Domino Problem
Dot Product 489
see also Fibonacci Number, Gomory's Theorem,
Hexomino, Pentomino, Polyomino, Tetromino,
Triomino
References
Dickau, R. M. "Fibonacci Numbers." http : //www .
prairienet , org/-pops/f ibboard . html.
Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer-
ican Book of Mathematical Puzzles & Diversions. New
York: Simon and Schuster, pp. 124-140, 1959.
Kraitchik, M. "Dominoes." §12.1.22 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 298-302, 1942.
Lei, A. "Domino." http://www.cs.ust.hk/-phxlipl/omino/
domino.html.
Madachy, J. S. "Domino Recreations." Madachy's Mathe-
matical Recreations. New York: Dover, pp. 209-219, 1979.
Domino Problem
see Wang's Conjecture
Donaldson Invariants
Distinguish between smooth MANIFOLDS in 4-D.
Donkin's Theorem
The product of three translations along the directed
sides of a TRIANGLE through twice the lengths of these
sides is the identity.
Donut
see Torus
Doob's Theorem
A theorem proved by Doob (1942) which states that any
random process which is both GAUSSIAN and MARKOV
has the following forms for its correlation function, spec-
tral density, and probability densities:
C y {r) = <r y 2 e- T/T *
G y (f)= 4r -" V " 2
pi(y)
P2(yi|jfe,r)
(27r/) 2 -h7>- 2
1
\/2n(Ty
y^Trtl-e- 2 -/-,)^
x exp
f [(w-v)-e- T/T -(yi-
\ 2(l-e- 2 ^)«V
y)f
where y is the Mean, a y the Standard Deviation,
and r r the relaxation time.
References
Doob, J. L. "Topics in the Theory of Markov Chains." Trans.
Amer. Math. Soc. 52, 37-64, 1942.
Dot
The "dot" • has several meanings in mathematics, in-
cluding Multiplication (a • b is pronounced "a times
6"), computation of a Dot PRODUCT (ab is pronounced
"a dot b"), or computation of a time Derivative (d is
pronounced "a dot").
see also Derivative, Dot Product, Times
Dot Product
The dot product can be defined by
X.Y=|X||Y|cosfi,
(1)
where 8 is the angle between the vectors. It follows
immediately that X ■ Y = if X is Perpendicular to
Y. The dot product is also called the INNER PRODUCT
and written (a, b). By writing
A X = A cos 9 A B x =B cos 8 B (2)
A y = AsiuOa By = B sin 8b } (3)
it follows that (1) yields
A-B = ABcos(6 a -0b)
= ^UB(cos 6a cos 9b + sin 9a sin 6b)
= A cos 9aB cos 8b + A sin 8aB sin 8b
= A x B x +A y B y . (4)
So, in general,
X • Y = xiyi + . . . + x n y n . (5)
The dot product is COMMUTATIVE
X-Y = Y X, (6)
Associative
(rX)-Y = r(X-Y), (7)
and Distributive
X>(Y + Z) = X^Y + X-Z. (8)
The Derivative of a dot product of Vectors is
dt
[ri(t)T 2 (t)] = n(t)
dr2 , dr
~dt
+ 5rT 2 (t). (9)
dt
490 Douady's Rabbit Fractal
The dot product is invariant under rotations
A' B' = AiBl = dijAjaikBk = {a^a^AjBk
= SjkAjBk = AjBj = A B, (10)
where EINSTEIN SUMMATION has been used.
The dot product is also defined for TENSORS A and B
by
A-B = A a B a . (11)
see also CROSS PRODUCT, INNER PRODUCT, OUTER
Product, Wedge Product
References
Arfken, G. "Scalar or Dot Product." §1.3 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 13-18, 1985.
Douady's Rabbit Fractal
A Julia Set with c = —0.123 + 0.745z, also known as
the Dragon Fractal.
see also San Marco Fractal, Siegel Disk Fractal
References
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 176, 1991.
Double Bubble
The planar double bubble (three circular arcs meeting
in two points at equal 120° ANGLES) has the minimum
Perimeter for enclosing two equal areas (Foisy 1993,
Morgan 1995).
see also Apple, Bubble, Double Bubble Conjec-
ture, Sphere-Sphere Intersection
References
Campbell, P. J. (Ed.). Reviews. Math. Mag. 68, 321, 1995.
Foisy, J.; Alfaro, M.; Brock, J.; Hodges, N.; and Zimba, J.
"The Standard Double Soap Bubble in R 2 Uniquely Min-
imizes Perimeter." Pacific J. Math. 159, 47-59, 1993.
Morgan, F. "The Double Bubble Conjecture." FOCUS 15,
6-7, 1995.
Peterson, I. "Toil and Trouble over Double Bubbles." Sci.
News 148, 101, Aug. 12, 1995.
Double Exponential Integration
Double Bubble Conjecture
Two partial SPHERES with a separating boundary
(which is planar for equal volumes) separate two vol-
umes of air with less Area than any other boundary.
The planar case was proved true for equal volumes by
J. Hass and R. Schlafy in 1995 by reducing the problem
to a set of 200,260 integrals which they carried out on
an ordinary PC.
see also DOUBLE BUBBLE
References
Haas, J. and Schlafy, R. "Double Bubbles Minimize."
Preprint, 1995.
Double Contraction Relation
A TENSOR t is said to satisfy the double contraction
relation when
im*,n c
ij ij OyriTi'
This equation is satisfied by
ro _ 2zz - xx - yy
t* 1 = t|(xz + zx) - ±t(y» - zy)
i ±2 = T§(xx + yy) ~ §*(xy - yx),
where the hat denotes zero trace, symmetric unit TEN-
SORS. These TENSORS are used to define the SPHERICAL
Harmonic Tensor.
see also SPHERICAL HARMONIC TENSOR, TENSOR
References
Arfken, G. "Alternating Series." Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, p. 140,
1985.
Double Cusp
see Double Point
Double Exponential Distribution
see FlSHER-TlPPETT DISTRIBUTION, LAPLACE DISTRI-
BUTION
Double Exponential Integration
An excellent NUMERICAL INTEGRATION technique used
by Maple V R4® (Waterloo Maple Inc.) for numerical
computation of integrals.
see also Integral, Integration, Numerical Inte-
gration
References
Davis, P. J. and Rabinowitz, P. Methods of Numerical Inte-
gration, 2nd ed. New York: Academic Press, p. 214, 1984.
Di Marco, G.; Favati, P.; Lotti, G.; and Romani, F. "Asymp-
totic Behaviour of Automatic Quadrature." J. Complexity
10, 296-340, 1994.
Mori, M. Developments in the Double Exponential Formula
for Numerical Integration. Proceedings of the Interna-
tional Congress of Mathematicians, Kyoto 1 990. New
York: Springer- Verlag, pp. 1585-1594, 1991.
Double Factorial
Mori, M. and Ooura, T. "Double Exponential Formulas for
Fourier Type Integrals with a Divergent Integrand." In
Contributions in Numerical Mathematics (Ed. R. P. Agar-
wal). World Scientific Series in Applicable Analysis, Vol. 2,
pp. 301-308, 1993.
Ooura, T. and Mori, M. "The Double Exponential Formula
for Oscillatory Functions over the Half Infinite Interval."
J. Corn-put. Appl. Math. 38, 353-360, 1991.
Takahasi, H. and Mori, M. "Double Exponential Formulas
for Numerical Integration." Pub. RIMS Kyoto Univ. 9,
721-741, 1974.
Toda, H. and Ono, H. "Some Remarks for Efficient Usage
of the Double Exponential Formulas." Kokyuroku RIMS
Kyoto Univ. 339, 74-109, 1978.
Double Gamma Function 491
For n Odd,
n!_ _ n(n- l)(n - 2) • • • (1)
n\\ ~ n(ra-2)(n-4)...(l)
= (n-l)(n-3)...(l) = (n-l)!!. (7)
For n Even,
n!_ _ ra(n-l)(n-2)---(2)
nil ~ n(n-2)(n-4)---(2)
= („-!)(„ -3)." (2) = (*-!)!!. (8)
Double Factorial
The double factorial is a generalization of the usual FAC-
TORIAL n! defined by
in • (n — 2) . . . 5 ■ 3 ■ 1 n odd
n- (n- 2)...6-4- 2 n even (1)
1 n=-l f 0.
For n = 0, 1, 2, . . . , the first few values are 1, 1, 2, 3, 8,
15, 48, 105, 384, ... (Sloane's A006882).
There are many identities relating double factorials to
Factorials. Since
(2n+l)!!2 n n!
= [(2n + l)(2n - 1) • • • l][2n][2(n - l)][2(n - 2)] • - • 2(1)
= [(2n + l)(2ra - 1) • • ■ l][2n(2n - 2)(2n - 4) ■ ■ • 2]
= (2n + l)(2n)(2n - l)(2n - 2)(2n - 3)(2n - 4) • • • 2(1)
= (2n+l)! f (2)
it follows that (2n+ 1)!! = ^7^. Since
(2n)!! = (2n)(2n-2)(2n-4)..-2
= [2(n)][2(n - l)][2(n - 2)] • • • 2 = 2 n n!, (3)
it follows that (2n)!! = 2 n n!. Since
(2n-l)!!2 n n!
= [(2n - l)(2n - 3) • • ■ l][2n][2(n - l)][2(n - 2)] • ■ • 2(1)
= (2n - l)(2n - 3) • • ■ l][2n(2n - 2)(2n - 4) • • • 2]
= 2n(2n - l)(2n - 2)(2n - 3)(2n - 4) • • • 2(1)
= (2n)!, (4)
Therefore, for any n,
it follows that
(2n-l)!! =
(2n)!
2 n nl '
(5)
Similarly, for n = 0, 1, . . . ,
( 9„ 1VI (- 1 )" (-l)r»2"n! ^
( ~ 2n " 1)!! = (2^I)TT = (2 n)! ' (6)
n!
i = (n ~ 1)!!
n! = n!!(n - 1)!!.
(9)
(10)
The Factorial may be further generalized to the Mul-
tifactorial
see also FACTORIAL, MULTIFACTORIAL
References
Sloane, N. J. A. Sequence A006882/M0876 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Double Folium
see BlFOLlUM
Double- Free Set
A SET of POSITIVE integers is double-free if, for any
integer #, the Set {x, 2x} <£_ S (or equivalent ly, if x 6 S
Implies 2x g S). Define
r(n) = max{5 : S C {1, 2, . . . , n} is double-free}.
Then an asymptotic formula is
r(n) ~ |n + 0(lnn)
(Wang 1989).
see also Triple-Free Set
References
Finch, S. "Favorite Mathematical Constants." http://vwv.
mathsoft.com/asolve/constant/triple/triple.html.
Wang, E. T. H. "On Double-Free Sets of Integers." Ars Corn-
bin. 28, 97-100, 1989.
Double Gamma Function
see DlGAMMA FUNCTION
492
Double Point
Doubly Magic Square
Double Point
A point traced out twice as a closed curve is traversed.
The maximum number of double points for a nondegen-
erate Quartic Curve is three. An Ordinary Double
Point is called a Node.
Arnold (1994) gives pictures of spherical and PLANE
Curves with up to five double points, as well as other
curves.
see also Biplanar Double Point, Conic Double
Point, Crunode, Cusp, Elliptic Cone Point,
Gauss's Double Point Theorem, Node (Alge-
braic Curve), Ordinary Double Point, Quadru-
ple Point Rational Double Point, Spinode, Tac-
node, Triple Point, Uniplanar Double Point
References
Aicardi, F. Appendix to "Plane Curves, Their Invariants,
Perestroikas, and Classifications." In Singularities & Bi-
furcations (V. I. Arnold). Providence, RI: Amer. Math.
Soc, pp. 80-91, 1994.
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 12-13, 1986.
Double Sixes
Two sextuples of Skew Lines on the general Cubic
Surface such that each line of one is Skew to one Line
in the other set. Discovered by Schlafli.
see also Boxcars, Cubic Surface, Solomon's Seal
Lines
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 11, 1986.
Double Sum
A nested sum over two variables. Identities involving
double sums include the following:
oo p
Lr/2J
ZLZ^ a ^ V ~ q = Z^ 2-*t a n,rn = ^ ^ a 5 ,r-2s, (l)
p=0 q=0
where
m = n=0 r—0 s =
IT/2J
is the Floor Function, and
[\r r
\i(r-l) r
even
odd
n n
y y x%Xj — n
2 / 2\
(2)
(3)
i=l j=\
Consider the sum
S{a,b,c\s) — y ^ (am 2 + bmn + en ) s (4)
(m,n)^(0,0)
over binary Quadratic FORMS. If S can be decom-
posed into a linear sum of products of DlRICHLET L-
Series, it is said to be solvable. The related sums
Si (a, 6, c,s)= Y^ (-l) m (am 2 + bmn + cn)~
(m,n)^(0,0)
S 2 (a, 6, c;s)= ]P (-l) n (am 2 + bmn + en 2 )'
(m,n)^(0,0)
(5)
(6)
5i, 2 (a, 6, c; s) = ^ (-l) m+n (am 2 + bmn + cn 2 )~ s
(m,n)^(0,0)
(7)
can also be defined, which gives rise to such impressive
Formulas as
«■,,,«;.)=-«!. (8)
A complete table of the principal solutions of all solvable
5(a, 6, c; s) is given in Glasser and Zucker (1980, pp. 126-
131).
see also Euler Sum
References
Glasser, M. L. and Zucker, I. J. "Lattice Sums in Theoretical
Chemistry." Theoretical Chemistry: Advances and Per-
spectives, Vol. 5. New York: Academic Press, 1980.
Zucker, I. J. and Robertson, M. M. "A Systematic Approach
to the Evaluation of Y\ /nn ,(am 2 +6mn + cn 2 )" 8 ." J.
Phys. A: Math. Gen. 9, 1215-1225, 1976.
Doublet Function
y = S'(x-a),
where S(x) is the Delta Function.
see also Delta Function
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 324, 1993.
Doubly Even Number
An even number N for which N = (mod 4). The first
few Positive doubly even numbers are 4, 8, 12, 16, ...
(Sloane's A008586).
see also Even Function, Odd Number, Singly Even
Number
References
Sloane, N. J. A. Sequence A008586 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Doubly Magic Square
see BlMAGIC SQUARE
Dougall-Ramanujan Identity
Dowker Notation 493
Dougall-Ramanujan Identity
Discovered by Ramanujan around 1910. Prom Hardy
(1959, pp. 102-103),
Dougall's Theorem
s s (n) (x + y + z + u + 2s + l) (n)
*)(n)
TT a? (w )
11 ( x + s + l)( n )
x,y,2,u
s
T(s 4- l)T(x + y + 2 + w + 5 + l)
r(a + s + l)r(y + z + u + 5 + l)
n
r(^ + u + 5 + i)
where
a (Tl) =a(a + l)-..(a + n- 1)
a (n ) = a(a — 1) ■ • • (a — n + 1)
, (1)
(2)
(3)
(here, the POCHHAMMER Symbol has been written
( n )). This can be rewritten as
s, 1 + |s, —x — y, — z, — u,x — y-\-z-\-u + 2s-\-l
7 F 6 [ §s,x + s + l,;y + s + l,z + s + l,u + s + l, ;1
— X — 1/ — 2 — U — S
1
n
r(s + l)r(x + y + 2 + u + s + l)
r(a + s + l)r(j/ + ;z + u + fi + l)
r(z + ^ + s + i)
(4)
In a more symmetric form, if n = 2ai + 1 = a-i + 03 +
0,4 + asj ^6 = 1 4- ffli/2, 07 — — n, and 6, = 1 + ai — a^+i
for i = 1, 2, . . . , 6, then
7^6
CL\ , a2 , CiZ j &4 , 05 , (26 , (27
&1,&2,&3,&4,&5,&6
(01 + l)n(ai - ^2 - as + l) n
(ai - a 2 + l)n(ai - a 3 + l) n
(ai — Q2 — a4 + l)n(fli — a3 ~ ^4 + l)n
(ai — a4 4- l) n (o>i — CL2 — az — a±-\- l) n '
(5)
where (a) n is the POCHHAMMER SYMBOL (Petkovsek ei
a/. 1996),
The identity is a special case of Jackson's Identity.
see also Dixon's Theorem, Dougall's Theo-
rem, Generalized Hypergeometric Function,
Hypergeometric Function, Jackson's Identity,
Saalschutz's Theorem
References
Dixon, A. C. "Summation of a Certain Series." Proc. London
Math. Soc. 35, 285-289, 1903.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
1959.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, pp. 43, 126-127, and 183-184, 1996.
5 F 4
\n 4 l,n, -x, ~y, -z 1 _
^ 71,03 + 71+ l,y + 7l+ 1, Z + 71 + 1 J
r(x + n + l)r(y + n + l)V(z + n + l)r(x + y + z + n + l)
T(n + l)r(x + y + n + l)Y{y + z + n + l)r(x + z + n + 1) '
where si^a, 6, c, <2, e; /, 5, /i, i; z) is a GENERALIZED HY-
PERGEOMETRIC Function and T(z) is the Gamma
Function.
see also DOUGALL-RAMANUJAN IDENTITY, GENERAL-
IZED HYPERGEOMETRIC FUNCTION
Doughnut
see Torus
Douglas-Neumann Theorem
If the lines joining corresponding points of two directly
similar figures are divided proportionally, then the Lo-
cus of the points of the division will be a figure directly
similar to the given figures.
References
Eves, H. "Solution to Problem E52L" Amer. Math. Monthly
50, 64, 1943.
Musselman, J. R. "Problem E521." Amer. Math. Monthly
49, 335, 1942.
Dovetailing Problem
see Cube Dovetailing Problem
Dowker Notation
A simple way to describe a knot projection. The advan-
tage of this notation is that it enables a Knot Diagram
to be drawn quickly.
For an oriented ALTERNATING KNOT with n crossings,
begin at an arbitrary crossing and label it 1. Now fol-
low the undergoing strand to the next crossing, and de-
note it 2. Continue around the knot following the same
strand until each crossing has been numbered twice.
Each crossing will have one even number and one odd
number, with the numbers running from 1 to 2n,
Now write out the Odd Numbers 1, 3, . . . , 2n — 1 in
a row, and underneath write the even crossing number
corresponding to each number. The Dowker NOTATION
is this bottom row of numbers. When the sequence of
even numbers can be broken into two permutations of
consecutive sequences (such as {4,6,2} {10,12,8}), the
knot is composite and is not uniquely determined by the
Dowker notation. Otherwise, the knot is prime and the
Notation uniquely defines a single knot (for amphichi-
ral knots) or corresponds to a single knot or its MIRROR
Image (for chiral knots).
For general nonalternating knots, the procedure is mod-
ified slightly by making the sign of the even numbers
494
Down Arrow Notation
Droz-Farny Circles
POSITIVE if the crossing is on the top strand, and NEG-
ATIVE if it is on the bottom strand.
These data are available only for knots, but not for links,
from Berkeley's gopher site.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 35-40, 1994.
Dowker, C. H. and Thistlethwaite, M. B. "Classification of
Knot Projections." TopoL Appl. 16, 19-31, 1983.
Down Arrow Notation
An inverse of the up Arrow Notation defined by
e .J, n = In n
e 44- n — m * n
e Hi n = In** n,
where In* n is the number of times the NATURAL LOG-
ARITHM must be iterated to obtain a value < e.
see also Arrow Notation
References
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 12 and 231-232, 1991,
Dozen
12.
see also Baker's Dozen, Gross
Dragon Curve
Nonintersecting curves which can be iterated to yield
more and more sinuosity. They can be constructed
by taking a path around a set of dots, representing
a left turn by 1 and a right turn by 0. The first-
order curve is then denoted 1. For higher order curves,
add a 1 to the end, then copy the string of digits
preceding it to the end but switching its center digit.
For example, the second-order curve is generated as
follows: (1)1 -> (1)1(0) -► 110, and the third as:
(110)1 -» (110)1(100) -> 1101100. Continuing gives
110110011100100... (Sloane's A014577). The OCTAL
representation sequence is 1, 6, 154, 66344, . . . (Sloane's
A003460). The dragon curves of orders 1 to 9 are illus-
trated below.
This procedure is equivalent to drawing a Right Angle
and subsequently replacing each RIGHT ANGLE with an-
other smaller Right Angle (Gardner 1978). In fact,
the dragon curve can be written as a LlNDENMAYER
System with initial string "FX", String Rewriting
rules "X" -> "X+YF+", "Y" -> "-FX-Y", and angle 90°.
see also Lindenmayer System, Peano Curve
References
Dickau, R. M. "Two-Dimensional L-Systems." http://
forum.swarthmore.edu/advanced/robertd/lsys2d.html.
Dixon, R. Mathographics. New York: Dover, pp. 180-181,
1991.
Dubrovsky, V. "Nesting Puzzles, Part I: Moving Oriental
Towers." Quantum 6, 53-57 (Jan.) and 49-51 (Feb.),
1996.
Dubrovsky, V. "Nesting Puzzles, Part II: Chinese Rings Pro-
duce a Chinese Monster." Quantum 6, 61-65 (Mar.) and
58-59 (Apr.), 1996.
Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, pp. 207-209 and 215-220, 1978.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 48—
53, 1991.
Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal
Images. New York: Springer- Verlag, p. 284, 1988.
Sloane, N. J. A. Sequences A014577 and A003460/M4300 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Vasilyev, N. and Gutenmacher, V. "Dragon Curves." Quan-
tum 6, 5-10, 1995.
Dragon Fractal
see Douady's Rabbit Fractal
Draughts
see Checkers
Drinfeld's Symmetric Space
A set of points which do not lie on any of a certain class
of Hyperplanes.
References
Teitelbaum, J. "The Geometry of p-adic Symmetric Spaces."
Not. Amer. Math. Soc. 42, 1120-1126, 1995.
Droz-Farny Circles
Droz-Farny Circles
Du Bois Raymond Constants 495
Draw a CIRCLE with center H which cuts the lines O2O3,
O3O1, and O1O2 (where Oi are the MIDPOINTS) at Pi,
Q\\ P2, Q2; and P 3 , Qz respectively, then
A1P1 = A2P2 = ^ 3 P 3 = A1Q1 = A2Q2 = A 3 Q 3 .
Conversely, if equal CIRCLES are drawn about the VER-
TICES of a Triangle, they cut the lines joining the Mid-
points of the corresponding sides in six points. These
points lie on a CIRCLE whose center is the ORTHOCEN-
TER. If r is the RADIUS of the equal CIRCLES centered
on the vertices A\, A 2 , and A3, and #0 is the Radius
of the Circle about H y then
Pi 2 =4R 2 +r 2
|(ai 2 +a 2 2 + a 3 2 ).
If the circles equal to the ClRCUMClRCLE are drawn
about the VERTICES of a triangle, they cut the lines
joining midpoints of the adjacent sides in points of a
Circle R 2 with center H and Radius
R 2 Z
r r>2 1 / 2 . 2 , 2
5P - ± (ai +a 2 + a 3
It is equivalent to the circle obtained by drawing cir-
cles with centers at the feet of the altitudes and passing
through the ClRCUMCENTER. These circles cut the cor-
responding sides in six points on a circle R' 2 whose center
isH.
1 / \ \ l
1 NY y^i
""--y^CX
V/-\ '
■ y — i-— ^v
Furthermore, the circles about the midpoints of the sides
and passing though H cut the sides in six points lying
on another equivalent circle R 2 whose center is O. In
summary, the second Droz-Farny circle passes through
12 notable points, two on each of the sides and two on
each of the lines joining midpoints of the sides.
References
Goormaghtigh, R. "Droz-Farny's Theorem." Scripta Math.
16, 268-271, 1950.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 256-258, 1929.
Drum
see Isospectral Manifolds
*ois Raymond Constants
2 4
The constants C n defined by
which are difficult to compute numerically. The first few
are
Ci « 455
C 2 « 0.1945
C 3 « 0.028254
C 4 « 0.00524054.
Rather surprisingly, the second Du Bois Raymond con-
stant is given analytically by
C 2 = \{e 2 - 7) = 0.1945280494...
496
Dual Basis
Dual Polyhedron
(Le Lionnais 1983).
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 23, 1983.
Plouffe, S. "Dubois-Raymond 2nd Constant." http://
lacim.uqam. ca/piDATA/dubois .txt.
Dual Basis
Given a CONTRAVARIANT BASIS {ei,...,e n }, its dual
COVARIANT basis is given by
-*a -* / -*a -* \ cat
e ■ ep = 9(e > <W = op,
where g is the METRIC and 5p is the mixed KRONECKER
Delta. In Euclidean Space with an Orthonormal
Basis,
e° = ej,
so the BASIS and its dual are the same.
Dual Bivector
A dual Bivector is defined by
X~ 1 -ycd
and a self-dual Bivector by
Dual Graph
The dual graph G* of a POLYHEDRAL GRAPH G has
Vertices each of which corresponds to a face of G and
each of whose faces corresponds to a Vertex of G. Two
nodes in G* are connected by an EDGE if the correspond-
ing faces in G have a boundary Edge in common.
Dual Map
see Pullback Map
Dual Polyhedron
By the Duality Principle, for every Polyhedron,
there exists another POLYHEDRON in which faces and
VERTICES occupy complementary locations. This POLY-
HEDRON is known as the dual, or RECIPROCAL. The
dual polyhedron of a PLATONIC SOLID or ARCHIMED-
EAN SOLID can be drawn by constructing EDGES tangent
to the Reciprocating Sphere (a.k.a. Midsphere and
Intersphere) which are PERPENDICULAR to the origi-
nal Edges.
The dual of a general solid can be computed by connect-
ing the midpoints of the sides surrounding each Ver-
tex, and constructing the corresponding tangent POLY-
GON. (The tangent polygon is the polygon which is tan-
gent to the Circumcircle of the Polygon produced
by connecting the Midpoint on the sides surrounding
the given VERTEX.) The process is illustrated below for
the Platonic Solids. The Polyhedron Compounds
consisting of a POLYHEDRON and its dual are generally
very attractive, and are also illustrated below for the
Platonic Solids.
The Archimedean Solids and their duals are illus-
trated below.
#
e
© w
The following table gives a list of the duals of the PLA-
TONIC Solids and Kepler-Poinsot Solids together
with the names of the POLYHEDRON-dual COMPOUNDS.
Polyhedron
Dual
Csaszar polyhedron
cube
cuboctahedron
dodecahedron
great dodecahedron
great icosahedron
great stellated dodec.
icosahedron
octahedron
small stellated dodec.
Szilassi polyhedron
tetrahedron
Szilassi polyhedron
octahedron
rhombic dodecahedron
icosahedron
small stellated dodec.
great stellated dodec.
great icosahedron
dodecahedron
cube
great dodecahedron
Csaszar polyhedron
tetrahedron
polyhedron compound
cube
dodecahedron
great dodecahedron
great icosahedron
great stellated dodec.
icosahedron
octahedron
small stellated dodec.
tetrahedron
cube-octahedron compound
dodec. -icosahedron compound
great dodecahedron-small
stellated dodec. compound
great icosahedron-great
stellated dodec. compound
great icosahedron-great
stellated dodec. compound
dodec.-icosahedron compound
cube-octahedron compound
great dodec. -small
stellated dodec. compound
stella octangula
Dual Scalar
Duffing Differential Equation 497
see also Duality Principle, Polyhedron Com-
pound, Reciprocating Sphere
References
^ Weisstein, E. W. "Polyhedron Duals." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/Duals.m.
Wenninger, M. Dual Models. Cambridge, England: Cam-
bridge University Press, 1983.
Dual Scalar
Given a third RANK TENSOR,
V ijk =det[A B C],
where det is the DETERMINANT, the dual scalar is de-
fined as
V = -jyUjkVijk,
where e^fe is the Levi-Civita Tensor.
see also Dual Tensor, Levi-Civita Tensor
Dual Solid
see Dual Polyhedron
Dual Tensor
Given an antisymmetric second RANK TENSOR Cij, a
dual pseudotensor d is defined by
d = 2 e *ifcQ?fc)
(i)
where
Ci =
C23
C31
(2)
Cjk =
"
_ C31
C12
2
— C23
— C31
C23
(3)
see also Dual 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.
Dual Voting
A term in SOCIAL CHOICE THEORY meaning each alter-
native receives equal weight for a single vote.
see also ANONYMOUS, MONOTONIC VOTING
Duality Principle
All the propositions in PROJECTIVE GEOMETRY occur
in dual pairs which have the property that, starting from
either proposition of a pair, the other can be immedi-
ately inferred by interchanging the parts played by the
words "point" and "line." A similar duality exists for
Reciprocation (Casey 1893).
see also Brianchon's Theorem, Conservation of
Number Principle, Desargues' Theorem, Dual
Polyhedron, Pappus's Hexagon Theorem, Pas-
cal's Theorem, Permanence of Mathematical
Relations Principle, Projective Geometry, Re-
ciprocation
References
Casey, J. "Theory of Duality and Reciprocal Polars." Ch. 13
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. f rev. enl. Dublin: Hodges, Figgis, &; Co., pp. 382-
392, 1893.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 107-110, 1990.
Duality Theorem
Dual pairs of Linear Programs are in "strong duality"
if both are possible. The theorem was first conceived by
John von Neumann. The first written proof was an Air
Force report by George Dantzig, but credit is usually
given to Tucker, Kuhn, and Gale.
see also Linear Programming
Duffing Differential Equation
The most general forced form of the Duffing equation is
x + 8x + (f3x s ± w 2 x) = A sin(urt + <j>). (1)
If there is no forcing, the right side vanishes, leaving
x + Sx + {fix 3 ± u 2 x) = 0. (2)
If 6 = and we take the plus sign,
x + vq 2 x + j3x s = 0. (3)
This equation can display chaotic behavior. For > 0,
the equation represents a "hard spring," and for (3 < 0,
it represents a "soft spring." If f3 < 0, the phase portrait
curves are closed. Returning to (1), take (3 = 1, coo = 1,
A = 0, and use the minus sign. Then the equation is
x + 8x + (x — x) —
(4)
(Ott 1993, p. 3). This can be written as a system of
first-order ordinary differential equations by writing
y = x — x — Sy.
(5)
(6)
498 Duffing Differential Equation
The fixed points of these differential equations
x = y = 0,
so y = 0, and
y — x — x — Sy = x(l — x ) —
giving x — 0, ±1. Differentiating,
x = y = x — x — 5y
y = (1 — 3x )£ — Jy
Duodecillion
la;
1 - 3a: 2 -5 y
(7)
(8)
(9)
(10)
(11)
Examine the stability of the point (0,0):
A(A + <5)-l = A 2 + A<$-l = (12)
0-A
1
1
-8 -A
A£'°>=!(-<5±vW4).
(13)
But J 2 > 0, so A^°' 0) is real. Since y/8 2 +4 > \8\, there
will always be one POSITIVE ROOT, so this fixed point
is unstable. Now look at (±1, 0).
0-A
-2
1
-J- A
= A(A + <S) + 2 = \ 7 ' + \8 + 2 = (14)
A (± 1 ,o)^i M±v /^r^ ) .
(15)
For £ > 0, 5R[A^ 1,0 *] < 0, so the point is asymptoti-
cally stable. If 6 — 0, A^ 1 ' 05 = ±iy/2, so the point is
linearly stable. If 6 e (-2^2,0), the radical gives an
Imaginary Part and the Real Part is > 0, so the
point is unstable. If 8 = — 2\/2, \± = v 7 ^, which
has a Positive Real Root, so the point is unstable.
If 8 < -2 a/2, then |<S| < y/8 2 - 8, so both Roots are
POSITIVE and the point is unstable. Summarizing,
{asymptotically stable 8 >
linearly stable (superstable) 8 — (16)
unstable 8 < 0.
Now specialize to the case 8 — 0, which can be integrated
by quadratures. In this case, the equations become
x-y
y-x
(17)
(18)
Differentiating (17) and plugging in (18) gives
x ~ y = x — x . (19)
Multiplying both sides by x gives
so we have an invariant of motion /i,
ft O O I A *
Solving for x 2 gives
dx
~di
(I) -»+*-&
t
-/*■/•
cfa
V^/i + a^ + fa: 2
Note that the invariant of motion h satisfies
. __ dh _ dh
dx dy
dh ^ 3
~^ = ~X + X = -y,
so the equations of the Duffing oscillator are given by
the Hamiltonian System
(21)
(22)
(23)
(24)
(25)
(26)
(27)
X ~ By
v = -Sh.
(28)
References
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, 1993.
Duhamel's Convolution Principle
Can be used to invert a LAPLACE TRANSFORM.
Dumbbell Curve
2 2/4 6\
y — a (x — x ).
see also Butterfly Curve, Eight Curve, Piriform
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.
Duodecillion
In the American system, 10
see also Large Number
xx — xx 4- xx —
(20)
Dupin's Cy elide
Dupin's Cyclide
see Cyclide
Dupin's Indicatrix
A pair of conies obtained by expanding an equation in
Monge's Form z = F(x,y) in a Maclaurin Series
z = z(0,0) + zix + z 2 y
+ ^(znx 2 -h 2zi 2 xy + Z22V 2 ) + .-.
= |(&ii^ 2 + 26i2^y + 6 2 22/ 2 ).
This gives the equation
6na: 2 + 2bi 2 xy + b 2 2y 2 = ±1.
Amazingly, the radius of the indicatrix in any direction
is equal to the SQUARE ROOT of the RADIUS OF CUR-
VATURE in that direction (Coxeter 1969).
References
Coxeter, H. S. M. "Dupin's Indicatrix" §19.8 in Introduction
to Geometry, 2nd ed. New York: Wiley, pp. 363-365, 1969.
Dupin's Theorem
In three mutually orthogonal systems of the surfaces, the
Lines of Curvature on any surface in one of the sys-
tems are its intersections with the surfaces of the other
two systems.
Duplication of the Cube
see Cube Duplication
Duplication Formula
see Legendre Duplication Formula
Durand's Rule
The Newton-Cotes Formula
Diirer's Magic Square 499
Diirer's Conchoid
f
** x-i
f(x) dx
= MIA + 15/2 + h + • • ■ + /n-2 + £/»-! + f /„).
see also Bode's Rule, Hardy's Rule, Newton-
Cotes Formulas, Simpson's 3/8 Rule, Simpson's
Rule, Trapezoidal Rule, Weddle's Rule
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 127, 1987.
These curves appear in Diirer's work Instruction in Mea-
surement with Compasses and Straight Edge (1525) and
arose in investigations of perspective. Diirer constructed
the curve by drawing lines QRP and P'QR of length 16
units through Q(g, 0) and R(r, 0), where q-\-r = 13. The
locus of P and P' is the curve, although Diirer found
only one of the two branches of the curve.
The Envelope of the lines QRP and P'QR is a
Parabola, and the curve is therefore a Glissette of
a point on a line segment sliding between a PARABOLA
and one of its TANGENTS.
Diirer called the curve "Muschellini," which means CON-
CHOID. However, it is not a true CONCHOID and so is
sometimes called Durer'S Shell Curve. The Carte-
sian equation is
2y 2 (x 2 + y 2 ) - 2by 2 (x + y) + (b 2 - 3a 2 )y 2 - a 2 x 2
+ 2a 2 b(x + y)+a 2 (a 2 - b 2 ) = 0.
The above curves are for (a, 6) = (3,1), (3,3), (3,5).
There are a number of interesting special cases. If 6 = 0,
the curve becomes two coincident straight lines x = 0.
For a = 0, the curve becomes the line pair x — 6/2,
x = —6/2, together with the CIRCLE x + y = b. If
a = 6/2, the curve has a CUSP at (— 2a, a).
New
References
Lawrence, J. D. A Catalog of Special Plane Curves.
York: Dover, pp. 157-159, 1972.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 163, 1967.
MacTutor History of Mathematics Archive. "Diirer's Shell
Curves." http: // www - groups . des . st - and .ac.uk/
-hi story /Curve s/Durers .html.
Diirer's Magic Square
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
Diirer's magic square is a MAGIC SQUARE with MAGIC
Constant 34 used in an engraving entitled Melencolia
I by Albrecht Diirer (The British Museum). The en-
graving shows a disorganized jumble of scientific equip-
ment lying unused while an intellectual sits absorbed in
500
Diirer's Shell Curve
Dymaxion
thought. Diirer's magic square is located in the upper
left-hand corner of the engraving. The numbers 15 and
14 appear in the middle of the bottom row, indicating
the date of the engraving, 1514.
References
Boyer, C. D. and Merzbach, U. C. A History of Mathematics,
New York: Wiley, pp. 296-297, 1991.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, p. 24, 1975.
Rivera, C. "Melancholia." http://www.sci.net.mx/
-crivera/melancholia . htm.
Diirer's Shell Curve
see Durer's Conchoid
Durfee Polynomial
Let F(n) be a family of Partitions of n and let F(n, d)
denote the set of PARTITIONS in F(n) with DURFEE
SQUARE of size d. The Durfee polynomial of F(n) is
then defined as the polynomial
P F , n = ^2\F(n,d)\y d
where < d < yfn.
see also DURFEE SQUARE, PARTITION
References
Canfield, E. R.; Corteel, S.; and Savage, C. D. "Durfee Poly-
nomials." Electronic J. Combinatorics 5, No. 1, R32,
1—21, 1998. http://www.combinatorics.org/VolumeJj/
v5iltoc.html#R32.
Durfee Square
The length of the largest-sized SQUARE contained within
the Ferrers Diagram of a Partition.
see also Durfee Polynomial, Ferrers Diagram,
Partition
Dvoretzky's Theorem
Each centered convex body of sufficiently high dimen-
sion has an "almost spherical" fc-dimensional central sec-
tion.
Dyad
Dyads extend VECTORS to provide an alternative de-
scription to second Rank TENSORS. A dyad D(A,B)
of a pair of VECTORS A and B is defined by £>(A, B) =
AB. The Dot Product is defined by
A BC = (A B)C
AB C = A(B C),
and the COLON PRODUCT by
AB : CD = C AB D = (A C)(B D).
Dyadic
A linear POLYNOMIAL of DYADS AB + CD + . . . con-
sisting of nine components Aij which transform as
ityi OXyyx OXji
E
h^ dx\ dx'j
h'itij dx\ dx'j
h'ihn dx'i dx n .
hmh'j dx m dx'j mn
(1)
(2)
(3)
Dyadics are often represented by Gothic capital letters.
The use of dyadics is nearly archaic since TENSORS per-
form the same function but are notationally simpler.
A unit dyadic is also called the Idemfactor and is de-
fined such that
I A = A. (4)
In Cartesian Coordinates,
I = icx + yy + zz,
and in SPHERICAL COORDINATES
1 = Vr.
(5)
(6)
see also Dyad, Tetradic
References
Arfken, G. "Dyadics." §3.5 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 137—
140, 1985.
Morse, P. M. and Feshbach, H. "Dyadics and Other Vector
Operators." §1.6 in Methods of Theoretical Physics, Part
L New York: McGraw-Hill, pp. 54-92, 1953.
Dyck's Theorem
see von Dyck's Theorem
Dye's Theorem
For any two ergo die measure-preserving transformations
on nonatomic PROBABILITY SPACES, there is an ISO-
MORPHISM between the two Probability Spaces car-
rying orbits onto orbits.
Dymaxion
Buckminster Fuller's term for the CUBOCTAHEDRON.
see also CUBOCTAHEDRON, MECON
References
Morse, P. M. and Feshbach, H. "Dyadics and Other Vector
Operators." §1.6 in Methods of Theoretical Physics, Part
I. New York: McGraw-Hill, pp. 54-92, 1953.
Dynamical System
Dynamical System
A means of describing how one state develops into an-
other state over the course of time. Technically, a dy-
namical system is a smooth action of the reals or the In-
tegers on another object (usually a Manifold). When
the reals are acting, the system is called a continuous
dynamical system, and when the INTEGERS are acting,
the system is called a discrete dynamical system. If /
is any CONTINUOUS FUNCTION, then the evolution of a
variable x can be given by the formula
x n+ i = f(x n ). (1)
This equation can also be viewed as a difference equation
2>n + l X n — J\Xn) X n ^ \ )
Dynkin Diagram 501
so defining
g(x) = f(x) -x
(3)
gives
x n +i — x n = g(x n ) * 1,
(4)
which can be read "as n changes by 1 unit, x changes by
g(x)" This is the discrete analog of the Differential
Equation
x(n) =g(x{n)). (5)
see also ANOSOV DlFFEOMORPHISM, AnOSOV FLOW,
Axiom A Diffeomorphism, Axiom A Flow, Bifur-
cation Theory, Chaos, Ergodic Theory, Geo-
desic Flow
References
Aoki, N. and Hiraide, K. Topological Theory of Dynamical
Systems. Amsterdam, Netherlands: North- Holland, 1994.
Golubitsky, M. Introduction to Applied Nonlinear Dynamical
Systems and Chaos. New York: Springer- Verlag, 1997.
Guckenheimer, J. and Holmes, P. Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields, 3rd
ed. New York: Springer- Verlag, 1997.
Lichtenberg, A. and Lieberman, M. Regular and Stochastic
Motion, 2nd ed. New York: Springer- Verlag, 1994.
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, 1993.
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, 1990.
Strogatz, S. H. Nonlinear Dynamics and Chaos, with Appli-
cations to Physics, Biology, Chemistry, and Engineering.
1994.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, 1989.
Dynkin Diagram
A diagram used to describe Chevalley Groups.
see also COXETER-DYNKIN DIAGRAM
References
Jacobson, N. Lie Algebras. New York: Dover, p. 128, 1979.
503
E
The base of the NATURAL Logarithm, named in honor
of Euler. It appears in many mathematical contexts
involving LIMITS and DERIVATIVES, and can be defined
by
e = lim
x— »-oo
or by the infinite sum
00
=y--
(2)
The numerical value of e is
e = 2.718281828459045235360287471352662497757.
(Sloane's A001113).
(3)
Euler proved that e is IRRATIONAL, and Liouville proved
in 1844 that e does not satisfy any QUADRATIC EQUA-
TION with integral COEFFICIENTS. Hermite proved e to
be TRANSCENDENTAL in 1873. It is not known if ir + e
or 7r/e is Irrational. However, it is known that n + e
and 7r/e do not satisfy any POLYNOMIAL equation of de-
gree < 8 with Integer Coefficients of average size
10 9 (Bailey 1988, Borwein et al. 1989).
The special case of the Euler Formula
e tx = cos a? -\-is\nx
with x = 7r gives the beautiful identity
e i7r + 1 = 0,
(4)
(5)
an equation connecting the fundamental numbers i, Pi,
e, 1, and (Zero).
Some Continued FRACTION representations of e in-
clude
e = 2 +
(6)
1 +
2+-
3+ —
= [2,1,2,1,1,4,1,1,6,...]
(Sloane's A003417) and
^j = [2,6,10,14,...]
e + 1
e-l = [1,1,2, 1,1,4, 1,1,6,...]
l(e-l) = [0,l,6,10,14,...]
V~e = [1,1,1,1,5,1,1,1,9,1,...].
(7)
(8)
(9)
(10)
(11)
The first few convergents of the CONTINUED FRAC-
TION are 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, ...
(Sloane's A007676 and A007677).
Using the RECURRENCE RELATION
a n = n(a n -i 4- 1)
with a± — a -1 , compute
11(1 + a„- 1 ).
(12)
(13)
The result is e a . Gosper gives the unusual equation
connecting 7v and e,
oo
nn -f y/n 2/ K 2 — 9
12e 3
= -0.040948222.
(14)
Rabinowitz and Wagon (1995) give an ALGORITHM for
computing digits of e based on earlier DIGITS, but a
much simpler SPIGOT Algorithm was found by Sales
(1968). Around 1966, MIT hacker Eric Jensen wrote
a very concise program (requiring less than a page of
assembly language) that computed e by converting from
factorial base to decimal.
Let p(n) be the probability that a random ONE-TO-ONE
function on the Integers 1, . . . , n has at least one
Fixed Point. Then
lim p(n) = V l > =1--
fc=l
Stirling's Formula gives
(n\) 1/ri _
0.6321205588....
(15)
lim
(16)
Castellanos (1988) gives several curious approximations
to e,
:2 +
54 2 + 41 2
80 2
(7r 4 +7r 5 ) 1/6
271801
99990
150
87 3 + 12 5 \ 1/5
83 3 }
300 4 - 100 4 - 1291 2 + 9 2
1097
91 5
55 5 +311 3 -
68 5
U^\
1/7
(17)
(18)
(19)
(20)
(21)
(22)
504
E n -Function
which are good to 6, 7, 9, 10, 12, and 15 digits respec-
tively.
Examples of e MNEMONICS (Gardner 1959, 1991) in-
clude:
"By omnibus I traveled to Brooklyn" (6 digits).
"To disrupt a playroom is commonly a practice of
children" (10 digits).
"It enables a numskull to memorize a quantity of
numerals" (10 digits).
"I'm forming a mnemonic to remember a function in
analysis" (10 digits).
"He repeats: I shouldn't be tippling, I shouldn't be
toppling here!" (11 digits).
"In showing a painting to probably a critical or ven-
omous lady, anger dominates. O take guard, or she
raves and shouts" (21 digits). Here, the word "O"
stands for the number 0.
A much more extensive mnemonic giving 40 digits is
"We present a mnemonic to memorize a constant
so exciting that Euler exclaimed: '!' when first it
was found, yes, loudly '!'. My students perhaps will
compute e, use power or Taylor series, an easy sum-
mation formula, obvious, clear, elegant!"
(Barel 1995). In the latter, 0s are represented with "!".
A list of e mnemonics in several languages is maintained
by A. P. Hatzipolakis.
Scanning the decimal expansion of e until all n-digit
numbers have occurred, the last appearing is 6, 12, 548,
1769, 92994, 513311, . . . (Sloane's A032511). These end
at positions 21, 372, 8092, 102128, 1061613, 12108841,
see also CARLEMAN'S INEQUALITY, COMPOUND INTER-
EST, de Moivre's Identity, Euler Formula, Expo-
nential Function, Hermite-Lindemann Theorem,
Natural Logarithm
References
Bailey, D. H. "Numerical Results on the Transcendence of
Constants Involving 7r, e, and Euler's Constant." Math.
Comput. 50, 275-281, 1988.
Barel, Z. "A Mnemonic for e." Math. Mag. 68, 253, 1995.
Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ra-
manujan, Modular Equations, and Approximations to Pi
or How to Compute One Billion Digits of Pi." Amer. Math.
Monthly 96, 201-219, 1989.
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. 201 and 250-254, 1996.
Finch, S. "Favorite Mathematical Constants." http://wwv.
mathsoft.com/asolve/constant/e/e.html.
Gardner, M. "Memorizing Numbers." Ch. 11 in The Scien-
tific American Book of Mathematical Puzzles and Diver-
sions. New York: Simon and Schuster, pp. 103 and 109,
1959.
Gardner, M. Ch. 3 in The Unexpected Hanging and Other
Mathematical Diversions. Chicago, IL: Chicago University
Press, p. 40, 1991.
Hatzipolakis, A. P. "PiPhilology." http://users.hol.gr/
"xpolakis/piphil .html.
Hermite, C. "Sur la fonction exponent ielle." C. R. Acad. Sci.
Paris 77, 18-24, 74-79, and 226-233, 1873.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 47, 1983.
Maor, E. e: The Story of a Number. Princeton, NJ: Prince-
ton University Press, 1994.
Minkus, J. "A Continued Fraction." Problem 10327. Amer.
Math. Monthly 103, 605-606, 1996.
Mitchell, U. G. and Strain, M. "The Number e." Osiris 1,
476-496, 1936.
Olds, CD. "The Simple Continued Fraction Expression of
e." Amer. Math. Monthly 77, 968-974, 1970.
Plouffe, S. "PloufFe's Inverter: Table of Current Records for
the Computation of Constants." http://lacim.uqam.ca/
pi/records. html.
Rabinowitz, S. and Wagon, S. "A Spigot Algorithm for the
Digits of 7T." Amer. Math. Monthly 102, 195-203, 1995.
Sales, A. H. J. "The Calculation of e to Many Significant
Digits." Computer J. 11, 229-230, 1968.
Sloane, N. J. A. Sequences A032511, A001113/M1727,
A003417/M0088, A007676/M0869, and A007677/M2343
in "An On-Line Version of the Encyclopedia of Integer Se-
quences."
e-Divisor
d is called an e-divisor (or EXPONENTIAL DIVISOR) of
n~px *P2 '"Pr
if d\n and
- bi &2
b r
d = Pl 0l p2°
where bj\a,j with 1 < j <r.
see also e-PERFECT Number
References
Guy, R K. "Exponential-Perfect Numbers." §B17 in Un-
solved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 73, 1994.
Straus, E. G. and Subbarao, M. V. "On Exponential Divi-
sors." Duke Math. J. 41, 465-471, 1974.
E n -Punction
The En{x) function is defined by the integral
En{x)
[
e~ xt dt
t n
(1)
and is given by the Mathematical (Wolfram Research,
Champaign, IL) function ExpIntegralE[n,x]. Defining
t = 7]~ x so that dt = —rf 2 dr},
E„(aO = /
Jo
e" x/T
V
'drj
(2)
E„(0) =
1
n —
1'
(3)
satisfies the RECURRENCE RELATIONE
El* (a) =
-En-
-iW
(4)
nEn+iO) =
— x
e -
■ lEn
.(*).
(5)
E n -Function
Equation (4) can be derived from
/°° — tx
(6)
E^a:) = — / dt = / — dt
nK ' dx J 1 *» J 1 dx\ t» J
= - / t- — dt
Ji tn
f°° e~ tx
=-y ^*=-e.-.w, (7)
and (5) using integrating by parts, letting
1
u = — dv = e x dt
t n
(8)
fi g
du = — eft v = , (9)
En (#) = I udv = uv — I vdu
= __£^_ n f°° e~ tx d
e~ ta: dt
__r_^_l°°_ f°° e^dx
= a;En(») = e s -nE„+i(4 ( 10 )
Solving (10) for 7iEn(z) gives (5). An asymptotic ex-
pansion gives
(n-l)!En(a)
= (-^r -1 Ei(x) + e- x J2 ~ 2 ( n ~ s ~ 2 ) ! (-z) 3 , (11)
e-Multiperfect Number 505
where 7 is the Euler-Mascheroni Constant.
Ei(0) = oo (15)
Eii(ix) = — ci(a;) -j- isi(x), (16)
where ci(cc) and si(x) are the COSINE INTEGRAL and
Sine Integral.
see also COSINE INTEGRAL, Et-FUNCTION, EXPONEN-
TIAL Integral, Gompertz Constant, Sine Inte-
gral
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Exponential In-
tegral and Related Functions." Ch. 5 in Handbook of Math-
ematical Functions with Formulas, Graphs, and Mathe-
matical Tables, 9th printing. New York: Dover, pp. 227-
233, 1972.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Exponential Integrals." §6.3 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 215-219, 1992.
Spanier, J. and Oldham, K. B. "The Exponential Integral
Ei(x) and Related Functions." Ch. 37 in An Atlas of Func-
tions. Washington, DC: Hemisphere, pp. 351-360, 1987.
^-Function
A function which arises in Fractional Calculus.
Et(v,a)
1 at I v-l
=7— re / X i
r (") Jo
— ax j j_u at / ,\
e dx = t e 7(1/, at),
(i)
where 7 is the incomplete Gamma Function and T the
complete Gamma Function. The E t function satisfies
the Recurrence Relation
E t {v,a) — aE t {v + I, a) +
t"
I> + 1)-
A special value is
Et{O t a) = e at .
(2)
(3)
En fa) =
X
1 _ n n(n + 1)
x x*
(12)
The special case n = 1 gives
... f°° e~ tx dt r e~ u du
(13)
where ei(a;) is the EXPONENTIAL INTEGRAL, which is
also equal to
El ( a .) = _ 7 _lnx-f;^f^ 1 (14)
n = l
see also E^-FUNCTION
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Exponential In-
tegral and Related Functions." Ch. 5 in Handbook of Math-
ematical Functions with Formulas, Graphs, and Mathe-
matical Tables, 9th printing. New York: Dover, pp. 227-
233, 1972.
e-Multiperfect Number
A number n is called a k e-perfect number if a e (n) = kn y
where a e (n) is the Sum of the e-DlVlSORS of n.
see also e-DlVISOR, e-PERFECT NUMBER
References
Guy, R. K. "Exponential-Perfect Numbers." §B17 in Un-
solved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 73, 1994.
506
e-Perfect Number
Eccentric Anomaly
e-Perfect Number
A number n is called an e-perfect number if a e (n) = 2n,
where a e {n) is the Sum of the e-DlviSORS of n. If m
is SQUAREFREE, then <r e (m) = m. As a result, if n is
e-perfect and m is Squarefree with m _L 6, then mn
is e-perfect. There are no Odd e-perfect numbers.
see also e-DlVISOR
References
Guy, R. K. "Exponential-Perfect Numbers." §B17 in Un-
solved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 73, 1994.
Subbarao, M. V. and Suryanarayan, D. "Exponential Perfect
and Unitary Perfect Numbers." Not. Amer. Math. Soc.
18, 798, 1971.
Ear
A Principal Vertex x» of a Simple Polygon P is
called an ear if the diagonal [a:t-i,£i+i] that bridges xi
lies entirely in P. Two ears Xi and Xj are said to overlap
if
int[xi_i,Xi,Xi+i] flint[xj_i,a;j,Xj+i] = 0.
The Two-Ears Theorem states that, except for Tri-
angles, every SIMPLE POLYGON has at least two
nonoverlapping ears.
see also Anthropomorphic Polygon, Mouth, Two-
Ears Theorem
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 122, 31-35, 1991.
Early Election Results
Let Jones and Smith be the only two contestants in an
election that will end in a deadlock when all votes for
Jones (J) and Smith (5) are counted. What is the EX-
PECTATION VALUE of Xk = \S - J\ after k votes are
counted? The solution is
/v \_ 2iV (Lfe/2 1 j)([fc/2j 1 -i)
\ A */ ~ 72N\
( k(2N-k) ( N \ 2 (2N\~ 1
2N U/2/ \ k )
k(2N-k+l) { N \2(2N\-
2N \(k-l)/2) \k-l)
for k even
for k odd.
References
Handelsman, M. B. Solution to Problem 10248. "Early Re-
turns in a Tied Election." Amer. Math. Monthly 102,
554-556, 1995.
Eban Number
The sequence of numbers whose names (in English) do
not contain the letter "e" (i.e., "e" is "banned"). The
first few eban numbers are 2, 4, 6, 30, 32, 34, 36, 40, 42,
44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004,
. . . (Sloane's A006933); i.e., two, four, six, thirty, etc.
References
Sloane, N. J. A. Sequence A006933/M1030 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Eberhart's Conjecture
If q n is the nth prime such that M qn is a MERSENNE
Prime, then
q n - (3/2)".
It was modified by Wagstaff (1983) to yield
*>~(2 e ~Y,
where 7 is the Euler-Mascheroni CONSTANT.
References
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 332-333, 1989.
Wagstaff, S. S. "Divisors of Mersenne Numbers." Math.
Corn-put. 40, 385-397, 1983.
Eccentric
Not Concentric.
see also CONCENTRIC, CONCYCLIC
Eccentric Angle
The angle 6 measured from the Center of an Ellipse
to a point on the ELLIPSE.
see also Eccentricity, Ellipse
Eccentric Anomaly
The Angle obtained by drawing the Auxiliary Cir-
cle of an ELLIPSE with center O and FOCUS F, and
drawing a LINE PERPENDICULAR to the SEMIMAJOR
Axis and intersecting it at A. The Angle E is then
defined as illustrated above. Then for an Ellipse with
Eccentricity e,
AF ^OF - AO-ae- acosE.
(1)
But the distance AF is also given in terms of the dis-
tance from the Focus r = FP and the SUPPLEMENT of
the ANGLE from the SEMIMAJOR AXIS v by
AF = r cos(7r — v) = — r cos v.
(2)
Eccentricity
Equating these two expressions gives
a(cos E — e)
r = ,
COSf
which can be solved for cos v to obtain
afcos E — e)
cos v = .
Eckert IV Projection 507
(3)
(4)
To get E in terms of r, plug (4) into the equation of the
Ellipse
_ q(l-e 2 )
1 + e cos v
r(l + e cos v) = a(l — e )
(5)
(6)
r 1 +
ae cos E
')-
r + ae cos E — e 2 — a(l — e )
(7)
r = a(l - e 2 ) - eacosE + e 2 a = a(l -ecos£). (8)
Differentiating gives
r = aeE sin E. (9)
The eccentric anomaly is a very useful concept in or-
bital mechanics, where it is related to the so-called mean
anomaly M by Kepler's Equation
M = E - esmE.
(10)
M can also be interpreted as the Area of the shaded
region in the above figure (Finch).
see also Eccentricity, Ellipse, Kepler's Equation
References
Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed.,
rev. ed, Richmond, VA: Willmann-Bell, 1988.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/lpc/lpc.html.
Eccentricity
A quantity defined for a CONIC Section which can be
given in terms of Semimajor and SEMIMINOR Axes for
an Ellipse. For an Ellipse with Semimajor Axis a
and SEMIMINOR AXIS 6,
« s ,/i-£.
The eccentricity can be interpreted as the fraction of the
distance to the semimajor axis at which the FOCUS lies,
where c is the distance from the center of the CONIC
Section to the Focus. The table below gives the type
of CONIC Section corresponding to various ranges of
eccentricity e.
e
Curve
e =
< e < 1
e = l
e> 1
circle
ellipse
parabola
hyperbola
see also CIRCLE, CONIC SECTION, ECCEN-
TRIC Anomaly, Ellipse, Flattening, Hyperbola,
Oblateness, Parabola, Semimajor Axis, Semimi-
nor Axis
Eccentricity (Graph)
The length of the longest shortest path from a VERTEX
in a Graph.
see also DIAMETER (GRAPH)
Echidnahedron
ICOSAHEDRON STELLATION #4.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 65, 1971.
Eckardt Point
On the CLEBSCH DIAGONAL Cubic, all 27 of the com-
plex lines present on a general smooth Cubic Surface
are real. In addition, there are 10 points on the surface
where three of the 27 lines meet. These points are called
Eckardt points (Fischer 1986).
see also CLEBSCH DIAGONAL CUBIC, CUBIC SURFACE
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 11, 1986.
Eckert IV Projection
The equations are
V^ 4 + *)
(\-\ )(l + cosO)
y = 2
4 + 7T
sin#,
(i)
(2)
508
Eckert VI Projection
Edge (Polygon)
where 9 is the solution to
9 + sin 9 cos 9 + 2 sin 9 = (2 + |tt) sin 0. (3)
This can be solved iteratively using NEWTON'S METHOD
with #o = 0/2 to obtain
+ sin0cos0 + 2sin0-(2- |7r)sin0
A Q _
2cos0(l + cos0)
The inverse FORMULAS are
_! ,'0 + sin0cos0 + 2sin0
2+^7T
A = A +
7T\/4 + 7T X
1+COS0 '
where
(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. 253-258, 1987.
Eckert VI Projection
The equations are
_ (A-Aq)(1 + cos9)
y= WTt' (2)
where 9 is the solution to
<9 + sin<9 = (1+ |7r)sin0. (3)
This can be solved iteratively using Newton's Method
with 9o = <f> to obtain
Ag= fl + sinfl-(l + f7r)sin0
1 + cos 9 ' ^ }
The inverse FORMULAS are
. . _! (6 + sm0\ f .
^ = sin \ttfJ ()
A . Ao + ^|±Z|, (6 )
1 + cos 9
where
9=\y/2 + Hy. (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. 253-258, 1987.
Economized Rational Approximation
A Pade Approximation perturbed with a Chebyshev
Polynomial of the First Kind to reduce the leading
Coefficient in the Error.
Eddington Number
136 -2 256 « 1.575 x 10 79 .
According to Eddington, the exact number of protons
in the universe, where 136 was the RECIPROCAL of the
fine structure constant as best as it could be measured
in his time.
see also LARGE NUMBER
Edge-Coloring
An edge-coloring of a GRAPH G is a coloring of the
edges of G such that adjacent edges (or the edges bound-
ing different regions) receive different colors. BRELAZ'S
Heuristic Algorithm can be used to find a good, but
not necessarily minimal, edge-coloring.
see also BRELAZ'S HEURISTIC ALGORITHM, CHRO-
MATIC Number, ^-Coloring
References
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:
Assaults and Conquest. New York: Dover, p. 13, 1986.
Edge Connectivity
The minimum number of EDGES whose deletion from a
Graph disconnects it.
see also Vertex Connectivity
Edge (Graph)
For an undirected GRAPH, an unordered pair of nodes
which specify the line connecting them. For a DIRECTED
Graph, the edge is an ordered pair of nodes.
see also Edge Number, Null Graph, Tait Color-
ing, Tait Cycle, Vertex (Graph)
Edge Number
The number of Edges in a GRAPH, denoted \E\.
see also EDGE (GRAPH)
Edge (Polygon)
A Line Segment on the boundary of a Face, also called
a Side.
see also Edge (Polyhedron), Vertex (Polygon)
Edge (Polyhedron)
Edge (Polyhedron)
A Line Segment where two Faces of a Polyhedron
meet, also called a SIDE.
see also Edge (Polygon), Vertex (Polyhedron)
Edge (Polytope)
A 1-D Line Segment where two 2-D Faces of an n-D
Polytope meet, also called a Side.
see also Edge (Polygon), Edge (Polyhedron)
Edgeworth Series
Approximate a distribution in terms of a NORMAL DIS-
TRIBUTION. Let
4>(t)
then
/(t) = 0(*) + |f7itf> (3) <
+
[> (4 V)+ 10 6 ?>«]
+ ,
see also Cornish-Fisher Asymptotic Expansion,
Gram-Charlier 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. 935, 1972.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 108, 1951.
Edmonds' Map
A nonreflexible regular map of Genus 7 with eight VER-
TICES, 28 Edges, and eight HEPTAGONAL faces.
Efron's Dice
A B C D
4
4
4
4
3
11
10
9
2
3
3
3
3
3
3
3
1
8
8
8
7
2
6 6 2 2
2
5
6 6 6 6
5
1
5
5
1
5
1
12
4
4
4
4
12
Ehrhart Polynomial 509
A set of four nontransitive Dice such that the proba-
bilities of A winning against B, B against C, C against
D, and D against A are all 2:1. A set in which ties may
occur, in which case the DICE are rolled again, which
gives Odds of 11:6 is
A B C D
o
8 17 8
9
5
6 7 5 7
6
5
4
12
3
4
11
see also Dice, SlCHERMAN DICE
References
Gardner, M. "Mathematical Games: The Paradox of the
Nontransitive Dice and the Elusive Principle of Indiffer-
ence." Set. Amer. 223, 110-114, Dec. 1970.
Honsberger, R. "Some Surprises in Probability." Ch. 5 in
Mathematical Plums (Ed. R. Honsberger). Washington,
DC: Math. Assoc. Amer., pp. 94-97, 1979.
Egg
An Oval with one end more pointed than the other.
see also ELLIPSE, MOSS'S EGG, Oval, Ovoid, THOM'S
Eggs
Egyptian Fraction
see Unit Fraction
Ehrhart Polynomial
Let A denote an integral convex Polytope of Dimen-
sion n in a lattice M, and let /a(&) denote the number
of LATTICE Points in A dilated by a factor of the inte-
ger fc,
l A (k) = #(fcAnM) (l)
for k G Z + . Then /a is a polynomial function in k of
degree n with rational coefficients
l&(k) — a n k + a n
-ik n + . . . + a
(2)
called the Ehrhart polynomial (Ehrhart 1967, Pommer-
sheim 1993). Specific coefficients have important geo-
metric interpretations.
1. a n is the Content of A.
2. a n -! is half the sum of the CONTENTS of the (n — 1)-
D faces of A.
3. a = 1.
Let 52(A) denote the sum of the lattice lengths of the
edges of A, then the case n = 2 corresponds to PICK'S
Theorem,
l A (k) = Vol(A)fc 2 + |5 2 (A) + 1.
(3)
Let 53(A) denote the sum of the lattice volumes of the
2-D faces of A, then the case n = 3 gives
l A (k) = Vol(A)fc 3 + §5 3 (A)fc 2 + ai k + 1, (4)
510
Ei
Eigenvalue
where a rather complicated expression is given by Pom-
mersheim (1993), since cl\ can unfortunately not be in-
terpreted in terms of the edges of A. The Ehrhart poly-
nomial of the tetrahedron with vertices at (0, 0, 0), (a,
0, 0), (0, 6, 0), (0, 0, c) is
l&(k) = \abck 2, + \{ab + ac + be + d) k 2
-f
4^
1 f ac
12 It
be ab d 2
— + — + ~T
a c abc
+ l(a + b + c+A + B + C)-Aa(^,^j
„ (ac bB
H-fr?)]
fc + 1, (5)
where s(x,y) is a Dedekind Sum, A = gcd(6, c), B =
gcd(a, c), C = gcd(a, 6) (here, gcd is the Greatest
Common Denominator), and d = ABC (Pommer-
sheim 1993).
see also Dehn Invariant, Pick's Theorem
References
Ehrhart, E. "Sur une probleme de geometrie diophantine
lineaire." J. Reine angew. Math. 227, 1-29, 1967.
MacDonald, I. G. "The Volume of a Lattice Polyhedron."
Proc. Camb. Phil. Soc. 59, 719-726, 1963.
McMullen, P. "Valuations and Euler-Type Relations on Cer-
tain Classes of Convex Polytopes." Proc. London Math.
Soc. 35, 113-135, 1977.
Pommersheim, J. "Toric Varieties, Lattices Points, and
Dedekind Sums." Math. Ann. 295, 1-24, 1993.
Reeve, J. E. "On the Volume of Lattice Polyhedra." Proc.
London Math. Soc. 7, 378-395, 1957.
Reeve, J. E. "A Further Note on the Volume of Lattice Poly-
hedra." Proc. London Math. Soc. 34, 57-62, 1959.
Ei
see Exponential Integral, ^-Function
Eigenfunct ion
If L is a linear Operator on a Function Space, then /
is an eigenfunction for L and A is the associated EIGEN-
VALUE whenever Lf = A/.
see also Eigenvalue, Eigenvector
Eigenvalue
Let A be a linear transformation represented by a MA-
TRIX A. If there is a VECTOR X € E n ^ such that
AX = AX
(1)
for some Scalar A, then A is the eigenvalue of A with
corresponding (right) EIGENVECTOR X. Letting A be a
k x k Matrix,
(2)
an
ai2 •
■ Clik
a 2 i
G22 •
• a 2 k
dki
dk2 '
' Clkk
with eigenvalue A, then the corresponding EIGENVEC-
TORS satisfy
cm ai2
^21 CL22
dkl Ctk2
aifc"
"xi"
~Xi~
a>2k
x 2
= A
X2
CLkk _
_Xk_
_X k _
(3)
which is equivalent to the homogeneous system
an - A ai2
Q>21 0,22 ~~ A
Ofcl
Q>k2
Cilk
~xi~
"0"
a>2k
x 2
=
akk — A_
_x k _
_0_
(4)
Equation (4) can be written compactly as
(A - AI)X = 0, (5)
where I is the Identity Matrix.
As shown in Cramer's Rule, a system of linear equa-
tions has nontrivial solutions only if the DETERMINANT
vanishes, so we obtain the CHARACTERISTIC EQUATION
|A — Al| = 0.
(6)
If all k As are different, then plugging these back in
gives k — 1 independent equations for the k components
of each corresponding EIGENVECTOR. The EIGENVEC-
TORS will then be orthogonal and the system is said to
be nondegenerate. If the eigenvalues are n-fold Degen-
erate, then the system is said to be degenerate and the
Eigenvectors are not linearly independent. In such
cases, the additional constraint that the EIGENVECTORS
be orthogonal,
A-i ' J*-j — Ji.iJi.jUij ,
(7)
where Sij is the KRONECKER DELTA, can be applied to
yield n additional constraints, thus allowing solution for
the Eigenvectors.
Assume A has nondegenerate eigenvalues Ai, A2, • . . , A n
and corresponding linearly independent EIGENVECTORS
Xi, X 2 , . . . , Xfc which can be denoted
(8)
#11
£21
#fcl
#12
J
#22
j ■ • *
Xk2
-El*.
_X2k _
_Xkk_
Define the matrices composed of eigenvectors
P = [X X X 2 ••• X fc ]
#11
#21 '
* ' Xkl
#12
#22
• * Xk2
#lfc
#2fc
• • Xkk
(9)
Eigenvalue
and eigenvalues
D =
Ai
A 2
(10)
•■■ A*
where D is a DIAGONAL MATRIX. Then
AP = A[Xi X 2 •••
X fc ]
= [AXi AX 2 •■
• AX fc ]
— [AiXi A2X2
• • AfcXfc ]
Aia3n A 2 a;2i
* • ■ Afc^fci
A1X12 A 2 #22
* ' * AfeXfc2
Ai^ifc A 2 a; 2 fc • • • \k%kk
#11
#21
Xkl
=
X\2
X-22
Xk2
_Xlk
%2k
Xkk_
= PD,
A =
PDP
_i
'Ai
A 2
Xk
(11)
(12)
Furthermore,
A 2 = (PDP" 1 )(PDP- 1 ) = PD(P- 1 P)DP" 1
= PD 2 P"\ (13)
By induction, it follows that for n > 0,
A n - PD n P-\ (14)
The inverse of A is
A" 1 = (POP" 1 )" 1 - PD-'P -1 , (15)
where the inverse of the DIAGONAL MATRIX D is triv-
ially given by
(16)
r Ai_1
D "-i
A2- 1 ..
••■ Afc" 1
Equation (14) therefore holds for both Positive and
Negative n.
A further remarkable result involving the matrices P and
D follows from the definition
.a s y*A^ = f* PD"P- 1
_ 2-^i n! ^—t n!
= P
V°°„D n
p-i = Pc Dp-i.
(17)
Eigenvalue 511
Since D is a Diagonal Matrix,
D
d = sr_ = y^ —
Z-~d n! L—j n\
xr
A 2 n
EZo^
... Afc
,* ...
EL^J
e Aa
L o o
(18)
e can be found using
D n
Ai n
X 2 n
A fc n J
(19)
Assume we know the eigenvalue for
AX = AX. (20)
Adding a constant times the IDENTITY MATRIX to A,
(A + c l)X = (A 4- c)X = A'X, (21)
so the new eigenvalues equal the old plus c. Multiplying
A by a constant c
(cA)X = c(AX) = A'X,
(22)
so the new eigenvalues are the old multiplied by c.
Now consider a Similarity Transformation of A.
Let |A| be the Determinant of A, then
|Z" 1 AZ - Al| = |Z- X (A - AI)Z|
= |Z||A-AI||Z- 1 | = |A-AI|, (23)
so the eigenvalues are the same as for A.
see also BRAUER'S THEOREM, CONDITION NUMBER,
Eigenfunction, Eigenvector, Frobenius Theo-
rem, Gersgorin Circle Theorem, Lyapunov's
First Theorem, Lyapunov's Second Theorem, Os-
trowski's Theorem, Perron's Theorem, Perron-
Frobenius Theorem, Poincare Separation Theo-
rem, Random Matrix, Schur's Inequalities, Stur-
mian Separation Theorem, Sylvester's Inertia
Law, Wielandt's Theorem
References
Arfken, G. "Eigenvectors, Eigenvalues." §4.7 in Mathemati-
cal Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 229-237, 1985.
512 Eigenvector
Eight Curve
Nash, J. C. "The Algebraic Eigenvalue Problem." Ch. 9 in
Compact Numerical Methods for Computers: Linear Alge-
bra and Function Minimisation, 2nd ed. Bristol, England:
Adam Hilger, pp. 102-118, 1990.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Eigensystems." Ch. 11 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 449-489, 1992.
Eigenvector
A right eigenvector satisfies
AX = AX,
(1)
where X is a column VECTOR. The right EIGENVALUES
therefore satisfy
|A-AI| = 0. (2)
A left eigenvector satisfies
XA = AX,
where X is a row VECTOR, so
(3)
(XA) T = A £ X T
A T X T = A L X T ,
(4)
(5)
where X T is the transpose of X. The left EIGENVALUES
satisfy
|A T - \ L \\ = |A T - \ L \ T \ = |(A - Ail) T | = |(A - Ail)|,
(6)
(since |A| = |A T |) where |A| is the Determinant of
A. But this is the same equation satisfied by the right
Eigenvalues, so the left and right Eigenvalues are
the same. Let Xj? be a Matrix formed by the columns
of the right eigenvectors and Xl be a MATRIX formed
by the rows of the left eigenvectors. Let
Dee
(7)
Then
AX* = X*D X x ,A = DX i , (8)
X L AX R = XiXflD X l AXh = DXiXfl, (9)
X L X R D = DX^X*.
(10)
But this equation is of the form CD = DC where D is a
Diagonal Matrix, so it must be true that C = X L X R
is also diagonal. In particular, if A is a Symmetric Ma-
trix, then the left and right eigenvectors are transposes
of each other. If A is a Self-Adjoint Matrix, then
the left and right eigenvectors are conjugate HERMITIAN
Matrices.
Given a 3 x 3 MATRIX A with eigenvectors xi , X2 , and X3
and corresponding EIGENVALUES Ai, A2, and A3, then
an arbitrary VECTOR y can be written
y — 61x1 + 6 2 x 2 + &3X3.
Applying the MATRIX A,
(ii)
Ay = bi Axi + 6 2 Ax 2 + fc 3 Ax 3
/ A2 A3 \
= Al ( &1X1 + -T-&2X2 + T-&3X3 J , (12)
A n y = Ai n
61x1 +(^) n 6 2 x 2 + (Q"j,x,]. (13)
If Ai > A2, A3, it therefore follows that
lim A n y = Ai n 6ixi,
(14)
so repeated application of the matrix to an arbitrary vec-
tor results in a vector proportional to the EIGENVECTOR
having the largest EIGENVALUE.
see also ElGENFUNCTION, EIGENVALUE
References
Arfken, G. "Eigenvectors, Eigenvalues." §4.7 in Mathemati-
cal Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 229-237, 1985.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Eigensystems." Ch. 11 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 449-489, 1992.
Eight Curve
A curve also known as the Gerono Lemniscate. It is
given by Cartesian Coordinates
4 2/2 2\
x =0 [x -y ),
Polar Coordinates,
r 2 =a 2 sec 4 0cos(2(9),
and parametric equations
x = a sin t
y — a sin t cos t.
(i)
(2)
(3)
(4)
Eight-Point Circle Theorem
Eilenberg-Mac Lane-Steenrod-Milnor Axioms
Eight Surface
513
k
KJ
^
\
r
The Curvature and Tangential Angle are
_ 3sin£-f sin(3£)
* W " "2[cos 2 i + cos 2 (2t)]V2
<f>(t) = — tan _1 [cos£sec(2i)].
(5)
(6)
see also Butterfly Curve, Dumbbell Curve, Eight
Surface, Piriform
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. 124-126, 1972.
Lee, X. "Lemniscate of Gerono." http://www.best.com/
-xah/SpecialPlaneCurvesjdir/LemniscateOf Gerono jdir/
lemniscateOf Gerono .html.
MacTutor History of Mathematics Archive. "Eight Curve."
http : //www- groups . dcs . st-and . ac . uk/ -hi story /Curves
/Eight, html.
Eight-Point Circle Theorem
D
^ a c
Let ABCD be a Quadrilateral with Perpendicu-
lar Diagonals. The Midpoints of the sides (a, 6, c,
and d) determine a PARALLELOGRAM (the VARIGN0N
Parallelogram) with sides Parallel to the Diag-
onals. The eight-point circle passes through the four
Midpoints and the four feet of the Perpendiculars
from the opposite sides a , &', c' , and d'.
see also FEUERBACH'S THEOREM
References
Brand, L. "The Eight-Point Circle and the Nine-Point Cir-
cle." Amer. Math. Monthly 51, 84-85, 1944.
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 11-13, 1976.
The Surface of Revolution given by the parametric
equations
x(u,v) = cosusin(2v) (1)
y(u,v) = sinusin(2i;) (2)
z(u, v) — sinv (3)
for u € [0,2tt) and v e [-7r/2,7r/2].
see also Eight Curve
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 209-210 and 224,
1993.
Eikonal Equation
£(£)
Eilenberg-Mac Lane Space
For any Abelian Group G and any Natural Number
n, there is a unique SPACE (up to Homotopy type)
such that all HOMOTOPY GROUPS except for the nth are
trivial (including the 0th HOMOTOPY GROUPS, meaning
the SPACE is path-connected), and the nth HOMOTOPY
Group is Isomorphic to the Group G. In the case
where n = 1, the GROUP G can be non-ABELlAN as
well.
Eilenberg-Mac Lane spaces have many important appli-
cations. One of them is that every TOPOLOGICAL SPACE
has the HOMOTOPY type of an iterated FlBRATiON of
Eilenberg-Mac Lane spaces (called a POSTNIKOV SYS-
TEM). In addition, there is a spectral sequence relating
the COHOMOLOGY of Eilenberg-Mac Lane spaces to the
Homotopy Groups of Spheres.
Eilenberg-Mac Lane-Steenrod-Milnor
Axioms
see Eilenberg-Steenrod Axioms
514 Eilenberg-Steenrod Axioms
Eisenstein Integer
Eilenberg-Steenrod Axioms
A family of FUNCTORS H n (-) from the CATEGORY of
pairs of TOPOLOGICAL SPACES and continuous maps,
to the Category of Abelian Groups and group ho-
momorphisms satisfies the Eilenberg-Steenrod axioms if
the following conditions hold.
1. Long Exact Sequence of a Pair Axiom. For
every pair (X, A), there is a natural long exact se-
quence
. . . -* H n (A) -> H n (X) -» H n (X, A)
-+H n -!(A) ->...,
where the Map H n (A) -> H n (X) is induced by the
Inclusion Map A -»■ X and H n (X) -> H n (X, A) is
induced by the INCLUSION MAP (X,<j>) -> (X,A).
The Map H n (X,A) -» H n -i{A) is called the
Boundary Map.
2. Homotopy Axiom. If / : (X,A) -> (Y,B) is ho-
motopic to g : (X,A) — ¥ (Y,B), then their IN-
DUCED Maps /* : H n (X,A) -> H n (Y,B) and g* :
H n (X t A) -> H n (Y, B) are the same.
3. Excision Axiom. If X is a Space with Sub-
spaces A and U such that the Closure of A is
contained in the interior of U, then the INCLUSION
Map (X U,AU) -* (X, A) induces an isomorphism
H n (X U,AU)^H n {X,A).
4. Dimension Axiom. Let X be a single point space.
H n (X) = unless n — 0, in which case Hq(X) = G
where G are some GROUPS. The H are called the
Coefficients of the Homology theory H(-).
These are the axioms for a generalized homology the-
ory. For a cohomology theory, instead of requiring that
H(-) be a FUNCTOR, it is required to be a co-functor
(meaning the INDUCED MAP points in the opposite di-
rection). With that modification, the axioms are essen-
tially the same (except that all the induced maps point
backwards) .
see also Aleksandrov-Cech Cohomology
Ein Function
^. , x f* (l-e- l )dt „, x ,
Ein(z)=/ ^ — } — =Ei(z)+ln^ + 7 ,
where 7 is the Euler-Mascheroni Constant and Ei
is the E n -FUNCTI0N with n = 1.
see also E n -FUNCTION
Einstein Functions
The functions x 2 e x /(e x - l) 2 , x/(e x - 1), ln(l - e" x ),
and x/(e x - 1) - ln(l - e~ x ).
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Debye Func-
tions." §27.1 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 999-1000, 1972.
Einstein Summation
The implicit convention that repeated indices are
summed over so that, for example,
ai<ix = y CLidi.
Eisenstein Integer
The numbers a -b bus, where
is one of the ROOTS of z s — 1, the others being 1 and
u 2 =\(-l-iV3).
Eisenstein integers are members of the QUADRATIC
Field Q(^/ r 3 ), and the Complex Numbers Z[o>]. Ev-
ery Eisenstein integer has a unique factorization. Specif-
ically, any NONZERO Eisenstein integer is uniquely the
product of POWERS of -1, u>, and the "positive" EISEN-
STEIN PRIMES (Conway and Guy 1996). Every Eisen-
stein integer is within a distance |n|/\/3 of some multiple
of a given Eisenstein integer n.
Dorrie (1965) uses the alternative notation
J=i(l + iV3) (1)
= |(1-»V3). (2)
Eisenstein-Jacobi Integer
Elastica
515
for — {J 2 and — a>, and calls numbers of the form aj + bO
G-NUMBERS. and J satisfy
(3)
(4)
(5)
(6)
(7)
(8)
The sum, difference, and products of G numbers are also
G numbers. The norm of a G number is
j + o =
1
JO =
: 1
J 2 +o =
2 + J =
J 3 =
-1
o 3 -
-1.
N(aJ + bO) = a 2 + b 2 - ab.
(9)
The analog of FERMAT'S Theorem for Eisenstein inte-
gers is that a PRIME NUMBER p can be written in the
form
a — ab + b 2 = (a + buj)(a + 6a; )
IFF 3jp+ 1. These are precisely the PRIMES of the form
3m 2 + n 2 (Conway and Guy 1996).
see also EISENSTEIN PRIME, ElSENSTEIN UNIT, GAUS-
SIAN Integer, Integer
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 220-223, 1996.
Cox, D. A. §4A in Primes of the Form x 2 + ny 2 : Fer-
mat, Class Field Theory and Complex Multiplication. New
York: Wiley, 1989.
Dorrie, H. "The Fermat-Gauss Impossibility Theorem." §21
in 100 Great Problems of Elementary Mathematics: Their
History and Solutions. New York: Dover, pp. 96-104,
1965.
Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes."
§A16 in Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 33-36, 1994.
Riesel, H. Appendix 4 in Prime Numbers and Com-
puter Methods for Factorization, 2nd ed. Boston, MA:
Birkhauser, 1994.
Wagon, S. "Eisenstein Primes." Mathematica in Action.
New York: W. H. Freeman, pp. 278-279, 1991.
Eisenstein-Jacobi Integer
see Eisenstein Integer
Eisenstein Prime
••> (• - •} <••
VkSSS
••> {• ? •) <**
Let a; be the CUBE ROOT of unity (-1 + n/3)/2. Then
the Eisenstein primes are
1. Ordinary PRIMES CONGRUENT to 2 (mod 3),
2. 1 — a; is prime in Z[w],
3. Any ordinary Prime CONGRUENT to 1 (mod 3) fac-
tors as aa*, where each of a and a* are primes in
Z[u;] and a and a* are not "associates" of each other
(where associates are equivalent modulo multiplica-
tion by an EISENSTEIN UNIT).
References
Cox, D. A. §4A in Primes of the Form x 2 + ny 2 : Fer-
mat, Class Field Theory and Complex Multiplication. New
York: Wiley, 1989.
Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes."
§A16 in Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 33-36, 1994.
Wagon, S. "Eisenstein Primes." Mathematica in Action.
New York: W. H. Freeman, pp. 278-279, 1991.
Eisenstein Series
E r (t) - £'
(mt + n) 2r '
where the sum £' excludes m = n = 0,£s[£]>0, and r
is an Integer > 2. The Eisenstein series satisfies the
remarkable property
see also RAMANUJAN-ElSENSTEIN SERIES
Eisenstein Unit
The Eisenstein units are the EISENSTEIN INTEGERS ±1,
±u;, ±tt> 2 , where
w = §(-l+zV3)
w a = i(-i -*/§).
see also EISENSTEIN INTEGER, EISENSTEIN PRIME
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 220-223, 1996.
Elastica
The elastica formed by bent rods and considered in phys-
ics can be generalized to curves in a RlEMANNIAN MAN-
IFOLD which are a CRITICAL POINT for
F x (j)
J y
■A),
where k is the GEODESIC CURVATURE of 7, A is a REAL
Number, and 7 is closed or satisfies some specified
516
Elation
Elementary Symmetric Function
boundary condition. The curvature of an elastica must
satisfy
= 2k"(s) + k (s) + 2k(s)G(s) - A«(«),
where k is the signed curvature of 7, G(s) is the GAUS-
SIAN Curvature of the oriented Riemannian surface M
along 7, k" is the second derivative of n with respect to
s, and A is a constant.
References
Barros, M. and Garay, O. J. "Free Elastic Parallels in a Sur-
face of Revolution." Amer. Math. Monthly 103, 149-156,
1996.
Bryant, R. and Griffiths, P. "Reduction for Constrained Vari-
ational Problems and J(k 2 /s)ds." Amer. J. Math. 108,
525-570, 1986.
Langer, J. and Singer, D. A. "Knotted Elastic Curves in fi 3 ."
J. London Math. Soc. 30, 512-520, 1984.
Langer, J. and Singer, D. A, "The Total Squared of Closed
Curves." J. Diff. Geom. 20, 1-22, 1984.
Elation
A perspective COLLINEATION in which the center and
axis are incident.
see also Homology (Geometry)
Elder's Theorem
A generalization of Stanley's Theorem. It states that
the total number of occurrences of an INTEGER k among
all unordered Partitions of n is equal to the number
of occasions that a part occurs k or more times in a
Partition, where a Partition which contains r parts
that each occur k or more times contributes r to the
sum in question.
see also Stanley's Theorem
References
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer, pp. 8-9, 1985.
Election
see Early Election Results, Voting
Electric Motor Curve
see Devil's Curve
Element
If x is a member of a set A, then x is said to be an
element of A, written x G A. If x is not an element of
A, this is written x £ A. The term element also refers to
a particular member of a GROUP, or entry in a MATRIX.
Elementary Function
A function built up of compositions of the EXPONENTIAL
Function and the Trigonometric Functions and
their inverses by ADDITION, MULTIPLICATION, DIVI-
SION, root extractions (the Elementary Operations)
under repeated compositions. Not all functions are el-
ementary. For example, the NORMAL DISTRIBUTION
Function
•<•> -£!•-"■
dt
is a notorious example of a nonelementary function.
Nonelementary functions are called TRANSCENDENTAL
Functions.
see also ALGEBRAIC FUNCTION, ELEMENTARY OPER-
ATION, Elementary Symmetric Function, Trans-
cendental Function
References
Shanks, D, Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 145, 1993.
Elementary Matrix
The elementary Matrices are the Permutation Ma-
trix p^. and the SHEAR MATRIX e^.
Elementary Operation
One of the operations of ADDITION, SUBTRACTION,
Multiplication, Division, and root extraction.
see also ALGEBRAIC FUNCTION, ELEMENTARY FUNC-
TION
Elementary Symmetric Function
The elementary symmetric functions IT n on n variables
{rci, . . . , x n } are denned by
LTi = 22 Xi
l<i<n
112 = y xiXj
l<i<j<n
113 = 2_^ XiXjXk
l<i<j<k<n
114 = y XiXjXkXl
l<i<j<k<l<n
n n = Yl x i-
i<i<n
(i)
(2)
(3)
(4)
(5)
Alternatively, IIj- can be defined as the coefficient of
X n ~ j in the GENERATING FUNCTION
n (x+x^.
(6)
Elements
Elkies Point
517
The elementary symmetric functions satisfy the relation-
ships
n
^^ 2 = ni 2 -2ii2 (T)
2 = 1
Tl
]P xi 3 = n x 3 - 31I1II2 + an 3 (8)
i=l
Tl
]T xi 4 = iii 4 - 4n! 2 n 2 + 2n 2 2 + 4n x n 3 - 4n 4 (9)
z = l
(Beeler ct a/. 1972, Item 6).
see also FUNDAMENTAL THEOREM OF SYMMET-
RIC Functions, Newton's Relations, Symmetric
Function
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Elements
The classic treatise in geometry written by Euclid and
used as a textbook for more than 1,000 years in western
Europe. The Elements, which went through more than
2,000 editions and consisted of 465 propositions, are di-
vided into 13 "books" (an archaic word for "chapters").
Book
Contents
1
triangles
2
rectangles
3
Circles
4
polygons
5
proportion
6
similarity
7-10
number theory
11
solid geometry
12
pyramids
13
platonic solids
The elements started with 23 definitions, five POSTU-
LATES, and five "common notions," and systematically
built the rest of plane and solid geometry upon this foun-
dation. The five EUCLID'S POSTULATES are
1. It is possible to draw a straight LINE from any POINT
to another Point.
2. It is possible to produce a finite straight LINE con-
tinuously in a straight LINE.
3. It is possible to describe a Circle with any Center
and Radius.
4. All Right Angles are equal to one another.
5. If a straight Line falling on two straight Lines makes
the interior ANGLES on the same side less than two
Right Angles, the straight Lines (if extended in-
definitely) meet on the side on which the ANGLES
which are less than two RIGHT ANGLES lie.
(Dunham 1990). Euclid's fifth postulate is known as the
Parallel Postulate. After more than two millennia
of study, this POSTULATE was found to be independent
of the others. In fact, equally valid NON-EUCLIDEAN
Geometries were found to be possible by changing the
assumption of this POSTULATE. Unfortunately, Euclid's
postulates were not rigorously complete and left a large
number of gaps. Hilbert needed a total of 20 postulates
to construct a logically complete geometry.
see also Parallel Postulate
References
Casey, J. A Sequel to the First Six Books of the Elements of
Euclid, 6th ed. Dublin: Hodges, Figgis, & Co., 1892.
Dixon, R. Mathographics. New York: Dover, pp. 26-27, 1991.
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 30-83, 1990.
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.
Joyce, D. E. "Euclid's Elements." http://aleph0.clarku.
edu/-djoyce/java/elements /elements .html
Elevator Paradox
A fact noticed by physicist G. Gamow when he had an
office on the second floor and physicist M. Stern had
an office on the sixth floor of a seven-story building
(Gamow and Stern 1958, Gardner 1986). Gamow no-
ticed that about 5/6 of the time, the first elevator to
stop on his floor was going down, whereas about the
same fraction of time, the first elevator to stop on the
sixth floor was going up. This actually makes perfect
sense, since 5 of the 6 floors 1, 3, 4, 5, 6, 7 are above the
second, and 5 of the 6 floors 1, 2, 3, 4, 5, 7 are below the
sixth. However, the situation takes some unexpected
turns if more than one elevator is involved, as discussed
by Gardner (1986).
References
Gamow, G. and Stern, M. Puzzle Math. New York: Viking,
1958.
Gardner, M. "Elevators." Ch. 10 in Knotted Doughnuts and
Other Mathematical Entertainments. New York: W. H.
Freeman, pp. 123-132, 1986.
Elkies Point
Given POSITIVE numbers s ay Sb, and s cy the Elkies point
is the unique point Y in the interior of a TRIANGLE
A ABC such that the respective INRADII r a , n>, r c of
the TRIANGLES ABYC, ACYA, and AAYB satisfy r a :
Tb '• r c = s a : Sb : s c -
see also CONGRUENT INCIRCLES POINT, INRADIUS
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. and Elkies, N. "Problem 1238 and Solution."
Math. Mag. 60, 116-117, 1987.
518 Ellipse
Ellipse
A curve which is the Locus of all points in the Plane
the Sum of whose distances r\ and ti from two fixed
points F\ and F 2 (the Foci) separated by a distance of
2c is a given POSITIVE constant 2a (left figure). This re-
sults in the two-center Bipolar Coordinate equation
n + r 2 = 2a,
(i)
where a is the SEMIMAJOR AXIS and the ORIGIN of the
coordinate system is at one of the Foci. The ellipse
can also be defined as the LOCUS of points whose dis-
tance from the FOCUS is proportional to the horizontal
distance from a vertical line known as the DIRECTRIX
(right figure).
The ellipse was first studied by Menaechmus, investi-
gated by Euclid, and named by Apollonius. The FOCUS
and Directrix of an ellipse were considered by Pap-
pus. In 1602, Kepler believed that the orbit of Mars
was OVAL; he later discovered that it was an ellipse with
the Sun at one Focus. In fact, Kepler introduced the
word "FOCUS" and published his discovery in 1609. In
1705 Halley showed that the comet which is now named
after him moved in an elliptical orbit around the Sun
(MacTutor Archive).
A ray passing through a FOCUS will pass through the
other focus after a single bounce. Reflections not passing
through a FOCUS will be tangent to a confocal HYPER-
BOLA or Ellipse, depending on whether the ray passes
between the Foci or not. Let an ellipse lie along the
a;- AXIS and find the equation of the figure (1) where r\
and r 2 are at (-c,0) and (c,0). In CARTESIAN COOR-
DINATES,
y/(x + c) 2 + y 2 + y/(x - c) 2 + y 2 = 2a. (2)
Bring the second term to the right side and square both
sides,
{x+c) 2 +y 2 = 4a 2 -4ay/(x - c) 2 + y 2 + (x-cf+y 2 . (3)
Now solve for the SQUARE ROOT term and simplify
y/(x-c) 2 +y 2
~4a
1 / 2 , ,2,2 A 2 2 , o 2 2\
— — (x -\-2xc-\-c 4- y —4a — x + 2xc — c — y)
— — — -{Axe- 4a 2 ) = a x. (4)
4a a
Square one final time to clear the remaining SQUARE
Root,
2
x 2 - 2xc + c +y 2 = a - 2cx + ~x 2 . (5)
Grouping the x terms then gives
2 2
20 — C 2 2 2
x |_ y =a _ c ?
which can be written in the simple form
x
V
a' a' — c A
Defining a new constant
i2 _ 2 2
o — a — c
= 1.
(6)
(7)
(8)
puts the equation in the particularly simple form
2 2
? + &-'• »
The parameter b is called the Semiminor Axis by anal-
ogy with the parameter a, which is called the SEMIMA-
JOR Axis. The fact that b as defined above is actu-
ally the Semiminor Axis is easily shown by letting r\
and T2 be equal. Then two Right Triangles are pro-
duced, each with HYPOTENUSE a, base c, and height
b = yja 2 — c 2 . Since the largest distance along the MI-
NOR Axis will be achieved at this point, b is indeed the
Semiminor Axis.
If, instead of being centered at (0, 0), the Center of
the ellipse is at (xo, yo), equation (9) becomes
Qk-sq) 2 (y - yo) 2 1
6 2
(10)
As can be seen from the CARTESIAN EQUATION for the
ellipse, the curve can also be given by a simple paramet-
ric form analogous to that of a CIRCLE, but with the x
and y coordinates having different scalings,
x = a cost
y = b sin t .
(11)
(12)
The unit TANGENT VECTOR of the ellipse so parame-
terized is
x T (t) = -
yr(t) =
asint
yb 2 cos 2 t + a 2 sin 2 t
bcost
yb 2 cos 2 t + a 2 sin 2 t
(13)
(14)
A sequence of Normal and Tangent Vectors are
plotted below for the ellipse.
Ellipse
Ellipse 519
curve rotated by angle i
For an ellipse centered at the ORIGIN but inclined at
an arbitrary ANGLE to the x-AxiS, the parametric
equations are
cos#
— sin#
sin0
COS0
acosi
bsint
a cos 9 cos t + b sin 9 sin t
—a sin 9 cos t + b cos 9 sin t
(15)
In Polar Coordinates, the Angle 9' measured from
the center of the ellipse is called the Eccentric An-
gle. Writing r f for the distance of a point from the
ellipse center, the equation in Polar Coordinates is
just given by the usual
x ~ r cos 9
v — r sin 9 .
(16)
(17)
Here, the coordinates 9' and r' are written with primes
to distinguish them from the more common polar co-
ordinates for an ellipse which are centered on a focus.
Plugging the polar equations into the Cartesian equa-
tion (9) and solving for r' 2 gives
J 2
r =
o a
b 2 cos 2 9' + a 2 sin 2 9'
(18)
Define a new constant < e < 1 called the ECCENTRIC-
ITY (where e = is the case of a CIRCLE) to replace
b
^A/ 1 "^'
(19)
from which it ^also follows from (8) that
2 2 2 »2 _ 2
a e = a — o — c
c = ae
6 2 =a 2 (l-e 2 ).
Therefore (18) can be written as
q 2 (l^e 2 )
12
r =
r = a
1 - e 2 cos 2 9'
1-e 2
(20)
(21)
(22)
(23)
(24)
, l-e 2 cos 2 0'*
If e < 1, then
r' - a{\ - \e sin 2 9' - ^e 4 [5 + 3 cos(20')] sin 2 9' + . . .},
(25)
so ; f
Ar _ a — r x 2 . i a t / 0ft \
= & ±e sin 9 . (26)
a a
~cj= a{\ -e)
If r and 9 are measured from a FOCUS instead of from
the center, as they commonly are in orbital mechanics,
then the equations of the ellipse are
(27)
(28)
(29)
x = c -h r cos 9
y — r sin0,
and (9) becomes
(c + r cos 9) 2 r 2 sin 2 9 _
tf + — 62" ~
Clearing the DENOMINATORS gives
b 2 (c 2 + 2cr cos 9 + r 2 cos 2 9) + a V sin 2 9 = a 2 b 2 (30)
6 2 c 2 +2rc6 2 cos + 6 2 r 2 cos 2 + aV-aV cos 2 = aV.
(31)
Plugging in (21) and (22) to re-express b and c in terms
of a and e,
a 2 (l-e 2 )a 2 e 2 -f2aea 2 (l-e 2 )rcos(9 + a 2 (l-e 2 )r 2 cos 2 <9
+aV - aV cos 2 9 = a 2 [a 2 (l - e 2 )]. (32)
-r 2 + [ercostf - a(l - e 2 )] 2 = (33)
r = ±[ercos<? - o(l - e 2 )]. (34)
The sign can be determined by requiring that r must be
Positive. When e = 0, (34) becomes r = ±(— a), but
Simplifying,
520 Ellipse
since a is always POSITIVE, we must take the NEGATIVE
sign, so (34) becomes
r — a(l — e 2 ) — ercos#
r(l + e cos 9) — a(l — e )
_ a(l-e 2 )
(35)
(36)
(37)
1 + e cos 6 '
The distance from a FOCUS to a point with horizontal
coordinate x is found from
COS0 :
C + X
Plugging this into (37) yields
r + e(c + x) = a(l — e )
r = a(l — e ) — e(c + x).
(38)
(39)
(40)
Summarizing relationships among the parameters char-
acterizing an ellipse,
b — ayl — e 2 = y a 2
Va 2 - b 2
ae
a z a
(41)
(42)
(43)
The ECCENTRICITY can therefore be interpreted as the
position of the FOCUS as a fraction of the SEMIMAJOR
Axis.
In Pedal Coordinates with the Pedal Point at the
FOCUS, the equation of the ellipse is
2a
r
1.
(44)
To find the RADIUS OF CURVATURE, return to the para-
metric coordinates centered at the center of the ellipse
and compute the first and second derivatives,
(45)
(46)
X
= — asint
t
y
= b cos t
tt
X
— —a cost
ff
y
— — bsint.
(47)
(48)
Therefore,
#-
(x' 2 + y ,2 ) s/2
(a 2 sin 2 t + b 2 cos 2 t)^ 2
— asini(— 6sini) — (acos£)(bcos£)
(a 2 sin 2 t + b 2 cos 2 t) 3/2
a6(sin 2 t + cos 2 t)
(a 2 sin 2 t + b 2 cos 2 tf? 2
ab
(49)
Ellipse
Similarly, the unit TANGENT VECTOR is given by
T =
-a sin t
bcost J JtfsinH + Vcosn'
(50)
The Arc Length of the ellipse can be computed using
/ b 2
<(1 -sin 2 t) + —sin 2 id*
-•/■
= a \/l -e 2 sin 2 tdt = aE(t,e),
(51)
where E is an incomplete ELLIPTIC INTEGRAL OF THE
SECOND Kind. Again, note that Hsa parameter which
does not have a direct interpretation in terms of an AN-
GLE. However, the relationship between the polar angle
from the ellipse center 6 and the parameter t follows
from
= tan -1 (-) = tan -1 (- t&nt) . (52)
12 3 4 5 6
This function is illustrated above with shown as the
solid curve and t as the dashed, with b/a = 0.6. Care
must be taken to make sure that the correct branch
of the Inverse TANGENT function is used. As can be
seen, weaves back and forth around t, with crossings
occurring at multiples of 7r/2.
The Curvature and Tangential Angle of the ellipse
are given by
ab
(& 2 cos 2 £ + a 2 sin 2 r.) 3 /2
— tan" 1 ( - cos£] .
(53)
(54)
Ellipse
Ellipse 521
The entire PERIMETER p of the ellipse is given by setting
t = 2tt (corresponding to = 27r), which is equivalent to
four times the length of one of the ellipse's QUADRANTS,
p = aE(2ir, e) = 4a£(±7r, e) = 40,57(6), (55)
where £7(e) is a complete Elliptic Integral of the
Second Kind with Modulus k. The Perimeter
can be computed numerically by the rapidly converg-
ing Gauss-Kummer Series
p = ir(a + b) ^2 [ 2 ) h
: ^ + &)(i + ^ + M+5k fe +-) J ( 5 S)
where
fc =
a — b
a + b
(57)
and (2) is a BINOMIAL COEFFICIENT. Approximations
to the Perimeter include
(58)
p^7ri/2(a 2 + 6 2 )
w 7r[3(a + 6) - V( a + 3& )( 3a + b )] ( 5 ^)
3i
?r(a + b) ( 1 +
10 + V4 - 3i
where the last two are due to Ramanujan (1913-14),
(60)
U + &J '
(61)
and (60) is accurate to within ~ 3 ■ 2 17 t 5
The maximum and minimum distances from the FOCUS
are called the APOAPSIS and PERIAPSIS, and are given
by
?"+ — 7"apoapsis — Q>{L + 6)
r_ = periapsis = a(l — e).
(62)
(63)
The Area of an ellipse may be found by direct INTE-
GRATION
a r>b\J a 2 — x 2 j a
LI
a J — by/a 2 —x 2 /a
dydx ■
J —a
dx
26 (l
a \2
:\/a 2
x 2 + a sin
2 . -1 / Z
= a6[sin 1 1 — sin 1 ( — 1)] = a& — — I — 1 = 7ra&.
(64)
The AREA can also be computed more simply by making
the change of coordinates x = (b/a)x and y' = y from
the elliptical region i? to the new region R* . Then the
equation becomes
or a;' 2 + 1/ /2 = 6 2 , so #' is a Circle of Radius b. Since
dx 1 ~
the JACOBIAN is
a(s,y) I _
a(x',y') | ~
ax a y '
dx f dx'
dx dy
dy' dy'
(!)'
t o
1
a
6'
a
6'
(66)
(67)
The AREA is therefore
J J dxdy^JJ J^j|dx'd»'
= - / / dx dy — -(nb ) = 7ra6,
6 JJk' 6
(68)
as before. The Area of an arbitrary ellipse given by the
Quadratic Equation
ax + bxy + cy =1
2tt
(69)
(70)
V4ac — b 2
The Area of an Ellipse with semiaxes a and 6 with
respect to a Pedal Point P is
^=l7r(a 2 +fe 2 + |OP| 2 ).
(71)
The ellipse INSCRIBED in a given TRIANGLE and tangent
at its Midpoints is called the Midpoint Ellipse. The
Locus of the centers of the ellipses INSCRIBED in a TRI-
ANGLE is the interior of the Medial Triangle. New-
ton gave the solution to inscribing an ellipse in a convex
Quadrilateral (Dorrie 1965, p. 217). The centers of
the ellipses Inscribed in a Quadrilateral all lie on
the straight line segment joining the Midpoints of the
Diagonals (Chakerian 1979, pp. 136-139).
The Area of an ellipse with Barycentric Coordi-
nates (a,/3,7) Inscribed in a Triangle of unit Area
is
A - Try^l - 2a)(l - 20)(1 - 2 7 ). (72)
(Chakerian 1979, pp. 142-145).
The LOCUS of the apex of a variable CONE containing
an ellipse fixed in 3-space is a Hyperbola through the
Foci of the ellipse. In addition, the Locus of the apex
of a Cone containing that Hyperbola is the original
ellipse. Furthermore, the ECCENTRICITIES of the ellipse
and Hyperbola are reciprocals. The Locus of centers
522 Ellipse Caustic Curve
Ellipse Envelope
of a Pappus Chain of Circles is an ellipse. Surpris-
ingly, the locus of the end of a garage door mounted
on rollers along a vertical track but extending beyond
the track is a quadrant of an ellipse (the envelopes of
positions is an Astroid).
see also Circle, Conic Section, Eccentric
Anomaly, Eccentricity, Elliptic Cone, Ellip-
tic Curve, Elliptic Cylinder, Hyperbola, Mid-
point Ellipse, Parabola, Paraboloid, Quadratic
Curve, Reflection Property, Salmon's Theorem,
Steiner's Ellipse
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 126 and 198-199, 1987.
Casey, J. "The Ellipse." Ch. 6 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. 201-249, 1893.
Chakerian, G. D. "A Distorted View of Geometry." Ch. 7
in Mathematical Plums (Ed. R. Honsberger). Washington,
DC: Math. Assoc. Amer., 1979.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 75, 1996.
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, 1965.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 72-78, 1972.
Lee, X. "Ellipse." http : //www . best . com/-xah/Special
PlaneCurves_dir/Ellipse_dir/ellipse.html.
Lockwood, E. H. "The Ellipse." Ch. 2 in A Book of Curves.
Cambridge, England: Cambridge University Press, pp. 13-
24, 1967.
MacTutor History of Mathematics Archive. "Ellipse." http:
//www -groups . dcs . st-and . ac . uk/ -history /Curves/
Ellipse.html.
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.
At (oo,0),
_ cosr.[-l + 5r 2 - cos(2*)(l + r 2 )]
4r
y = sin t.
(9)
(10)
Ellipse Envelope
Consider the family of Ellipses
,2
x 2
-7 +
y
(l-c) 2
1 =
(1)
Ellipse Caustic Curve
For an ELLIPSE given by
x — r cos t
y = sin t
with light source at (s,0), the CAUSTIC is
N x
x — —
D x
y
Ny
(1)
(2)
(3)
(4)
where
N x = 2rz(3 - 5r 2 ) + (-6r 2 + 6r 4 - 3x 2 + 9r V) cos t
+ 6rx(l -r 2 )cos(2t)
+ (-2r 2 + 2r 4 - x 2 - rV) cos(3*) (5)
D x = 2r(l + 2r 2 + Ax 2 ) + 3x(l - 5r 2 ) cos t
+ (6r + 6r 3 ) cos(2t) + x(l - r 2 ) cos(3t) (6)
N y = $r(-l + r 2 -x 2 )sint (7)
D y = 2r(-l - r 2 - 4a: 2 ) + 3(~x + 5r 2 ) cost
+ 6r(l - r 2 ) cos(2£) + x(~l + r 2 ) cos(3£). (8)
for c € [0,1]. The PARTIAL DERIVATIVE with respect to
c is
+
(1 - c)t
=
(1 - C )s
Combining (1) and (3) gives the set of equations
£2" (l-c)2
1 1
L c 3 (l-c)3 J
1_
A
1_
A
i i
l i
~^ -&
i
i
where the DISCRIMINANT is
1 1
c 2 (l-c) 3 c 3 (l-c) 2 c 3 (l-c) 3
(2)
(3)
(4)
(5)
, (6)
Ellipse Evolute
Ellipsoid 523
so (5) becomes
Ellipse Involute
y
(l-c) 3
(7)
X ~ c COS t
y = (1 — c) sini.
Eliminating c then gives
x ,/s + i, v, = l l (8)
which is the equation of the Astroid. If the curve is
instead represented parametrically, then
(9)
(10)
Solving
dx dy dx dy
dt dc dc dt
= (— csin£)( — sint) — (cost)[(l — c) cost]
= c(sin 2 t + cos 2 i) - cos 2 1 = c - cos 2 1 = (11)
for c gives
c = cos £,
so substituting this back into (9) and (10) gives
x — (cos t) cos t = cos £
2/ = (1 — cos 2 i) sint = sin 3 t,
the parametric equations of the ASTROID.
see also Astroid, Ellipse, Envelope
Ellipse Evolute
(12)
(13)
(14)
The Evolute of an Ellipse is given by the parametric
equations
a 2 -b 2
3 ,
• COS t
y =
a
b — a
* 3 ,
sm c,
(i)
(2)
which can be combined and written
{axf* + (byf /3
3 .,2/3
(3)
= [(a 2 - b 2 ) cos 3 t) 2/3 + [(b 2 - a 2 ) sin 3 t]
= (a 2 -6 2 ) 2/3 (sin 2 <+cos 2 f) = (a 2 -6 2 ) 2 ' 3 = c 4 ' 3 ,
which is a stretched Astroid called the Lame Curve.
From a point inside the EVOLUTE, four NORMALS can
be drawn to the ellipse, but from a point outside, only
two NORMALS can be drawn.
see also Astroid, Ellipse, Evolute, Lame Curve
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 77, 1993.
From Ellipse, the Tangent Vector is
—a sint
6 cost
and the Arc Length is
(1)
s = a \/\-e 2 sin 2 tdt = aE(t,e), (2)
where E(t t e) is an incomplete Elliptic Integral of
the Second Kind. Therefore,
n = r - sT =
a cost
6 sint
- aeE(t,e)
—a sint
b cost
o{cos t + aeE(t, e) sin i]
6{sin t — aeE(t } e) cos t}.
(3)
(4)
Ellipse Pedal Curve
The pedal curve of an ellipse with a FOCUS as the PEDAL
Point is a Circle.
Ellipsoid
A Quadratic Surface which is given in Cartesian
Coordinates by
+ — + — = 1
^ b 2 c 2 '
(1)
where the semi-axes are of lengths a, 6, and c. In SPHER-
ICAL Coordinates, this becomes
r 2 cos 2 6 sin 2 <p r 2 sin 2 9 sin 2 <j> r 2 cos 2 <j>
tf b 2 c~ 2
1. (2)
524 Ellipsoid
The parametric equations are
x = acos#sin</> (3)
y = b sin 9 sin <f> (4)
z — ccos</>. (5)
The Surface Area (Bowman 1961, pp. 31-32) is
2tt6
S = 2irc* +
Va*
- T [{a 2 -c 2 )E{9) + c*Ql (6)
where E(9) is a Complete Elliptic Integral of the
Second Kind,
~2 J2
2 a — c
2 _
e 2 =
t 2
b z - C
a"
2 „2
b 2
ai
and 9 is given by inverting the expression
d = sn(0, fc),
(7)
(8)
(9)
(10)
where sn(0, k) is a JACOBI ELLIPTIC FUNCTION. The
Volume of an ellipsoid is
V = ^Tzabc.
(ii)
If two axes are the same, the figure is called a SPHEROID
(depending on whether c < a or c > a, an OBLATE
Spheroid or Prolate Spheroid, respectively), and if
all three are the same, it is a Sphere.
A different parameterization of the ellipsoid is the so-
called stereographic ellipsoid, given by the parametric
equations
(12)
(13)
(14)
Ellipsoid Geodesic
(Gray 1993).
The Support Function of the ellipsoid is
1/2
"=(544)" ■ <«>
and the GAUSSIAN CURVATURE is
h 4
K
a 2 b 2 c 2
(19)
(Gray 1993, p. 296).
see also CONVEX OPTIMIZATION THEORY, OBLATE
Spheroid, Prolate Spheroid, Sphere, Spheroid
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 131, 1987.
Bowman, F. Introduction to Elliptic Functions, with Appli-
cations. New York: Dover, 1961.
Fischer, G. (Ed.). Plate 65 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 60, 1986.
Gray, A. "The Ellipsoid" and "The Stereographic Ellipsoid."
§11.2 and 11.3 in Modern Differential Geometry of Curves
and Surfaces. Boca Raton, FL: CRC Press, pp. 215-217,
and 296, 1993.
Ellipsoid Geodesic
An Ellipsoid can be specified parametrically by
x = a cos u sin v
y — b sin u sin v
z = ccosv.
(1)
(2)
(3)
The GEODESIC parameters are then
P = sin 2 u(6 2 cos 2 u + a 2 sin 2 u) (4)
Q = \{b 2 - a 2 )sin(2u)sin(2tj) (5)
R = cos 2 v(a 2 cos 2 u-\-b 2 sin 2 u) + c sin v. (6)
When the coordinates of a point are on the QUADRIC
= 1 (7)
2 2 2
- + T + -
a o c
and expressed in terms of the parameters p and q of the
confocal quadrics passing through that point (in other
words, having a+p, b+p, c+p, and a + q, & + £, c + q for
the squares of their semimajor axes), then the equation
of a Geodesic can be expressed in the form
A third parameterization is the Mercator parameteriza-
tion
x(u, v) = asechvcosu
y{u,v) = bsechvsinu
z(ujv) = ctanhv
(15)
(16)
(17)
qdq
y/q(a + q)(b + q)(c + q)(6 + q)
pdp
'y/p(a+p)(b + p){c + p)(9+p)
0, (8)
Ellipsoidal Calculus
with an arbitrary constant, and the Arc Length el-
ement ds is given by
dq
Ellipsoidal Harmonic of the First Kind 525
A Lame function of degree n may be expressed as
m
(9 + a' i ) K H9 + b 2 r(6 + c 2 ) Kil [[(9-e p ), (3)
PI y/q(a + q){b + q)(c + q)(8 + q)
dp
Vp(a + p)(& + p)(c + p)(0+p)
, 0)
where upper and lower signs are taken together.
see also Oblate Spheroid Geodesic, Sphere Geo-
desic
References
Eisenhart, L. P. A Treatise on the Differential Geometry of
Curves and Surfaces. New York: Dover, pp. 236-241,
1960.
Forsyth, A. R. Calculus of Variations. New York: Dover,
p. 447, 1960.
Ellipsoidal Calculus
Ellipsoidal calculus is a method for solving problems
in control and estimation theory having unknown but
bounded errors in terms of sets of approximating
ellipsoidal-value functions. Ellipsoidal calculus has been
especially useful in the study of Linear Programming.
References
Kurzhanski, A. B. and Valyi, I. Ellipsoidal Calculus for Es-
timation and Control. Boston, MA: Birkhauser, 1996.
Ellipsoidal Coordinates
see CONFOCAL ELLIPSOIDAL COORDINATES
Ellipsoidal Harmonic
see Ellipsoidal Harmonic of the First Kind, El-
lipsoidal Harmonic of the Second Kind
Ellipsoidal Harmonic of the First Kind
The first solution to Lame's Differential Equation,
denoted E™(x) for m = 1, . . . , 2n + 1. They are also
called Lame Functions. The product of two ellipsoidal
harmonics of the first kind is a SPHERICAL HARMONIC.
Whittaker and Watson (1990, pp. 536-537) write
e p =
y
a 2 + e p b 2 + e p
n(0) = eie 2 ---0 m ,
+
c 2 + l
- 1
(1)
(2)
and give various types of ellipsoidal harmonics and their
highest degree terms as
1. n(9) : 2m
2. xU(e),yU(e),zU(e) :2m-\-l
3. yzU(&),zxU(Q),xyU(&) :2m + 2
4. xyzU(Q) : 2m + 3.
p=i
where Kj = or 1/2, 6{ are Real and unequal to each
other and to —a 2 , — 6 2 , and — c 2 , and
\n = m + K\ + K2 + «3*
(4)
Byerly (1959) uses the Recurrence RELATIONS to ex-
plicitly compute some ellipsoidal harmonics, which he
denotes by K(x), L(x), M(x), and N(x),
K (x) = l
L Q (x) =
M o (x) =
N o (x) =
Ki{x) = x
Li(a:) = Vx 2 ~ & 2
Mi (a:) = yx 2 — c 2
JVi(x) =
K?{x) = x 2 - Hb 2 + c 2 - v / (6 2 +c 2 ) 2 -36 2 c 2 ]
K* 2 (x) = x 2 - |[6 2 4- c + A /(6 2 +c 2 ) 2 -3fe 2 c 2 ]
L 2 {x) — xy/x 2 - b 2
M 2 {x) = xyx 2 — c 2
N 2 (x) = ^(x 2 -b 2 ){x 2 ~c 2 )
Kl x (x) = x 3 - \x[2(b 2 + c 2 )
- x /4(6 2 + c 2 ) 2 -156 2 c 2 ]
K% 2 {x) = x 3 -\x[2{b 2 + c)
+ v / 4(6 2 + c 2 ) 2 -156 2 c 2 ]
Lf (x) = y/x 2 -b 2 [x 2 - l{b 2 + 2c 2
- v / (6 2 + 2c 2 ) 2 -56 2 c 2 )]
Lf{x) = ^x 2 ~b 2 [x 2 - l(b 2 + 2c 2
+ v/(6 2 + 2c 2 ) 2 -56 2 c 2 )]
M^{x) = y/x 2 -c 2 [x 2 - \{2b 2 + c 2
- y/(2b 2 + c 2 ) 2 -5b 2 c 2 )}
M$ 2 (x) = yjx 2 -c 2 [x 2 - \{2b 2 + c 2
+ v /(26 2 +c 2 ) 2 -56 2 c 2 )]
M| 3 (x) = rzVV* -^ 2 )(^ 2 -c 2 )
see a/50 ELLIPSOIDAL HARMONIC OF THE SECOND
Kind, Stieltjes' Theorem
References
Byerly, W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
526 Ellipsoidal Harmonic of the Second Kind
Elliptic Alpha Function
with Applications to Problems in Mathematical Physics.
New York: Dover, pp. 254-258, 1959.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, ^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Ellipsoidal Harmonic of the Second Kind
Given by
*£(*) = (2m +I)£&(a0
t/ X
dx
(x' -&')(*» -C»)[£* (a?)] 2 '
Elliptic Alpha Function
Elliptic alpha functions relate the complete ELLIPTIC
Integrals of the First K(k r ) and Second Kinds
E(k r ) at Elliptic Integral Singular Values k r ac-
cording to
a(r) =
E'(k r ) 7T
K(k r ) A[K{k r )Y
+ Vr-
E(k r )y/r~
4[K(k r )} 2 ' v ' K(k r )
= ^ * VrQ dq # 4 ( q )
where # 3 (g) is a Theta Function and
k r = A*(r)
— -K\/r
(1)
(2)
(3)
(4)
(5)
and A*(r) is the ELLIPTIC LAMBDA FUNCTION. The
elliptic alpha function is related to the Elliptic Delta
Function by
oc{r) = \[^ - 5{r)]. (6)
It satisfies
a(4r) = (1 + k r ) 2 a(r) - 2^r~k r , (7)
and has the limit
lim \a(r) --]^s(y/r--) e -7 ^ (8)
r-Kx> L 7rJ \ 7T/
(Borwein et al 1989). A few specific values (Borwein
and Borwein 1987, p. 172) are
a(l) = J
a(2) = V2- 1
a(4) = 2(v / 2-l) 2
a(5) = §(>/5 - \/2V5 - 2 )
a(6) = 5^+6^-8^- 11
a(7)=J(V7-2)
a(8) = 2(10 + 7v / 2)(l - Vv8-2) 2
a(9) - J[3-3 S/4 V2(V3-1)]
a(10) = -103 + 72v / 2-46V5 + 33v / 10
a(12) = 264 + 154 V3 - 188 V^ - 108^
a(13) = §(>/l3- V74VT^258)
a(15) = J(>/l5- V5 - 1)
a(16)=^-V
V ' (2 1 / 4 + l) 4
a(18) = -3057 + 2163\/2 + 1764V3 - 1248^
a(22) = -12479 - 8824^2 + 3762vTT + 2661\/22
a(25)= f[l-25 1/4 (7-3v / 5)]
a(27)-3[|(v / 3 + l)-2 1/3 ]
a(30) = Hv / 30-(2 + v / 5) 3 (3 + v / i0) 2
X (-6 - 5y/2 - 3VE - 2v / 10 + Vg^/h + 40\/2 )
x [56 + 38\/2 + V / 30(2 + V / 5)(3+ vTo)]}
a(37) = 5(^37- (171 - 25v / 37)^A / 37-6]
a(49) = §
- V / 7[\ / 2 7 3 /4(330ll + l2477v / 7) -21(9567 + 3616^)]
a(46) = ^[v^ + (18 + 13v^+ y/ 661 + 468^ ) 2
x (18 + 13\/2 - 3\/2\/l47 + 104^ 4- \/661 + 468\/2 )
x (200+ 14v / 2 + 26\/23 + 18v / 46 + \/46\/ 661 + 468v/ 2)]
a(58) = [i(v / 29 + 5)] 6 (99V^9-444)(99v / 2-70- 13V29)
= 3(-40768961 + 28828008v^2 - 7570606^29
+ 5353227\/58)
8[2(\/8- 1) - (2 1/4 - l) 4 ]
a(64) =
V / v / 2 + l + 2 5 /8)4
J. Borwein has written an Algorithm which uses lat-
tice basis reduction to provide algebraic values for a(n).
see also ELLIPTIC INTEGRAL OF THE FIRST KIND, EL-
LIPTIC Integral of the Second Kind, Elliptic In-
tegral Singular Value, Elliptic Lambda 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, 1987.
Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanu-
jan, Modular Equations, and Approximations to Pi, or
How to Compute One Billion Digits of Pi." Amer. Math.
Monthly 96, 201-219, 1989.
Elliptic Cone
Elliptic Curve 527
# Weisstein, E. W. "Elliptic Singular Values." http://www.
astro.virginia.edu/-eww6n/math/notebooks/Elliptic
Singular.m.
Elliptic Cone
A Cone with Elliptical Cross-Section. The para-
metric equations for an elliptic cone of height h, SEMI-
MAJOR Axis a, and Semiminor Axis b are
x = (h — z)a cos
y = (h — z)bsinO
z = z,
where 6 e [0, 2tt) and z € [0, h].
see also Cone, Elliptic Cylinder, Elliptic
Paraboloid, Hyperbolic Paraboloid
References
Fischer, G. (Ed.). Plate 68 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 63, 1986.
Elliptic Cone Point
see Isolated Singularity
Elliptic Curve
Informally, an elliptic curve is a type of CUBIC CURVE
whose solutions are confined to a region of space which
is topologically equivalent to a TORUS. Formally, an
elliptic curve over a FIELD K is a nonsingular CUBIC
CURVE in two variables, f(X, Y) = 0, with a if -rational
point (which may be a point at infinity). The FIELD
K is usually taken to be the Complex Numbers C,
Reals K, Rationals Q, algebraic extensions of Q, p-
adic Numbers Q p , or a Finite Field.
By an appropriate change of variables, a general elliptic
curve over a FIELD of CHARACTERISTIC ^2,3
Ax 3 + Bx 2 y + Cxy 2 + Dy 3 + Ex 2
+Fxy + Gy 2 +Hx + Iy + J = 0, (1)
where A, B, . . . , are elements of K, can be written in
the form
(2)
y 2 = x 3 + ax + b,
where the right side of (2) has no repeated factors. If K
has Characteristic three, then the best that can be
done is to transform the curve into
(the x 2 term cannot be eliminated). If K has CHAR-
ACTERISTIC two, then the situation is even worse. A
general form into which an elliptic curve over any K
can be transformed is called the WEIERSTRAfi Form,
and is given by
y + ay — x +bx + cxy + dx + e, (4)
where a, 6, c, d, and e are elements of K. Luckily, Q,
R, and C all have CHARACTERISTIC zero.
Whereas CONIC SECTIONS can be parameterized by the
rational functions, elliptic curves cannot. The simplest
parameterization functions are ELLIPTIC FUNCTIONS.
Abelian Varieties can be viewed as generalizations
of elliptic curves.
If the underlying Field of an elliptic curve is algebraic-
ally closed, then a straight line cuts an elliptic curve at
three points (counting multiple roots at points of tan-
gency). If two are known, it is possible to compute the
third. If two of the intersection points are K- RATIONAL,
then so is the third. Let (xi, yi) and (#2,2/2) be two
points on an elliptic curve E with DISCRIMINANT
satisfying
A E = -16(4a 3 +276 2 )
A E ^0.
(5)
(6)
A related quantity known as the j-lNVARIANT of E is
defined as
o8o3 3
HE)= 23a
4a 3 + 27b 2 '
(7)
Now define
a:i-a:2
3si 2 +q
2yx
for xi 7^ X2
for Xi — #2-
Then the coordinates of the third point are
— X\ — X2
ys = A(#3 - xi) + yi.
(8)
(9)
(10)
y 2 — x 3 + ax 2 + bx + c
(3)
For elliptic curves over Q, Mordell proved that there are
a finite number of integral solutions. The MORDELL-
Weil Theorem says that the Group of Rational
528 Elliptic Curve
Elliptic Curve Group Law
Points of an elliptic curve over Q is finitely generated.
Let the ROOTS of y 2 be n, r2, and r$. The discriminant
is then
A = k(ri - r 2 ) 2 (n - r 3 ) 2 (r 2 - r 3 ) 2 . (11)
Swinnerton-Dyer, H. P. F. "Correction to: 'On 1-adic Rep-
resentations and Congruences for Coefficients of Modu-
lar Forms.*" In Modular Functions of One Variable,
Vol. 4, Proc. Internal. Summer School for Theoret. Phys.,
Univ. Antwerp, Antwerp, RUCA, July-Aug. 1972. Berlin:
Springer-Verlag, 1975.
The amazing Taniyama-Shimura Conjecture states
that all rational elliptic curves are also modular. This
fact is far from obvious, and despite the fact that the
conjecture was proposed in 1955, it was not proved until
1995. Even so, Wiles' proof surprised most mathemati-
cians, who had believed the conjecture unassailable. As
a side benefit, Wiles' proof of the Taniyama-Shimura
CONJECTURE also laid to rest the famous and thorny
problem which had baffled mathematicians for hundreds
of years, Fermat's Last THEOREM.
Curves with small CONDUCTORS are listed in Swinner-
ton-Dyer (1975) and Cremona (1997). Methods for com-
puting integral points (points with integral coordinates)
are given in Gebel et al. and Stroeker and Tzanakis
(1994).
see also Elliptic Curve Group Law, Fer-
mat's Last Theorem, Frey Curve, j-Invariant,
Minimal Discriminant, Mordell-Weil Theorem,
Ochoa Curve, Ribet's Theorem, Siegel's The-
orem, Swinnerton-Dyer Conjecture, Taniyama-
Shimura Conjecture, WeierstraB Form
References
Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal-
ity Proving." Math. Comput. 61, 29-68, 1993.
Cassels, J. W. S. Lectures on Elliptic Curves. New York:
Cambridge University Press, 1991.
Cremona, J. E. Algorithms for Modular Elliptic Curves, 2nd
ed. Cambridge, England: Cambridge University Press,
1997.
Cremona, J. E. "Elliptic Curve Data." ftp://euclid.ex.
ac.uk/pub/cremona/data/.
Du Val, P. Elliptic Functions and Elliptic Curves. Cam-
bridge: Cambridge University Press, 1973.
Gebel, J.; Petho, A.; and Zimmer, H. G. "Computing Integral
Points on Elliptic Curves." Acta Arith. 68, 171-192, 1994.
Ireland, K. and Rosen, M. "Elliptic Curves." Ch. 18 in A
Classical Introduction to Modern Number Theory, 2nd ed.
New York: Springer-Verlag, pp. 297-318, 1990.
Katz, N. M. and Mazur, B. Arithmetic Moduli of Elliptic
Curves. Princeton, NJ: Princeton University Press, 1985.
Knapp, A. W. Elliptic Curves. Princeton, NJ: Princeton
University Press, 1992.
Koblitz, N. Introduction to Elliptic Curves and Modular
Forms. New York: Springer-Verlag, 1993.
Lang, S. Elliptic Curves: Diophantine Analysis. Berlin:
Springer-Verlag, 1978.
Silverman, J. H. The Arithmetic of Elliptic Curves. New
York: Springer-Verlag, 1986.
Silverman, J. H. The Arithmetic of Elliptic Curves II. New
York: Springer-Verlag, 1994.
Silverman, J. H. and Tate, J. T. Rational Points on Elliptic
Curves. New York: Springer-Verlag, 1992.
Stroeker, R. J. and Tzanakis, N. "Solving Elliptic Diophan-
tine Equations by Estimating Linear Forms in Elliptic Log-
arithms." Acta Arith. 67, 177-196, 1994.
Elliptic Curve Factorization Method
A factorization method, abbreviated ECM, which com-
putes a large multiple of a point on a random Elliptic
Curve modulo the number to be factored N. It tends
to be faster than the POLLARD p FACTORIZATION and
Pollard p - 1 Factorization Method.
see also Atkin-Goldwasser-Kilian-Morain Cer-
tificate, Elliptic Curve Primality Proving, El-
liptic Pseudoprime
References
Atkin, A. O. L. and Morain, F. "Finding Suitable Curves
for the Elliptic Curve Method of Factorization." Math.
Comput. 60, 399-405, 1993.
Brent, R. P. "Some Integer Factorization Algorithms Using
Elliptic Curves." Austral. Comp. Sci. Comm. 8, 149-163,
1986.
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 . anu . edu . au/pub/Brent/1 15 . dvi . Z.
Brillhart, J.; Lehmer, D. E; 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, p. lxxxiii, 1988.
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.
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). Elsevier, pp. 673-715, 1990.
Lenstra, H. W. Jr. "Factoring Integers with Elliptic Curves."
Ann. Math. 126, 649-673, 1987.
Montgomery, P. L. "Speeding the Pollard and Elliptic Curve
Methods of Factorization." Math. Comput. 48, 243-264,
1987.
Elliptic Curve Group Law
The Group of an Elliptic Curve which has been
transformed to the form
2 3 . ,i
y = x + ax + b
is the set of /^-Rational Points, including the single
Point at Infinity. The group law (addition) is de-
fined as follows: Take 2 K-Rational Points P and Q.
Now 'draw' a straight line through them and compute
the third point of intersection R (also a .K'-RATIONAL
Point). Then
p+Q+R=0
gives the identity point at infinity. Now find the inverse
of R, which can be done by setting R = (a, b) giving
-R= (a, -b).
This remarkable result is only a special case of a more
general procedure. Essentially, the reason is that this
Elliptic Curve Primality Proving
Elliptic Cylindrical Coordinates 529
type of Elliptic Curve has a single point at infinity
which is an inflection point (the line at infinity meets
the curve at a single point at infinity, so it must be an
intersection of multiplicity three) .
Elliptic Curve Primality Proving
A class of algorithm, abbreviated ECPP, which provides
certificates of primality using sophisticated results from
the theory of ELLIPTIC CURVES. A detailed description
and list of references are given by Atkin and Morain
(1990, 1993).
Adleman and Huang (1987) designed an independent
algorithm using elliptic curves of genus two.
see also Atkin-Goldwasser-Kilian-Morain Cer-
tificate, Elliptic Curve Factorization Method,
Elliptic Pseudoprime
References
Adleman, L. M. and Huang, M. A. "Recognizing Primes in
Random Polynomial Time." In Proc. 19th STOC, New
York City, May 25-27 } 1986. New York: ACM Press,
pp. 462-469, 1987.
Atkin, A. O. L. Lecture notes of a conference, Boulder, CO,
Aug. 1986.
Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal-
ity Proving." Res. Rep. 1256, INRIA, June 1990.
Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal-
ity Proving." Math. Comput. 61, 29-68, 1993.
Bosma, W. "Primality Testing Using Elliptic Curves."
Techn. Rep. 85-12, Math. Inst., Univ. Amsterdam, 1985.
Chudnovsky, D. V. and Chudnovsky, G. V. "Sequences of
Numbers Generated by Addition in Formal Groups and
New Primality and Factorization Tests." Res. Rep. RC
11262, IBM, Yorktown Heights, NY, 1985.
Cohen, H. Cryptographie, factorisation et primalite:
Vutilisation des courbes elliptiques. Paris: C. R. J. Soc.
Math. France, Jan. 1987.
Kaltofen, E.; Valente, R.; and Yui, N. "An Improved Las
Vegas Primality Test." Res. Rep. 89-12, Rensselaer Poly-
technic Inst., Troy, NY, May 1989.
Elliptic Cylinder
A Cylinder with Elliptical Cross-Section. The
parametric equations for an elliptic cylinder of height /i,
Semimajor Axis a, and Semiminor Axis b are
x = a cos 9
y = bsinO
z = z,
where 9 € [0, 2tt) and z G [0, h].
see also Cone, Cylinder, Elliptic Cone, Elliptic
Paraboloid
Elliptic Cylindrical Coordinates
y
%
/
"^
h=3/2
h=1/
/ V y< \
r^J^ u
^
v=n \
(-a
°^Q
\\u=o r?
^V °\
1 v=0
m
1 v=2tc
«^l\
^
u=3/2
^6
/
/
u=2
\
The v coordinates are the asymptotic angle of confocal
Parabola segments symmetrical about the x axis. The
u coordinates are confocal Ellipses centered on the ori-
gin.
x — a cosh u cos v
y = a sinh u sin v
z = z,
(1)
(2)
(3)
where u € [0,oo), v £ [0,27r), and z G (-00,00). They
are related to CARTESIAN COORDINATES by
a 2 cosh 2 u a 2 sinh 2 u
= 1
a 2 cos 2 v a 2 sin 2 v
= 1.
The Scale Factors are
hi = a v cosh 2 u sin 2 v + sinh 2 u cos 2 v
cosh(2w) - cos(2v)
2
= ay sinh 2 u + sin 2 v
hi = ay sinh 2 u sin 2 v 4- sinh 2 u cos 2 v
cosh(2n) — cos(2v)
= ay sinh u + sin 2 v
h 3 = 1.
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
The Laplacian is
a 2 (sinh'
1 / r) 2 f) 2 \ f) 2
u + sin 2 v) \dtf + d^J + d*' (13)
530
Elliptic Delta Function
Elliptic Function
Let
qi =
coshi
XL
<22 =
cosv
Q3 =
z.
Then the
new
Scale Fact
h qi = a\
h q2 = a*
roRS ;
are
/ gi 2
-Q2 2
-1
It
-92 2
«1 2
h qz
= 1.
(14)
(15)
(16)
(17)
(18)
(19)
The Helmholtz Differential Equation is Separa-
ble.
see also CYLINDRICAL COORDINATES, HELMHOLTZ DIF-
FERENTIAL Equation — Elliptic Cylindrical Co-
ordinates
References
Arfken, G. "Elliptic Cylindrical Coordinates (u, v, z)." §2.7
in Mathematical Methods for Physicists, 2nd ed. Orlando,
FL: Academic Press, pp. 95-97, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 657, 1953.
Elliptic Delta Function
S(r) = yfr — 2a(r),
where a is the ELLIPTIC ALPHA FUNCTION.
see also Elliptic Alpha Function, Elliptic Inte-
gral Singular Value
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
$ Weisstein, E. W. "Elliptic Singular Values." http://www.
astro . Virginia. edu/-eww6n/math/notebooks /Elliptic
Singular. m.
Elliptic Exponential Function
The inverse of the Elliptic Logarithm
dt
eln (x) \
f
v X
v^ 3 + at 2 + bt
It is doubly periodic in the COMPLEX PLANE.
Elliptic Fixed Point (Differential Equations)
A Fixed Point for which the Stability Matrix is
purely Imaginary, A± = ±iu (for u; > 0).
see also DIFFERENTIAL EQUATION, FIXED POINT, HY-
PERBOLIC Fixed Point (Differential Equations),
Parabolic Fixed Point, Stable Improper Node,
Stable Node, Stable Spiral Point, Stable Star,
Unstable Improper Node, Unstable Node, Unsta-
ble Spiral Point, Unstable Star
References
Tabor, M. "Classification of Fixed Points." §1.4. b in Chaos
and Integrability in Nonlinear Dynamics: An Introduc-
tion. New York: Wiley, pp. 22-25, 1989.
Elliptic Fixed Point (Map)
A Fixed Point of a Linear Transformation (Map)
for which the rescaled variables satisfy
(5 - a) 2 + 407 < 0.
see also Hyperbolic Fixed Point (Map), Linear
Transformation, Parabolic Fixed Point
Elliptic Function
A doubly periodic function with periods 2u>i and 2u;2
such that
f(z + 2u>i) = f(z + 2w 2 ) = f(z),
(1)
which is Analytic and has no singularities except for
POLES in the finite part of the COMPLEX PLANE. The
ratio <jl)\Juj2 must not be purely real. If this ratio is real,
the function reduces to a singly periodic function if it is
rational and a constant if the ratio is irrational (Jacobi,
1835). u>i and ll>2 are labeled such that $>(uJ2/t*)i) > 0. A
"cell" of an elliptic function is defined as a parallelogram
region in the Complex Plane in which the function is
not multi-valued. Properties obeyed by elliptic functions
include
1. The number of POLES in a cell is finite.
2. The number of ROOTS in a cell is finite.
3. The sum of Residues in any cell is 0.
4. Liouville's Elliptic Function Theorem: An el-
liptic function with no POLES in a cell is a constant.
5. The number of zeros of f(z) — c (the "order") equals
the number of POLES of f(z).
6. The simplest elliptic function has order two, since a
function of order one would have a simple irreducible
Pole, which would need to have a Nonzero residue.
By property (3), this is impossible.
7. Elliptic functions with a single POLE of order 2 with
Residue are called WeierstraB Elliptic Func-
tions. Elliptic functions with two simple POLES
having residues ao and — ao are called JACOBI EL-
LIPTIC Functions.
8. Any elliptic function is expressible in terms of ei-
ther WeierstraB Elliptic Function or Jacobi
Elliptic Functions.
9. The sum of the Affixes of Roots equals the sum
of the Affixes of the Poles.
10. An algebraic relationship exists between any two el-
liptic functions with the same periods.
The elliptic functions are inversions of the ELLIPTIC IN-
TEGRALS. The two standard forms of these functions
are known as Jacobi Elliptic Functions and Weier-
straB Elliptic Functions. Jacobi Elliptic Func-
tions arise as solutions to differential equations of the
form
d 2 x
dt 2
A + Bx + Cx z + Dx*
(2)
Elliptic Functional
and WEIERSTRAfi ELLIPTIC FUNCTIONS arise as solu-
tions to differential equations of the form
cfx
dt 2
= A + Bx + Cx .
(3)
see also Elliptic Curve, Elliptic Integral, Jacobi
Elliptic Functions, Liouville's Elliptic Func-
tion Theorem, Modular Form, Modular Func-
tion, Neville Theta Function, Theta Function,
WEiERSTRAfi Elliptic Functions
References
Akhiezer, N. I. Elements of the Theory of Elliptic Functions.
Providence, RI: Amer. Math. Soc, 1990.
Bellman, R. E. A Brief Introduction to Theta Functions.
New York: Holt, Rinehart and Winston, 1961.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Bowman, F. Introduction to Elliptic Functions, with Appli-
cations. New York: Dover, 1961.
Byrd, P. F. and Friedman, M. D. Handbook of Elliptic In-
tegrals for Engineers and Scientists, 2nd ed., rev. Berlin:
Springer- Ver lag, 1971.
Cayley, A. An Elementary Treatise on Elliptic Functions,
2nd ed. London: G. Bell, 1895.
Chandrasekharan, K. Elliptic Functions. Berlin: Springer-
Verlag, 1985.
Du Val, P. Elliptic Functions and Elliptic Curves. Cam-
bridge, England: Cambridge University Press, 1973.
Dutta, M. and Debnath, L. Elements of the Theory of Ellip-
tic and Associated Functions with Applications. Calcutta,
India: World Press, 1965.
Eagle, A. The Elliptic Functions as They Should Be: An
Account, with Applications, of the Functions in a New
Canonical Form. Cambridge, England: Galloway and
Porter, 1958.
Greenhill, A. G. The Applications of Elliptic Functions. Lon-
don: Macmillan, 1892.
Hancock, H. Lectures on the Theory of Elliptic Functions.
New York: Wiley, 1910.
Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum
Ellipticarum. Regiomonti, Sumtibus fratrum Borntraeger,
1829.
King, L. V. On the Direct Numerical Calculation of Elliptic
Functions and Integrals. Cambridge, England: Cambridge
University Press, 1924.
Lang, S. Elliptic Functions, 2nd ed. New York: Springer-
Verlag, 1987.
Lawden, D. F. Elliptic Functions and Applications. New
York: Springer Verlag, 1989.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 427 and 433-434,
1953.
Murty, M. R. (Ed.). Theta Functions. Providence, RI: Amer.
Math. Soc, 1993.
Neville, E. H. Jacobian Elliptic Functions, 2nd ed. Oxford,
England: Clarendon Press, 1951.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Elliptic Func-
tion Identities." §1.8 in A=B. Wellesley, MA: A. K. Peters,
pp. 13-15, 1996.
Whittaker, E. T. and Watson, G. N. Chs. 20-22 in A Course
of Modern Analysis, 4th ed. Cambridge, England: Univer-
sity Press, 1943.
Elliptic Functional
see Coercive Functional
Elliptic Helicoid 531
Elliptic Geometry
A constant curvature NON-EUCLIDEAN GEOMETRY
which replaces the PARALLEL POSTULATE with the
statement "through any point in the plane, there exist
no lines PARALLEL to a given line." Elliptic geometry is
sometimes also called Riemannian GEOMETRY. It can
be visualized as the surface of a SPHERE on which "lines"
are taken as GREAT CIRCLES. In elliptic geometry, the
sum of angles of a TRIANGLE is > 180°.
see also Euclidean Geometry, Hyperbolic Geom-
etry, Non-Euclidean Geometry
Elliptic Group Modulo p
E{a, b)/p denotes the elliptic GROUP modulo p whose el-
ements are 1 and oo together with the pairs of INTEGERS
(x, y) with < x, y < p satisfying
y 2 = x -+■ ax + b (mod p)
with a and b Integers such that
4a 3 + 276 2 =£0 (modp).
Given (a?i, j/i), define
(xi,yi) = (x u yiY (modp).
The Order h of E(a, b)/p is given by
*=1 + E
x + ax
*)
+ i
(i)
(2)
(3)
(4)
where (a? 3 -J- ax + b/p) is the Legendre Symbol,
although this FORMULA quickly becomes impractical.
However, it has been proven that
p + 1 - 2y/p < h(E(a, b)/p) < p + 1 + 2 y/p. (5)
Furthermore, for p a Prime > 3 and and Integer n in
the above interval, there exists a and b such that
h(E{a,b)/p) = n,
(6)
and the orders of elliptic GROUPS mod p are nearly uni-
formly distributed in the interval.
Elliptic Helicoid
532 Elliptic Hyperboloid
Elliptic Integral
A generalization of the HELICOID to the parametric
equations
x(u t v) = av cos u
y{u,v) = bvsinu
z(U)V) = cu.
see also HELICOID
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 264, 1993.
Elliptic Hyperboloid
The elliptic hyperboloid is the generalization of the HY-
PERBOLOID to three distinct semimajor axes. The ellip-
tic hyperboloid of one sheet is a RULED SURFACE and
has Cartesian equation
and parametric equations
~2 2 Jl
x y z
and parametric equations
x(u, v) = a v 1 + u 2 cos v
y(u } v) — b \J 1 + u 2 sin v
z(y,jv) = cu
for v E [0,27r), or
or
x(u, v) = a(cos u^v sin u)
y(u, v) = fe(sin u rb v cos u)
z(u,v) = ±ct;,
x(u, v) — a cosh v cos u
y(u,v) — b cosh v sin u
z(u,v) — csinhv.
(i)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
The two-sheeted elliptic hyperboloid oriented along the
£-AxiS has Cartesian equation
2 2
1 ~T" o
-1,
(11)
and parametric equations
x = a sinh u cos v
(12)
y = 6 sinh it sin?;
(13)
z = c ± coshti.
(14)
The two-sheeted elliptic hyperboloid oriented along the
a;- Axis has Cartesian equation
V
(15)
x — a cosh u cosh v
(16)
y — b sinh u cosh v
(17)
z = csinhu.
(18)
see also Hyperboloid, Ruled Surface
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 296-297, 1993.
Elliptic Integral
An elliptic integral is an INTEGRAL of the form
/
A{x)+B(x)y/S{x)
C{x) + D{x)y/S{x)
A{x) dx
Bixh/S^'
dx,
/
(i)
(2)
where A, B, C, and D are POLYNOMIALS in x and S is
a POLYNOMIAL of degree 3 or 4. Another form is
/
R(w,x) dx,
(3)
where i^isa RATIONAL FUNCTION of x and y, w 2 is a
function of x CUBIC or QUADRATIC in #, R(w, x) con-
tains at least one ODD POWER of w, and w has no
repeated factors.
Elliptic integrals can be viewed as generalizations of the
Trigonometric Functions and provide solutions to
a wider class of problems. For instance, while the Arc
Length of a Circle is given as a simple function of the
parameter, computing the Arc Length of an Ellipse
requires an elliptic integral. Similarly, the position of a
pendulum is given by a Trigonometric Function as
a function of time for small angle oscillations, but the
full solution for arbitrarily large displacements requires
the use of elliptic integrals. Many other problems in
electromagnetism and gravitation are solved by elliptic
integrals.
A very useful class of functions known as Elliptic
FUNCTIONS is obtained by inverting elliptic integrals (by
analogy with the inverse trigonometric functions). EL-
LIPTIC Functions (among which the Jacobi Elliptic
Functions and Weierstrass Elliptic Function are
the two most common forms) provide a powerful tool for
analyzing many deep problems in Number Theory, as
well as other areas of mathematics.
All elliptic integrals can be written in terms of three
"standard" types. To see this, write
R(w,x)
P(w,x) _ wP(w,x)Q(—w,x)
Q(w t x) wQ{w,x)Q{—w,x)'
(4)
Elliptic Integral
But since w 2 = f(x),
Q(w y x)Q(-w y x) = Qi(wjx) = Qi(w,x), (5)
then
wP{w, x)Q(-w, x) = A + Bx + Cw + Da; 2 + £wz
+F™ 2 + G™ 2 x + i2V:r
= (A 4- Bx + £>z 2 + Fw 2 + Gw 2 a;)
+xu(c + Ex + i^Vcc + . . .)
= Pi(x) + wP 2 (x), (6)
^^) = ft( !^^ (a) = ^ + ftw- m
wQi(w)
w
But any function J R 2 (x) dx can be evaluated in terms
of elementary functions, so the only portion that need
be considered is
fRi(x)
J u } dx. (8)
w
Now, any quartic can be expressed as SiS 2 where
Si = aix + 2feix + ci (9)
S 2 = a 2 z 2 + 26 2 z + c 2 . (10)
The Coefficients here are real, since pairs of Com-
plex Roots are Complex Conjugates
[x - {R + Ii)][x - {R- Ii)]
= x 2 + x(-P + H-R-H) + (R 2 - I 2 i)
= x 2 -2Rx + {R 2 + I 2 ). (11)
If all four Roots are real, they must be arranged so as
not to interleave (Whittaker and Watson 1990, p. 514).
Now define a quantity A such that Si + AS 2
(01 - Aa 2 )z 2 - (26i - 2b 2 X)x + (ci - Ac 2 ) (12)
is a Square Number and
2 v /(ai-Aa 2 )(ci-A 2 ) - 2(&i - b 2 X) (13)
(a! - Aa 2 )(ci - Ac 2 ) - (61 - A6 2 ) 2 = 0. (14)
Call the ROOTS of this equation Ai and A 2 , then
-i2
*Si — Ai*S 2 —
V (ai - Aia 2 )x 4- \Jc\ - \c 2
Ci — A1C2
= (ai - Aia 2 ) \x+ ,, .
V V ai — Aia 2
= (ai — Aia 2 )(x — a) (15)
Si — A 2 S 2 — y (ai — Aia 2 )x + Y c i — Ac 2
= (ai - Aia 2 ) x 4-
ci — A 2 c 2
ai — A 2 a 2
(ai - A 2 a 2 )(x - (3) .
(16)
Elliptic Integral 533
Taking (15)-(16) and A 2 (l) - Ai(2) gives
S 2 (A 2 - Ai) = (ai - Aia 2 )(x - a) 2
- {d! - \ 2 a 2 ){x - 0) 2 (17)
Si(A 2 - Ai) = A 2 (ai - Xia 2 )(x - a) 2
-Ai(ai-A a a 2 )(a;-/9 2 ). (18)
Solving gives
Si = a^X^l {x _ a)2 _ opAp (x _ /?)2
A 2 — Ai A 2 — Ai
= Ai(x - a) 2 + Bi(x ~ P) 2 (19)
A 2 (ai — Aia 2 ) , , 2 Ai(ai — A 2 a 2 ) ( . 2
S 2 = _ (x-a) (x-P)
A 2 — Ai A 2 — Ai
= A 2 (x-a) 2 +B 2 (x-0) 2 , (20)
so we have
w 2 = S1S2
= [Ai(x - a) 2 + Bi (x - f3f][A 2 {x - a) 2 + B 2 (x - 0) 2 ].
(21)
Now let
t =
x-p
dy = [(x - py 1 -(x- a)(x - py 2 ) dx
__ (x — p) — (x — a)
(22)
a-P
dx
dx,
{x-P) 2 '
w 2 = (x-/3) 4
(23)
— V
+ Si
= (x-0) 4 (A 1 t 2 +B 1 )(A 2 t 2 + B 2 ), (24)
and
w = (x - Pf yJ(A x t* + B 1 )(A 2 t 2 + B 2 )
(25)
da;
a-/?
rft
1
Now let
a: — p
(26)
(27)
f R 1 (x)dx _ r R 3 (t)dt
J w J y/iArf + B^AiP + Bi)'
534 Elliptic Integral
Rewriting the Even and Odd parts
R3{t) + R 3 (-t) = 2R 4 (t 2 )
Rz{t)~R 3 (-t) = 2tR 5 (t 2 ),
(29)
(30)
Elliptic Integral
can be computed analytically (Whittaker and Watson
1990, p. 453) in terms of the Weierstrad Elliptic
Function with invariants
92 — 0,00,4 — 4aia3 + 3ct2
(39)
gives
Rs{t) = ±(R eV en - Rodd) = R4{t 2 ) + tR 5 (t 2 ), (31)
so we have
r Ri(x)dx _ r R 4 (t 2 )dt
J w ~ J y/{A 1 f2 + B 1 )(A 2 ti+B 2 )
f R 5 (t 2 )tdt
+ y JiA^+B!
9z = aoa 2 a 4 — 2a\a 2 a$ — a 4 ai — 03 oq. (40)
If a = xq is a root of f(x) = 0, then the solution is
x = xo + \f(xo)[p(z\g2,9s) - ^/"(so)] -1 . (41)
For an arbitrary lower bound,
x = a+
j {Arf + B,){Arf 7W) (32) v%)p'(*) + £/'(«)[?(*) - £/»] + £/(«)/'»
Letting
dtt = 2£ dt
reduces the second integral to
R*>(u) du
II
2 J ^/(.4iu + .Bi)( J 4 2 u + .B2)'
(33)
(34)
(35)
which can be evaluated using elementary functions.
The first integral can then be reduced by INTEGRA-
TION BY PARTS to one of the three Legendre elliptic
integrals (also called Legendre- J acobi Elliptic INTE-
GRALS), known as incomplete elliptic integrals of the
first, second, and third kind, denoted F(</>, k), E(<fi,k),
and II(n; 0, &), respectively (von Karman and Biot 1940,
Whittaker and Watson 1990, p. 515). If <f> = tt/2, then
the integrals are called complete elliptic integrals and
are denoted K(k), E(k), U(n;k).
Incomplete elliptic integrals are denoted using a MOD-
ULUS k, Parameter m = k 2 , or Modular Angle
a = sin -1 k. An elliptic integral is written I(<fr \m) when
the Parameter is used, 1(0, k) when the Modulus is
used, and I(<f>\a) when the Modular Angle is used.
Complete elliptic integrals are defined when <f> = tt/2
and can be expressed using the expansion
(1 - * 2 sin 2 *)-/» = £ ( ^k^k 2n -n 2 " 6. (36)
n=0
An elliptic integral in standard form
dx
F
J a
fm
(37)
where
f{x) = a 4 x + azx + a 2 x + a\x + ao, (38)
%(*) - £/"(«)] 2 " £/(<*)/ (o) (a)
(42)
where p{z) = p(z;g 2} g3) is a WeierstraB Elliptic
Function.
