEEE Trans. Info. Th. 22, 644-654, 1976.

Hellman, M. E. "The Mathematics of Public-Key Cryptog-
raphy." Sci. Amer. 241, 130-139, Aug. 1979.

Rivest, R.; Shamir, A.; and Adleman, L. "A Method for
Obtaining Digital Signatures and Public-Key Cryptosys-
tems." MIT Memo MIT/LCS/TM-82, 1982.

Wagon, S. "Public-Key Encryption." §1.2 in Mathematica in
Action. New York: W. H. Freeman, pp. 20-22, 1991.

Puiseaux's Theorem

Puzzle 1461

Puiseaux's Theorem

The whole neighborhood of any point yi of an alge-
braic PLANE CURVE may be uniformly represented by
a certain finite number of convergent developments in
Power Series,

2

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 207, 1959.

Pullback Map

A pullback is a general CATEGORICAL operation appear-
ing in a number of mathematical contexts, sometimes
going under a different name. If T : V — > W is a
linear transformation between VECTOR SPACES, then
T* : W* -> V* (usually called TRANSPOSE MAP or
DUAL Map because its associated matrix is the MATRIX
Transpose of T) is an example of a pullback map.

In the case of a Diffeomorphism and Differentiable
MANIFOLD, a very explicit definition can be formu-
lated. Given an r-form a on a Manifold M2, de-
fine the r-form T* (a) on Mi by its action on an r-
tuple of tangent vectors (Xi,...,X r ) as the number
T*(a)(Xi,...,X r ) = a(T+X u ...,T*X r ). This defines
a map on r-forms and is the pullback map.

see also CATEGORY

Pulse Function

see Rectangle Function

Purser's Theorem

Pursuit Curve

Let t, u, and v be the lengths of the tangents to a CIRCLE
C from the vertices of a TRIANGLE with sides of lengths
a, 6, and c. Then the condition that C is tangent to the
ClRCUMCIRCLE of the TRIANGLE is that

±at ± bu ± cv — 0.

The theorem was discovered by Casey prior to Purser's
independent discovery.

see also Casey's Theorem, Circumcircle

If A moves along a known curve, then P describes a pur-
suit curve if P is always directed toward A and A and P
move with uniform velocities. These were considered in
general by the French scientist Pierre Bouguer in 1732.
The case restricting A to a straight line was studied by
Arthur Bernhart (MacTutor Archive). It has CARTE-
SIAN Coordinates equation

y = ex — In x.

see also APOLLONIUS PURSUIT PROBLEM, MICE PROB-
LEM

References

Bernhart, A. "Curves of Pursuit." Scripta Math. 20, 125-

141, 1954.
Bernhart, A. "Curves of Pursuit-II." Scripta Math. 23, 49-

65, 1957.
Bernhart, A. "Polygons of Pursuit." Scripta Math. 24, 23-

50, 1959.
Bernhart, A. "Curves of General Pursuit." Scripta Math.

24, 189-206, 1959.
MacTutor History of Mathematics Archive. "Pursuit Curve."

http: //www-groups .dcs. st-and.ac.uk/-history/Curves

/Pursuit .html.
Yates, R. C. "Pursuit Curve." A Handbook on Curves and

Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 170-

171, 1952.

Push

An action which adds a single element to the top of a
Stack, turning the STACK (ai, a2, - - - , a n ) into (ao, ai,

02, • ■ ■ , O n ).

see also POKE MOVE, POP, STACK

Puzzle

A mathematical Problem, usually not requiring ad-
vanced mathematics, to which a solution is desired.
Puzzles frequently require the rearrangement of exist-
ing pieces (e.g., 15 Puzzle) or the filling in of blanks
(e.g., crossword puzzle).

see also 15 Puzzle, Baguenaudier, Caliban Puzzle,
Conway Puzzle, Cryptarithmetic, Dissection
Puzzles, Icosian Game, Pythagorean Square
Puzzle, Rubik's Cube, Slothouber-Graatsma
Puzzle, T-Puzzle

References

Bogomolny, A. "Interactive Mathematics Miscellany and
Puzzles." http://www.cut-the-knot.com/.

1462 Pyramid

Pyramidal Number

Dudeney, H. E. Amusements in Mathematics. New York:

Dover, 1917.
Dudeney, H. E. The Canterbury Puzzles and Other Curious

Problems, 7th ed. London: Thomas Nelson and Sons, 1949.
Dudeney, H. E. 536 Puzzles & Curious Problems. New York:

Scribner, 1967.
Fujii, J. N. Puzzles and Graphs. Washington, DC: National

Council of Teachers, 1966.

Pyramid

A Polyhedron with one face a Polygon and all the
other faces TRIANGLES with a common VERTEX. An n-
gonal regular pyramid (denoted Y n ) has EQUILATERAL
Triangles, and is possible only for n = 3, 4, 5. These
correspond to the TETRAHEDRON, SQUARE PYRAMID,
and PENTAGONAL Pyramid, respectively. A pyramid
therefore has a single cross-sectional shape in which the
length scale of the CROSS- SECTION scales linearly with
height. The AREA at a height z is given by

Pyramidal Frustum

A(z) = A b

(«

(i)

Let s be the slant height, pi the top and bottom base
PERIMETERS, and At the top and bottom AREAS. Then
the Surface Area and Volume of the pyramidal frus-
tum are given by

S=±{P1+ P 2)S

V =\h{A l +A 2 + %fMA 2 ).
see also CONICAL FRUSTUM, FRUSTUM, PYRAMID,

Spherical Segment, Truncated Square Pyramid

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,

28th ed, Boca Raton, FL: CRC Press, p. 128, 1987.
Dunham, W. Journey Through Genius: The Great Theorems

of Mathematics. New York: Wiley, pp. 3-4, 1990.

where Ab is the base AREA and h is the pyramid height.
The VOLUME is therefore given by

ph ph

V= A{z) dz = A b
Jo Jo

!Ldz =£{&*):

\A b h.

(2)

These results also hold for the CONE, TETRAHEDRON
(triangular pyramid), SQUARE PYRAMID, etc.

The CENTROID is the same as for the CONE, given by

z=\h.

(3)

The Surface Area of a pyramid is

Pyramidal Number

A FlGURATE Number corresponding to a configuration
of points which form a pyramid with r-sided REGULAR
POLYGON bases can be thought of as a generalized pyra-
midal number, and has the form

P;=i(n+l)(2 P ;+n) = in(n + l)[(r-2)n+(5-r)].

(1)
The first few cases are therefore

P n 3 = §n(n + l)(n + 2) (2)

j£ = |n(n + l)(2n + l) (3)

P n 5 = in 2 (n + 1), (4)

\ps,

(4)

where s is the SLANT HEIGHT and p is the base PERI-
METER. Joining two PYRAMIDS together at their bases
gives a BlPYRAMiD, also called a Dipyramid.

see also Bipyramid, Elongated Pyramid, Gyro-

ELONGATED PYRAMID, PENTAGONAL PYRAMID, PYRA-
MID, Pyramidal Frustum, Square Pyramid, Tet-
rahedron, Truncated Square Pyramid

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,

28th ed. Boca Raton, FL: CRC Press, p. 128, 1987.
Hart, G. W. "Pyramids, Dipyramids, and Trapezohe-

dra." http://www.li.net/-george/virtual-polyhedra/

pyramids-info.html.

so r = 3 corresponds to a Tetrahedral Number Te ny
and r = 4 to a Square Pyramidal Number P n .

The pyramidal numbers can also be generalized to 4-D
and higher dimensions (Sloane and Plouffe 1995).

see also HEPTAGONAL PYRAMIDAL NUMBER, HEXAGO-
NAL Pyramidal Number, Pentagonal Pyramidal
Number, Square Pyramidal Number, Tetrahe-
dral Number

References

Conway, J. H. and Guy, R. K. "Tetrahedral Numbers" and
"Square Pyramidal Numbers" The Book of Numbers. New
York: Springer-Verlag, pp. 44-49, 1996.

Sloane, N. J. A. and Plouffe, S. "Pyramidal Numbers." Ex-
tended entry for sequence M3382 in The Encyclopedia of
Integer Sequences. San Diego, CA: Academic Press, 1995.

Pyritohedron
Pyritohedron

An irregular DODECAHEDRON composed of identical ir-
regular Pentagons.

see also Dodecahedron, Rhombic Dodecahedron,
Trigonal Dodecahedron

References

Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 63, 1990.

Pythagoras's Constant

The number

y/2 = 1.4142135623...,

which the Pythagoreans proved to be IRRATIONAL. The
Babylonians gave the impressive approximation

r- „ 24 51 10

1.41421296296296...

(Guy 1990, Conway and Guy 1996, pp. 181-182).

see also Irrational Number, Octagon, Pythago-
ras's Theorem, Square

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New

York: Springer- Verlag, p. 25 and 181-182, 1996.
Guy, R. K. "Review: The Mathematics of Plato's Academy."

Amer. Math. Monthly 97, 440-443, 1990.
Shanks, D. Solved and Unsolved Problems in Number Theory,

4th ed. New York: Chelsea, p. 126, 1993.

Pythagoras's Theorem

Proves that the DIAGONAL d of a SQUARE with sides of
integral length s cannot be RATIONAL. Assume d/s is
rational and equal to p/q where p and q are INTEGERS
with no common factors. Then

s 2 +s 2

2s\

SO

©■■(J)'-*

and p 2 = 2q 2 , so p 2 is even. But if p 2 is Even, then p
is Even. Since p/q is denned to be expressed in lowest
terms, q must be Odd; otherwise p and q would have the
common factor 2. Since p is EVEN, we can let p = 2r,
then 4r 2 = 2q 2 . Therefore, q 2 = 2r 2 , and q 2 , so g must
be Even. But q cannot be both Even and Odd, so
there are no d and s such that d/s is RATIONAL, and
d/s must be Irrational.

Pythagorean Fraction 1463

In particular, PYTHAGORAS'S CONSTANT y/2 is IRRA-
TIONAL. Conway and Guy (1996) give a proof of this
fact using paper folding, as well as similar proofs for <j>
(the Golden Ratio) and \/3 using a Pentagon and
Hexagon.

see also Irrational Number, Pythagoras's Con-
stant, Pythagorean Theorem

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New

York: Springer- Verlag, pp. 183-186, 1996.
Pappas, T. "Irrational Numbers & the Pythagoras Theorem."

The Joy of Mathematics. San Carlos, CA: Wide World

Publ./Tetra, pp. 98-99, 1989.

Pythagoras Tree

A Fractal with symmetric

and asymmetric

forms.

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 67-77
and 111-113, 1991.
# Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-evw6n/math/notebooks/Fractal.m.

Pythagorean Fraction

Given a PYTHAGOREAN TRIPLE (a, 6,c), the fractions
a/b and b/a are called Pythagorean fractions. Diophan-
tus showed that the Pythagorean fractions consist pre-
cisely of fractions of the form (p 2 — q 2 )/(2pq).

References

Conway, J. H. and Guy, R. K. "Pythagorean Fractions."

In The Book of Numbers. New York: Springer- Verlag,

pp. 171-173, 1996.

1464 Pythagorean Quadruple

Pythagorean Theorem

Pythagorean Quadruple

Positive Integers a, 6, c, and d which satisfy

a 2 + b 2 +c 2 = d 2 .

(1)

For Positive Even a and 6, there exist such Integers
c and d\ for Positive Odd a and 6, no such Integers
exist (Oliverio 1996). Oliverio (1996) gives the following
generalization of this result. Let S = (ai, . . . ,a n -2),
where at are INTEGERS, and let T be the number of
Odd Integers in S. Then Iff T ^ 2 (mod 4), there
exist Integers a n _i and a n such that

2 2 2

&1 + «2 + . ■ ■ + Cln-1 = Q>n

A set of Pythagorean quadruples is given by

(2)

a = 2mp

(3)

b — 2np

(4)

c = p 2 — (m 2 + n 2 )

(5)

<Z = p 2 + (m 2 +n 2 ),

(6)

where m,

n, and p are INTEGERS,

m-\- n + p = 1 (mod 2) ,

(7)

and

{m,n,p) - 1

(8)

(Mordell 1969). This does not, however, generate all so-
lutions. For instance, it excludes (36, 8, 3, 37). Another
set of solutions can be obtained from

a = 2mp + 2nq
b = 2np — 2mq

2,2 / 2 . 2\

c = p + q — [rn +n )
d~p + q + (m +n)

(9)
(10)

(11)
(12)

(Carmichael 1915).

see also Euler Brick, Pythagorean Triple

References

Carmichael, R. D. Diophantine Analysis. New York: Wiley,

1915.
Mordell, L. J. Diophantine Equations. London: Academic

Press, 1969.
Oliverio, P. "Self- Generating Pythagorean Quadruples and

JV-tuples." Fib. Quart. 34, 98-101, 1996.

Pythagorean Square Puzzle

Combine the two above squares on the left into the single

large square on the right.

see also Dissection, T-Puzzle

Pythagorean Theorem

For a Right Triangle with legs a and b and Hy-
potenuse c,

a)

2 , i2

a + 6

Many different proofs exist for this most fundamental of
all geometric theorems.

A clever proof by DISSECTION which reassembles two
small squares into one larger one was given by the Ara-
bian mathematician Thabit Ibn Qurra (Ogilvy 1994,
Frederickson 1997).

Another proof by DISSECTION is due to Perigal (Pergial
1873, Dudeney 1970, Madachy 1979, Ball and Coxeter

1987).

The Indian mathematician Bhaskara constructed a
proof using the following figure.

b

b-a

c /

a

c*

a b-a a

Several similar proofs are shown below.

a b b

c 2 +4(|a6) = (a + 6) 2

(2)

Pythagorean Theorem

Pythagorean Theorem 1465

c 2 + 2ob = a 2 + lab + & 2

2 2 , i2

c = a -J- o .

(3)
(4)

Similarly,

In the above figure, the Area of the large Square is
four times the AREA of one of the TRIANGLES plus the
Area of the interior Square. From the figure, d = b—a,

A = 4(f aft) + d 2 = 2ab + (b - a) 2 = 2a6 + 6 2 - 2a6 + a 2

2 . ,2 2

= a +o = c .

(5)

Perhaps the most famous proof of all times is Euclid's
geometric proof. Euclid's proof used the figure below,
which is sometimes known variously as the BRIDE'S
Chair, Peacock's Tail, or Windmill.

H

b

D L E

Let AABC be a Right Triangle, HCAFG,
OCBKH, and HABED be squares, and CL\\BD. The
Triangles AFAB and ACAD are equivalent except
for rotation, so

2AFAB = 2&CAD.

(6)

Shearing these TRIANGLES gives two more equivalent
Triangles

2ACAD = C3ADLM. (7)

Therefore,

HACGF = UDADLM.

(8)

UBC = 2AABK = 2ABCE = OBL (9)

a 2 + 6 2 = ex + c(c — x) = c .

(10)

Heron proved that AK, CL, and BF intersect in a point
(Dunham 1990, pp. 48-53).

Heron's Formula for the Area of the Triangle, con-
tains the Pythagorean theorem implicitly. Using the
form

K = \y/2a 2 b 2 -f- 2a 2 c 2 + ab 2 c 2 - (a 4 + b 4 + c 4 ) (11)

and equating to the Area

K = \ab (12)

gives

\a 2 b 2 = 2a 2 b 2 + 2aV + a&V - (a 4 + 6 4 + c 4 ). (13)

Rearranging and simplifying gives

(14)

2 , .2 2

a +o = c ,

the Pythagorean theorem, where K is the AREA of
a Triangle with sides a, b, and c (Dunham 1990,
pp. 128-129).

A novel proof using a TRAPEZOID was discovered by
James Garfield (1876), later president of the United
States, while serving in the House of Representatives
(Pappas 1989, pp. 200-201; Bogomolny).

^trapezoid = \ ^J [bases] • [altitude]
= |(a + 6)(a + 6)

- \ab+\ab+ \c 2 .

(15)

Rearranging,

|(a 2 + 2a6 + 6 2 ) = a6+|c 2

(16)

a + 2ab + b 2 = 2ab 4- <?

(17)

2 . ,2 2

a +o = c .

(18)

1466 Pythagorean Theorem

Pythagorean Triple

An algebraic proof (which would not have been accepted
by the Greeks) uses the Euler Formula. Let the sides
of a TRIANGLE be a, 6, and c, and the PERPENDICULAR
legs of RIGHT TRIANGLE be aligned along the real and
imaginary axes. Then

a + bi — ce .
Taking the Complex Conjugate gives

a — bi = ce
Multiplying (19) by (20) gives

2 . , 2 2

a + b = c .

Another algebraic proof proceeds by similarity.

x

(19)

(20)

(21)

b yd

It is a property of RIGHT TRIANGLES, such as the one
shown in the above left figure, that the Right Trian-
gle with sides #, a, and d (small triangle in the left
figure; reproduced in the right figure) is similar to the
Right Triangle with sides d, 6, and y (large trian-
gle in the left figure; reproduced in the middle figure),
giving

(22)

(23)

(24)
(25)

c = x + y —

y
b

_ b
c

y =

b 2

c

c c

a 2

+ b 2
c

2 , i2

- a +b

Because this proof depends on proportions of poten-
tially Irrational Numbers and cannot be translated
directly into a GEOMETRIC CONSTRUCTION, it was not
considered valid by Euclid.

see also Bride's Chair, Cosines Law, Peacock's
Tail, Pythagoras's Theorem, Windmill

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 87-88,

1987.
Bogomolny, A. "Pythagorean Theorem." http://www.cut-

t he -knot . com/pythagor as /index. html.
Dixon, R. "The Theorem of Pythagoras." §4.1 in Matho-

graphics. New York: Dover, pp. 92-95, 1991.
Dudeney, H. E. Amusements in Mathematics. New York:

Dover, p. 32, 1958.
Dunham, W. "Euclid's Proof of the Pythagorean Theorem."

Ch. 2 in Journey Through Genius: The Great Theorems

of Mathematics. New York: Wiley, 1990.
Frederickson, G. Dissections: Plane and Fancy. New York:

Cambridge University Press, pp. 28-29, 1997.

Garfield, J. A. "Pons Asinorum." New England J. Educ. 3,
161, 1876,

Loomis, E. S. The Pythagorean Proposition: Its Demonstra-
tion Analyzed and Classified and Bibliography of Sources
for Data of the Four Kinds of "Proofs." Reston, VA: Na-
tional Council of Teachers of Mathematics, 1968.

Machover, M. "Euler's Theorem Implies the Pythagorean
Proposition." Amer. Math. Monthly 103, 351, 1996.

Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 17, 1979.

Ogilvy, C. S. Excursions in Mathematics. New York: Dover,
p. 52, 1994.

Pappas, T. "The Pythagorean Theorem," "A Twist to the
Pythagorean Theorem," and "The Pythagorean Theorem
and President Garfield." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, pp. 4, 30, and 200-
201, 1989.

Perigal, H. "On Geometric Dissections and Transforma-
tions." Messenger Math. 2, 103-106, 1873.

Project Mathematics! The Theorem of Pythagoras. Video-
tape (22 minutes). California Institute of Technology.
Available from the Math. Assoc. Amer.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 123-127, 1993.

Yancey, B. F. and Calderhead, J. A. "New and Old Proofs
of the Pythagorean Theorem." Amer. Math. Monthly 3,
65-67, 110-113, 169-171, and 299-300, 1896.

Yancey, B. F. and Calderhead, J. A. "New and Old Proofs
of the Pythagorean Theorem." Amer. Math. Monthly 4,
11-12, 79-81, 168-170, 250-251, and 267-269, 1897.

Yancey, B. F. and Calderhead, J. A. "New and Old Proofs
of the Pythagorean Theorem." Amer. Math. Monthly 5,
73-74, 1898.

Yancey, B. F. and Calderhead, J. A. "New and Old Proofs
of the Pythagorean Theorem." Amer. Math. Monthly 6,
33-34 and 69-71, 1899.

Pythagorean Triad

see Pythagorean Triple

Pythagorean Triangle

see PYTHAGOREAN TRIPLE, RIGHT TRIANGLE

Pythagorean Triple

A Pythagorean triple is a TRIPLE of POSITIVE INTE-
GERS a, 6, and c such that a RIGHT TRIANGLE exists
with legs a, 6 and HYPOTENUSE c. By the PYTHAGO-
REAN Theorem, this is equivalent to finding POSITIVE
Integers a, 6, and c satisfying

2 , i2 2

a + b = c

(i)

The smallest and best-known Pythagorean triple is
(a,6,c) = (3,4,5).

It is usual to consider only "reduced" (or "primitive")
solutions in which a and b are RELATIVELY Prime, since
other solutions can be generated trivially from the prim-
itive ones. For primitive solutions, one of a or 6 must be
EVEN, and the other ODD (Shanks 1993, p. 141), with
c always ODD. In addition, in every primitive Pythag-
orean triple, one side is always Divisible by 3 and one
by 5.

Pythagorean Triple

Given a primitive triple (ao,&o,Co), three new primitive
triples are obtained from

where

(ai,6i,

ci) = (ao»

bo,co

)U

(2)

(a2,&2,C2) = (ao,6o,Co)A

(3)

(03,63^3) = (ao,&o,Co)D,

(4)

u =

" 1 2 2 "

-2 -1 -2

_ 2 2 3 .

(5)

A =

'1 2 2"

2 1 2

.2 2 3.

(6)

D =

"-I -2
2 1
_ 2 2

-2"
2
3 _

(7)

Roberts (1977) proves that (a,b,c) is a primitive Py-
thagorean triple Iff

M.C) = (3,4,5)M,

(8)

where M is a FINITE PRODUCT of the MATRICES U, A,
D. It therefore follows that every primitive Pythagorean
triple must be a member of the Infinite array

( 5, 12, 13)

(3, 4, 5) (21, 20, 29)

(15, 8, 17)

For any Pythagorean triple, the PRODUCT of the two
nonhypotenuse LEGS (i.e., the two smaller numbers) is
always DIVISIBLE by 12, and the Product of all three
sides is DIVISIBLE by 60. It is not known if there are
two distinct triples having the same PRODUCT. The
existence of two such triples corresponds to a NONZERO
solution to the DlOPHANTINE EQUATION

7, 24, 25)
55, 48, 73)
45, 28, 53)

39,

80,

89)

119,

120,

169).

(9)

77,

36,

85)

33,

56,

65)

65,

72,

97)

35,

12,

37)

/ 4 4\ / 4 4\

xy(x - y ) = zw(z - w )

(10)

(Guy 1994, p. 188).

Pythagoras and the Babylonians gave a formula for gen-
erating (not necessarily primitive) triples:

(2m, (m 2 -l),(m 2 + l)),
and Plato gave

(2m 2 ,(m 2 -l) 2 ,(m 2 + l) 2 ).

(11)

(12)

Pythagorean Triple 1467

A general reduced solution (known to the early Greeks)
is

(v 2 -u,2uv,u +v 2 ), (13)

where u and v are RELATIVELY PRIME (Shanks 1993,
p. 141). Let F n be a FIBONACCI NUMBER. Then

(KF n+3l 2F n+ iF n+2 ,F n+1 2 +F n+ 2 2 ) (14)

is also a Pythagorean triple.

For a Pythagorean triple (a, 6, c),

ft(o) + ft(6) = ft(c),

(15)

where P 3 is the PARTITION FUNCTION P (Garfunkel
1981, Honsberger 1985). Every three-term progression
of SQUARES r 2 , s 2 , t 2 can be associated with a Pythag-
orean triple (X, Y, Z) by

r^X -Y
t = X + Y

(16)

(17)
(18)

(Robertson 1996).

The Area of a Triangle corresponding to the Pythag-
orean triple (u 2 — v 2 ,2uv y u 2 + v 2 ) is

A = \{u - v 2 )(2uv) = uv{u - v 2 ). (19)

Fermat proved that a number of this form can never be
a Square Number.

To find the number L p (s) of possible primitive TRI-
ANGLES which may have a Leg (other than the HY-
POTENUSE) of length s, factor s into the form

8 = Pi * ' ■ Pn

The number of such TRIANGLES is then

* for 5 = 2 (mod 4)

o(s)= [ 2 n-

1 otherwise,

(20)

(21)

i.e., for Singly EVEN s and 2 to the power one less
than the number of distinct prime factors of s otherwise
(Beiler 1966, pp. 115-116). The first few numbers for
s = 1, 2, . . . , are 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0,
2, . . . (Sloane's A024361). To find the number of ways
L(s) in which a number s can be the LEG (other than
the Hypotenuse) of a, primitive or nonprimitive Right
TRIANGLE, write the factorization of s as

S = 2 Q ° Pl " 1 ---pn

(22)

Then

1468 Pythagorean Triple

Pythagorean Triple

M-) =

|[(2a 1 + l)(2a 2 + l)---(2a n + l)-l]

for a =
|[(2o - l)(2oi + l)(2a 2 + 1) • • • (2a n + 1) - 1]

for ao > 2

(23)

(Beiler 1966, p. 116). The first few numbers for s — 1,
2, . . . are 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, . . . (Sloane's
A046079).

To find the number of ways H p (s) in which, a number s
can be the HYPOTENUSE of a primitive RIGHT TRIAN-
GLE, write its factorization as

S = 2 O0 ( Pl O1 •■•p„ a ")(<7i 6l --V r ),

(24)

where the ps are of the form Ax — 1 and the qs are of the
form Ax + 1. The number of possible primitive RIGHT
Triangles is then

„ i x f 2 r_1 for n = ;
Hp(s) = < rt

I otherwise,

= and ao =

(25)

The first few Primes of the form 4x + 1 are 5, 13, 17,
29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, ...
(Sloane's A002144), so the smallest side lengths which
are the hypotenuses of 1, 2, 4, 8, 16, ... primitive right
triangles are 5, 65, 1105, 32045, 1185665, 48612265, . . .
(Sloane's A006278). The number of possible primitive
or nonprimitive Right Triangles having s as a Hy-
potenuse is

fT(a) = -|[(26i + l)(26a + l)-

• (2b r + 1) - 1] (26)

(Beiler 1966, p. 117). The first few numbers for s = 1,
2, . . . are 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0,
0, ... (Sloane's A046080).