A generalized elliptic integral can be denned by the func-
tion
T(a,b)
= 2 r /2
~ n Jo y/d-
-\f:i
do
\/a 2 cos 2 + b 2 sin 2 9
d6
cosOVa 2 +b 2 t&n 2 9
(Borwein and Borwein 1987). Now let
t = btan6
But
so
dt = 6 sec d9.
:0 = y/l + tan 2 ,
(43)
(44)
(45)
(46)
(47)
dt = sec 6 d9 = v/l + tan 2 dO
cos 6 cos 6
cos#
dO
cos 6
VWt 2 ,
and
d<9 d£
COS0 y/b 2 + £ 2 '
and the equation becomes
eft
(48)
(49)
T(a,6) = - /
W-oc vV+* 2 )(& 2 + < 2 )
VV+i 2 )(& 2 +t 2 )
(50)
Elliptic Integral
Elliptic Integral 535
Now we make the further substitution u = \{t — ab/t).
The differential becomes
du= \{\ + ab/t 2 )dt, (51)
but 2u~t- ab/t, so
2u/t = 1 - ab/t 2 (52)
ab/t 2 = 1 - 2u/t (53)
and
1 + ab/t 2 = 2 - 2u/t = 2(1 - u/t). (54)
However, the left side is always positive, so
1 + ab/t 2 = 2 - 2u/t = 2|1 - u/*l (55)
and the differential is
dt=j-^-r. (56)
We need to take some care with the limits of integration.
Write (50) as
/oo n0~ /»oo
f{t)dt= / f{t)dt+ / }{t)dt. (57)
-oo «/ — oo »/0+
Now change the limits to those appropriate for the u
integration
But
/oo />oo /»oo
g(u) du+ I g(u) du = 2 / c/(u) du,
■oo J — oo «/ — oo
(58)
so we have picked up a factor of 2 which must be in-
cluded. Using this fact and plugging (56) in (50) there-
fore gives
T(a, b) ■
Now note that
du
|l- f | ^a 2 b 2 + (a 2 + b 2 )t 2 +t 4 '
2 t 4 ~2abt 2 +a 2 b 2
U = 4?
4u 2 t 2 = r 4 - 2a& 2 + 2ata 2
a 2 b 2 +t 4 =4u 2 t 2 +2abt 2 .
Plug (62) into (59) to obtain
(59)
(60)
(61)
(62)
T(a,6) = -
du
* J-oo |l - f I V /4 ^ 2 ^ 2 + 2a6i 2 + (a 2 + & 2 )i 2
= - / , 63
^7-00 |t-u| V / 4u 2 + (a-h6) 2
2ut — t-ab
t 2 -2ut-ab =
t= \{2u± yj±u 2 + Aab) =u± y/v? + ab,
t-u = ±yfu 2 + a6,
(64)
(65)
(66)
(67)
and (63) becomes
T{a,b) = ~r -— =
* J-oo y/[4u 2 + (a + b) 2 ]{u 2 + ab)
du
t/ — c
We have therefore demonstrated that
T(o,6) = r(|(a + 6),Va6).
. (68)
We can thus iterate
fli+i = \{ai + bi)
bi+i = yaibi,
(69)
(70)
(71)
as many times as we wish, without changing the value of
the integral. But this iteration is the same as and there-
fore converges to the Arithmetic-Geometric Mean,
so the iteration terminates at ai = bi = M(a ,&o), and
we have
T{a , b ) - T(M{a , b ), M(a , bo))
=i r dt
7rM(a ,&o) [ an \M(a ,b ) J
Hi)]
7rM(a ,6o)
1
M(a ,6o)'
(72)
Complete elliptic integrals arise in finding the arc length
of an ELLIPSE and the period of a pendulum. They also
arise in a natural way from the theory of THETA FUNC-
TIONS. Complete elliptic integrals can be computed us-
ing a procedure involving the Arithmetic- Geometric
Mean. Note that
T(
\/a 2 cos 2 + b 2 sin 2
d9
a,6) = ^'/
* Jo
= *[
* J° cnJcos 2 6+(±) 2 sin 2 8
= a r /2 dfl
^ ^-(l-^sin^'
(73)
536 Elliptic Integral
Elliptic Integral
So we have
T(a,b) = —K l - — = — — _ , (74)
an \ a 2 J M(a,b)
where K{k) is the complete ELLIPTIC INTEGRAL OF THE
FIRST Kind. We are free to let a = ao = 1 and b ~ bo =
k' ', so
K(V^)= 2 -K (k) = M ± iry (75)
since k = \/l — k' 2 ^ so
iC(fc):
(76)
2M(l,fc')"
But the Arithmetic-Geometric Mean is defined by
a% = |(a<_i+6i_i) (77)
6* = \/a*-i^-i ( 78 )
_ f |(ai_i - 6i_i) £ >
i = 0,
(79)
where
2 2
1 * __ Cn ^ C n
so we have
4a ri +i - 4M(a ,6o)'
K{k) = -^-,
(80)
(81)
where a,N is the value to which a n converges. Similarly,
taking instead a = 1 and b f = k gives
*'(*> = ^
2a',
(82)
Borwein and Borwein (1987) also show that defining
/>?r/2
t/(a, 6) = | / Va 2 cos 2 +6 2 sin 2 6>d<9 = a£' ( - J
(83)
leads to
2U(a n +i,b n +i) - C/(a n , 6„) = a n b n T(a nj b n ), (84)
iT(fc) - E(k) _ i 2 ( 2 , o2 ^ 2 , , on 2 ^
A-(ifc) 2 *
Kco 2 + 2ci 2 + 2 2 c 2 2 + ..- + 2 n c n 2 ) (85)
for ao = 1 and bo = k\ and
Jf'(fc) - E'(k) _ 1( ,2,~,2, ,2 , 2 , J
= 2V C + 2c x |2c 2 +... + 2 c n ).
#'(*)
(86)
The elliptic integrals satisfy a large number of identities.
The complementary functions and moduli are defined by
K'(k) = K(y/l~k 2 ) = K(k'). (87)
Use the identity of generalized elliptic integrals
T{a,b) = T{\{a + b),yfab) (88)
to write
a + b
(a + b) 2
JL_ K ( [*+E
a + b \\ (o +
2 K ( a ~ b )
a + b \a + b)
2ab
6) 2
(89)
K\ Wl
6M 2 .. / 1 - *
-if
2 J - 1 , 6 " I , , » I • ( 90 )
Define
a * ) 1 + ~a V 1 + °
and use
(91)
A; = v^I - A;' 2 , (92)
so
K ^ = ih K {\+^)- w
Now letting / = (1 - k')/{l + fc') gives
/(l + A;') = 1 - k' => k'{l + 1) = 1 - I (94)
k ' = TTl W
k =^^=]HM :
2 7T7> ( 96 )
(l + l) 2
l + l'
and
*<■ + ">-*(' + &)-£
1 + J
-0
1
_ l + r
1
Writing fc instead of /,
K(k)
fc+i v i+fc /'
(97)
(98)
Elliptic Integral
Similarly, from Borwein and Borwein (1987),
*<*> = H** (rrl) + t*<*> o»)
E(k) = (l + k')E(\^\-k'K(k). (100)
Expressions in terms of the complementary function can
be derived from interchanging the moduli and their com-
plements in (93), (98), (99), and (100).
1+k \l+k)
(101)
* , ™-itf*(i^)-itf*'(t^)-
and
(102)
E'(k) = (l + k)E' (—-j-kK'(k) (103)
^)=(^)^'(^)4^). (104)
Taking the ratios
K'(k)_K {TTk) _l K [TW)
K(k) K ^ *K(\&)
(105)
gives the MODULAR EQUATION of degree 2. It is also
true that
K(x) =
(1 + V^ 7 ) 2
K
1- tfl^
1+ ^1-x 4
(106)
see also Abelian Integral, Amplitude, Argument
(Elliptic Integral), Characteristic (Elliptic
Integral), Delta Amplitude, Elliptic Function,
Elliptic Integral of the First Kind, Elliptic In-
tegral of the Second Kind, Elliptic Integral
of the Third Kind, Elliptic Integral Singular
Value, Heuman Lambda Function, Jacobi Zeta
Function, Modular Angle, Modulus (Elliptic
Integral), Nome, Parameter
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Elliptic Inte-
grals." Ch. 17 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 587-607, 1972.
Elliptic Integral of the First Kind 537
Arfken, G. "Elliptic Integrals." §5.8 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 321-327, 1985.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Hancock, H. Elliptic Integrals. New York: Wiley, 1917.
King, L. V. The Direct Numerical Calculation of Elliptic
Functions and Integrals. London: Cambridge University
Press, 1924.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Elliptic Integrals and Jacobi Elliptic Func-
tions." §6.11 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 254-263, 1992.
Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I.
Integrals and Series, Vol. 1: Elementary Functions. New
York: Gordon & Breach, 1986.
Timofeev, A. F. Integration of Functions. Moscow and
Leningrad: GTTI, 1948.
von Karman, T. and Biot, M. A. Mathematical Methods in
Engineering: An Introduction to the Mathematical Treat-
ment of Engineering Problems. New York: McGraw-Hill,
p. 121, 1940.
Whit taker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Elliptic Integral of the First Kind
Let the MODULUS k satisfy < k 2 < 1. (This may
also be written in terms of the PARAMETER m = k 2 or
MODULAR Angle a = sin" 1 k.) The incomplete elliptic
integral of the first kind is then defined as
F(<j>,k) =
Jo \f\
dO
yjl - k 2 sin 2 6
Let
; sin0
dt = cos 9d9 = y/l - t 2 d0
(1)
(2)
(3)
F{<}>,k)
/»sin <f>
Jo
J dt
y/1 - k 2 t 2 y/l^¥
/»sin <p
dt
^(l-kH 2 )(l-t 2 )
Let
v = tan 6
dv = sec 2 9 d0 = (1 + v 2 ) d9 t
so the integral can also be written as
/•tan 4>
(4)
(5)
(6)
F(4>,k)
Jo A /i_ fc 2 7 ^l + ^ 2
I
1/
VT+v 2 ^{\ + v 2 ) - k 2 v 2
/»tan <p
~ Jo J(l+v
dv
y/(l+v*){l + k'v*)'
(7)
(8)
538 Elliptic Integral of the First Kind
Elliptic Integral of the First Kind
where k' 2 = 1 - k 2 is the complementary MODULUS.
The integral
V2j
dO
•\/cos 6 — cos 0o
(9)
which arises in computing the period of a pendulum, is
also an elliptic integral of the first kind. Use
cos0- l-2sin (±0)
sin(i0)
1 — cos t
(10)
(11)
to write
a/cos - cos 0o = yl - 2sin 2 (|0) -cos0 o
= ^l-cos9 Jl - z 2 — r sin 2 (§0)
y i - cos O
= v^sin^ 00)^1 - csc 2 (|0 o ) sin 2 (§0),
(12)
so
2 J smae )Jl - csc*aeo)sm 2 ae)
Now let
sin(§0) = sin(i0o)sin0, (14)
so the angle is transformed to
= sin
'(^).
(15)
which ranges from to 7r/2 as varies from to 0o-
Taking the differential gives
§cos(§0)d0 = sin(§0 o )cos<M0, (16)
\Jl- sin 2 (|0 o ) sin 2 <pd9 = sin(f O ) cos 4>d<f>. (17)
Plugging this in gives
/•tt/2
■ r , ■
Jo v /l-sin 2 (|0 o )si
-I
sin(|0 o )cos0d0
o \A -sin 2 (|0 o )sin 2 <£
sin 2 (j) sin(§0 o )^/l - sin 2
X(sin(±0 o )), (18)
v^X
d0
\/2 ,/ Vcos — cos 0o
A-(sin(l* )). (19)
Making the slightly different substitution <fr = 0/2, so
dO = 2 d<f> leads to an equivalent, but more complicated
expression involving an incomplete elliptic function of
the first kind,
/ = 2^^csc(|0o) r__^__
V2V2 V2 J ^/l-csc 2 (|0 o )sin 2
= csc(|0 o )F(|0 o ,csc(i0o)). (20)
Therefore, we have proven the identity
csca:F(a;,cscx) = K(sinx). (21)
The complete elliptic integral of the first kind, illus-
trated above as a function of m — A; 2 , is defined by
K(k)=F(±n,k)
(22)
|^3 2 (3)
(2n-l)!!,_ 2 „7r (2n-l)!!
2 (2n)H
(2n)\l
-k* n ±
IE
(2n.-l)!!
(2n)!!
n=0
= i7r 9 Ji(i,i,l;fe 2 )
2 " *■* i \2 > 2
7T
2V1 - k 2
P-i
/2
1 + fc 2
1-fc 2
where
g _ e -^K'(k)/K{k)
(24)
(25)
(26)
(27)
(28)
is the Nome (for \q\ < 1), 2 i 71 i(a, 6; c; x) is the Hyperge-
ometric Function, and P n (x) is a Legendre Poly-
nomial. K(k) satisfies the Legendre Relation
E{k)K'{k) + E\k)K{k) - K(k)K f (k) = §tt, (29)
Elliptic Integral of the Second Kind
Elliptic Integral of the Second Kind 539
where E(k) and K{k) are complete elliptic integrals of
the first and Second Kinds, and E'(k) and K'(k) are
the complementary integrals. The modulus k is often
suppressed for conciseness, so that E(k) and K(k) are
often simply written E and K, respectively.
The Derivative of K(k) is
dK = f 1 dt_
E(k) K{k)
^(l~t 2 )(l-k f H 2 ) k(l-k 2 )
(30)
(31)
so
-»p-» , >(£ + f)-<i-*'>(»£ + *)<»>
(Whittaker and Watson 1990, pp. 499 and 521).
see also Amplitude, Characteristic (Elliptic
Integral), Elliptic Integral Singular Value,
Gauss's Transformation, Landen's Transforma-
tion, Legendre Relation, Modular Angle, Mod-
ulus (Elliptic Integral), Parameter
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Elliptic Inte-
grals." Ch. 17 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 587-607, 1972.
Spanier, J. and Oldham, K. B. "The Complete Elliptic In-
tegrals K(p) and £(p)" and "The Incomplete Elliptic In-
tegrals F(p;<j)) and E(p;<p)." Chs. 61-62 in An Atlas of
Functions. Washington, DC: Hemisphere, pp. 609-633,
1987.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, 4$h e d- Cambridge, England: Cambridge Uni-
versity Press, 1990.
Elliptic Integral of the Second Kind
Let the MODULUS k satisfy < k 2 < 1. (This may
also be written in terms of the Parameter m = k 2 or
Modular Angle a = sin" 1 k.) The incomplete elliptic
integral of the second kind is then defined as
E(4>,k)= J y/l -k 2 sin 2 OdO.
Jo
A generalization replacing sin 6 with sinh 6 gives
-iE{i&-k)= J y/l -k 2 sinh 2 Odd.
Jo
(i)
(2)
To place the elliptic integral of the second kind in a
slightly different form, let
t = sin
dt = cos 0d0= y/l - t 2 dO,
(3)
(4)
so the elliptic integral can also be written as
dt
/•sin <p
Jo
-f
Jo
1 - kH 2
vT^l 2 "
sin0 l i-k 2 t 2
i-t 2
dt
(5)
10-10
lo-io
10-10
The complete elliptic integral of the second kind, illus-
trated above as a function of the PARAMETER m, is de-
fined by
(2n-l)!!
(2n)!!)
2n
= a^aFif-^, f,l;fe )
-/
Jo
dn udu,
(6)
(7)
(8)
(9)
where 2-Fi(a, 6; c; x) is the Hypergeometric FUNC-
TION and dnw is a Jacobi Elliptic Function. The
complete elliptic integral of the second kind satisfies the
Legendre Relation
E(k)K'(k) + E'{k)K(k) - K(k)K'(k) = \ 7T, (10)
where E and K are complete ELLIPTIC INTEGRALS OF
the FIRST and second kinds, and E l and K' are the
complementary integrals. The Derivative is
dE _ E(k) - K(k)
dk k
(id
(Whittaker and Watson 1990, p. 521). If k r is a singular
value (i.e.,
k T = A», (12)
540 Elliptic Integral of the Third Kind
Elliptic Integral Singular Value
where A* is the Elliptic Lambda Function), and
K{k r ) and the Elliptic Alpha Function a(r) are
also known, then
E{k) =
K{k)
mm 2
a(r)
+ K(k). (13)
see also ELLIPTIC INTEGRAL OF THE FIRST KIND, EL-
LIPTIC Integral of the Third Kind, Elliptic In-
tegral Singular Value
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Elliptic Inte-
grals." Ch. 17 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 587-607, 1972.
Spanier, J. and Oldham, K. B. "The Complete Elliptic In-
tegrals K(p) and E(p)" and "The Incomplete Elliptic In-
tegrals F(p;<f>) and E(p; </>)." Chs. 61 and 62 in An Atlas
of Functions. Washington, DC: Hemisphere, pp. 609-633,
1987.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Elliptic Integral of the Third Kind
Let < k 2 < 1. The incomplete elliptic integral of the
third kind is then defined as
II(n; <£, k)
-s:
-a
dO
(1 - nsin 2 6)\/l-k 2 sin 2 e
a * dt
(1)
,(2)
/o (l-nt 2 )^/(l-* a )(l-A a t a )
where n is a constant known as the CHARACTERISTIC.
The complete elliptic integral of the second kind
U(n\m) = II(n; \*\m) (3)
is illustrated above.
see also ELLIPTIC INTEGRAL OF THE FIRST KIND, EL-
LIPTIC Integral of the Second Kind, Elliptic In-
tegral Singular Value
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Elliptic Inte-
grals" and "Elliptic Integrals of the Third Kind." Ch. 17
and §17.7 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 587-607, 1972.
Elliptic Integral Singular Value
When the MODULUS k has a singular value, the complete
elliptic integrals may be computed in analytic form in
terms of Gamma Functions. Abel (quoted in Whit-
taker and Watson 1990, p. 525) proved that whenever
K\k) _ a + 6y^
K(k) " c + <Vn'
(1)
where a, 6, c, d, and n are INTEGERS, K(k) is a com-
plete Ellipti c Inte gral of the First Kind, and
K'{k) = K(-\/l — k 2 ) is the complementary complete
Elliptic Integral of the First Kind, then the
Modulus k is the Root of an algebraic equation with
Integer Coefficients.
A Modulus k T such that
K\k T )
K(k r )
= y/r,
(2)
is called a singular value of the elliptic integral. The
Elliptic Lambda Function A*(r) gives the value of
k r . Selberg and Chowla (1967) showed that K(\*(r))
and E(\*(r)) are expressible in terms of a finite number
of Gamma Functions. The complete Elliptic Inte-
grals of the Second Kind E(k r ) and E'(k r ) can be
expressed in terms of K(k r ) and K'(k r ) with the aid of
the Elliptic Alpha Function a(r).
The following table gives the values of K(k r ) for small
integral r in terms of GAMMA FUNCTIONS.
K(k x ) =
K{k 2 ) =
K{k 3 )
K{k 4 )
K(k 5 )
K(k 6 ) :
r 2 q)
4v^
>A/2 + ir(i)r(j)
2 13 / 4 0i :
3 1/4 r 3 (f)
2 7 / 3 7T
(V2 + i)r 2 (i)
(v / 5 + 2) 1/4
v®
£)r(A)r(£)r<A)
1607T
\/{V2- l)(V3 + V / 2)(2 + v / 3)
V^
&)r(A)r(£)r<ii)
384tt
r(|)r(f)r(f)
71/4 . 4n
2^+ y/\ + Sv/2 ( v / 2 + i)^ 4 r(|)r(f )
4V2 8^
Elliptic Integral Singular Value
K{k 9 )
K(k 10 )
K(k lx )
K(k 12 )
K(k 13 )
^(2 + 3V2 + \/5)
/ r(A)r(A
)r(jL)r(a)r(j})r(a)r(S)r(S)
2560tt 3
[2 + (17 + 3V^3) 1/3 - (3v/33 - 17) 1/3 ] 2
„ r(A)r(ft)r(ft)r(jL)r(ft)
Hl/4 1447r 2
3 1/4 (y2 + 1)(a/3 + v/2)>/ 2 - V3r 3 (|)
2 i3/ 37r
K(fci 5 )
^(fcie)
^(^17)
(18 + 5X/13) 174
V6656tt 5
x Vr(g)r(fj)r(g)r(|i)r(S)r(|i)
(v^+i)r(i)r(i)r(i)r(X)
240tt
(2^ 4 + i) 2 r 2 (i)
= C t
2 9 / 2 n /tt
r(£)r(X)r(x )r (Ai)r(|§)V' 4
tf (fc 25 ) =
r (A) r (S) r (S) r (|f) .
x[r(fi)r(§|)r(§i)r(fi)r(|f)]^ 4
V5 + 2r 2 (i)
20
v^ '
where F(z) is the Gamma Function and C\ is an alge-
braic number (Borwein and Borwein 1987, p. 298).
Borwein and Zucker (1992) give amazing expressions for
singular values of complete elliptic integrals in terms of
Central Beta Functions
/J(p) = 5(p,p).
(3)
Furthermore, they show that K (k n ) is always expressible
in terms of these functions for n = 1,2 (mod 4). In such
cases, the T functions appearing in the expression are of
the form T{t/4n) where 1 < t < (2n- 1) and {t, An) = 1.
The terms in the numerator depend on the sign of the
Kronecker Symbol {t/An}. Values for the first few n
K{k 1 ) = 2~ 2 l3{\)
K{k 2 )
_ ,-13/4
■0(1)
K(k 3 ) = 2- 4/3 3- 1/4 /3(|) = 2- 5/3 3" 3/4 /3(i)
K(k 5 ) = 2 -33/20 5 -5/ 8(11 + 5 ^5 } i/4 M ± nm I )
= 2 - 29/2 o 5 -3/8 (1 + y5 )1 /4 sin( ^ 7r)/3( ^ )
#(*«) = 2- 47 / 12 3" 3/4 (^ - l)(%/3 + l)/?(i)
= 2- 43/12 3" 1/4 (v^-l)/3(^)
K(k r ) = 2 • 7~ 3/4 sin(iTr) sin(f w )B(i, f )
_ 9 -2/T 7 -V4^(Mi)
Elliptic Integral Singular Value 541
tf(fcio)
_ 9 -61/20r-l/4
5- 1/4 (V5-2) 1/2 (v / 10 + 3)
= 2" 15/4 5" 3/4 (\/5 - 2) 1 / 2 ^l^Miif^
0(t)0(^)
/9(*40)
0(1)
ff(fc n ) = it • 2" 7/11 sin^Tr) sin( ^7r)B(i, ±)
K(k 13 ) = 2- 3 13- 5/8 (5V / 13+ 18) 1/4
x [tan(^ 7 r)tan(A 7r)tan (^ 7r )]V 2 0(|)0(5l)
"U2'
# (M = \/a/4\/2 + 2 + \/2 + V ^2V^ -1 • 2- 13/4 7 _3/8
'W^T)toa(gir)
tan(gjr)
1/4 //J(&)/J(g)/J(i)
0(ti)
^(fc 15 ) = 2- 1 3- 3 / 4 5- 7 / 12 S(^,A )
_ 2- 2 3- 3/4 5- 3 / 4 (V5-l)^(^)/?(^)
0(1)
A-(fciT) = C 2
0(^)0(^(0(^)0(^)0(i)0(i)l 1/4
68 ^ V 68 V^ V 68 /^ V 68 ^ V 68 /^ V 6 8 *
0(l»)0(i)
^68/^V68
where R is the REAL ROOT of
3
x
Ax = 4 =
(4)
and C2 is an algebraic number (Borwein and Zucker
1992). Note that K(ku) is the only value in the above
list which cannot be expressed in terms of CENTRAL
Beta Functions.
Using the ELLIPTIC ALPHA FUNCTION, the ELLIPTIC
Integrals of the Second Kind can also be found
from
E-
A^fK
+
a(r)
7^
K
E' = — +a(r)K,
and by definition,
K' = Ky/n.
(5)
(6)
(7)
see also Central Beta Function, Elliptic Alpha
Function, Elliptic Delta Function, Elliptic In-
tegral of the First Kind, Elliptic Integral
of the Second Kind, Elliptic Lambda Function,
Gamma Function, Modulus (Elliptic Integral)
References
Abel, N. J. fur Math 3, 184, 1881. Reprinted in Abel, N. H.
Oeuvres Completes (Ed. L. Sylow and S. Lie). New York:
Johnson Reprint Corp., p. 377, 1988.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 139 and 298, 1987.
Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evalua-
tion of the Gamma Function at Rational Values of Small
542 Elliptic Integral Singular Value
Elliptic Integral Singular Value
Denominator." IMA J. Numerical Analysis 12, 519-526, and
1992.
Bowman, F. Introduction to Elliptic Functions, with Appli-
cations. New York: Dover, pp. 75, 95, and 98, 1961.
Glasser, M. L. and Wood, V. E. "A Closed Form Evaluation so
of the Elliptic Integral." Math. Comput. 22, 535-536,
1971.
Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J.
Reine. Angew. Math. 227, 86-110, 1967.
# Weisstein, E. W. "Elliptic Singular Values." http://www.
astro . Virginia. edu/-eww6n/math/notebooks /Elliptic
Singular .m.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 524-528, 1990.
Wrigge, S. "An Elliptic Integral Identity." Math. Comput.
27, 837-840, 1973.
Zucker, I. J. "The Evaluation in Terms of T-Functions of the
Periods of Elliptic Curves Admitting Complex Multiplica-
tion." Math. Proc. Cambridge Phil Soc. 82, 111-118,
1977.
Elliptic Integral Singular Value — k\ L e t
The first Singular Value k u corresponding to
K'(k 1 ) = K(k 1 ),
is given by
(i)
1 sin(7rx)
T(l - x) ?r
T(x),
(10)
_J__ I = Si ^iilr(i) = J-rri) (ii)
Therefore,
K
y/2j 4vy/2 40F ' K }
Now consider
■(*H«* «»
2tdt= -2udu
(14)
(15)
dt= --udu = u{l-u 2 )~ 1/2 du, (16)
z
As shown in Lemniscate Function,
Let
then
K
(±)= r *
\V2j J /(!_(«(! i
y/{l -*)(!-&)
V2
f
Jo
dt
VT-t*
u = t
3 / 4 ,
du = 4f dt = 4u J/ * dt
dt = \u~ 3/i du,
(2)
(3)
(4)
(5)
(6)
(7)
^ s(U) = 3Mi>^, (8)
tin £$&•*-««•*
-L
yj(i + " 2 )
u{l - u 2 )~ 1/2 du
jl r i i+u2
V2J0 Vi-« 2
du.
Now note that
1
( 1 , J y = (i+u 2 ) 2
{l + v?){l-u 2 ) 1-u 2
El T2
(17)
(1 + tt 2 ) 2 _l + u 2
^, (18)
i / 1 \ 1 r 1 u 2 du
-^bj + ^y 7r^- (19)
where B(a 1 b) is the Beta FUNCTION and T(z) is the
Gamma Function. Now use
nh) = yft
(9)
Now let
t = u
dt = 4u 3 dt*,
(20)
(21)
Elliptic Integral Singular Value
Elliptic Integral Singular Value 543
But
Jo
dt
(i-ty 1/2 dt
ir/3 i\ _ £U_)£ii2
S U '^~ 4T(f) "
[r(l)]- 1 = tfr(i)]- 1
r(f) = 7 rV2[r(i)]- 1
r(i) = >/5F,
(22)
(23)
(24)
(25)
Elliptic Integral Singular Value — £3
The third SINGULAR VALUE fc 3 , corresponding to
K f (k 3 ) = V3K(k 3 ),
is given by
fc3 = sin (S) = * ( ^ _ ^ ) -
As shown by Legendre,
VSF r(i)
JC(*s)
2 - 33/4 r(|)
(i)
(2)
(3)
(Whittaker and Watson 1990, p. 525). In addition,
so
du _ 1 ttV2 ■ 40T _ v / 2tt ;
,3/2
yr^? 4 r»(i) H(i)
(26) = i^rrf 1+ ^£(i
) , 2r(|)
e) r (s)
(4)
£ Ui
r(\) + «
3/2
4.
r(|)
+
80F P(i)
r(|)
r(f) r(f)
Summarizing (12) and (27) gives
K
K'
E
E' ±
1
j_
1
r 2 (|)
40r
r 2 (|)
4v^f
r 2 (i)
4Z + »
,3/2
8A H(i)
_£!(i
r 2 G) + r
3/2
8v^ P(l) -
(27)
Elliptic Integral Singular Value — k2
The second Singular Value &2, corresponding to
K'(k 2 ) = V2K(k 2 ),
(1)
is given by
k 2 = tan (-J
= v^-l,
(2)
& 2 = ^2(^-1).
(3)
For this modulus,
. E{V2-l) = \^
[Hi) , r(f)"
[r(I) + r(|)j
■ . ' (4)
and
47 ( fcs ) = A zs,,u ^ + „ K K (**)•
4 JC'(*s) 2^3
Summarizing,
^[|(V^-v / 2)] =
V5F r(J)
2-3 3 /4 r (|)
K'[\(V6-s/2)] = ^K
E[\(Ve-V2)}
1/2
>/5F r(i)
2-3 1 /4 r (|)
4VV5y K V5,/r(f) r(i)
,Y
I) + 2r(|)
B'[i(V6-V^)]
2
3 3/4
r(|) , VS-irm
+
r(i)^ 2-33/4 r(f)
(5)
(6)
(7)
(8)
(9)
(Whittaker and Watson 1990).
see also Theta Function
References
Rainanujan, S. "Modular Equations and Approximations to
ir." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, 4th ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 525-527 and 535, 1990.
544 Elliptic Lambda Function
Elliptic Lambda Function
Elliptic Lambda Function
The A Group is the Subgroup of the Gamma Group
with a and d Odd; b and c Even. The function
,t4
\(t) = \(q)
k\q)
Mi)
Mi)
(i)
where
q = e^ (2)
is a A-Modular Function and #* are Theta Func-
tions.
A*(r) gives the value of the MODULUS k r for which the
complementary and normal complete ELLIPTIC INTE-
GRALS of the First Kind are related by
K'(k r )
K{k r )
It can be computed from
= VF.
\'(r) = k(q) =
* 2 («)'
where
— 7Tv/r
9 = e v ,
(3)
(4)
(5)
and fii is a Theta Function.
From the definition of the lambda function,
A*(r') = A*(i)=V'(r). (6)
For all rational r, K(X*(r)) and E(X*(r)) are expressi-
ble in terms of a finite number of GAMMA FUNCTIONS
(Selberg and Chowla 1967). A*(r) is related to the Ra-
MANUJAN g- AND G-FUNCTIONS by
A» = \{^l + G~ 12 - Vl-G" 12 )
Special values are
A*(&) = (13v/58 - 99)(v^ + l) 6
r(f) = (v^0-3)(V2 + l) 2
A*(|) = (2-V^)(v^+v / 3)
A*(!)-(V3-^) 2 (^+l) 2
A*(2) = V2-1
A*(3)=^(v / 3-l)
A* (4) = 3-2\/2
A* (5) = \ (y/y/E-1- a/3 - V5 )
A*(6) = (2-v / 3)(v / 3»\/2)
A*(7)=Jv^(3-v^)
A* (8) = (y/2 + 1 - VW2T2)
A*(9) = |(v / 2-3 1/4 )(v / 3-l)
A* (10) = (v / 10-3)(V2-l) 2
(7)
(8)
A*(ll) = ±V6( V / l + 2x 11 -4xu- 1
-V / ll + 2a?n -4x11-0
A* (12) = (\/3 - \/2 ) 2 (\/2 - l) 2
= 15 -W2 + 8^3-6^6
A*(13) = §(\/5\/l3-17- \/l9-5\/l3)
A*(14) = -11 -8V2- 2(^+2) V5T4V2
+ \/ll + 8^2(2 + 2\/2 + v / 2V5 + 4v / 2)
A*(15) = ^\/2(3 - V5 )(a/5 - \/3 )(2 - v^)
(2 i/4 _ 1)2
A*(16):
(21/4 + 1)2
A*(17) = i\/2(V42 + l(h/l7
-i3\/-3 + vTrVs + vTr
-3vfr a/-3 + VrfVs + \/i7
-y -38 - iovTt -t- 13\/-3 + y/rry/s + vTr
+3VTr \/-3 + vT7>/5 + \/l7)
A*(18) = (\/2-l) 3 (2-\/3) 2
A* (22) = (3VTT - 7\/2 )(10 - 3>/ll )
A* (30) = (VS - V2) 2 (2 - v / 3)(v / 6- V / 5)(4 - \/l5)
A*(34)-(v / 2-l) 2 (3v / 2-\/T7)
x(v / 297 + 72\/l7- V^296 + 72vTr)
A + (42)^(V^-l) 2 (2-\/3) 2 (\/7-^)(8-3\/7)
A* (58) = (13\/58 - 99)(\/2 - l) 6
A*(210) = (\/2 - 1) 2 (2 - V3)(V7 - V5) 2 (8 - 3\/7)
x(vTo - 3) 2 (4 - v / 15) 2 (\/i5 - VT4)(6 - a/35),
vhere
In addition,
an = (17 + 3\/33) 1/3 .
A*(2') = VW2-2
A*(3') = Jv^(>/3 + l)
A*(4') = 2 1/4 (2^-2)
A* (5') = i (\/^-l + \/3-v^)
A*(r) = |v^(3 + V7)
A*(9 , ) = |(V / 2 + 3 1/4 )(v / 3-l)
A*(12') = 2\/-208 + 147V^2 - 120^3 + 85^6.
see also ELLIPTIC ALPHA FUNCTION, ELLIPTIC INTE-
GRAL of the First Kind, Modulus (Elliptic In-
tegral), RAMANUJAN g- AND G-FUNCTIONS, THETA
Function
Elliptic Logarithm
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 139 and 298, 1987.
Bowman, F. Introduction to Elliptic Functions, with Appli-
cations. New York: Dover, pp. 75, 95, and 98, 1961.
Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J.
Reine. Angew. Math. 227, 86-110, 1967.
Watson, G. N, "Some Singular Moduli (1)." Quart. J. Math.
3, 81-98, 1932.
Elliptic Logarithm
A generalization of integrals of the form
dt
Vt 2 + at '
which can be expressed in terms of logarithmic and in-
verse trigonometric functions to
eln (x)
f
dt
Vt 3 + at 2 + bt
The inverse of the elliptic logarithm is the ELLIPTIC EX-
PONENTIAL Function.
Elliptic Modular Function
tp(z)
^2 4 (0,z)
1 1/8
tf3 4 (0,2\
where $ is a Theta FUNCTION. A special case is
¥? (_ e -^) = (4v / 3-7) 1/8 .
see also MODULAR FUNCTION
Elliptic Paraboloid
A Quadratic Surface which has Elliptical Cross-
Section. The elliptic paraboloid of height /i, SEMIMA-
jor Axis a, and Semiminor Axis b can be specified
parametrically by
x = a\fu cos v
y = byfu sin v
z = u.
Elliptic Plane 545
for v E [0,27r) and u € [0,/i].
see also Elliptic Cone, Elliptic Cylinder,
Paraboloid
References
Fischer, G. (Ed.). Plate 66 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 61, 1986.
Elliptic Partial Differential Equation
A second-order Partial Differential Equation, i.e.,
one of the form
Au xx + 2Bu xy + Cuyy + Du x + Eu y + F = 0, (1)
is called elliptic if the MATRIX
z =
A B
B C
(2)
is Positive Definite. Laplace's Equation and
POISSON'S EQUATION are examples of elliptic partial
differential equations. For an elliptic partial differen-
tial equation, BOUNDARY CONDITIONS are used to give
the constraint u{x,y) = g{x,y) on dfl, where
U XX + Uyy = f{u X ,Uy,U,X,y)
(3)
holds in Q.
see also Hyperbolic Partial Differential Equa-
tion, Parabolic Partial Differential Equation,
Partial Differential Equation
Elliptic Plane
The Real Projective Plane with elliptic Metric
where the distance between two points P and Q is de-
fined as the Radian Angle between the projection of
the points on the surface of a SPHERE (which is tangent
to the plane at a point S) from the Antipode N of the
tangent point.
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 94, 1969.
546 Elliptic Point
EUison-Mendes-France Constant
Elliptic Point
A point p on a Regular Surface M e R 3 is said
to be elliptic if the Gaussian Curvature K(p) >
or equivalently, the PRINCIPAL CURVATURES m and K2
have the same sign.
see also Anticlastic, Elliptic Fixed Point (Dif-
ferential Equations), Elliptic Fixed Point
(Map), Gaussian Curvature, Hyperbolic Point,
Parabolic Point, Planar Point, Synclastic
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 280, 1993.
Elliptical Projection
see Mollweide Projection
Elliptic Pseudoprime
Let E be an ELLIPTIC CURVE denned over the FIELD of
Rational Numbers Q(V-d) having equation
y 2 = x s + ax + b
with a and b INTEGERS. Let P be a point on E with inte-
ger coordinates and having infinite order in the additive
group of rational points of E, and let n be a Compos-
ite Natural Number such that (-d/n) = -1, where
(-d/n) is the Jacobi Symbol. Then if
(n + 1)P = (mod n) ,
n is called an elliptic pseudoprime for (E,P).
see also Atkin-Goldwasser-Kilian-Morain Cer-
tificate, Elliptic Curve Primality Proving,
Strong Elliptic Pseudoprime
References
Balasubramanian, R. and Murty, M. R. "Elliptic Pseudo-
primes. II." Submitted.
Gordon, D. M. "The Number of Elliptic Pseudoprimes."
Math. Comput. 52, 231-245, 1989.
Gordon, D. M. "Pseudoprimes on Elliptic Curves." In
Theorie des nombres (Ed. J. M. DeKoninck and
C. Levesque). Berlin: de Gruyter, pp. 290-305, 1989.
Miyamoto, I. and Murty, M. R. "Elliptic Pseudoprimes."
Math. Comput. 53, 415-430, 1989.
Ribenboim, P. The New Book of Prime Number Records, 3rd
ed. New York: Springer- Verlag, pp. 132-134, 1996.
Elliptic Rotation
Leaves the CIRCLE
2.2 1
invariant .
x — x cos — y sin t
y — x sin + y sin 6
Elliptic Theta Function
see Neville Theta Function, Theta Function
Elliptic Torus
A generalization of the ring TORUS produced by stretch-
ing or compressing in the z direction. It is given by the
parametric equations
x(u, v) = (a + b cos v) cos u
y(u, v) = (a + b cos v) sin u
z(u t v) = csint;.
see also TORUS
References
Gray, A. "Tori." §11.4 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 218-220, 1993.
Elliptic Umbilic Catastrophe
A Catastrophe which can occur for three control fac-
tors and two behavior axes.
see also HYPERBOLIC UMBILIC CATASTROPHE
Ellipticity
Given a SPHEROID with equatorial radius a and polar
radius c,
a > c (oblate spheroid)
see also Equiaffinity
\l c a s a . a<c (prolate spheroid)
see also FLATTENING, OBLATE SPHEROID, PROLATE
Spheroid, Spheroid
Ellison— Mendes- France Constant
J2~ ln (^) = |0nx) 2 + 7 lnx + D + O(x- 1 ),
n<x
where 7 is the Euler-Mascheroni Constant, and
D = 2.723...
is the Ellision-Mendes-Prance constant.
References
Ellison, W. J. and Mendes-France, M. Les nombres premiers.
Paris: Hermann, 1975.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 47, 1983.
Elongated Cupola,
Elongated Square Dipyramid 547
Elongated Cupola
A n-gonal CUPOLA adjoined to a 2n-gonal PRISM.
see also Elongated Pentagonal Cupola, Elon-
gated Square Cupola, Elongated Triangular
Cupola
Elongated Dipyramid
see also ELONGATED PENTAGONAL DIPYRAMID, ELON-
GATED Square Dipyramid, Elongated Triangular
Dipyramid
Elongated Dodecahedron
A Space-Filling Polyhedron and Parallelohe-
dron.
References
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, pp. 29-30 and 257, 1973.
Elongated Gyrobicupola
see Elongated Pentagonal Gyrobicupola, Elon-
gated Square Gyrobicupola, Elongated Trian-
gular Gyrobicupola
Elongated Pentagonal Gyrobirotunda
see Johnson Solid
Elongated Pentagonal Gyrocupolarotunda
see Johnson Solid
Elongated Pentagonal Orthobicupola
see Johnson Solid
Elongated Pentagonal Orthobirotunda
see Johnson Solid
Elongated Pentagonal Orthocupolarotunda
see Johnson Solid
Elongated Pentagonal Pyramid
see Johnson Solid
Elongated Pentagonal Rotunda
A Pentagonal Rotunda adjoined to a decagonal
Prism which is Johnson Solid J 2 i-
Elongated Gyrocupolarotunda
see Elongated Pentagonal Gyrocupolarotunda
Elongated Orthobicupola
see Elongated Pentagonal Orthobicupola,
Elongated Triangular Orthobicupola
Elongated Orthobirotunda
see Elongated Pentagonal Orthobirotunda
Elongated Orthocupolarotunda
see Elongated Pentagonal Orthocupolarotun-
da
Elongated Pentagonal Cupola
see Johnson Solid
Elongated Pentagonal Dipyramid
see Johnson Solid
Elongated Pyramid
An n-gonal PYRAMID adjoined to an n-gonal PRISM.
see also Elongated Pentagonal Pyramid, Elon-
gated Square Pyramid, Elongated Triangular
Pyramid, Gyroelongated Pyramid
Elongated Rotunda
see Elongated Pentagonal Rotunda
Elongated Square Cupola
see Johnson Solid
Elongated Square Dipyramid
see Johnson Solid
Elongated Pentagonal Gyrobicupola
see Johnson Solid
548 Elongated Square Gyrobicupola
Elongated Square Gyrobicupola
<
A
V
>
A nonuniform Polyhedron obtained by rotating the
bottom third of a SMALL Rhombicuboctahedron
(Ball and Coxeter 1987, p. 137). It is also called
Miller's Solid, the Miller-Askinuze Solid, or
the Pseudorhombicuboctahedron, and is Johnson
Solid J37.
see also SMALL RHOMBICUBOCTAHEDRON
References
Askinuze, V. G. "O cisle polupravil'nyh mnogogrannikov."
Math. Prosvesc. 1, 107-118, 1957.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 137—
138, 1987.
Cromwell, P. R. Polyhedra. New York: Cambridge University
Press, pp. 91-92, 1997.
Elongated Square Pyramid
see JOHNSON SOLID
Elongated Triangular Cupola
see Johnson Solid
Elongated Triangular Dipyramid
see Johnson Solid
Elongated Triangular Gyrobicupola
see Johnson Solid
Elongated Triangular Orthobicupola
see Johnson Solid
Elongated Triangular Pyramid
see Johnson Solid
Elsasser Function
E(y.
/1/2
exp
-1/2
2iryu sinh(27T7/)
cosh(27ry) — cos(27rz)
dx.
Encoding
Embeddable Knot
A KNOT K is an n-embeddable knot if it can be placed
on a Genus n standard embedded surface without
crossings, but K cannot be placed on any standardly
embedded surface of lower GENUS without crossings.
Any Knot is an n-embeddable knot for some n. The
Figure-of-Eight Knot is a 2-Embeddable Knot. A
knot with BRIDGE NUMBER b is an n-embeddable knot
where n < b.
see also TUNNEL NUMBER
Embedding
see Extrinsic Curvature, Hyperboloid Embed-
ding, Injection, Sphere Embedding
Empty Set
The Set containing no elements, denoted 0. Strangely,
the empty set is both Open and CLOSED for any Set X
and Topology. A Groupoid, Semigroup, Quasi-
group, Ringoid, and Semiring can be empty. A
Monoid, Group, and Rings must have at least one
element, while DIVISION RINGS and FIELDS must have
at least two elements.
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 266, 1996.
Enantiomer
Two objects which are MIRROR Images of each other
are called enantiomers. The term enantiomer is synony-
mous with Enantiomorph.
see also Amphichiral Knot, Chiral, Disymmetric,
Handedness, Mirror Image, Reflexible
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.
Enantiomorph
see Enantiomer
Encoding
An encoding is a way of representing a number or expres-
sion in terms of another (usually simpler) one. However,
multiple expressions can also be encoded as a single ex-
pression, as in, for example,
(a,6) = |[(a + 6) 2 + 3a + 6]
which encodes a and b uniquely as a single number.
a
b
(a, 6)
1
1
1
2
2
3
1
2
4
2
5
see also CODE, CODING THEORY
Endogenous Variable
Enneadecagon 549
Endogenous Variable
An economic variable which is independent of the
relationships determining the equilibrium levels, but
nonetheless affects the equilibrium.
see also Exogenous Variable
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 458, 1980.
Endomorphism
A SURJECTIVE MORPHISM from an object to itself. In
Ergodic Theory, let X be a Set, F a Sigma Alge-
bra on X and m a PROBABILITY MEASURE. A MAP
T : X — > X is called an endomorphism or MEASURE-
Preserving Transformation if
1. T is SURJECTIVE,
2. T is Measurable,
3. m(T' l A) = m{A) for all AeF.
An endomorphism is called ERGODIC if it is true that
T~ 1 A — A Implies m(A) — or 1, where T' 1 A — {x £
X : T(x) € A}.
see also Measurable Function, Measure-Preserv-
ing Transformation, Morphism, Sigma Algebra,
SURJECTIVE
Endrafi surfaces are a pair of OCTIC SURFACES which
have 168 Ordinary Double Points. This is the max-
imum number known to exist for an OCTIC SURFACE,
although the rigorous upper bound is 174. The equa-
tions of the surfaces X^r are
64(x 2
2 )(y 2 - w 2 )[( x + yf-2w 2 ]
[(x-y) 2 -2w 2 ]-{~A(l±V2)(x 2 +y 2 ) 2
+ [8(2 ± V2)z 2 + 2(2 ± 7V2)w 2 ](x 2 + y 2 )
-16z 4 + 8(1 =F 2\/2 )z 2 w 2 - (1 4- 12\/2 )w 4 } 2 = 0,
where w is a parameter taken as w = 1 in the above
plots. All Ordinary Double Points of X£ are real,
while 24 of those in Xg are complex. The surfaces were
discovered in a 5-D family of octics with 112 nodes, and
are invariant under the GROUP D$ <g> Z2-
see also OCTIC SURFACE
References
Endrafi, S. "Octics with 168 Nodes." http:// www .
mathematik.uni-mainz .de/AlgebraischeGeometrie/docs
/Eendrassoctic . shtml.
Endrafi, S. "Flachen mit vielen Doppelpunkten." DMV-
Mitteilungen 4, 17-20, 4/1995.
Endrafi, S. "A Proctive Surface of Degree Eight with 168
Nodes." J. Algebraic Geom. 6, 325-334, 1997.
Energy
The term energy has an important physical meaning in
physics and is an extremely useful concept. A much
more abstract mathematical generalization is defined as
follows. Let O be a Space with Measure jjl > and
let $(P, Q) be a real function on the Product Space
fix Q. When
(/i ) n«)= j[ ®{P,Q)dti{Q)dv{P)
= J \(P,v)dv{P)
exists for measures ^/, v > 0, (fi, v) is called the Mutual
Energy and (/z, fi) is called the Energy.
see also DlRICHLET ENERGY, MUTUAL 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.
Engel's Theorem
A finite- dimensional LIE ALGEBRA all of whose elements
are ad-NiLPOTENT is itself a Nilpotent Lie Algebra.
Enneacontagon
A 90-sided POLYGON.
Enneacontahedron
A ZONOHEDRON constructed from the 10 diameters of
the Dodecahedron which has 90 faces, 30 of which
are RHOMBS of one type and the other 60 of which are
RHOMBS of another. The enneacontahedron somewhat
resembles a figure of Sharp.
see also Dodecahedron, Rhomb, Zonohedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 142-
143, 1987.
Sharp, A. Geometry Improv'd. London, p. 87, 1717.
Enneadecagon
A 19-sided POLYGON, sometimes also called the En-
NEAKAIDECAGON.
550 Enneagon
Enriques Surfaces
Enneagon
see NONAGON
Enneagonal Number
see Nonagonal Number
Enneakaidecagon
see Enneadecagon
Enneper's Surfaces
The Enneper surfaces are a three-parameter family of
surfaces with constant curvature. In general, they are
described by elliptic functions. However, special cases
which can be specified parametrically using Elemen-
tary Function include the Kuen Surface, Rembs'
Surfaces, and Sievert's Surface. The surfaces
shown above can be generated using the ENNEPER- WEI-
ERSTRAfi Parameterization with
(Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 43,
1986.
Enneper, A. "Analytisch-geometrische Untersuchungen."
Nachr. Konigl. Gesell. Wissensch. Georg- Augustus- Univ.
Gottingen 12, 258-277, 1868.
Fischer, G. (Ed.). Plate 92 in Mathematische Mod-
elle/ Mathematical Models, Bildband/Photograph Volume.
Braunschweig, Germany: Vieweg, p. 88, 1986.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 265, 1993.
Maeder, R. The Mathematica Programmer. San Diego, CA:
Academic Press, pp. 150-151, 1994.
Nordstrand, T. "Enneper's Minimal Surface." http://www.
uib . no/people/nf ytn/enntxt . htm.
Reckziegel, H. "Enneper's Surfaces." §3.4.4 in Mathemati-
cal Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 37-39, 1986.
Wolfram Research "Mathematica Version 2.0 Graphics
Gallery." http : // www . mathsource . com / cgi - bin / Math
Source/Applications/Graphics/3D/0207-155.
Enneper- Weierstrafi Parameterization
Gives a parameterization of a MINIMAL SURFACE.
3R
/
9 2 )
/(I
if(l+9 2 )
2fg
dC.
/(C) = i
9(0 = C-
(1)
(2)
Letting z = re t<p and taking the Real Part give
x = R[re i4> - f r 3 e w ] (3)
y = $t[ire i<f3 + \ir Z e Zi4
5R[rV
2 2i</>i
(4)
(5)
see also Minimal Surface
References
Dickson, S. "Minimal Surfaces." Mathematica J. 1, 38-40,
1990.
do Carmo, M. P. Mathematical Models from the Collections
of Universities and Museums (Ed. G. Fischer). Braun-
schweig, Germany: Vieweg, p. 41, 1986.
Weierstrafi, K". "Uber die Flachen deren mittlere Krummung
uberall gleich null ist." Monatsber. Berliner Akad., 612-
625, 1866.
where r 6 [0,1] and (j> 6 [— 7r,7r). Letting z = u + iv
instead gives the figure on the right,
Enormous Theorem
see Classification Theorem
13, 2
x = u — g-u -\~uv
y = — v — U V + \v
2 2
z = u — V
(6)
(7)
(8)
(do Carmo 1986, Gray 1993, Nordstrand). This surface
has a HOLE in its middle. Nordstrand gives the implicit
form
(y 2 -x
\ 2z
- + l* 2 +
-6
\{y 2 -* 2 )
4z
References
•)'
-!(*'+„' + §*') +
8 2\
0. (9)
Dickson, S. "Minimal Surfaces." Mathematica J. 1, 38-40,
1990.
do Carmo, M. P. "Enneper's Surface." §3.5C in Mathematical
Models from the Collections of Universities and Museums
Enriques Surfaces
An Enriques surface X is a smooth compact complex
surface having irregularity q(X) = and nontrivial
canonical sheaf i^x such that K x = Ox (Endrafl).
Such surfaces cannot be embedded in projective 3-space,
but there nonetheless exist transformations onto singu-
lar surfaces in projective 3-space. There exists a family
of such transformed surfaces of degree six which passes
through each edge of a TETRAHEDRON twice. A sub-
family with tetrahedral symmetry is given by the two-
parameter (r, c) family of surfaces
r , / 2 2 2 , 2 2 2
JrXoXiX2Xs + C{Xq X\ X2 + Xq X\ X$
+XQ X2 Xz +Xl X2 X3 = U
and the polynomial f r is a sphere with radius r,
f r = (3 - r)(x 2 + X! 2 -r x 2 2 + x 3 2 )
-2(1 + r)(XQXi + #o£2 + XqXs + #1#2 + #1^3 + $2X3)
Entire Function
Envelope 551
(Endrafi).
References
Angermiiller, G. and Barth, W. "Elliptic Fibres on Enriques
Surfaces." Compos. Math. 47, 317-332, 1982.
Barth, W. and Peters, C. "Automorphisms of Enriques Sur-:
faces," Invent Math. 73, 383-411, 1983.
Barth, W. P.; Peters, C. A.; and van de Ven, A. A. Compact
Complex Surfaces. New York: Springer- Verlag, 1984.
Barth, W. "Lectures on K3- and Enriques Surfaces." In Al- '
gebraic Geometry, Sitges (Barcelona) 1983, Proceedings
of a Conference Held in Sitges (Barcelona), Spain, Octo-
ber 5-12, 1983 (Ed. E. Casas-Alvero, G. E. Welters, and
S. Xambo-Descamps). New York: Springer- Verlag, pp. 21-
ST, 1983.
Endrafi, S. "Enriques Surfaces." http:// wv , mathematik .
uni - mainz . de / Algebraische Geometrie / docs /
enriques . shtml.
Enriques, F. Le superficie algebriche. Bologna, Italy:
Zanichelli, 1949.
Enriques, F. "Sulla classificazione." Atti Accad. Naz. Lincei
5, 1914,
Hunt, B. The Geometry of Some Special Arithmetic Quo-
tients. New York: Springer- Verlag, p. 317, 1996.
Entire Function
If a function is ANALYTIC on C*, where C* denotes the
extended Complex Plane, then it is said to be entire.
see also ANALYTIC FUNCTION, HOLOMORPHIC FUNC-
TION, Meromorphic
Entringer Number
The Entringer numbers E{n, k) are the number of PER-
MUTATIONS of {1,2,..., ra + 1}, starting with k + 1,
which, after initially falling, alternately fall then rise.
The Entringer numbers are given by
£7(0,0) = 1
E(n,0) =
together with the RECURRENCE RELATION
E(n, k) = E(n, k + 1) + E(n - 1, n - k).
The numbers E(n) — E(n, n) are the Secant and Tan-
gent Numbers given by the Maclaurin Series
sec x + tana;
see also Alternating Permutation, Boustrophe-
don Transform, Euler Zigzag Number, Permuta-
tion, Secant Number, Seidel-Entringer-Arnold
Triangle, Tangent Number, Zag Number, Zig
Number
References
Entringer, R. C. "A Combinatorial Interpretation of the Eu-
ler and Bernoulli Numbers." Nieuw. Arch. Wisk, 14, 241-
246, 1966.
Millar, J,; Sloane, N. J. A.; and Young, N. E. "A New Op-
eration on Sequences: The Boustrophedon Transform." J.
Combin. Th. Ser. A 76, 44-54, 1996.
Poupard, C. "De nouvelles significations enumeratives des
nombres d'Entringer." Disc. Math. 38, 265-271, 1982.
Entropy
In physics, the word entropy has important physical im-
plications as the amount of "disorder" of a system. In
mathematics, a more abstract definition is used. The
(Shannon) entropy of a variable X is defined as
i?(X) = -^p(x)ln[p(x)],
where p(x) is the probability that X is in the state x,
and plnp is defined as if p = 0. The joint entropy of
variables Xi, . . . , X n is then defined by
H{X\, . ... , X n )
- ~ Z^ " ' z2 p ( Xl > * * * > Xn ^ ln fc( Xi > • • • > *»)]•
see also KOLMOGOROV ENTROPY, KOLMOGOROV-SlNAI
Entropy, Maximum Entropy Method, Metric En-
tropy, Ornstein's Theorem, Redundancy, Shan-
non Entropy, Topological Entropy
References
Ott, E. "Entropies." §4.5 in Chaos in Dynamical Systems.
New York: Cambridge University Press, pp. 138-144, 1993,
Entscheidungsproblem
see Decision Problem
Enumerative Geometry
Schubert's application of the Conservation OF Num-
ber Principle.
see also CONSERVATION OF NUMBER PRINCIPLE, DUAL-
ITY Principle, Hilbert's Problems, Permanence
of Mathematical Relations Principle
References
Bell, E. T. The Development of Mathematics , 2nd ed. New
York: McGraw-Hill, p. 340, 1945.:
Envelope
The envelope of a one-parameter family of curves given
implicitly by
U(x,y,c) = 0, (1)
or in parametric form by (/(£, c),<7(£, c)), is a curve
which touches every member of the family. For a curve
represented by (/(£, c), g(t, c)), the envelope is found by
solving
df Og df dg
=
dt dc dc dt '
For a curve represented implicitly, the envelope is given
by simultaneously solving
dc
U(x,y y c) — 0.
(3)
(4)
552 Envelope Theorem
Epicycloid
see also Astroid, Cardioid, Catacaustic, Caustic,
Cayleyian Curve, Durer's Conchoid, Ellipse En-
velope, Envelope Theorem, Evolute, Glissette,
Hedgehog, Kiepert's Parabola, Lindelof's The-
orem, Negative Pedal Curve
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 33-34, 1972.
Lee, X. "Envelope." http://www.best.com/-xah/Special
PlaneCurves_dir/Envelope_dir/envelope.html.
Yates, R. C. "Envelopes." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 75-80,
1952.
Envelope Theorem
Relates Evolutes to single paths in the Calculus OF
Variations. Proved in the general case by Darboux and
Zermelo (1894) and Kneser (1898). It states: "When a
single parameter family of external paths from a fixed
point O has an ENVELOPE, the integral from the fixed
point to any point A on the ENVELOPE equals the inte-
gral from the fixed point to any second point B on the
Envelope plus the integral along the envelope to the
first point on the ENVELOPE, Joa = Job + Jba"
References
Kimball, W. S. Calculus of Variations by Parallel Displace-
ment. London: Butterworth, p. 292, 1952.
Envyfree
An agreement in which all parties feel as if they have
received the best deal.
Epicycloid
The path traced out by a point P on the Edge of a
Circle of Radius b rolling on the outside of a Circle
of Radius a.
It is given by the equations
x = (a + 6)cos0 — 6 cos ( — - — <j>\ (1)
y = (a + b) sin0 — b sin ( — - — <j> J (2)
x 2 = (a 4- 6) 2 cos 2 — 26(a + 6)cos0cos f — — <j>)
+ 6 W (£+*,) (3)
y 2 = (a + b) 2 sin 2 <f> — 2b(a -f b) sin <f> sin I — - — <j)\
,,2.2 ( a + h A
+ 6 sin (— 7— <Pj
(4)
x 2 + y 2 = ( a + 6) 2 +6 2
+
26(a + b) jcos U | + lj (jA cos<j>
sin[(^ + l)0]sin^}. (5)
But
so
cosacos/3 -f- sinasin/? = cos(a — /?), (6)
= (a + bf + b 2 - 2b(a + 6) cos U^ + l) - <p
= (a + bf + b 2 - 2b(a + b) cos (~4>) . (7)
Note that <j> is the parameter here, not the polar angle.
The polar angle from the center is
y (a + 6)sin</>-6sin(^)
tanp = — = — — — r-. (o)
x (a + 6)cos0-6cos(^^)
To get n CUSPS in the epicycloid, b = a/n, because then
n rotations of 6 bring the point on the edge back to its
starting position.
'(' + £)'♦ G)"-'(i)(* + :)~H
• + § + ;? + ;?- (:)(^)-H
cos(n</>)
n 2 + 2n + 2 _ 2(w + 1)
n 2 n 2
^ [(n 2 + 2n + 2) - 2(n + 1) cos(n0)] ,
q(^)sin0-^sin[(n + l)0]
a(^)cos0-^cos[(n + l)0]
(n + 1) sin (f) — sin[(n + !)</>]
(9)
tan# =
(n + 1) cos (ft — cos[(n + 1)0]
(10)
Epicycloid — 1 - Cusped
Epicycloid Involute 553
An epicycloid with one cusp is called a CARDIOID, one
with two cusps is called a NEPHROID, and one with five
cusps is called a RANUNCULOID.
n-epicycloids can also be constructed by beginning with
the Diameter of a Circle, offsetting one end by a se-
ries of steps while at the same time offsetting the other
end by steps n times as large. After traveling around
the CIRCLE once, an n-cusped epicycloid is produced,
as illustrated above (Madachy 1979).
Epicycloids have TORSION
and satisfy
«2 ^2
- + £-
a 2 b 2
1,
(11)
(12)
where p is the RADIUS OF CURVATURE (1/k).
see also Cardioid, Cyclide, Cycloid, Epicycloid —
1-CuSPED, HYPOCYCLOID, NEPHROID, RANUNCULOID
References
Bogomolny, A. "Cycloids." http://www.cut-the-knot.com/
pythagoras/cycloids .html.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 160-164 and 169, 1972.
Lee, X. "Epicycloid and Hypocycloid." http://www.best,
com/~xah/SpecialPlaneCurvesjdir/EpiHypocycloid^dir/
epiHypocycloid.html.
MacTutor History of Mathematics Archive. "Epicycloid."
http : //www-groups . dcs . st-and. ac .uk/~history/Curves
/Epicycloid. html.
Madachy, J. S* Madachy's Mathematical Recreations. New
York: Dover, pp. 219-225, 1979.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 50-52, 1991.
Yates, R. C. "Epi- and Hypo-Cycloids." A Handbook on
Curves and Their Properties, Ann Arbor, MI: J. W. Ed-
wards, pp. 81-85, 1952.
Epicycloid — 1-Cusped
A 1-cusped epicycloid has b = a, so n = 1. The radius
measured from the center of the large circle for a 1-
cusped epicycloid is given by EPICYCLOID equation (9)
with n = 1 so
7,2 = % [("* + 2n + 2) - 2(n + 1) cos(n0)]
= a 2 [(l 2 + 2 • 1 + 2) - 2(1 + 1) cos(l - <£)]
= a 2 (5 — 4cos0)
and
tan#
v — a-y/5 - 4cos<£,
2 sin — sin(2</>)
2cos<£ — cos(2<£) *
The 1-cusped epicycloid is just an offset Cardioid.
Epicycloid — 2-Cusped
see Nephroid
Epicycloid Evolute
(1)
(2)
(3)
^
y
\
/
\
/
\
1
\
s* S 1 ^
X \
/ /
/ /
W\ 1
/
\ /
/ V
\ /
\s
I /
\
\ f
/ \
\ v
. / \
\ \
J 1
**- \~ '
s I
\
/
/
The Evolute of the Epicycloid
x = (a + b) cos t - b cos ( — - — J t
y— (a + b) sin t — b sin f — - — J t
is another EPICYCLOID given by
^^{ (a+6)cosi+6cos [(^H}
a +
hi {a+b)
sin t + b cos
[m-]}'
Epicycloid Involute
. - - j
- - _ A.
554 Epicycloid Pedal Curve
Epitrochoid
The Involute of the Epicycloid
x = (a + b) cos t — b cos ( — - — ) t
y = (a + b) sin t — b sin ( — - — ) t
is another Epicycloid given by
a + 26
a + 2b f,
{(a + 6)
a + 26
cos £ + 6 cos
sin £ + b cos
Epicycloid Pedal Curve
The Pedal Curve of an Epicycloid with Pedal
Point at the center, shown for an epicycloid with four
cusps, is not a ROSE as claimed by Lawrence (1972).
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, p. 204, 1972.
Epicycloid Radial Curve
/
\ \ 1
t / JL
\ \ 1
\
\
V /
^- .-^
I
The Radial Curve of an Epicycloid is shown above
for an epicycloid with four cusps. It is not a ROSE, as
claimed by Lawrence (1972).
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, p. 202, 1972.
Epimenides Paradox
A Paradox, also called the Liar's Paradox, at-
tributed to the philosopher Epimenides in the sixth cen-
tury BC. "All Cretans are liers. . . One of their own po-
ets has said so." A sharper version of the paradox is the
EUBULIDES PARADOX, "This statement is false."
see also EUBULIDES PARADOX, SOCRATES' PARADOX
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 17, 1989.
Epimorphism
A SURJECTIVE MORPHISM.
Epispiral
A plane curve with polar equation
a
cos(n0) *
There are n sections if n is Odd and 2n if n is EVEN.
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 192-193, 1972.
Epispiral Inverse Curve
The Inverse Curve of the Epispiral
r = a sec(nt)
with INVERSION CENTER at the origin and inversion ra-
dius k is the Rose
k cosint)
r = .
a
Epitrochoid
The Roulette traced by a point P attached to a CIR-
CLE of radius b rolling around the outside of a fixed
Epitrochoid Evolute
Equal Detour Point 555
Circle of radius a. These curves were studied by
Diirer (1525), Desargues (1640), Huygens (1679), Leib-
niz, Newton (1686), L'Hospital (1690), Jakob Bernoulli
(1690), la Hire (1694), Johann Bernoulli (1695), Daniel
Bernoulli (1725), Euler (1745, 1781). An epitrochoid ap-
pears in Diirer's work Instruction in Measurement with
Compasses and Straight Edge (1525). He called epitro-
choids SPIDER LINES because the lines he used to con-
struct the curves looked like a spider.
The parametric equations for an epitrochoid are
x = m cos t ■
- hcos [-rtj
y = msint — /isin ( — 1\ ,
where m = a + b and h is the distance from P to the
center of the rolling CIRCLE. Special cases include the
LiMAgON with a = b, the Circle with a = 0, and the
Epicycloid with h = b.
see also EPICYCLOID, HYPOTROCHOID, SPIROGRAPH
References
New
Lawrence, J. D. A Catalog of Special Plane Curves.
York: Dover, pp. 168-170, 1972.
Lee, X. "Epitrochoid." http://www.best.com/-xah/Special
PlaneCurves_dir/Epitrochoid-dir/epitrochoid.html.
Lee, X. "Epitrochoid and Hypotrochoid Movie Gallery."
http://www.best.com/~xah/SpecialPlaneCurves_dir/
EpiHypoTMovieGalleryjdir/epiHypoTMovieGallery.html.
Epitrochoid Evolute
(7V^
^vC
3&~
A^ j
\ J
i V
-<y
v^S
/^
Epsilon
In mathematics, a small Positive Infinitesimal quan-
tity whose LIMIT is usually taken to be 0. The late
mathematician P. Erdos also used the term "epsilons"
to refer to children.
Epsilon- Neighborhood
see Neighborhood
Epstein Zeta Function
where g and h are arbitrary VECTORS, the SUM runs
over a d- dimensional LATTICE, and 1 = — g is omitted if
g is a lattice Vector.
see also Zeta Function
References
Glasser, M. L. and Zucker, I. J. "Lattice Sums in Theoretical
Chemistry." Theoretical Chemistry: Advances and Per-
spectives, Vol. 5. New York: Academic Press, pp. 69-70,
1980.
Shanks, D. "Calculation and Applications of Epstein Zeta
Functions." Math. Comput. 29, 271-287, 1975.
Equal
Two quantities are said to be equal if they are, in some
well-defined sense, equivalent. Equality of quantities a
and b is written a = b.
A symbol with three horizontal line segments (=) re-
sembling the equals sign is used to denote both equality
by definition (e.g., A = B means A is Defined to be
equal to B) and CONGRUENCE (e.g., 13 = 12 (mod 1)
means 13 divided by 12 leaves a Remainder of 1 — a
fact known to all readers of analog clocks).
see also CONGRUENCE, DEFINED, DIFFERENT, EQUAL
BY DEFINITION, EQUALITY, EQUIVALENT, ISOMOR-
PHISM
Equal by Definition
see Defined
Equal Detour Point
The center of an outer Soddy Circle. It has TRIANGLE
Center Function
a = 1 +
2A
= sec(| A) cos(^B) cos(f C) + 1.
(«;*) = £
a(b + c — a)
Given a point Y not between A and i?, a detour of length
\AY\ + \YB\ - \AB\
is made walking from A to B via Y , the point is of equal
detour if the three detours from one side to another via
Y are equal. If ABC has no ANGLE > 2sin~ 1 (4/5),
then the point given by the above Trilinear COORDI-
NATES is the unique equal detour point. Otherwise, the
Isoperimetric POINT is also equal detour.
References
Kimberling, C. "Central Points and Central Lines in the
' Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Isoperimetric Point and Equal Detour
Point." http : //www . evansville . edu/-ck6/tcenters/
recent/isoper.html.
Veldkamp, G. R. "The Isoperimetric Point and the Point(s) of
Equal Detour." Amer. Math. Monthly 92, 546-558, 1985.
k(i + g)] s/2:
556 Equal Parallelians Point
Equidistant Cylindrical Projection
Equal Parallelians Point
The point of intersection of the three LINE SEGMENTS,
each parallel to one side of a TRIANGLE and touching
the other two, such that all three segments are of the
same length. The Trilinear Coordinates are
bc(ca + ab — be) : ca(ab + be — ea) : ab(bc + ca — ab).
References
Kimberling, C. "Equal Parallelians Point." http://www.
evansville . edu/-ck6/tcenters/recent/eqparal .html.
Equality
A mathematical statement of the equivalence of two
quantities. The equality "A is equal to B" is written
A = B.
see also Equal, Inequality
Equally Likely Outcomes Distribution
Let there be a set S with N elements, each of them
having the same probability. Then
P(S) = pl\jE t \=Y i P(E i )
N
= P(Ei)^2l = NP{Ei).
1 = 1
Using P(S) = 1 gives
P{Ei)
N'
see also UNIFORM DISTRIBUTION
Equi-Brocard Center
The point Y for which the TRIANGLES BYC, CYA, and
AYB have equal Brocard Angles.
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Equiaffinity
An AREA-preserving AFFINITY. Equiaffinities include
the Elliptic Rotation, Hyperbolic Rotation, Hy-
perbolic Rotation (Crossed), and Parabolic Ro-
tation.
Equiangular Spiral
see Logarithmic Spiral
Equianharmonic Case
The case of the WeierstraB Elliptic Function with
invariants g 2 = and 03 = 1.
see also Lemniscate Case, Pseudolemniscate Case
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Equianharmonic
Case (g 2 — 0, g 3 — 1)." §18.13 in Handbook of Mathemat-
ical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing. New York: Dover, p. 652, 1972.
Equichordal Point
A point P for which all the CHORDS passing through P
are of the same length. It satisfies
px + py = [const] ,
where p is the CHORD length. It is an open question
whether a plane convex region can have two equichordal
points.
see also Equichordal Problem, Equiproduct
Point, Equireciprocal Point
Equichordal Problem
Is there a planar body bounded by a simple closed curve
and star-shaped with respect to two interior points p
and q whose point X-rays at p and q are both constant?
Rychlik (1997) has answered the question in the nega-
tive.
see also Equichordal Point
References
Rychlik, M. "The Equichordal Point Problem." Elec. Res.
Announcements Amer. Math. Soc. 2, 108-123, 1996.
Rychlik, M. "A Complete Solution to the Equichordal Prob-
lem of Pujiwara, Blaschke, Rothe, and Weitzenbock." In-
vent. Math. 129, 141-212, 1997.
Equidecomposable
The ability of two plane or space regions to be Dis-
sected into each other.
Equidistance Postulate
PARALLEL lines are everywhere equidistant. This POS-
TULATE 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.
Equidistant Cylindrical Projection
see Cylindrical Equidistant Projection
Equidistributed Sequence
Equilateral Triangle 557
Equidistributed Sequence
A sequence of REAL NUMBERS {x n } is equidistributed
if the probability of finding x n in any subinterval is pro-
portional to the subinterval length.
see also Weyl's Criterion
References
Kuipers, L. and Niederreiter, H. Uniform Distribution of Se-
quences. New York: Wiley, 1974.
Polya, G. and Szego, G. Problems and Theorems in Analysis
I. New York: Springer-Verlag, p. 88, 1972.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 155-156, 1991.
Equilateral Hyperbola
see Rectangular Hyperbola
Equilateral Triangle
An equilateral triangle is a Triangle with all three
sides of equal length s. An equilateral triangle also has
three equal 60° ANGLES.
An equilateral triangle can be constructed by TRISECT-
ING all three ANGLES of any TRIANGLE (MORLEY'S
Theorem). Napoleon's Theorem states that if three
equilateral triangles are drawn on the Legs of any TRI-
ANGLE (either all drawn inwards or outwards) and the
centers of these triangles are connected, the result is an-
other equilateral triangle.
Given the distances of a point from the three corners of
an equilateral triangle, a, 6, and c, the length of a side
s is given by
3(o 4 + 6 4 + c 4 + s 4 ) = (a 2 + b 2 + c 2 + s 2 ) 2 (1)
(Gardner 1977, pp. 56-57 and 63). There are infinitely
many solutions for which a, 6, and c are INTEGERS. In
these cases, one of a, 6, c, and s is DIVISIBLE by 3, one
by 5, one by 7, and one by 8 (Guy 1994, p. 183).
The Altitude h of an equilateral triangle is
±V3s,
where s is the side length, so the Area is
A=\sh=\
V3s 2 .
(2)
(3)
The INRADIUS r, ClRCUMRADIUS R, and AREA A can
be computed directly from the formulas for a general
regular POLYGON with side length s and n = 3 sides,
r = 3 scot (f) = 5 stan (i) =l^ s ( 4 )
J*=iscBc(!) = i«sec(!) = i^* (5)
A = Ins 2 cot (j)=^/3» 2 . (6)
The Areas of the Incircle and Circumcircle are
A 2 1 2
A r — nr = Y2 ns
(7)
Ar = 7rR = |7TS .
(8)
R
Let any Rectangle be circumscribed about an Equi-
lateral Triangle. Then
X + Y = Z,
(9)
where X, Y > and Z are the AREAS of the triangles in
the figure (Honsberger 1985).
Begin with an arbitrary TRIANGLE and find the ExCEN-
tral Triangle. Then find the Excentral Triangle
of that triangle, and so on. Then the resulting triangle
approaches an equilateral triangle. The only Rational
TRIANGLE is the equilateral triangle (Conway and Guy
1996). A Polyhedron composed of only equilateral
triangles is known as a DELTAHEDRON.
The largest equilateral triangle which can be inscribed
in a Unit Square (left) has side length and area
A=\y/Z.
(10)
(11)
The smallest equilateral triangle which can be inscribed
(right) is oriented at an angle of 15° and has side length
and area
s = sec(15°) = V6- \/2
,4 = 2^3-3
(12)
(13)
558 Equilibrium Point
Equireciprocal Point
(Madachy 1979).
see also Acute Triangle, Deltahedron, Equilic
Quadrilateral, Fermat Point, Gyroelongated
Square Dipyramid, Icosahedron, Isogonic Cen-
ters, Isosceles Triangle, Morley's Theorem,
Octahedron, Pentagonal Dipyramid, Right Tri-
angle, Scalene Triangle, Snub Disphenoid, Tet-
rahedron, Triangle, Triangular Dipyramid, Tri-
augmented Triangular Prism, Viviani's Theorem
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 121, 1987.
Conway, J. H. and Guy, R. K. "The Only Rational Triangle."
In The Book of Numbers. New York: Springer-Verlag,
pp. 201 and 228-239, 1996.
Dixon, R. Mathographics. New York: Dover, p. 33, 1991.
Gardner, M. Mathematical Carnival: A New Round- Up of
Tantalizers and Puzzles from Scientific American. New
York: Vintage Books, 1977.
Guy, R. K. "Rational Distances from the Corners of a
Square." §D19 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer-Verlag, pp. 181-185, 1994.
Honsberger, R. "Equilateral Triangles," Ch. 3 in Mathemat-
ical Gems I. Washington, DC: Math. Assoc. Amer., 1973.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 19-21, 1985.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 115 and 129-131, 1979.
Equilibrium Point
An equilibrium point in Game Theory is a set of strate-
gies {x\, .,.,x n } such that the zth payoff function i*Tj(x)
is larger or equal for any other ith. strategy, i.e.,
Ki(Xl,...,X n ) > l^i(xi,...,Xi_l,a?i,£i+l,...,£n).
see Nash Equilibrium
Equilic Quadrilateral
A. Quadrilateral in which a pair of opposite sides
have the same length and are inclined at 60° to each
other (or equivalently, satisfy (A) + {B} = 120°). Some
interesting theorems hold for such quadrilaterals. Let
ABCD be an equilic quadrilateral with AD = BC and
(A) + {B) = 120°. Then
1. The Midpoints P, Q, and R of the diagonals and
the side CD always determine an Equilateral
Triangle.
2. If Equilateral Triangle PCD is drawn out-
wardly on CD, then APAB is also an EQUILATERAL
Triangle.
3. If Equilateral Triangles are drawn on AC, DC,
and DB away from AB, then the three new VER-
TICES P, Q, and R are COLLINEAR.
See Honsberger (1985) for additional theorems. •
References
Garfunkel, J. "The Equilic Quadrilateral." Pi Mu Epsilon
J., 317-329, Fall 1981.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 32-35, 1985.
Equinumerous
Let A and B be two classes of POSITIVE integers. Let
A(n) be the number of integers in A which are less than
or equal to n, and let B(n) be the number of integers in
B which are less than or equal to n. Then if
A(n) ~ B(n) t
A and B are said to be equinumerous.
The four classes of Primes 8k + 1, 8k + 3, 8k + 5, 8k + 7
are equinumerous. Similarly, since 8k -f- 1 and 8k + 5 are
both of the form 4fc + l, and 8k + 3 and 8k + 7 are both of
the form 4A: + 3, Ak + 1 and 4fc + 3 are also equinumerous.
see also Bertrand's Postulate, Choquet Theory,
Prime Counting Function
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 21-22 and 31-32, 1993.
Equipollent
Two statements in LOGIC are said to be equipollent if
they are deducible from each other. Two SETS with the
same CARDINAL Number are also said to be equipol-
lent. The term EQUIPOTENT is sometimes used instead
of equipollent.
Equip otent
see EQUIPOLLENT
Equipotential Curve
A curve in 2-D on which the value of a function /(x,y)
is a constant. Other synonymous terms are ISARITHM
and ISOPLETH,
see also Lemniscate
Equiproduct Point
A point, such as interior points of a disk, such that
(px)(py) = [const],
where p is the CHORD length.
see also EQUICHORDAL POINT, EQUIRECIPROCAL
Point
Equireciprocal Point
A point, such as the FOCI of an ELLIPSE, which satisfies
1 1 r i
1 = [const ,
px py
where p is the CHORD length.
see also EQUICHORDAL POINT, EQUIPRODUCT POINT
Equirectangular Projection
Equirectangular Projection
A Map Projection, also called a Rectangular Pro-
jection, in which the horizontal coordinate is the lon-
gitude and the vertical coordinate is the latitude.
Equiripple
A distribution of ERROR such that the ERROR remaining
is always given approximately by the last term dropped.
Equitangential Curve
see Tractrix
Equivalence Class
An equivalence class is defined as a SUBSET of the form
{x£l: xRa}> where a is an element of X and the NO-
TATION "xRy" is used to mean that there is an Equiv-
alence Relation between x and y. It can be shown
that any two equivalence classes are either equal or dis-
joint, hence the collection of equivalence classes forms a
partition of X. For all a,5el, we have aRb Iff a and
b belong to the same equivalence class.
A set of Class Representatives is a Subset of X
which contains EXACTLY ONE element from each equiv-
alence class.
For n a POSITIVE INTEGER, and a, 6 INTEGERS, consider
the CONGRUENCE a = b (mod n), then the equivalence
classes are the sets {. . . , — 2n, — n, 0, n, 2n, ...'}, {. . . ,
1 - 2n, 1 - ra, 1, 1 + n, 1 + 2n, . . . } etc. The standard
Class Representatives are taken to be 0, 1, 2, ...,
n-1.
see also CONGRUENCE, COSET
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 56-57, 1993.
Equivalence Problem
see Metric Equivalence Problem
Equivalence Relation
An equivalence relation on a set X is a SUBSET of X x
X, i.e., a collection R of ordered pairs of elements of
X, satisfying certain properties. Write "xRy" to mean
(x, y) is an element of R, and we say u x is related to y"
then the properties are
1. Reflexive: aRa for all a € X,
2. Symmetric: aRb IMPLIES bRa for all a, 6 G X
Eratosthenes Sieve 559
3. Transitive: aRb and bRc imply aRc for all a, 6, c G X,
where these three properties are completely indepen-
dent. Other notations are often used to indicate a rela-
tion, e.g., a = b or a ~ b.
see also Equivalence Class, Teichmuller Space
References
Stewart, L and Tall, D. The Foundations of Mathematics.
Oxford, England: Oxford University Press, 1977.
Equivalent
If A => B and B => A (i.e, A => B/\B ^ A, where => de-
notes IMPLIES), then A and B are said to be equivalent,
a relationship which is written symbolically as A <=> B
or A ^ B. However, if A and B are "equivalent by
definition" (i.e., A is DEFINED to be £), this is writ-
ten A = B, a notation which conflicts with that for a
Congruence.
see also Defined, Iff, Implies
Equivalent Matrix
An mxn MATRIX A is said to be equivalent to another
m x n Matrix B Iff
B = PAQ
for P and Q any mxn and nxn MATRICES, respectively.
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1103, 1979.
Eratosthenes Sieve
1 2 3 J 5 | 7 | 9 l|0 1 2 3 | 5 g 7 g | 1^0
11 lb 13 ik 15 1J6 17 lis 19 2J0 11 1| 13 ]L 1J5 1J6 17 lfe 19 2b
21 2J2 23 2k 25 2J6 27 2J8 29 3J0 2|l 2b 23 ij 25 2J6 2J7 2J8 29 ifc
31 3J2 33 3k 35 3J6 37 3J8 39 4J0 31 i 3? ty 35 $ 37 3 I 8 3 1 9 4°
41 it 43 4J4 4l5 4 e 47 4fe 49 5 I°
1 2 3 J 5 fl 7 j J JJ
11 it 13 « if * 17 1! 19 ii
3} i 2 23 ii 2 i 5 f i 1 i» 29 &
31 ¥ 3 1 3 * si II 37 3 1 8 f ii
41 |g 43 4^ jj 4^6 47 y 4l9 y
An Algorithm for making tables of Primes. Sequen-
tially write down the INTEGERS from 2 to the highest
number n you wish to include in the table. Cross out
all numbers > 2 which are divisible by 2 (every second
number). Find the smallest remaining number > 2. It
is 3. So cross out all numbers > 3 which are divisible
by 3 (every third number). Find the smallest remaining
number > 3. It is 5. So cross out all numbers > 5 which
are divisible by 5 (every fifth number).
Continue until you have crossed out all numbers divisi-
ble by [ y /n\ J where [x\ is the FLOOR FUNCTION. The
numbers remaining are Prime. This procedure is illus-
trated in the above diagram which sieves up to 50, and
41 4J2 43 4k 45 4J5 47 4 8 49 5 1°
1 2 3 i 5 a 7 \ \ a
11 || 13 $ J! $ 17 \l 19 g
2|l 2J2 23 ik 2[5 2J6 2|7 2J8 29 jM
31 $ $ f f \l 37 £ 3J9 j|
41 4| 43 4J4 4fc 4 6 47 if 49 it
560
Erdos-Anning Theorem
Erdos-Selfridge Function
therefore crosses out Primes up to |_\/50j = 7. If the
procedure is then continued up to n, then the number
of cross-outs gives the number of distinct PRIME factors
of each number.
References
Conway, J. H. and Guy, R, K. The Book of Numbers, New
York: Springer- Verlag, pp. 127-130, 1996.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 20-21, 1996.
Erdos-Anning Theorem
If an infinite number of points in the Plane are all sep-
arated by Integer distances, then all the points lie on
a straight Line.
Erdos-Kac Theorem
A deeper result than the Hardy-Ramanujan Theo-
rem. Let JV(x, a, b) be the number of Integers in [3, x]
such that inequality
u)(n) — In Inn
a < = — < b
V In In n
holds, where u>(n) is the number of different PRIME fac-
tors of n. Then
limiV(,,a,6)= ( * + ^ fe^^dt.
x ^°° V2tt J a
The theorem is discussed in Kac (1959).
References
Kac, M. Statistical Independence in Probability, Analysis and
Number Theory. New York: Wiley, 1959.
Riesel, H. "The Erdos-Kac Theorem." Prime Numbers and
Computer Methods for Factorization, 2nd ed. Boston,
MA: Birkhauser, pp. 158-159, 1994.
Erdos-Mordell Theorem
If O is any point inside a TRIANGLE AABC, and P, Q,
and R are the feet of the perpendiculars from O upon
the respective sides BC, CA, and AB, then
OA + OB-rOC> 2(OP + OQ + OR).
Oppenheim (1961) and Mordell (1962) also showed that
OAxOBxOC> (OQ + OR)(OR + OP)(OP + OQ).
References
Bankoff, L. "An Elementary Proof of the Erdos-Mordell The-
orem." Amer. Math. Monthly 65, 521, 1958.
Brabant, H. "The Erdos-Mordell Inequality Again." Nieuw
Tijdschr. Wisk. 46, 87, 1958/1959.
Casey, J. A Sequel to the First Six Books of the Elements
of Euclid, 6th ed. Dublin: Hodges, Figgis, & Co., p. 253,
1892.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 9, 1969.
Erdos, P. "Problem 3740." Amer. Math. Monthly 42, 396,
1935.
Fejes-Toth, L, Lagerungen in der Ebene auf der Kugel und
im Raum. Berlin: Springer, 1953.
Mordell, L. J. "On Geometric Problems of Erdos and Oppen-
heim." Math. Gaz. 46, 213-215, 1962.
Mordell, L. J. and Barrow, D. F. "Solution to Problem 3740."
Amer. Math. Monthly 44, 252-254, 1937.
Oppenheim, A. "The Erdos Inequality and Other Inequalities
for a Triangle." Amer. Math. Monthly 68, 226-230 and
349, 1961.
Veldkamp, G. R. "The Erdos-Mordell Inequality." Nieuw
Tijdschr. Wisk. 45, 193-196, 1957/1958.
Erdos Number
An author's Erdos number is 1 if he has co-authored a
paper with Erdos, 2 if he has co-authored a paper with
someone who has co-authored a paper with Erdos, etc.
References
Grossman, J. and Ion, P. "The Erdos Number Project."
http : //www . acs . Oakland . edu/-grossman/erdoshp . html.
Erdos Reciprocal Sum Constants
see A-Sequence, 52-Sequence, Nonaveraging Se-
quence
Erdos-Selfridge Function
The Erdos-Selfridge function g(k) is defined as the least
integer bigger than k + 1 such that all prime factors of
( sC fc fc) ) exceed k (Ecklund et al. 1974). The best lower
bound known is
g(k) > exp
( ln3fc l/2
l^lnlnfc
(Granville and Ramare 1996). Scheidler and Williams
(1992) tabulated g(k) up to k = 140, and Lukes et al.
(1997) tabulated g(k) for 135 < k < 200. The values for
n = 2, 3, . . . are 4, 7, 7, 23, 62, 143, 44, 159, 46, 47,
174, 2239, . . . (Sloane's A046105).
see also Binomial Coefficient, Least Prime Fac-
tor
References
Ecklund, E. F. Jr.; Erdos, P.; and Selfridge, J. L. "A New
Function Associated with the prime factors of L?J . Math.
Comput. 28, 647-649, 1974.
Erdos, P.; Lacampagne, C. B.; and Selfridge, J. L. "Estimates
of the Least Prime Factor of a Binomial Coefficient." Math.
Comput. 61, 215-224, 1993,
Granville, A. and Ramare, O. "Explicit Bounds on Exponen-
tial Sums and the Scarcity of Squarefree Binomial Coeffi-
cients." Mathematika 43, 73-107, 1996.
Lukes, R. F.; Scheidler, R.; and Williams, H. C. "Further
Tabulation of the Erdos-Selfridge Function." Math. Com-
put. 66, 1709-1717, 1997.
Scheidler, R. and Williams, H. C. "A Method of Tabulat-
ing the Number- Theoretic Function g{k)." Math. Comput.
59, 251-257, 1992.
Sloane, N. J. A. Sequence A046105 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Erdos Square free Conjecture
Erf 561
Erdos Squarefree Conjecture
The Central Binomial Coefficient ( 2 ™) is never
SQUAREFREE for n > 4. This was proved true for all suf-
ficiently large n by Sarkozy's Theorem. Goetgheluck
(1988) proved the Conjecture true for 4 < n <
2 4220 51 84 and y ardi (1Q91) for 4 < n < 2774840978^ The
conjecture was proved true in its entirely by Granville
and Ramare (1996).
see also Central Binomial Coefficient
References
Erdos, P. and Graham, R. L. Old and New Problems
and Results in Combinatorial Number Theory. Geneva,
Switzerland: L'Enseignement Mathematique Universite de
Geneve, Vol. 28, p. 71, 1980.
Goetgheluck, P. "Prime Divisors of Binomial Coefficients."
Math. Comput 51, 325-329, 1988.
Granville, A, and Ramare, O. "Explicit Bounds on Exponen-
tial Sums and the Scarcity of Squarefree Binomial Coeffi-
cients. " Mathematika 43, 73-107, 1996.
Sander, J. W. "On Prime Divisors of Binomial Coefficients."
Bull London Math. Soc. 24, 140-142, 1992.
Sander, J. W. "A Story of Binomial Coefficients and Primes."
Amer. Math. Monthly 102, 802-807, 1995.
Sarkozy, A. "On Divisors of Binomial Coefficients. I." J.
Number Th. 20, 70-80, 1985.
Vardi, I. "Applications to Binomial Coefficients." Com-
putational Recreations in Mathematica. Reading, MA:
Addison- Wesley, pp. 25-28, 1991.
Erdos- Szekeres Theorem
Suppose a, b £ N, n = ab + 1, and xi, . . . , x n is a
sequence of n Real Numbers. Then this sequence con-
tains a MONOTONIC increasing (decreasing) subsequence
of a + 1 terms or a MONOTONIC decreasing (increasing)
subsequence of b + 1 terms. Dilworth'S Lemma is a
generalization of this theorem.
see also COMBINATORICS
Erf
1 ■ ^ :
0.5 /
-4 -2 I 2 " 4
-0/5
*S _]_
|Erf z|
Olmtz
The "error function" encountered in integrating the
Gaussian Distribution.
erf(*) =
f
2 / - t
1 J 2
e dt
1 — erfc(z)
V^iih z 2 ),
(i)
(2)
(3)
where ERFC is the complementary error function and
7(x,a) is the incomplete Gamma Function. It can
also be defined as a Maclaurin Series
2 ^(-l) n z 2n+1
^ ^ n\(2n + 1) "
71 =
Erf has the values
erf (0) =
erf(oo) = 1.
It is an Odd Function
erf(— z) = — erf(z),
and satisfies
erf(z) + erfc(z) = 1.
(4)
(5)
(6)
(7)
(8)
Erf may be expressed in terms of a CONFLUENT HYPER-
GEOMETRIC FUNCTION OF THE FIRST KlND M as
erf(*) = ^=M(l f, -z 2 ) = ^e~* M(l, §,z 2 ). (9)
Erf is bounded by
} < e* 2 r e"' 2 dt < ) (10)
x + v^+2 j x - x + J&Tl
Its Derivative is
d n
dzn erf(z) = (-l)"- 1 !^^, (11)
where H n is a Hermite Polynomial. The first De-
rivative is
(12)
d „, , 2 _-j2/ 2
^ erf(2) = ^ C
and the integral is
/
erf (z) dz — z erf (z) +
v^F
(13)
562 Erf
For x <C 1, erf may be computed from
erf(x) = — ^= I e~ l dt
-ef
V* Jo
V*Jo h »■
(14)
dt
dt
_2
I ^^ X
fc=0
2fc + l/
x^ +1 (-l) fe
fc!(2]fe + l)
1 3 , J^ I
r^\~ 3 X ~+~ 10^ ~~ 42"*- "T 216 d
rV*^ 3^ ' 10 ^ 42 ^ ~"~ ™
320 X ^ ' * V
1 +
2x 2
(2^) 2
1-3 ' 1-3- 5
+
f'-' dt )
- ^
(Acton 1990). For x > 1,
erf (x) = — p= ( / e _t dt
= 1 - -= / e"* dt.
vW*
Using Integration by Parts gives
+ ..
(15)
(16)
(17)
(18)
°° c- dt
t 2
2a; + 4
2a; 4a; 3
^^-^l 1 -^--)
(19)
(20)
and continuing the procedure gives the ASYMPTOTIC
Series
erf (x) — 1
v^
(x
- 1 l„-3
+ la;- 5 -^a;- 7 + ^x- 9 + ...
A Complex generalization of erf is defined as
w (z) = e erfc(— iz
2% 2i_
V™ V^ Jo
1 + — + — / c*
dt
~ * J-oo Z-t ~ IT J
e~ l dt
z 2 - t 2 *
(21)
(22)
(23)
(24)
Erfc
see also DAWSON'S INTEGRAL, ERFC, ERFI, GAUSSIAN
Integral, Normal Distribution Function, Prob-
ability Integral
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Error Function"
and "Repeated Integrals of the Error Function." §7.1-
7.2 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 297-300, 1972.
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., p. 16, 1990.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 568-569, 1985.
Spanier, J. and Oldham, K. B. "The Error Function erf(a;)
and Its Complement erfc(x)." Ch. 40 in An Atlas of Func-
tions. Washington, DC: Hemisphere, pp. 385-393, 1987.
Erfc
The "complementary error function"
2 f°° _ 2
erfc(ic) = —= I e l
= 1 - erf (a)
= Vtt7(!,z 2 ),
dt
(1)
(2)
(3)
where 7 is the incomplete Gamma Function. It has
the values
erfc(0) = 1
erfc (00) =
erfc (—a;) = 2 — erfc (a;)
I
erfc (a;) dx = —=
V*
,«, v . 2-y/2
erfc (x)dx — ■=—.
v*-
(4)
(5)
(6)
(7)
(8)
A generalization is obtained from the differential equa-
tion
d2 V , *dy
-£ + 2 Z f z -2ny = 0.
(9)
The general solution is then
y = Aerfcin(^) + Beifci n (—z), (10)
where erfci n (z) is the erfc integral. For integral n > 1,
erfci n (z) =/■••/ erfc(z) dz
_ 2 V(t-«)» c .
V* Jo n -
dt.
(11)
(12).
Erfi
Error 563
The definition can be extended to n = —1 and using
erfci_i(z) = — — e
erfcio(z) — erfc(z).
(13)
(14)
see also Erf, Erfi
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Repeated Inte-
grals of the Error Function." §7.2 in Handbook of Mathe-
matical Functions with Formulas, Graphs, and Mathemat-
ical Tables, 9th printing. New York: Dover, pp. 299-300,
1972.
Press, W. H.; 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.
Spanier, J. and Oldham, K. B. "The Error Function
erf(ai) and Its Complement erfc(x)" and "The exp(x) and
erfc(y / x) and Related Functions." Chs. 40 and 41 in
An Atlas of Functions. Washington, DC: Hemisphere,
pp. 385-393 and 395-403, 1987.
Erfi
erfi(z) = — ierf(i^).
see also Erf, Erfc
Ergodic Measure
An Endomorphism is called ergodic if it is true that
T -1 A = A Implies m(A) = or 1, where T~ X A = {x €
X : T(x) e A}. Examples of ergodic endomorphisms
include the MAP X -» 2x mod 1 on the unit interval
with Lebesgue Measure, certain Automorphisms of
the TORUS, and "Bernoulli shifts" (and more generally
"Markov shifts").
Given a Map T and a Sigma Algebra, there may be
many ergodic measures. If there is only one ergodic
measure, then T is called uniquely ergodic. An example
of a uniquely ergodic transformation is the MAP x »->■ x-\-
a mod 1 on the unit interval when a is irrational. Here,
the unique ergodic measure is LEBESGUE MEASURE.
Ergodic Theory
Ergodic theory can be described as the statistical and
qualitative behavior of measurable group and semigroup
actions on MEASURE SPACES. The GROUP is most com-
monly N, M, M + , and Z.
Ergodic theory had its origins in the work of Boltzmann
in statistical mechanics. Its mathematical origins are
due to von Neumann, Birkhoff, and Koopman in the
1930s. It has since grown to be a huge subject and
has applications not only to statistical mechanics, but
also to number theory, differential geometry, functional
analysis, etc. There are also many internal problems
(e.g., ergodic theory being applied to ergodic theory)
which are interesting.
see also AMBROSE-KAKUTANI THEOREM, BlRKHOFF'S
ERGODIC THEOREM, DYE'S THEOREM, DYNAMICAL
System, Hopf's Theorem, Ornstein's Theorem
References
Billingsley, P. Ergodic Theory and Information. New York:
Wiley, 1965.
Cornfeld, I.; Fomin, S.; and Sinai, Ya. G. Ergodic Theory.
New York: Springer- Verlag, 1982.
Katok, A. and Hasselblatt, B. An Introduction to the Mod-
ern Theory of Dynamical Systems. Cambridge, England:
Cambridge University Press, 1996.
Nadkarni, M. G. Basic Ergodic Theory. India: Hindustan
Book Agency, 1995.
Parry, W. Topics in Ergodic Theory. Cambridge, England:
Cambridge University Press, 1982.
Smorodinsky, M. Ergodic Theory, Entropy. Berlin: Springer-
Verlag, 1971.
Walters, P. Ergodic Theory: Introductory Lectures. New
York: Springer- Verlag, 1975.
Ergodic Transformation
A transformation which has only trivial invariant SUB-
SETS is said to be invariant.
Erlanger Program
A program initiated by F. Klein in an 1872 lecture to
describe geometric structures in terms of their group
Automorphisms.
References
Klein, F. "Vergleichende Betrachtungen uber neuere ge-
ometrische Forschungen." 1872.
Yaglom, I. M. Felix Klein and Sophus Lie: Evolution of the
Idea of Symmetry in the Nineteenth Century. Boston, MA:
Birkhauser, 1988.
ErmakofFs Test
The series ^ /(w) for a monotonic nonincreasing f(x)
is convergent if
iim :; , J < i
and divergent if
iim ;> } > 1-
References
Bromwich, T. J. Pa and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, p. 43, 1991.
Error
The difference between a quantity and its estimated or
measured quantity.
see also ABSOLUTE ERROR, PERCENTAGE ERROR, REL-
ATIVE Error
564 Error- Correcting Code
Error- Correcting Code
An error-correcting code is an algorithm for expressing
a sequence of numbers such that any errors which are
introduced can be detected and corrected (within cer-
tain limitations) based on the remaining numbers. The
study of error-correcting codes and the associated math-
ematics is known as CODING THEORY.
Error detection is much simpler than error correction,
and one or more "check" digits are commonly embedded
in credit card numbers in order to detect mistakes. Early
space probes like Mariner used a type of error-correcting
code called a block code, and more recent space probes
use convolution codes. Error-correcting codes are also
used in CD players, high speed modems, and cellular
phones. Modems use error detection when they compute
Checksums, which are sums of the digits in a given
transmission modulo some number. The ISBN used to
identify books also incorporates a check DIGIT.
A powerful check for 13 DIGIT numbers consists of the
following. Write the number as a string of DIGITS
aia,2a$ • ■ ■ ai3- Take ai + ^3 + ... + a\z and double. Now
add the number of DIGITS in Odd positions which are
> 4 to this number. Now add a^ + 0,4 + . . . + a\2- The
check number is then the number required to bring the
last DIGIT to 0. This scheme detects all single DIGIT
errors and all TRANSPOSITIONS of adject DIGITS except
and 9.
see also Checksum, Coding Theory, Galois Field,
Hadamard Matrix, ISBN
References
Conway, J. H. and Sloane, N. J. A. "Error-Correcting Codes."
§3.2 in Sphere Packings, Lattices, and Groups, 2nd ed.
New York: Springer- Verlag, pp. 75-88, 1993.
Gallian, J. "How Computers Can Read and Correct ID Num-
bers." Math Horizons, pp. 14-15, Winter 1993.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 119-121, 1994.
MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-
Correcting Codes. Amsterdam, Netherlands: North-
Holland, 1977.
Error Curve
see Gaussian Function
Error Function
see Erf, Erfc
Error Function Distribution
A Normal Distribution with Mean 0.
p / „\ n ~h 2 x 2
y/7T
The Characteristic Function is
*(i) = e- ta /< 4fca >.
(1)
(2)
Error Propagation
The Mean, Variance, Skewness, and Kurtosis are
The CUMULANTS are
/x =
2 1
7i =0
72 = 0.
Kl =0
_ 1
K2 ~2h?
K n =
(3)
(4)
(5)
(6)
(7)
(8)
(9)
for n > 3.
Error Propagation
Given a FORMULA y = f(x) with an ABSOLUTE ERROR
in x of dx, the Absolute Error is dy. The RELATIVE
ERROR is dy/y. If x = f(u,v), then
dx dx
Xi - x — (m - u) — + (Vi ~ v) — + . . . , (1)
ou ov
JV
i
- j£t £[<-.>• (£)' +<«-«>' (I)"
i L
+ 3 (m -«)(„-«) (||) (|) +...]• (2)
The definitions of Variance and Covariance then give
N
°» 2 = w^i £ (t,i - €)2
i=l
N
(3)
(4)
(5)
i=l
** =a " (to) + <T " (to)
+ M£)(S) + - w
If u and v are uncorrelated, then <r uv = so
/r 2 - t 2
&X — &U
foxy. 2
(to) + <T " •
(7)
Error Propagation
Ethiopian Multiplication 565
Now consider addition of quantities with errors. For
x = aw± bv, dx/du — a and dx/dv = ±6, so
(8)
2 2 2,,2 2, t 2
cr x = a (T u -r o a v ± laocr U v ■
For division of quantities with x = ±au/v, dx/du =
±a/v and dx/dv — ^au/v 2 , so
2 2 2
2 a 2 a w a au 2
U J <J V V V Z
-,2„,2 2
(9)
f — V - — — - 2 4- a U U O f^ f ^"\ 2
\ # / v 2 a 2 u 2 u v 4 a 2 u 2 \vJ \v 2 )
-(t)' + (t)*-'(v)(v)- < 10 >
For exponentiation of quantities with
x =
±bu / lna\±6ti ±b(lna)u
a — i c j — c ,
(ii)
dx
du
= ±6(lna)e =tMnau = ±6(lna)x,
(12)
so
ex = cr u 6(lna)a;
(13)
— = 6 In acr^ .
X
(14)
If a = e,
then
cr x
— = b(T u .
X
(15)
For Logarithms of quantities with x = aln(±6tt),
dx/du = a(±6)/(±bu) = a/it, so
2 [a
<Tx = ^U ~TT
(16)
cr x = a— . (17)
u
For multiplication with x = ±auv, dx/du = ±av and
dx/dv = ±au, so
2 22 2.22 2,02 2 /i o\
era; = a v a u -\- a u a v -\-2a uva uv (18)
/cr x \ 2 _ a 2 v 2 2 & 2 u 2 2 2a 2 uv 2
I J — ^ ^ nO'u T ~~Z 7. ^CT V -\- - - -<J UV
\ x / a z u A v z a £ u z v z a^u^v*
-(t)' + (t) ,+ '(v)(v)-'»>
For Powers, with x = au ±b , dx/du =
= ia&u* 6 " 1 =
±bx/u, so
J.2^2
2 2^ X
0"x = (7^ —
tt 2
(20)
(21)
see a/so ABSOLUTE ERROR, PERCENTAGE ERROR, REL-
ATIVE 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.
Bevington, P. R. Data Reduction and Error Analysis for the
Physical Sciences, New York: McGraw-Hill, pp. 58-64,
1969.
Escher's Map
f( z ) ^ z (l+cos/3+isin/3)/2^
Escribed Circle
see EXCIRCLE
Essential Singularity
A Singularity a for which f(z)(z - a) n is not Differ-
ENTIABLE for any INTEGER n > 0.
see also Picard's Theorem, WeierstraB-Casorati
Theorem
Estimate
An estimate is an educated guess for an unknown quan-
tity or outcome based on known information. The mak-
ing of estimates is an important part of statistics, since
care is needed to provide as accurate an estimate as
possible using as little input data as possible. Often, an
estimate for the uncertainty AE of an estimate E can
also be determined statistically. A rule that tells how to
calculate an estimate based on the measurements con-
tained in a sample is called an Estimator.
see also BIAS (ESTIMATOR), ERROR, ESTIMATOR
References
Iyanaga, S. and Kawada, Y. (Eds.). "Statistical Estimation
and Statistical Hypothesis Testing." Appendix A, Table 23
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, pp. 1486-1489, 1980.
Estimator
An estimator is a rule that tells how to calculate an
Estimate based on the measurements contained in a
sample. For example, the "sample Mean" Average x
is an estimator for the population Mean \i.
The mean square error of an estimator is defined by
MSE = ((0-9) 2 ).
Let B be the BIAS, then
MSE=([(6-(0)) + B(0)} 2 )
= (0 ~ (#)) 2 ) + B 2 0) = V0) + S 2 (0),
where V is the estimator VARIANCE.
see also Bias (Estimator), Error, Estimate, k-
Statistic
Eta Function
see Dedekind Eta Function, Dirichlet Eta Func-
tion, Theta Function
Ethiopian Multiplication
see Russian Multiplication
566
Etruscan Venus Surface
Euclid's Theorems
Etruscan Venus Surface
A 3-D shadow of a 4-D Klein Bottle.
see also IDA SURFACE
References
Peterson, I. Islands of Truth: A Mathematical Mystery
Cruise. New York: W. H. Freeman, pp. 42-44, 1990.
Eubulides Paradox
The PARADOX "This statement is false," stated in the
fourth century BC. It is a sharper version of the EPI-
MENIDES PARADOX, "All Cretans are liers. ..One of
their own poets has said so."
see also Epimenides Paradox, Socrates 1 Paradox
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 17, 1989.
Euclid's Axioms
see Euclid's Postulates
Euclid's Elements
see Elements
Euclid's Fifth Postulate
see Euclid's Postulates
Euclid Number
The nth Euclid number is defined by
E n = l + Y[ Pi ,
where pi is the ith PRIME. The first few E n are 3,
7, 31, 211, 2311, 30031, 510511, 9699691, 223092871,
6469693231, . . . (Sloane's A006862). The largest fac-
tor of E n are 3, 7, 31, 211, 2311, 509, 277, 27953, ...
(Sloane's A002585). The n of the first few Prime Euclid
numbers E n are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457,
616, 643, . . . (Sloane's A014545) up to a search limit of
700. It is not known if there are an Infinite number of
Prime Euclid numbers (Guy 1994, Ribenboim 1996).
see also Smarandache Sequences
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, 1994.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer-Verlag, 1996.
Sloane, N. J. A. Sequences A014544, A006862/M2698, and
A002585/M2697 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 35-37, 1991.
Euclid's Postulates
1. A straight LINE SEGMENT can be drawn joining any
two points.
2. Any straight LINE Segment can be extended indef-
initely in a straight Line.
3. Given any straight Line Segment, a Circle can
be drawn having the segment as Radius and one
endpoint as center.
4. All Right Angles are congruent.
5. If two lines are drawn which intersect a third in such
a way that the sum of the inner angles on one side
is less than two Right Angles, then the two lines
inevitably must intersect each other on that side if
extended far enough. This postulate is equivalent to
what is known as the PARALLEL POSTULATE.
Euclid's fifth postulate cannot be proven as a theorem,
although this was attempted by many people. Euclid
himself used only the first four postulates ("Absolute
GEOMETRY") for the first 28 propositions of the Ele-
ments, but was forced to invoke the PARALLEL POSTU-
LATE on the 29th. In 1823, Janos Bolyai and Nicolai
Lobachevsky independently realized that entirely 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.)
see also Absolute Geometry, Circle, Elements,
Line Segment, Non-Euclidean Geometry, Paral-
lel Postulate, Pasch's Theorem, Right Angle
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, pp. 88-92, 1989.
Euclid's Principle
see Euclid's Theorems
Euclid's Theorems
A theorem sometimes called "Euclid's First Theorem"
or Euclid's Principle states that if p is a Prime
and p\ab, then p\a or p\b (where | means Divides). A
COROLLARY is that p\a n => p\a (Conway and Guy 1996).
The Fundamental Theorem of Arithmetic is an-
other Corollary (Hardy and Wright 1979).
Euclid's Second Theorem states that the number of
Primes is Infinite. This theorem, also called the In-
finitude of Primes theorem, was proved by Euclid in
Proposition IX. 20 of the Elements. Ribenboim (1989)
gives nine (and a half) proofs of this theorem. Eu-
clid's elegant proof proceeds as follows. Given a finite
sequence of consecutive PRIMES 2, 3, 5, . . . , p, the num-
ber
iV = 2-3-5-"p+l, (1)
known as the zth Euclid Number when p = pi is the 2th
Prime, is either a new Prime or the product of Primes.
Euclid's Theorems
Euclidean Algorithm 567
If N is a Prime, then it must be greater than the pre-
vious PRIMES, since one plus the product of PRIMES
must be greater than each Prime composing the prod-
uct. Now, if A?" is a product of PRIMES, then at least
one of the PRIMES must be greater than p. This can be
shown as follows. If N is COMPOSITE and not greater
than p, then one of its factors (say F) must be one of
the PRIMES in the sequence, 2, 3, 5, . . . , p. It therefore
DIVIDES the product 2 • 3 • 5 • • -p. However, since it is
a factor of TV, it also Divides N. But a number which
Divides two numbers a and b < a also Divides their
difference a — 6, so F must also divide
JV-(2.3*5 • • -p) = (2-3-5 • • -p+l)-(2-3-5 ■ ■ -p) = 1. (2)
However, in order to divide 1, F must be 1, which is
contrary to the assumption that it is a PRIME in the
sequence 2, 3, 5, It therefore follows that if N
is composite, it has at least one factor greater than p.
Since N is either a Prime greater than p or contains a
factor greater than p, a Prime larger than the largest
in the finite sequence can always be found, so there are
an infinite number of PRIMES. Hardy (1967) remarks
that this proof is "as fresh and significant as when it
was discovered" so that "two thousand years have not
written a wrinkle" on it.
A similar argument shows that p\ ± 1 is PRIME, and
1.3.5.7".p+l
(3)
see also Divide, Euclid Number, Prime Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 60, 1987.
Conway, J. H. and Guy, R. K. "There are Always New
Primes!" In The Book of Numbers. New York: Springer-
Verlag, pp. 133-134, 1996.
Cosgrave, J. B. "A Remark on Euclid's Proof of the Infinitude
of Primes." Amer. Math. Monthly 96, 339-341, 1989.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 22, 1996.
Dunham, W. "Great Theorem: The Infinitude of Primes."
Journey Through Genius: The Great Theorems of Mathe-
matics. New York: Wiley, pp. 73-75, 1990.
Guy, R. K. §A12 in Unsolved Problems in Number Theory.
New York: Springer- Verlag, 1981.
Guy, R. K. "The Strong Law of Small Numbers." Amer.
Math. Monthly 95, 697-712, 1988.
Hardy, G. H. A Mathematician's Apology. Cambridge, Eng-
land: Cambridge University Press, 1992.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 3-12, 1989.
Euclidean Algorithm
An Algorithm for finding the Greatest Common Di-
visor of two numbers a and 6, also called Euclid's al-
gorithm. It is an example of a P-Problem whose time
complexity is bounded by a quadratic function of the
length of the input values (Banach and Shallit). Let
a — bq-\-r, then find a number u which DIVIDES both a
and b (so that a = su and 6 = tu) } then u also DIVIDES
r since
must be either Prime or be divisible by a PRIME > p.
Kummer used a variation of this proof, which is also a
proof by contradiction. It assumes that there exist only
a finite number of PRIMES N = pi, p2 7 ■ . ■ , p T - Now
consider N — 1. It must be a product of PRIMES, so it
has a Prime divisor pi in common with N. Therefore,
Pi\N — (N — 1) = 1 which is nonsense, so we have proved
the initial assumption is wrong by contradiction.
It is also true that there are runs of Composite Num-
bers which are arbitrarily long. This can be seen by
defining
3
(4)
2 = 1
n -^ = il*'
where j\ is a FACTORIAL. Then the j — 1 consecutive
numbers n + 2, n 4- 3, . . . , n + j are COMPOSITE, since
n + 2 = (1 • 2 ■ ■ ■ j) + 2 = 2(1 • 3 • 4 • • • n + 1) (5)
n + 3= (l-2---j) + 3 = 3(l-2.4.5---n+l) (6)
n + i = (1.2-..j)+i=i[l-2...(i-l) + l]. (7)
Guy (1981, 1988) points out that while p\p2 • • -p n + 1 is
not necessarily PRIME, letting q be the next Prime after
P1P2 • • • pn 4- 1, the number q — p\p 2 ■ - - p n + 1 is almost
always a Prime, although it has not been proven that
this must always be the case.
r = a — bq = su — qtu = (s — qt)u.
(i)
Similarly, find a number v which Divides b and r (so
that b = s'v and r = t'v), then v DIVIDES a since
a — bq + r — svq + t'v = (s'q + t')v.
(2)
Therefore, every common Divisor of a and 6 is a com-
mon Divisor of 6 and r, so the procedure can be iterated
as follows
a = bqi + n
(3)
b — q2Ti + V2
(4)
fl = q$T2 + 7*3
(5)
7*n-2 = qnTn-1 + V n
(6)
r n -i = q n +ir nj
(7)
where r n is GCD(a, b) = (a, 6). Lame showed that the
number of steps needed to arrive at the Greatest Com-
mon Divisor for two numbers less than N is
log 10 <p log 10 <t>
(8)
568 Euclidean Algorithm
where <j> is the G OLDEN MEAN, or < 5 times the number
of digits in the smaller number. Numerically, Lame's
expression evaluates to
Euclidean Geometry
steps < 4.785 log 10 N + 1.6723.
(9)
As shown by Lame's THEOREM, the worst case occurs
when the Algorithm is applied to two consecutive Fi-
bonacci Numbers. Heilbronn showed that the aver-
age number of steps is 121n2/7r 2 log 10 n = 0.843 log 10 n
for all pairs (n, b) with b < n. Kronecker showed that
the shortest application of the ALGORITHM uses least
absolute remainders. The Quotients obtained are dis-
tributed as shown in the following table (Wagon 1991).
Quotient %
41.5
17.0
9.3
For details, see Uspensky and Heaslet (1939) or Knuth
(1973). Let T(ra,n) be the number of divisions required
to compute GCD(m,n) using the Euclidean algorithm,
and define T(m, 0) = if m > 0. Then
m/ , fH-T(n,
T(m,») = | 1 + T j Bf
m mod n) for m > n
(10)
, m) for ?n < n.
Define the functions
T(n) = ~ V T(m,n) (11)
n *> — '
0<m<n
V J 0<m<n
GCD(m,n) = l
(13)
l<m<N
Kn<N
where <j> is the TOTIENT FUNCTION, T(n) is the average
number of divisions when n is fixed and m chosen at
random, r(n) is the average number of divisions when
n is fixed and m is a random number coprime to n, and
A(N) is the average number of divisions when m and
n are both chosen at random in [l,iV]. Norton (1990)
showed that
m / x 121n2
Tin) = —^
A(d)
-- E*f
d\n
d\r,
where A is the von Mangoldt Function and C is
Porter's Constant. Porter (1975) showed that
T ( n ) = il^- 2 In n + C + C(n- 1/6 + c), (15)
and Norton (1990) proved that
A(N)
12 In 2
InJV-i + VW
+C-| + 0(AT- 1 / 6+e ). (16)
There exist 22 Quadratic Fields in which there is a
Euclidean algorithm (Inkeri 1947).
see also Ferguson-Forcade Algorithm
References
Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1:
Efficient Algorithms. Cambridge, MA: MIT Press, 1996.
Courant, R. and Robbins, H. "The Euclidean Algorithm."
§2.4 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. 42-51, 1996.
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 69-70, 1990.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/porter/porter.html.
Inkeri, K. "Uber den Euklidischen Algorithmus in quadrati-
schen Zahlkorpern." Ann. Acad. Sci. Fennicae. Ser. A. I.
Math.-Phys. 1947, 1-35, 1947.
Knuth, D. E. The Art of Computer Programming, Vol 1:
Fundamental Algorithms, 2nd ed. Reading, MA: Addison-
Wesley, 1973.
Knuth, D, E. The Art of Computer Programming, Vol 2:
Seminumerical Algorithms, 2nd ed. Reading, MA:
Addison- Wesley, 1981.
Norton, G. H. "On the Asymptotic Analysis of the Euclidean
Algorithm." J. Symb. Comput. 10, 53-58, 1990.
Porter, J. W. "On a Theorem of Heilbronn." Mathematika
22, 20-28, 1975.
Uspensky, J. V. and Heaslet, M. A. Elementary Number The-
ory. New York: McGraw-Hill, 1939.
Wagon, S. "The Ancient and Modern Euclidean Algorithm"
and "The Extended Euclidean Algorithm." §8.1 and 8.2
in Mathematica in Action. New York: W. H. Freeman,
pp. 247-252 and 252-256, 1991.
Euclidean Construction
see Geometric Construction
Euclidean Geometry
A Geometry in which Euclid's Fifth Postulate
holds, sometimes also called PARABOLIC GEOMETRY.
2-D Euclidean geometry is called PLANE GEOMETRY,
and 3-D Euclidean geometry is called SOLID GEOME-
TRY. Hilbert proved the Consistency of Euclidean ge-
ometry.
see also ELLIPTIC GEOMETRY, GEOMETRIC CONSTRUC-
TION, Geometry, Hyperbolic Geometry, Non-
Euclidean Geometry, Plane Geometry
+C + - V 4>(d)0(d- 1/6+€ ), (14) References
Alt shiller- Court, N. College Geometry: A Second Course in
Plane Geometry for Colleges and Normal Schools, 2nd ed.,
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.
Euclidean Group
Euler's Addition Theorem
569
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.
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.
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.
Johnson, R. A. Advanced Euclidean Geometry. New York:
Dover, 1960.
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.
Euclidean Number
A Euclidean number is a number which can be obtained
by repeatedly solving the Quadratic EQUATION. Eu-
clidean numbers, together with the RATIONAL NUM-
BERS, can be constructed using classical GEOMETRIC
Constructions. However, the cases for which the val-
ues of the Trigonometric Functions Sine, Cosine,
Tangent, etc., can be written in closed form involv-
ing square roots of Real Numbers are much more re-
stricted.
see also ALGEBRAIC INTEGER, ALGEBRAIC NUMBER,
CONSTRUCTIBLE NUMBER, RADICAL INTEGER
References
Conway, J. H. and Guy, R. K. "Three Greek Problems."
In The Book of Numbers. New York: Springer- Verlag,
pp. 192-194, 1996.
Euclidean Plane
The 2-D Euclidean Space denoted R 2 .
see also Complex Plane, Euclidean Space
Euclidean Group
The Group of Rotations and Translations.
see also Rotation, Translation
References
Lomont, J. S. Applications of Finite Groups. New York:
Dover, 1987.
Euclidean Metric
The Function / : R n x R n -► R that assigns to any
two VECTORS (#i, . . . , x n ) and (yi, . . . ,y n ) the number
\^(xi - yi) 2 + . . . + (x n - 2/n) 2 ,
and so gives the "standard" distance between any two
Vectors in R n .
Euclidean Motion
A Euclidean motion of R n is an Affine TRANSFORMA-
TION whose linear part is an ORTHOGONAL TRANSFOR-
MATION.
see also RIGID MOTION
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 105, 1993.
Euclidean Norm
see L2-N0RM
Euclidean Space
Euclidean n-space is the Space of all n-tuples of REAL
Numbers, (21, #2, . ■ ■ , x n ) and is denoted R n . R 71 is a
Vector Space and has Lebesgue Covering Dimen-
sion n. Elements of R 71 are called n- VECTORS. R 1 = R
is the set of Real Numbers (i.e., the Real Line), and
R 2 is called the Euclidean Plane. In Euclidean space,
COVARIANT and CONTRAVARIANT quantities are equiv-
alent so e 3 = ej.
see also Euclidean Plane, Real Line, Vector
References
Gray, A. "Euclidean Spaces." §1.1 in Modern Differential
Geometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 2-4, 1993.
Eudoxus's Kampyle
see Kampyle of Eudoxus
Euler's 6n + 1 Theorem
Every Prime of the form 6n + 1 can be written in the
form x 2 + 3y 2 .
Euler's Addition Theorem
Let g(x) = (1 - x 2 )(l - k 2 x 2 ). Then
F
Jo
where
Vd( x ) Jo y/9{?)
o\/g(a) +aJg{b)
c =
VI - k 2 a 2 b 2
570 Euler Angles
Euler Angles
Euler Angles
According to EULER'S ROTATION THEOREM, any RO-
TATION may be described using three ANGLES. If the
Rotations are written in terms of Rotation Matri-
ces B, C, and D, then a general ROTATION A can be
written as
A = BCD. (1)
The three angles giving the three rotation matrices are
called Euler angles. There are several conventions for
Euler angles, depending on the axes about which the
rotations are carried out. Write the MATRIX A as
an
ai 2
ai3~
0,21
^22
d23
an
Ol2
ai3_
A =
In the so-called "^-convention," illustrated above,
(2)
D =
cos <p sin <p "
— sin <p cos
1.
(3)
C =
"10 "
cos 9 sin 9
_0 — sin# cos#_
(4)
B =
cost/' sin?/>
— sin ip cos -0
1
j
(5)
an
= cos ip cos (p — cos 9 sin <p sin ip
ai 2
= cos ip sin <p + cos 9 cos <p sin ip
ai3
= sin ip sin 6
«21
— — sin ip cos <p — cos 9 sin (p cos
*
fl22
= — sin ip sin <f> + cos 9 cos (p cos
i>
«23
= cos ip sin 9
031
= sin 9 sin <j>
«32
— — sin 9 cos </>
A33
= cos*
9
To obtain the components of the ANGULAR VELOCITY
u> in the body axes, note that for a Matrix
[Ax A 2
(6)
it is true that
an Oi2 ai3
a21 «22 ^23
_a3i a32 a33^
Q>llU>x + Ol2Uy + OisUJz
0,2lOJ x + CL22Wy + «23^
^31^0; + ft32^y + ^33^2
(7)
Aiu^ + A 2 ^ + A 3 a; z . (8)
Now, uj z corresponds to rotation about the <p axis, so
look at the u> z component of Aa;,
u)<p = Aiu; z
sin ip sin 9
cos ip sin
cos 9
(9)
The line of nodes corresponds to a rotation by 9 about
the £-axis, so look at the u>£ component of Bu>,
Me = Biuj£ = Bi0 =
COS'0
— sin^
(10)
Similarly, to find rotation by ip about the remaining axis,
look at the uty component of Bo;,
Uty = Baoty = B 3 t/> :
Combining the pieces gives
i>.
(ii)
sin ip sin 9<p + cos V>0
cos ip sin 00 — sin ip
cos9<p + ip.
(12)
For more details, see Goldstein (1980, p. 176) and Lan-
dau and Lifschitz (1976, p. 111).
The cc-convention Euler angles are given in terms of the
Cayley-Klein Parameters by
/ a 1/2 7 1/4 \
^-2zln^ 1/4(i + ^ )1/4 J,
2zln ±
ia l/2 7 l/4
■/3V4(l + ^ 7 )l/4
/ a 1/2 /3 1/4 \
Ip = -2iln ±-T7T7 — „ „„ ,
-2iln ±
" 7^4(1 +0 7 )i/4
(9 = ±2 cos -1 (iy'l + 07).
In the "y-convention,"
Therefore,
Ipx
sin X = cos y
cos <p x — — sin 0y
sin ip x — — cos ip y
cos V'x = sin ip y
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
Euler Angles
D =
B =
and A is given by
— sin cos
— cos — sin
1
1
cos 6 sin
- sin cos 9
sin ip — cos ip
cos ip simp
1
(22)
(23)
(24)
an = — sin ip sin + cos 9 cos cos ip
ai2 = sin ip cos + cos 9 sin cos ip
ai3 = — cos ip sin
a2i = — cos ip sin — cos cos sin ip
a 2 2 = cos ip cos ^ — cos 9 sin sin -0
a 2 3 = sin ?/> sin 9
&zi = sin cos
a32 = sin 9 sin
^33 = cos 9,
In the "xyz" (pitch-roll-yaw) convention, 9 is pitch, ip
is roll, and is yaw.
D =
cos0 sin
— sin cos
0"
1.
(25)
C =
"cos# — sin#"
1
_ sin 9 cos 9
(26)
B =
"1
cos ip
_ — sin ip
sin?/?
cos^
(27)
and A is given by
an = cos 9 cos
a\i = cos 9 sin
ai3 == — sin 9
021 = sin ip sin cos <p -
- cos ip sin
0,22 = sin -0 sin sin <p + cos -0 cos
a23 = cos sin ip
a 31 = cos sin cos + sin -0 sin
«32 = cos ip sin 9 sin -
- sin ip cos
033 = cos
0COS-0.
Euler Angles 571
Using Euler Parameters (which are Quaternions),
an arbitrary Rotation Matrix can be described by
2 2 2 2
an — eo + ci — e2 — e3
ai2 = 2(eie 2 + e e 3 )
a i3 = 2(eie 3 - e e 2 )
a 2 i = 2(eie 2 - e e 3 )
2 2 2 2
«22 = eo — ei + e2 — e3
a 2 3 = 2(e 2 e 3 +e ei)
a 3 i = 2(eie 3 + e e 2 )
«32 = 2(e2e 3 - eoei)
2 2 2 2
^33 = eo — ei — e2 + e3
(Goldstein 1960, p. 153).
If the coordinates of two pairs of n points x; and x^ are
known, one rotated with respect to the other, then the
Euler rotation matrix can be obtained in a straightfor-
ward manner using LEAST SQUARES FITTING. Write
the points as arrays of vectors, so
[xi ... <]=A[xi ... x„], (30)
Writing the arrays of vectors as matrices gives
X' = AX (31)
X'X T = AXX T , (32)
and solving for A gives
A = X'X T (XX T )-\ (33)
However, we want the angles 9, 0, and ip, not their com-
binations contained in the Matrix A. Therefore, write
the 3 x 3 Matrix
A =
/l(0,0,</O /2(0,0,^> /3(Mitf)
/ 4 (0,0,</O MM,^) MM,^)
LM0,0,VO MM.V0 MM.VO.
(34)
as a 1 x 9 VECTOR
fi(0,<fi,i>)
MeAA)
(35)
Now set up the matrices
A set of parameters sometimes used instead of angles
are the EULER PARAMETERS e , ei, ei and e 3 , defined
by
eo = cos
ei
e 2
G3j
= Asin(^)
(28)
(29)
a/i e/i fl/i
dh I A/a I 9f 9 I
d0
<*0
df.
(36)
Using Nonlinear Least Squares Fitting then gives
solutions which converge to (0,0,0).
572 Euler-Bernoulli Triangle
Euler 's Circle
see also CAYLEY-KLEIN PARAMETERS, EULER PARAM-
ETERS, Euler's Rotation Theorem, Infinitesimal
Rotation, Quaternion, Rotation, Rotation Ma-
trix
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 198-200, 1985.
Goldstein, H. "The Euler Angles" and "Euler Angles in Alter-
nate Conventions." §4-4 and Appendix B in Classical Me-
chanics, 2nd ed. Reading, MA: Addison- Wesley, pp. 143-
148 and 606-610, 1980.
Landau, L. D. and Lifschitz, E. M. Mechanics, 3rd ed. Ox-
ford, England: Pergamon Press, 1976.
Euler-Bernoulli Triangle
see Seidel-Entringer-Arnold Triangle
Euler Brick
^ab^
\fc
d b?
-~-2*<^y'abc
— ♦^
"\
Xr
\
A Rectangular Parallelepiped ("Brick") with in-
tegral edges a > b > c and face diagonals dij given by
dab = V a 2 + b 2
d ac = V o? + c 2
d bc = vV + c 2 .
(1)
(2)
(3)
The problem is also called the Brick, DIAGONALS
Problem, Perfect Box, Perfect Cuboid, or Ra-
tional Cuboid problem.
Euler found the smallest solution, which has sides a =
240, b = 117, and c = 44 and face DIAGONALS d a b =
267, d ac = 244, and d bc = 125. Kraitchik gave 257
cuboids with the Odd edge less than 1 million (Guy
1994, p. 174). F. Helenius has compiled a list of the 5003
smallest (measured by the longest edge) Euler bricks.
The first few are (240, 117, 44), (275, 252, 240), (693,
480, 140), (720, 132, 85), (792, 231, 160), ... (Sloane's
A031173, A031174, and A031175). Parametric solutions
for Euler bricks are also known.
No solution is known in which the oblique SPACE DIAG-
ONAL
' (4)
d abc = ^Jo? + 6 2 + c 2
is also an INTEGER. If such a brick exists, the smallest
side must be at least 1,281,000,000 (R. Rathbun 1996).
Such a solution is equivalent t;o solving the DlOPHAN-
tine Equations
(5)
(6)
(7)
(8)
A 2
+ B 2
= C 2
A 2
+ D 2
= E 2
B 2
+ D 2
= F 2
A solution with integral Space Diagonal and two out
of three face di agonals is a = 672, b = 153, and c = 104,
giving dab = 3V52777, d ac = 680, d bc — 185, and d abc =
697. A solution giving integral space and face diagonals
with only a single nonintegral EDGE is a = 18720, b =
V211773121, and c = 7800, giving d ab = 23711, d ac =
20280, d bc = 16511, and d abc = 24961.
see also Cuboid, Cyclic Quadrilateral, Diag-
onal (Polyhedron), Parallelepiped, Pythago-
rean Quadruple
References
Guy, R. K. "Is There a Perfect Cuboid? Four Squares whose
Sums in Pairs are Square. Four Squares whose Differences
are Square." §D18 in Unsolved Problems in Number The-
ory, 2nd ed. New York: Springer-Verlag, pp. 173-181,
1994.
Helenius, F. First 1000 Primitive Euler Bricks, notebooks/
EulerBricks.dat.
Leech, J. "The Rational Cuboid Revisited." Amer. Math.
Monthly 84, 518-533, 1977. Erratum in Amer. Math.
Monthly 85, 472, 1978.
Sloane, N. J. A. Sequences A031173, A031174, and A031175
in "An On-Line Version of the Encyclopedia of Integer Se-
quences."
Rathbun, R. L. Personal communication, 1996.
Spohn, W. G. "On the Integral Cuboid." Amer. Math.
Monthly 79, 57-59, 1972.
Spohn, W. G. "On the Derived Cuboid." Canad. Math. Bull.
17, 575-577, 1974.
Wells, D. G. The Penguin Dictionary of Curious and Inter-
esting Numbers. London: Penguin, p. 127, 1986.
Euler Chain
A Chain (Graph) whose Edges consist of all graph
Edges.
Euler Characteristic
Let a closed surface have Genus g. Then the Polyhe-
dral Formula becomes the Poincare Formula
X = V -E + F = 2-2g,
(1)
where % ls tne Euler characteristic, sometimes also
known as the EULER-POINCARE CHARACTERISTIC. In
terms of the Integral Curvature of the surface X,
//
Kda — 2tix
(2)
The Euler characteristic is sometimes also called the Eu-
ler Number. It can also be expressed as
X =P0 -Pl +P2,
where pi is the ith Betti Number of the space.
see also Chromatic Number, Map Coloring
Euler's Circle
see Nine-Point Circle
(3)
B 2 + E 2 = G 2
(1)1-
Euler's Conjecture
Euler's Conjecture
g(k) = 2 k +
where g(k) is the quantity appearing in Waring'S
Problem, and [x\ is the Floor Function.
see also Waring's Problem
Euler Constant
see e, EULER-MASCHERONI CONSTANT, MACLAURIN-
Cauchy Theorem
Euler's Criterion
Let p — 2m + 1 be an Odd Prime and a a Positive
Integer with p\a. Then
a m = 1 (mod p) (1)
Iff there exists an Integer t such that
p = t (mod p) .
In other words,
o {p - 1)/2 = -(modp),
V
where (a/p) is the Legendre Symbol.
see also QUADRATIC RESIDUE
References
Rosen, K. H. Ch. 9 in Elementary Number Theory and Its
Applications, 3rd ed. Reading, MA: Addis on- Wesley, 1993.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 33-37, 1993.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 293, 1991.
Euler Curvature Formula
k = K\ cos 2 + K2 sin #,
where k is the normal CURVATURE in a direction making
an ANGLE 6 with the first principle direction,
Euler Differential Equation
The general nonhomogeneous equation is
(2)
(3)
„ 2 d 2 y
dy
x~ —^ + az— + (3y = S(x)
dx 2 dx
The homogeneous equation is
x y" + Oixy + 0y =
y + -y + -=-y = 0.
X X*
Euler Differential Equation 573
Now attempt to convert the equation from
y" + p(z)y + Q(x)y = (4)
to one with constant COEFFICIENTS
(5)
dz z dz
by using the standard transformation for linear Second-
Order Ordinary Differential Equations. Com-
paring (3) and (5), the functions p(x) and q(x) are
/ \ - a - 1
p(x) — — = ax
x
q(x) = 4 = Px~\
X*
Let B = p and define
z = B~ x/ * I \fqjx)dx^0- 112 J y/p:
(6)
(7)
■"7
x~ 2 dx
I
= I x dx = \nx.
(8)
Then A is given by
= q'(x) + 2p{x)q(x) 1/2
" 2[g(x)]3/2
_ ~2/3x- 3 + 2(ax- 1 )(f3x- 2 ) al/2
2{/3x- 2 ) 3 / 2
F
a — 1,
(9)
which is a constant. Therefore, the equation becomes a
second-order ODE with constant COEFFICIENTS
Define
§+<-.>*+*-*
n = § (-A+\/A 2 -4B)
= i [l-a+-v/(a-l) a -^]
= | (-A-y/A*-4B^
r 2
1-a- V(a-l) 2 -4/3]
(10)
(11)
(12)
and
a=i(l-a) (13)
6= 1^/4/3- (a -l) 2 . (14)
The solutions are
(1)
r Cl e ri *+c 2 e r2Z («-l) 2 >4/3
2/=< (ci + cazje" (a - l) 2 = 4/3
1 e az [ci cos(fcz) + c 2 sin(^)] (a - l) 2 < 4/3.
(15)
In terms of the original variable ce,
(2)
(3)
(C!\x\^ +c 2 |x|^ («~1) 2 >4/3
y= ^ (cx+Caln^Dlxl (a - l) 2 = 4/3
I |x| a [ci cos(61n |x|) + c 2 sin(61n |x|)] (a - l) 2 < 4/3.
(16)
574 Euler's Displacement Theorem
Euler Formula
Euler's Displacement Theorem
The general displacement of a rigid body (or coordinate
frame) with one point fixed is a ROTATION about some
axis. Furthermore, a ROTATION may be described in
any basis using three Angles.
see also Euclidean Motion, Euler Angles, Rigid
Motion, Rotation
Euler's Distribution Theorem
For signed distances,
AB CD + AC -DB + AD BC = 0,
since
(b - a)(d -c) + (c- a)(b - d) + (d - a)(c - b) = 0.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle, Boston,
MA: Houghton Mifflin, p. 3, 1929.
Euler Equation
see also Euler Differential Equation, Euler For-
mula, Euler-Lagrange Differential Equation
Euler's Factorization Method
Works by expressing AT as a QUADRATIC FORM in two
different ways. Then
N =.a 2 +b 2 =-c +d 2 ,
(1)
a » _ c 2 = d 2 _ b 2 (2)
(a-c)(o + c) = (d-6)(d + 6). (3)
Let k be the GREATEST COMMON DIVISOR of a - c and
d - b so
a — c = kl (4)
d~b = km (5)
(Z,m) = l, (6)
(where (l,m) denotes the Greatest Common Divisor
of / and m), and
l{a + c) - m(d + b).
But since (l } m) = 1, m\a 4- c and
a + c = ran,
which gives
b + d = ln y
so we have
[(ifc) 2 + (^) 2 ](Z 2 +m 2 )=i(fc 2 +n 2 )(Z 2 +m 2 )
= \[{knf + (kl) 2 + (nm) 2 + (nl) 2 ]
= i[(d - 6) 2 + (a - cf + (a + c) 2 + (d + b) 2 }
(7)
(8)
(9)
= \{2a 2 + 26 2 +2c 2 + 2d 2 )
= \(2N + 2N) = N.
see also Prime Factorization Algorithms
(10)
Euler's Finite Difference Transformation
A transformation for the acceleration of the convergence
of slowly converging Alternating Series,
E(-u'* = E£
A fe a
2 n+i '
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1163,
1980.
Euler Formula
The Euler formula states
cosx + i since,
(i)
where i is the Imaginary Number. Note that the Eu-
ler Polyhedral Formula is sometimes also called
the Euler formula, as is the EULER Curvature For-
mula. The equivalent expression
ix = ln(cos x + i sin x)
(2)
had previously been published by Cotes (1714). The
special case of the formula with x = n gives the beautiful
identity
e i7r + 1 = 0, (3)
an equation connecting the fundamental numbers i, Pi,
e, 1, and (Zero).
The Euler formula can be demonstrated using a series
expansion
{i X y
n!
= £
Mr* 2
(2n)!
+ *
(-1>
n—0 " ' n=\
cosz -\-ismx.
(2n-
n-\J2n-\
(2n-l)!
(4)
It can also be proven using a COMPLEX integral. Let
z = cos + i sin 9 (5)
dz = (— sin# + zcosfl) dO = i(cos6 + isinO) d9 = izdO
(6)
/?"/'
\nz = i9,
d0
z = e l = cos 9 + i sin 9.
(7)
(8)
(9)
see also de Moivre's Identity, Euler Polyhedral
Formula
Euler Four-Square Identity
Euler Identity 575
References
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988.
Conway, J. H. and Guy, R. K. "Euler 's Wonderful Rela-
tion." The Book of Numbers. New York: Springer- Verlag,
pp. 254-256, 1996.
Cotes, R. Philosophical Transactions 29, 32, 1714.
Euler, L. Miscellanea Berolinensia 7, 179, 1743.
Euler, L. Introductio in Analysin Infinitorum, Vol. 1. Lau-
sanne, p. 104, 1748.
Euler Four-Square Identity
The amazing polynomial identity
(ai 2 + a 2 2 + az 2 + a4 2 )(&i 2 + b 2 2 4- b$ 2 + b$ 2 )
= (ai&i — a 2 b 2 — Cbzbz — a^b^)
+ (ai&2 H- a 2 b\ + fl3&4 — a>4:b$)
-\-{a\bz — a 2 b± + £3&i + 0462)
-h(ai&4 4- a 2 b% — ^362 + &4b\) ,
communicated by Euler in a letter to Goldbach on April
15, 1705. The identity also follows from the fact that the
norm of the product of two QUATERNIONS is the product
of the norms (Conway and Guy 1996).
see also Fibonacci Identity, Lagrange's Four-
square Theorem
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New-
York: Springer- Verlag, p. 232, 1996.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 8, 1996.
Euler's Graeco- Roman Squares Conjecture
Euler conjectured that there do not exist GRAECO-
Roman Squares (now known as Euler Squares) of
order n — Ak + 2 for k — 1, 2, Such squares were
found to exist in 1959, refuting the CONJECTURE.
see also Euler Square, Latin Square
Euler Graph
A GRAPH containing an EULERIAN CIRCUIT. An undi-
rected Graph is Eulerian Iff every Vertex has Even
Degree. A Directed Graph is Eulerian Iff ev-
ery Vertex has equal Indegree and Outdegree. A
planar BIPARTITE GRAPH is DUAL to a planar Euler
graph and vice versa. The number of Euler graphs with
n nodes are 1, 1, 2, 3, 7, 16, 54, 243, ... (Sloane's
A002854).
References
Sloane, N. J. A. Sequence A002854/M0846 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Euler's Homogeneous Function Theorem
Let f{x,y) be a Homogeneous Function of order n
so that
f(tx,ty)=t n (x,y). (1)
Then define x' ~ xt and y' = yt. Then
ni j{x,y) - +
dx f dt dy' dt
dx'^ y dy' X ~d{xt) ' »d{yt)'
df df df df , n .
Let t=l. then
X d-x +V d-y =nf{x ' y) -
(3)
This can be generalized to an arbitrary number of vari-
ables
z,|£=n/(x), (4)
where Einstein SUMMATION has been used.
Euler's Hypergeometric Transformations
Pi .b-l/-, ,\ C -6-l
2 i ? i(a,6;c;z)
l (1-
ty
tz) a
dt, (1)
where 2 Fi(a y b;c;z) is a HYPERGEOMETRIC FUNCTION.
The solution can be written using the Euler's transfor-
mations
t-¥t
t-*l-t
t^> (l-z-tz)~
1-t
1 -tz
(2)
(3)
(4)
(5)
in the equivalent forms
2 F 1 (a y b;c; z) = (1 - z)~ a 2 F 1 (a,c - b;c;z/(z - 1)) (6)
= (l-z)- b 2 F 1 (c-a y b;c ] z/(z-l)) (7)
= (1 - z) c - a ~ b 2 Fi (c - a, c - 6; c; z). (8)
see also Hypergeometric Function
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 585-591, 1953.
Euler Identity
For \z\ < 1,
n^+^ii 1 -* 3 '" 1 )" 1 -
P =i
q = l
see also JACOBI TRIPLE PRODUCT, g-SERIES
576
Euler's Idoneal Number
Euler-Lagrange Differential Equation
Euler's Idoneal Number
see Idoneal Number
Euler Integral
Euler integration was defined by Schanuel and subse-
quently explored by Rota, Chen, and Klain. The Euler
integral of a FUNCTION / : R. — > R (assumed to be
piecewise-constant with finitely many discontinuities) is
the sum of
/(*)-£[/(*+) + /(*-)]
over the finitely many discontinuities of /. The n-D
Euler integral can be defined for classes of functions
W 1 — ► R. Euler integration is additive, so the Euler
integral of / + g equals the sum of the Euler integrals of
/ and g.
see also Euler Measure
Euler-Jacobi Pseudoprime
An Euler-Jacobi pseudoprime is a number n such that
2 (n-l)/2^ 2 (modn)>
n
The first few are 561, 1105, 1729, 1905, .2047, 2465, . . .
(Sloane's A006971).
see also Pseudoprime
References
Sloane, N. J. A. Sequence A006971/M5461 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Euler L- Function
A special case of the Artin L-Function for the Poly-
nomial x 2 + 1. It is given by
«•>- n r^
p odd prime
x~(p)p -5 '
where
={-.
-(?)
for p = 1 (mod 4)
for p = 3 (mod 4)
where (-1/p) is a Legendre Symbol.
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Euler-Lagrange Differential Equation
A fundamental equation of CALCULUS OF VARIATIONS
which states that if J is defined by an INTEGRAL of the
form
(1)
J
/ f{x,y,y)dx,
where
y =
_ dy
dt'
(2)
then J has a Stationary VALUE if the Euler-Lagrange
differential equation
dj_
dy
dt \dy)
(3)
is satisfied. If time Derivative NOTATION is replaced
instead by space variable notation, the equation be-
comes
0. (4)
df d df
dy dx dy x
In many physical problems, f x (the PARTIAL DERIVA-
TIVE of / with respect to x) turns out to be 0, in which
case a manipulation of the Euler-Lagrange differential
equation reduces to the greatly simplified and partially
integrated form known as the Beltrami Identity,
'-<=*■
(5)
For three independent variables (Arfken 1985, pp. 924-
944), the equation generalizes to
df
d df
dx du x
d df
dy du y
d df
dz du z
= 0.
(6)
Problems in the CALCULUS OF VARIATIONS often can
be solved by solution of the appropriate Euler-Lagrange
equation.
To derive the Euler-Lagrange differential equation, ex-
SJ = S fL(q,q,t)dt= f f^6q+^6q)
dt
I
dL dL d(6q)
dq Q + dq dt
dt,
(7)
since Sq = d(5q)/dt. Now, integrate the second term by
Parts using
dL
dq
dv = d(Sq)
du = i\jk) dt v=dq '
(8)
(9)
Euler-Lagrange Differential Equation
Euler Line 577
so
j%*g-f%«*>
dL
8q
Sq
2 _ r (^- e —
dt ) 6q. (10)
Combining (7) and (10) then gives
I *2 fta
SJ
§N>f (i-^)-- M
But we are varying the path only, not the endpoints, so
8q(ti) = Sq(t2) = and (11) becomes
SJ
=m-m^ <•*
We are finding the STATIONARY VALUES such that S J —
0. These must vanish for any small change 8q, which
gives from (12),
This is the Euler-Lagrange differential equation.
The variation in J can also be written in terms of the
parameter « as
SJ
= /[/(*,
y + nv,y + kv) - f(x, y, y)] dt
= nh + \nh + \nh + £* 4 l4 + . . . , (14)
where
v = Sy
v = Sy
and the first, second, etc., variations are
A= J{vfy+Vfy)dt
h = / (^ 2 /y V + ZVVfyy + t? 2 /yy) dt
(15)
(16)
(17)
(18)
-/<•■
/l/J/I/ + 3V u/yyjj + 3VV fyyy + U fyyy) dt
(19)
/yyi/y + ^ v ^/yyj/y + 6^ V f y
+ ^V^fyyyy + V^ fyyyy) dt. (20)
The second variation can be re-expressed using
~(^ 2 A) = *; 2 A + 2™A, (21)
But
(22)
[u 2 A]£ = 0.
Now choose A such that
fyy {fyy + A) — (f yy + A)
and z such that
fyy dz
/yy+A--^
2 dt
so that z satisfies
fyy z "+" /yy-Z — (/yy — Jyy) z = U.
It then follows that
(23)
(24)
(25)
(26)
'-=/'«(* + ^) , *=/'«(*-iS) > -
(27)
see a/50 BELTRAMI IDENTITY, BRACHISTOCHRONE
Problem, Calculus of Variations, Euler-La-
grange Derivative
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, 1985.
Forsyth, A. R. Calculus of Variations. New York: Dover,
pp. 17-20 and 29, 1960.
Morse, P. M. and Feshbach, H. "The Variational Integral
and the Euler Equations." §3.1 in Methods of Theoretical
Physics, Part I. New York: McGraw-Hill, pp. 276-280,
1953.
Euler-Lagrange Derivative
The derivative
SL
~ dq dt ydq J
appearing in the Euler-Lagrange Differential
Equation.
Euler Line
578
Euler Line
Euler-Maclaurin Integration Formulas
The line on which the Orthocenter iJ, Centroid M,
ClRCUMCENTER O, DE LONGCHAMPS POINT L, NlNE-
Point Center F, and the Tangential Triangle
ClRCUMCIRCLE O t of a Triangle lie. The Incenter
lies on the Euler line only if the TRIANGLE is an ISOS-
CELES Triangle. The Euler line consists of all points
with Trilinear Coordinates a : f3 : 7 which satisfy
a j3 7
cos A cos B cos C
cos B cos C cos C cos .A cos A cos £
which simplifies to
= 0, (i)
a cos ^4(cos 2 B — cos 2 C) + (3 cos £(cos 2 C - cos 2 A)
+7 cos C(cos 2 A - cos 2 B) = 0. (2)
This can also be written
a sin(2 A) sin(B - C) + /3 sin(2£) sin(c7 - A)
4-7sin(2C)sin(;4- B)
0. (3)
The Euler line may also be given parametrically by
P(\) = Q + \H
(4)
(Oldknow 1996).
A
Center
-2
point at infinity
-1
de Longchamps point L
circumcenter O
1
centroid G
2
nine-point center F
00
orthocenter H
The Orthocenter is twice as far from the Centroid
as is the ClRCUMCENTER. The ClRCUMCENTER O,
Nine-Point Center F, Centroid G, and Orthocen-
ter H form a HARMONIC RANGE.
The Euler line intersects the SODDY LINE in the DE
Longchamps Point, and the Gergonne Line in the
Evans Point. The Isotomic Conjugate of the Eu-
ler line is called Jerabek'S HYPERBOLA (Casey 1893,
Vandeghen 1965).
see also CENTROID (TRIANGLE), ClRCUMCENTER,
Evans Point, Gergonne Line, Jerabek's Hyper-
bola, de Longchamps Point, Nine-Point Center,
Orthocenter, Soddy Line, Tangential Triangle
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 rev. enl. ed. Dublin: Hodges, Figgis, Sz Co., 1893.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 18-20, 1967.
Dorrie, H. "Euler's Straight Line." §27 in 100 Great Problems
of Elementary Mathematics: Their History and Solutions.
New York: Dover, pp. 141-142, 1965.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 117-119, 1990.
Oldknow, A. "The Euler- Gergonne- Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
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.
Euler-Lucas Pseudoprime
Let U(P,Q) and V(P,Q) be LUCAS SEQUENCES gener-
ated by P and Q, and define
D = P 2 - 4Q.
Then
f U (n - (D / n ))/ 2 = (mod n) when (Q/n) = 1
\ V( n -(D/ n ))/2 = D (mod n) when (Q/n) = -1,
where (Q/n) is the Legendre Symbol. An Odd Com-
posite Number n such that (n,QD) = 1 (i.e., n and
QD are Relatively Prime) is called an Euler-Lucas
pseudoprime with parameters (P, Q).
see also Pseudoprime, Strong Lucas Pseudoprime
References
Ribenboim, P. "Euler-Lucas Pseudoprimes (elpsp(P, Q)) and
Strong Lucas Pseudoprimes (slpsp(P, Q))." §2.X.C in The
New Book of Prime Number Records. New York: Springer-
Verlag, pp. 130-131, 1996,
Euler's Machin-Like Formula
The Machin-Like Formula
I 7 r = tan- 1 (|) + tan- 1 (|).
The other 2-term MACHIN-LlKE FORMULAS are
Hermann's Formula, Hutton's Formula, and
Machin's Formula.
see also INVERSE TANGENT
Euler-Maclaurin Integration Formulas
The first Euler-Maclaurin integration formula is
Jo
/(*)<**=§[/(!) + /(0)]
-E74t^[/ (2P " 1) ( 1 )-^ P_1) (°)]
p=l
(2p)l
+
(2*)! i '
f {2q) (x)B 2q (x)dx y (1)
where B n are BERNOULLI NUMBERS. Sums may be con-
verted to INTEGRALS by inverting the FORMULA to ob-
tain
"■ /»T1
£/(*») = /
/(*)dx +§[/(!) + /(n)]
+ ff [/'(») -/'(!)] + •••• ( 2 )
Euler-Mascheroni Constant
Euler-Mascheroni Constant 579
For a more general case when f(x) is tabulated at n
values /i , J2 , . . . , / n ,
J X\
f(x) dx = h[\h + / 3 + / 3 + . . . + / n _! + §/n]
B 2fc /i :
E^Sr[/» (2fc " 1) -^ (al, " 1) ]- 0)
k=i
(2*)!
The Euler-Maclaurin formula is implemented in
Mathematical (Wolfram Research, Champaign, IL) as
the function NSum with option Method->Integrate.
The second Euler-Maclaurin integration formula is used
when f(x) is tabulated at n values fs/2t /s/2> - • • ■>
fn-l/2 :
f(x) dx = h[f B /2 + /b/2 + / 7 /2 + ■ . • + /n-3/2
fc = l
(2*)!
+ ^i-Ew (1 - 2 " 2H1)[/ " <2k " 1) - /i<2fc " 1)l '
(4)
see also Sum, Wynn's Epsilon 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. 16 and 806, 1972.
Arfken, G. "Bernoulli Numbers, Euler-Maclaurin Formula."
§5.9 in Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 327-338, 1985.
Borwein, J. M.; Borwein, P. B.; and Dilcher, K. "Pi, Eu-
ler Numbers, and Asymptotic Expansions." Amer. Math.
Monthly 96, 681-687, 1989.
Vardi, I. "The Euler-Maclaurin Formula." §8.3 in Com-
putational Recreations in Mathematica. Reading, MA:
Addison-Wesley, pp. 159-163, 1991.
Euler-Mascheroni Constant
The Euler-Mascheroni constant is denoted 7 (or some-
times C) and has the numerical value
7 « 0.577215664901532860606512090082402431042. . .
(1)
(Sloane's A001620). The Continued Fraction of
the Euler-Mascheroni constant is [0, 1, 1, 2, 1, 2, 1,
4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] (Sloane's
A002852). The first few CONVERGENTS are 1, 1/2, 3/5,
4/7, 11/19, 15/26, 71/123, 228/395, 3035/5258, 15403/
26685, ... (Sloane's A046114 and A046115). The po-
sitions at which the digits 1, 2, ... first occur in the
Continued Fraction are 2, 4, 9, 8, 11, 69, 24, 14,
139, 52, 22, ... (Sloane's A033149). The sequence of
largest terms in the CONTINUED FRACTION is 1, 2, 4,
13, 40, 49, 65, 399, 2076, . . . (Sloane's A033091), which
occur at positions 2, 4, 8, 10, 20, 31, 34, 40, 529, ...
(Sloane's A033092).
It is not known if this constant is IRRATIONAL, let alone
Transcendental. However, Conway and Guy (1996)
are "prepared to bet that it is transcendental," although
they do not expect a proof to be achieved within their
lifetimes.
The Euler-Mascheroni constant arises in many integrals
Jo
-/
Jo
-f
Jo
e x In x dx
VI -e-* xj
— ( e~ x ) dx.
x Vl + x J
and sums
\ n J Z^ 2 n (n + :
l (n + l)
= lim
n— ► oo
n n
E^-^-^+E^
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
where £(z) is the Riemann ZetA FUNCTION and B n
are the BERNOULLI NUMBERS. It is also given by the
Euler Product
n
= lim I I =-,
n-+oo Inn XJ - 1 — -
(10)
where the product is over PRIMES p. Another connection
with the PRIMES was provided by Dirichlet's 1838 proof
that the average number of DIVISORS of all numbers
from 1 to n is asymptotic to
Er=i g o(*)
- Inn + 27- 1
(11)
(Conway and Guy 1996). de la Vallee Poussin (1898)
proved that, if a large number n is divided by all PRIMES
< n, then the average amount by which the QUOTIENT
is less than the next whole number is 7.
580 Euler-Mascheroni Constant
Euler-Mascheroni Constant
Infinite Products involving 7 also arise from the G-
Function with Positive Integer n. The cases G{2)
and G(S) give
ne— '<»»>(i-4)
n=l
OO
n<*"l'+D"-
gl+T/2
/2tt
n ^3 + 2 7
2tt
(12)
(13)
The Euler-Mascheroni constant is also given by the lim-
its
7 = Iim &lzi
s-»l S — 1
= -r'(i)
- r (i)
= lira
x— >oo
(14)
(15)
(16)
(Le Lionnais 1983).
The difference between the nth convergent in (6) and 7
is given by
n 1 f°
£__i nn _ 7= /
fc=i ,/n
^-M cfx, (17)
where [x\ is the FLOOR FUNCTION, and satisfies the
Inequality
7T, TT <> r- m ^-7<7^ (18)
v ' fc=i
(Young 1991). A series with accelerated convergence is
00
7 =§-ln2-£(-ir^[C(m)-l] (19)
m=2
(Flajolet and Vardi 1996). Another series is
nLlgnJ
z — ' n
(20)
(Vacca 1910, Gerst 1969), where Lg is the LOGARITHM
to base 2. The convergence of this series can be greatly
improved using Euler's CONVERGENCE IMPROVEMENT
transformation to
oo fc — 1
fc = l j=0 V j )
where (£) is a BINOMIAL COEFFICIENT (Beeler et ol.
1972, Item 120, with k — j replacing the undefined i).
Bailey (1988) gives
2 n
OO „ 771
E2 mn v^ 1 / 1 \
m=0 t =
(22)
which is an improvement over Sweeney (1963).
The symbol 7 is sometimes also used for
7' = e 1 w 1.781072 (23)
(Gradshteyn and Ryzhik 1979, p. xxvii).
Odena (1982-1983) gave the strange approximation
(0.11111111) 1/4 = 0.577350 . . . , (24)
and Castellanos (1988) gave
(X)V9 = 0.57721521... (25)
( 520 52^ 22 ) = °' 5772156634 • ■ • ( 26 )
'80 3 +92\ 1/6
61 4
990 3 - 55 3 - 79 2 - 4 2
70 5
= 0.57721566457... (27)
= 0.5772156649015295....
(28)
No quadratically converging algorithm for computing 7
is known (Bailey 1988). 7,000,000 digits of 7 have been
computed as of Feb. 1998 (Plouffe).
see also Euler Product, Mertens Theorem,
Stieltjes Constants
References
Bailey, D. H. "Numerical Results on the Transcendence of
Constants Involving n, e, and Euler's Constant." Math.
Comput. 50, 275-281, 1988.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Brent, R. P. "Computation of the Regular Continued Frac-
tion for Euler's Constant." Math. Comput. 31, 771—777,
1977.
Brent, R. P. and McMillan, E. M. "Some New Algorithms for
High- Precision Computation of Euler's Constant." Math.
Comput. 34, 305-312, 1980.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988.
Conway, J. H. and Guy, R. K. "The Euler-Mascheroni Num-
ber." In The Book of Numbers. New York: Springer-
Verlag, pp. 260-261, 1996.
de la Vallee Poussin, C.-J. Untitled communication. Annates
de la Soc. Set. Bruxelles 22, 84-90, 1898.
DeTemple, D. W. "A Quicker Convergence to Euler's Con-
stant." Amer. Math. Monthly 100, 468-470, 1993.
Dirichlet, G. L. J. fur Math. 18, 273, 1838.
Finch, S. "Favorite Mathematical Constants." http://wvv.
mathsoft.com/asolve/constant/euler/euler.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.
Gerst, I. "Some Series for Euler's Constant." Amer. Math.
Monthly 76, 273-275, 1969.
Glaisher, J. W. L. "On the History of Euler's Constant."
Messenger of Math. 1, 25-30, 1872.
Euler-Mascheroni Integrals
Euler Number 581
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.
Knuth, D. E. "Euler's Constant to 1271 Places." Math. Corn-
put. 16, 275-281, 1962.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 28, 1983.
Plouffe, S. "PloufTe's Inverter: Table of Current Records for
the Computation of Constants." http://lacim.uqam.ca/
pi/records. html.
Sloane, N. J. A. Sequences A033091, A033092, A046114,
A046115, A001620/M3755, and A002852/M0097 in "An
On-Line Version of the Encyclopedia of Integer Sequences."
Sweeney, D. W. "On the Computation of Euler's Constant."
Math. Comput. 17, 170-178, 1963.
Vacca, G. "A New Series for the Eulerian Constant." Quart.
J. Pure Appl. Math. 41, 363-368, 1910.
Young, R. M. "Euler's Constant." Math. Gaz. 75, 187-190,
1991.
Euler-Mascheroni Integrals
Define
In = ("I)'
poo
"/ On*)"
Jo
e z dz.
(1)
then
poo
/<,= / e- z dz = {-e- z }? = (0 + l) = l (2)
Jo
h
poo
/ (lnz)e-"
dz = 7
J a =7 3 + J* 2
h = 7 3 + |7?r 2 + 2C(3)
J 4 = 7 4 + 7 V-A 7r 4 + 87C(3))
(3)
(4)
(5)
(6)
where 7 is the EULER-MASCHERONI CONSTANT and f(3)
is Apery's Constant.
Euler Measure
Define the Euler measure of a polyhedral set as the Eu-
LER INTEGRAL of its indicator function. It is easy to
show by induction that the Euler measure of a closed
bounded convex Polyhedron is always 1 (independent
of dimension), while the Euler measure of a d-D relative-
open bounded convex POLYHEDRON is ( — l) d .
Euler Number
The Euler numbers, also called the SECANT NUMBERS
or Zig Numbers, are defined for |x| < tt/2 by
sech;
1 =
2!
+
4!
6!
2!
+
4!
E 3 X
6!
+ ...,
(1)
(2)
where sech is the HYPERBOLIC SECANT and sec is the
Secant. Euler numbers give the number of Odd Al-
ternating Permutations and are related to Genoc-
chi Numbers. The base e of the Natural Logarithm
is sometimes known as Euler's number.
Some values of the Euler numbers are
e; = 5
e; = 6i
El = 1,385
Et = 50,521
E; = 2,702,765
Ej = 199,360,981
El = 19,391,512,145
E; = 2,404,879,675,441
Eiq = 370,371,188,237,525
Ei! = 69,348,874,393,137,901
E{ 2 = 15,514,534,163,557,086,905
(Sloane's A000364). The first few Prime Euler num-
bers E n occur for n = 2, 3, 19, 227, 255, . . . (Sloane's
A014547) up to a search limit of n = 1415.
The slightly different convention defined by
E 2n = {-l) n E* n (3)
E 2n +i = (4)
is frequently used. These are, for example, the Euler
numbers computed by the Mathematical (Wolfram Re-
search, Champaign, IL) function EulerE[n], This defi-
nition has the particularly simple series definition
sech x — 1 = >
k=o
and is equivalent to
E k x h
(5)
E n = 2 n E n {\), (6)
where E n (x) is an Euler POLYNOMIAL.
To confuse matters further, the Euler Characteris-
tic is sometimes also called the "Euler number."
see also Bernoulli Number, Eulerian Number, Eu-
ler Polynomial, Euler Zigzag Number, Genocchi
Number
References
Abramowitz, M, and Stegun, C. A. (Eds.). "Bernoulli
and Euler Polynomials and the Euler-Maclaurin Formula."
§23.1 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 804-806, 1972.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, pp. 110-111, 1996.
Guy, R. K. "Euler Numbers." §B45 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
p. 101, 1994.
Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent,
Euler, and Bernoulli Numbers." Math. Comput. 21, 663-
688, 1967.
Sloane, N. J. A. Sequences A014547 and A000364/M4019 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Spanier, J. and Oldham, K. B. "The Euler Numbers, E n ."
Ch. 5 in An Atlas of Functions. Washington, DC: Hemi-
sphere, pp. 39-42, 1987.
582
Euler Parameters
Euler's Polygon Division Problem
Euler Parameters
The four parameters eo, ei, e-z, and e$ describing a finite
rotation about an arbitrary axis. The Euler parameters
are defined by
eo = cos
"ei
e =
e2
.63
(!)
nsin(|)
(1)
(2)
and are a QUATERNION in scalar-vector representation
(e , e) = e + e\x + e 2 j + e$k. (3)
Because Euler's Rotation THEOREM states that an
arbitrary rotation may be described by only three pa-
rameters, a relationship must exist between these four
quantities
2 . 2,2.2.2-,
e + e • e = e + ei + e 2 + e3 =1
(4)
(Goldstein 1980, p. 153). The rotation angle is then
related to the Euler parameters by
, 2-, 2 2222 /r\
cos <p = 2eo — 1 = eo — e ■ e = eo — ei — e2 — e,z (o)
nsin</> = 2eeo- (6)
The Euler parameters may be given in terms of the Eu-
ler Angles by
e = cos[|(0 + il>)]cos(l$)
ei=sin[§(^-V)]siii(^)
e 2 = cos[|(0 — V')] sin(|^)
e 3 = sin[f(0 + ^)]cos(±0)
(7)
(8)
(9)
(10)
(Goldstein 1980, p. 155).
Using the Euler parameters, the ROTATION FORMULA
becomes
r' = r(eo 2 -ei 2 -e2 2 -e 3 2 )4-2e(e-r) + (rxn)sin^ (11)
and the ROTATION MATRIX becomes
(12)
where the elements of the matrix are
a»j = Sij(eo - ekek) + 2e%ej + 2e»jfceoefe. (13)
Here, EINSTEIN SUMMATION has been used, Sij is the
Kronecker Delta, and €i jk is the Permutation
fx'l
' X~
y'
= A
y
U'J
_z _
Symbol. Written out explicitly, the matrix elements
are
2.2 2 2 /-. A \
an = eo + ei — 62 — 63 (14)
012 = 2(eie 2 + e e 3 ) (15)
ai3 = 2(eie 3 -e e 2 ) (16)
a 2 i = 2(eie 2 - e e 3 ) (17)
2 2.2 2 / 1Q \
^22 = e - ei + e 2 - e 3 (18)
fl23 = 2(e 2 e 3 + e ei) (19)
a 3 i =2(eie 3 + eoe 2 ) (20)
a 3 2 = 2(e 2 e 3 - e ei) (21)
2 2 2 , rt 2 / no \
a 3 3 = eo — ei — e 2 + e 3 . (22)
see a/50 Euler Angles, Quaternion, Rotation Ma-
trix
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 198-200, 1985.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA:
Addison-Wesley, 1980.
Landau, L. D. and Lifschitz, E. M. Mechanics, 3rd ed. Ox-
ford, England: Pergamon Press, 1976.
Euler's Pentagonal Number Theorem
00 00
J](l - x n ) = Y, (-l) n x n(3n+1)/2 , (1)
n = l n= — 00
where n(3n + l)/2 are generalized PENTAGONAL Num-
bers. Related equalities are
°° / i\n^n(n + l)/2 + i
no -*>-e '-i£ *„_..)' <*>
nd-^r^Err
(-l) n x n ( n + 1)/2 t n
r
(3)
see also Partition Function P, Pentagonal Num-
ber
Euler's Phi Function
see TOTIENT FUNCTION
Euler-Poincare Characteristic
see Euler Characteristic
Euler's Polygon Division Problem
The problem of finding in how many ways E n a PLANE
convex POLYGON of n sides can be divided into TRI-
ANGLES by diagonals. Euler first proposed it to Chris-
tian Goldbach in 1751, and the solution is the CATALAN
Number E n = C n -2.
see also Catalan Number, Catalan's Problem
References
Guy, R. K. "Dissecting a Polygon Into Triangles." Bull.
Malayan Math. Soc. 5, 57-60, 1958.
Euler Polyhedral Formula
Euler's Quadratic Residue Theorem 583
Euler Polyhedral Formula
see Polyhedral Formula
Euler Polynomial
A Polynomial E n {x) given by the sum
2e xt A,,, J"
e' + 1 "^ n\
(1)
Euler polynomials are related to the Bernoulli Num-
bers by
E n -l(x)
2 n
MH 1 ) -*•(!)]
= |[B.W-2-B n (|)]
(2)
(3)
E„_ 2 (x) = 2 ( " ) £ U ) K 2 """ " l)5n- k S fc (x)],
(4)
where (™) is a BINOMIAL COEFFICIENT. Settings = 1/2
and normalizing by 2 n gives the EULER NUMBER
E n — 2 E n ( 2 ) •
(5)
Call E' n = £„(0), then the first few terms are -1/2, 0,
1/4, -1/2, 0, 17/8, 0, 31/2, 0, .... The terms are the
same but with the SIGNS reversed if x — 1. These values
can be computed using the double sum
£ n (0) = 2-"£
3 = 1
( _ 1)J+ n +1/ g/n + l\
• (6)
The Bernoulli Numbers B n for n > 1 can be ex-
pressed in terms of the E n by
B n —
2(2" -1)*
(7)
see also BERNOULLI POLYNOMIAL, EULER NUMBER,
Genocchi Number
References
Abramowitz, M. and Stegun, C. A. (Eds.), "Bernoulli
and Euler Polynomials and the Euler-Maclaurin Formula."
§23.1 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 804-806, 1972.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.
Spanier, J. and Oldham, K. B. "The Euler Polynomials
E n (x)." Ch. 20 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 175-181, 1987.
Euler Polynomial Identity
see Euler Four-Square Identity
Euler Power Conjecture
see Euler's Sum of Powers Conjecture
Euler Product
For a > 1,
«')-E?-nirr.
n=l p P*
where ((z) is the RlEMANN ZETA FUNCTION.
e 7 = lim I I =-,
n-J-oo Inn XJ - 1 —
;=i Pi
where the product is over Primes p, where 7 is the
Euler-Mascheroni Constant.
see also Dedekind Function
Euler Pseudoprime
Euler pseudoprimes to a base a are Odd COMPOSITE
numbers such that (a, n) = 1 and the JACOBI SYMBOL
satisfies
(£) = <-i>/» (mod „).
No Odd COMPOSITE number is an Euler pseudoprime
for all bases a RELATIVELY PRIME to it. This class in-
cludes some Carmichael Numbers and all Strong
Pseudoprimes to base a. An Euler pseudoprime is
pseudoprime to at most 1/2 of all possible bases less
than itself. The first few Euler pseudoprimes are 341,
561, 1105, 1729, 1905, 2047, ... (Sloane's A006970).
see also Pseudoprime, Strong Pseudoprime
References
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.
Sloane, N. J. A. Sequence A006970/M5442 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Euler's Quadratic Residue Theorem
A number D that possesses no common divisor with a
prime number p is either a Quadratic Residue or non-
residue of p, depending whether £>( p_1 " 2 is congruent
mod p to ±1.
584 Euler Quartic Conjecture
Euler Quartic Conjecture
Euler conjectured that there are no POSITIVE INTEGER
solutions to the quartic DlOPHANTINE EQUATION
A 4 + B 4 = C 4 + D 4 .
This conjecture was disproved by N. D. Elkies in 1988,
who found an infinite class of solutions.
see also Diophantine Equation — Quartic
References
Berndt, B. C. and Bhargava, S. "Ramanujan — For Low-
brows." Amer. Math. Monthly 100, 644-656, 1993.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 139-140, 1994.
Lander, L. J.; Parkin, T. R.; and Selfridge, J. L. "A Survey of
Equal Sums of Like Powers." Math. Comput. 21, 446-459,
1967.
Ward, M. "Euler 's Problem on Sums of Three Fourth Pow-
ers." Duke Math. J. 15, 827-837, 1948.
Euler Sum
where (£) are BINOMIAL COEFFICIENTS. The POSITIVE
terms in the series can be converted to an ALTERNATING
Series using
oo oo
Xa = $^ _1 ) r lwr >
r=l r=l
where
W r = V r + 2v 2r + 4^4r + Sv^r + • - •
(3)
(4)
see also Alternating 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. 16, 1972.
Euler's Rotation Theorem
An arbitrary ROTATION may be described by only three
parameters.
see also Euler Angles, Euler Parameters, Rota-
tion Matrix
Euler's Rule
The numbers 2 n pq and 2 n r are AMICABLE NUMBERS if
the three INTEGERS
p = 2 m (2 n ~ m + l)-l
g = 2 m (2 n_m + l) -1
r _ 2 n+mj 2 n-m +1 j2 _ 1
are all Prime numbers for some Positive Integer m
satisfying 1 < m < n - 1 (Dickson 1952, p. 42). How-
ever, there are exotic Amicable Numbers which do
not satisfy Euler's rule, so it is a SUFFICIENT but not
Necessary condition for amicability.
see also AMICABLE NUMBERS
References
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, 1952.
Euler's Series Transformation
Accelerates the rate of CONVERGENCE for an ALTER-
NATING Series
S = £(-l)'«,
s=0
= Uq — U! + U2
. . - n n _i + 2_^
(-i) 2
2 *+i
[A S M (1)
Euler's Spiral
see CORNU SPIRAL
Euler Square
A square ARRAY made by combining n objects of two
types such that the first and second elements form LATIN
SQUARES. Euler squares are also known as GRAECO-
Latin Squares, Graeco-Roman Squares, or Latin-
GRAECO Squares. For many years, Euler squares were
known to exist for n = 3, 4, and for every Odd n except
n = 3fc. Euler's Graeco-Roman Squares Conjec-
ture maintained that there do not exist Euler squares
of order n — 4k -j- 2 for k = 1, 2, — However, such
squares were found to exist in 1959, refuting the Con-
jecture.
see also Latin Rectangle, Latin Square, Room
Square
References
Beezer, R. "Graeco-Latin Squares." http://buzzard.ups.
edu/squares .html.
Kraitchik, M. "Euler (Graeco-Latin) Squares." §7.12 in
Mathematical Recreations. New York: W. W. Norton,
pp. 179-182, 1942.
Euler Sum
In response to a letter from Goldbach, Euler considered
Double Sums of the form
oo
«(m,n) = £(l + i + ... + i) (fc+1)- (1)
k = l
OO
= 5>+Mfc+i)n*+ir n (2)
for n Even and A the Forward Difference operator
A k u n = £(-!)'
m=0
■o-
+ fe — my
(2)
with m > 1 and n > 2 and where 7 is the EULER-
Mascheroni Constant and V(x) = Vo(^) *s the
DiGAMMA Function. Euler found explicit formulas in
Euler Sum
terms of the RlEMANN ZETA FUNCTION for s(l,n) with
n > 2, and E. Au-Yeung numerically discovered
£( 1+ | + '" + *) fc " 2 = TC(4), (3)
fc = l
where ((z) is the RlEMANN Zeta Function, which was
subsequently rigorously proven true (Borwein and Bor-
wein 1995). Sums involving k~ n can be re-expressed in
terms of sums the form (k + l)~ n via
Euler Sum 585
where Sh and s a have the special forms
Jfc=i
00
a = ^{ln2 + i(-ir
(14)
OO
*[M\n + |) - Vo(|n + l)]} m (k + l)~ m . (15)
Analytic single or double sums over £(z) can be con-
structed for
= £
A 1
+ ^T + ---+ (fc + i)*
n — is
(fc + 1) "" «*(!.«) = i"C(«+l)-i ^ C(n-*)C(fc + l) (16)
OO OO
= E( 1+ ^+-+i^)(*+ i r m +E*" (m4 " )
(4)
Sh(2, n) = f n(n + l)C(n + 2) + C(2)C(n)
k=i
= ^(m, n) + C(m + n)
and
jt=i
= s h (2, n) + 2s fc (l, n + 1) + C(" + 2), (5)
where <th is denned below.
Bailey et al. (1994) subsequently considered sums of the
forms
oo
a h (m, n) = ^ (l + ~ + . . . + ±) " (* + l)" n (6)
E°° r i f~i) fe+i i m
[l-- + ... + ^— j (fc + l)-» (7)
fc=i
o fc (m,n) = 53(l+i+... + i) m (-i)*+»(* + I)"" (8)
fc = l
E^ / 1 f-l 1 ) fe + 1 \ m
J( = l
(9)
OO
fffc (m,„) = 53(l + -L + ... + -L) (* + !)-» (10)
fc = l
E^ / 1 (-l) fe + 1 \
{ 1 -^ + --- + -^~) {k + irn (11)
_ 2 n 53 C(n_fc)C(fc + 2)
fe =
ti-2 fc-1
(17)
s h (2,2n - 1) = ±(2n 2 - 7n - 3)C(2n + 1) + C(2)C(2n - 1)
n-2
n-2
+ |53C(2* + 1) 53 C(2j + l)C(2n - 1 - 2fc - 2j)
fc=l 3=1
(18)
tr fc (l,n) = a fc (l I n) (19)
<7 fc (2, 2n - 1) = -i(2n 2 +n+ l)£(2n + 1) + C(2)C(2n - 1)
n-l
+ ^ 2*C(* + l)C(2n - 2fc) (20)
fc=i
a h (m even,n odd) = M ( m n ) - 1 £( m + n) + C(m)C(n)
C(2j - l)C(m + n - 2j + 1)
(21)
CTh("T- odd,n even) = — ^
( ) + 1
C(m + n)
OO
a h (m,n) = ^ (l + ±- + . . . + ±-"j (-l) fe+1 (* + 1)"
♦EIO+O'
xC(2j-l)C(m + n-2j + l),
(22)
(12)
where (^) is a Binomial Coefficient. Explicit for-
mulas inferred using the PSLQ Algorithm include
* a {m,n) = 53 (l - J- + . .. + ^-^-) (- 1 ) fc+1 ( fe + !)"".
fc = l
(13)
s /l (2,2) = |C(4) + |[C(2)] 2
(23)
(24)
586
Euler Sum
Euler Totient Function
a/.(2,4) =
|C(6)
— 37
TV
|C(2)C(4) + |[C(2)] 3
- [C(3)] 2
[C(3)] 2
22680
¥C(5) + C(2)C(3)
«h(3,2)
Sh (3,3) = -f|C(6) + 2[C(3)] 2
s h (3,4) = m C(7 ) - M C(3 ) C (4) + 2C(2)C(5)
Sh(3,6)
«fc(4,2)
s h (4,3)
s h (4,5)
= WC(9) - f C(4)C(5) - f C(3)C(6)
+ [C(3)] 3 + 3C(2)C(7)
= W<(6) + 3[C(3)] 2
= -TC(7) + fC(3)C(4)-5C(2)C(5)
= -fC(9) + fC(4)C(5) + fC(3)C(6)
- f[C(3)] 3 -
= 1 fi s C(7) + 33C(3)C(4) + ^C(2)C(5)
7C(2)C(7)
s h (5,2)
fl/l (5,4) = 2§0C(») + 66<(4K(5)
^(3)C(6)
+
-C(2)C(7)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
5[C(3)] 3
(35)
Sh(6,3)
«h(7,2)
= -3S|5C(9) - 243C(4)C(5) + ^C(3)C(6)
+ f[C(3)] 3 -^C(2)C(7)
(36)
L C(9) + ^FC(4)C(5) + ^CWCW
■56[C(3)] 3 +^C(2)C(7),
(37)
(38)
a«(2,2) = 6Li4(I) + |0n2) 4 - f C(4) + |C(2)(ln2)
*„(2,3) = 4Li 5 (±) - ^(ln2) 5 - gC(5) - ^C(4)ln2
+ iC(3)(ln2) 2 + |C(2)(ln2) 3 -!C(2)C(3),
(39)
s a (3,2) = -24Li B (f ) + 61n2Li 4 (±) + ^(ln2) 5 + ^££(5)
- ^C(4)ln2 + |C(2)(ln2) 3 + §C(2)C(3), (40)
where Li n is a POLYLOGARITHM, and C,{z) is the RlE-
MANN ZETA Function (Bailey and Plouffe). Of these,
only 5^(3,2), Sh(3, 3) and the identities for s a (m,n),
ah(m,n) and a a {m,n) have been rigorously established.
References
Bailey, D. and Plouffe, S. "Recognizing Numerical
Constants." http: //www. cecm. sfu. ca/organics/papers/
bailey/.
Bailey, D. H.; Borwein, J. M.: and Girgensohn, R. "Experi-
mental Evaluation of Euler Sums." Exper. Math. 3,17-30,
1994.
Berndt, B. C. Ramanujan's Notebooks: Part I. New York:
Springer- Verlag, 1985.
Borwein, D. and Borwein, J. M. "On an Intriguing Integral
and Some Series Related to C(4)*" Proc. Amer. Math. Soc.
123, 1191-1198, 1995.
Borwein, D.; Borwein, J. M.; and Girgensohn, R. "Explicit
Evaluation of Euler Sums." Proc. Edinburgh Math. Soc.
38, 277-294, 1995.
de Doelder, P. J. "On Some Series Containing V(x) - \I>(y)
and (*(#) — ^(y)) 2 for Certain Values of x and y." J.
Comp. Appl Math. 37, 125-141, 1991.
Euler's Sum of Powers Conjecture
Euler conjectured that at least n nth POWERS are re-
quired for n > 2 to provide a sum that is itself an nth
POWER. The conjecture was disproved by Lander and
Parkin (1967) with the counterexample
27 5 + 84 5 + HO 5 + 133 5 = 144 5 .
see also DlOPHANTINE EQUATION
References
Lander, L. J. and Parkin, T. R. "A Counterexample to Eu-
ler's Sum of Powers Conjecture." Math. Comput. 21,101-
103, 1967.
a*(2,2) = -2Li 4 (|) - ^(ln2) 4 + §§C(4) - K(3)ln2
+ K(2)(ln2) 2
(41)
a fc (2,3) = -4Li 5 (i) - 4(ln2)Li 4 (±) - ^(m2) 5 + ^C(5)
- K(3)(ln2) 2 + |C(2)(ln2) 3 + |C(2)C(3)
(42)
a h (3,2)^6Li 5 (|)-f6(ln2)Li 4 (f) + |(ln2) 5 -fC(5)
+ fC(3)(ln2) 2 -C(2)(ln2) 3 -f|C(2)C(3),
(43)
and
a Q (2,2) = -4Li 4 (4) - i(ln2) 4 + §C(4) + JC(3)(ln2)
-2C(2)(ln2) 2
(44)
a a (2,3) = 4(ln2)Li 4 (±) + ±(ln2) 5 -f§C(5)
+ ^C(4)(ln2)-C(2)(ln2) 3 + |C(2)C(3)
a o (3,2)=30Li 5 (f)-i(ln2) 6 -^C(5)
+ ^C(4)(ln2) + fC(3)(ln2) 2
3,3,
(45)
-fC(2)(ln2) 3 + |C(2)C(3)
(46)
Euler 's Theorem
A generalization of Fermat's Little Theorem. Euler
published a proof of the following more general theorem
in 1736. Let <p(n) denote the Totient Function. Then
. a* (n) = 1 (mod n)
for all a Relatively Prime to n.
see also CHINESE HYPOTHESIS, EULER'S DISPLACE-
MENT Theorem, Euler's Distribution Theorem,
Fermat's Little Theorem, Totient Function
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 21 and 23-25, 1993.
Euler Totient Function
see Totient Function
Euler's Totient Rule
Euler Zigzag Number 587
Euler's Totient Rule
The number of bases modp in which 1/p has cycle length
/ is the same as the number of Fractions 0/(p — 1),
l/(p - 1), . . . , (p - 2)/(p - 1) which have least DENOM-
INATOR /.
see also Totient Function
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 167-168, 1996.
Euler's Transform
A technique for SERIES CONVERGENCE IMPROVEMENT
which takes a convergent alternating series
/( — 1) o-k = Go — a\ + 0,2
for k E [2,n], where A n , k are EULERIAN NUMBERS.
1
1 1
1 4 1
1 11 11 1
1 26 66 26 1
1 57 302 302 57 1
1 120 1191 2416 1191 120 1.
The numbers 1, 1, 1, 1, 4, 1, 1, 11, 11, 1, . . . are Sloane's
A008292. Amazingly, the Z-TRANSFORMS of t n
(1)
T n z L ' T n z x-K> dx n
\z-e~' T )
into a series with more rapid convergence to the same
value to
_ ^ (-l) fc A fe q
k=0
where the FORWARD DIFFERENCE is defined by
a*.- £-(-ir(£)
CLk-
(2)
(3)
(Abramowitz and Stegun 1972; Beeler et at 1972, Item
120).
see also FORWARD 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. 16, 1972.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Euler Transformation
see Euler's Finite Difference Transformation,
Euler's Hypergeometric Transformations, Eu-
ler's Transform
Euler's Triangle
The triangle of numbers A n ^ given by
and the Recurrence Relation
A n+ i, k = kA n , k + (n + 2 - k)A ntk -i
are generators for Euler's triangle.
see also Clark's Triangle, Eulerian Number,
Leibniz Harmonic Triangle, Number Trian-
gle, Pascal's Triangle, Seidel-Entringer- Ar-
nold Triangle, Z-Transform
References
Sloane, N. J. A. Sequence A008292 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Euler Triangle Formula
Let O and J be the ClRCUMCENTER and Incenter of a
Triangle with Circumradius R and Inradius r. Let
d be the distance between O and I. Then
Euler Walk
see Eulerian Trail
Euler Zigzag Number
The number of ALTERNATING PERMUTATIONS for n ele-
ments is sometimes called an Euler zigzag number. De-
note the number of ALTERNATING PERMUTATIONS on
n elements for which the first element is k by E(n,k).
Then £(1,1) and
{0 for k > n or k < 1
E(n, k + 1) otherwise.
+E(n- l,n -k)
see also Alternating Permutation, Entringer
Number, Secant Number, Tangent Number
References
Ruskey, F. "Information of Alternating Permutations."
http:// sue . esc . uvic . ca / - cos / inf / perm /
Alternating . html.
Sloane, N. J. A. Sequence A000111/M1492 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
588
Eulerian Circuit
Evans Point
Eulerian Circuit
An Eulerian Trail which starts and ends at the same
Vertex. In other words, it is a Cycle which uses each
Edge exactly once. The term Eulerian Cycle is also
used synonymously with Eulerian circuit. For technical
reasons, Eulerian circuits are easier to study mathemat-
ically than are Hamiltonian Circuits. As a gener-
alization of the Konigsberg Bridge Problem, Euler
showed (without proof) that a Connected Graph has
an Eulerian circuit Iff it has no Vertices of Odd De-
gree.
see also Euler Graph, Hamiltonian Circuit
Eulerian Cycle
see Eulerian Circuit
Eulerian Integral of the First Kind
Legendre and Whittaker and Watson's (1990) term for
the Beta Function.
References
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, Jfth ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Eulerian Integral of the Second Kind
U(z
n)= f (l-^Yt z ' 1 dt = n z f (I-tJV^t
z(z + l)---(z + n)
Eulerian Number
The number of PERMUTATIONS of length n with k < n
RUNS, denoted (£), A ntk , or A{n,k). The Eulerian
numbers are given explicitly by the sum
3=0
Making the definition
&n,l = 1
bl,n = 1
(2)
(3)
together with the RECURRENCE RELATION
b n ,k = nb n ,k-i + kb n ~i,k (4)
for n > k then gives
^^=6fc,n-fc + l- (5)
The arrangement of the numbers into a triangle gives
Euler's Triangle, whose entries are 1, 1, 1, 1, 4, 1,
1, 11, 11, 1, ... (Sloane's A008292). Therefore, they
represent a sort of generalization of the BINOMIAL CO-
EFFICIENTS where the denning RECURRENCE RELATION
weights the sum of neighbors by their row and column
numbers, respectively.
The Eulerian numbers satisfy
£«>-•
(6)
Eulerian numbers also arise in the surprising context of
integrating the SlNC FUNCTION, and also in sums of the
form
E^^'-M't^EO""-*. m
where Li m (^) is the POLYLOGARITHM function.
see also Combination Lock, Euler Number, Eu-
ler's Triangle, Euler Zigzag Number, Polylog-
arithm, Sinc Function, Worpitzky's Identity, Z-
Transform
References
Carlitz, L. "Eulerian Numbers and Polynomials." Math.
Mag. 32, 247-260, 1959.
Foata, D. and Schutzenberger, M.-P. Theorie Geometrique
des Polynomes Euleriens. Berlin: Springer- Verlag, 1970.
Kimber, A. C. "Eulerian Numbers." Supplement to Encyclo-
pedia of Statistical Sciences. (Eds. S. Kotz, N. L. Johnson,
and C. B. Read). New York: Wiley, pp. 59-60, 1989.
Salama, I. A. and Kupper, L. L. "A Geometric Interpretation
for the Eulerian Numbers." Amer. Math. Monthly 93, 51-
52, 1986.
Sloane, N. J. A. Sequence A008292 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Eulerian Trail
A Walk on the Edges of a Graph which uses each
Edge exactly once. A Connected Graph has an Eu-
lerian trail IFF it has at most two VERTICES of ODD
Degree.
see also Eulerian Circuit
Eutactic Star
An orthogonal projection of a CROSS onto a 3-D Sub-
SPACE. It is said to be normalized if the CROSS vectors
are all of unit length.
see also Hadwiger's Principal Theorem
Evans Point
The intersection of the Gergonne Line and the Euler
LINE. It does not appear to have a simple parametric
representation.
References
Oldknow, A. "The Euler- Gergonne- Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
Eve
Evolute
589
Eve
see Apple, Root, Snake, Snake Eyes, Snake Oil
Method, Snake Polyiamond
Even Function
A function f(x) such that f(x) — f(—x). An even func-
tion times an Odd Function is odd.
Even Number
An Integer of the form N = 2n, where n is an Inte-
ger. The even numbers are therefore . . . , —4, —2, 0, 2,
4, 6, 8, 10, . . . (Sloane's A005843). Since the even num-
bers are integrally divisible by two, N = (mod 2) for
even N. An even number N for which N = 2 (mod 4)
is called a Singly Even Number, and an even num-
ber N for which N = (mod 4) is called a DOUBLY
Even Number. An integer which is not even is called
an Odd Number. The Generating Function of the
even numbers is
= 2x + 4x 2 + 6x 3 + 8x 4 +
(x-1)
see also Doubly Even Number, Even Function,
Odd Number, Singly Even Number
References
Sloane, N. J. A. Sequence A005843/M0985 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Eventually Periodic
A Periodic Sequence such as {1, 1, 1, 2, 1, 2, 1, 2,
1, 2, 1, 1, 2, 1, ...} which is periodic from some point
onwards.
see also Periodic Sequence
Everett's Formula
U = (1 - P)f0 + P/l + ^2*0 + *W? + E A 8i
+F A 6t + E*8t + F*8l + ..., (1)
for p G [0, 1], where S is the Central Difference and
E 2n = G 2 n — G2n + l = #2n — B 2n +1 (2)
F 2n = (?2ti+1 = Bin + #2n + l, (3)
where G k are the Coefficients from Gauss's Back-
ward Formula and Gauss's Forward Formula and
B k are the Coefficients from Bessel's Finite Dif-
ference Formula. The E k s and F k s also satisfy
for
E 2n (p) = F 2n (q)
F 2n (p) = E 2n (q),
q=l-p.
(4)
(5)
(6)
see also Bessel's Finite Difference 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. 880-881, 1972.
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., pp. 92-93, 1990.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 433, 1987.
Everett Interpolation
see Everett's Formula
Eversion
A curve on the unit sphere S 2 is an eversion if it has no
corners or cusps (but it may be self- intersecting). These
properties are guaranteed by requiring that the curve's
velocity never vanishes. A mapping cr : S 1 — > S 2 forms
an immersion of the Circle into the SPHERE Iff, for
ail e e R,
:*(0]
>0.
Smale (1958) showed it is possible to turn a SPHERE
inside out (Sphere Eversion) using eversion.
see also SPHERE EVERSION
References
Smale, S. "A Classification of Immersions of the Two-
Sphere." Trans. Amer. Math. Soc. 90, 281-290, 1958.
Evolute
An evolute is the locus of centers of curvature (the en-
velope) of a plane curve's normals. The original curve
is then said to be the INVOLUTE of its evolute. Given
a plane curve represented parametrically by (f(t),g(t)),
the equation of the evolute is given by
x = / — Rsinr
y = g + R cos r,
(i)
(2)
where (x y y) are the coordinates of the running point, R
is the Radius of Curvature
R
(f t2 + g t2 ) 3/2
(3)
f'g" - f'g'
and r is the angle between the unit TANGENT VECTOR
1
■4, X
T = — T =
and the x-AxiS,
X'l y/f' 2 +9 t2
cos T = T * X
sin r = T ■ y.
(4)
(5)
(6)
590 Exact Covering System
Exact Trilinear Coordinates
Combining gives
x = f-
y = 9 +
(f' 2 +9' 2 )9'
fa" - fa'
(/ ,2 + g ,2 V
fa" - f'g' '
(7)
(8)
The definition of the e volute of a curve is independent
of parameterization for any differentiate function (Gray
1993). If E is the evolute of a curve i", then / is said to
be the INVOLUTE of E. The centers of the OSCULATING
CIRCLES to a curve form the evolute to that curve (Gray
1993, p. 90).
The following table lists the evolutes of some common
curves.
Curve
Evolute
astroid
cardioid
cayley's sextic
circle
cycloid
deltoid
ellipse
epicycloid
hypocycloid
limagon
logarithmic spiral
nephroid
parabola
tractrix
astroid 2 times as large
cardioid 1/3 as large
nephroid
point (0, 0)
equal cycloid
deltoid 3 times as large
Lame curve
enlarged epicycloid
similar hypocycloid
circle catacaustic
for a point source
equal logarithmic spiral
nephroid 1/2 as large
Neile's parabola
catenary
see also Involute, Osculating Circle
References
Cayley, A. "On Evolutes of Parallel Curves." Quart. J. Pure
Appl. Math. 11, 183-199, 1871.
Dixon, R. "String Drawings." Ch. 2 in Mathographics. New
York: Dover, pp. 75-78, 1991.
Gray, A. "Evolutes." §5.1 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 76-80, 1993.
Jeffrey, H. M. "On the Evolutes of Cubic Curves." Quart. J.
Pure Appl. Math. 11, 78-81 and 145-155, 1871.
Lawrence, J. D, A Catalog of Special Plane Curves. New
York: Dover, pp. 40 and 202, 1972.
Lee, X. "Evolute." http: //www. best . com/-xah/Special
PlaneCurves_dir/Evolute_dir/e volute . html.
Lockwood, E. H. "Evolutes and Involutes." Ch. 21 in A Book
of Curves. Cambridge, England: Cambridge University
Press, pp. 166-171, 1967.
Yates, R. C. "Evolutes." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 86-92,
1952.
Exact Covering System
A system of congruences a* mod rii with 1 < i < k is
called a Covering System if every Integer y satisfies
y = di (mod n) for at least one value of i. A cover-
ing system in which each integer is covered by just one
congruence is called an exact covering system.
see also Covering System
References
Guy, R. K. "Exact Covering Systems." §F14 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 253-256, 1994.
Exact Differential
A differential of the form
df = P(x, y) dx + Q(x, y) dy
(1)
is exact (also called a Total Differential) if / df is
path-independent. This will be true if
d f = d i dx+ ify d y>
so P and Q must be of the form
But
dp _ d 2 f
dy dydx
dQ = d 2 f
dx dxdy '
dP dQ
dy dx '
(2)
(3)
(4)
(5)
(6)
see also Pfaffian Form, Inexact Differential
Exact Period
see Least Period
Exact Trilinear Coordinates
The Trilinear Coordinates a : /3 : 7 of a point P
relative to a TRIANGLE are PROPORTIONAL to the di-
rected distances a' : b' : c' from P to the side lines (i.e,
a' : b' : c' — ka : b' = k/3 : kj). Letting k be the
constant of proportionality,
k =
2A
aa + bf3 + cj *
where A is the Area of AABC and a, 6, and c are the
lengths of its sides. When the trilinears are chosen so
that k = 1, the coordinates are known as exact trilinear
coordinates.
see also TRILINEAR COORDINATES
Exactly One
Excentral Triangle 591
Exactly One
"Exactly one" means "one and only one," sometimes
also referred to as "JUST One." J. H. Conway has
also humorously suggested "onee" (one and only one)
by analogy with Iff (if and only if), "twoo" (two and
only two), and "threee" (three and only three). This
refinement is sometimes needed in formal mathematical
discourse because, for example, if you have two apples,
you also have one apple, but you do not have exactly
one apple.
In 2-valued LOGIC, exactly one is equivalent to the ex-
clusive or operator XOR,
P(E) XOR P(F) = P{E) + P(F) - 2P(E n F).
see also IFF, PRECISELY UNLESS, XOR
Exactly When
see IFF
Excenter
The center Ji of an ExciRCLE. There are three excen-
ters for a given TRIANGLE, denoted Ji, J2, Jz> The
Incenter J and excenters J; of a TRIANGLE are an
Orthocentric System.
OI 2 + OJi 2 + OJ 2 2 + OJ 3 2 =
12R 2
where O is the ClRCUMCENTER, Ji are the excenters,
and R is the ClRCUMRADIUS (Johnson 1929, p, 190),
Denote the MIDPOINTS of the original TRIANGLE Mi,
Ma, and M 3 . Then the lines J1M1, J2M2, and J3M3
intersect in a point known as the MlTTENPUNKT.
see also CENTROID (ORTHOCENTRIC SYSTEM), EXCEN-
ter-excenter circle, excentral triangle, ex-
circle, Incenter, Mittenpunkt
References
Dixon, R, Mathographics. New York: Dover, pp. 58-59, 1991.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Excenter- Excenter Circle
Given a Triangle AAiA 2 A 3 , the points Ai, /, and J\
lie on a line, where I is the INCENTER and J\ is the EX-
CENTER corresponding to A\. Furthermore, the circle
with J2J3 as the diameter has Q as its center, where
P is the intersection of A± Ji with the ClRCUMCIRCLE
of Ai .42^4.3 and Q is the point opposite P on the ClR-
CUMCIRCLE. The circle with diameter J 2 Jz also passes
through A2 and A3 and has radius
r = |ai csc(|ai) = 2i^cos(|o:i).
It arises because the points J, Ji, J2, and J3 form an
Orthocentric System.
see also EXCENTER, INCENTER-EXCENTER CIRCLE,
Orthocentric System
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 185-186, 1929.
Excentral Triangle
The Triangle J = AJ1J2J3 with Vertices corre-
sponding to the Excenters of a given Triangle A,
also called the Tritangent Triangle.
Beginning with an arbitrary TRIANGLE A, find the ex-
central triangle J, Then find the excentral triangle J'of
that TRIANGLE, and so on. Then the resulting TRIAN-
GLE J (oo) approaches an EQUILATERAL TRIANGLE.
592
Excess
Exclusive Or
Call T the TRIANGLE tangent externally to the EXCIR-
CLES of A. Then the INCENTER It of K coincides with
the ClRCUMCENTER Cj of TRIANGLE AJi J 2 J3, where
Ji are the EXCENTERS of A. The INRADIUS tt of the
Incircle of T is
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 11-13, 1967.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 176-177 and 182-194, 1929.
r T = 2R + r = ±(r + n + r 2 + r 3 ),
where R is the CIRCUMRADIUS of A, r is the INRADIUS,
and n are the Exradii (Johnson 1929, p. 192).
see also EXCENTER, EXCENTER-EXCENTER CIRCLE,
EXCIRCLE, MlTTENPUNKT
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Excess
see KURTOSIS
Excess Coefficient
see KURTOSIS
Excessive Number
see Abundant Number
Excircle
Given a TRIANGLE, extend two nonadjacent sides. The
CIRCLE tangent to these two lines and to the other side
of the TRIANGLE is called an ESCRIBED CIRCLE, or ex-
circle. The Center Ji of the excircle is called the Ex-
CENTER and lies on the external ANGLE BISECTOR of
the opposite ANGLE. Every TRIANGLE has three excir-
cles, and the Trilinear Coordinates of the Excen-
TERS are -1 : 1 : 1, 1 : -1 : 1, and 1 : 1 : -1. The
Radius n of the excircle i is called its Exradius.
Given a TRIANGLE with INRADIUS
Altitudes of the excircles, and r,
Exradii). Then
r, let hi be the
their RADII (the
111 1111
ir + TT + ^r^ — + - + — = -
h\ ri2 ris ri V2 r$ r
(Johnson 1929, p. 189).
see also Excenter, Excenter-Excenter Circle,
Excentral Triangle, Feuerbach's Theorem,
Nagel Point, Triangle Transformation Princi-
ple
Excision Axiom
One of the Eilenberg-Steenrod Axioms which states
that, if X is a SPACE with SUBSPACES A and U such that
the CLOSURE of A is contained in the interior of (7, then
the Inclusion Map (X U,A U) -¥ (X, A) induces an
isomorphism H n (X U^AU)—^ H n (X^A).
Excluded Middle Law
A law in (2- valued) LOGIC which states there is no third
alternative to Truth or FALSEHOOD. In other words,
every statement must be either A or not- A This fact no
longer holds in Three- Valued Logic or Fuzzy Logic
Excludent
A method which can be used to solve any QUADRATIC
Congruence. This technique relies on the fact that
solving
x 2 = b (mod p)
is equivalent to finding a value y such that
b + py = x 2 .
Pick a few small moduli m. If y mod m does not make
b-\-py a quadratic residue of m, then this value of y may
be excluded. Furthermore, values of y > p/4 are never
necessary.
Excludent Factorization Method
Also known as the difference of squares. It was first
used by Fermat and improved by Gauss. Gauss looked
for Integers x and y satisfying
y 2 =x 2 - N (mod E)
for various moduli E. This allowed the exclusion of
many potential factors. This method works best when
factors are of approximately the same size, so it is some-
times better to attempt mN for some suitably chosen
value of m.
see also Prime Factorization Algorithms
Exclusive Or
see XOR
Exeter Point
Exp 593
Exeter Point
Define A' to be the point (other than the VERTEX A)
where the Median through A meets the Circumcir-
CLE of ABC, and define B f and C similarly. Then the
Exeter point is the PERSPECTIVE CENTER of the TRI-
ANGLE A'B'C and the Tangential Triangle, It has
Triangle Center Function
a = a(6 4 -he — a ).
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Exeter Point." http://www.evansville .
edu/-ck6/t centers/recent /exeter. html.
Kimberling, C. and Lossers, O. P. "Problem 6557 and Solu-
tion." Amer. Math. Monthly 97, 535-537, 1990.
Exhaustion Method
The method of exhaustion was a iNTEGRAL-like limiting
process used by Archimedes to compute the AREA and
Volume of 2-D Lamina and 3-D Solids.
see also Integral, Limit
Existence
If at least one solution can be determined for a given
problem, a solution to that problem is said to exist. Fre-
quently, mathematicians seek to prove the existence of
solutions and then investigate their UNIQUENESS.
see also Exists, Unique
Existential Closure
A class of processes which attempt to round off a domain
and simplify its theory by adjoining elements.
see also Model Completion
References
Kenneth, M. "Domain Extension and the Philosophy of
Mathematics." J. Philos. 86, 553-562, 1989.
Exists
If there exists an A, this is written 3A. Similarly, A
does not exit is written flA.
see also EXISTENCE, FOR ALL, QUANTIFIER
Exmedian
The line through the Vertex of a TRIANGLE which is
Parallel to the opposite side.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 176, 1929.
Exmedian Point
The point of intersection of two Exmedians.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 176, 1929.
Exogenous Variable
An economic variable that is related to other economic
variables and determines their equilibrium levels.
see also ENDOGENOUS VARIABLE
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 458, 1980.
Exotic M 4
Donaldson (1983) showed there exists an exotic smooth
Differential Structure on M 4 . Donaldson's result
has been extended to there being precisely a CONTIN-
UUM of nondiffeomorphic DIFFERENTIAL STRUCTURES
onR 4 .
see also EXOTIC SPHERE
References
Donaldson, S. K. "Self-Dual Connections and the Topology
of Smooth 4-Manifold." Bull Amer. Math. Soc. 8, 81-83,
1983.
Monastyrsky, M. Modern Mathematics in the Light of the
Fields Medals. Wellesley, MA: A. K. Peters, 1997.
Exotic Sphere
Milnor (1963) found more than one smooth struc-
ture on the 7-D HYPERSPHERE. Generalizations have
subsequently been found in other dimensions. Using
SURGERY theory, it is possible to relate the number of
DlFFEOMORPHISM classes of exotic spheres to higher ho-
motopy groups of spheres (Kosinski 1992). Kervaire and
Milnor (1963) computed a list of the number N(d) of dis-
tinct (up to DlFFEOMORPHISM) DIFFERENTIAL STRUC-
TURES on spheres indexed by the DIMENSION d of the
sphere. For d = 1, 2, . . . , assuming the PoiNCARE CON-
JECTURE, they are 1, 1, 1, > 1, 1, 1, 28, 2, 8, 6, 992,
1, 3, 2, 16256, 2, 16, 16, ... (Sloane's A001676). The
status of d = 4 is still unresolved: at least one exotic
structure exists, but it is not known if others do as well.
The only exotic Euclidean spaces are a CONTINUUM of
Exotic
structures.
see also Exotic E , Hypersphere
References
Kervaire, M. A. and Milnor, J. W. "Groups of Homotopy
Spheres: I." Ann. Math. 77, 504-537, 1963.
Kosinski, A. A. §X.6 in Differential Manifolds. Boston, MA:
Academic Press, 1992.
Milnor, J. "Topological Manifolds and Smooth Manifolds."
Proc. Intemat. Congr. Mathematicians (Stockholm, 1962)
Djursholm: Inst. Mittag-Leffler, pp. 132-138, 1963.
Milnor, J. W. and Stasheff, J. D. Characteristic Classes.
Princeton, NJ: Princeton University Press, 1973.
Monastyrsky, M. Modem Mathematics in the Light of the
Fields Medals. Wellesley, MA: A. K. Peters, 1997.
Novikov, S. P. (Ed.). Topology I. New York: Springer-Verlag,
1996.
Sloane, N. J. A. Sequence A001676/M5197 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Exp
see Exponential Function
594 Expansion
Exponential Distribution
Expansion
An Affine Transformation in which the scale is in-
creased. It is the opposite of a Dilation (Contrac-
tion).
see also DILATION
Expansive
Let (f> be a MAP. Then <j) is expansive if the DISTANCE
d(<f> n x, <t> n y) < S for all n € Z, then x = y. Equivalently,
<fi is expansive if the orbits of two points x and y are
always very close.
Expectation Value
For one discrete variable,
X
For one continuous variable,
(f(x)) = Jf(x)P(x)dx.
(1)
(2)
The expectation value satisfies
{ax + by) = a{x) + b{y) (3)
(a) = a (4)
(£*) = £<*>■ < 5 )
For multiple discrete variables
(f{x u ...,x n )} = ^ P(zi,...,zn)- (6)
xi,...,x n
For multiple continuous variables
(f(x u ... y x n ))
— lf(x 1 ,...,Xn)P(xi i ... J X n )dx 1 ---dx n . (7)
The (multiple) expectation value satisfies
{{x - fj, x )(y - fly)} = {xy - fi x y - fax + fj, x fi y )
= (xy) - tl x fi y - [lyVx + Vxtty
= (xy) - (x) (y) , (8)
where //» is the MEAN for the variable L
see also MEAN
Experimental Design
see Design
Exploration Problem
see Jeep Problem
Exponent
The POWER p in an expression a p .
Exponent Laws
The laws governing the combination of EXPONENTS
(Powers) are
rn n m m+n
X - X — X
(i)
rn
x m — n
= X
x n
(2)
/ m\n mrx
(x ) = X
(3)
(xy) m = x m y m
(4)
(A n -*1
(5)
\yj y n
x n
(6)
/ \ ~ n , ■. -r,
-) =m n .
(7)
.y,
where quantities in the DENOMINATOR are taken to be
nonzero. Special cases include
and
x — x
i
X = 1
(8)
(9)
for x ^ 0. The definition 0° = 1 is sometimes used
to simplify formulas, but it should be kept in mind that
this equality is a definition and not a fundamental math-
ematical truth.
see also EXPONENT, POWER
Exponent Vector
Let pi denote the ith PRIME, and write
'IK*
Then the exponent vector is v(m) = (fi, f2, . . .)•
see also Dixon's Factorization Method
References
Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996,
Exponential Digital Invariant
see Narcissistic Number
Exponential Distribution
Exponential Distribution
Given a PoiSSON DISTRIBUTION with rate of change
A, the distribution of waiting times between successive
changes (with k = 0) is
D{x) = P(X <x) = l- P(X > x)
(As)°e- A * A ,
0! "
(1)
(2)
e~ Xx dx
P{x) = D'(x) = \e- Xx ,
which is normalized since
/ P(x) dx = X
Jo Jo
= -[c- A -]S° = -(0-l) = l. (3)
This is the only MEMORYLESS RANDOM DISTRIBU-
TION. Define the MEAN waiting time between successive
changes as 6 = A -1 . Then
P( X) = {I
\e~ x/e x>0
(4)
x < 0.
The Moment-Generating Function is
M(t) = f°° e tx (i) e-" e dx=- ^ e -^- et) * /6 dx
~(l-0t)x/9
1
M f (t) -
M"(t) -
1-Ot J 1-Ot
(i - oty
2d 2
(i - ety '
(5)
(6)
(7)
R(t) = In M(t) = - ln(l - Ot) (8)
R'(t) =
R"(t) =
l-et
o 2
(1 - Ot) 2
H = R'{0) = 8
a 2 =R"(0) = 2 .
The SKEWNESS and KURTOSIS are given by
7i = 2
72 = 6.
(9)
(10)
(11)
(12)
(13)
(14)
The Mean and Variance can also be computed directly
/•OO -. POO
(x)= / P(x)dx=- / xe~ x/3 dx. (15)
Jo *Jq
Use the integral
/
xe ax dx — — (aa; — 1)
(16)
Exponential Distribution 595
to obtain
<«> = !
-'•(-f)i:
= -s(0-l) = s.
Now, to find
<*>-tf
x 2 e- x/s ^,
use the integral
fx 2 e-* /a
dx — — r(2 — 2az + a 2 x 2 )
a 6
(17)
(18)
(19)
<* 2 >
-x/s
LR) v s s
H
= -s 2 (0-2) = 2s 2
giving
<r 2 = <z 2 ) - (x) 2
o 2 2 2
cr = y var(x) = s.
(20)
(21)
(22)
If a generalized exponential probability function is de-
fined by
JW)(*) = ie-**-"'" 5 , (23)
then the CHARACTERISTIC FUNCTION is
m = —
ipt 1
(24)
and the Mean, Variance, Skewness, and Kurtosis
are
Ai = a + /3
2
-/3 2
7i
= 2
72
= 6.
(25)
(26)
(27)
(28)
see also DOUBLE EXPONENTIAL DISTRIBUTION
References
Balakrishnan, N. and Basu, A. P. The Exponential Distri-
bution: Theory, Methods, and Applications. New York:
Gordon and Breach, 1996.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 534-535, 1987.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 119, 1992.
596 Exponential Divisor
Exponential Integral
Exponential Divisor
see e-DlVlSOR
Exponential Function
7
6
5
4
3
2
The exponential function is defined by
exp(;c) = e x , (1)
where e is the constant 2.718. ... It satisfies the identity
exp(a; + y) = exp(z) exp(y).
(2)
If z = x + iy,
If
then
e z = e x+iy = e x e iy = e x {cosy + ismy). (3)
a + bi = e x + iy , (4)
y = tan -1 (-) (5)
x = In i b esc tan~ ( — ) \ >
= In < a sec tan -1 f - 1 \> . (6)
IExp z |
Re[z? 2
The above plot shows the function e 1 '*.
see also EULER FORMULA, EXPONENTIAL RAMP, FOUR-
IER Transform — Exponential Function, Sigmoid
Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Exponential
Function." §4.2 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, p. 69-71, 1972.
Fischer, G. (Ed.). Plates 127-128 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 124-125, 1986.
Spanier, J. and Oldham, K. B. "The Exponential Function
exp(6x + c)" and "Exponentials of Powers exp(— aa;")."
Chs. 26-27 in An Atlas of Functions. Washington, DC:
Hemisphere, pp. 233-261, 1987.
Yates, R. C. "Exponential Curves." A Handbook on Curves
and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 86-97, 1952.
Exponential Function (Truncated)
see Exponential Sum Function
Exponential Inequality
For c < 1,
x c < l + c(a- 1).
For c > 1,
x c > l + c(ar- T).
Exponential Integral
Let Ei(x) be the £?„,- FUNCTION with n = 1,
Ei(x)
= r e~ tx dt = r e~ u du
' k t ~k
(1)
Then define the exponential integral ei(x) by
Ei(a) = -ei(-aO, ( 2 )
where the retention of the — ei(— x) NOTATION is a his-
torical artifact. Then ei(x) is given by the integral
ei(x)
r e-'dt
J — x
(3)
This function is given by the Mathematical (Wolfram
Research, Champaign, IL) function ExpIntegralEifx].
The exponential integral can also be written
ei(ix) = ci(x) + isi(x), (4)
where ci(z) and si(ar) are Cosine and Sine Integral.
Exponential Map
The real ROOT of the exponential integral occurs at
0.37250741078. . . , which is not known to be expressi-
ble in terms of other standard constants. The quantity
-eei(-l) = 0.596347362... is known as the Gompertz
Constant.
see also COSINE INTEGRAL, £ n -FUNCTION, GOMPERTZ
Constant, Sine Integral
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 566-568, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 434-435, 1953.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Exponential Integrals." §6.3 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 215-219, 1992.
Spanier, J. and Oldham, K. B. "The Exponential Integral
Ei(a:) and Related Functions." Ch. 37 in An Atlas of Func-
tions. Washington, DC: Hemisphere, pp. 351-360, 1987.
Exponential Map
On a Lie Group, exp is a Map from the Lie Algebra
to its Lie Group. If you think of the Lie Algebra as
the Tangent Space to the identity of the Lie Group,
exp(t;) is defined to be h(l), where h is the unique LIE
Group Homeomorphism from the Real Numbers to
the Lie Group such that its velocity at time is v.
On a RlEMANNlAN MANIFOLD, exp is a Map from the
Tangent Bundle of the Manifold to the Manifold,
and exp(ij) is denned to be /i(l), where h is the unique
Geodesic traveling through the base-point of v such
that its velocity at time is v.
The three notions of exp (exp from COMPLEX ANALY-
SIS, exp from LlE GROUPS, and exp from Riemannian
geometry) are all linked together, the strongest link be-
ing between the LIE GROUPS and Riemannian geometry
definition. If G is a compact LIE GROUP, it admits a left
and right invariant RIEMANNIAN METRIC. With respect
to that metric, the two exp maps agree on their common
domain. In other words, one-parameter subgroups are
geodesies. In the case of the Manifold S 1 , the Cir-
cle, if we think of the tangent space to 1 as being the
Imaginary axis (y-Axis) in the Complex Plane, then
ex P Riemannian geometry ( V ) ~ eX PLie Groups \ V )
= CXp complex ana iy S j s [V J,
and so the three concepts of the exponential all agree in
this case.
see also Exponential Function
Exponential Matrix
see Matrix Exponential
Exponential Sum Formulas 597
Exponential Ramp
The curve
-i ax
■ = 1 - e .
see also Exponential Function, Sigmoid Function
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 158, 1993.
Exponential Sum Formulas
N-l i _ ^ Nx _ e iNx/2 / e -iNx/2 _ e iNx/2\
EiNx _ i — e
1 — q}X
_ e ix/2 ( e -ix/2 _ e ix/2\
sm(±Nx) ix(JV -i)/2
• (i \ e >
sin^a)
where
AT-l
— r
r
El -'
n~0
has been used. Similarly,
(i)
(2)
AT-l
En inx 1 p e
P ~ 1 - pe*
1 _ p x e iNx _ (l-p N e iNx )(l-pe~ ix )
(1 - pe ix )(l - pe~ ix )
pe z
1 _ p N e iN X _ pe -ix ^ pN + l e ix(N-l)
1 -p(e ix +e~ ix ) +p 2
p N + l e ix(N-l) _ p N e iNx + 1 _ pe ~i*
1 — 2p cos x 4- p 2
(3)
This gives
1 — pe
En inx v \ ^ n inx *■ P e
p e = lim y p e = ■ — 5-.
JV-kx) ^— ' 1 — 2p cos x + p
n=0 n=0
(4)
By looking at the Real and Imaginary Parts of these
FORMULAS, sums involving sines and cosines can be ob-
tained.
598 Exponential Sum Function
Exponential Sum Function
es n (x) = exp n (x) = ^
see also Gamma Function
Exradius
Exsecant
x
m!
The Radius of an Excircle. Let a Triangle have
exradius r a (sometimes denoted p a ), opposite side of
length a, AREA A, and SEMIPERIMETER s. Then
\s — a)
_ s(s — c)(s — b)
(i)
. n (2)
s — a
= 4i?sin(|ai)cos(|a2)cos(|a 3 ) (3)
(Johnson 1929, p. 189) where R is the ClRCUMRADIUS.
Let r be the INRADIUS, then
AR = r a + n + r c — r
T a Tb T c T
(4)
(5)
(6)
rr a Tbr c = A .
Some fascinating Formulas due to Feuerbach are
7*27*3 + TzVi 4" 7*1 T3 = S (7)
7 , (7 , 2 r 3 + r 3 ri + rir > 2 ) = sA = 7*1^7*3 (8)
r(ri + 7*2 + 7*3) = a 2 a 3 + a 3 ai + aia 2 - s 2 (9)
tti +rr*2 H-T-rs +rir 2 +7*2r3 +r3ri = a 2 a3 -\-a$a\ 4-aia 2
(10)
T'27'3 + VzT\ + rir 2 — 7Ti — 7T2 — Tr3 — ^ (<^1 + <^2 + ^3 )
(11)
(Johnson 1929, pp. 190-191).
see also CIRCLE, ClRCUMRADIUS, EXCIRCLE, INRADIUS,
Radius
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Mackay, J. S. "Formulas Connected with the Radii of the In-
circle and Excircles of a Triangle." Proc. Edinburgh Math.
Soc. 12, 86-105.
Mackay, J. S. "Formulas Connected with the Radii of the In-
circle and Excircles of a Triangle." Proc. Edinburgh Math.
Soc. 13, 103-104.
Extension Problem
■i,
exsec x = sec x
where sec a* is the Secant.
see also Coversine, Haversine, Secant, Versine
References
Abramowitz, M. and Stegun, C A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 78, 1972.
Extended Cycloid
see Prolate Cycloid
Extended Goldbach Conjecture
see Goldbach Conjecture
Extended Greatest Common Divisor
see Greatest Common Divisor
Extended Mean- Value Theorem
Let the functions / and g be DlFFERENTIABLE on the
Open Interval (a, 6) and Continuous on the Closed
Interval [a, 6]. If g'(x) ^ for any x e (a,b), then
there is at least one point c 6 (a, b) such that
f'(c) = f(b)-f(a)
g'(c) g(b)-g(a)'
see also MEAN- VALUE THEOREM
Extended Riemann Hypothesis
The first quadratic nonresidue mod p of a number is
always less than 2(lnp) 2 .
see also RIEMANN HYPOTHESIS
References
Bach, E. Analytic Methods in the Analysis and Design
of Number- Theoretic Algorithms. Cambridge, MA: MIT
Press, 1985.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 295, 1991.
Extension
The definition of a SET by enumerating its members.
An extensional definition can always be reduced to an
Intentional one.
see also Intension
References
Russell, B. "Definition of Number." Introduction to Mathe-
matical Philosophy. New York: Simon and Schuster, 1971.
Extension Problem
Given a SuBSPACE A of a SPACE X and a MAP from A
to a Space Y, is it possible to extend that Map to a
Map from X to Y?
see also LIFTING PROBLEM
Extensions Calculus
Exterior Derivative
599
Extensions Calculus
see Exterior Algebra
Extent
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.
see also N-Cluster
Exterior
That portion of a region lying "outside" a specified
boundary.
see also INTERIOR
Exterior Algebra
The Algebra of the Exterior Product, also called
an Alternating Algebra or Grassmann Algebra.
The study of exterior algebra is also called AuSDEHN-
ungslehre and Extensions Calculus. Exterior al-
gebras are GRADED ALGEBRAS.
In particular, the exterior algebra of a Vector Space
is the DIRECT Sum over k in the natural numbers of the
Vector Spaces of alternating fc-forms on that Vector
Space. The product on this algebra is then the wedge
product of forms. The exterior algebra for a VECTOR
Space V is constructed by forming monomials u, v /\w,
x A y A z, etc., where u, v, w, x, y> and z are vectors
in V and A is asymmetric multiplication. The sums
formed from linear combinations of the MONOMIALS are
the elements of an exterior algebra.
References
Forder, H. G. The Calculus of Extension. Cambridge, Eng-
land: Cambridge University Press, 1941.
Lounesto, P. "Counterexamples to Theorems Published and
Proved in Recent Literature on Clifford Algebras, Spinors,
Spin Groups, and the Exterior Algebra." http://www.hit.
f i/-lounesto/counterexamples.htm.
Exterior Angle Bisector
interior angle
bisector
exterior angle
bisection
The exterior bisector of an ANGLE is the LINE or Line
Segment which cuts it into two equal Angles on the
opposite "side" as the ANGLE.
For a Triangle, the exterior angle bisector bisects the
Supplementary Angle at a given Vertex. It also di-
vides the opposite side externally in the ratio of adjacent
sides.
see also ANGLE BISECTOR, ISODYNAMIC POINTS
Exterior Angle Theorem
In any TRIANGLE, if one of the sides is extended, the
exterior angle is greater than both the interior and op-
posite angles.
References
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, p. 41, 1990.
Exterior Derivative
Consider a DIFFERENTIAL fc-FORM
oj 1 — b\ dx\ + &2 dx2.
Then its exterior derivative is
duj 1 = dbi A dx\ + d&2 A dx 2%
(i)
(2)
where A is the WEDGE PRODUCT. Similarly, consider
u> — 6i(xi, X2) dx± + b 2 (xi, #2) dx2~
(3)
Then
du; 1 = dbi A dx\ + dbi A dx2
= -w—dxi + ^—dx 2
\ OX\ OX2 /
db 2 , , db 2 j
dxi + — — aX2
0x2
A dx\
(db 2
\dX!
) A dx2 .
Denote the exterior derivative by
Dt = — A t
ox
Then for a 0-form t,
m*
dt
dx»"
for a 1-form t,
lDt ) =I(*!L_0V\
and for a 2-form t,
(Dt) ijk
x fdt 2 3 , dt 3 i . dti 2 \
jCijfc
+
3 l3k Kdx 1 ' dx 2 ' dx 3 J
where e ijk is the PERMUTATION TENSOR.
The second exterior derivative is
d ( d \ ( d
ox \ox J \ox ox J
(4)
(5)
(6)
(7)
(8)
(9)
which is known as Poincare's Lemma.
see also DIFFERENTIAL fc-FORM, POINCARE'S LEMMA,
Wedge Product
600
Exterior Dimension
Extreme Value Distribution
Exterior Dimension
A type of Dimension which can be used to characterize
Fat Fractals.
see also Fat Fractal
References
Grebogi, C; McDonald, S. W.; Ott, E.; and Yorke, J. A.
"Exterior Dimension of Fat Fractals." Phys. Let. A 110,
1-4, 1985.
Grebogi, C.; McDonald, S. W.; Ott, E.; and Yorke, J. A,
Erratum to "Exterior Dimension of Fat Fractals." Phys.
Let. A 113, 495, 1986.
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, p. 98, 1993.
Exterior Product
see Wedge Product
Exterior Snowflake
A Fractal.
see also Flowsnake Fractal, Koch Antisnow-
flake, Koch Snowflake, Pentaflake
References
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 193-195, 1991.
^ Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
Extra Strong Lucas Pseudoprime
Given the LUCAS SEQUENCE U n {b,-l) and 14(6,-1),
define A = b 2 — 4. Then an extra strong Lucas pseu-
doprime to the base b is a Composite Number n =
2 r s 4- (A/n), where s is Odd and (n, 2A) = 1 such
that either U s = (mod n) and V s = ±2 (mod n), or
V 2 t 3 = (mod n) for some t with < t < r — 1. An
extra strong Lucas pseudoprime is a STRONG LUCAS
Pseudoprime with parameters (6, -1). Composite n
are extra strong pseudoprimes for at most 1/8 of possi-
ble bases (Grantham 1997).
see also LUCAS PSEUDOPRIME, STRONG LUCAS PSEU-
DOPRIME
References
Grantham, J. "Frobenius Pseudoprimes." http: //www.
dark . net /pub/grantham/pseudo/pseudo . ps
Grantham, J. "A Frobenius Probable Prime Test with
High Confidence." 1997. http://www.clark.net/pub/
grantham/pseudo/pseudo2.ps
Jones, J. P. and Mo, Z. "A New Primality Test Using Lucas
Sequences." Preprint.
Extrapolation
see RICHARDSON EXTRAPOLATION
Extremal Coloring
see Extremal Graph
Extremal Graph
A two-coloring of a Complete Graph K n of n nodes
which contains exactly the number N = (R + -B)min
of Monochromatic Forced Triangles and no more
(i.e., a minimum of R + B where R and B are the num-
bers of red and blue Triangles). Goodman (1959)
showed that for an extremal graph,
N(n) = J J
^ 3
\m(m- l)(m-2)
2m(m- l)(4m + 1)
32m(m+ l)(4m- 1)
for n = 2m
for n = Am + 1
for n = 4m + 3.
This is sometimes known as GOODMAN'S FORMULA.
Schwenk (1972) rewrote it in the form
N(n)=Q-[InL|(n-l) 2 jj,
sometimes known as Schwenk's Formula, where [zj
is the FLOOR Function. The first few values of N(n)
for n = 1, 2, . . . are 0, 0, 0, 0, 0, 2, 4, 8, 12, 20, 28, 40,
52, 70, 88, . . . (Sloane's A014557).
see also BlCHROMATIC GRAPH, BLUE-EMPTY GRAPH,
Goodman's Formula, Monochromatic Forced
Triangle, Schwenk's Formula
References
Goodman, A. W. "On Sets of Acquaintances and Strangers
at Any Party." Amer. Math. Monthly 66, 778-783, 1959.
Schwenk, A. J. "Acquaintance Party Problem." Amer. Math.
Monthly 79, 1113-1117, 1972.
Sloane, N. J. A. Sequence A014557 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Extremals
A field of extremals is a plane region which is Simply
CONNECTED by a one-parameter family of extremals.
The concept was invented by Weierstraft.
Extreme and Mean Ratio
see Golden Mean
Extreme Value Distribution
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let M n denote the "extreme" (i.e., largest) ORDER
Statistic X^ for a distribution of n elements Xi taken
from a continuous UNIFORM DISTRIBUTION. Then the
distribution of the M n is
r o if x < o
P(M n < x) = I x n if < x < 1
(1)
[l if £B > 1,
and the Mean and VARIANCE are
n
(2)
M ~ n + 1
2 n
(3)
a ~ (n+l) 2 (n + 2)'
Extreme Value Distribution
Extremum
601
If Xi are taken from a STANDARD NORMAL DISTRIBU-
TION, then its cumulative distribution is
F(
x) = -i=y* e- t2 / 2 d<= ! + <&(*),
(4)
where $(x) is the Normal Distribution Function.
The probability distribution of M n is then
P(M n < x) = [F(x)] n
V2i
f
J — c
[*"(*)]
n-1 -t^/2
dt.
(5)
The Mean fi(n) and Variance cr 2 (n) are expressible in
closed form for small n,
M(2) = 4=
M3) =
20F
(6)
(7)
(8)
0)
(10)
and
<r 2 (l)
<T 2 (2) = 1 •
<x 2 (3) =
4tt - 9 + 2y/3
4tt
^ 2 (4) = 1+ — -[M4)]
<r 2 (5) = l +
5V3 5A/3
47T
2tt 2
(11)
(12)
(13)
(14)
sin-^D-MS)] 2 . (15)
No exact expression is known for fi(6) or cr 2 (6), but there
is an equation connecting them
[M6)] 2 + ^(6) = l + -^ +
573 15\/3
47T
2tt 2
sin- 1 ^). (16)
An analog to the Central Limit THEOREM states that
the asymptotic normalized distribution of M n satisfies
one of the three distributions
(17)
„, ^ (0 if y < .,„.
P ^ = {«p(-y-) < 18 >
P(») = {f
P(y) = eM-e~ V )
. exp(-y~
exp[-(-y) a ] ify<0
if y > 0,
(19)
also known as GuMBEL, Frechet, and WEIBULL DISTRI-
BUTIONS, respectively.
see also Fisher-Tippett Distribution, Order
Statistic
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.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/extval/extval.html.
Gibbons, J. D. Nonparametric Statistical Inference. New-
York: McGraw-Hill, 1971.
Extreme Value Theorem
If a function / is continuous on a closed interval [a, 6],
then / has both a MAXIMUM and a MINIMUM on [a, b].
If / has an extreme value on an open interval (a, 6),
then the extreme value occurs at a Critical Point.
This theorem is sometimes also called the WeierstraB
Extreme Value Theorem.
Extremum
A Maximum or Minimum. An extremum may be Lo-
cal (a.k.a. a RELATIVE EXTREMUM; an extremum in a
given region which is not the overall Maximum or Min-
imum) or Global. Functions with many extrema can
be very difficult to Graph. Notorious examples include
the functions cos(l/x) and sin(l/x) near x =
and sin(e +9 ) near and 1.
The latter has
^_ 1
t 2
e
7T
+ 1 = 19058 - 2579 + 1 = 16480
extrema in the CLOSED INTERVAL [0,1] (Mulcahy 1996).
see also GLOBAL EXTREMUM, GLOBAL MAXIMUM,
Global Minimum, Kuhn-Tucker Theorem, La-
grange Multiplier, Local Extremum, Local
Maximum, Local Minimum, Maximum, Minimum
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.
Mulcahy, C. "Plotting and Scheming with Wavelets." Math.
Mag. 69, 323-343, 1996.
Tikhomirov, V. M. Stories About Maxima and Minima.
Providence, RI: Amer. Math. Soc, 1991.
602 Extremum Test Extrinsic Curvature
Extremum Test
Consider a function f(x) in 1-D. If f(x) has a relative
extremum at xo, then either f'(xo) = or / is not
Differentiable at xo. Either the first or second De-
rivative tests may be used to locate relative extrema
of the first kind.
A Necessary condition for f(x) to have a Minimum
(Maximum) at x is
/'(xo) - 0,
and
f"(x ) > (/"(xo) < 0).
A Sufficient condition is f(xo) = and f"(xo) >
(/"(so) < 0). Let f(x ) = 0, /"(xo) = 0, . . . ,
/< n >(so) = 0, but f (n+1) (x ) # 0. Then f(x) has a Rel-
ative Maximum at x if n is Odd and / (n+1) (x ) < 0,
and f(x) has a Relative Minimum at x if n is Odd
and / (n+1) (x ) > 0. There is a Saddle Point at x if
n is Even.
see also Extremum, First Derivative Test, Rela-
tive Maximum, Relative Minimum, Saddle Point
(Function), Second Derivative Test
Extrinsic Curvature
A curvature of a SUBMANIFOLD of a MANIFOLD which
depends on its particular EMBEDDING. Examples of ex-
trinsic curvature include the CURVATURE and TORSION
of curves in 3-space, or the mean curvature of surfaces
in 3-space.
see also Curvature, Intrinsic Curvature, Mean
Curvature
F -Distribution
F
F-Distribution
Arises in the testing of whether two observed samples
have the same VARIANCE. Let Xm 2 and %n 2 be inde-
pendent variates distributed as CHI-SQUARED with m
and n Degrees OF FREEDOM. Define a statistic Fn,™
as the ratio of the dispersions of the two distributions
Facet
603
F n , ri
Xn 2 /n
Xm 2 /m'
(1)
This statistic then has an F-distribution with probabil-
ity function and cumulative distribution
Fn,m,\X) :
T(^)n n
n/2-1
r(f)r(f) (m + ra )(^)/ 2
m m/2 n n/2 a ,n/2-l
(m + nx)( n+m )/ 2 B(|n, \m)
m
(2)
(3)
= /(l;I m ;I„)-j(-^_ ; I ra ;J„) | (4)
where r(z) is the GAMMA FUNCTION, B(a,b) is the
Beta Function, and I(a,b\x) is the Regularized
Beta Function. The Mean, Variance, Skewness
and Kurtosis are
see also Beta Function, Gamma Function, Regu-
larized Beta Function, Snedecor's F-Distribu-
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. 946-949, 1972.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Incomplete Beta Function, Student's Distribu-
tion, F-Distribution, Cumulative Binomial Distribution."
§6.2 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 219-223, 1992.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, pp. 117-118, 1992.
F-Polynomial
see KAUFFMAN POLYNOMIAL F
F-Ratio
The Ratio of two independent estimates of the Vari-
ance of a Normal Distribution.
see also F-DlSTRIBUTION, NORMAL DISTRIBUTION,
Variance
F-Ratio Distribution
see F-DlSTRIBUTION
m-2
.2/
2 2m 2 (m + n- 2)
a = —
7i
72
n(m-2) 2 (m-4)
2(ro + 2n - 2) / 2(m - 4)
m — 6 y n(m + n — 2)
12(-16 + 20m - 8m 2 + m 3 + 44n)
+
n(m — 6)(m — 8)(n + m — 2)
12(-32mn + 5m 2 n - 22n 2 + 5mn 2 )
n(ra — 6)(m — 8)(ra -f m — 2)
(5)
(6)
(7)
(8)
The probability that F would be as large as it is if the
first distribution has a smaller variance than the second
is denoted Q(F n>rn ).
The noncentral F-distribution is given by
P(x) = e ~ A / 2 +( An l a: )/[ 2 ( Tl 2+riix)]
x n 1 ni/2 n 2 n2/2 x ni/2 - 1 (n 2 + mz)- (ni+Tl2)/2
r(|n0r(i + fn 2 )L-;r(-^fe))
B(|n 1} |n 2 )r[|(ni+n 2 )]
Fabry Imbedding
A representation of a Planar Graph as a planar
straight line graph such that no two EDGES cross.
Face
(9)
The intersection of an n-D POLYTOPE with a tan-
gent Hyperplane. 0-D faces are known as Vertices
(nodes), 1-D faces as Edges, (n — 2)-D faces as Ridges,
and (n — 1)-D faces as FACETS.
see also EDGE (POLYHEDRON), FACET, POLYTOPE,
Ridge, Vertex (Polyhedron)
Facet
An (n - 1)-D FACE of an n-D Polytope. A procedure
for generating facets is known as Faceting.
where T(z) is the GAMMA FUNCTION, B(a,0) is the
Beta Function, and L^(z) is an associated La-
guerre Polynomial.
604
Faceting
Factorial
Faceting
Using a set of corners of a SOLID that lie in a plane to
form the VERTICES of a new POLYGON is called faceting.
Such Polygons may outline new Faces that join to
enclose a new SOLID, even if the sides of the POLYGONS
do not fall along EDGES of the original SOLID.
References
Holden, A. Shapes, Space, and Symmetry.
Columbia University Press, p. 94, 1971.
New York:
Factor
A factor is a portion of a quantity, usually an INTE-
GER or POLYNOMIAL. The determination of factors is
called Factorization (or sometimes "Factoring"). It
is usually desired to break factors down into the smallest
possible pieces so that no factor is itself factorable. For
Integers, the determination of factors is called Prime
FACTORIZATION. For large quantities, the determina-
tion of all factors is usually very difficult except in ex-
ceptional circumstances.
see also Divisor, Factorization, Greatest Prime
Factor, Least Prime Factor, Prime Factoriza-
tion Algorithms
Factor Base
The primes with Legendre Symbol (n/p) = 1 (less
than N = 7r(d) for trial divisor d) which need be consid-
ered when using the QUADRATIC SIEVE FACTORIZATION
Method.
see also DIXON'S FACTORIZATION METHOD
References
Morrison, M. A. and Brillhart, J. "A Method of Factoring
and the Factorization of F 7 . n Math. Comput. 29, 183-
205, 1975.
Factor (Graph)
A 1-factor of a Graph with n Vertices is a set of n/2
separate Edges which collectively contain all n of the
Vertices of G among their endpoints.
Factor Group
see Quotient Group
Factor Level
A grouping of statistics.
Factor Ring
see Quotient Ring
Factor Space
see Quotient Space
Factorial
The factorial n! is defined for a POSITIVE INTEGER n as
z! = f n • (n - 1) ■ • ■ 2 • 1 n = 1, 2, . . .
" ~ 1 1 n = 0.
(1)
The first few factorials for n = 0, 1, 2, .. . are 1, 1, 2,
6, 24, 120, . . . (Sloane's A000142). An older Notation
for the factorial is [n_ (Dudeney 1970, Gardner 1978,
Conway and Guy 1996).
As n grows large, factorials begin acquiring tails of trail-
ing Zeros. To calculate the number of trailing Zeros
for n!, use
^ 5*
where
fe=i
Kmax —
Inn
Tn~5.
(2)
(3)
and [x\ is the FLOOR FUNCTION (Gardner 1978, p. 63;
Ogilvy and Anderson 1988, pp. 112-114). For n = 1, 2,
. . . , the number of trailing zeros are 0, 0, 0, 0, 1, 1, 1,
1, 1, 2, 2, 2, 2, 2, 3, 3, . . . (Sloane's A027868). This is a
special application of the general result that the POWER
of a PRIME p dividing n\ is
o(n) = J2
(4)
(Graham et al. 1994, Vardi 1991). Stated another way,
the exact Power of a Prime p which divides n! is
n — sum of digits of the base-p representation of n
By noting that
p-l
n! = r(n + l),
(5)
(6)
where T(n) is the GAMMA FUNCTION for INTEGERS n,
the definition can be generalized to COMPLEX values
z\ = T(z + l)
Jo
(7)
This defines z\ for all Complex values of z, except when
z is a Negative Integer, in which case z\ = oo. Us-
ing the identities for GAMMA FUNCTIONS, the values of
{\n)\ (half integral values) can be written explicitly
(n-i)! = ^(2n-l)!!
(8)
(9)
(10)
(11)
where nil is a DOUBLE FACTORIAL.
Factorial
For Integers s and n with s < n,
(s-n)l _ (-l) n - s (2n~2s)\
(2s -2n)! ~ (n-s)\
The Logarithm of 2! is frequently encountered
(12)
»') = i|»
7TZ
sin(7rz)
7 Z^ 2n + l
7> = 1
2n+l
(13)
-i"
7TJZ
sin(7rz)
-H\H)
+(1 7 )z £[C(2» + 1) l] 2n+1
Tl = l
(14)
= ln
lim , . , . , . n
n->oo (2 + l)(z + 2) • • • (z + n)
(15)
= lim [ln(n!) + zlnra-ln(z + l)
n— J- 00
- ln(z + 2) - ... - ln(jz 4- n)]
(16)
n=l
(17)
00
= -72 + H(-l)"^-C(n)
^ — ' n
(18)
= -ln(l + «) + «(l-7)
+5
— \
(-i)"[C0
n
(19)
where 7 is the Euler-Mascheroni Constant, £ is the
Riemann Zeta Function, and F n is the Polygamma
Function. The factorial can be expanded in a series
^ = V2^ z+1/ V*(l + ^z
1 ~-i
12 x
+ ^" a - BiHo^" 3 + ■••). (20)
Stirling's Series gives the series expansion for ln(z!),
B 2
ln(z!) = \ ln(27r) +(z+\)\nz-z +
2z
+ ...+
■ + ■■■
2n(2n-l)z 2n ~ 1
I ln(2?r) + (z + |) In 2 - 2 + ^ _1
?_ z - 3 4. -J^z -5 - f2n
360" ' 1260
where £ n is a BERNOULLI NUMBER.
Factorial
Identities satisfied by sums of factorials include
00
Ei- =e = 2.718281828...
(-1)
EL— L = g" 1 = 0.3678794412 . . .
k\
k=0
00
^ w " /o(2) = 2 - 279585302 • • ■
fc=0 ^ ''
^ (Jfc!) 2
Jo (2) = 0.2238907791.
605
(22)
(23)
(24)
(25)
00 ,
Y^ T^prj = cosh 1 = 1.543080635 . . . (26)
E
(a*)
J
(-1)"
(2*0!
cos 1 = 0.5403023059 . . . (27)
OO
E /ftI 1 ,„ = sinh 1 = 1.175201194 . . . (28)
(2fc + l)! v }
(2fc + l)
E (~ 1 )
(2/fe + l)!
= sin 1 = 0.8414709848 . . . (29)
(Spanier and Oldham 1987), where I is a MODIFIED
Bessel Function of the First Kind, J is a Bessel
Function of the First Kind, cosh is the Hyper-
bolic Cosine, cos is the Cosine, sinh is the Hyper-
bolic Sine, and sin is the Sine.
Let h be the exponent of the greatest POWER of a PRIME
p dividing n!. Then
*=E
i=l
(30)
Let g be the number of Is in the BINARY representation
of n. Then
g + h = n (31)
(Honsberger 1976). In general, as discovered by Legen-
dre in 1808, the POWER m of the PRIME p dividing n!
is given by
-E
n — (np + n\ -f . . . + tin)
p-1
(32)
where the INTEGERS m, . . . , tin are the digits of n in
base p (Ribenboim 1989).
The sum-of-factorials function is defined by
n
E(n) = £fc!
fc=i
_ -e + ei(l) + m + E 2 „+i(-l)r(n + 2)
_ -e + ei(l) + ft[E 3 »+i(-l)]r(n + 2)
,(33)
, (34)
606
Factorial
Factorial
where ei(l) « 1.89512 is the Exponential Integral,
E n is the E^-FUNCTION, and i is the IMAGINARY NUM-
BER. The first few values are 1, 3, 9, 33, 153, 873,
5913, 46233, 409113, . . . (Sloane's A007489). S(n) can-
not be written as a hypergeometric term plus a constant
(Petkovsek et al. 1996). However the sum
E'(n) = ]TA:fc! = (ri + l)!-l
(35)
(Sloane's A014597). The first few values for which the
alternating Sum
5>i) n -'a
(36)
is Prime are 3, 4, 5, 6, 7, 8, 41, 59, 61, 105, 160, ...
(Sloane's A014615, Guy 1994, p. 100). The only known
factorials which are products of factorial in an Arith-
metic Sequence are
has a simple form, with the first few values being 1, 5,
23, 119, 719, 5039, . . . (Sloane's A033312).
The numbers n! + l are prime for n = 1, 2, 3, 11, 27,
37, 41, 73, 77, 116, 154, . . . (Sloane's A002981), and the
numbers n\ — 1 are prime for n = 3, 4, 6, 7, 12, 14, 30,
32, 33, 38, 94, 166, ... (Sloane's A002982). In general,
the power-product sequences (Mudge 1997) are given by
S±(n) = (n!) fc ± 1. The first few terms of S+(n) are 2,
5, 37, 577, 14401, 518401, ... (Sloane's A020549), and
S£(n) is PRIME for n - 1, 2, 3, 4, 5, 9, 10, 11, 13, 24,
65, 76, ... (Sloane's A046029). The first few terms of
S~(n) are 0, 3, 35, 575, 14399, 518399, ... (Sloane's
A046030), but S% '(n) is PRIME for only n = 2 since
S~(n) = (n!) 2 -l= (n! + l)(n!-l)for n> 2. The first
few terms of S^(n) are 0, 7, 215, 13823, 1727999,
and the first few terms of Sf(n) are 2, 9, 217, 13825,
1728001, . . . (Sloane's A19514).
There are only four INTEGERS equal to the sum of the
factorials of their digits. Such numbers are called FAC-
TORIONS. While no factorial is a SQUARE NUMBER,
D. Hoey listed sums < 10 12 of distinct factorials which
give SQUARE NUMBERS, and J. McCranie gave the one
additional sum less than 21! = 5.1 x 10 19 :
= 3"
= 5 2
= 11 2
= 12 2
= 27 2
= 29 2
= 71 2
= 72 2
= 213 2
= 215 2
= 603 2
= 635 2
= 1917 2
1! + 2! + 3! + 7! + 8! + 9! 4- 10! + 11! + 12!
+ 13! + 14! + 15! = 1183893 2
0! + l! + 2
l! + 2! + 3
l! + 4
l! + 5
4! + 5
l! + 2! + 3! + 6
l! + 5! + 6
l! + 7
4! + 5! + 7
l! + 2! + 3! + 7! + 8
1! + 4! + 5! + 6! + 7! + 8
l! + 2! + 3! + 6! + 9
l! + 4! + 8! + 9
1! + 2! + 3! + 6! + 7! + 8! + 10
0!1! = 1!
1!2! = 2!
0!1!2! = 2!
6!7! = 10!
1!3!5! = 6!
1!3!5!7! = 10!
(Madachy 1979).
There are no identities of the form
i !no! • • • a r \
(37)
for r > 2 with a* > a 3 > 2 for i < j for n < 18160
except
9! = 7!3!3!2!
(38)
10! = 7!6! = 7!5!3!
(39)
16! = 14!5!2!
(40)
(Guy 1994, p. 80).
There are three numbers less than 200,000 for which
(n- 1)! + 1 = (modn 2 ) ,
(41)
namely 5, 13, and 563 (Le Lionnais 1983). BROWN
Numbers are pairs (m, n) of Integers satisfying the
condition of Brocard's Problem, i.e., such that
n! + l = m 2 .
(42)
Only three such numbers are known: (5, 4), (11, 5), (71,
7). Erdos conjectured that these are the only three such
pairs (Guy 1994, p. 193).
see also ALLADI-GRINSTEAD CONSTANT, BROCARD'S
Problem, Brown Numbers, Double Factorial,
Factorial Prime, Factorion, Gamma Function,
Hyperfactorial, Multifactorial, Pochhammer
Symbol, Primorial, Roman Factorial, Stirling's
Series, Subfactorial, Superfactorial
References
Conway, J. H. and Guy, R. K. "Factorial Numbers." In The
Book of Numbers. New York: Springer- Verlag, pp. 65-66,
1996.
Dudeney, H. E. Amusements in Mathematics. New York:
Dover, p. 96, 1970.
Factorial Moment
Factorial Sum
607
Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical
Magic Show: More Puzzles, Games, Diversions, Illusions
and Other Mathematical Sleight- of- Mind from Scientific
American. New York: Vintage, pp. 50-65, 1978.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Factorial
Factors." §4.4 in Concrete Mathematics: A Foundation
for Computer Science. Reading, MA: Addison- Wesley,
pp. 111—115, 1990.
Guy, R. K. "Equal Products of Factorials," "Alternating
Sums of Factorials," and "Equations Involving Factorial
n." §B23, B43, and D25 in Unsolved Problems in Number
Theory, 2nd ed. New York: Springer- Verlag, pp. 80, 100,
and 193-194, 1994.
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., p. 2, 1976.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 56, 1983.
Leyland, P. ftp:// sable . ox . ac . uk/ pub /math /factors/
factorial- . Z and ftp : // sable . ox . ac . uk / pub / math /
f actors/f actorial+ . Z.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 174, 1979.
Mudge, M. "Not Numerology but Numeralogy!" Personal
Computer World, 279-280, 1997.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, 1988.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 86, 1996.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Gamma Function, Beta Function, Factorials,
Binomial Coefficients." §6.1 in Numerical Recipes in FOR-
TRAN: The Art of Scientific Computing, 2nd ed. Cam-
bridge, England: Cambridge University Press, pp. 206-
209, 1992.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 22-24, 1989.
Sloane, N. J. A. Sequences A014615, A014597, A033312,
A020549, A000142/M1675, and A007489/M2818 in "An
On-Line Version of the Encyclopedia of Integer Sequences."
Spanier, J. and Oldham, K. B. "The Factorial Function n!
and Its Reciprocal." Ch. 2 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 19-33, 1987.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 67, 1991.
Factorial Moment
x
where
x (r) =x(x-l)---{x-r + l).
References
Borning, A. "Some Results for fc! + 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 M ! + 1 and 2 • 3 ■ 5 ■ - p+ 1." Math. Comput. 38,
639-643, 1982.
Caldwell, C. K. "On the Primality of iV!±l and 2-3-5 •■ -p±
1." Math. Comput 64, 889-890, 1995.
Dubner, H. "Factorial and Primorial Primes." J. Rec. Math.
19, 197-203, 1987.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 7, 1994.
Sloane, N. J. A. Sequences A002981/M0908 and A002982/
M2321 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Temper, M. "On the Primality of fc! + 1 and -3 ■ 5 ■ ■ -p + 1."
Math. Comput. 34, 303-304, 1980.
Factorial Sum
Sums with unity NUMERATOR and FACTORIALS in the
Denominator which can be expressed analytically in-
clude
v -
-^ (n + i - k)\(n - i)\
£
2 ii\(l,-n;l+n-fc;-l)-l
T(l + n)r(l + n - k)
(n + t-l)!(n-i)l 2I\1 +n)T(l + n)
(1)
(2)
£
(n + i)!(n- i)\
^
2r(i+n)r(l + n) 2r 2 (l + n)
V I
^ (n + i + l)!(n- i)\
i = X
^ 1
2r(§ + n)r(l + n) 2r(l + n)r(2 + n) '
(3)
(4)
where 2 ^i(a,6;c;^) is a HYPERGEOMETRIC FUNCTION
and T(z) is a GAMMA FUNCTION.
Sums with i in the NUMERATOR having analytic solu-
tions include
Factorial Number
see FACTORIAL
Factorial Prime
A Prime of the form n! ± 1. n\ + 1 is Prime for 1, 2,
3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427,
872, 1477, . . . (Sloane's A002981) up to a search limit
4850. n! - 1 is Prime for 3, 4, 6, 7, 12, 14, 30, 32, 33,
38, 94, 116, 324, 379, 469, 546, 974, 1963, 3507, 3610,
. . . (Sloane's A002982) up to a search limit of 4850.
^ (n + i
■k)\(n-i)\
_ n a i r i(2 1 l-n;2-fe + n;-l)
"" (1 - k + n)r(l + n)T(l - k + n)
Zs ( n + i- l)!(n-i)!
2T(n)
^
+
2r(i+n) r(l + n)
v -
■^-' (n + i)l(n - i
{n + i)\(n-i)\ 2r 2 (l + n)
(5)
(6)
(7)
608 Factoring
Fair Game
y i
^ (n + i + l)!(n-i)!
2r(l + n)
r(2 + n)
(n 2 + 3n + 2)0F
2r(f +n)
(8)
A sum with i 2 in the NUMERATOR is
n 2
v "
*-* {n + i - k)\(n-i)\
(1 - k + ra)(2 - fc + n)r(l + n)r(l - fc -j- n)
x[(2 - A; + n) 2 i<i(2,l - n; 2 - fc + n;-l)
+2(n - 1) 2 Fi(3, 2 - n; 3 - k + n; -1)], (9)
where 2^1 ( a > &; c; z) is the Hypergeometric FUNC-
TION.
Sums of factorial POWERS include
>P ( n! ) 2 _ 4 2tt
(2n)!
9v/3
(10)
^(3n)! 7
[P(t) + Q(t) cos" 1 #(*)]<&, (11)
where
P(t) =
«(*) =
2(8 + 7* 2 -7t 3 )
(4_^2 +t 3)2
4t(l - t)(5 + i 2 ~i 3 )
(12)
(13)
(4 - t 2 + t3)2 JV /( 1 _ t )( 4 _ i 2 +t 3)
H(t) = l-|(* 2 -* 3 ) (14)
(Beeler ei a/. 1972, Item 116).
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Factoring
see Factorization
Fact or ion
A factorion is an Integer which is equal to the sum of
Factorials of its digits. There are exactly four such
numbers:
1-1!
(1)
2 = 2!
(2)
145 - 1! + 4! + 5!
(3)
40, 585 = 4! + 0! + 5! + 8! + 5!
(4)
(Gardner 1978, Madachy 1979, Pickover 1995). The fac-
torion of an n-digit number cannot exceed n ■ 9! digits.
see also Factorial
References
Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical
Magic Show: More Puzzles, Games, Diversions, Illusions
and Other Mathematical Sleight-of-Mind from Scientific
American. New York: Vintage, pp. 61 and 64, 1978.
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, p. 167, 1979.
Pickover, C. A. "The Loneliness of the Factorions." Ch. 22 in
Keys to Infinity. New York: W. H. Freeman, pp. 169-171
and 319-320, 1995.
Factorization
The finding of FACTORS (DIVISORS) of a given INTEGER,
Polynomial, etc. Factorization is also called Factor-
ing.
see also Factor, Prime Factorization Algorithms
Fagnano's Point
The point of coincidence of P and P' in Fagnano's
Problem.
Fagnano's Problem
In a given AcuTE-angled Triangle A ABC, Inscribe
another TRIANGLE whose PERIMETER is as small as pos-
sible. The answer is the PEDAL TRIANGLE of AABC.
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 88-89, 1967.
Fagnano's Theorem
If P{x,y) and P(x' } y f ) are two points on an ELLIPSE
S4 = 1 > w
with Eccentric Angles <f> and 0' such that
tan (j) tan <j) = - (2)
a
and A = P(a, 0) and B = P(0, b). Then
arc BP + arc BP' = ^^ . (3)
This follows from the identity
E(u, k) + E(v, k) - E(k) = k 2 sn(u, k) sn(v, ft), (4)
where E(u, k) is an incomplete ELLIPTIC INTEGRAL OF
the Second Kind, E(k) is a complete Elliptic Inte-
gral of the Second Kind, and sn(v,fc) is a Jacobi
Elliptic Function. If P and P' coincide, the point
where they coincide is called Fagnano's Point.
Fair Game
A GAME which is not biased toward any player.
see also Game, Martingale
Fairy Chess
Fano's Axiom 609
Fairy Chess
A variation of CHESS involving a change in the form of
the board, the rules of play, or the pieces used. For
example, the normal rules of chess can be used but with
a cylindrical or MOBIUS Strip connection of the edges.
see also Chess
References
Kraitchik, M. "Fairy Chess." §12.2 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 276-279, 1942.
Fallacy
A fallacy is an incorrect result arrived at by appar-
ently correct, though actually specious reasoning. The
most common example of a mathematical fallacy is the
"proof" that 1 — 2 as follows. Let a = 6, then
ab = a
ab — b — a — b
b(a-b) = (a + b){a-b)
b = a + b
6 = 26
1 = 2.
The incorrect step is division by a — 6 (equal to 0), which
is invalid. Ball and Coxeter (1987) give other such ex-
amples in the areas of both arithmetic and geometry.
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 41-45
and 76-84, 1987.
Pappas, T. "Geometric Fallacy & the Fibonacci Sequence."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, p. 191, 1989.
with y = 0, using yi = /(xi), and solving for x n there-
fore gives the iteration
Xn — 1 X\ x ( \
f{x n -i) - f{Xl)
see also Brent's Method, Ridders' Method, Se-
cant Method
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.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Secant Method, False Position Method, and
Ridders' Method." §9.2 in Numerical Recipes in FOR-
TRAN: The Art of Scientific Computing, 2nd ed. Cam-
bridge, England: Cambridge University Press, pp. 347-
352, 1992.
Faltung (Form)
Let A and B be bilinear forms
A = A(x, y) = ^^ y^.OijXiyi
B = B(x, y) = ^ Yl hi i XiVi
and suppose that A and B are bounded in [p,p ; ] with
bounds M and N. Then
where the series
fij = 2, a ikbkj
False
A statement which is rigorously not TRUE. Regular
two-valued LOGIC allows statements to be only TRUE
or false, but FUZZY LOGIC treats "truth" as a contin-
uum which can have a value between and 1.
see also Alethic, Fuzzy Logic, Logic, True, Truth
Table, Undecidable
is absolutely convergent, is called the faltung of A and
B. F is bounded in [p,p'], and its bound does not exceed
MN.
References
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 210-211, 1988.
False Position Method
An Algorithm for finding Roots which uses the point
where the linear approximation crosses the axis as the
next iteration and keeps the same initial point for each
iteration. Using the two-point form of the line
/(sn-i) -/Qgi) ,„
y-yi = (x n
x n -i - Xi
Xi)
Faltung (Function)
see Convolution
Fan
A Spread in which each node has a FINITE number of
children.
see also SPREAD (Tree)
Fano's Axiom
The three diagonal points of a Complete Quadrilat-
eral are never COLLINEAR.
610 Fano Plane
Fano Plane
Farey Sequence
The 2-D Projective Plane over GF{2) ("of order
two"), illustrated above. It is a BLOCK DESIGN with
v = 7, k = 3, A = 1, t = 3, and 6 = 7, and is also the
Steiner Triple System S(7).
The Fano plane also solves the TRANSYLVANIA LOT-
TERY, which picks three numbers from the Integers
1-14. Using two Fano planes we can guarantee match-
ing two by playing just 14 times as follows. Label the
Vertices of one Fano plane by the Integers 1-7, the
other plane by the Integers 8-14. The 14 tickets to
play are the 14 lines of the two planes. Then if (a, 6, c)
is the winning ticket, at least two of a, 6, c are either in
the interval [1, 7] or [8, 14]. These two numbers are on
exactly one line of the corresponding plane, so one of
our tickets matches them.
The Lehmers (1974) found an application of the Fano
plane for factoring INTEGERS via QUADRATIC FORMS.
Here, the triples of forms used form the lines of
the Projective Geometry on seven points, whose
planes are Fano configurations corresponding to pairs of
residue classes mod 24 (Lehmer and Lehmer 1974, Guy
1975, Shanks 1985). The group of AUTOMORPHISMS
(incidence-preserving BlJECTIONS) of the Fano plane is
the Simple Group of Order 168 (Klein 1870).
see also Design, Projective Plane, Steiner Triple
System, Transylvania Lottery
References
Guy, R. "How to Factor a Number." Proc. Fifth Manitoba
Conf. on Numerical Math., 49-89, 1975.
Lehmer, D. H. and Lehmer, E. "A New Factorization Tech-
nique Using Quadratic Forms." Math. Corn-put. 28, 625-
635, 1974.
Shanks, D, Solved and Unsolved Problems in Number Theory,
3rd ed. New York: Chelsea, pp. 202 and 238, 1985.
Far Out
A word used by Tukey to describe data points which are
outside the outer Fences.
References
Tukey, J. W. Explanatory Data Analysis.
Addison- Wesley, p. 44, 1977.
Reading, MA:
Far-Out Point
For a TRIANGLE with side lengths a, 6, and c, the far-out
point has Triangle Center Function
a = a(b + c
a — b
2 ).
As a : b : c approaches 1:1:1, this point moves out
along the Euler Line to infinity.
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C; Lyness, R. C; and Veldkamp, G. R. "Prob-
lem 1195 and Solution." Crux Math. 14, 177-179, 1988.
Farey Sequence
The Farey sequence F n for any POSITIVE INTEGER n
is the set of irreducible RATIONAL Numbers a/b with
< a < b < n and (a, b) = 1 arranged in increasing
order.
* = {?>!}
x * 1 1 »■ 2 ' 1 J
rO 1 1 2 li
U' 3' 2' 3' 1 J
fO 1 1 1 2 3 1\
1l»4»3»2 ) 3»4»lJ
(0 1
ll' 5
F 3
F 4
F 5
i i
4> 3'
2 13
5' 2' 5'
4' 5
.*>■
(i)
(2)
(3)
(4)
(5)
There is always an Odd number of terms, and the mid-
dle term is always 1/2. Let p/g, p f /q', and p"/q" be
three successive terms in a Farey series. Then
(6)
(7)
These two statements are actually equivalent.
The number of terms N(n) in the Farey sequence for
the Integer n is
qp'
1 1
-pq = 1
v'
P + P"
q'
q + q"
N(n) = 1 + XT ^ (A;) = 1 + * (n) '
(8)
fe = l
where </>(k) is the TOTIENT FUNCTION and $(n) is the
SUMMATORY FUNCTION of <p(k), giving 2, 3, 5, 7, 11,
13, 19, ... (Sloane's A005728). The asymptotic limit
for the function N(n) is
JV(n).
3n 2
0.3039635509*2
(9)
(Vardi 1991, p. 155). For a method of computing a suc-
cessive sequence from an existing one of n terms, insert
the MEDIANT fraction (a + b)/(c + d) between terms
a/c and b/d when c + d < n (Hardy and Wright 1979,
pp. 25-26; Conway and Guy 1996).
Ford Circles provide a method of visualizing the
Farey sequence. The Farey sequence F n defines a sub-
tree of the STERN-BROCOT Tree obtained by pruning
unwanted branches (Graham et at. 1994).
see also Ford Circle, Mediant, Rank (Sequence),
Stern-Brocot Tree
Farey Series
Fast Fourier Transform 611
References
Beiler, A. H. "Farey Tails." Ch. 16 in Recreations in the The-
ory of Numbers: The Queen of Mathematics Entertains.
New York: Dover, 1966.
Conway, J. H. and Guy, R. K. "Farey Fractions and Ford
Circles." The Book of Numbers. New York: Springer-
Verlag, pp. 152-154 and 156, 1996.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 155-
158, 1952.
Farey, J. "On a Curious Property of Vulgar Fractions." Lon-
don, Edinburgh and Dublin Phil. Mag. 47, 385, 1816.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science, 2nd
ed. Reading, MA: Addison- Wesley, pp. 118-119, 1994.
Guy, R. K. "Mahler's Generalization of Farey Series." §F27
in Unsolved Problems in Number Theory, 2nd ed. New
York: Springer- Verlag, pp. 263-265, 1994.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.
Sloane, N. J. A. Sequences A005728/M0661, A006842/
M0041, and A006843/M0081 in "An On-Line Version of
the Encyclopedia of Integer Sequences."
Sylvester, J. J. "On the Number of Fractions Contained in
Any Farey Series of Which the Limiting Number is Given."
London, Edinburgh and Dublin Phil. Mag. (5th Series) 15,
251, 1883.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 155, 1991.
# Weisstein, E. W. "Plane Geometry." http: //www. astro.
Virginia. edu/~eww6n/math/notebooks/PlaneGeonietry.m.
Farey Series
see Farey Sequence
Farkas's Lemma
The Inequality (/o,#) < follows from
(/i,aj)<0,...,(/„ ) x)<0
IFF there exist Nonnegative numbers Ai, . . . , A n with
The inverse is then given by
,_i _ (-A,xA + B)
(A,B)~
2^ ^kfk = /o-
This LEMMA is used in the proof of the KUHN-TUCKER
Theorem.
see also Kuhn-Tucker Theorem, Lagrange Multi-
plier
Faro Shuffle
see Riffle Shuffle
Fast Fibonacci Transform
For a general second-order recurrence equation
/ n+ l = Xf n +J//n-l, (l)
define a multiplication rule on ordered pairs by
(A, B)(C, D) = (AD + BC + xAC, BD + yAC). (2)
B 2 + xAB - yA 2 '
and we have the identity
(/l,»/o)(l,0)" = (/»+!, Vfn)
(Beeler et al. 1972, Item 12).
(3)
(4)
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Fast Fourier Transform
The fast Fourier transform (FFT) is a DISCRETE FOUR-
IER Transform Algorithm which reduces the num-
ber of computations needed for N points from 27V 2 to
2N\gN, where Lg is the base-2 LOGARITHM. If the
function to be transformed is not harmonically related
to the sampling frequency, the response of an FFT looks
like a SlNC FUNCTION (although the integrated Power
is still correct). ALIASING (Leakage) can be reduced by
Apodization using a Tapering Function. However,
Aliasing reduction is at the expense of broadening the
spectral response.
FFTs were first discussed by Cooley and Tukey (1965),
although Gauss had actually described the critical fac-
torization step as early as 1805 (Gergkand 1969, Strang
1993). A Discrete Fourier Transform can be
computed using an FFT by means of the DANIELSON-
Lanczos Lemma if the number of points N is a Power
of two. If the number of points N is not a Power of
two, a transform can be performed on sets of points cor-
responding to the prime factors of N which is slightly
degraded in speed. An efficient real Fourier transform
algorithm or a fast Hartley Transform (Bracewell
1965) gives a further increase in speed by approximately
a factor of two. Base-4 and base-8 fast Fourier trans-
forms use optimized code, and can be 20-30% faster
than base-2 fast Fourier transforms. PRIME factoriza-
tion is slow when the factors are large, but discrete Four-
ier transforms can be made fast for TV — 2, 3, 4, 5, 7,
8, 11, 13, and 16 using the WlNOGRAD TRANSFORM
Algorithm (Press et al 1992, pp. 412-413, Arndt).
Fast Fourier transform algorithms generally fall into
two classes: decimation in time, and decimation in fre-
quency. The Cooley-Tukey FFT ALGORITHM first re-
arranges the input elements in bit-reversed order, then
builds the output transform (decimation in time). The
612
Fast Fourier Transform
Fatou's Theorems
basic idea is to break up a transform of length N into
two transforms of length N/2 using the identity
JV-l N/2-1
-2Tvink/N _ V~^ -2ni(2n)k/N
° — / ^ a 2nc
n=0 n—
N/2-1
J2 ane
+ 2_^ »2n+ie
)k/N
Fat Fractal
A Cantor Set with Lebesgue Measure greater than
0.
see also Cantor Set, Exterior Derivative, Frac-
tal, Lebesgue Measure
References
Ott, E. "Fat Fractals." §3.9 in Chaos in Dynamical Systems.
New York: Cambridge University Press, pp. 97-100, 1993.
N/2-1
even -2irink/(N/2)
Fatou Dust
see Fatou Set
N/2-1
. -27Tlfc/N V"^ Odd -27T27lfc/(N/2)
-\-e J ^ €L n e ,
sometimes called the DANIELSON-LANCZOS LEMMA.
The easiest way to visualize this procedure is perhaps
via the FOURIER Matrix.
The Sande-Tukey ALGORITHM (Stoer and Burlisch
1980) first transforms, then rearranges the output values
(decimation in frequency).
see also Danielson-Lanczos Lemma, Discrete
Fourier Transform, Fourier Matrix, Fourier
Transform, Hartley Transform, Number Theo-
retic Transform, Winograd Transform
References
Arndt, J. "FFT Code and Related Stuff." http://www.jjj.
de/fxt/.
Bell Laboratories. "Netlib FFTPack." http://netlib.bell-
labs . com/netlib/f f tpack/.
Blahut, R. E. Fast Algorithms for Digital Signal Processing.
New York: Addison- Wesley, 1984.
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, 1965.
Brigham, E. O. The Fast Fourier Transform and Applica-
tions. Englewood Cliffs, NJ: Prentice Hall, 1988.
Cooley, J. W. and Tukey, O. W. "An Algorithm for the Ma-
chine Calculation of Complex Fourier Series." Math. Corn-
put. 19, 297-301, 1965.
Duhamel, P. and Vetterli, M, "Fast Fourier Transforms: A
Tutorial Review." Signal Processing 19, 259-299, 1990.
Gergkand, G. D. "A Guided Tour of the Fast Fourier Trans-
form." IEEE Spectrum, pp. 41-52, July 1969.
Lipson, J. D. Elements of Algebra and Algebraic Computing.
Reading, MA: Addison- Wesley, 1981.
Nussbaumer, H. J. Fast Fourier Transform and Convolution
Algorithms, 2nd ed. New York: Springer- Verlag, 1982.
Papoulis, A. The Fourier Integral and its Applications. New
York: McGraw-Hill, 1962.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Fast Fourier Transform." Ch. 12 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 490-529, 1992.
Stoer, J. and Burlisch, R. Introduction to Numerical Analy-
sis. New York: Springer- Verlag, 1980.
Strang, G. "Wavelet Transforms Versus Fourier Transforms,"
Bull. Amer. Math. Soc. 28, 288-305, 1993.
Van Loan, C. Computational Frameworks for the Fast Four-
ier Transform. Philadelphia, PA: SIAM, 1992.
Walker, J. S. Fast Fourier Transform, 2nd ed. Boca Raton,
FL: CRC Press, 1996.
Fatou's Lemma
If a Sequence {f n } of Nonnegative measurable func-
tions is defined on a measurable set E y then
/lim inf f n dfi < lim inf / f n dfi.
n— ^oo n— >oo J
References
Zeidler, E, Applied Functional Analysis: Applications to
Mathematical Physics. New York: Springer- Verlag, 1995.
Fatou Set
A set consisting of the complementary set of complex
numbers to a Julia Set.
see also Julia Set
References
Schroeder, M. Fractals, Chaos, Power Laws. New York:
W. H. Freeman, p. 39, 1991.
Fatou's Theorems
Let f(6) be LEBESGUE INTEGRABLE and let
f(^) = iJj(t) 1 _ 2r ; o - t r2 _ e)+r2 dt (i)
be the corresponding PoiSSON INTEGRAL. Then AL-
MOST Everywhere in -n < < tt,
lim f(r,0) = f(O).
r— *-0 _
(2)
Let
F(z) = co + ciz + c 2 z 2 + . . . + c n z n + . . . (3)
be regular for \z\ < 1, and let the integral
± f \F(re i$ )\ 2 de
(4)
be bounded for r < 1. This condition is equivalent to
the convergence of
M 2 + |c 1 | 2 + ... + |c„| 2 + ....
(5)
Faulhaber's Formula
Then almost everywhere in — 7r < 8 < tv,
lim F{re l6 ) = F(e ie ).
(6)
Furthermore, F(e ie ) is measurable, |F(e^)| 2 is Lebes-
gue Integrable, and the Fourier Series of F(e ie )
is given by writing z = e l .
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 274, 1975.
Faulhaber's Formula
In a 1631 edition of Academiae Algebrae, 3. Faulhaber
published the general formula for the SUM of pth POW-
ERS of the first n Positive Integers,
E''-^D-') , "( p t 1 )
fc=l i=l V 7
B p +i-in, (1)
where S ip is the Kronecker Delta, (") is a Binomial
Coefficient, and B t is the ith Bernoulli Number.
Computing the sums for p = 1, . . . , 10 gives
£)fc=i(n a + n)
fc=l
n
^/c 2 = |(2n 3 + 3n 2 +n)
fc=i
Y / k 3 = \{n* + 2n 3 +n 2 )
fc = l
n
Y^k 4 = ±(6n + 15n 4 + 10n 3 - n)
fc=i
n
^ fc 5 = ^(2n 6 + 6n 5 + 5n 4 - n 2 )
(2)
(3)
(4)
(5)
(6)
^2 k 6 = h ( 6 " 7 + 21n 6 + 21n 5 - 7ra 3 + n) (7)
n
Y^ k 7 = £(3n 8 + 12n 7 + 14n 6 - 7n 4 + 2n 2 ) (8)
fc = l
n
^2 k * = & ( i0n ° + 45n8 + 6 ° n7 - 42 ™ 5
fc=i
+ 20n 3 - 3n) (9)
n
^ fc 9 = ^(2n 10 + 10n 9 + 15n 8 - 14n 6
fc = l
+ 10n 4 - 3n 2 ) (10)
n
^2 k 10 = ± (en 11 + 33n 10 + 55n 9 - 66n 7
Feigenbaum Constant 613
see a/so Power, Sum
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 106, 1996.
Favard Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let T n (x) be an arbitrary trigonometric POLYNOMIAL
T n {x) = \ao + < yjfafc cos(fca;) + b k sin(fcx)]
where the Coefficients are real. Let the rth deriva-
tive of T n (x) be bounded in [—1, 1], then there exists a
Polynomial T n (x) for which
!/(*)- T n (x)|<
K r
(n+1)''
for all x, where K r is the rth Favard constant, which is
the smallest constant possible.
K r
E
(-i) fe
2k + 1
fe-| r+1
These can be expressed by
K -If A(r + 1
) for r odd
) for r even,
where A is the DlRICHLET LAMBDA FUNCTION and is
the Dirichlet Beta Function. Explicitly,
K
=
1
Ki
=
¥
K 2
=
I- 2
K 3
=
1 J>
+ 66n 5 -33n 3 + 5n).
(11)
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/f avard/f avard.html.
Kolmogorov, A. N. "Zur Grossenordnung des Restgliedes
Fourierscher reihen differenzierbarer Funktionen." Ann.
Math. 36, 521-526, 1935.
Zygmund, A. G. Trigonometric Series, Vols. 1-2, 2nd ed.
New York: Cambridge University Press, 1959.
Feigenbaum Constant
A universal constant for functions approaching CHAOS
via period doubling. It was discovered by Feigenbaum
in 1975 and demonstrated rigorously by Lanford (1982)
and Collet and Eckmann (1979, 1980). The Feigenbaum
constant S characterizes the geometric approach of the
bifurcation parameter to its limiting value. Let fik be
the point at which a period 2 k cycle becomes unstable.
614 Feigenbaum Constant
Feigenbaum Constant
Denote the converged value by fj,oo • Assuming geometric
convergence, the difference between this value and fik is
denoted
r
lim ^oo -Mfc = -~£, (1)
where T is a constant and S is a constant > 1. Solving
for 5 gives
5= lim M " +1 ~ Mn (2)
n^-oo /X n + 2 — /in + 1
(Rasband 1990, p. 23). For the LOGISTIC EQUATION,
(5 = 4.669216091...
(3)
T = 2.637...
(4)
//oo = 3.5699456 ....
(5)
Amazingly, the Feigenbaum constant S « 4.669 is "uni-
versal" (i.e., the same) for all 1-D MAPS f(x) if f(x) has
a single locally quadratic MAXIMUM. More specifically,
the Feigenbaum constant is universal for 1-D MAPS if
the Schwarzian Derivative
Dsc
~ f'(x)
(6)
is NEGATIVE in the bounded interval (Tabor 1989,
p. 220). Examples of maps which are universal in-
clude the Henon Map, Logistic Map, Lorenz Sys-
tem, Navier-Stokes truncations, and sine map x n +i —
asin(7ra; n ). The value of the Feigenbaum constant can
be computed explicitly using functional group renormal-
ization theory. The universal constant also occurs in
phase transitions in physics and, curiously, is very nearly
equal to
7r + tan- 1 (e 7r ) = 4.669201932.
(7)
The CIRCLE Map is not universal, and has a Feigenbaum
constant of S ^ 2.833. For an AREA-PRESERVING 2-D
Map with
y n +i = g{x n ,y n ),
(8)
(9)
the Feigenbaum constant is S = 0.7210978 . . . (Tabor
1989, p. 225). For a function of the form
f(x) = l-a\x\ n
(10)
with a and n constant and n an INTEGER, the Feigen-
baum constant for various n is given in the following
table (Briggs 1991, Briggs et al. 1991), which updates
the values in Tabor (1989, p. 225).
n
2 5.9679
4 7.2846
6 8.3494
8 9.2962
An additional constant a, defined as the separation of
adjacent elements of PERIOD DOUBLED ATTRACTORS
from one double to the next, has a value
d n
lim
n— >oo (Xn+1
-2.502907875 .
(11)
for "universal" maps (Rasband 1990, p. 37). This value
may be approximated from functional group renormal-
ization theory to the zeroth order by
1-cT
1-a"
[l-a-^l-a" 1 )] 2 '
(12)
which, when the QUINTIC EQUATION is numerically
solved, gives a = -2.48634. . ., only 0.7% off from the
actual value (Feigenbaum 1988).
see also Attractor, Bifurcation, Feigenbaum
Function, Linear Stability, Logistic Map, Pe-
riod Doubling
References
Briggs, K. "A Precise Calculation of the Feigenbaum Con-
stants." Math. Comput. 57, 435-439, 1991.
Briggs, K.; Quispel, G.; and Thompson, C. "Feigenvalues for
Mandelsets." J. Phys. A: Math. Gen. 24 3363-3368, 1991.
Briggs, K.; Quispel, G.; and Thompson, C. "Feigenvalues
for Mandelsets." http : //epideml3 .plant sci . cam. ac .uk/
-kbriggs/.
Collett, P. and Eckmann, J.-P. "Properties of Continuous
Maps of the Interval to Itself." Mathematical Problems
in Theoretical Physics (Ed. K. Osterwalder). New York:
Springer- Verlag, 1979.
Collett, P. and Eckmann, J.-P. Iterated Maps on the Interval
as Dynamical Systems. Boston, MA: Birkhauser, 1980.
Eckmann, J.-P. and Wittwer, P. Computer Methods and
Borel Summability Applied to Feigenbaum's Equations.
New York: Springer- Verlag, 1985.
Feigenbaum, M. J. "Presentation Functions, Fixed Points,
and a Theory of Scaling Function Dynamics." J. Stat.
Phys. 52, 527-569, 1988.
Finch, S. "Favorite Mathematical Constants." http://vww.
mathsof t . com/asolve/constant/f gnbaum/f gnbaum.html.
Finch, S. "Generalized Feigenbaum Constants." http: //www
.mathsoft.com/asolve/constant/fgnbaum/gerieral.html.
Lanford, O. E. "A Computer-Assisted Proof of the Feigen-
baum Conjectures." Bull. Amer. Math. Soc. 6, 427-434,
1982.
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, 1990.
Stephenson, J. W. and Wang, Y. "Numerical Solution of
Feigenbaum's Equation." Appl. Math. Notes 15, 68-78,
1990.
Stephenson, J. W. and Wang, Y. "Relationships Between the
Solutions of Feigenbaum's Equations." Appl. Math. Let. 4,
37-39, 1991.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, 1989.
Feigenbaum Function
Feller-Levy Condition 615
Feigenbaum Function
Consider an arbitrary 1-D Map
Xn+l = F(x n )
(1)
at the onset of CHAOS. After a suitable rescaling, the
Feigenbaum function
M-n^F^m*^** 1 '^ (2)
is obtained. This function satisfies
9(9(x)) = --9(<xx),
a
(3)
with a — 2.50290 . . ., a quantity related to the FEIGEN-
BAUM Constant.
see also Bifurcation, Chaos, Feigenbaum Con-
stant
References
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.
Feit-Thompson Conjecture
Concerns PRIMES p and q for which p q — 1 and q p — 1
have a common factor. The only (p,g) pair with both
values less than 400,000 is (17, 3313), with a common
factor 112,643.
References
Wells, D. G. The Penguin Dictionary of Curious and Inter-
esting Numbers. London: Penguin, p. 17, 1986.
Feit-Thompson Theorem
Every FINITE SIMPLE GROUP (which is not CYCLIC) has
Even Order, and the Order of every FINITE SIMPLE
noncommutative group is Doubly Even, i.e., divisible
by 4 (Feit and Thompson 1963).
see also Burnside Problem, Finite Group, Order
(Group), Simple Group
References
Feit, W. and Thompson, J. G. "Solvability of Groups of Odd
Order." Pacific J. Math. 13, 775-1029, 1963.
Fejes Toth's Integral
2*(n + l)JJ {x) \ sin(H J
gives the nth Cesaro Mean of the Fourier Series of
f(x).
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, Rl:
Amer. Math. Soc, p. 12, 1975.
Fejes Toth's Problem
How can n points be distributed on a Unit SPHERE such
that they maximize the minimum distance between any
pair of points? In 1943, Fejes Toth proved that for N
points, there always exist two points whose distance d
is
d<
ttN
6(N - 2)
and that the limit is exact for N = 3, 4, 6, and 12.
For two points, the points should be at opposite ends of
a Diameter. For four points, they should be placed at
the Vertices of an inscribed Tetrahedron. There is
no best solution for five points since the distance can-
not be reduced below that for six points. For six points,
they should be placed at the Vertices of an inscribed
Octahedron. For seven points, the best solution is
four equilateral spherical triangles with angles of 80°.
For eight points, the best dispersal is not the VERTICES
of the inscribed Cube, but of a square Antiprism with
equal Edges. The solution for nine points is eight equi-
lateral spherical triangles with angles of cos _1 (l/4). For
12 points, the solution is an inscribed ICOSAHEDRON.
The general problem has not been solved.
see also Thomson Problem
References
Ogilvy, C. S. Excursions in Mathematics. New York: Dover,
p. 99, 1994.
Ogilvy, C. S. Solved by L. Moser. "Minimal Configuration
of Five Points on a Sphere." Problem E946. Amer. Math.
Monthly 58, 592, 1951.
Schiitte, K. and van der Waerden, B. L. "Auf welcher Kiigel
haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins
Plata?" Math. Ann. 123, 96-124, 1951.
Whyte, L. L. "Unique Arrangement of Points on a Sphere."
Amer. Math. Monthly 59, 606-611, 1952.
Feller's Coin- Tossing Constants
see Coin Tossing
Feller-Levy Condition
Given a sequence of independent random variates Xi,
X2, . . . , if crk 2 = var(Xjk) and
then
- (°> 2 \
= max — -
k<n ys n z J
lim p n = 0.
This means that if the LlNDEBERG CONDITION holds
for the sequence of variates Xl, . . . , then the Variance
of an individual term in the sum S n of Xk is asymp-
totically negligible. For such sequences, the LlNDEBERG
Condition is Necessary as well as Sufficient for
the Lindeberg-Feller Central Limit Theorem to
hold.
References
Zabell, S. L. "Alan Turing and the Central Limit Theorem."
Amer. Math. Monthly 102, 483-494, 1995.
616
Fence
Fermat-Euler Theorem
Fence
Values one Step outside the Hinges are called inner
fences, and values two steps outside the HINGES are
called outer fences. Tukey calls values outside the outer
fences FAR Out.
see also Adjacent Value
References
Tukey, J. W. Explanatory Data
Addison- Wesley, p. 44, 1977.
Analysis. Reading, MA:
Fence Poset
A Partial Order defined by (i
Odd i.
see also Partial Order
1, j), (z + 1, j)for
References
Ruskey, F. "Information on Ideals of Partially Ordered
Sets." http:// sue . esc . uvic . ca / - cos / inf / pose /
Ideals.html.
Ferguson-Forcade Algorithm
A practical algorithm for determining if there exist in-
tegers ai for given real numbers xi such that
a\X\ + a%X2 + . . . + a n x n = 0,
or else establish bounds within which no such Integer
Relation can exist (Ferguson and Forcade 1979). A
nonrecursive variant of the original algorithm was sub-
sequently devised by Ferguson (1987). The Ferguson-
Forcade algorithm has shown that there are no algebraic
equations of degree < 8 with integer coefficients having
Euclidean norms below certain bounds for e/ir, e + 7r,
ln7r, 7, e 7 , 7/e, 7/71-, and In 7, where e is the base for
the Natural Logarithm, -k is Pi, and 7 is the Euler-
Mascheroni Constant (Bailey 1988).
Constant
Bound
e/7r
6.1030 x 10 14
e + 7r
2.2753 x 10 18
ln7r
8.7697 x 10 9
7
3.5739 x 10 9
e 7
1.6176 x 10 17
7/e
1.8440 x 10 11
7/tt
6.5403 x 10 9
In 7
2.6881 x 10 10
see also CONSTANT PROBLEM, EUCLIDEAN ALGO-
RITHM, Integer Relation, PSLQ Algorithm
References
Bailey, D. H. "Numerical Results on the Transcendence of
Constants Involving 7r, e, and Euler's Constant." Math.
Comput. 50, 275-281, 1988.
Ferguson, H. R. P. "A Short Proof of the Existence of Vector
Euclidean Algorithms." Proc. Amer. Math. Soc. 97, 8-10,
1986.
Ferguson, H. R. P. "A Non-inductive GL(n, Z) Algorithm
that Constructs Linear Relations for n Z-Linearly Depen-
dent Real Numbers." J. Algorithms 8, 131-145, 1987.
Ferguson, H. R. P. and Forcade, R. W. "Generalization of the
Euclidean Algorithm for Real Numbers to All Dimensions
Higher than Two." Bull Amer. Math. Soc. 1, 912-914,
1979.
Fermat An + 1 Theorem
Every Prime of the form An + 1 is a sum of two Square
Numbers in one unique way (up to the order of Sum-
MANDS). The theorem was stated by Fermat, but the
first published proof was by Euler.
see also Sierpinski's Prime Sequence Theorem,
Square Number
References
Hardy, G. H. and Wright, E. M. "Some Notation." Th. 251 in
An Introduction to the Theory of Numbers, 5th ed. Oxford,
England: Clarendon Press, 1979.
Fermat 's Algorithm
see Fermat's Factorization Method
Fermat Compositeness Test
Uses Fermat's Little Theorem
Fermat's Congruence
see Fermat's Little Theorem
Fermat Conic
A Plane Curve of the form y = x n . For n > 0, the
curve is a generalized PARABOLA; for n < it is a gen-
eralized Hyperbola.
see also Conic Section, Hyperbola, Parabola
Fermat's Conjecture
see Fermat's Last Theorem
Fermat Difference Equation
see Pell Equation
Fermat Diophantine Equation
see Fermat Difference Equation
Fermat Equation
The Diophantine Equation
x n + y n = z n .
The assertion that this equation has no nontrivial solu-
tions for n > 2 is called Fermat's Last Theorem.
see also Fermat's Last Theorem
Fermat-Euler Theorem
see Fermat's Little Theorem
Fermat's Factorization Method
Fermat's Last Theorem 617
Fermat's Factorization Method
Given a number n, look for INTEGERS x and y such that
n = x 2 — y 2 . Then
n = (x - y)(x + y)
(i)
and n is factored. Any Odd NUMBER can be represented
in this form since then n = ab, a and b are Odd, and
a = x + y
b — x — y.
Adding and subtracting,
a + b = 2x
a-b = 2y,
so solving for x and y gives
x=i(a + b)
y =Ua-b).
(2)
(3)
(4)
(5)
(6)
(7)
Therefore,
x 2 -y 2 = \{{a + bf-{a-bf] = ab. (8)
As the first trial for x, try x\ |"v^]j wnere \ x ] 1S the
Ceiling Function. Then check if
Axi = xi — n
(9)
is a Square Number. There are only 22 combinations
of the last two digits which a Square Number can
assume, so most combinations can be eliminated. If Axi
is not a SQUARE NUMBER, then try
£ 2 = xi + 1,
(10)
so
Ax 2 = x 2 2 - n = (xi + l) 2 - n = X\ 2 4- 2ei + 1 — 71
= Aa;i +2xi + l. (11)
Continue with
A#3 = X3 2 - n = (rc2 + l) 2 - n = X2 2 + 2^2 + 1 - n
= Ax 2 + 2x 2 + 1 = Ax 2 + 2xi + 3, (12)
so subsequent differences are obtained simply by adding
two.
Maurice Kraitchik sped up the Algorithm by looking
for x and y satisfying
x = y (mod n) ,
(13)
i.e., n\(x 2 — y 2 ). This congruence has uninteresting
solutions x = ±2/ (mod n) and interesting solutions
x ^ ±y (mod n). It turns out that if n is Odd and DI-
VISIBLE by at least two different PRIMES, then at least
half of the solutions to x 2 = y 2 (mod n) with xy Co-
PRIME to n are interesting. For such solutions, (n, x — y)
is neither n nor 1 and is therefore a nontrivial factor of
n (Pomerance 1996). This ALGORITHM can be used to
prove primality, but is not practical. In 1931, Lehmer
and Powers discovered how to search for such pairs using
Continued Fractions. This method was improved
by Morrison and Brillhart (1975) into the CONTINUED
Fraction Factorization Algorithm, which was the
fastest Algorithm in use before the Quadratic Sieve
Factorization Method was developed.
see also Prime Factorization Algorithms, Smooth
Number
References
Lehmer, D. H. and Powers, R. E. "On Factoring Large Num-
bers." Bull. Amer. Math. Soc. 37, 770-776, 1931.
Morrison, M. A. and Brillhart, J. "A Method of Factoring
and the Factorization of F 7 . n Math. Comput. 29, 183-
205, 1975.
Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996.
Fermat's Last Theorem
A theorem first proposed by Fermat in the form of a
note scribbled in the margin of his copy of the ancient
Greek text Arithmetica by Diophantus. The scribbled
note was discovered posthumously, and the original is
now lost. However, a copy was preserved in a book pub-
lished by Fermat's son. In the note, Fermat claimed to
have discovered a proof that the DlOPHANTINE EQUA-
TION x n + y n — z n has no INTEGER solutions for n > 2.
The full text of Fermat's statement, written in Latin,
reads "Cubum autem in duos cubos, aut quadrato-
quadratum in duos quadratoquadratos & generaliter
nullam in infinitum ultra quadratum potestatem in duos
eiusdem nominis fas est diuidere cuius rei demonstra-
tionem mirabilem sane detexi. Hanc marginis exiguitas
non caperet." In translation, "It is impossible for a cube
to be the sum of two cubes, a fourth power to be the
sum of two fourth powers, or in general for any number
that is a power greater than the second to be the sum
of two like powers. I have discovered a truly marvelous
demonstration of this proposition that this margin is too
narrow to contain."
As a result of Fermat's marginal note, the proposition
that the DlOPHANTINE EQUATION
x n + y n
(1)
where x, y, z, and n are INTEGERS, has no NONZERO so-
lutions for n > 2 has come to be known as Fermat's Last
Theorem. It was called a "THEOREM" on the strength of
Fermat's statement, despite the fact that no other math-
ematician was able to prove it for hundreds of years.
618
FermaVs Last Theorem
FermaVs Last Theorem
Note that the restriction n > 2 is obviously necessary
since there are a number of elementary formulas for gen-
erating an infinite number of PYTHAGOREAN TRIPLES
(#,y, z) satisfying the equation for n = 2,
2,2 2
x + y - z .
(2)
A first attempt to solve the equation can be made by
attempting to factor the equation, giving
(z n/2 + y n/2 ){z n/2 - y n/2 ) = x n . (3)
Since the product is an exact POWER,
,«/2
n/2
2q n
J z n/2 +2/ n/2
\ z n/2 _ ^n/2
= 2p"
Solving for y and z gives
(4)
f z n/2 = 2 "- 2 p" + q " ( 2 n/ 2 = p n + 2 n ~ 2 q n
\ y n ' 2 = 2 n - 2 p n - q n ° r \y n/2 = p n -2 n ~ 2 q n ,
(5)
which give
J z = (2 n - V + <f) 2/n / z = (p n + 2 n - V) 2/ "
\ y = (2 n - 2 p n - q n ) 2/n ° T \y={p n -2 n - 2 q n ) 2 / n .
(6)
However, since solutions to these equations in RATIONAL
NUMBERS are no easier to find than solutions to the
original equation, this approach unfortunately does not
provide any additional insight.
It is sufficient to prove Fermat's Last Theorem by con-
sidering Prime Powers only, since the arguments can
otherwise be written
(x m ) p + (y m ) P - (* m ) P ,
so redefining the arguments gives
x p + y p = z p .
(7)
(8)
The so-called "first case" of the theorem is for expo-
nents which are RELATIVELY PRIME to x, y> and z
(p\x, y, z) and was considered by Wieferich. Sophie Ger-
main proved the first case of Fermat's Last Theorem for
any ODD Prime p when 2p+ 1 is also a PRIME. Legen-
dre subsequently proved that if p is a PRIME such that
4p +1, Sp + 1, 10p + 1 ? 14p + 1, or 16p + 1 is also a
PRIME, then the first case of Fermat's Last Theorem
holds for p. This established Fermat's Last Theorem for
p < 100. In 1849, Kummer proved it for all REGULAR
Primes and Composite Numbers of which they are
factors (Vandiver 1929, Ball and Coxeter 1987).
Rummer's attack led to the theory of Ideals, and Van-
diver developed Vandiver's Criteria for deciding if
a given IRREGULAR PRIME satisfies the theorem. Gen-
occhi (1852) proved that the first case is true for p if
(p,p - 3) is not an Irregular Pair. In 1858, Rum-
mer showed that the first case is true if either (p } p — 3)
or (p,p — 5) is an IRREGULAR Pair, which was subse-
quently extended to include (p, p - 7) and (p, p - 9) by
Mirimanoff (1905). Wieferich (1909) proved that if the
equation is solved in integers RELATIVELY Prime to an
Odd Prime p, then
2 P_1 = 1 (modp 2 ).
(9)
(Ball and Coxeter 1987). Such numbers are called
Wieferich Primes. Mirimanoff (1909) subsequently
showed that
3 P_1 = 1 (modp 2 ) (10)
must also hold for solutions RELATIVELY PRIME to an
Odd Prime p, which excludes the first two Wieferich
PRIMES 1093 and 3511. Vandiver (1914) showed
5 P_1 = 1 (modp 2 ),
and Frobenius extended this to
ll p - 1 ,17 p - 1 = l (modp 2 ),
(11)
(12)
It has also been shown that if p were a PRIME of the
form 6x — 1, then
7 P-1 ,13 P
,19 p
= 1 (mod p ) ,
(13)
which raised the smallest possible p in the "first case" to
253,747,889 by 1941 (Rosser 1941). Granville and Mon-
agan (1988) showed if there exists a PRIME p satisfying
Fermat's Last Theorem, then
q?- 1 = 1 (modp 2 )
(14)
for q = 5, 7, 11, ..., 71. This establishes that
the first case is true for all PRIME exponents up to
714,591,416,091,398 (Vardi 1991).
The "second case" of Fermat's Last Theorem (for
p\x,yj z) proved harder than the first case.
Euler proved the general case of the theorem for n = 3,
Fermat n = 4, Dirichlet and Lagrange n = 5, In 1832,
Dirichlet established the case n = 14. The n = 7 case
was proved by Lame (1839), using the identity
(X + Y + Z) 7 - (X 7 + Y 7 4- Z 7 )
= 7(X + Y)(X + Z)(Y + Z)
x [(X 2 + Y 2 + Z 2 + XY + XZ + YZ) 2
+ XYZ{X + Y + Z)}. (15)
Although some errors were present in this proof, these
were subsequently fixed by Lebesgue (1840). Much ad-
ditional progress was made over the next 150 years, but
FermaVs Last Theorem
FermaVs Last Theorem
619
no completely general result had been obtained. Buoyed
by false confidence after his proof that Pi is TRANSCEN-
DENTAL, the mathematician Lindemann proceeded to
publish several proofs of Fermat's Last Theorem, all of
them invalid (Bell 1937, pp. 464-465). A prize of 100,000
German marks (known as the Wolfskel Prize) was also
offered for the first valid proof (Ball and Coxeter 1987,
p. 72).
A recent false alarm for a general proof was raised by
Y. Miyaoka (Cipra 1988) whose proof, however, turned
out to be flawed. Other attempted proofs among both
professional and amateur mathematicians are discussed
by vos Savant (1993), although vos Savant erroneously
claims that work on the problem by Wiles (discussed
below) is invalid. By the time 1993 rolled around, the
general case of Fermat's Last Theorem had been shown
to be true for all exponents up to 4 x 10 6 (Cipra 1993).
However, given that a proof of Fermat's Last Theo-
rem requires truth for all exponents, proof for any fi-
nite number of exponents does not constitute any sig-
nificant progress towards a proof of the general theorem
(although the fact that no counterexamples were found
for this many cases is highly suggestive).
In 1993, a bombshell was dropped. In that year,
the general theorem was partially proven by Andrew
Wiles (Cipra 1993, Stewart 1993) by proving the
Semistable case of the Taniyama-Shimura Conjec-
ture. Unfortunately, several holes were discovered in
the proof shortly thereafter when Wiles' approach via
the Taniyama-Shimura Conjecture became hung up
on properties of the Selmer Group using a tool called
an "Euler system." However, the difficulty was circum-
vented by Wiles and R. Taylor in late 1994 (Cipra 1994,
1995ab) and published in Taylor and Wiles (1995) and
Wiles (1995). Wiles' proof succeeds by (1) replacing
Elliptic Curves with Galois representations, (2) re-
ducing the problem to a Class Number Formula, (3)
proving that FORMULA, and (4) tying up loose ends that
arise because the formalisms fail in the simplest degen-
erate cases (Cipra 1995a).
The proof of Fermat's Last Theorem marks the end of a
mathematical era. Since virtually all of the tools which
were eventually brought to bear on the problem had yet
to be invented in the time of Fermat, it is interesting to
speculate about whether he actually was in possession
of an elementary proof of the theorem. Judging by the
temerity with which the problem resisted attack for so
long, Fermat's alleged proof seems likely to have been
illusionary.
see also abc Conjecture, Bogomolov-Miyaoka-
Yau Inequality, Mordell Conjecture, Pythag-
orean Triple, Ribet's Theorem, Selmer Group,
Sophie Germain Prime, Szpiro's Conjecture,
Taniyama-Shimura Conjecture, Vojta's Conjec-
ture, Waring Formula
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 69-73,
1987.
Beiler, A. H. "The Stone Wall." Ch. 24 in Recreations in
the Theory of Numbers: The Queen of Mathematics En-
tertains. New York: Dover, 1966.
Bell, E. T. Men of Mathematics. New York: Simon and
Schuster, 1937.
Bell, E. T. The Last Problem. New York: Simon and Schus-
ter, 1961.
Cipra, B. A. "Fermat Theorem Proved." Science 239, 1373,
1988.
Cipra, B. A. "Mathematics — Fermat's Last Theorem Finally
Yields." Science 261, 32-33, 1993.
Cipra, B. A. "Is the Fix in on Fermat's Last Theorem?" Sci-
ence 266, 725, 1994.
Cipra, B. A. "Fermat's Theorem — At Last." What's Hap-
pening in the Mathematical Sciences, 1995-1996, Vol. 3.
Providence, RI: Amer. Math. Soc, pp. 2-14, 1996.
Cipra, B. A. "Princeton Mathematician Looks Back on Fer-
mat Proof." Science 268, 1133-1134, 1995b.
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.
Cox, D. A. "Introduction to Fermat's Last Theorem." Amer.
Math. Monthly 101, 3-14, 1994.
Dickson, L. E. "Fermat's Last Theorem, ax r + by 3 = cz t , and
the Congruence x n -\-y n = z n (mod p)." Ch. 26 in History
of the Theory of Numbers, Vol. 2: Diophantine Analysis.
New York: Chelsea, pp. 731-776, 1952.
Edwards, H. M. Fermat's Last Theorem: A Genetic Intro-
duction to Algebraic Number Theory. New York: Springer-
Verlag, 1977.
Edwards, H. M. "Fermat's Last Theorem." Set Amer., Oct.
1978.
Granville, A. "Review of BBC's Horizon Program, 'Fermat's
Last Theorem'." Not. Amer. Math. Soc. 44, 26-28, 1997.
Granville, A. and Monagan, M. B. "The First Case of Fer-
mat's Last Theorem is True for All Prime Exponents up
to 714,591,416,091,389." Trans. Amer. Math. Soc. 306,
329-359, 1988.
Guy, R. K. "The Fermat Problem." §D2 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 144-146, 1994.
Hanson, A. "Fermat Project." http://www.cica. indiana.
edu/projects/Fermat/.
Kolata, G. "Andrew Wiles: A Math Whiz Battles 350- Year-
Old Puzzle." New York Times, June 29, 1993.
Lynch, J. "Fermat's Last Theorem." BBC Horizon tele-
vision documentary. http : //www . bbc . co . uk/horizon/
fermat. shtml.
Lynch, J. (Producer and Writer). "The Proof." NOVA tele-
vision episode. 52 mins. Broadcast by the U. S. Public
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Mirimanoff, D. "Sur le dernier theoreme de Fermat et le
criterium de wiefer." Enseiggnement Math. 11, 455-459,
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Mordell, L. J. Fermat's Last Theorem. New York: Chelsea,
1956.
Murty, V, K. (Ed.). Fermat's Last Theorem: Proceedings of
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Ribenboim, P. Lectures on Fermat's Last Theorem. New
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620
Fermat J s Lesser Theorem
Fermat's Little Theorem
Ribet, K. A. and Hayes, B. "Fermat's Last Theorem
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Ribet, K. A. and Hayes, B. Correction to "Fermat's Last
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May/June 1994.
Rosser, B. "On the First Case of Fermat's Last Theorem."
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Rosser, B. "A New Lower Bound for the Exponent in the
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Soc. 46, 299-304, 1940.
Rosser, B. "An Additional Criterion for the First Case of
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109-110, 1941.
Shanks, D. Solved and Unsolved Problems in Number Theory,
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Singh, S. Fermat's Enigma: The Quest to Solve the World's
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1997.
Stewart, I. "Fermat's Last Time-Trip." Sci. Amer. 269,
112-115, 1993.
Taylor, R. and Wiles, A. "Ring-Theoretic Properties of Cer-
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van der Poorten, A. Notes on Fermat's Last Theorem. New
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Vandiver, H. S. "On Fermat's Last Theorem." Trans. Amer.
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Vandiver, H. S. Fermat's Last Theorem and Related Topics
in Number Theory. Ann Arbor, MI: 1935.
Vandiver, H. S. "Fermat's Last Theorem: Its History and
the Nature of the Known Results Concerning It." Amer.
Math. Monthly, 53, 555-578, 1946.
Vardi, I. Computational Recreations in Mathematica. Read-
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vos Savant, M. The World's Most Famous Math Problem.
New York: St. Martin's Press, 1993.
Wieferich, A. "Zum let zt en Fermat'schen Theorem." J. reine
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Wiles, A. "Modular Elliptic- Curves and Fermat's Last The-
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Fermat's Lesser Theorem
see Fermat's Little Theorem
Fermat's Little Theorem
If p is a PRIME number and a a NATURAL NUMBER,
then
a p = a (mod p) . (1)
Furthermore, if p\a (p does not divide a), then there
exists some smallest exponent d such that
The theorem is easily proved using mathematical IN-
DUCTION. Suppose p\a p — a. Then examine
a d - 1 = (mod p)
and d divides p — 1. Hence,
a?' 1 -1 = (modp).
(2)
(3)
This is a generalization of the CHINESE HYPOTHESIS
and a special case of Euler's THEOREM. It is sometimes
called Fermat's Primality Test and is a Necessary
but not Sufficient test for primality. Although it was
presumably proved (but suppressed) by Fermat, the first
proof was published by Euler in 1749.
(a + l) p -(a+l).
From the BINOMIAL THEOREM,
(4)
(5)
Rewriting,
(a+ l)P-a'-l=fj)a'- l +Q«'- a + ...+ ^ 1 ^a.
/(6)
But p divides the right side, so it also divides the left
side. Combining with the induction hypothesis gives
that p divides the sum
[(a+ l) p - a p - 1] + (a p - a) = (a + if - (a + 1), (7)
as assumed, so the hypothesis is true for any a. The
theorem is sometimes called FERMAT'S SIMPLE THEO-
REM. Wilson's Theorem follows as a Corollary of
Fermat's Little Theorem.
Fermat's little theorem shows that, if p is PRIME, there
does not exists a base a < p with {a,p) = 1 such that
a p ~ 1 — 1 possesses a nonzero residue modulo p. If such
base a exists, p is therefore guaranteed to be compos-
ite. However, the lack of a nonzero residue in Fermat's
little theorem does not guarantee that p is PRIME. The
property of unambiguously certifying composite num-
bers while passing some PRIMES make Fermat's little
theorem a COMPOSITENESS Test which is sometimes
called the Fermat Compositeness Test. Composite
Numbers known as Fermat Pseudoprimes (or some-
times simply "PSEUDOPRIMES") have zero residue for
some as and so are not identified as composite. Worse
still, there exist numbers known as Carmichael NUM-
BERS (the smallest of which is 561) which give zero
residue for any choice of the base a Relatively Prime
top. However, Fermat's Little Theorem Converse
provides a criterion for certifying the primality of a num-
ber.
A number satisfying Fermat's little theorem for some
nontrivial base and which is not known to be composite
is called a PROBABLE Prime. A table of the small-
est Pseudoprimes P for the first 100 bases a follows
(Sloane's A007535).
Fermat's Little Theorem Converse
Fermat Number 621
a P
a
P
a
P
a
P
a P
2 341
22
69
42
205
62
63
82 91
3 91
23
33
43
77
63
341
83 105
4 15
24
25
44
45
64
65
84 85
5 124
25
28
45
76
65
133
85 129
6 35
26
27
46
133
66
91
86 87
7 25
27
65
47
65
67
85
87 91
8 9
28
87
48
49
68
69
88 91
9 28
29
35
49
66
69
85
89 99
10 33
30
49
50
51
70
169
90 91
11 15
31
49
51
65
71
105
91 115
12 65
32
33
52
85
72
85
92 93
13 21
33
85
53
65
73
111
93 301
14 15
34
35
54
55
74
75
94 95
15 341
35
51
55
63
75
91
95 141
16 51
36
91
56
57
76
77
96 133
17 45
37
45
57
65
77
95
97 105
18 25
38
39
58
95
78
341
98 99
19 45
39
95
59
87
79
91
99 145
20 21
40
91
60
341
80
81
100 259
21 55
41
105
61
91
81
85
see also Binomial Theorem, Carmichael Number,
Chinese Hypothesis, Composite Number, Compos-
iteness Test, Euler's Theorem, Fermat's Little
Theorem Converse, Fermat Pseudoprime, Mod-
ulo Multiplication Group, Pratt Certificate,
Primality Test, Prime Number, Pseudoprime,
Relatively Prime, Totient Function, Wieferich
Prime, Wilson's Theorem, Witness
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 61, 1987.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 141-142, 1996.
Courant, R. and Robbins, H. "Fermat's Theorem." §2.2 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. 37-38, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 20, 1993.
Sloane, N. J. A. Sequence A007535/M5440 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Fermat's Little Theorem Converse
The converse of FERMAT'S LITTLE THEOREM is also
known as Lehmer's THEOREM. It states that, if an
Integer x is Prime to m and x m_1 = 1 (mod m)
and there is no Integer e < m — 1 for which x e =
1 (mod m), then m is Prime. Here, x is called a Wit-
ness to the primality of m. This theorem is the basis
for the Pratt Primality Certificate.
see also FERMAT'S LITTLE THEOREM, PRATT CERTIFI-
CATE, Primality Certificate, Witness
References
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, p. 96, 1994.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 278-279, 1991.