Therefore, the total number of ways in which s may be
either a Leg or Hypotenuse of a Right Triangle is
given by

T(a) = L(8) + H(8). (27)

The values for s = 1, 2, . . . are 0, 0, 1, 1, 2, 1, 1, 2, 2,
2, 1, 4, 2, 1, 5, 3, . . . (Sloane's A046081). The smallest
numbers s which may be the sides of T general RIGHT
Triangles for T = 1, 2, ... are 3, 5, 16, 12, 15, 125,
24, 40, . . . (Sloane's A006593; Beiler 1966, p. 114).

There are 50 Pythagorean triples with Hypotenuse
less than 100, the first few of which, sorted
by increasing c, are (3,4,5), (6,8,10), (5,12,13),
(9, 12, 15), (8, 15, 17), (12, 16, 20), (15, 20, 25), (7, 24, 25),
(10, 24, 26), (20, 21, 29), (18, 24, 30), (16, 30, 34),
(21,28,35), ... (Sloane's A046083, A046084, and
A046085). Of these, only 16 are primitive triplets^
with Hypotenuse less than 100: (3,4,5), (5,12,13),
(8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41),
(28, 45, 53), (11, 60, 61), (33, 56, 65), (16, 63, 65),
(48,55,73), (36,77,85), (13,84,85), (39,80,89), and

(65,72,97) (Sloane's A046086, A046087, and A046088).
Of these 16 primitive triplets, seven are twin triplets (de-
fined as triplets for which two members are consecutive
integers). The first few twin triplets, sorted by increas-
ing c, are (3,4,5), (5,12,13), (7,24,25), (20,21,29),
(9,40,41), (11,60,61), (13,84,85), (15,112,113), ....

Let the number of triples with Hypotenuse less than N
be denoted A(iV), and the number of twin triplets with
Hypotenuse less than N be denoted A 2 (iV). Then, as
the following table suggests and Lehmer (1900) proved,
the number of primitive solutions with HYPOTENUSE
less than N satisfies

lim

A(7V)
N

2tt

0.159155.

(28)

N

A(N)

A(N)/N

A 2 (iV)

100

16

0.1600

7

500

80

0.1600

17

1000

158

0.1580

24

2000

319

0.1595

34

3000

477

0.1590

41

4000

639

0.1598

47

5000

792

0.1584

52

10000

1593

0.1593

74

Considering twin triplets in which the LEGS are consecu-
tive, a closed form is available for the rth such pair. Con-
sider the general reduced solution (u — v , 2uv, u +v ),
then the requirement that the LEGS be consecutive in-
tegers is

2uv±l. (29)

2 2

U — V

Rearranging gives

Defining

(u-v) 2 ~2v 2 = ±1.

u — x + y
v = y

then gives the Pell Equation

x 2 - 2y 2 = 1.

Solutions to the Pell EQUATION are given by

_ (i + v^) p + (i-v^r

(l + V^) 7 --(l-y / 2) r
2\/2

(30)

(31)
(32)

(33)

(34)
(35)

Pythagorean Triple

Pythagorean Triple 1469

so the lengths of the legs X r and Y r and the Hy-
potenuse Z r are

v 2 2 2

X T — U — V = X

2xy

^2 + l) 2r+1 - (y/2 - l) 2r+1

4

Y T

= luv = 2xj/ + 2y

V2 + l) 2r+1

-(V2 -

. l)2r+l

Z r

2,2 2

4
+ 2xy + 2y 2

(x/2 + 1) 2 '+

' + (V2

_ l)2r+l

2\/2

+ K-l)" (36)

- §(-l) r (37)

(38)

Denoting the length of the shortest LEG by A T then gives
(V2 + l) ar+1 -(VS-l) 2r+1 1

A,
Z r =

4 2

(v / 2 + l) 2r+1 + (y / 2-l) 2r+1
2^2

(39)
(40)

(Beiler 1966, pp. 124-125 and 256-257), which cannot be
solved exactly to give r as a function of Z T . However, the
approximate number of leg-leg twin triplets A%(N) = r
less than a given value of Z r = N can be found by noting
that the second term in the DENOMINATOR of Z r is a
small number to the power 1 + 2r and can therefore be
dropped, leaving

N= Z r >

(V2 + l) 1+2r
2^/2

N > (1 + 2r) ln(\/2 + 1) - In (2 a/2 ).
Solving for r = A£ (ra) gives

L lnJV + ln(2>/2)-ln(v^+l)

2V ; 21n(V5+l)

IniV

(41)
(42)

21n(l + V2)_
0.567 In N.

(43)
(44)

The first few Leg-Leg triplets are (3, 4, 5), (20, 21, 29),
(119, 120, 169), (696, 697, 985), ... (Sloane's A046089,
A046090, and A046091).

Leg-Hypotenuse twin triples {a,b,c) = (v 2 -
u 2 , 2uv,u 2 -h v 2 ) occur whenever

u + v 2 = 2uw + 1

(u-v) = 1,

(45)
(46)

that is to say when v = u -f 1, in which case the Hy-
potenuse exceeds the Even Leg by unity and the twin
triplet is given by (1 + 2u, 2u(l + «),! + 2u(l 4- u)). The

number of leg- hypotenuse triplets with hypotenuse less
than N is therefore given by

A?(JV)=[i(^V^T-l)J,

(47)

where [x\ is the FLOOR FUNCTION. The first few Leg-
HYPOTENUSE triples are (3, 4, 5), (5, 12, 13), (7, 24,
25), (9, 40, 41), (11, 60, 61), (13, 84, 85), ... (Sloane's
A005408, A046092, and A046093).

The total number of twin triples A2(N) less than N is
therefore approximately given by

A 2 (N) = A?(N) + A$(N) - 1 (48)

« [\y/2N - 1 + 0.5671n JV - 1.5J , (49)

where one has been subtracted to avoid double counting
of the leg-leg-hypotenuse double-twin (3,4,5).

There is a general method for obtaining triplets of Py-
thagorean triangles with equal AREAS. Take the three
sets of generators as

2 2

mi = r + rs + s

(50)

2 2
ni = r — s

(51)

2 2
7712 = r + rs -\- s

(52)

ri2 = 2rs + s 2

(53)

ms = r + 2rs

(54)

ri3 = r 2 + rs 4- s 2 .

(55)

Then the RIGHT TRIANGLE generated by each triple
{m 2 — n 2 , 2mi7ii , m 2 + n^ 2 ) has common Area

A = rs(2r + a)(r + 2s)(r + a)(r - s)(r 2 + rs + s 2 ) (56)

(Beiler 1966, pp. 126-127). The only EXTREMUM of this
function occurs at (r } s) — (0,0). Since A(r,s) = for
r = 5, the smallest Area shared by three nonprimitive
Right Triangles is given by (r,s) = (1,2), which re-
sults in an area of 840 and corresponds to the triplets
(24, 70, 74), (40, 42, 58), and (15, 112, 113) (Beiler 1966,
p. 126). The smallest Area shared by three primitive
Right Triangles is 13123110, corresponding to the
triples (4485, 5852, 7373), (1380, 19019, 19069), and
(3059, 8580, 9109) (Beiler 1966, p. 127).

One can also find quartets of RIGHT TRIANGLES with
the same Area. The Quartet having smallest known
area is (111, 6160, 6161), (231, 2960, 2969), (518, 1320,
1418), (280, 2442, 2458), with Area 341,880 (Beiler
1966, p. 127). Guy (1994) gives additional information.

It is also possible to find sets of three and four Pythago-
rean triplets having the same PERIMETER (Beiler 1966,

1470 Pythagorean Triple

Pythagorean Triple

pp. 131-132). Lehmer (1900) showed that the number
of primitive triples N(p) with Perimeter less than p is

lim N(p)

p— J-OO

p\n2

0.070230...

(57)

In 1643, Fermat challenged Mersenne to find a Pythag-
orean triplet whose HYPOTENUSE and Sum of the LEGS
were SQUARES. Fermat found the smallest such solu-
tion:

with

X = 4565486027761
Y = 1061652293520
Z = 4687298610289,

Z = 2165017^
X + y = 2372159 2 .

(58)
(59)
(60)

(61)
(62)

A related problem is to determine if a specified INTEGER
N can be the Area of a Right Triangle with rational
sides. 1, 2, 3, and 4 are not the Areas of any Rational-
sided Right Triangles, but 5 is (3/2, 20/3, 41/6), as
is 6 (3, 4, 5). The solution to the problem involves the
Elliptic Curve

Garfunkel, J. Pi Mu Epsilon X, p. 31, 1981.

Guy, R. K. "Triangles with Integer Sides, Medians, and
Area." §D21 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 188-190, 1994.

Hindin, H. "Stars, Hexes, Triangular Numbers, and Pythag-
orean Triples." J. Recr. Math. 16, 191-193, 1983-1984.

Honsberger, R. Mathematical Gems HI. Washington, DC:
Math. Assoc. Amer., p. 47, 1985.

Koblitz, N. Introduction to Elliptic Curves and Modular
Forms, 2nd ed. New York: Springer- Verlag, pp. 1-50,
1993.

Kraitchik, M. Mathematical Recreations. New York:
W. W. Norton, pp. 95-104, 1942.

Kramer, K. and Tunnell, J. "Elliptic Curves and Local Ep-
silon Factors." Comp. Math. 46, 307-352, 1982.

Lehmer, D. N, "Asymptotic Evaluation of Certain Totient
Sums." Amer. J. Math. 22, 294-335, 1900.

Roberts, J. Elementary Number Theory: A Problem Ori-
ented Approach. Cambridge, MA: MIT Press, 1977.

Robertson, J. P. "Magic Squares of Squares." Math. Mag.
69, 289-293, 1996.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 121 and 141, 1993.

Sloane, N. J. A. Sequences A006278, A046079, A002144/
M3823, and A006593/M2499 in "An On-Line Version of
the Encyclopedia of Integer Sequences."

Taussky-Todd, O. "The Many Aspects of the Pythagorean
Triangles." Linear Algebra and Appl. 43, 285-295, 1982.

y 2 = x 3 - N 2 X.

(63)

A solution (a, 6, c) exists if (63) has a Rational solu-
tion, in which case

1 2

x = z c

y=Ua 2 -b 2 )c

(64)
(65)

(Koblitz 1993). There is no known general method for
determining if there is a solution for arbitrary A/", but a
technique devised by J. Tunnell in 1983 allows certain
values to be ruled out (Cipra 1996).

see also HERONIAN TRIANGLE, PYTHAGOREAN QUAD-
RUPLE, Right Triangle

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 57-59,
1987.

Beiler, A H. "The Eternal Triangle." Ch. 14 in Recreations
in the Theory of Numbers: The Queen of Mathematics
Entertains. New York: Dover, 1966.

Cipra, B. "A Proof to Please Pythagoras." Science 271,
1669, 1996.

Courant, R. and Robbins, H. "Pythagorean Numbers and
Fermat's Last Theorem." §2.3 in Supplement to Ch. 1 in
What is Mathematics? : An Elementary Approach to Ideas
and Methods, 2nd ed. Oxford, England: Oxford University
Press, pp. 40-42, 1996.

Dickson, L. E. "Rational Right Triangles." Ch. 4 in History
of the Theory of Numbers, Vol. 2: Diophantine Analysis.
New York: Chelsea, pp. 165-190, 1952.

Dixon, R. Mathographics. New York: Dover, p. 94, 1991.

Q

The Field of Rational Numbers.
see cdso C t C\ I, N, R, Z

q- Analog

A g-analog, also called a g-EXTENSION or q-
GENERALIZATION, is a mathematical expression param-
eterized by a quantity q which generalizes a known ex-
pression and reduces to the known expression in the
limit q -» 1. There are g-analogs of the Factorial,
Binomial Coefficient, Derivative, Integral, Fi-
bonacci NUMBERS, and so on. Koornwinder, Suslov,
and Bustoz, have even managed some kind of q- Fourier
analysis.

The g-analog of a mathematical object is generally called
the "^-object", hence ^-Binomial Coefficient, q-
Factorial, etc. There are generally several ^-analogs
if there is one, and there is sometimes even a multibasic
analog with independent gi, #2,

see also g?-Analog, qr-BETA Function, ^-Binomial
Coefficient, ^-Binomial Theorem, ^-Cosine, q-
Derivative, ^-Factorial, ^-Gamma Function, q-
Series, q-Sine, 5-Vandermonde Sum

References

Exton, H. q~ Hyper geometric Functions and Applications.
New York: Halstead Press, 1983.