Fermat-Lucas Number
A number of the form 2 n + 1 obtained by setting x = 1
in a Fermat-Lucas Polynomial. The first few are 3,
5, 9, 17, 33, . . . (Sloane's A000051).
see also Fermat Number (Lucas)
References
Sloane, N. J. A. Sequence A000051/M0717 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Fermat Number
A Binomial Number of the form F n = 2 2n + 1. The
first few for n = 0, 1, 2, . . . are 3, 5, 17, 257, 65537,
4294967297, ... (Sloane's A000215). The number of
Digits for a Fermat number is
D(n) = L [log(2 2 " + 1)] + lj « [log(2 2 ") + lj
= [2"log2 + lJ. (1)
Being a Fermat number is the NECESSARY (but not SUF-
FICIENT) form a number
N n = 2" + 1
(2)
must have in order to be Prime. This can be seen by
noting that if N n = 2™ + 1 is to be PRIME, then n cannot
have any ODD factors b or else N n would be a factorable
number of the form
2 n + l = (2 a ) b + l
■ + !]• (3)
Therefore, for a Prime N ni n must be a Power of 2.
Fermat conjectured in 1650 that every Fermat number
is Prime, but only Composite Fermat numbers F n
are known for n > 5. Eisenstein (1844) proposed as
a problem the proof that there are an infinite number
of Fermat primes (Ribenboim 1996, p. 88), but this has
not yet been achieved. An anonymous writer proposed
that numbers of the form 2 2 + 1, I* + 1, 2 22 +1 were
PRIME. However, this conjecture was refuted when Sel-
fridge (1953) showed that
+ 1 =
+ 1
(4)
is Composite (Ribenboim 1996, p. 88). Numbers of the
form a 2 + b 2 are called generalized Fermat numbers
(Ribenboim 1996, pp. 359-360).
Fermat numbers satisfy the RECURRENCE RELATION
F m =F F 1 ---F m - 1 + 2. (5)
F n can be shown to be Prime iff it satisfies Pepin's
Test
3 (F»-i)/a = _! ( mo dF n ). (6)
622 Fermat Number
Pepin's Theorem
= -1 (mod F n )
is also Necessary and Sufficient.
(7)
In 1770, Euler showed that any FACTOR of F n must have
the form
.,71+1
K + l,
(8)
where K is a POSITIVE INTEGER. In 1878, Lucas in-
creased the exponent of 2 by one, showing that FACTORS
of Fermat numbers must be of the form
If
2 L + 1.
F =PlP2 ---Pr
(9)
(10)
is the factored part of F n = FC (where C is the cofactor
to be tested for primality), compute
A = 3"
F n -1
(mod F n )
B = 3 F_1 (mod F n )
R-A-B (mod C).
(11)
(12)
(13)
Then if R = 0, the cofactor is a PROBABLE PRIME to
the base 3^; otherwise C is Composite.
In order for a POLYGON to be circumscribed about a
Circle (i.e., a Constructible Polygon), it must
have a number of sides N given by
N = 2 k F .-Fn,
(14)
where the F n are distinct Fermat primes. This is equiv-
alent to the statement that the trigonometric func-
tions sin(&7r/iV), cos(k7r/N), etc., can be computed in
terms of finite numbers of additions, multiplications,
and square root extractions iff N is of the above form.
The only known Fermat PRIMES are
Fo = 3
F 1= 5
F 2 = 17
F 3 = 257
F 4 = 65537
and it seems unlikely that any more exist.
Factoring Fermat numbers is extremely difficult as a re-
sult of their large size. In fact, only F$ to Fn have been
Fermat Number
complete factored, as summarized in the following table.
Written out explicitly, the complete factorizations are
F 5 = 641 • 6700417
F 6 = 274177 • 67280421310721
F 7 = 59649589127497217 ■ 5704689200685129054721
F 8 = 1238926361552897 • 93461639715357977769163 • ■ •
• • • 558199606896584051237541638188580280321
F 9 = 2424833 • 74556028256478842083373957362004- • •
• • ■ 54918783366342657 ■ P99
Fio = 45592577 • 6487031809 • 46597757852200185 • • •
• • • 43264560743076778192897 ■ P252
Fu = 319489 • 974849 • 167988556341760475137
• 3560841906445833920513 • P564.
Here, the final large Prime is not explicitly given since
it can be computed by dividing F n by the other given
factors.
F
Digits
Facts.
Digits
Reference
5
10
2
3, 7
Euler 1732
6
20
2
6, 14
Landry 1880
7
39
2
7, 22
Morrison and
Brillhart 1975
8
78
2
16, 62
Brent and Pollard 1981
9
155
3
7
', 49, 99
Manasse and Lenstra
(In Cipra 1993)
10
309
4
8,
10,
40, 252
Brent 1995
11
617
5
6, 6,
21,
22, 564
Brent 1988
Tables of known factors of Fermat numbers are given by
Keller (1983), Brillhart et al (1988), Young and Buell
(1988), Riesel (1994), and Pomerance (1996). Young
and Buell (1988) discovered that F 20 is COMPOSITE,
and Crandall et al. (1995) that F 22 is COMPOSITE. A
current list of the known factors of Fermat numbers is
maintained by Keller, and reproduced in the form of a
Mathematical notebook by Weisstein. In these tables,
since all factors are of the form k2 n +l, the known factors
are expressed in the concise form (fe, n). The number of
factors for Fermat numbers F n for n = 0, 1, 2, . . . are
1,1,1, 1,1,2,2,2,2,3,4,5,....
see also CULLEN NUMBER, PEPIN'S TEST, PEPIN'S
Theorem, Pocklington's Theorem, Polygon,
Proth's Theorem, Selfridge-Hurwitz Residue,
Woodall Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
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Amer, Math. Soc. Abstracts 1, 565, 1980.
Brent, R. P. "Factorisation of F10." http://cslab.anu.edu.
au/~rpb/F10.html.
Brent, R. P "Factorization of the Tenth and Eleventh
Fermat Numbers." Submitted to Math. Corn-put.
ftp : //nimbus . arm . edu.au/pub/Brent/rpbl61tr . dvi . Z.
Fermat Number
Fermat Point 623
Brent, R. P. and Pollard, J. M. "Factorization of the Eighth
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Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.;
and Tuckerman, B. Factorizations of b 71 ± 1, b = 2,
3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence,
RI: Amer. Math. Soc, pp. lxxxvii and 2—3 of Update 2.2,
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Cipra, B. "Big Number Breakdown." Science 248, 1608,
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Conway, J. H. and Guy, R. K. "Fermat's Numbers." In The
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Cormack, G. V. and Williams, H. C. "Some Very Large
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Courant, R. and Robbins, H. What is Mathematics?: An
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Twenty-Second Fermat Number is Composite." Math.
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Dixon, R. Mathographics. New York: Dover, p. 53, 1991.
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Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996.
Ribenboim, P. "Fermat Numbers" and "Numbers k x 2 n ±l."
§2.6 and 5.7 in The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 83-90 and 355-360, 1996.
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Basel: Birkhauser, pp. 384-388, 1994.
Robinson, R. M. "A Report on Primes of the Form k ■ 2 n -f 1
and on Factors of Fermat Numbers." Proc. Amer. Math.
Soc. 9, 673-681, 1958.
Selfridge, J. L. "Factors of Fermat Numbers." Math. Com-
put. 7, 274-275, 1953.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 13 and 78-80, 1993.
Sloane, N. J. A. Sequence A000215/M2503 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
$fr Weisstein, E. W. "Fermat Numbers." http: //www. astro.
virginia.edu/~eww6n/math/notebooks/Fermat .m.
Wrathall, C. P. "New Factors of Fermat Numbers." Math.
Comput. 18, 324-325, 1964.
Young, J. and Buell, D. A, "The Twentieth Fermat Number
is Composite." Math. Comput. 50, 261-263, 1988.
Fermat Number (Lucas)
A number of the form 2 n — 1 obtained by setting x = 1
in a Fermat Polynomial is called a Mersenne Num-
ber.
see also Fermat-Lucas Number, Mersenne Number
Fermat Point
Also known as the first ISOGONIC CENTER and the TOR-
ricelli Point. In a given Acute Triangle AABC,
the Fermat point is the point X which minimizes the
sum of distances from ^4, £?, and C,
\AX\ + \BX\ + \CX\.
(1)
This problem is called Fermat's Problem or
Steiner'S PROBLEM (Courant and Robbins 1941) and
was proposed by Fermat to Torricelli. Torricelli's solu-
tion was published by his pupil Viviani in 1659 (Johnson
1929).
If all Angles of the Triangle are less than 120°
(27r/3), then the Fermat point is the interior point X
from which each side subtends an Angle of 120°, i.e.,
LBXC = LCXA = LAXB = 120°
(2)
The Fermat point can also be constructed by drawing
Equilateral Triangles on the outside of the given
Triangle and connecting opposite Vertices. The
624 Fermat's Polygonal Number Theorem
Fermat Polynomial
three diagonals in the figure then intersect in the Fer-
mat point. The TRIANGLE CENTER FUNCTION of the
Fermat point is
a = csc(^4+ |?r) (3)
— bc[c a +(c + a — b ) ][a b — (a -\-b — c ) ]
x [4A- v^(6 2 +c 2 -a 2 )]. (4)
The Antipedal Triangle is Equilateral and has
Area
(5)
A' = 2A Tl + cotajcot (^)
where tv is the Brocard Angle.
Given three Positive Real Numbers Z, m, n, the "gen-
eralized" Fermat point is the point P of a given ACUTE
Triangle AABC such that
IPA + m- PB + n- PC
(6)
is a minimum (Greenberg and Robertello 1965, van de
Lindt 1966, Tong and Chua 1995)
see also ISOGONIC CENTERS
References
Courant, R. and Robbins, H. What is Mathematics? , 2nd ed.
Oxford, England: Oxford University Press, 1941.
Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.
London: Hodgson, p. 107, 1913.
Greenberg, I. and Robertello, R. A. "The Three Factory
Problem." Math. Mag. 38, 67-72, 1965.
Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., pp. 24-34, 1973.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 221-222, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, p. 174, 1994.
Kimberling, C. "Fermat Point." http://vvw .evansville.
edu/~ck6/t cent ers/class/f ermat.html.
MowafTaq, H. "An Advanced Calculus Approach to Finding
the Fermat Point." Math. Mag. 67, 29-34, 1994.
Pottage, J. Geometrical Investigations. Reading, MA:
Addison- Wesley, 1983.
Spain, P. G. "The Fermat Point of a Triangle." Math. Mag.
69, 131-133, 1996.
Tong, J. and Chua, Y. S. "The Generalized Fermat's Point."
Math. Mag. 68, 214-215, 1995.
van de Lindt, W. J. "A Geometrical Solution of the Three
Factory Problem." Math. Mag. 39, 162-165, 1966.
Fermat's Polygonal Number Theorem
In 1638, Fermat proposed that every Positive Integer
is a sum of at most three TRIANGULAR NUMBERS, four
Square Numbers, five Pentagonal Numbers, and
n n-POLYGONAL NUMBERS. Fermat claimed to have a
proof of this result, although Fermat's proof has never
been found. Gauss proved the triangular case, and noted
the event in his diary on July 10, 1796, with the notation
* * ETRHKA
num — A + A -f- A.
This case is equivalent to the statement that every num-
ber of the form 8ra + 3 is a sum of three Odd SQUARES
(Duke 1997). More specifically, a number is a sum of
three Squares Iff it is not of the form 4 6 (8m + 7) for
6 > 0, as first proved by Legendre in 1798.
Euler was unable to prove the square case of Fermat's
theorem, but he left partial results which were subse-
quently used by Lagrange. The square case was finally
proved by Jacobi and independently by Lagrange in
1772. It is therefore sometimes known as LAGRANGE'S
Four-Square Theorem. In 1813, Cauchy proved the
proposition in its entirety.
see also Fifteen Theorem, Vinogradov's Theo-
rem, Lagrange's Four-Square Theorem, War-
ing's Problem
References
Cassels, J. W. S. Rational Quadratic Forms. New York: Aca-
demic Press, 1978.
Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane,
N. J. A. "The Primary Pretenders." Acta Arith. 78, 307-
313, 1997.
Duke, W. "Some Old Problems and New Results about Quad-
ratic Forms." Not. Amer. Math. Soc. 44, 190-196, 1997.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 143-144, 1993.
Smith, D. E. A Source Book in Mathematics. New York:
Dover, p. 91, 1984.
Fermat Polynomial
The Polynomials obtained by setting p(x) = 3a and
q(x) = -2 in the LUCAS POLYNOMIAL SEQUENCES. The
first few Fermat polynomials are
^i(a) = l
F 2 (x) = 3a
JF 3 (a) = 9a 2 -2
F 4 (x) = 27x 3 -12a
r 5 (x) = 81a 4 - 54a 2 + 4,
and the first few Fermat-Lucas polynomials are
/i(a) = 3a
f 2 (x) = 9x 2 -A
f z (x) = 27a 3 - 18a
/ 4 (a) = 81a 4 -72a 2 +8
fa(x) = 243a 5 - 270a 3 + 60a.
Fermat and Fermat-Lucas POLYNOMIALS satisfy
J r «(l) = J r n
/n(l) = /»
where Tn are FERMAT NUMBERS and /„ are Fermat-
Lucas Numbers.
Fermat's Primality Test
Format's Spiral 625
Fermat's Primality Test
see Fermat's Little Theorem
Fermat Prime
A Fermat Number F n = 2 2n + 1 which is Prime.
see also Constructible Polygon, Fermat Number
Fermat's Problem
In a given ACUTE TRIANGLE AABC, locate a point
whose distances from A, B, and C have the smallest
possible sum. The solution is the point from which each
side subtends an angle of 120°, known as the FERMAT
Point.
see also Acute Triangle, Fermat Point
WlEFERlCH Primes 1093 and 3511 (Lehmer 1981, Cran-
dall 1986).
see also Wieferich Prime
References
Crandall, R. Projects in Scientific Computation. New York:
Springer- Verlag, 1986.
Lehmer, D. H. "On Fermat's Quotient, Base Two." Math.
Comput. 36, 289-290, 1981.
Fermat's Right Triangle Theorem
The Area of a Rational Right Triangle cannot be
a Square Number. This statement is equivalent to "a
Congruum cannot be a Square Number."
see also Congruum, Rational Triangle, Right
Triangle, Square Number
Fermat Pseudoprime
A Fermat pseudoprime to a base a, written psp(a), is a
Composite Number n such that a n_1 = 1 (mod n)
(i.e., it satisfies Fermat's Little Theorem, some-
times with the requirement that n must be ODD; Pomer-
ance et al. 1980). psp(2)s are called Poulet NUMBERS
or, less commonly, SARRUS NUMBERS or FERMATIANS
(Shanks 1993). The first few Even psp(2)s (including
the PRIME 2 as a pseudoprime) are 2, 161038, 215326,
... (Sloane's A006935).
If base 3 is used in addition to base 2 to weed out po-
tential Composite Numbers, only 4709 Composite
NUMBERS remain < 25 x 10 9 . Adding base 5 leaves 2552,
and base 7 leaves only 1770 COMPOSITE NUMBERS.
see also Fermat's Little Theorem, Poulet Num-
ber, Pseudoprime
References
Pomerance, C; Selfridge, J. L.; and Wagstaff, S. S. "The
Pseudoprimes to25-10 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, p. 115, 1993.
Sloane, N. J. A. Sequence A006935/M2190 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Fermat Quotient
The Fermat quotient for a number a and a PRIME base
p is defined as
a p ~ l - 1
q p (a) = ?——±. (1)
If p\ab t then
q p {ab) = q P (a) + q p (b)
gp(p±l) = =Fl
(2)
(3)
,(4)
all (mod p). The quantity q p {2) = (2 P l - l)/p is
known to be SQUARE for only two PRIMES: the so-called
Fermat's Sigma Problem
Solve
a(x 3 )=y 2
and
where a is the DIVISOR FUNCTION.
see also Wallis's Problem
Fermat's Simple Theorem
see Fermat's Little Theorem
Fermat's Spiral
An Archimedean Spiral with m = 2 having polar
equation
r ■
~a6 xl \
discussed by Fermat in 1636 (MacTutor Archive). It is
also known as the PARABOLIC SPIRAL. For any given
POSITIVE value of 0, there are two corresponding values
of r of opposite signs. The resulting spiral is therefore
symmetrical about the line y — — x. The CURVATURE is
k(9)
3a _L n 2 fi
(£+« 2 ')
3/2 '
References
Dixon, R. Mathographics. New York: Dover, p. 121, 1991.
626 Fermat Spiral Inverse Curve
Feuerbach's Conic Theorem
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 69-70, 1993.
Lee, X. "Equiangular Spiral." http://www.best.com/-xah/
SpecialPlaneCuxvesjdir/EquiangularSpiraljdir/
equiangularSpiral .html.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 175, 1967.
MacTutor History of Mathematics Archive. "Fermat' s Spi-
ral." http: //www-groups .dcs . st-and.ac .uk/ -history/
Curves/Fermats .html.
Wells, D. The Penguin Dictionary of Curious and Interesting
Geometry. Middlesex, England: Penguin Books, 1991.
Fermat Spiral Inverse Curve
The Inverse Curve of Fermat's Spiral with the ori-
gin taken as the INVERSION CENTER is the LlTUUS.
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 186-187, 1972.
Fermat Sum Theorem
The only whole number solution to the DlOPHANTINE
Equation
y
x 2 + 2
is y = 3, x — ±5. This theorem was offered as a problem
by Fermat, who suppressed his own proof.
Fermat's Theorem
A Prime p can be represented in an essentially unique
manner in the form x 2 + y 2 for integral x and y Iff
p = 1 (mod 4) or p — 2. It can be restated by letting
Q(x,y) = x 2 +y 2 ,
then all RELATIVELY Prime solutions (x,y) to the prob-
lem of representing Q(x,y) = m for m any INTEGER
are achieved by means of successive applications of the
Genus Theorem and Composition Theorem. There
is an analog of this theorem for ElSENSTElN INTEGERS.
see also Eisenstein Integer, Square Number
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 142-143, 1993.
Fermat's Two- Square Theorem
see Fermat's Theorem
Fermat ian
see Poulet Number
Fermi-Dirac Distribution
A distribution which arises in the study of half-integral
spin particles in physics,
k 8
P(k)
e fc-M + 1'
Its integral is
r °° k s dk
f
Jo
e fc-M _|_ 1
= eT(« + l)*(-e /1 ) a + l,l),
where $ is the LERCH TRANSCENDENT.
Fern
see Barnsley's Fern
Ferrari's Identity
(a 2 -f 2ac - 2bc - b 2 ) 4 + (6* - 2ab - 2ac - c 2 ) 4
^c 2 +2ab + 2bc- a 2 ) 4 = 2{a 2 +b 2 + c 2 -- ab + ac + bc) 4 .
see also Diophantine Equation — Quartic
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 96-97, 1994.
Ferrers Diagram
see Young Diagram
Ferrers' Function
An alternative name for an associated Legendre POLY-
NOMIAL.
see also Legendre Polynomial
References
Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, p. 246, 1991.
Ferrier's Prime
According to Hardy and Wright (1979), the largest
PRIME found before the days of electronic computers
is the 44-digit number
F=£(2 148 + l)
= 20988936657440586486151264256610222593863921,
which was found using only a mechanical calculator.
References
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 16-22, 1979.
Feuerbach Circle
see Nine-Point Circle
Feuerbach's Conic Theorem
The LOCUS of the centers of all CONICS through the
Vertices and Orthocenter of a Triangle (which
are RECTANGULAR Hyperbolas when not degenerate),
is a Circle through the Midpoints of the sides, the
points half way from the ORTHOCENTER to the VER-
TICES, and the feet of the Altitude.
see also Altitude, Conic Section, Feuerbach's
Theorem, Kiepert's Hyperbola, Midpoint, Or-
thocenter, Rectangular Hyperbola
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 198, 1959.
Feuerbach Point
Feuerbach Point
The point F at which the INCIRCLE and Nine-Point
Circle are tangent. It has Triangle Center Func-
tion
a = 1 -cos(B-C).
see also Feuerbach's Theorem
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 200, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Salmon, G. Conic Sections, 6th ed. New York: Chelsea,
p. 127, 1954.
Feuerbach's Theorem
1. The CIRCLE which passes through the feet of the
Perpendiculars dropped from the Vertices of
any TRIANGLE on the sides opposite them passes
also through the Midpoints of these sides as well
as through the MIDPOINT of the segments which join
the VERTICES to the point of intersection of the PER-
PENDICULAR (a Nine-Point Circle).
2. The Nine-Point Circle of any Triangle is Tan-
gent internally to the INCIRCLE and TANGENT ex-
ternally to the three ExciRCLES.
see also Excircle, Feuerbach Point, Incircle,
Midpoint, Nine-Point Circle, Perpendicular,
Tangent
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 117-119, 1967.
Dixon, R. Mathographics. New York: Dover, p. 59, 1991.
Fiber Space 627
Feynman Point
The sequence of six 9s which begins at the 762th decimal
place of Pi,
tt = 3.14159 ... 134 999999 837 ... .
see also Pi
FFT
see Fast Fourier Transform
Fiber
A quantity F corresponding to a FIBER BUNDLE, where
the Fiber Bundle is a Map / : E -> £, with E the
Total Space of the Fiber Bundle and B the Base
Space of the Fiber Bundle.
see also FIBER BUNDLE, WHITNEY SUM
Fiber Bundle
A fiber bundle (also called simply a Bundle) with
Fiber F is a Map / : E -> B where E is called the To-
tal Space of the fiber bundle and B the Base Space
of the fiber bundle. The main condition for the MAP to
be a fiber bundle is that every point in the Base Space
b e B has a Neighborhood U such that /~ 1 (C/) is
Homeomorphic to U x F in a special way. Namely, if
h: f~\U)^UxF
is the HOMEOMORPHISM, then
proj^ oh = / (/ -i (l7) |,
where the MAP proj^ means projection onto the U com-
ponent. The homeomorphisms h which "commute with
projection" are called local TRIVIALIZATIONS for the
fiber bundle /. In other words, E looks like the product
B x F (at least locally), except that the fibers / _1 (x)
for x € B may be a bit "twisted."
Examples of fiber bundles include any product B xF — >
B (which is a bundle over B with FIBER F), the MOBIUS
Strip (which is a fiber bundle .over the CIRCLE with
Fiber given by the unit interval [0,1]; i.e, the Base
Space is the Circle), and § 3 (which is a bundle over S 2
with fiber § ) . A special class of fiber bundle is the VEC-
TOR Bundle, in which the Fiber is a Vector Space.
see also Bundle, Fiber Space, Fibration
Fiber Space
A fiber space, depending on context, means either a
Fiber Bundle or a Fibration.
see also FIBER BUNDLE, FlBRATION
628 Fibonacci Dual Theorem
Fibonacci Matrix
Fibonacci Dual Theorem
Let F n be the nth FIBONACCI NUMBER. Then the se-
quence {Fn}ZL2 = {!> 2, 3, 5, 8, ...} is COMPLETE,
even if one is restricted to subsequences in which no two
consecutive terms are both passed over (until the desired
total is reached; Brown 1965, Honsberger 1985).
see also Complete Sequence, Fibonacci Number.
References
Brown, J. L. Jr. "A New Characterization of the Fibonacci
Numbers." Fib. Quart 3, 1-8, 1965.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., p. 130, 1985.
Fibonacci Hyperbolic Cosine
Let
V> = 1 + = §(3 + V5 ) « 2.618034 (1)
where <f> is the GOLDEN RATIO, and
a = ln0^ 0.4812118.
Then define
_ ^+i/ 2 + ^" (:c+1/2)
cFh(*) _ ^
0(2x+l) + ^-(2x+l)
= V5
2
= — =cosh[(2x + l)a).
v5
This function satisfies
cFh(-x) = cFh(« - 1).
(2)
(3)
(4)
(5)
(6)
For n G Z, cFh(n) = F 2n +i where F n is a FIBONACCI
Number.
References
Trzaska, Z. W. "On Fibonacci Hyperbolic Trigonometry and
Modified Numerical Triangles." Fib. Quart. 34, 129-138,
1996.
Fibonacci Hyperbolic Cotangent
cFh(a;)
ctFh(x)
sFh(z) :
where cFh(x) is the Fibonacci Hyperbolic Cosine
and sFh(:r) is the Fibonacci Hyperbolic Sine.
References
Trzaska, Z. W. "On Fibonacci Hyperbolic Trigonometry and
Modified Numerical Triangles." Fib. Quart. 34, 129-138,
1996.
Fibonacci Hyperbolic Sine
Let
i> = 1 + (f> = |(3 + VE ) « 2.618034 (1)
where is the Golden Ratio, and
a = \n<f>K 0.4812118.
(2)
Then define
(3)
02* _ 0-2x
(4)
2
= — — sinh[2xa].
v5
(5)
For n e Z, sFh(n) = F 2n where F n is a FIBONACCI
Number. The function satisfies
sFh(-z) = -sFh(a;).
(6)
References
Trzaska, Z. W. "On Fibonacci Hyperbolic Trigonometry and
Modified Numerical Triangles." Fib. Quart. 34, 129-138,
1996.
Fibonacci Hyperbolic Tangent
_ sFh(x)
tFh(z)
cFh(x) '
where sFh(x) is the FIBONACCI HYPERBOLIC SINE and
cFh(x) is the Fibonacci Hyperbolic Cosine.
References
Trzaska, Z. W. "On Fibonacci Hyperbolic Trigonometry and
Modified Numerical Triangles." Fib. Quart. 34, 129-138,
1996.
Fibonacci Identity
Since
\(a + ib){c + id)\ = \a + ib\ \c + di\ (1)
I (ac - bd) + i{bc + ad)\ = y/a 2 + b*y/c' + <P i (2)
it follows that
(a 2 + b 2 ){c 2 +d 2 ) = {ac-bd) 2 + {bc+adf - e 2 + / 2 . (3)
This identity implies the 2-D CAUCHY-SCHWARZ Sum
Inequality.
see also Cauchy-Schwarz Sum Inequality, Euler
Four-Square Identity
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 9, 1996.
Fibonacci Matrix
A Square Matrix related to the Fibonacci Num-
bers. The simplest is the FIBONACCI Q-MATRIX.
Fibonacci n-Step Number
Fibonacci Number
629
Fibonacci n-Step Number
An n-step Fibonacci sequence is given by defining Fk
for k < 0, Fi = F 2 = 1, F 3 = 2, and
Fk = /^ F n -i
(i)
for A; > 3. The case n = 1 corresponds to the degener-
ate 1, 1, 2, 2, 2, 2 . . . , n = 2 to the usual FIBONACCI
NUMBERS 1, 1, 2, 3, 5, 8, . . . (Sloane's A000045), n = 3
to the Tribonacci Numbers 1, 1, 2, 4, 7, 13, 24, 44,
81, ... (Sloane's A000073), n = 4 to the Tetranacci
Numbers 1, 1, 2, 4, 8, 15, 29, 56, 108, ... (Sloane's
A000078), etc.
The limit limfc-^oo Fk/Fk-i is given by solving
x n {2-x) = l (2)
for x and taking the Real Root x > 1. If n = 2, the
equation reduces to
s 2 (2-a:) = 1
2:r 2 4- 1 = (x - l)(x 2 - x - 1) = 0,
giving solutions
The ratio is therefore
M(l±>/5).
(3)
(4)
(5)
»= £(l + \/5) = 0=1.618..., (6)
which is the Golden Ratio, as expected. Solutions
for n = 1, 2, ... are given numerically by 1, 1.61803,
1.83929, 1.92756, 1.96595, . . . , approaching 2asn->oo.
see also Fibonacci Number, Tribonacci Number
References
Sloane, N. J. A. Sequences A000045/M0692, A000073/
M1074, and A000078/M1108 in "An On-Line Version of
the Encyclopedia of Integer Sequences."
Fibonacci Number
The sequence of numbers defined by the U n in the LUCAS
Sequence. They are companions to the Lucas NUM-
BERS and satisfy the same RECURRENCE RELATION,
Fn = F n -2 + F n -
(1)
for n = 3, 4, . . . , with F x = F 2 = 1. The first few
Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, ...
(Sloane's A000045). The Fibonacci numbers give the
number of pairs of rabbits n months after a single pair
begins breeding (and newly born bunnies are assumed
to begin breeding when they are two months old).
The ratios of alternate Fibonacci numbers are given by
the convergents to 0~ 2 , where cj> is the Golden Ratio,
and are said to measure the fraction of a turn between
successive leaves on the stalk of a plant (PhyllOTAXIs):
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 Cox-
eter 1987). The Fibonacci numbers are sometimes called
Pine Cone Numbers (Pappas 1989, p. 224)
Another Recurrence RELATION for the Fibonacci
numbers is
"n + l
F n (l + v/5) + l
- [<f>F n + ij , (2)
where [zj is the FLOOR FUNCTION and <p is the GOLDEN
RATIO. This expression follows from the more general
Recurrence Relation that
0. (3)
The Generating Function for the Fibonacci numbers
is
F n
F n +i
Fn+k
F n +k+i
i^n + fc + 2
Fn+2k
Fn + k(k-l) + l
^n + A:Cfc-l) + 2 *
- • F n+k 2
*■ — ' 1 — x — X
(4)
Yuri Matijasevic (1970) proved that the equation n =
F 2 m is a DlOPHANTlNE Equation. This led to the proof
of the impossibility of the tenth of Hilbert's Problems
(does there exist a general method for solving DlOPHAN-
TlNE Equations?) by Julia Robinson and Martin Davis
in 1970.
The Fibonacci number F n +i gives the number of ways
for 2 x 1 Dominoes to cover a 2 x n Checkerboard,
as illustrated in the following diagrams (Dickau).
630 Fibonacci Number
Fibonacci Number
The number of ways of picking a Set (including the
Empty Set) from the numbers 1, 2, . . . , n without
picking two consecutive numbers is F n+2 . The num-
ber of ways of picking a set (including the Empty Set)
from the numbers 1, 2, . . . , n without picking two con-
secutive numbers (where 1 and n are now consecutive)
is L n = F n +i + JFn-i, where L n is a LUCAS Number.
The probability of not getting two heads in a row in n
tosses of a Coin is F n +2/2 n (Honsberger 1985, pp. 120-
122). Fibonacci numbers are also related to the number
of ways in which n Coin TOSSES can be made such
that there are not three consecutive heads or tails. The
number of ideals of an n-element Fence Poset is the
Fibonacci number F n .
Sum identities are
^Ffc=F n + 2 -l.
(5)
fc=i
F x 4 F 3 4 F 5 + . . . 4 F 2k +i = F 2k+2 (6)
1 4 F 2 4 F 4 + F 6 4 . . . 4 F 2k = F 2k +i (7)
/ ^ Fk = F n F n
77T 771 2 77) 2
*T2n — i*n + l — * n-1
Fzn = F n +i + F n 4 F n -i .
(8)
(9)
(10)
Additional Recurrence Relations are Cassini's
Identity
F n -iF n +i — F n = (—1)"
and the relations
F 2n+1 = 1 + F 2 4 F 4 + • - ■ + F 2n (12)
F n+ i 2 =4F n F n ^+F n _ 2 2 (13)
(Brousseau 1972),
Fn+m = Fn-lFm 4 F n F m + i (14)
F( k + \)n — F n -iFkn + FnFkn+1 (15)
(Honsberger 1985, p. 107),
F n = F t F n -i+i + F/_iF„_(, (16)
so if / = n - I + 1, then 2Z = n + 1 and / = (n 4 l)/2
F„ = F
n = **(n+l)/2 "I" ^(n-l)/2
+ Vi)
Letting fc = (n - l)/2,
F 2 k + 1 — Fk + l + Fjfc
F n +2 — -Pn+i = F n F n +3
(17)
(18)
(19)
Sum Formulas for F n include
F --L
.iK) + **(0 + -
Cesaro derived the FORMULAS
fc=o x 7
fe=0 x '
(21)
(22)
(23)
(24)
(Honsberger 1985, pp. 109-110). Additional identities
can be found throughout the Fibonacci Quarterly jour-
nal. A list of 47 generalized identities are given by Hal-
ton (1965).
In terms of the Lucas NUMBER L nj
F 2n = F n L n
F 2n {L 2n — 1) = Fq u
Fm+p + (~~1) Fm-p = FpL n
a+4n
(25)
(26)
(27)
/ ^ Fk = F a +4n+2 — F a +2 — F 2n L a + 2 n+2 (28)
k — a+1
( n ) (Honsberger 1985, pp. 111-113). A remarkable identity
exp{Lix+±L 2 x 4 + ±L z x * + ...) = Fi 4F 2 z4F 3 a: +. . .
(29)
(Honsberger 1985, pp. 118-119). It is also true that
and
5F„ a =L„ a -4(-l)"
Ln — { — 1) Ln+a
F n 2 - (-l)°F n + a 2
for a Odd, and
L n + L n + a — 8(— l) n
= 5
= 5
(30)
(31)
(32)
F n 2 - F„_i 2 + 3F n _ 2 2 4 2F n _ 2 F n _ 3 . (20)
Fn 4 F n + a
for a Even (Freitag 1996).
The equation (1) is a LINEAR RECURRENCE SEQUENCE
X n = Ax x -i + BXn-2 Tl > 3, (33)
so the closed form for F n is given by
a — p
Fn =
a-0
(34)
Fibonacci Number
Fibonacci Number 631
where a and j3 are the roots of x 2 = Ax + B. Here,
^4 = B — 1, so the equation becomes
x — x — 1 = 0,
which has ROOTS
x = ±(1±VE).
The closed form is therefore given by
•Tn — p= •
(35)
(36)
(37)
This is known as BlNET'S FORMULA. Another closed
form is
V5\ 2
V5.
(38)
where [x] is the NlNT function.
From (1), the RATIO of consecutive terms is
F„ =1 + ^2 =1+ 1
F n -1 Fn-1
= 1 +
F n - 2
1 +
1 = fn Ei
F n-2
= [1,1,. ..,1],
(39)
which is just the first few terms of the Continued
Fraction for the Golden Ratio <j>. Therefore,
(40)
lim —2-
6 1
The "Shallow Diagonals" of Pascal's Triangle
sum to Fibonacci numbers (Pappas 1989),
y^, k x (-irVP 2 (l,2 J l-n;|(3-n) J 2-|n;-i)
?r(2-3n + n 2 )
F n+ U (41)
where zF 2 {a, 6,c; d, e; z) is a GENERALIZED HYPERGEO-
metric Function.
The sequence of final digits in Fibonacci numbers re-
peats in cycles of 60. The last two digits repeat in 300,
the last three in 1500, the last four in 15,000, etc.
£
(-ir
F n F n
= 2-\/5
+2
(42)
(Clark 1995). A very curious addition of the Fibonacci
numbers is the following addition tree,
1
1
2
3
5
8
13
21
34
55
89
0112359550561...
which is equal to the fractional digits of 1/89,
v^ F n 1_
2-^t 10 7l+1 89*
(43)
n=Q
For n > 3, F n |F m Iff n|ra. L n \L m Iff n divides
into 7n an EVEN number of times. (F m ,F n ) = F^ m ^
(Michael 1964; Honsberger 1985, pp. 131-132). No Odd
Fibonacci number is divisible by 17 (Honsberger 1985,
pp. 132 and 242). No Fibonacci number > 8 is ever
of the form p — 1 or p + 1 where p is a PRIME number
(Honsberger 1985, p. 133).
Consider the sum
Sk
k k / \
= y l =y ^ 1 —\.
^-^ F n -iFn+i £—i V F n -iF n F n F n +i I
■n—O. n— 9 x /
n=2 n=2
This is a Telescoping Sum, so
1
Sk = 1
Fk+iFk+2
thus
S = lim Sk = 1
k—^oo
(44)
(45)
(46)
(Honsberger 1985, pp. 134-135). Using Binet's FOR-
MULA, it also follows that
p\n+r
F n+r = a n+r - /T +r = a n+r 1 ~ (f )
F n ~ a n - (3 n ~ a n l _ (£)
where
a=i(l + V5)
/3=§(1-a/5)
— , (47)
(48)
(49)
632
Fibonacci Number
Fibonacci Number
Fn+r
r
= a .
CO
F n
n+l^n+2
(50)
(51)
(Honsberger 1985, pp. 138 and 242-243). The MlLLIN
Series has sum
oo
F 2
(52)
(Honsberger 1985, pp. 135-137).
The Fibonacci numbers are Complete. In fact, drop-
ping one number still leaves a COMPLETE SEQUENCE,
although dropping two numbers does not (Honsberger
1985, pp. 123 and 126). Dropping two terms from the
Fibonacci numbers produces a sequence which is not
even WEAKLY COMPLETE (Honsberger 1985, p. 128).
However, the sequence
K = F n - (-1)"
(53)
is Weakly Complete, even with any finite subse-
quence deleted (Graham 1964). {F n 2 } is not Com-
plete, but {F n 2 } + {F n 2 } are. 2 N ~ 1 copies of {F n N }
are COMPLETE.
For a discussion of SQUARE Fibonacci numbers, see
Cohn (1964), who proved that the only SQUARE Num-
ber Fibonacci numbers are 1 and F12 = 144 (Cohn 1964,
Guy 1994). Ming (1989) proved that the only Trian-
gular Fibonacci numbers are 1, 3, 21, and 55. The
Fibonacci and LUCAS NUMBERS have no common terms
except 1 and 3. The only Cubic Fibonacci numbers are
1 and 8.
(FnF n+ 3,2F n+1 F Tl+ 2,F 2 n+3 = F n+1 2 + F n+2 2 ) (54)
is a Pythagorean Triple.
F 4n 2 + 8F 2n (F 2n + F Qn ) = (3F 4n ) 2 (55)
is always a Square Number (Honsberger 1985, p. 243).
In 1975, James P. Jones showed that the Fibonacci num-
bers are the Positive Integer values of the Polynom-
ial
p{x, y) = -y 5 + 2y 4 z + y V - 2yV - y(x 4 - 2) (56)
for Gaussian Integers x and y (Le Lionnais 1983). If
n and k are two POSITIVE Integers, then between n k
and n fc+1 , there can never occur more than n Fibonacci
numbers (Honsberger 1985, pp. 104-105).
Every F n that is Prime has a Prime n, but the converse
is not necessarily true. The first few PRIME Fibonacci
numbers are for n = 3, 4, 5, 7, 11, 13, 17, 23, 29, 43,
47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, ...
(Sloane's A001605; Dubner and Keller 1998). Gardner's
statement that F531 is prime is incorrect, especially since
531 is not even PRIME (Gardner 1979, p. 161). It is not
known if there are an Infinite number of Fibonacci
primes.
The Fibonacci numbers F nj are SQUAREFUL for n = 6,
12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ... , 300,
306, 312, 324, 325, 330, 336, . . . (Sloane's A037917) and
Squarefree for n = 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13,
. . . (Sloane's A037918). The largest known SQUAREFUL
Fibonacci number is F336, and no SQUAREFUL Fibonacci
numbers F p are known with p PRIME.
see also CASSINl'S IDENTITY, FAST FIBONACCI TRANS-
FORM, Fibonacci Dual Theorem, Fibonacci n-
Step Number, Fibonacci Q-Matrix, Generalized
Fibonacci Number, Inverse Tangent, Linear Re-
currence Sequence, Lucas Sequence, Near No-
ble Number, Pell Sequence, Rabbit Constant,
Stolarsky Array, Tetranacci Number, Tri-
bonacci Number, Wythoff Array, Zeckendorf
Representation, Zeckendorf's Theorem
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 56-57,
1987.
Basin, S. L. and Hoggatt, V. E. Jr. "A Primer on the Fi-
bonacci Sequence." Fib. Quart 1, 1963.
Basin, S. L. and Hoggatt, V. E. Jr. "A Primer on the Fi-
bonacci Sequence— Part II." Fib. Quart 1, 61-68, 1963.
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 Silverman, R. D. "Ta-
bles of Fibonacci and Lucas Factorizations." Math. Corn-
put 50, 251-260 and S1-S15, 1988.
Brook, M. "Fibonacci Formulas." Fib. Quart 1, 60, 1963.
Brousseau, A. "Fibonacci Numbers and Geometry." Fib.
Quart 10, 303-318, 1972.
Clark, D. Solution to Problem 10262. Amer. Math. Monthly
102, 467, 1995.
Cohn, J. H. E. "On Square Fibonacci Numbers." J. London
Math. Soc. 39, 537-541, 1964.
Conway, J. H. and Guy, R. K. "Fibonacci Numbers." In The
Book of Numbers. New York: Springer-Verlag, pp. 111-
113, 1996.
Coxeter, H. S. M. "The Golden Section and Phyllotaxis."
Ch. 11 in Introduction to Geometry, 2nd ed. New York:
Wiley, 1969.
Dickau, R. M. "Fibonacci Numbers." http : //www .
prairienet . org/~pops/f ibboard.html.
Dubner, H. and Keller, W. "New 'Fibonacci and Lucas
Primes." Math. Comput 1998.
Freitag, H. Solution to Problem B-772. "An Integral Ratio."
Fib. Quart 34, 82, 1996.
Gardner, M. Mathematical Circus: More Puzzles, Games,
Paradoxes and Other Mathematical Entertainments from
Scientific American. New York: Knopf, 1979.
Graham, R. "A Property of Fibonacci Numbers." Fib.
Quart 2, 1-10, 1964.
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.
Fibonacci Polynomial
Fibonacci Pseudoprime 633
Halton, J. H. "On a General Fibonacci Identity." Fib. Quart.
3, 31-43, 1965.
Hoggatt, V. E. Jr. The Fibonacci and Lucas Numbers.
Boston, MA: Houghton Mifflin, 1969.
Hoggatt, V. E. Jr. and Ruggles, I. D. "A Primer on the Fi-
bonacci Sequence — Part III." Fib. Quart 1, 61-65, 1963.
Hoggatt, V. E. Jr. and Ruggles, I. D. "A Primer on the Fi-
bonacci Sequence — Part IV." Fib. Quart 1, 65-71, 1963.
Hoggatt, V. E. Jr. and Ruggles, I. D. "A Primer on the Fi-
bonacci Sequence— Part V." Fib. Quart. 2, 59-66, 1964.
Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M. "A Primer
for the Fibonacci Numbers: Part XII." Fib, Quart 11,
317-331, 1973.
Honsberger, R. "A Second Look at the Fibonacci and Lucas
Numbers." Ch. 8 in Mathematical Gems III. Washington,
DC: Math. Assoc. Amer., 1985.
Knott, R. "Fibonacci Numbers and the Golden Section."
http : // www . mcs . surrey .ac.uk/ Personal / R . Knott /
Fibonacci/fib. html.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 146, 1983.
Leyland, P. ftp : //sable . ox . ac .uk/pub/math/f actors/
f ibonacci.Z.
Matijasevic, Yu. V. "Solution to of the Tenth Problem of
Hilbert." Mat. Lapok 21, 83-87, 1970.
Matijasevich, Yu. V. Hubert's Tenth Problem. Cambridge,
MA: MIT Press, 1993.
Michael, G. "A New Proof for an Old Property." Fib. Quart.
2, 57-58, 1964.
Ming, L. "On Triangular Fibonacci Numbers." Fib. Quart.
27, 98-108, 1989.
Ogilvy, C. S. and Anderson, J. T. "Fibonacci Numbers."
Ch. 11 in Excursions in Number Theory. New York:
Dover, pp. 133-144, 1988.
Pappas, T. "Fibonacci Sequence," "Pascal's Triangle, the Fi-
bonacci Sequence Sc Binomial Formula," "The Fibonacci
Trick," and "The Fibonacci Sequence & Nature." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 28-29, 40-41, 51, 106, and 222-225, 1989.
Schroeder, M. Fractals, Chaos, Power Laws: Minutes from
an Infinite Paradise. New York: W. H. Freeman, pp. 49-
57, 1991.
Sloane, N. J. A. Sequences A037917, A037918, A000045/
M0692, and A001605/M2309 in "An On-Line Version of
the Encyclopedia of Integer Sequences."
Vorob'ev, N. N. Fibonacci Numbers. New York: Blaisdell
Publishing Co., 1961.
Fibonacci Polynomial
corresponding w POLYNOMIALS are called LUCAS POLY-
NOMIALS.) The Fibonacci polynomials are defined by
the Recurrence Relation
The W Polynomials obtained by setting p(x) — x and
q(x) = 1 in the LUCAS POLYNOMIAL SEQUENCE. (The
F n +i{x) = xF n (x) + F n „ 1 (x),
(1)
with Fi(x) — 1 and F 2 (x) = x. They are also given by
the explicit sum formula
L(«-1)/2J
'• w - § try
,n~2j-l
(2)
where [x\ is the FLOOR FUNCTION and (^) is a BINO-
MIAL Coefficient, The first few Fibonacci polynomi-
als are
F 1 (x) = l
F 2 (x) = x
F 3 (x) = x 2 + l
F 4 (x) = x 3 + 2x
F 5 (x) = x 4 + 3x 2 + l.
The Fibonacci polynomials are normalized so that
F n (l) = F n , (3)
where the F n s are FIBONACCI NUMBERS.
The Fibonacci polynomials are related to the MORGAN-
VOYCE POLYNOMIALS by
F 2n +i{x) =b n (x 2 ) (4)
F 2 n + n2{x) = xB n (x 2 ) (5)
(Swamy 1968).
see also Brahmagupta Polynomial, Fibonacci
Number, Morgan-Voyce Polynomial
References
Swamy, M. N. S. "Further Properties of Morgan-Voyce Poly-
nomials." Fib. Quart 6, 167-175, 1968.
Fibonacci Pseudoprime
Consider a LUCAS SEQUENCE with P > and Q = ±1.
A Fibonacci pseudoprime is a Composite Number n
such that
V n = P (mod n) .
There exist no EVEN Fibonacci pseudoprimes with pa-
rameters P = 1 and Q = -1 (Di Porto 1993) or P =
Q = 1 ( Andre- Jeannin 1996). Andre- Jeannin (1996)
also proved that if (P,Q) # (1,-1) and (P,Q) ^ (1,1),
then there exists at least one Even Fibonacci pseudo-
prime with parameters P and Q.
see also PSEUDOPRIME
References
Andre-Jeannin, R. "On the Existence of Even Fibonacci
Pseudoprimes with Parameters P and Q." Fib. Quart.
34, 75-78, 1996.
Di Porto, A. "Nonexistence of Even Fibonacci Pseudoprimes
of the First Kind." Fib. Quart 31, 173-177, 1993.
Ribenboim, P. "Fibonacci Pseudoprimes." §2.X.A in The
New Book of Prime Number Records, 3rd ed. New York:
Springer- Verlag, pp. 127-129, 1996.
634 Fibonacci Q-Matrix
Fields Medal
Fibonacci Q-Matrix
A Fibonacci Matrix of the form
M =
771 1
1
If U and V are defined as BlNET FORMS
U n = mU n -i + U n ~i (U = 0,Ui = 1)
V n = mV n -i + V n - 2 (V Q = 2, Vi = m),
then
M" 1 = M-ml
U n Un-1
1
1 — m
Defining
then
Q =
>2
F1
F
=
"l
1
r
Q"
i
F n
n-1
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(Honsberger 1985, pp. 106-107).
see also BlNET FORMS, FIBONACCI NUMBER
References
Honsberger, R. "A Second Look at the Fibonacci and Lucas
Numbers." Ch. 8 in Mathematical Gems III. Washington,
DC: Math. Assoc. Amer., 1985.
Fibonacci Sequence
see Fibonacci Number
Fibration
If / : E ->- B is a FIBER BUNDLE with B a PARACOM-
pact TOPOLOGICAL SPACE, then / satisfies the Homo-
topy Lifting Property with respect to all Topolog-
ical Spaces. In other words, if g : [0, 1] x X -> B is
a HOMOTOPY from go to g\ , and if g' is a LIFT of the
Map go with respect to /, then g has a Lift to a Map
g' with respect to /. Therefore, if you have a Homo-
TOPY of a Map into B, and if the beginning of it has a
Lift, then that LIFT can be extended to a LIFT of the
HOMOTOPY itself.
A fibration is a MAP between TOPOLOGICAL SPACES
/ : E -* B such that it satisfies the Homotopy Lifting
Property.
see also FIBER BUNDLE, FIBER SPACE
Field
A field is any set of elements which sat; the FIELD
AXIOMS for both addition and multiplica - >n and is a
commutative DIVISION ALGEBRA. An archaic word for
a field is RATIONAL DOMAIN. A field with a finite num-
ber of members is known as a FINITE FlEI " r GALOIS
FIELD.
Because the identity condition must be different for ad-
dition and multiplication, every field must have at least
two elements. Examples include the COMPLEX NUM-
BERS (C), Rational Numbers (Q), and Real Num-
bers (K), but not the Integers (Z), which form a
RING. It has been proven by Hilbert and Weierstrafi
that all generalizations of the field concept to triplets of
elements are equivalent to the field of COMPLEX NUM-
BERS.
see also ADJUNCTION, ALGEBRAIC NUMBER FIELD,
Coefficient Field, Cyclotomic Field, Field Ax-
ioms, Field Extension, Function Field, Galois
Field, Mac Lane's Theorem, Module, Number
Field, Quadratic Field, Ring, Skew Field, Vec-
tor Field
Field Axioms
The field axioms are generally written in additive and
multiplicative pairs.
Name
Addition
Multiplication
Commutivity
o + 6 = b + a
ab = ba
Associativity
(a + b) + c = a + (6 + c)
(ab)c = a(bc)
Distributivity
a(b + c) — ab -\- ac
(o + b)c — ac + 6c
Identity
a+0=a=0+a
a • 1 — a = 1 ■ a
Inverses
o+ (-a) = = (-a) +a
aa~ x = 1 = a~ x a
if o /
see also Algebra, Field
Field Extension
A Field L is said to be a field extension of field K
if K is a Subfield of L. This is denoted L/K (note
that this Notation conflicts with that of a Quotient
Group). The Complex Numbers are a field extension
of the Real Numbers, and the Real Numbers are a
field extension of the RATIONAL NUMBERS,
see also Field
Fields Medal
The mathematical equivalent of the Nobel Prize (there
is no Nobel Prize in mathematics) which is awarded by
the International Mathematical Union every four years
to one or more outstanding researchers, usually under
40 years of age. The first Fields Medal was awarded in
1936.
see also Burnside Problem, Mathematics Prizes,
Poincare Conjecture, Roth's Theorem, Tau
Conjecture
Fifteen Theorem
Figurate Number 635
References
MacTutor History of Mathematics Archives. "The
Fields Medal." http : //www-groups . dcs . st-and . ac . uk/
-history/Societies/FieldsMedal.html.
Monastyrsky, M. Modern Mathematics in the Light of the
Fields Medals. Wellesley, MA: A. K. Peters, 1997.
Fifteen Theorem
A theorem due to Conway et al. (1997) which states
that, if a POSITIVE definite QUADRATIC Form with in-
tegral matrix entries represents all natural numbers up
to 15, then it represents all natural numbers. This the-
orem contains LAGRANGE'S FOUR-SQUARE THEOREM,
since every number up to 15 is the sum of at most four
Squares.
see also Integer- Matrix Form, Lagrange's Four-
square Theorem, Quadratic Form
References
Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane,
N. J. A. "The Primary Pretenders." Acta Arith. 78, SOT-
SIS, 1997.
Duke, W. "Some Old Problems and New Results about Quad-
ratic Forms." Not Amer. Math. Soc. 44, 190-196, 1997.
Figurate Number
Name
Formula
A number which can be represented by a regular geo-
metrical arrangement of equally spaced points. If the
arrangement forms a REGULAR POLYGON, the number
is called a POLYGONAL NUMBER. The polygonal num-
bers illustrated above are called triangular, square, pen-
tagonal, and hexagon numbers, respectively. Figurate
numbers can also form other shapes such as centered
polygons, L-shapes, 3-dimensional solids, etc. The fol-
lowing table lists the most common types of figurate
numbers.
biquadratic
centered cube
centered pentagonal
centered square
centered triangular
cubic
decagonal
gnomic
heptagonal
heptagonal pyramidal
hex
hexagonal
hexagonal pyramidal
octagonal
octahedral
pentagonal
pentagonal pyramidal
pent at ope
pronic number
rhombic dodecahedral
square
stella octangula
tetrahedral
triangular
truncated octahedral
truncated tetrahedral
(2n-l)(n 2 -n + 1)
|(5n 2 -5n + 2)
n 2 + {n~ l) 2
§(3n 2 -3n + 2)
n 3
An 2 - 3n
2n- 1
|n(5n-3)
|n(n+l)(5n-2)
3n 2 - 3n + 1
n(2n- 1)
\n{n +l)(4n-l)
n(3n - 2)
±n(2n 2 + 1)
§n(3n - 1)
fn 2 (n+l)
^n(n+l)(n + 2)(n + 3)
n(n + l)
(2n-l)(2n 2 - 2n + 1)
n 2
n(2n 2 - 1)
|n(n + l)(n + 2)
|n(n+l)
16n 3 - 33n 2 + 24n - 6
|n(23n 2 - 27n + 10)
An n-D FIGURATE NUMBER can be defined by
f r =
(rs + m — s)(r + m — 2)
ml{r~ 1)! '
see also BIQUADRATIC NUMBER, CENTERED CUBE
Number, Centered Pentagonal Number, Cen-
tered Polygonal Number, Centered Square
Number, Centered Triangular Number, Cubic
Number, Decagonal Number, Figurate Number
Triangle, Gnomic Number, Heptagonal Number,
Heptagonal Pyramidal Number, Hex Number,
Hex Pyramidal Number, Hexagonal Number,
Hexagonal Pyramidal Number, Nexus Number,
Octagonal Number, Octahedral Number, Pen-
tagonal Number, Pentagonal Pyramidal Num-
ber, Pentatope Number, Polygonal Number,
Pronic Number, Pyramidal Number, Rhombic
Dodecahedral Number, Square Number, Stella
Octangula Number, Tetrahedral Number, Tri-
angular Number, Truncated Octahedral Num-
ber, Truncated Tetrahedral Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 30-62, 1996.
Dickson, L. E. "Polygonal, Pyramidal, and Figurate Num-
bers." Ch. 1 in History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, pp. 1-39, 1952.
636 Figurate Number Triangle
Goodwin, P. "A Polyhedral Sequence of Two." Math. Gaz.
69, 191-197, 1985.
Guy, R. K. "Figurate Numbers." §D3 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 147-150, 1994.
Kraitchik, M. "Figurate Numbers." §3.4 in Mathematical
Recreations. New York: W. W. Norton, pp. 66-69, 1942.
Figurate Number Triangle
A Pascal's Triangle written in a square grid and
padded with zeroes, as written by Jakob Bernoulli
(Smith 1984). The figurate number triangle therefore
has entries
<*=(;).
where i is the row number, j the column number, and
(*.) a Binomial Coefficient. Written out explicitly
(beginning each row with j = 0),
1
1
1
1
2
1
1
3
3
1
1
4
6
4
1
1
5
10
10
5
1
1
6
15
20
15
6
1
1
7
21
35
35
21.
7
Then we have the sum identities
3=0
i
^2 a H = 2* - 1
3 = 1
n
En-h i
<Hj = a(n+l),(j + l) = J^J a "
ra + 1
see also Binomial Coefficient, Figurate Number,
Pascal's Triangle
References
Smith, D. E. A Source Book in Mathematics. New York:
Dover, p. 86, 1984.
Figure Eight Knot
see Figure-of-Eight Knot
Figure Eight Surface
see Eight Surface
Filon's Integration Formula
Figure-of-Eight Knot
The knot 04 oi, which is the unique Prime Knot of
four crossings, and which is a 2-EMBF,DDABLE Knot.
It is Amphichiral. It is also known as tht l,em-
ish Knot and Savoy Knot, and it has Braid Yord
0~1<J2 <7iO~2
References
Owen, P. Knots. Philadelphia, PA: Courage, o. 16, 1993.
Figures
A number x is said to have "n figures" if it takes n
DIGITS to express it. The number of figures is therefore
equal to one more than the POWER of 10 in the Sci-
entific Notation representation of the number. The
word is most frequently used in reference to monetary
amounts, e.g., a "six-figure salary" would fall in the
range of $100,000 to $999,999.
see also Digit, Scientific Notation, Significant
Figures
Filon's Integration Formula
A formula for Numerical Integration,
Jxn
f(x) cos(tx) dx
= h{a(th)[f2n sin(tx 2 n) - /o sin(ta;o)]
+0(th)C 2 n + l{th)C 2n - X + ^th 4 S' 2n -l} - Ai, (1)
where
C 2 n = ^/2tC0s(££ 2 i) ~ ^[f2nCOs(tX2n) + /o COs(fcCo)]
(2)
n
Cin-i = y ^ f2i-l COs(tX2i-l)
i=l
//n 1 sin(20) 2 sin 2
a{e) = o + ~w W~
l + cos 2 sin(2<9)
m = 2
e 2 e 3
sin cos 6 >
fsinO cos0\
and the remainder term is
R n = ± n h*f w (t) + 0(th 7 ).
(3)
(4)
(5)
(6)
(7)
(8)
Filter
Finite Difference 637
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 890-891, 1972.
Tukey, J. W. In On Numerical Approximation: Proceedings
of a Symposium Conducted by the Mathematics Research
Center, United States Army, at the University of Wis-
consin, Madison, April 21-23, 1958 (Ed. R. E, Langer).
Madison, WI: University of Wisconsin Press, p. 400, 1959.
Filter
Formally, a filter is defined in terms of a Set X and a
Set <£ of Subsets of X. Then $ is called a filter if
1. X £$ y
2. the Empty Set £ $,
3. Ac B C X and Ae$ Implies B e <£,
4. and A, B € 3> Implies AUB e$.
Informally, a filter is a function or procedure which re-
moves unwanted parts of a signal. The concept of fil-
tering and filter functions is particularly useful in en-
gineering. One particularly elegant method of filtering
Fourier Transforms a signal into frequency space,
performs the filtering operation there, then transforms
back into the original space (Press et al. 1992).
see also Savitzky-Golay Filter, Wiener Filter
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Digital Filtering in the Time Domain." §13.5
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 551-556, 1992.
Fine's Equation
n
(l-^Xi-^Xi-^Xi-g 12 ")
(1 - q n )(l - q 24n )
Finite Difference
The finite difference is the discrete analog of the Deriv-
ative. The finite FORWARD DIFFERENCE of a function
f p is defined as
A/„ = / P+1 - /„, (1)
and the finite Backward Difference as
V/ P = / P - /p_i
(2)
If the values are tabulated at spacings /i, then the nota-
tion
f P = f(xo+ph) = f(x) (3)
is used. The kth FORWARD DIFFERENCE would then
be written as A k f p , and similarly, the kth BACKWARD
Difference as V fc / P .
However, when f p is viewed as a discretization of the
continuous function f(x), then the finite difference is
sometimes written
Af(x) = f(x + I) - /(* - I) = 2 Ij(x) * /(*), (4)
where * denotes Convolution and l\{x) is the odd Im-
pulse Pair. The finite difference operator can therefore
be written
(5)
A = 21-
1"
An nth POWER has a constant nth finite difference. For
example, take n = 3 and make a DIFFERENCE TABLE,
A A2
A 3
2 8 ' 12 " A 4
7 " A 3
3 27 t; 18 J
4 64 I] 24 6
5 125 61
(6)
1 + 2_^ fi,5,7,n(^i 24)^, The A 3 column is the constant 6.
where E is the sum of the DIVISORS of N CONGRUENT
to 1, 5, 7, and 11 (mod 24) minus the sum of DIVISORS
of TV Congruent to -1, -5, -7, and -11 (mod 24).
see also ^-Series
Finite
A Set which contains a Nonnegative integral number
of elements is said to be finite. A Set which is not finite
is said to be Infinite. A finite or Countably Infi-
nite Set is said to be Countable. While the meaning
of the term "finite" is fairly clear in common usage, pre-
cise definitions of FINITE and INFINITE are needed in
technical mathematics and especially in Set Theory.
see also Countable Set, Countably Infinite Set,
Infinite, Set Theory, Uncountably Infinite Set
Finite difference formulas can be very useful for extrap-
olating a finite amount of data in an attempt to find the
general term. Specifically, if a function /(n) is known at
only a few discrete values n = 0, 1, 2, . . . and it is de-
sired to determine the analytical form of /, the following
procedure can be used if / is assumed to be a Polynom-
ial function. Denote the nth value in the SEQUENCE of
interest by a n . Then define b n as the Forward Dif-
ference A n = a n+ i — a n , c n as the second Forward
Difference A n = 6 n+ i — b ni etc., constructing a table
as follows
a = /(0) oi=/(l) a 2 = /(2) ... a p = f(p)
bo = a\ — ao b± = a 2 — a± ... b p -\ = a p — a p -\
Co = 62 — b\
(7)
638
Finite Difference
Finite Field
Continue computing do, eo, etc., until a value is ob-
tained. Then the POLYNOMIAL function giving the val-
ues a n is given by
«»>-t-(:)
ao-\-b Q n-\-
con(n
1) don(n-l)(n-2)
2-3
(8)
• (9)
When the notation Ao = ao, Aq = &o, etc., is used,
this beautiful equation is called NEWTON'S FORWARD
Difference FORMULA. To see a particular example,
consider a Sequence with first few values of 1, 19, 143,
607, 1789, 4211, and 8539. The difference table is then
given by
1 19 143 607 1789 4211 8539
18 124 464 1182 2422 4328
106 340 718 1240 1906
234 378 522 666
144 144 144
Reading off the first number in each row gives a = 1,
bo = 18, co = 106, d = 234, e = 144. Plugging these
in gives the equation
f(n) = 1 + 18n + 53n(n - 1) + 39n(n - l)(n - 2)
+6n(n - l)(n - 2)(n - 3), (10)
which simplifies to f(n) = 6n 4 + 3n 3 + 2n 2 +7n + l, and
indeed fits the original data exactly!
Beyer (1987) gives formulas for the derivatives
un d n f{x +ph) _, n d n f p _ d n f p
dx n
= h n
dx n dp"
(11)
(Beyer 1987, pp. 449-451) and integrals
f f{x)dx = h J f p dp (12)
J XQ JO
(Beyer 1987, pp. 455-456) of finite differences.
Finite differences lead to Difference Equations, fi-
nite analogs of Differential Equations. In fact,
UMBRAL CALCULUS displays many elegant analogs of
well-known identities for continuous functions. Com-
mon finite difference schemes for Partial Differen-
tial Equations include the so-called Crank-Nicholson,
Du Fort-Frankel, and Laasonen methods.
see also Backward Difference, Bessel's Finite
Difference Formula, Difference Equation, Dif-
ference Table, Everett's Formula, Forward
Difference, Gauss's Backward Formula, Gauss's
Forward Formula, Interpolation, Jackson's
Difference Fan, Newton's Backward Differ-
ence Formula, Newton-Cotes Formulas, New-
ton's Divided Difference Interpolation For-
mula, Newton's Forward Difference Formula,
Quotient-Difference Table, Steffenson's For-
mula, Stirling's Finite Difference Formula, Um-
bral Calculus
References
Abramowitz, M. and Stegun, C.^A. (Eds.). "Differences."
§25.1 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 877-878, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 429-515, 1987.
Boole, G. and Moulton, J. F. A Treatise on the Calculus of
Finite Differences, 2nd rev. ed. New York: Dover, 1960.
Conway, J. H. and Guy, R. K. "Newton's Useful Little For-
mula." In The Book of Numbers. New York: Springer-
Verlag, pp. 81-83, 1996.
Iyanaga, S. and Kawada, Y. (Eds.). "Interpolation." Ap-
pendix A, Table 21 in Encyclopedic Dictionary of Mathe-
matics. Cambridge, MA: MIT Press, pp. 1482-1483, 1980.
Jordan, K. Calculus of Finite Differences, 2nd ed. New York:
Chelsea, 1950.
Levy, H. and Lessman, F. Finite Difference Equations. New
York: Dover, 1992.
Milne- Thomson, L. M. The Calculus of Finite Differences.
London: Macmillan, 1951.
Richardson, C. H. An Introduction to the Calculus of Finite
Differences. New York: Van Nostrand, 1954.
Spiegel, M. Calculus of Finite Differences and Differential
Equations. New York: McGraw-Hill, 1971.
Finite Field
A finite field is a Field with a finite Order (number
of elements), also called a GALOIS Field. The order of
a finite field is always a Prime or a Power of a Prime
(Birkhoffand Mac Lane 1965). For each Prime Power,
there exists exactly one (up to an ISOMORPHISM) fi-
nite field GF(p n ), often written as ¥ p n in current us-
age, GF(p) is called the Prime Field of order p, and
is the Field of Residue Classes modulo p, where the
p elements are denoted 0, 1, . . . , p — 1. a — bin GF(p)
means the same as a = b (mod p). Note, however, that
2x2 = (mod 4) in the Ring of residues modulo 4,
so 2 has no reciprocal, and the RING of residues mod-
ulo 4 is distinct from the finite field with four elements.
Finite fields are therefore denoted GF(p n ), instead of
GF(pi ■ ■ -pn) for clarity.
The finite field GF(2) consists of elements and 1 which
satisfy the following addition and multiplications tables.
+
1
1
1
1
X
1
1
1
If a subset S of the elements of a finite field F satisfies
the above Axioms with the same operators of F, then S
Finite Field
Finite Group 639
is called a SUBFIELD. Finite fields are used extensively
in the study of Error-Correcting Codes.
When n > 1, GF(p") can be represented as the Field
of Equivalence Classes of Polynomials whose Co-
efficients belong to GF(p). Any Irreducible Poly-
nomial of degree n yields the same FIELD up to an ISO-
MORPHISM. For example, for GF(2 3 ), the modulus can
be taken as cc 3 +a; 2 + l = 0, x 3 +x-\-l, or any other Irre-
ducible Polynomial of degree 3. Using the modulus
x 3 + x + 1, the elements of GF(2 3 )— -written 0, x°, a: 1 ,
... — can be represented as Polynomials with degree
less than 3. For instance,
x 3 = —x — 1 = x + 1
4 __
1 = Z + X + 1
x(x ) = x(x + 1) = X + X
5 _ / 2 . \ _ 3 , 2 _ 2
X = x(x + X) = X + X = x -
x 6 = x(x 2 +x + l) = x 3 + x 2 + x = x 2 -l = x 2 + l
x = x(x + 1) = X + x = —1 = 1 = Xq.
Now consider the following table which contains several
different representations of the elements of a finite field.
The columns are the power, polynomial representation,
triples of polynomial representation COEFFICIENTS (the
vector representation), and the binary INTEGER corre-
sponding to the vector representation (the regular rep-
resentation).
Representation
Power
Polynomial
Vector Regular
(000)
x°
1
(001)
1
x 1
X
(010)
2
x l
x 2
(100)
4
x 3
x + 1
(011)
3
x 4
x 2 + X
(110)
6
x 5
x 2 + X + 1
(111)
7
x 6
x 2 + 1
(101)
5
The set of POLYNOMIALS in the second column is closed
under ADDITION and Multiplication modulo x 3 + x-\-
1, and these operations on the set satisfy the Axioms
of finite field. This particular finite field is said to be
an extension field of degree 3 of GF(2), written GF(2 3 ),
and the field GF(2) is called the base field of GF(2 3 ). If
an Irreducible Polynomial generates all elements in
this way, it is called a PRIMITIVE IRREDUCIBLE POLY-
NOMIAL. For any PRIME or PRIME POWER q and any
Positive Integer n, there exists a Primitive Irre-
ducible Polynomial of degree n over GF(q).
For any element c of GF(q), c q = c, and for any Non-
zero element d of GF(g), d q ~ l = 1. There is a small-
est Positive Integer n satisfying the sum condition
n ■ 1 = in GF(g), which is called the characteristic
of the finite field GF(g). The characteristic is a PRIME
NUMBER for every finite field, and it is true that
over a finite field with characteristic p.
see also Field, Hadamard Matrix, Ring, Subfield
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 73—75,
1987.
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra,
3rd ed. New York: Macmillan, p. 413, 1965.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, p. viii,
1952.
Finite Game
A Game in which each player has a finite number of
moves and a finite number of of choices at each move.
see also Game, Zero-Sum Game
References
Dresner, M. The Mathematics of Games of Strategy: Theory
and Applications. New York: Dover, p. 2, 1981.
Finite Group
A Group of finite Order. Examples of finite groups are
the Modulo Multiplication Groups and the Point
Groups. The Classification Theorem of finite Sim-
ple Groups states that the finite Simple Groups can
be classified completely into one of five types.
There is no known FORMULA to give the number of pos-
sible finite groups as a function of the Order h. It is
possible, however, to determine the number of Abelian
Groups using the Kronecker Decomposition The-
orem, and there is at least one A B ELIAN Group for
every finite order h.
The following table gives the numbers and names of the
first few groups of Order h. In the table, Nna denotes
the number of non- Abelian groups, Na denotes the num-
ber of Abelian Groups, and N the total number of
groups. In addition, Z n denotes an CYCLIC GROUP of
Order n, A n an Alternating Group, D n a Dihe-
dral Group, Q 8 the group of the Quaternions, T
the cubic group, and <g> a Direct Product.
h
Name
N NA
N A
N
1
<e>
1
1
2
z 2
1
1
3
z*
1
1
4
Z 2 ® Z<2, z 4
2
2
5
z 5
1
1
6
Z 6 ,D 3
1
1
2
7
Z 7
1
1
8
Z 2 <g> Z 2 <g> Z 2 , Z 2 ® Z 4 , Z 8 , Q 8 , D A
3
2
5
9
Z$ <g> Z 3 , Zq
2
2
10
Z 1Q ,D 5
1
1
2
11
Z 1X
1
1
12
Z 2 ® Z 6 ,Z 12 ,A 4 ,D 6 ,T
2
3
5
13
Z13
1
1
14
Z 14 ,D 7
1
1
2
15
Zis
1
1
(x + y) p = x p + y p
640 Finite Group
Finite Group
Miller (1930) gave the number of groups for orders 1-
100, including an erroneous 297 as the number of groups
of Order 64. Senior and Lunn (1934, 1935) subse-
quently completed the list up to 215, but omitted 128
and 192. The number of groups of ORDER 64 was cor-
rected in Hall and Senior (1964). James et al. (1990)
found 2328 groups in 115 ISOCLINISM families of OR-
DER 128, correcting previous work, and O'Brien (1991)
found the number of groups of ORDER 256. The number
of groups is known for orders up to 1000, with the pos-
sible exception of 512 and 768. Besche and Eick (1998)
have determined the number of finite groups of orders
less than 1000 which are not powers of 2 or 3. These
numbers appear in the Magma® database. The num-
bers of nonisomorphic finite groups 7V of each ORDER h
for the first few hundred orders are given in the following
table (Sloane's A000001— the very first sequence).
The number of ABELIAN GROUPS of ORDER h is denoted
N A (Sloane's A000688). The smallest order for which
there exist n = 1, 2, , . . nonisomorphic groups are 1, 4,
75, 28, 8, 42, . . . (Sloane's A046057). The incrementally
largest number of nonisomorphic finite groups are 1, 2,
5, 14, 15, 51, 52, 267, 2328, ... (Sloane's A046058),
which occur for orders 1, 4, 8, 16, 24, 32, 48, 64, 128,
... (Sloane's A046059).
h N
N A
h
N
JV A
h
N
N A
h
N
N A
1 1
1
51
1
1
101
1
1
151
1
1
2 1
1
52
5
2
102
4
1
152
12
3
3 1
1
53
1
1
103
1
1
153
2
2
4 2
2
54
15
3
104
14
3
154
4
1
5 1
1
55
2
1
105
2
1
155
2
1
6 2
1
56
13
3
106
2
1
156
18
2
7 1
1
57
2
1
107
1
1
157
1
1
8 5
3
58
2
1
108
45
6
158
2
1
9 2
2
59
1
1
109
1
1
159
1
1
10 2
1
60
13
2
110
6
1
160
238
7
11 1
1
61
1
1
111
2
1
161
1
1
12 5
2
62
2
1
112
43
5
162
55
5
13 1
1
63
4
2
113
1
1
163
1
1
14 2
1
64
267
11
114
6
1
164
5
2
15 1
1
65
1
1
115
1
1
165
2
1
16 14
5
66
4
1
116
5
2
166
2
1
17 1
1
67
1
1
117
4
2
167
1
1
18 5
2
68
5
2
118
2
1
168
57
3
19 1
1
69
1
1
119
1
1
169
2
2
20 5
2
70
4
1
120
47
3
170
4
1
21 2
1
71
1
1
121
2
2
171
5
2
22 2
1
72
50
6
122
2
1
172
4
2
23 1
1
73
1
1
123
1
1
173
1
1
24 15
3
74
2
1
124
4
2
174
4
1
25 2
2
75
3
2
125
5
3
175
2
2
26 2
1
76
4
2
126
16
2
176
42
5
27 5
3
77
1
1
127
1
1
177
1
1
28 4
2
78
6
1
128
2328
15
178
2
1
29 1
1
79
1
1
129
2
1
179
1
1
30 4
1
80
52
5
130
4
1
180
37
4
31 1
1
81
15
5
131
1
1
181
1
1
32 51
7
82
2
1
132
10
2
182
4
1
33 1
1
83
1
1
133
1
1
183
2
1
34 2
1
84
15
2
134
2
1
184
12
3
35 1
1
85
1
1
135
5
3
185
1
1
36 14
4
86
2
1
136
15
3
186
6
1
37 1
1
87
1
1
137
1
1
187
1
1
38 2
1
88
12
3
138
4
1
188
4
2
39 2
1
89
1
1
139
1
1
189
13
3
40 14
3
90
10
2
140
11
2
190
4
1
41 1
1
91
1
1
141
1
1
191
1
1
42 6
1
92
4
2
142
2
1
192
1543
11
43 1
1
93
2
1
143
1
1
193
1
1
44 4
2
94
2
1
144
197
10
194
2
1
45 2
2
95
1
1
145
1
1
195
2
1
46 2
1
96
230
7
146
2
1
196
17
4
47 1
1
97
1
1
147
6
2
197
1
1
48 52
5
98
5
2
148
5
2
198
10
2
49 2
2
99
2
2
149
1
1
199
1
1
50 2
2
100
16
4
150
13
2
200
52
6
Finite Group
Finite Group— £> 3 641
h
N
N A
h
N
N A
/t
N
N A
h
N
N A
201
2
1
251
1
1
301
2
1
351
14
3
202
2
1
252
46
4
302
2
1
352
195
7
203
2
1
253
2
1
303
1
1
353
1
1
204
12
2
254
2
1
304
42
5
354
4
1
205
2
1
255
1
1
305
2
1
355
2
1
206
2
1
256
56092
22
306
10
2
356
5
2
207
2
2
257
1
1
307
1
1
357
2
1
208
51
5
258
6
1
308
9
2
358
2
1
209
1
1
259
1
1
309
2
1
359
1
1
210
12
1
260
15
2
310
6
1
360
162
6
211
1
1
261
2
2
311
1
1
361
2
2
212
5
2
262
2
1
312
61
3
362
2
1
213
1
1
263
1
1
313
1
1
363
3
2
214
2
1
264
39
3
314
2
1
364
11
2
215
1
1
265
1
1
315
4
2
365
1
1
216
177
9
266
4
1
316
4
2
366
6
1
217
1
1
267
1
1
317
1
1
367
1
1
218
2
1
268
4
2
318
4
1
368
42
5
219
2
1
269
1
1
319
1
1
369
2
2
220
15
2
270
30
3
320
1640
11
370
4
1
221
1
1
271
1
1
321
1
1
371
1
1
222
6
1
272
54
5
322
4
1
372
15
2
223
1
1
273
5
1
323
1
1
373
1
1
224
197
7
274
2
1
324
176
10
374
4
1
225
6
4
275
4
2
325
2
2
375
7
3
226
2
1
276
10
2
326
2
1
376
12
3
227
1
1
277
1
1
327
2
1
377
1
1
228
15
2
278
2
1
328
15
3
378
60
3
229
1
1
279
4
2
329
1
1
379
1
1
230
4
1
280
40
3
330
12
1
380
11
2
231
2
1
281
1
1
331
1
1
381
2
1
232
14
3
282
4
1
332
4
2
382
2
1
233
1
1
283
1
1
333
5
2
383
1
1
234
16
2
284
4
2
334
2
1
384
20169
15
235
1
1
285
2
1
335
1
1
385
2
1
236
4
2
286
4
1
336
228
5
386
2
1
237
2
1
287
1
1
337
1
1
387
4
2
238
4
1
288
1045
14
338
5
2
388
5
2
239
1
1
289
2
2
339
1
1
389
1
1
240
208
5
290
4
1
340
15
2
390
12
1
241
1
1
291
2
1
341
1
1
391
1
1
242
5
2
292
5
2
342
18
2
392
44
6
243
67
7
293
1
1
343
5
3
393
1
1
244
5
2
294
23
2
344
12
3
394
2
1
245
2
2
295
1
1
345
1
1
395
1
1
246
4
1
296
14
3
346
2
1
396
30
4
247
1
1
297
5
3
347
1
1
397
1
1
248
12
3
298
2
1
348
12
2
398
2
1
249
1
1
299
1
1
349
1
1
399
5
1
250
15
3
300
49
4
350
10
2
400
221
10
see also Abelian Group, Abel's Theorem, Ab-
hyankar's Conjecture, Alternating Group,
Burnside's Lemma, Burnside Problem, Chevalley
Groups, Classification Theorem, Composition
Series, Dihedral Group, Group, Jordan-Holder
Theorem, Kronecker Decomposition Theorem,
Lie Group, Lie-Type Group, Linear Group, Mod-
ulo Multiplication Group, Order (Group), Or-
thogonal Group, p-Group, Point Groups, Simple
Group, Sporadic Group, Symmetric Group, Sym-
plectic Group, Twisted Chevalley Groups, Uni-
tary Group
References
Arfken, G. "Discrete Groups." §4.9 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 243-251, 1985.
Artin, E. "The Order of the Classical Simple Groups."
Coram. Pure Appl. Math. 8, 455-472, 1955.
Aschbacher, M. Finite Group Theory. Cambridge, England:
Cambridge University Press, 1994.
Ball, W. W. R. and Coxeter, H, S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 73—75,
1987.
Besche and Eick. "Construction of Finite Groups." To Ap-
pear in J. Symb. Comput.
Besche and Eick. "The Groups of Order at Most 1000." To
Appear in J. Symb. Comput.
Conway, J. H.; Curtis, R. T.; Norton, S. R; 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, 1985.
Hall, M. Jr. and Senior, J. K. The Groups of Order 2 n (n <
6). New York: Macmillan, 1964.
James, R.; Newman, M. F.; and O'Brien, E. A. "The Groups
of Order 128." J. Algebra 129, 136-158, 1990.
Miller, G. A. "Determination of All the Groups of Order 64."
Amer. J. Math. 52, 617-634, 1930.
O'Brien, E. A. "The Groups of Order 256." J. Algebra 143,
219-235, 1991.
O'Brien, E. A. and Short, M. W. "Bibliography on Classifi-
cation of Finite Groups." Manuscript, Australian National
University, 1988.
Senior, J. K. and Lunn, A. C. "Determination of the Groups
of Orders 101-161, Omitting Order 128." Amer. J. Math.
56, 328-338, 1934.
Senior, J. K. and Lunn, A. C. "Determination of the Groups
of Orders 162-215, Omitting Order 192." Amer. J. Math.
57, 254-260, 1935.
Simon, B. Representations of Finite and Compact Groups.
Providence, RI: Amer. Math. Soc, 1996.
Sloane, N. J. A. Sequences A000001/M0098 and A000688/
M0064 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
University of Sydney Computational Algebra Group. "The
Magma Computational Algebra for Algebra, Number The-
ory and Geometry." http://www.maths.usyd.edu.au:
8000/u/magma/.
$$ Weisstein, E. W. "Groups." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Groups.m.
Wilson, R. A. "ATLAS of Finite Group Representation."
http : //f or . mat . bham . ac . uk/atlas .
Finite Group — D$
D,
The Dihedral Group D 3 is one of the two groups of
Order 6. It the non- Abelian group of smallest Order.
Examples of D 3 include the POINT GROUPS known as
C 3 h, C 3v , 5 3 , £>3, the symmetry group of the EQUILAT-
ERAL Triangle, and the group of permutation of three
642 Finite Group— D 3
objects. Its elements Ai satisfy Af — 1, and four of
its elements satisfy Ai 2 = 1, where 1 is the Identity
Element. The Cycle Graph is shown above, and the
Multiplication Table is given below.
D Z
1
A
B
C
D
E
1
1
A
B
C
D
E
A
A
1
D
E
B
C
B
B
E
1
D
C
A
C
C
D
E
1
A
B
D
D
C
A
B
E
1
E
E
B
C
A
1
D
The CONJUGACY CLASSES are {1}, {A,B,C}
A^AA^A
B^AB^C
C'^AC = B
D^AD = C
E~ X AE = B,
and {£>, E},
DA~ L D = E
B~ 1 DB = D.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
A reducible 2-D representation using Real Matrices
can be found by performing the spatial rotations corre-
sponding to the symmetry elements of C 3v . Take the
z-AxiS along the C 3 axis.
I = R 2 (Q) =
1
1
(8)
A = R x (l*) =
cos(|7r) sin(|7r)
_-sin(§7r) cos( 5 7r)_
=
\~\ -1^31
JV5 ~\ _
(9)
B = R X (± V ) =
(10)
C = Rc(7T) =
-i (
L
(11)
D = R d (tv) ^CB =
(12)
E = l
1e{k) = C
7A =
2 2
1
"2 .
(13)
To find the irreducible representation, note that there
are three CONJUGACY CLASSES. Rule 5 requires that
there be three irreducible representations satisfying
/ i = / 1 2 +/ 2 2 +/ 3 2 =,6,
so it must be true that
l\ = h = 1, J3 = 2.
(14)
(15)
Finite Group — Dz
By rule 6, we can let the first representation have all Is.
X> 3
1
A
B
C
D
E
Ti
1
1
1
1
1
1
To find representation orthogonal to the totally symmet-
ric representation, we must have three +1 and three —1
Characters. We can also add the constraint that the
components of the Identity Element 1 be positive.
The three C ON JUG AC Y Classes have 1, 2, and 3 ele-
ments. Since we need a total of three +ls and we have
required that a +1 occur for the Conjugacy Class of
Order 1, the remaining +ls must be used for the el-
ements of the Conjugacy Class of Order 2, i.e., A
and B.
D 3
l
A
B
C
D
E
Ti
l
1
1
1
1
1
r 2
l
1
1
-1
-1
-1
Using the rule 1, we see that
1 2 + 1 2 +X3 2 (1) = 6,
(16)
so the final representation for 1 has CHARACTER 2. Or-
thogonality with the first two representations (rule 3)
then yields the following constraints:
1 • 1 • 2 + 1 • 2 • X 2 + 1 • 3 • xs = 2 + 2 X2 + 3x3 =
(17)
1 • 1 • 2 + 1 • 2 • X2 + (-1) • 3 • xs = 2 + 2x2 - 3x3 = 0.
(18)
Solving these simultaneous equations by adding and
subtracting (18) from (17), we obtain X2 = -1, X3 = 0.
The full Character Table is then
D 3
l
A
B
C
D
E
Ti
i
1
1
1
1
1
r 2
l
1
1
-1
-1
-1
r 3
2
-1
-1
Since there are only three CONJUGACY CLASSES, this
table is conventionally written simply as
D 3
i
A = B
C = D = E
Ti
l
1
1
r 2
i
1
-1
r 3
2
-1
Writing the irreducible representations in matrix form
then yields
1 =
ri o
0"
1
1
1
r x
2
-\yft
|V3
-\ o
1
1
(19)
(20)
Finite Group — D A
c =
D
E:
- _ 1
§v/3 C
]
) 0"
)
.
1_
"-1
"
1
1
. o o
-1_
I
-|\/3 /
"|
1
-1.
r 1
2
|\/3 ■
-\ ° °
1
-1.
(21)
(22)
(23)
(24)
see also Dihedral Group, Finite Group — D 4i Fi-
nite Group — Z 6
Finite Group — D4
The Dihedral Group D± is one of the two non-Abelian
groups of the five groups total of Order 8. It is some-
times called the octic group. Examples of D4 include the
symmetry group of the SQUARE. The CYCLE GRAPH is
shown above.
see also Dihedral Group, Finite Group — £> 3 , Fi-
nite Group— Z 8) Finite Group— Z 2 <8> Z 2 ® Z 2 , Fi-
nite Group — Z 2 <8> Z 4 , Finite Group — Z 8 ,
Finite Group — (e)
The unique (and trivial) group of ORDER 1 is denoted
(e). It is (trivially) AbeLIAN and CYCLIC. Examples
include the POINT GROUP C\ and the integers modulo
1 under addition.
(e)
The only class is {!}.
Finite Group
Finite Group— Z 2 <8> Z 2 643
One of the three Abelian groups of the five groups to-
tal of ORDER 8. The group Q 8 has the MULTIPLICA-
TION Table of ±l,i,j, fe, where 1, t, j, and k are the
QUATERNIONS. The CYCLE GRAPH is shown above.
see also Finite Group — L> 4 , Finite Group — Z 2 <8>
Z 2 ® Z 2 , Finite Group — Z 2 ® Z 4 , Finite Group —
Z 8 , Quaternion
Finite Group — Z 2
The unique group of ORDER 2. Z 2 is both ABELIAN and
Cyclic Examples include the Point Groups C a , d,
and C 2 , the integers modulo 2 under addition, and the
Modulo Multiplication Groups M 3 , M 4 , and M 6 .
The elements Ai satisfy Ai 2 — 1, where 1 is the IDEN-
TITY Element. The Cycle Graph is shown above,
and the MULTIPLICATION TABLE is given below.
z 2
1
A
1
1
A
A
A
1
The CONJUGACY CLASSES are {1} and {^4}. The irre-
ducible representation for the C 2 group is {1,-1}.
Finite Group — Z 2 <£) Z 2
z 2 m 2
IP
One of the two groups of Order 4. The name of this
group derives from the fact that it is a DIRECT PROD-
UCT of two Z 2 SUBGROUPS. Like the group Z 4 , Z 2 <g> Z 2
is an Abelian Group. Unlike Z4, however, it is not
Cyclic. In addition to satisfying Ai 4 = 1 for each
element Ai, it also satisfies Ai 2 = 1, where 1 is the
Identity Element. Examples of the Z 2 <g> Z 2 group
include the Viergruppe, Point Groups D 2 , C 2 h, and
C 2v , and the MODULO MULTIPLICATION GROUPS M 8
and Mi 2 . That M 8 , the Residue Classes prime to 8
given by {1, 3, 5, 7}, are a group of type Z 2 <§> Z 2 can
be shown by verifying that
1 3^ = 9 == 1 5^
25 = 1 V
and
3-5 = 15 = 7 3-7 = 21 = 5
= 49 = 1 (mod 8) (1)
5-7 = 35-3 (mod 8).
(2)
644
Finite Group — Z 2 <8> Z 2
Finite Group — Z 2 ® Z 2
Z 2 ® Z 2 is therefore a MODULO MULTIPLICATION
Group.
The Cycle Graph is shown above, and the multiplica-
tion table for the Z 2 ® Z 2 group is given below.
Z 2 <8> Z 2
1
A
B
C
1
1
A
B
C
A
A
1
C
B
B
B
C
1
A
C
C
B
A
1
The Conjugacy Classes are {1}, {A},
{*},
A~ 1 AA = A
B~ 1 AB = A
C' X AC = A,
A^BA^B
(3)
(4)
(5)
(6)
(7)
and {C}.
Now explicitly consider the elements of the C 2 v POINT
Group.
c 2v
E
c 2
<T V
&V
E
E
c 2
(?v
<
c 2
c 2
E
t
a v
<T V
a v
<
E
c 2
*l
a' v
<7 V
c 2
E
In terms of the VlERGRUPPE elements
V
I
v 1
v 2
v 3
I
Vi
v 2
v 3
v 4
Vi
Vi
I
v 3
v 2
v 2
v 2
v 3
I
Vi
v 3
v 3
v 2
Vi
I
A reducible representation using 2-D Real MATRICES
(8)
(9)
(10)
(11)
Another reducible representation using 3-D Real MA-
TRICES can be obtained from the symmetry elements of
the D 2 group (1, C 2 (z), C 2 (y), and C 2 {x)) or C 2v group
1 =
1
1
A =
"-1 "
-1
B =
"o l"
1
C =
" -l"
-1
(1, C 2 , cr v , and a' v ). Place the C 2 axis along the z-axis,
a v in the x-y plane, and a' v in the y-z plane.
1 = E = E:
A = R x (tv) = <j v =
C^R z (tv) = C 2 =
B = R y (ir) = cr v =
10
10
1
1
0-10
1
-10
0-10
1
-10
10
1
(12)
(13)
(14)
(15)
In order to find the irreducible representations, note
that the traces are given by x(l) = 3,x(^2) = — 1,
and x(°v) = xfav) = 1- Therefore, there are at least
three distinct Conjugacy Classes. However, we see
from the MULTIPLICATION TABLE that there are actu-
ally four Conjugacy Classes, so group rule 5 requires
that there must be four irreducible representations. By
rule 1, we are looking for POSITIVE INTEGERS which
satisfy
(16)
h 2 + h 2
+ l 3 2 + h 2 = 4.
The only combination which will work is
l\ = l 2 = ^3 = l 4
(17)
so there are four one-dimensional representations. Rule
2 requires that the sum of the squares equal the ORDER
h = 4, so each 1-D representation must have CHAR-
ACTER ±1. Rule 6 requires that a totally symmetric
representation always exists, so we are free to start off
with the first representation having all Is. We then use
orthogonality (rule 3) to build up the other representa-
tions. The simplest solution is then given by
c 2v
1
c 2
<T V
*' v
Ti
1
1
1
1
r 2
1
-1
-1
1
r 3
1
-1
1
-1
r 4
1
1
-1
-1
These can be put into a more familiar form by switching
Ti and T 3 , giving the CHARACTER TABLE
c 2v
1
c 2
<T V
<
r 3
1
-1
1
-1
r 2
1
-1
-1
1
Ti
1
1
1
1
r 4
1
1
-1
-1
Finite Group— Z 2 ®Z 2 ®Z 2
Finite Group— Z 4 645
The matrices corresponding to this representation are
now
1= (18)
Finite Group — Z3
"10
0"
1
1
_0
1.
"-1
-1
1
_
1
"1
-1
1
_0
-1
"-1
1
1
.
-
-1
(19)
*-= 7 " « (20)
(21)
which consist of the previous representation with an ad-
ditional component. These matrices are now orthogonal,
and the order equals the matrix dimension. As before,
x(<*v) = x(<r'v)-
see also Finite Group — Z 4
Finite Group
One of the three Abelian groups of the five groups total
of Order 8. Examples include the Modulo Multi-
plication Group M 24 . The elements Ai of this group
satisfy Ai 2 = 1, where 1 is the IDENTITY ELEMENT.
The Cycle Graph is shown above.
see also Finite Group — Z> 4 , Finite Group — Q 8 , Fi-
nite Group — Z 2 <8> Z 4 , Finite Group — Z 8
Finite Group — Z2
One of the three Abelian groups of the five groups to-
tal of Order 8. Examples include the Modulo Mul-
tiplication Groups M15, M i6 , M 20 , and M 30 . The
elements Ai of this group satisfy Ai 4 = 1, where 1 is the
Identity Element, and four of the elements satisfy
Ai 2 - 1. The Cycle Graph is shown above.
see also Finite Group — Z> 4 , Finite Group — Q 8 , Fi-
nite Group — Z 2 ® Z 2 % Z 2 , Finite Group — Z 8
The unique group of Order 3. It is both Abelian
and Cyclic. Examples include the Point Groups <7 3
and £>3 and the integer modulo 3. The elements Ai
of the group satisfy A{ 3 = 1 where 1 is the Identity
Element. The Cycle Graph is shown above, and the
Multiplication Table is given below.
Z3
1
A
B
1
1
A
B
A
A
B
1
B
B
1
A
The Conjugacy Classes are {1}, {A},
A^AA = A
B~ X AB = A,
and {B},
A~ X BA = B
B^BB^B.
The irreducible representation (CHARACTER TABLE) is
therefore
r
1
A
B
Ti
1
1
1
r 2
1
1
-1
r 3
1
-1
1
Finite Group — Z4
One of the two groups of ORDER 4. Like Z 2 <g> Z 2 , it is
ABELIAN, but unlike Z 2 <g>Z 2 , it is a Cyclic Examples
include the Point Groups Ca and 5 4 and the Modulo
Multiplication Groups M 5 and M 10 . Elements Ai
of the group satisfy Ai 4 = 1, where 1 is the IDENTITY
Element, and two of the elements satisfy Ai 2 = 1.
The Cycle Graph is shown above. The Multipli-
cation Table for this group may be written in three
(i) ^(2)
7 (3)
equivalent ways — denoted here by Z\ \Z\ , and Z\
by permuting the symbols used for the group elements.
z™
1
A
B
C
1
1
A
B
C
A
A
B
C
1
B
B
C
1
A
C
C
1
A
B
646 Finite Group— Z 4
Finite Group — Z 6
The Multiplication Table for Z^ 2) is obtained from
Z4 by interchanging A and B.
7(2)
1
A
B
C
1
1
A
B
c
A
A
1
C
B
B
B
C
A
1
C
C
B
1
A
The Multiplication Table for Z^ is obtained from
Z± by interchanging A and C.
zT
1
A
B
C
1
1
A
B
C
i4
A
C
1
B
£
B
1
C
A
C
c
B
A
1
The Conjugacy Classes of Z 4 are {1}, {A},
A~ l AA = A
B~ l AB = A
C 1 AC = A,
(1)
(2)
(3)
Finite Group — Z$
The unique GROUP of ORDER 5, which is ABELIAN. Ex-
amples include the POINT GROUP C5 and the integers
mod 5 under addition. The elements Ai satisfy Ai 5 = 1,
where 1 is the Identity Element. The Cycle Graph
is shown above, and the MULTIPLICATION TABLE is il-
lustrated below.
Z5
1
A
B
C
D
1
1
A
B
C
D
A
A
B
C
D
1
B
B
C
D
1
A
C
C
D
1
A
B
D
D
1
A
B
C
The Conjugacy Classes are {1}, {A}, {B}, {C}, and
{D}.
{B},
Finite Group — Z 6
A~ X BA = B
B~ X BB = B
C~ X BC = B,
(4)
(5)
(6)
and {C}.
The group may be given a reducible representation using
Complex Numbers
1 = 1
A = i
B = -l
C = -L
or Real Matrices
A =
B =
C ==
"i o"
1
"o -l"
1
"-1
-
" o r
1
-1
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
One of the two groups of ORDER 6 which, unlike £>3,
is Abelian. It is also a Cyclic. It is isomorphic to
Z 2 <g> Z 3 . Examples include the POINT GROUPS Cq and
Sq, the integers modulo 6 under addition, and the MOD-
ULO Multiplication Groups M 7 , M 9 , and M14. The
elements Ai of the group satisfy Ai 6 — 1, where 1 is
the Identity Element, three elements satisfy A* 3 = 1,
and two elements satisfy Ai 2 = 1. The CYCLE GRAPH is
shown above, and the MULTIPLICATION Table is given
below.
Z 6
1
A
B
C
D
E
1
1
A
B
C
D
E
A
A
1
E
D
B
C
B
B
E
1
A
C
D
C
C
D
A
1
E
B
D
D
B
C
E
1
A
E
E
C
D
B
A
1
The Conjugacy Classes are {1}, {A}, {B}, {C},
{D}, and {E}.
see also Finite Group — JD 3
see also FINITE GROUP— Z 2 <S> Z 2
Finite Group — Z 7
Finsler Space 647
Finite Group — Z7
The unique Group of Order 7. It is Abelian and
Cyclic. Examples include the Point Group C 7 and
the integers modulo 7 under addition. The elements Ai
of the group satisfy Ai 7 — 1, where 1 is the Identity
Element. The Cycle Graph is shown above.
z 7
1
A
B
C
D
E
F
1
1
A
B
C
D
E
F
A
A
B
C
D
E
F
1
B
B
C
D
E
F
1
A
C
C
D
E
F
1
A
B
D
D
E
F
1
A
B
C
E
E
F
1
A
B
C
D
F
F
1
A
B
C
D
E
The Conjugacy Classes are {1}, {A}, {B}, {C},
{D}, {£}, and {F}.
Finite Group
One of the three Abelian groups of the five groups total
of Order 8. An example is the residue classes modulo
17 which Quadratic Residues, i.e., {1, 2, 4, 8, 9, 13,
15, 16} under multiplication modulo 17. The elements
Ai satisfy Ai 8 = 1, four of them satisfy A, 4 = 1, and two
satisfy A? ~ 1. The Cycle Graph is shown above.
see also Finite Group — £> 4 , Finite Group — Q 8 , Fi-
nite Group — Z 2 (8> Z 4 , Finite Group — Z 2 <8> Z 2 <g> Z 2
Finite Mathematics
The branch of mathematics which does not involve infi-
nite sets, limits, or continuity.
see also COMBINATORICS, DISCRETE MATHEMATICS
References
Hildebrand, F. H. and Johnson, C. G. Finite Mathematics.
Boston, MA: Prindle, Weber, and Schmidt, 1970.
Kemeny, J. G.; Snell, J. L.; and Thompson, G. L. Introduc-
tion to Finite Mathematics, 3rd ed. Englewood Cliffs, NJ:
Prentice-Hall, 1974.
Marcus, M. A Survey of Finite Mathematics. New York:
Dover, 1993.
Finite Simple Group
see Simple Group
Finite Simple Group Classification Theorem
see Classification Theorem
Finite-to-One Factor
A Map ip : M -> M, where M is a MANIFOLD, is a
finite-to-one factor of a Map \I> : X — > X if there exists
a continuous Onto Map n : X -» M such that ip o n =
7r o ^ and n~ 1 (x) C X is finite for each x G M.
Finsler Geometry
The geometry of FINSLER SPACE.
Finsler Manifold
see Finsler Space
Finsler Metric
A continuous real function L(x,y) defined on the TAN-
GENT Bundle T(M) of an n-D Differentiable Man-
ifold M is said to be a Finsler metric if
1. L(x,y) is Differentiable at x ^ y,
2. L(x,\y) = \\\L(x,y) for any element (x,y) G T(M)
and any Real Number A,
3. Denoting the METRIC
! 8»[L(s,y)] a
9* 3 {*>V)- 2 dyidyj ,
then gij is a POSITIVE DEFINITE MATRIX.
A DIFFERENTIABLE MANIFOLD M with a Finsler metric
is called a Finsler Space.
see also Differentiable Manifold, Finsler Space,
Tangent Bundle
References
lyanaga, S. and Kawada, Y. (Eds.). "Finsler Spaces." §161
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, p. 540-542, 1980.
Finsler Space
A general space based on the Line Element
ds = F(x , . . . , x n \ dx , . . . , dx n ),
with F(x,y) > for y ^ a function on the TAN-
GENT Bundle T(M), and homogeneous of degree 1 in
y. Formally, a Finsler space is a DIFFERENTIABLE MAN-
IFOLD possessing a FINSLER METRIC. Finsler geometry
is RlEMANNIAN GEOMETRY without the restriction that
the Line Element be quadratic of the form
F = gij(x) dx 1, dx J .
A compact boundaryless Finsler space
Minkowskian Iff it has "flag curvature."
locally
648 Finsler-Hadwiger Theorem
Fischer Groups
see also FlNSLER METRIC, HODGE'S THEOREM, RlE-
mannian Geometry, Tangent Bundle
References
Akbar-Zadeh, H. "Sur les espaces de Finsler a courbures sec-
tionnelles constantes." Acad. Roy. Belg. Bull. CI. Sci. 74,
281-322, 1988.
Bao, D.; Chern, S.-S.; and Shen, Z. (Eds.). Finsler Geome-
try. Providence, RI: Amer. Math. Soc, 1996.
Chern, S.-S. "Finsler Geometry is Just Riemannian Geome-
try without the Quadratic Restriction." Not. Amer. Math.
Soc. 43, 959-963, 1996.
Iyanaga, S. and Kawada, Y. (Eds,). "Finsler Spaces," §161
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, p. 540-542, 1980.
Finsler-Hadwiger Theorem
Let the SQUARES \3ABCD and dAB'C'D' share a com-
mon VERTEX A. The midpoints Q and S of the segments
B'D and BD' together with the centers of the original
squares R and T then form another square OQRST.
This theorem is a special case of the FUNDAMENTAL
Theorem of Directly Similar Figures (Detemple
and Harold 1996).
see also FUNDAMENTAL THEOREM OF DIRECTLY SIMI-
LAR Figures, Square
References
Detemple, D. and Harold, S. "A Round-Up of Square Prob-
lems." Math. Mag. 69, 15-27, 1996.
Finsler, P. and Hadwiger, H. "Einige Relationen im Dreieck."
Comment. Helv. 10, 316-326, 1937.
Fisher, J. C; Ruoff, D.; and Shileto, J. "Polygons and Poly-
nomials." In The Geometric Vein: The Coxeter Fest-
schrift. New York: S p ringer- Ver lag, 321-333, 1981.
First- Countable Space
A Topological Space in which every point has a
countable BASE for its neighborhood system.
First Curvature
see Curvature
First Derivative Test
f\x) < o, 1
/"Cr)>0\
f(x) =
A
fix) > / \/'(jc) <
stationary point
minimum
maximum
Suppose f(x) is Continuous at a Stationary Point
xo-
1. If f(x) > on an Open Interval extending left
from xo and f(x) < on an OPEN INTERVAL ex-
tending right from Xo, then / has a RELATIVE MAX-
IMUM (possibly a Global Maximum) at x .
2. If /'(#) < on an Open Interval extending left
from xo and f(x) > on an Open Interval ex-
tending right from xo, then / has a RELATIVE MIN-
IMUM (possibly a Global Minimum) at xo.
3. If f'(x) has the same sign on an OPEN INTERVAL
extending left from x and on an OPEN INTERVAL
extending right from xo, then / does not have a Rel-
ative Extremum at x .
see also Extremum, Global Maximum, Global
Minimum, Inflection Point, Maximum, Minimum,
Relative Extremum, Relative Maximum, Rela-
tive Minimum, Second Derivative Test, Station-
ary 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.
First Digit Law
see Benford's Law
First Digit Phenomenon
see Benford's Law
First Multiplier Theorem
Let D be a planar Abelian DIFFERENCE Set and t be
any Divisor of n. Then Ms a numerical multiplier of
£>, where a multiplier is defined as an automorphism a
of G which takes D to a translation g + D of itself for
some g £ G. If a is of the form a : x — > tx for t 6 Z
relatively prime to the order of G, then a is called a
numerical multiplier.
References
Gordon, D. M. "The Prime Power Conjecture is True
for n < 2, 000, 000." Electronic J. Combinatorics 1,
R6, 1-7, 1994. http://vvv.combinatorics.Org/VolumeJ./
volumel.html#R6.
Fischer's Baby Monster Group
see Baby Monster Group
Fischer Groups
The Sporadic Groups Fi 2 2, Fi 2 3, and Fi 2 ^ These
groups were discovered during the investigation of 3-
Transposition Groups.
see also Sporadic Group
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http : //for . mat . bham . ac . uk/ atlas /F22 . html, F23 . html,
and F24.html.
Fish Bladder
Fisher Index
649
Fish Bladder
see Lens
Fisher-Behrens Problem
The determination of a test for the equality of MEANS
for two Normal Distributions with different Vari-
ances given samples from each. There exists an ex-
act test which, however, does not give a unique answer
because it does not use all the data. There also exist
approximate tests which do not use all the data.
see also NORMAL DISTRIBUTION
References
Fisher, R. A. "The Fiducial Argument in Statistical Infer-
ence." Ann. Eugenics 6, 391-398, 1935.
Kenney, J. F. and Keeping, E. S. "The Behrens-Fisher Test."
§9.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton,
NJ: Van Nostrand, pp. 257-260 and 261-264, 1951.
Sukhatme, P. V. "On Fisher and Behrens' Test of Signifi-
cance of the Difference in Means of Two Normal Samples."
Sankhya 4, 39, 1938.
Fisher's Block Design Inequality
A balanced incomplete Block Design (t>, fc, A, r,
exists only or b > v (or, equivalently, r > k).
see also BRUCK-RYSER-CHOWLA THEOREM
b)
References
Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to
Design Theory." Ch. 1 in Contemporary Design Theory: A
Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson).
New York: Wiley, pp. 1-12, 1992.
Fisher's Estimator Inequality
Given T an Unbiased Estimator of so that (T) = 0.
Then
var(T) > = ,
where var is the VARIANCE.
Fisher's Exact Test
A Statistical Test used to determine if there are non-
random associations between two CATEGORICAL VARI-
ABLES. Let there exist two such variables X and Y,
with 77i and n observed states, respectively. Now form
an n x m Matrix in which the entries a,ij represent the
number of observations in which x = i and y = j. Cal-
culate the row and column sums Ri and Cj , respectively,
and the total sum
^ = E^ = E^
(which is a Hypergeometric Distribution). Now
find all possible Matrices of Nonnegative Integers
consistent with the row and column sums Ri and Cj.
For each one, calculate the associated P- VALUE using
(0) (where the sum of these probabilities must be 1).
Then the P- Value of the test is given by the sum of all
P- Values which are < P cr it.
The test is most commonly applied to a 2 x 2 MATRICES,
and is computationally unwieldy for large m or n.
As an example application of the test, let X be a journal,
say either Mathematics Magazine or Science, and let Y
be the number of articles on the topics of mathematics
and biology appearing in a given issue of one of these
journals. If Mathematics Magazine has five articles on
math and one on biology, and Science has none on math
and four on biology, then the relevant matrix would be
Math. Mag. Science
math 5 Ri = 5
biology 1 4 R2 = 5
C x = 6 C 2 = 4 N = 10.
Computing P crit gives
■Pcrit —
5! 2 6!4!
= 0.0238,
10!(5!0!1!4!)
and the other possible matrices and their Ps are
4 1
2 3
"2
3"
4
1
"l
4"
5
P = 0.2381
P = 0.4762
P = 0.2381
P = 0.0238,
which indeed sum to 1, as required. The sum of P- values
less than or equal to P cr it = 0.0238 is then 0.0476 which,
because it is less than 0.05, is SIGNIFICANT. Therefore,
in this case, there would be a statistically significant
association between the journal and type of article ap-
pearing.
Fisher Index
The statistical Index
Pb = VKP^,
of the Matrix. Then calculate the conditional Likeli-
hood (P- Value) of getting the actual matrix given the
particular row and column sums, given by
Pcrit —
{Rx\R2\---Rm\){C 1 \C 2 \"-C ri \)
where P L is Laspeyres' Index and Pp is Paasche's
Index.
see also INDEX
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 66, 1962.
650
Fisher Kurtosis
Fisher Kurtosis
- K - ^ 4
72 = 02 = «
M2 2
^4
where /a» is the ith MOMENT about the MEAN and a —
y/jj£ is the Standard Deviation.
see also Fisher Skewness, Kurtosis, Pearson Kur-
tosis
Fisher Sign Test
A robust nonparametric test which is an alternative to
the Paired £-Test. This test makes the basic assump-
tion that there is information only in the signs of the dif-
ferences between paired observations, not in their sizes.
Take the paired observations, calculate the differences,
and count the number of +s n+ and — s n_, where
N = n+ + n-
is the sample size. Calculate the BINOMIAL COEFFI-
CIENT
£)■
Then B/2 N gives the probability of getting exactly this
many +s and — s if Positive and Negative values are
equally likely. Finally, to obtain the P- VALUE for the
test, sum all the COEFFICIENTS that are < B and divide
by 2".
see also Hypothesis Testing
Fisher Skewness
7i =
M3
V>Z
M2
3/2
where \x% is the i Moment about the Mean, and a =
yftii is the Standard Deviation.
see also Fisher Kurtosis, Moment, Skewness,
Standard Deviation
Fisher's Theorem
Let A be a sum of squares of n independent normal
standardized variates xi , and suppose A = B + C where
B is a quadratic form in the Xz, distributed as CHI-
Squared with h Degrees of Freedom. Then C is
distributed as x 2 with n — h DEGREES OF FREEDOM
and is independent of B. The converse of this theorem
is known as COCHRAN'S THEOREM.
see also Chi-Squared Distribution, Cochran's
THEOREM
Fisher-Tippett Distribution
Fisher-Tippett Distribution
/
f
Also called the Extreme Value Distribution and
Log-Weibull Distribution. It is the limiting distri-
bution for the smallest or largest values in a large sample
drawn from a variety of distributions.
P(x) =
D(x) = ,
p (a-*)/6- e <— *>/*
_ e (a-x)/b
These can be computed directly be defining
/a — x\
b\nz
dz = — - exp I — - — 1 dx.
z = exp
x = a — b In z
(i)
(2)
(3)
(4)
(5)
Then the MOMENTS are
/oo
x n P(x) dx
■oo
= \ r * n exp (^j) exp[-e (o - x)/ V*
J — OO
(a-blnz) n e~ z dz
>
(a-b]nz) n e~ z dz
f
J oo
-/
Jo
=t,(lYv kan ~ kbk f°° {]nz)ke ~ Zdz
i,—n \ / •'O
= E
n k )a n - k b k I(k),
(6)
where I(k) are Euler-Mascheroni Integrals. Plug-
ging in the Euler-Mascheroni Integrals I(k) gives
/io = 1
fii = a + bj
H2 = a + 2aby + b 2 (-y 2 + \iz 2 )
fi 3 = a 3 + 3a 2 by + iab 2 ^ 2 + \* 2 )
+ 6 3 [ 7 3 +|7T 2 + 2C(3)]
[14 = a 4 + 4a 3 by + 6a 2 b 2 (-y 2 + \n 2 )
+ 4a6 3 [7 3 + 57T 2 +2C(3)]
+ 6 4 [7 4 + 7V + ^7r 4 + 8 7 C(3)],
(7)
(8)
(9)
(10)
(11)
Fisher's z-Distribution
Five Cubes 651
where 7 is the Euler-Mascheroni Constant and £(3)
is Apery's Constant. The Mean, Variance, Skew-
NESS, and KURTOSIS are therefore
giving
(j, — a + 67
2 2 1 2,2
(T ~ 111 — ll\ = g7T
71 = F
6 3 7T 3
r {a 3 + 3a 2 67 + 3a6 2 (7 2 + \-k 2 )
+ 6 3[ 7 3 + I 77r 2 + 2C(3)]}
72-^-3
36 {a 4 +4a 3 6 7 + a 2 6 2 (67 2 + 7r 2 )
(12)
(13)
(14)
6V
+ 4a6 3 [ 7 3 + | 77 r 2 + 2C(3)]
+ 6 4 [7 4 + 7 2 7r 2 + ^7r 4 + 87C(3)]}. (15)
The Characteristic Function is
<t>(t) = r(l-ij3t)e ioc \
(16)
where T(z) is the Gamma Function. The special case
of the Fisher-Tippett distribution with a = 0, b = 1 is
called Gumbel's Distribution.
see also Euler-Mascheroni Integrals, Gumbel's
Distribution
Fisher's z-Distribution
9(z)
2ni
l/2 n2 n 2 /2
B (^ «a) (me 2 * + n 2 )( ni+ni >/ 2
(1)
(Kenney and Keeping 1951). This general distribution
includes the Chi-Squared Distribution and Stu-
dent's ^-DISTRIBUTION as special cases. Let u 2 and v 2
be Independent Unbiased Estimators of the Vari-
ance of a Normally Distributed variate. Define
= -(;)- !-(?)•
Then let
Ns 2 2
n 2
(2)
(3)
so that n\F/ri2 is a ratio of Chi-Squared variates
niF x 2 (m)
ri2 X 2 (^2)'
(4)
which makes it a ratio of Gamma Distribution vari
ates, which is itself a Beta Prime Distribution vari
ate,
AY)
7(¥)
= 0'(^>¥).
(5)
f(F)
cgr"- 1 (!+*£)
-(ni+n 2 )/2 n^
™2
The Mean is
and the Mode is
{F) =
n 2
n 2 -2'
712 71 1 — 2
n 2 + 2 m
(6)
(7)
(8)
see also Beta Distribution, Beta Prime Distri-
bution, Chi-Squared Distribution, Gamma Dis-
tribution, Normal Distribution, Student's t-
Distribution
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
PL 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 180-181,
1951.
Fisher's ^'-Transformation
Let r be the Correlation Coefficient. Then defin-
ing
z = tanh 1 r
\ tanh p,
gives
var(z )
1 4-p-
n 2n 2
7i =
72 =
p\p 2 -&\
n 3 / 2
32 - 3p 4
16N
(1)
(2)
(3)
(4)
(5)
(6)
where n = N — 1.
see also CORRELATION COEFFICIENT
Fitting Subgroup
The unique smallest Normal Nilpotent Subgroup
of H , denoted F(H). The generalized fitting subgroup
is defined by F*(H) = F(H)E(H), where E(H) is the
commuting product of all components of i?, and F is
the fitting subgroup of H.
Five Cubes
see Cube 5-Compound
652 Five Disks Problem
Five Disks Problem
Given five equal DISKS placed symmetrically about a
given center, what is the smallest Radius r for which the
Radius of the circular Area covered by the five disks
is 1? The answer is r = <p - 1 = 1/0 = 0.6180340. . .,
where <j> is the GOLDEN RATIO, and the centers a of the
disks i = 1, .... 5 are located at
Ci -
The Golden Ratio enters here through its connection
with the regular PENTAGON. If the requirement that the
disks be symmetrically placed is dropped (the general
Disk Covering Problem), then the Radius for n =
5 disks can be reduced slightly to 0.609383. . . (Neville
1915).
see also Arc, Disk Covering Problem, Flower of
Life, Seed of Life
References
Ball, W. W. R. and Coxeter, H. S. M. "The Five-Disc Prob-
lem." In Mathematical Recreations and Essays, 13th ed.
New York: Dover, pp. 97-99, 1987.
Neville, E. H. "On the Solution of Numerical Functional
Equations, Illustrated by an Account of a Popular Puz-
zle and of its Solution." Proc. London Math. Soc. 14,
308-326, 1915.
Five Tetrahedra Compound
see Tetrahedron 5-Compound
Fixed
When referring to a planar object, "fixed" means that
the object is regarded as fixed in the plane so that it
may not be picked up and flipped. As a result, MIRROR
IMAGES are not necessarily equivalent for fixed objects.
see also FREE, MIRROR IMAGE
Fixed Element
see Fixed Point (Map)
Fixed Point (Map)
Fixed Point
A point which does not change upon application of a
Map, system of Differential Equations, etc.
see also Fixed Point (Differential Equations),
Fixed Point (Map), Fixed Point Theorem
References
Shashkin, Yu. A. Fixed Points. Providence, RI: Amer. Math.
Soc, 1991.
Fixed Point (Differential Equations)
Points of an AUTONOMOUS system of ordinary differen-
tial equations at which
^ = fi(xi,. . .,a3 n ) =
—^ — jn\El-> * * • ) x n) — 0.
If a variable is slightly displaced from a Fixed Point, it
may (1) move back to the fixed point ("asymptotically
stable" or "superstable" ) , (2) move away ("unstable"),
or (3) move in a neighborhood of the fixed point but
not approach it ("stable" but not "asymptotically sta-
ble"). Fixed points are also called CRITICAL POINTS
or Equilibrium Points. If a variable starts at a point
that is not a CRITICAL Point, it cannot reach a critical
point in a finite amount of time. Also, a trajectory pass-
ing through at least one point that is not a CRITICAL
POINT cannot cross itself unless it is a CLOSED CURVE,
in which case it corresponds to a periodic solution.
A fixed point can be classified into one of several classes
using Linear Stability analysis and the resulting Sta-
bility Matrix.
see also Elliptic Fixed Point (Differential Equa-
tions), Hyperbolic Fixed Point (Differential
Equations), Stable Improper Node, Stable Node,
Stable Spiral Point, Stable Star, Unstable Im-
proper Node, Unstable Node, Unstable Spiral
Point, Unstable Star
Fixed Point (Map)
A point x* which is mapped to itself under a Map G, so
that x* — G(x*). Such points are sometimes also called
Invariant Points, or Fixed Elements (Woods 1961).
Stable fixed points are called elliptical. Unstable fixed
points, corresponding to an intersection of a stable and
unstable invariant Manifold, are called Hyperbolic
(or Saddle). Points may also be called asymptotically
stable (a.k.a. superstable).
see also CRITICAL POINT, INVOLUNTARY
References
Shashkin, Yu. A. Fixed Points. Providence, RI: Amer. Math.
Soc, 1991.
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, p. 14,
1961.
Fixed Point Theorem
Fletcher Point 653
Fixed Point Theorem
If g is a continuous function g(x) E [a, b] FOR ALL x £
[a, 6], then g has a Fixed Point in [a, b]. This can be
proven by noting that
g(a) > a g{b) < b
g(a) - a > g(b) - 6 < 0.
Since g is continuous, the Intermediate Value THE-
OREM guarantees that there exists a c € [a, 6] such that
fl ( c ) _ c = 0,
so there must exist a c such that
g(c) = c,
so there must exist a Fixed Point e [a, 6].
see also Banach Fixed Point Theorem, Brouwer
Fixed Point Theorem, Kakutani's Fixed Point
Theorem, Lefshetz Fixed Point Formula, Lef-
shetz Trace Formula, Poincare-Birkhoff Fixed
Point Theorem, Schauder Fixed Point Theorem
Fixed Point (Transformation)
see Fixed Point (Map)
Flag
A collection of FACES of an n-D POLYTOPE or simplicial
Complex, one of each Dimension 0, 1, . . . , n-1, which
all have a common nonempty Intersection. In normal
3-D, the flag consists of a half-plane, its bounding RAY,
and the Ray's endpoint.
Flag Manifold
For any SEQUENCE of INTEGERS < m < . . . < n fc ,
there is a flag manifold of type (m, . . . , nk) which is
the collection of ordered pairs of vector SuBSPACES of
M nfe (Vi, . . . , y fc ) with dim(Vi) = m and V- a Subspace
of Vi+i. There are also COMPLEX flag manifolds with
Complex subspaces of C nfc instead of REAL SUBSPACES
of a Real rik -space. These flag manifolds admit the
structure of MANIFOLDS in a natural way and are used
in the theory of LIE GROUPS.
see also GRASSMANN MANIFOLD
References
Lu, J.-H. and Weinstein, A. "Poisson Lie Groups, Dressing
Transformations, and the Bruhat Decomposition." J. Diff.
Geom. 31, 501-526, 1990.
Flat
A set in K formed by translating an affine subspace or
by the intersection of a set of Hyperplanes.
Flat Norm
The flat norm on a CURRENT is defined by
HS)
= /{AreaT
+ vo\R:S-T = dR},
where dR is the boundary of R.
see also Compactness Theorem, Current
References
Morgan, F. "What Is a Surface?" Amer. Math. Monthly 103,
369-376, 1996.
Flat Space Theorem
If it is possible to transform a coordinate system to a
form where the metric elements g^ u are constants inde-
pendent of x^, then the space is flat.
Flat Surface
A Regular Surface and special class of Minimal
Surface for the Gaussian Curvature vanishes ev-
erywhere. A Tangent Developable, Generalized
Cone, and Generalized Cylinder are all flat sur-
faces.
see also Minimal Surface
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 280, 1993.
Flattening
The flattening of a Spheroid (also called Oblateness)
is denoted e or /. It is defined as
f £=£ = 1-
■ = J a
— 1 c—a c
\ a a
- oblate
a
1 prolate,
where c is the polar Radius and a is the equatorial
Radius.
see also Eccentricity, Ellipsoid, Oblate Spher-
oid, Prolate Spheroid, Spheroid
Flemish Knot
see FlGURE-OF-ElGHT KNOT
Fletcher Point
654 Flexible Polyhedron
Flip Bifurcation
The intersection of the Gergonne Line and the Soddy
Line. It has Trilinear Coordinates given by
»-'-l(l + i . + 7
Ge,
where / is the Incenter, Ge the Gergonne Point,
and d, e, and / are the lengths of the sides of the Con-
tact Triangle ADEF.
see also Contact Triangle, Gergonne Line, Ger-
gonne Point, Soddy Line
References
Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
Flexible Polyhedron
mountain fold
■valley fold
The Rigidity Theorem states that if the faces of a
convex POLYHEDRON are made of metal plates and the
Edges are replaced by hinges, the Polyhedron would
be Rigid. The theorem was stated by Cauchy (1813),
although a mistake in this paper went unnoticed for
more than 50 years. Concave polyhedra need not be
Rigid, and such nonrigid polyhedra are called flexible
polyhedra. Connelly (1978) found the first example of a
reflexible polyhedron, consisting of 18 triangular faces.
A flexible polyhedron with only 14 triangular faces and
9 vertices (shown above), believed to be the simplest
possible composed of only triangles, was subsequently
found by Steffen (Mackenzie 1998). There also exists
a six-vertex eight-face flexible polyhedron (Wunderlich
and Schwabe 1986, Cromwell 1997).
Connelly et al. (1997) proved that a flexible polyhedron
must keep its Volume constant (Mackenzie 1998).
see also POLYHEDRON, QUADRICORN, RIGID, RIGIDITY
Theorem
References
XVIe
Cauchy, A. L. "Sur les polygons et le polyheders.'
CahierlX, 87-89, 1813.
Connelly, R. "A Flexible Sphere." Math. Intel 1, 130-131,
1978.
Connelly, R.; Sabitov, I.; and Walz, A. "The Bellows Conjec-
ture." Contrib. Algebra Geom. 38, 1-10, 1997.
Cromwell, P. R. Polyhedra. New York: Cambridge University
Press, 1997.
Mackenzie, D. "Polyhedra Can Bend But Not Breathe." Sci-
ence 279, 1637, 1998.
Wunderlich, W. and Schwabe, C. "Eine Familie von
geschlossen gleichflachigen Polyhedern, die fast beweglich
sind." Elem. Math. 41, 88-98, 1986.
Flexagon
An object created by FOLDING a piece of paper along
certain lines to form loops. The number of states pos-
sible in an n-FLEXAGON is a CATALAN NUMBER. By
manipulating the folds, it is possible to hide and reveal
different faces.
see also Flexatube, Folding, Hexaflexagon, Tet-
RAFLEXAGON
References
Crampin, J. "On Note 2449." Math. Gazette 41, 55-56, 1957.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 205-207, 1989.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 62-84, 1979.
Gardner, M. "Hexaflexagons." Ch. 1 in The Scientific Amer-
ican Book of Mathematical Puzzles & Diversions. New
York: Simon and Schuster, 1959.
Gardner, M. Ch. 2 in The Second Scientific American Book
of Mathematical Puzzles & Diversions: A New Selection.
New York: Simon and Schuster, pp. 24-31, 1961.
Maunsell, F. G. "The Flexagon and the Hexaflexagon."
Math. Gazette 38, 213-214, 1954.
Oakley, C. O. and Wisner, R. J. "Flexagons." Amer. Math.
Monthly 64, 143-154, 1957.
Wheeler, R. F. "The Flexagon Family." Math. Gaz. 42, 1-6,
1958.
Flexatube
\i/
A FLEXAGON-like structure created by connecting the
ends of a strip of four squares after folding along 45°
diagonals. Using a number of folding movements, it is
possible to flip the flexatube inside out so that the faces
originally facing inward face outward. Gardner (1961)
illustrated one possible solution, and Steinhaus (1983)
gives a second.
see also FLEXAGON, HEXAFLEXAGON, TETRAFLEXA-
GON
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 205, 1989.
Gardner, M. The Second Scientific American Book of Math-
ematical Puzzles & Diversions: A New Selection. New
York: Simon and Schuster, pp. 29-31, 1961.
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 177-181 and 190,
1983.
Flip Bifurcation
Let /:MxIR^IRbea one-parameter family of C 3
maps satisfying
d£
dx
/(0,0) =
^i— 0,x=0
Floor Function
Floquet Analysis 655
dx 2
dx s
<0
p — 0,a;=0
<0.
M = 0,x =
Then there are intervals (yni,0), (0,^2), and e > such
that
1. If \i € (0,^2), then / M (x) has one unstable fixed point
and one stable orbit of period two for x G (— e, e), and
2. If /i G (^i,0), then f^(x) has a single stable fixed
point for x £ ( — e, e).
This type of BIFURCATION is known as a flip bifurcation.
An example of an equation displaying a flip bifurcation
is
f(x) = jjl — x — x .
see also BIFURCATION
References
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, pp. 27-30, 1990.
Floor Function
\x\ Ceiling
[x] Nint (Round)
[x\ Floor
The function [x\ is the largest INTEGER < x, shown as
the dashed curve in the above plot, and also called the
Greatest Integer Function. In many computer lan-
guages, the floor function is called the INTEGER PART
function and is denoted int(x). The name and sym-
bol for the floor function were coined by K. E. Iverson
(Graham et al. 1990).
Unfortunately, in many older and current works (e.g.,
Shanks 1993, Ribenboim 1996), the symbol [x] is used
instead of [x\ . Because of the elegant symmetry of the
floor function and Ceiling Function symbols [as J and
\x] , and because [x] is such a useful symbol when inter-
preted as an IVERSON BRACKET, the use of [x] to denote
the floor function should be deprecated. In this work,
the symbol [x] is used to denote the nearest integer NlNT
function since it naturally falls between the [asj and \x]
symbols.
see also CEILING FUNCTION, FRACTIONAL PART, INT,
Iverson Bracket, Nint
References
Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Integer
Functions." Ch. 3 in Concrete Mathematics: A Foun-
dation for Computer Science. Reading, MA: Addison-
Wesley, pp. 67-101, 1990.
Iverson, K. E. A Programming Language. New York: Wiley,
p. 12, 1962.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 180-182, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 14, 1993.
Spanier, J. and Oldham, K. B. "The Integer- Value lnt(s) and
Fractional- Value frac(z) Functions." Ch. 9 in An Atlas of
Functions. Washington, DC: Hemisphere, pp. 71-78, 1987.
Floquet Analysis
Given a system of periodic Ordinary Differential
Equations of the form
X
r
-1
d
y
-1
dt
v x
$xx
$yy
[v V _
_$ X y
®yy
v x
y J
(i)
the solution can be written as a linear combination of
functions of the form
r *(*) 1
" xo '
y(t)
2/o
v x (t)
^x0
UwJ
_VyQ_
e^P^t),
(2)
where Pp(f) is a function periodic with the same period
T as the equations themselves. Given an Ordinary
Differential Equation of the form
x + g{t)x = 0,
(3)
where g(t) is periodic with period T, the ODE has a
pair of independent solutions given by the REAL and
Imaginary Parts of
x(t) = w(t)e ilp(t)
x = (w + iwip)e
x — [w + iwip + i(wip + wij) + iwtp 2 )]e
= [(w — wip ) 4- i(2wip 4- wip)]e .
Plugging these into (3) gives
w + 2iwip + w(g + {$ — ip 2 ) = 0,
so the Real and Imaginary Parts are
w 4 w(g — tp ) —
2wip 4 wip — 0.
From (9),
— + K = 2- (into + -rAHn
w ip at at
= —\n(ipw 2 ) = 0.
at
(4)
(5)
(6)
(7)
(8)
(9)
(10)
656 Floquet's Theorem
Flype
Integrating gives
i>
(ii)
where C is a constant which must equal 1, so ip is given
by
The REAL solution is then
x(t) — w(t) cos[tft(t)],
x — w cos ib — wib sin ib — w wip sin ip
w
(13)
= w — — w-
X 1
■ sin^ = w sini/'
(14)
and
1 = cos tp -f sin tp = x w 4-
•\2 _
( . x \
w X
V w J
x w + (wx — wx) = /(x, i, t) }
(15)
which is an integral of motion. Therefore, although
w(t) is not explicitly known, an integral / always ex-
ists. Plugging (10) into (8) gives
w + g(t)w- — =Q y (16)
which, however, is not any easier to solve than (3).
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 727, 1972.
Binney, J. and Tremaine, S. Galactic Dynamics. Princeton,
NJ: Princeton University Press, p. 175, 1987.
Lichtenberg, A. and Lieberman, M. Regular and Stochastic
Motion. New York: Springer- Verlag, p. 32, 1983.
Margenau, H. and Murphy, G. M. The Mathematics of Phys-
ics and Chemistry, 2 vols. Princeton, NJ: Van Nostrand,
1956-64.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 556-557, 1953.
Floquet's Theorem
see Floquet Analysis
Flow
An ACTION with G = R. Flows are generated by VEC-
TOR Fields and vice versa.
see also Action, Ambrose-Kakutani Theorem,
Anosov Flow, Axiom A Flow, Cascade, Geodesic
Flow, Semiflow
Flow Line
A flow line for a map on a VECTOR FIELD F is a path
a{t) such that <r'(t) = F(<r(t)).
Flower
see Daisy, Flower op Life, Rose
Flower of Life
One of the beautiful arrangements of CIRCLES found at
the Temple of Osiris at Abydos, Egypt (Rawles 1997).
The CIRCLES are placed with six- fold symmetry, forming
a mesmerizing pattern of CIRCLES and LENSES.
see also Five Disks Problem, Reuleaux Triangle,
Seed of Life, Venn Diagram
References
Rawles, B. Sacred Geometry Design Sourcebook: Universal
Dimensional Patterns. Nevada City, CA: Elysian Pub.,
p. 15, 1997.
Wein, J. "The Flower of Life." http://www2.cruzio.com/
-flower.
$ Weisstein, E. W. "Flower of Life." http: //www. astro.
Virginia. edu/~eww6n/math/notebooks/Flower0f Lif e.m.
Flowsnake
see Peano-Gosper Curve
Flowsnake Fractal
see Gosper Island
Floyd's Algorithm
An algorithm for finding the shortest path between two
Vertices.
see also Dukstra's Algorithm
Fluent
Newton's term for a variable in his method of FLUXIONS
(differential calculus).
Fluxion
The term for DERIVATIVE in Newton's CALCULUS.
Flype
A 180° rotation of a TANGLE.
see also Flyping Conjecture, Tangle
Flyping Conjecture
Foliation 657
Flyping Conjecture
Also called the Tait FLYPING Conjecture. Given two
reduced alternating projections of the same knot, they
are equivalent on the SPHERE IFF they are related by a
series of Flypes. It was proved by Menasco and This-
tlethwaite (1991). It allows all possible REDUCED alter-
nating projections of a given ALTERNATING KNOT to be
drawn.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 164-165, 1994.
Menasco, W. and Thistlethwaite, M. "The Tait Flyping Con-
jecture." Bull. Amer. Math. Soc. 25, 403-412, 1991.
Stewart, I. The Problems of Mathematics, 2nd ed. Oxford,
England: Oxford University Press, pp. 284-285, 1987.
Focus
A point related to the construction and properties of
Conic Sections.
see also Ellipse, Ellipsoid, Hyperbola, Hyper-
boloid, Parabola, Paraboloid, Reflection Prop-
erty
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 141-144, 1967.
Fold Bifurcation
Let /:MxE-ylbea one-parameter family of C 2
Map satisfying
/(0,0) =
frl
\_OX J u=0,i=0
dx 2
>0
J ^=0,aj=0
>o,
J ju=0,x=0
then there exist intervals (/ii,0), (0,/^) and e > such
that
1. If fJb £ (^1,0), then fn(x) has two fixed points in
(— e, e) with the positive one being unstable and the
negative one stable, and
2. If (X £ (0,//2), then fp(x) has no fixed points in
(-e,e).
This type of BIFURCATION is known as a fold bifurca-
tion, sometimes also called a Saddle-Node BIFURCA-
TION or Tangent Bifurcation. An example of an
equation displaying a fold bifurcation is
2
X = fU, — X
(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. 145-149, 1997.
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, pp. 27-28, 1990.
Fold Catastrophe
A Catastrophe which can occur for one control factor
and one behavior axis.
Folding
The points accessible from c by a single fold which leaves
ai , . . . , a n fixed are exactly those points interior to or on
the boundary of the intersection of the CIRCLES through
c with centers at a;, for i = 1, . . . , n. Given any three
points in the plane a, 6, and c, there is an Equilateral
Triangle with Vertices x, y, and z for which a, &, and
c are the images of x, y, and z under a single fold. Given
any four points in the plane a, 6, c, and d y there is some
Square with Vertices x, y, z, and w for which a, 6, c,
and d are the images of x, y, z, and w under a sequence
of at most three folds. Also, any four collinear points
are the images of the VERTICES of a suitable SQUARE
under at most two folds. Every five (six) points are the
images of the Vertices of suitable regular Pentagon
(Hexagon) under at most five (six) folds. The least
number of folds required for n > 4 is not known, but
some bounds are. In particular, every set of n points is
the image of a suitable REGULAR n-gon under at most
F(n) folds, where
F(n) <
H(3«-
2) for n even
3) for n odd.
The first few values are 0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15,
17, 18, 20, 21, . . . (Sloane's A007494).
see also Flexagon, Map Folding, Origami
References
Sabinin, P. and Stone, M. G. "Transforming n-gons by Fold-
ing the Plane." Amer. Math. Monthly 102, 620-627, 1995.
Sloane, N. J. A. Sequence A007494 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
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 F is called a foliation of M of codimension
c (with < c < n) if there EXISTS a COVER of M by
Open Sets U, each equipped with a Homeomorphism
h : U —> M. n or h : U — >■ M+ which throws each nonempty
component of F a D U onto a parallel translation of the
standard Hyperplane IR n_c in R n . Each F a is then
called a Leaf and is not necessarily closed or compact.
see also LEAF (FOLIATION), REEB FOLIATION
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 284, 1976.
658 Folium
Folium
The word "folium" means leaf-shaped. The polar equa-
tion is
r = cos 0(4a sin 2 — b) .
If b > 4a, it is a single folium. If b = 0, it is a BlFOLIUM.
If < 6 < 4a, it is a Trifolium. The simple folium is
the Pedal Curve of the Deltoid where the Pedal
Point is one of the Cusps.
see also Bifolium, Folium of Descartes, Kepler's
Folium, Quadrifolium, Rose, Trifolium
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 152-153, 1972.
MacTutor History of Mathematics Archive. "Folium." http:
// www - groups . dcs . st - and .ac.uk/ -history / Curves /
Folium.html.
Folium of Descartes
A plane curve proposed by Descartes to challenge Fer-
mat's extremum-finding techniques. In parametric form,
V :
3at
1 + t 3
3at 2
1 + t 3 '
(1)
(2)
The curve has a discontinuity at t = — 1. The left wing
is generated as t runs from —1 to 0, the loop as t runs
from to oo, and the right wing as t runs from — oo to
-1.
The Curvature and Tangential Angle of the folium
of Descartes, illustrated above, are
2(1 + i 3 ) 4
«(*) =
3(1 + 4 t 2 _ 4£ 3 _ 4t 5 + 4t 6 + £8)3/2
<t>(t)
7r + tan
\t 4 -2tj
tan
2i 3
t A -2t
(3)
(4)
Fontene Theorems
Converting the parametric equations to POLAR COOR-
DINATES gives
a (3a*) 2 (l + t 2 )
T (1 + * 3 ) 2
= tan" 1 (-) = tan x t,
d6 =
dt
1 + t 2 '
The Area enclosed by the curve is
A = ± I r 2 dQ
(5)
(6)
(7)
1 + t 2
3 2/ St dt
= * a J o^W-
(8)
Now let u = 1 + t 3 so du = 3t 2 dt
^§- 2 f ^=§^]:=§° 2 (-o + i)=f
2
a .
(9)
In Cartesian Coordinates,
, , = (3rf)'(l-H») = (Sat) ' axy
x + y
(1 + i 3 ) 3 (1 + t 3 ) 2
(MacTutor Archive). The equation of the ASYMPTOTE
is
y=-a-x. (11)
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 59-62, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 106-109, 1972.
MacTutor History of Mathematics Archive, "Folium of
Descartes." http : // www - groups . dcs . st - and .ac.uk/
-hist ory/Curves/Foliumd. html.
Stroeker, R. J. "Brocard Points, Circulant Matrices, and
Descartes' Folium." Math. Mag. 61, 172-187, 1988.
Yates, R. C. "Folium of Descartes." In A Handbook on
Curves and Their Properties. Ann Arbor, MI: J. W. Ed-
wards, pp. 98-99, 1952.
Follows
see Succeeds
Fontene Theorems
1. If the sides of the Pedal Triangle of a point
P meet the corresponding sides of a Triangle
AO1O2O3 at Xi, X 2 , and X 3 , respectively, then
PiXi, P2X2, P3X3 meet at a point L common to
the CIRCLES O1O2O3 and P1P2P3. In other words,
L is one of the intersections of the NlNE-PoiNT CIR-
CLE of A1A2A3 and the Pedal Circle of P.
2. If a point moves on a fixed line through the ClRCUM-
CENTER, then its PEDAL CIRCLE passes through a
fixed point on the Nine-Point Circle.
Foot
Form
659
3. The Pedal Circle of a point is tangent to the
Nine-Point Circle Iff the point and its Isogo-
nal Conjugate lie on a Line through the Ortho-
center. Feuerbach's Theorem is a special case
of this theorem.
see also Circumcenter, Feuerbach's Theorem,
Isogonal Conjugate, Nine-Point Circle, Ortho-
center, Pedal Circle
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 245-247, 1929.
Foot
see Perpendicular Foot
For All
If a proposition P is true for all J3, this is written PVB.
see also Almost All, Exists, Quantifier
Forcing
A technique in Set Theory invented by P. Cohen
(1963, 1964, 1966) and used to prove that the AXIOM OF
Choice and Continuum Hypothesis are independent
of one another in ZERMELO-FRAENKEL Set THEORY.
see also Axiom of Choice, Continuum Hypothesis,
Set Theory, Zermelo-Fraenkel Set Theory
References
Cohen, P. J. "The Independence of the Continuum Hypoth-
esis." Proc. Nat. Acad. Sci. U. S. A. 50, 1143-1148, 1963.
Cohen, P. J. "The Independence of the Continuum Hypothe-
sis. II." Proc. Nat Acad. Sci. U. S. A. 51, 105-110, 1964.
Cohen, P. J. Set Theory and the Continuum Hypothesis. New
York: W. A. Benjamin, 1966.
Ford Circle
Pick any two INTEGERS h and k, then the CIRCLE of
RADIUS l/(2k 2 ) centered at (h/k, l/(2k 2 )) is known as
a Ford circle. No matter what and how many /is and ks
are picked, none of the Ford circles intersect (and all are
tangent to the x-Axis). This can be seen by examining
the squared distance between the centers of the circles
with (h,k) and (/i',fc'),
d 2 =(~
+
(—- —
\2k' 2 2k 2 )
Let s be the sum of the radii
S — 7*1 + T2
2k 2 2k n '
then
d 2
2 (h'k - hk'y - 1
■ s = -
k 2 k' 2
(2)
(3)
But (h'k - k'h) 2 > 1, so d 2 - s 2 > and the dis-
tance between circle centers is > the sum of the CIR-
CLE RADII, with equality (and therefore tangency) IFF
\h'k-k'h\ - 1. Ford circles are related to the FAREY
Sequence (Conway and Guy 1996).
see also Adjacent Fraction, Farey Sequence,
Stern-Brocot Tree
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.
Ford, L. R. "Fractions." Amer. Math. Monthly 45, 586-601,
1938.
Pickover, C. A. "Fractal Milkshakes and Infinite Archery,"
Ch. 14 in Keys to Infinity. New York: W. H. Freeman,
pp. 117-125, 1995.
Rademacher, H. Higher Mathematics from an Elementary
Point of View. Boston, MA: Birkhauser, 1983.
Ford's Theorem
Let a, 6, and k be INTEGERS with k > 1. For j = 0, 1,
2, let
i=0
i=j (mod 3)
Then
(1)
2(a 2 +a6 + 6 2 ) 2fc = (5 -5 1 ) 4 + (5i-5 2 ) 4 + (5 2 -5o) 4 .
see also BHARGAVA'S THEOREM, DlOPHANTINE
Equation — Quartic
References
Berndt, B. C. Ramanujan f s Notebooks, Part IV. New York:
Springer- Verlag, pp. 100-101, 1994.
Forest
A Graph without any Circuits (Cycles), which
therefore consists only of TREES. A forest with k com-
ponents and n nodes has n — k EDGES.
Fork
see Tree
Form
see Canonical Form, Cusp Form, Differential
fc-FoRM, Form (Geometric), Form (Polynomial),
Modular Form, Normal Form, Pfaffian Form,
Quadratic Form
660 Form (Geometric)
Fortunate Prime
Form (Geometric)
A 1-D geometric object such as a PENCIL or RANGE.
Form (Polynomial)
A Homogeneous Polynomial in two or more vari-
ables.
see also Disconnected Form, A;-Form
Formal Logic
see Symbolic Logic
Formosa Theorem
see Chinese Remainder Theorem
Formula
A mathematical equation or a formal logical expression.
The correct Latin plural form of formula is "formu-
lae," although the less pretentious-sounding "formulas"
is used more commonly.
see also Archimedes' Recurrence Formula, Bayes'
Formula, Benson's Formula, Bessel's Finite Dif-
ference Formula, Bessel's Interpolation For-
mula, Bessel's Statistical Formula, Binet's For-
mula, Binomial Formula, Brahmagupta's For-
mula, Brent-Salamin Formula, Bretschneider's
Formula, Brioschi Formula, Calderon's For-
mula, Cardano's Formula, Cauchy's Formula,
Cauchy's Cosine Integral Formula, Cauchy
Integral Formula, Chasles-Cayley-Brill For-
mula, Chebyshev Approximation Formula, Chris-
toffel-Darboux Formula, Christoffel For-
mula, Clausen Formula, Clenshaw Recurrence
Formula, Descartes-Euler Polyhedral For-
mula, Descartes' Formula, Dirichlet's Formula,
Dixon-Ferrar Formula, Dobinski's Formula, Du-
plication Formula, Enneper-WeierstraB Param-
eterization, Euler Curvature Formula, Euler
Formula, Euler-Maclaurin Integration Formu-
las, Euler Polyhedral Formula, Euler Triangle
Formula, Everett's Formula, Exponential Sum
Formulas, Faulhaber's Formula, Frenet Formu-
las, Gauss's Backward Formula, Gauss-Bonnet
Formula, Gauss's Formula, Gauss's Forward
Formula, Gauss Multiplication Formula, Gauss-
Salamin Formula, Girard's Spherical Excess
Formula, Goodman's Formula, Gregory's For-
mula, Grenz-Formel, Grinberg Formula, Hal-
ley's Irrational Formula, Halley's Rational
Formula, Hansen-Bessel Formula, Heron's For-
mula, Hook Length Formula, Jacobi Ellip-
tic Functions, Jensen's Formula, Jonah For-
mula, Kac Formula, Kneser-Sommerfeld For-
mula, Rummer's Formulas, Laisant's Recur-
rence Formula, Landen's Formula, Lefshetz
Fixed Point Formula, Lefshetz Trace For-
mula, Legendre Duplication Formula, Legen-
dre's Formula, Lehmer's Formula, Lichnerowicz
Formula, Lichnerowicz-Weitzenbock Formula,
Lobachevsky's Formula, Logarithmic Binomial
Formula, Ludwig's Inversion Formula, Machin's
Formula, Machin-Like Formulas, Mehler's Bes-
sel Function Formula, Mehler's Hermite Poly-
nomial Formula, Meissel's Formula, Mensura-
tion Formula, Mobius Inversion Formula, Mor-
ley's Formula, Newton's Backward Differ-
ence Formula, Newton-Cotes Formulas, New-
ton's Forward Difference Formula, Nichol-
son's Formula, Pascal's Formula, Pick's For-
mula, Poincare Formula, Poisson's Bessel Func-
tion Formula, Poisson's Harmonic Function
Formula, Poisson Sum Formula, Polyhedral
Formula, Prosthaphaeresis Formulas, Quadra-
tic Formula, Quadrature Formulas, Rayleigh's
Formulas, Riemann's Formula, Rodrigues For-
mula, Rotation Formula, Schlafli's Formula,
Schroter's Formula, Schwenk's Formula, Seg-
ner's Recurrence Formula, Serret-Frenet For-
mulas, Sherman-Morrison Formula, Sommer-
feld's Formula, Sonine-Schafheitlin Formula,
Steffenson's Formula, Stirling's Finite Dif-
ference Formula, Stirling's Formula, Strassen
Formulas, Thiele's Interpolation Formula,
Wallis Formula, Watson's Formula, Watson-
Nicholson Formula, Weber's Formula, Weber-
Sonine Formula, Weyrich's Formula, Woodbury
Formula
References
Carr, G. S. Formulas and Theorems in Pure Mathematics.
New York: Chelsea, 1970.
Spiegel, M. R. Mathematical Handbook of Formulas and Ta-
bles. New York: McGraw-Hill, 1968.
Tallarida, R. J. Pocket Book of Integrals and Mathematical
Formulas, 3rd ed. Boca Raton, FL: CRC Press, 1992.
Fortunate Prime
2000
■
1500
i
il
ii
1000
. 1
I iJkiu
1
h
500
jJW
yifF vww
^y yu
20 40 60 80 100 120 140
k
Let
X fc EEl+ Pfc #,
where pk is the fcth Prime and p# is the Primorial,
and let qk be the Next Prime (i.e., the smallest Prime
greater than Xk),
qk =Pl + iv(X k ) = />l +7r (l+p fe #),
Forward Difference
Four-Color Theorem
661
where rc{n) is the PRIME COUNTING FUNCTION. Then
R. F. Fortune conjectured that Fk = qk ~ Xk + 1 is
Prime for all k. The first values of Fk are 3, 5, 7, 13,
23, 17, 19, 23, ... (Sloane's A005235), and all known
values of Fk are indeed Prime (Guy 1994). The indices
of these primes are 2, 3, 4, 6, 9, 7, 8, 9, 12, 18, ... . In
numerical order with duplicates removed, the Fortunate
primes are 3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71,
79, 89, . . . (Sloane's A046066).
see also ANDRICA'S Conjecture, PRIMORIAL
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 7, 1994.
Sloane, N. J. A. Sequences A046066 and A005235/M2418 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Forward Difference
The forward difference is a Finite Difference defined
by
A/ P = /p+i - / P . (1)
Higher order differences are obtained by repeated oper-
ations of the forward difference operator, so
A 2 /p = A p 2 = A(A„) = A(/ p+1 - f v )
= A p+ i — A p = /p+2 - 2/p+i + f p . (2)
In general,
A^A^£(-l) m (*W- m , (3)
m=0 ^ '
where (^) is a Binomial Coefficient.
Newton's Forward Difference Formula expresses
f p as the sum of the nth forward differences
/ p -/o+pAo + |rP(p+l)A? + ^p(p+l)(p + 2)Ag + ...
(4)
where Aq is the first nth difference computed from the
difference table.
see also Backward Difference, Central Differ-
ence, Difference Equation, Divided Difference,
Reciprocal 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. 877, 1972.
Fountain
An (riyk) fountain is an arrangement of n coins in rows
such that exactly k coins are in the bottom row and each
coin in the (i -f l)st row touches exactly two in the ith
row.
References
Berndt, B. C. Ramanujan's Notebooks, Part III. New York:
Springer- Verlag, p. 79, 1985.
Four Coins Problem
A
Given three coins of possibly different sizes which are
arranged so that each is tangent to the other two, find
the coin which is tangent to the other three coins. The
solution is the inner SODDY CIRCLE.
see also Apollonius Circles, Apollonius' Prob-
lem, Arbelos, Bend (Curvature), Circumcircle,
Coin, Descartes Circle Theorem, Hart's Theo-
rem, Pappus Chain, Soddy Circles, Sphere Pack-
ing, Steiner Chain
References
Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
Four-Color Theorem
The four-color theorem states that any map in a PLANE
can be colored using four-colors in such a way that re-
gions sharing a common boundary (other than a sin-
gle point) do not share the same color. This prob-
lem is sometimes also called GUTHRIE'S PROBLEM after
F. Guthrie, who first conjectured the theorem in 1853.
The CONJECTURE was then communicated to de Mor-
gan and thence into the general community. In 1878,
Cayley wrote the first paper on the conjecture.
Fallacious proofs were given independently by Kempe
(1879) and Tait (1880). Kempe's proof was accepted for
a decade until Heawood showed an error using a map
with 18 faces (although a map with nine faces suffices
to show the fallacy). The Heawood Conjecture pro-
vided a very general result for map coloring, showing
that in a Genus Space (i.e., either the Sphere or
Plane), six colors suffice. This number can easily be
reduced to five, but reducing the number of colors all
the way to four proved very difficult.
Finally, Appel and Haken (1977) announced a computer-
assisted proof that four colors were SUFFICIENT. How-
ever, because part of the proof consisted of an exhaus-
tive analysis of many discrete cases by a computer, some
mathematicians do not accept it. However, no flaws
have yet been found, so the proof appears valid. A
potentially independent proof has recently been con-
structed by N. Robertson, D. P. Sanders, P. D. Seymour,
and R. Thomas.
662
Four-Color Theorem
Four-Vertex Theorem
Martin Gardner (1975) played an April Fool's joke by
(incorrectly) claiming that the map of 110 regions illus-
trated below requires five colors and constitutes a coun-
terexample to the four-color theorem.
E 6^ r
&
o
^
HE
E SW
r5&
■ -■ T ^ r TVi
o
I , I , I , I ,-L
E = E S:
i i r i i i i i i
see also Chromatic Number, Heawood Conjec-
ture, Map Coloring, Six-Color Theorem
References
Appel, K. and Haken, W. "Every Planar Map is Four-
Colorable, I and II." Illinois J. Math. 21, 429-567, 1977.
Appel, K. and Haken, W. "The Solution of the Four-Color
Map Problem." Sci. Amer. 237, 108-121, 1977.
Appel, K. and Haken, W. Every Planar Map is Four-
Colorable. Providence, RJ: Amer. Math. Soc, 1989.
Barnette, D. Map Coloring, Polyhedra, and the Four-Color
Problem. Providence, RI: Math. Assoc. Amer., 1983.
Birkhoff, G. D. "The Reducibility of Maps." Amer. Math. J.
35, 114-128, 1913.
Chartrand, G. "The Four Color Problem." §9.3 in Introduc-
tory Graph Theory. New York: Dover, pp. 209-215, 1985.
Coxeter, H. S. M. "The Four-Color Map Problem, 1840-
1890." Math. Teach., Apr. 1959.
Franklin, P. The Four-Color Problem. New York: Scripta
Mathematica, Yeshiva College, 1941.
Gardner, M. "Mathematical Games: The Celebrated Four-
Color Map Problem of Topology." Sci. Amer. 203, 218-
222, Sep. 1960.
Gardner, M. "The Four-Color Map Theorem." Ch. 10
in Martin Gardner's New Mathematical Diversions from
Scientific American. New York: Simon and Schuster,
pp. 113-123, 1966.
Gardner, M. "Mathematical Games: Six Sensational Discov-
eries that Somehow or Another have Escaped Public At-
tention." Sci. Amer. 232, 127-131, Apr. 1975.
Gardner, M. "Mathematical Games: On Tessellating the
Plane with Convex Polygons." Sci Amer. 232, 112-117,
Jul. 1975.
Kempe, A. B. "On the Geographical Problem of Four-
Colors." Amer. J. Math. 2, 193-200, 1879.
Kraitchik, M. §8.4.2 in Mathematical Recreations. New York:
W. W. Norton, p. 211, 1942.
Ore, 0. The Four-Color Problem. New York: Academic
Press, 1967.
Pappas, T. "The Four-Color Map Problem: Topology Turns
the Tables on Map Coloring." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 152-153,
1989.
Robertson, N.; Sanders, D. P.; and Thomas, R. "The Four-
Color Theorem." http://www.math.gatech.edu/~thonias/
FC/f ourcolor.html.
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:
Assaults and Conquest. New York: Dover, 1986.
Tait, P. G. "Note on a Theorem in Geometry of Position."
Trans. Roy. Soc. Edinburgh 29, 657-660, 1880.
Four Travelers Problem
Let four Lines in a Plane represent four roads in Gen-
eral Position, and let one traveler Ti be walking along
each road at a constant (but not necessarily equal to any
other traveler's) speed. Say that two travelers Ti and Tj
have "met" if they were simultaneously at the intersec-
tion of their two roads. Then if Ti has met all other
three travelers (T 2 , T 3 , and T 4 ) and T 2 , in addition to
meeting Ti, has met T3 and T4, then T3 and T4 have
also met!
References
Bogomolny, A. "Four Travellers Problem." http://www.cut-
the-knot . com/gproblems .html.
Four- Vector
A four-element vector
(i)
which transforms under a LORENTZ TRANSFORMATION
like the POSITION FOUR- VECTOR. This means it obeys
a' M = A£a"
%p • bfj, = a^b^
dp • 6 M
(2)
(3)
(4)
where A£ is the LORENTZ TENSOR. Multiplication of
two four- vectors with the METRIC g^ u gives products of
the form
g^x" = (x ) 2 - (x 1 ) 2 - (x 2 ) 2
(x 3 ) 2 .
(5)
In the case of the POSITION FOUR- VECTOR, x° — ct
(where c is the speed of light) and this product is an
invariant known as the spacetime interval.
see also Gradient Four- Vector, Lorentz Trans-
formation, Position Four- Vector, Quaternion
References
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.
Four- Vertex Theorem
A closed embedded smooth Plane Curve has at least
four vertices, where a vertex is defined as an extremum
of Curvature.
see also CURVATURE
References
Tabachnikov, S. "The Four- Vertex Theorem Revisited — Two
Variations on the Old Theme." Amer. Math. Monthly 102,
912-916, 1995.
Fourier-Bessel Series
Fourier-Bessel Series
see bessel function fourier expansion, schlo-
milch's Series
Fourier-Bessel Transform
see Hankel Transform
Fourier Cosine Series
If f(x) is an EVEN FUNCTION, then b n = and the
Fourier Series collapses to
f(x) = \ao + 22 an cos ( nx )>
(1)
where
oo = -/ f(x)dx=- f{x)dx (2)
* J-* n JO
i r
a n = — f(x) cos(nx) dx
7T /
J _ 7T
2 r
= — I f(x)cos(nx)dx, (3)
^ Jo
where the last equality is true because
f(x) cos(nx) = /(— x) cos(— nx). (4)
Letting the range go to L y
a n
= j;J f{x)dx (5)
= lf f(x)cos(^)dx. (6)
see also EVEN FUNCTION, FOURIER COSINE TRANS-
FORM, Fourier Series, Fourier Sine Series
Fourier Cosine Transform
The Fourier cosine transform is the REAL PART of the
full complex Fourier Transform,
?cos[f{x)] = Sl[r[f(x)]\.
see also Fourier Sine Transform, Fourier Trans-
form
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "FFT of Real Functions, Sine and Cosine
Transforms." §12.3 in Numerical Recipes in FORTRAN:
The Art of Scientific Computing, 2nd ed. Cambridge, Eng-
land: Cambridge University Press, pp. 504-515, 1992.
Fourier Integral
see Fourier Transform
Fourier-Mellin Integral 663
Fourier Matrix
The nx n SQUARE MATRIX F n with entries given by
F jk = e
2Trijk/n
(i)
for j, k = 1, 2, . . . , n, and normalized by 1/y/n to make
it a Unitary. The Fourier matrix F2 is given by
v/2
1 1
1 i 2
(2)
and the F4 matrix by
1111
1 i i 2 i 3
1 i 2 i 4 i Q
1 i 3 i 6 i 9
1 1
1 i
1 -1
1 -i
1 1
1 i 2
1 1
1 i 2
In general,
F2™ =
with
F„
F„/2
In D n
l» -D„
Fn
F„
even-odd
shuffle
(3)
(4)
'n/2 D n /2
a/2
n/2
■n/2
'n/2
D n/2
-D»/2j
"n/2
F n
/2
n/2
even-odd
0, 2 (mod 4)
even-odd
1,3 (mod 4)
(5)
where l n is the n x n IDENTITY Matrix. Note that the
factorization (which is the basis of the Fast Fourier
Transform) has two copies of F2 in the center factor
Matrix.
see also FAST FOURIER TRANSFORM, FOURIER TRANS-
FORM
References
Strang, G. "Wavelet Transforms Versus Fourier Transforms."
Bull. Amer. Math. Soc. 28, 288-305, 1993.
Fourier-Mellin Integral
The inverse of the LAPLACE TRANSFORM
F(i) = £- 1 [/( S )]=^ ? / e^f{s)ds
J y — too
f(s) = C[F(t)]= / F(t)e- St dt.
Jo
see also Bromwich Integral, Laplace Transform
664 Fourier Series
Fourier Series
Fourier series are expansions of Periodic Functions
f(x) in terms of an infinite sum of SlNES and COSINES
oo oo
f(x) = V^ a n cos(nx) + \^ b n sin(nx). (1)
n=0 n=0
Fourier series make use of the ORTHOGONALITY rela-
tionships of the Sine and Cosine functions, which can
be used to calculate the coefficients a n and b n in the
sum. The computation and study of Fourier series is
known as HARMONIC ANALYSIS.
To compute a Fourier series, use the integral identities
sin(mx) s'm(nx) dx = 7rJ mTl for n, m / (2)
F
J —7
cos(mx) cos(nx) dx = 7r£ mn for n,m / (3)
/sin(rnx) cos(nx) dx = (4)
TV
/TV
sin(rax) dx — (5)
-7T
cos(mx) dx = 0,
(6)
where S mn is the Kronecker Delta. Now, expand
your function f(x) as an infinite series of the form
oo oo
f(x) = Y^ a n cos(nx) + Y^ b n sin(nx)
n=0 n—0
oo oo
= \av + \J a n cos(nx) -f ^ b n sin(nx), (7)
n=l 71=1
where we have relabeled the ao = 2a term for future
convenience but left a n — a! n . Assume the function is
periodic in the interval [— 7r, 7r]. Now use the orthogo-
nality conditions to obtain
//(#) dx
TV
/IT [" OO °°
\^ a n cos(nx) -f > 6 n sin(nx) + |ao
■^ |_n = l rx = l
°° /»7r /*'
— / / [ a n cos(nx) + b n sin(nx)] dx + \a>o /
„ — -\ J — 7T ^ —
oo
= ^(0 + 0) + 7ra = 7va (8)
dx
dx
Fourier Series
and
//(x) sin(mx) dx
■TV
/TV [" °° °°
\J a n cos(nx) + \^ b n sin(nx) + |a
■^ |_n=l n=l
x sin (ma:) dx
= / / [an cos(nx) sin(mx) + &„ sin(nx) sin(mx)] dx
+ |ao / sin(rax)dx
</ — TV
OO
= ^(0 + b n 7r5mn) + = 7T& n , (9)
n = l
SO
/TV pTV r °°
/(x) cos(mx) dx = / N^a n cos(nx)
■f ^-^ U=i
OO
4- V^ 6 n sin(nx) + |ao cos(mx) dx
n=l
= / / [a n cos(nx) cos(mx)
+fo n sin(nx) cos(mx)] dx + |ao / cos(mx) dx
•/ — 7T
OO
= ^(a n 7r<5 mn + 0) + = 7ra n . (10)
n=l
Plugging back into the original series then gives
ao = -/ /(x)dx (11)
^ J -TV
i A 7 "
a n = — / /(x) cos(nx) dx (12)
1 r
b n = — I /(x) sin(nx) dx (13)
7T /
J — 7T
for n = 1, 2, 3, .... The series expansion converges to
the function / (equal to the original function at points
of continuity or to the average of the two limits at points
of discontinuity)
f=<
' \ [lim MX0 - f{x) + \im x ^ XQ + f(x)]
for — 7T < Xo < TT /. .x
§ [lim a ._,_ ff + f(x) + lim^-^. /(x)]
k for Xo = ~ 7T,7T
if the function satisfies the Dirichlet Conditions.
Fourier Series
Fourier Series 665
Now examine
Near points of discontinuity, a "ringing" known as the
Gibbs Phenomenon, illustrated below, occurs. For
a function f(x) periodic on an interval [—L,L], use a
change of variables to transform the interval of integra-
tion to [—1,1]. Let
dx =
L
ndx'
(15)
(16)
Solving for x' , x f = Lx/tt. Plugging this in gives
/./ /x i V^ I mrx f \
f(x ) = ^ao + 2_^ Qn cos I ~~7~ )
* / /
El T17TX
o n sin
' a ° = i$- L f( x ') dx '
^ = r/^/(«')cos(^) dx'
(18)
If a function is EVEN so that f(x) ~ /( — x), then
/(as)sin(nx) is ODD. (This follows since sin(ncc) is ODD
and an EVEN FUNCTION times an ODD FUNCTION is an
Odd Function.) Therefore, b n = for all n. Simi-
larly, if a function is ODD so that f(x) = /(— x), then
f(x) cos(nx) is Odd. (This follows since cos(nx) is Even
and an Even Function times an Odd Function is an
Odd Function.) Therefore, a n = for all n.
Because the Sines and Cosines form a Complete
Orthogonal Basis, the Superposition Principle
holds, and the Fourier series of a linear combination of
two functions is the same as the linear combination of
the corresponding two series. The COEFFICIENTS for
Fourier series expansions for a few common functions
are given in Beyer (1987, pp. 411-412) and Byerly (1959,
p. 51).
The notion of a Fourier series can also be extended to
Complex Coefficients. Consider a real-valued func-
tion f(x). Write
nx) - Yl A - ei "
(19)
e 1Tnx dx
f(x)e- imx dx= I" I J2 A n e inx \
OO /.7T
= ^2 A n e i(n - m)l dx
n= — oo ~ 7r
y A n I {cos[(n — m)x] -f 2sin[(n — m):c]} dx
n--oo J-*
oo
= ^ A n 2ird mn = 2k A m , (20)
m= — oo
An = h I f W e ~ inx dx -
J — 7T
(21)
The Coefficients can be expressed in terms of those
in the Fourier Series
f (x)[cos(nx) — isin(nx)] dx
~ f™ f(x)[cos(nx) + 2sin(ncc)] dx n <
ife I- f(x)[cos(nx) — zs'm(nx)] dx n >
\{a n + ib n ) n <
f a n =
\{a n - ib n ) n > 0.
(22)
For a function periodic in [— L, L], these become
oo
f(x)= J2 A ne H2 ™* /L) (23)
--If"
L J-L/2
f{x)e
— i{2-Knx/L)
dx.
(24)
These equations are the basis for the extremely impor-
tant FOURIER Transform, which is obtained by trans-
forming A n from a discrete variable to a continuous one
as the length L -~> oo.
see also Dirichlet Fourier Series Conditions,
Fourier Cosine Series, Fourier Sine Series,
Fourier Transform, Gibbs Phenomenon, Lebes-
gue Constants (Fourier Series), Legendre Se-
ries, Riesz-Fischer Theorem, Schlomilch's Series
References
Arfken, G. "Fourier Series." Ch. 14 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 760-793, 1985.
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, 1987.
Brown, J. W. and Churchill, R, V. Fourier Series and Bound-
ary Value Problems, 5th ed. New York: McGraw-Hill,
1993.
Byerly, W. E. An Elementary Treatise on Fourier's Series,
and Spherical, Cylindrical, and Ellipsoidal Harmonics,
666
Fourier Series — Power Series
Fourier Series — Triangle
with Applications to Problems in Mathematical Physics.
New York: Dover, 1959.
Carslaw, H. S. Introduction to the Theory of Fourier's Series
and Integrals, 3rd ed. } rev. and enl. New York: Dover,
1950.
Davis, H. F. Fourier Series and Orthogonal Functions. New
York: Dover, 1963.
Dym, H. and McKean, H. P. Fourier Series and Integrals.
New York: Academic Press, 1972.
Folland, G. B. Fourier Analysis and Its Applications. Pacific
Grove, CA: Brooks/Cole, 1992.
Groemer, H. Geometric Applications of Fourier Series and
Spherical Harmonics. New York: Cambridge University-
Press, 1996.
Korner, T. W. Fourier Analysis. Cambridge, England: Cam-
bridge University Press, 1988.
Korner, T. W. Exercises for Fourier Analysis. New York:
Cambridge University Press, 1993.
Lighthill, M. J. Introduction to Fourier Analysis and Gen-
eralised Functions. Cambridge, England: Cambridge Uni-
versity Press, 1958.
Morrison, N. Introduction to Fourier Analysis. New York:
Wiley, 1994.
Sansone, G. "Expansions in Fourier Series." Ch. 2 in Or-
thogonal Functions, rev. English ed. New York: Dover,
pp. 39-168, 1991.
Fourier Series — Power Series
For f(x) = x k on the INTERVAL [-L, L) and periodic
with period 2L, the FOURIER SERIES is given by
1 /* k fn r Kx\ ,
a n — — I x cos I —r~ J dx
2L k
1 + Ar
1-P2
1 + ^fc
2 ,b 122
2 2 -(3+*r 4
1 f k . (wkx\
b n — y x sin I — — I dx
2mrL k
2 + k ]
l + U
i 2 + ^
2 n 12 2
2 u;-^ n
where 1-^2(0; &, c; x) is a generalized Hypergeometric
Function.
Fourier Series — Right Triangle
Consider a string of length 2L plucked at the right end,
then
if x /nivx\ _
an = iJ 2i cos {-ir) dx
[2n7rcos(mr) — sin(mr)] sin(n7r)
=
-ijf
x . /mrx\ _
sin I — — 1 dx
2L V L
_ — 2n7rcos(2n7r) + sin(2n7r) _ 1
2n 2 7T 2 717T*
The Fourier series is therefore
*t \ 1 1 r 1 • fnirx\
see also Fourier Series
Fourier Series — Square Wave
Consider a square wave of length 2L. Since the function
is ODD, ao — a n = 0, and
2 f . { nizx\ ,
= lJ ^{-t-)**
4 . 2/1 x 4 /O n
= — sin {^nir) = — <!
717T ^ n7r I 1 n
even
odd.
The Fourier series is therefore
'<*> = * 22 n sin (ir)
see also Fourier Series, Square Wave
Fourier Series — Triangle
0.5 1
Let a string of length 2L have a y-displacement of unity
when it is pinned an z-distance which is (l/m)th of the
way along the string. The displacement as a function of
x is then
Jrn{x) = < m /_£ \ 21
L 1-m \2L 1 ) m
< x < 2L.
Fourier Series — Triangle Wave
Fourier-Stieltjes Transform 667
The Coefficients are therefore
p2L/m r 2L
ao = i
= i
" p2L//m pZLt x OO
6 n =
m [l-m-cos(27rn)+mcos(^ L )]
2(m-l)n 2 7r 2
m 2 [cos(^)^l]
2(m- l)n 2 7r 2
m [msin (^) - sin(27rra)]
2(m-l)n 2 7r 2
m sm l^rJ
2(m - l)n 2 7r 2 '
The Fourier series is therefore
2
fm(x) = | +
2 ' 2(m-l)7T 2
71=1
.HS rin «
sin .
n^ \ L J
If m = 2, then a n and 6 n simplify to
4 . 2/i x 4 TO n = 0,2, ...
n 2 7r 2 2 n 2 ?r 2 11 n — 1, ci, . . .
giving
OO
71 = 1,3,5,...
see a/50 Fourier Series
Fourier Series — Triangle Wave
Consider a triangle wave of length 2L. Since the function
is Odd, ao = a n = 0, and
+ ^
■/
J Li
i- z («- a <
iL) sin f — — J dx > da;
/r/2 L
32 /l \ • 3/1 \
-r— r-cos(in7r)sin (47171-)
7r 2 n 2 4 4
(0 n = 0, 4, ...
i n = l, 5, ...
n = 2, 6, ...
-\ n = 3 ; 7,...
_8_ r (_i)(— i)/ 2 for n odd
7r 2 n 2 \ 1
The Fourier series is therefore
8 ^ (_!)Cn-l)/2
ti = 1,3,5,...
see also FOURIER SERIES
n*
(n7rx\
17)'
Fourier Sine Series
If f(x) is an Odd FUNCTION, then a n = and the
Fourier Series collapses to
f(x) = y^& n sin(naQ,
(1)
where
* Jo
b n = — I f(x)sm(nx)dx = — I f(x)sin(nx) dx
* J-„ *" Jo
(2)
for n = 1, 2, 3, The last EQUALITY is true because
f(x)sin(nx) = [—f(—x)][—sin(—nx)]
— f(—x) sin(— nx). (3)
Letting the range go to L,
b n = -j I f{x)sax \JY~j dx *
(4)
see also Fourier Cosine Series, Fourier Series,
Fourier Sine Transform
Fourier Sine Transform
The Fourier sine transform is the IMAGINARY PART of
the full complex FOURIER TRANSFORM,
Fsin[f(x)] - 9W(:c)]].
see also FOURIER COSINE TRANSFORM, FOURIER
TRANSFORM
References
Press, W. H.; Flannery, B. R; Teukolsky, S. A.; and Vet-
terling, W. T. "FFT of Real Functions, Sine and Cosine
Transforms." §12.3 in Numerical Recipes in FORTRAN:
The Art of Scientific Computing, 2nd ed. Cambridge, Eng-
land: Cambridge University Press, pp. 504-515, 1992.
Fourier-Stieltjes Transform
Let f(x) be a positive definite, measurable function on
the Interval (—00,00). Then there exists a monotone
increasing, real-valued bounded function a(t) such that
/(*):
J — c
e itx da{t)
■
for n even.
for "ALMOST All" x. If a(t) is nondecreasing and
bounded and f(x) is denned as above, then f(x) is called
the Fourier-Stieltjes transform of a(£), and is both con-
tinuous and positive definite.
see also FOURIER TRANSFORM, LAPLACE TRANSFORM
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 618, 1980.
668 Fourier Transform
Fourier Transform
Fourier Transform
The Fourier transform is a generalization of the COM-
PLEX Fourier Series in the limit as L — y oo. Replace
the discrete A n with the continuous F(k) dk while let-
ting n/L — > k. Then change the sum to an INTEGRAL,
and the equations become
f(x) = I F(k)e 2ixikx dk
-f
/oo
/(x)e -3 *""
■OO
dx.
(i)
(2)
Here,
/oo
f(x)e- 2 " ikx dx (3)
-oo
is called the forward (—i) Fourier transform, and
/oo
F(k)e 2nikx dk (4)
-OO
is called the inverse (+i) Fourier transform. Some au-
thors (especially physicists) prefer to write the trans-
form in terms of angular frequency u> = 27rz/ instead of
the oscillation frequency v. However, this destroys the
symmetry, resulting in the transform pair
/oo
h{t)e- iwt dt (5)
-oo
h(t) = T-^Hly)] = — / H(u)e wt du>. (6)
J — oo
In general, the Fourier transform pair may be defined
using two arbitrary constants A and B as
F(u>) = A I /(t)e Biut dt
(7)
f(t)
2-kA f K
«/ — oo
)e~ Biut dw. (8)
The Mathematical program (Wolfram Research, Cham-
paign, IL) calls A the $FourierOverallConstant and B
the $FourierFrequencyConstant, and defines A = B =
1 by default. Morse and Feshbach (1953) use B = 1 and
A = l/\/27r. In this work, following Bracewell (1965,
pp. 6-7), A = 1 and B = — 2n unless otherwise stated.
Since any function can be split up into Even and Odd
portions E(x) and 0(x),
f{x) = l[f( x )+f(- x )] + i[f(x)-f(-x)] = E(x)+0(x),
(9)
a Fourier transform can always be expressed in terms of
the Fourier Cosine Transform and Fourier Sine
Transform as
/oo
E(x) cos(27rkx) dx
-oo
/oo
0(x)sm(27vkx)dx. (10)
-oo
A function f(x) has a forward and inverse Fourier trans-
form such that
/(*)
;) continuous
+ /(*-)]
for f(x) discontinuous at x y
roo^ e 2nikx \J^f(x)e' 2lvikx
for f(x) continuous at x
(11)
provided that
1. j^° \f{x) | dx exists.
2. Any discontinuities are finite.
3. The function has bounded variation. A SUFFI-
CIENT weaker condition is fulfillment of the LlP-
schitz Condition.
The smoother a function (i.e., the larger the number of
continuous DERIVATIVES), the more compact its Fourier
transform.
The Fourier transform is linear, since if f(x) and g(x)
have Fourier Transforms F(k) and G(k), then
/'
af(x) + bg(x)]e- 27rikx dx
/oo
f{x)e- 27,ikx dx + b
-OO
/oo
g(x)e~ 27rzkx dx
■oo
= F(k) + G(k). (12)
Therefore,
T[af(x)+bg(x)} = a^[/(x)]+6^[ 5 (a;)] = aF(k)+bG(k).
(13)
The Fourier transform is also symmetric since F(k) =
F[f(x)] implies F(-k) = F[f(x)].
Let f*g denote the CONVOLUTION, then the transforms
of convolutions of functions have particularly nice trans-
forms,
r\f*9] = r\f\r\s] (14)
nf9] = F[f]*ng} (is)
F[Hf) + F(9)] = f*9 (16)
F[nf)*H9)\ = f9- (17)
Fourier Transform
The first of these is derived as follows
-III
-2-Kikx r/ l
f(x')g(x — x)dx dx
-oo </-oo
/»co /»oo
-2-Kikx' ,
e f(x)dx]
oo »/ —oo
— 2irik(x — x')
x |e " x ~ ~ * g(x — x)dx\
l°° e~ 2 * ikx> f{x)dx'
_J — CO
X
F
y — c
-2-Kikx / ii\ i 11
g(x ) dx
where x" = x — x' .
(18)
There is also a somewhat surprising and extremely im-
portant relationship between the AUTOCORRELATION
and the Fourier transform known as the WlENER-
Khintchine Theorem. Let T[f{x)\ = F(k), and F*
denote the Complex Conjugate of F, then the Four-
ier Transform of the Absolute Square of F(k) is
given by
n\F(k)\ 2
/OO
■oo
(r)f(T + x)dT. (19)
The Fourier transform of a Derivative f(x) of a func-
tion f(x) is simply related to the transform of the func-
tion f(x) itself. Consider
nf(x)] = r /'
J — oo
Now use Integration by Parts
vdu= [uv] — / udv
{x)e~ 27rikx dx. (20)
(21)
with
then
?[f'{*)]
du = f'(x)dx v = e- 2 * ikx (22)
u = f(x) dv — —2nike~ 7ri x dx, (23)
J — C
[f(x)e- 2nik *]~ 00 - I f(x)(-2irike- 2 " ikx dx).
J —OO
(24)
The first term consists of an oscillating function times
f(x). But if the function is bounded so that
lim f(x) =
x— >-±oo
(25)
Fourier Transform 669
(as any physically significant signal must be), then the
term vanishes, leaving
/oo
f(x)e- 2 " ihx dx = 2mkT[f{x)}.
-co
(26)
This process can be iterated for the nth Derivative to
yield
F[f {n) (x)} = (2irik) n ?[f(x)}. (27)
The important MODULATION THEOREM of Fourier
transforms allows ^ r [cos(27rA;o^)/(ic)] to be expressed in
terms of FF[f{x)\ = F(k) as follows,
/CO
f(x) cos(27vkox)e~ 2 ™ kx dx
-co
/oo
f ^ e 2nik x e -2*ik* dx
■CO
/CO
f{x)e- 2 " i '">*e- 2l ' ikx dx
■OO
/OO
f{x)e-^ k - ko)x dx
■CO
■»/
J — C
H / f(x)e
— 2Tri(k-\-ko)x
dx
= l[F(k - ko) + F(k + ko)]. (28)
Since the Derivative of the Fourier Transform is
given by
F'(fc) = ±?[f{x)] = r(-2«x)/(
J — OO
x)e- 2 " ikx dx,
it follows that
/CO
xf(x) dx.
-oo
Iterating gives the general Formula
F (n) (0)
Mn
F
x n f(x) dx
(29)
(30)
(31)
(-27ri)"'
The Variance of a Fourier Transform is
<T f 2 = ((xf - (xf)) 2 ) , (32)
and it is true that
<7f+9 = <*f +^3- ( 33 )
If f(x) has the FOURIER TRANSFORM F(k), then the
Fourier transform has the shift property
F
J — C
f{x - x )e- 2 " ikx dx
-F
J —c
f(x - IO ) e - a - < (—'>) fc e- 2, " (fc " o) d(x - x )
= e~ 2l,ik:co F(k), (34)
670 Fourier Transform
so f(x - Xo) has the FOURIER TRANSFORM
F[f(x-x )] = e- 27TikX0 F{k). (35)
If f(x) has a FOURIER TRANSFORM F(k), then the Four-
ier transform obeys a similarity theorem.
«/ — c
f{ax)e- 2 " ikx dx
= A / f(ax)e- 2,,i< -' Lx)(k/a) d(ax)
W J-oo
-H'(:)- w
so f(ax) has the FOURIER TRANSFORM H^F (£).
The "equivalent width" of a Fourier transform is
W e =
IZof( x ) dx _ F(0)
The "autocorrelation width" is
[/*/*]o
n //*<**
(37)
. (38)
where f *g denotes the CROSS-CORRELATION of / and
9-
Any operation on f(x) which leaves its Area unchanged
leaves F(0) unchanged, since
f
J — c
f(x)dx = f[f(0)] = F(0).
(39)
In 2-D, the Fourier transform becomes
/oo /»oo
/ /(fcx, feje" 2 "^^^^^ dfe dA; y
-oo J — oo
(40)
/oo y»oo
/ F( L x,y)e a * i V" x+h « y) dxdy. (41)
■oo «/ — oo
Similarly, the n-D Fourier transform can be denned for
k, x e W 1 by
F(x)
■£-/
/(k)e- 2,rik - x d n k (42)
/(k)
/:■/
F(x)e 27rik - X ^ n x. (43)
Fourier Transform — J
Hankel Transform, Hartley Transform, Inte-
gral Transform, Laplace Transform, Struc-
ture Factor, Winograd Transform
References
Arfken, G. "Development of the Fourier Integral," "Fourier
Transforms — Inversion Theorem," and "Fourier Transform
of Derivatives." §15.2-15.4 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp, 794-
810, 1985.
Blackman, R. B. and Tukey, J. W. The Measurement of
Power Spectra, From the Point of View of Communica-
tions Engineering. New York: Dover, 1959.
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, 1965.
Brigham, E. O. The Fast Fourier Transform and Applica-
tions. Englewood Cliffs, NJ: Prentice Hall, 1988.
James, J. F. A Student's Guide to Fourier Transforms with
Applications in Physics and Engineering. New York:
Cambridge University Press, 1995.
Korner, T. W. Fourier Analysis. Cambridge, England: Cam-
bridge University Press, 1988.
Morrison, N. Introduction to Fourier Analysis. New York:
Wiley, 1994.
Morse, P. M. and Feshbach, H. "Fourier Transforms." §4.8
in Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 453-471, 1953.
Papoulis, A. The Fourier Integral and Its Applications. New
York: McGraw-Hill, 1962.
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.
Sansone, G. "The Fourier Transform." §2.13 in Orthogonal
Functions, rev. English ed. New York: Dover, pp. 158-168,
1991.
Sneddon, I. N. Fourier Transforms. New York: Dover, 1995.
Sogge, C. D. Fourier Integrals in Classical Analysis. New
York: Cambridge University Press, 1993.
Spiegel, M. R. Theory and Problems of Fourier Analysis with
Applications to Boundary Value Problems. New York:
McGraw-Hill, 1974.
Strichartz, R. Fourier Transforms and Distribution Theory.
Boca Raton, FL: CRC Press, 1993.
Titchmarsh, E. C. Introduction to the Theory of Fourier In-
tegrals, 3rd ed. Oxford, England: Clarendon Press, 1948.
Tolstov, G. P. Fourier Series. New York: Dover, 1976.
Walker, J. S. Fast Fourier Transforms, 2nd ed. Boca Raton,
FL: CRC Press, 1996.
Fourier Transform — 1
The Fourier Transform of the Constant Function
f(x) — 1 is given by
f
J — c
F[l]= I e- 27rihx dx = 5(k),
according to the definition of the Delta Function.
see also Delta Function
OO t/ — oo
see also Autocorrelation, Convolution, Discrete
Fourier Transform, Fast Fourier Transform,
Fourier Series, Fourier-Stieltjes Transform,
Fourier Transform — 1/x
Fourier Transform — Gaussian 671
Fourier Transform — 1/x
The Fourier Transform of the function 1/x is given
by
?(-!-)=-* r - — dx
V TVXj 7T / X
_ DT ^ f°° cos(27rfcx) — isin(27rA;a;)
/ x
<J — OO
dx
-{
-g /" " n(3 .' fcg) d» forfc<0
w f « jin^-*.! dx fo r jfc > o
—2 for fc <
i for fe > 0,
which can also be written as the single equation
j:(-JL)=i[l-2ff(-fc)], (2)
where if (x) is the Heaviside Step Function. The
integrals follow from the identity
8111(2***) dx = [~ B in(27rto) d(27rifcx)
T 00 sin(27rfca0 . f°
/ — * — dx= /
Jo ^ Jo
-f
Jo
2-ivkx
sine zdz = ~tv. (3)
Fourier Transform — Cosine
/oo
■oo
e 27rifex — I dx
r e -2ni(k-k )x , e -27rz(fc + fc )a ; l ^
where <S(x) is the Delta Function.
see also COSINE, FOURIER TRANSFORM — SlNE
Fourier Transform — Delta Function
The FOURIER TRANSFORM of the DELTA FUNCTION is
given by
y>oo
T[8{x - x )] = / 8{x- x )e' 2irikx dx = e ~ 2irikXQ .
see also DELTA FUNCTION
Fourier Transform — Exponential Function
The Fourier Transform of e~ k °^ is given by
}\x\^-2TTikx ^
— 2itikx — 2irkox i
/oo
e -*oh
■oo
/0 /»o
e~ 2nikx e 27Txk °dx + I
-oo «/0
= / [cos{2nkx)~isin{27vkx)}e 2nko;c dx
J — oo
+ / [cos(27rfc;r) - isin(27rfcz)]e~ 27rfcoa: dx. (1)
Jo
Now let u = —x so du = — dx, then
/■oo
jr[ e -^o|x|] = / [ cos (2 7 zku) + isin{2izku)]e-^ kQU du]
Jo
/>oo
+ / [cos(2ttA;w) - isin{27rku)]e~ 2 " rk ° u du]
Jo
y»oo
= 2 / cos(27rfcu)e- 27rfcou du, (2)
Jo
which, from the Damped Exponential Cosine Inte-
gral, gives
,pr -27rfcoMl - X fc °
(3)
J tt^+Ajo 2 '
which is a LORENTZIAN FUNCTION.
see also Damped Exponential Cosine Integral,
Exponential Function, Lorentzian Function
Fourier Transform — Gaussian
The Fourier Transform of a Gaussian Function
f(x) = e~ ax is given by
/°° 2
e -ax e ikx dx
■oo
f°° _ ax 2
= I e [cos(kx) + i sin(kx)] dx
J —oo
/oo /«oo
e~ ax cos(kx) dx + i / e~ ax sin(kx)dx.
■oo «/ — oo
The second integrand is EVEN, so integration over a
symmetrical range gives 0. The value of the first inte-
gral is given by Abramowitz and Stegun (1972, p. 302,
equation 7.4.6)
F(*) = fa
-k 2 /4a
so a Gaussian transforms to a GAUSSIAN.
see also Gaussian Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
1972.
672 Fourier Transform — Heaviside Step Function
Fractal
Fourier Transform — Heaviside Step Function
F[H(x)}
-2irikx
H(x) dx
8{k)
trk
where H(x) is the HEAVISIDE STEP FUNCTION and S(k)
is the Delta Function.
see also HEAVISIDE STEP FUNCTION
Fourier Transform — Lorentzian Function
T
1
-2-KtkXQ— T-K\k\
_k{x-xo) 2 + {\T)\
see also LORENTZIAN FUNCTION
Fourier Transform — Ramp Function
Let R(x) be the Ramp Function, then the Fourier
Transform of R(x) is given by
J —c
e 7rr x R{x) dx — 7viS t (27rk)
£ix 2 k 2 '
where 5 r (x) is the DERIVATIVE of the DELTA FUNCTION.
see also Ramp Function
Fourier Transform — Rectangle Function
Let n(:r) be the RECTANGLE FUNCTION, then the
Fourier Transform is
.F[n(a;)] = sinc(7rA;),
where sinc(z) is the SlNC FUNCTION.
see also Rectangle Function, Sinc Function
Fourier Transform — Sine
T[sm(2ivk\
•ox)] = /
J — c
— 2ivzknx I c c
2iriuQt — 2-jvikQX
l i I \ e - 2 ^i(k-k )x _j_ e -27ri(fc+/e )a;i ^
2z
+ e~
dx
= ±i[5{k + k )-5{k-k )],
where S(x) is the Delta FUNCTION.
see also FOURIER TRANSFORM — COSINE, SINE
Fox's i7-Function
A very general function defined by
H(z) _ „-.« [J (a 1 ,a 1 ) 1 ...,(a p ,a 1 ,)l
__i_ r nr-i r(bj - - a<) nr-i r(i - a > + a * s)
^(l-^+ft.jn^rCa,-
° ds,
where < m < qr, < n < p, otj, j3j > 0, and a,j,bj are
Complex Numbers such that the pole of T(b 3 -0js) for
j = 1, 2, . . . , m coincides with any POLE of T(l — aj +
ajs) for j = 1, 2, . . . , n. In addition C, is a CONTOUR
in the complex s-plane from a? — zoo to a; -Moo such that
(bj + fc)//3j and (aj — 1 — k)/otj lie to the right and left
of C, respectively.
see also MacRobert's ^-Function, Meijer'S G-
FUNCTION
References
Carter, B. D. and Springer, M. D. "The Distribution of Prod-
ucts, Quotients, and Powers of Independent i/-Functions."
SIAM J. Appl. Math. 33, 542-558, 1977.
Fox, C. "The G and i7-Functions as Symmetrical Fourier
Kernels." Trans. Amer. Math. Soc. 98, 395-429, 1961.
Frac
see Fractional Part
Fractal
An object or quantity which displays Self-Similarity,
in a somewhat technical sense, on all scales. The object
need not exhibit exactly the same structure at all scales,
but the same "type" of structures must appear on all
scales. A plot of the quantity on a log-log graph versus
scale then gives a straight line, whose slope is said to be
the Fractal Dimension. The prototypical example
for a fractal is the length of a coastline measured with
different length Rulers. The shorter the Ruler, the
longer the length measured, a PARADOX known as the
Coastline Paradox.
see also BACKTRACKING, BARNSLEY'S FERN, BOX
Fractal, Butterfly Fractal, Cactus Fractal,
Cantor Set, Cantor Square Fractal, Carotid-
Kundalini Fractal, Cesaro Fractal, Chaos
Game, Circles-and-Squares Fractal, Coastline
Paradox, Dragon Curve, Fat Fractal, Fa-
tou Set, Flowsnake Fractal, Fractal Dimen-
sion, H-Fractal, Henon Map, Iterated Func-
tion System, Julia Fractal, Kaplan- Yorke Map,
Koch Antisnowflake, Koch Snowflake, Levy
Fractal, Levy Tapestry, Lindenmayer System,
Mandelbrot Set, Mandelbrot Tree, Menger
Sponge, Minkowski Sausage, Mira Fractal, New-
ton's Method, Pentaflake, Pythagoras Tree,
Rabinovich-Fabrikant Equation, San Marco
Fractal, Sierpinski Carpet, Sierpinski Curve,
Sierpinski Sieve, Star Fractal, Zaslavskii Map
Fractal
Fractal Process 673
References
Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed.
Boston, MA: Academic Press, 1993.
Bogomolny, A. "Fractal Curves and Dimension." http://
www.cut-the-knot.com/do_youJmow/dimension.html.
Brandt, C; Graf, S.; and Zahle, M. (Eds.). Fractal Geometry
and Stochastics. Boston, MA: Birkhauser, 1995.
Bunde, A. and Havlin, S. (Eds.). Fractals and Disordered
Systems, 2nd ed. New York: Springer- Verlag, 1996.
Bunde, A. and Havlin, S. (Eds.). Fractals in Science. New
York: Springer- Verlag, 1994.
Devaney, R. L. Complex Dynamical Systems: The Mathe-
matics Behind the Mandelbrot and Julia Sets. Providence,
RI: Amer. Math. Soc, 1994.
Devaney, R. L. and Keen, L. Chaos and Fractals: The Math-
ematics Behind the Computer Graphics. Providence, RI:
Amer. Math. Soc, 1989.
Edgar, G. A. Classics on Fractals. Reading, MA: Addison-
Wesley, 1994.
Eppstein, D. "Fractals." http:// www . ics . uci . edu / -
eppstein/ junkyard/fractal. html.
Falconer, K, J. The Geometry of Fractal Sets, 1st pbk. ed.,
with corr. Cambridge, England Cambridge University
Press, 1986.
Feder, J. Fractals. New York: Plenum Press, 1988.
Giffin, N. "The Spanky Fractal Database." http://spanky.
triumf . ca/www/welcomel .html.
Hastings, H. M. and Sugihara, G. Fractals: A User's Guide
for the Natural Sciences. New York: Oxford University
Press, 1994.
Kaye, B. H. A Random Walk Through Fractal Dimensions,
2nd ed. New York: Wiley, 1994.
Lauwerier, H. A. Fractals: Endlessly Repeated Geometrical
Figures. Princeton, NJ: Princeton University Press, 1991.
Mandelbrot, B. B. Fractals: Form, Chance, & Dimension.
San Francisco, CA: W. H. Freeman, 1977.
Mandelbrot, B. B. The Fractal Geometry of Nature. New
York: W. H. Freeman, 1983.
Massopust, P. R. Fractal Functions, Fractal Surfaces, and
Wavelets. San Diego, CA: Academic Press, 1994.
Pappas, T. "Fractals — Real or Imaginary." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 78-79, 1989.
Peitgen, H.-O.; Jurgens, H.; and Saupe, D. Chaos and Frac-
tals: New Frontiers of Science. New York: Springer-
Verlag, 1992.
Peitgen, H.-O. and Richter, D. H. The Beauty of Frac-
tals: Images of Complex Dynamical Systems. New York:
Springer- Verlag, 1986.
Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal
Images. New York: Springer- Verlag, 1988.
Pickover, C. A. (Ed.). The Pattern Book: Fractals, Art, and
Nature. World Scientific, 1995.
Pickover, C. A. (Ed.). Fractal Horizons: The Future Use of
Fractals. New York: St. Martin's Press, 1996.
Rietman, E. Exploring the Geometry of Nature: Computer
Modeling of Chaos, Fractals, Cellular Automata, and Neu-
ral Networks. New York: McGraw-Hill, 1989.
Russ, J. C. Fractal Surfaces. New York: Plenum, 1994.
Schroeder, M. Fractals, Chaos, Power Law: Minutes from
an Infinite Paradise. New York: W. H. Freeman, 1991.
Sprott, J. C. "Sprott's Fractal Gallery." http://sprott .
physics . wise , edu/f ractals , htm.
Stauffer, D. and Stanley, H. E. From Newton to Mandelbrot,
2nd ed. New York: Springer- Verlag, 1995.
Stevens, R. T. Fractal Programming in C. New York: Henry
Holt, 1989.
Takayasu, H. Fractals in the Physical Sciences. Manchester,
England: Manchester University Press, 1990.
Taylor, M. C. "sci. fractals FAQ." http://www.mta.ca/
~mctaylor/sci .f ractals-f aq.
Tricot, C. Curves and Fractal Dimension. New York:
Springer-Verlag, 1995.
Triumf Mac Fractal Programs, http://spanky.triumf.ca/
pub/fractals/programs/MAC/.
Vicsek, T. Fractal Growth Phenomena, 2nd ed. Singapore:
World Scientific, 1992.
# Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
Yamaguti, M.; Hat a, M.; and Kigami, J. Mathematics of
Fractals. Providence, RI: Amer. Math. Soc, 1997.
Fractal Dimension
The term "fractal dimension" is sometimes used to refer
to what is more commonly called the CAPACITY DI-
MENSION (which is, roughly speaking, the exponent D
in the expression n(e) — e~ D , where n(e) is the min-
imum number of OPEN SETS of diameter e needed to
cover the set). However, it can more generally refer
to any of the dimensions commonly used to character-
ize fractals (e.g., CAPACITY DIMENSION, CORRELATION
Dimension, Information Dimension, Lyapunov Di-
mension, MlNKOWSKI-BOULIGAND DIMENSION).
see also Box Counting Dimension, Capacity Di-
mension, Correlation Dimension, Fractal Di-
mension, Hausdorff Dimension, Information
Dimension, Lyapunov Dimension, Minkowski-
Bouligand Dimension, Pointwise Dimension, q-
Dimension
References
Rasband, S. N. "Fractal Dimension." Ch. 4 in Chaotic Dy-
namics of Nonlinear Systems. New York: Wiley, pp. 71-
83, 1990.
Fractal Land
see Carotid-Kundalini Fractal
Fractal Process
A 1-D MAP whose increments are distributed according
to a Normal Distribution. Let y(t-At) and y(t+Ai)
be values, then their correlation is given by the Brown
Function
r = 2 2 "- 1 - 1.
When H = 1/2, r = and the fractal process corre-
sponds to 1-D Brownian motion. If H > 1/2, then
r > and the process is called a PERSISTENT PRO-
CESS. If H < 1/2, then r < and the process is called
an Antipersistent Process.
see also ANTIPERSISTENT PROCESS, PERSISTENT PRO-
CESS
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, 1993.
674 Fractal Sequence
Fractional Calculus
Fractal Sequence
Given an INFINITIVE SEQUENCE {x n } with associated
array a(i,j), then {x n } is said to be a fractal sequence
1. If i + 1 = x n , then there exists m < n such that
2. If h < ij then, for every j, there is exactly one k such
that a(i,j) < a(h,k) < a(i,j + 1).
(As i and j range through JV, the array A = a(i,j),
called the associative array of as, ranges through all of
AT.) An example of a fractal sequence is 1, 1, 1, 1, 2, 1,
2, 1,3,2,1,3,2, 1,3,....
If {x n } is a fractal sequence, then the associated array is
an INTERSPERSION. If as is a fractal sequence, then the
Upper-Trimmed Subsequence is given by X(x) = x,
and the Lower-Trimmed Subsequence V(x) is an-
other fractal sequence. The Signature of an Irra-
tional Number is a fractal sequence.
see also Infinitive Sequence
References
Kimberling, C "Fractal Sequences and Interspersions." Ars
Combin. 45, 157-168, 1997.
Fractal Valley
see Carotid-Kundalini Function
Fraction
A RATIONAL NUMBER expressed in the form a/6, where
a is called the NUMERATOR and b is called the DENOM-
INATOR. A Proper Fraction is a fraction such that
a/b < 1, and a Lowest Terms Fraction is a fraction
with common terms canceled out of the NUMERATOR
and Denominator.
The Egyptians expressed their fractions as sums (and
differences) of Unit Fractions. Conway and Guy
(1999) give a table of Roman NOTATION for fractions, in
which multiples of 1/12 (the Uncia) were given separate
names.
see also Adjacent Fraction, Anomalous Can-
cellation, Continued Fraction, Denominator,
Egyptian Fraction, Farey Sequence, Golden
Rule, Half, Lowest Terms Fraction, Medi-
ant, Numerator, Proper Fraction, Pythago-
rean Fraction, Quarter, Rational Number, Unit
Fraction
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 22-23, 1996.
Courant, R. and Robbins, H. "Decimal Fractions. Infinite
Decimals." §2.2.2 in What is Mathematics?: An Elemen-
tary Approach to Ideas and Methods, 2nd ed. Oxford, Eng-
land: Oxford University Press, pp. 61-63, 1996.
Fractional Calculus
Denote the nth DERIVATIVE D n and the n-fold INTE-
GRAL D~ n . Then
Jo
m)ds.
(i)
Now, if
D~ n f(t)
(n
1 t*
-D'X
(*-0" _1 /(£)^ (2)
is true for n, then
D- {n+1) f(t) = D~ l
^jfc-O-'AO*
-jfef'- "- 1
f{i)di
dx.
(3)
Interchanging the order of integration gives
D- (n+l >/(«) = - f\t - 0*7(0 «• (4)
n * Jo
But (2) is true for n = 1, so it is also true for all n by
INDUCTION. The fractional integral of f(t) can then be
defined by
D-f(t)
i» i
(t-ervm.
(5)
where T(u) is the Gamma Function.
The fractional integral can only be given in terms of
elementary functions for a small number of functions.
For example,
D -" rX = r[x X + t+lf + " ^ A >-!,,> 0(6)
D~ v e a
= FTT^* / a"" 1 *"" dx = Ek(v, a), (7)
r W Jo
where E t (v,a) is the ^-Function. The fractional de-
rivative of / (if it exists) can be defined by
D»f(t) = D Tn [D- (Tn - fl) f(t)].
(8)
An example is
,p.X
D»t
r(A + i)
T(A + m-M+l)
r(A + i) A _ M
r(A-At + i)
for A > -l,n >
(9)
D"E t (v,a)=Et{v-p,a) for i> >0, p ^ 0. (10)
Fractional Derivative
Fractran
675
It is always true that, for /a, v > 0,
D-^D~ u f{t) = D _Cm+v
but noi always true that
see also Derivative, Integral
(11)
(12)
References
Love, E. R. "Fractional Derivatives of Imaginary Order." J.
London Math. Soc. 3, 241-259, 1971.
McBride, A. C. Fractional Calculus. New York: Halsted
Press, 1986.
Miller, K. S. "Derivatives of Noninteger Order." Math. Mag.
68, 183-192, 1995.
Nishimoto, K. Fractional Calculus. New Haven, CT: Univer-
sity of New Haven Press, 1989.
Spanier, J. and Oldhan, K. B. The Fractional Calculus. New-
York: Academic Press, 1974.
Fractional Derivative
see Fractional Calculus
Fractional Differential Equation
The solution to the differential equation
[D 2v + aD v + bD°]y(t) =
f e a (t) - ep(t)
for a ^
y(t) = {
where
for a = ^
r(2t0
for «
= 0,
Q= -
v
9-1
ee{t) = Y,P q ~ k ~ lE t(- k v,0 q ),
fe=0
E t (a,x) is the ^-FUNCTION, and T(n) is the Gamma
Function.
References
Miller, K. S. "Derivatives of Noninteger Order." Math. Mag.
68, 183-192, 1995.
Fractional Fourier Transform
A ^-Transform with
2-nia/N
z = e '
for a ^ ±1. This transform can be used to detect fre-
quencies which are not Integer multiples of the lowest
Discrete Fourier Transform frequency.
see also z-TRANSFORM
References
Graham, R. L.; Knuth, D. E.; and Patashnik, 0. Concrete
Mathematics, 2nd ed. Reading, MA: Addison- Wesley,
1994.
Fractional Integral
see Fractional Calculus
Fractional Part
The function giving the fractional (nonintegral) part of
a number and denned as
frac(a^
^ f x - [x\
\x-[x\
x>0
x < 0,
where [x\ is the FLOOR FUNCTION.
see also CEILING FUNCTION, FLOOR FUNCTION, NlNT,
Round, Truncate, Whole Number
References
Spanier, J. and Oldham, K. B. "The Integer- Value lnt(x) and
Fractional- Value frac(a;) Functions." Ch. 9 in An Atlas of
Functions. Washington, DC: Hemisphere, pp. 71-78, 1987.
Fractran
Fractran is an algorithm applied to a given list /i, /2,
. . . , fk of Fractions. Given a starting Integer N t the
Fractran algorithm proceeds by repeatedly multiplying
the integer at a given stage by the first element fi given
an integer PRODUCT. The algorithm terminates when
there is no such fi.
The list
17^^^29779577^111315155
91' 85' 51' 38' 33' 29' 23' 19' 17' 13' 11' 2 ' 7' 1
with starting integer N = 2 generates a sequence 2,
15, 825, 725, 1925, 2275, 425, 390, 330, 290, 770, ....
Conway (1987) showed that the only other powers of 2
which occur are those with PRIME exponent: 2 2 , 2 3 , 2 5 ,
References
Conway, J. H. "Unpredictable Iterations." In Proc. Number
Theory Conf., Boulder, CO, pp. 49-52, 1972.
Conway, J. H. "Fractran: A Simple Universal Programming
Language for Arithmetic." Ch. 2 in Open Problems in
Communication and Computation (Ed. T. M. Cover and
B. Gopinath). New York: Springer- Verlag, pp. 4-26, 1987.
676
Framework
Fredholm Integral Equation of the Second Kind
Framework
Consider a finite collection of points p = (pi, • ■ ■ ,Pn),
pi e R d Euclidean Space (known as a Configura-
tion) and a graph G whose VERTICES correspond to
pairs of points that are constrained to stay the same
distance apart. Then the graph G together with the
configuration p, denoted G(p), is called a framework.
see also Bar (Edge), Configuration, Rigid
Franklin Magic Square
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).
see also Magic Square, Panmagic Square
References
Madachy, J. S. "Magic and Antimagic Squares." Ch. 4 in
Madachy 's Mathematical Recreations. New York: Dover,
pp. 103-113, 1979.
Pappas, T. "The Magic Square of Benjamin Franklin." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 97, 1989.
Fransen-Robinson Constant
f™ dx
F~ I ^-r = 2.8077702420...,
/°
Jo
dx
where F(x) is the Gamma Function. The above plots
show the functions T(x) and l/V(x).
see also GAMMA FUNCTION
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/fran/fran.html.
Fransen, A. "Accurate Determination of the Inverse Gamma
Integral." BIT 19, 137-138, 1979.
Fransen, A. "Addendum and Corrigendum to 'High-Precision
Values of the Gamma Function and of Some Related Co-
efficients."' Math. Comput. 37, 233-235, 1981.
Fransen, A. and Wrigge, S. "High-Precision Values of the
Gamma Function and of Some Related Coefficients."
Math. Comput 34, 553-566, 1980.
Plouffe, S. "Fransen-Robinson Constant." http://lacim.
uqam.ca/piDATA/f ransen.txt.
Frechet Bounds
Any bivariate distribution function with marginal dis-
tribution functions F and G satisfies
mzx{F{x) + G(y) - 1, 0} < H(x y y) < mm{F(z), G(y)}.
Frechet Derivative
A function / is Frechet differentiate at a if
lim
/(*) - /(a)
exists. This is equivalent to the statement that has a
removable DISCONTINUITY at a, where
0(z) =
/(») - /(«)
Every function which is Frechet differentiable is also
Caratheodory differentiable.
see also CARATHEODORY DERIVATIVE, DERIVATIVE
Frechet Space
A complete metrizable SPACE, sometimes also with the
restriction that the space be locally convex.
Fredholm Integral Equation of the First
Kind
An INTEGRAL EQUATION of the form
/oo
K{x,t)<f>(t)dt
-oo
)
<K*)
2* ;_ km
duj.
see also FREDHOLM INTEGRAL EQUATION OF THE SEC-
OND Kind, Integral Equation, Volterra Inte-
gral Equation of the First Kind, Volterra In-
tegral Equation of the Second Kind
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 865, 1985.
Fredholm Integral Equation of the Second
Kind
An Integral Equation of the form
/oo
K{x,t)<t>{t)dt
■oo
1 r F(t)e-^-
dt
\/%K\K{t)
see also Fredholm Integral Equation of the
First Kind, Integral Equation, Neumann Se-
ries (Integral Equation), Volterra Integral
Free
Freiman's Constant 677
Equation of the First Kind, Volterra Integral
Equation of the Second Kind
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 865, 1985.
Press, W. H.; Flannery, B. P.; Teukolsky, S.A.; and Vet-
terling, W. T. "Fredholm Equations of the Second Kind."
§18.1 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 782-785, 1992.
Free
When referring to a planar object, "free" means that the
object is regarded as capable of being picked up out of
the plane and flipped over. As a result, Mirror Images
are equivalent for free objects.
A free abstract mathematical object is generated by n
elements in a "free manner," i.e., such that the n ele-
ments satisfy no nontrivial relations among themselves.
To make this more formal, an algebraic GADGET X is
freely generated by a SUBSET G if, for any function
/ : G — ► Y where Y is any other algebraic Gadget,
there exists a unique HOMOMORPHISM (which has dif-
ferent meanings depending on what kind of GADGETS
you're dealing with) g : X -> Y such that g restricted
to G is /.
If the algebraic GADGETS are VECTOR SPACES, then
G freely generates X Iff G is a BASIS for X. If the
algebraic GADGETS are ABELIAN GROUPS, then G freely
generates X Iff X is a DIRECT Sum of the INTEGERS,
with G consisting of the standard BASIS.
see also Fixed, Gadget, Mirror Image, Rank
Free Group
The generators of a group G are defined to be the small-
est subset of group elements such that all other elements
of G can be obtained from them and their inverses. A
GROUP is a free group if no relation exists between its
generators (other than the relationship between an el-
ement and its inverse required as one of the defining
properties of a group). For example, the additive group
of whole numbers is free with a single generator, 1.
see also FREE SEMIGROUP
Free Semigroup
A SEMIGROUP with a noncommutative product in which
no PRODUCT can ever be expressed more simply in terms
of other ELEMENTS.
see also Free Group, Semigroup
Free Variable
An occurrence of a variable in a LOGIC Formula which
is not inside the scope of a Quantifier.
see also BOUND, SENTENCE
Freemish Crate
An IMPOSSIBLE FIGURE box which can be drawn but
not built.
References
Fineman, M. The Nature of Visual Illusion. New York:
Dover, p. 120-122, 1996.
Jablan, S. "Are Impossible Figures Possible?" http://
members .tripod. com/ -modular ity/kulpa. htm.
Pappas, T. "The Impossible Tribar." The Joy of Mathemat-
ics. San Carlos, CA: Wide World Publ./Tetra, p. 13, 1989.
Freeth's Nephroid
A Strophoid of a Circle with the Pole O at the Cen-
ter of the CIRCLE and the fixed point P on the CIR-
CUMFERENCE of the Circle. In a paper published by
the London Mathematical Society in 1879, T. J. Freeth
described it and various other STROPHOIDS (MacTutor
Archive). If the line through P Parallel to the y-Axis,
cuts the Nephroid at A, then Angle AOP is 37r/7, so
this curve can be used to construct a regular HEPTAGON.
The POLAR equation is
r = a[l + 2sin(§0)].
see also STROPHOID
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 175 and 177-178, 1972.
MacTutor History of Mathematics Archive. "Freeth's
Nephroid." http : // www - groups . dcs . st - and .ac.uk/
-history/Curves/Freeths .html.
Freiman's Constant
The end of the last gap in the LAGRANGE SPECTRUM,
given by
F =
2221564096 4- 293748\/462
491993569
= 4.5278295661 ....
Real Numbers greater, than F are members of the
Markov Spectrum.
678 French Curve
Fresnel Integrals
see also Lagrange Spectrum, Markov Spectrum
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 188-189, 1996.
French Curve
French curves are plastic (or wooden) templates having
an edge composed of several different curves. French
curves are used in drafting (or were before computer-
aided design) to draw smooth curves of almost any de-
sired curvature in mechanical drawings. Several typical
French curves are illustrated above.
see also CORNU SPIRAL
Frenet Formulas
Also known as the Serret-Frenet Formulas
Tl
r o
K
01
[T]
N
=
— K
r
N
B
— r
B
where T is the unit TANGENT Vector, N is the unit
Normal Vector, B is the unit Binormal Vector,
r is the TORSION, k is the CURVATURE, and x denotes
dx/ds.
see also Centrode, Fundamental Theorem of
Space Curves, Natural Equation
References
Frenet, F. "Sur les courbes a double courbure." These.
Toulouse, 1847. Abstract in J. de Math. 17, 1852.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 126, 1993.
Kreyszig, E. "Formulae of Frenet." §15 in Differential Ge-
ometry. New York: Dover, p. 40-43, 1991.
Serret, J. A. "Sur quelques formules relatives a la theorie des
courbes a double courbure." J. de Math. 16, 1851.
Frequency Curve
see Gaussian Function
Fresnel's Elasticity Surface
A Quartic Surface given by
where
r = ya 2 x 2 + b 2 y 2 -f- c 2 z 2 ,
2 /2 . /2 , t2
r = x +y + z ,
also known as Fresnel's Wave Surface. It was intro-
duced by Fresnel in his studies of crystal optics.
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 16, 1986.
Fischer, G. (Ed.). Plates 38-39 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 38-39, 1986.
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 304, 1993.
Fresnel Integrals
In physics, the Fresnel integrals are most often defined
by
C{u) + iS{
u)= f e ilzx2/2 dx
Jo
pu nu
I co$(\irx 2 )dx + i I sm(^7rx 2 )dx, (1)
Jo Jo
7vx ) dx
pu
C(u)= / cos(i
Jo
S(u) = / sm(~7Tx 2 )dx.
Jo
They satisfy
C(±oo) = -i
S(±oo) = i.
Related functions are defined as
cost dt
sin t 2 dt
cost
at
An asymptotic expansion for x ^> 1 gives
Cf ( u )*o + — siri(i™ 2 )
1
i 2x
S(u) « C0S(~7TU
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
Therefore, as u -+ oo, C(u) = 1/2 and S(u) - 1/2. The
Fresnel integrals are sometimes alternatively denned as
x(t)
!/(*) =
= / COs(f
Jo
I sin(u
Jo
2 )dv
2 )dv.
(12)
(13)
Fresnel's Wave Surface
Friday the Thirteenth 679
Letting x = v 2 so dx = 2vdv = 2y/xdv, and dv
x-V 3 dx/2
x(t) =
y(t)
JO
1 / -1/2 •
= 2 / X S1
Jo
' 2 cos a; da;
sinxdx.
(14)
(15)
In this form, they have a particularly simple expan-
sion in terms of SPHERICAL BESSEL FUNCTIONS OF THE
First Kind. Using
(16)
(17)
Mx) = ——
X
m(x) = -j-i(x) = --
where m(x) is a SPHERICAL BESSEL FUNCTION OF THE
Second Kind
s(* 2 ) = -I / n 1 (x)x 1/2 dx
Jo
= \ f j- 1 (x)x 1/2 dx = x 1/2 f^j 2n (x) (18)
i / j (^)x 1/2 da:
Jo
y(0 =
= X 1/2 Y^hn+l{x).
(19)
see a/so Cornu Spiral
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Fresnel Inte-
grals." §7.3 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 300-302, 1972.
Leonard, I. E. "More on Fresnel Integrals." Amer. Math.
Monthly 95, 431-433, 1988.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Fresnel Integrals, Cosine and Sine Integrals."
§6.79 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 248-252, 1992.
Spanier, J. and Oldham, K. B. "The Fresnel Integrals S(x)
andC(ai)." Ch. 39 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 373-383, 1987.
Fresnel's Wave Surface
see Fresnel's Elasticity Surface
Frey Curve
Let a? + b p = <? be a solution to Fermat'S Last THE-
OREM. Then the corresponding Frey curve is
y 2 = x{x-a p ){x + b p ).)
(1)
Frey showed that such curves cannot be MODULAR, so if
the Taniyama-Shimura Conjecture were truey Frey
curves couldn't exist and Fermat's Last Theorem
would follow with b Even and a = -1 (mod 4). Frey
curves are SEMISTABLE. Invariants include the DIS-
CRIMINANT
{a p - 0) 2 {-b p - 0)[a p - {-b) p } 2 = a 2p b 2p c 2p . (2)
The Minimal Discriminant is
A = 2-VW P , (3)
the Conductor is
n = n '. ( 4 )
l\abc
and the ^-INVARIANT is
. _ 2 s (a 2p + b 2p + oFWf _ 2 8 (c 2p - b p (?f
a 2 Pb 2 Pc 2 P
(abc) 2 ?
(5)
see also Elliptic Curve, Fermat's Last Theorem,
Taniyama-Shimura Conjecture
References
Cox, D. A. "Introduction to Fermat's Last Theorem." Amer.
Math. Monthly 101, 3-14, 1994.
Gouvea, F. Q. "A Marvelous Proof." Amer. Math. Monthly
101, 203-222, 1994.
Frey Elliptic Curve
see Frey Curve
Friday the Thirteenth
The Gregorian calendar follows a pattern of leap years
which repeats every 400 years. There are 4,800 months
in 400 years, so the 13th of the month occurs 4,800 times
in this interval. The number of times the 13th occurs
on each weekday is given in the table below. As shown
by Brown (1933), the thirteenth of the month is slightly
more likely to be on a Friday than on any other day.
Day
Number of 13s
Fraction
Sunday
687
14.31%
Monday
685
14.27%
Tuesday
685
14.27%
Wednesday
687
14.31%
Thursday
684
14.25%
Friday
688
14.33%
Saturday
684
14.25%
see also 13, Weekday
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New Ydrk: Dover, p. 27, 1987.
Brown, B. H. "Solution to Problem E3B." Amer. Math.
Monthly 40, 607, 1933.
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. 14-15, 1992.
680
Friend
Frobenius Method
Friend
A friend of a number n is another number m such that
(ra, n) is a FRIENDLY PAIR.
see also Friendly Pair, Solitary Number
References
Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly
Integers." Amer. Math. Monthly 84, 65-66, 1977.
Friendly Giant Group
see Monster Group
Friendly Pair
Define
2j(n) = ,
where cr(n) is the Divisor Function. Then a Pair of
distinct numbers (k, m) is a friendly pair (and k is said
to be a Friend of m) if
E(fc) = S(m).
For example, 4320 and 4680 are a friendly pair, since
cr(4320) = 15120, <r(4680) = 16380, and
E(4320) = ^ = l
S(4680) = *gjj? = \
Numbers which do not have FRIENDS are called
Solitary Numbers. Solitary Numbers satisfy
(<x(n),n) = 1, where (a, 6) is the Greatest Common
Divisor of a and b.
see also Aliquot Sequence, Friend, Solitary Num-
ber
References
Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly
Integers." Amer. Math. Monthly 84, 65-66, 1977.
Frieze Pattern
Frobenius Map
A map x ^ x p where p is a Prime.
Frobenius Method
If xo is an ordinary point of the ORDINARY DIFFEREN-
TIAL Equation, expand y in a Taylor Series about
xo, letting
y = } j a n x n .
(1)
Plug y back into the ODE and group the COEFFICIENTS
by Power. Now, obtain a Recurrence Relation for
the nth term, and write the TAYLOR SERIES in terms of
the a n s. Expansions for the first few derivatives are
oo
y = ^2 a *x n ( 2 )
n=0
oo oo
y = ^2 na nX n ~ l = ^(n + l)a n +ix n (3)
n=l
n=0
y" = ^n{n~ l)a n x n ~ 2 = ^(n + 2)(n + l)a n+2 a; n .
n = 2 n =
(4)
If Xo is a regular singular point of the ORDINARY DIF-
FERENTIAL Equation,
P(x)y" + Q{x)y + R(x)y = 0,
(5)
solutions may be found by the Frobenius method or
by expansion in a Laurent Series. In the Frobenius
method, assume a solution of the form
y = x k ^^ a "
(6)
so that
An arrangement of numbers at the intersection of two
sets of perpendicular diagonals such that a + d = fe+c+1
(for an additive frieze pattern) or ad = be + 1 (for a
multiplicative frieze pattern) in each diamond.
References
Conway, J. H. and Coxeter, H. S. M. "Triangulated Polygons
and Frieze Patterns." Math. Gaz. 57, 87-94, 1973.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, pp. 74-76 and 96-97, 1996.
Frobenius-Konig Theorem
The Permanent of an n x n Matrix with all entries
either or 1 is IFF the MATRIX contains an r x s
submatrix of 0s with r + s — n + 1. This result follows
from the Konig-Egevary Theorem.
see also Konig-Egevary Theorem, Permanent
y = xk £ a ^ n = J2 anxU+k W
71 = 71 =
OO
y = y £ d a n {n + k)x k+n - 1 (8)
n=0
oo
y" = J2^(n + k)(n + k-l)x k+n - 2 . (9)
71 —
Now, plug y back into the ODE and group the COEFFI-
CIENTS by POWER to obtain a recursion FORMULA for
the a n th term, and then write the TAYLOR SERIES in
terms of the a n s. Equating the ao term to will pro-
duce the so-called INDICIAL EQUATION, which will give
the allowed values of k in the Taylor Series.
Fuchs's Theorem guarantees that at least one Power
series solution will be obtained when applying the Fro-
benius method if the expansion point is an ordinary,
Frobenius-Peron Equation
FrullanVs Integral 681
or regular, SINGULAR POINT. For a regular SINGULAR
Point, a Laurent Series expansion can also be used.
Expand y in a Laurent Series, letting
y = c~ n x~ n + ... + c- 1 x~ 1 +c +Cix + ... + c n x n +
(10)
Plug y back into the ODE and group the COEFFICIENTS
by POWER. Now, obtain a recurrence FORMULA for the
c n th term, and write the TAYLOR EXPANSION in terms
of the c„s.
see also FUCHS'S THEOREM, ORDINARY DIFFERENTIAL
Equation
References
Arfken, G. "Series Solutions — Frobenius' Method." §8.5 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 454-467, 1985.
Frobenius-Peron Equation
p„+i(x) = / p n (y)S[x - M(y)} dy,
where 5(x) is a Delta Function, M(x) is a map, and
p is the Natural Density.
References
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, p. 51, 1993.
Frobenius Pseudoprime
Let f(x) be a Monic Polynomial of degree d with
discriminant A. Then an Odd Integer n with
(n,/(0)A) = 1 is called a Frobenius pseudoprime with
respect to f{x) if it passes a certain algorithm given
by Grantham (1996). A Frobenius pseudoprime with
respect to a Polynomial f(x) e Z[x] is then a compos-
ite Frobenius probably prime with respect to the POLY-
NOMIAL x — a.
While 323 is the first Lucas PSEUDOPRIME with respect
to the Fibonacci polynomial x 2 — x — 1, the first Froben-
ius pseudoprime is 5777. If f(x) = x 3 —rx 2 +sx — l, then
any Frobenius pseudoprime n with respect to f(x) is
also a Perrin Pseudoprime. Grantham (1997) gives a
test based on Frobenius pseudoprimes which is passed by
Composite Numbers with probability at most 1/7710.
see also PERRIN PSEUDOPRIME, PSEUDOPRIME,
Strong Frobenius Pseudoprime
References
Grantham, J. "Frobenius Pseudoprimes." 1996. http://
www.clark.net/pub/grantham/pseudo/pseudo.ps
Grantham, J. "A Frobenius Probable Prime Test with
High Confidence." 1997. http : //www . dark . net/pub/
grantham/pseudo/pseudo2.ps
Grantham, J. "Pseudoprimes/Probable Primes." http://
www . dark . net /pub/grantham/pseudo.
Frobenius Theorem
Let A = an be a Matrix with Positive Coefficients
so that aij > for all i, j = 1, 2, . . . , n, then A has a
Positive Eigenvalue A , and all its Eigenvalues lie
on the Closed Disk
\z\ < Ao.
see also CLOSED DISK, OSTROWSKl'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.
Frobenius Triangle Identities
Let C l ,m be a PADE Approximant. Then
C(l+i)/mS(l-i)/m - Cl/(m+i)Sl/(m+i)
= Cl/mSl/m (1)
Cl/(M+1)S(L+1)/M - C(l+i)/mSl/(M+1)
= C(l+i)/(m+i)&Sl/m (2)
C(l+i)/mSl/m — Cl/mS(l+i)/m
= C(l+i)/(m+i)&Sl/(m-i) ( 3 )
Cl/(m+i)$l/m — Cl/mSl/(m+i)
= C(I,+1)/(M+1)3S(L-1)/M) (4)
where
S l /m = G(x)P L (x) + H(x)Q M (x) (5)
and C is the C-Determinant.
see also C-Determinant, Pade Approximant
References
Baker, G. A. Jr. Essentials of Pade Approximants in Theo-
retical Physics. New York: Academic Press, p. 31, 1975.
Frontier
see Boundary
Frullani's Integral
If f(x) is continuous and the integral converges,
f f{ax) ~ f{bx) dx = [/(o) - /<«,)] m (J) .
References
Spiegel, M. R. Mathematical Handbook of Formulas and Ta-
bles. New York: McGraw-Hill, 1968.
682
Frustum
Fuhrmann Triangle
Frustum
The portion of a solid which lies between two PARALLEL
PLANES cutting the solid. Degenerate cases are obtained
for finite solids by cutting with a single Plane only.
see also Conical Frustum, Pyramidal Frustum,
Spherical Segment
Fubini Principle
If the average number of envelopes per pigeonhole is
a, then some pigeonhole will have at least a envelopes.
Similarly, there must be a pigeonhole with at most a
envelopes.
see also PIGEONHOLE PRINCIPLE
Fuchsian System
A system of linear differential equations
dy
dz
A(z)y,
with A(z) an Analytic n x n Matrix, for which the
Matrix A(z) is Analytic in C\{ai, . . . ,a N } and has
a POLE of order 1 at a,j for j = 1, . . . , N. A system
is Fuchsian IFF there exist n x n matrices i?i, . . . , Bjv
with entries in Z such that
A(z)
N ^
£~^ z — a,-
j'=i
i>
Fuchs's Theorem
At least one POWER SERIES solution will be obtained
when applying the Frobenius Method if the expan-
sion point is an ordinary, or regular, SINGULAR POINT.
The number of ROOTS is given by the ROOTS of the
Indicial Equation.
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 462-463, 1985.
Fuhrmann Circle
Mac
M BC
The Circumcircle of the Fuhrmann Triangle.
see also Fuhrmann Triangle, Mid-Arc Points
References
Fuhrmann, W. Synthetische Beweise Planimetrischer Satze.
Berlin, p. 107, 1890.
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.
Fuhrmann's Theorem
R
N P
Let the opposite sides of a convex CYCLIC HEXAGON be
a, a', fc, &', c, and c', and let the DIAGONALS e, /, and g
be so chosen that a, a , and e have no common VERTEX
(and likewise for 6, &', and /), then
efg = aae -f- bb' f + ccg + abc + ab'c .
This is an extension of PTOLEMY'S THEOREM to the
Hexagon.
see also Cyclic Hexagon, Hexagon, Ptolemy's
Theorem
References
Fuhrmann, W. Synthetische Beweise Planimetrischer Satze.
Berlin, p. 61, 1890.
Johnson, FL A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 65-66, 1929.
Fuhrmann Triangle
Mbc
The Fuhrmann triangle of a Triangle AABC is the
Triangle AF c F b F a formed by reflecting the Mid-
Arc Points Mas, Mac, Mbc about the lines AB, AC,
Full Reptend Prime
Function
683
and BC. The ClRCUMCIRCLE of the Fuhrmann triangle
is called the FUHRMANN CIRCLE, and the lines F A M B c,
F b Mac, and F c M A b Concur at the ClRCUMCENTER
0.
see also Fuhrmann Circle, Mid-Arc Points
References
Fuhrmann, W. Synthetische Beweise Planimetrischer Satze.
Berlin, p. 107, 1890.
Johnson, R. A. Modem Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 228-229, 1929.
Full Reptend Prime
A Prime p for which 1/p has a maximal period Decimal
Expansion of p— 1 Digits. The first few numbers with
maximal decimal expansions are 7, 17, 19, 23, 29, 47,
59, 61, 97, . . . (Sloane's A001913).
References
Sloane, N. J. A. Sequence A001913/M4353 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Full Width at Half Maximum
The full width at half maximum (FWHM) is a param-
eter commonly used to describe the width of a "bump"
on a curve or function. It is given by the distance be-
tween points on the curve at which the function reaches
half its maximum value. The following table gives the
analytic and numerical full widths for several common
curves.
Function Formula FWHM
1
~W
2y/2c
Bartlett
Blackman " 0.810957a
Connes f 1 - ^ J \A
Cosine cosJSf) fa
Gaussian e -* 2 /(2<r 2 ) 2 V2\n2a
Hamming 1.05543a
Hanning
Lorentzian
Welch
|r
r
y/2a
1-4
see
also Apodization Function
, Maximum
Fu
Her Dome
see
Geodesic Dome
Function
A way of associating unique objects to every point in a
given Set. A function from A to B is an object / such
that for every a G A, there is a unique object /(a) G B.
Examples of functions include sinx, x, # 2 , etc. The term
Map is synonymous with function.
Poincare remarked with regard to the proliferation of
pathological functions, "Formerly, when one invented a
new function, it was to further some practical purpose;
today one invents them in order to make incorrect the
reasoning of our fathers, and nothing more will ever be
accomplished by these inventions."
see also Abelian Function, Absolute Value, Ack-
ermann Function, Airy Functions, Algebraic
Function, Algebroidal Function, Alpha Func-
tion, Andrew's Sine, Anger Function, Apodi-
zation Function, Apparatus Function, Argu-
ment (Function), Artin L-Function, Automor-
phic Function, Bachelier Function, Barnes G-
Function, Bartlett Function, Basset Func-
tion, Bateman Function, Bei, Ber, Bernoulli
Function, Bessel Function of the First Kind,
Bessel Function of the Second Kind, Bessel
Function of the Third Kind, Beta Function,
Beta Function (Exponential), Binomial Coeffi-
cient, Blackman Function, Blancmange Func-
tion, Boolean Function, Bourget Function,
Boxcar Function, Brown Function, Cal, Can-
tor Function, Carmichael Function, Carotid-
Kundalini Function, Ceiling Function, Center
Function, Central Beta Function, Character-
istic Function, Chebyshev Function, Circular
Functions, Clausen Function, Comb Function,
Complete Functions, Complex Conjugate, Com-
putable Function, Concave Function, Conflu-
ent Hypergeometric Function, Confluent Hy-
pergeometric Function of the First Kind, Con-
fluent Hypergeometric Function of the Second
Kind, Confluent Hypergeometric Limit Func-
tion, Conical Function, Connes Function, Con-
stant Function, Contiguous Function, Continu-
ous Function, Convex Function, Copula, Cose-
cant, Cosine, Cosine Apodization Function, Co-
tangent, Coulomb Wave Function, Coversine,
Cube Root, Cubed, Cumulant-Generating Func-
tion, Cumulative Distribution Function, Cun-
ningham Function, Cylinder Function, Cylin-
drical Function, Debye Functions, Decreas-
ing Function, Dedekind Eta Function, Dedekind
Function, Delta Function, Digamma Function,
Dilogarithm, Dirac Delta Function, Dirich-
let Beta Function, Dirichlet Eta Function,
Dirichlet Function, Dirichlet Lambda Func-
tion, Distribution Function, Divisor Function,
Double Gamma Function, Doublet Function,
e^-function, et-function, elgenfunction, eln
Function, Einstein Functions, Elementary Func-
tion, Elliptic Alpha Function, Elliptic Delta
Function, Elliptic Exponential Function, El-
liptic Function, Elliptic Functional, Elliptic
Lambda Function, Elliptic Modular Function,
Elliptic Theta Function, Elsasser Function, En-
tire Function, Epstein Zeta Function, Erdos-
Selfridge Function, Erf, Error Function, Ex-
ponential Ramp, Euler L-Function, Even Func-
tion, Exponential Function, Exponential Func-
tion (Truncated), Exponential Sum Function,
Exsecant, Floor Function, Fox's H-Function,
684
Function
Function
Function Space, G-Function, Gamma Function,
Gate Function, Gaussian Function, Gegen-
bauer Function, Generalized Function, Gener-
alized Hyperbolic Functions, Generalized Hy-
pergeometric Function, Generating Function,
Gordon Function, Green's Function, Growth
Function, Gudermannian Function, ^-Function,
Haar Function, Hamming Function, Hankel
Function, Hankel Function of the First Kind,
Hankel Function of the Second Kind, Hann
Function, Hanning Function, Harmonic Func-
tion, Haversine, Heaviside Step Function, Hecke
l-function, hemicylindrical function, hemi-
SPHERICAL Function, Heuman Lambda Function,
Hh Function, Hilbert Function, Holonomic
Function, Homogeneous Function, Hurwitz Zeta
Function, Hyperbolic Cosecant, Hyperbolic Co-
sine, Hyperbolic Cotangent, Hyperbolic Func-
tions, Hyperbolic Secant, Hyperbolic Sine, Hy-
perbolic Tangent, Hyperelliptic Function, Hy-
pergeometric Function, Identity Function, Im-
plicit Function, Implicit Function Theorem, In-
complete Gamma Function, Increasing Func-
tion, Infinite Product, Instrument Function,
Int, Inverse Cosecant, Inverse Cosine, In-
verse Cotangent, Inverse Function, Inverse Hy-
perbolic Functions, Inverse Secant, Inverse
Sine, Inverse Tangent, j-Function, Jacobi El-
liptic Functions, Jacobi Function of the First
Kind, Jacobi Function of the Second Kind, Ja-
cobi Theta Function, Jacobi Zeta Function,
Jinc Function, Joint Probability Density Func-
tion, Jonquiere's Function, ^-Function, Kei,
Kelvin Functions, Ker, Koebe Function, L-
Function, Lambda Function, Lambda Hypergeo-
metric Function, Lambert's W-Function, Lame
Function, Legendre Function of the First Kind,
Legendre Function of the Second Kind, Lem-
niscate Function, Lemniscate Function, Length
Distribution Function, Lerch Transcendent,
Levy Function, Linearly Dependent Func-
tions, Liouville Function, Lipschitz Function,
Logarithm, Logarithmically Convex Function,
Logit Transformation, Lommel Function, Lya-
punov Function, MacRobert's £7-Function, Man-
goldt Function, Mathieu Function, Measur-
able Function, Meijer's G-Function, Meromor-
phic, Mertens Function, Mertz Apodization
Function, Mittag-Leffler Function, Mobius
Function, Mobius Periodic Function, Mock
Theta Function, Modified Bessel Function of
the First Kind, Modified Bessel Function of
the Second Kind, Modified Spherical Bessel
Function, Modified Struve Function, Modular
Function, Modular Gamma Function, Modular
Lambda Function, Moment-Generating Func-
tion, Monogenic Function, Monotonic Func-
tion, Mu Function, Multiplicative Function,
Multivalued Function, Multivariate Function,
Neumann Function, Nint, Nu Function, Null
Function, Numeric Function, Oblate Spher-
oidal Wave Function, Odd Function, Omega
Function, One-Way Function, Parabolic Cyl-
inder Function, Partition Function P, Par-
tition Function Q, Parzen Apodization Func-
tion, Pearson-Cunningham Function, Pearson's
Function, Periodic Function, Planck's Radi-
ation Function, Plurisubharmonic Function,
Pochhammer Symbol, Poincare-Fuchs-Klein Au-
tomorphic Function, Poisson-Charlier Func-
tion, POLYGAMMA FUNCTION, POLYGENIC FUNC-
TION, Polylogarithm, Positive Definite Func-
tion, Potential Function, Power, Prime Count-
ing Function, Prime Difference Function, Prob-
ability Density Function, Probability Distribu-
tion Function, Prolate Spheroidal Wave Func-
tion, Psi Function, Pulse Function, q-Beta Func-
tion, Q-FUNCTION, g-GAMMA FUNCTION, QUASIPERI-
ODIC FUNCTION, RADEMACHER FUNCTION, RAMANU-
jan Function, Ramanujan g- and G- Functions,
Ramanujan Theta Functions, Ramp Function,
Rational Function, Real Function, Rectan-
gle Function, Regular Function, Regularized
Gamma Function, Restricted Divisor Function,
Riemann Function, Riemann-Mangoldt Func-
tion, Riemann-Siegel Functions, Riemann Theta
Function, Riemann Zeta Function, Ring Func-
tion, Sal, Sampling Function, Scalar Function,
Schlomilch's Function, Secant, Sequency Func-
tion, Sgn, Shah Function, Siegel Modular Func-
tion, Sigma Function, Sigmoid Function, Sign,
Sinc Function, Sine, Smarandache Function,
Spence's Function, Spherical Bessel Function
of the First Kind, Spherical Bessel Function
of the Second Kind, Spherical Hankel Function
of the First Kind, Spherical Hankel Function
of the Second Kind, Spherical Harmonic, Spher-
oidal Wavefunction, Sprague-Grundy Function,
Square Root, Squared, Step Function, Struve
Function, Sturm Function, Summatory Func-
tion, Symmetric Function, TAK Function, Tan-
gent, Tapering Function, Tau Function, Tetra-
choric Function, Theta Function, Toroidal
Function, Toronto Function, Total Function,
Totient Function, Totient Valence Function,
Transcendental Function, Transfer Function,
Trapdoor Function, Triangle Center Function,
Triangle Function, Tricomi Function, Trigono-
metric Functions, Uniform Apodization Func-
tion, Univalent Function, Vector Function,
Versine, von Mangoldt Function, V^-Function,
Walsh Function, Weber Functions, WeierstraB
Elliptic Function, WeierstraB Function, Weier-
straB Sigma Function, WeierstraB Zeta Func-
tion, Weighting Function, Welch Apodization
Function Field
Fundamental Discriminant
685
Function, Whittaker Function, Wiener Func-
tion, Window Function, Xi Function, Zeta Func-
tion
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Miscellaneous
Functions." Ch. 27 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 997-1010, 1972.
Arfken, G. "Special Functions." Ch. 13 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 712-759, 1985.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Special Functions." Ch. 6 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 205-265, 1992.
Function Field
see Algebraic Function Field
Function Space
/(/) is the collection of all real- valued continuous func-
tions defined on some interval /. p n ^ (/) is the collection
of all functions £ f(I) with continuous nth Deriva-
tives. A function space is a TOPOLOGICAL VECTOR
SPACE whose "points" are functions.
see also Functional, Functional Analysis, Oper-
ator
Functional
A mapping between FUNCTION SPACES if the range is
on the Real Line or in the Complex Plane.
see also Coercive Functional, Current, Ellip-
tic Functional, Generalized Function, Lax-
Milgram Theorem, Operator, Riesz Representa-
tion Theorem
Functional Analysis
A branch of mathematics concerned with infinite dimen-
sional spaces (mainly FUNCTION SPACES) and mappings
between them-. The SPACES may be of different, and pos-
sibly Infinite, Dimensions. These mappings are called
Operators or, if the range is on the Real line or in
the Complex Plane, Functionals.
see also FUNCTIONAL, OPERATOR
References
Balakrishnan, A. V. Applied Functional Analysis, 2nd ed.
New York: Springer- Verlag, 1981.
Berezansky, Y. M.; Us, G. F.; and Sheftel, Z. G. Functional
Analysis, Vol. 1. Boston, MA: Birkhauser, 1996.
Berezansky, Y. M.; Us, G. F.; and Sheftel, Z. G. Functional
Analysis, Vol. 2. Boston, MA: Birkhauser, 1996.
Birkhoff, G. and Kreyszig, E. "The Establishment of Func-
tional Analysis." Historia Math. 11, 258-321, 1984.
Hutson, V. and Pym, J. S. Applications of Functional Anal-
ysis and Operator Theory. New York: Academic Press,
1980.
Kreyszig, E. Introductory Functional Analysis with Applica-
tions. New York: Wiley, 1989.
Yoshida, K. Functional Analysis and Its Applications. New
York: Springer- Verlag, 1971.
Zeidler, E. Nonlinear Functional Analysis and Its Applica-
tions. New York: Springer- Verlag, 1989,
Zeidler, E. Applied Functional Analysis: Applications to
Mathematical Physics. New York: Springer- Verlag, 1995.
Functional Calculus
An early name for CALCULUS OF VARIATIONS.
Functional Derivative
A generalization of the concept of the DERIVATIVE to
Generalized Functions.
Functor
A function between CATEGORIES which maps objects to
objects and MORPHISMS to MORPHISMS. Functors exist
in both covariant and contravariant types.
see also Category, Eilenberg-Steenrod Axioms,
MORPHISM, SCHUR FUNCTOR
Fundamental Class
The canonical generator of the nonvanishing HOMO-
LOGY Group on a Topological Manifold.
see also CHERN NUMBER, PONTRYAGIN NUMBER,
Stiefel- Whitney Number
Fundamental Continuity Theorem
Given two POLYNOMIALS of the same order in one vari-
able where the first p COEFFICIENTS (but not the first
p — 1) are and the COEFFICIENTS of the second ap-
proach the corresponding COEFFICIENTS of the first as
limits, then the second Polynomial will have exactly p
roots that increase indefinitely. Furthermore, exactly k
Roots of the second will approach each Root of mul-
tiplicity k of the first as a limit.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 4, 1959.
Fundamental Discriminant
—D is a fundamental discriminant if D is a POSITIVE
Integer which is not Divisible by any square of an
Odd Prime and which satisfies D = 3 (mod 4) or D =
4,8 (mod 16).
see also DISCRIMINANT
References
Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal-
ity Proving." Math. Comput. 61, 29-68, 1993.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, p. 294, 1987.
Cohn, H. Advanced Number Theory. New York: Dover, 1980.
Dickson, L. E. History of the Theory of Numbers, Vols. 1-3.
New York: Chelsea, 1952.
686
Fundamental Forms
Fundamental Forms
Fundamental Forms
There are three types of so-called fundamental forms.
The most important are the first and second (since the
third can be expressed in terms of these). The fun-
damental forms are extremely important and useful in
determining the metric properties of a surface, such
as Line Element, Area Element, Normal Curva-
ture, Gaussian Curvature, and Mean Curvature.
Let M be a REGULAR SURFACE with v p ,w p points on
the Tangent Space M p of M. Then the first funda-
mental form is the Inner Product of tangent vectors,
I(Vp,Wp)=Vp-W p .
(1)
For MeM 3 , the second fundamental form is the sym-
metric bilinear form on the TANGENT SPACE M p ,
II(vp,Wp) = £(vp).Wp
(2)
where S is the SHAPE OPERATOR. The third fundamen-
tal form is given by
III(v PJ w p ) = S(v p ).S(w p ).
The first and second fundamental forms satisfy
(3)
I(ox„ + 6x„, ax„ + fcx„) = Eo 2 + 2Fob + Gb 2 (4)
II(ax„ + 6x„,ax„ + 6x„) = ea 2 + 2 fab + gb , (5)
and so their ratio is simply the Normal Curvature
II(v p )
k(v p ) =
I(v P )
(6)
for any nonzero TANGENT VECTOR. The third funda-
mental form is given in terms of the first and second
forms by
in -2ini + in = o, (7)
where H is the MEAN CURVATURE and K is the GAUS-
SIAN Curvature.
The first fundamental form (or LINE ELEMENT) is given
explicitly by the RlEMANNIAN METRIC
ds 2 = E du + 2F dudv + G dv 2
(8)
It determines the ARC LENGTH of a curve on a surface.
The coefficients are given by
(9)
(10)
(11)
ht — x uu —
du
i
dx dx
r — Xu-u — — * —
OU ov
G = X-y-y =
ax
dv
2
The coefficients are also denoted g uu = E, 9uv = F,
and g vv = G. In CURVILINEAR COORDINATES (where
F = 0), the quantities
h u = y/^Z = VE (12)
K = y/g^ = Vg (13)
are called Scale Factors.
The second fundamental form is given explicitly by
where
e du 2 -h 2/ du dv + g dv
i
i
dudv
d 2 x t
dv 2 '
(14)
(15)
(16)
(17)
and Xj are the Direction Cosines of the surface nor-
mal. The second fundamental form can also be written
e = -N„
* X-u — IN * X<uu
(18)
/=-N,
' X-ji — -L ^ * J\.iiv —
N v „
* x vu
= -N.
•x v
(19)
g = -N v
* X-y — IN * yiyy i
(20)
where N is the NORMAL VECTOR, or
QetJ^Xu-uXiiXi) j
€ ~~ y/EG-F 2
QGtlXyyX'uX'y J
f =
9 =
x/EG - F 2
U6t^XuDX u Xu j
VEG-F 2 ''
(21)
(22)
(23)
see also Arc Length, Area Element, Gaussian
Curvature, Geodesic, Kahler Manifold, Line of
Curvature, Line Element, Mean Curvature, Nor-
mal Curvature, Riemannian Metric, Scale Fac-
tor, Weingarten Equations
References
Gray, A. "The Three Fundamental Forms." §14.6 in Modern
Differential Geometry of Curves and Surfaces, Boca Ra-
ton, FL: CRC Press, pp. 251-255, 259-260, 275-276, and
282-291, 1993.
Fundamental Group
Fundamental Theorem of Curves 687
Fundamental Group
The fundamental group of a Connected Set S is the
Quotient Group of the Group of all paths with initial
and final points at a given point P and the SUBGROUP of
all paths HOMOTOPIC to the degenerate path consisting
of the point P.
The fundamental group of the Circle is the Infinite
CYCLIC GROUP. Two fundamental groups having dif-
ferent points P are ISOMORPHIC. If the fundamental
group consists only of the identity element, then the set
S is simply connected.
see also Milnor's Theorem
Fundamental Homology Class
see also FUNDAMENTAL CLASS
Fundamental Lemma of Calculus of
Variations
If
I
J a
M(x)h(x)dx =
V h(x) with Continuous second Partial Deriva-
tives, then
M{x) =
on the Open Interval (a, 6).
Fundamental Theorem of Arithmetic
Any POSITIVE INTEGER can be represented in exactly
one way as a PRODUCT of PRIMES. The theorem is
also called the UNIQUE FACTORIZATION THEOREM. The
fundamental theorem of algebra is a COROLLARY of the
first of Euclid's Theorems (Hardy and Wright 1979).
see also EUCLID'S THEOREMS, INTEGER, PRIME NUM-
BER
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. 23, 1996.
Hardy, G. H. and Wright, E, M. "Statement of the Funda-
mental Theorem of Arithmetic," "Proof of the Fundamen-
tal Theorem of Arithmetic," and "Another Proof of teh
Fundamental Theorem of Arithmetic." §1.3, 2.10 and 2.11
in An Introduction to the Theory of Numbers, 5th ed. Ox-
ford, England: Clarendon Press, pp. 3 and 21, 1979.
Fundamental Theorems of Calculus
The first fundamental theorem of calculus states that,
if / is Continuous on the Closed Interval [a, b] and
F is the Antiderivative (Indefinite Integral) of /
on [a, 6], then
J a
fix) dx = F(b) - F(a).
(1)
Fundamental System
A set of Algebraic Invariants for a Quantic such
that any invariant of the QUANTIC is expressible as a
POLYNOMIAL in members of the set. In 1868, Gordan
proved the existence of finite fundamental systems of al-
gebraic invariants and covariants for any binary Qu AN-
TIC. In 1890, Hilbert (1890) proved the Hilbert Basis
Theorem, which is a finiteness theorem for the related
concept of Syzygies.
see also Hilbert Basis Theorem, Syzygy
References
Hilbert, D. "Uber die Theorie der algebraischen Formen."
Math. Ann. 36, 473-534, 1890.
Fundamental Theorem of Algebra
Every POLYNOMIAL equation having Complex Coef-
ficients and degree > 1 has at least one Complex
Root. This theorem was first proven by Gauss. It is
equivalent to the statement that a Polynomial P(z)
of degree n has n values of z (some of them possi-
bly degenerate) for which P(z) = 0. An example of
a Polynomial with a single Root of multiplicity > 1
is z 2 — 2z + 1 = (z — l)(z — 1), which has z = 1 as a
ROOT of multiplicity 2.
see also Degenerate, Polynomial
References
Courant, R. and Robbins, H. "The Fundamental Theorem
of Algebra." §2.5.4 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 101-103, 1996.
The second fundamental theorem of calculus lets / be
Continuous on an Open Interval 7 and lets a be any
point in J. If F is defined by
then
F(x)= f f(t)dt,
J a
F'(x) = f{x)
(2)
(3)
at each point in I.
The complex fundamental theorem of calculus states
that if f(z) has a Continuous Antiderivative F(z) in
a region R containing a parameterized curve 7:2 = z(t)
for a < t < /?, then
t/'Y
fiz)dz = Fizil3))-F{zia)).
(4)
see also CALCULUS, DEFINITE INTEGRAL, INDEFINITE
Integral, Integral
Fundamental Theorem of Curves
The Curvature and Torsion functions along a Space
Curve determine it up to an orientation-preserving
Isometry.
688 Fundamental Theorem of Directly Similar Figures
Fundamental Unit
Fundamental Theorem of Directly Similar
Figures
Let Fo and F\ denote two directly similar figures in the
plane, where P\ 6E Fi corresponds to Pq 6 Fo under
the given similarity. Let r £ (0,1), and define F r =
{(1 - r)P + rPi : Fo £ F }. Then F r is also directly
similar to Fo.
see also FlNSLER-HADWIGER THEOREM
References
Detemple, D, and Harold, S. "A Round-Up of Square Prob-
lems." Math. Mag. 69, 15-27, 1996.
Eves, H. Solution to Problem E521. Airier. Math. Monthly
50, 64, 1943.
Fundamental Theorem of Gaussian
Quadrature
The Abscissas of the N point Gaussian Quadrature
Formula are precisely the ROOTS of the ORTHOGONAL
Polynomial for the same Interval and Weighting
Function.
see also Gaussian Quadrature
Fundamental Theorem of Genera
rjwfd)-
X-d)l,
where u(d) is the genus of forms and h(—d) is the CLASS
Number of an Imaginary Quadratic Field.
References
Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary
Quadratic Fields with Small Odd Class Number." http: //
www.math.uiuc.edu/Algebraic-Number-Theory/0009/.
Cohn, H. Advanced Number Theory. New York: Dover,
p. 224, 1980.
Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT:
Yale University Press, 1966.
Fundamental Theorem of Plane Curves
Two unit-speed plane curves which have the same Cur-
vature differ only by a EUCLIDEAN MOTION.
see also Fundamental Theorem of Space Curves
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 103 and 110-111,
1993.
Fundamental Theorem of Space Curves
If two single-valued continuous functions k(s) (CURVA-
TURE) and t(s) (Torsion) are given for s > 0, then
there exists EXACTLY One SPACE CURVE, determined
except for orientation and position in space (i.e., up to
a Euclidean Motion), where s is the Arc Length,
k is the Curvature, and r is the Torsion.
see also Arc Length, Curvature, Euclidean Mo-
tion, Fundamental Theorem of Plane Curves,
Torsion (Differential Geometry)
References
Gray, A. "The Fundamental Theorem of Space Curves." §7.7
in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 123 and 142-145, 1993.
Struik, D. J. Lectures on Classical Differential Geometry.
New York: Dover, p. 29, 1988.
Fundamental Theorem of Symmetric
Functions
Any symmetric polynomial (respectively, symmetric ra-
tional function) can be expressed as a POLYNOMIAL (re-
spectively, Rational Function) in the Elementary
Symmetric Functions on those variables.
see also Elementary Symmetric Function
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 2, 1959.
Herstein, I. N. Noncommutative Rings. Washington, DC:
Math. Assoc. Amer., 1968.
Fundamental Unit
In a real Quadratic Field, there exists a special Unit
r} known as the fundamental unit such that all units p
are given by p — ±77"% for m = 0, ±1, ±2, The
following table gives the fundamental units for the first
few real quadratic fields.
Fundamental Theorem of Projective
Geometry
A PROJECTIVITY is determined when three points of one
RANGE and the corresponding three points of the other
are given.
see also Projective Geometry
Funnel
FWHM 689
d
V(d)
d
V(d)
2
I + a/2
51
50 + 7^51
3
2 + v/3
53
§(7 + ^53)
5
£(1 + V5)
55
89+12\/55
6
5 + 2v/6
57
151 + 20^
7
8 + 3\/7
58
99 + 13\/58
10
3 + vTo
59
530 + 69\/59
11
10 + 3VTI
61
|(39 + 5\/61)
13
1(3 + ^)
62
63 + 8\/62
14
15 + 4 V / 14
65
8 + v^
15
4 + ^
66
65 + 8^/66
17
4+x/Tf
67
48842 + 5967\/67
19
170 + 39\/l9
69
|(25 + 3a/69)
21
f(5 + v / 21)
70
251 + 30^70
22
197 + 42^
71
3480 + 413^
23
24 + 5\/2^
73
1068 + 125^
26
5 + \/26
74
43 + 5a/74
29
f(5 + v^9)
77
I(9 + x/77)
30
11 + 2^30
78
53 + 6\/78
31
1520 + 273\/3l
79
80 + 9\/79
33
5 + 4\/33
82
9 + \/82
34
35 + 6^34
83
82 + 9^
35
6 + \/3o~
85
K9 + X/85)
37
6 + \/37
86
10405 + 1122v/86
38
37 + 6^
87
28 + 3N/87
39
25 + 4v"39
89
501 + 54\/89
41
32 + 5v/41
91
1574 + 165\/91
42
13 + 2V/42
93
|(29 + 3v^3)
43
3482 + 531\/43
95
39 + 4^
46
24335 + 3588\/46
97
5604 + 569\/97
47
48 + 7v/47
and the parametric equations
see also Quadratic Field, Unit
References
Cohn, H. "Fundamental Units" and "Construction of Funda-
mental Units." §6.4 and 6.5 in Advanced Number Theory.
New York: Dover, pp. 98-102, and 261-274, 1980.
$ Weisstein, E. W. "Class Numbers." http: //www. astro.
Virginia. edu/~eww6n/math/notebooks/ClassNumbers.m.
Funnel
x(r,9) = rcos9
y(r, 0) — r sinO
z(r,6) = lnr.
(2)
(3)
(4)
see also GABRIEL'S HORN, PSEUDOSPHERE, SINCLAIR'S
Soap Film Problem
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 325-327, 1993.
Fuss's Problem
see Bicentric Polygon
Futile Game
A GAME which permits a draw ("tie") when played
properly by both players.
Fuzzy Logic
An extension of two- valued LOGIC such that statements
need not be True or False, but may have a degree of
truth between and 1. Such a system can be extremely
useful in designing control logic for real-world systems
such as elevators.
see also Alethic, False, Logic, True
References
McNeill, D. and Freiberger, P. Fuzzy Logic. New York: Si-
mon & Schuster, 1993.
Nguyen, H. T. and Walker, E. A. A First Course in Fuzzy
Logic. Boca Raton, FL: CRC Press, 1996.
Yager, R. R. and Zadeh, L. A. (Eds.) An Introduction to
Fuzzy Logic Applications in Intelligent Systems. Boston,
MA: Kluwer, 1992.
Zadeh, L. and Kacprzyk, J. (Eds.). Fuzzy Logic for the Man-
agement of Uncertainty. New York: Wiley, 1992.
FWHM
see Full Width at Half Maximum
The funnel surface is a REGULAR SURFACE defined by
the Cartesian equation
z= |ln(x 2 +y 2 )
(1)
g-Function
G
g- Function
see RAMANUJAN g- AND G-FUNCTIONS
0.95
Defined in Whittaker and Watson (1990, p. 264) and
also called the Barnes G-Function.
G{z + 1} = (27rr /2 e -[,(, + D + ^ 2 ]/2
n[(> + i)".
-z + z 2 /(2n)
■ (1)
where 7 is the Euler-Mascheroni Constant. This is
an Analytic Continuation of the G function defined
in the construction of the Glaisher-Kinkelin Con-
stant
G(n+1) =
K{n + lY
(2)
which has the special values
G(n)
if n = 0,-1,-2,...
1 if n = 1 (3)
0!l!2!-.'(n-2)! if n = 2,3,4,.. .
for INTEGER n. This function is what Sloane and
Plouffe (1995) call the SUPERFACTORIAL, and the first
few values for n = 1, 2, ... are 1, 1, 1, 2, 12, 288,
34560, 24883200, 125411328000, 5056584744960000, . . .
(Sloane's A000178).
The G-function is the reciprocal of the Double Gamma
Function. It satisfies
G(z+l) = r(z)G(z)
(n!) n
G(n + 1)
l x -2 2 -3 3 -
(4)
(5)
G-Space
691
g'{z + \) 1 ( . x _r>(*)
(6)
In
^ Z l\ = f nzcot(nz)dz zln(27r)
[G{l + z)\ J
(7)
has the special values
G(i) = A-V'ir-VVW*
(8)
G(l) = 1,
(9)
where
: exp
C'(2) ln(27r) 7
2n 2 12 2
= 1.28242713 . . .
(10)
The G-function can arise in spectral functions in math-
ematical physics (Voros 1987).
An unrelated pair of functions are denoted g n and G n
and are known as RAMANUJAN g- AND G-FUNCTIONS.
see also Euler-Mascheroni Constant, Glaisher-
Kinkelin Constant, ^-Function, Meijer's G-
FUNCTION, RAMANUJAN g- AND G-FUNCTIONS, SUPER-
FACTORIAL
References
Barnes, E. W. "The Theory of the G-Function." Quart J.
Pure Appl. Math. 31, 264-314, 1900.
Glaisher, J. W. L. "On a Numerical Continued Product."
Messenger Math. 6, 71-76, 1877.
Glaisher, J. W. L. "On the Product 1 1 2 2 3 3 • -n n ." Messen-
ger Math. 7, 43-47, 1878.
Glaisher, J. W. L. "On Certain Numerical Products." Mes-
senger Math. 23, 145-175, 1893.
Glaisher, J. W. L. "On the Constant which Occurs in the
Formula for 1 1 2 2 3 3 • * • n n ." Messenger Math. 24, 1-16,
1894.
Kinkelin. "Uber eine mit der Gammafunktion verwandte
Transcendente und deren Anwendung auf die Integralrech-
nung." J. Reine Angew. Math. 57, 122-158, 1860.
Sloane, N. J. A. Sequence A000178/M2049 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Voros, A. "Spectral Functions, Special Functions and the Sel-
berg Zeta Function." Commun. Math. Phys. 110, 439-
465, 1987.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, ^.th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
G-Number
see Eisenstein Integer
G- Space
A G-space is a special type of HAUSDORFF Space. Con-
sider a point x and a HOMEOMORPHISM of an open
Neighborhood V of x onto an Open Set of W 1 . Then
a space is a G-space if, for any two such NEIGHBOR-
HOODS V' and V", the images of V U V" under the
different HOMEOMORPHISMS are ISOMETRIC If n = 2,
the HOMEOMORPHISMS need only be conformal (but not
necessarily orientation-preserving) .
see also Green Space
692 Gabriel's Horn
Gabriel's Horn
The Surface of Revolution of the function y = 1/x
about the x-axis for x > 1. It has FINITE VOLUME
V
poo /»oo j
= t[-J]~=*[0-(-1)] = *,
but Infinite Surface Area, since
f l + y ,2 dx
/oo
f°° f°° dx
> 2?r / ydx = 27r — = 27r[lnz]f >
= 27r[lnoo — 0] = oo.
This leads to the paradoxical consequence that while
Gabriel's horn can be filled up with 7r cubic units of
paint, an INFINITE number of square units of paint are
needed to cover its surface!
see also FUNNEL, PSEUDOSPHERE
Gabriel's Staircase
The Sum
E fcrfc = 7T
valid for < r < 1.
(l-r)*'
Gadget
A term of endearment used by Algebraic Topolo-
GISTS when talking about their favorite power tools such
as Abelian Groups, Bundles, Homology Groups,
Homotopy Groups, ^-Theory, Morse Theory, Ob-
structions, stable homotopy theory, VECTOR SPACES,
etc.
see also Abelian Group, Algebraic Topology,
Bundle, Free, Homology Group, Homotopy
Group, A;-Theory, Obstruction, Morse Theory,
Vector Space
Gallows
Gale-Ryser Theorem
Let p and q be PARTITIONS of a POSITIVE INTEGER,
then there exists a (0,l)-matrix A such that c(A) = p,
r(A) = q IFF q is dominated by p* .
References
Brualdi, R. and Ryser, H. J. §6.2.4 in Combinatorial Matrix
Theory. New York: Cambridge University Press, 1991.
Krause, M. "A Simple Proof of the Gale-Ryser Theorem."
Amer. Math. Monthly 103, 335-337, 1996.
Robinson, G. §1.4 in The Representation Theory of the Sym-
metric Group. Toronto, Canada: University of Toronto
Press, 1961.
Ryser, H. J, "The Class A(R, S)." Combinatorial Mathemat-
ics. Buffalo, NY: Math. Assoc. Amer., pp. 61-65, 1963.
Galilean Transformation
A transformation from one reference frame to another
moving with a constant VELOCITY v with respect to
the first for classical motion. However, special relativ-
ity shows that the transformation must be modified to
the Lorentz Transformation for relativistic motion.
The forward Galilean transformation is
and the inverse transformation is
1
0]
"n
~v
1
X
1
y
1_
_z _
~t~
X
y
_z_
1 01
[t'l
v 1
x'
10
v'
1.
lyJ
see also Lorentz Transformation
Gall's Stereographic Projection
A Cylindrical Projection which projects the equa-
tor onto a tangent cylinder which intersects the globe at
± 45° . The transformation equations are
x = A
y = t&n{\4>),
where A is the LONGITUDE and <j> the LATITUDE.
see also STEREOGRAPHIC PROJECTION
References
Dana, P. H. "Map Projections." http://www.utexas.edu/
depts/grg/gcraft/notes/mapproj/mapproj.html.
Gallows
Schroeder (1991) calls the CEILING FUNCTION symbols
[~ and "] the "gallows" because of their similarity in ap-
pearance to the structure used for hangings.
see also CEILING FUNCTION
References
Schroeder, M. Fractals, Chaos, Power Laws: Minutes from
an Infinite Paradise. New York: W. H. Freeman, p. 57,
1991.
Gallucci's Theorem
Game Expectation 693
Gallucci's Theorem
If three SKEW LINES all meet three other Skew Lines,
any Transversal to the first set of three meets any
Transversal to the second set of three.
see also Skew Lines, Transversal Line
Galoisian
An algebraic extension E of F for which every Irre-
ducible Polynomial in F which has a single Root in
E has all its ROOTS in E is said to be Galoisian. Ga-
loisian extensions are also called algebraically normal.
Galois Extension Field
The splitting Field for a separable Polynomial over a
Finite Field K, where L is a Field Extension of K,
Galois Field
see Finite Field
Galois Group
Let L be a Field Extension of K, denoted L/K, and
let G be the set of Automorphisms of L/K, that is,
the set of AUTOMORPHISMS a of L such that o~(x) = x
for every x G K, so that K is fixed. Then G is a GROUP
of transformations of L, called the Galois group of L/K .
The Galois group of (C/M) consists of the Identity EL-
EMENT and Complex Conjugation. These functions
both take a given REAL to the same real.
see also Abhyankar's Conjecture, Finite Group,
Group
References
Jacobson, N. Basic Algebra I, 2nd ed. New York: W.
Freeman, p. 234, 1985.
H.
Galois Imaginary
A mathematical object invented to solve irreducible
Congruences of the form
F(x) = (mod p) ,
where p is PRIME.
Galois's Theorem
An algebraic equation is algebraically solvable IFF its
Group is SOLVABLE. In order that an irreducible equa-
tion of Prime degree be solvable by radicals, it is Nec-
essary and Sufficient that all its Roots be rational
functions of two ROOTS.
see also Abel's Impossibility Theorem, Solvable
Group
Galois Theory
If there exists a One-to-One correspondence between
two Subgroups and Subfields such that
G(E(G')) = G 1
E(G(E , )) = E , 1
then E is said to have a Galois theory.
Gambler's Ruin
Let two players each have a finite number of pennies
(say, m for player one and ri2 for player two) . Now, flip
one of the pennies (from either player), with each player
having 50% probability of winning, and give the penny
to the winner. If the process is repeated indefinitely, the
probability that one or the other player will eventually
lose all his pennies is unity. However, the chances that
the individual players will be rendered penniless are
Pi =
P 2 =
Til
rii + H2
ri2
n-i -h ri2
see also Coin Tossing, Martingale, Saint Peters-
burg Paradox
References
Cover, T. M. "Gambler's Ruin: A Random Walk on the Sim-
plex." §5.4 in In Open Problems in Communications and
Computation. (Ed. T. M. Cover and B. Gopinath). New
York: Springer- Verlag, p. 155, 1987.
Hajek, B. "Gambler's Ruin: A Random Walk on the Sim-
plex." §6.3 in In Open Problems in Communications and
Computation. (Ed. T. M. Cover and B. Gopinath). New
York: Springer- Verlag, pp. 204-207, 1987.
Kraitchik, M. "The Gambler's Ruin." §6.20 in Mathematical
Recreations. New York: W. W. Norton, p. 140, 1942.
Game
A game is defined as a conflict involving gains and losses
between two or more opponents who follow formal rules.
The study of games belongs to a branch of mathematics
known as Game Theory.
see also Game Theory
Game Expectation
Let the elements in a PAYOFF Matrix be denoted a,ij,
where the is are player A's Strategies and the js are
player B's STRATEGIES. Player A can get at least
mm aij
(i)
for STRATEGY i. Player B can force player A to get
no more than maxj< m a^ for a STRATEGY j. The best
Strategy for player A is therefore
mm mm a
tj)
and the best STRATEGY for player B is
min max a^ .
(2)
(3)
694 Game of Life
In general,
min min a^ < min max aij .
(4)
Equality holds only if a Saddle Point is present, in
which case the quantity is called the VALUE of the game.
see also Game, Payoff Matrix, Saddle Point
(Game), Strategy, Value
Game of Life
see Life
Game Matrix
see Payoff Matrix
Game Theory
A branch of Mathematics and Logic which deals with
the analysis of GAMES (i.e., situations in which parties
are involved in situations where their interests conflict).
In addition to the mathematical elegance and complete
"solution" which is possible for simple games, the prin-
ciples of game theory also find applications to compli-
cated games such as cards, checkers, and chess, as well
as real-world problems as diverse as economics, property
division, politics, and warfare.
see also Borel Determinacy Theorem, Cate-
gorical Game, Checkers, Chess, Decision The-
ory, Equilibrium Point, Finite Game, Futile
Game, Game Expectation, Go, Hi-Q, Impartial
Game, Mex, Minimax Theorem, Mixed Strat-
egy, Nash Equilibrium, Nash's Theorem, Nim,
Nim- Value, Partisan Game, Payoff Matrix, Peg
Solitaire, Perfect Information, Saddle Point
(Game), Safe, Sprague-Grundy Function, Strat-
egy, Tactix, Tit-for-Tat, Unsafe, Value, Wyth-
off's Game, Zero-Sum Game
References
Berlekamp, E. R.; Conway, J. H; and Guy, R. K. Winning
Ways, For Your Mathematical Plays, Vol. 1: Games in
General. London: Academic Press, 1982.
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.
Dresner, M. The Mathematics of Games of Strategy: Theory
and Applications. New York: Dover, 1981.
Eppstein, D. "Combinatorial Game Theory." http://wvw.
ics.uci.edu/-eppstein/cgt/.
Gardner, M. "Game Theory." Ch. 3 in Mathematical
Magic Show: More Puzzles, Games, Diversions, Illusions
and Other Mathematical Sleight- of- Mind from Scientific
American. New York: Vintage, 1978.
Karlin, S. Mathematical Methods and Theory in Games, Pro-
gramming, and Economics, 2 Vols. Vol. 1: Matrix Games,
Programming, and Mathematical Economics. Vol. 2: The
Theory of Infinite Games. New York: Dover, 1992.
Kuhn, H. W. (Ed.). Classics in Game Theory. Princeton,
NJ: Princeton University Press, 1997.
McKinsey, J. C. C Introduction to the Theory of Games.
New York: McGraw-Hill, 1952.
Gamma Distribution
Neumann, J. von and Morgenstern, O. Theory of Games and
Economic Behavior, 3rd ed. New York: Wiley, 1964,
Packel, E. The Mathematics of Games and Gambling. Wash-
ington, DC: Math. Assoc. Amer., 1981.
Straffin, P. D. Jr. Game Theory and Strategy. Washington,
DC: Math. Assoc. Amer., 1993.
Vajda, S. Mathematical Games and How to Play Them. New
York: Routledge, 1992.
Walker, P. "An Outline of the History of Game The-
ory." http : //william-king . www . drexel . edu/t op/class/
histf.html.
Williams, J. D. The Compleat Strategyst, Being a Primer on
the Theory of Games of Strategy. New York: Dover, 1986.
Gamma Distribution
A general type of statistical DISTRIBUTION which is re-
lated to the Beta Distribution and arises naturally in
processes for which the waiting times between POISSON
DISTRIBUTED events are relevant. Gamma distributions
have two free parameters, labeled a and 0, a few of which
are illustrated above.
Given a POISSON DISTRIBUTION with a rate of change A,
the Distribution Function D(x) giving the waiting
times until the hth change is
D(x) = P(X <x) = l- P{X > x)
->-£
= 1-e"
(Xx) k e' Xx
,^(A*) fc
(i)
for x > 0. The probability function P(x) is then ob-
tained by differentiating D(x),
P(x) = D'(x)
= Ae~
X "E
(Ax)
— e
~Xx
Ae x + Ae
fc=0
-A^(Ax) fc
y4 fc(Ax) fc - 1 A
^ fc!
= \e~ Xx - \e- Xx
E
h-l
E
fc!
k(Xx)
— e
. Xx ^k(\x) k
k=l
fc-1 /\«\fc
EK[AX)
fc=l
(Xx) k
k\
k\
Ae"- 1 - £
(Ax)*
(A*)*
= Ae" Ax i 1
1-
(fc-1)! k\
(Ax)*- 1 ]! _ XjXxf- 1
(ft-l)!j/- (A-l)! '
(2)
Gamma Distribution
Gamma Distribution 695
Now let a = h and define = 1/A to be the time between
changes. Then the above equation can be written
P(x)
< ,.-i t -.;»
\ r(c)e«
lo
< x < oo /3\
x < 0.
The Characteristic Function describing this distri-
bution is
4>{t) = (l - ity, (4)
and the Moment-Generating Function is
e tx x a - 1 e- x/e dx
M{t) = f
Jo
r(a)0 Q
00 a .—i e -(i-»0-/« dx
(5)
In order to find the Moments of the distribution, let
(l~0t)x
V =—9 —
dy = — — dx,
(6)
(7)
0*
(1 - 0t)« '
and the logarithmic Moment- Generating function is
R(t) = In M(t) = -q ln(l - 9t) (9)
a0
i?'(t) =
R"(t) =
i — <9e
afl 2
(1 - 0t) 2 "
(10)
(11)
The Mean, Variance, Skewness, and Kurtosis are
then
li = R'(Q) = a9
(12)
a 2 = ^(0) = a(9 2
(13)
2
7i = ~7=
(14)
6
72
a
(15)
The gamma distribution is closely related to other statis-
tical distributions. If Xi, X 2 , . . . , X n are independent
random variates with a gamma distribution having pa-
rameters (ai,0), (a 2 ,0), -.., (ot n ,9), then $^™ =1 ^ is
distributed as gamma with parameters
a = Y, ai
(16)
(17)
Also, if Xi and X2 are independent random variates
with a gamma distribution having parameters (ai , 0)
and (o 2 , 0), then X 1 /(X 1 +X 2 ) is a BETA DISTRIBUTION
variate with parameters (0:1,0:2). Both can be derived
as follows.
P(x,y) =
1
r(ai)r(a 2 )
e *i+** Xl "*-i X2 ~*-\
Let
v =
xi + # 2
then the JACOBIAN is
U = Xi + X2 Xi = UV
Xi
X2 — 1l(l — v),
V U
1 - V —u
= -w,
(18)
(19)
(20)
(21)
g(u,v) dudv = f(x,y)dxdy = f(x,y)ududv. (22)
(8) fl (u,t>)
r(ai)r(a 2 )
1
1 r(ai)r(a a )
«1-1, ."J-l/1 _„1 a 2- 1
e" u (u») ,,, " 1 B 0, - 1 (l-«)
-w, "1+02-1 ai-l/i \a 2 -l
v ai - l (l-v)
(23)
The sum X\ + X 2 therefore has the distribution
f(u) = f(xi+x 2 ) = / g(u,v)dv
r(a! + o 2 ) '
(24)
which is a gamma distribution, and the ratio Xi/(Xi +
X 2 ) has the distribution
h(v) = h( — ^ — )= / g{u 1 v)du
\x 1 +x 2 J J
B(ai,a2)
(25)
where B is the Beta Function, which is a Beta Dis-
tribution.
If X and Y are gamma variates with parameters a% and
o 2 , the X/Y is a variate with a BETA PRIME DISTRI-
BUTION with parameters a\ and a 2 . Let
u = x + y v = — ,
(26)
696 Gamma Distribution
then the Jacobian is
z + y _ (l + v) 2
1 1
1 X
v d
dxdy ■
(l + u) :
■ dudv
(27)
(28)
g{u,v) =
r(oi)r(oa)
Vl + W
\l + v) (1 + V
r(ai)r(a 2 )
(l + «) 2
(29)
The ratio X/Y therefore has the distribution
f°° v ai ~ l (l + v)~ ai ~ a2
h{v) = J g(u,v)du= ^-—^ ,
(30)
which is a Beta Prime Distribution with parameters
(ai,a 2 ).
The "standard form" of the gamma distribution is given
by letting y = x/9, so dy = dx/9 and
P(y) dy = %,.*„„ dx = [Uy J,„,Z (* d y)
y a - x e-* J
so the Moments about are
T(a)0 a
(31)
v r = — -, / ex dx = v = ^
(32)
where (a) r is the Pochhammer Symbol. The MO-
MENTS about ii — in are then
Mi = a
JJ>2 = a
^3 = 2a
/j,4 = 3a + 6a.
The Moment-Generating Function is
and the Cumulant-Generating Function is
K(t) = aln(l - t) = a (t + \t 2 + |t 3 + . . .) , (38)
so the Cumulants are
k t = aT{r). (39)
(33)
(34)
(35)
(36)
(37)
Gamma Function
If x is a NORMAL variate with MEAN ft and STANDARD
Deviation a, then
y =
2<7 2
(40)
is a standard gamma variate with parameter a = 1/2.
see also BETA DISTRIBUTION, CHI-SQUARED DISTRIBU-
TION
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 534, 1987.
Gamma Function
u
4 -2
A
2 4
Im [Gamma z]
| Gamma z |
The complete gamma function is defined to be an exten-
sion of the Factorial to Complex and Real Number
arguments. It is Analytic everywhere except at z = 0,
-1, -2, It can be defined as a Definite Integral
for R[z] > (Euler's integral form)
T(z)
r(*) =
/•oo
= / f-'e-'dt
Jo
poo
jfhfi)]'"'*
(i)
(2)
(3)
Integrating (1) by parts for a Real argument, it can
be seen that
poo
r(x) = / t'-Vdt
Jo
poo
=-[-t"- 1 e- t ]g° + / (x- l)t—V* dt
Jo
poo
= (x - 1) / ^"V eft = (x - l)T(x - 1).
Jo
(4)
Gamma Function
Gamma Function 697
If x is an INTEGER n = 1, 2, 3, . . . then
r(n) = (n - l)r(n - 1) = (n - l)(n - 2)F(n - 2)
= (n-l)(n-2)-..l = (n-1)!, (5)
so the gamma function reduces to the Factorial for a
Positive Integer argument.
Binet's Formula is
In
r(a) = (a-|)lna-a+|ln(27r) + 2 /
Jo
dz
L
(6)
for R[a] > (Whittaker and Watson 1990, p. 251). The
gamma function can also be defined by an INFINITE
PRODUCT form (Weierstrafi Form)
T(z)
CO
(7)
where 7 is the Euler-Mascheroni Constant. This
can be written
T(z) = - exp
£
(-'>'" x -
where
S! = 7
s k = C(fc)
(8)
(9)
(10)
for k > 2, where £ is the RlEMANN Zeta FUNCTION
(Finch). Taking the logarithm of both sides of (7),
00
(11)
Differentiating,
T'(z)
r'(z) = -r(z)
^-r + Ef-^- 1 )
(12)
(13)
(14)
= r(z)*(z) = r(zWo(z)
r'(i) = -r(i)-{i + 7 + [(i-i) + (i-i)
+ - + (^TT-D + -]}
= -(1 + 7-1) = -7 (15)
i> ) = -r<„){I +7 + [(^-i) + (^-l)
(rb-5) + -]}
+ 1
where $(z) is the Digamma Function and Vo(z) is
the POLYGAMMA FUNCTION, nth derivatives are given
in terms of the POLYGAMMA FUNCTIONS ip„, ip n -i, ■■■,
il>o-
The minimum value Xo of T(x) for REAL POSITIVE x =
xq is achieved when
r'(x ) = T{x )i>oM =
^o(a;o) = 0,
(17)
(18)
This can be solved numerically to give x = 1.46163. . .
(Sloane's A030169), which has CONTINUED FRAC-
TION [1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, ...]
(Sloane's A030170). At xo, T(xo) achieves the value
0.8856031944... (Sloane's A030171), which has Con-
tinued Fraction [0, 1, 7, 1, 2, 1, 6, 1, 1, ...] (Sloane's
A030172).
The Euler limit form is
1
r(*)
lim e^+Va+.-.+i/m-innOzj
n— j-oo J
m
!L»_n{('+0«-""}
n— 1
-jA [(• + ;)>=)'
T(z) = lim
l-23---n
(19)
(20)
z(z + l)(z + 2)---(z + n)
The Lanczos Approximation for z > is
r(z + 1) = (z + 7 + i)* +1/ V +7+1/2 ^
L z+1 z + 2 z+n J
The complete gamma function T(x) can be generalized
to the incomplete gamma function T(x,o) such that
T(x) = r(;c,0). The gamma function satisfies the re-
currence relations
T(l + z) = zr(z)
r(i - z) = -zr(-z).
Additional identities are
r(i)r(-x) = -
x sin(7rx)
r(x)r(i-x) = —
(22)
(23)
(24)
(25)
sin (7m;)
ln[r(aj+iy+l)] = ln(a; 2 + t/ 2 ) + itan" 1 (^\
+ ]n[T(x + iy)] (26)
|(^)!| 2 =
sinh(7rx)
(27)
= -(n-l)!^ + 7-E^ (16) l^ + ^^V^^yn^^^ 7 -^
698 Gamma Function
Gamma Function
For integral arguments, the first few values are 1, 1,
2, 6, 24, 120, 720, 5040, 40320, 362880, . . . (Sloane's
A000142). For half integral arguments,
r(|) = V5f
r(f) = |V5F
r(|) = fVSF.
In general, for m a Positive Integer m = 1, 2,
1 ■ 3 ■ 5 • • • (2m - 1)
(29)
(30)
(31)
r(|+m) =
-y/lT
(2m -1)!! r-
— i — — vV
/ -i \rncyrn
r ^-")= 1.3 ( 5^.(L-l) ^
(2m -1)!!
(2m - 1)
For »M = -|,
IH+*»)!| 2 = *
cosh(7ry)
(32)
(33)
(34)
Gamma functions of argument 2z can be expressed using
the Legendre Duplication Formula
V(2z) = (27v)~ 1/2 2 2z ' l/2 r(z)T(z + |). (35)
Gamma functions of argument 3z can be expressed using
a triplication FORMULA
r(3z) = (27r)- 1 3 3 ^- 1/2 r(z)r(z + §)r(* + f ). (36)
The general result is the GAUSS MULTIPLICATION FOR-
MULA
T(z)T{z+ i) • • • T{z+ 2±1) = (27r) (n - 1)/2 n 1/2 - Tlz r(n^.
(37)
The gamma function is also related to the RlEMANN
Zeta Function £ by
r (|) *-nw - r (ifi) ^-'"cd - .). (38)
Borwein and Zucker (1992) give a variety of identities
relating gamma functions to square roots and Elliptic
Integral Singular Values fc n , i.e., Moduli k n such
that
K'(k n )
K(k n )
= \/n,
(39)
where K(k) is a complete ELLIPTIC INT EGRAL OF THE
First Kind and K'(k) = K(k') = K(y/l^W) is the
complementary integral.
r(|) = 2 7/9 3- 1/12 7r 1/3 [K(fc 3 )] 1/3 (40)
r(i)-2 7 r 1/4 [K(A ;i )] 1/2 (41)
r(i) = 2- 1/3 3 1/2 7r- 1/2 [r(|)] 2 (42)
r(i)r(|) = (y/2- l) 1/2 2 13/ V /2 tf(fc 2 ) (43)
r(|)
r(i)
2(v^ + i) 1/2 7r- 1/4 [i<r(fc 1 )] 1/2
(44)
r(i) = 2- 1/4 3 3/8 (v^ + l) 1/2 7r- 1/2 r(i)r(i) (45)
i / 2 - i / 2 £iiZ
r(|)
(46)
(47)
r(^) = 2 1 / 4 3" 1/8 (^-i) 1/ V
ritSS! =4-3^ 4 (V3 + V2)7r- 1 ^(fc 1 ) (48)
1 U4M V24/
r !gjp!g! = 2 25 / 18 3 1/3 (v / 2 + l)*" 1 ' 8 ^*,)] 8 ' 8
(49)
r(£)r(&)r(£)r(£)
= 384(V2 + l)(\/3 - V2)(2 - \Z3)7r[Ar(fc 6 )] 2 (50)
r(i) = 2- 7 / 10 5 1/4 (v / 5 + i) 1/2 *-* /2 r(§)r(§) (51)
r(^) = 2- 3 / B (>/5-l) 1 r
1/2 £(i>
r(|)
(52)
^"^"j 1 ^ = 2 • 3 1/2 5 1/e sin(^7r)[r(i)] 2 (53)
r(i)r(i)T(i)
r(£)
= 2 2 .3 2/5 sin(i7r)sin(£7r)[r(i)] 2
(54)
r(
&)r(£)r(£)
r(£)
2 -3/2 3 -l/E
; 5 1/4 (v/5-
D 1/2 [r(I)] 2
sin(^Tr)
(55)
1 1 15 J
(56)
(57)
r ( 2 o) r (2o) -O-lcl/4/^/c . i\
r(&)r(&)
r(&)r(&)
= 2 4 / 5 (10-2v^) 1 / 2 7r- 1 sin(^^)sin(^)[r(|)] 2
(58)
r(&)r(&)
r(&)r(&)
Gamma Function
2 3 / 5 (10 + 2V5) 1 ' 2 *- 1 sin( ±k) sin(^7r)[r(§)] 2
(59)
= 160(VE ~ 2) 1/ \{K(k 5 )}\
A few curious identities include
8
n=l v '
V 5 a -l V
[r(|)] 4
16tt 2 3 2 - 1 5 2 7 2 - 1
r'(i) r'(i)
r(i) r(i)
21n2
(60)
(61)
(62)
(63)
(Magnus and Oberhettinger 1949, p. 1). Ramanujan
also gave a number of fascinating identities:
T 2 (n+1)
T(n + xi + l)r(n - xi + 1)
=n
1 +
(n + A:) 2
(64)
<f>(m,n)<f>(n i m i
r 3 (m + i)r 3 (n + i)
T(2m + n + l)r(2n + m + 1)
cosh[7r(m + n)\/3] — cos[7r(m — n)]
27r 2 (m 2 + ran + n 2 )
where
0(m, n)
OO r
■nHrS)'
, (65)
(66)
fc^i L J fc=i L
_ ^(jn) cosh(7m\/3) - cos(7rn)
r[|(»+i)]
2n+2 7r 3/2 r
(67)
(Berndt 1994).
The following Asymptotic Series is occasionally use-
ful in probability theory (e.g., the 1-D RANDOM Walk):
r(J+|) n t. ii
128J 2
21
1024J 3 32768J 4
+
...) (68)
(Graham et al. 1994). This series also gives a nice
asymptotic generalization of STIRLING NUMBERS OF
the First Kind to fractional values.
It has long been known that r(|)7T~ 1/4 is TRANSCEN-
DENTAL (Davis 1959), as is T(\) (Le Lionnais 1983), and
Chudnovsky has apparently recently proved that T(|)
is itself Transcendental.
Gamma Function 699
The upper incomplete gamma function is given by
/»oo
r(a,s)= / t a ~ 1 e" t dt = l-7(a,x), (69)
where 7 is the lower incomplete gamma function. For a
an Integer n
n ' 1 „-
r(n, x) = (n - l)\e~ x ^ |- = (n - l)\e~ x es n _i(x),
3=0
(70)
where es is the EXPONENTIAL SUM FUNCTION. The
lower incomplete gamma function is given by
<y(a 1 x) = T(a)-r(a 1 x)= / e - *^ -1
Jo
= a~ x a e~ x iFi(l\ 1 + a\x)
= a~ x a iFi(a; 1 + a; — x),
dt
(71)
where 1F1 (a; 6; x) is the CONFLUENT HYPERGEOMETRIC
FUNCTION OF THE FIRST KIND. For a an INTEGER n,
7(n, x) = (n - 1)! I 1 - e * ^ ~
= (n»l)![l-es„_i(a:)].
(72)
The function T(a, z) is denoted Gamma[a,z] and
the function 7(0, z) is denoted Gamma[a,0,z] in
Mathematical (Wolfram Research, Champaign, IL).
see also Digamma Function, Double Gamma Func-
tion, Fransen-Robinson Constant G-Function,
Gauss Multiplication Formula, Lambda Func-
tion, Legendre Duplication Formula, Mu Func-
tion, Nu Function, Pearson's Function, Poly-
gamma Function, Regularized Gamma Function,
Stirling's Series
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Gamma (Facto-
rial) Function" and "Incomplete Gamma Function." §6,1
and 6.5 in Handbook of Mathematical Functions with For-
mulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 255-258 and 260-263, 1972.
Arfken, G. "The Gamma Function (Factorial Function)."
Ch. 10 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 339-341 and 539-572,
1985.
Artin, E. The Gamma Function. New York: Holt, Rinehart,
and Winston, 1964.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 334-342, 1994.
Borwein, J. M. and Zucker, I. J. "Elliptic Integral Evalua-
tion of the Gamma Function at Rational Values of Small
Denominator." IMA J. Numerical Analysis 12, 519-526,
1992.
Davis, H. T. Tables of the Higher Mathematical Functions.
Bloomington, IN: Principia Press, 1933.
Davis, P. J. "Leonhard Euler's Integral: A Historical Profile
of the Gamma Function." Amer. Math. Monthly 66, 849-
869, 1959.
700
Gamma Group
Gauss's Backward Formula
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/fran/fran.html.
Graham, R. L.; Knuth, D. E.; and Patashnik, 0. Answer to
problem 9.60 in Concrete Mathematics: A Foundation for
Computer Science. Reading, MA: Addison- Wesley, 1994.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 46, 1983.
Magnus, W. and Oberhettinger, F. Formulas and Theorems
for the Special Functions of Mathematical Physics. New
York: Chelsea, 1949.
Nielsen, H. Die Gammafunktion. New York: Chelsea, 1965.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Gamma Function, Beta Function, Fac-
torials, Binomial Coefficients" and "Incomplete Gamma
Function, Error Function, Chi-Square Probability Func-
tion, Cumulative Poisson Function." §6.1 and 6.2 in Nu-
merical Recipes in FORTRAN: The Art of Scientific Com-
puting, 2nd ed. Cambridge, England: Cambridge Univer-
sity Press, pp. 206-209 and 209-214, 1992.
Sloane, N. J. A. Sequences A030169, A030170, A030171,
A030172, and A000142/M1675 in "An On-Line Version of
the Encyclopedia of Integer Sequences."
Spanier, J. and Oldham, K. B. "The Gamma Function T(x)"
and "The Incomplete Gamma 7(1/; x) and Related Func-
tions." Chs. 43 and 45 in An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 411-421 and 435-443, 1987.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, 4th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Gamma Group
The gamma group T is the set of all transformations w
of the form
, N at -\-b
w{t) = WTd>
where a, 6, c, and d are INTEGERS and ad — be = 1.
see also KLEIN'S ABSOLUTE INVARIANT, LAMBDA
Group, Theta Function
References
Borwein, J. M. and Borwein, P. B. Pi & the ACM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 127-132, 1987.
Gamma- Modular
see Modular Gamma Function
Gamma Statistic
where K r are CUMULANTS and a is the STANDARD De-
viation.
see also KURTOSIS, Skewness
Garage Door
see Astroid
Garding's Inequality
Gives a lower bound for the inner product (Lu, w), where
L is a linear elliptic REAL differential operator of order
m, and u has compact support.
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Garman-Kohlhagen Formula
V t = e- yT S t iV(di) - e- rr KN(d 2 ),
where N is the cumulative NORMAL DISTRIBUTION and
log(f) + (r-y±K)r
di,d2
CTy/r
If y = 0, this is the standard form of the Black-Scholes
formula.
see also BLACK-SCHOLES THEORY
References
Garman, M. B. and Kohlhagen, S. W. "Foreign Currency
Option Values." J. International Money and Finance 2,
231-237, 1983.
Price, J. F. "Optional Mathematics is Not Optional." Not.
Amer. Math. Soc. 43, 964-971, 1996.
Gate Function
Bracewell's term for the RECTANGLE FUNCTION.
References
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, 1965.
Gauche Conic
see Skew Conic
Gaullist Cross
A Cross also called the Cross of Lorraine or Patri-
archal Cross.
see also CROSS, DISSECTION
Gauss's Backward Formula
fp = fo+pb~-l/2-rG2$Q+G3°'-l/2 + Gl ^0+^5^-1/2 + * * • »
for p e [0, 1], where S is the CENTRAL DIFFERENCE and
G
f p + n\
Gauss-Bodenmiller Theorem
Gauss-Bonnet Theorem
701
where (£) is a Binomial Coefficient.
see also CENTRAL DIFFERENCE, GAUSS'S FORWARD
Formula
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 433, 1987.
Gauss-Bodenmiller Theorem
The Circles on the Diagonals of a Complete Quad-
rilateral as Diameters are Coaxal. Furthermore,
the ORTHOCENTERS of the four TRIANGLES of a COM-
PLETE Quadrilateral are Collinear on the Radi-
cal Axis of the Coaxal Circles.
see also COAXAL CIRCLES, COLLINEAR, COMPLETE
Quadrilateral, Diagonal (Polygon), Orthocen-
ter, Radical Axis
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 172, 1929.
Gauss-Bolyai-Lobachevsky Space
A non-Euclidean space with constant NEGATIVE GAUS-
SIAN Curvature.
see also Lobachevsky-Bolyai-Gauss Geometry,
Non-Euclidean Geometry
Gauss-Bonnet Formula
The Gauss-Bonnet formula has several formulations.
The simplest one expresses the total Gaussian Cur-
vature of an embedded triangle in terms of the total
Geodesic Curvature of the boundary and the Jump
Angles at the corners.
More specifically, if M is any 2-D Riemannian Mani-
fold (like a surface in 3-space) and if T is an embedded
triangle, then the Gauss-Bonnet formula states that the
integral over the whole triangle of the GAUSSIAN Cur-
vature with respect to Area is given by 27r minus the
sum of the Jump Angles minus the integral of the Geo-
desic Curvature over the whole of the boundary of the
triangle (with respect to Arc Length),
KdA = 2ir ~^2 a i- K 9
J Jt JdT
(1)
where K is the GAUSSIAN CURVATURE, dA is the AREA
measure, the ais are the JUMP Angles of dT, and n g
is the Geodesic Curvature of dT, with ds the Arc
Length measure.
The next most common formulation of the Gauss-
Bonnet formula is that for any compact, boundary less
2-D Riemannian Manifold, the integral of the Gaus-
sian Curvature over the entire Manifold with re-
spect to Area is 27r times the Euler CHARACTERISTIC
of the Manifold,
J Jm
KdA = 2nx(M).
(2)
This is somewhat surprising because the total GAUSSIAN
CURVATURE is differential-geometric in character, but
the Euler Characteristic is topological in character
and does not depend on differential geometry at all. So
if you distort the surface and change the curvature at
any location, regardless of how you do it, the same total
curvature is maintained.
Another way of looking at the Gauss-Bonnet theorem for
surfaces in 3-space is that the Gauss Map of the surface
has Degree given by half the Euler Characteristic
of the surface
// K dA = 2ttx(M) - ^2 a * - / K 9 ds >
JjM JdM
(3)
which works only for ORIENTABLE SURFACES. This
makes the Gauss-Bonnet theorem a simple consequence
of the POINCARE-HOPF INDEX THEOREM, which is a
nice way of looking at things if you're a topologist, but
not so nice for a differential geometer. This proof can
be found in Guillemin and Pollack (1974). Millman
and Parker (1977) give a standard differential-geometric
proof of the Gauss-Bonnet theorem, and Singer and
Thorpe (1996) give a Gauss's Theorema Egregium-
inspired proof which is entirely intrinsic, without any
reference to the ambient EUCLIDEAN Space.
A general Gauss-Bonnet formula that takes into account
both formulas can also be given. For any compact 2-D
Riemannian Manifold with corners, the integral of
the Gaussian Curvature over the 2-Manifold with
respect to Area is 2-rr times the EULER CHARACTERIS-
TIC of the Manifold minus the sum of the Jump An-
gles and the total GEODESIC CURVATURE of the bound-
ary.
References
Chavel, I. Riemannian Geometry: A Modern Introduction.
New York: Cambridge University Press, 1994.
Guillemin, V. and Pollack, A. Differential Topology. Engle-
wood Cliffs, NJ: Prentice- Hall, 1974.
Millman, R. S. and Parker, G. D. Elements of Differential
Geometry. Prentice-Hall, 1977.
Reckziegel, H. In Mathematical Models from the Collections
of Universities and Museums (Ed. G. Fischer). Braun-
schweig, Germany: Vieweg, p. 31, 1986.
Singer, I. M. and Thorpe, J. A. Lecture Notes on Elemen-
tary Topology and Geometry. New York: Springer- Verlag,
1996.
Gauss-Bonnet Theorem
see Gauss-Bonnet Formula
702 Gauss's Circle Problem
Gauss's Circle Problem
t # • •
• <- — m • • • • • ■ ■■< . • »
• : " • • •
Count the number of LATTICE POINTS N(r) inside the
boundary of a Circle of Radius t with center at the
origin. The exact solution is given by the SUM
N(r) = l + 4|rj +4^T Wr 2 - :
(1)
The first few values are 1, 5, 13, 29, 49, 81, 113, 149, . . .
(Sloane's A000328).
Gauss showed that
N(r) ^irr 2 + J5(r),
where
(2)
\E(r)\ < 2V2nr. (3)
Writing \E(r)\ < Cr 6 , the best bounds on are 1/2 <
< 46/73 « 0.630137 (Huxley 1990). The problem
has also been extended to CONICS and higher dimen-
sions. The limit 1/2 was obtained by Hardy and Landau
(1915), and the limit 46/73 improves previous values of
24/37 as 0.64864 (Cheng 1963) and 34/53 x 0.64150
(Vinogradov), and 7/11 & 0.63636.
see also CIRCLE LATTICE POINTS
References
Cheng, J. R. "The Lattice Points in a Circle." Sci. Sinica
12, 633-649, 1963.
Cilleruello, J. "The Distribution of Lattice Points on Circles."
J. Number Th. 43, 198-202, 1993.
Guy, R. K. "Gaun's Lattice Point Problem." §F1 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 240-2417, 1994.
Huxley, M. N. "Exponential Sums and Lattice Points." Proc.
London Math. Soc. 60, 471-502, 1990.
Huxley, M. N. "Corrigenda: 'Exponential Sums and Lattice
Points'." Proc. London Math. Soc. 66, 70, 1993.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 24, 1983.
Sloane, N. J. A. Sequence A000328/M3829 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
$ Weisstein, E. W. "Circle Lattice Points." http:// www .
astro . Virginia . edu/ ~eww6n/ math /notebooks /Circle
LatticePoints.m.
Gauss's Class Number Conjecture
In his monumental treatise Disquisitiones Arithmeticae,
Gauss conjectured that the Class Number h(-d) of
an Imaginary quadratic field with Discriminant -d
tends to infinity with d. A proof was finally given by
Heilbronn (1934), and Siegel (1936) showed that for any
e > 0, there exists a constant c e > such that
h{-d) > c e d 1/2 ~ e
Gauss's Class Number Problem
as d — > oo. However, these results were not effective
in actually determining the values for a given m of a
complete list of fundamental discriminants — d such that
h(-d) = m, a problem known as GAUSS'S CLASS Num-
ber Problem.
Goldfeld (1976) showed that if there exists a "Weil
curve" whose associated DlRICHLET L- SERIES has a zero
of at least third order at s = 1, then for any e > 0, there
exists an effectively computable constant c € such that
h(-d) >c E (lnd) l - e .
Gross and Zaiger (1983) showed that certain curves must
satisfy the condition of Goldfeld, and Goldfeld's proof
was simplified by Oesterle (1985).
see also Class Number, Gauss's Class Number
Problem, Heegner Number
References
Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imaginary
Quadratic Fields with Small Odd Class Number." http://
www . math . uiuc . edu/Algebraic-Number-Theory/0009/.
Bocherer, S. "Das GauS'sche Klassenzahlproblem." Mitt..
Math. Ges. Hamburg 11, 565-589, 1988.
Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT:
Yale University Press, 1966.
Goldfeld, D. M. "The Class Number of Quadratic Fields
and the Conjectures of Birch and Swinnerton-Dyer." Ann.
Scuola Norm. Sup. Pisa 3, 623-663, 1976.
Gross, B. and Zaiger, D. "Points de Heegner et derivees de
fonctions L." C. R. Acad. Sci. Paris 297, 85-87, 1983.
Heilbronn, H. "On the Class Number in Imaginary Quadratic
Fields." Quart. J. Math. Oxford Ser. 25, 150-160, 1934.
Oesterle, J. "Nombres de classes des corps quadratiques
imaginaires."„As*erigue 121-122, 309-323, 1985.
Siegel, C. L. "Uber die Klassenzahl quadratischer Zahlkor-
per." Acta. Arith. 1, 83-86, 1936.
Gauss's Class Number Problem
For a given m, determine a complete list of fundamen-
tal Discriminants — d such that the Class Number
is given by h(-d) = m. Heegner (1952) gave a solution
for m = 1, but it was not completely accepted due to a
number of apparent gaps. However, subsequent exam-
ination of Heegner's proof show it to be "essentially"
correct (Conway and Guy 1996). Conway and Guy
(1996) therefore call the nine values of n(—d) having
h(—d) = 1 where — d is the DISCRIMINANT correspond-
ing to a Quadratic Field a + by/^n (n = -1, -2, -3,
-7, -11, -19, -43, -67, and -163; Sloane's A003173)
the Heegner Numbers. The Heegner Numbers have
a number of fascinating properties.
Stark (1967) and Baker (1966) gave independent proofs
of the fact that only nine such numbers exist; both
proofs were accepted. Baker (1971) and Stark (1975)
subsequently and independently solved the generalized
class number problem completely for m = 2. Oesterle
(1985) solved the case m — 3, and Arno (1992) solved
the case m = 4. Wagner (1996) solve the cases n = 5, 6,
and 7. Arno et al. (1993) solved the problem for ODD
m satisfying 5 < m < 23.
Gauss's Constant
Gauss's Equation (Radius Derivatives) 703
see also CLASS NUMBER, GAUSS'S CLASS NUMBER
Conjecture, Heegner Number
References
Arno, S. "The Imaginary Quadratic Fields of Class Number
4." Acta Arith. 40, 321-334, 1992.
Arno, S.; Robinson, M. L.; and Wheeler, F. S. "Imag-
inary Quadratic Fields with Small Odd Class Num-
ber." Dec. 1993. http://www.math.uiuc.edu/Algebraic-
Number-Theory/0009/.
Baker, A. "Linear Forms in the Logarithms of Algebraic
Numbers. I." Mathematika 13, 204-216, 1966.
Baker, A. "Imaginary Quadratic Fields with Class Number
2." Ann. Math. 94, 139-152, 1971.
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.
Goldfeld, D. M. "Gauss' Class Number Problem for Imagi-
nary Quadratic Fields." Bull. Amer. Math. Soc. 13, 23-
37, 1985.
Heegner, K. "Diophantische Analysis und Modulfunktionen."
Math. Z. 56, 227-253, 1952.
Heilbronn, H. A. and Linfoot, E. H. "On the Imaginary Quad-
ratic Corpora of Class-Number One." Quart. J. Math.
(Oxford) 5, 293-301, 1934.
Lehmer, D. H. "On Imaginary Quadratic Fields whose Class
Number is Unity." Bull. Amer. Math. Soc, 39, 360, 1933.
Montgomery, H. and Weinberger, P. "Notes on Small Class
Numbers." Acta. Arith. 24, 529-542, 1974.
Oesterle, J. "Nombres de classes des corps quadratiques
imaginaires." Asterique 121-122, 309-323, 1985.
Oesterle, J. "Le probleme de Gauss sur le nombre de classes."
Enseign Math. 34, 43-67, 1988.
Serre, J.-R A = b 2 - 4ac." Math. Medley 13, 1-10, 1985.
Shanks, D. "On Gauss's Class Number Problems." Math.
Comput. 23, 151-163, 1969.
Sloane, N. J. A. Sequence A003173/M0827 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. "On Complex Quadratic Fields with Class Num-
ber Two." Math. Comput. 29, 289-302, 1975.
Wagner, C. "Class Number 5, 6, and 7," Math. Comput. 65,
785-800, 1996.
Gauss's Constant
The Reciprocal of the Arithmetic-Geometric
Mean of 1 and \/2,
(i)
(2)
(3)
— t=- = — l , dx
,V2) * Jo VT^*
= 2 r /2 <m_
n Jo y/l + si
sin 2 6
References
Finch, S. "Favorite Mathematical Constants." http://wvv.
mathsof t , com/ asolve/constant/gauss/gauss .html.
Gauss's Criterion
Let p be an Odd Prime and b a Positive Integer not
divisible by p. Then for each Positive Odd Integer
2k — 1 < p, let n be
r k = (2k - 1)6 (modp)
with < rjt < p, and let t be the number of EVEN r»s.
Then
(b/p) = (-I)',
where (b/p) is the LEGENDRE SYMBOL.
References
Shanks, D. "Gauss's Criterion." §1.17 in Solved and Unsolved
Problems in Number Theory, ^th ed. New York: Chelsea,
pp. 38-40, 1993.
Gauss's Double Point Theorem
If a sequence of Double Points is passed as a Closed
CURVE is traversed, each DOUBLE POINT appears once
in an Even place and once in an ODD place.
References
Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 61-66,
1957.
Gauss Equations
If x is a regular patch on a REGULAR SURFACE in M 3
with normal N, then
1 ?
x uu
: rnX u + ruX v +eN
= r} 2 x u + r? 2 x v + /n
X vv = T^Xu + TJ 2 ^-v + ffN,
(1)
(2)
(3)
r[r(i)] 2
(2tt) 3 /2 l V4
0.83462684167..., (4)
where e, /, and g are coefficients of the second Funda-
mental Form and r*- are Christoffel Symbols of
the Second Kind.
see also Christoffel Symbol of the Second Kind,
Fundamental Forms, Mainardi-Codazzi Equa-
tions
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 398-400, 1993,
Gauss's Equation (Radius Derivatives)
Expresses the second derivatives of r in terms of the
Christoffel Symbol of the Second Kind.
where K(k) is the complete Elliptic Integral of the
First Kind and T(z) is the Gamma Function.
see also Arithmetic-Geometric Mean, Gauss-
Kuzmin-Wirsing Constant
r »j = rfj-r* + (Tij ■ n)n.
704
Gauss's Formula
Gauss-Jacobi Mechanical Quadrature
Gauss's Formula
where R and S are Homogeneous Polynomials in x
and y with integral COEFFICIENTS.
see also AURIFEUILLEAN FACTORIZATION, GAUSS ? S
Backward Formula, Gauss's Forward Formula
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 105, 1993.
Gauss's Formulas
Let a Spherical Triangle have sides a, 6, and c with
A, B, and C the corresponding opposite angles. Then
8in[§(q-6)] = B in[j(A-B)]
sin(ic) cos(|C)
sm[±(a + b)} = cos[±(A-B)}
sin(|c) sin(|C)
cos[±(a-b)] = Bm[±{A + B)]
cos(|c) cos(|C)
cos[^(a + 6)] cos[|(i4 + J B)]
cos(|c)
sin(|C)
(1)
(2)
(3)
(4)
see also SPHERICAL TRIGONOMETRY
Gauss's Forward Formula
fp = /o + P^l/2 + ^2^0 + ^3^1/2 + G4S0 + ^5^1/2 + • • • »
for p G [0, 1], where S is the Central Difference and
p + n — I s
C?2n =
^2n+l ~
2n
p + n
2n+lJ'
where (") is a BINOMIAL COEFFICIENT.
see also Central Difference, Gauss's Backward
Formula
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 433, 1987.
Gauss's Harmonic Function Theorem
If a function (f> is HARMONIC in a SPHERE, then the value
of <f> at the center of the Sphere is the Arithmetic
Mean of its value on the surface.
Gauss's Hypergeometric Theorem
for 3R[c - a - b] > 0, where 2^1 ( a > 6; c; x) is a HYPERGE-
OMETRIC Function. If a is a Negative Integer — n,
this becomes
2 ,F , i(-n,6;c;l) =
(C - b) n
which is known as the Vandermonde Theorem.
see also Generalized Hypergeometric Function,
Hypergeometric Function
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, pp. 42 and 126, 1996.
Gauss's Inequality
If a distribution has a single MODE at p,Q, then
P(|x- M o|>Ar)<^,
where
2 _ 2 . / \2
T —a + (ji - fJL ) .
Gauss's Interpolation Formula
2n
f(x)^t n (x) = J2f^k(x),
k=0
where t n {x) is a trigonometric POLYNOMIAL of degree n
such that t n (xk) = fk for k = 0, . . . , 2n, and
aw =
sin[|(a; - x )] • - sin[f (s - x k ~i)}
sin[|(x fc -a*)] •••sin[§(z fc - x k -i)]
sin[i(x - Xfc+i)] • • *sin[|(x - rc 2 n)]
sin[i(x fc - ajfe+i)] ■ ■ -sin[|(a;fc - x 2n )]
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 881, 1972.
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, pp. 442-443, 1987.
Gauss-Jacobi Mechanical Quadrature
If xi < X2 < . . . < x n denote the zeros of p n (x), there
exist Real Numbers Ai,A2, . . . , A n such that
v a
p(x)da(x) = Xip(xi) + A 2 p(z2) + ... + Xnp(x n ),
for an arbitrary POLYNOMIAL of order 2n — 1 and the
X' n s are called Christoffel Numbers. The distribu-
tion da(x) and the INTEGER n uniquely determine these
numbers A^.
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 47, 1975.
Gauss-Jordan Elimination
Gauss's Lemma
705
Gauss-Jordan Elimination
A method for finding a Matrix Inverse. To apply
Gauss-Jordan elimination, operate on a Matrix
[A l] =
Define
an
din
1
■
•
CL21
• ' CL2n
1 .
•
Ctnl
where I is the Identity Matrix, to obtain a Matrix
of the form
1 • • • bn ■ • • 6m
1 *•• 621 ••• b 2n
■L Unl * * • n n
The Matrix
B =
'feu
621
bnl
&2n
is then the Matrix Inverse of A. The procedure is
numerically unstable unless PIVOTING (exchanging rows
and columns as appropriate) is used. Picking the largest
available element as the pivot is usually a good choice.
see also Gaussian Elimination, LU Decomposition,
Matrix Equation
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Gauss-Jordan Elimination" and "Gaussian
Elimination with Backsubstitution." §2.1 and 2.2 in Nu-
merical Recipes in FORTRAN: The Art of Scientific Com-
puting, 2nd ed. Cambridge, England: Cambridge Univer-
sity Press, pp. 27-32 and 33-34, 1992.
Gauss-Kummer Series
71 = ^ '
where 2-Fi(a,6;c;x) is a Hypergeometric Function.
This can be derived using RUMMER'S QUADRATIC
Transformation.
Gauss-Kuzmin- Wirsing Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let xo be a random number from [0, 1] written as a
simple Continued Fraction
xq = +
(1)
a\ +
a 2 +
x n = +
CLn+l +
Cln+2 ~\-
1
dn+3 + * •
CCn-1
X n -1
(2)
Gauss (1800) showed that if F(n y x) is the probability
that x n < x, then
lim F(n,x) =
ln(l + x)
In 2 '
(3)
Kuzmin (1928) published the first proof, which was sub-
sequently improved by Levy (1929). Wirsing (1974)
showed, among other results, that
lim
71— ►OO
F(n,x)
_ ln(l+x)
In 2
(-A)«
*(*),
(4)
as + .
where A = 0.3036630029... and *(sc) is an analytic
function with *(0) = *(1) = 0. This constant is con-
nected to the efficiency of the EUCLIDEAN ALGORITHM
(Knuth 1981).