<7-Beta Function

A g-ANALOG of the Beta Function

B(a,b)= f t a_1
Jo

(i-ty^dt^

T(a)T(b)

r(a + fe)'
where T(z) is a GAMMA FUNCTION, is given by

B,(a,6)= f\ b -\qt ]q ) a ^d(a y t)^ ^^
Jo T q {a + b)

where T q (a) is a g-GAMMA FUNCTION and (a;q) n is a
g-SERlES coefficient (Andrews 1986, pp. 11-12).

see also g-FACTORIAL, g-GAMMA FUNCTION

References

Andrews, G. E. q- Series: Their Development and Applica-
tion in Analysis, Number Theory, Combinatorics, Phys-
ics, and Computer Algebra. Providence, RI: Amer. Math.
Soc, 1986.

q-Binomial Theorem 1471

g-Binomial Coefficient

A ^-Analog for the Binomial Coefficient, also
called the Gaussian Coefficient. It is given by

n — m ■*""" J- q

m J (q)m(q)n-m " 1 - tf

3 i=0

where

<*>» s n i

i-«"

sjk-\-7Tl

For example, the first few g-binomial coefficients are

1 + 9

= 1-? 2

:).-©.-£*->♦•♦'

(2)

(3)
(4)

1-9 4

= ^- = l + q + q *+q J (5)

1-3

4^ „(l_- 5 3 )(l-^) =(1+g)(1 + g + g2) . (6)

1 V3y l-q

q \ / q

2j g (l- q )(l- q i)

From the definition, it follows that

a-1

(7)

In the LIMIT q — > 1, the g-binomial coefficient collapses
to the usual Binomial Coefficient.

see also Cauchy Binomial Theorem, Gaussian
Polynomial

g-Binomial Theorem

The q- ANALOG of the BINOMIAL THEOREM

( i- gr = i. M+ ^- 1 ) ^- w ("7 1 j(y 2 ) Jg » + ...

1-2

1-2-3

is given by

i-4Wr z

l-

1 - q n z 1 - q n 1 - q

1- Z ~

n i „n — 1 „2

+

l-5f 1 - £ 1-g 2 g" + (n-l)

-...±

gn(n+l)/2 *

Written as a ^-Series, the identity becomes

1472 q-Cosine

where

(1 - aq m )

(a,g)n= 11 {1 _ a ^ +n)

(Heine 1847, p. 303; Andrews 1986). The Cauchy Bi-
nomial Theorem is a special case of this general the-
orem.

see also Binomial Series, Binomial Theorem, Cau-
chy Binomial Theorem, Heine Hypergeometric

Series, Ramanujan Psi Sum

References

Andrews, G, E. q-Series: Their Development and Applica-
tion in Analysis, Number Theory, Combinatorics, Phys-
ics, and Computer Algebra. Providence, RI: Amer. Math.
Soc, p. 10, 1986.

Heine, E. "Untersuchungen iiber die Reihe 1 + ^~J ui- 9 -*) *

x + (i-g)(i-q2 )(1 _^ )( i_^+i) a; -I-..- J- reine angew.
Math. 34, 285-328, 1847.

q- Cosine

The ^-Analog of the Cosine function, as advocated by

R. W. Gosper, is defined by

cos q (z,q) =

<Mo,p)'

where $2(2, v) 1S a Theta Function and p is defined
via

(lnp)(\nq) = tv .

This is a period 2x, EVEN FUNCTION of unit ampli-
tude with double and triple angle formulas and addition
formulas which are analogous to ordinary SINE and CO-
SINE. For example,

coSq(2z,g) = cosg 2 ^,^ 2 ) - sin g 2 (z,g 2 ),

where sm q (z,a) is the g-SlNE, and it q is qr-Pl. The q-
cosine also satisfies

2 {<ko) -

Er = -oo(-i) n q (n+tt)a

see also gr-FACTORIAL, gr-SlNE

References

Gosper, R. W, "Experiments and Discoveries in q-
Trigonometry." Unpublished manuscript.

g-Derivative

The ^-Analog of the Derivative, defined by

fix) - f(qx)

(=).«■>- ^

q x

q-Dimension

For example,

d \ . sin a: — sin(qx)

smx =

( d\ .

— sin x -
\dxj q x — qx

(JL\ i~~- lng-In(qg) _ ln [\)

(

dx) a

x — qx (1 — q)x

d \ 2_£ — q x
dx) Q

= (1 + «)a;

\dX/ Q

x — qx

x 3 -q 3 x s 2 2

= (1 + 9 + 5 )z *

q x — qx

In the LIMIT <? — > 1, the g-derivative reduces to the usual
Derivative.

see also DERIVATIVE

g-Dimension

D q =

L ^_ ]im lnI(q A

l-q<->o ln(i),

where

i{q,t) = y^A*« q i

(1)

(2)

e is the box size, and m* is the NATURAL MEASURE. If
qi > £2, then

D qi <D q2 . (3)

The Capacity Dimension (a.k.a. Box Counting Di-
mension) is given by q = 0,

D =

1 ^(e^i)

^— lim — ^— = ^

linx 1 ^™. (4)

1 - e-+o — lne e-*-o lne

If all puis are equal, then the CAPACITY DIMENSION is
obtained for any q. The INFORMATION DIMENSION is
defined by

D± = lim D q = lim

lim e ^o

-rs^

>M.«1

lne

1-9

m (E^V)

= lim lim , .,

t->og-^i lne(<7 — 1)

(5)

But

/N(e) \ /*(«) \

lim In { ^ Mi* ) = In I 5^ Mi 1 = ln * = °> ( 6 )

so use L'Hospital's Rule

D 1 = liml J-lim^^*" 1

Therefore,

e-j-o V lne q-¥i ^2fii q
. =1 m*1hm<

Di = lim

lne

(7)

(8)

D 2 is called the Correlation Dimension. The q-
dimensions satisfy

D q+1 < D q .
see also Fractal Dimension

(9)

Q.E.D.

Q-Matrix 1473

Q.E.D.

An abbreviation for the Latin phrase "quod erat demon-
strandum" ("that which was to be demonstrated"), a
NOTATION which is often placed at the end of a mathe-
matical proof to indicate its completion.

g-Extension

see q- ANALOG

q- Factorial

The gf-ANALOG of the Factorial (by analogy with the
qr-GAMMA FUNCTION). For a an integer, the g-factorial
is defined by

faq(a, q) = 1(1 + q){l + q + q 2 ) ■ ■ ■ (1 + q + . . . + q a ~ l ).

A reflection formula analogous to the GAMMA FUNC-
TION reflection formula is given by

cosq(7ra) = sinq[7r(| — a)]

KqQ

(a-l/2)(a+l/2)

faq(a- ±, ? 2 )faq(-(a + |),? 2 )'

where cos q (z) is the ^-COSINE, sin q (z) is the gr-SlNE, and
-K q is g-Pi.

see also q-Beta Function, ^-Cosine, ^-Gamma
Function, g-Pi, ^-Sine

References

Gosper, R. W. "Experiments and Discoveries in q-
Trigonometry." Unpublished manuscript.

Q-Function

Let

then

q=e = e ,

Qo^f[(l~q 2n )

n=l

oo

n=l

oo

Q 3 sTJ(i + , a »- x )

OO

Q sS TJ(l_^-i).

(1)

(2)
(3)
(4)
(5)

The Q-functions are sometimes written using a lower-
case q instead of a capital Q. The Q-functions also sat-
isfy the identities

see also JACOBI IDENTITIES, g-SERIES

References

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 55 and 63-85, 1987.

Tannery, J. and Molk, J. Elements de la Theorie des Fonc-
tions Elliptiques, 4 vols, Paris: Gauthier-Villars et fils,
1893-1902.

Whit taker, E. T. and Watson, G. N. A Course in Modern
Analysis, 4th ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 469-473 and 488-489, 1990.

q- Gamma Function

A ^-Analog of the Gamma Function defined by

r - Ms (^ (i -«>'■■■ «

where {x.q)^ is a qr-SERIES. The g-gamma function
satisfies

lim T q (x) = r(a>) (2)

(Andrews 1986).

A curious identity for the functional equation

f(a - b)f(a - c)f(a - d)f(a - e) - f(b)f(c)f(d)f(e)
= q b f(*)f(a - b - c)f(a - b - d)f(a - b - e), (3)

where

is given by

6 + c + d + e = 2a

(4)

fsin(fta)

/(a) = < i

^ r Q (a)r fl (l-a)

forg=l

for < q < 1, W

for any k.

see also <y-BETA FUNCTION, g-FACTORIAL

References

Andrews, G. E. "W. Gosper's Proof that lim,_n- T q (x) =
r(x)." Appendix A in q-Series: Their Development and
Application in Analysis, Number Theory, Combinatorics,
Physics^ and Computer Algebra. Providence, HI: Amer.
Math. Soc, p. 11 and 109, 1986.

Wenchang, C. Problem 10226 and Solution. "A q-

Trigonometric Identity." Amer. Math. Monthly 103, 175-
177, 1996.

^-Generalization

see g-ANALOG

q-Hypergeometric Series

see Heine Hypergeometric Series

QoQi = Qo(q 2 )
QoQs = Qo(q 1/2 )
Q 2 Q 3 = Qs(g 2 )

(6)
(7)
(8)

Q-Matrix

see Fibonacci Q-Matrix

Q1Q2 = Qi(q 1/2 ).

(9)

1474 Q-Number

q-Sine

Q-Number

see HOFSTADTER'S Q-Sequence

g-Pi

The q- ANALOG of Pi n q can be defined by taking a =

in the g-FACTORIAL

faq(«, q) = 1(1 + q)(l + q + q 2 ) • • - (1 + q + - . • + <7 a_1 ),

giving

1 = sin g (|7r) =

faq 2 (-i,^) g i/4'

where sin g (z) is the g-SlNE. Gosper has developed an
iterative algorithm for computing tt based on the alge-
braic RECURRENCE RELATION

4^4 ^ (q 2 + l)W (g 4 + l)7T g2 2

q 4 + 1 7T q 2 7T q 4

Q-Polynomial

see BLM/Ho Polynomial

g-Product

see Q-Function

g-Series

A SERIES involving coefficients of the form

where 2 <t>i (a, b; c; g, z) is a Heine Hypergeometric Se-
ries. Other g-series identities, e.g., the JACOBI IDEN-
TITIES, Rogers-Ramanujan Identities, and Heine
Hypergeometric Identity

20i(a,6;c;g,z) = — ; r — 2 <M C /M; az\ g,6),

(c;gj 00 (z;gJoo

(7)

seem to arise out of the blue.

see also BORWEIN CONJECTURES, FINE'S EQUATION,

Gaussian Coefficient, Heine Hypergeometric
Series, Jackson's Identity, Jacobi Identities,
Mock Theta Function, q- Analog, ^-Binomial
Theorem, g-CosiNE, ^-Factorial, Q-Function, q-
Gamma Function, g-SiNE, Ramanujan Psi Sum, Ra-
manujan Theta Functions, Rogers-Ramanujan
Identities

References

Andrews, G. E. q-Series: Their Development and Applica-
tion in Analysis, Number Theory, Combinatorics, Phys-
ics, and Computer Algebra. Providence, RI: Amer. Math.
Soc, 1986.

Berndt, B. C. "g-Series." Ch. 27 in Ramanujan f s Notebooks,
Part IV. New York: Springer-Verlag, pp. 261-286, 1994.

Gasper, G. and Rahman, M. Basic Hypergeometric Series.
Cambridge, England: Cambridge University Press, 1990.

Gosper, R. W. "Experiments and Discoveries in q-
TVigonometry." Unpublished manuscript.

Q-Signature

see Signature (Recurrence Relation)

(a) n = (a;q) n = JJ

(l-aq k )

k=0

71-1

k=0

(Andrews 1986). The symbols

[n] = l + < ? + < ? 2 + ... + ^- 1
[„]! = [„][„- !]...[!]

are sometimes also used when discussing g-series.

a)

(1 - aq k + n )
H(l ~aq k ) (2)

(3)
(4)

There are a great many beautiful identities involving
g-series, some of which follow directly by taking the q-
Analog of standard combinatorial identities, e.g., the
4-Binomial Theorem

E

(a;q) n z n _ {az;q) Q

(q\q)n (z;q) a

(5)

(\z\ < 1, \q\ < 1; Andrews 1986, p. 10) and q-

Vandermonde Sum

2</>i(a,q n ;c i q,q)

a n (c/a,q)n
(c;q) n

(6)

g-Sine

The g- Analog of the. Sine function, as advocated by
R. W. Gosper, is defined by

• / V #l(*>P)

sm g (z,g) = — -j -,

where $i(z,p) is a THETA FUNCTION and p is defined
via

(lnp)(mg) = 7r 2 .

This is a period 27r, Odd FUNCTION of unit amplitude
with double and triple angle formulas and addition for-
mulas which are analogous to ordinary Sine and CO-
SINE. For example,

sin g (2z,g) = (g -f 1)— - cos q (z, g 2 ) sin q (z, g 2 ),

Pg2

where cos q (z y a) is the g-CosiNE, and ix q is g-Pl.
see also g-CosiNE, g-FACTORlAL

References

Gosper, R. W. "Experiments and Discoveries in q-
Trigonometry." Unpublished manuscript.

q-Vandermonde Sum

Quadratic Curve 1475

<7-Vandermonde Sum

2 4>i(a,q~ n \c;q,q) =

Quadrant

a n (c/a,q) n

(c; g)n

where 2 <M a > 6; c; g, z) is a HEINE Hypergeometric Se-
ries.

see also Chu-Vandermonde Identity, Heine Hyper-
geometric Series

References

Andrews, G. E. q-Series: Their Development and Applica-
tion in Analysis, Number Theory, Combinatorics, Phys-
ics, and Computer Algebra. Providence, RI: Amer. Math.
Soc, pp. 15-16, 1986.

QR Decomposition

Given a MATRIX A, its QiZ-decomposition is of the form

A = QR,

where R is an upper Triangular Matrix and Q is an
Orthogonal Matrix, i.e., one satisfying

Q T Q - 1,

where I is the IDENTITY MATRIX. This matrix decom-
position can be used to solve linear systems of equations.
see also CHOLESKY DECOMPOSITION, LU DECOMPOSI-
TION, Singular Value Decomposition

References

Householder, A. S. The Numerical Treatment of a Single
Non-Linear Equations. New York: McGraw-Hill, 1970.

Nash, J. C. Compact Numerical Methods for Computers:
Linear Algebra and Function Minimisation, 2nd ed. Bris-
tol, England: Adam Hilger, pp. 26-28, 1990.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "QR Decomposition." §2.10 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 91-95, 1992.

Stewart, G. W. "A Parallel Implementation of the QR Al-
gorithm." Parallel Comput. 5, 187-196, 1987. ftp://
thales . cs .umd.edu/pub/reports/piqra.ps.

Quadrable

A plane figure for which QUADRATURE is possible is said
to be quadrable.

Quadrangle

A plane figure consisting of four points, each of which is
joined to two other points by a Line SEGMENT (where
the line segments may intersect). A quadrangle may
therefore be Concave or Convex; if it is Convex, it
is called a Quadrilateral.

see also COMPLETE QUADRANGLE, CYCLIC QUADRAN-
GLE, Quadrilateral

jc < 0, y >
Quadrant 2

Quadrant 3
x < 0, y <

x > 0, y >
Quadrant 1

Quadrant 4

x> 0, y <

One of the four regions of the Plane defined by the four
possible combinations of Signs (+,+), (+,—), (— ,+)>
and (-,-) for (x,y).
see also Octant, x-Axis, y-Axis

References

Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 73, 1996.

Quadratfrei

see Squarefree

Quadratic Congruence

A Congruence of the form

ax 2 + bx + c = (mod m) ,

where a, 6, and c are INTEGERS. A general quadratic
congruence can be reduced to the congruence

x 2 = q (mod p)

and can be solved using ExCLUDENTS, although solution
of the general polynomial congruence

Q>mX

' + ... + a 2 x 2 + aix + a = (mod n)

is intractable.

see also Congruence, Excludent, Linear Congru-
ence

Quadratic Curve

The general 2-variable quadratic equation can be writ-
ten

ax 2 + 2bxy + cy 2 + 2dx + 2fy + g = 0. (1)

Define the following quantities:

(2)

(3)
(4)
(5)

a

b

d

A =

b

c

f

d

f

9

a

h

J =

b

c

I = t

2 +

c

K =

a
d

d

9

+

c

f

f

9

1476 Quadratic Curve

Quadratic Curve

Then the quadratics are classified into the types sum-
marized in the following table (Beyer 1987). The real
(nondegenerate) quadratics (the ELLIPSE, HYPERBOLA,
and PARABOLA) correspond to the curves which can be
created by the intersection of a PLANE with a (two-
NAPPES) Cone, and are therefore known as CONIC SEC-
TIONS.

Curve

A

A/I K

coincident lines

ellipse (imaginary) ^ > >

ellipse (real) ^ > <

hyperbola / <

intersecting lines (imaginary) >

intersecting lines (real) <

parabola ^

parallel lines (imaginary) >

parallel lines (real) <

It is always possible to eliminate the xy cross term by a
suitable ROTATION of the axes. To see this, consider ro-
tation by an arbitrary angle 0. The Rotation Matrix

X

cos sin

t

X

y _

— sin cos 9

w

x' cos + y' sin 9

—x 1 sin 9 + y' cos 9

(6)

x = x cos + 2/' sin (7)

y = —x sin + y cos (8)

xy — —x cos sin + x'y'(cos — sin 0)

+ y' 2 cos0sin0 (9)

x = x cos + 2xy cos sin + y sin (10)
y ~ — x sin — 2xy sin cos + y cos 0. (11)

Plugging these into (1) gives

a(x cos + 2x y cos + y sin 0)

+2b(x cos0 + y'sin0)(— x'sin0 + y cos0)

+c(z' 2 sin 2 - 2xy cos sin + y 2 cos 2 0)

-\-2d(x cos0 + y sin0)

+2f(-x sin + y cos 0) + g = 0. (12)

Rewriting,

a{x 2 cos 2 + 2xy cos + y' 2 sin 2 0)
-\-2b{—x cos sin — xy sin 0+rry cos 0+y cos sin 0)

+c(x' 2 sin 2 — 2:r'y' cos sin + y' 2 cos 2 0)

+2d(x' cos 9 + y' sin 0)

+2/(-;z'sin0 + y'cos0) +£ = 0. (13)

Grouping terms,

x (a cos + c sin 9 — 2b cos sin 0)

+#y [2a cos sin - 2c sin cos + 26(cos 2 - sin 2 0)]
+y' 2 (a sin 2 + c cos 2 + 26 cos sin 0)
+z'(2dcos0 - 2/sin0) + y'(-2dsin0 + 2/cos0)
+9 = 0. (14)

Comparing the COEFFICIENTS with (1) gives an equa-
tion of the form

ax' 2 + 2b'xy f + c'y 2 + 2dV + 2/'y' +5=0, (15)

where the new Coefficients are

= a cos 2 0-26 cos sin + c sin 2 9 (16)

= 6(cos 2 - sin 2 0) + (a - c) sin cos (17) ■
= a sin 2 + 26 sin cos + ccos 2 (18)

d' = d cos — / sin
/' = — d sin + / cos
9 =9-

(19)
(20)
(21)

The cross term 2b , x'y' can therefore be made to vanish

by setting

b' = 6(cos 2 — sin 2 0) — (c — a) sin cos

= 6 cos(20) - \ (c - a) sin(20) = 0. (22)

For b' to be zero, it must be true that

cot(20) = ^=K. (23)

The other components are then given with the aid of the
identity

cos[cot (x)] =

yfl + x 2

by defining

K

VTTk^'

sin0 =

COS0 =

1-L

1 + L

Rotating by an angle

therefore transforms (1) into

aa; +cy +2ax+2/y+p=0.

(24)

(25)

(26)
(27)

(28)
(29)

Quadratic Curve

Completing the Square,

a' (x 12 + ^-x\ + c' L' 2 + *£y>\ + g' = (30)

a I x +

d'

+ d[y+ f -

j/2 w2

-9' + ^. (3D

Defining x" = x' + d'/a', y" = y' + f'/c', and g"
-g' + d' 2 /a! + f' 2 /c' gives

/ tt2 . / //2 //

ax -\- cy — g

(32)

If <j" z/i 0, then divide both sides by g" ' . Defining a" =

a! /g" and c" = c f /g" then gives

// //2 . // "2 -,

ax + c y = 1.

(33)

Therefore, in an appropriate coordinate system, the
general Conic Section can be written (dropping the
primes) as

ax 2 + cy 2 = 1 a,c,^/0
ax 2 + cy 2 = a, c / 0, g = 0.

(34)

Consider an equation of the form ax 2 + 2bxy + cy 2 — 1
where 6^0. Re-express this using t\ and £2 in the form

ax + 2bxy-\-cy — t\x +£22/ - (35)

Therefore, rotate the COORDINATE SYSTEM

cos V sin U
— sin B cos #

(36)

Quadratic Curve 1477

Prom (41) and (42),

a-c _ {ti -£ 2 )cos(20)

b ' |(£i -£ 2 )sin(2<9)
the same angle as before. But

cos(2^) = cos[cot- 1 (^r^)l
= cob [tan" 1 (^)]

= 2cot(20), (43)

so

\A+(^) 2 '

tl-*2

Rewriting and copying (41),

= ^(a - c) 2 + 46 2
h + <2 = a + c.

Adding (46) and (47) gives

h = §[a4-c+- v /(a-c) 2 +46 2 ]

(44)

(45)

(46)
(47)

(48)

( 2 = a + c-t 1 = \[a + c- v'(a-c) 2 + 46 2 ].

(49)

Note that these ROOTS can also be found from

(t - £i)(£ - £ 2 ) = t 2 - t(*i + £ 2 ) + ht 2 = (50)

ax + 2bxy + cy = t\X -\-t2y

= ti(x cos + 2ajy cos 6 sin + y sin 0)

-h £2 (x 2 sin 2 — 2 zy sin cos 6 + y 2 cos 2 0)
= z 2 (£i cos 2 + £2 sin 2 0) + 2xycos0sin0(ti — £2)

and

+ y (£1 sin -|- £2 cos 0)

a = ti cos + £2 sin

(37)

(38)

b = (*i — h ) cos sinS = |(fi - i 2 )sin(20) (39)

(40)

c = ti sin + £2 cos 0.

Therefore,

a + c = (£1 cos 2 6> + £2 sin 2 0) + (£1 sin 2 + £ 2 cos 2 0)

= £i + £2 (41)

a — c = £i cos + £2 sin 6 — £1 sin + £2 cos

= (£1 - £2)(cos 2 - sin 2 6) = (£1 - £ 2 ) cos(20).

(42)

£ 2 -£(a + c) + I{(a + c) 2 - [(a - c) 2 + 46 2 ]}
= £ 2 - £(a + c)

+ |[a 2 + 2ac + c 2 - a + 2ac - c - 46 2 ]
= £ 2 - £(a + c) + (ac - 6 2 ) = (a - £)(c - £) - b 2

a — t b
b c-t

= (a-t)(c-t) -b 2 = 0. (51)

The original problem is therefore equivalent to looking
for a solution to

a b \ \ x _ \ x

by cy

][:]-[?]■

which gives the simultaneous equations

1 aa; 2 + feasy = £a; 2
\ bxy + cy 2 = £i/ 2 .

(52)
(53)

(54)

1478 Quadratic Curve

Quadratic Curve

Let X be any point (x,y) with old coordinates and
{x\y f ) be its new coordinates. Then

and

r

X

= x+-

y '.

A

T

y

= x_

_y _

+ 2bxy + cy 2 = t + x+t-y 2 = 1 (55)

(56)
(57)

If t+ and t_ are both > 0, the curve is an Ellipse. If
£4. and t- are both < 0, the curve is empty. If t+ and
t- have opposite SIGNS, the curve is a HYPERBOLA. If
either is 0, the curve is a Parabola.

To find the general form of a quadratic curve in POLAR
COORDINATES (as given, for example, in Moult on 1970),
plug x = rcos0 and y = rsin0 into (1) to obtain

ar cos + 2br cos sin + cr sin

+2dr cos + 2/r sin + 5 = (58)

(a cos 2 + 26 cos sin + csin 8)

+ -(dcos0 + /sin0) + 4 =0- ( 59 )
r r J

Define u = 1/r. For p 7^ 0,we can divide through by 2g y

\u + -(dcos$ + fsm6)u
9

+ — (a cos 2 + 26 cos sin + c sin 2 0) = 0. (60)
2g

Applying the QUADRATIC FORMULA gives

u = --cos0- -sin0±\/#, (61)

9 9

where

R =

(rfcos0 + /sin0) 2

G)

4( *") ( — ) (acos 2 + 26cos0sin0 + csin 2 0)

= 4 cos 2 e + 2 -% cos0sin6> + 4 sin 2 9

(a cos 2 + 26cos<9 sin + csin 2 0). (62)

9

Using the trigonometric identities

sin(20) = 2 sin cos 0,

(63)
(64)

it follows that

R =

a f

+ - cos 2

+ l^- b -)M28) +

W ~9)

i[l + cos(2«)]

d 2 -ag- f 2 + eg

df-bg\ f 2 - eg

+ sin(26>)

d 2 - ag - f + w cos{20) + df-bg ^^

+

Defining

d 2

2ff 2
ag-f + cg + if - leg

2fl 2

(65)

9

(66)

g

(67)

c _ df-bg
9 2

(68)

d 2 - f + cg-ag

(69)

e, _ d 2 + f 2 -ag-cg

h - 0„2

(70)

then gives the equation

u=- =Asin9 + Bcos8± JC sin(20) + D cos(20) + E
r

(71)

(Moulton 1970). If g = 0, then (59) becomes instead

1 _ a cos 2 + 26 cos sin + c sin 2

r ~~ ~~~ ~ 2(dcos0 + /sin0) ~~"~*

(72)

Therefore, the general form of a quadratic curve in polar
coordinates is given by

A sin + B cos

for £ ^

u = < ± \/ C sin(20) + D cos(20) + £7

a cos 2 6+2b cos fl sin fl-f-c sin 2 fl r a

2(dcos0+/sin0) iUI i/ U '

(73)
see also CONIC SECTION, DISCRIMINANT (QUADRATIC

Curve), Elliptic Curve

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 200-201, 1987.

Casey, J. "The General Equation of the Second Degree."
Ch. 4 in A Treatise on the Analytical Geometry of the
Point, Line, Circle, and Conic Sections, Containing an
Account of Its Most Recent Extensions, with Numerous
Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, 8c
Co., pp. 151-172, 1893.