References
Babenko, K. I. "On a Problem of Gauss." Soviet Math. DokL
19, 136-140, 1978.
Daude, H.; Flajolet, P.; and Vallee, B. "An Average-Case
Analysis of the Gaussian Algorithm for Lattice Reduc-
tion." Submitted.
Durner, A. "On a Theorem of Gauss-Kuzmin-Levy."
Arch. Math. 58, 251-256, 1992.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.c0m/asolve/constant/kuzmin/ku2min.html.
Flajolet, P. and Vallee, B. "On the Gauss-Kuzmin-Wirsing
Constant." Unpublished memo. 1995, http://pauillac.
inria . fr / algo / flajolet / Publications / gauss -
kuzmin . ps .
Knuth, D. E. The Art of Computer Programming, Vol. 2:
Seminumerical Algorithms, 2nd ed. Reading, MA:
Addison-Wesley, 1981.
MacLeod, A. J. "High-Accuracy Numerical Values of the
Gauss-Kuzmin Continued Fraction Problem." Computers
Math. Appl. 26, 37-44, 1993.
Wirsing, E. "On the Theorem of Gauss-Kuzmin-Levy and
a Frobenius-Type Theorem for Function Spaces." Acta
Arith. 24, 507-528, 1974.
Gauss-Laguerre Quadrature
see Laguerre-Gauss Quadrature
Gauss's Lemma
Let the multiples m, 2m, . . . , [(p— l)/2]m of an INTEGER
such that p\m be taken. If there are an Even number
r of least Positive Residues mod p of these numbers
> p/2, then m is a QUADRATIC RESIDUE of p. If r is
Odd, 771 is a Quadratic Nonresidue. Gauss's lemma
can therefore be stated as (m\p) = ( — l) r , where (m\p)
is the Legendre Symbol. It was proved by Gauss as
a step along the way to the Quadratic Reciprocity
Theorem.
See also QUADRATIC RECIPROCITY THEOREM
706
Gauss's Machin-Like Formula
Gauss's Theorem
Gauss's Machin-Like Formula
The Machin-Like Formula
±tt = 12 cot" 1 18 + 8 cot -1 57- 5 cot -1 239.
Gauss-Manin Connection
A connection denned on a smooth Algebraic Variety
defined over the COMPLEX NUMBERS.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 81, 1980.
Gauss Map
The Gauss map is a function from an ORIENTABLE SUR-
FACE in Euclidean Space to a Sphere. It associates
to every point on the surface its oriented Normal VEC-
TOR. For surfaces in 3-space, the Gauss map of the
surface has DEGREE given by half the EULER CHARAC-
TERISTIC of the surface
// K dA = 2ttx{M) -^2<*i- K g ds y
J J M J OM
which works only for ORIENTABLE SURFACES.
see also Curvature, Nirenberg's Conjecture,
Patch
References
Gray, A. "The Local Gauss Map" and "The Gauss Map via
Mathematical §10.3 and §15.3 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 193-194 and 310-316, 1993.
Gauss's Mean- Value Theorem
Let f(z) be an Analytic Function in \z - a\ < R.
Then
for < r < R,
-I
f(z + re ie )dd
Gauss Measure
The standard Gauss measure of a finite dimensional
Real Hilbert Space H with norm || • \\h has the
Borel Measure
Hh (dh) = (V^)- dim(H) exp(±\\h\\ 2 H )\ H (dh),
where Ah is the Lebesgue Measure on H.
Gauss Multiplication Formula
(2n7r) {n - 1)/2 n 1/2 - nz T(nz)
= r{ Z )r(z+i)T(z+l)---T(z+^)
n-1
where T(z) is the GAMMA FUNCTION.
see also Gamma Function, Legendre Duplication
Formula, Polygamma 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. 256, 1972.
Gauss Plane
see Complex Plane
Gauss's Polynomial Theorem
If a Polynomial
f(x) = x N + dx"- 1 + C 2 x N ~ 2 + . . . + C N
with integral COEFFICIENTS is divisible into a product
of two Polynomials f = ^<t>
</> = x n +0 1 x n - 1 + ... + n ,
then the COEFFICIENTS of this POLYNOMIAL are INTE-
GERS.
see also Abel's Irreducibility Theorem, Abel's
Lemma, Kronecker's Polynomial Theorem, Poly-
nomial, Schoenemann's Theorem
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 119,
1965.
Gauss's Reciprocity Theorem
see Quadratic Reciprocity Theorem
Gauss-Salamin Formula
see Brent-Salamin Formula
Gauss's Test
If u n > and given B(n) a bounded function of n as
n — > oo, express the ratio of successive terms as
u n __ h B(n)
J- ~f~ _ i~ 9
n^»'i
U n +1
The SERIES converges for h >1 and diverges for h < 1.
see also CONVERGENCE TESTS
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 287-288, 1985.
Gauss's Theorem
see DIVERGENCE THEOREM
nj
Gauss's Theorema Egregium
Gaussian Bivariate Distribution 707
Gauss's Theorema Egregium
Gauss's theorema egregium states that the Gaussian
Curvature of a surface embedded in 3-space may
be understood intrinsically to that surface. "Resi-
dents" of the surface may observe the GAUSSIAN CUR-
VATURE of the surface without ever venturing into full
3-dimensional space; they can observe the curvature of
the surface they live in without even knowing about the
3-dimensional space in which they are embedded.
In particular, GAUSSIAN CURVATURE can be measured
by checking how closely the Arc Length of small Ra-
dius CIRCLES correspond to what they should be in EU-
CLIDEAN Space, 2-n-r. If the Arc Length of Circles
tends to be smaller than what is expected in EUCLID-
EAN Space, then the space is positively curved; if larger,
negatively; if the same, Gaussian Curvature.
Gauss (effectively) expressed the theorema egregium by
saying that the Gaussian Curvature at a point is
given by —R(v, w)v,w, where R is the RlEMANN TEN-
SOR, and v and w are an orthonormal basis for the Tan-
gent Space.
see also Christoffel Symbol of the Second Kind,
Gauss Equations, Gaussian Curvature
References
Gray, A. "Gauss's Theorema Egregium." §20.2 in Modern
Differential Geometry of Curves and Surfaces. Boca Ra-
ton, FL: CRC Press, pp. 395-397, 1993.
Reckziegel, H. In Mathematical Models from the Collections
of Universities and Museums (Ed. G. Fischer). Braun-
schweig, Germany: Vieweg, pp. 31—32, 1986.
Gauss's Transformation
If
and
(1 + x sin a) sin f3 — (1 + x) sin a,
then
(i +
x) r_d±_ = r_
Jo \/l - x 2 sin 2 (j) Jo [\.
^-^ 2 ^ 2 <t> Jo y/l-^sin 2 .
see also Elliptic Integral of the First Kind, Lan-
den's Transformation
Gaussian Approximation Algorithm
see Arithmetic-Geometric Mean
Gaussian Bivariate Distribution
The Gaussian bivariate distribution is given by
P(xi,x 2 )
exp
2(1 -P 2 )
2ircr\CT2 y 1 — p 2
where
_ (Xl ~Ml) 2 2p(xi -/il)(x 2 - jl2) , (X2~fl2)
(1)
CTl-
<Ti<T2
+
OV
(2)
_ / x (xix 2 ) - (xi) (x 2 ) , q ,
p = cov{x ly x 2 ) — ^ * — - (3)
aio~2
is the Covariance. Let Xi and X 2 be normally and
independently distributed variates with MEAN and
Variance 1. Then define
Y\ = /xi + criiXx + (T12X2
Y2 = fl2 + &2\X\ + C2 2 X 2 .
(4)
(5)
These new variates are normally distributed with MEAN
Mi and p2t Variance
2 _ 2 , 2
CT\ =: <Jn -+- (7i2
2 2 2
0~2 = &21 + CT 2 2 ,
and Covariance
Fl2 = CTH0T21 + CT120-22-
(6)
(7)
(8)
(9)
(10)
O'\0~2 <T\0^2
The joint probability density function for x\ and x 2 is
f(x u x 2 ) dx! dx 2 = l- e -(*i 2 +*2 2 )/ 2 dx! dx 2 . (11)
However, from (4) and (5) we have
The Cova
RIANCE matrix is
Vij =
2
<7l pCTiCT2
2
P&1 <7 2 0~2
5
where
v 12
n =
_ <7llO"21 + &1 2 <J2 2
2/i "Mi
2/2 - P>2
0*11 Cl2
0*21 CT22
Now, if
then this can be inverted to give
(Til CTl2
0*21 CT22
(12)
(13)
Xl
X2
Cll (T12
-1
yi -mi
0*21 0*22
2/2 - ^2
1
(722
— (721
— 0"12
(7n
2/1
2/2
-Mi
-M2_
Cl 1<T22 — <7l
2*721
(14)
Therefore,
2 . 2 [0-22(2/1 - Mi) ~ 0-12(2/2 - Ma)]
xi + x 2 = —
+ -
(criia 22 — cri2<T2i) 2
Q'2i(2/i - Mi) + <7i 1(3/2 - M2)] 2
(cnO-22 — <7i2<72l) 2
(15)
708 Gaussian Bivariate Distribution
Expanding the Numerator gives
o-22 2 (yi - Mi) 2 - 2<Ti2<x 22 (yi - Mi) (3/2 - Ma)
+<7i2 2 Q/2 - M2) 2 + C2i 2 (yi - Mi) 2
-2crii<X2i(yi - Mi) (2/2 - M2) + cni 2 (?/2 - M2) ,
(16)
(xi + X2 )(criicr 2 2 - cr 12 <7 2 i)
- (?/i -Ml) 2 (^"21 2 +^22 2 )
-2(t/i — Ml)(?/2 - M2)(^11^21 + CT 12 CT22)
~h(V2 - M2) 2 (crn 2 + cri2 2 )
= <?2 2 (yi - Ml) 2 - 2(2/1 - Ml) (2/2 - M2)(M0"1<7"2)
+ CTl 2 (2/2 -M2) 2
2 2
= &1 &2
But
1
(2/i -Mi) 2 2p(yi -Mi)(ya - M2)
<n'
<Ti(T2
+
^r 2 t 2
(7i (J2
(2/2 -M2) 2
<T2 2
(17)
1- P 2 !_ v j2 % <nW-Vi2 2
~ 2 ~ 2
<7i (72
(cTn 2 + <Ti2 2 )(<J21 2 + CT 2 2 2 ) - (<ru<T21 + C"l2Cr 22 ) 2
(18)
The Denominator is
00 2 2 22 22 22
<7*11 (721 + CTn <J22 + C12 (721 + <7l2 (7 2 2 — ^11 (721
— 2(7iiCri2<T2l0'22 — °"12 ^"22 = (o r H<7 , 22 ~ 0'l2^'2l) 5 (19)
&1 &2
1 — p 2 (crii(T22 — Cri2CT2l)
(20)
and
~ 2 ^L n, 2 "^
Xl + X2 =
1-M 2
(2/1 - Mi) 2 2p(yi - Mi) (2/2 -M2) , (2/2 -M2)
(7l'
CTi<T2
+
0V
Solving for xi and X2 and defining
P =
<j\<ji
^
gives
xi
X2 =
&11CT22 — Cri20~21
^22(2/1 - Mi) - ^12(2/2 - M2)
p'
-<T2l(2/l ~ Ml) + 0"ll(y2 ~ M2)
P'
(21)
(22)
(23)
(24)
Gaussian Bivariate Distribution
The Jacobian is
, /xi,x 2 \ __
\yuv* )
dx-i dx 1
dyi dyi
8x2 dx2
dyi dy 2
1 ,
P 2
°"22
(T 2 1
p'
— CTi2CT2l)
cr 12
p'
1
1
p'
<Tl<T2^/l ~ P 2
Therefore,
dx\ dx2 =
dyidy
2
<J\(T2
V^ 7 ?
(25)
(26)
and
1 -(xi 2 +z 2 2 )/2
2tt
where
dx\ dx2
27vai(T2yl — p 2
e- v/2 d yi dy 2 , (27)
1-P 2
(yi -mi) 2 _ 2p(yi -Mi)(y2 -M2) (y2 - M2) 2
(7i 2 (7l(T2 <72 2
Now, if
then
(7ii CTi2
(721 (722
= 0,
(7n(7i2 = <Tl2(721
(28)
(29)
(30)
2/i = Mi + ^li »i + <Ti2a:2
(31)
, <7i2(721 , <Jii(J2lXi +CTi20"2lX2
y2 = M2 H X 2 = M2 H
<7ll
1 a21 / 1 \
= M2 H (CTllXi + CT12X2J,
(711
(32)
2/i = Mi + ^3
(33)
, 0*21
y2 = M2 H x 3)
£711
(34)
X 3 =
Cll ,
= 2/1 -Mi = (j/2 -
C"21
~M2)-
(35)
where
The Characteristic Function is given by
/oo /»oo
/ e i(tlXl+t2X2) P(xi,x 2 )dx 1 rfx2
-00 t/ —00
/oo />oo
/ e* (tlxl+t2X2) exp
-OO </ — OO
2(1 -P 2 )
dxi dX2,
(36)
Gaussian Bivariate Distribution
where
(xi - fti) 2 2p(xi - pi)(x 2 - M2) (a?2 - IteY
<J\ 2 <J\<Jl <J 2 2
and
JV =
2-KG\U2 yj\ — p 2
Now let
U — Xi — pi
W = X2 — P-2.
(37)
(38)
(39)
(40)
Then
4>(h,t 2 ) = N
*£(■
ito'w
e exp
2(l-p 2 )^ 2 2
00
x / e v e tlU dudw, (41)
where
- 1 1 r
2p<T\W
N'
'(1-P 2 )l
i(t 1 /2 1 +i 2 M2)
(T2
27ra 1 a 2 ^/l ~ p 2
Complete the Square in the inner integral
2 2pa 1 w
(42)
r r 1 1
<T2
|e ilt
du
// 1 r pio-i^i 2 !
*{w<^(^f )>"'""»■ (43)
Rearranging to bring the exponential depending on w
outside the inner integral, letting
v — u — p-
1
&2
and writing
e Hu — cos(tiu) 4- isin(tiu)
(44)
(45)
gives
0(*i
x exp
J — c
itn-W
e exp
2^(1 - P 2) -
\L
2a 2 2 (l-p 2 )
exp
2t7 2 2 (l-p 2 )
^ cos \ti (v
+
CT2
4-isinL (^+^^)]} dvdw - ( 46 )
Gaussian Bivariate Distribution 709
Expanding the term in braces gives
)S[t\v) cos I I — sm(t\v) sin [ 1
4-i
\ CT2tl / \ CT2
V 0"2 / V <7*2tl /J
■ / \ / p(TiW\ , . . ( pG\Wt\ \\
$m{t\v) cos I 1 4- cos(ii^) sm I I
[cos(tiv) + 2sin(tit;)]
= exp f 1\ ) [cos(iiv) + isin(tiv)]. (47)
But e ax sin(6;c) is Odd, so the integral over the sine
term vanishes, and we are left with
/oo
e it2W exp
-oo
x exp
w*
2<7 2 2
2 2
p W
/oo
■oo
2<7 2 2 (l-p 2 )
exp
exp
\i£wh] dw
v
2<ri 2 (l -p 2 )
L CT2 J
cos(ii^) du
= Nj exp zti; I ^2 + ti f p — J J exp
/oo
■oo
«T
dw
2<T! 2 (l-p 2 )
2(72 2
cos(iiv) du. (48)
Now evaluate the Gaussian Integral
/oo poo
e ikx e -ax 2 dx= € ~ax 2 cog ( fc;r ) dx
-oo «/ — oo
a
(49)
to obtain the explicit form of the CHARACTERISTIC
Function,
0(*i,t 2 ) =
P »(*1M1+*2M2)
2 7T 0-1 (72 y 1 — p 2
x <^ a 2 V^7rexp --U2+p — *ij 2cr 2 2 >
x | ( r lv /27r(l-p 2 )exp [-|*i 2 2<7i 2 (l - p 2 )] |
= c <( * l " l+taM) exp{-|[t 2 2 <T 2 2 + 2p<T 1 <T 2 tlt 2
+p 2 ai 2 h 2 4- (1 - p 2 )o-i 2 ti 2 ]}
= exp[i(*i/xi + £2^2)
-§(<ti V + 2pa 1 a 2 t 1 t 2 + <ri 2 ti 2 )]. (50)
Let zi and 22 be two independent Gaussian variables
with Means pi = and c; 2 = 1 for i = 1, 2. Then
the variables ai and a2 defined below are Gaussian bi-
variates with unit Variance and CROSS-CORRELATION
Coefficient p:
at
1 + P
21 +
f?«
(51)
710
Gaussian Brackets
Gaussian Curvature
a 2
i + P
i-p
z 2 .
The conditional distribution is
1
P(X 2 \X!)
where
<r a y/2ir{l - ft)
exp
(^ 2 -M /2 ) 2
2^ 2
p! 2 = fi 2 + p — (xi -Mi)
0~l
(7 2= CT2 y 1 — p2 -
The marginal probability density is
/oo
P(xi,S2)rf2Cl
■oo
(52)
(53)
(54)
(55)
CT2
v/2?
exp
(x 2 - p-2) 2
2<7 2 2
(56)
see also Box-Muller Transformation, Gaussian
Distribution, McMohan's Theorem, Normal Dis-
tribution
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 936-937, 1972.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 118, 1992.
Gaussian Brackets
Published by Gauss in Disquisitiones Arithmetical
They are defined as follows.
[ax] = ai
[ai,a 2 ] = [ai]a 2 + [ ]
(1)
(2)
(3)
[ai,a2, . - . , a n ] = [ai,a2, - - ■ , a n _i]a n
+ [ai,a 2 , . . .,a n _ 2 ]. (4)
Gaussian brackets are useful for treating CONTINUED
Fractions because
1
ai +
[Q2,Qn]
[ai,a n ]
(5)
a 2 +
a 3 4- . . . + —
The Notation [x] conflicts with that of Gaussian
Polynomials and the Nint function.
References
Herzberger, M. Modern Geometrical Optics. New York: In-
terscience Publishers, pp. 457-462, 1958.
Gaussian Coefficient
see qr-BlNOMIAL COEFFICIENT
Gaussian Coordinate System
A coordinate system which has a METRIC satisfying
gu = -1 and dgij/dxj = 0.
Gaussian Curvature
An intrinsic property of a space independent of the co-
ordinate system used to describe it. The Gaussian cur-
vature of a Regular Surface in
formally defined as
jr(p) = idet(S(p)),
at a point p is
(i)
where S is the SHAPE Operator and det denotes the
Determinant.
If x : U -> M 3 is a Regular Patch, then the Gaussian
curvature is given by
K =
eg
EG-F 2 '
(2)
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). The
Gaussian curvature can be given entirely in terms of the
first Fundamental Form
ds
E du +2Fdudv + G dv 2
and the DISCRIMINANT
g = EG~F 2
by
K -
1
7$
dv
y/9 v 2 \ d fy/g 2 V
'Y ill )~di\E' 112 )
(3)
(4)
(5)
where T^ are the CONNECTION COEFFICIENTS. Equiv-
alently,
K
^23
where
&23 =
&33 =
E
F
1 dE
2 du
E F
F G
1 dE 1 dG
2 dv 2 dv
dF _ idE
du 2 dv
1 d 2 E d 2 F
T? dF. _ 1 dG
r dv 2 dv.
&33
1 dE
!&
2 du
2 dv 2 dudv
\ d 2 G
2 du 2 '
(6)
(7)
(8)
Gaussian Curvature
Gaussian Distribution 711
Writing this out,
K
2s
d 2 F
9 <?E _ d 2 G
dudv dv 2 du 2
G
[du V
)
2—- —\ - (-Y
F [dEdG
dv du
^OEdG
dv du
4p2 L du dv
+ u°* _ f) ( 2 £ _ f)
V du dv J \ dv du J
_E_\dG SdF _ dE\ _ /dG\
4p 2 \dv V du dv) \du)
(9)
The Gaussian curvature is also given by
K =
[|x„| a |x.| a -(x,-x 1 ,) a ] a
(Gray 1993, p. 285), as well as
[NN1N2] _ {"[NTTilj
K =
V9
V9
(10)
(11)
where e ij is the Levi-Civita Symbol, N is the unit
Normal Vector and T is the unit Tangent Vector,
The Gaussian curvature is also given by
IT R 1
(12)
where R is the CURVATURE SCALAR, m and K 2 the
Principal Curvatures, and #1 and R 2 the Princi-
pal Radii of Curvature. For a Monge Patch with
z = h(u, t>),
K :
(l + h u 2 + h v 2 ) 2 '
(13)
The Gaussian curvature K and Mean Curvature H
satisfy
H 2 > K, (14)
with equality only at Umbilic Points, since
H 2 -K 2 = \{k 1 -k 2 ) 2 . (15)
If p is a point on a REGULAR SURFACE M C M 3 and
v p and w p are tangent vectors to M at p, then the
Gaussian curvature of M at p is related to the Shape
Operator S by
S(v p ) x S(w p ) = ^(p)v p x w p
(16)
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 DwZ)
2JZp
K
(17)
(Gray 1993, pp. 291-292).
For a SPHERE, the Gaussian quadrature is K = 1/a 2 .
For Euclidean Space, the Gaussian quadrature is
K = 0. For Gauss-Bolyai-Lobachevsky Space, the
Gaussian quadrature is K — —1/a 2 . A Flat Surface
is a Regular Surface and special class of Minimal
SURFACE on which Gaussian curvature vanishes every-
where.
A point p on a Regular Surface M € M 3 is classified
based on the sign of K(p) as given in the following table
(Gray 1993, p. 280), where S is the Shape Operator.
Sign
Point
K(p) >
K(p) <
K(p) = but S(p) ^
K(p) = and S(p) =
elliptic point
hyperbolic point
parabolic point
planar point
A surface on which the Gaussian curvature K is every-
where Positive is called Synclastic, while a surface
on which K is everywhere NEGATIVE is called Anti-
CLASTIC. Surfaces with constant Gaussian curvature
include the Cone, Cylinder, Kuen Surface, Plane,
PSEUDOSPHERE, and SPHERE. Of these, the CONE and
Cylinder are the only Flat Surfaces of Revolu-
tion.
see also Anticlastic, Brioschi Formula, Devel-
opable Surface, Elliptic Point, Flat Surface,
Hyperbolic Point, Integral Curvature, Mean
Curvature, Metric Tensor, Minimal Surface,
Parabolic Point, Planar Point, Synclastic, Um-
bilic Point
References
Geometry Center. "Gaussian Curvature." http://vww.geom
. umn . edu / zoo / diffgeom / surf space / concepts /
curvatures/gauss-curv.html.
Gray, A. "The Gaussian and Mean Curvatures" and "Sur-
faces of Constant Gaussian Curvature." §14.5 and Ch. 19
in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 279-285 and 375-387,
1993.
Gaussian Differential Equation
see Hypergeometric Differential Equation
Gaussian Distribution
712
Gaussian Distribution
Gaussian Distribution
The Gaussian probability distribution with Mean jjl and
Standard Deviation a is a Gaussian Function of
the form
P(x)
1
ry/2^
-(z-/x) 2 /2<r 2
(1)
where P(x) dx gives the probability that a variate with
a Gaussian distribution takes on a value in the range
[x, x + dx]. This distribution is also called the NORMAL
Distribution or, because of its curved flaring shape,
the BELL Curve. The distribution P(x) is properly
normalized for x £ ( — 00,00) since
F
•J — c
P(x)dx = 1.
(2)
The cumulative Distribution Function, which gives
the probability that a variate will assume a value < x y
is then
/X -j px
P(x)dx= — ^ I e
■oo ctV2tt J^^
dx.
(3)
Gaussian distributions have many convenient properties,
so random variates with unknown distributions are of-
ten assumed to be Gaussian, especially in physics and
astronomy. Although this can be a dangerous assump-
tion, it is often a good approximation due to a surprising
result known as the Central Limit Theorem. This
theorem proves that the Mean of any set of variates with
any distribution having a finite MEAN and VARIANCE
tends to the Gaussian distribution. Many common at-
tributes such as test scores, height, etc., follow roughly
Gaussian distributions, with few members at the high
and low ends and many in the middle.
Making the transformation
X — pi
(4)
so that dz — dz/a gives a variate with unit VARIANCE
and Mean
P(x) dx
1
\Z2tt
-z 2 /2
dz }
(5)
known as a standard NORMAL DISTRIBUTION. So de-
fined, z is known as a z-Score).
The Normal Distribution Function gives the prob-
ability that a standard normal variate assumes a value
in the interval [0, z].
• w =£jf'~' / **=*« f (£)
(6)
Here, Erf is a function sometimes called the error func-
tion. Neither <&(z) nor Erf can be expressed in terms of
finite additions, subtractions, multiplications, and root
extractions, and so both must be either computed nu-
merically or otherwise approximated. The value of a for
which P(x) falls within the interval [—a, a] with a given
probability P is called the P Confidence Interval.
The Gaussian distribution is also a special case of the
Cht-Squared Distribution, since substituting
(x-n?
so that
dz
1 2(3! - H)
dx :
&
dx
(7)
(8)
(where an extra factor of 1/2 has been added to dz since
z runs from to 00 instead of from -00 to 00), gives
P(x) dx
l_ e -(z/<7)/2
2?T
(;)"'"*(;)-
-^a^'""(~r"^^
which is a CHI-SQUARED DISTRIBUTION in z/cr with
r — 1 (i.e., a Gamma Distribution with a — 1/2 and
(9 = 2).
Cramer showed in 1936 that if X and Y are Indepen-
dent variates and X + Y has a Gaussian distribution,
then both X and Y must be Gaussian (CRAMER'S THE-
OREM).
The ratio X/Y of independent Gaussian-distributed
variates with zero MEAN is distributed with a Cauchy
Distribution. This can be seen as follows. Let X and
Y both have MEAN and standard deviations of cr x and
tj y , respectively, then the joint probability density func-
tion is the Gaussian Bivariate Distribution with
P = 0,
f(x,y)
2na x o'y
-[x 2 /(2<T x 2 )+y 2 /(2<r y 2 ))
(10)
From Ratio Distribution, the distribution of U
Y/X is
\x\f(x 1 ux)dx
P(u) = f
J — 00
= ->-[
27V(J X <Jy J _
= / zexp -x - — - + - — -
TT^xO-y J |_ \2<T X 2 2<T y , 2 J _
x 2 /(2* x 2 ) + u 2 x 2 /(2<r y 2 )}
dx.
(ii)
But
f
Jo
dx
L 2a
00 1 1
(12)
Gaussian Distribution
Gaussian Distribution 713
and
P(u)
(TxO'y
KCTxCTy 9 ( 1 , u 2 \ 7T U 2 (T X 2 + (7 y 2
*«'+(£)
(13)
which is a CAUCHY DISTRIBUTION with MEAN /x =
and full width
(14)
r= 2oj,
The Characteristic Function for the Gaussian dis-
tribution is
4>(t) = e irnt ~ a f /2 , (15)
and the Moment-Generating Function is
/OO f x
_e _ (x _„ )W dx
= — == / exp -! -— — \x 2 - 2(/i + o 2 t)x + fi 2 ] \dx.
(16)
Completing the Square in the exponent,
-L[ x 2 -2(ii + <T 2 t)x + ti 2 ]
= A fl* - (" + ff2 *)l a + [M 2 " (M + ^t) 2 ]} • (17)
Let
y = X - (jA + (7 t)
dy = dec
1
The integral then becomes
M(t)
— r
exp
2 2ua 2 £ + <r 4 t 2
"«V + J?
(18)
(19)
(20)
dy
1 f°°
— — / exp[-ay 2 -\- [it -\- \a 2 t 2 ]dy
aV27t J_ QO
1 Ht + cr 2 t 2
cr\/27r
•c
dy
1 /7T ^£+^2*2/2
crV^i- V a
\/2^ 2 7T
a\/27r
^t+o- 2 t 2 /2 _ ^t + cr 2 t 2 /2
(21)
M'{t) = (/* + <r t)e'
,2,x u* + ff 3 * 2 /2
(22)
M"(t) = Set**'**'' 2 + e^+^^ifi + ta 2 ) 2 , (23)
A* = M'(0) = /i
ff 2 = M"(0) - [M'(0)] 2
/ 2 , 2\ 2 2
These can also be computed using
R(t) = ln[M (*)] = [it + §a 2 * 2
^ , (t) = A i + cr 2 ^
ii"(t) = <r 2 ,
yielding, as before,
fj, = ij'(0) =/i
(T 2 = fl»(o) = a'
(24)
(25)
(26)
(27)
(28)
(29)
(30)
The moments can also be computed directly by comput-
ing the Moments about the origin fi' n = (x n ),
1 f°°
-(x- M ) a /2^ a te.
Now let
du =
X — jJL
V2a
dx
y/2<r
x = any 2 + //,
giving
V2cr f n _ u 2 if
jx n = —r== / x e du= ~-=
Mo = 1
//i = —= I xe~ u du
VW_oo
1 /*°°
= — = / (V2au + n)e~ u du
= [V2aH x {l) + fJ,H (l)] = (0 + /*) = A*
(31)
(32)
(33)
(34)
(35)
(36)
(37)
M2
2 -u* j
a; e cm
1 Z" 00 2
= -= / (2<rV + 2\/2crAm + ^ 2 )e" u
du
[2<7 2 tf 2 (l) + 2V2a^Hi(l) + M 2 ^o(l)]
2 (38)
= (2cr 2 !+0 + /i 2 ) = ^ 2 +<r
M3 = -7= /
3 -u^ 1
714 Gaussian Distribution
1 /*°°
= — / (2V2 crV +6 fj,<r 2 u 2
= [2V2<r*H 3 (l)+6n* 2 H 2 (l)
+ 3\/2/i 2 <^i(l) + M 3 ^fo(l)]
= (0 + 6/xV§ + + M 3 ) = M(M 2 + 3<r 2 ) (39)
- — r
3 -ir
cfu
(4<tV + 8a/2/x<jV
+ 12MV 2 u 2 +4v / 2M 3 ^ + // 4 )e" u du
= [4<r 4 fr 4 (l) + 8V2mo" 3 ^3(1) + 12/xVi7 2 (l)
+ 4v / 2^ 3 <ri?i(l)+M 4 ^o(l)]
= (4cr 4 f + + 12/zV| + + /x 4 )
= ^ 4 +6mV 2 + 3^t 4 ; (40)
where H n (a) are GAUSSIAN INTEGRALS.
Now find the Moments about the Mean,
Mi = (41)
M2-M2-(Mi) 2 = (^+^)-^ = ^ 2 (42)
Us = /x 3 - Sfj&fii + 2(/xi) 3
= /x(m 2 + 3<r 2 ) - 3(a 2 + m 2 )m + V = (43)
^4 = /x 4 - 4/X3/xi + 6p 2 (pi) 2 - 3Qui) 4
= (m 4 + 6m 2 o- 2 + 3cr 4 ) - 4(m 3 + 3/i<t 2 )m
+ 6(m 2 +^ 2 )m 2 "3m 4
= 3(j ,
(44)
so the Variance, Standard Deviation, Skewness,
and KURTOSIS are given by
var(ar) = ^2 = cr
stdv (x) = y/vax(x) = a
M3
<T 3
71
72 ^_ 3 =^-3 = 0.
(45)
(46)
(47)
(48)
The Variance of the Sample Variance s 2 for a sample
taken from a population with a Gaussian distribution is
N 3
[(■/V-1)(// 4 + 6aiV + 3<t'
a (N-l)[(N-l)ti-(N-3)n' 2
var^s ) =
_ /( ^_ 3)( ^ + (T 2 )2]
_ 2(JV - 1)(m 4 + 2p?N<T 2 + JVcr 4 )
~ AJV 3 '
/2
(49)
Gaussian Distribution
If ^ = 0, this expression simplifies to
raK ,^«^ = ?£fci>, (60)
and the STANDARD ERROR is
V 2 i N ~ !)
[standard error]
JV
(51)
The Cumulant-Generating Function for a Gaus-
sian distribution is
K{h) = \n(e Ulh e a2h2/2 ) = nh + §aV, (52)
«1 = ^1
(53)
2
At 2 = CT
(54)
K r = for r > 2.
(55)
For Gaussian variates, k t = for r > 2, so the variance
of ^-Statistic k 3 is
«6 9«2«4 9/€3 , 6«2
N(N - l)(JV-2)
6/c 2 3
JV(JV-l)(JV-2)'
Also,
var(&4) =
var(yi) =
var(# 2 ) =
24k 2 4 N(N - l) 2
(JV - 3)(iV - 2)(JV + 3)(JV + 5)
6iV(iV - 1)
(JV-2)(JV + l)(JV + 3)
24iV(iV-l) 2
{N - 3)(JV - 2)(JV + 3)(N + 5)
where
9i =
k 2
3/2
_ &4
52 = fc?-
If -P(a?) is a Gaussian distribution, then
(56)
(57)
(58)
, (59)
(60)
(61)
D(*) = ^
1 + erf
(62)
so variates w% with a Gaussian distribution can be gener-
ated from variates yi having a UNIFORM DISTRIBUTION
in (0,1) via
ti = aV2evf 1 {2y i -l) + y,>
(63)
Gaussian Distribution — Linear Combination.
However, a simpler way to obtain numbers with a Gaus-
sian distribution is to use the Box-MULLER TRANSFOR-
MATION.
The Gaussian distribution is an approximation to the
Binomial Distribution in the limit of large numbers,
Pirn)
y/2irNpq
exp
(m - Npf
2Npq
(64)
where ni is the number of steps in the POSITIVE direc-
tion, N is the number of trials (TV = n± + 722), and p
and q are the probabilities of a step in the POSITIVE
direction and NEGATIVE direction (q = 1 — p).
The differential equation having a Gaussian distribution
as its solution is
dy _ j/(m ~ x)
dx cr 2
since
In
dy
y
yo
y = yoe
■ dx
-2>-*> 2
-(x-M) 2 /2o- 2
(65)
(66)
(67)
(68)
This equation has been generalized to yield more compli-
cated distributions which are named using the so-called
Pearson System.
see also BINOMIAL DISTRIBUTION, CENTRAL LIMIT
Theorem, Erf, Gaussian Bivariate Distribution,
Logit Transformation, Normal Distribution,
Normal Distribution Function, Pearson System,
Ratio Distribution, z-Score
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 533-534, 1987.
Kraitchik, M. "The Error Curve." §6.4 in Mathematical
Recreations. New York: W. W. Norton, pp. 121-123, 1942.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 109-111, 1992.
Gaussian Distribution — Linear Combination
of Variates
If x is Normally Distributed with Mean h and
VARIANCE a 2 , then a linear function of x.
y — ax + b,
(i)
is also Normally Distributed. The new distribution
has Mean a^i + b and Variance a 2 cr 2 , as can be derived
using the Moment-Generating Function
t(ax + b)\ _ J,b / e *tx\ _ e tb e tiat + <r 2 (at) 2 /2
M(t) = (e t{ax+b) )=e tb (e
tb+fi,at + <T 2 a 2 t 2 /2 __ (b-\-afi)t~{-a 2 a 2 t 2 /2 /r>\
Gaussian Elimination 715
which is of the standard form with
li = b -+- ajjb
il 2 2
a = a a .
(3)
(4)
For a weighted sum of independent variables
n
y = ^2 aiXi > (5)
i=i
the expectation is given by
M{t) = (e yt ) = /exp j t^aiXi J \
n n
= Y[{e atx ) = Y[exp(aifnt + faWt 2 ). (6)
Setting this equal to
gives
expQu£ + \cr 2 t 2 )
1=1
n
- 2 — Y^„ 2 ~ 2
cr — > a,i Ci
(7)
(8)
(9)
Therefore, the Mean and VARIANCE of the weighted
sums of n RANDOM VARIABLES are their weighted sums.
If Xi are Independent and Normally Distributed
with Mean and Variance cr 2 , define
y
>i — y ^ CijXjj
(10)
where c obeys the ORTHOGONALITY CONDITION
CikCjk = Sij, (11)
with S the KRONECKER DELTA. Then y» are also in-
dependent and normally distributed with Mean and
Variance a 2 .
Gaussian Elimination
A method for solving MATRIX EQUATIONS of the form
Ax = b.
(1)
716 Gaussian Function
Starting with the system of equations
an 0,12
a21 tt22
afci a^2
dlk'
~Xi~
[61]
&2k
X 2
=
b 2
&kk _
_Xh _
M.
compose the augmented MATRIX equation
Oil
ai2 *
* * aifc
61 1
~Xl
021
a22 *
* * Cb2k
62
X2
Q>k\
Q>ki
o,kk
bk_
_x k
(2)
(3)
Gaussian Function
Solving,
-(x -^) 2 /2<r 2 _ 2" 1
■ln2
{xq - fj,) 2
2<r 2
(x -m) 2 = 2cr 2 ln2
xq = ±aV '2 In 2 + ^,
(4)
(5)
(6)
(7)
The Full Width at Half Maximum is therefore given
by
FWHM = x+-x-= 2^2 In 2 a « 2.3548a. (8)
Then, perform Matrix operations to put the aug-
mented Matrix into the form
r On a i2
a 22
aifc
a 2fe
a fcfc
&L
-x[-
L x kJ
(4)
Solve for a' kk , then substitute back in to obtain solutions
for n = 1, 2, . . . , k - 1,
^ = a 7 : ( 6 ' ~ 5Z a ^'
(5)
j=t+i
see also Gauss-Jordan Elimination, LU Decompo-
sition, Matrix Equation, Square Root Method
Gaussian Function
In 1-D, the Gaussian function is the function from the
Gaussian Distribution,
f(x).
1
-(x-tx) 2 /2<r 2
<r\/27r
(1)
sometimes also called the FREQUENCY CURVE. The
Full Width at Half Maximum (FWHM) for a Gaus-
sian is found by finding the half-maximum points xq.
The constant scaling factor can be ignored, so we must
solve
e -(* -M) 2 /^ 2 = lf( Xm ^) ( 2 )
But /(a? max ) occurs at x max — ^, so
e -(..-M)V^ = i m = I.
(3)
In 2-D, the circular Gaussian function is the distribu-
tion function for uncorrelated variables x and y having
a Gaussian Bivariate Distribution and equal Stan-
dard Deviation a = <r x = <r y}
f(x,y)
1 -[(x-^) 2 + (y-M„) 2 ]/2<r 2
27R7 2
(9)
The corresponding elliptical Gaussian function corre-
sponding to cr x ^ <Ty is given by
f^ y ) = _L_ e -K*-"-> a /a<'. a +[<v-M.) a /a»„ a ]. (10)
Re [Gaussian z
The above plots show the real and imaginary parts of
(2ir)~ 1 ^ 2 e~ z together with the complex absolute value
|(2,r)-
-l/2 e -x a
0.(14
The Gaussian function can also be used as an Apodi-
zation Function, shown above with the corresponding
Instrument Function.
Gaussian Hypergeometric Series
The Hypergeometric Function is also sometimes
known as the Gaussian function.
see also Erf, Erfc, Fourier Transform — Gauss-
ian, Gaussian Bivariate Distribution, Gaussian
Distribution, Normal Distribution
References
MacTutor History of Mathematics Archive. "Frequency
Curve." http : //www-groups . dcs . st-and . ac . uk/ -history
/Curves /Frequency . html .
Gaussian Hypergeometric Series
see Hypergeometric Function
Gaussian Integer
A Complex Number a+bi where a and b are Integers.
The Gaussian integers are members of the QUADRATIC
Field Q(y/—i). The sum, difference, and product of
two Gaussian integers are Gaussian integers, but a +
bi\c + di only if there is an e + fi such that
(a + bi)(e 4- fi) = (ae — bf) + (a/ + be)i = c + di.
Gaussian INTEGERS can be uniquely factored in terms
of other Gaussian Integers up to Powers of i and
rearrangements .
The norm of a Gaussian integer is defined by
n(x 4- iy) — x + y .
Gaussian Primes are Gaussian integers a-j-ib for which
n(a + ib) — a 2 +b 2 is Prime and a a Prime Integer a
such that a = 3 (mod 4).
1. If 2\n(x + iy), then 1 + i and 1 — i\x + iy. These
factors are equivalent since — i(i — 1) = i + 1. For
example, 2 = (1 + i)(l — i) is not a Gaussian prime.
2. Iin(x + iy) = 3 (mod 4) \n(x + iy), then n(a + zb)\x-\-
iy.
3. If n(x + iy) = 1 (mod 4) \n(x + iy) y then a + ib or
b + ia\x + iy. If both do, then n(a 4- ib)\x + iy.
The Gaussian primes with \a\, \b\ < 5 are given by — 5 —
4z, -5 - 2z, -5 + 2i, -5 + 4z, -4 - 5i, -4 - i, -4 +
-4 + 5i, -3 - 2i, -3, -3 4- 2z, -2 - 5i, -2 - 3z, -2 -
-2 + i, -2 + 3i, -2 + 5i, -l-4i, -l-2i, -1-i, -1 + i,
-1 + 2i, -1 + 4i, -3z, 32, 1 — 4i, 1 — 2i, 1 — *, 1 4-
1 + 2i, 1 + 4i, 2 - 52, 2 - 32, 2 - z, 2 4- 2, 2 + 3i, 2 4- 5i,
3-2i, 3, 3 + 2i, 4-5i, 4-i, 4 + i, 4 + 5i, 5-4i, 5 - 2i,
5 + 22, 5 + 42.
Every Gaussian integer is within |n|/\/2 of a multiple of
a Gaussian integer n.
see also Complex Number, Eisenstein Integer,
Gaussian Prime, Integer, Octonion
References
Conway, J. H. and Guy, R. K. "Gauss's Whole Numbers."
In The Book of Numbers. New York: Springer- Verlag,
pp. 217-223, 1996.
Shanks, D. "Gaussian Integers and Two Applications." §50
in Solved and Unsolved Problems in Number Theory, ^th
ed. New York: Chelsea, pp. 149-151, 1993.
Gaussian Integral 717
Gaussian Integral
The Gaussian integral, also called the Probability
Integral, is the integral of the 1-D Gaussian over
(—00,00). It can be computed using the trick of com-
bining two 1-D Gaussians
/ /»oo /*oo
= \ / e ~ (x2+y2)d y dx W
y J — 00 J — 00
and switching to POLAR COORDINATES,
/oo / p2ir po
• 00 y Jo Jo
f*2TT /»00
e~ x ~dx=\l I / e' r2 rdrdO
y/2n[-±e-<*r]~ = yft. (2)
However, a simple proof can also be given which does
not require transformation to POLAR COORDINATES
(Nicholas and Yates 1950).
The integral from to a finite upper limit a can be given
by the CONTINUED FRACTION
Jo
e x dx
yfH 1 2 3 4
2 a+ 2a+ a+ 2a + . . .
The general class of integrals of the form
/•oo
I n (a) = I e~~ ax x n dx
(3)
In{a) = f
Jo
can be solved analytically by setting
— -1/2
x = a ' y
dx = a ' dy
2 2
y = ax .
(4)
(5)
(6)
(7)
Then
I n (a) = a~ 1/2 [
Jo
= a" (1+n)/2 f
Jo
e -y ( -Va)» dy
e v y"dy.
(8)
For n — 0, this is just the usual Gaussian integral, so
T („\ V^T -1/2 1 /^
Jo(o) = — a = -^-.
(9)
For 71 = 1, the integrand is integrable by quadrature,
Jo
h(a) = a" 1 / e~ y ydy = a-\-\e~ y ]g° = fa" 1 .
(10)
718 Gaussian Integral
To compute I n (a) for n > 1, use the identity
-^/„- a (a)
e" a * x n ~ 2 dx
2 -ax* n — 2 i
-x e x ax
da Jo
-f
Jo
= / e~ ax x n dx^I n {a). (11)
Jo
For n = 2s EVEN,
7 -W=(-|;) 7 -'M = (-D a '-<
A 71 / 2 /z A 71 / 2
SO
a-.-.' ^. _ (!_" |)_! _ (2* - 1)!! /tF (13)
J Q 2a*+ 1 /* 2«+ 1 a- V a
If n - 2s + 1 is Odd, then
7 -w =(-£)'-' w=(-D''-<<«
£ \(n-l)/2
^(n-l)/2 j ^(n-l)/2
= aa(«-D/a /l(a) = ^ ^(n-D/a " 1 ' ( 14 )
2da( n ~ 1 )/ 2
f
Jo
x 2s+1 e~ a * dx :
5!
2a s + 1 '
The solution is therefore
* V rf* - J 2"£+iWa VT for n even
i e ^ X " da:= lim^ for .odd.
first few values are
therefore
Jo (a)
~" 2 V a
Ji(a)
_ X
~ 2a
Ja(a)
1 /tF
~ 4a V a
Js(a)
1
~ 2a 2
/ 4 (a)
3 /tF
~ 8a 2 V a
A(a)
1
~ a 3
/e(o)
15 /tF
16a 3 V a
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
Gaussian Polynomial
A related, often useful integral is
H n (a) = ~ f e—Vdx,
which is simply given by
H n = { ^^ for »» even
*- for n odd.
(24)
(25)
References
Nicholas, C. B. and Yates, R. C. "The Probability Integral."
Amer. Math. Monthly 57, 412-413, 1950.
Gaussian Integral (Linking Number)
see Linking Number
Gaussian Joint Variable Theorem
Also called the Multivariate Theorem. Given an
Even number of variates from a Normal Distribu-
tion with Means all 0,
(£l£2> = (xi) {x 2 ) ,
(1)
= (xix 2 ) (x 3 x 4 ) + (xix z ) (x 2 x 4 ) + (xix 4 ) (x 2 x 3 } , (2)
etc. Given an Odd number of variates,
(xi) = 0, (3)
{xix 2 xz) = 0, (4)
etc.
Gaussian Mountain Range
see Carotid-Kundalini Function
Gaussian Multivariate Distribution
see also Gaussian Bivariate Distribution, Joint
Theorem, Multivariate Theorem
Gaussian Polynomial
Defined by
« S T
(i)
for integral /, and
"^{njLi 12 !^ for0<fc<n ( 2)
^ otherwise.
Unfortunately, the NOTATION conflicts with that of
Gaussian Brackets and the Nearest Integer
Gaussian Prime
Gaussian Quadrature 719
Function. Gaussian Polynomials satisfy the iden-
tities
"n+1
Jfe + 1
n
fe+1
1
„n+l
1 - q n ~ k
n + 1
fc + 1
n + 1
A:
1-9"
-fc+i
i - «*+ i
(3)
(4)
For 5 = 1, the Gaussian polynomial turns into the Bi-
nomial Coefficient.
see also Binomial Coefficient, Gaussian Coeffi-
cient, qr-SERIES
Gaussian Prime
■-■■ ^ v^^fw^^^*/^^--:
"•>&
'^v-
.~-:."*Z.y><.-f^:h-y
Gaussian primes are GAUSSIAN INTEGERS a + ib for
which n(a + ib) - a 2 -\- b 2 is Prime and a a Prime
INTEGER a such that a = 3 (mod 4). The above plot
of the Complex Plane shows the Gaussian primes as
filled squares.
see also ElSENSTEIN INTEGER, GAUSSIAN INTEGER
References
Guy, R. K. "Gaussian Primes. Eisenstein-Jacobi Primes."
§A16 in Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 33-36, 1994.
Wagon, S. "Gaussian Primes." §9.4 in Mathematica in Ac-
tion. New York: W. H. Freeman, pp. 298-303, 1991.
Gaussian Quadrature
Seeks to obtain the best numerical estimate of an inte-
gral by picking optimal ABSCISSAS xi at which to eval-
uate the function f(x). The FUNDAMENTAL THEOREM
of Gaussian Quadrature states that the optimal Ab-
scissas of the m-point Gaussian Quadrature For-
mulas are precisely the roots of the orthogonal POLY-
NOMIAL for the same interval and WEIGHTING FUNC-
TION. Gaussian quadrature is optimal because it fits all
Polynomials up to degree 2m exactly. Slightly less op-
timal fits are obtained from Radau Quadrature and
Laguerre Quadrature.
W(x)
Interval xi Are Roots Of
1 (-1,1) P n (x)
e~ f (0,oo) L n (x)
2
e~* ( — 00,00) H n (x)
(l-t 2 )- 1/2 (-1,1) T n (x)
(1-< 2 ) 1/2 (-1,1) U n (x)
r 1 ' 2 (0,1) *- 1/2 P 2 n+l(^)
-1/2
(0,1)
Pn{Vi)
To determine the weights corresponding to the Gaus-
sian Abscissas, compute a Lagrange Interpolating
Polynomial for f(x) by letting
*"(») = Y[(X - Xj)
(1)
J=l
(where Chandrasekhar 1967 uses F instead of 7r), so
7V f (Xj) =
dx
= n^ ~ x{>> '
(2)
Then fitting a LAGRANGE INTERPOLATING POLYNOM-
IAL through the m points gives
tt(x)
r-f (x-XjjTr'ixj)
3=1
(3)
for arbitrary points Xi . We are therefore looking for a set
of points Xj and weights Wj such that for a WEIGHTING
Function W(x),
m
^^WjfiXj),
(4)
j=i
with Weight
1 fvrnv*,. ( 5 )
The weights Wj are sometimes also called the CHRIS-
toffel Number (Chandrasekhar 1967). For orthogo-
nal Polynomials <j>j{x) with j=l, . . . , n,
^(x) = Ajir(x)
(6)
(Hildebrand 1956, p. 322), where A n is the COEFFI-
CIENT of x n in 4>n{x), then
M*i) Ja
W(x)
<j>{x)
dx
-4n+l7n
A n <f>n(Xj)<f) n +i(x) J
(7)
720 Gaussian Quadrature
where
7m = J[(j> m {x)} 2 W(x)dx.
Using the relationship
A ( \ ^n+l^n-1 In , , \
<Pn+l(Xi) = —5 <p n -i{Xi)
A n 7n-l
(Hildebrand 1956, p. 323) gives
An
7n-l
^.n-1 $i(£j)^n-l(?j)
(8)
(9)
(10)
(Note that Press et al 1992 omit the factor A n /A n -i.)
In Gaussian quadrature, the weights are all POSITIVE.
The error is given by
/( an) (fl r>
l^(a;)[7r(x)] 2 dx:
7n /< 2 ">(0
A> 2 (2n)!
where a < £ < 6 (Hildebrand 1956, pp. 320-321).
Other curious identities are
(11)
fc=0
[0»(s)] a
Am+ljrr
f>'m+l(x)<i>rn{x) ~ <t>' m (x)<l> m +l(x)] (12)
and
E
[^fc(s)] 2 _ ^m0m(^t)0Tn + l(^) _ _1_
Ik
^4m + l7rr
(13)
(Hildebrand 1956, p. 323).
In the Notation of Szego (1975), let xi n < •< x nn be
an ordered set of points in [a, 6], and let Ai n , . . . , A nn be
a set of Real Numbers. If f(x) is an arbitrary function
on the Closed Interval [a, 6], write the Mechanical
Quadrature as
Qn(f) = / ^Knf(Xyn).
(14)
Here x U n are the Abscissas and \ un are the Cotes
Numbers.
see also Chebyshev Quadrature, Chebyshev-
Gauss Quadrature, Chebyshev-Radau Quadra-
ture, Fundamental Theorem of Gaussian Quad-
rature, Hermite-Gauss Quadrature, Jacobi-
Gauss Quadrature, Laguerre-Gauss Quadra-
ture, Legendre-Gauss Quadrature, Lobatto
Quadrature, Mehler Quadrature, Radau Quad-
rature
References
Abramowitz, M. and Stegun, C. A. (Eds.), Handbook
of Mathematical Functions with Formulas, Graphs, and
Gaussian Sum
Mathematical Tables, 9th printing. New York: Dover,
pp. 887-888, 1972.
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., p. 103, 1990.
Arfken, G. "Appendix 2: Gaussian Quadrature." Mathemat-
ical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 968-974, 1985.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed,
Boca Raton, FL: CRC Press, p. 461, 1987.
Chandrasekhar, S. An Introduction to the Study of Stellar
Structure. New York: Dover, 1967.
Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 319-323, 1956.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Gaussian Quadratures and Orthogonal Poly-
nomials." §4.5 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 140—155, 1992.
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, pp. 37-48 and 340-349, 1975.
Whittaker, E. T. and Robinson, G. The Calculus of Observa-
tions: A Treatise on Numerical Mathematics, J^th ed. New
York: Dover, pp. 152-163, 1967.
Gaussian Sum
g-l
S(p,g) = 5>-™ a */*, (1)
where p and q are RELATIVELY PRIME INTEGERS. If
(n^n 1 ) = 1, then
S(m,nri) = S{mri \n)S(mn,n). (2)
Gauss showed
iV 1 *'* = V=T^ (3)
r—
for Odd q. A more general result was obtained by
Schaar. For p and q of opposite PARITY (i.e., one is
Even and the other is Odd), Schaar's Identity states
9-1
./a2-*t
— nirp/q
„-«/4
v/9
e
r=0 v * r=0
Vp
p-1
nir 2 q/p
(4)
Such sums are important in the theory of QUADRATIC
Residues.
see also Kloosterman's Sum, Schaar's Identity,
Singular Series
References
Evans, R. and Berndt, B. "The Determination of Gauss
Sums." Bull. Amer. Math. Soc. 5, 107-129, 1981.
Katz, N. M. Gauss Sums, Kloosterman Sums, and Mon-
odromy Groups. Princeton, NJ: Princeton University
Press, 1987.
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, pp. 132—134,
1994.
Gear Graph
General Prismatoid 721
Gear Graph
A Wheel Graph with a Vertex added between each
pair of adjacent VERTICES.
Gegenbauer Function
see Ultraspherical Function
Gegenbauer Polynomial
see Ultraspherical Polynomial
Gelfond-Schneider Constant
The number 2^ = 2.66514414 . . . which is known to be
Transcendental by Gelfond's 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. 107, 1996.
Gelfond-Schneider Theorem
see Gelfond's Theorem
Gelfond's Theorem
Also called the Gelfond-Schneider Theorem. a b is
Transcendental if
1. a is Algebraic ^ 0, 1 and
2. b is Algebraic and Irrational.
This provides the solution to the seventh of Hilbert's
Problems.
see also Algebraic Number, Hilbert's Problems,
Irrational Number, Transcendental Number
References
Baker, A. Transcendental Number Theory. London: Cam-
bridge University Press, 1990.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 107, 1996.
Genaille Rods
Numbered rods which can be used to perform multipli-
cation.
see also Napier's Bones
References
Gardner, M. "Napier's Bones." Ch. 7 in Knotted Dough-
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, 1986.
Genera
see Fundamental Theorem of Genera
General Linear Group
The general linear group GL n (q) is the set of n x n Ma-
trices with entries in the FIELD F^ which have NON-
ZERO Determinant.
see also Langlands Reciprocity, Projective Gen-
eral Linear Group, Projective Special Linear
Group, Special Linear 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). n §2.1 in Atlas of Finite Groups:
Maximal Subgroups and Ordinary Characters for Simple
Groups. Oxford, England: Clarendon Press, p. x, 1985.
General Orthogonal Group
The general orthogonal group GO n (q>F) is the SUB-
GROUP of all elements of the Projective General
Linear Group, that fix the particular nonsingular
Quadratic Form F. The determinant of such an ele-
ment is ±1.
see also PROJECTIVE GENERAL LINEAR 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),
PGOn(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.
General Position
An arrangement of points with no three COLLINEAR, or
of lines with no three concurrent.
see also Ordinary Line, Near-Pencil
References
Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903-909, 1989.
General Prismatoid
A solid such that the Area A y of any section parallel to
and a distance y from a fixed PLANE can be expressed
as
A y = ay 3 4- by 2 + cy + d.
The volume of such a solid is the same as for a PRISMA-
TOID,
V= \h(Ai+AM + A 2 ).
Examples include the CONE, CYLINDER, PRISMATOID,
Sphere, and Spheroid.
see also PRISMATOID, PRISMOID
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 132, 1987.
722 General Unitary Group
Generalized Fibonacci Number
General Unitary Group
The general unitary group GU n (q) is the SUBGROUP of
all elements of the General Linear Group GL(q 2 )
that fix a given nonsingular Hermitian form. This is
equivalent, in the canonical case, to the definition of
GU n as the group of Unitary Matrices.
References
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker,
R. A.; and Wilson, R A. "The Groups GU n (q) t SU n (q),
PGU n {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.
Generalized Fibonacci Number
A generalization of the FIBONACCI NUMBERS defined
by 1 = G\ = G 2 = ... = G c -i and the Recurrence
Relation
G n = G n -1 + Gn-c> (1)
These are the sums of elements on successive diagonals
of a left-justified Pascal's Triangle beginning in the
left-most column and moving in steps of c - 1 up and
1 right. The case c = 2 equals the usual FIBONACCI
Number. These numbers satisfy the identities
G\ + C?2 H~ Gz + ■ . • + G n — Gn+3 ~ 1
(2)
Generalized Cone
A Ruled Surface is called a generalized cone if it can
be parameterized by x(u,v) = p + vy(u), where p is
a fixed point which can be regarded as the vertex of
the cone. A generalized cone is a REGULAR SURFACE
wherever uyxy' ^ 0. The above surface is a generalized
cylinder over a Cardioid. A generalized cone is a Flat
Surface.
see also Cone
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 341-342, 1993.
Generalized Cylinder
A Ruled Surface is called a generalized cylinder if it
can be parameterized by x(ti,v) = vp + y(«), where p
is a fixed point. A generalized cylinder is a Regular
Surface wherever y' x p / 0, The above surface is
a generalized cylinder over a CARDIOID. A generalized
cylinder is a Flat Surface,
see also Cylinder
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 341-342, 1993.
C?3 + G& + Gq + . . . + Gzk = Gzk+i — 1 (3)
Gi + G 4 + G 7 + . . . + GWi = Gsm-2 (4)
Gt. + G$ + G$ + . . . + t?3fc + 2 = Gsk + 3 (5)
(Bicknell- Johnson and Spears 1996). For the special
case c = 3,
G n -\- w = G w — 2G n + Gw-sGn + l + Gu, — lCr n +2* (6)
Bicknell- Johnson and Spears (1996) give many further
identities.
Horadam (1965) defined the generalized Fibonacci num-
bers {w n } as w n = w n {a^ 6;p, g), where a, 6, p, and q are
Integers, w — a, w\ = 6, and w n = v w -n.-\ — qw n -2
for n > 2. They satisfy the identities
W n W n+2 r - eq n U r = Wn+r (7)
4w n W n + l 2 W n+ 2 + {Vjq n ) 2 = (WnWn + 2 + W n +\ 2 ) 2 (8)
= w n +2 4 + eq n (p 2 + q)w n+2 2 + e 2 q 2n+1 p 2 (9)
4w n W n + l'Wn + 2'Wn+4'Wn + 5'Wn+6
+e 2 q 2n {w n U 4 U 5 - w n+1 U 2 Us - WnUiUs) 2
= (Wn + lWn+2Wn+6 + ^n^n+4^n+5 ) , (10)
where
e = pab — qa — b
U n = u>n(0, l;p,qr).
(ii)
(12)
The final above result is due to Morgado (1987) and is
called the Morgado Identity.
Another generalization of the Fibonacci numbers is de-
noted x n . Given x\ and X2^ define the generalized Fi-
bonacci number by x n ~ x n -2 + x n ~i for n > 3,
/ v X n = Xn + 2 — X2
(13)
Generalized Function
Generalized Hyperbolic Functions 723
y ^Xn = 11^7
(14)
X n 2 ~ Xn-xXn + 2 = (-l) n (aS 2 ~ X\ - ^l^), (15)
where the plus and minus signs alternate.
see also Fibonacci Number
References
Bicknell, M. "A Primer for the Fibonacci Numbers, Part
VIII: Sequences of Sums from Pascal's Triangle." Fib.
Quart. 9, 74-81, 1971.
Bicknell-Johnson, M. and Spears, C. P. "Classes of Identities
for the Generalized Fibonacci Numbers G n = G n _i+<j? n _ c
for Matrices with Constant Valued Determinants." Fib.
Quart. 34, 121-128, 1996.
Dujella, A. "Generalized Fibonacci Numbers and the Prob-
lem of Diophantus." Fib. Quart. 34, 164-175, 1996.
Horadam, A. F. "Generating Functions for Powers of a Cer-
tain Generalized Sequence of Numbers." Duke Math. J.
32, 437-446, 1965.
Horadam, A. F. "Generalization of a Result of Morgado."
Portugaliae Math. 44, 131-136, 1987.
Horadam, A. F. and Shannon, A. G. "Generalization of Iden-
tities of Catalan and Others." Portugaliae Math. 44, 137-
148, 1987.
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.
Generalized Function
The class of all regular sequences of PARTICULARLY
Well-Behaved Functions equivalent to a given reg-
ular sequence (sometimes also called a DISTRIBUTION
or Functional). A generalized function p{x) has the
properties
/oo /»oo
p(x)f(x)dx = - / p(x)f'(x)dx
-oo J — oo
/oo po
p^f(x)dx = (-l) n /
-oo J — c
f(x)dx = (~l) n I p(x)f (n) (x)dx.
The Delta Function is a generalized function.
see also Delta Function
Generalized Helicoid
The SURFACE generated by a twisted curve C when ro-
tated about a fixed axis A and, at the same time, dis-
placed Parallel to A so that the velocity of displace-
ment is always proportional to the Angular Velocity
of Rotation.
see also Generalized Helix, Helicoid, Helix
References
do Carmo, M. P.; Fischer, G.; Pinkall, U.; and Reckziegel, H.
"General Helicoids." §3.4.3 in Mathematical Models from
the Collections of Universities and Museums (Ed. G. Fis-
cher). Braunschweig, Germany: Vieweg, pp. 36-37, 1986.
Fischer, G. (Ed.). Plate 89 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 85, 1986.
Kreyszig, E. Differential Geometry. New York: Dover, p. 88,
1991.
Generalized Helix
The GEODESICS on a general cylinder generated by lines
Parallel to a line / with which the Tangent makes a
constant ANGLE.
see also Helix
Generalized Hyperbolic Functions
In 1757, V. Riccati first recorded the generalizations of
the Hyperbolic Functions defined by
^) = c £p^)!*" fc+r .
(i)
for r = 0, . . . , n — 1, where a is COMPLEX, and where
the normalization is taken so that
F n %(0) = 1.
(2)
This is called the a-hyperbolic function of order n of the
kth kind. The functions F^ r satisfy
and
where
J£, r (*) = (#5)- r (tf5*)
/<*>(*) = a f{x),
(3)
(4)
/( fc >(0) = (° fe ^ r ' 0<fe<n-l, ^
In addition,
± p« M _ / *Er-i (*) for < r < n - 1
dx ^ rW -\<„_,W forr = 0. W
The functions give a generalized EULER Formula
Tl-l
e v5 =^(^a)'J^, r (*). (7)
Since there are n nth roots of a, this gives a system of
n linear equations. Solving for F^ r gives
n-l
Fl r {x) = -(v^)" r yu; n - rfc expK fc ifex), (8)
fc=0
where
oj n = exp
is a Primitive Root of Unity,
The Laplace Transform is
(9)
/»00 n - r — l r
/ e- st FZJat) dt = — .
Jo s" + aa n
(10)
724 Generalized Hypergeometric Function
Generalized Hypergeometric Function
The generalized hyperbolic function is also related to
the Mittag-Leffler Function E y (x) by
F*, (x) = E n (x n ).
(11)
The values n = 1 and n = 2 give the exponential and
circular/hyperbolic functions (depending on the sign of
a), respectively.
F^oix) = cosh(\/ax)
sinh(yfax)
f?a*) =
yfc
(12)
(13)
(14)
For a = 1, the first few functions are
*i,o(aO = e x
F 2 l ,o(x) = cosh a;
F^i{x) — sinhz
Flo{x) = \[e x +2e- x/2 cos(|v / 3x)]
Fi tl (x) = \[e x + 2e- a:/2 cos(±\/3z + |tt)]
^(x) = \[e x + 2e" x/2 cos(f v^x - |tt)]
^4,0 0*0 = | (cosh x + cos z)
^4,i 0*0 — § (sinh z + sin a:)
F^ 2 {x) = \ (cosh x — cos a:)
^4,3 ( x ) — |(sinha; — sinz).
see also HYPERBOLIC FUNCTIONS, MlTTAG-LEFFLER
Function
References
Kaufman, H. "A Biographical Note on the Higher Sine Func-
tions." Scripta Math. 28, 29-36, 1967.
Muldoon, M, E. and Ungar, A. A. "Beyond Sin and Cos."
Math. Mag. 69, 3-14, 1996.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, 1996.
Ungar, A. "Generalized Hyperbolic Functions." Amer. Math.
Monthly 89, 688-691, 1982.
Ungar, A. "Higher Order Alpha-Hyperbolic Functions." In-
dian J. Pure. Appl. Math. 15, 301-304, 1984.
Generalized Hypergeometric Function
The generalized hypergeometric function is given by a
Hypergeometric Series, i.e., a series for which the
ratio of successive terms can be written
afc+i _ P(k) _ {k + ai)(k + a 2 ) - • - (k + a P )
a k ~ Q(k) ~ (k + b 1 )(k + b 2 )--(k + b q )(k + l) X '
(1)
(The factor of k + 1 in the DENOMINATOR is present for
historical reasons of notation.) The resulting generalized
hypergeometric function is written
/ ^CLkX
k=0
, u> p _
\X
(6i
(ai)fc(a 2 )fc *•• (a P )fc x k
(bi) k b(b 2 ) k .-.(b q ) k fc!'
(2)
(3)
where (a) k is the POCHHAMMER SYMBOL or RISING
Factorial
(a) h = ^ + *) =a(a+ i)...( a + A ._i). (4 )
T(a)
If the argument x — 1, then the function is abbreviated
pJTq
&1 j 0,2 ■ • - , Clp
6i, 62, • • . ,b q
O-l , 0,2 * ■ ■ 5 tip
&1,&2, . . . , &q
;x
(5)
2 Fi(a, 6;c;z) is "the" Hypergeometric Function,
and iFi{a\b;z) = M(z) is the Confluent Hypergeo-
metric Function. A function of the form oFi(; b; z) is
called a Confluent Hypergeometric Limit Func-
tion.
The generalized hypergeometric function
ai,a 2 ,.. . , a p+ i t
p+1 P [ 6i,6 a ,...,fcp ''
is a solution to the DIFFERENTIAL EQUATION
(6)
[0(0 + 6- 1)»-(0 + &p-1)
-^(0 + ai)(# + a 2 ) • ■ ■ (0 + a p+ i)]y = 0, (7)
where
dz
The other linearly independent solution is
1 + ai — 61 , 1 — 0,2 — &2 j
. . . , 1 + a p+ i — 61 #
2 — 61, 1 — 62 — &i, ••• ,
1 - 6 P - bi
;*
(8)
(9)
A generalized hypergeometric equation is termed "well
posed" if
1 + ai - 61 + a 2 ~ . . . = b p + a p+ i. (10)
Many sums can be written as generalized hypergeomet-
ric functions by inspection of the ratios of consecutive
terms in the generating Hypergeometric Series. For
example, for
/(n) = £(-!)* ( 2 ;) 2 , (ID
the ratio of successive terms is
a h+l _ (-l) fc+1 ( t +i) 2 _ (k-2nf
CLk
(-i)*(> fc ») 2 (k+ir'
(12)
Generalized Hypergeometric Function
yielding
-2n,-2n
f(n) = 2 Fi
2 Fi(-2n,-2n;l;-l)
(13)
(Petkovsek 1996, pp. 44-45).
Gosper (1978) discovered a slew unusual hypergeo-
metric function identities, many of which were sub-
sequently proven by Gessel and Stanton (1982). An
important generalization of Gosper's technique, called
Zeilberger's Algorithm, in turn led to the powerful
machinery of the WlLF-ZElLBERGER Pair (Zeilberger
1990).
Special hypergeometric identities include GAUSS'S HY-
PERGEOMETRIC Theorem
2 F 1 (a,b;c;l) =
F(c)F(c-a-b)
r(c-a)T(c-b)
for U[c - a - b] > 0, RUMMER'S FORMULA
r(£& + i)r(&-a + i)
2 Fi(a,6;c;-l)
r(6+l)r(|6-a+l)
(14)
(15)
where a — b + c — 1 and b is a positive integer,
Saalschutz's Theorem
3F2(a,b,c;d,e]l) = , -, — (16)
d\ c \(d — a — b)
\c\
{ord-\-e — a-\-b-\-c-\-l with c a negative integer and
(a) n the Pochhammer Symbol, Dixon's Theorem
zF 2 {a, 6, c;d, e; 1)
(la)!(a-b)!(a- c )!(lq-6-c)!
a\{\a - b)\(\a - c)!(o - b - c)
(17)
where 1 + a/2 — b — c has a positive REAL PART, d -
a - b + 1, and e = a - c + 1, the CLAUSEN FORMULA
4^3
abed
e f 9
;i
(2a)| d |(a + b)| d |(26)| d |
(2a + 26)| df a jd |6|d|
(18)
for a+5+c-d = 1/2, e = a+6+1/2, 0+/ = d+1 = 6+5,
d a nonpositive integer, and the DOUGALL-RAMANUJAN
Identity
7^6
Q>1 , <%2 j &Z j &4 , 0,5 , a6 , CI7
;l
&1, &2, &3,&4, &5, &6
(fli + l)n(fli - ^2 - az + l)n
(ai — a 2 + l) n (ai — 03 + l) n
(ai - a2 — a4 + l)n(ai — a3 — a* + l) n
(ai - a4 + l)n(ai — ^2 - a 3 - a 4 + l) n '
(19)
where n — 2a\ + 1 = a 2 + az + a4 + as, a6 = 1 -f ai/2,
a 7 = — n, and 6» = 1 + a\ — a;+i for i = 1, 2, . . . , 6. For
all these identities, (a) n is the POCHHAMMER SYMBOL.
Generalized Matrix Inverse 725
Gessel (1994) found a slew of new identities using WlLF-
ZEILBERGER PAIRS, including the following:
\-a -6,n + l,n + c + l,2n-a-6+l,n+|(3-a-6)l
5p *[n-a -6-c+l,n-a-6+l,27i + 2,n+|(l-a-b) ; J
-0 (20)
-3n,f
c, 3n + 2 3
1.1 -3c
3^2
-36,-
*.|(i-
-3n, :
2 v* 3 ™).4
— & — n ' 3
2 + 5 71 ' 3' ~~ '"» ^ /A ^ ^ . _?_
"41,1 J 27
» S " ^ 2
+
n +
-n,2n + 2
3' 5'
(C+ 3)71(3)71
(1
C)n{ s )n
(I - &)»
(£ + &)»
( 2 )^ \ 2 )^
(21)
(22)
(23)
(Petkovsek et a/. 1996, pp. 135-137).
see also CARLSON'S THEOREM, CLAUSEN FOR-
MULA, Confluent Hypergeometric Function,
Confluent Hypergeometric Limit Function,
Dixon's Theorem, Dougall-Ramanujan Identity,
Dougall's Theorem, Gosper's Algorithm, Heine
Hypergeometric Series, Hypergeometric Func-
tion, Hypergeometric Identity, Hypergeomet-
ric Series, Jackson's Identity, Kummer's The-
orem, Ramanujan's Hypergeometric Identity,
Saalschutz's Theorem, Saalschutzian, Sister
Celine's Method, Thomae's Theorem, Watson's
Theorem, Whipple's Transformation, Wilf-Zeil-
berger Pair, Zeilberger's Algorithm
References
Bailey, W. N. Generalised Hypergeometric Series. Cam-
bridge, England: Cambridge University Press, 1935.
Dwork, B, Generalized Hypergeometric Functions. Oxford,
England: Clarendon Press, 1990.
Exton, H. Multiple Hypergeometric Functions and Applica-
tions. New York: Wiley, 1976.
Gessel, I. "Finding Identities with the WZ Method." Theo-
ret. Comput. Sci. To appear.
Gessel, I. and Stanton, D. "Strange Evaluations of Hyperge-
ometric Series." SIAM J. Math. Anal. 13, 295-308, 1982.
Gosper, R. W. "Decision Procedures for Indefinite Hyper-
geometric Summation." Proc. Nat. Acad. Sci. USA 75,
40-42, 1978.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, 1996.
Saxena, R. K. and Mathai, A. M. Generalized Hypergeomet-
ric Functions with Applications in Statistics and Physical
Sciences. New York: Springer- Verlag, 1973.
Slater, L. J. Generalized Hypergeometric Functions. Cam-
bridge, England: Cambridge University Press, 1966.
Zeilberger, D. "A Fast Algorithm for Proving Terminating
Hypergeometric Series Identities." Discrete Math. 80,
207-211, 1990.
Generalized Matrix Inverse
see MOORE-PENROSE GENERALIZED MATRIX INVERSE
726
Generalized Mean
Gentle Diagonal
Generalized Mean
A generalized version of the MEAN
w^fjzy
1/*
(i)
with parameter t which gives the GEOMETRIC MEAN,
Arithmetic Mean, and Harmonic Mean as special
cases:
(2)
lim m(t) = G
t-»o
m(l) = A
m(-l) = H.
(3)
(4)
see also Mean
Generalized Remainder Method
An algorithm for computing a UNIT FRACTION.
Generating Function
A Power Series
CO
f(x) = y ^a n x n
whose Coefficients give the Sequence {a , a x ,
. . . }. The Mathematical (Wolfram Research, Cham-
paign, IL) function DiscreteMath'RSolve'PowerSum
gives the generating function of a given expression, and
ExponentialPowerSum gives the exponential generating
function.
Generating functions for the first few powers are
1 :
n :
n :
^3 .
x(x 2 +4x + l)
(x-1) 4
s(x + l)(x 2 + 10x + l)
(x-l)5
= x 4- 2a? 2 + 3z 3 + 4a; 4 + . .
= x + 4x 2 + 9x 3 + 16x 4 + .
= x + Sx 2 + 27a: 3 + . . .
= z + 16x 2 + 81z 3 + ....
see a/50 Moment-Generating Function, Recur-
rence Relation
References
Wilf, H. S. Generatingfunctionology, 2nd ed. New York:
Academic Press, 1990.
Generation
In population studies , the direct offspring of a refer-
ence population (roughly) constitutes a single genera-
tion. For a Cellular Automaton, the fundamental
unit of time during which the rules of reproduction are
applied once is called a generation.
Generator (Digitadition)
An Integer used to generate a Digitadition. A num-
ber can have more than one generator. If a number has
no generator, it is called a SELF NUMBER.
Generator (Group)
An element of a Cyclic GROUP, the POWERS of which
generate the entire GROUP.
References
Arfken, G. "Generators." §4.11 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 261-
267, 1985.
Genetic Algorithm
An adaptive ALGORITHM involving search and optimiza-
tion first used by John Holland. Holland created an elec-
tronic organism as a binary string ("chromosome"), and
then used genetic and evolutionary principles of fitness-
proportionate selection for reproduction (including ran-
dom crossover and mutation) to search enormous solu-
tion spaces efficiently. So-called genetic programming
languages apply the same principles, using an expres-
sion tree instead of a bit string as the "chromosome."
see also Cellular Automaton
Genocchi Number
A number given by the GENERATING FUNCTION
2t
7 = I>A-
1 ^-— ' n!
It satisfies G\ = 1, G3
coefficients are given by
G 7
., and even
2 2n )B 2n
G 2 n = 2(1
= 2n£ 2n _i(0),
where B n is a Bernoulli Number and E n (x) is an
Euler Polynomial. The first few Genocchi numbers
for n Even are -1, 1, -3, 17, -155, 2073, . . . (Sloane's
A001469).
see also Bernoulli Number, Euler Polynomial
References
Comtet, L. Advanced Combinatorics: The Art of Finite and
Infinite Expansions, rev. enl. ed.ordrecht, Netherlands:
Reidel, p. 49, 1974.
Kreweras, G. "An Additive Generation for the Genocchi
Numbers and Two of its Enumerative Meanings." Bull.
Inst. Combin. Appl. 20, 99-103, 1997.
Kreweras, G. "Sur les permutations comptees par les nombres
de Genocchi de 1-iere et 2-ieme espece." Europ. J. Comb.
18, 49-58, 1997.
Sloane, N. J. A. Sequence A001469/M3041 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Gentle Diagonal
see Pascal's Triangle
Gentle Giant Group
Gentle Giant Group
see Monster Group
Genus (Curve)
One of the Plucker Characteristics, defined by
p = f(n-l)(n-2)-(<5 + K) = |(m-l)(m-2)-(r + t),
where m is the class, n the order, 8 the number of nodes,
k the number of CUSPS, i the number of stationary tan-
gents (Inflection Points), and r the number of Bi-
TANGENTS.
see also Riemann Curve Theorem
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p, 100, 1959.
Genus (Knot)
The least genus of any SEIFERT Surface for a given
Knot. The Unknot is the only Knot with genus 0.
Genus (Surface)
A topologically invariant property of a surface defined
as the largest number of nonintersecting simple closed
curves that can be drawn on the surface without sepa-
rating it. Roughly speaking, it is the number of HOLES
in a surface.
see also Euler Characteristic
Genus Theorem
A Diophantine Equation
2 2
x -\-y = p
can be solved for p a Prime Iff p = 1 (mod 4) or p = 2.
The representation is unique except for changes of sign
or rearrangements of x and y.
see also Composition Theorem, Fermat's Theorem
Geocentric Latitude
An Auxiliary Latitude given by
(j) g = tan" [(1 — e ) tan0].
The series expansion is
(j) g = - e2 sin(20) + \e 2 2 sin(40) + \e 2 z sin(60) + . . . ,
where
e 2 =
2-e 2 *
see also Latitude
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, pp. 17-18, 1987.
Geodesic 727
Geodesic
Given two points on a surface, the geodesic is defined
as the shortest path on the surface connecting them.
Geodesies have many interesting properties. The NOR-
MAL VECTOR to any point of a GEODESIC arc lies along
the normal to a surface at that point (Weinstock 1974,
p. 65).
Furthermore, no matter how badly a SPHERE is dis-
torted, there exist an infinite number of closed geodes-
ies on it. This general result, demonstrated in the early
1990s, extended earlier work by Birkhoff, who proved
in 1917 that there exists at least one closed geodesic
on a distorted sphere, and Lyusternik and Schirelmann,
who proved in 1923 that there exist at least three closed
geodesies on such a sphere (Cipra 1993).
For a surface g(x,y,z) = 0, the geodesic can be found
by minimizing the Arc Length
.= jds= J ^dx 2 +dy 2 +dz 2 .
But
dx dx
dx = -pr-du + Tr-dv
du ov
f dx^
(i)
(2)
and similarly for dy 2 and dz 2 . Plugging in,
-/{[(IHS9"+(I)>'
[" dx dx dy dy . dz dz 1
l du dv du dv du dv J
+ [(!)'+ (SO* +(!)"H
This can be rewritten as
L = / ^P + 2Qv' + Rv' 2 du
= / y/Pu' 2 + 2Qu f +Rdv,
where
u
dv
du
du
dv
and
_ dx dx dy dy dz dz
~~ du dv du dv du dv
«-(s) , + (S)' + (s)'-
■ (4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
728 Geodesic
Geodesic Curvature
Taking derivatives,
dL
dv
i-(P + 2Qv' + Rv t2 )- 1/2
(£+& + &•) <■»
dL
dv
7 = \{P + 2Qv' + /to' 2 ) ' (2Q + 2Rv'), (13)
so the Euler-Lagrange Differential Equation
then gives
ov ov ov
Q + Rv'
2^/P + 2Qv' + Rv' 2 du \^JP + 2Qv' + Rv' 2
= 0.
(14)
In the special case when P, Q, and R are explicit func-
tions of u only,
Q + Rv'
y/P + 2Qv f + Rv' 2
Q 2 + 2Qito' + R 2 v' 2
■ Ci
: Ci
(15)
P + 2Qv' + Rv' 2
v f2 R(R - ci 2 ) + 2v'Q(R - ci 2 ) + (Q 2 - Pc x 2 ) - (17)
[2Q( Cl 2 - R)
2R(R- Cl 2 )
±^4Q 2 {R - ci 2 ) 2 - m{R - d 2 )(Q 2 - Pci 2 ) ]. (18)
Now, if P and R are explicit functions of u only and
= o,
U ~ 2 J R(#-ci 2 )
— ci
R{R-d 2 )
V = C1 J
R(R-d 2 )
du.
(19)
(20)
In the case Q = where P and i? are explicit functions
of v only, then
dv ^ u dv a
2 v / PTP^ 72 " du Ky/FTRv 12
Rv'
= 0, (21)
\/P + Rv' 2
+ H)
x t/(2ifc;V)
(P + ifo/ 2 ) 3 /2
= (22)
£♦<-"■♦££-• <»>
Rv 1 '
VP + Pv /2
\/P + P^' 2 = Ci
(24)
Rv' 2 - (P + ifr/ 2 ) = ci y/P + ito' 2 (25)
(-^) =P + Pv' 2 (26)
p2 -- 2p ,2
(27)
Pci 2
and
U=C! I J
R
P 2 -Ci 2 P
<it;.
(28)
For a surface of revolution in which y = g(x) is rotated
about the :r-axis so that the equation of the surface is
2,2 2/ \
y +z =g (x),
the surface can be parameterized by
x — u
y — g(u) cosv
z = </(u)sinv.
(16) The equation of the geodesies is then
y/l + \g'{u)]*du
f yi + ^c
v = a I ==
9{u)^[g{u)] 2 -cS
(29)
(30)
(31)
(32)
(33)
see also ELLIPSOID GEODESIC, GEODESIC CURVATURE,
Geodesic Dome, Geodesic Equation, Geodesic
Triangle, Great Circle, Harmonic Map, Oblate
Spheroid Geodesic, Paraboloid Geodesic
References
Cipra, B. What's Happening in the Mathematical Sciences,
Vol. 1. Providence, Rl: Amer. Math. Soc, pp. 27, 1993.
Weinstock, R. Calculus of Variations, with Applications to
Physics and Engineering. New York: Dover, pp. 26—28
and 45-46, 1974.
Geodesic Curvature
For a unit speed curve on a surface, the length of the
surface-tangential component of acceleration is the geo-
desic curvature k q . Curves with k 9 = are called
Geodesics. For a curve parameterized as ct(i) =
x(u(£),v(£)), the geodesic curvature is given by
k 9 = y/EG-Ftl-rliU* + T\ 2 v' z - {2T\ 2 - T\ x )u n v
+(2ri 2 -rL)uV 2 +«%'-«"«'],
where E, F, and G are coefficients of the first FUNDA-
MENTAL Form and r£- are Christoffel Symbols of
the Second Kind.
see also Geodesic
References
Gray, A. "Geodesic Curvature." §20.5 in Modern Differential
Geometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 402-407, 1993.
Geodesic Dome
Geodesic Dome
Geometric Construction 729
A Triangulation of a Platonic Solid or other
Polyhedron to produce a close approximation to a
Sphere. The nth order geodesation operation replaces
each polygon of the polyhedron by the projection onto
the ClRCUMSPHERE of the order n regular tessellation
of that polygon. The above figure shows geodesations
of orders 1 to 3 (from top to bottom) of the TETRA-
HEDRON, Cube, Octahedron, Dodecahedron, and
ICOSAHEDRON (from left to right).
R. Buckminster Fuller designed the first geodesic dome
(i.e., geodesation of a HEMISPHERE). Fuller's dome was
constructed from an ICOSAHEDRON by adding ISOSCE-
LES Triangles about each Vertex and slightly reposi-
tioning the Vertices. In such domes, neither the VER-
TICES nor the centers of faces necessarily lie at exactly
the same distances from the center. However, these con-
ditions are approximately satisfied.
In the geodesic domes discussed by KnifFen (1994), the
sum of Vertex angles is chosen to be a constant. Given
a Platonic Solid, let e = 2e/v be the number of
Edges meeting at a Vertex and n be the number of
EDGES of the constituent POLYGON. Call the angle of
the old Vertex point A and the angle of the new Ver-
tex point F. Then
A = B
2eA = nF
2A + F = 180°.
Solving for A gives
nA 2e' , n . / e
2A + — A = 2A 1+ —
n \ n
A = 90 c
180°
e' -f- n'
and
F= ^ A = 180°-^-.
n e' 4- n
The Vertex sum is
S = nF = 180 c
e' + n
(i)
(2)
(3)
(4)
(5)
(6)
(7)
Solid
■ /
V
e'
n
A
F
£
tetrahedron
3
3
45°
90°
270°
cube
24
14
3
4
5lf °
81 f °
308f °
octahedron
4
3
38f °
108 1°
308|°
dodecahedron
60
32
3
5
56±°
71±°
4
337 \ °
icosahedron
5
3
33 f °
118f °
4
337±°
see also Triangular Symmetry Group
References
Kenner, H. Geodesic Math and How to Use It. Berkeley, CA:
University of California Press, 1976.
KnifFen, D. "Geodesic Domes for Amateur Astronomers."
Sky and Telescope, pp. 90-94, Oct. 1994.
Pappas, T. "Geodesic Dome of Leonardo da Vinci." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 81, 1989.
Geodesic Equation
or
dr 2
0.
see also GEODESIC
Geodesic Flow
A type of FLOW technically denned in terms of the TAN-
GENT Bundle of a Manifold.
see also Dynamical System
Geodesic Triangle
A Triangle formed by the arcs of three Geodesics on
a smooth surface.
see also Integral Curvature
Geodetic Latitude
see Latitude
Geographic Latitude
see Latitude
Geometric Construction
In antiquity, geometric constructions of figures and
lengths were restricted to use of only a Straightedge
and COMPASS. Although the term "RULER" is some-
times used instead of "STRAIGHTEDGE," no markings
which could be used to make measurements were al-
lowed according to the Greek prescription. Furthermore,
the "Compass" could not even be used to mark off dis-
tances by setting it and then "walking" it along, so the
Compass had to be considered to automatically collapse
when not in the process of drawing a CIRCLE.
Because of the prominent place Greek geometric con-
structions held in Euclid's Elements, these constructions
730
Geometric Construction
Geometric Construction
are sometimes also known as EUCLIDEAN CONSTRUC-
TIONS. Such constructions lay at the heart of the Geo-
metric Problems of Antiquity of Circle Squar-
ing, Cube Duplication, and Trisection of an An-
gle. The Greeks were unable to solve these problems,
but it was not until hundreds of years later that the
problems were proved to be actually impossible under
the limitations imposed.
Simple algebraic operations such as a + 6, a — 6, ra
(for r a Rational Number), a/b, ab, and y/x can be
performed using geometric constructions (Courant and
Robbins 1996). Other more complicated constructions,
such as the solution of APOLLONIUS' PROBLEM and the
construction of Inverse Points can also accomplished.
^Aine segment bisector
One of the simplest geometric constructions is the con-
struction of a Bisector of a Line Segment, illustrated
above.
N { r N 5 F O E AT 3
Pentagon 17-gon
The Greeks were very adept at constructing POLYGONS,
but it took the genius of Gauss to mathematically de-
termine which constructions were possible and which
were not. As a result, Gauss determined that a se-
ries of POLYGONS (the smallest of which has 17 sides;
the HEPTADECAGON) had constructions unknown to
the Greeks. Gauss showed that the CONSTRUCTIBLE
POLYGONS (several of which are illustrated above) were
closely related to numbers called the FERMAT PRIMES.
Wernick (1982) gave a list of 139 sets of three located
points from which a TRIANGLE was to be constructed.
Of Wernick's original list of 139 problems, 20 had not
yet been solved as of 1996 (Meyers 1996).
It is possible to construct Rational Numbers and
Euclidean Numbers using a Straightedge and
COMPASS construction. In general, the term for a
number which can be constructed using a COMPASS
and Straightedge is a Constructible Number.
Some Irrational Numbers, but no Transcenden-
tal Numbers, can be constructed.
It turns out that all constructions possible with a COM-
PASS and Straightedge can be done with a Compass
alone, as long as a line is considered constructed when
its two endpoints are located. The reverse is also true,
since Jacob Steiner showed that all constructions pos-
sible with Straightedge and Compass can be done
using only a straightedge, as long as a fixed CIRCLE and
its center (or two intersecting CIRCLES without their
centers, or three nonintersecting Circles) have been
drawn beforehand. Such a construction is known as a
Steiner Construction.
Geometrography is a quantitative measure of the
simplicity of a geometric construction. It reduces ge-
ometric constructions to five types of operations, and
seeks to reduce the total number of operations (called
the "Simplicity") needed to effect a geometric con-
struction.
Dixon (1991, pp. 34-51) gives approximate construc-
tions for some figures (the HEPTAGON and Nonagon)
and lengths (Pi) which cannot be rigorously con-
structed. Ramanujan (1913-14) and Olds (1963) give
geometric constructions for 355/113 « it. Gardner
(1966, pp. 92-93) gives a geometric construction for
3+ ^ = 3.1415929... ^ tt.
Constructions for n are approximate (but inexact) forms
of Circle Squaring.
see also Circle Squaring, Compass, Constructible
Number, Constructible Polygon, Cube Duplica-
tion, Elements, Fermat Prime, Geometric Prob-
lems of Antiquity, Geometrography, Mascher-
oni Construction, Napoleon's Problem, Neu-
sis Construction, Plane Geometry, Polygon,
Poncelet-Steiner Theorem, Rectification, Sim-
plicity, Steiner Construction, Straightedge,
Trisection
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 96-97,
1987.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 191-202, 1996.
Courant, R. and Robbins, H. "Geometric Constructions. The
Algebra of Number Fields." Ch. 3 in What is Mathemat-
ics?: An Elementary Approach to Ideas and Methods, 2nd
ed. Oxford, England: Oxford University Press, pp. 117-
164, 1996.
Dantzig, T. Number, The Language of Science. New York:
Macmillan, p. 316, 1954.
Dixon, R. Mathographics. New York: Dover, 1991.
Geometric Distribution
Geometric Distribution 731
Eppstein, D. "Geometric Models." http://www . ics . uci .
edu/-eppstein/ junkyard/model. html.
Gardner, M. "The Transcendental Number Pi." Ch. 8 in
Martin Gardner 's New Mathematical Diversions from Sci-
entific American. New York: Simon and Schuster, 1966.
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,
Herterich, K. Die Konstruktion von Dreiecken. Stuttgart:
Ernst Klett Verlag, 1986.
Krotenheerdt, O. "Zur Theorie der Dreieckskonstruktio-
nen." Wissenschaftliche Zeitschrift der Martin- Luther-
Univ. Halle- Wittenberg, Math. Naturw. Reihe 15, 677—
700, 1966.
Meyers, L. F. "Update on William Wernick's 'Triangle Con-
structions with Three Located Points.'" Math. Mag. 69,
46-49, 1996.
Olds, C. D. Continued Fractions. New York: Random House,
pp. 59-60, 1963.
Petersen, J. "Methods and Theories for the Solution of Prob-
lems of Geometrical Constructions." Reprinted in String
Figures and Other Monographs. New York: Chelsea, 1960.
Plouffe, S.. "The Computation of Certain Numbers Us-
ing a Ruler and Compass." Dec. 12, 1997. http://www.
research. at t . com/ ~nj as /sequences/ J IS/ compass .html.
Posamentier, A. S. and Wernick, W. Advanced Geometric
Constructions. Palo Alto, CA: Dale Seymour Pub., 1988.
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart. J. Pure. Appl. Math. 45, 350-372, 1913-1914.
Wernick, W. "Triangle Constructions with Three Located
Points." Math. Mag. 55, 227-230, 1982.
Geometric Distribution
A distribution such that
P(n) = q n ~ l p = p(l -p) n_1 ,
(1)
where q ~ 1 — p and for n = 1, 2, The distribution
is normalized since
E p w = Ert>^£* n = i
P— = i = i
q p
(2)
The Moment-Generating Function is
tf(t)=p[l-(l-p)e"]-\ (3)
M(t) = {e ) — y, e PQ —V y, e Q
M'{t) =p
oo t
t ST^t t ^n pe
Tl —
-(l_ e * 9 ) e «_ c t(_ e « g) -
pe
M"(t)=p
(1-^)2
pie* - ge 2t + ge 2t ) __
(1-e'g) 2 (1-e'g) 2
(l-e^) 2 e t -e t 2(l-e t g)(-e t g)
(4)
(5)
= P
= P
= P
(1-e'g) 4
(1 - 2e t q + e 2t g 2 )e l + 2ge 2t (l - e*q)
(1 -c*g) 4
e l -2e 2t q + e* t q 2 + 2qe 2t -2q 2 e u
(l-e'g) 4
e t -q 2 e 3t _ pe* (1 - q 2 e 2t )
M"'(t) =
Therefore,
(l-e*g) 4 (1-e'g) 4
pe^l + qe*)
(l-e*g) 3
pe* [1 + 4e'(l -p) + e 2t {\-p) 2 }
(1-e* + e'p) 4
M'(0)=/ii=/i:
P = P_ = 1
(1 - q) 2 p 2 p
(l-\-q) 6 p 3 p 2
M'"(0) = ^= (6 - 6p + p2)
(6)
(7)
(8)
(9)
(10)
M(4)(0) = ^ = (^2KV+ifc^), (11 )
and
M2 =M2 - Oi) 2 =
_L - 1 ~' p
r>2 ~~ -n2
= Q_
P 2
^ 3 = Ms - 3^2^i + 2(//i) 3
= 6-6p + p ! _ 3 2-pl +
p 6 p z p \p
_ 6-6p + p 2 -3(2-p) + 2
^ (p-l)(p-2)
= (p-2)(-p 2 + 12p-12) ^~Q p + v 2 l
p 4 p 3 p
+6 ^n 2 -3^ 4
(12)
(13)
P \P ) \P
(p - 1)(-P 2 + 9p - 9)
(14)
732
Geometric Distribution
so the Mean, Variance, Skewness, and Kurtosis are
given by
M
1
P
(15)
2
= /*2
5
P 2
(16)
7i
a 3
(P-1)(P
p3
- 2, i
( p>
V 1 -P
\ 3/2
(P
(1-
-1)(p-2) .
-p)y/l ~P
2-
-P
-P
2-p
(17)
72
_ ^ 4
~ <J 4
-3=^"
l)("P 2
p4d-
+ 9p-
B 4
" 9 )-3
-9 + 9p-p 2
(P-1)
3
P 2
-6p + 6
MM
1-P v '
In fact, the moments of the distribution are given ana-
lytically in terms of the POLYLOGARITHM function,
oo
Mfc = Yi p ^ nk = y^p( i ~p) n ~ in '
n=l
n=l
n-1 k _ pLi-fe(l -p)
1-P
(19)
For the case p = 1/2 (corresponding to the distribu-
tion of the number of COIN TOSSES needed to win in
the Saint Petersburg Paradox) this formula imme-
diately gives
Mi = 2 (20)
/i' 3 = 6 (21)
p' 3 = 26 (22)
fi' 4 = 150, (23)
so the Mean, Variance, Skewness, and Kurtosis in
this case are
M = 2
a 2 = 2
7i = |V2
72 = ^-
(24)
(25)
(26)
(27)
The first Cumulant of the geometric distribution is
i-p
«i =
p
(28)
and subsequent Cumulants are given by the Recur-
rence Relation
Kr+l = (1-P)
dp
(29)
see also Saint Petersburg Paradox
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 531-532, 1987.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 118, 1992.
Geometric Problems of Antiquity
Geometric Mean
l/n
G= Y[ai
Hoehn and Niven (1985) show that
G{a\ + c, <i2 + c, . . . , a n + c) > c + G(ai, a2, . . . , a„)
for any POSITIVE constant c.
see also Arithmetic Mean, Arithmetic-Geometric
Mean, Carleman's Inequality, Harmonic Mean,
Mean, Root-Mean-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. 10, 1972.
Hoehn, L. and Niven, I. "Averages on the Move." Math.
Mag. 58, 151-156, 1985.
Geometric Mean Index
The statistical INDEX
Pg =
nfe)'
vq~\ 1/S^o
where p n is the price per unit in period n> q n is the
quantity produced in period n, and v n = Pnq-n the value
of the n units.
see also INDEX
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 69, 1962.
Geometric Probability Constants
see Cube Point Picking, Cube Triangle Picking
Geometric Problems of Antiquity
The Greek problems of antiquity were a set of geometric
problems whose solution was sought using only COM-
PASS and Straightedge:
1. Circle Squaring.
2. Cube Duplication.
3. Trisection of an Angle.
Only in modern times, more than 2,000 years after they
were formulated, were all three ancient problems proved
insoluble using only Compass and Straightedge.
Another ancient geometric problem not proved impos-
sible until 1997 is Alhazen's Billiard Problem.
As Ogilvy (1990) points out, constructing the general
Regular Polyhedron was really a "fourth" unsolved
problem of antiquity.
Geometric Progression
see also Alhazen's Billiard Problem, Circle
Squaring, Compass, Constructible Number, Con-
structible polygon, cube duplication, ge-
OMETRIC Construction, Regular Polyhedron,
Straightedge, Trisection
References
Conway, J. H. and Guy, R. K. "Three Greek Problems."
In The Book of Numbers. New York: Springer- Verlag,
pp. 190-191, 1996.
Courant, R. and Robbins, H. "The Unsolvability of the Three
Greek Problems." §3.3 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 117-118 and 134-
140, 1996.
Ogilvy, C S. Excursions in Geometry. New York: Dover,
pp. 135-138, 1990.
Pappas, T. "The Impossible Trio." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 130-132,
1989.
Jones, A.; Morris, S.; and Pearson, K. Abstract Algebra and
Famous Impossibilities. New York: Springer- Verlag, 1991.
Geometric Progression
see Geometric Sequence
Geometric Sequence
A geometric sequence is a SEQUENCE {a k }, k = 1, 2,
. . . , such that each term is given by a multiple r of the
previous one. Another equivalent definition is that a
sequence is geometric IFF it has a zero BIAS. If the
multiplier is r, then the kth term is given by
Geometrography 733
a k = rak-i
r ak-2 — dor .
Without loss of generality, take ao = 1, giving
a k =r .
Geometric Series
A geometric series ]P & a k is a series for which the ratio of
each two consecutive terms a k +i/a k is a constant func-
tion of the summation index k, say r. Then the terms
a k are of the form a k = aor fc , so a k +i/a k = r. If {a*,},
with k = 1, 2, . .., is a GEOMETRIC SEQUENCE with
multiplier — 1 < r < 1 and ao = 1, then the geometric
series
S n — 2_^ a * =
£'
a)
is given by
S n = J2 r * = 1 + r + r * + ■ • • + r n i ( 2 )
fc=0
rS n =r + r 2 +r 3 + ... + r n+1 . (3)
Subtracting
(l-r)5 n -(l + r + r 2 + ... + r n )
-(r + r 2 +r 3 + ... + r n+1 )
= l-r
n+l
1*,'
— r
s.-E-'-^
,n+l
As n — > oo, then
oo
S = Soo = > r k = .
^-^ 1 - r
(4)
(5)
(6)
see also Arithmetic Series, Gabriel's Staircase,
Harmonic Series, Hypergeometric Series, Wheat
and Chessboard Problem
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 10, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 278-279, 1985.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 8, 1987.
Courant, R. and Robbins, H. "The Geometric Progression."
§1.2.3 in What is Mathematics? : An Elementary Approach
to Ideas and Methods, 2nd ed. Oxford, England; Oxford
University Press, pp. 13-14, 1996.
Pappas, T. "Perimeter, Area Sz the Infinite Series." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 134-135, 1989.
Geometrization Conjecture
see Thurston's Geometrization Conjecture
Geometrography
A quantitative measure of the simplicity of a Geomet-
ric Construction which reduces geometric construc-
tions to five steps. It was devised by E. Lemoine.
Si Place a Straightedge's Edge through a given
Point,
S2 Draw a straight Line,
Ci Place a Point of a Compass on a given Point,
C 2 Place a POINT of a COMPASS on an indeterminate
Point on a Line,
C 3 Draw a CIRCLE.
Geometrography seeks to reduce the number of opera-
tions (called the "SIMPLICITY") needed to effect a con-
struction. If the number of the above operations are
denoted mi, ra2, rti, 712, and 713, respectively, then the
Simplicity is mi +m2+n\+ri2+nz and the symbol is
mi Si -\-m2S2 +n\C\ -\-n2C2 +TI3C3. It is apparently an
unsolved problem to determine if a given GEOMETRIC
Construction is of the smallest possible simplicity.
734 Geometry
Gergonne Line
see also SIMPLICITY
References
De Temple, D. W. "Carlyle Circles and the Lemoine Simplic-
ity of Polygonal Constructions." Amer. Math. Monthly 98,
97-108, 1991.
Eves, H. An Introduction to the History of Mathematics, 6th
ed. New York: Holt, Rinehart, and Winston, 1990.
Geometry
Geometry is the study of figures in a SPACE of a
given number of dimensions and of a given type. The
most common types of geometry are PLANE GEOMETRY
(dealing with objects like the Line, CIRCLE, TRIANGLE,
and Polygon), Solid Geometry (dealing with objects
like the Line, Sphere, and Polyhedron), and Spher-
ical Geometry (dealing with objects like the Spher-
ical Triangle and Spherical Polygon).
Historically, the study of geometry proceeds from a
small number of accepted truths (AXIOMS or POSTU-
LATES), then builds up true statements using a system-
atic and rigorous step-by-step PROOF. However, there
is much more to geometry than this relatively dry text-
book approach, as evidenced by some of the beautiful
and unexpected results of PROJECTIVE GEOMETRY (not
to mention Schubert's powerful but questionable Enu-
merative Geometry).
Formally, a geometry is defined as a complete locally
homogeneous RlEMANNlAN METRIC. In R 2 , the possible
geometries are Euclidean planar, hyperbolic planar, and
elliptic planar. In R 3 , the possible geometries include
Euclidean, hyperbolic, and elliptic, but also include five
other types.
see also Absolute Geometry, Affine Geometry,
Coordinate Geometry, Differential Geometry,
Enumerative Geometry, Finsler Geometry, In-
versive Geometry, Minkowski Geometry, Nil Ge-
ometry, Non-Euclidean Geometry, Ordered Ge-
ometry, Plane Geometry, Projective Geometry,
Sol Geometry, Solid Geometry, Spherical Ge-
ometry, Thurston's Geometrization Conjecture
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, 1952.
Brown, K. S. "Geometry." http://www.seanet.com/
-ksbrown/igeometr . htm.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, 1969.
Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Prob-
lems in Geometry. New York: Springer- Verlag, 1994.
Eppstein, D. "Geometry Junkyard." http://www.ics.uci.
edu/~eppstein/ junkyard/.
Eppstein, D. "Many- Dimensional Geometry." http://www.
ics.uci.edu/-eppstein/junkyard/highdim.html.
Eppstein, D. "Planar Geometry." http://www.ics.uci.edu
/-eppstein/ junkyard/2d. html.
Eppstein, D. "Three-Dimensional Geometry." http://www.
ics.uci.edu/-eppstein/junkyard/3d.html.
Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA:
Allyn and Bacon, 1972.
Geometry Center, http://www.geom.umn.edu.
Ghyka, M. C. The Geometry of Art and Life, 2nd ed. New
York: Dover, 1977.
Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago,
IL: The Open Court Publishing Co., 1921.
Johnson, R. A. Advanced Euclidean Geometry: An Elemen-
tary Treatise on the Geometry of the Triangle and the Cir-
cle. New York: Dover, 1960.
King, J. and Schattschneider, D. (Eds.). Geometry Turned
On: Dynamic Software in Learning, Teaching and Re-
search. Washington, DC: Math. Assoc. Amer., 1997.
Klein, F. Famous Problems of Elementary Geometry and
Other Monographs. 082840108X New York: Dover, 1956.
Melzak, Z. A. Invitation to Geometry. New York: Wiley,
1983.
Moise, E. E. Elementary Geometry from an Advanced Stand-
point, 3rd ed. Reading, MA: Addison- Wesley, 1990.
Ogilvy, C. S. "Some Unsolved Problems of Modern Geom-
etry." Ch. 11 in Excursions in Geometry. New York:
Dover, pp. 143-153, 1990.
Simon, M. Uber die Entwicklung der Element argeometrie im
XIX Jahrhundert. Berlin, pp. 97-105, 1906.
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, 1961.
Gergonne Line
The perspective line for the Contact Triangle
ADEF and its Tangential Triangle AABC. It is
determined by the Nobbs Points D' , £', and F f . In
addition to the NOBBS POINTS, the FLETCHER POINT
and EVANS POINT also lie on the Gergonne line where
it intersects the SODDY LINE and EULER Line, respec-
tively. The D and D f coordinates are given by
D = B + 1-C
e
D f = B- ^C,
e
so BDCD' form a Harmonic Range. The equation of
the Gergonne line is
a 8 7
h — + -
d^ e ^ f
0.
see also Contact Triangle, Euler Line, Evans
Point, Fletcher Point, Nobbs Points, Soddy
Line, Tangential Triangle
References
Oldknow, A. "The Euler- Gergonne- Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
Gergonne Point
Gergonne Point
The common point of the CONCURRENT lines from the
Tangent points of a Triangle's Incircle to the op-
posite Vertices. It has Triangle Center Function
a = [a(b + c - a)]" 1 = \ sec 2 A.
It is the Isotomic Conjugate Point of the Nagel
Point. The Contact Triangle and Tangential
Triangle are perspective from the Gergonne point.
see also Gergonne Line
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.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited,
New York: Random House, pp. 11-13, 1967.
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. 22, 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 216, 1929.
Kimberling, C. "Gergonne Point." http://www.evansvi.lle.
edu/-ck6/tcenters/class/gergonne.html.
Germain Primes
see Sophie Germain Prime
Gerono Lemniscate
see Eight Curve
Gersgorin Circle Theorem
Gives a region in the Complex Plane containing all
the Eigenvalues of a Complex Square Matrix. Let
\xk\ = max{|xi| : 1 < i < n} >
and define
Ri = X^l a tfl-
(1)
(2)
Then each Eigenvalue of the Matrix A of order n is
in at least one of the disks
Ghost 735
The theorem can be made stronger as follows. Let r be
an Integer with 1 < r < n, then each Eigenvalue of
A is either in one of the disks Ti
{z:\z-
or in one of the regions
>«l < s)
- 1 '},
jz:^|z-a«|<^fl;L
(4)
(5)
where SJ r l * is the sum of magnitudes of the r— 1 largest
off-diagonal elements in column j.
References
Buraldi, R. A. and Mellendorf, S. "Regions in the Complex
Plane Containing the Eigenvalues of a Matrix." Amer.
Math. Monthly 101, 975-985, 1994.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1120-1121, 1979.
Taussky-Todd, O. "A Recurring Theorem on Determinants."
Amer. Math. Monthly 56, 672-676, 1949.
Ghost
; JL_ ;
20 40 60 80
Frequency (Hz)
If the sampling of an interferogram is modulated at a
definite frequency instead of being uniformly sampled,
spurious spectral features called "ghosts" are produced
(Brault 1985). Periodic ruling or sampling errors intro-
duce a modulation superposed on top of the expected
fringe pattern due to uniform stage translation. Be-
cause modulation is a multiplicative process, spurious
features are generated in spectral space at the sum and
difference of the true fringe and ghost fringe frequencies,
thus throwing power out of its spectral band.
Ghosts are copies of the actual spectrum, but appear
at reduced strength. The above shows the power spec-
trum for a pure sinusoidal signal sampled by translat-
ing a Fourier transform spectrometer mirror at constant
speed. The small blips on either side of the main peaks
are ghosts.
In order for a ghost to appear, the process producing it
must exist for most of the interferogram. However, if
the ruling errors are not truly sinusoidal but vary across
the length of the screw, a longer travel path can reduce
their effect.
see also JITTER
{z:\z- au\ < Ri}.
(3)
736
Gibbs Constant
Gill's Method
References
Brault, J. W. "Fourier Transform Spectroscopy." In High
Resolution in Astronomy: 15th Advanced Course of
the Swiss Society of Astronomy and Astrophysics (Ed.
A. Benz, M. Huber, and M. Mayor). Geneva Observatory,
Sauverny, Switzerland, 1985.
Gibbs Constant
see WlLBRAHAM-GlBBS CONSTANT
Gibbs Effect
see Gibbs Phenomenon
Gibbs Phenomenon
An overshoot of FOURIER SERIES and other ElGEN-
FUNCTION series occurring at simple DISCONTINUITIES.
It can be removed with the LANCZOS a FACTOR.
see also Fourier Series
References
Arfken, G. "Gibbs Phenomenon." §14.5 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 783-787, 1985.
Foster, J. and Richards, F. B. "The Gibbs Phenomenon for
Piecewise-Linear Approximation." Amer. Math, Monthly
98, 47-49, 1991.
Gibbs, J. W. "Fourier Series." Nature 59, 200 and 606, 1899.
Hewitt, E. and Hewitt, R. "The Gibbs- Wilbraham Phenom-
enon: An Episode in Fourier Analysis." Arch. Hist. Exact
Sci. 21, 129-160, 1980.
Sansone, G. "Gibbs' Phenomenon." §2.10 in Orthogonal
Functions, rev. English ed. New York: Dover, pp. 141—
148, 1991.
Gigantic Prime
A Prime with 10,000 or more decimal digits. As of
Nov. 15, 1995, 127 were known.
see also TITANIC PRIME
References
Caldwell, C. "The Ten Largest Known Primes." http://www.
utm.edu/research/primes/largest .html#largest.
Gilbrat's Distribution
A Continuous Distribution in which the Loga-
rithm of a variable x has a NORMAL DISTRIBUTION,
P(x)
V2n
-(In s) 2 ^
It is a special case of the Log Normal Distribution
P(x) =
SV2i
-(lnx-M) 2 /2S 2
with S = 1 and M = 0.
see also LOG NORMAL DISTRIBUTION
Gilbreath's Conjecture
Let the DIFFERENCE of successive Primes be defined by
d n = Pn+i - Pn, and d n by
in -{ \d k n l
1 J
+1 -^
for k = 1
*" 1 | for k> 1.
N. L. Gilbreath claimed that d* = 1 for all k (Guy 1994).
It has been verified for k < 63419 and all Primes up to
7r(10 13 ), where -k is the Prime Counting Function.
References
Guy, R. K. "Gilbreath's Conjecture." §A10 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 25-26, 1994.
Gill's Method
A formula for numerical solution of differential equa-
tions,
y n +i =Vn-h §[&i + (2 - V2 )k 2
-r(2-rV2)k 3 +k 4 ) + 0(h 5 ),
where
k\ = hf{x n ,y n )
ki — hf(xn + §/i, y n + | fei)
*3 = hf(x n + \Ky n + \{-l + y/2)ki + (1 - \V2)k 2 )
fc 4 ^hf{x n + h,y n - fv / 2fc 2 + (l + §\Z2)fc 3 ).
see also Adams' Method, Milne'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,
p. 896, 1972.
Gingerbreadman Map
Gingerbreadman Map
-2024
A 2-D piecewise linear MAP defined by
Xn+l = 1 - Vn + \Xn\
Vn+1 = %n-
The map is chaotic in the filled region above and stable
in the six hexagonal regions. Each point in the interior
hexagon defined by the vertices (0, 0), (1, 0), (2, 1), (2,
2), (1, 2), and (0, 1) has an orbit with period six (except
the point (1, 1), which has period 1). Orbits in the other
five hexagonal regions circulate from one to the other.
There is a unique orbit of period five, with all others
having period 30. The points having orbits of period
five are (-1, 3), (-1, -1), (3, -1), (5, 3), and (3, 5),
indicated in the above figure by the black line. However,
there are infinitely many distinct periodic orbits which
have an arbitrarily long period.
References
Devaney, R. L. "A Piecewise Linear Model for the Zones of
Instability of an Area Preserving Map." Physica D 10,
387-393, 1984.
Peitgen, H.-O. and Saupe, D. (Eds.). "A Chaotic Ginger-
breadman." §3.2.3 in The Science of Fractal Images. New-
York: Springer- Verlag, pp. 149-150, 1988.
Girard's Spherical Excess Formula
Let a Spherical Triangle A have angles A, B, and
C. Then the SPHERICAL EXCESS is given by
A = A + B + C -7T.
see also Angular Defect, L'Huilier's Theorem,
Spherical Excess
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 94-95, 1969.
Giuga Number 737
Girko's Circular Law
Let A be Eigenvalues of a set of Random nxn Matri-
ces. Then \/y/n is uniformly distributed on the Disk.
References
Girko, V. L. Theory of Random Determinants. Boston, MA:
Kluwer, 1990.
Girth
The length of the shortest CYCLE in a GRAPH.
Girth Example
tetrahedron
cube, Kz^
Petersen graph
Giuga's Conjecture
If n > 1 and
n | 1 n-l + 2 n-l + _ + („ _ ^n-l + Xj
is n necessarily a PRIME? In other words, defining
n-l
s n — y k ,
does there exist a COMPOSITE n such that s n =
— 1 (mod n)? It is known that s n = — 1 (mod n) Iff
for each prime divisor p of n, (p — l)\(n/p — 1) and
p\(n/p—l) (Giuga 1950, Borwein et al 1996); therefore,
any counterexample must be SQUAREFREE. A compos-
ite Integer n satisfies s n = — 1 (mod n) IFF it is both
a Carmichael Number and a Giuga Number. Giuga
showed that there are no exceptions to the conjecture up
to 10 1000 . This was later improved to 10 1700 (Bedocchi
1985) and 10 13800 (Borwein et al 1996).
see also Argoh's Conjecture
References
Bedocchi, E. "The Z(\/l4) Ring and the Euclidean Algo-
rithm." Manuscripta Math. 53, 199-216, 1985.
Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen-
sohn, R. "Giuga's Conjecture on Primality." Amer. Math.
Monthly 103, 40-50, 1996.
Giuga, G. "Su una presumibile propertieta caratteristica dei
numeri primi." 1st. Lombardo Sci. Lett. Rend. A 83, 511—
528, 1950.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 20-21, 1989.
Giuga Number
Any Composite Number n with p\{n/p - 1) for all
Prime Divisors p of n. n is a Giuga number Iff
n-l
^]V (Tl) = -1 (modn)
fc^=i
where <j> is the Totient Function and Iff
p ax p
p\n p\n
738 Giuga Sequence
Glaisher-Kinkelin Constant
n is a Giuga number Iff
nB<p(n) = — 1 (mod n) ,
where B k is a Bernoulli Number and 4> is the To-
TIENT FUNCTION. Every counterexample to Giuga's
conjecture is a contradiction to Argoh'S CONJECTURE
and vice versa. The smallest known Giuga numbers are
30 (3 factors), 858, 1722 (4 factors), 66198 (5 factors),
2214408306, 24423128562 (6 factors), 432749205173838,
14737133470010574, 550843391309130318 (7 factors),
244197000982499715087866346,
554079914617070801288578559178
(8 factors), . . . (Sloane's A007850).
It is not known if there are an infinite number of Giuga
numbers. All the above numbers have sum minus prod-
uct equal to 1, and any Giuga number of higher order
must have at least 59 factors. The smallest Odd Giuga
number must have at least nine Prime factors.
see also ARGOH'S CONJECTURE, BERNOULLI NUMBER,
TOTIENT FUNCTION
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.
Sloane, N. J. A. Sequence A007850 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Giuga Sequence
A finite, increasing sequence of INTEGERS {m, . . . , n m }
such that
i=l i=X
A sequence is a Giuga sequence Iff it satisfies
rii\(ni • ■ -rii-i * n;+i
i)
for i — 1, ..., m. There are no Giuga sequences of
length 2, one of length 3 ({2, 3, 5}), two of length 4
({2, 3, 7, 41} and {2, 3, 11, 13}), 3 of length 5 ({2,
3, 7, 43, 1805}, {2, 3, 7, 83, 85}, and {2, 3, 11, 17,
59}), 17 of length 6, 27 of length 7, and hundreds of
length 8. There are infinitely many Giuga sequences.
It is possible to generate longer Giuga sequences from
shorter ones satisfying certain properties.
see also Carmichael Sequence
References
Borwein, D.; Borwein, J. M.; Borwein, P. B.; and Girgen-
sohn, R. "Giuga's Conjecture on Primality." Amer. Math.
Monthly 103, 40-50, 1996.
Glaisher-Kinkelin Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Define
K{n + \) = tfl 1 2 2 $ z --n n (1)
^ + } ~ K(n + 1) \0!l!2!...(n-l)! ifn>0.
(2)
where G is the G-FUNCTION and K is the if-FUNCTiON.
Then
K(n+l)
lim
n ^L ~ n n*/2+ n /2+l/2 e -n*/4
G(n + 1)
= A
pl/12
n~+oo n n 2 /2-l/12( 27r )n/2 e -3n2/4
(3)
(4)
(5)
where £(z) is the RlEMANN ZETA FUNCTION, 7r is Pi,
and 7 is the EULER-MASCHERONI CONSTANT (Kinkelin
1860, Glaisher 1877, 1878, 1893, 1894). Glaisher (1877)
also obtained
where
A = exp
C'(2) ln(2*Q 7
2ir 2 12 2
= 1.28242713..
A = 2 r/36 7r- 1/6 exp
{Mr
ln[r(x + 1)] <te
Glaisher (1894) showed that
/ A i2 \ ^ 2 /6
1 l/l 2 l/4 3 l/9 4 l/16 5 l/25 _ / Ji \
\ 27re^ )
1 1/1 3 1/9 5 1/25 7 1/49 9 1/81 .
1 l/l 5 l/125gl/729_ #
3l/27yl/343^^1/1331 . . .
-(
2 /8
(6)
(7)
(8)
25/32^1/32 e 3/32+7/48+
s/4)
where
C(3) 1
C(5)
A + C(7)
+ ....
(9)
(10)
3-4- 5 4 3 5-6 -74 5 7-8-9 4 7
see also G-Function, Hyperfactorial, K-Function
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/glshkn/glshkn.html.
Glaisher, J. W. L. "On a Numerical Continued Product."
Messenger Math. 6, 71-76, 1877.
Glaisher, J. W. L. "On the Product 1 1 2 2 3 3 ■ * *n n ." Messen-
ger Math. 7, 43-47, 1878.
Glaisher, J. W. L. "On Certain Numerical Products." Mes-
senger Math. 23, 145-175, 1893.
Glaisher, J. W. L. "On the Constant which Occurs in the
Formula for 1 1 2 2 3 3 • • • n n ." Messenger Math. 24, 1-16,
1894.
Kinkelin. "Uber eine mit der Gammafunktion verwandte
Transcendent e und deren Anwendung auf die Integralrech-
nung." J. Reine Angew. Math. 57, 122-158, 1860.
Glide
Gnomic Number 739
Glide
A product of a Reflection in a line and Translation
along the same line.
see also Reflection, Translation
Glissette
The locus of a point P (or the envelope of a line) fixed in
relation to a curve C which slides between fixed curves.
For example, if C is a line segment and P a point on
the line segment, then P describes an Ellipse when C
slides so as to touch two ORTHOGONAL straight LINES.
The glissette of the LINE SEGMENT C itself is, in this
case, an Astroid.
see also ROULETTE
References
Besant, W. H. Notes on Roulettes and Glissettes, 2nd enl.
ed. Cambridge, England: Deighton, Bell & Co., 1890.
Lockwood, E. H. "Glissettes." Ch. 20 in A Book of
Curves. Cambridge, England: Cambridge University-
Press, pp. 160-165, 1967.
Yates, R. C. "Glissettes." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 108-112,
1952.
Global C(G;T) Theorem
If a Sylow 2-SuBGROUP T of G lies in a unique max-
imal 2-local P of 6?, then P is a "strongly embedded"
Subgroup of G, and G is known.
Global Extremum
A Global Minimum or Global Maximum.
see also LOCAL EXTREMUM
Global Maximum
The largest overall value of a set, function, etc., over its
entire range.
see also Global Minimum, Local Maximum, Maxi-
mum
Global Minimum
The smallest overall value of a set, function, etc., over
its entire range.
see also Global Maximum, Kuhn-Tucker Theorem,
Local Minimum, Minimum
Globe
A SPHERE which acts as a model of a spherical (or el-
lipsoidal) celestial body, especially the Earth, and on
which the outlines of continents, oceans, etc. are drawn.
see also Latitude, Longitude, Sphere
Glove Problem
Let there be m doctors and n < m patients, and let all
run possible combinations of examinations of patients
by doctors take place. Then what is the minimum num-
ber of surgical gloves needed G(m, n) so that no doctor
must wear a glove contaminated by a patient and no
patient is exposed to a glove worn by another doctor?
In this problem, the gloves can be turned inside out and
even placed on top of one another if necessary, but no
"decontamination" of gloves is permitted. The optimal
solution is
{2 m—n—2
i(m + l) n = 1, m= 2fc + 1
[^(ro) + |nl otherwise,
where \x~\ is the Ceiling Function (Vardi 1991). The
case m = n = 2 is straightforward since two gloves have
a total of four surfaces, which is the number needed for
mn = 4 examinations.
References
Gardner, M. Aha! Aha! Insight. New York: Scientific Amer-
ican, 1978.
Gardner, M. Science Fiction Puzzle Tales. New York:
Crown, pp. 5, 67, and 104-150, 1981.
Hajnal, A. and Lovasz, L. "An Algorithm to Prevent the
Propagation of Certain Diseases at Minimum Cost." §10.1
in Interfaces Between Computer Science and Operations
Research (Ed. J. K. Lenstra, A. H. G. Rinnooy Kan, and
P. van Emde Boas). Amsterdam: Matematisch Centrum,
1978.
Orlitzky, A. and Shepp, L. "On Curbing Virus Propagation."
Exercise 10.2 in Technical Memo. Bell Labs, 1989.
Vardi, I. "The Condom Problem." Ch. 10 in Computational
Recreations in Mathematica. Redwood City, CA: Addison-
Wesley, p. 203-222, 1991.
Glue Vector
A VECTOR specifying how layers are stacked in a LAM-
INATED Lattice.
Gnomic Number
A FlGURATE Number of the form g n = 2n - 1 which
are the areas of square gnomons, obtained by removing
a Square of side n — 1 from a Square of side n,
9n = n 2 - (n - l) 2 = 2n - 1.
The gnomic numbers are therefore equivalent to the
Odd Numbers, and the first few are 1, 3, 5, 7, 9, 11,
... (Sloane's A005408). The Generating Function
for the gnomic numbers is
x(l + x)
= x + 3x -f 5cc + 7x 4- • • • .
see also ODD NUMBER
References
Sloane, N. J. A. Sequence A005408/M2400 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
740 Gnomic Projection
Gnomic Projection
A nonconformal projection from a Sphere's center in
which ORTHODROMES are straight LINES.
cos<£sin(A — Ao)
cose
cos <f>± sin <f> — sin <fi± cos <f> cos(A — Ao)
(i)
(2)
where
cose = sin 0i sin</> -f- cos <j>\ cos</>cos(A — Ao). (3)
The inverse FORMULAS are
<j) = sin" 1 (coscsin0i + y sin c cose cos <f>i) (4)
A = Ao + tan -
cos (f>i — y sin 0i
(5)
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 150-153, 1967.
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 164-168, 1987.
Gnomon
A shape which, when added to a figure, yields another
figure Similar to the original.
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 123, 1993.
Gnomon Magic Square
A 3 x 3 array of numbers in which the elements in each
2x2 corner have the same sum.
see also MAGIC SQUARE
Go
There are estimated to be about 4.63 x 10 170 possible
positions on a 19 x 19 board (Flammenkamp). The num-
ber of n-move Go games are 1, 362, 130683, 47046242,
... (Sloane's A007565).
GobeVs Sequence
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Item 96, Feb. 1972.
Flammenkamp, A. "A Short, Concise Ruleset of Go."
http : //www . minet . uni- j ena . de/-achim/gorules . html.
Kraitchik, M. "Go." §12.4 in Math em atical Recreations. New
York: W. W. Norton, pp. 279-280, 1942.
Sloane, N. J. A. Sequence A007565/M5447 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
GobePs Sequence
Consider the Recurrence Relation
1 + xq 2 + xi 2 + . . . + x n ~i 2
(1)
with xo = 1. The first few iterates of x n are 1, 2, 3,
5, 10, 28, 154, . . . (Sloane's A003504). The terms grow
extremely rapidly, but are given by the asymptotic for-
mula
x n w (n 2 +2n-l+4n _1 -21n" 2 +137n" 3 -...)^ 2n , (2)
where
C = 1.04783144757641122955990946274313755459. . .
(3)
(Zagier). It is more convenient to work with the trans-
formed sequence
s n = 2 + X! 2 + x 2 + . . . + x n -i 2 - nx n , (4)
which gives the new recurrence
Sn 2
Sn+1 — S n + ^V (5)
with initial condition s\ = 2. Now, s n -\-i will be nonin-
tegral IFF n\s n . The smallest p for which s p ^ (mod
p) therefore gives the smallest nonintegral s p +i. In ad-
dition, since p\s p , x p = s p /p is also the smallest nonin-
tegral x p .
For example, we have the sequences {s n (mod fc)} n=1 :
2,6 = 2,1=0,0,0
2,6,15 = 1,| =0,0,0,0
2,6,15 = 4,f = 7,^=8,^ =0,0,...,
(mod 5) (6)
(mod 7) (7)
(mod 11). (8)
Testing values of k shows that the first nonintegral x n
is £43. Note that a direct verification of this fact is
impossible since
X43 « 5.4093 x 10
178485291567
(9)
(calculated using the asymptotic formula) is much too
large to be computed and stored explicitly.
Goblet Illusion
Godel Number 741
A sequence even more striking for remaining integral
over many terms is the 3-G6bel sequence
l + xo 3 + xi 3 + ... + x n _i 3
(10)
The first few terms of this sequence are 1, 2, 5, 45, 22815,
... (Sloane's A005166).
The Gobel sequences can be generalized to k powers by
x n = . (11)
see also SOMOS SEQUENCE
References
Guy, R. K. "The Strong Law of Small Numbers." Amer,
Math. Monthly 95, 697-712, 1988.
Guy, R. K. "A Recursion of Gobel." §E15 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 214-215, 1994.
Sloane, N. J. A. Sequences A003504/M0728 and A005166/
M1551 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Zaiger, D. "Solution: Day 5, Problem 3." http: //www-
groups . dcs . st - and .ac.uk/ -John/ Zagier / Solution
5. 3. html.
Goblet Illusion
An ILLUSION in which the eye alternately sees two black
faces, or a white goblet.
References
Fineman, M. The Nature of Visual Illusion. New York:
Dover, pp. Ill and 115, 1996.
Rubin, E. Synoplevede Figurer. Copenhagen, Denmark:
Gyldendalske, 1915.
What's Up with Kids Magazine. "Reversible Goblet."
http : //wuwk . spurtek . com/COI jreversible_goblet . htm.
Godel's Completeness Theorem
If T is a set of AXIOMS in a first-order language, and a
statement p holds for any structure M satisfying T, then
p can be formally deduced from T in some appropriately
denned fashion.
see also Godel's Incompleteness Theorem
Godel's Incompleteness Theorem
Informally, Godel's incompleteness theorem states that
all consistent axiomatic formulations of NUMBER THE-
ORY include undecidable propositions (Hofstadter 1989).
This is is sometimes called Godel's first incompleteness
theorem, and answers in the negative HlLBERT's Prob-
lem asking whether mathematics is "complete" (in the
sense that every statement in the language of Number
THEORY can be either proved or disproved). Formally,
Godel's theorem states, "To every ^-consistent recursive
class k of FORMULAS, there correspond recursive class-
signs r such that neither (v Gen r) nor Neg(i; Gen r)
belongs to Flg(«), where v is the FREE VARIABLE of r"
(Godel 1931).
A statement sometimes known as Godel's second incom-
pleteness theorem states that if NUMBER THEORY is
consistent, then a proof of this fact does not exist us-
ing the methods of first-order PREDICATE CALCULUS.
Stated more colloquially, any formal system that is in-
teresting enough to formulate its own consistency can
prove its own consistency IFF it is inconsistent.
Gerhard Gentzen showed that the consistency and com-
pleteness of arithmetic can be proved if "transfinite" in-
duction is used. However, this approach does not allow
proof of the consistency of all mathematics.
see also GODEL'S COMPLETENESS THEOREM,
Hilbert's Problems, Kreisel Conjecture, Natu-
ral Independence Phenomenon, Number Theory,
Richardson's Theorem, Undecidable
References
Barrow, J. D. Pi in the Sky: Counting, Thinking, and Being.
Oxford, England: Clarendon Press, p. 121, 1992.
Godel, K. "Uber Formal Unentscheidbare Satze der Prin-
cipia Mathematica und Verwandter Systeme, L" Monat-
shefte fur Math. u. Physik 38, 173-198, 1931.
Godel, K. On Formally Undecidable Propositions of Prin-
cipia Mathematica and Related Systems. New York:
Dover, 1992.
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 17, 1989.
Kolata, G. "Does Godel's Theorem Matter to Mathematics?"
Science 218, 779-780, 1982.
Smullyan, R. M. Godel's Incompleteness Theorems. New
York: Oxford University Press, 1992.
Whitehead, A. N. and Russell, B. Principia Mathematica.
New York: Cambridge University Press, 1927.
Godel Number
A Godel number is a unique number associated to a
statement about arithmetic. It is formed as the Prod-
uct of successive PRIMES raised to the POWER of the
number corresponding to the individual symbols that
comprise the sentence. For example, the statement
(3x)(x — sy) that reads "there EXISTS an x such that x
is the immediate successor of y" is coded
(2 8 )(3 4 )(5 13 )(7 9 )(11 8 )(13 13 )(17 5 )(19 7 )(23 16 )(29 9 ),
742 Goldbach Conjecture
Golden Ratio
where the numbers in the set (8, 4, 13, 9, 8, 13, 5, 7, 16,
9) correspond to the symbols that make up (3x)(x =
see also GODEL'S INCOMPLETENESS THEOREM
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 18, 1989.
Goldbach Conjecture
Goldbach's original conjecture, written in a 1742 letter
to Euler, states that every INTEGER > 5 is the SUM
of three PRIMES. As re-expressed by Euler, an equiv-
alent of this CONJECTURE (called the "strong" Gold-
bach conjecture) asserts that all POSITIVE EVEN INTE-
GERS > 4 can be expressed as the Sum of two PRIMES.
Schnirelmann (1931) proved that every EVEN number
can be written as the sum of not more than 300,000
Primes (Dunham 1990), which seems a rather far cry
from a proof for four Primes! The strong Goldbach
conjecture has been shown to be true up to 4 x 10 11
by Sinisalo (1993). Pogorzelski (1977) claimed to have
proven the Goldbach conjecture, but his proof is not
generally accepted (Shanks 1993).
The conjecture that all Odd numbers > 9 are the SUM
of three Odd Primes is called the "weak" Goldbach
conjecture. Vinogradov proved that all Odd Integers
starting at some sufficiently large value are the Sum
of three PRIMES (Guy 1994). The original "sufficiently
ol5 16. 573
large" N > 3 = e e was subsequently reduced to
e el1 ' 503 by Chen and Wang (1989). Chen (1973, 1978)
also showed that all sufficiently large EVEN NUMBERS
are the sum of a PRIME and the PRODUCT of at most
two Primes (Guy 1994, Courant and Robbins 1996).
It has been shown that if the weak Goldbach conjec-
ture is false, then there are only a FINITE number of
exceptions.
Other variants of the Goldbach conjecture include the
statements that every EVEN number > 6 is the SUM of
two Odd Primes, and every Integer > 17 the sum of
exactly three distinct PRIMES. Let R(n) be the number
of representations of an Even INTEGER n as the sum of
two Primes. Then the "extended" Goldbach conjecture
states that
R(n) ~ 2Ii 2
dx
k=2
Pk\n
(lnx) 2
where II2 is the TWIN PRIMES CONSTANT (Halberstam
and Richert 1974).
If the Goldbach conjecture is true, then for every number
m, there are PRIMES p and q such that
<P(p) + <j>(q) = 2m,
where <p(x) is the TOTIENT FUNCTION (Guy 1994,
p. 105).
Vinogradov (1937ab, 1954) proved that every suffi-
ciently large Odd Number is the sum of three Primes,
and Estermann (1938) proves that almost all Even
Numbers are the sums of two Primes.
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 64, 1987.
Chen, J.-R. "On the Representation of a Large Even Number
as the Sum of a Prime and the Product of at Most Two
Primes." Sci. Sinica 16, 157-176, 1973.
Chen, J.-R. "On the Representation of a Large Even Number
as the Sum of a Prime and the Product of at Most Two
Primes, II." Sci, Sinica 21, 421-430, 1978.
Chen, J.-R. and Wang, T.-Z. "On the Goldbach Problem."
Acta Math. Sinica 32, 702-718, 1989.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 30—31, 1996.
Devlin, K. Mathematics: The New Golden Age. London:
Penguin Books, 1988.
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, p. 83, 1990.
Estermann, T. "On Goldbach's Problem: Proof that Almost
All Even Positive Integers are Sums of Two Primes." Proc.
London Math. Soc. Ser. 2 44, 307-314, 1938.
Guy, R K. "Goldbach's Conjecture." §C1 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 105-107, 1994.
Hardy, G. H. and Littlewood, J. E. "Some Problems of Parti-
tio Numerorum (V): A Further Contribution to the Study
of Goldbach's Problem." Proc. London Math. Soc. Ser. 2
22, 46-56, 1924.
Halberstam, H. and Richert, H.-E. Sieve Methods. New York:
Academic Press, 1974.
Pogorzelski, H. A. "Goldbach Conjecture." J. Reine Angew.
Math. 292, 1-12, 1977.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 30-31 and 222, 1985.
Sinisalo, M. K. "Checking the Goldbach Conjecture up to
4.10 11 ." Math. Comput. 61, 931-934, 1993.
Vinogradov, I. M. "Representation of an Odd Number as
a Sum of Three Primes." Comtes rendus (Doklady) de
VAcademie des Sciences de VU.R.S.S. 15, 169-172, 1937a.
Vinogradov, I. "Some Theorems Concerning the Theory of
Primes." Recueil Math. 2, 179-195, 1937b.
Vinogradov, I. M. The Method of Trigonometrical Sums in
the Theory of Numbers. London: Interscience, p. 67, 1954.
Yuan, W. Goldbach Conjecture. Singapore: World Scientific,
1984.
Golden Mean
see Golden Ratio
Golden Ratio
A number often encountered when taking the ratios
of distances in simple geometric figures such as the
Decagon and Dodecagon. It is denoted 0, or some-
times r (which is an abbreviation of the Greek "tome,"
meaning "to cut"). <j> is also known as the DIVINE PRO-
PORTION, Golden Mean, and Golden Section and is
a Pisot-Vijayaraghavan Constant. It has surpris-
ing connections with CONTINUED FRACTIONS and the
Golden Ratio
Golden Ratio 743
Euclidean Algorithm for computing the Greatest
Common Divisor of two Integers.
Given a RECTANGLE having sides in the ratio 1 : <fi, <j>
is defined such that partitioning the original RECTAN-
GLE into a Square and new RECTANGLE results in a
new Rectangle having sides with a ratio 1 : <f>. Such
a Rectangle is called a Golden Rectangle, and
successive points dividing a Golden Rectangle into
Squares lie on a Logarithmic Spiral. This figure is
known as a Whirling Square.
L
L&l)
(Sloane's A000012). Another infinite representation in
terms of a Continued Square Root is
= y i + V i + Vi + VTT.
.(io)
Ramanujan gave the curious Continued Fraction
identities
= 1 +
(\/^75)e 27r /5 1
+
1 +
1 +
1 +
1 + ...
(11)
This means that
= </>
(1)
(2)
4> 2 _ $ - i = o.
So, by the QUADRATIC EQUATION,
* = |(1 ± VTT4) = |(1 + y/%) (3)
= 1.618033988749894848204586834365638117720. . .
(4)
(Sloane's A001622).
x
1
A B C
A geometric definition can be given in terms of the above
figure. Let the ratio x = AB/BC. The NUMERATOR
and Denominator can then be taken as AB = x and
BC = 1 without loss of generality. Now define the posi-
tion of B by
RH AR
(5)
BC __ AB
AB ~ AC
Plugging in gives
x 1 + x
x l -x-1 = 0,
(6)
(7)
which can be solved using the QUADRATIC EQUATION
to obtain
1 ± Jl 2 - (-4) w r -
<f> = X = V V =1(1 + y/%). (8)
<j> is the "most" IRRATIONAL number because it has a
Continued Fraction representation
.^:£M,1,:,.].
(9)
/ . V^ , \ e 27r/ y/5
-2ic>/5
1 +
-4irV5
(12)
1 + -
-6irV5
1 +
-8irV5
1+-
1+-
-IOttV'S
1 + ...
(Ramanathan 1984).
The legs of a GOLDEN TRIANGLE are in a golden ra-
tio to its base. In fact, this was the method used by
Pythagoras to construct <p. Euclid used the following
construction.
F G
Draw the SQUARE DABDC, call E the MIDPOINT of
AG, so that AE = EC = x. Now draw the segment
BE> which has length
sy/2 2 + l 2 = xVb,
(13)
and construct EF with this length. Now construct
FG ■= EF, then
744
Golden Ratio
Golden Ratio
The ratio of the ClRCUMRADlUS to the length of the side
of a Decagon is also 0,
In addition,
7 = |csc(^)=I(l + ^) = 0.
(15)
Similarly, the legs of a GOLDEN TRIANGLE (an ISOSCE-
LES Triangle with a Vertex Angle of 36°) are in
a Golden Ratio to the base. Bisecting a Gaullist
Cross also gives a golden ratio (Gardner 1961, p. 102).
Kl+<t>)
In the figure above, three TRIANGLES can be INSCRIBED
in the RECTANGLE U3ABCD of arbitrary aspect ratio
1 : r such that the three Right Triangles have equal
areas by dividing AB and BC in the golden ratio. Then
Kaade = \ -r(l + 0)-l= \r<f> 2
K&bef = \ • r<p ■ <t> = \r<t>
K AC df = |(l + 0)-r-=|r0 2 ,
(16)
(17)
(18)
which are all equal.
The golden ratio also satisfies the Recurrence Rela-
tion
4> n
Taking n — gives
+ <
■ + 1
= l + <
(19)
(20)
(21)
But this is the definition equation for 4> (when the root
with the plus sign is used). Squaring gives
= |(5 + 2^5 + 1) = i(6 + 2%/5)= 1(3 + ^5)
4> 3 = (<t>° + V = 0V 1 + (0 1 ) 2 = cj> 1 + <f>\
(22)
(23)
and so on.
For the difference equations
< x = 1
) X n = 1 +
Xn — 1
for n= 1,2,3,
is also given by
<j> = lim x n .
(24)
(25)
,. F n
•= hm ,
n— >-oo r n — \
(26)
where F n is the nth FIBONACCI NUMBER.
The Substitution Map
-^ 01
1 -»0
gives
0->01 -+010-^01001 -»...,
giving rise to the sequence
0100101001001010010100100101 . . .
(27)
(28)
(29)
(30)
(Sloane's A003849). Here, the zeros occur at positions
1, 3, 4, 6, 8, 9, 11, 12, ... (Sloane's A000201), and
the ones occur at positions 2, 5, 7, 10, 13, 15, 18, ...
(Sloane's A001950). These are complementary Beatty
Sequences generated by \n<t>\ and \n<t> 2 \. The se-
quence also has many connections with the FIBONACCI
Numbers.
Salem showed that the set of PlSOT-VlJAYARAGHAVAN
CONSTANTS is closed, with <fi the smallest accumulation
point of the set (Le Lionnais 1983).
see also BERAHA CONSTANTS, DECAGON, FIVE DISKS
Problem, Golden Ratio Conjugate, Golden Tri-
angle, ICOSIDODECAHEDRON, NOBLE NUMBER, PEN-
TAGON, Pentagram, Phi Number System, Secant
Method
References
Boyer, C. B. History of Mathematics. New York: Wiley,
p. 56, 1968.
Coxeter, H. S. M. "The Golden Section, Phyllotaxis, and
Wythoff's Game." Scripta Mathematica 19, 135-143,
1953.
Dixon, R. Mathographics. New York: Dover, pp. 30-31 and
50, 1991.
Finch, S. "Favorite Mathematical Constants." http://vww.
mathsof t . com/asolve/constant/cntf rc/cntf re .html.
Finch, S. "Favorite Mathematical Constants." http://wvw.
mathsoft . com/ asolve/constant/gold/gold. html.
Gardner, M. "Phi: The Golden Ratio." Ch. 8 in The Second
Scientific American Book of Mathematical Puzzles & Di-
versions, A New Selection. New York: Simon and Schus-
ter, 1961.
Gardner, M. "Notes on a Fringe- Watcher: The Cult of the
Golden Ratio." Skeptical Inquirer 18, 243-247, 1994.
Herz-Fischler, R. A Mathematical History of the Golden
Number. New York: Dover, 1998.
Huntley, H. E. The Divine Proportion. New York: Dover,
1970.
Knott, R. "Fibonacci Numbers and the Golden Section."
http:// www . mes . surrey .ac.uk/ Personal / R.Knott /
Fibonacci/fib. html.
Le Lionnais, F. Les nombres remarquables . Paris: Hermann,
p. 40, 1983.
Markowsky, G. "Misconceptions About the Golden Ratio."
College Math. J. 23, 2-19, 1992.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 122-134, 1990.
Golden Ratio Conjugate
Golomb Constant 745
Pappas, T. "Anatomy & the Golden Section." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 32-33, 1989.
Ramanathan, K. G. "On Ramanujan's Continued Fraction."
Acta. Arith. 43, 209-226, 1984.
Sloane, N. J. A. Sequences A003849, A000012/M0003,
A000201/M2322, A001622/M4046, and A001950/M1332
in "An On-Line Version of the Encyclopedia of Integer Se-
quences."
Golden Ratio Conjugate
The quantity
</> c = A = _ 1 = v \~ 1 « 0.6180339885, (1)
<p 2
where <f> is the GOLDEN RATIO. The golden ratio con-
jugate is sometimes also called the Silver Ratio. A
quantity similar to the FEIGENBAUM CONSTANT can be
found for the nth CONTINUED FRACTION representation
[ao,ai,a2, . ..J.
Taking the limit of
S n =
CTn — <?n~~l
<7n — 0"n+l
gives
6 = lim = 1 + = 2 + 0c.
n— J'oo
see also GOLDEN RATIO, SILVER RATIO
(2)
(3)
(4)
Golden Rectangle
Given a RECTANGLE having sides in the ratio 1 : 0, the
GOLDEN RATIO <f> is defined such that partitioning the
original RECTANGLE into a SQUARE and new RECTAN-
GLE results in a new RECTANGLE having sides with a
ratio 1 : 0. Such a RECTANGLE is called a golden rect-
angle, and successive points dividing a golden rectangle
into Squares lie on a Logarithmic Spiral.
see also Golden Ratio, Logarithmic Spiral, Rect-
angle
References
Pappas, T. "The Golden Rectangle." The Joy of Mathemat-
ics. San Carlos, CA: Wide World Publ./Tetra, pp. 102-
106, 1989.
Golden Rule
The mathematical golden rule states that, for any FRAC-
TION, both Numerator and Denominator may be
multiplied by the same number without changing the
fraction's value.
see also DENOMINATOR, FRACTION, NUMERATOR
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 151, 1996.
Golden Section
see Golden Ratio
Golden Theorem
see Quadratic Reciprocity Theorem
Golden Triangle
An Isosceles Triangle with Vertex angles 36°.
Such Triangles occur in the Pentagram and
Decagon. The legs are in a Golden Ratio to the
base. For such a TRIANGLE,
sin(18°)=sin(^) = M (1)
& = 2asin(^7r) = 2a ^ = |a(V5 - 1) (2)
(3)
0. (4)
,To'v - -<* 4 - 2 K
b + l= |a(V5 + l)
6 + a \/5 + l
see also DECAGON, GOLDEN RATIO, ISOSCELES TRIAN-
GLE, Pentagram
References
Pappas, T. "The Pentagon, the Pentagram & the Golden
Triangle." The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, pp. 188-189, 1989.
Goldschmidt Solution
The discontinuous solution of the SURFACE OF REVOLU-
TION Area minimization problem for surfaces connect-
ing two Circles. When the Circles are sufficiently
far apart, the usual Catenoid is no longer stable and
the surface will break and form two surfaces with the
Circles as boundaries.
see also CALCULUS OF VARIATIONS, SURFACE OF REV-
OLUTION
Golomb Constant
see Golomb-Dickman Constant
746
Golomb-Dickman Constant
Golomb-Dickman Constant
Golomb-Dickman Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let II be a PERMUTATION of n elements, and let at be
the number of CYCLES of length i in this PERMUTATION.
Picking II at RANDOM gives
/ oo \ n
\ j = l I i = l
oo \ n
. i=i
lim P(ai = 0) =
1
(2)
(3)
(Shepp and Lloyd 1966, Wilf 1990). Goncharov (1942)
showed that
lim P{aj = k) = ^e- l >ir\ (4)
which is a POISSON DISTRIBUTION, and
lim P
71— +00
^2a 3 -Inn (lnn)~ 1/2 <
*(*),
(5)
which is a Normal Distribution, 7 is the Euler-
Mascheroni Constant, and $ is the Normal Dis-
tribution Function. Let
M(a) = max a 3 -
l<j<oo
m(a) = min a,-.
l<j<oo J
Golomb (1959) derived
A = lim \ M (°0/ =0.6243299885.
n— >oo Tl
(6)
(7)
(8)
which is known as the Golomb CONSTANT or Golomb-
Dickman constant. Knuth (1981) asked for the con-
stants b and c such that
lim n b [{M(a))-Xn~ §A] = c,
71— J-OO
(9)
and Gourdon (1996) showed that
J-„7
(M(a)) = A(n+i)-^ + ^-
|(-l) n
_A2_7 _j_ I(_1^ _|_ I «l + 2n , 1 ,-2 + n
3840 C T 8\ / T 6J ~ 6^
where
• _ 2tti/3
7 = e .
(10)
(11)
A can be expressed in terms of the function f(x) defined
by f(x) = 1 for 1 < x < 2 and
(12)
dec
f(x-l)
x-l
for a? > 2, by
Shepp and Lloyd (1966) derived
h * 2
(13)
f°° ( f°° e~ y \
A = / exp I —x — I dy I
Jo V J* y )
= i exp (rs) dx -
Mitchell (1968) computed A to 53 decimal places.
(14)
Surprisingly enough, there is a connection between A
and Prime Factorization (Knuth and Pardo 1976,
Knuth 1981, pp. 367-368, 395, and 611). Dickman
(1930) investigated the probability P(x,n) that the
largest Prime Factor p of a random Integer between
1 and n satisfies p < n x for x € (0, 1). He found that
F(x) = lim P(x
,n) = {/o^(r
M
if x > 1
f if0<z<l.
(15)
Dickman then found the average value of x such that
p = n x , obtaining
{i = lim (x) = lim ( ) — / x —— dx
n-j-oo n^oo \lnn/ J dx
- / F (Y^~t) rf* = 0.62432999, (16)
which is A.
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/golomb/golomb.html.
Gourdon, X. 1996. http://www.mathsoft.com/asolve/
constant /golomb/gourdon. html.
Knuth, D. E. The Art of Computer Programming, Vol. 1:
Fundamental Algorithms, 2nd ed. Reading, MA: Addison-
Wesley, 1973.
Knuth, D. E. The Art of Computer Programming, Vol. 2:
Seminumerical Algorithms, 2nd ed. Reading, MA:
Addison- Wesley, 1981.
Knuth, D. E. and Pardo, L. T. "Analysis of a Simple Fac-
torization Algorithm." Theor. Comput. Sci. 3, 321-348,
1976.
Mitchell, W. C. "An Evaluation of Golomb 's Constant."
Math. Comput. 22, 411-415, 1968.
Purdom, P. W. and Williams, J. H. "Cycle Length in a Ran-
dom Function." Trans. Amer. Math. Soc. 133, 547-551,
1968.
Shepp, L. A. and Lloyd, S. P. "Ordered Cycle Lengths in
Random Permutation." Trans. Amer. Math. Soc. 121,
350-557, 1966.
Wilf, H. S. Generatingfunctionology, 2nd ed. New York:
Academic Press, 1993.
Golomb Ruler
Gompertz Constant 747
Golomb Ruler
A Golomb ruler is a set of NONNEGATIVE integers such
that all pairwise POSITIVE differences are distinct. The
optimum Golomb ruler with n marks is the Golomb
ruler having the smallest possible maximum element
("length"). The set (0, 1, 3, 7) is an order four Golomb
ruler since its differences are (1 = 1-0, 2 = 3 — 1,
3 = 3-0, 4 = 7-3, 6 = 7- 1, 7 = 7-0), all of which are
distinct. However, the optimum 4- mark Golomb ruler is
(0, 1, 4, 6), which measures the distances (1, 2, 3, 4, 5,
6) (and is therefore also a PERFECT Ruler).
The lengths of the optimal n-mark Golomb rulers for
n = 2, 3, 4, . . . are 1, 3, 6, 11, 17, 25, 34, ... (Sloane's
A003022, Vanderschel and Garry). The lengths of the
optimal n-mark Golomb rulers are not known for n > 20.
see also Perfect Difference Set, Perfect Ruler,
Ruler, Taylor's Condition, Weighings
References
Atkinson, M, D,; Santoro, N.; and Urrutia, J. "Integer
Sets with Distinct Sums and Differences and Carrier Fre-
quency Assignments for Nonlinear Repeaters." IEEE
Trans. Comm. 34, 614-617, 1986.
Colbourn, C. J. and Dinitz, J. H. (Eds.) CRC Handbook
of Combinatorial Designs. Boca Raton, FL: CRC Press,
p. 315, 1996.
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.
Lam, A. W. and D. V. Sarwate, D. V. "On Optimum Time
Hopping Patterns." IEEE Trans. Comm. 36, 380-382,
1988.
Robinson, J. P. and Bernstein, A. J. "A Class of Binary Re-
current Codes with Limited Error Propagation." IEEE
Trans. Inform. Th. 13, 106-113, 1967.
Sloane, N. J. A. Sequence A003022/M2540 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Vanderschel, D. and Garry, M. "In Search of the Optimal 20
& 21 Mark Golomb Rulers." http://members.aol.com/
golomb20/.
Golygon
A golygon can be formed if there exists an EVEN Inte-
ger n such that
X "!""! " ! i " r 4
♦ -■♦ --♦■■ - ♦■ - t ♦ -i
(MM!
M
~"
r
, — 9
1
t
!
► — ii
1
1
> ,
> <
A Plane path on a set of equally spaced LATTICE
Points, starting at the Origin, where the first step
is one unit to the north or south, the second step is two
units to the east or west, the third is three units to the
north or south, etc., and continuing until the ORIGIN is
again reached. No crossing or backtracking is allowed.
The simplest golygon is (0, 0), (0, 1), (2, 1), (2, -2),
(-2, -2), (-2, -7), (-8, -7), (-8, 0), (0, 0).
±l±3±...±(n-l) =
±2±4±...±n =
(1)
(2)
(Vardi 1991). Gardner proved that all golygons are of
the form n = 8k. The number of golygons of length n
(Even), with each initial direction counted separately,
is the Product of the Coefficient of x n /8 in
(l + xXl + sV-U + x"- 1 ), (3)
with the Coefficient of a^"/^ 1 )/ 8 i n
(l + aO(l + s 2 )---(l + x n/2 ). (4)
The number of golygons N(n) of length 8n for the first
few n are 4, 112, 8432, 909288, . . . (Sloane's A006718)
and is asymptotic to
JV(n).