Moulton, F. R. "Law of Force in Binary Stars" and "Geo-
metrical Interpretation of the Second Law." §58 and 59 in
An Introduction to Celestial Mechanics, 2nd rev. ed. New
York: Dover, pp. 86 89, 1970.

Quadratic Effect

Quadratic Field 1479

Quadratic Effect

see Prime Quadratic Effect

Quadratic Equation

A quadratic equation is a second-order POLYNOMIAL

ax + bx + c = 0,

(1)

2 b

x H — x

with a / 0. The roots x can be found by COMPLETING
the Square:

J,

(2)
(3)
(4)

( as+ s) 1

-2 + i-

a 4a 2

c
a

2 l2

4ac

4a 2

b ±Vb 2 - Aac

x-{ = — .

2a 2a

Solving for x then gives

-b ± Vb 2 - Aac

x = .

2a

This is the QUADRATIC FORMULA.

(5)

An alternate form is given by dividing (1) through by
x 2 :

a+- + 4=0 (6)

x x 2

V x 2 ex J
Therefore,

4ac b — Aac

Ac Ac

Ac

(8)

(9)

(10)

,- _. (11)

-b ± Vb 2 - Aac

This form is helpful if b 2 ^> 4ac, in which case the usual
form of the Quadratic Formula can give inaccurate
numerical results for one of the ROOTS. This can be
avoided by denning

X

b

2c

_ ± v^

— Aac
2c

1

b±VW

- Aac

X

X =

2c
2c

6 + sgn(6)y6 2 - 4ac

(12)

so that b and the term under the SQUARE ROOT sign
always have the same sign. Now, if b > 0, then

-= -|(6+ \/b 2 -Aac)

(13)

-2 b - Vb 2 - Aac _ -2(6 - \/& 2 - 4ac )

q b + Vb 2 - 4ac b - Vb 2 - Aac b 2 - (b 2 - Aac)

-2(6- V& 2 - 4ac ) _ -6+ Vb 2 - 4ac

4ac

2ac

(14)

a —6 — V& 2 — Aac

xi = - =

a 2a

c — 6 + \/& 2 — 4ac

x 2 = - = .

q 2a

(15)
(16)

Similarly, if b < 0, then

= -|(6- v /6 2 -4ac) = £(-&+ \/& 2 -4ac) (17)

-6 + V& 2 - 4ac 6 + Vb 2 - Aac
b + V& 2 - 4ac

b + Vb 2 - 4ac _ 2(&+V& 2 -4oc)
-b 2 + (b 2 - 4ac)

-2ac

-6 — Vb 2 — 4ac
2a^ '

(18)

»! = * =

a

-b + Vb 2 - 4ac
2a

(19)

c

x 2 = - =
5

— b — Vb 2 — 4ac
2a

(20)

Therefore, the ROOTS are always given by xi = q/a and
x 2 = c/q.

see also Carlyle Circle, Conic Section, Cubic
Equation, Quartic Equation, Quintic Equation,
Sextic Equation

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 17, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 9, 1987.

Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 91-92, 1996,

King, R. B. Beyond the Quartic Equation. Boston, MA:
Birkhauser, 1996.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Quadratic and Cubic Equations." §5.6
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 178-180, 1992.

Spanier, J. and Oldham, K. B. "The Quadratic Function
ax 2 + bx + c and Its Reciprocal." Ch. 16 in An Atlas
of Functions. Washington, DC: Hemisphere, pp. 123-131,
1987.

Quadratic Field

An Algebraic Integer of the form a + by/D where D
is Squarefree forms a quadratic field and is denoted
Q(y/D). If D > 0, the field is called a REAL QUAD-
RATIC FIELD, and if D < 0, it is called an IMAGINARY
Quadratic Field. The integers in Q(v / i) are sim-
ply called "the" INTEGERS. The integers in Q(V^T)
are called GAUSSIAN INTEGERS, and the integers in
Q(v / -3) are called Eisenstein Integers. The Al-
gebraic Integers in an arbitrary quadratic field do

1480 Quadratic Form

Quadratic Formula

not necessarily have unique factorizations. For exam-
ple, the fields Q(v / — 5) and Q(\/— 6) are not uniquely-
factorable, since

21 = 3-7 = (l + 2V^5)(l-2 v /r 5) (1)

6 = -V6(V^6) = 2-3, (2)

although the above factors are all primes within these
fields. All other quadratic fields Q(VD) with |D| < 7
are uniquely factorable.

Quadratic fields obey the identities

(a 4- bVB) ± (c + dy/D) = (a ± c) + (6 ± d)VD, (3)
(a + bVD) (c + dVD) = (ac + bdD) + (ad + be) \/D, (4)

and

a + b\/D ac — bdD be — ad r=r
c + dy/D = c 2 - d?D + c 2 - d 2 £) •

(5)

The Integers in the real field Q(\/T> ) are of the form
r + sp, where

f \/5 for D = 2 or D = 3 (mod 4) ( ,

9 \±(-i + y/D) forl> = l (mod 4). W

There exist 22 quadratic fields in which there is a EU-
CLIDEAN Algorithm (Inkeri 1947).

see also ALGEBRAIC INTEGER, ElSENSTEIN INTEGER,

Gaussian Integer, Imaginary Quadratic Field,

Integer, Number Field, Real Quadratic Field

References

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 153-154, 1993.

Quadratic Form

A quadratic form involving n REAL variables xi, #2, • ■ ■ ,
x n associated with the n x n MATRIX A = a^ is given
by

Q(xi,X2,. • ■ ,X n ) = CLijXiXjj (1)

where EINSTEIN SUMMATION has been used. Letting
x be a VECTOR made up of xi, . . . , x n and x T the
Transpose, then

equivalent to

Q(x) = x T Ax,

Q(x) = (x,Ax)

(2)

(3)

It is always possible to express an arbitrary quadratic
form

Q(x) = ctijXiXj (5)

in the form

Q(x) = (x,Ax), (6)

where A = an is a Symmetric Matrix given by

_ J eta i

= 3

*3-

(7)

Any Real quadratic form in n variables may be reduced
to the diagonal form

Q( X ) = AiZi + X 2 X 2 + • ■ ■ + KXrx

(8)

with Ai > A2 > . . . > A n by a suitable orthogonal
point-transformation. Also, two real quadratic forms
are equivalent under the group of linear transformations
Iff they have the same Rank and SIGNATURE.

see also DISCONNECTED FORM, INDEFINITE QUAD-
RATIC Form, Inner Product, Integer-Matrix
Form, Positive Definite Quadratic Form, Posi-
tive Semidefinite Quadratic Form, Rank (Quad-
ratic Form), Signature (Quadratic Form), Syl-
vester's Inertia Law

References

Buell, D. A. Binary Quadratic Forms: Classical Theory and
Modern Computations. New York: Springer- Verlag, 1989.

Conway, J. H. and Fung, F. Y. The Sensual (Quadratic)
Form. Washington, DC: Math. Assoc. Amer., 1998.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1104-106, 1979.

Lam, T. Y. The Algebraic Theory of Quadratic Forms. Read-
ing, MA: W. A. Benjamin, 1973.

Quadratic Formula

The formula giving the Roots of a Quadratic Equa-
tion

(i)

ax +bx + c =

_ -b ± Vb 2 - Aac
~ 2a

An alternate form is given by

2c

-b ± Vb 2 - 4ac

see also QUADRATIC EQUATION

(2)

(3)

in Inner Product notation. A Binary Quadratic
FORM has the form

Q{?i v) = anx 2 + 2a 12 xy + a 2 2y 2 • (4)

Quadratic Integral

Quadratic Irrational Number 1481

Quadratic Integral

To compute integral of the form

/

dx

a + bx + ex 2 '

(i)

Quadratic Invariant

Given the Binary Quadratic Form

ax + 2bxy + cy

with Discriminant b 2 - ac, let

Complete the Square in the Denominator to ob-
tain

r dx i r

J a + bx + cx 2 ~ c J ( x +±)

dx

■, + bx + cx* cj (.+ £)'+(*_£)■

Let u = x + 6/ 2c, Then define

(2)

x = pX + qY
y-rX + sY.

(1)

(2)
(3)

Then

.2 a, b 1 . ,2\

a(pX + gY) 2 + 2b(pX + qY)(rX + sY) + c(rX + sYf
= AX 2 + 2BXY + CY 2 , (4)

4^ 9 '

(3)

where

where

q = 4ac- b 2 (4)

is the Negative of the Discriminant. If q < 0, then

1

A =fc^

Now use Partial Fraction Decomposition,

(5)

du

1

cj (u + A)(u-A)

A — ap 2 + 2bpr + cr 2

5 — apq + 6(ps + qr) 4- crs

C = aq 2 + 26gs + cs 2 ,

5 2 - AC = [a 2 p 2 q 2 + b 2 (ps + gr) 2 + cVs 2

+ 2abpq(ps + gr) + 2acpqrs + 2bcrs(ps + qr)]

(5)
(6)
(7)

= c / (^+1 + T^a) du ~ (ap2 + 2bpr + C|,2 )( a ? 2 + 2b 1 s + " 2 )

^ . . 9 9 2, 9 9 2. ~, 2 .,222.

\u-\- A u- A)

Ax(u- A) + A 2 (u + A)

v? -A 2
(A 1 +A 2 )u + A(A 2 -A 1 )
u 2 - A 2

(6)

, (7)

so A 2 + Ai — => A 2 = -Ai and A(A 2 - Ax) =
-2^4i = 1 =^> A x = -1/(2A). Plugging these in,

l /* /__l l_ J- 1_A ,

c J V 2A« + A 2A«-AJ U

2,4c

[- ln(u + A) + ln(u - A)]

2,4c \u + A)

1 ln ^+^-^v^g \

= ln

2c ' 2c ^

/ 2cz + 6 - v 7 "^

-q \2cx-\-b+-

(8)

for g < 0. Note that this integral is also tabulated in
Gradshteyn and Ryzhik (1979, equation 2.172), where
it is given with a sign flipped.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.

2 2 2 , .2 2 2 . OI 2 , *2 2 2 , 2 2 2

— apq +b p s +26 pqrs + b q r +c r s

+ 2abp 2 qs + 2abpq 2 r + 2acpqrs + 2bcprs + 2bcqr 2 s

— a 2 p 2 q 2 - 2abp 2 qs — acp 2 s 2 — 2abpq 2 r — Ab 2 pqrs

ol 2 22 l2 222

— 2bcprs — acq r — 2bcqr s — c r s

— b 2 p 2 s 2 — 2b 2 pqrs + b 2 q 2 r 2 + 2acpqrs — acp 2 s 2

2 2

— acq r

= p 2 s 2 (b 2 — ac) + q 2 r 2 (b 2 — ac) — 2pqrs(b 2 — ac)

= (6 2 — ac)(p s — 2pgrs + q r )

= (ps — rq) (6 — ac), (8)

Surprisingly, this is the same discriminant as before, but
multiplied by the factor (ps — rq) 2 . The quantity ps — rq
is called the MODULUS.

see also ALGEBRAIC INVARIANT

Quadratic Irrational Number
An Irrational Number of the form

p±Vd

Q '

where P and Q are INTEGERS and D is a SQUARE-
FREE Integer. Quadratic irrational numbers are some-
times also called QUADRATIC SURDS. In 1770, Lagrange
proved that that any quadratic irrational has a CONTIN-
UED Fraction which is periodic after some point.

see also Continued Fraction, Quadratic Surd

1482 Quadratic Map

Quadratic Map

A 1-D MAP often called "the" quadratic map is defined

X n +l = Xn + C. (1)

This is the real version of the complex map defining
the Mandelbrot Set. The quadratic map is called
attracting if the JACOBIAN J < 1, and repelling if J > 1.
Fixed Points occur when

x™ = \z w ? + c

( x Wf-x w +c =

(2)

(3)
(4)

Period two Fixed Points occur when

Xn+2 — #n+l + C = (x n + c) + C

— x n + 2cx n + (c + c) — x n (5)

x 4 + 2x 2 -x + {cx 2 +c) = Oc 2 -z + c)(z 2 +;r-hl + c) =

(6)
x% ) = ±[l±y/l-4(l + c)] = ±(l±V=3=te). (7)

Period three Fixed Points occur when

3 , / 2

z + z b + (3c + IK + (2c + l)aT + (c + 3c + l)z

+(c + l) 2 z + (c 3 + 2c 2 + c + 1) - 0. (8)

The most general second-order 2-D MAP with an elliptic
fixed point at the origin has the form

x = x cos a - y sin a + a2o£ + duxy H~ ao2y (9)
y = zsina + ycosa -f b2ox + bnxy + bo2y . (10)

The map must have a DETERMINANT of 1 in order to be
AREA preserving, reducing the number of independent
parameters from seven to three. The map can then be
put in a standard form by scaling and rotating to obtain

x — xcosa — y sin a + x 2 sin a (11)

y = x sin a + y cos a — x cos a. (12)

The inverse map is

x — x cos a + y sin a (13)

y = —x sin a + y cos a: -j- (x cos a -f y sin a) . (14)

The Fixed Points are given by

Xi sin a + 2xi cos a — Xj_i — a^i+i — (15)

see also Bogdanov Map, Henon Map, Logistic
Map, Lozi Map, Mandelbrot Set

Quadratic Reciprocity Theorem

Quadratic Mean

see Root-Mean-Square

Quadratic Reciprocity Law

see Quadratic Reciprocity Theorem

Quadratic Reciprocity Relations

-1

= (-1)

(P"l)/2

- ) = (-l)^- 1 )/ 8

f i (-»

[(p-l)/2][(flf-l)/2]

(1)

(2)
(3)

where (|) is the Legendre Symbol.

see also Quadratic Reciprocity Theorem

Quadratic Reciprocity Theorem

Also called the AUREUM THEOREMA (GOLDEN THEO-
REM) by Gauss. If p and q are distinct Odd Primes,
then the CONGRUENCES

x = q (mod p)
x = p (mod q)

are both solvable or both unsolvable unless both p and q
leave the remainder 3 when divided by 4 (in which case
one of the Congruences is solvable and the other is
not). Written symbolically,

-(-1)

(p~l)(q-l)/4

where

p \ _ j 1 for x 2 = p (mod q) solvable for x
q J ~ \ — 1 for x 2 = p (mod q) not solvable for x

is known as a LEGENDRE SYMBOL. Legendre was the
first to publish a proof, but it was fallacious. Gauss
was the first to publish a correct proof. The quadratic
reciprocity theorem was Gauss's favorite theorem from
Number Theory, and he devised many proofs of it over
his lifetime.

see also Jacobi Symbol, Kronecker Symbol, Leg-
endre Symbol, Quadratic Residue, Reciprocity
Theorem

References

Courant, R. and Robbins, H. What is Mathematics? : An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 39, 1996.

Ireland, K. and Rosen, M. "Quadratic Reciprocity." Ch. 5 in
A Classical Introduction to Modern Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 50-65, 1990.

Nagell, T. "Theory of Quadratic Residues." Ch. 4 in Intro-
duction to Number Theory. New York: Wiley, 1951.

Riesel, H. "The Law of Quadratic Reciprocity." Prime Num-
bers and Computer Methods for Factorization, 2nd ed.
Boston, MA: Birkhauser, pp. 279-281, 1994.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 42-49, 1993.

Quadratic Recurrence

Quadratic Recurrence

N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.

A quadratic recurrence is a RECURRENCE RELATION on
a SEQUENCE of numbers {x n } expressing x n as a second
degree polynomial in Xk with k < n. For example,

X n = X n -\X n -2 (1)

is a quadratic recurrence. Another simple example is

(X n -l)

(2)

with xq = 2, which has solution x n = 2 2 . Another ex-
ample is the number of "strongly" binary trees of height
< n, given by

Vn = (j/n-l) + 1

with yo = 1. This has solution

where

= exp

y n = I c 2 " I ,

^2- J - 1 ln(l + yr 2 )

(3)
(4)

1.502836801.

(5)

and |_zj is the FLOOR FUNCTION (Aho and Sloane 1973).
A third example is the closest strict underapproximation
of the number 1,

xi

(6)

where 1 < Z\ < . . . < z n are integers. The solution is
given by the recurrence

Z n = (z n -l) — Z n ~i + 1,

with z\ = 2. This has a closed solution as

Z n =

dT +

i\

(7)

(8)

where

We

exp

I j=i

ln[l + (2^-l)- 2 ]

1.2640847353... (9)

(Aho and Sloane 1973). A final example is the well-
known recurrence

(Cn-l)

M

(10)

with co — used to generate the MANDELBROT SET.
see also MANDELBROT SET, RECURRENCE RELATION

References

Aho, A. V. and Sloane, N. J. A. "Some Doubly Exponential

Sequences." Fib. Quart. 11, 429-437, 1973.
Finch, S. "Favorite Mathematical Constants." http://www.

mathsof t . com/asolve/constant/quad/quad.html.

Quadratic Residue 1483

Quadratic Residue

If there is an INTEGER x such that

x 2 = q (mod p) ,

(i)

then q is said to be a quadratic residue of x mod p. If
not, q is said to be a quadratic nonresidue of x mod
p. For example, 4 2 = 6 (mod 10), so 6 is a quadratic
residue (mod 10). The entire set of quadratic residues
(mod 10) are given by 1, 4, 5, 6, and 9, since

1 (mod 10) 2 2 = 4 (mod 10) 3 2 = 9 (mod 10)

2 = 6 (mod 10) 5 2 = 5 (mod 10) 6=6 (mod 10)
2 = 9 (mod 10) 8 2 = 4 (mod 10) 9 2 = 1 (mod 10)

making the numbers 2, 3, 7, and 8 the quadratic non-
residues (mod 10).

A list of quadratic residues for p < 29 is given below
(Sloane's A046071), with those numbers < p not in the
list being quadratic nonresidues of p.

p

Quadratic Residues

1

(none)

2

1

3

1

4

1

5

1,4

6

1,3,

4

7

1,2,

4

8

1,4

9

1,4,

7

10

1,4,

5,

6,

9

11

1,3,

4,

5,

9

12

1,4,

9

13

1,3,

4,

9,

10, 12

14

1,2,

4,

7,

8, 9, 11

15

1,4,

6,

9,

10

16

1,4,

9

17

1,2,

4,

8,

9, 13, 15, 16

18

1,4,

7,

9,

10, 13, 16

19

1,4,

5,

6,

7, 9, 11, 16, 17

20

1,4,

5,

9,

16

The UNITS in the integers (mod rc),
Squares are the quadratic residues.

which are

Given an Odd Prime p and an Integer a, then the
Legendre Symbol is given by

_ ( I if a is a quadratic residue mod p

-1 otherwise.

If

„(p-1)/2

±1 (mod p) ,

(2)

(3)

1484 Quadratic Residue

Quadratic Sieve Factorization Method

then r is a quadratic residue (+) or nonresidue (— ). This
can be seen since if r is a quadratic residue of p, then
there exists a square x 2 such that r = x 2 (mod p), so

r (p-i)/2

00

,2Up-1)/2

^P- 1

(modp), (4)

and x p l is congruent to 1 (modp) by Fermat's Little
Theorem, x is given by

q k+1 (mod p)

for p = 4k + 3
q k+1 (mod p)

for p = 8k + 5 and g 2fe+1 == 1 (mod p)
(4^(2+1) ( m odp)

for p = 8k + 5 and <? 2fe+1 = -1 (mod p) .

(5)

More generally, let q be a quadratic residue modulo an
Odd Prime p. Choose h such that the Legendre Sym-
bol (h 2 — 4q/p) = — 1. Then defining

Vi=h

V 2 = h 2 - 2q

Vi = hVi-i - qVi-2

for i > 3,

gives

V 2i = V* 2 - 2q*
V 2i+ i = ViV i+1 - hrj ,

and a solution to the quadratic CONGRUENCE is

x =.y (p+1)/2 (^-g— ) ^ mod P ^'

(6)
(7)
(8)

(9)
(10)

(11)

The following table gives the PRIMES which have a given
number d as a quadratic residue.

d

Primes

-6

24^ + 1,5,7,11

-5

20ft + 1,3, 7, 9

-3

6/c + l

-2

8k + 1,3

-1

4fc + l

2

8ft ±1

3

12ft ±1

5

10ft ±1

6

24ft ±1,5

Finding the Continued Fraction of a Square Root
y/D and using the relationship

Qn

D

Q n ~i
for the nth CONVERGENT P n /Qn gives

P n 2 = -Q n Q n -! (modD).

(12)

(13)

Therefore, — Q n Qn-i is a quadratic residue of D. But
since Q\ — 1, — Q2 is a quadratic residue, as must be
— Q2Q3. But since — Q2 is a quadratic residue, so is Q3,
and we see that (—l) n ~ 1 Q n are all quadratic residues
of D. This method is not guaranteed to produce all
quadratic residues, but can often produce several small
ones in the case of large D, enabling D to be factored.

The number of SQUARES s(n) in Z n is related to the
number q(n) of quadratic residues in Z„. by

q(p n ) = s(p n ) - s(p n ' 2 )

(14)

for n > 3 (Stangl 1996). Both q and s are Multiplica-
tive Functions.

see also Euler's Criterion, Multiplicative Func-
tion, Quadratic Reciprocity Theorem, Riemann
Hypothesis

References

Burton, D. M. Elementary Number Theory, ^th ed. New-
York: McGraw-Hill, p. 201, 1997.

Courant, R. and Robbins, H. "Quadratic Residues." §2.3 in
Supplement to Ch. 1 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England; Oxford University Press, pp. 38-40, 1996.

Guy, R. K. "Quadratic Residues. Schur's Conjecture" and
"Patterns of Quadratic Residues." §F5 and F6 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 244-248, 1994.

Niven, I. and Zuckerman, H. An Introduction to the Theory
of Numbers, 4th ed. New York: Wiley, p. 84, 1980.

Rosen, K. H. Ch. 9 in Elementary Number Theory and Its
Applications, 3rd ed. Reading, MA: Addison- Wesley, 1993.

Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 63-66, 1993.

Sloane, N. J. A. Sequence A046071 in "An On-Line Version
of the Encyclopedia of Integer Sequences."

Stangl, W. D. "Counting Squares in Z n ." Math. Mag. 69,
285-289, 1996.

Wagon, S. "Quadratic Residues." §9.2 in Mathematica in
Action. New York: W. H. Freeman, pp. 292-296, 1991.

Quadratic Sieve Factorization Method

A procedure used in conjunction with DIXON'S FACTOR-
IZATION Method to factor large numbers. The rs are
chosen as

Lv^J+fc, (i)

where k = 1, 2, ... and [x\ is the Floor Function.
We are then looking for factors p such that

n = r 2 (mod p) ,

(2)

which means that only numbers with Legendre Sym-
bol (n/p) = 1 (less than N = 7r(d) for trial divisor d)
need be considered. The set of PRIMES for which this
is true is known as the Factor Base. Next, the Con-
gruences

x 2 = n (mod p) (3)

must be solved for each p in the Factor Base. Fi-
nally, a sieve is applied to find values of f(r) = r — n

Quadratic Surd

Quadratrix of Hippias 1485

which can be factored completely using only the FAC-
TOR Base. Gaussian Elimination is then used as in
Dixon's Factorization Method in order to find a
product of the /(r)s, yielding a Perfect Square.

The method requires about exp ( -\/log n log log n ) steps,
improving on the CONTINUED FRACTION FACTORIZA-
TION Algorithm by removing the 2 under the Square
Root (Pomerance 1996). The use of multiple Polyno-
mials gives a better chance of factorization, requires a
shorter sieve interval, and is well-suited to parallel pro-
cessing.

see also Prime Factorization Algorithms, Smooth

Number

References

Alford, W. R. and Pomerance, C. "Implementing the Self
Initializing Quadratic Sieve on a Distributed Network."
In Number Theoretic and Algebraic Methods in Com-
puter Science, Proc. Internat. Moscow Conf., June-July
1993 (Ed. A. J. van der Poorten, I. Shparlinksi, and
H. G. Zimer). World Scientific, pp. 163-174, 1995.

Brent, R. P. "Parallel Algorithms for Integer Factorisation."
In Number Theory and Cryptography (Ed. J. H. Lox-
ton). New York: Cambridge University Press, 26-37, 1990.
ftp : //nimbus . aim. edu. au/pub/Brent/115 . dvi . Z.

Bressoud, D. M. Ch. 8 in Factorization and Prime Testing.
New York: Springer- Verlag, 1989.

Gerver, J. "Factoring Large Numbers with a Quadratic
Sieve." Math. Comput 41, 287-294, 1983.

Lenstra, A. K. and Manasse, M. S. "Factoring by Electronic
Mail." In Advances in Cryptology — Eurocrypt '89 (Ed.
J. -J. Quisquarter and J. Vandewalle). Berlin: Springer-
Verlag, pp. 355-371, 1990.

Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996.

Pomerance, C.; Smith, J. W.; and Tuler, R. "A Pipeline Ar-
chitecture for Factoring Large Integers with the Quadratic
Sieve Method." SI AM J. Comput 17, 387-403, 1988.

Quadratic Surd

see Quadratic Irrational Number

Quadratic Surface

There are 17 standard-form quadratic surfaces.
general quadratic is written

The

ax 2 + by 2 + cz 2 + 2fyz + 2gzx + 2hxy

+2px + 2qy + 2rz + d = 0. (1)

Define

'a h

9'

e =

h b
.9 f

f
c _

(2)

'a h

9

P~

E =

h b
9 f

f
c

r

(3)

,V q

r

d]

p3 = rank e

(4)

p4 = rank E

(5)

A = c

let E,

(6)

and fci, kz, as k$ are the roots of

a — x h g

h b — x f
g f c- x

= 0.

Also define

i. _ f 1 if the signs of
~~ I otherwise.

nonzero ks are the same

(7)

(8)

sgn

Surface

Equation

P3

P4

(A)

k

coincident planes

x 2 =

1

1

ellipsoid (S)

£ + £ + £ = -!