3-2 8
7rn 2 (4n + 1)
(5)
(Sallows et al. 1991, Vardi 1991).
see also LATTICE PATH
References
Dudeney, A. K. "An Odd Journey Along Even Roads Leads
to Home in Golygon City," Sci. Amer. 263, 118-121, July
1990.
Sallows, L. C. F. "New Pathways in Serial Isogons." Math.
Intell. 14, 55-67, 1992.
Sallows, L.; Gardner, M.; Guy, R. K.; and Knuth, D. "Serial
Isogons of 90 Degrees." Math Mag. 64, 315-324, 1991.
Sloane, N. J. A. Sequence A006718/M3707 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Vardi, I. "American Science." §5.3 in Computational Recre-
ations in Mathematica. Redwood City, CA: Addison-
Wesley, pp. 90-96, 1991.
Gomory's Theorem
Regardless of where one white and one black square are
deleted from an ordinary 8x8 CHESSBOARD, the reduced
board can always be covered exactly with 31 DOMINOES
(of dimension 2x1).
see also CHESSBOARD
Gompertz Constant
f°° e~ u
Jo * + u
du = -e ei(-l) = 0.596347362
where ei(x) is the Exponential Integral. Stieltjes
showed it has the CONTINUED FRACTION representation
~ 2^4^6-8- "'■'
see also EXPONENTIAL INTEGRAL
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 29, 1983.
748 Gompertz Curve
Goodstein Sequence
Gompertz Curve
The function defined by
y - ab qX .
It is used in actuarial science for specifying a simpli-
fied mortality law. Using s(x) as the probability that a
newborn will achieve age z, the Gompertz law (1825) is
s(x) = exp[— m(c x — 1)],
for c > 1, x > 0.
see also Life Expectancy, Logistic Growth
Curve, Makeham 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.
Gompertz, B. "On the Nature of the Function Expressive
of the Law of Human Mortality." Phil. Trans. Roy. Soc.
London, 1825.
Gonal Number
see Polygonal Number
Good Path
see p-Good Path
Good Prime
A Prime p n is called "good" if
2
Pn -> Pn~iPn-\-i
for all 1 < i < n - 1 (there is a typo in Guy 1994 in
which the is are replaced by Is). There are infinitely
many good primes, and the first few are 5, 11, 17, 29,
37, 41, 53, . . . (Sloane's A028388).
see also Andrica's Conjecture, Polya Conjecture
References
Guy, R. K. "'Good' Primes and the Prime Number Graph."
§A14 in Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 32-33, 1994.
Sloane, N. J. A. Sequence A028388 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Schwenk (1972) rewrote the equation in the form
R + B ■
(j)-LMi(»-D a JJ.
where (£) is a BINOMIAL COEFFICIENT and [x\ is the
Floor Function.
see also Blue-Empty Graph, Extremal Graph,
Monochromatic Forced Triangle
References
Goodman, A. W. "On Sets of Acquaintances and Strangers
at Any Party." Amer. Math. Monthly 66, 778-783, 1959.
Schwenk, A. J. "Acquaintance Party Problem." Amer. Math.
Monthly 79, 1113-1117, 1972.
Goodstein Sequence
Given a HEREDITARY REPRESENTATION of a number
n in BASE, let B[b](n) be the NONNEGATIVE INTEGER
which results if we syntactically replace each b by b + 1
(i.e., B[b] is a base change operator that 'bumps the
base' from b up to 6+ 1). The Hereditary Represen-
tation of 266 in base 2 is
266 = 2 8 + 2 3 + 2
= 2 22+1 +2 2+1 + 2,
so bumping the base from 2 to 3 yields
^a^ 1
2 3+l
S[2](266) = 3 J +3° +± +3.
Now repeatedly bump the base and subtract 1,
Go (266) = 266 = 2 2
+ 2 J+i + 2
Gi(266) = £[21(266) - 1 = 3 + 3^ + 2
G 2 (266) = B[3](Gi) - 1 = 4 4 * +1 + 4 4+1 + 1
G 3 (266) = S[4](G 2 ) - 1 = 5 5 + 5 5+1
G 4 (266) = B[5](G 3 ) - 1 = 6 e6+1 + 6 6+1 - 1
= 6 66+1 +5- 6 6 + 5- 6 5 + ... + 5-6 + 5
G 5 (266) = B[6](G 4 )-1
-7
7 7 + l
+ 5- 7 7 + 5- 7 5 + ... + 5-7 + 4,
Goodman's Formula
A two-coloring of a COMPLETE GRAPH K n of n nodes
which contains exactly the number of MONOCHROMATIC
Forced Triangles and no more (i.e., a minimum of
R + B where R and B are the number of red and blue
Triangles) is called an Extremal Graph. Goodman
(1959) showed that for an extremal graph,
etc. Starting this procedure at an INTEGER n gives the
Goodstein sequence {Gfe(n)}. Amazingly, despite the
apparent rapid increase in the terms of the sequence,
Goodstein's Theorem states that Gk(n) is for any
n and any sufficiently large k.
see also Goodstein's Theorem, Hereditary Repre-
sentation
{~m(m — l)(m — 2) for n = 2m
|m(m - l)(4m +1) for n = 4m + 1
|m(m + l)(4m - 1) for n = 4m + 3.
Goodstein's Theorem
Gosper's Algorithm 749
Goodstein's Theorem
For all n, there exists a k such that the kth term of
the Goodstein Sequence Gk(n) = 0. In other words,
every GOODSTEIN SEQUENCE converges to 0.
The secret underlying Goodstein's theorem is that the
Hereditary Representation of n in base b mimics
an ordinal notation for ordinals less than some number.
For such ordinals, the base bumping operation leaves the
ordinal fixed whereas the subtraction of one decreases
the ordinal. But these ordinals are well-ordered, and
this allows us to conclude that a Goodstein sequence
eventually converges to zero.
Goodstein's theorem cannot be proved in Peano
Arithmetic (i.e., formal Number Theory).
see also Natural Independence Phenomenon,
Peano Arithmetic
Googol
A Large Number equal to 10 100 , or
10000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000.
see also Googolplex, Large Number
References
Kasner, E. and Newman, J. R. Mathematics and the Imagi-
nation. Redmond, WA: Tempus Books, pp. 20-27, 1989.
Pappas, T. "Googol & Googolplex." The Joy of Mathemat-
ics. San Carlos, CA: Wide World Publ./Tetra, p. 76, 1989.
Googolplex
A Large Number equal to 10 lol °°.
see also Googol, Large Number
References
Kasner, E. and Newman, J. R. Mathematics and the Imagi-
nation. Redmond, WA: Tempus Books, pp. 23-27, 1989.
Pappas, T. "Googol & Googolplex." The Joy of Mathemat-
ics. San Carlos, CA: Wide World Publ./Tetra, p. 76, 1989.
Gordon Function
Another name for the Confluent Hypergeometric
Function of the Second Kind, defined by
G(a\c\z) - e '™EM ( r ( 1 ~ c ) [--« , sin[7r(q - c)]
ls(a\c\z) - e r(a) | r(i _ a) ]e + ^^
x iFi(a; c; z) - 2 ^}° " 1 ^" c iFi(o - c + 1; 2 - c; z)
1 [c — a)
where F(x) is the GAMMA Function and iFi(a; b\ z) is
the Confluent Hypergeometric Function of the
First Kind.
see also CONFLUENT HYPERGEOMETRIC FUNCTION OF
the Second Kind
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 671-672, 1953.
Gorenstein Ring
An algebraic Ring which appears in treatments of du-
ality in Algebraic Geometry. Let A be a local Ar-
tinian Ring with m C A its maximal Ideal. Then
A is a Gorenstein ring if the ANNIHILATOR of m has
Dimension 1 as a Vector Space over K = A/m.
see also Cayley-Bacharach Theorem
References
Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach
Theorems and Conjectures." Bull. Amer. Math. Soc. 33,
295-324, 1996.
Gosper's Algorithm
An Algorithm for finding closed form Hypergeomet-
RIC IDENTITIES The algorithm treats sums whose suc-
cessive terms have ratios which are Rational FUNC-
TIONS. Not only does it decide conclusively whether
there exists a hypergeometric sequence z n such that
tn
■ Zn-\-l Zn )
but actually produces z n if it exists. If not, it pro-
duces X]fc=o *"■ ^ n outnne °f the algorithm follows
(Petkovsek 1996):
1. For the ratio r(n) = t n +i/t n which is a Rational
Function of n.
2. Write
r(n)
a(n) c(n + 1)
b(n) c(n) '
where a(n), 6(n), and c(n) are polynomials satisfying
GCD(a(n),6(n + /i) = 1
for all nonnegative integers h.
3. Find a nonzero polynomial solution x(n) of
a(n)x(n + 1) — b(n — l)x(n) = c(n),
if one exists.
4. Return b(n — l)x(n)/c(n)t n and stop.
Petkovsek et al. (1996) describe the algorithm as "one of
the landmarks in the history of computerization of the
problem of closed form summation." Gosper's algorithm
is vital in the operation of Zeilberger's Algorithm
and the machinery of WlLF-ZEILBERGER PAIRS.
see also HYPERGEOMETRIC IDENTITY, SlSTER CELINE'S
Method, Wilf-Zeilberger Pair, Zeilberger's Al-
gorithm
References
Gessel, I. and Stanton, D. "Strange Evaluations of Hyperge-
ometric Series." SI AM J. Math. Anal 13, 295-308, 1982.
Gosper, R. W. "Decision Procedure for Indefinite Hypergeo-
metric Summation." Proc. Nat. Acad. Sci. USA 75, 40-42,
1978.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science, 2nd
ed. Reading, MA: Addis on- Wesley, 1994.
750 Gosper Island
Graceful Graph
Lafron, J. C. "Summation in Finite Terms." In Computer Al-
gebra Symbolic and Algebraic Computation, 2nd ed. (Ed.
B. Buchberger, G. E. Collins, and R. Loos). New York:
Springer- Verlag, 1983.
Paule, P. and Schorn, M. "A Mathematica Version of Zeil-
berger's Algorithm for Proving Binomial Coefficient Iden-
tities." J. Symb. Comput. 20, 673-698, 1995.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Gosper's Al-
gorithm." Ch. 5 in A=B. Wellesley, MA: A. K. Peters,
pp. 73-99, 1996.
Zeilberger, D. "The Method of Creative Telescoping." J.
Symb. Comput 11, 195-204, 1991.
Gosper Island
A modification of the KOCH Snowflake which has
Fractal Dimension
jD =
2 In 3
InT
1.12915.
The term "Gosper island" was used by Mandelbrot
(1977) because this curve bounds the space filled by the
Peano-Gosper Curve; Gosper and Gardner use the
term Flowsnake Fractal instead. Gosper islands can
Tile the Plane.
see also Koch Snowflake, Peano-Gosper Curve
References
Mandelbrot, B. B. Fractals: Form, Chance, & Dimension.
San Francisco, CA: W. H. Freeman, Plate 46, 1977.
Gosper's Method
see Gosper's Algorithm
Graceful Graph
A Labelled Graph which can be "gracefully num-
bered" is called a graceful graph. Label the nodes
with distinct NONNEGATIVE INTEGERS. Then label the
EDGES with the absolute differences between node val-
ues. If the EDGE numbers then run from 1 to e, the
graph is gracefully numbered. In order for a graph to
be graceful, it must be without loops or multiple EDGES.
K 4 = T
I 4
9 3
Petersen
Golomb showed that the number of EDGES connecting
the EVEN-numbered and ODD-numbered sets of nodes
is |_(e + 1)/2J , where e is the number of EDGES. In ad-
dition, if the nodes of a graph are all of EVEN ORDER,
then the graph is graceful only if [(e -f- 1)/2J is Even.
The only ungraceful simple graphs with < 5 nodes are
shown below.
M
There are exactly e! graceful graphs with e EDGES
(Sheppard 1976), where e!/2 of these correspond to
different labelings of the same graph. Golomb (1974)
showed that all complete bipartite graphs are graceful.
Caterpillar Graphs; Complete Graphs K 2) K 3i
K a = W± = T (and only these; Golomb 1974); CYCLIC
GRAPHS C n when n = or 3 (mod 4), when the num-
ber of consecutive chords k = 2, 3, or n — 3 (Koh and
Punim 1982), or when they contain a P^ chord (Delorme
et ah 1980, Koh and Yap 1985, Punnim and Pabhapote
1987); Gear Graphs; Path Graphs; the Petersen
Graph; Polyhedral Graphs T = K 4 = W4, C, 0,
D, and J (Gardner 1983); Star Graphs; the Thomsen
Graph (Gardner 1983); and Wheel Graphs (Prucht
1988) are all graceful.
Some graceful graphs have only one numbering, but oth-
ers have more than one. It is conjectured that all trees
are graceful (Bondy and Murty 1976), but this has only
Graceful Graph
Gradient 751
been proved for trees with < 16 VERTICES. It has also
been conjectured that all unicyclic graphs are graceful.
An excellent on-line resource is Brundage (http://www*
math. . washingt on . edu/ "brundage/ oldgr acef ul/) .
see also Harmonious Graph, Labelled Graph
References
Abraham, J. and Kotzig, A. "All 2-Regular Graphs Consist-
ing of 4-Cycles are Graceful." Disc. Math, 135, 1-24,
1994.
Abraham, J. and Kotzig, A. "Extensions of Graceful Valu-
ations of 2-Regular Graphs Consisting of 4-Gons." Ars
Combin. 32, 257-262, 1991.
Bloom, G. S, and Golomb, S. W. "Applications of Numbered
Unidirected Graphs." Proc. IEEE 65, 562-570, 1977.
Bolian, L. and Xiankun, Z. "On Harmonious Labellings of
Graphs." Ars Combin. 36, 315-326, 1993.
Brualdi, R. A. and McDougal, K. F. "Semibandwidth of Bi-
partite Graphs and Matrices." Ars Combin. 30, 275-287,
1990.
Brundage, M. "Graceful Graphs." http://www.math.
washington.edu/firundage/oldgraceful/.
Cahit, I. "Are All Complete Binary Trees Graceful?" Amer.
Math. Monthly 83, 35-37, 1976.
Delorme, C; Maheo, M.; Thuillier, H.; Koh, K. M.; and
Teo, H. K. "Cycles with a Chord are Graceful." J. Graph
Theory 4, 409-415, 1980.
Prucht, R. W. and Gallian, J. A. "Labelling Prisms." Ars
Combin. 26, 69-82, 1988.
Gallian, J. A. "A Survey: Recent Results, Conjectures, and
Open Problems in Labelling Graphs." J. Graph Th. 13,
491-504, 1989.
Gallian, J. A. "Open Problems in Grid Labeling." Amer.
Math. Monthly 97, 133-135, 1990.
Gallian, J. A. "A Guide to the Graph Labelling Zoo." Disc.
Appl. Math. 49, 213-229, 1994.
Gallian, J. A.; Prout, J.; and Winters, S. "Graceful and Har-
monious Labellings of Prism Related Graphs." Ars Com-
bin. 34, 213-222, 1992.
Gardner, M. "Golomb's Graceful Graphs." Ch. 15 in Wheels,
Life, and Other Mathematical Amusements. New York:
W. H. Freeman, pp. 152-165, 1983.
Golomb, S. W. "The Largest Graceful Subgraph of the Com-
plete Graph." Amer. Math. Monthly 81, 499-501, 1974.
Guy, R. "Monthly Research Problems, 1969-75." Amer.
Math. Monthly 82, 995-1004, 1975.
Guy, R. "Monthly Research Problems, 1969-1979." Amer.
Math. Monthly 86, 847-852, 1979.
Guy, R. K. "The Corresponding Modular Covering Problem.
Harmonious Labelling of Graphs." §C13 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 127-128, 1994.
Huang, J. H. and Skiena, S. "Gracefully Labelling Prisms."
Ars Combin. 38, 225-242, 1994.
Koh, K. M. and Punnim, N. "On Graceful Graphs: Cycles
with 3-Consecutive Chords." Bull Malaysian Math. Soc.
5, 49-64, 1982.
Jungreis, D. S. and Reid, M. "Labelling Grids." Ars Combin.
34, 167-182, 1992.
Koh, K. M. and Yap, K. Y. "Graceful Numberings of Cycles
with a P 3 -Chord." Bull Inst. Math. Acad. Sinica 13, 41-
48, 1985.
Morris, P. A. "On Graceful Trees." http:// www . math .
Washington, edu /-brundage /math /graceful /source/ on _
graceful-trees .ps.
Moulton, D. "Graceful Labellings of Triangular Snakes." Ars
Combin. 28, 3-13, 1989.
Murty, U. S. R. and Bondy, J. A. Graph Theory with Appli-
cations. New York: North Holland, p. 248, 1976.
Punnim, N. and Pabhapote, N. "On Graceful Graphs: Cycles
with a JVChord, k > 4." Ars Combin. A 23, 225-228,
1987.
Rosa, A. "On Certain Valuations of the Vertices of a Graph."
In Theory of Graphs, International Symposium, Rome,
July 1966. New York: Gordon and Breach, pp. 349-355,
1967.
Sheppard, D. A. "The Factorial Representation of Balanced
Labelled Graphs." Discr. Math. 15, 379-388, 1976.
Sierksma, G. and Hoogeveen, H. "Seven Criteria for Integer
Sequences Being Graphic." J. Graph Th. 15, 223-231,
1991.
Slater, P. J. "Note on fc-Graceful, Locally Finite Graphs." J.
Combin. Th. Ser. B 35, 319-322, 1983.
Snevily, H. S. "New Families of Graphs That Have a-
Labellings," Preprint.
Snevily, H. S. "Remarks on the Graceful Tree Conjecture."
Preprint.
Xie, L. T. and Liu, G. Z. "A Survey of the Problem of Grace-
ful Trees." Qufu Shiyuan Xuebao 1, 8-15, 1984.
Graded Algebra
If A is a graded module and there EXISTS a degree-
preserving linear map <j> : A ® A — ¥ A, then (A, <j>) is
called a graded algebra.
References
Jacobson, N. Lie Algebras. New York: Dover, p. 163, 1979.
Gradian
A unit of angular measure in which the angle of an entire
CIRCLE is 400 gradians. A RIGHT ANGLE is therefore
100 gradians.
see also DEGREE, RADIAN
Gradient
The gradient is a VECTOR operator denoted V and
sometimes also called DEL or NABLA. It most often is
applied to a real function of three variables / (ui , ui , u% ) ,
and may be denoted
V/ = grad(f).
(1)
For general Curvilinear Coordinates, the gradient
is given by
hi dui hi OU2 hs 0U3
which simplifies to
(3)
in Cartesian Coordinates.
The direction of Vf is the orientation in which the DI-
RECTIONAL Derivative has the largest value and |V/|
is the value of that DIRECTIONAL DERIVATIVE. Further-
more, if V/ / 0, then the gradient is PERPENDICULAR
to the Level Curve through (xq, yo) if z = /(#, y) and
Perpendicular to the level surface through (#0, yo, ^0)
if F(a:,jM) = 0.
752 Gradient Four-Vector
In Tensor notation, let
ds 2 = g^ dx^ 2 (4)
be the LINE Element in principal form. Then
1
Ve* a e> = V a e> =
yfg^dxt
-ep.
For a Matrix A,
V|Ax| =
(Ax) T A
|Ax| '
(5)
(6)
For expressions giving the gradient in particular coordi-
nate systems, see Curvilinear Coordinates.
see also CONVECTIVE DERIVATIVE, CURL, DIVER-
GENCE, Laplacian, Vector Derivative
References
Arfken, G. "Gradient, V" and "Successive Applications of
V." §1.6 and 1.9 in Mathematical Methods for Physicists,
3rd ed. Orlando, FL: Academic Press, pp. 33-37 and 47-
51, 1985.
Gradient Four- Vector
The 4-dimensional version of the GRADIENT, encoun-
tered frequently in general relativity and special relativ-
ity, is
I An
cjt
d _l
d y
_d_
dz J
which can be written
where D 2 is the D'ALEMBERTIAN OPERATOR.
see also d'Alembertian Operator, Gradient, Ten-
sor, Vector
References
Morse, P. M. and Feshbach, H. "The Differential Operator
V." §1.4 in Methods of Theoretical Physics, Part L New
York: McGraw-Hill, pp. 31-44, 1953.
Gradient Theorem
[
(V/)-ds = /(6)-/(a),
where V is the GRADIENT, and the integral is a LINE
Integral. It is this relationship which makes the defi-
nition of a scalar potential function / so useful in gravi-
tation and electromagnetism as a concise way to encode
information about a VECTOR FIELD.
see also Divergence Theorem, Green's Theorem,
Line Integral
Graeffe's Method
Graeco-Latin Square
see EULER SQUARE
Graeco- Roman Square
see Euler Square
Graeffe's Method
A RoOT-finding method which proceeds by multiplying
a Polynomial f(x) by f(-x) and noting that
f(x) = (x - ai)(x - a 2 ) • • • (x - a n ) (1)
f(-x) = (-l) n (x + ai)(x + a 2 ) • • ■ (ai + a„) (2)
so the result is
f(x)f(-x) = (-l) n (x 2 - ai 2 )(x 2 - a 2 2 ) • • • (x 2
Repeat v times, then write this in the form
y +o\y
+ ... + &„
(3)
(4)
where y = x 2u . Since the coefficients are given by New-
ton's Relations
&i = -(yi + 2/2 + ■ - - + y n ) (5)
&2 = (2/12/2 + yiys + . . . + y n -iy n ) (6)
6„ = (-l) n yi^-.-y„, (7)
and since the squaring procedure has separated the
roots, the first term is larger than rest. Therefore,
giving
61 « -yi
(8)
&2 ~ 2/12/2
(9)
b n « (~l) n yiy2 •
■■yn,
(10)
y\ ~ -61
(11)
62
(12)
bn
(13)
-1
Solving for the original roots gives
. 21// r~
ai « V bl
a2 ~ y~*r
(14)
(15)
(16)
This method works especially well if all roots are real.
References
von Karman, T. and Biot, M. A. "Squaring the Roots
(Graeffe's Method)." §5.8.c in Mathematical Methods in
Engineering: An Introduction to the Mathematical Treat-
ment of Engineering Problems. New York: McGraw-Hill,
pp. 194-196, 1940.
Graham's Biggest Little Hexagon
Gram's Inequality 753
Graham's Biggest Little Hexagon
0.9891
0.343771
The largest possible (not necessarily regular) HEXAGON
for which no two of the corners are more than unit
distance apart. In the above figure, the heavy lines
are all of unit length. The AREA of the hexagon is
A = 0.674981 . . ., where A is a ROOT of
4096A 10 - 8192A 9 - 3008A 8 - 30, 848A 7 + 21, 056^ 6
+146,496A 5 - 221,360A 4 + X2S2A 3 + 144, 464^ 2
-78,4884 + 11,993 = 0.
see also Calabi's Triangle
References
Conway, J. H. and Guy, R. K. "Graham's Biggest Little Hex-
agon." In The Book of Numbers. New York: Springer-
Verlag, pp. 206-207, 1996.
Graham, R. L. "The Largest Small Hexagon." J. Combin.
Th. Ser. A 18, 165-170, 1975.
Graham's Number
The smallest dimension of a HYPERCUBE such that if the
lines joining all pairs of corners are two-colored, a Pla-
nar Complete Graph K$ of one color will be forced.
That an answer exists was proved by R. L. Graham and
B. L. Rothschild. The actual answer is believed to be 6,
but the best bound proved is
64 <
3|3,
where f is stacked Arrow Notation. It is less than
3 -» 3 -> 3 -» 3, where Chained Arrow Notation
has been used.
see also Arrow Notation, Chained Arrow Nota-
tion, Skewes Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 61-62, 1996.
Gardner, M. "Mathematical Games." Sci. Amer. 237, 18-
28, Nov. 1977.
Gram-Charlier Series
Approximates a distribution in terms of a NORMAL DIS-
TRIBUTION. Let
0(t) ^ J=e"' 2 ' 2
2tt
then
f(t) = </>(t) + SW 3) (*) + £t^ w W + • - ■ ■
,< 4 )f
see also Edgeworth Series
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 107-108,
1951.
Gram Determinant
The Determinant
<3(/l,/2,... »/n)
Jfidt Jhhdt .•• Jfifndt
Jf2fidt jfidt ... f hUdt
fflfndt Jfifndt
Jfldt
see also Gram-Schmidt Orthonormalization,
Wronskian
References
Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, p. 2, 1991.
Gram's Inequality
Let fi(x), . . . , f n (x) be Real Integrable Functions
over the Closed Interval [a, 6], then the Determi-
nant of their integrals satisfies
J"f 1 2 (x)dx j b a h{x)f 2 {x)dx
J h{x)h(x)dx J f 2 2 (x)dx
J a f n (x)fi{x)dx j a f n (x)f 2 (x)dx
j"f 1 (x)f n (x)dx
J*f 2 (x)f n (x)dx
Jj n (x)f n (x)dx
> 0,
see also Gram-Schmidt Orthonormalization
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.
754
Gram Matrix
Grain-Schmidt Orthonormalization
Gram Matrix
Given m points with n-D vector coordinates v», let M
be the n x m matrix whose jth column consists of the
coordinates of the vector Vj, with j = 1, . . . , m. Then
define the m x m Gram matrix of dot products chj =
Vj • Vj as
A = M T M,
where A T denotes the Transpose. The Gram matrix
determines the vectors Vj up to ISOMETRY.
Gram- Schmidt Orthonormalization
A procedure which takes a nonorthogonal set of LIN-
EARLY Independent functions and constructs an Or-
thogonal BASIS over an arbitrary interval with respect
to an arbitrary WEIGHTING FUNCTION w(x). Given an
original set of linearly independent functions {u n }> let
{ifin} denote the orthogonalized (but not normalized)
functions and {<j> n } the orthonormalized functions.
and
ipo(x) = ui(x)
4>o(x) =
ipo(x)
J J ipo 2 (x)w(x)dx
Take
if>i(x) = ui(x) + aio<t>o(x),
where we require
(i)
(2)
(3)
/ tl)i<t>owdx~ / ui<poW dx + aio / ^wdx = 0. (4)
By definition,
/•
>o w dx = 1 ,
aio — ~ I ui<j>owdx.
The first orthogonalized function is therefore
tf>i = ui(x) -
/
u\ <j)$w dx
00,
and the corresponding normalized function is
A/fipi 2 wdx
By mathematical induction, it follows that
, ( . $i{x)
J J ipi 2 wdx
where
ipi(x) = Ui + aiQ(po -f an(f>i . . . + ai y i-i(f>i-i
(5)
(6)
(7)
(8)
(9)
(10)
ay = - J Ui<pj
wdx.
(11)
If the functions are normalized to Nj instead of 1, then
(12)
/
J a
[4>j(x)] wdx = Nj
<t>i{ x ) = N i
ipi(x)
Jhh 2
2 w dx
J Ui<j>jW dx
(13)
(14)
ORTHOGONAL POLYNOMIALS are especially easy to gen-
erate using Gram-Schmidt Orthonormalization.
Use the notation
(xi\xj) = (xi\w\xj) = / Xi(x)xj(x)w(x)dx, (15)
J a
where w(x) is a Weighting Function, and define the
first few Polynomials,
p (x) = l
Pi 0*0 =
(xpo\po)
(Po\Po)
Po-
(16)
(17)
As denned, p and pi are Orthogonal Polynomials,
as can be seen from
(po\pi) =
(xp \po)
(Po\po)
Po
{xpo) _i^M {po)
(po\po)
= (xpo) - (xpo) = 0,
Now use the RECURRENCE RELATION
Pi+i(a) :
{xpi\pi)
{Pi\pi)
Pi
(Pi\pi)
{Pi-l\pi-l}
(18)
Pi-i (19)
to construct all higher order POLYNOMIALS.
To verify that this procedure does indeed produce OR-
THOGONAL Polynomials, examine
(xpi\pi)
(Pi+i\Pi)
Pi
(Pi\Pi)
{Pi\Pi)
(Pi-i|p*-i>
(xPi\Pi)
(Pi\pi)
Pi-l
= (xPi\Pi) -
(Pi\pi)
Pi
Pi
(Pi\pi)
(Pi-l\pi-l)
(pi\pi)
(Pi-llpi-l)
(Pi\pi)
(Pi-llpi-l>
(Pi-l\pi)
(Pi-l\pi)
(Pi-l\pj-l)
(Pj_ 2 |Pj-2>
(Pj-2\Pj~l)
= - = (-&$!&<*>&) = o,
(PolPo)
(20)
Gram Series
Graph (Graph Theory) 755
since (po|pi) = 0. Therefore, all the POLYNOMIALS pi(x)
are orthogonal. Many common ORTHOGONAL POLYNO-
MIALS of mathematical physics can be generated in this
manner. However, the process is numerically unstable
(Golub and van Loan 1989).
see also Gram Determinant, Gram's Inequality,
Orthogonal Polynomials
Graph (Function)
References
Arfken, G. "Gram-Schmidt Orthogonalization."
Mathematical Methods for Physicists, 3rd ed.
FL: Academic Press, pp. 516-520, 1985.
Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd
ed. Baltimore, MD: Johns Hopkins, 1989.
§9.3 in
Orlando,
Gram Series
R(x)
* + £
(Ins)*
JfeA:!C(fc + 1) '
where £ is the Riemann Zeta Function. This approx-
imation to the Prime Counting Function is 10 times
better than Li(#) for x < 10 9 but has been proven to be
worse infinitely often by Littlewood (Ingham 1990). An
equivalent formulation due to Ramanujan is
G(x) =
4y (-D*
_ ( lnx Y
7T Z-, B 2 k(2k- 1) V 2tt )
7r(x)
(Berndt 1994), where B 2 k is a BERNOULLI NUMBER.
The integral analog, also found by Ramanujan, is
t/0
QnxYdt
tr(* + i)c(* + i)
7r(x)
J(x)
(Berndt 1994).
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 124-129, 1994.
Gram, J. P. "Unders0gelser angaaende Maengden af Primtal
under en given Graeense." K. Videnskab. Selsk. Skr. 2,
183-308, 1884.
Ingham, A. E. Ch. 5 in The Distribution of Prime Numbers.
New York: Cambridge, 1990.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 74, 1991.
Granny Knot
A Composite Knot of seven crossings consisting of a
KNOT SUM of TREFOILS. The granny knot has the same
Alexander Polynomial (x 2 ~ x + 1) 2 as the Square
Knot.
1 2
Technically, the graph of a function is its RANGE (a.k.a.
image). Informally, given a FUNCTION /(a?i, . . . , x n ) de-
fined on a DOMAIN U, the graph of / is defined as a
Curve or Surface showing the values taken by / over
U (or some portion of £/),
graph /(x) = {(x,F(x)) G M 2 : x G U]
graph /(xi, . . . , x n ) = {(xi, . . . , x n , f(xi , . . . , x n ))
£R n+1 :(xi x n )6E/}.
A graph is sometimes also called a Plot.
Good routines for plotting graphs use adaptive algo-
rithms which plot more points in regions where the
function varies most rapidly (Wagon 1991, Math Works
1992, Heck 1993, Wickham-Jones 1994).
see also Curve, Extremum, Graph (Graph The-
ory), Histogram, Maximum, Minimum
References
Cleveland, W. S. The Elements of Graphing Data, rev. ed.
Summit, NJ: Hobart, 1994.
Heck, A. Introduction to Maple, 2nd ed. New York: Springer-
Verlag, pp. 303-304, 1993.
Math Works. Matlab Reference Guide. Natick, MA: The
Math Works, p. 216, 1992.
Tufte, E. R. The Visual Display of Quantitative Information.
Cheshire, CN: Graphics Press, 1983.
Tufte, E. R. Envisioning Information. Cheshire, CN: Graph-
ics Press, 1990.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 24-25, 1991.
Wickham-Jones, T. Computer Graphics with Mathematica.
Santa Clara, CA: TELOS, pp. 579-584, 1994.
Yates, R. C. "Sketching." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 188-205,
1952.
Graph (Graph Theory)
1 •
A A A
:: l: u k
756 Graph (Graph Theory)
Graph (Graph Theory)
A mathematical object composed of points known as
VERTICES or NODES and lines connecting some (possibly
empty) SUBSET of them, known as Edges. The study
of graphs is known as Graph Theory. Graphs are 1-D
COMPLEXES, and there are always an EVEN number of
Odd NODES in a graph. The number of nonisomorphic
graphs with v NODES is given by the POLYA ENUMER-
ATION Theorem. The first few values for n = 1, 2, . . . ,
are 1, 2, 4, 11, 34, 156, 1044, . . . (Sloane's A000088; see
above figure).
Graph sums, differences, powers, and products can be
defined, as can graph eigenvalues.
Before applying POLYA ENUMERATION THEOREM, de-
fine the quantity
(i)
where p\ is the FACTORIAL of p, and the related poly-
nomial
z,.(s)=$>j,n/* g, \
(2)
where the ji — (ji, • . . , j P )* are all of the p- VECTORS
satisfying
Ji + 2j2 + 3j3 + ...+pj p =p. (3)
For example, for p = 3, the three possible values of j are
ji = (3, 0, 0), since (1 ■ 3) + (2 • 0) + (3 • 0) = 3,
giving ^^ (i33!)(2 ^ !)(30()!) ^l (4)
j 2 = (1, 1, 0), since (1 ■ 1) + (2 • 1) + (3 • 0) = 3,
giving Aj a = (111!)(2 ^ !)(300!) = 3, (5)
j 3 = (0,0, 1), since (1 ■ 0) + (2 • 0) + (3 ■ 1) = 3
3!
giving fcj 3 =
(1°0!)(2°0!)(3 1 1!)
Therefore,
Z 3 (S) = f 1 3 + 3fif 2 + 2f 3 .
2. (6)
(7)
For small p, the first few values of Z P (S) are given by
Z 2 (S) = f 1 2 + f 2
Z 3 (S) = f 1 3 +3f 1 f 2 + 2f 3
Z A {S)
Zs(S)
= h + 6/i 2 / 2 + 3/ 2 2 + 8/1/3 + 6/ 4
= /i 5 + IO/1V2 + 15/i/2 2 + 20/i 2 /3
+ 20/2/3 + 30/i/4 + 24/ 5
(8)
(9)
(10)
(11)
Za{S) = /1 6 + 15/! 4 /2 + 45/i 2 /2 2 + 15/ 2 3
+ 40/i 3 /3 + I2O/1/2/3 + 40/ 3 2
+ 90/i 2 / 4 + 90/2/4 + 144/i/s + 120/e (12)
Z 7 (S) = f x 7 + 21/i 5 /2 + 105/! 3 /2 2 + IO5/1/2 3
+ 70/i 4 /3 + 420/! 2 / 2 / 3 + 210/ 2 2 / 3
+ 280/i/s 2 + 210/i 3 / 4 + 63O/1/2/4
+ 420/3/4 + 504/i 2 / 5 + 504/2/5
+ 840/i/ 6 + 720/r.
Application of the Polya Enumeration Theorem
then gives the formula
l(p-l)/2J
z(*)=i5> n 52 „ + i— +(2 - +x) (^ +i )
P ' U) n=0
IP/2J
Yl II 9lcm(,,t)
j q j r GCD(q,r)
, (14)
q=l r=q+l
where [^J is the FLOOR FUNCTION, („") is a BINOMIAL
Coefficient, LCM is the Least Common Multiple,
GCD is the Greatest Common Divisor, and the Sum
(j) is over all ji satisfying the sum identity described
above. The first few generating functions Z P (R) are
Z 2 (R) = 2 9l (15)
Z 3 (R) = gi 3 + 3gi92 + 2g 3 (16)
Z A {R) = 9i 6 + 9<?i V + 8 53 2 + 69294 (17)
Z*(R) = 91 10 + lOfin 4 ^ 3 + 15 9l 2 92 4 + 20^ l5 r 3 3
+ 30g 2 g 4 2 + 24g 5 2 + 20 9l g 3 g 6 (18)
Z 6 {R) = 9l 15 + V5gi 7 g 2 4 + Q0 9l 3 g 2 6 + 40 9l 3 g 3 4
+ 40p 3 5 + 180 9 i 92 g4 S + 144£5 3
+ 120gig 2 g 3 2 g 6 + 120p 3 p6°
Z 7 (R) = 9l 21 + 21 gi 11 92 * + 105gi 5 g 2 8 (19)
+ 105<?i 3 <? 2 9 + 70 gi 6 gz 5 + 2S0gs 7
4- 210gi 3 g 2 g4 4 + 630gig 2 2 g4 4
+ 504#i£ 5 4 -f 420g 1 2 g 2 2 g 3 3 g6
+ 210 9l 2 g 2 2 g 3 g 6 2 + SA0g 3 g 6 S + 720g 7 3
+ 504g l9 5 2 9 io + 420g 2 g 3 g 4 g 12 . (20)
Letting gi = 1 + x i then gives a POLYNOMIAL Si(x),
which is a GENERATING FUNCTION for (i.e., the terms
of x 1 give) the number of graphs with i EDGES. The
total number of graphs having i edges is 5t(l). The first
few Si{x) are
S 2 = 1 + x
£3 = 1 + x + x 2 + x 3
S 4 = 1 + x + 2x 2 + 3x 3 + 2x 4 + x 5 + x 6
5 5 = 1 + x + 2x 2 + 4x 3 + Qx 4 + 6x 5 + 6x 6
+ 4x 7 + 2:r 8 + x 9 + :
10
5 6 = 1 + x + 2x 2 + 5x 3 + 9x 4 + 15x 5
+ 21z 6 -f 24x 7 + 24a; 8 + 21a; 9
12
(13)
+ 15a; 10 + Qx 11 + 5x
. r, 13 . 14 . 15
+ 2x + x + x
S 7 = 1 + x + 2x 2 + 5x 3 + 10a: 4 + 21a; 5
+ 21a; 6 + 24x 7 + 41x 6 + 65x 7 + 97x 8
(21)
(22)
(23)
(24)
(25)
Graph (Graph Theory)
Graph Two-Coloring 757
+ 131z 9 + 148x 10 + 148X 11
+ 131a: 12 + 97z 13 + 65x 14 + 41x 15
+ 21x 16 + 10z 17 + 5x 18 + 2x 19 + x 20 + x
(26)
giving the number of graphs with n nodes as 1, 2, 4, 11,
34, 156, 1044, . . . (Sloane's A000088). King and Palmer
(cited in Read 1981) have calculated S n up to n = 24,
for which
5 24 = 195, 704, 906, 302, 078, 447, 922, 174, 862, 416, • • •
• • • 726, 256, 004, 122, 075, 267, 063, 365, 754, 368. (27)
see also Bipartite Graph, Caterpillar Graph,
Cayley Graph, Circulant Graph, Cocktail
Party Graph, Comparability Graph, Comple-
ment Graph, Complete Graph, Cone Graph, Con-
nected Graph, Coxeter Graph, Cubical Graph,
de Bruijn Graph, Digraph, Directed Graph,
Dodecahedral Graph, Euler Graph, Extremal
Graph, Gear Graph, Graceful Graph, Graph
Theory, Hanoi Graph, Harary Graph, Harmo-
nious Graph, Hoffman-Singleton Graph, Icos-
ahedral Graph, Interval Graph, Isomorphic
Graphs, Labelled Graph, Ladder Graph, Lattice
Graph, Matchstick Graph, Minor Graph, Moore
Graph, Null Graph, Octahedral Graph, Path
Graph, Petersen Graphs, Planar Graph, Ran-
dom Graph, Regular Graph, Sequential Graph,
Simple Graph, Star Graph, Subgraph, Super-
graph, Superregular Graph, Sylvester Graph,
Tetrahedral Graph, Thomassen Graph, Tourna-
ment, Triangular Graph, Turan Graph, Tutte's
Graph, Universal Graph, Utility Graph, Web
Graph, Wheel Graph
References
Bogomolny, A. "Graph Puzzles." http://wvw.cut-the-
knot . com/do_you_know/graphs2.html.
Fujii, J. N. Puzzles and Graphs. Washington, DC: National
Council of Teachers, 1966.
Harary, F. "The Number of Linear, Directed, Rooted, and
Connected Graphs." Trans. Avner. Math. Soc. 78, 445-
463, 1955.
Pappas, T. "Networks." The Joy of Mathematics. San Car-
los, CA: Wide World Publ./Tetra, pp. 126-127, 1989.
Read, R. "The Graph Theorists Who Count— and What
They Count." In The Mathematical Gardner (Ed.
D. Klarner). Boston, MA: Prindle, Weber, and Schmidt,
pp. 326-345, 1981.
Sloane, N. J. A. Sequences A000088/M1253 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.
Graph Theory
The mathematical study of the properties of the formal
mathematical structures called GRAPHS.
see also ADJACENCY MATRIX, ADJACENCY RELA-
TION, Articulation Vertex, Blue-Empty Color-
ing, Bridge (Graph), Chromatic Number, Chro-
matic Polynomial, Circuit Rank, Crossing Num-
ber (Graph), Cycle (Graph), Cyclomatic Num-
ber, Degree, Diameter (Graph), Dijkstra's Al-
gorithm, Eccentricity, Edge-Coloring, Edge
Connectivity, Eulerian Circuit, Eulerian Trail,
Factor (Graph), Floyd's Algorithm, Girth,
Graph Two-Coloring, Group Theory, Hamilton-
ian Circuit, Hasse Diagram, Hub, Indegree, Inte-
gral Drawing, Isthmus, Join (Graph), Local De-
gree, Monochromatic Forced Triangle, Outde-
gree, Party Problem, Polya Enumeration Theo-
rem, Polya Polynomial, Radius (Graph), Ramsey
Number, Re-Entrant Circuit, Separating Edge,
Tait Coloring, Tait Cycle, Traveling Sales-
man Problem, Tree, Tutte's Theorem, Unicursal
Circuit, Valency, Vertex Coloring, Walk
References
Berge, C. The Theory of Graphs. New York: Wiley, 1962.
Bogomolny, A. "Graphs." http://www.cut-the-knot.com/
do_you_knov/graphs .html.
Bo Hob as, B. Graph Theory: An Introductory Course. New
York: Springer- Verlag, 1979.
Chartrand, G. Introductory Graph Theory. New York:
Dover, 1985.
Foulds, L. R. Graph Theory Applications. New York:
Springer- Verlag, 1992.
Chung, F. and Graham, R. Erdos on Graphs: His Legacy of
Unsolved Problems. New York: A. K. Peters, 1998.
Grossman, I. and Magnus, W. Groups and Their Graphs.
Washington, DC: Math. Assoc. Amer., 1965.
Harary, F. Graph Theory. Reading, MA: Addis on- Wesley,
1994.
Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A
Comprehensive Introduction, 2nd ed. San Diego, CA: Aca-
demic Press, 1994.
Ore, 0. Graphs and Their Uses. New York: Random House,
1963.
Ruskey, F. "Information on (Unlabelled) Graphs." http://
sue . esc . uvic . ca/-cos/inf /grap/Graphlnf o . html.
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:
Assaults and Conquest. New York: Dover, 1986.
Skiena, S. S. Implementing Discrete Mathematics: Combi-
natorics and Graph Theory with Mathematica. Redwood
City, CA: Addis on- Wesley, 1988.
Trudeau, R. J. Introduction to Graph Theory. New York:
Dover, 1994.
Graph Two-Coloring
Assignment of each Edge of a Graph to one of two
color classes ("red" or "green").
see also Blue-Empty Graph, Monochromatic
Forced Triangle
758 Graphical Partition
Gray Code
Graphical Partition
A graphical partition of order n is the DEGREE SE-
QUENCE of a Graph with n/2 Edges and no isolated
VERTICES. For n = 2, 4, 6, . . . , the number of graphical
partitions is 1, 2, 5, 9, 17, . . . (Sloane's A000569).
References
Barnes, T. M. and Savage, C. D. "A Recurrence for Count-
ing Graphical Partitions." Electronic J. Combinatorics
2, Rll, 1-10, 1995. http://www.combinatorics.org/
Volume_2/volume2 . html#Rl 1 .
Barnes, T. M. and Savage, C. D. "Efficient Generation of
Graphical Partitions." Submitted.
Ruskey, F. "Information on Graphical Partitions." http://
sue . esc . uvic . ca / - cos / inf / nump / Graphical
Part it ion . html.
Sloane, N. J. A. Sequence A000569 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
di, which is kept the same since do is assumed to be a
0. The resulting number g\ g^ • ■ ■ g n -i g n is the reflected
binary Gray code.
To convert a binary reflected Gray code g± gi * * • g n ~i g n
to a Binary number, start again with the nth digit, and
compute
S n = ^9i (mod 2).
If S n is 1, replace g n by 1 - g n \ otherwise, leave it the
unchanged. Next compute
S n _i ee ^2/ 9i ( mod 2 ) }
Grassmann Algebra
see Exterior Algebra
Grassmann Coordinates
An (m + 1)-D Subspace W of an (n + 1)-D Vector
Space V can be specified by an (m + 1) x (n-fl) MATRIX
whose rows are the coordinates of a Basis of W. The set
of all (ZX\) (m + 1) x (m + 1) Minors of this Matrix
are then called the Grassmann coordinates of w (where
(fj is a Binomial Coefficient).
see also Chow Coordinates
References
Wilson, W. S.; Chern, S. S.; Abhyankar, S. S.; Lang, S.; and
Igusa, J.-I. "Wei-Liang Chow." Not Amer. Math. Soc.
43, 1117-1124, 1996.
Grassmann Manifold
A special case of a Flag MANIFOLD. A Grassmann
manifold is a certain collection of vector SUBSPACES of
a VECTOR Space. In particular, G n ,k is the Grass-
mann manifold of fc-dimensional subspaces of the VEC-
TOR SPACE M n . It has a natural MANIFOLD structure
as an orbit-space of the Stiefel Manifold V ny k of
orthonormal fc-frames in K™. One of the main things
about Grassmann manifolds is that they are classifying
spaces for VECTOR BUNDLES.
Gray Code
An encoding of numbers so that adjacent numbers have
a single DIGIT differing by 1. A BINARY Gray code with
n Digits corresponds to a Hamiltonian Path on an
n-D Hypercube (including direction reversals). The
term Gray code is often used to refer to a "reflected"
code, or more specifically still, the binary reflected Gray
code.
To convert a BINARY number d\ di ■ • ■ d n -i d n to its cor-
responding binary reflected Gray code, start at the right
with the digit d n (the nth, or last, Digit). If the d n -\
is 1, replace d n by 1 — d n ; otherwise, leave it unchanged.
Then proceed to d n -\. Continue up to the first Digit
and so on. The resulting number d\ dv ■ * • d n -i d n is
the BINARY number corresponding to the initial binary
reflected Gray code.
The code is called reflected because it can be generated
in the following manner. Take the Gray code 0, 1. Write
it forwards, then backwards: 0, 1, 1, 0. Then append Os
to the first half and Is to the second half: 00, 01, 11, 10.
Continuing, write 00, 01, 11, 10, 10, 11, 01, 00 to obtain:
000, 001, 011, 010, 110, 111, 101, 100, ... (Sloane's
A014550). Each iteration therefore doubles the number
of codes. The Gray codes corresponding to the first few
nonnegative integers are given in the following table.
20
11110
40
111100
1
1
21
11111
41
111101
2
11
22
11101
42
111111
3
10
23
11100
43
111110
4
110
24
10100
44
111010
5
111
25
10101
45
111011
6
101
26
10111
46
111001
7
100
27
10110
47
111000
8
1100
28
10010
48
101000
9
1101
29
10011
49
101001
10
1111
30
10001
50
101011
11
1110
31
10000
51
101010
12
1010
32
110000
52
101110
13
1011
33
110001
53
101111
14
1001
34
110011
54
101101
15
1000
35
110010
55
101100
16
11000
36
110110
56
100100
17
11001
37
110111
57
100101
18
11011
38
110101
58
100111
19
11010
39
110100
59
100110
The binary reflected Gray code is closely related to the
solution of the TOWERS OF HANOI as well as the Bague-
NAUDIER.
see also Baguenaudier, Binary, Hilbert Curve,
Morse-Thue Sequence, Ryser Formula, Towers
of Hanoi
Great Circle
Great Circle
759
References
Gardner, M. "The Binary Gray Code." Ch. 2 in Knotted
Doughnuts and Other Mathematical Entertainments. New
York: W. H. Freeman, 1986.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Gray Codes." §20.2 in Numerical Recipes
in FORTRAN: The Art of Scientific Computing, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 886-888, 1992,
Sloane, N. J. A. Sequence A014550 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 111-112 and 246,
1991.
Great Circle
The shortest path between two points on a Sphere,
also known as an ORTHODROME. To find the great cir-
cle (GEODESIC) distance between two points located at
Latitude S and Longitude A of (<5i,Ai) and (£ 2 ,A 2 )
on a Sphere of Radius a, convert Spherical Coor-
dinates to Cartesian Coordinates using
' cos Xi cos Si
sin Xi cos Si
sin St
(1)
(Note that the LATITUDE S is related to the COLATI-
tude <t> of Spherical Coordinates by S = 90° - </>,
so the conversion to Cartesian COORDINATES replaces
sin<£ and cos(f> by cos J and sin£, respectively.) Now
find the Angle a between ri and r 2 using the Dot
Product,
cos a = ri ■ f 2
= cos Si cos S2 (sin Ai sin A2 + cos Ai cos A2 )
+ sin Si sin S2
= cos£i cos £2 cos(Ai — A2) + sin^i sin<$2. (2)
The great circle distance is then
d = acos~~ [cos<5i cosfe cos(Ai — A2) + sin<5i sinfo]. (3)
For the Earth, the equatorial Radius is a « 6378 km, or
3963 (statute) miles. Unfortunately, the FLATTENING of
the Earth cannot be taken into account in this simple
derivation, since the problem is considerable more com-
plicated for a Spheroid or Ellipsoid (each of which
has a Radius which is a function of Latitude).
The equation of the great circle can be explicitly com-
puted using the GEODESIC formalism. Writing
u = X
V = S = ^7T ~ (j)
(4)
(5)
gives the P, Q, and R parameters of the GEODESIC
(which are just combinations of the PARTIAL DERIVA-
TIVES) as
~~ du dv du dv du dv
"-(£)'+(£)'+(!)'-•• < 8 >
The GEODESIC differential equation then becomes
cosfsin u + 2cost>sin vv -fcosW — sinW = 0. (9)
However, because this is a special case of Q = with P
and R explicit functions of v only, the GEODESIC solu-
tion takes on the special form
=1
R
P 2 - a 2 P
dv
dv
-/-
dv
2 sin 4 v — c\ 2 sin 2 v
tovjte?
sin v — 1
= —tan
M 2
+ C 2
(10)
(Gradshteyn and Ryzhik 1979, p. 174, eqn. 2.599.6),
which can be rewritten as
. _i / cotu - ,.,_,.
v = — sin I — — I + C2- (11)
.vSr
It therefore follows that
(sinc2)asin^cosn — (cosc2)asint;sinn
acosv
^/W ::
= 0. (12)
This equation can be written in terms of the CARTESIAN
Coordinates as
x sin C2 — y cos C2
4W-
= 0, (13)
which is simply a PLANE passing through the center of
the Sphere and the two points on the surface of the
Sphere.
760
Great Cubicuboctahedron
Great Ditrigonal Icosidodecahedron
see also GEODESIC, GREAT SPHERE, LOXODROME, Ml-
kusinski's Problem, Orthodrome, Point-Point
Distance — 2-D, Sphere
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.
Weinstock, R. Calculus of Variations, with Applications to
Physics and Engineering. New York: Dover, pp. 26-28
and 62-63, 1974.
Great Cubicuboctahedron
The Uniform Polyhedron U 14 whose Dual Poly-
hedron is the Great Hexacronic Icositetrahe-
DRON. It has Wythoff Symbol 34 | |. Its faces are
8{3}+6{4}+6{f }. It is a Faceted version of the Cube.
The ClRCUMRADlUS of a great cubicuboctahedron with
unit edge length is
R= \\/h-2\/2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 118-119, 1989.
Great Deltoidal Hexecontahedron
The Dual of the Great Rhombicosidodecahedron
(Uniform).
Great Deltoidal Icositetrahedron
The Dual of the Great Rhombicuboctahedron
(Uniform).
Great Dirhombicosidodecacron
The Dual of the Great Dirhombicosidodecahe-
dron.
Great Dirhombicosidodecahedron
The Uniform Polyhedron U75 whose Dual is the
Great Dirhombicosidodecacron. This Polyhe-
dron is exceptional because it cannot be derived from
Schwarz Triangles and because it is the only UNI-
FORM Polyhedron with more than six Polygons sur-
rounding each Vertex (four SQUARES alternating with
two Triangles and two Pentagrams). It has Wyth-
off Symbol | § § 3 §. Its faces are 40{3} + 60{4} +
24{|}, and its ClRCUMRADlUS for unit edge length is
R=\y/2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 200-203, 1989.
Great Disdyakis Dodecahedron
The Dual of the Great Truncated Cuboctahe-
dron.
Great Disdyakis Triacontahedron
The Dual of the Great Truncated Icosidodecahe-
dron.
Great Ditrigonal Dodecacronic
Hexecontahedron
The Dual of the Great Ditrigonal Dodecicosido-
decahedron.
Great Ditrigonal Dodecicosidodecahedron
The Uniform Polyhedron C/42 whose Dual is the
Great Ditrigonal Dodecacronic Hexecontahe-
dron. It has Wythoff Symbol 35 | |. Its faces are
20{3} + 12{5} + 12{^}, and its ClRCUMRADlUS for unit
edge length is
iZ = ^V / 34-6\/5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 125, 1989.
Great Ditrigonal Icosidodecahedron
Great Dodecacronic Hexecontahedron
Great Dodecicosacron
761
The Uniform Polyhedron L7 47 whose Dual is the
Great Triambic Icosahedron. It has Wythoff
Symbol §[35. Its faces are 20{3} + 12{5}, and its
ClRCUMRADlUS for unit edge length is
R=\y/l.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 135-136, 1989.
Great Dodecacronic Hexecontahedron
The Dual of the Great Dodecicosidodecahedron.
Great Dodecadodecahedron
see Dodecadodecahedron
Great Dodecahedron
The Uniform Polyhedron L7 35 which is the Dual
of the Small Stellated Dodecahedron and one of
the Kepler-Poinsot Solids. Its faces are 12{5}. Its
Schlafli Symbol is {5, §}, and its Wythoff Symbol
is § I 2 5. Its faces are 12{5}. Its ClRCUMRADlUS for unit
edge length is
Ifi 1 /** 1 /**
5 1/4 y/2(l + y/b),
where <f> is the GOLDEN RATIO.
see also GREAT ICOSAHEDRON, GREAT STELLATED DO-
DECAHEDRON, Kepler-Poinsot Solid, Small Stel-
lated Dodecahedron
References
Fischer, G. (Ed.). Plate 105 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 104, 1986.
Great Dodecahedron- Small Stellated
Dodecahedron Compound
A Polyhedron Compound in which the Great Do-
decahedron is interior to the Small Stellated Do-
decahedron.
see also POLYHEDRON COMPOUND
Great Dodecahemicosacron
The Dual of the Great Dodecahemicosahedron.
Great Dodecahemicosahedron
The Uniform Polyhedron U 6 5 whose Dual is the
Great Dodecahemicosacron. It has Wythoff
Symbol § § | §. Its faces are 12{§ } + 6{^}. It is a
Faceted Dodecadodecahedron. The Circumra-
DIUS for unit edge length is
R
\Vz.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 106-107, 1989.
Great Dodecahemidodecacron
The Dual of the Great Dodecahemidodecahedron.
Great Dodecahemidodecahedron
The Uniform Polyhedron U 7 o whose Dual is the
Great Dodecahemidodecacron. It has Wythoff
Symbol § § | f . Its faces are 12{f } 4- 6{^}. Its CiR-
CUMRADIUS for unit edge length is
R = <t>-\
where <j> is the Golden Ratio.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 165, 1989.
Great Dodecicosacron
The Dual of the Great Dodecicosahedron.
762 Great Dodecicosahedron
Great Dodecicosahedron
Great Icosicosidodecahedron
Great Icosahedron
The Uniform Polyhedron Um whose Dual is the
One of the KEPLER-POINSOT SOLIDS whose Dual is
the Great Stellated Dodecahedron. Its faces are
20{3}. It is also Uniform Polyhedron C/53 and has
Great Dodecicosacron.
3
It has Wythoff Symbol Wythoff Symbol 3 § | f . Its faces are 20{3} + 12{§}+
Its faces are 20{6} + 12{™}- Its ClRCUMRA-
DIUS for unit edge length is
R= 1^34-6^.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 156-157, 1989.
Great Dodecicosidodecahedron
The Uniform Polyhedron E/ei whose Dual is the
Great Dodecacronic Hexecontahedron. Its
Wythoff Symbol is 2 f 1 3. Its faces are 20{6}+12{§ },
and its Circumradius for unit edge length is
R= i\/58-18\/5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 148, 1989.
Great Hexacronic Icositetrahedron
The Dual of the Great Cubicuboctahedron.
Great Hexagonal Hexecontahedron
The Dual of the Great Snub Dodecicosidodecahe-
dron.
Great Icosacronic Hexecontahedron
The Dual of the Great Icosicosidodecahedron.
12{^}. Its Circumradius for unit edge length is
R= |v / H-4\/5.
see also GREAT DODECAHEDRON, GREAT ICOSAHE-
DRON, Great Stellated Dodecahedron, Kepler-
Poinsot Solid, Small Stellated Dodecahedron,
Truncated Great Icosahedron
References
Fischer, G. (Ed.). Plate 106 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 105, 1986.
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 154, 1989.
Great Icosahedron- Great Stellated
Dodecahedron Compound
A Polyhedron Compound most easily constructed by
adding the Vertices of a Great Icosahedron to a
Great Stellated Dodecahedron.
see also Polyhedron Compound
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 132-133, 1989.
Great Icosicosidodecahedron
Great Icosidodecahedron
Great Quasitruncated Icosidodecahedron 763
The Uniform Polyhedron U^s whose Dual is the
Great Icosacronic Hexecontahedron. It has
Wythoff Symbol | 5 | 3. Its faces are 20{3} + 20{6} +
12{5}. Its Circumradius for unit edge length is
#= W34-6\/5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 137-139, 1989.
Great Icosidodecahedron
A Uniform Polyhedron U 54 whose Dual is the
Great Rhombic Triacontahedron (also called the
Great Stellated Triacontahedron). It is a Stel-
lated Archimedean Solid. It has Schlafli Sym-
bol I | \ and Wythoff Symbol 2 |3 f . Its faces are
20{3} + 12{§}. Its Circumradius for unit edge length
Great Inverted Pentagonal Hexecontahedron
The Dual of the Great Inverted Snub Icosidodec-
ahedron.
Great Inverted Retrosnub
Icosidodecahedron
see Great Retrosnub Icosidodecahedron
Great Inverted Snub Icosidodecahedron
The Uniform Polyhedron U$g whose Dual is the
Great Inverted Pentagonal Hexecontahedron.
It has Wythoff Symbol | 23 f . Its faces are 80{3} +
12{f }. For unit edge length, it has CIRCUMRADIUS
1 / 8-2 2 / 3 -16s + 2 1 / 3 s 2
2 V 8 • 2 2 / 3 - 10z + 2 1 /3 X 2
= 0.816080674799923,
where <j> is the GOLDEN RATIO.
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 124, 1989.
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 147, 1989.
Great Icosihemidodecacron
The Dual of the Great Icosihemidodecahedron.
Great Icosihemidodecahedron
where
The Uniform Polyhedron Un whose Dual is the
Great Icosihemidodecacron. It has Wythoff
SYMBOL § 3 | §. Its faces are 20{3} + 6{f}. For unit
edge length, its CIRCUMRADIUS is
where <f> is the Golden Ratio.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 164, 1989.
= (49 - 27^ + 3vW93 - 49\/5 )
1/3
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 179, 1989.
Great Pentagonal Hexecontahedron
The Dual of the Great Snub Icosidodecahedron.
Great Pentagrammic Hexecontahedron
The Dual of the Great Retrosnub Icosidodecahe-
dron.
Great Pentakis Dodecahedron
The Dual of the Small Stellated Truncated Do-
decahedron.
Great Quasitruncated Icosidodecahedron
see Great Truncated Icosidodecahedron
764 Great Retrosnub Icosidodecahedron
Great Retrosnub Icosidodecahedron
The Uniform Polyhedron t/74, also called the Great
Inverted Retrosnub Icosidodecahedron, whose
Dual is the Great Pentagrammic Hexecontahe-
dron. It has Wythoff Symbol |2 § f. Its faces are
80{3} + 12{f }. For unit edge length, it has ClRCUMRA-
DIUS
R
-I l 2 ~ x
" 2 V 1-a
0.5800015,
where x is the smaller NEGATIVE root of
x 3 +2x 2 -<j>~ 2 =0,
with <f> the Golden Mean.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 189-193, 1989.
Great Rhombic Triacontahedron
A ZONOHEDRON which is the DUAL of the GREAT ICOS-
IDODECAHEDRON. It is also called the Great Stel-
lated Triacontahedron.
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 126, 1989.
Great Rhombicosidodecahedron
(Archimedean)
Great Rhombicuboctahedron
An Archimedean Solid also known as the Rhom-
bitruncated Icosidodecahedron. It is sometimes
improperly called the TRUNCATED ICOSIDODECAHE-
DRON, a name which is inappropriate since TRUNCATION
would yield RECTANGULAR instead of SQUARE. The
great rhombicosidodecahedron is also UNIFORM POLY-
HEDRON U 2 8> Its DUAL is the DlSDYAKIS TRIACON-
TAHEDRON, also called the Hexakis Icosahedron. It
has Schlafli Symbol t{J} and Wythoff Symbol
235 |. The Inradius, Midradius, and ClRCUMRADius
for a = 1 are
r = 2^1 (105 + 6^5 )\/31 + 12Vb « 3.73665
p = |v / 30 + 12v / 5^ 3.76938
R=± V31 + 12V5 w 3.80239.
see also Small Rhombicosidodecahedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 137,
1987.
Great Rhombicosidodecahedron (Uniform)
The Uniform Polyhedron Uq 7 , also called the
Quasirhombicosidodecahedron, whose Dual is the
Great Deltoidal Hexecontahedron. It has
Schlafli Symbol r'| I }. It has Wythoff Symbol
3 § | 2. Its faces are 20{3} + 30{4} + 12{§}. For unit
edge length, its ClRCUMRADius is
R
= ^v / H-4v / 5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 162-163, 1989.
Great Rhombicuboctahedron
(Archimedean)
Great Rhombicuboctahedron (Uniform)
Great Rhombihexahedron
765
An Archimedean Solid sometimes (improperly) called
the Truncated Cuboctahedron and also called the
Rhombitruncated Cuboctahedron. Its Dual is
the Disdyakis Dodecahedron, also called the Hex-
akis Octahedron. It has Schlafli Symbol tj^}.
It is also Uniform Polyhedron Un and has Wyth-
OFF Symbol 2 34 |. Its faces are 8{6} + 12{4} + 6{8}.
The Small Cubicuboctahedron is a Faceted ver-
sion. The Inradius, Midradius, and Circumradius
for unit edge length are
■= ^(14+\/2)\/l3 + 6\/2:
: 2.26303
' 2.20974
p= 1^12 + 6^2 J
R=\ V / 13ToV2 « 2.31761.
Additional quantities are
t — tan(|7r) = -s/2 - 1
/ = 2* = 2(a/2 -1)
h= l + /sin(|7r) = 3 - y/2.
see also SMALL RHOMBICUBOCTAHEDRON,
Truncated Cuboctahedron
References
Ball, W. W. R. and Coxeter, H. S.'M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 138,
1987.
Great Rhombicuboctahedron (Uniform)
Great
The Uniform Polyhedron U 17 , also known as
the Quasirhombicuboctahedron, whose Dual is
the Great Deltoidal Icositetrahedron. It has
Schlafli Symbol r'{§} and Wythoff Symbol § 4 1 2.
Its faces are 8{3} + 20{4}. Its Circumradius for unit
edge length is
R = ±\/5-2\/2.
Great Rhombidodecacron
The Dual of the Great Rhombidodecahedron.
Great Rhombidodecahedron
The Uniform Polyhedron U73 whose Dual is the
Great Rhombidodecacron. It Wythoff Symbol
3
Its faces are 30{4} + 12{^}. Its CIRCUM-
RADIUS for unit edge length is
R= |\/ll-4\/5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 168-170, 1989.
Great Rhombihexacron
The Dual of the Great Rhombihexahedron.
Great Rhombihexahedron
The Uniform Polyhedron U21 whose Dual is the
Great Rhombihexacron. It has Wythoff Symbol
3
Its faces are 12{4} + 6{f }. Its CIRCUMRADIUS
n 4 2
z 3 4
for unit edge length is
R= |\/5- 2v/2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 159-160, 1989.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 132-133, 1989.
766 Great Snub Dodecicosidodecahedron
Great Snub Dodecicosidodecahedron
The Uniform Polyhedron U 6 4 whose Dual is the
Great Hexagonal Hexecontahedron. It has
Wythoff Symbol 1 3 § §. Its faces are 80{3} + 24{§}.
Its ClRCUMRADIUS for unit edge length is
R=±y/2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 183-185, 1989.
Great Snub Icosidodecahedron
The Uniform Polyhedron t/ 57 whose Dual is the
Great Pentagonal Hexecontahedron. It has
Wythoff Symbol |23§. Its faces are 80{3} + 12{§}.
For unit edge length, it has ClRCUMRADIUS
»=*i/i5f
: 0,6450202,
where x is the most NEGATIVE ROOT of
x 3 + 2x 2 - (j>~ 2 = 0,
with 4> the Golden Ratio.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 186-188, 1989.
Great Sphere
The great sphere on the surface of a Hypersphere is
the 3-D analog of the GREAT CIRCLE on the surface of
a Sphere. Let 2h be the number of reflecting Spheres,
and let great spheres divide a HYPERSPHERE into g 4-D
TETRAHEDRA. Then for the POLYTOPE with Schlafli
Symbol {p,q,r},
64/i 1ft n 4 4
= 12-p-2g-r+- + -.
9 p r
see also Great Circle
Great Stellated Truncated Dodecahedron
Great Stellapentakis Dodecahedron
The Dual of the Great Truncated Icosahedron.
Great Stellated Dodecahedron
One of the Kepler-PoinsOT Solids whose DUAL is the
Great Icosahedron. Its Schlafli Symbol is {§,3}.
It is also Uniform Polyhedron U52 and has Wyth-
off Symbol 3 | 2 |. Its faces are
RADIUS for unit edge length is
12{§}. Its Circum-
R = f v^" 1 - \VS(VE - 1).
The easiest way to construct it is to make 12 TRIANGU-
LAR Pyramids
with side length <f> = (l + y/b)/2 (the Golden Ratio)
times the base and attach them to the sides of an Icos-
ahedron.
see also Great Dodecahedron, Great Icosahe-
dron, Great Stellated Truncated Dodecahe-
dron, Kepler-Poinsot Solid, Small Stellated
Dodecahedron
References
Fischer, G. (Ed.). Plate 104 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 103, 1986.
Great Stellated Triacontahedron
see Great Rhombic Triacontahedron
Great Stellated Truncated Dodecahedron
The Uniform Polyhedron t7 66 , also called the Qua-
sitruncated Great Stellated Dodecahedron,
whose Dual is the Great Triakis Icosahedron. It
has Schlafli Symbol t'{|,3} and Wythoff Symbol
Great Triakis Icosahedron
Greater Than/Less Than Symbol 767
23 | §. Its faces are 20{3}-h 12{^}. Its C irc um radius Great Stellapentakis Dodecahedron. It has
for unit edge length is SCHLAFLI SYMBOL t{3, §} and WYTHOFF SYMBOL
2 § | 3. Its faces are 20{6} 4- 12{§}. Its ClRCUMRADlUS
for unit edge length is
R= ±v / 74-30v / 5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 161, 1989.
Great Triakis Icosahedron
The Dual of the Great Stellated Truncated Do-
decahedron.
Great Triakis Octahedron
The Dual of the Stellated Truncated Hexahe-
dron.
see also Small Triakis Octahedron
Great Triambic Icosahedron
The Dual of the Great Ditrigonal Icosidodeca-
hedron.
Great Truncated Cuboctahedron
The Uniform Polyhedron L/20, also called the Qua-
sitruncated Cuboctahedron, whose Dual is the
Great Disdyakis Dodecahedron. It has Schlafli
Symbol t'{| } and Wythoff Symbol 2 3 f |. Its Cir-
CUMRADIUS for unit edge length is
R = |\/l3-6\/2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 145-146, 1989.
R= |\/58-18\/5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 148, 1989.
Great Truncated Icosidodecahedron
The Uniform Polyhedron t/lss, also called the Great
Quasitruncated Icosidodecahedron, whose Dual
is the Great Disdyakis Triacontahedron. It has
Schlafli Symbol t'j | j and Wythoff Symbol
2 3 § |. Its faces are 20{6} + 30{4} + 12{f }. Its ClR-
CUMRADlUS for unit edge length is
R= W31-12a/5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 166-167, 1989.
Greater
A quantity a is said to be greater than 6 if a is larger
than 6, written a > b. If a is greater than or EQUAL
to 6, the relationship is written a > b. If a is Much
Greater than 6, this is written a > b. Statements
involving greater than and LESS than symbols are called
Inequalities.
see also Equal, Greater Than/Less Than Symbol,
Inequality, Less, Much Greater
Great Truncated Icosahedron
The Uniform Polyhedron E/55, also called the Trun-
cated Great Icosahedron, whose Dual is the
Greater Than/Less Than Symbol
When applied to a system possessing a length R at which
solutions in a variable r change character (such as the
gravitational field of a sphere as r runs from the interior
to the exterior), the symbols
r> = max(r, R)
r < = min(r, R)
are sometimes used.
see also Equal, Greater, Less
768 Greatest Common Denominator
Greatest Prime Factor
Greatest Common Denominator
see Greatest Common Divisor
Greatest Common Divisor
The greatest common divisor of a and b GCD(a, 6),
sometimes written (a, &), is the largest DIVISOR com-
mon to a and b. Symbolically, let
i
(i)
(2)
Then the greatest common divisor is given by
(a,b) = 1 [[pr ll(ai ' M , (3)
i
where min denotes the MINIMUM. The GCD is DIS-
TRIBUTIVE
(ma, mb) = m(a, b) (4)
(ma, mb, mc) = m(a, b, c),
and Associative
(5)
(ab,cd) = (a,c)(b, d)
(a,b,c) = ((a,b),c) = (a,(b,c)) (6)
a d \ ( c b
(a,c)' {b,d)J Ua,c)' (b,d)
(7)
If a = ai (a, b) and b ~ bi (a, b) , then
(a, b) = (ai(a ; fe),6i(a,6)) = (a, 6)(ai, &i), (8)
so (ai , &i ) = 1 and a± and 61 are said to be Relatively
Prime. The GCD is also Idempotent
(a, a) = a,
Commutative
(a, b) = (6, a),
and satisfies the ABSORPTION LAW
[a, (a, 6)] = a.
(9)
(10)
(11)
The probability that two INTEGERS picked at random
are RELATIVELY PRIME is [C(2)] _1 = 6/71- 2 , where ((z) is
the Riemann Zeta Function. Polezzi (1997) observed
that (m,n) = k y where k is the number of LATTICE
Points in the Plane on the straight Line connecting
the Vectors (0, 0) and (m,n) (excluding (m, n) itself).
This observation is intimately connected with the prob-
ability of obtaining RELATIVELY PRIME integers, and
also with the geometric interpretation of a REDUCED
Fraction y/x as a string through a Lattice of points
with ends at (1,0) and (x,y). The pegs it presses against
(xi,yt) give alternate Convergents y%/xi of the Con-
tinued Fraction for y/x, while the other Conver-
gents are obtained from the pegs it presses against with
the initial end at (0, 1).
Knuth showed that
(2"-l,9«-l) = 2
(p.«)
(12)
for p, q Prime.
The extended greatest common divisor of two INTEGERS
m and n can be defined as the greatest common divisor
of m and n which also satisfies the constraint g = rm +
sn for r and s given INTEGERS. It is used in solving
Linear Diophantine Equations,
see also Bezout Numbers, Euclidean Algorithm,
Least Prime Factor
References
Polezzi, M. "A Geometrical Method for Finding an Explicit
Formula for the Greatest Common Divisor." Amer. Math.
Monthly 104, 445-446, 1997.
Greatest Common Divisor Theorem
Given m and n, it is possible to choose c and d such that
cm + dn is a common factor of m and n.
Greatest Common Factor
see Greatest Common Divisor
Greatest Integer Function
see Floor Function
Greatest Lower Bound
see INFIMUM, LEAST UPPER BOUND
Greatest Prime Factor
20 40 60 80 100
For an INTEGER n > 2, let gpf(x) denote the greatest
prime factor of n, i.e., the number pk in the factorization
■Pi ai '-Pk ak ,
with pi < pj for i < j. For n = 2, 3, . . . , the first
few are 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, ...
(Sloane's A006530). The greatest multiple prime factors
Grebe Point
Green's Function
769
for SQUAREFUL integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2,
2, 3, ... (Sloane's A046028).
see also Distinct Prime Factors, Factor, Least
Common Multiple, Least Prime Factor, Man-
goldt Function, Prime Factors, Twin Peaks
References
Erdos, P. and Pomerance, C. "On the Largest Prime Factors
of n and n + 1." Aequationes Math. 17, 211-321, 1978.
Guy, R. K. "The Largest Prime Factor of n." §B46 in Un-
solved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 101, 1994.
Heath-Brown, D. R. "The Largest Prime Factor of the Inte-
gers in an Interval." Sci. China Ser. A 39, 449-476, 1996.
Mahler, K. "On the Greatest Prime Factor of ax m + 6y n ."
Nieuw Arch. Wiskunde 1, 113-122, 1953.
Sloane, N. J. A. Sequence A006530/M0428 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Grebe Point
see Lemoine Point
Greedy Algorithm
An algorithm used to recursively construct a Set of ob-
jects from the smallest possible constituent parts.
Given a Set of k Integers (oi, a 2 , . . . , a*;) with a\ <
CL2 < . . • < afc, a greedy algorithm can be used to find a
Vector of coefficients (ci, c 2 , . . . , c k ) such that
E
Cidi = c • a = n,
(i)
where ca is the DOT PRODUCT, for some given INTEGER
n. This can be accomplished by letting Ci = for i = 1,
...,& — 1 and setting
Ck
-L=J
(2)
Now define the difference between the representation
and n as
A = n - c • a. (3)
If A = at any step, a representation has been found.
Otherwise, decrement the NONZERO ai term with least
z, set all aj = for j < i, and build up the remaining
terms from
for j = i - 1, . . . , 1 until A = or all possibilities have
been exhausted.
For example, McNUGGET NUMBERS are numbers which
are representable using only (01,02,03) = (6,9,20).
Taking n = 62 and applying the algorithm iteratively
gives the sequence (0, 0, 3), (0, 2, 2), (2, 1, 2), (3, 0,
2), (1, 4, 1), at which point A = 0. 62 is therefore a
McNugget Number with
If any Integer n can be represented with ct = or
1 using a sequence (ai, 02, •■•)) then this sequence is
called a COMPLETE SEQUENCE.
A greedy algorithm can also be used to break down arbi-
trary fractions into Unit Fractions in a finite number
of steps. For a Fraction a/6, find the least INTEGER
x\ such that I/jei < a/6, i.e.,
xi =
r*i
(6)
where \x] is the CEILING FUNCTION. Then find the
least Integer X2 such that I/X2 < a/6 — 1/sci. Iterate
until there is no remainder. The ALGORITHM gives two
or fewer terms for 1/n and 2/n, three or fewer terms for
3/n, and four or fewer for 4/n.
Paul Erdos and E. G. Strays have conjectured that the
Diophantine Equation
4 1 11
n a c
(7)
always can be solved, and W. Sierpiriski conjectured that
(8)
5
n
111
a b c
can be solved.
see also Complete Sequence, Integer Rela-
tion, Levine-O'Sullivan Greedy Algorithm, Mc-
Nugget Number, Reverse Greedy Algorithm,
Square Number, Sylvester's Sequence, Unit
Fraction
References
Greek Cross
An irregular DODECAHEDRON CROSS in the shape of a
Plus Sign.
see also CROSS, DISSECTION, DODECAHEDRON, LATIN
Cross, Plus Sign, Saint Andrew's Cross
Greek Problems
see Geometric Problems of Antiquity
Green's Function
Let
L = D n + a n -i{t)D n ' r + . . . + ai(t)D + a {t) (1)
be a differential OPERATOR in 1-D, with a»(t) CONTINU-
OUS for i — 0, 1, . . . , n — 1 on the interval 7, and assume
we wish to find the solution y(t) to the equation
62 = (1-6) + (4 -9) + (1*20).
(5)
Ly(t) = h(t),
(2)
770 Green's Function
Green's Function — Helmholtz,
where h(t) is a given CONTINUOUS on /. To solve equa-
tion (2), we look for a function g : C n (I) \-> C(I) such
that L(g(h)) = h, where
y(t) = g(h(t)). (3)
This is a CONVOLUTION equation of the form
y = g*h, (4)
so the solution is
y(t)
f g(t-
J t Q
x)h(x) dx,
(5)
where the function g(t) is called the Green's function for
L on /.
Now, note that if we take h(t) ~ S(t) y then
y(*)= / g(t-x)S(x)dx = g(t), (6)
so the Green's function can be defined by
Lg(t) = 6(t).
(7)
However, the Green's function can be uniquely deter-
mined only if some initial or boundary conditions are
given.
For an arbitrary linear differential operator L in 3-D,
the Green's function G(r, r') is defined by analogy with
the 1-D case by
LG(r,r') = <5(r-r'). (8)
The solution to L<f> = f is then
<Kr) = yG(r,r')/(rVV. (9)
Explicit expressions for <3(r, r') can often be found in
terms of a basis of given eigenfunctions n (i*i) by ex-
panding the Green's function
oo
G(ri ) r 2 ) = 5^a n (P2)0n(ri) (10)
n=0
and Delta Function,
oo
<5 3 (r 1 -r 2 ) = ^6„^„(r 1 ). (11)
n =
Multiplying both sides by <f>m(^2) and integrating over
ri space,
/ 0m(r 2 )5 3 (ri-r 2 )fi 3 ri = ]j^&n / <j>m{^2)4>n{vi) d 3 ri
n —
(12)
0m (r 2 ) = 2_^ bn$nm. = &m, (13)
<5 3 (r! - F 2 ) = 5^0n(Pl)^n(P 3 ). (14)
By plugging in the differential operator, solving for the
a n s, and substituting into G, the original nonhomoge-
neous equation then can be solved.
References
Arfken, G. "Nonhomogeneous Equation — Green's Func-
tion," "Green's Functions — One Dimension," and "Green's
Functions — Two and Three Dimensions." §8.7 and §16.5-
16.6 in Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 480-491 and 897-924,
1985.
Green's Function — Helmholtz Differential
Equation
The inhomogeneous HELMHOLTZ DIFFERENTIAL EQUA-
TION is
vV(r) + *V(r)=p(r), (1)
where the Helmholtz operator is defined as L = V + k .
The Green's function is then defined by
(V 2 + fc 2 )G(r 1 ,r 2 ) = <5 3 (r 1 -r 2 ).
(2)
Define the basis functions <t>„ as the solutions to the
homogeneous Helmholtz Differential Equation
V 2 4>n{T)+k n 2 4>„{T)=0.
(3)
The Green's function can then be expanded in terms of
the n s,
oo
G(ri,r a ) = ^ a n (r 2 )^ n (n), (4)
n=0
and the DELTA FUNCTION as
oo
* 8 (ri-r a ) = $3 *»(*i)M»*)- (5)
71 =
Plugging (4) and (5) into (2) gives
oo oo
/] an(r 2 )<£n(ri) + k 2 ^J a n (r 2 )<i) n (ri)
,n=0 J n~
oo
= J3^»(ri)^»(ra). (6)
71 =
Using (3) gives
oo oo
~Z-, aTl ( r2 )k™ 2 <M r ) + fc 2 y^a n (r2)<ftn(ri)
n=0 n=0
oo
= 53*»0n)*»fo) (7)
Green's Function — Poisson's Equation
oo oo
J2 an(r 2 )0n(r 1 )(A; 2 - k n 2 ) = ^ ^(n)^ n (r 2 ). (8)
n=0 n=0
This equation must hold true for each n, so
a n (r 2 )0 n (n)(A; 2 - fc n 2 ) = 0n(ri)0 n (r 2 ) (9)
a n (r 2 )
and (4) can be written
AC /C71
(10)
(11)
The general solution to (1) is therefore
ip(n) = / G(ri,r2)p(r 2 )d 3 r2
/
U
1 (ri)0 n (r 2 )p(r 2 )
<2 3 r 2 . (12)
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 529-530, 1985.
Green's Function — Poisson's Equation
Poisson's Equation equation is
V z 4> = 4ttp,
(1)
where <j> is often called a potential function and p a den-
sity function, so the differential operator in this case is
L = V 2 . As usual, we are looking for a Green's function
G f (ri,r 2 ) such that
V 3 G(ri ) r 2 ) = * 3 (ri-r 2 ).
But from Laplacian,
7 2 / 1
r- r
-47n5 3 (r — r'),
so
G(r.r'):
and the solution is
47r|r — r'
(r) = J G(r, r')[4,rp(r')] «*V = -J ^^.
(2)
(3)
(4)
(5)
Expanding G(ri,r 2 ) in the Spherical Harmonics Y"
gives
G(ri,r 2 )
ob I
1 r'
Green's Identities 771
where r< and r> are GREATER Than/Less THAN SYM-
BOLS. This expression simplifies to
1=0 >
where Pi are LEGENDRE POLYNOMIALS, and cos 7 =
ri ■ r 2 . Equations (6) and (7) give the addition theorem
for LEGENDRE POLYNOMIALS.
In Cylindrical Coordinates, the Green's function is
much more complicated,
00
f
Jo
Imikp^Kmikp^e
im(01-<£ 2 )
cos[k(zi — Z2)]dk,
(8)
where I m (x) and K m (x) are Modified Bessel Func-
tions of the First and Second Kinds (Arfken 1985).
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 485-486, 905, and 912,
1985.
Green's Identities
Green's identities are a set of three vector deriva-
tive/integral identities which can be derived starting
with the vector derivative identities
v • (vv<£) = ipv 2 <p + (WO • (V0) (1)
and
V ■ (<f>Vi>) = 0V 2 V + (V0) • (VV), (2)
where V- is the DIVERGENCE, V is the GRADIENT, V 2
is the Laplacian, and a-b is the Dot Product. From
the Divergence Theorem,
f(V-F)dV= f
Jv J s
(V-F)dV= / F-da.
(3)
Plugging (2) into (3),
J <f>(ViP) • da = / [<£VV + (V0) ■ (VVO] dV. (4)
J s Jv
This is Green's first identity.
Subtracting (2) from (1),
V - {4>Vtp - ipV</>) = <j>V 2 ip - ^V 2 ^. (5)
Therefore,
/ (</>V 2 V> - ^V 2 0) dV = / {<pVip - ipV<f>) • da. (6)
Jv J s
772
Greene's Method
Grenz-Formel
This is Green's second identity.
Let u have continuous first PARTIAL DERIVATIVES and
be HARMONIC inside the region of integration. Then
Green's third identity is
Gregory's Formula
A series FORMULA for Pi found by Gregory and Leibniz,
u(x,y) = ±£[ln(±)^-u-^lnQ.)
ds (7)
(Kaplan 1991, p. 361).
References
Kaplan, W. Advanced Calculus, ^ih ed. Reading, MA:
Addison- Wesley, 1991.
Greene's Method
A method for predicting the onset of widespread CHAOS.
It is based on the hypothesis that the dissolution of an
invariant torus can be associated with the sudden change
from stability to instability of nearly closed orbits (Ta-
bor 1989, p. 163).
see also OVERLAPPING RESONANCE METHOD
References
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, 1989.
Green Space
A G- Space provides local notions of harmonic, hyper-
harmonic, and superharmonic functions. When there
exists a nonconstant superharmonic function greater
than 0, it is a called a Green space. Examples are R n
(for n > 3) and any bounded domain of W 1 .
Green's Theorem
Green's theorem is a vector identity which is equivalent
to the Curl Theorem in the Plane. Over a region D
in the plane with boundary dD,
f^f{x,y)dx + g(z,y)dy = JJ^-^dxdy
J F-ds= [J (V x¥)-kdA.
JdD J J D
If the region D is on the left when traveling around dD y
then Area of D can be computed using
JdL
A = J / xdy — ydx.
IdD
see also Curl Theorem, Divergence Theorem
References
Arfken, G. "Gauss's Theorem." §1.11 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 57-61, 1985.
7T , 1 1
It converges very slowly, but its convergence can be ac-
celerated using certain transformations, in particular
where £(z) is the RlEMANN Zeta FUNCTION (Vardi
1991).
see also MACHIN'S FORMULA, MACHIN-LlKE FORMU-
LAS, Pi
References
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, pp. 157-158, 1991.
Gregory Number
A number
t x = tan _1 (^) = cot -1 #,
where x is an Integer or Rational Number, tan -1 x
is the Inverse Tangent, and cot _1 x is the Inverse
Cotangent. Gregory numbers arise in the determina-
tion of MACHIN-LlKE FORMULAS. Every Gregory num-
ber t x can be expressed uniquely as a sum of t n s where
the ns are ST0RMER NUMBERS.
References
Conway, J. H. and Guy, R. K. "Gregory's Numbers" In The
Book of Numbers. New York: Springer-Verlag, pp. 241-
242, 1996.
Grelling's Paradox
A semantic PARADOX, also called the HETEROLOGICAL
Paradox, which arises by defining "heterological" to
mean "a word which does not describe itself." The word
"heterological" is therefore heterological IFF it is not.
see also RUSSELL'S PARADOX
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, pp. 20-21, 1989.
Grenz-Formel
An equation derived by Kronecker:
]TV + y* + <*.")- = 4C(.)i(«) + s 2 _V ( d.-i 2)
(s) Z^ Z_f u 2 — 2 /
i, "0
+ r(
where
i,y/Wd(y+y-' L ) »-2
y" dy,
u>|n
r(n) = 4\Jsin(|7rrf),
d\n
Griffiths Points
Groemer Packing 773
((z), is the RlEMANN ZETA FUNCTION, rj(z) is the
Dirichlet Eta Function, T(z) is the Gamma Func-
tion, and the primed sum omits infinite terms (Selberg
and Chowla 1967).
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 296-297, 1987.
Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J.
Reine. Angew. Math. 227, 86-110, 1967.
Griffiths Points
"The" Griffiths point is the fixed point in GRIFFITHS'
Theorem. Given four points on a Circle and a line
through the center of the CIRCLE, the four correspond-
ing Griffiths points are COLLINEAR (Tabov 1995).
The points
Gr = I + AGe
Gr =/-4Ge,
are known as the first and second Griffiths points, where
I is the INCENTER and Ge is the GERG0NNE POINT.
see also Gergonne Point, Griffiths' Theorem, In-
center, Oldknow Points, Rigby Points
References
Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
Tabov, J. "Four Collinear Griffiths Points." Math. Mag. 68,
61-64, 1995.
Griffiths' Theorem
When a point P moves along a line through the ClR-
CUMCENTER of a given TRIANGLE A, the ClRCUMCIR-
cle of the Pedal Triangle of P with respect to A
passes through a fixed point (the Griffiths Point) on
the Nine-Point Circle of A.
see also ClRCUMCENTER, GRIFFITHS POINTS, NlNE-
Point Circle, Pedal Triangle
Grimm's Conjecture
Grimm conjectures that ifn+1, n + 2, ..., n-\-k are all
Composite Numbers, then there are distinct Primes
Pij such that pi j \(n + j) for 1 < j < k.
References
Guy, R. K. "Grimm's Conjecture." §B32 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, p. 86, 1994.
Grinberg Formula
A formula satisfied by all Hamiltonian Circuits with
n nodes. Let fj be the number of regions inside the
circuit with j sides, and let gj be the number of regions
outside the circuit with j sides. If there are d interior
diagonals, then there must be d + 1 regions
Any region with j sides is bounded by j EDGES, so such
regions contribute jfj to the total. However, this counts
each diagonal twice (and each Edge only once). There-
fore,
2/ 2 + 3/3 + ... + nf n = 2d + n.
Take (2) - 2 x (1),
h + 2/4 + 3/5 + . . . + (n - 2)/„ = n -
Similarly,
9s + 2g 4 + . . . + (n - 2)g n = n - 2,
(2)
(3)
(4)
(/3-p3) + 2(/ 4 -04) + 3(/ 5 -05) + . • . + (Tl-2)(/ n - 5 „) = 0.
(5)
Grobner Basis
A Grobner basis for a system of POLYNOMIAL equations
is an equivalence system that possesses useful proper-
ties. It is very roughly analogous to computing an Or-
THONORMAL BASIS from a set of BASIS VECTORS and
can be described roughly as a combination of Gaus-
sian Elimination (for linear systems) and the Euclid-
ean Algorithm (for Univariate Polynomials over
a Field).
Grobner bases are useful in the construction of sym-
bolic algebra algorithms. The algorithm for computing
Grobner bases is known as BUCHBERGER'S ALGORITHM.
see also BUCHBERGER'S ALGORITHM, COMMUTATIVE
Algebra
References
Adams, W. W. and Loustaunau, P. An Introduction to
Grobner Bases. Providence, RI: Amer. Math. Soc, 1994.
Becker, T. and Weispfennig, V. Grobner Bases: A Compu-
tational Approach to Commutative Algebra. New York:
Springer- Verlag, 1993.
Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and
Algorithms: An Introduction to Algebraic Geometry and
Commutative Algebra, 2nd ed. New York: Springer-
Verlag, 1996.
Eisenbud, D. Commutative Algebra with a View toward Al-
gebraic Geometry. New York: Springer- Verlag, 1995.
Mishra, B. Algorithmic Algebra. New York: Springer- Verlag,
1993.
Groemer Packing
A honeycomb- like packing that forms HEXAGONS.
see also GROEMER THEOREM
References
Stewart, I. "A Bundling Fool Beats the Wrap." Sci. Amer.
268, 142-144, 1993.
[# regions in interior] = d + 1 = fa + fz + . . . + f n . (1)
774
Groemer Theorem
Grothendieck's Theorem
Groemer Theorem
Given n Circles and a Perimeter p, the total Area
of the Convex Hull is
^Convex Hull = 2y/S{n - 1) + p(l - \ V?> ) + 7r{VS ~ 1).
Furthermore, the actual Area equals this value IFF the
packing is a GROEMER PACKING. The theorem was
proved in 1960 by Helmut Groemer.
see also CONVEX HULL
Gronwall's Theorem
Let a(n) be the DIVISOR Function. Then
lim
°"( w )
oo n In Inn
where 7 is the Euler-Mascheroni Constant. Ra-
manujan independently discovered a less precise version
of this theorem (Berndt 1994). Robin (1984) showed
that the validity of the inequality
cr(n) < e 7 nlnln7i
for n > 5041 is equivalent to the RlEMANN HYPOTHESIS.
References
Berndt, B. C. Ramanujan's Notebooks: Part I. New York:
Springer- Verlag, p. 94, 1985.
Gronwall, T. H. "Some Asymptotic Expressions in the The-
ory of Numbers." Trans. Amer. Math. Soc. 37, 113-122,
1913.
Nicholas, J.-L. "On Highly Composite Numbers." In Ra-
manujan Revisited: Proceedings of the Centenary Confer-
ence (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin).
Boston, MA: Academic Press, pp. 215-244, 1988.
Robin, G. "Grandes Valeurs de la foction somme des diviseurs
et hypothese de Riemann." J. Math. Pures Appl. 63, 187—
213, 1984.
Gross
A Dozen Dozen, or the Square Number 144.
see also 12, Dozen
Grossencharacter
In the original formulation, a quantity associated with
ideal class groups. According to Chevalley's formula-
tion, a Grossencharacter is a Multiplicative Char-
acter of the group of Adeles that is trivial on the
diagonally embedded fc x , where A; is a number Field.
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Grossman's Constant
Define the sequence ao = 1, a\ = x, and
a n +2 = —
1 + a n +i
for n > 0. Janssen and Tjaden (1987) showed that
this sequence converges for exactly one value of x,
x ~ 0.73733830336929..., confirming Grossman's con-
jecture.
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ asolve/constant/grssmn/grssmn. html,
Janssen, A. J. E. M. and Tjaden, D. L. A. Solution to Prob-
lem 86-2. Math. Intel. 9, 40-43, 1987.
Grothendieck's Majorant
The best known majorant of Grothendieck's constant.
Let A be an n x n Real Square Matrix such that
£
(i)
aijXii/j
1 <l,J <7l
in which Xi and yj have REAL ABSOLUTE VALUES <
1. Grothendieck has shown there exists a number Kg
independent of A and n satisfying
£
(2)
l<i,j<n
in which the vectors xi and y$ have a norm < 1 in
Hilbert Space. The Grothendieck constant is the
smallest Real Number for which this inequality has
been proven. Krivine (1977) showed that
1.676... <K G < 1.782...,
and has postulated that
TV
K G
1.7822139.
(3)
(4)
21n(l + \/2)
It is related to KHINTCHINE'S CONSTANT.
References
Krivine, J. L. "Sur las constante de Grothendieck." C. R. A.
S. 284, 8, 1977.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 42, 1983.
Grothendieck'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 £i,I?2 € S y then there exists a
B 3 e S such that B 3 D B 1 and B 3 3 B 2 . Then E s is
complete Iff algebraic linear functional f(y) of F that
is weakly continuous on every B € S is expressed as
/(y) = {xjy) for some x 6 E. When Es is not com-
plete, the space of all linear functionals satisfying this
condition gives the completion Es of Es-
see also MACKEY'S THEOREM
References
Iyanaga, S. and Kawada, Y. (Eds.). "Grothendieck's Theo-
rem." §407L in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, p. 1274, 1980.
Ground Set
Ground Set
A Partially Ordered Set is defined as an ordered
pair P = (X, <). Here, X is called the GROUND Set of
P and < is the PARTIAL ORDER of P.
see also Partial Order, Partially Ordered Set
Group
A group G is defined as a finite or infinite set of
Operands (called "elements") A, B,C, ... that may be
combined or "multiplied" via a BINARY OPERATOR to
form well-defined products and which furthermore sat-
isfy the following conditions:
1. Closure: If A and B are two elements in G, then the
product AB is also in G.
2. Associativity: The defined multiplication is associa-
tive, i.e., for all A, B, C e G, (AB)C = A(BC).
3. Identity: There is an IDENTITY ELEMENT / (a.k.a.
1, E, or e) such that I A = AI ~ A for every element
AeG.
4. Inverse: There must be an inverse or reciprocal of
each element. Therefore, the set must contain an
element B = A' 1 such that AA~ X = A' 1 A = I for
each element of G.
A group is therefore a MONOID for which every element
is invertible. A group must contain at least one element.
The study of groups is known as Group Theory. If
there are a finite number of elements, the group is called
a FINITE Group and the number of elements is called
the Order of the group.
Since each element A, B, C, .. . , X, and Y is a mem-
ber of the Group, Group property 1 requires that the
product
D-ABC-XY (1)
must also be a member. Now apply D to
Y- 1 X- 1 ---C- 1 B~ 1 A-\
D{Y- 1 X- X ---C- 1 B~ 1 A- 1 )
= {ABC ■ ■ • XY^Y^X- 1 • • • C^B^A- 1 ). (2)
But
ABC • • • XYY^X' 1 • • • C^B^A' 1
= ABC ■ • • XIX- 1 • • • C" 1 ^" 1 ^" 1
= ABC ■ ■ - C^B^A' 1 = ... = AA' 1 = J, (3)
so
I = DiY^X' 1 ■ • • C^B^A- 1 ), (4)
which means that
D' 1 = Y^X' 1 ■ ■ • C" 1 ^- 1 ^" 1
(5)
Group 775
An Irreducible Representation of a group is a rep-
resentation for which there exists no UNITARY TRANS-
FORMATION which will transform the representation
MATRIX into block diagonal form. The IRREDUCIBLE
Representation has some remarkable properties. Let
the Order of a Group be h, and the dimension of the
ith representation (the order of each constituent matrix)
be k (a Positive Integer). Let any operation be de-
noted R, and let the mth row and nth column of the
matrix corresponding to a matrix R in the ith IRRE-
DUCIBLE Representation be Ti(R) mn . The following
properties can be derived from the GROUP ORTHOGO-
NALITY Theorem,
h
2_^^i(R)mnTj(R) rn ' n t* = SijS,
ij mm t0 nn t
■S n
(7)
1. The Dimensionality Theorem:
h = Y,h 2 = h 2 + h 2 + h 2 + ... = £* 3 (I), (8)
i i
where each h must be a Positive Integer and x 1S
the Character (trace) of the representation.
2. The sum of the squares of the CHARACTERS in any
Irreducible Representation i equals h,
h=j2^ 2 ^-
3. Orthogonality of different representations
J2xi(R)xKR) = for Mi.
(9)
(10)
4. In a given representation, reducible or irreducible,
the Characters of all Matrices belonging to op-
erations in the same class are identical (but differ
from those in other representations).
5. The number of Irreducible Representations of
a GROUP is equal to the number of CONJUGACY
Classes in the Group. This number is the dimen-
sion of the r MATRIX (although some may have zero
elements).
6. A one-dimensional representation with all Is (totally
symmetric) will always exist for any GROUP.
7. A 1-D representation for a GROUP with elements ex-
pressed as Matrices can be found by taking the
Characters of the Matrices.
8. The number a. of IRREDUCIBLE REPRESENTATIONS
Xi present in a reducible representation c is given by
and
^JE^)^)-
(ii)
{ABC ■ ■ ■ XY)- 1 = Y~ 1 X~ 1 ---C~ 1 B- 1 A- 1 . (6)
776 Group
Group Ring
where h is the ORDER of the GROUP and the sum
must be taken over all elements in each class. Writ-
ten explicitly,
ai = lj2 x W Xi 'W nR > (12)
where \i i$ the CHARACTER of a single entry in
the Character Table and ur is the number of
elements in the corresponding CONJUGACY CLASS.
see also Abelian Group, Adele Group, Affine
Group, Alternating Group, Artinian Group, As-
chbacher's Component Theorem, ^-Theorem,
Baby Monster Group, Betti Group, Bimonster,
Bordism Group, Braid Group, Brauer Group,
Burnside Problem, Center (Group), Central-
izer, Character (Group), Character (Multi-
plicative), Chevalley Groups, Classical Groups,
Cobordism Group, Cohomotopy Group, Compo-
nent, Conjugacy Class, Coset, Conway Groups,
Coxeter Group, Cyclic Group, Dihedral Group,
Dimensionality Theorem, Dynkin Diagram, El-
liptic Group Modulo p, Engel's Theorem, Eu-
clidean Group, Feit-Thompson Theorem, Finite
Group, Fischer Groups, Fischer's Baby Mon-
ster Group, Fundamental Group, General Lin-
ear Group, General Orthogonal Group, Gen-
eral Unitary Group, Global C(G\T) Theo-
rem, Groupoid, Group Orthogonality Theorem,
Hall-Janko Group, Hamiltonian Group, Harada-
Norton Group, Heisenberg Group, Held Group,
Hermann-Mauguin Symbol, Higman-Sims Group,
Homeomorphic Group, Hypergroup, Icosahedral
Group, Irreducible Representation, Isomorphic
Groups, Janko Groups, Jordan-Holder The-
orem, Kleinian Group, Kummer Group, Im-
balance Theorem, Lagrange's Group Theo-
rem, Local Group Theory, Linear Group,
Lyons Group, Mathieu Groups, Matrix Group,
McLaughlin Group, Mobius Group, Modular
Group, Modulo Multiplication Group, Mon-
odromy Group, Monoid, Monster Group, Mul-
liken Symbols, Neron-Severi Group, Nilpotent
Group, Noncommutative Group, Normal Sub-
group, NORMALIZER, O'NAN GROUP, OCTAHEDRAL
Group, Order (Group), Orthogonal Group, Or-
thogonal Rotation Group, Outer Automor-
phism Group, p-Group, p'-Group, p-Layer, Point
Groups, Positive Definite Function, Prime
Group, Projective General Linear Group, Pro-
jective General Orthogonal Group, Projec-
tive General Unitary Group, Projective Spe-
cial Linear Group, Projective Special Or-
thogonal Group, Projective Special Unitary
Group, Projective Symplectic Group, Pseu-
dogroup, Quasigroup, Quasisimple Group, Qu-
asithin Theorem, Quasi-Unipotent Group, Rep-
resentation, Residue Class, Rubik's Cube, Rud-
valis Group, Schonflies Symbol, Schur Mul-
tiplier, Semisimple, Signalizer Functor Theo-
rem, Selmer Group, Semigroup, Simple Group,
Solvable Group, Space Groups, Special Lin-
ear Group, Special Orthogonal Group, Spe-
cial Unitary Group, Sporadic Group, Stochas-
tic Group, Strongly Embedded Theorem, Sub-
group, Subnormal, Support, Suzuki Group, Sym-
metric Group, Symplectic Group, Tetrahe-
dral Group, Thompson Group, Tightly Embed-
ded, Tits Group, Triangular Symmetry Group,
Twisted Chevalley Groups, Unimodular Group,
Unipotent, Unitary Group, Viergruppe, von
Dyck's Theorem
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 237-276, 1985.
Farmer, D. Groups and Symmetry. Providence, RI: Amer.
Math. Soc, 1995.
# Weisstein, E. W. "Groups." http://www. astro. Virginia.
edu/~eww6n/math/notebooks/Groups.m.
Weyl, H. The Classical Groups: Their Invariants and Rep-
resentations. Princeton, NJ: Princeton University Press,
1997.
Wybourne, B. G. Classical Groups for Physicists. New York:
Wiley, 1974.
Group Convolution
The convolution of two COMPLEX-valued functions on a
Group G is defined as
{a*b){g) = ^2a(k)b(k- 1 g)
keG
where the SUPPORT (set which is not zero) of each func-
tion is finite.
References
Weinstein, A. "Groupoids: Unifying Internal and External
Symmetry." Not Amer. Math. Soc. 43, 744-752, 1996.
Group Orthogonality Theorem
Let T be a representation for a GROUP of Order h, then
/ 1 i\H) mn l jyfijm'n' = i OijVmm'Unn' •
r V lil i
The proof is nontrivial and may be found in Eyring et
al. (1944).
References
Eyring, H.; Walker, J.; and Kimball, G. E. Quantum Chem-
istry. New York: Wiley, p. 371, 1944.
Group Ring
The set of sums ^ a x x ranging over a multiplicative
GROUP and a, are elements of a FIELD with all but a
finite number of ai = 0.
Group Theory
Growth Spiral 777
Group Theory
The study of GROUPS. Gauss developed but did not
publish parts of the mathematics of group theory, but
Galois is generally considered to have been the first to
develop the theory. Group theory is a powerful formal
method for analyzing abstract and physical systems in
which Symmetry is present and has surprising impor-
tance in physics, especially quantum mechanics.
see also FINITE GROUP, GROUP, PLETHYSM, SYMME-
TRY
References
Arfken, G. "Introduction to Group Theory." §4.8 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 237-276, 1985.
Burnside, W. Theory of Groups of Finite Order, 2nd ed. New
York: Dover, 1955.
Burrow, M. Representation Theory of Finite Groups. New
York: Dover, 1993.
Carmichael, R. D. Introduction to the Theory of Groups of
Finite Order, New York: Dover, 1956.
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, 1985.
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, 1990.
Dixon, J. D. Problems in Group Theory. New York: Dover,
1973.
Grossman, I. and Magnus, W. Groups and Their Graphs.
Washington, DC: Math. Assoc. Amer., 1965.
Hamermesh, M. Group Theory and Its Application to Phys-
ical Problems. New York: Dover, 1989.
Lomont, J. S. Applications of Finite Groups. New York:
Dover, 1987.
Magnus, W.; Karrass, A.; and Solitar, D. Combinatorial
Group Theory: Presentations of Groups in Terms of Gen-
erators and Relations. New York: Dover, 1976.
Robinson, D. J. S. A Course in the Theory of Groups, 2nd
ed. New York: Springer- Verlag, 1995.
Rose, J. S. A Course on Group Theory. New York: Dover,
1994.
Rotman, J. J. An Introduction to the Theory of Groups, ith
ed. New York: Springer- Verlag, 1995.
Groupoid
There are at least two definitions of "groupoid" cur-
rently in use.
The first type of groupoid is an algebraic structure on
a Set with a Binary Operator. The only restriction
on the operator is CLOSURE (i.e., applying the BINARY
Operator to two elements of a given set S returns
a value which is itself a member of 5). Associativity,
commutativity, etc., are not required (Rosenfeld 1968,
pp. 88-103). A groupoid can be empty. The numbers of
nonisomorphic groupoids of this type having n elements
are 1, 1, 10, 3330, 178981952, ... (Sloane's A001329),
and the numbers of nonisomorphic and nonantiisimor-
phic groupoids are 1, 7, 1734, 89521056, ... (Sloane's
A001424). An associative groupoid is called a SEMI-
GROUP.
The second type of groupoid is an algebraic structure
first defined by Brandt (1926) and also known as a VIR-
TUAL GROUP. A groupoid with base B is a set G with
mappings a and f3 from G onto B and a partially defined
binary operation (g, h) h> gh, satisfying the following
four conditions:
1. gh is defined only when (3(G) = a(h) for certain
maps a and (3 from G onto K. with a : (x, 7, y) »-»■ x
and : (x,7,y) ^ y.
2. Associativity: If either (gh)k or g(hk) is defined,
then so is the other and (gh)k = g(hk).
3. For each g in G, there are left and right IDENTITY
Elements X g and p g such that X g g = g — gp g .
4. Each g in G has an inverse g^ 1 for which gg^ 1 = \ g
and g~ x g = p g
(Weinstein 1996). A groupoid is a small CATEGORY with
every morphism invertible.
see also Binary Operator, Inverse Semigroup, Lie
Algebroid, Lie Groupoid, Monoid, Quasigroup,
Semigroup, Topological Groupoid
References
Brandt, W. "Uber eine Verallgemeinerung des Gruppen-
griffes." Math. Ann. 96, 360-366, 1926.
Brown, R. "Prom Groups to Groupoids: A Brief Survey."
Bull. London Math. Soc. 19, 113-134, 1987.
Brown, R. Topology: A Geometric Account of General To-
pology, Homotopy Types, and the Fundamental Groupoid.
New York: Halsted Press, 1988.
Higgins, P. J. Notes on Categories and Groupoids. London:
Van Nostrand Reinhold, 1971.
Ramsay, A.; Chiaramonte, R.; and Woo, L. "Groupoid
Home Page." http://amath-vww. Colorado. edu : 80/math/
researchgroups/groupoids/groupoids . shtml.
Rosenfeld, A. An Introduction to Algebraic Structures. New
York: Holden-Day, 1968.
Sloane, N. J. A. Sequences A001329/M4760 and A001424/
M4465 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Weinstein, A. "Groupoids: Unifying Internal and External
Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996.
Growth
A general term which refers to an increase (or decrease
in the case of the oxymoron "NEGATIVE growth") in a
given quantity.
see also GROWTH FUNCTION, GROWTH SPIRAL
Growth Function
see Block Growth
Growth Spiral
see Logarithmic Spiral
778 Grundy's Game
Gyrobirotunda
Grundy's Game
A special case of NlM played by the following rules.
Given a heap of size n, two players alternately select a
heap and divide it into two unequal heaps. A player loses
when he cannot make a legal move because all heaps
have size 1 or 2. Flammenkamp gives a table of the ex-
tremal Sprague- Grundy Values for this game. The
first few values of Grundy's game are 0, 0, 0, 1, 0, 2, 1,
0, 2, ... (Sloane's A002188).
References
Flammenkamp, A. "Sprague-Grundy Values of Grundy's
Game." http:// www . minet . uni - jena . de / - achim /
grundy.html.
Sloane, N. J. A. Sequence A002188/M0044 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Grundy-Sprague Number
see Nim- Value
Gudermannian Function
Denoted either "f(x) or gd(x).
gd(cc) = tan - (sinhir) — 2 tan" (e x ) — \-k (1)
gd _1 (cc) = ln[tan(^7r + ~x)] — ln(secx + tancc). (2)
The derivatives are given by
d
dx
gd(#) = sechrc
dx
gd 1 (x) = secz.
(3)
(4)
Guldinus Theorem
see Pappus's Centroid Theorem
Gumbel's Distribution
A special case of the Fisher-Tippett Distribution
with a = 0, 6 = 1. The Mean, Variance, Skewness,
and KURTOSIS are
2
7i
12a/6C(3)
~, — 12
72 - T .
where 7 is the EULER-MASCHERONI CONSTANT, and
£(3) is Apery's Constant.
see also FlSHER-TlPPETT DISTRIBUTION
Guthrie's Problem
The problem of deciding if four-colors are sufficient to
color any map on a plane or Sphere.
see also FOUR-COLOR THEOREM
Gutschoven's Curve
see Kappa Curve
Guy's Conjecture
Guy's conjecture, which has not yet been proven or dis-
proven, states that the CROSSING Number for a Com-
plete Graph of order n is
where [x\ is the FLOOR FUNCTION, which can be rewrit-
ten
J ^4 n ( n ~~ 2) 2 ( n — 4) for n even
\±(n- l) 2 (n - 3) 2 for n odd.
The first few values are 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, . . .
(Sloane's A000241).
see also CROSSING NUMBER (GRAPH)
References
Sloane, N. J. A. Sequence A000241/M2772 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Gyrate Bidiminished
Rhombicosidodecahedron
see Johnson Solid
Gyrate Rhombicosidodecahedron
see Johnson Solid
Gyrobicupola
A Bicupola in which the bases are in opposite orienta-
tions.
see also BlCUPOLA, PENTAGONAL GYROBICUPOLA,
Square Gyrobicupola
Gyrobifastigium
Johnson Solid J26, consisting of two joined triangular
Prisms,
Gyrobirotunda
A Birotunda in which the bases are in opposite orien-
tations.
Gyrocupolarotunda
Gyroelongated Triangular Cupola 779
Gyrocupolarotunda
A CUPOLAROTUNDA in which the bases are in opposite
orientations.
see also Orthocupolarotunda
Gyroelongated Cupola
A n-gonal Cupola adjoined to a 2n-gonal Antiprism.
see also Gyroelongated Pentagonal Cupola, Gy-
roelongated Square Cupola, Gyroelongated
Triangular Cupola
Gyroelongated Dipyramid
see Gyroelongated Pyramid, Gyroelongated
Square Dipyramid
Gyroelongated Pentagonal Bicupola
Johnson Solid J 4 e, which consists of a Pentagonal
Rotunda adjoined to a decagonal Antiprism.
Gyroelongated Pentagonal Birotunda
see Johnson Solid
Gyroelongated Pentagonal Cupola
see Johnson Solid
Gyroelongated Pentagonal Cupolarotunda
see Johnson Solid
Gyroelongated Pentagonal Pyramid
see Johnson Solid
Gyroelongated Pentagonal Rotunda
see Johnson Solid
Gyroelongated Pyramid
An n-gonal pyramid adjoined to an n-gonal Antiprism.
see also ELONGATED PYRAMID, GYROELONGATED DI-
PYRAMID, Gyroelongated Pentagonal Pyramid,
GYROELONGATED SQUARE DIPYRAMID, GYROELON-
GATED Square Pyramid
Gyroelongated Rotunda
see Gyroelongated Pentagonal Rotunda
Gyroelongated Square Cupola
see Johnson Solid
Gyroelongated Square Dipyramid
One of the eight convex Deltahedra. It consists of
two oppositely faced SQUARE PYRAMIDS rotated 45° to
each other and separated by a ribbon of eight side-to-
side Triangles. It is Johnson Solid J 17 .
Call the coordinates of the upper PYRAMID bases (± 1,
± 1, hx) and of the lower (±y/2, 0, -fti) and (0, ±\/2,
-hi). Call the Pyramid apexes (0, 0, ±(hi + ft 2 )).
Consider the points (1, 1, 0) and (0, 0, hi + /12). The
height of the Pyramid is then given by
y/l 2 + l 2 + h 2 2 = V% + h 2 2 = 2
h 2 = V2.
(1)
(2)
Now consider the points (1, 1, /ii) and (\/2, 0, — hi).
The height of the base is given by
(1 - V2) 2 + l 2 + (2/ti) 2 = 1 - 2V2 + 2 + 1 + 4/n 2
= 4 - 2\/2 + 4/ii 2 = 2 2 = 4 (3)
4/n 2 = 2y/l
(4)
, 2 v2 1 -l/2
(5)
ft! = 2" 1/4
(6)
ft 2 = 2 1 ' 2 .
(7)
Gyroelongated Square Pyramid
see Johnson Solid
Gyroelongated Triangular Bicupola
see Johnson Solid
Gyroelongated Triangular Cupola
see Johnson Solid
Gyroelongated Square Bicupola
see Johnson Solid
h-Cobordism
H
Haar Function
781
/i-Cobordism
An /i-cobordism is a Cobordism W between two MANI-
FOLDS Mi and M 2 such that W is SIMPLY CONNECTED
and the inclusion maps Mi — > W and M 2 — >- W are
HOMOTOPY equivalences.
/i-Cobordism Theorem
If W is a Simply Connected, Compact Manifold
with a boundary that has two components, Mi and M2,
such that inclusion of each is a Homotopy equivalence,
then W is DlFFEOMORPHIC to the product Mi x [0, 1]
for dim (Mi) > 5. In other words, if M and M' are two
simply connected Manifolds of Dimension > 5 and
there exists an h-COBORDISM W between them, then
W is a product M x I and M is DlFFEOMORPHIC to
M'.
The proof of the /i-cobordism theorem can be accom-
plished using Surgery. A particular case of the h-
cobordism theorem is the Poincare CONJECTURE in
dimension n > 5. Smale proved this theorem in 1961.
see also DlFFEOMORPHISM, POINCARE CONJECTURE,
Surgery
References
Smale, S. "Generalized Poincare's Conjecture in Dimensions
Greater than Four." Ann. Math. 74, 391-406, 1961.
H-Fractal
M MKiSKiS
h nraffl
The Fractal illustrated above.
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 1-2,
1991.
# Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/~eww6n/math/notebooks/Fractal.m.
iif-Function
see Fox's ^-Function
H-Spread
The difference H 2 - H lt where Hi and H 2 are HINGES.
It is the same as the INTERQUARTILE RANGE for N = 5,
9, 13, . . . points.
see also HlNGE, INTERQUARTILE RANGE, STEP
References
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, p. 44, 1977.
H- Transform
A 2-D generalization of the Haar Transform which is
used for the compression of astronomical images. The
algorithm consists of dividing the 2 N x 2 N image into
blocks of 2 x 2 pixels, calling the pixels in the block
aoo, aiO) aoi> and an. For each block, compute the four
coefficients
ho = 2 (aii + aio + aoi 4- aoo)
h x = 2( an + ai ° — ao1 — a °°)
h y = 2 ( a n ~" a io + a oi — aoo)
h c = 2-(an — aio — aoi + aoo).
Construct a 2 N ~ X x 2 N ~ X image from the ho values, and
repeat until only one ho value remains. The H-transform
can be performed in place and requires about 16iV 2 /3
additions for an N x N image.
see also HAAR TRANSFORM
References
Capaccioli, M.; Held, E. V.; Lorenz, H.; Richter, G. M.; and
Ziener, R. "Application of an Adaptive Filtering Technique
to Surface Photometry of Galaxies. I. The Method Tested
on NGC 3379." Astron. Nachr. 309, 69-80, 1988.
Fritze, K.; Lange, M.; Mostle, G.; Oleak, H.; and Richter,
G. M. "A Scanning Microphotometer with an On-Line
Data Reduction for Large Field Schmidt Plates." Astron.
Nachr. 298, 189-196, 1977.
Richter, G. M. "The Evaluation of Astronomical Pho-
tographs with the Automatic Area Photometer." Astron.
Nachr. 299, 283-303, 1978.
White, R. L.; Postman, M.; and Lattanzi, M. G. "Com-
pression of the Guide Star Digitised Schmidt Plates." In
Digitised Optical Sky Surveys: Proceedings of the Con-
ference on "Digitised Optical Sky Surveys" held in Edin-
burgh, Scotland, 18-21 June 1991 (Ed. H. T. MacGillivray
and E. B. Thompson). Dordrecht, Netherlands: Kluwer,
pp. 167-175, 1992.
Haar Function
ll 1 lh
-1
l—i
-1
-1
782
Haar Function
Hadamard's Inequality
Define
and
4>(x) = { -l
i:
0<x< \
\<X<1
otherwise
(1)
(2)
i> jk {x)=i;{2 3 x-k),
where the FUNCTIONS plotted above are
-000 = tp(x)
ipio = ip(2x)
t^n = iP(2x - 1)
1P20 = i/>(4x)
^21 = 1p(4x - 1)
^21 = i>{±x - 2)
V>2i =^(4x-3).
Then a FUNCTION f{x) can be written as a series ex-
pansion by
2^-
f( X ) = C + ^ X^ C ^3k{^)-
j=0 k=0
(3)
The Functions ^ and ^ are all Orthogonal in
[0,1], with
L
cf>(x)<f>jk{x)dx =
/
<j?jk{x)<pi m (x) dx = 0.
(4)
(5)
These functions can be used to define WAVELETS. Let a
FUNCTION be defined on n intervals, with n a POWER of
2. Then an arbitrary function can be considered as an
n- VECTOR f, and the Coefficients in the expansion
b can be determined by solving the MATRIX equation
f = W„b
(6)
for b, where W is the MATRIX of ip basis functions. For
example,
W 4 =
'1 1
1
"
1 1
-1
1 -1
1
1
1 -1
1
-1_
'1
1
"1
1
1 -1
1
1
-1
1
1
1
1
-1
(7)
The Wavelet Matrix can be computed in 0(n) steps,
compared to G(n\gn) for the FOURIER Matrix.
see also Wavelet, Wavelet Transform
References
Haar, A. "Zur Theorie der orthogonalen Funktionensys-
teme." Math. Ann. 69, 331-371, 1910.
Strang, G. "Wavelet Transforms Versus Fourier Transforms."
Bull. Amer. Math. Soc. 28, 288-305, 1993.
Haar Integral
The Integral associated with the Haar Measure.
see also HAAR MEASURE
Haar Measure
Any locally compact Hausdorff topological group has a
unique (up to scalars) NONZERO left invariant measure
which is finite on compact sets. If the group is Abelian
or compact, then this measure is also right invariant and
is known as the Haar measure.
Haar Transform
A 1-D transform which makes use of the Haar Func-
tions.
see H-TRANSFORM, HAAR FUNCTION
References
Haar, A. "Zur Theorie der orthogonalen Funktionensys-
teme." Math. Ann. 69, 331-371, 1910.
Haberdasher's Problem
x^W
With four cuts, Dissect an Equilateral Triangle
into a SQUARE. First proposed by Dudeney (1907) and
discussed in Gardner (1961, p. 34) and Stewart (1987,
p. 169). The solution can be hinged so that the three
pieces collapse into either the TRIANGLE or the SQUARE.
see also DISSECTION
References
Gardner, M. The Second Scientific American Book of Math-
ematical Puzzles & Diversions: A New Selection. New
York: Simon and Schuster, 1961.
Stewart, I. The Problems of Mathematics, 2nd ed. Oxford,
England: Oxford University Press, 1987.
Hadamard Design
A Symmetric Block Design (4n + 3, n + 1, n) which
is equivalent to a HADAMARD Matrix of order An +
4. It is conjectured that Hadamard designs exist from
all integers n > 0, but this has not yet been proven.
This elusive proof (or disproof) remains one of the most
important unsolved problems in COMBINATORICS.
References
Dinitz, J. H. and Stinson, D. R. "A Brief Introduction to
Design Theory." Ch. 1 in Contemporary Design Theory: A
Collection of Surveys (Ed. J. H. Dinitz and D. R. Stinson).
New York: Wiley, pp. 1-12, 1992.
Hadamard's Inequality
Let A = an be an arbitrary n x n nonsingular MATRIX
with Real elements and Determinant |A|, then
iAi 2 <n E a -
Hadamard Matrix
Hadamard Matrix 783
see also Hadamard'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. 1110, 1979.
Hadamard Matrix
44, 52, 60, 68, 76, 84, 92, and 100 cannot be built up
from lower order Hadamard matrices.
i-
A class of Square Matrix invented by Sylvester (1867)
under the name of ANALLAGMATIC PAVEMENT. A Had-
amard matrix is a SQUARE MATRIX containing only Is
and —Is such that when any two columns or rows are
placed side by side, Half the adjacent cells are the same
Sign and half the other (excepting from the count an L-
shaped "half-frame" bordering the matrix on two sides
which is composed entirely of Is). When viewed as pave-
ments, cells with Is are colored black and those with —Is
are colored white. Therefore, the n x n Hadamard ma-
trix H™ must have n(n — l)/2 white squares (—Is) and
n(n + l)/2 black squares (Is).
This is equivalent to the definition
H n H r
i\ n
(i)
where l n is the n x n IDENTITY MATRIX. A Hadamard
matrix of order 4n + 4 corresponds to a HADAMARD
Design {An + 3, 2n + 1, n).
Paley's Theorem guarantees that there always exists
a Hadamard matrix H n when n is divisible by 4 and of
the form 2 e {q Tn + 1), where p is an ODD PRIME. In such
cases, the MATRICES can be constructed using a PALEY
Construction. The Paley Class k is undefined for
the following values of m < 1000: 92, 116, 156, 172,
184, 188, 232, 236, 260, 268, 292, 324, 356, 372, 376,
404, 412, 428, 436, 452, 472, 476, 508, 520, 532, 536,
584, 596, 604, 612, 652, 668, 712, 716, 732, 756, 764,
772, 808, 836, 852, 856, 872, 876, 892, 904, 932, 940,
944, 952, 956, 964, 980, 988, 996.
Sawade (1985) constructed H268. It is conjectured (and
verified up to n < 428) that H n exists for all n DIVISIBLE
by 4 (van Lint and Wilson 1993). However, the proof
of this Conjecture remains an important problem in
Coding Theory. The number of Hadamard matrices of
order An are 1, 1, 1, 5, 3, 60, 487, . . . (Sloane's A007299).
If H n and H m are known, then H nm can be obtained by
replacing all Is in H m by H n and all -Is by — H n . For
n < 100, Hadamard matrices with n — 12, 20, 28, 36,
H 2 =
H 4 =
1 1
-1 1
-
1 1"
" 1 1"
-
H2 H2
— H2 H2
=
-1 1
1 1
-1 1
1 1"
-1 1
-1 1
1111]
-11-11
-1-111
1 -
1
-1
1
(2)
(3)
hU can be similarly generated from H4. Hadamard ma-
trices can also be expressed in terms of the WALSH
Functions Cal and Sal
Cal(0,*)'
Sal(4,t)
Sal(2,t)
Cal(2,£)
Sal(l,£)
Cal(3,t)
Cal(l,t)
L Sal(3, t) ,
H 8 =
(4)
Hadamard matrices can be used to make Error-
Correcting Codes.
see also Hadamard Design, Paley Construction,
Paley's Theorem, Walsh Function
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. New
York: Cambridge University Press, 1986.
Colbourn, C. J. and Dinitz, J. H. (Eds.) "Hadamard Matrices
and Designs." Ch. 24 in CRC Handbook of Combinatorial
Designs. Boca Raton, FL: CRC Press, pp. 370-377, 1996.
Geramita, A. V. Orthogonal Designs: Quadratic Forms and
Hadamard Matrices. New York: Marcel Dekker, 1979.
Golomb, S. W. and Baumert, L. D. "The Search for Hada-
mard Matrices." Amer. Math. Monthly 70, 12-17, 1963.
Hall, M. Jr. Combinatorial Theory, 2nd ed. New York: Wi-
ley, p. 207, 1986.
Hedayat, A. and Wallis, W. D. "Hadamard Matrices and
Their Applications." Ann. Stat. 6, 1184-1238, 1978.
Kimura, H. "Classification of Hadamard Matrices of Order
28." Disc. Math. 133, 171-180, 1994.
Kimura, H. "Classification of Hadamard Matrices of Order
28 with Hall Sets." Disc. Math. 128, 257-269, 1994.
Kitis, L. "Paley's Construction of Hadamard Matrices."
http : // www . mathsource . com / cgi - bin / Math Source /
Applications/Mathematics/0205-760.
Ogilvie, G. A. "Solution to Problem 2511." Math. Questions
and Solutions 10, 74-76, 1868.
Paley, R. E. A. C. "On Orthogonal Matrices." J. Math. Phys.
12, 311-320, 1933.
Ryser, H. J. Combinatorial Mathematics. Buffalo, NY:
Math. Assoc. Amer., pp. 104-122, 1963.
784
Hadamard's Theorem
Hafner-Sarnak-McCurley Constant
Sawade, K. "A Hadamard Matrix of Order-268." Graphs
Combinatorics 1, 185-187, 1985.
Seberry, J. and Yamada, M. "Hadamard Matrices, Sequences,
and Block Designs." Ch. 11 in Contemporary Design
Theory: A Collection of Surveys (Eds. J. H. Dinitz and
D. R. Stinson). New York: Wiley, pp. 431-560, 1992.
Sloane, N. J. A, Sequence A007299/M3736 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Spence, E. "Classification of Hadamard Matrices of Order 24
and 28." Disc. Math 140, 185-243, 1995.
Sylvester, J. J. "Thoughts on Orthogonal Matrices, Simulta-
neous Sign-Successions, and Tessellated Pavements in Two
or More Colours, with Applications to Newton's Rule, Or-
namental Tile- Work, and the Theory of Numbers." Phil
Mag. 34, 461-475, 1867.
Sylvester, J. J. "Problem 2511." Math. Questions and Solu-
tions 10, 74, 1868.
van Lint, J. H. and Wilson, R. M. A Course in Combina-
torics. New York: Cambridge University Press, 1993.
Hadamard's Theorem
Let |A| be an n x n DETERMINANT with COMPLEX (or
Real) elements a^, then |A| ^ if
>
X)i a ^'i
3 = 1
see also HADAMARD'S INEQUALITY
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1110, 1979.
Hadamard Transform
A Fast Fourier TRANSFORM-like Algorithm which
produces a hologram of an image.
Hadamard- Vallee Poussin Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
The sum of RECIPROCALS of PRIMES diverges, but
lim
n— +oo
^i--ln(lnn)
oo r / \ i
V LA-^ + JL
■i +
= Ci.
0.2614972128.
(1)
where 7r(n) is the Prime Counting Function and
7 is the Euler-Mascheroni Constant (Le Lionnais
1983). Hardy and Wright (1985) show that, if w(n) is
the number of distinct PRIME factors of n, then
lim
n— >oo
n
— } w(k) — ln(lnn)
n ^— '
= Ci.
(2)
Furthermore, if Q(n) is the total number of Prime fac-
tors of n, then
lim
Tl— K30
n
l£n(fc)-ln(lnn)
n £ — '
oo
= d + V* —r^- — — = 1.0346538819 .... (3)
Pk{Pk-l)
Similarly,
rr(n)
i im y^_ lnn = _ 7 _ry^
\ fc = l / 3=2 fc = l
= -C 2 = -1.3325822757 .... (4)
References
Finch, S. "Favorite Mathematical Constants." http://wvw.
mathsof t . c om/ as o 1 ve/ const ant /hdmrd/hdmrd. html.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1985.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 24, 1983.
Rosser, J. B. and Schoenfeld, L. "Approximate Formulas for
Some Functions of Prime Numbers." ///. J. Math. 6, 64-
94, 1962.
Hadwiger's Principal Theorem
The VECTORS ±ai , . . . , ±a n in a 3-space form a nor-
malized Eutactic Star Iff Tx = x for all x in the
3-space.
Hadwiger Problem
What is the largest number of subcubes (not necessarily
different) into which a CUBE cannot be divided by plane
cuts? The answer is 47.
see also Cube Dissection
Hafner-Sarnak-McCurley Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Given two randomly chosen INTEGER n x n matrices,
what is the probability D(n) that the corresponding de-
terminants are coprime? Hafner et ah (1993) showed
that
Din) = J] 1
l-f[(l- Pk -i)
j=i
(1)
where the product is over PRIMES. The case D(l) is just
the probability that two random INTEGERS are coprime,
D{1)= -r =0.6079271019....
7T
(2)
Hahn-Banach Theorem
Eall-Janko Group 785
Vardi (1991) computed the limit
<j = lim D(n) = 0.3532363719.
(3)
The speed of convergence is roughly
and Vardi 1996).
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/hafner/hafner.html.
Flajolet, P. and Vardi, I. "Zeta Function Expan-
sions of Classical Constants." Unpublished manu-
script. 1996. http://pauillac.inria.fr/algo/flajolet/
Publications/landau. ps.
Hafner, J. L.; Sarnak, P.; and McCurley, K. "Relatively
Prime Values of Polynomials." In Contemporary Math-
ematics Vol. 143 (Ed. M. Knopp and M. Seingorn). Prov-
idence, RI: Amer. Math. Soc, 1993.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, 1991.
Hahn-Banach Theorem
A linear FUNCTIONAL defined on a SUBSPACE of a VEC-
TOR Space V and which is dominated by a sublinear
function defined on V has a linear extension which is
also dominated by the sublinear function.
References
Zeidler, E. Applied Functional Analysis: Applications to
Mathematical Physics. New York: Springer- Verlag, 1995.
Hailstone Number
Sequences of Integers generated in the COLLATZ
Problem. For example, for a starting number of 7,
the sequence is 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10,
5, 16, 8, 4, 2, 1, 4, 2, 1, Such sequences are called
hailstone sequences because the values typically rise and
fall, somewhat analogously to a hailstone inside a cloud.
While a hailstone eventually becomes so heavy that it
falls to ground, every starting INTEGER ever tested has
produced a hailstone sequence that eventually drops
down to the number 1 and then "bounces" into the small
loop 4, 2, 1,
see also COLLATZ PROBLEM
References
Schwartzman, S. The Words of Mathematics: An Etymolog-
ical Dictionary of Mathematical Terms Used in English.
Washington, DC: Math. Assoc. Amer., 1994.
Hairy Ball Theorem
There does not exist an everywhere NONZERO VECTOR
Field on the 2-Sphere § 2 . This implies that some-
where on the surface of the Earth, there is a point with
zero horizontal wind velocity.
Half- Closed Interval
An Interval in which one endpoint is included but not
the other. A half-closed interval is denoted [a, 6) or (a, b]
and is also called a HALF-OPEN INTERVAL.
0.57 n (Flajolet see also Closed Interval, Open Interval
Half-Normal Distribution
A Normal Distribution with Mean and Standard
Deviation 1/0 limited to the domain [0, oo).
P{x) = ™ e -*>*>,«
7T
D{.
X) = erf (^)-
The Moments are
Mi
—
t
M2
=
IT
2P
M3
=
7T
M4
4< 4 '
(1)
(2)
(3)
(4)
(5)
(6)
so the Mean, Variance, Skewness, and Kurtosis are
"=B
V)
2 2 7 1 " ~~ 2
° =M2 Mi = 2f2
(8)
^= 2 S
(9)
72 = 0.
(10)
see also Normal Distribution
Half-Open Interval
see Half-Closed Interval
Hall-Janko Group
The Sporadic Group HJ, also denoted J 2 .
see also Janko Groups
Half
The Unit Fraction 1/2.
see also Quarter, Square Root, Unit Fraction
786 Halley's Irrational Formula
Halley's Method
Halley's Irrational Formula
A RoOT-finding ALGORITHM which makes use of a
third-order TAYLOR SERIES
f(x) = f{x n )+f{x n ){x-X n )+\f"(x n ){x-Xn) 2 +
(1)
A Root of f(x) satisfies f(x) = 0, so
« f{x n ) + f'(x n )(x n +l - X n ) + | /" (x n ) (x n +l ~ X n )
(2)
Using the QUADRATIC EQUATION then gives
-/'(*») ± y/[f'(Xn)]*-2f(x n )f"(Xn)
Xn-\-\ — x n ~r
/"(*«)
Picking the plus sign gives the iteration function
1 f. 2f{x)f"{x)
1 V [/'(-)]*
Cf(x) = x -
/"(»)
/'(*)
(3)
(4)
This equation can be used as a starting point for deriving
Halley's Method.
If the alternate form of the Quadratic Equation is
used instead in solving (2), the iteration function be-
comes instead
Cf(x) = x
2/(x)
/'(*) ± y/[f'(x)]> - 2f(x)f"(x)
(5)
This form can also be derived by setting n = 2 in
Laguerre's Method. Numerically, the Sign in the
Denominator is chosen to maximize its Absolute
Value. Note that in the above equation, if f"(x) — 0,
then Newton's Method is recovered. This form of
Halley's irrational formula has cubic convergence, and
is usually found to be substantially more stable than
NEWTON'S Method. However, it does run into diffi-
culty when both f(x) and f'(x) or f'(x) and f"(x) are
simultaneously near zero.
see also Halley's Method, Laguerre's Method,
Newton's Method
References
Qiu, H. "A Robust Examination of the Newton-Raphson
Method with Strong Global Convergence Properties."
Master's Thesis. University of Central Florida, 1993.
Scavo, T. R. and Thoo, J. B. "On the Geometry of Halley's
Method." Amer. Math. Monthly 102, 417-426, 1995.
Halley's Method
Also known as the TANGENT HYPERBOLAS METHOD
or Halley's Rational Formula. As in Halley's
Irrational Formula, take the second-order Taylor
Polynomial
f(x) = f{x n )+f'{x n ){x-Xr l )+\f"{Xr l )(x-Xn) +
(i)
A Root of f(x) satisfies f(x) = 0, so
« f(x n ) + f'(x n )(Xn+l - X n ) + %f"(Xn)(x n +l ~ X n f '*
(2)
Now write
= f(x„) + (x„+i - x n )[f'(x n ) + ^f"(x n )(x n+ i - x n )],
giving
, = _ /(£»)
" +1 " /'(*„) + i/»(z„)(Xn+i - X») '
Using the result from Newton's Method,
(3)
(4)
Xn+l X n
/'(*«)*
gives
Xn+l — Xn
2f(x n )f'(x n )
2[/'(*»)] a -/(*»)/"(*»)'
so the iteration function is
2/(x)/'(x)
Hf{x) = x ■
2[/'(i)] a -/(*)/"(*)'
(5)
(6)
(7)
This satisfies Hf(a) = H'J(a) = where a is a ROOT,
so it is third order for simple zeros. Curiously, the third
derivative
*/'(«) =
f"{a) 3 |7"(«)lM (R .
/'(a) 2 [/'(a) J j W
is the Schwarzian Derivative. Halley's method may
also be derived by applying Newton's Method to
ff~ 1/2 . It may also be derived by using an Osculat-
ing Curve of the form
Taking derivatives
/ \ \X Xfi ) ~\- c
a(x — x n ) + o
(9)
j3 1
(10)
f(, ) _b-ac
J (An) — , 2
(H)
,„, . 2a(oc-6)
/ (x„) = ^
(12)
which has solutions
/"(*»)
b =
2[f>(x n )]*-f(x n )f"(x n )
2/'(x»)
2[/'(^n)] 2 -/(x„)/"(x„)
2f(x n )f'(x n )
2[/'(Zn)] 2 - /(*„)/"(*„)'
(13)
(14)
(15)
Halley's Rational Formula
Hamiltonian Circuit 787
so at a Root, y(x n +i) — and
(16)
which is Halley's method.
see also HALLEY'S IRRATIONAL FORMULA, LAGUERRE'S
Method, Newton's Method
References
Scavo, T. R. and Thoo, J. B. "On the Geometry of Halley's
Method." Amer. Math. Monthly 102, 417-426, 1995.
Halley's Rational Formula
see Halley's Method
Halphen Constant
see One-Ninth Constant
Halphen's Transformation
A curve and its polar reciprocal with regard to the fixed
Conic have the same Halphen transformation.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 346-347, 1959.
Halting Problem
The determination of whether a Turing Machine will
come to a halt given a particular input program. This
problem is formally UNDECIDABLE, as first proved by
Turing.
see also BUSY BEAVER, CHAITIN'S CONSTANT, TURING
Machine, Undecidable
References
Chaitin, G. J. "Computing the Busy Beaver Function." §4.4
in Open Problems in Communication and Computation
(Ed. T. M. Cover and B. Gopinath). New York: Springer-
Verlag, pp. 108-112, 1987.
Davis, M. "What It a Computation." In Mathematics Today:
Twelve Informal Essays (Ed. L. A. Steen). New York:
Springer- Verlag, pp. 241-267, 1978.
Penrose, R. The Emperor 's New Mind: Concerning Comput-
ers, Minds, and the Laws of Physics. Oxford, England:
Oxford University Press, pp. 63—66, 1989.
Ham Sandwich Theorem
The volumes of any n n-D solids can always be simulta-
neously bisected by a (n — 1)-D Hyperplane. Proving
the theorem for n = 2 (where it is known as the PAN-
CAKE Theorem) is simple and can be found in Courant
and Robbins (1978). The theorem was proved for n > 3
by Stone and Tukey (1942).
see also PANCAKE THEOREM
References
Chinn, W. G. and Steenrod, N. E. First Concepts of Topol-
ogy. Washington, DC: Math. Assoc. Amer., 1966.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods. Oxford, Eng-
land: Oxford University Press, 1978.
Davis, P. J. and Hersh, R. The Mathematical Experience.
Boston, MA: Houghton Mifflin, pp. 274-284, 1981.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 67-69, 1975.
Stone, A. H. and Tukey, J. W. "Generalized 'Sandwich' The-
orems." Duke Math. J. 9, 356-359, 1942.
Hamilton's Equations
The equations defined by
8H
dp
3H
(1)
(2)
where x = dx/dt and H is the so-called Hamiltonian, are
called Hamilton's equations. These equations frequently
arise in problems of celestial mechanics. Another formu-
lation related to Hamilton's equation is
P =
dL
dq'
(3)
where L is the so-called Lagrangian.
References
Morse, P. M. and Feshbach, H. "Hamilton's Principle and
Classical Dynamics." §3.2 in Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 280-301, 1953.
Hamilton's Rules
The rules for the Multiplication of Quaternions.
see also QUATERNION
Hamiltonian Circuit
A closed loop through a GRAPH that visits each node
exactly once and ends adjacent to the initial point. The
Hamiltonian circuit is named after Sir William Rowan
Hamilton, who devised a puzzle in which such a path
along the EDGES of an ICOSAHEDRON was sought (the
Icosian Game).
All PLATONIC Solids have a Hamiltonian circuit, as
do planar 4-connected graphs. However, no foolproof
method is known for determining whether a given gen-
eral GRAPH has a Hamiltonian circuit. The number of
Hamiltonian circuits on an n-HYPERCUBE is 2, 8, 96,
43008, . . . (Sloane's A006069, Gardner 1986, pp. 23-
24).
see also Chvatal's Theorem, Dirac's Theo-
rem, Euler Graph, Grinberg Formula, Ham-
iltonian Graph, Hamiltonian Path, Icosian
Game, Kozyrev-Grinberg Theory, Ore's Theo-
rem, Posa's Theorem, Smith's Network Theorem
References
Chartrand, G. Introductory Graph Theory. New York:
Dover, p. 68, 1985.
Gardner, M. "The Binary Gray Code." In Knotted Dough-
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, pp. 23-24, 1986.
Sloane, N. J. A. Sequence A006069/M1903 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
788 Hamiltonian Cycle
Hamming Function
Hamiltonian Cycle
see Hamiltonian Circuit
Hamiltonian Graph
A Graph possessing a Hamiltonian Circuit.
see also HAMILTONIAN CIRCUIT, HAMILTONIAN PATH
References
Chartrand, G. Introductory Graph Theory. New York:
Dover, p. 68, 1985.
Chartrand, G.; Kapoor, S. F.; and Kronk, H. V. "The Many
Facets of Hamiltonian Graphs." Math, Student 41, 327-
336, 1973.
Hamiltonian Group
A non-Abelian GROUP all of whose SUBGROUPS are self-
conjugate.
References
Carmichael, R. D. "Hamiltonian Groups." §31 in Introduc-
tion to the Theory of Groups of Finite Order. New York:
Dover, p. 113-116, 1956.
Hamiltonian Map
Consider a 1-D Hamiltonian Map of the form
H{p,q) = \p 2 + V{q),
which satisfies Hamilton's Equations
. OH
OH
V- ~
Now, write
where
8q-
(gi+l ~ 1i)
At '
q i+ i = q(t + At).
Then the equations of motion become
9i+l = Qi + PiAt
Pi+i = Pi — At
8V
dqt
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Note that equations (7) and (8) are not AREA-
Preserving, since
d(gi+ljPi+l)
8{quV%)
-At
dqi 2
1 + (Aty
dqi*
7^1.
(9)
At 1
However, if we take instead of (7) and (8),
qt+i =qi-\~PiAt (10)
p i+1 = Pi -At[ £- ) (11)
(dqi)
9(quPi)
i -**&m
At
1 + (At)
Q = Qi + l
, d 2 V
dq^
(12)
which is Area-Preserving.
Hamiltonian Path
A loop through a GRAPH that visits each node exactly
once but does not end adjacent to the initial point. The
number of Hamiltonian paths on an n-HYPERCUBE is
0, 0, 48, 48384, . . . (Sloane's A006070, Gardner 1986,
pp. 23-24).
see also Hamiltonian Circuit, Hamiltonian Graph
References
Gardner, M. "The Binary Gray Code." In Knotted Dough-
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, pp. 23-24, 1986.
Sloane, N. J. A. Sequence A006070/M5295 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hamiltonian System
A system of variables which can be written in the form
of Hamilton's Equations.
Hammer-Aitoff Equal- Area Projection
A Map PROJECTION whose inverse is defined using the
intermediate variable
Z = Jl-(± X )2-(±y)2.
Then the longitude and latitude are given by
A = 2 tan *
2(2z :
M
4> = sin 1 {yz).
Hamming Function
0.1
0/5
-0.25
-0.5
0.0
0.
.0(35
i$W
An Apodization Function chosen to minimize the
height of the highest sidelobe. The Hamming function
is given by
A(x) = 0.54 + 0.46 cos
firx\
It/'
(i)
Its Full Width at Half Maximum is 1.05543a. The
corresponding INSTRUMENT FUNCTION is
i(k) =
a(1.08 - 0.64a 2 fc 2 ) sinc(27rafc)
1 - 4a 2 A; 2
(2)
Handedness
This Apodization Function is close to the one pro-
duced by the requirement that the APPARATUS FUNC-
TION goes to at ka = 5/4. From APODIZATION FUNC-
TION, a general symmetric apodization function A(x)
can be written as a FOURIER Series
oo
A(x) = a + 2 ^ a n cos ( — — J , (3)
n = l
where the COEFFICIENTS satisfy
oo
a + 2^a n = l. (4)
n = l
The corresponding apparatus function is
oo
I(t) = 2b{a sinc(27rA;fe) + ^[sinc(27rA;6 -f nir)
n = l
+ sinc(27rA;6-n7r)]}. (5)
To obtain an APODIZATION FUNCTION with zero at ka -
3/4, use
a + 2ai = 1, (6)
so
ao sinc(|7r) + ai[sinc(|7r) + sinc(|7r) = (7)
(l-2ai)^-«i (£ + £)= (l-2a 1 )i-a 1 (i + i) = °
(8)
°i(7 + 3 + !) = S (»)
7-3
ai - a + i + i -
5 ~ 7 ' 3
= ~ « 0.2283
2-3-7H-3-5 + 5-7
ao = 1 — 2ai
92-2-21 92-42
50
92
25
46
92
0.5435.
92
(10)
(11)
The FWHM is 1.81522, the peak is 1.08, the peak Neg-
ative and Positive sidelobes (in units of the peak) are
-0.00689132 and 0.00734934, respectively.
see also APODIZATION FUNCTION, HANNING FUNC-
TION, Instrument Function
References
Blackman, R. B. and Tukey, J. W. "Particular Pairs of Win-
dows." In The Measurement of Power Spectra, From
the Point of View of Communications Engineering. New-
York: Dover, pp. 98-99, 1959.
Handedness
Objects which are identical except for a mirror reflection
are said to display handedness and to be CHIRAL.
see also Amphichiral, Chiral, Enantiomer, Mir-
ror Image
Hankel Function
Handkerchief Surface
789
A surface given by the parametric equations
x(u,v) = u
y{u,v) — v
z(u 9 v) = \u +uv 2 + 2{u -v 2 ).
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 628, 1993.
Handle
Handles are to Manifolds as Cells are to CW-
Complexes. If M is a Manifold together with a
(k — 1)- Sphere S*" 1 embedded in its boundary with a
trivial Tubular Neighborhood, we attach a fc-handle
to M by gluing the tubular NEIGHBORHOOD of the
^ _1 to the Tubular Neighborhood
(k - 1)-Sphere §*
of the standard (k - 1)-Sphere § fc ~ x in the dim(M)-
dimensional DISK.
In this way, attaching a fc-handle is essentially just the
process of attaching a fattened-up fc-DlSK to M along
the (k - 1)-Sphere S fe_1 . The embedded Disk in this
new Manifold is called the fc-handle in the Union of
M and the handle.
see also HANDLEBODY, SURGERY, TUBULAR NEIGH-
BORHOOD
Handlebody
A handlebody of type (n, k) is an n-D Manifold that
is attained from the standard n-DlSK by attaching only
k-D Handles.
see also Handle, Heegaard Splitting, Surgery
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 46, 1976.
Hankel Function
A Complex function which is a linear combination of
Bessel Functions of the First and Second Kinds.
see also HANKEL FUNCTION OF THE FIRST KIND, HAN-
KEL Function of the Second Kind, Spherical
Hankel Function of the First Kind, Spherical
Hankel Function of the Second Kind
790
Hankel Function of the First Kind
Hankel Transform
References
Arfken, G. "Hankel Functions." §11.4 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 604-610, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 623-624, 1953.
Hankel Function of the First Kind
H£\z) = J n {z) + iY n {z),
where J n (z) is a Bessel Function of the First Kind
and Y n (z) is a Bessel Function of the Second
Kind. Hankel functions of the first kind can be rep-
resented as a Contour Integral using
fff'M
t7r Jo
p (*/2)(t-l/t)
dt.
[upper half plane]
see also Debye's Asymptotic Representation,
Watson-Nicholson Formula, Weyrich's Formula
References
Arfken, G. "Hankel Functions." §11.4 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 604-610, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 623-624, 1953.
Hankel Function of the Second Kind
Hg\z) = Jn(z)-iY n (z),
where J n (z) is a Bessel Function of the First
Kind and Y n (z) is a Bessel Function of the Sec-
ond Kind. Hankel functions of the second kind can be
represented as a CONTOUR INTEGRAL using
"PM-c/
(z/2)(t-l/t)
oo [lower half plane]
t n+1
■dt.
see also Watson-Nicholson Formula
References
Arfken, G. "Hankel Functions." §11.4 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 604-610, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 623-624, 1953.
Hankel's Integral
Jm{x) =
2™- 1 v^ : r(m+§)
: I cos(xt)(l - t 2 ) m ~ l/2 dt,
Jo
where J m (x) is a Bessel Function of the First
Kind and T(z) is the Gamma Function. Hankel's in-
tegral can be derived from SONINE'S INTEGRAL.
see also Poisson Integral, Sonine's Integral
Hankel Matrix
A Matrix with identical values for each element in a
given diagonal. Define H n to be the Hankel matrix with
leading column made up of the INTEGERS 1, . . . , n, then
H 2 =
"l 2~
2
[1 2 3]
H 3 =
2 3
.3
0_
Hankel Transform
Equivalent to a 2-D FOURIER TRANSFORM with a radi-
ally symmetric KERNEL, and also called the FOURIER-
Bessel Transform.
/oo />oo
/ f(r)e- 2 ^ ux+vv) dxdy.
-oo J — oo
(1)
Let
x + iy = re
u + iv = qe
id
i<p
so that
x = r cos V
y = r sin 6
r = y/ x 2 + \
u = q cos <fi
v = q sin <j>
q ~ yu 2 4- v 2 .
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Then
<?(<?)
/»00 /»27T
= / / f( r ) e - 27rirq ( cos * cos *+ sin ^ sin 9 ) r dr d6
Jo Jo
/•oo p2tt
= / /( r )e- 2 * ir ' cos ( 9 -*V(fr^
Jo t/0
/»oo p2ir — <f>
= I I f(r)e- 2 " iTqcose rdrd0
Jo J — <j
/»00 /»27T
/»QO pZTT
/ / f(r)e- 2wir,coae rdrd0
Jo Jo
r /(r) [ f
Jo iJo
'I
Jo
-2-irirq cos 6
'0 L</0
27T / f(r)Jo(2-jrqr)rdr }
(10)
Hann Function
where Jq(z) is a zeroth order Bessel Function of the
First Kind. Therefore, the Hankel transform pairs are
/
g(k) = / f(x)Jo(kx)xdx
I
Jo
f(x) = / g(k)Jo(kx)kdk.
(ii)
(12)
see also Bessel Function of the First Kind, Four-
ier Transform, Laplace Transform
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. 244-250, 1965.
Hann Function
see Hanning Function
Hanning Function
An Apodization Function, also called the Hann
Function, frequently used to reduce ALIASING in
Fourier Transforms. The illustrations above show
the Hanning function, its INSTRUMENT FUNCTION, and
a blowup of the Instrument Function sidelobes. The
Hanning function is given by
/<.)--*(=)- i-i~ (=). w
The Instrument Function for Hanning apodization
can also be written
a[sinc(27rA;a) + \ sinc(27rfca — 7r)+ \ sinc(27r&a + 7r)]. (2)
Its Full Width at Half Maximum is a. It has Ap-
paratus Function
A{x) = £ [i - § cos (^)] e~ 2 ^ dx
J —a
1 / e -2*ikx dx .
e -2*ikx dx
= ±(A 1 +A 2 ).
The first integral is
(3)
J — C
T i -2nikx , sin(27rA:a) .
I 1 = J e ax = = 2asmc(27rA;a). (4)
7T/C
Hanning Function
The second integral can be rewritten
h = J cos (!^)
i/ —a
+ I COS ( ! ?)
= r\ 0S (TE.y e ^* +e -*«i>>* )dx
= 2 / cos ( — I cos(2;rfca:) dx
{sin (f - 27rifc) x sin (f + 2-nk) x
791
e dx
e dx
sin(7r — 27rka) sin(7r -f 2nka)
7r — 2nka it + 2nka
sin(27rA;a) sin(27rA;a)
(5)
1 - 2ka 1 + 2ka
= a[sinc(7r — 2nka) + sinc(7r + 2nka)].
Combining (4) and (5) gives
A(x) = a[sinc(27rfca) + \ sinc(7r — 2irka)
+ f sinc(7r + 27rfca)]. (6)
To find the extrema, define x = 2nka and rewrite (6) as
A(x) = a[smx + | sinc(a: — 7r) + | sinc(x + ?r)]. (7)
Then solve
dA _ 7r (—x cosa; + 3x sina; + 7r x cos x - 7r sin a;)
dx X 2 (7T 2 — x 2 ) 2
= (8)
to find the extrema. The roots are x = 7.42023
and 10.7061, giving a peak NEGATIVE sidelobe of
-0.026708 and a peak POSITIVE sidelobe (in units of
a) of 0.00843441. The peak in units of a is 1, and the
full-width at half maximum is given by setting (7) equal
to 1/2 and solving for #, yielding
Xi /2 — 27T&i/2tt = 7T.