3

4

+

1

ellipsoid (3ft)

£ + £ + ^ = 1

3

4

-

1

elliptic cone (3)

S+£-£=o

3

3

1

elliptic cone (3ft)

z * - *L + vL

Z a 2 ^ b 2

3

3

elliptic cylinder (£$)

— 4- *i — — i

a 2 -r b 2 — J-

2

3

1

elliptic cylinder (3ft)

si + xL - i

a i -r b 2 — -t

2

3

1

elliptic paraboloid

z — — 4- ^~

Z a 2 ^ b 2

2

4

-

1

hyperbolic cylinder

£■ - 4 - -l

2

3

hyperbolic paraboloid

z- £ - si

* _ b 2 a*

2

4

+

hyperboloid of one sheet

*1 + vl _ *i - !

a 2 T b 2 c 2 — -L

3

4

+

hyperboloid of two sheets

-J + ST ~ ^2 ~ "I

3

4

-

intersecting planes (S)

a 2 * b 2 U

2

2

1

intersecting planes (3ft)

si - ^ -0

2

2

parabolic cylinder

x 2 + 2rz —

1

3

parallel planes (S)

^. 2 „ 2

x = — a

1

2

parallel planes (3ft)

™2 „2

x = a

1

2

see also CUBIC SURFACE, ELLIPSOID, ELLIPTIC CONE,

Elliptic Cylinder, Elliptic Paraboloid, Hyper-
bolic Cylinder, Hyperbolic Paraboloid, Hyper-
boloid, Plane, Quartic Surface, Surface

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 210-211, 1987.

Quadratrix of Hippias

The quadratrix was discovered by Hippias of Elias in 430
BC, and later studied by Dinostratus in 350 BC (Mac-
Tutor Archive). It can be used for ANGLE TRISECTION
or, more generally, division of an Angle into any inte-
gral number of equal parts, and CIRCLE SQUARING. In
Polar Coordinates,

Trp = 2r6 esc 0,

so

p7r sin 6

r = —e—'

1486 Quadrature

Quadricorn

which is proportional to the COCHLEOID.

References

Lawrence, J. D. A Catalog of Special Plane Curves. New

York: Dover, pp. 195 and 198, 1972.
Lee, X. "Quadratrix of Hippias." http://www . best . com/-

xah/SpecialPlaneCurvesjdir/QuadratrixOf Hippias _dir/

quadratrixOf Hippias .html.
MacTutor History of Mathematics Archive. "Quadratrix of

Hippias." http: // www - groups . dcs . st - and .ac.uk/

-history/Curves/Quadratrix . html.

Quadrature

The word quadrature has (at least) three incompati-
ble meanings. Integration by quadrature either means
solving an INTEGRAL analytically (i.e., symbolically in
terms of known functions), or solving of an integral
numerically (e.g., Gaussian Quadrature, Quadra-
ture Formulas). The word quadrature is also used
to mean SQUARING: the construction of a square using
only Compass and Straightedge which has the same
Area as a given geometric figure. If quadrature is pos-
sible for a Plane figure, it is said to be Quadrable.

For a function tabulated at given values xi (so the AB-
SCISSAS cannot be chosen at will), write the function <j>
as a sum of ORTHONORMAL FUNCTIONS Pj satisfying

J a

pi(x)pj(x)W(x)dx = Si

3=0

(1)

(2)

and plug into

ob

m

(3)

j=i

giving

2_]o > jPj(x)W(x) dx = 2_, Wi

- j=0 t=l

j=o

(4)

But we wish this to hold for all degrees of approximation,
so

Pj(x)W(x)dx — a,j y ^Wjpj(xj)

/ pj(x)W(x) dx = / ^ w t pj(xi).
Setting i — in (1) gives

«/ a

Po(x)pj(x)W(x) dx = Sqj.

(5)
(6)

(7)

The zeroth order orthonormal function can always be

taken as po(x) = 1, so (7) becomes

J a

p j {x)W(x)dx^S 0j .(8)

n

= ^2wiPj(xi), (9)

where (6) has been used in the last step. We therefore
have the MATRIX equation

Po(xi)

Pi(xi)

.Pn~l(xi)

Po(x n )
Pl(x n )

Pn-l(Xn),

r W\ '

"1"

W2

=

_W n _

_0_

, (io)

which can be inverted to solve for the wis (Press et al.
1992).

see also CALCULUS, CHEBYSHEV-GAUSS QUADRATURE,

Chebyshev Quadrature, Derivative, Fundamen-
tal Theorem of Gaussian Quadrature, Gauss-
Jacobi Mechanical Quadrature, Gaussian Quad-
rature, Hermite-Gauss Quadrature, Hermite
Quadrature, Jacobi-Gauss Quadrature, Jacobi
Quadrature, Laguerre-Gauss Quadrature, La-
guerre Quadrature, Legendre-Gauss Quadra-
ture, Legendre Quadrature, Lobatto Quadra-
ture, Mechanical Quadrature, Mehler Quadra-
ture, Newton-Cotes Formulas, Numerical Inte-
gration, Radau Quadrature, Recursive Mono-
tone Stable Quadrature

References

Abramowitz, M. and Stegun, C. A. (Eds.). "Integration."
§25.4 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 885-897, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 365-366, 1992.

Quadrature Formulas

see Newton-Cotes Formulas

Quadric

An equation of the form

+

y

■ + ■

a 2 + o b 2 + e c 2 + e

where 9 is said to be the parameter of the quadric.

Quadricorn

A Flexible Polyhedron due to C. Schwabe (with the
appearance of having four horns) which flexes from one
totally flat configuration to another, passing through in-
termediate configurations of positive VOLUME.
see also FLEXIBLE POLYHEDRON

Quadrifolium
Quadrifolium

The ROSE with n — 2. It has polar equation

r = asin(20),
and Cartesian form

/ 2 , 2\3 A 2 2 2

566 a/50 BlFOLIUM, FOLIUM, ROSE, TRIFOLIUM

Quadrilateral

A A

B B

A four-sided POLYGON sometimes (but not very often)
also known as a TETRAGON. If not explicitly stated, all
four Vertices are generally taken to lie in a Plane. If
the points do not lie in a Plane, the quadrilateral is
called a Skew Quadrilateral.

For a planar convex quadrilateral (left figure above),
let the lengths of the sides be a, 6, c, and d, the
Semiperimeter s, and the Diagonals p and q. The
Diagonals are Perpendicular Iff a 2 + c 2 = b 2 + d 2 .
An equation for the sum of the squares of side lengths
is

a 2 + b 2 + c 2 + d 2 = p 2 + q + Ax\ (1)

where x is the length of the line joining the MIDPOINTS
of the Diagonals. The Area of a quadrilateral is given
by

K = \\pq sin

= \(b 2 +d 2 -a 2 -c 2 )tan<9

= \\/4p 2 q 2 - (6 2 + d 2 - a 2 - c 2 ) 2
= y^s - a) (s - 6) (a - c)(s - d) - a&cd cos 2 [§(A + 5)],

(5)

(2)
(3)
(4)

Quadriplanar Coordinates 1487

where (4) is known as Bretschneider's Formula
(Beyer 1987).

A special type of quadrilateral is the Cyclic Quadri-
lateral, for which a Circle can be circumscribed so
that it touches each Vertex. For Bicentric quadri-
laterals, the ClRCUMCIRCLE and INCIRCLE satisfy

2r 2 (R 2 -s 2 ) = (R 2 -s 2 ) 2

a 2 2

4r s ,

(6)

where R is the ClRCUMRADIUS, r in the INRADIUS, and
s is the separation of centers. A quadrilateral with two
sides Parallel is called a Trapezoid.

There is a relationship between the six distances di2,
di3t di4, d23, d24, and d34 between the four points of a
quadrilateral (Weinberg 1972):

0:

= di2 d34 + di3 d24 + di4 d23 + C?23 d\
+ ^24^13 + ^34^12

-h d 12 d 2 3d3 1 + d 12 d24d 41

d 13 d3 4 d 41

+ dzzd^d^ ~ ^12^23^34 — di3d32d24

t2 j2 j2 j2 j2 j2 t2 j2 t2

— 012024043 — 014042023 — OX3O34O42

— di4d43d32 — d 2 3d 31 di4 — d2idi3d34

i2 t2 i2 t2 j2 t2 t2 t2 t2

— O24O41O13 — 021014043 — 03iOi2024

— 03202idj4.

(7)

see also Bimedian, Brahmagupta's Formula, Bret-
schneider's Formula, Complete Quadrilateral,
Cyclic-Inscriptable Quadrilateral, Cyclic
Quadrilateral, Diamond, Eight-Point Circle
Theorem, Equilic Quadrilateral, Fano's Axiom,
Leon Anne's Theorem, Lozenge, Orthocentric
Quadrilateral, Parallelogram, Ptolemy's The-
orem, Rational Quadrilateral, Rhombus, Skew
Quadrilateral, Trapezoid, Varignon's Theorem,
von Aubel's Theorem, Wittenbauer's Parallel-
ogram

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables.

28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.
Routh, E. J. "Moment of Inertia of a Quadrilateral." Quart.

J. Pure Appl. Math. 11, 109-110, 1871.
Weinberg, S. Gravitation and Cosmology: Principles and

Applications of the General Theory of Relativity. New

York: Wiley, p. 7, 1972.

Quadrillion

In the American system, 10 15 .

see also LARGE NUMBER

Quadriplanar Coordinates

The analog of TRILINEAR COORDINATES for TETRAHE-
DRA.

See also TETRAHEDRON, TRILINEAR COORDINATES

1488 Quadruple

Quarter

References

Alt shiller- Court, N. Modern Pure Solid Geometry. New
York: Macmillan, 1935.

Mitrinovic, D. S.; Pecaric, J. E.; and Volenec, V. Ch. 19
in Recent Advances in Geometric Inequalities. Dordrecht,
Netherlands: Kluwer, 1989.

Woods, F. S. Higher Geometry: An Introduction to Ad-
vanced Methods in Analytic Geometry. New York: Dover,
pp. 193-196, 1961.

Quadruple

A group of four elements, also called a QUADRUPLET or
Tetrad.

see also Amicable Quadruple, Diophantine Quad-
ruple, Monad, Pair, Prime Quadruplet, Py-
thagorean Quadruple, Quadruplet, Quintuplet,
Tetrad, Triad, Triple, Twins, Vector Quadru-
ple Product

Quadruple Point

A point where a curve intersects itself along four arcs.
The above plot shows the quadruple point at the ORIGIN
of the QUADRIFOLIUM (x 2 + y 2 f - 4z V = 0.
see also Double Point, Triple Point

References

Walker, R. J. Algebraic Curves. New York: Springer- Verlag,
pp. 57-58, 1978.

Quadruplet

see Quadruple

Quadtree

A Tree having four branches at each node. Quadtrees
are used in the construction of some multidimensional
databases (e.g., cartography, computer graphics, and
image processing). For a d-D tree, the expected num-
ber of comparisons over all pairs of integers for success-
ful and unsuccessful searches are given analytically for
d ~ 2 and numerically for d > 3 by Finch.

References

Finch, S. "Favorite Mathematical Constants." http://wwv.
mathsof t . c om/asolve/const ant /qdt/qdt .html.

Flajolet, P.; Gonnet, G.; Puech, C; and Robson, J. M. "Ana-
lytic Variations on Quadtrees." Algorithmica 10, 473—500,
1993.

Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 11—
13, 1991.

Quantic

An m-ary n-ic polynomial (i.e., a HOMOGENEOUS POLY-
NOMIAL with constant COEFFICIENTS of degree n in m
independent variables) .

see also Algebraic Invariant, Fundamental Sys-
tem, p-adic Number, Syzygies Problem

Quantifier

One of the operations Exists 3 or FOR ALL V.

see also Bound, Exists, For All, Free

Quantization Efficiency

Quantization is a nonlinear process which generates ad-
ditional frequency components (Thompson et al. 1986).
This means that the signal is no longer band-limited, so
the Sampling Theorem no longer holds. If a signal is
sampled at the Nyquist Frequency, information will
be lost. Therefore, sampling faster than the NYQUIST
FREQUENCY results in detection of more of the signal
and a lower signal-to- noise ratio [SNR]. Let (3 be the
OVERSAMPLING ratio and define

VQ

SNR,

-quant

SNR.

un quant

Then the following table gives values of t]q for a number
of parameters.

Quantization
Levels

VQ

VQ

((3 = 2)

2
3

4

0.64
0.81
0.88

0.74
0.89
0.94

The Very Large Array of 27 radio telescopes in Socorro,
New Mexico uses three-level quantization at /3 = 1, so
t)q = 0.81.

References

Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr.
Fig. 8.3 in Interferometry and Synthesis in Radio Astron-
omy. New York: Wiley, p. 220, 1986.

Quantum Chaos

The study of the implications of CHAOS for a system
in the semiclassical (i.e., between classical and quantum

mechanical) regime.

References

Ott, E. "Quantum Chaos." Ch. 10 in Chaos in Dynamical

Systems. New York: Cambridge University Press, pp. 334-

362, 1993.

Quarter

The Unit Fraction 1/4, also called one-fourth.

the value of KOEBE'S CONSTANT.
see also Half, Quartile

It is