(9)
Therefore, with L = 2a, the Full Width at Half
Maximum is
FWHM = 2A; 1/2 = - = %.
a L
(10)
see also APODIZATION FUNCTION, HAMMING FUNC-
TION
792 Hanoi Graph
Hanoi Graph
A Graph H n arising in conjunction with the Towers
OF Hanoi problem. The above figure is the Hanoi graph
see also Towers of Hanoi
Hanoi Towers
see Towers of Hanoi
Hansen-Bessel Formula
J n ( Z )= — J e izco*t e i n (t-«/2) dt
-f
J — 7T
* Jo
e izcost cos(nt)dt
i r
— I cos(z sin t — nt) dt
* Jo
for n = 0, 1, 2, ... , where J n (z) is a Bessel Function
of the First Kind.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1472,
1980.
Hansen Chain
An Addition Chain for which there is a Subset H of
members such that each member of the chain uses the
largest element of H which is less than the member.
see also Addition Chain, Brauer Chain, Hansen
Number
References
Guy, R. K. "Addition Chains. Brauer Chains. Hansen
Chains." §C6 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 111-113, 1994.
Hansen Number
A number n for which a shortest chain exists which is a
Hansen Chain is called a Hansen number.
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 111-112, 1994.
Harary Graph
Hansen's Problem
A Surveying Problem: from the position of two
known but inaccessible points A and £?, determine the
position of two unknown accessible points P and P' by
bearings from A, B, P' to P and A, B y P to P f .
see also Surveying Problems
References
Dorrie, H. "Annex to a Survey." §40 in 100 Great Problems
of Elementary Mathematics: Their History and Solutions.
New York: Dover, pp. 193-197, 1965.
Happy Number
Let the sum of the Squares of the Digits of a Pos-
itive Integer so be represented by s\. In a similar
way, let the sum of the SQUARES of the DIGITS of s\ be
represented by S2, and so on. If some s» = 1 for i > 1,
then the original INTEGER s is said to be happy.
Once it is known whether a number is happy (or not),
then any number in the sequence si, S2, $3, • • • wm a ^ so
be happy (or not). A number which is not happy is
called Unhappy. Unhappy numbers have Eventually
Periodic sequences of Si 4, 16, 37, 58, 89, 145, 42, 20,
4, . . . which do not reach 1.
Any Permutation of the Digits of an Unhappy or
happy number must also be unhappy or happy. This
follows from the fact that Addition is Commutative.
The first few happy numbers are 1, 7, 10, 13, 19, 23, 28,
31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, . . .
(Sloane's A007770). These are also the numbers whose
2-Recurring Digital Invariant sequences have pe-
riod 1.
see also Kaprekar Number, Recurring Digital In-
variant , Unhappy Number
References
Dudeney, H. E. Problem 143 in 536 Puzzles & Curious Prob-
lems. New York: Scribner, pp. 43 and 258-259, 1967.
Guy, R. K. "Happy Numbers." §E34 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 234-235, 1994.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 163-165, 1979.
Schwartzman, S. The Words of Mathematics: An Etymolog-
ical Dictionary of Mathematical Terms Used in English.
Washington, DC: Math. Assoc. Amer., 1994.
Sloane, N. J. A. Sequence A007770 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Harada-Norton Group
The Sporadic Group HN.
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http : //for . mat . bham . ac . uk/atlas/HN . html.
Harary Graph
The smallest fe-connected GRAPH with n VERTICES.
Hard Hexagon Entropy Constant
Hardy-Littlewood Conjectures 793
Hard Hexagon Entropy Constant
N. B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
A constant related to the HARD SQUARE ENTROPY
Constant. This constant is given by
K h = lim [G(N)] 1/N = 1.395485972 . . . , (1)
where G(N) is the number of configurations of nonat-
tacking Kings on an n x n chessboard with regular
hexagonal cells, where N = n 2 . Amazingly, Kh is al-
gebraic and given by
Kh — K1K2K3K4, (2)
where
« 1 =4- 1 3 5/4 ll- 5/12 c- 2
= [1 - Vl - c + \] 2 + c + 2 a/1 + c + c 2 ] 2
«3
K4
(3)
K 2 = ll-Vl-c+Y^ + c + 2V 1 + c + c "J" ( 4 )
= [-l-VT^l:+y2 + c + 2^1 + c + c 2 } 2 (5)
[VT^Ti +y2 + a + 2y/l + a + a 2 ] _1/2 (6)
(7)
c^d + fa^+l) 1 / 3 -^-!) 1 / 3 ]} 1 / 3 . (9)
(Baxter 1980, Joyce 1988).
References
Baxter, R. J. "Hard Hexagons: Exact Solution." J. Physics
A 13, 1023-1030, 1980.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/square/square.html.
Joyce, G. S. "On the Hard Hexagon Model and the Theory
of Modular Functions." Phil. Trans. Royal Soc. London A
325, 643-702, 1988.
Plouffe, S. "Hard Hexagons Constant." http://lacim.uqam.
ca/piDATA/hardhex .html.
Hard Square Entropy Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let F(n ) be the number of binary nxn MATRICES with
no adjacent Is (in either columns or rows). Define N =
n , then the hard square entropy constant is defined by
k= lim [F(N)] 1/N = 1.503048082....
N— >oo
The quantity In k arises in statistical physics (Baxter
et al. 1980, Pearce and Seaton 1988), and is known as
the entropy per site of hard squares. A related constant
known as the HARD HEXAGON ENTROPY CONSTANT
can also be defined.
References
Baxter, R. J.; Enting, I. G.; and Tsang, S. K. "Hard-Square
Lattice Gas." J. Statist. Phys. 22, 465-489, 1980.
Finch, S. "Favorite Mathematical Constants." http://vww.
mathsoft.com/asolve/constant/square/square.html.
Pearce, P. A. and Seaton, K. A. "A Classical Theory of Hard
Squares." J. Statist. Phys. 53, 1061-1072, 1988.
Hardy's Inequality
Let {a n } be a Nonnegative Sequence and f(x) a
NONNEGATIVE integrable FUNCTION. Define
*. = £
oo
&k
&k
and
n*)= [ x f(t)dt
Jo
/•oo
G(x)= f(t)dt,
J x
(i)
(2)
(3)
(4)
and take p > 1. For sums,
00 A « / \ P °°
E(*)<(,-h)B->- <*>
n=l V / n=l
(unless all a n =0), and for integrals,
\F(x)
f
Jo
dx <
I
[f(x)] p dx (6)
(unless / is identically 0).
References
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 239-243, 1988.
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities
Involving Functions and Their Integrals and Derivatives.
New York: Kluwer, 1991.
Opic, B. and Kufner, A. Hardy-Type Inequalities. Essex,
England: Longman, 1990.
Hardy-Littlewood Conjectures
The first Hardy-Littlewood conjecture is called the k-
Tuple Conjecture. It states that the asymptotic
number of PRIME CONSTELLATIONS can be computed
explicitly.
The second Hardy-Littlewood conjecture states that
ir(x + y) ~tt(x) < 7r(y)
for all x and y, where tt(x) is the PRIME COUNTING
FUNCTION. Although it is not obvious, Richards (1974)
proved that this conjecture is incompatible with the first
Hardy-Littlewood conjecture.
see also Prime Constellation, Prime Counting
Function
References
Richards, I. "On the Incompatibility of Two Conjectures
Concerning Primes." Bull Amer. Math. Soc. 80, 419-
438, 1974.
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, pp. 61-62
and 68-69, 1994.
794 Hardy-Littlewood Constants
Harmonic Addition Theorem
Hardy-Littlewood Constants
see Prime Constellation
Hardy-Littlewood Tauberian Theorem
Let a n > and suppose
£
a n e
1
Hardy-Ramanujan Theorem
Let v(n) be the number of Distinct Prime Factors
of n. If \&(as) tends steadily to infinity with as, then
In In x — ^(x)yAnhix < w(n) < In In x + ^(a^Vlnlna:
for Almost All numbers n < as. "Almost All"
means here the frequency of those Integers n in the
interval 1 < n < x for which
as a-»0 + . Then
E
a n ~ as
as x — ► oo.
see also Tauberian Theorem
References
Berndt, B. C, Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 118-119, 1994.
Hardy-Littlewood k- Tuple Conjecture
see Prime Patterns Conjecture
Hardy-Ramanujan Number
The smallest nontrivial Taxicab Number, i.e., the
smallest number representable in two ways as a sum of
two CUBES. It is given by
1729 = r + 12*
9 3 + 10 3 .
The number derives its name from the following story
G. H. Hardy told about Ramanujan. "Once, in the taxi
from London, Hardy noticed its number, 1729. He must
have thought about it a little because he entered the
room where Ramanujan lay in bed and, with scarcely a
hello, blurted out his disappointment with it. It was, he
declared, 'rather a dull number,' adding that he hoped
that wasn't a bad omen. 'No, Hardy,' said Ramanujan,
' it is a very interesting number. It is the smallest number
expressible as the sum of two [POSITIVE] cubes in two
different ways'" (Hofstadter 1989, Kanigel 1991, Snow
1993).
see also Diophantine Equation — Cubic, Taxicab
Number
References
Guy, R. K. "Sums of Like Powers. Euler's Conjecture." §D1
in Unsolved Problems in Number Theory, 2nd ed. New
York: Springer- Verlag, pp. 139-144, 1994.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 68, 1959.
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 564, 1989.
Kanigel, R. The Man Who Knew Infinity: A Life of the
Genius Ramanujan. New York: Washington Square Press,
p. 312, 1991.
Snow, C P. Foreword to Hardy, G. H. A Mathematician's
Apology, reprinted with a foreword by C. P. Snow. New
York: Cambridge University Press, p. 37, 1993.
|u>(n) — In In as | > ^(a^Vlnlnas
approaches as x — ► oo.
see also DISTINCT PRIME FACTORS, ERDOS-KAC THE-
OREM
Hardy's Rule
Let the values of a function /(as) be tabulated at in-
tervals equally spaced by h about aso, so that /_ 3 —
/(aso — 3/fc), f-2 — /(aso — 2/i), etc. Then Hardy's rule
gives the approximation to the integral of /(as) as
px +3h
JxQ-Zh
f(x) dx = I i 5 /i(28/_ 3 + 162/_ 2 + 22/o + 162/ 2
+28/ 3 ) + db/> 7 [2/ (4) (6) - A 3 / (8) «i)],
where the final term gives the error, with £i,£2 € [aso —
3/i, aso 4- 3/i].
see also Bode's Rule, Durand's Rule, Newton-
Cotes Formulas, Simpson's 3/8 Rule, Simpson's
Rule, Trapezoidal Rule, Weddle's Rule
Harmonic Addition Theorem
To convert an equation of the form
to the form
f(0) = acos0-t-bsm8
f(0) = ccos(9 + 8),
(1)
(2)
expand (2) using the trigonometric addition formulas to
obtain
f{6) — c cos cos 8 — c sin 6 sin 5. (3)
Now equate the COEFFICIENTS of (1) and (3)
a = c cos S
b = — csinj,
tan£
and we have
„ 2 _L h 2 — ^ 2
a + b — c ,
c= y/a 2 + b 2 .
(4)
(5)
(6)
(7)
(8)
(9)
Harmonic Analysis
Given two general sinusoidal functions with frequency
ipi = Ax sin(u;£ + <5i)
ip2 = A2 sin{wt + (52 ),
(10)
(11)
their sum ip can be expressed as a sinusoidal function
with frequency u>
ip = tpi + ip2 = Ai[sin(ujt) cos Sj + sin Si cos(u>£)]
+ A2[sin(a;t) cos J2 + sin $2 cos(ivt)]
— [Ai cos 61 + A2 cos fo] sin(a;i)
+ [Ai sin^i + A2 sinfo] cos(u;£). (12)
Now, define
A cos 5 = A x cos^x + A 2 cos<$2 (13)
A sin J = Aisin^i + A 2 sm£ 2 . (14)
Then (12) becomes
A cos S sin(ojt) + A sin S cos(ujt) = Asm(u>t + J). (15)
Square and add (13) and (14)
A 2 = Ai 2 + A 2 2 + 2A X A 2 cos(S 2 - 61). (16)
Also, divide (14) by (13)
A\ sin Si + A2 sin £ 2
tan 5 =
A± cos Ji + A 2 cos <5 2 '
(17)
(18)
tp — Asin(ujt + 5),
where A and 5 are defined by (16) and (17).
This procedure can be generalized to a sum of n har-
monic waves, giving
71
j/} — ^2 Ai cos(uit + Si) = Acos(wt + S), (19)
where
j > i i = l
and
a 2 = ]p A i 2 + 2 ^2 ^2 AiAj cos ( Si ~ Sj ^ ( 20 )
i i = l
tanS = —
j™ =1 Ai cos 6i'
(21)
Harmonic Analysis
see also FOURIER SERIES
Harmonic Coordinates 795
Harmonic Brick
A right-angled PARALLELEPIPED with dimensions a x
ab x abc, where a, 6, and c are INTEGERS.
see also Brick, de Bruijn's Theorem, Euler Brick
Harmonic Conjugate Function
The harmonic conjugate to a given function u(x,y) is a
function v(x,y) such that
f(x,y) = u{x,y) + v(x,y)
is Complex Differentiable (i.e., satisfies the
Cauchy-Riemann Equations). It is given by
v(z) = / uxdy — uy dx.
Harmonic Conjugate Points
W X Y Z
Given Collinear points W, X, Y, and Z, Y and Z are
harmonic conjugates with respect to W and X if
\WY\ \WZ\
\YX\
\zx\
The distances between such points are said to be in HAR-
MONIC Ratio, and the Line Segment depicted above
is called a Harmonic Segment.
Harmonic conjugate points are also defined for a Tri-
angle. If W and X have Trilinear Coordinates
a : : 7 and 0/ : /?' : 7', then the TRILINEAR COORDI-
NATES of the harmonic conjugates are
Y = a + a : /3 + : 7 + 7'
Z = a — a : — /3 : 7 — 7
(Kimberling 1994).
see ateo HARMONIC RANGE, HARMONIC RATIO
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 13-14, 1990.
Phillips, A. W. and Fisher, I. Elements of Geometry. New
York: American Book Co., 1896.
Wells, D. The Penguin Dictionary of Curious and Interesting
Geometry. New York: Viking Penguin, p. 92, 1992.
Harmonic Coordinates
Satisfy the condition
T A = g^Y^ x = 0,
or equivalently,
dx r
(V99 XK ) = 0.
(1)
(2)
796 Harmonic Decomposition
Harmonic Function
It is always possible to choose such a system. Using the
d'Alembertian Operator,
tf* = (9 X «M;« = 9 XK A^. ~ T A ^T- (3)
dx^dx* dx x
But since T A = for harmonic coordinates,
□V = o.
(4)
Sloane, N. J. A. Sequences A007340/M4299 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.
Zachariou, A. and Zachariou, E. "Perfect, Semi-Perfect and
Ore Numbers." Bull Soc. Math. Grece (New Ser.) 13,
12-22, 1972.
Harmonic Equation
see Laplace's Equation
Harmonic Decomposition
A Polynomial function in the elements of x can be
uniquely decomposed into a sum of harmonic POLYNO-
MIALS times Powers of |x|.
Harmonic Divisor Number
A number n for which the Harmonic Mean of the Di-
visors of n, i.e., nd(n)/a-(n)j is an Integer, where d(n)
is the number of POSITIVE integral DIVISORS of n and
a(n) is the DIVISOR FUNCTION. For example, the divi-
sors of n = 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70,
and 140, giving
d(140) = 12
a(140) = 336
140d(140) __ 140 • 12 _
<r(140)
335
so 140 is a harmonic divisor number. Harmonic divisor
numbers are also called ORE NUMBERS. Garcia (1954)
gives the 45 harmonic divisor numbers less than 10 7 .
The first few are 1, 6, 140, 270, 672, 1638, . . . (Sloane's
A007340).
For distinct PRIMES p and q, harmonic divisor numbers
are equivalent to EVEN PERFECT NUMBERS for numbers
of the form p v q. Mills (1972) proved that if there exists
an Odd Positive harmonic divisor number n, then n
has a prime-PoWER factor greater than 10 7 .
Another type of number called "harmonic" is the HAR-
MONIC Number.
see also DIVISOR FUNCTION, HARMONIC NUMBER
References
Edgar, H. M. W. "Harmonic Numbers." Amer. Math.
Monthly 99, 783-789, 1992.
Garcia, M. "On Numbers with Integral Harmonic Mean."
Amer. Math. Monthly 61, 89-96, 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- Verlag, pp. 45-53, 1994.
Mills, W. H. "On a Conjecture of Ore." Proceedings of the
1972 Number Theory Conference. University of Colorado,
Boulder, pp. 142-146, 1972.
Ore, 0. "On the Averages of the Divisors of a Number."
Amer. Math. Monthly 55, 615-619, 1948.
Pomerance, C. "On a Problem of Ore: Harmonic Numbers."
Unpublished manuscript, 1973.
Harmonic Function
Any real- valued function u(x,y) with continuous sec-
ond Partial Derivatives which satisfies Laplace's
Equation
V 2 u(x,y)^0 (1)
is called a harmonic function. Harmonic functions are
called Potential Functions in physics and engineer-
ing. Potential functions are extremely useful, for exam-
ple, in electromagnetism, where they reduce the study
of a 3-component Vector Field to a 1-component
Scalar Function. A scalar harmonic function is
called a SCALAR Potential, and a vector harmonic
function is called a VECTOR POTENTIAL.
To find a class of such functions in the PLANE, write the
Laplace's Equation in Polar Coordinates
1 1
u rr H — u r H — ^uee = U,
r r z
and consider only radial solutions
1
Urr ~i U-p
r
0.
(2)
(3)
dv 1
— + -v^Q
dr r
dv
v
dr
r
This is integrable by quadrature, so define v = du/dr,
(4)
(5)
(6)
(7)
(8)
(9)
in (J)-*,
A r
_ du _ A
dr r
du = A — ,
r
so the solution is
u = A In r.
(10)
Ignoring the trivial additive and multiplicative con-
stants, the general pure radial solution then becomes
u = \n[(x-a) 2 + (y-bf] 1/2 = \ In [(* - af + (y - b) 2 ] .
(11)
Harmonic-Geometric Mean
Harmonic Logarithm 797
Other solutions may be obtained by differentiation, such
as
(12)
(13)
(x - a)^ + (y- b) 2
y -b
(x - a) 2 + (y - fc) 2 '
u = e siny
v — e x cosy,
and
tan
_1 (— )■
\x - aj
(14)
(15)
(16)
Harmonic functions containing azimuthal dependence
include
u = r n cos(n6)
v — r n sin(n#).
The Poisson Kernel
u(r,B,M) =
R 2 ~r 2
R 2 -2rRcos(0-<t>) + r 2
(17)
(18)
(19)
is another harmonic function.
see also SCALAR POTENTIAL, VECTOR POTENTIAL
References
Ash, J. M. (Ed.) Studies in Harmonic Analysis. Washing-
ton, DC: Math. Assoc. Amer., 1976.
Axler, S.; Pourdon, P.; and Ramey, W. Harmonic Function
Theory. Springer- Verlag, 1992.
Benedetto, J. J. Harmonic Analysis and Applications. Boca
Raton, FL: CRC Press, 1996.
Cohn, H. Conformal Mapping on Riemann Surfaces. New
York: Dover, 1980.
Harmonic- Geometric Mean
Let
OCn + l —
2a n /3 n
Otn + fin
fin+l = yOLnfin,
then
if(a ,/?o) = lim an
M{ao~\f3o~ l y
where M is the Arithmetic-Geometric Mean.
see also Arithmetic Mean, Arithmetic-Geometric
Mean, Geometric Mean, Harmonic Mean
Harmonic Homology
A Perspective Collineation with center O and axis
o not incident is called a HOMOLOGY. A HOMOLOGY
is said to be harmonic if the points A and A! on a line
through O are harmonic conjugates with respect to O
and o-a. Every PERSPECTIVE COLLINEATION of period
two is a harmonic homology.
see also HOMOLOGY (GEOMETRY), PERSPECTIVE
Collineation
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 248, 1969.
Harmonic Logarithm
For all Integers n and Nonnegative Integers t, the
harmonic logarithms A„ (x) of order t and degree n are
defined as the unique functions satisfying
1. \U(x) = (\n X y,
2. A)l (x) has no constant term except Aq \x) = 1,
where the "ROMAN SYMBOL" |_»1 is defined by
w-{r
n for n ^
for n =
(Roman 1992). This gives the special cases
Al 0) (x) = {f
where H n is a Harmonic Number
for n >
for n <
(i)^ _ J x n {\nx — H n ) for n >
for n < 0,
~-ti
The harmonic logarithm has the INTEGRAL
A( 1 »(x)«i c =-!- T A< 1 »(i).
/•
The harmonic logarithm can be written
\W(x)=ln\\D- n (lr l x) t ,
(1)
(2)
(3)
(4)
(5)
(6)
where D is the DIFFERENTIAL OPERATOR, (so D n is
the nth Integral). Rearranging gives
D k W(x)
Mi
[n — k
«*£*(*).
(7)
This formulation gives an analog of the Binomial The-
orem called the Logarithmic Binomial Formula.
Another expression for the harmonic logarithm is
A?>(*) = x" J^-l^OicWGnaO*"', (8)
798 Harmonic Map
Harmonic Number
where (t)j = t(t - 1) ■ • • (t — j + 1) is a POCHHAMMER
SYMBOL and c$ is a two-index Harmonic Number
(Roman 1992).
see also LOGARITHM, ROMAN FACTORIAL
References
Loeb, D. and Rota, G.-C. "Formal Power Series of Logarith-
mic Type." Advances Math. 75, 1-118, 1989.
Roman, S. "The Logarithmic Binomial Formula." Amer.
Math. Monthly 99, 641-648, 1992.
Harmonic Map
A harmonic map between RlEMANNlAN MANIFOLDS can
be viewed as a generalization of a GEODESIC when the
domain Dimension is one, or of a Harmonic Function
when the range is a EUCLIDEAN SPACE.
see also BOCHNER IDENTITY, EUCLIDEAN SPACE, GEO-
DESIC, Harmonic Function, Riemannian Manifold
References
Burstal, F.; Lemaire, L.; and Rawnsley, J. "Harmonic
Maps Bibliography." http: //www. bath. ac.uk/~masfeb/
harmonic .html.
Eels, J. and Lemaire, L. "A Report on Harmonic Maps."
Bull London Math. Soc. 10, 1-68, 1978.
Eels, J. and Lemaire, L. "Another Report on Harmonic
Maps." Bull London Math. Soc. 20, 385-524, 1988.
Harmonic Mean
The harmonic mean H(#i, . . , , x n ) of n points Xi (where
i = l, . . . , n) is
_L = I \^ 1
H ~~ n 2-^f Xi '
The special case of n = 2 is therefore
or
1 _ Xi + X2
H ~
(1)
(2)
(3)
2cci#2
The VOLUME-to-SURFACE Area ratio for a cylindrical
container with height h and radius r and the Mean
Curvature of a general surface are related to the har-
monic mean.
Hoehn and Niven (1985) show that
H(ai+c, a 2 + c, ...,a n +c) > c + H(ai,a2, ■ ■ ■ ,a n ) (4)
for any POSITIVE constant c.
see also Arithmetic Mean, Arithmetic-Geometric
Mean, Geometric Mean, Harmonic-Geometric
Mean, Root-Mean-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. 10, 1972.
Hoehn, L. and Niven, I. "Averages on the Move." Math.
Mag. 58, 151-156, 1985.
Harmonic Mean Index
The statistical Index
Ph =
E
poqo
E
P0 2 90
Pn
where p n is the price per unit in period n, q n is the
quantity produced in period n, and v n = PnQn the value
of the n units.
see also INDEX
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 69, 1962.
Harmonic Number
A number of the form
"-Si-
This can be expressed analytically as
H„ = 7 + ipo(n+ 1),
(1)
(2)
where 7 is the Euler-Mascheroni Constant and
ty(x) = Vo(z) is the DlGAMMA FUNCTION. The number
formed by taking alternate signs in the sum also has an
analytic solution
H n
_\M-D*
fc=X
(3)
ln2+|(-l)"[Vo(in+i)-Vo(in + l)]. (4)
The first few harmonic numbers H n are 1, 3/2, 11/6,
25/12, 137/60, ... (Sloane's A001008 and A002805).
The Harmonic Number H n is never an Integer (ex-
cept for #i), which can be proved by using the strong
triangle inequality to show that the 2-ADIC VALUE of H n
is greater than 1 for n > 1. The harmonic numbers have
Odd Numerators and Even Denominators. The
nth harmonic number is given asymptotically by
H n ~ In n + 7 +
In
(5)
where 7 is the Euler-Mascheroni Constant (Con-
way and Guy 1996). Gosper gave the interesting identity
z*Hi
Ez Hi z v^
kkl
= e*[ln* + r(0 ) z)+7], (6)
Harmonic Number
Harmonic Range 799
where r(0,z) is the incomplete GAMMA FUNCTION and
7 is the Euler-Mascheroni Constant. Borwein and
Borwein (1995) show that
plus the recurrence relation
^ (n
H n
(n + iy
H n
4 f W 360
11 >rr 4
n
¥C(4) = Ms* 4
H n
E$ = K( 4 ) = ^ 4 '
(7)
(8)
(9)
where £(z) is the Riemann Zeta Function. The first
of these had been previously derived by de Doelder
(1991), and the last by Euler (1775). These identities
are corollaries of the identity
i [' * 2 {ln[2 cos(faO]} 2 dx = £<(4) = ^tt 4 (10)
71 Jo
(Borwein and Borwein 1995). Additional identities due
to Euler are
no
H n
£fr = 2 « 3 )
(ii)
oo m — 2
2 E ^ = ("»+2)C("»+l)-X; C(m-n)C(n+l) (12)
Tl = l 71 = 1
for m = 2, 3, ... (Borwein and Borwein 1995), where
C(3) is Apery's Constant. These sums are related to
so-called Euler Sums.
Conway and Guy (1996) define the second harmonic
number by
n
Hi 2) =Y,Hi = (n+l)(fl-„+i-l) = (n+l)(ff n+1 -#i),
i=l
(13)
the third harmonic number by
H^^±H^^( n+ 2 2 )(H n+2 -H 2 ), (14)
i = l ^ '
and the nth harmonic number by
^> = ( n +^ 1 ) (jff „ +fc _ 1 - fffc
i)-
(15)
A slightly different definition of a two-index harmonic
number cL is given by Roman (1992) in connection with
the Harmonic Logarithm. Roman (1992) defines this
by
,(°)
.W)
f 1 for n >
10 for n <
for j =
for j ^
( 1 for j =
10
(16)
(17)
crW^^+n^
(18)
For general n > and j > 0, this is equivalent to
2 = 1
and for n > 0, it simplifies to
^^^(^(-l)'- 1 ^. (20)
For n < 0, the harmonic number can be written
c^ = (-1Y [n]ls(-nj) y (21)
where [n]\ is the ROMAN FACTORIAL and s is a STIR-
LING Number of the First Kind.
A separate type of number sometimes also called a "har-
monic number" is a Harmonic Divisor Number (or
Ore Number).
see also Apery's Constant, Euler Sum, Harmonic
Logarithm, Harmonic Series, Ore Number
References
Borwein, D. and Borwein, J. M. "On an Intriguing Integral
and Some Series Related to C(4)." Proc. Amer. Math. Soc.
123, 1191-1198, 1995.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 143 and 258-259, 1996.
de Doelder, P. J. "On Some Series Containing *(cc) - *(y)
and (*(x) — *(y)) 2 for Certain Values of x and y" J.
Comp. Appl. Math. 37, 125-141, 1991.
Roman, S. "The Logarithmic Binomial Formula." Amer.
Math. Monthly 99, 641-648, 1992.
Sloane, N. J. A. Sequences A001008/M2885 and A002805/
Ml 5 89 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Harmonic Progression
see Harmonic Series
Harmonic Range
A B
C
D
2 1 3
A set of four COLLINEAR points ^4, B y C, and D ar-
ranged such that
AB : BC = 2 : 1
AD: DC = 6:3.
Hardy (1967) uses the term HARMONIC SYSTEM OF
Points to refer to a harmonic range.
see also Euler Line, Gergonne Line, Harmonic
Conjugate Points, Soddy Line
References
Hardy, G. H. A Course of Pure Mathematics, 10th ed. Cam-
bridge, England: Cambridge University Press, pp. 99 and
106, 1967.
800
Harmonic Ratio
Harnack's Inequality
Harmonic Ratio
see Harmonic Conjugate Points
Harmonic Segment
see Harmonic Conjugate Points
Harmonic Series
The Sum
oo
(1)
is called the harmonic series. It can be shown to DI-
VERGE using the INTEGRAL TEST by comparison with
the function 1/x. The divergence, however, is very slow.
In fact, the sum
Ei (»)
V
taken over all Primes also diverges. The generalization
of the harmonic series
«»>-£=
(3)
is known as the RlEMANN Zeta FUNCTION.
The sum of the first few terms of the harmonic series is
given analytically by the nth HARMONIC NUMBER
H n
n
^I= 7 + Vo(n+l),
(4)
j=i
where 7 is the Euler-Mascheroni CONSTANT and
$(x) = ip (x) is the Digamma Function. The number
of terms needed to exceed 1, 2, 3, ... are 1, 4, 11, 31,
83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397,
. . . (Sloane's A004080). Using the analytic form shows
that after 2.5 x 10 8 terms, the sum is still less than 20.
Furthermore, to achieve a sum greater than 100, more
than 1.509 x 10 43 terms are needed!
Progressions of the form
1
1
a\ 01+d ai+ Id
(5)
are also sometimes called harmonic series (Beyer 1987).
The modified harmonic series, given by the sum
00
•^ Pk
(6)
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 279-280, 1985.
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.
Boas, R. P. and Wrench, J. W. "Partial Sums of the Harmonic
Series." Amer. Math. Monthly 78, 864-870, 1971.
Honsberger, R. "An Intriguing Series." Ch. 10 in Mathe-
matical Gems II. Washington, DC: Math. Assoc. Amer.,
pp. 98-103, 1976.
Sloane, N. J. A. Sequence A004080 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Harmonic System of Points
see Harmonic Range
Harmonious Graph
A connected LABELLED GRAPH with n EDGES in which
all Vertices can be labeled with distinct Integers
(mod n) so that the sums of the PAIRS of numbers at the
ends of each Edge are also distinct (mod n). The LAD-
DER Graph, Fan, Wheel Graph, Petersen Graph,
Tetrahedral Graph, Dodecahedral Graph, and
ICOSAHEDRAL GRAPH are all harmonious (Graham and
Sloane 1980).
see also GRACEFUL GRAPH, LABELLED GRAPH,
Postage Stamp Problem, Sequential Graph
References
Gallian, J. A. "Open Problems in Grid Labeling." Amer.
Math. Monthly 97, 133-135, 1990.
Gardner, M. Wheels, Life, and other Mathematical Amuse-
ments. New York: W. H. Freeman, p. 164, 1983.
Graham, R. L. and Sloane, N. "On Additive Bases and Har-
monious Graphs." SIAM J. Algebraic Discrete Math. 1,
382-404, 1980.
Guy, R. K. "The Corresponding Modular Covering Problem.
Harmonious Labelling of Graphs." §C13 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 127-128, 1994.
Harnack's Inequality
Let D = D(zq,R) be an Open Disk, and let u be a
Harmonic Function on D such that u(z) > for all
z € D. Then for all z 6 D, we have
see also LlOUVILLE'S CONFORMALITY THEOREM
References
Flanigan, F.
Variables:
J. "Harnack's Inequality." §2.5.1 in Complex
Harmonic and Analytic Functions. New York:
Dover, pp. 88-90, 1983.
where pk is the fcth Prime, diverges.
see also Arithmetic Series, Bernoulli's Paradox,
Book Stacking Problem, Euler Sum, Zipf's Law
Harnack's Theorems
Hartley Transform 801
Harnack's Theorems
Harnack's first theorem states that a real irreducible
curve of order n cannot have more than
|(n-l)(n-2)-^5i( 5i -l) + l
circuits (Coolidge 1959, p. 57).
Harnack's second theorem states that there exists a
curve of every order with the maximum number of cir-
cuits compatible with that order and with a certain num-
ber of double points, provided that number is not per-
missible for a curve of lower order (Coolidge 1959, p. 61).
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, 1959.
Harshad Number
A Positive Integer which is Divisible by the sum of
its Digits, also called a Niven Number (Kennedy et
al. 1980). The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
12, 18, 20, 21, 24, ... (Sloane's A005349). Grundman
(1994) proved that there is no sequence of more than
20 consecutive Harshad numbers, and found the small-
est sequence of 20 consecutive Harshad numbers, each
member of which has 44,363,342,786 digits.
Grundman (1994) defined an n-Harshad (or n-Niven)
number to be a Positive Integer which is Divisible
by the sum of its digits in base n > 2. Cai (1996) showed
that for n = 2 or 3, there exists an infinite family of
sequences of consecutive n-Harshad numbers of length
2n.
References
Cai, T. "On 2-Niven Numbers and 3-Niven Numbers." Fib.
Quart 34, 118-120, 1996.
Cooper, C. N. and Kennedy, R. E. "Chebyshev's Inequality
and Natural Density." Amer. Math. Monthly 96, 118-124,
1989.
Cooper, C. N. and Kennedy, R. "On Consecutive Niven Num-
bers." Fib. Quart. 21, 146-151, 1993.
Grundman, H. G. "Sequences of Consecutive n-Niven Num-
bers." Fib. Quart. 32, 174-175, 1994.
Kennedy, R.; Goodman, R.; and Best, C. "Mathematical Dis-
covery and Niven Numbers." MATYC J. 14, 21-25, 1980.
Sloane, N. J. A. Sequence A005349/M0481 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Vardi, L "Niven Numbers." §2.3 in Computational Recre-
ations in Mathematica. Redwood City, CA: Addison-
Wesley, pp. 19 and 28-31, 1991.
Hart's Inversor
A linkage which draws the inverse of a given curve. It
can also convert circular to linear motion. The rods
satisfy AB = CD and EC = DA, and O, P, and P'
remain Collinear. Coxeter (1969, p. 428) shows that
if AO = fxAB, then
OP x OP' = /x(l - fi)(AD 2 - AB 2 ).
see also PEAUCELLIER INVERSOR
References
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods. Oxford, Eng-
land: Oxford University Press, p. 157, 1978.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 82-83, 1969.
Rademacher, H, and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 124-129,
1957.
Hart's Theorem
Any one of the eight APOLLONIUS CIRCLES of three
given Circles is Tangent to a Circle C, as are the
other three APOLLONIUS CIRCLES having (1) like con-
tact with two of the given CIRCLES and (2) unlike con-
tact with the third,
see also APOLLONIUS CIRCLES
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 127-128, 1929.
Hartley Transform
An Integral Transform which shares some features
with the Fourier Transform, but which (in the dis-
crete case), multiplies the KERNEL by
cos \-ir)- sm \nr)
(i)
instead of
.-Mkn/N = _ (^n\ _ ,.;„ f^H\
V N
j-isin^— J. (2)
The Hartley transform produces Real output for a
Real input, and is its own inverse. It therefore can have
computational advantages over the DISCRETE FOURIER
Transform, although analytic expressions are usually
more complicated for the Hartley transform.
The discrete version of the Hartley transform can be
written explicitly as
N-l
| COS
= R.F[a]-9tf r [a],
iV — 1
(2nkn\ . /2irkn
(4)
where T denotes the FOURIER TRANSFORM. The Hart-
ley transform obeys the CONVOLUTION property
U[a * b] k = \{A k B k - A k B k + A k B k + A k B k ), (5)
802 Hartley Transform
where
ao = ao
a n / 2 = a n / 2
a>k = Cin-k
(6)
(7)
(8)
(Arndt). Like the Fast Fourier Transform, there is
a "fast" version of the Hartley transform. A decimation
in time algorithm makes use of
ttifM = n n/2 [a even ] + XH n/2 [a° dd ] (9)
K ight W - -Hn/ 2 [a even ] - XH n/2 [a odd ], (10)
where X denotes the sequence with elements
a n cos (^ J - d n sin (^ J . (11)
A decimation in frequency algorithm makes use of
n /even r l n i r left . nehti
n n [a] =n n/2 [a + a J,
(12)
H° n dd [a] = U n/2 [X(a lett - o right )]. (13)
The Discrete Fourier Transform
JV-l
A h = T[a\ = J2 e- 2 " ikn/N a n
(14)
n=0
can be written
N-l
A k
A- k
£
n=0
-2nikn/N
2-Kikn/N
(15)
-Ei
n=0 ^_
1 -i 1+i
l+i 1-i
cos(^) sin(^)
-sin(2=Sn) cos (3=^)
T" 1
H
l+i 1-i
1-i l+i
so
F = T _1 HT.
, (16)
(17)
see also Discrete Fourier Transform, Fast Four-
ier Transform, Fourier Transform
References
Arndt, J. "The Hartley Transform (HT)." Ch. 2 in "Remarks
on FFT Algorithms." http://www.jjj.de/fxt/.
Bracewell, R. N. The Fourier Transform and Its Applica-
tions. New York: McGraw-Hill, 1965.
Bracewell, R. N. The Hartley Transform. New York: Oxford
University Press, 1986.
Hasse Diagram
HashLife
A Life ALGORITHM that achieves remarkable speed by-
storing subpatterns in a hash table, and using them to
skip forward, sometimes thousands of generations at a
time. HashLife takes tremendous amounts of memory
and can't show patterns at every step, but can quickly
calculate the outcome of a pattern that takes millions of
generations to complete.
References
Hensel, A. "A Brief Illustrated Glossary of Terms in Con-
way's Game of Life." http://www.cs.jhu.edu/-callahan/
glossary.html.
Hasse's Algorithm
see COLLATZ PROBLEM
Hasse's Conjecture
Define the Zeta Function of a Variety over a Num-
ber FIELD by taking the product over all PRIME IDEALS
of the Zeta Functions of this Variety reduced mod-
ulo the PRIMES. Hasse conjectured that this product
has a MEROMORPHIC continuation over the whole plane
and a functional equation.
References
Lang, S. "Some History of the Shimura-Taniyama Conjec-
ture." Not. Amer. Math. Soc. 42, 1301-1307, 1995.
Hasse-Davenport Relation
Let F be a FINITE FIELD with q elements, and let F s
be a Field containing F such that [F s : F] — s. Let %
be a nontrivial MULTIPLICATIVE CHARACTER of F and
x' — X ° Np 3 / F a character of F s . Then
{-g{x)Y = -g(x),
where g(x) is a GAUSSIAN SUM.
see also Gaussian Sum, Multiplicative Character
References
Ireland, K. and Rosen, M. "A Proof of the Hasse-Davenport
Relation." §11.4 in A Classical Introduction to Modern
Number Theory, 2nd ed. New York: Sp ringer- Verlag,
pp. 162-165, 1990.
Hasse Diagram
A graphical rendering of a PARTIALLY ORDERED Set
displayed via the Cover relation of the Partially Or-
dered Set with an implied upward orientation. A point
is drawn for each element of the POSET, and line seg-
ments are drawn between these points according to the
following two rules:
1. If x < y in the poset, then the point corresponding
to x appears lower in the drawing than the point
corresponding to y.
2. The line segment between the points corresponding
to any two elements x and y of the poset is included
in the drawing IFF # covers y or y covers x.
Hasse diagrams are also called Upward Drawings.
Hasse-Minkowski Theorem
HausdorfF Measure 803
Hasse-Minkowski Theorem
Two nonsingular forms are equivalent over the rationals
Iff they have the same Determinant and the same
p-SlGNATURES for all p.
Hasse Principle
A collection of equations satisfies the Hasse principle if,
whenever one of the equations has solutions in R and
all the Q , then the equations have solutions in the RA-
TIONALS Q. Examples include the set of equations
ax 2 + bxy + cy 2
with a, 6, and c INTEGERS, and the set of equations
2 . 2
x + y = a
for a rational. The trivial solution x = y = is usu-
ally not taken into account when deciding if a collec-
tion of homogeneous equations satisfies the Hasse princi-
ple. The Hasse principle is sometimes called the LOCAL-
Global Principle.
see also LOCAL FIELD
Hasse's Resolution Modulus Theorem
The Jacobi Symbol (a/y) = x(y) as a Character can
be extended to the KRONECKER SYMBOL (f(a)/y) =
X*(y) so that x*(y) = x(y) whenever %{y) 7^ 0. When
y is Relatively Prime to /(a), then x*{v) ^ °i
and for NONZERO values x*(yi) = X*(yi) I pF 2/i —
y% mod + /(a)- In addition, \f(a)\ is the minimum value
for which the latter congruence property holds in any
extension symbol for x{v)-
see also Character (Number Theory), Jacobi Sym-
bol, Kronecker Symbol
References
Cohn, H. Advanced Number Theory. New York: Dover,
pp. 35-36, 1980.
Hat
The hat is a caret-shaped symbol most commonly used
to denote a Unit Vector (v) or an Estimator (x).
see also ESTIMATOR, UNIT VECTOR
Haupt-Exponent
The smallest exponent e for which b e = 1 (mod p),
where b and p are given numbers, is the haupt-
exponent of b (mod p). The number of bases having
a haupt-exponent e is <£(e), where <j){e) is the TOTIENT
FUNCTION. Cunningham (1922) published the haupt-
exponents for primes to 25409 and bases 2, 3, 5, 6, 7,
10, 11, and 12.
see also Complete Residue System, Residue Index
References
Cunningham, A. Haupt- Exponents, Residue Indices, Primi-
tive Roots. London: F. Hodgson, 1922.
HausdorfF Axioms
Describe subsets of elements a: in a NEIGHBORHOOD Set
E of x. The Neighborhood is assumed to satisfy:
1. There corresponds to each point x at least one
Neighborhood U(x), and each Neighborhood
U(x) contains the point x.
2. If U(x) and V(x) are two NEIGHBORHOODS of the
same point x, there must exist a NEIGHBORHOOD
W(x) that is a subset of both.
3. If the point y lies in U(x), there must exist a NEIGH-
BORHOOD U(y) that is a SUBSET of U(x).
4. For two different points x and y, there are two corre-
sponding Neighborhoods U(x) and U(y) with no
points in common.
Hausdorff-Besicovitch Dimension
see Capacity Dimension
HausdorfF Dimension
Let A be a Subset of a Metric Space X. Then the
Hausdorff dimension D(A) of A is the INFIMUM of d >
such that the d-dimensional HAUSDORFF MEASURE of
A is 0. Note that this need not be an Integer.
In many cases, the Hausdorff dimension correctly de-
scribes 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).
see also Capacity Dimension, Fractal Dimension,
Minkowski-Bouligand Dimension
References
Federer, H. Geometric Measure Theory. New York:
Springer- Verlag, 1969.
Hausdorff, F. "Dimension und aufieres Mafi." Math. Ann.
79, 157-179, 1919.
Ott, E. "Appendix: Hausdorff Dimension." Chaos in Dy-
namical Systems. New York: Cambridge University Press,
pp. 100-103, 1993.
Schroeder, M. Fractals, Chaos, Power Laws: Minutes from
an Infinite Paradise. New York: W. H. Freeman, pp. 41-
45, 1991.
HausdorfF Measure
Let X be a Metric Space, A be a Subset of X, and d
a number > 0. The d-dimensional Hausdorff measure of
A, H d (A), is the Infimum of Positive numbers y such
that for every r > 0, A can be covered by a countable
family of closed sets, each of diameter less than r, such
that the sum of the dth POWERS of their diameters is
less than y. Note that H d (A) may be infinite, and d
need not be an Integer.
References
Federer, H. Geometric Measure Theory. New York:
Springer- Verlag, 1969.
Ott, E. Chaos in Dynamical Systems. Cambridge, England:
Cambridge University Press, p. 103, 1993.
804
Hausdorff Paradox
Heat Conduction Equation
Hausdorff Paradox
For n > 3, there exist no additive finite and invariant
measures for the group of displacements in M n .
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 49, 1983.
Hausdorff Space
A Topological Space in which any two points have
disjoint NEIGHBORHOODS.
Haversine
hav(^) ~ \ vers(z) = |(1 — cosz),
where vers(z) is the VERSINE and cos is the COSINE.
Using a trigonometric identity, the haversine is equal to
hav(z)
: sin {\z).
see also COSINE, COVERSINE, EXSECANT, VERSINE
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 78, 1972.
Heads Minus Tails Distribution
A fair COIN is tossed 2n times. Let D = \H - T\ be
the absolute difference in the number of heads and tails
obtained. Then the probability distribution is given by
P(D = 2k)
rami
l 2 (j) an (n + V
) * = 1,2,...,
where P(D = 2k — 1) = 0. The most probable value of
D is D = 2, and the expectation value is
(D)
2 2n-l '
see also BERNOULLI DISTRIBUTION, COIN, COIN TOSS-
ING
References
Handelsman, M. B. Solution to Problem 436, "Distribut-
ing 'Heads' Minus 'Tails.'" College Math. J. 22, 444-446,
1991.
Heap
A SET of TV members forms a heap if it satisfies d[j/2\ >
aj for 1 < [j/2\ < j < iV, where [x\ is the FLOOR
Function,
see also HEAPSORT
Heapsort
An N\gN Sorting Algorithm which is not quite as
fast as QUICKSORT. It is a "sort-in-place" algorithm
and requires no auxiliary storage, which makes it par-
ticularly concise and elegant to implement.
see also QUICKSORT, SORTING
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Heapsort." §8.3 in Numerical Recipes
in FORTRAN: The Art of Scientific Computing, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 327-329, 1992.
Heart Surface
A heart-shaped surface given by the Sextic Equation
/o 2 . o 2 . 2 -,\3 12 3 2 3 ^
(2x +2y +z -1) - yqX z -yz = 0.
see also BONNE PROJECTION, PIRIFORM
http : //www . uib . no/people/
References
Nordstrand, T. "Heart
nf ytn/hearttxt .htm.
Heat Conduction Equation
A diffusion equation of the form
£-«r-
(i)
Physically, the equation commonly arises in situations
where k is the thermal diffusivity and T the tempera-
ture.
The 1-D heat conduction equation is
dT
d 2 T
(2)
dt dx 2
This can be solved by SEPARATION OF VARIABLES using
T(x,t) = X(x)T(t). (3)
Then
dt dx 2
(4)
Heat Conduction Equation
Dividing both sides by kXT gives
J_dT _ J^c^X _
kT dt ~ X dx 2
1_
(5)
where each side must be equal to a constant. Antic-
ipating the exponential solution in T, we have picked
a negative separation constant so that the solution re-
mains finite at all times and A has units of length. The
T solution is
T(t)^Ae~ Kt/x \
and the X solution is
X(x) = Ccos (J~\ + £>sin (j) .
The general solution is then
T(x,t) =T(t)X(x)
= Ae- Kt ' x2 [Ccos(|)+2?sin(|)"
= .-/*' [D«e) + E-ng)
(6)
(7)
(8)
If we are given the boundary conditions
T(0,t) = (9)
and
T(L,t) = 0,
then applying (9) to (8) gives
(10)
Esin [ — ) =0=^-=n7T^X= — , (12)
Dcos(^) = 0=> D = Q,
and applying (10) to (8) gives
sin (x) =0 ^ =
so (8) becomes
T„(z,t) = Ke-^^Sin (^) . (13)
Since the general solution can have any n,
T(x,t) = Y / c n sm( r ^)e-^^ 2t . (14)
n = l
Now, if we are given an initial condition T(x, 0), we have
oo
T(x,0) = ^c n sin(^). (15)
Heat Conduction Equation — Disk 805
Multiplying both sides by sin(m7r;r/L) and integrating
from to L gives
/ sin f — — J T(x y 0)dx
= / y^ j c n sin ( — — j sin ( — — j dx. (16)
**° n=l
Using the ORTHOGONALITY of sin(na;) and sin(mx),
oo r L oo
El . / r mrx\ . frmrxX _ v~^ i <•
c n I sin f — — j sin I — — ) dx = 2_^ pd mn Cn
n = l ^° n = l
= I^ Cm = | sin (^) T(x, 0) dx, (17)
c n = ^ j sin(^)T(x,0)dx. (18)
If the boundary conditions are replaced by the require-
ment that the derivative of the temperature be zero at
the edges, then (9) and (10) are replaced by
dx
ar
dx
(0,t)
(L,t)
0.
(19)
(20)
Following the same procedure as before, a similar answer
is found, but with sine replaced by cosine:
( U ) 7W) = X>cos(^)e-«
(mr/L) 2 t
(21)
where
2 / / 7717TX \
dT(x,Q)
dx
dx. (22)
Heat Conduction Equation — Disk
To solve the Heat CONDUCTION Equation on a 2-D
disk of radius R = 1, try to separate the equation using
T(r,9,t) = R(r)e(e)T(t).
(1)
Writing the 9 and r terms of the Laplacian in SPHER-
ICAL Coordinates gives
2 _ <?R 2dR 1 d 2
dr 2 r dr r 2 d9 2
so the Heat Conduction Equation becomes
(2)
ROd*T d>R QT+ 2_d_R GT+ ^<?e RT (3)
k dt 2 dr 2
r dr
d0 2
806 Heaviside Calculus
Multiplying through by r 2 /RQT gives
kT dt 2 ~ R dr 2 + R dr + dO 2 6 '
The term can be separated.
d 2 e i
d# 2 e
-n(n+ 1),
(4)
(5)
which has a solution
G(0) = A cos 1^(71+ 1)0 + 5sin y^n + 1) 6
The remaining portion becomes
f_d*T _r^d 2 R 2rdR__
kT dt 2 ~ Rdr 2+ Rdr n ^ n+1 ^
Dividing by r 2 gives
1 d 2 T _ 1 d 2 R 2 d.R n(n + 1) _
(6)
(7)
kT dt 2 R dr 2 rR dr
A^
. (8)
where a NEGATIVE separation constant has been chosen
so that the t portion remains finite
T{t)=Ce~ Kt/x \
The radial portion then becomes
1 d 2 R 2 dR n(n + l) 1
R dr 2 + rR dr
+ *=°
d 2 R
n dR
r
+ 2r —
+
dr 2
dr
— -n(n + 1)
R = 0,
(9)
(10)
(11)
which is the SPHERICAL BESSEL DIFFERENTIAL EQUA-
TION. If the initial temperature is T(r, 0) = and the
boundary condition is T(l,t) = 1, the solution is
*-^ a n J 1 (a n )
(12)
where a n is the nth Positive zero of the Bessel Func-
tion of the First Kind J .
Heaviside Calculus
A method of solving differential equations using Four-
ier Transforms and Laplace Transforms.
Heaviside Step Function
Heaviside Step Function
-1 -0.5 0.5 1
A discontinuous "step" function, also called the Unit
Step, and defined by
{0 x <
1 x > 0.
(1)
It is related to the Boxcar Function. The Deriva-
tive is given by
■^H(x) = 8(x),
(2)
where S(x) is the Delta Function, and the step func-
tion is related to the Ramp FUNCTION R(x) by
4~R(x) = -H{x).
dx
(3)
Bracewell (1965) gives many identities, some of which
include the following. Letting * denote the Convolu-
tion,
H(x)*f(x)= / f{x)dx l
(4)
/oo
H(u)H(T - u) du (5)
■oo
/*oo
= H(0) / H(T - u) du
Jo
= H(0)H(T) J du = TH{T). (6)
Jo
Additional identities are
*W»M-{ZjJ) HI <"
H(ax + b) = H (x+-\ H(a) + H (-x - -) H(-a)
-\H(-x-±) a<0. (8)
see also FOURIER TRANSFORM, LAPLACE TRANSFORM The step function obeyg the integra] identities
/b pb
H(u-u )f(u)du = H{u ) f{u)du (9)
-a J un
f
J —a
H(ui — u)f{u) du = H{u\)
■o
)du (10)
Heawood Conjecture
/H(u - u )H(ui - u)f(u) du
-a
= H{uo)H( Ul ) f * f(u)du. (11)
J UQ
The Heaviside step function can be defined by the fol-
lowing limits,
= \ limerfcf-f)
= _Llim / t^e'^^du
= lim / t' 1 sine f - 1 du
.-m\-I(l-e-"
= -o/V lA (^)^
a; >0
) x <
(12)
(13)
(14)
(15)
(16)
(17)
(18)
where A is the one-argument TRIANGLE FUNCTION and
Si(x) is the SINE INTEGRAL.
The Fourier Transform of the Heaviside step func-
tion is given by
J — o
— 2-nikx
H(x) dx ■
1
2 L
'M-S.
(19)
where 5(k) is the DELTA FUNCTION.
see also Boxcar Function, Delta Function, Four-
ier Transform— Heaviside Step Function, Ramp
Function, Ramp Function, Rectangle Function,
Square Wave
References
Br ace well, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, 1965.
Spanier, J. and Oldham, K. B. "The Unit-Step u(x — a)
and Related Functions." Ch. 8 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 63-69, 1987.
Heawood Conjecture
The bound for the number of colors which are SUFFI-
CIENT for Map Coloring on a surface of Genus #,
x(9)=llC?+V**9+i:
is the best possible, where [zj is the Floor Function.
x(g) is called the Chromatic Number, and the first
few values for g = 0, 1, . . . are 4, 7, 8, 9, 10, 11, 12, 12,
13, 13, 14, . . . (Sloane's A000934).
Hedgehog 807
The fact that x(#) is ^ so NECESSARY was proved by
Ringel and Youngs (1968) with two exceptions: the
Sphere (Plane), and the Klein Bottle (for which
the Heawood Formula gives seven, but the correct
bound is six). When the Four-Color THEOREM was
proved in 1976, the Klein Bottle was left as the only
exception. The four most difficult cases to prove were
g = 59, 83, 158, and 257.
see also Chromatic Number, Four-Color Theo-
rem, Map Coloring, Six-Color Theorem, Torus
Coloring
References
Ringel, G. Map Color Theorem. New York: Springer- Verlag,
1974.
Ringel, G. and Youngs, J. W. T. "Solution of the Heawood
Map-Coloring Problem." Proc. Nat Acad. Sci. USA 60,
438-445, 1968.
Sloane, N. J. A. Sequence A000934/M3292 in "An On-Line
Version of the Encyclopedia of Integer Sequences,"
Wagon, S. "Map Coloring on a Torus." §7.5 in Mathematica
in Action. New York: W. H. Freeman, pp. 232-237, 1991.
Hebesphenomegacorona
see Johnson Solid
Hecke Algebra
An associative Ring, also called a Hecke Ring, which
has a technical definition in terms of commensurable
Subgroups.
Hecke L- Function
A generalization of the EULER L-Function associated
with a GROSSENCHARACTER.
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Hecke Operator
A family of operators on each Space of Modular
Forms. Hecke operators Commute with each other.
Hecke Ring
see Hecke Algebra
Hectogon
A 100-sided POLYGON.
Hedgehog
An envelope parameterized by its GAUSS Map. The
parametric equations for a hedgehog are
x = p(6) cos S + p(6) sin
y = p(9)sm8 + p{9)cos9.
A plane convex hedgehog has at least four VERTICES
where the Curvature has a stationary value. A plane
808 Heegaard Diagram
Height
convex hedgehog of constant width has at least six VER-
TICES (Martinez-Maure 1996).
References
Langevin, R.; Levitt, G.; and Rosenberg, H. "Herissons et
Multiherissons (Enveloppes parametrees par leu applica-
tion de Gauss." Warsaw: Singularities, 245-253, 1985.
Banach Center Pub. 20, PWN Warsaw, 1988.
Martinez-Maure, Y. "A Note on the Tennis Ball Theorem."
Amer. Math. Monthly 103, 338-340, 1996.
Heegaard Diagram
A diagram expressing how the gluing operation that
connects the HANDLEBODIES involved in a HEEGAARD
Splitting proceeds, usually by showing how the merid-
ians of the Handlebody are mapped.
see also Handlebody, Heegaard Splitting
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 239, 1976.
Heegaard Splitting
A Heegaard splitting of a connected orientable 3-
MANIFOLD M is any way of expressing M as the
Union of two (3,1)-Handlebodies along their bound-
aries. The boundary of such a (3,l)-HANDLEBODY is an
orientable SURFACE of some GENUS, which determines
the number of HANDLES in the (3,l)-HANDLEBODlES.
Therefore, the HANDLEBODIES involved in a Heegaard
splitting are the same, but they may be glued together
in a strange way along their boundary. A diagram show-
ing how the gluing is done is known as a Heegaard
Diagram.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, p. 255, 1994.
Heegner Number
The values of -d for which Quadratic Fields
Q(y/—d ) are uniquely factorable into factors of the form
a + by/^d. Here, a and b are half-integers, except
for d = 1 and 2, in which case they are Integers.
The Heegner numbers therefore correspond to DISCRIM-
INANTS -d which have CLASS Number h(-d) equal to
1, except for Heegner numbers —1 and — 2, which corre-
spond to d = —4 and —8, respectively.
The determination of these numbers is called GAUSS'S
Class Number Problem, and it is now known that
there are only nine Heegner numbers: —1, —2, -3, -7,
-11, -19, -43, -67, and -163 (Sloane's A003173), cor-
responding to discriminants -4, -8, -3, —7, -11, -19,
—43, —67, and —163, respectively.
Heilbronn and Linfoot (1934) showed that if a larger d
existed, it must be > 10 9 . Heegner (1952) published a
proof that only nine such numbers exist, but his proof
was not accepted as complete at the time. Subsequent
examination of Heegner's proof show it to be "essen-
tially" correct (Conway and Guy 1996).
The Heegner numbers have a number of fascinating
connections with amazing results in PRIME NUMBER
theory. In particular, the ^-FUNCTION provides stun-
ning connections between e, 7T, and the ALGEBRAIC
Integers. They also explain why Euler's Prime-
Generating Polynomial n 2 -n+41 is so surprisingly
good at producing PRIMES.
see also Class Number, Discriminant (Binary
Quadratic Form), Gauss's Class Number Prob-
lem, j-Function, Prime-Generating Polynomial,
Quadratic Field
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.
Heegner, K. "Diophantische Analysis und Modulfunktionen."
Math. Z. 56, 227-253, 1952.
Heilbronn, H. A. and Linfoot, E. H. "On the Imaginary Quad-
ratic Corpora of Class-Number One." Quart J. Math.
(Oxford) 5, 293-301, 1934.
Sloane, N. J. A. Sequence A003173/M0827 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Heesch Number
The Heesch number of a closed plane figure is the max-
imum number of times that figure can be completely
surrounded by copies of itself. The determination of the
maximum possible (finite) Heesch number is known as
Heesch's Problem. The Heesch number of a Trian-
gle, Quadrilateral, regular Hexagon, or any other
shape that can TlLE or TESSELLATE the plane, is in-
finity. Conversely, any shape with infinite Heesch num-
ber must tile the plane (Eppstein). The largest known
(finite) Heesch number is 3, and corresponds to a tile
invented by R. Ammann (Senechal 1995).
References
Eppstein, D. "Heesch's Problem." http://www.ics.uci.
edu/~eppstein/ junky ard/heesch/.
Fontaine, A. "An Infinite Number of Plane Figures with
Heesch Number Two." J. Comb. Th. A 57, 151-156, 1991.
Senechal, M. Quasicrystals and Geometry. New York: Cam-
bridge University Press, 1995.
Heesch's Problem
How many times can a shape be completely surrounded
by copies of itself without being able to TlLE the en-
tire plane, i.e., what is the maximum (finite) HEESCH
Number?
References
Eppstein, D. "Heesch's Problem." http://www.ics.uci.
edu/-eppstein/junkyard/heesch/.
Height
The vertical length of an object from top to bottom.
see also LENGTH (SlZE), WIDTH (SlZE)
Heilbronn Triangle Problem
Heine Hypergeometric Series 809
Heilbronn Triangle Problem
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Given any arrangement of n points within a UNIT
SQUARE, let H n be the smallest value for which there is
at least one TRIANGLE formed from three of the points
with AREA < H n . The first few values are
Heine-Borel Theorem
If a Closed Set of points on a line can be covered by a
set of intervals so that every point of the set is an interior
point of at least one of the intervals, then there exist a
finite number of intervals with the covering property.
Heine Hypergeometric Series
H 3 = \
H 4 = \
H 8 > \(2-y/3)
Hn > ±
H x2 > ^3
His > 0.030
#14 > 0.022
#15 > 0.020
#i 6 > 0.0175.
Komlos et al. (1981, 1982) have shown that there are
constants c such that
clnn
n<
<#n<
,8/7
for any e > and all sufficiently large n.
Using an Equilateral Triangle of unit Area instead
gives the constants
h 3 = 1
/i 5 = 3 - 2\/2
h 6 = |.
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/hlb/hlb.html.
Goldberg, M. "Maximizing the Smallest Triangle Made by N
Points in a Square." Math. Mag. 45, 135-144, 1972.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer-Verlag, pp. 242-244, 1994.
Komlos, J.; Pintz, J.; and Szemeredi, E. "On Heilbronn's
Triangle Problem." J. London Math. Soc. 24, 385-396,
1981.
Komlos, J.; Pintz, J.; and Szemeredi, E. "A Lower Bound for
Heilbronn's Triangle Problem." J. London Math. Soc. 25,
13-24, 1982.
Roth, K. F. "Developments in Heilbronn's Triangle Prob-
lem." Adv. Math. 22, 364-385, 1976.
ai,a 2 , . . . ,o; r
0u-.., P.
\z
E
{0Ll\q)n{ci2\q)n • •• {0Cr\q)n
(q',q) n ((3i\q) n "-{0 s ]q) n
*", (1)
where
(a;q) n = (1 - a)(l - aq)(l - aq 2 ) • • • (1 - aq n 1 ),(2)
(a;q) = l. (3)
In particular,
2ipi(a,b\c;q 1 z) = }]
{a;q)n(b\q)nz n
(q\q)n(c;q) n
(4)
(Andrews 1986, p. 10). Heine proved the transformation
formula
2 0i(a,6;c;g,z) = — — 2 0i(c/M; az\ q,b),
(c]q)oc{z\q)oo
(5)
and Rogers (1893) obtained the formulas
20i (a, b; c;q,z)
_ (c/b',q)oo(bz',q) c
-201 (6, abz/c\ bz\ q, c/b) (6)
(z;q)oo(c;q) e
20i (a, 6, c; q, z)
= (abz/c; q)oo(z\ q)oo2<f>i(c/a, c/b; c; q, abz/c) (7)
(Andrews 1986, pp. 10-11).
see also ^-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, p. 10, 1986.
Heine, E. "Uber die Reihe 1 + ^"'V^'V x
+ (ff a -l)(q tt + 1 -l)(g J -l)(q g + 1 -l) 2 , » J reine anQew
+ (g_i)(q2_ 1)(q7 _ 1)(g7 +i_ 1) x -t-.... J. retne angew.
Math. 32, 210-212, 1846.
Heine, E. "Untersuchungen uber die Reihe 1 + *V~ q )^}~\} •
, (l~q a )(l-q a + 1 )(l-g (3 )(l-q (3 + 1 ) 2 , " T
Math. 34, 285-328, 1847.
Heine, E. Theorie der Kugelfunctionen und der verwandten
Functionen, Vol. 1. Berlin: Reimer, 1878.
Rogers, L. J. "On a Three-Fold Symmetry in the Elements
of Heine's Series." Proc. London Math. Soc. 24, 171-179,
1893.
810 Heisenberg Group
Helicoid
Heisenberg Group
The Heisenberg group H n in n COMPLEX variables is
the GROUP of all (z y t) with z G C n and t € R having
multiplication
(w, t)(z, t') = {w + z, t + t' + 3[w T z])
where iu T is the conjugate transpose. The Heisenberg
group is Isomorphic to the group of Matrices
1 z
1
T §N 2 + «*
z
and satisfies
(*,t) _1 = (-*.-*)■
Every finite-dimensional unitary representation is trivial
on Z and therefore factors to a REPRESENTATION of the
quotient C n .
see also NIL GEOMETRY
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not Amer. Math. Soc. 43, 537-549, 1996.
Heisenberg Space
The boundary of COMPLEX HYPERBOLIC 2-SPACE.
see also HYPERBOLIC SPACE
Held Group
The Sporadic Group He.
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat.bham.ac.uk/atlas/He.html.
Helen of Geometers
see Cycloid
Helicoid
The Minimal Surface having a Helix as its bound-
ary. It is the only Ruled Minimal Surface other than
the Plane (Catalan 1842, do Carmo 1986). For many
years, the helicoid remained the only known example of
a complete embedded Minimal Surface of finite topol-
ogy with infinite Curvature. However, in 1992 a sec-
ond example, known as Hoffman's Minimal Surface
and consisting of a helicoid with a HOLE, was discovered
(Sci. News 1992).
The equation of a helicoid in CYLINDRICAL COORDI-
NATES is
z = cO. (1)
In Cartesian Coordinates, it is
*—(!)• »
It can be given in parametric form by
x = u cos v
y — u sin v
Z = CUj
(3)
(4)
(5)
which has an obvious generalization to the ELLIPTIC
Helicoid. The differentials are
dx = cos v du — u sin v dv
dy — sin v du + u cos v dv
dz = leu dy,
so the Line Element on the surface is
ds = dx +dy + dz
(6)
(7)
(8)
= cos 2 v du — 2u sin v cos vdudv -\- u sin v dv
+ sin 2 v du 2 + 2u sin v cos v du dv + u cos 2 v dv 2
+ 4c 2 u 2 du 2
= (l + 4c 2 u)du +udv 2 i
and the METRIC components are
(9)
g-u.ii = 1 4- 4c u
(10)
9uv =
(11)
g vv = u .
(12)
Prom Gauss's Theorema Egregium, the Gaussian
Curvature is then
K
4c 2
(13)
(l + 4c 2 n 2 ) 2 *
The Mean Curvature is
H = 0, (14)
and the equation for the LINES OF CURVATURE is
u = ±csinh(v — k). (15)
Helix
Helix
811
The helicoid can be continuously deformed into a
CATENOID by the transformation
makes a constant ANGLE with a fixed line. The helix is
a Space Curve with parametric equations
x(u, v) = cos a sinh v sin u -\- sin a cosh v cos u (16)
y(u, v) = — cos a sinh v cos u + sin a cosh v sin u (17)
z(u, v) = ticosa -f vsina, (18)
#
rcos£
y = r sin £
z = c£,
(i)
(2)
(3)
where a = corresponds to a helicoid and a = 7r/2 to
a Catenoid.
If a twisted curve C (i.e., one with TORSION r ^ 0)
rotates about a fixed axis A and, at the same time, is
displaced parallel to A such that the speed of displace-
ment is always proportional to the angular velocity of
rotation, then C generates a GENERALIZED HELICOID.
See also CALCULUS OF VARIATIONS, CATENOID, ELLIP-
TIC Helicoid, Generalized Helicoid, Helix, Hoff-
man's Minimal Surface, Minimal Surface
References
Catalan E. "Sur les surfaces regleess dont l'aire est un mini-
mum." J. Math. Pure Appl 7, 203-211, 1842.
do Carmo, M. P. "The Helicoid." §3.5B in Mathematical
Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 44-45, 1986.
Fischer, G. (Ed.). Plate 91 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 87, 1986.
Geometry Center. "The Helicoid." http://www.geom.umn.
edu/zoo/diffgeom/surf space/helicoid/.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 264, 1993.
Kreyszig, E. Differential Geometry. New York: Dover, p. 88,
1991.
Meusnier, J. B. "Memoire sur la courbure des surfaces."
Mem. des savans etrangers 10 (lu 1776), 477-510, 1785.
Peterson, I. "Three Bites in a Doughnut." Sci. News 127,
168, Mar. 16, 1985.
"Putting a Handle on a Minimal Helicoid." Sci. News 142,
276, Oct. 24, 1992.
Wolfram, S. The Mathematica Book, 3rd ed. Champaign, IL:
Wolfram Media, p. 164, 1996.
Helix
where c is a constant. The CURVATURE of the helix is
given by
« = ^T^' (4)
and the LOCUS of the centers of CURVATURE of a helix
is another helix. The ARC LENGTH is given by
(5)
s = y/ x 'i + y'2 + z n dt = ^r 2
+ c 2 t.
The TORSION of a helix is given by
1
— rsint —rcost
rsint
rcost —rsint
c
—r cost
T ~ r 2( r 2 +c 2)
C
r^ + c 2
r 2 +c 2
(6)
(7)
A helix is also called a Curve of Constant Slope.
It can be defined as a curve for which the TANGENT
which is a constant. In fact, Lancret's Theorem
states that a NECESSARY and SUFFICIENT condition for
a curve to be a helix is that the ratio of CURVATURE to
Torsion be constant. The Osculating Plane of the
helix is given by
z\ — r cos t 22 — t sin t z$ — ct
— r sin t r cos t c = (8)
—rcost —rsint
Z\c sin t — accost + (z$ — ct)r = 0. (9)
The Minimal Surface of a helix is a Helicoid.
see also Generalized Helix, Helicoid, Spherical
Helix
References
Geometry Center. "The Helix." http://www.geom.umn.edu/
zoo/dif fgeom/ surf space/helicoid/helix. html.
Gray, A. "The Helix and Its Generalizations." §7.5 in Mod-
ern Differential Geometry of Curves and Surfaces. Boca
Raton, FL: CRC Press, pp. 138-140, 1993.
Isenberg, C. Plate 4.11 in The Science of Soap Films and
Soap Bubbles. New York: Dover, 1992.
Pappas, T. "The Helix — Mathematics & Genetics." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 166-168, 1989.
Wolfram, S. The Mathematica Book, 3rd ed. Champaign, IL:
Wolfram Media, p. 163, 1996.
812 Helly Number
Helmholtz Differential Equation
Helly Number
Given a Euclidean n-space,
H n = n + 1.
see also EUCLIDEAN Space, Helly'S Theorem
Helly's Theorem
If F is a family of more than n bounded closed convex
sets in Euclidean n-space W 1 , and if every H n (where
H n is the Helly Number) members of F have at least
one point in common, then all the members of F have
at least one point in common.
see also Caratheodory's Fundamental Theorem,
Helly Number
Helmholtz Differential Equation
A Partial Differential Equation which can be
written in a SCALAR version
vV + fcV = o,
or Vector form,
V 2 A + fc 2 A:
0,
(1)
(2)
where V 2 is the LAPLACIAN. When k = 0, the
Helmholtz differential equation reduces to LAPLACE'S
EQUATION. When k 2 < 0, the equation becomes the
space part of the diffusion equation.
The Helmholtz differential equation can be solved by
Separation of Variables in only 11 coordinate sys-
tems, 10 of which (with the exception of CONFOCAL
Paraboloidal Coordinates) are particular cases of
the Confocal Ellipsoidal system: Cartesian, Con-
focal Ellipsoidal, Confocal Paraboloidal, Con-
ical, Cylindrical, Elliptic Cylindrical, Oblate
Spheroidal, Paraboloidal, Parabolic Cylindri-
cal, Prolate Spheroidal, and Spherical Coordi-
nates (Eisenhart 1934). Laplace's EQUATION (the
Helmholtz differential equation with k = 0) is separa-
ble in the two additional BlSPHERlCAL COORDINATES
and TOROIDAL COORDINATES.
If Helmholtz's equation is separable in a 3-D coordinate
system, then Morse and Feshbach (1953, pp. 509-510)
show that
/ll/l2/l3 J. / X / X
, 2 = fn{Un)9n{Ui,Uj),
fin
(3)
where i ^ j ^ n. The Laplacian is therefore of the
form
v2 = wbr{ pi(U2 ' U3) ^[ /l(Ul) ^r]
ft r
/2(«2)
+32(1*1, ^3)-^ —
0U2
d_
' du 2
+93(u u u 2 ) —
f3 ^]}'
which simplifies to
d
^2 1 d r, , N 8 1
+-
— 1
du 2 \
■hhik[ Mu3) ik]- (5)
Such a coordinate system obeys the Robertson Con-
dition, which means that the STACKEL DETERMINANT
is of the form
S =
hihzhz
fi(u 1 )f2(u 2 )h(u 3 )'
(6)
Coordinate System
Variables
Solution Functions
Cartesian
circular cylindrical
conical
ellipsoidal
elliptic cylindrical
oblate spheroidal
parabolic
parabolic cylindrical
paraboloidal
prolate spheroidal
spherical
X(x)Y(y)Z(z)
R(r)@(9)Z(z)
A{\)M(fi)N(v)
U(u)V(v)Z(z)
A(\)M(n)N(v)
U(u)V{v)B($)
A(\)M(ti)N(v)
R(r)S(6)${<f>)
exponential, circular,
hyperbolic
Bessel, exponential,
circular
ellipsoidal harmonics,
power
ellipsoidal harmonics
Mathieu, circular
Legendre, circular
Bessel, circular
Parabolic cylinder,
Bessel, circular
Baer functions, circular
Legendre, circular
Legendre, power,
circular
see also LAPLACE'S EQUATION, POISSON'S EQUATION,
Separation of Variables, Spherical Bessel Dif-
ferential Equation
References
Eisenhart, L. P. "Separable Systems in Euclidean 3-Space."
Physical Review 45, 427-428, 1934.
Eisenhart, L. P. "Separable Systems of Stackel." Ann. Math.
35, 284-305, 1934.
Eisenhart, L. P. "Potentials for Which Schroedinger Equa-
tions Are Separable." Phys. Rev. 74, 87-89, 1948.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 125-126 and 509-
510, 1953.
Helmholtz Differential Equation — Bipolar
Coordinates
In Bipolar Coordinates, the Helmholtz Differ-
ential Equation is not separable, but Laplace's
Equation is.
see also Laplace's Equation — Bipolar Coordi-
nates
Helmholtz Differential Equation — Cartesian
Coordinates
In 2-D Cartesian Coordinates, attempt Separa-
tion of Variables by writing
(4)
F{x,y) = X(x)Y(y),
(1)
Helmholtz Differential Equation
Helmholtz Differential Equation 813
then the Helmholtz Differential Equation be-
comes
(2)
gr + gx + fxr-o.
Dividing both sides by X Y gives
+ T7~r^+k = 0.
X dx 2 Y dy 2
(3)
This leads to the two coupled ordinary differential equa-
tions with a separation constant m 2 ,
1 d'X 2
m
X dx 2
Y dy 2 ~ ( + >'
(4)
(5)
where X and Y could be interchanged depending on the
boundary conditions. These have solutions
Ji. — Ji-mG -\- X? m e
(6)
(7)
Y = C m e i ^ m2+fc2 y + D m e- i ^+^ y
= E m sin(\/ro 2 + k 2 y) + F m cos( yjm? +k 2 y).
The general solution is then
oo
F(x,y) = ^2(A m e mx + B m e- mx )
771 = 1
x[E m sin(v / m 2 + k 2 y) + F m cos(\/m 2 + A; 2 y)]. (8)
In 3-D Cartesian Coordinates, attempt Separa-
tion of Variables by writing
F(x t y,z) = X{x)Y{y)Z(z),
(9)
then the Helmholtz DIFFERENTIAL Equation be-
comes
d ^ YZ+d ^ xz+d ^ XY+k2XY =°- w
Dividing both sides by XYZ gives
1 d 2 X 1 d 2 Y 1 d 2 Z
+
Y dy* + Z ~dz* + * " °'
X da; 2
This leads to the three coupled differential equations
ii)
i fix a
X dx 2
LiX.
Y dy 2
jS = -(* a + ^+«» a ).
(12)
(13)
(14)
where X, y, and Z could be permuted depending on
boundary conditions. The general solution is therefore
F(x,y,z)
oo oo
= Y, Y,( Aie '* + Bie- lx )(C m e my + D m e~ my )
1 = 1 m = l
x {E lrn e-^ k2+l2+7n2 z + F /m e^ fc2+/2+m2 *). (15)
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 501-502, 513-514
and 656, 1953.
Helmholtz Differential Equation — Circular
Cylindrical Coordinates
In Cylindrical Coordinates, the Scale Factors
are h T = 1, he = f, h z = 1 and the separation functions
are /i(r) = r, f 2 (0) = 1, fz{z) = 1, so the STACKEL DE-
TERMINANT is 1. Attempt SEPARATION OF VARIABLES
by writing
F(r,e,z)=R(r)@(0)Z{z), (1)
then the Helmholtz Differential Equation be-
dr 1 r dr r z dv z dz z
Now divide by RQZ,
r 2 d 2 R r_dR\ ^9^ ^_ r l_ tf
~R~dr^ + Rdr ) + d6 2 6 + dz 2 Z +
(2)
~ + * 2 = °, W
so the equation has been separated. Since the solution
must be periodic in G from the definition of the circular
cylindrical coordinate system, the solution to the second
part of (3) must have a Negative separation constant
d 2 & 1
d0 2 &
-(fc 2 +m 2 ),
which has a solution
0(0) = C m e-^ fca+m " + D m e^ k2+m2e .
Plugging (5) back into (3) gives
(4)
(5)
r 2 d 2 R r dR
R dr 2 + R dr ~
2 d 2 Z r 2
=
(«)
1 d 2 R 1 dR
R'dr 2 + rR ~dr "
m 2 d 2 Z 1
" r 2 + dz 2 Z
-0.
(7)
The solution to the second part of (7) must not be sinu-
soidal at ±00 for a physical solution, so the differential
equation has a Positive separation constant
d 2 Z 1
dz 2 Z
W
814 Helmholtz Differential Equation
and the solution is where
Z{z)=E n e- n *+F n e nx . (9)
Plugging (9) back into (7) and multiplying through by
R yields
d 2
dr 2
~ 2 r dr Y r 2 J
1 d 2 R 1 ldR
n 2 dr 2 (nr) n dr
d 2 R 1 dR
d(nr) 2 (nr) d(nr)
1-
1-
(nr) 2
m
(nr) 2
R =
R = 0.
(10)
(11)
(12)
This is the Bessel Differential Equation, which
has a solution
R(r) = AmnJm(nr) + B mn Y m (nr),
(13)
where J n (x) and Y n (x) are BESSEL FUNCTIONS OF THE
First and Second Kinds, respectively. The general
solution is therefore
F(r,6,z)
CO oo
= ^2 ^\ArnnJm,(nr) + BmnYm(nr)]
m=0 n=0
< {C m e- iVk2+m2 * + D m e^ k2+m2 e )(E n e- nz + F n e nz ).
(14)
Actually, the Helmholtz Differential Equation is
separable for general k of the form
k\r,e,z) = f(r) + ^ + h(z) + k' 2 . (15)
see also CYLINDRICAL COORDINATES, HELMHOLTZ DIF-
FERENTIAL Equation
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 514 and 656-657,
1953.
Helmholtz Differential Equation — Confocal
Ellipsoidal Coordinates
Using the NOTATION of Byerly (1959, pp. 252-253), LA-
PLACE'S Equation can be reduced to
V 2 F = ( ^-^)g + (A 2 -, 2 )S+(A 2 V)S = 0,
'da 2
d(3 2
d<y 2
(1)
Helmholtz Differential Equation
d\
J c V /(A2-62)(A2- C 2)
B = c
I
dfi
b vV - mW - *> 2 )
= F
(2)
(3)
Jo JW^'
y/{b 2 - V 2 ){c 2 - I/ 2 )
In terms of a, /?, and 7,
A = cdc ( a, - J
fi = 6nd ( j3.
v — 6sn
(4)
(5)
(6)
(7)
Equation (1) is not separable using a function of the
form
F = L(a)M(f3)N(y),
but it is if we let
_1^L
Lda 2
1 d 2 M
= J2 ckuk
M dp 2
1 d 2 N
N d 7 2
These give
ao = —60 = Co
a>2 = —62 = C2,
(8)
(9)
(10)
(ID
(12)
(13)
and all others terms vanish. Therefore (1) can be broken
up into the equations
-r-z = (a + a 2 X 2 )L
da A
d 2 M
df3 2
d?N
dy
-(ao + a,2ti )M
2 = (a + a 2 v 2 )N.
(14)
(15)
(16)
Helmholtz Differential Equation
Helmholtz Differential Equation 815
For future convenience, now write
a = -(& 2 + c 2 )p
a2 = m(m + 1),
(17)
(18)
then
d*L
dOL 2
d 2 M
dp 2
£N
dy
[m{m + 1)A 2 - (6 2 + c 2 )p]L = (19)
+ [m{m + 1)m 2 - (b 2 + c 2 )p\M = (20)
2 [m(m + 1> 2 - (6 2 + c 2 )p]N = 0. (21)
Now replace a, /3, and 7 to obtain
(A2 _ 62)(A2 _ c >)g + A(A ^ + A *_ c ^
-[m(m + 1)A 2 - (b 2 + c 2 )p]L - (22)
-[m(m + l)^ 2 - (& 2 + c 2 )p}M = (23)
— + ,(^ _ fe + I/ _ c) __
-[m(m + l)^ 2 - (b 2 + c 2 )p]A^ = 0. (24)
(i/ 2 - & 2 )(* 2 - c 2 )^f + K^ " b 2 + 1/ - O
Each of these is a Lame's Differential Equation,
whose solution is called an ELLIPSOIDAL HARMONIC.
Writing
L(A) = £*,(A)
M(A) = ££(/*)
JV(A) = ES,(*0
(25)
(26)
(27)
gives the solution to (1) as a product of Ellipsoidal
Harmonics E^x).
F = E^(X)Ef H (fi)EUu).
(28)
References
Arfken, G. "Confocal Ellipsoidal Coordinates (£1,^2, £3)-"
§2.15 in Mathematical Methods for Physicists, 2nd ed. Or-
lando, FL: Academic Press, pp. 117-118, 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, pp. 251-258, 1959.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 663, 1953.
Helmholtz Differential Equation — Confocal
Paraboloidal Coordinates
As shown by Morse and Feshbach (1953), the
Helmholtz Differential Equation is separable in
Confocal Paraboloidal Coordinates.
see also CONFOCAL PARABOLOIDAL COORDINATES
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 664, 1953.
Helmholtz Differential Equation — Conical
Coordinates
In Conical Coordinates, Laplace's Equation can
be written
d 2 V
da 2
df3 2
where
&V ., 2 2,d f, 2 dV\
J a VV~a 2 )(
VV-a 2 )(6 2 - M 2 )
dv
h \/{a 2 ~ v
(Byerly 1959). Letting
V = U(u)R(r)
breaks (1) into the two equations,
>dR^
dr
( r ^) =m(m+1) *
g + +m ( m+1)(M >_^ = O.
Solving these gives
R(r)
Ar m + Br-
Uiu) = EMEUv),
0, (1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
where E^ are ELLIPSOIDAL HARMONICS. The regular
solution is therefore
V = Ar m EUn)E^(u).
0)
However, because of the cylindrical symmetry, the so-
lution Elrn{v)E™{v) is an mth degree Spherical Har-
monic
References
Arflcen, G. "Conical Coordinates (£i,£ 2 ,&)-" §2.16 in Math-
ematical Methods for Physicists, 2nd ed. Orlando, FL:
Academic Press, pp. 118-119, 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. 263, 1959.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 514 and 659,
1953.
Helmholtz Differential Equation — Elliptic
Cylindrical Coordinates
In Elliptic Cylindrical C oordinates, th e Scale
FACTORS are h u = h v — avsinh 2 it + sin 2 ^, h z = 1,
816 Helmholtz Differential Equation
Helmholtz Differential Equation
and the separation functions are fa (u) = fa (v) =
fa(z) = 1, giving a STACKEL DETERMINANT of S -
a 2 (sin 2 v + sinh 2 u). The Helmholtz differential equation
is
1 (d 2 F d 2 F\ d 2 F 2
a 2 (sinh 2 u + sin 2 *;) \du 2 ^ dv 2 ) ^ 8z 2 ^
(1)
Attempt Separation of Variables by writing
F(u,v,z) = U(u)V{v)Z(z) } (2)
then the Helmholtz Differential Equation be-
comes
Now use
sinh 2 u + sin 2 v V dv? dv 2
r d 2 U
d 2 V
+UV^+k 2 UVZ = 0. (3)
Now divide by UVZ to give
1
1 d 2 U 1 d 2 V
sinh 2 u + sin 2 v V U dv? V dv 2
1 d Z T o n / lS
Separating the Z part,
Z dz 2
= -(r + rn
1 d 2 U 1 d'V
sinh 2 u + sin 2 t> V t/ dv? V dv 2
(5)
(6)
sinh u — |[1 — cosh(2u)]
sin 2 v = |[1 — cos(2v)]
(14)
(15)
to obtain
^-{c+im 2 [l-cosh(2u)]}tf = (16)
du z
^ + {c + im 2 [l - cob(2i;)]}V = 0. (17)
Regrouping gives
d 2 U
dv?
d 2 V
[(c+|m J )- \m z 2cosh(2u)]U = (18)
dv 2 + K c + l m ) - I m 2 cos(2v)]F = 0. (19)
Let b= \m 2 + c and g = ^m 2 , then these become
d 2 <7
du 2
dt; 2
-[6-2gcosh(2u)]t/^0 (20)
+ [b-2qcos(2v)]V = 0. (21)
Here, (21) is the Mathieu DIFFERENTIAL EQUA-
TION and (20) is the modified MATHIEU DIFFERENTIAL
Equation. These solutions are known as Mathieu
Functions.
see also Elliptic Cylindrical Coordinates, Math-
ieu Differential Equation, Mathieu Function
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 514 and 657,
1953.
d Z , 2 2\r7
-^ = -{k +m)Z,
(7)
which has the solution
Z(z) = ,4cos( \4 2 + m? z) + B sm(^k 2 +m?z). (8)
Rewriting (6) gives
ld 2 U 2 . 2 \ / 1 d 2 V
us*-™ sinh u r\v^
which can be separated into
1 d 2 U
U dv 2
— m sinh u = c
so
1 d 2 V 2 . 2
c + — ~—r- - m sin v — 0,
V dv 2.
, — (c + m sinh n)C7 =
dv?
d V , / 2 - 2 w, n
, + (c — m sin t>) 1/ = 0.
cfo 2
— m sin v I — 0,
(9)
(10)
(11)
(12)
(13)
Helmholtz Differential Equation — Oblate
Spheroidal Coordinates
As shown by Morse and Feshbach (1953) and Arfken
(1970), the Helmholtz Differential Equation is
separable in Oblate Spheroidal Coordinates.
References
Arfken, G. "Oblate Spheroidal Coordinates (u,u, (p)" §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. 662, 1953.
Helmholtz Differential Equation — Parabolic
Coordinates
The Scale Factors are h u = h v = %/u 2 + v 2 , h e = uv
and the separation functions are fi{u) = u y fa(v) = v,
fa(0) = 1, given a Stackel Determinant of S = v? +
v 2 . The LAPLACIAN is
1
i of d 2 F
v? -\- v 2 \u du dv?
ldF d 2 F \
v dv dv 2 J
+ ^^^ 2 = o- (i)
Helmholtz Differential Equation
Attempt Separation of Variables by writing
F(u,v,z) = U(u)V(v)e(e), (2)
then the Helmholtz Differential Equation be-
comes
u du du z
u 2 + v 2
G
Now divide by UVQ
dV d 2 V
dv dv 2
+ k 2 UVO = 0. (3)
u 2 + v 2
1 (\^L tR\ 1 (\^L d2y \
U \u du du 2 J V \v dv dv 2 J
+hTF + t ' = °- W
Helmholtz Differential Equation 817
2 d 2 u ^du a
u -— + - (c + k )U =
du 2 du
v-- rT + — + (c-k d )V = 0.
dv 2 dv
(13)
(14)
Separating the @ part,
References
Arfken, G. "Parabolic Coordinates (£,t/,0)." §2.12 in Math-
ematical Methods for Physicists, 2nd ed. Orlando, FL:
Academic Press, pp. 109-111, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 514-515 and 660,
1953.
Helmholtz Differential Equation — Parabolic
Cylindrical Coordinates
In Parabolic Cylindrical Coordinates, the Scale
Factors are h u = h v = Vu 2 -f v 2 , h z = 1 and the
separation functions are fi(u) = f2{v) = fs(z) — 1,
giving Stackel Determinant of S = u 2 + v 2 . The
Helmholtz Differential Equation is
O fO
__ = _(^ +m 3)
(5)
d 2 F d 2 F \ d 2 F 2
u 2 + v 2 V du 2 + dv 2 ) + dz 2 + " ( }
U 2 + V 2
17 V u du du 2 I V \ v dv dv 2
d 2 Q
-{k 2 +m 2 )6,
k\ (6)
(?)
which has solution
0(0) = Acos(\/* 2 + m 2 0) + Bsin{yJk 2 +m 2 6), (8)
and
U \u du du 2 J
+
A fl^K d2y
V V v dv dv 2
2 t 2
rf 2 t/
ldI7
u du du 2
A (\®L
V \y dv
d 2 V
dv 2
0. (10)
This can be separated
Attempt Separation of Variables by writing
F{u,v,z) = U(u)V{v)Z(z),
(2)
then the Helmholtz Differential Equation be-
comes
vz^IL + uz^X
u z + v 2 \ du 2 dv 2
+ UV
d 2 Z
Divide by UVZ,
1 / 1 d 2 U 1_
u 2 +v 2 \U du 2 + V
Separating the Z part,
d 2 V
dv 2
dz 2
+k 2 UVZ = Q. (3)
+ |g + ^=0. (4)
1 d 2 U 1 dV
+
u 2 +v 2 \U du 2 V dv 2
1 d 2 U 1 dV
A; 2 =
p A ,» + v^-*'^ + ^ = '
d 2 Z
dz 2
which has solution
- -{k 2 +m 2 )Z,
(5)
(6)
(7)
(8)
A (\®?_ d 2 V
V \ v dv dv 2
-c,
(12)
Z{z) = A cos( y^ 2 + m 2 z) + £ sin( ^Jk 2 +m 2 z) y (9)
818
and
Helmholtz Differential Equation
{h£-"H&-**)-°- <"»
This can be separated
1 d 2 U j2 2
u*J' ku = c
1 d 2 V
V dv 2
- k 2 v 2 = -c,
so
av 2
(11)
(12)
(13)
(14)
These are the WEBER DIFFERENTIAL EQUATIONS, and
the solutions are known as Parabolic Cylinder
Functions.
see also PARABOLIC CYLINDER FUNCTION, PARABOLIC
Cylindrical Coordinates, Weber Differential
Equations
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 515 and 658,
1953.
Helmholtz Differential Equation — Polar
Coordinates
In 2-D Polar Coordinates, attempt Separation of
Variables by writing
F(r,0) = R(r)S(0) 1
(1)
then the Helmholtz Differential Equation be-
comes
d 2 R l^Q — — J? k 2 R®
dr 2 r dr r 2 dO 2
Divide both sides by RQ
(2)
r 2 d 2 R r dR\ ( 1 d 2 Q , 2 \ „
R^ + R^) + {e^ + k )= Q - ^
The solution to the second part of (3) must be periodic,
so the differential equation is
d 2 e 1
dd 2
which has solutions
= —(A; + m ),
(4)
0(0) = ClC «V* a +'» a * + (4e -'\/* J +-»'«
= c 3 sin(y / fc 2 + m 2 6) + c 4 cos(y/k 2 + m 2 6).
(5)
Helmholtz Differential Equation
Plug (4) back into (3)
r 2 R" + TR 1 -m 2 R = 0.
(6)
This is an EULER DIFFERENTIAL EQUATION with ex = 1
and f3 = —m 2 . The roots are r = ±m. So for m = 0,
r = and the solution is
R{r) = ci + C2lnr.
(7)
But since lnr blows up at r = 0, the only possible phys-
ical solution is R(r) — c\. When m > 0, r — ±m, so
R(r) = cir m + c 2 r
(8)
But since r _TTl blows up at r = 0, the only possible
physical solution is Rm(r) = cir m . The solution for R
is then
J Rm(r) = c m r m (9)
for m = 0, 1, ... and the general solution is
F(r,6) = ]T[a m r m sin( y/k 2 + m 2 (9)
m~0
+ fe m r m cos(v / A; 2 + m 2 ^)].
(10)
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 502-504, 1953.
Helmholtz Differential Equation — Prolate
Spheroidal Coordinates
As shown by Morse and Feshbach (1953) and Arfken
(1970), the Helmholtz Differential Equation is
separable in Prolate Spheroidal Coordinates.
References
Arfken, G. "Prolate Spheroidal Coordinates (u,v, y?)." §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.
Helmholtz Differential Equation — Spherical
Coordinates
In Spherical Coordinates, the Scale Factors are
h r = 1, he = rsin0, h<j> = r, and the separation func-
tions are /i(r) = r 2 , f 2 {9) = 1, fs((f>) = sin<£, giving a
Stackel Determinant of S = 1. The Laplacian is
r 2 Q T
H)
-r
1
d 2
r 2 sin 2 <j> d9 2
+
1
d
r 2 sin <j> d<j>
( sin ^) •
(i)
To solve the Helmholtz Differential Equation
in Spherical Coordinates, attempt Separation of
Variables by writing
F{r,e,<j>) = R{r)®{9)*{4>)-
(2)
Helmholtz Differential Equation
Then the HELMHOLTZ DIFFERENTIAL EQUATION be-
comes
j!g $e + ^*e+ 1 ^r
dr 2 v dr r 2 sin 2 <j> d6 2
cos0 d$ 1 d 2 $
H sin d<p r J d<p z
Now divide by J£6$,
r 2 sin 2 (j>^d 2 R 2 r 2 sin 2 „ drt
$#0 dr 2 r &R9 dr
1 r 2 sin 2 <K„d 2 , cos<£ r 2 sin 2 0d$^ o
r 2 sin 2 ()6 $#9 d<9 2 r 2 sin<£ $e#
1 r 2 sin 2 <£cZ 2 $
+
r 2 $RQ d<j> 2
QR^O (4)
( r 2 sin 2 <j>d 2 R 2r sin 2 (f> dR\ (\ d 2 G
\ R dr 2 + # dry) + \B dO 2
(cos sin (j) d3> sin 2 <£ d 2 3>
1 # + $ #2"
0. (5)
The solution to the second part of (5) must be sinusoidal,
so the differential equation is
d 2 S 1 2
^0 = - m '
(6)
which has solutions which may be defined either as a
Complex function with m = — oo, . . . , oo
8(0) = ilme*
(7)
or as a sum of Real sine and cosine functions with m =
— oo, . . . , oo
0(0) = Sm sin(ro0) + Cm. cos{mQ).
Plugging (6) back into (7),
cos0sin</>\ d<&
(8)
v 2 d 2 R 2r dR _ 1 / 2 , cos0sin</> \
~R~dr~ 2 ~^ ~R~dr~ ~ sin 2 ^ \ $ /
sin 2 <j) d 2 <$>
The radial part must be equal to a constant
r 2 d 2 R 2r dR
R dr 2 + R dr
1(1 + 1)
= 0- (9)
(10)
nd R n dR T , , „ , „ ,
r 2 TY +2r- r =//+l£
dr 2 dr
(ii)
But this is the Euler Differential Equation, so we
try a series solution of the form
R=J2a n r n+c .
(12)
Helmholtz Differential Equation 819
Then
r 2 ^(n+c)(n+c-l)a„r n+c - 2 +2r^(n+c)a n r n+c - 1
n=0 n=0
oo
-Z(Z + l)^a n r n+c = (13)
n=0
oo oo
J^(n + c){n + c- l)a n r n+c + 2 ^(n + c)a n r n+c
n—O n~0
OO
-1(1 + 1)^2 a n r n+c = (14)
oo
J^[(n + c)(n + c + 1) - 1(1 + l)]a„r" +<= = 0. (15)
This must hold true for all POWERS of r. For the r c
term (with n = 0),
c(c +!) = /(/ + !),
(16)
which is true only if c — /, -I - 1 and all other terms
vanish. So a n = for n ^ /, — / — 1. Therefore, the
solution of the R component is given by
Rtir) = Air 1 + Bir- 1 - 1 . (17)
Plugging (17) back into (9),
sm 2 sin ^ $ d(j> $ d0 2
Slll(p
J(J + 1)
sin <j>
* = 0, (19)
which is the associated Legendre Differential
Equation for x — cos <j> and m = 0, . . . , /. The general
Complex solution is therefore
J^ Yl (^ir l +B l r' l ' 1 )Pr(cos4>)e- ime
1=0 m=-l
oo I
-Y.Y1 ( Air ' + Btr~'~ 1 )Yr(0, <t>), (20)
1=0 m=-l
where
Yr(e,<p) =Pr (cos <j,)e
-irrtB
(21)
are the (Complex) Spherical Harmonics. The gen-
eral Real solution is
J] £(V +B l r- l - 1 )Pr(cos<j>)
i=0 m =
x [Sm sm(m6) + Cm cos(m0)}. (22)
820 Helmholtz Differential Equation
Helmholtz Differential Equation
Some of the normalization constants of P™ can be ab-
sorbed by Sm and C m , so this equation may appear in
the form
]r J2( Airl +B l r- i - i )pr i {cos <j>)
x [Sr sin(m(9) + CT cos(m<9)]
oo I
1=0 m=0
xtfrY^frft + CrY™^ ($,<!>)], (23)
where
Yr (o) (0, <t>) = P™ (cos 6) sin(m(9) (24)
y f m(e) ((9,0) = iT (cos 0)cos(m0) (25)
are the Even and Odd (real) Spherical Harmonics.
If azimuthal symmetry is present, then 0(0) is constant
and the solution of the $ component is a LEGENDRE
POLYNOMIAL Pi(cos<j)). The general solution is then
oo
F(r,<f>) = ^2(Air l + JB/r- I_1 )P/(cos0). (26)
Actually, the equation is separable under the more gen-
eral condition that k 2 is of the form
*( r>M ) = /( P ) + M + -^ + fc «. (27)
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 514 and 658, 1953.
Helmholtz Differential Equation — Spherical
Surface
On the surface of a SPHERE, attempt SEPARATION OF
Variables in Spherical Coordinates by writing
F(M) = e(0)$(0),
(1)
then the Helmholtz Differential Equation be-
comes
d 2 §,
1 ^e $+ cos0^ 0+ a. e + fc2e$ = o (2)
sin 2 <f> dO 2 sin <fi d<j) d<f> 2
Dividing both sides by $0,
/ cos <f> sin <f> d$ sm 2 (j)d 2 ^\ I 1 d 2 Q 2
^ $ d^ + $ ~d$> ) + \e¥ +
which can now be separated by writing
d 2 e i
d6 2 e
-(k 2 +m 2 ).
--0,
(3)
(4)
The solution to this equation must be periodic, so m
must be an Integer. The solution may then be defined
either as a COMPLEX function
9(0) - Ame 1 ^ 2 *™ 26 + B m c-^ 2+maj (5)
for m — — oo, . . . , oo, or as a sum of REAL sine and
cosine functions
0(0) = Sm sm(y / k 2 +m 2 0) + C m cos(y/k 2 + m 2 0)
for m = 0, . . . , oo. Plugging (4) into (3) gives
cos <f> sin 4> d<& sin 2 <j> d 2 <& 2
(6)
+
+ m* = (7)
sin <p snr
(8)
which is the LEGENDRE DIFFERENTIAL EQUATION for
x — cos 4> with
m 2 = 1(1+1), (9)
giving
I 2 + I - rn =
1= |(-l±\/l + 4m 2 ).
(10)
(11)
Solutions are therefore LEGENDRE POLYNOMIALS with
a Complex index. The general Complex solution is
then
oo
F(6,<l>)= £ Pi(cos(j>)(A m e ime + B m e- ime ), (12)
m= — oo
and the general REAL solution is
oo
F(9,<j>) = ^ P f (cos <ft)[S m sin(m<9) + C m cos(mfl)].
m=0
(13)
Note that these solutions depend on only a single vari-
able m. However, on the surface of a sphere, it is usual to
express solutions in terms of the SPHERICAL HARMON-
ICS derived for the 3-D spherical case, which depend on
the two variables / and m.
Helmholtz Differential Equation — Toroidal
Coordinates
The Helmholtz Differential Equation is not sep-
arable.
see Laplace's Equation— Toroidal Coordinates
Helmholtz's Theorem
Hemispherical Function 821
Helmholtz's Theorem
Any Vector Field v satisfying
[V-V]oo=0
[V x v]oo -0
Hemisphere
(i)
(2)
may be written as the sum of an IRROTATIONAL part
and a Solenoidal part,
v= -V0 + V x A,
where for a VECTOR FIELD F,
A
= -/ . v : F ,dV
Jv
4tt\t' - r|
f V xF
7 dV
(3)
(4)
(5)
see also Irrotational Field, Solenoidal Field,
Vector Field
References
Arfken, G. "Helmholtz's Theorem." §1.15 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 78-84, 1985.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1084, 1980.
Helson-Szego Measure
An absolutely continuous measure on dD whose density
has the form exp(x + j/), where x and y are real- valued
functions in L°°, \\y\\oo < ar/2, exp is the EXPONENTIAL
Function, and ||y|| is the Norm.
Hemicylindrical Function
A function S n (z) which satisfies the Recurrence Re-
lation
S n - 1 {z)-S n +i(z) = 2S' n (z)
together with
S 1 {z) = -S' (z)
is called a hemicylindrical function.
References
Sonine, N. "Recherches sur les fonctions cylindriques et le
developpement des fonctions continues en series." Math.
Ann. 16, 1-9 and 71-80, 1880.
Watson, G. N. "Hemi-Cylindrical Functions." §10.8 in A
Treatise on the Theory of Bessel Functions, 2nd ed. Cam-
bridge, England: Cambridge University Press, p. 353,
1966.
Half of a Sphere cut by a Plane passing through its
Center. A hemisphere of Radius r can be given by
the usual Spherical Coordinates
x = r cos sin <f>
y = r sin sin <f>
z = r cos 0,
(1)
(2)
(3)
where 9 e [0,2tt) and <f> € [0,7r/2], All Cross-Sections
passing through the z-axis are SEMICIRCLES.
The Volume of the hemisphere is
V ■
The weighted mean of z over the hemisphere is
7T / [V —
Jo
z )dz
2 3
^7rr .
(z)
Jo
z(r 2 -z 2 )dz
(4)
(5)
(6)
The CENTROID is then given by
z {Z) 3 r
(Beyer 1987).
see also Semicircle, Sphere
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 133, 1987.
Hemispherical Function
822 Hempel's Paradox
The hemisphere function is defined as
H{x,y)
^ a — x 2 — y 2 for ^Jx 2 + y 2 < a
^ for \Jx 2 + y 2 > a.
Watson (1966) defines a hemispherical function as a
function S which satisfies the RECURRENCE RELATIONS
Sn-i(z)-S n +i(z)=2S n '(z)
with
Si(*) = -SS(z).
see also CYLINDER FUNCTION, CYLINDRICAL FUNC-
TION
References
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
p. 353, 1966.
Hempel's Paradox
A purple cow is a confirming instance of the hypothesis
that all crows are black.
References
Carnap, R. Logical Foundations of Probability. Chicago, IL:
University of Chicago Press, pp. 224 and 469, 1950.
Gardner, M. The Scientific American Book of Mathematical
Puzzles & Diversions. New York: Simon and Schuster,
pp. 52-54, 1959.
Goodman, N. Ch. 3 in Fact, Fiction, and Forecast. Cam-
bridge, MA: Harvard University Press, 1955.
Hempel, C. G. "A Purely Syntactical Definition of Confirma-
tion." J. Symb. Logic 8, 122-143, 1943.
Hempel, C. G. "Studies in Logic and Confirmation." Mind
54, 1-26, 1945.
Hempel, C. G. "Studies in Logic and Confirmation. II." Mind
54, 97-121, 1945.
Hempel, C. G. "A Note on the Paradoxes of Confirmation."
Mind 55, 1946.
Hosiasson-Lindenbaum, J. "On Confirmation." J. Symb.
Logic 5, 133-148, 1940.
Whiteley, C. H. "Hempel's Paradoxes of Confirmation."
Mind 55, 156-158, 1945.
Hendecagon
see Undecagon
Henneberg's Minimal Surface
Henon-Heiles Equation
A double algebraic surface of 15th order and fifth class
which can be given by parametric equations
#(u, v) = 2 sinh u cos v - | sinh(3u) cos(3v) (1)
y(ujv) = 2sinhtxsinv — | sinh(3u) sin(3t;) (2)
z(u,v) = 2cosh(2u)cos(2v).
(3)
It can also be obtained from the Enneper-Weierstrad
Parameterization with
/ = 2 - 2z~
9 = z.
(4)
(5)
see also Minimal Surface
References
Eisenhart, L. P. A Treatise on the Differential Geometry of
Curves and Surfaces. New York: Dover, p. 267, 1960.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 446-448, 1993.
Nitsche, J. C. C. Introduction to Minimal Surfaces. Cam-
bridge, England: Cambridge University Press, p. 144,
1989.
Henon Attractor
see Henon Map
Henon-Heiles Equation
A nonlinear nonintegrable HAMILTONIAN SYSTEM with
OV
' dx
OV
dy '
where
V{x,y) = \{x 2 +y z + 2x*y-ly 6 )
1
3 ?
V(r,0) = fr 2 + ^r 3 sin(3<9).
The energy is
E = V(x,v) + $(x a +v 2 ).
(1)
(2)
(3)
(4)
(5)
E = 1/12
0.4
0.2
'" •■-^"''•N
0.0
- '; 1 i':<Y<55")Vi ■
-0.2
- A ' K: M'^ :/ ■
-0.4
Esl/8
0.4
y^^yi^f^'^r-;::.-^
: V '''^^c&y^r' : '- : -^ '''■-■
0.2
>t :'! -v ;.;■;■"■" : 'ii ; . , >-:-'- > "';"- ••""-- ■'-•-/^:' : -■■>•
y °°
\ r , % *^l\\-%>i : ^Z'T?\: : ?'^- : : ■■>*
■■: •■ ^-^-/-.rv;:' -'-■ .<£?;>..: '■■■.'■■■•.■■ -'
-0.2
■v*. ^■■■V- <S> . ■>."■'" ■■.■-'■■■ ■■■'•'.■■■. .-
-0.4
' S\ -- '■■>:,• ..-,
The above plots are Surfaces OF Section for E =
1/12 and E — 1/8. The Hamiltonian for a generalized
Henon-Heiles potential is
H = \{Vx 2 + p y 2 + Ax 2 + By 2 ) + Dx 2 y - \Cy\ (6)
Henon Map
Heptacontagon 823
The equations of motion are integrable only for
1. D/C = 0,
2. D/C = -l t A/B = l,
3. D/C = -1/6, and
4. D/C = -l/16 9 A/B = l/6.
References
Gleick, J. Chaos: Making a New Science, New York: Pen-
guin Books, pp. 144-153, 1988.
Henon, M. and Heiles, C. "The Applicability of the Third In-
tegral of Motion: Some Numerical Experiments." Astron.
J. 69, 73-79, 1964.
Henon Map
-1 o 1
A quadratic 2-D MAP given by the equations
#n+i - 1 - ax n 2 + y n (1)
y n +i = /3x n (2)
x n+1 — x n cos a - (y n - x n ) sin a (3)
2/n+i = z n sina-r- (y n - x n 2 )cosa. (4)
The above map is for a = 1.4 and f3 = 0.3. The Henon
map has CORRELATION EXPONENT 1.25 ± 0.02 (Grass-
berger and Procaccia 1983) and CAPACITY DIMENSION
1.261±0.003 (Russell et al. 1980). Hitzl and Zele (1985)
give conditions for the existence of periods 1 to 6.
see also Bogdanov Map, Lozi Map, Quadratic Map
References
Dickau, R. M. "The Henon Attractor." http:// forum .
swarthmore.edu/advanced/robertd/henon. html.
Gleick, J. Chaos: Making a New Science. New York: Pen-
guin Books, pp. 144-153, 1988.
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.
Hitzl, D, H, and Zele, F. "An Exploration of the Henon Quad-
ratic Map." Physica D 14, 305-326, 1985.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 128-
133, 1991.
Peitgen, H.-O. and Saupe, D. (Eds.). "A Chaotic Set in the
Plane." §3.2,2 in The Science of Fractal Images. New
York: Springer- Verlag, pp. 146-148, 1988.
Russell, D. A.; Hanson, J. D.; and Ott, E. "Dimension of
Strange Attractors." Phys. Rev. Let. 45, 1175-1178, 1980.
Hensel's Lemma
An important result in VALUATION THEORY which gives
information on finding roots of POLYNOMIALS. Hensel's
lemma is formally stated as follow. Let (K, | ■ |) be a com-
plete non-Archimedean valuated field, and let R be the
corresponding Valuation Ring. Let f(x) be a Poly-
nomial whose Coefficients are in R and suppose ao
satisfies
|/(oo)| < |/'(a )| 2
(1)
where /' is the (formal) Derivative of /. Then there
exists a unique element a G R such that f(a) = and
■ao| <
/(ao)
/'(ao)
(2)
Less formally, if f(x) is a POLYNOMIAL with "INTEGER"
Coefficients and /(ao) is "small" compared to /'(ao),
then the equation f(x) = has a solution "near" ao. In
addition, there are no other solutions near ao, although
there may be other solutions. The proof of the Lemma
is based around the Newton-Raphson method and relies
on the non- Archimedean nature of the valuation.
Consider the following example in which Hensel's lemma
is used to determine that the equation x 2 = — 1 is solv-
able in the 5-adic numbers Q 5 (and so we can embed
the Gaussian Integers inside Q 5 in a nice way). Let
K be the 5-adic numbers Q 5 , let f(x) = x 2 + 1, and let
a = 2. Then we have /(2) = 5 and /'(2) = 4, so
l/(2)| 5
<i/'(2)ir
(3)
and the condition is satisfied. Hensel's lemma then tells
us that there is a 5-adic number a such that a 2 4- 1 =
and
|a-2|5<=|!|B = £. (4)
Similarly, there is a 5-adic number b such that b 2 + 1
and
|&-3| B <=|¥|. = i. (5)
Therefore, we have found both the square roots of —1 in
Q 5 . It is possible to find the roots of any POLYNOMIAL
using this technique.
Henstock-Kurzweil Integral
see HK INTEGRAL
Heptacontagon
A 70-sided Polygon.
824 Heptadecagon
Heptadecagon
The Regular Polygon of 17 sides is called the Hep-
tadecagon, or sometimes the Heptakaidecagon.
Gauss proved in 1796 (when he was 19 years old)
that the heptadecagon is CONSTRUCTIBLE with a COM-
PASS and Straightedge. Gauss's proof appears in
his monumental work Disquisitiones Arithmeticae. The
proof relies on the property of irreducible Polynomial
equations that ROOTS composed of a finite number of
SQUARE ROOT extractions only exist when the order of
the equation is a product of the form 2 a 3 b F c • Fj- ■ ■ F e ,
where the F n are distinct PRIMES of the form
Fn-2 2 +1,
known as Fermat Primes. Constructions for the regu-
lar Triangle (3 1 ), Square (2 2 ), Pentagon (2 2 * + 1),
Hexagon (2 1 3 1 ), etc., had been given by Euclid, but
constructions based on the Fermat Primes > 17 were
unknown to the ancients. The first explicit construction
of a heptadecagon was given by Erchinger in about 1800.
17-gon
The following elegant construction for the heptadecagon
(Yates 1949, Coxeter 1969, Stewart 1977, Wells 1992)
was first given by Richmond (1893).
1. Given an arbitrary point O, draw a CIRCLE centered
on O and a DIAMETER drawn through O.
2. Call the right end of the Diameter dividing the Cir-
cle into a Semicircle P .
3. Construct the Diameter Perpendicular to the
original Diameter by finding the Perpendicular
Bisector OB.
4. Find J a Quarter the way up OB.
5. Join JP and find E so that LOJE is a QUARTER of
IOJP .
6. Find F so that LEJF is 45°.
7. Construct the SEMICIRCLE with DIAMETER FP Q .
Heptadecagon
8. This Semicircle cuts OB at K.
9. Draw a SEMICIRCLE with center E and Radius EK.
10. This cuts the extension of OPq at Nq.
11. Construct a line PERPENDICULAR to OP through
N s .
12. This line meets the original SEMICIRCLE at P 3 -
13. You now have points Po and P$ of a heptadecagon.
14. Use Po and P3 to get the remaining 15 points of the
heptadecagon around the original CIRCLE by con-
structing P , Ps, Pe, P9, P12, Pis> Pi, P4, P 7) P10,
P13, Pie, P 2) Ps, Ps, P11, and P x4 .
15. Connect the adjacent points Pj.
This construction, when suitably streamlined, has Sim-
plicity 53. The construction of Smith (1920) has a
greater SIMPLICITY of 58. Another construction due to
Tietze (1965) and reproduced in Hall (1970) has a Sim-
plicity of 50. However, neither Tietze (1965) nor Hall
(1970) provides a proof that this construction is cor-
rect. Both Richmond's and Tietze's constructions re-
quire extensive calculations to prove their validity. De
Temple (1991) gives an elegant construction involving
the Carlyle Circles which has Geometrography
symbol 85i + 4S 2 + 22Ci + 11C 3 and Simplicity 45.
The construction problem has now been automated to
some extent (Bishop 1978).
see also 257-gon, 65537-gon, Compass, Con-
structible Polygon, Fermat Number, Fer-
mat Prime, Regular Polygon, Straightedge,
Trigonometry Values — 7r/17
References
Archibald, R. C. "The History of the Construction of the
Regular Polygon of Seventeen Sides." Bull Amer. Math.
Soc. 22, 239-246, 1916.
Archibald, R. C. "Gauss and the Regular Polygon of Seven-
teen Sides." Amer. Math. Monthly 27, 323-326, 1920,
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 95-96,
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. 201 and 229-230, 1996.
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. "Gauss Extends Euclid." §1.4 in Mathographics.
New York: Dover, pp. 52-54, 1991.
Gauss, C. F. §365 and 366 in Disquisitiones Arithmeticae.
Leipzig, Germany, 1801. New Haven, CT: Yale University
Press, 1965.
Hall, T. Carl Friedrich Gauss: A Biography. Cambridge,
MA: MIT Press, 1970.
Klein, F. Famous Problems of Elementary Geometry and
Other Monographs. New York: Chelsea, 1956.
Ore, 0. Number Theory and Its History. New York: Dover,
1988.
Rademacher, H. Lectures on Elementary Number Theory.
New York: Blaisdell, 1964.
Heptagon
Herbrand's Theorem 825
Richmond, H. W. "A Construction for a Regular Polygon of
Seventeen Sides." Quart J. Pure Appl. Math. 26, 206-
207, 1893.
Smith, L. L. "A Construction of the Regular Polygon of Sev-
enteen Sides." Amer. Math. Monthly 27, 322-323, 1920.
Stewart, I. "Gauss." Sci. Amer. 237, 122-131, 1977.
Tietze, H. Famous Problems of Mathematics. New York:
Graylock Press, 1965.
Wells, D. The Penguin Dictionary of Curious and Interesting
Geometry. New York: Viking Penguin, 1992.
Yates, R. C. Geometrical Tools. St. Louis, MO: Educational
Publishers, 1949.
Heptagon
The unconstructible regular seven-sided POLYGON, il-
lustrated above, has Schlafli Symbol {7}.
Although the regular heptagon is not a Constructible
POLYGON, Dixon (1991) gives several close approxima-
tions. While the ANGLE subtended by a side is 360°/7 «
51.428571°, Dixon gives constructions containing an-
gles of 2 sin" 1 (V3/4) ~ 51.317812°, tan" 1 ^) «
51.340191°, and 30° + sin- 1 (( v / 3 - l)/2) w 51.470701°.
Madachy (1979) illustrates how to construct a heptagon
by folding and knotting a strip of paper.
see also Edmonds' Map, Trigonometry Values —
tt/7
References
Courant, R. and Robbins, H. "The Regular Heptagon."
§3.3.4 in What is Mathematics?: An Elementary Approach
to Ideas and Methods, 2nd ed. Oxford, England: Oxford
University Press, pp. 138-139, 1996.
Dixon, R. Mathographics. New York: Dover, pp. 35-40, 1991.
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 59-61, 1979.
Heptagonal Number
A FiGURATE Number of the form n(5n — 3)/2. The first
few are 1, 7, 18, 34, 55, 81, 112, . . . (Sloane's A000566).
The Generating Function for the heptagonal num-
bers is
x(4x +1) „o „„ s ~ , 4
-A — ^-/ - x + 7x 2 + lSx 3 + 34a; 4 + . . . .
(1 - x) 6
Heptagonal Pyramidal Number
A Pyramidal Number of the form n(n + l)(5n - 2)/6,
The first few are 1, 8, 26, 60, 115, ... (Sloane's
A002413). The Generating Function for the hep-
tagonal pyramidal numbers is
X ^ 4X + J = x + Sx 2 + 26x 3 + 60z 4 + . . . .
(x - l) 4
References
Sloane, N. J. A. Sequence A002413/M4498 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Heptahedron
The regular heptahedron is a one-sided surface made
from four TRIANGLES and three QUADRILATERALS. It is
topologically equivalent to the Roman SURFACE (Wells
1991). While all of the faces are regular and ver-
tices equivalent, the heptahedron is self-intersecting and
is therefore not considered an Archimedean SOLID.
There are three semiregular heptahedra: the pentago-
nal and pentagrammic Prisms, and a Faceted Octa-
hedron (Holden 1991).
References
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
p. 95, 1991.
Wells, D. The Penguin Dictionary of Curious and Interesting
Geometry. New York: Viking Penguin, p. 98, 1992.
Heptakaidecagon
see HEPTADECAGON
Heptaparallelohedron
see CUBOCTAHEDRON
Heptomino
The heptominoes are the 7-POLYOMINOES. There are
108 different heptominoes.
see also Herschel, Pi Heptomino, Polyomino
Herbrand's Theorem
Let an ideal class be in A if it contains an Ideal whose
Zth power is PRINCIPAL. Let i be an Odd INTEGER
1 < i < I and define j by i + j = 1. Then Ai = (e). If
i > 3 and l\Bj, then Ai = (e).
References
Ireland, K. and Rosen, M. "Herbrand's Theorem." §15.3 in
A Classical Introduction to Modern Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 241-248, 1990.
References
Sloane, N. J. A. Sequence A000566/M4358 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
826 Hereditary Representation
Hermite Differential Equation
Hereditary Representation
The representation of a number as a sum of powers of a
Base 6, followed by expression of each of the exponents
as a sum of powers of 6, etc., until the process stops. For
example, the hereditary representation of 266 in base 2
266 = 2 8 + 2 3 + 2
= 2 22+1 +2 2+1 + 2.
see also Goodstein Sequence
Heredity
A property of a Space which is also true of each of
its SUBSPACES. Being "Countable" is hereditary, but
having a given GENUS is not.
Hermann's Formula
The Machin-Like Formula
i 7 r = 2tan- 1 (i)-tan- 1 (i).
The other 2-term MACHlN-LlKE FORMULAS are Eu-
ler's Machin-Like Formula, Hutton's Formula,
and Machin's Formula.
Hermann Grid Illusion
A regular 2-D arrangement of squares separated by ver-
tical and horizontal "canals." Looking at the grid pro-
duces the illusion of gray spots in the white AREA be-
tween square VERTICES. The illusion was noted by Her-
mann (1870) while reading a book on sound by J. Tyn-
dall.
References
Fineman, M. The Nature of Visual Illusion.
Dover, pp. 139-140, 1996.
Hermann-Hering Illusion
New York:
The illusion in view by staring at the small black dot
for a half minute or so, then switching to the white dot.
The black squares appear stationary when staring at
the white dot, but a fainter grid of moving squares also
appears to be present.
Hermann-Mauguin Symbol
A symbol used to represent the point and space groups
(e.g., 2/ra3). Some symbols have abbreviated form. The
equivalence between Hermann-Mauguin symbols ( "crys-
tallography symbol") and Schonflies Symbols for the
Point Groups is given by Cotton (1990).
see also Point Groups
References
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 379, 1990.
Hermit Point
see Isolated Point
Hermite Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
The Hermite constant is denned for DIMENSION n as the
value
SUp^mhla- /(Xl,#2,...,£n)
7n ~ [discriminant^)] 1 /"
(Le Lionnais 1983). In other words, they are given by
On
7 " = 4 fe) '
where S n is the maximum lattice PACKING DENSITY for
Hypersphere Packing and V n is the Content of the
n-HYPERSPHERE, The first few values of (7 n ) n are 1,
4/3, 2, 4, 8, 64/3, 64, 256, .... Values for larger n are
not known.
For sufficiently large n,
1
2?re
< 7n < 1.744.
2ne
see also Hypersphere Packing, Kissing Number,
Sphere Packing
References
Finch, S. "Favorite Mathematical Constants." http://wvv.
mathsof t . com/asolve/constant/hermit/hermit .html.
Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices,
and Groups, 2nd ed. New York: Springer- Verlag, p. 20,
1993.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 38, 1983.
Hermite Differential Equation
ax 2 ax
0.
(1)
This differential equation has an irregular singularity at
oo. It can be solved using the series method
E
(n + 2)(n + l)a n+2 £ n - ^ 2na n x n + ]P Xa n x n =
n=l n=0
(2)
Hermite Differential Equation
oo
(2a2 + Aa 4 ) + y^[(n + 2)(n + l)a n +2 -2na n + \a n ]x n = 0.
n=l
(3)
Therefore,
a 2 = -^ (4)
and
fln+2
Aao
~2~
2n- A
(n + 2)(n+l) an
for n = 1, 2, Since (4) is just a special case of (5)
2n-A
(5)
«n + 2 =
(n + 2)(n+l) an
(6)
for n = 0, 1, The linearly independent solutions are
then
3/i = ao
. A 2 (4 - A)A 4
1_ 2! X --^I - X
(8-A)(4-A)A a;6 _
2/2 = fll
6!
• (2-A) 3 (6-A)(2-A) g
*+ 3! + 5!
(7)
(8)
If A = 4n = 0, 4, 8, . . . , then y\ terminates with the
Power x a , and y\ (normalized so that the Coeffi-
cient of x n is 2 n ) is the regular solution to the equation,
known as the HERMITE POLYNOMIAL. If A = 4n+2 = 2,
6, 10, ... , then y 2 terminates with the Power x x , and
y 2 (normalized so that the Coefficient of x n is 2 n )
is the regular solution to the equation, known as the
Hermite Polynomial.
If A = 0, then Hermite's differential equation becomes
y" - 2xy' = 0, (9)
which is of the form P 2 (aOy" + Pi(x)y* = and so has
solution
v = */
dx
■■ Ci
exp (/ g- dx)
/dx
exp J — 2xdx
+ C 2
+ C 2
/dx f x 2 7
—^ + c 2 = ci / e da? + c 2 .
(10)
Hermite-Gauss Quadrature 827
Hermite-Gauss Quadrature
Also called Hermite Quadrature. A Gaussian
Quadrature over the interval (—00, 00) with Weight-
ing Function W{x) = e~ x . The Abscissas for quad-
rature order n are given by the roots of the Hermite
Polynomials H n (x)> which occur symmetrically about
0. The Weights are
■"■n + l7n
7n-l
(1)
where A n is the COEFFICIENT of x n in H n (x). For HER-
MITE Polynomials,
A — 2 n
■fin — ^ }
(2)
so
Ai + 1 ^
(3)
Addition
ally,
7n = A2"n!,
(4)
so
2" +1 n!v^
1 _ H n+ i(xi)Hk(xi)
2™(n-l)! v ^ :
~ fl„-l(*0#»(*0'
(5)
Using the RECURRENCE RELATION
H' n {x) = 2nH n -i(x) = 2xH n {x) - H n +i{x) (6)
yields
H' n {xi) = 2nif„_!(x0 = -H n +i(xi) (7)
and gives
u>i =
2 n+1 n! v / ^ _ 2" +1 n!v^F
[HUxi)] 2 [H n+1 (xiW
The error term is
E
1 } -Vn A2n
2 n (2ra)!
/(a ») (0>
(8)
(9)
Beyer (1987) gives a table of ABSCISSAS and weights up
to n=12.
n Xi
Wi
2 ±0.707107 0.886227
3 1.18164
±1.22474 0.295409
4 ±0.524648 0.804914
±1.65068 0.0813128
5 0.945309
±0.958572 0.393619
±2.02018 0.0199532
828 Hermite Interpolation
Hermite Polynomial
The ABSCISSAS and weights can be computed analyti-
cally for small n.
n
Xi
Wi
2
±\V2
h^
3
|v^
±§V6
\^
4
±v^
4(3- s/6)
±v^
V*
4(3+^)
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 464, 1987.
Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 327-330, 1956.
Hermite Interpolation
see Hermite's Interpolating Fundamental Poly-
nomial
Hermite's Interpolating Fundamental
Polynomial
Let l(x) be an nth degree POLYNOMIAL with zeros at
a?i, . . . , Xm. Then the fundamental POLYNOMIALS are
K i} (x)
I'M
M*)Y
and
h {2 \x) = (x- x v )[l v (x)] 2 .
They have the properties
ft <1) '„(*».) = o
h m {x^=0
h (2) ' (x M ) = S vlt .
(1)
(2)
(3)
(4)
(5)
(6)
Now let /i,
expansion
/„ and f{ , . . . , f' v be values. Then the
W n (x) = J2f„hl 1) (x) + J2flh w (x)
(7)
gives the unique HERMITE'S INTERPOLATING FUNDA-
MENTAL Polynomial for which
W n (x v ) = /„
(8)
(9)
If f u = o, these are called Step Polynomials. The
fundamental Polynomials satisfy
and
Also,
Y.^hi^ix) + Y^ h » ) ( x ) = x - ( n )
h[, ' (x) da{x) = \ u
(12)
h ( 1 }\x)da(x)=0
(13)
xh„(x) da(x) =
(14)
hl 2) (x) da{x) =
(15)
h w ' v da{x) = \ v
(16)
i:
f
J a
i:
i
I xh {2)l u {x)dx = \vXv, (17)
J a
for v — 1, . . . , n.
References
Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 314-319, 1956.
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, pp. 330-332, 1975.
Hermite-Lindemann Theorem
The expression
Aie ai +A 2 e a2 +A 3 e OC3 +...,
in which the Coefficients A{ differ from zero and in
which the exponents ai are ALGEBRAIC NUMBERS dif-
fering from each other, cannot equal zero.
see also Algebraic Number, Constant Problem,
Integer Relation, Lindemann-WeierstraB Theo-
rem
References
Dorrie, H. "The Hermite-Lindemann Transcendence Theo-
rem." §26 in 100 Great Problems of Elementary Mathe-
matics: Their History and Solutions. New York: Dover,
pp. 128-137, 1965.
Hermite Polynomial
M 35 ) + - ■ • + M®) = 1
(10)
Hermite Polynomial
A set of Orthogonal Polynomials. The Hermite
polynomials H n (x) are illustrated above for x € [0,1]
and n = 1, 2, . . . , 5.
The Generating Function for Hermite polynomials
exp(2z*-t 2 ) = £^
H n {x)t n
I
(1)
Using a TAYLOR SERIES shows that,
-[■
-^- (!)"«-'■-*•
(2)
Since df(x - t)/dt = -df(x - t)/dx,
= (_l)» e - s *Le-'.
-(x-t) 2
dx n
Now define operators
Ox - -e — e
ax
-x 2 /2
It follows that
- a-2 d .3.2 d/
O2/ = e
- 2 /2
(-|)^-"1
xf + xf-^- = 2xf-f,
dx dx
Oi = O2,
and
ax
which means the following definitions are equivalent:
(3)
(4)
(5)
(6)
(7)
(8)
(9)
exp^-t 2 )^^
H n (x)t n
1
H n (x) = (-ire* 2 £;e-* 2
(10)
(11)
ff n (x)^e^ 2 (x-£)ne-^ 2 . (12)
The Hermite POLYNOMIALS are related to the derivative
of the Error Function by
H n{z y={-lf^e* 2 £^erf(z). (13)
Hermite Polynomial 829
They have a contour integral representation
-t 2 +2tx t - n -i dt.
H ^ = £ij'
(14)
They are orthogonal in the range (— oo, oo) with respect
to the Weighting Function e~ x
f
J —c
H n (x)H m (x)e- x dx = 5mn2 n n\V^-
Define the associated functions
u ^ x ) = v^Sf H ^ ax ^ e a2x2/2
These obey the orthogonality conditions
, du
(15)
(16)
£
^±1 m = n+l
Un ^~dx~ dx = j ~ a \/l m = n ~ 1 (17)
v otherwise
U m (x)u n (x) dx = Smn
(18)
f
J — c
/
i-/2±r m = n + l
u m (x)xu n (x) dx = < I /| m = n-l ( 19 )
v otherwise
2n + l
2a 2
Um{x)x 2 U n (x)dx= < V(n+l)(n+2)
m = n + 2
e x HaHpH-f dx = V^
7n ^ n ^ n±2
(20)
2 3 a\/3ljl
(s - a)\(s - j3)\(s - y)V
(21)
if a + f3 + 7 = 2s is Even and 5 > a, s > /3, and s > 7.
Otherwise, the last integral is (Szego 1975, p. 390).
They also satisfy the RECURRENCE RELATIONS
H n +i = 2xH n (x) - 2nH n - 1 (x)
H t n (x) = 2nH n - 1 (x).
The Discriminant is
n
D n = 2 3n{n - 1)/2 Y[u u
u-l
(Szego 1975, p. 143).
An interesting identity is
£ f"Wx)ff„_„(y) = 2" /2 i/ n [2" 1/2 (x + »)]. (25)
>,— n v /
(22)
(23)
(24)
830 Hermite Polynomial
The first few POLYNOMIALS are
H {x) = 1
Hi(x) = 2x
H 2 (x) =4x 2 -2
H z (x) = Sx s - 12s
H A (x) = 16s 4 - 4Sx 2 + 12
if 5 (s) = 32s 5 - 160s 3 + 120s
2*6 (x) = 64s 6 - 480s 4 + 720s 2 - 120
if 7 (z) = 128s 7 - 1344s 5 + 3360s 3 - 1680s
H s (x) = 256s 8 - 3594s 6 + 13440s 4 - 13440s 2
+ 160
H 9 (x) = 512s 9 - 9216s 7 + 48384s 5 - 80640s 3
+ 30240s
H 10 (x) = 1024s 10 - 23040s 8 + 161280s 6 - 403200s 4
+ 302400s 2 - 30240.
A class of generalized Hermite POLYNOMIALS 7^(s) sat-
isfying
e mxt - trn =J2ln(x)t n
(26)
n=0
was studied by Subramanyan (1990). A class of related
Polynomials defined by
hn,m = 7™ g) (27)
and with GENERATING FUNCTION
oo
e 2xi - tm =Y J hn, m {x)t n (28)
n=0
was studied by Djordjevic (1996). They satisfy
H n (x) = n\h n , 2 (x). (29)
A modified version of the Hermite Polynomial is
sometimes denned by
(30)
He n (x) = H n [ ~- ).
see also Mehler's Hermite Polynomial Formula,
Weber Functions
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. "Hermite Functions." §13.1 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 712-721, 1985.
Hermitian Matrix
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. 49-508, 1987.
Djordjevic, G. "On Some Properties of Generalized Hermite
Polynomials." Fib, Quart. 34, 2-6, 1996.
Hermite, C. "Sur un nouveau developpement en serie de
fonctions." Compt. Rend. Acad. Sci. Paris 58, 93-100
and 266-273, 1864. Reprinted in Hermite, C. Oeuvres
completes, Vol 2. Paris, pp. 293-308, 1908.
Hermite, C. Oeuvres completes, Vol. 3. Paris, p. 432, 1912.
Iyanaga, S. and Kawada, Y. (Eds.). "Hermite Polynomials."
Appendix A, Table 20. IV in Encyclopedic Dictionary of
Mathematics. Cambridge, MA: MIT Press, pp. 1479-1480,
1980.
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 Hermite Polynomials
H n (x)" Ch. 24 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 217-223, 1987,
Subramanyan, P. R. "Springs of the Hermite Polynomials."
Fib. Quart. 28, 156-161, 1990.
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, 1975.
Hermite Quadrature
see Hermite-Gauss Quadrature
Hermite's Theorem
e is Transcendental.
Hermitian Form
A combination of variables x and y given by
axx* + bxy* + b*x*y + cyy* ,
where x* and y* are COMPLEX CONJUGATES.
Hermitian Matrix
If a Matrix is Self- Adjoint, it is said to be a Hermi-
tian matrix. Therefore, a Hermitian MATRIX is defined
as one for which
A = A f , (1)
where f denotes the Adjoint Matrix. Hermitian Ma-
trices have Real Eigenvalues with Orthogonal
Eigenvectors. For Real Matrices, Hermitian is the
same as symmetrical. Any MATRIX C which is not Her-
mitian can be expressed as the sum of two Hermitian
matrices
(2)
C = i(C + C t ) + i(C-C t ).
Let U be a Unitary Matrix and A be a Hermitian
matrix. Then the Adjoint Matrix of a Similarity
Transformation is
(UAir 1 ) 1 = [(UA)(u- 1 )] t = (itVcua)*
= (U t ) t (A t U t )-UAU t -UAU~ 1 . (3)
Hermitian Operator
The specific matrix
H(z,y,z) =
z x + iy
x — iy —z
xP 1 +yP 2 +zP 3 , (4)
where Pi are Pauli Spin Matrices, is sometimes called
"the" Hermitian matrix.
see also Adjoint Matrix, Hermitian Operator,
Pauli Spin Matrices
References
Arfken, G. "Hermitian Matrices, Unitary Matrices." §4.5 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 209-217, 1985.
Hermitian Operator
A Hermitian Operator L is one which satisfies
pb pb
I v*Ludx = I
J a J a
uLv* dx.
(1)
As shown in Sturm-Liouville Theory, if L is Self-
Adjoint and satisfies the boundary conditions
[V*pu] x =a = [V*pu] x=b ,
(2)
then it is automatically Hermitian. Hermitian operators
have Real Eigenvalues, Orthogonal Eigenfunc-
TIONS, and the corresponding ElGENFUNCTIONS form a
Complete set when L is second-order and linear. In
order to prove that Eigenvalues must be Real and
ElGENFUNCTIONS ORTHOGONAL, consider
Lui + XiWUi — 0. (3)
Assume there is a second Eigenvalue A-, such that
Luj + Xjwuj =
(4)
Luj* -+- \j*wuj* = 0.
(5)
Now multiply (3) by Uj* and (5) by m
Uj*Lui -\-Uj*\iWUi =
(6)
UiLuj* + Ui\j*wuj* —
(7)
Uj* Lui — UiLuj* = (Xj* — Xi)wuiUj* .
(8)
Now integrate
Ab pb pb
pb pb pb
I Uj* Lui — I UiLuj* = (Xj* — At) / wuiUj*
Ja J a J a-
But because L is Hermitian, the left side vanishes.
r b
(9)
(A,
-A*)/
J a
wuiUj* = 0.
(10)
Hermitian Operator 831
If EIGENVALUES A* and Xj are not degenerate, then
J b wuiUj* = 0, so the ElGENFUNCTIONS are ORTHOG-
ONAL. If the Eigenvalues are degenerate, the Eigen-
FUNCTIONS are not necessarily orthogonal. Now take
i = j.
(A.
i* — Aj) / wuiUi* = 0.
J a
(ii)
The integral cannot vanish unless m — 0, so we have
Ai* — At and the Eigenvalues are real.
For a Hermitian operator O,
In integral notation,
(12)
(!</>)* ipdx = (f>*Ai>dx. (13)
Given Hermitian operators A and £?,
(<p\ABiP) = (A<f>\Bil>) = (BA</>\i/>) = (<l)\BA<il>y .
(14)
Because, for a Hermitian operator A with EIGENVALUE
(WAi/,) = <AV#) (15)
a(il>\1>) = a'(1>\1>). (16)
Therefore, either (V#) = or a = a*. But (V#> =
Iff ip = 0, so
<</#> + o, (17)
for a nontrivial ElGENFUNCTION. This means that
a — a : , namely that Hermitian operators produce REAL
expectation values. Every observable must therefore
have a corresponding Hermitian operator. Furthermore,
(ll)n\All> m ) = (A0n|V>m)
am (ifrntym) = On * (lpn\lprn} = CL n {lpn\lp?n) ,
since a n = a n * . Then
(a m - a n ) (ip n \fpm) =
For a m ^ a n (i.e., ip n =£ ip m ) y
{lpn\lpm)=0.
For a m = a n (i.e., Vn = V>m)»
(V'nlV'm) - {lpn\ll>n} = 1.
Therefore,
(18)
(19)
(20)
(21)
(22)
(23)
832 Heron's Formula
Heron's Formula
so the basis of ElGENFUNCTlONS corresponding to a Her-
mitian operator are ORTHONORMAL. Given two Hermi-
tian operators A and B,
(ASy = b ] A^ = bA = Ab + [b, A], (24)
the operator AB equals (AB)\ and is therefore Hermi-
tian, only if
[B,A]=0. (25)
Given an arbitrary operator A,
(V>i|(i + Ai)ih) = ((A 1 +A)1> 1 \fo)
= ((A + Alfalfa), (26)
so A + A* is Hermitian.
(rl>i\i{A - i f )^ 2 ) = <-i(i f - i)Vi|V>2)
= (i(A-A*)1> 1 \th), (27)
so i(A — A*) is Hermitian. Similarly,
(V-iKil 1 )^) = (AVil^t^) = ((ii^v-il^) ,
(28)
so AA^ is Hermitian.
Define the Hermitian conjugate operator A^ by
(iV#) = (iflity) ■ (29)
For a Hermitian operator, A = A^ . Furthermore, given
two Hermitian operators A and E,
(ifa\(AB?il>i) = ((i%|^i) = (S^ilAVi)
= {rh\&£il> 1 ), (30)
so
(AB) f =B t i t . (31)
By further iterations, this can be generalized to
(AB-.-Z) 1 " = Z t ---.B t i t . (32)
see a/so Adjoint Operator, Hermitian Matrix,
Self-Adjoint Operator, Sturm-Liouville The-
ory
References
Arfken, G. "Hermitian (Self- Adjoint) Operators." §9.2 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 504-506 and 510-516, 1985.
Heron's Formula
Gives the AREA of a TRIANGLE in terms of the lengths
of the sides a, 6, and c and the Semiperimeter
Heron's formula then states
s= |(a + 6 + c).
(1)
A = y/s(s — a)(s ~ b)(s — c).
(2)
Expressing the side lengths a, 6, and c in terms of the
radii a', &', and c } of the mutually tangent circles cen-
tered on the Triangle vertices (which define the Soddy
Circles),
+ c
b — a +c
c = a + b ,
gives the particularly pretty form
A = y/a'b'c'ia' + 6' + c')
(3)
(4)
(5)
(6)
The proof of this fact was discovered by Heron (ca. 100
BC-100 AD), although it was already known to Arch-
imedes prior to 212 BC (Kline 1972). Heron's proof
(Dunham 1990) is ingenious but extremely convoluted,
bringing together a sequence of apparently unrelated
geometric identities and relying on the properties of
Cyclic Quadrilaterals and Right Triangles.
Heron's proof can be found in Proposition 1.8 of his work
Metrica. This manuscript had been lost for centuries
until a fragment was discovered in 1894 and a complete
copy in 1896 (Dunham 1990, p. 118). More recently,
writings of the Arab scholar Abu'l Raihan Muhammed
al-Biruni have credited the formula to Heron's predeces-
sor Archimedes (Dunham 1990, p. 127).
A much more accessible algebraic proof proceeds from
the Law of Cosines,
cos A =
b + c — a
2bc *
(7)
Then
. . V-a 4 - b 4 - c 4 4- 26 2 c 2 + 2c 2 a 2 + 2a 2 b 2 /Q ,
sin A = — , (8)
2fec
giving
A = |6csin^4
(9)
= \ V~ a4 - & 4 - c 4 + 26 2 c 2 + 2c 2 a 2 + 2a 2 6 2 (10)
= ±[(a + b + c)(-a + b + c)(a-b + c)(a + b-c)] 1/2
(11)
- y/s(8-a){8-b)(8-c) (12)
(Coxeter 1969). Heron's formula contains the Pythag-
orean Theorem.
see also Brahmagupta's Formula, Bretschneider's
Formula, Heronian Tetrahedron, Heronian Tri-
angle, Soddy Circles, SSS Theorem, Triangle
Heron Triangle
Hessenberg Matrix 833
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New-
York: Wiley, p. 12, 1969.
Dunham, W. "Heron's Formula for Triangular Area." Ch. 5
in Journey Through Genius: The Great Theorems of
Mathematics. New York: Wiley, pp. 113-132, 1990.
Kline, M. Mathematical Thought from Ancient to Modern
Times. New York: Oxford University Press, 1972.
Pappas, T. "Heron's Theorem." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, p. 62, 1989.
Heron Triangle
see Heronian Triangle
Heronian Tetrahedron
A Tetrahedron with Rational sides, Face Areas,
and VOLUME. The smallest examples have pairs of op-
posite sides (148, 195, 203), (533, 875, 888), (1183, 1479,
1804), (2175, 2296, 2431), (1825, 2748, 2873), (2180,
2639, 3111), (1887, 5215, 5512), (6409, 6625, 8484), and
(8619, 10136, 11275).
see also HERON'S FORMULA, HERONIAN TRIANGLE
References
Guy, R. K. "Simplexes with Rational Contents." §D22 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 190-192, 1994.
Heronian Triangle
A Triangle with Rational side lengths and Ratio-
nal Area. Brahmagupta gave a parametric solution
for integer Heronian triangles (the three side lengths and
area can be multiplied by their Least Common Multi-
ple to make them all INTEGERS): side lengths c(a 2 +6 2 ),
b(a 2 + c 2 ), and (b + c)(a 2 - be), giving SEMIPERIMETER
and Area
s = a{b + c)
A = abc(a + b)(a — be).
The first few integer Hernonian triangles, sorted by in-
creasing maximal side lengths, are (3, 4, 5), (6, 8, 10), (5,
12, 13), (9, 12, 15), (4, 13, 15), (13, 14, 15), (9, 10, 17),
. . . (Sloane's A046128, A046129, and A046130), having
areas 6, 24, 30, 54, 24, 84, 36, . . . (Sloane's A046131).
Schubert (1905) claimed that Heronian triangles with
two rational MEDIANS do not exist (Dickson 1952). This
was shown to be incorrect by Buchholz and Rathbun
(1997), who discovered six such triangles.
see also HERON'S FORMULA, MEDIAN (TRIANGLE), PY-
THAGOREAN Triple, Triangle
References
Buchholz, R. H. On Triangles with Rational Altitudes, Angle
Bisectors or Medians. Doctoral Dissertation. Newcastle,
England: Newcastle University, 1989.
Buchholz, R. H. and Rathbun, R. L. "An Infinite Set of Heron
Triangles with Two Rational Medians." Amer. Math.
Monthly 104, 107-115, 1997.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, pp. 199 and
208, 1952.
Guy, R. K. "Simplexes with Rational Contents." §D22 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 190-192, 1994.
Kraitchik, M. "Heronian Triangles." §4.13 in Mathematical
Recreations. New York: W. W. Norton, pp. 104-108, 1942.
Schubert, H. "Die Ganzzahligkeit in der algebraischen Ge-
ometric" In Festgabe J^8 Versammlung d. Philologen und
Schulmdnner zu Hamburg. Leipzig, Germany, pp. 1—16,
1905.
Wells, D. G. The Penguin Dictionary of Curious and Inter-
esting Puzzles. London: Penguin Books, p. 34, 1992.
Herschel
A Heptomino shaped like the astronomical symbol for
Uranus (which was discovered by William Herschel).
Herschfeld's Convergence Theorem
For real, Nonnegative terms x n and Real p with <
p < 1, the expression
um xo + ( Xl + (x 2 + (. . . + {x k yy) p y
k— )-oo
converges Iff (x n ) p is bounded.
see also Continued Square Root
References
Herschfeld, A. "On Infinite Radicals," Amer. Math. Monthly
42, 419-429, 1935.
Jones, D. J. "Continued Powers and a Sufficient Condition
for Their Convergence." Math. Mag. 68, 387-392, 1995.
Hesse's Theorem
If two pairs of opposite VERTICES of a COMPLETE
Quadrilateral are pairs of Conjugate Points, then
the third pair of opposite VERTICES is likewise a pair of
Conjugate Points.
Hessenberg Matrix
A matrix of the form
"an
ai2
ai3
Ol(n-l)
Q>\n
O21
^22
«23
<*>2(n-l)
CL27i
a32
^33
fl3(n-l)
tt3n
<U3
»4(n-l)
&4n
0>5(n-l)
asn
G(n-l)(n-l)
0(n-l)n
.
a n (n-i)
Q> nn
Refer
ences
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Reduction of a General Matrix to Hessenberg
Form." §11.5 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 476-480, 1992.
834 Hessian Covariant
Hessian Covariant
TT \ i in t tt n
tt = \aa a \a x n-2a x n~2 0, x n-2 = u.
The nonsingular inflections of a curve are its nonsingular
intersections with the Hessian.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 79, 95-98, and 151-161, 1959.
Hessian Determinant
The Determinant
Hf(x,y) =
9 2 f a 2 /
!h? dxdy
d 2 f d 2 f
dydx dy 1
appearing in the Second Derivative TEST as D =
Hf(x t y).
see also SECOND DERIVATIVE TEST
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1112-1113, 1979.
Heteroclinic Point
If intersecting stable and unstable MANIFOLDS (SEP-
ARATRICES) emanate from FIXED POINTS of different
families, they are called heteroclinic points.
see also HOMOCLINIC POINT
Heterogeneous Numbers
Two numbers are heterogeneous if their PRIME factors
are distinct.
see also Distinct Prime Factors, Homogeneous
Numbers
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 146, 1983.
Heuristic
A heterosquare is an n x n Array of the integers from
1 to n 2 such that the rows, columns, and diagonals have
different sums. (By contrast, in a MAGIC SQUARE, they
have the same sum.) There are no heterosquares of order
two, but heterosquares of every ODD order exist. They
can be constructed by placing consecutive INTEGERS in
a SPIRAL pattern (Pults 1974, Madachy 1979).
An ANTIMAGIC SQUARE is a special case of a het-
erosquare for which the sums of rows, columns, and main
diagonals form a Sequence of consecutive integers.
see also ANTIMAGIC SQUARE, MAGIC SQUARE, TALIS-
MAN Square
References
Duncan, D. "Problem 86." Math. Mag. 24, 166, 1951.
Fults, J. L. Magic Squares. Chicago, IL: Open Court, 1974.
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 101-103, 1979.
$$ Weisstein, E. W. "Magic Squares." http: //www. astro.
virginia.edu/-eww6n/math/notebooks/MagicSquares.rn.
Heuman Lambda Function
A °^l™) = F Kn S + l K (m)Z(cf>\l - m),
A (1 — m) 7r
where <f> is the AMPLITUDE, m is the PARAMETER, Z is
the Jacobi Zeta Function, and F{<j>\rn) and K(m)
are incomplete and complete ELLIPTIC INTEGRALS OF
the First Kind.
see also ELLIPTIC INTEGRAL OF THE FlRST KlND, JA-
cobi Zeta 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. 595, 1972.
Heun's Differential Equation
Heterological Paradox
see Grelling's Paradox
Heteroscedastic
A set of Statistical Distributions having different
Variances.
see also HOMOSCEDASTIC, VARIANCE
d 2 w
dx 2
where
s
e
x —
a
dw
dx
i
a/3x — q
x(x
-l)(x-
a)
a + /3-
-7-
-6-
-6+1
= 0.
w = 0,
Heterosquare
9 8 7
2 16
3 4 5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
16
15
References
Erdelyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi,
F. G. Higher Transcendental Functions, Vol. 3. Krieger,
pp. 57-62, 1981.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, p. 576, 1990.
Heuristic
(1) Based on or involving trial and error. (2) Convincing
without being rigorous.
Hex Game
Hexadecimal
835
Hex Game
A two-player Game. There is a winning strategy for
the first player if there is an even number of cells on
each side; otherwise, there is a winning strategy for the
second player.
References
Gardner, M. Ch. 8 in The Scientific American Book of Math-
ematical Puzzles & Diversions. New York, NY: Simon and
Schuster, 1959.
Hex Pyramidal Number
A Figurate Number which is equal to the Cubic
Number n 3 . The first few are 1, 8, 27, 64, . . . (Sloane's
A000578).
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 42-44, 1996.
Sloane, N. J. A. Sequence A000578/M4499 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hex Number
The Centered Hexagonal Number given by
H n = 1 -f 6T n = 2H n -i - H n -2 + 6 = Zn - 3n + 1,
where T n is the nth Triangular Number. The first
few hex numbers are 1, 7, 19, 37, 61, 91, 127, 169, . . .
(Sloane's A003215). The Generating Function of
the hex numbers is
Hexa
see POLYHEX
Hexabolo
A 6-Polyabolo.
Hexacontagon
A 60-sided Polygon.
Hexacronic Icositetrahedron
see Great Hexacronic Icositetrahedron, Small
Hexacronic Icositetrahedron
Hexad
A Set of six.
see also MONAD, QUARTET, QUINTET, TETRAD, TRIAD
,(x 2 + 4 a; + l) =g + 7a;2 + 19x 3 + 3 7a; 4 +
(1 — X) 3
The first TRIANGULAR hex numbers are 1 and 91, and
the first few SQUARE ones are 1, 169, 32761, 6355441, . . .
(Sloane's A006051). SQUARE hex numbers are obtained
by solving the DlOPHANTINE EQUATION
3x 2 + l = y 2 .
The only hex number which is SQUARE and TRIANGU-
LAR is 1. There are no CUBIC hex numbers.
see also MAGIC HEXAGON, CENTERED SQUARE NUM-
BER, Star Number, Talisman Hexagon
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 41, 1996.
Gardner, M. "Hexes and Stars." Ch. 2 in Time Travel and
Other Mathematical Bewilderments. New York: W. H.
Freeman, 1988.
Hindin, H. "Stars, Hexes, Triangular Numbers, and Pythag-
orean Triples." J. Recr. Math. 16, 191-193, 1983-1984.
Sloane, N. J. A. Sequences A003215/M4362 and A006051/
M5409 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Hex (Polyhex)
see Polyhex
Hexadecagon
A 16-sided Polygon, sometimes also called a Hex-
akaidecagon.
see also POLYGON, REGULAR POLYGON, TRIGONOME-
TRY Values — tt/16
Hexadecimal
The base 16 notational system for representing REAL
NUMBERS. The digits used to represent numbers using
hexadecimal NOTATION are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A,
B, C, D, E, and F.
see also Base (Number), Binary, Decimal, Meta-
drome, Octal, Quaternary, Ternary, Vigesimal
References
$ Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/~eww6n/math/notebooks/Bases.m.
836 Hexafiexagon
Hexagon
Hexafiexagon
A FLEXAGON made by folding a strip into adjacent
Equilateral Triangles. The number of states possi-
ble in a hexafiexagon is the Catalan Number C 4 = 42,
see also Flexagon, Flexatube, Tetraflexagon
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 205-207, 1989.
Gardner, M. Ch. 1 in The Scientific American Book of Math-
ematical Puzzles & Diversions. New York: Simon and
Schuster, 1959.
Gardner, M. Ch. 2 in The Second Scientific American Booh
of Mathematical Puzzles & Diversions: A New Selection.
New York: Simon and Schuster, 1961.
Maunsell, F. G. "The Flexagon and the Hexafiexagon."
Math. Gazette 38, 213-214, 1954.
Wheeler, R. F. "The Flexagon Family" Math. Gaz. 42, 1-6,
1958.
Hexagon
A six-sided Polygon. In proposition IV.15, Euclid
showed how to inscribe a regular hexagon in a CIRCLE.
The Inradius r, Circumradius R, and Area A can
be computed directly from the formulas for a general
regular POLYGON with side length s and n — 6 sides,
r=iscot(^-J = iV3s
(1)
R = |scsc 1 — J = s
(2)
A = fns 2 cot (J) = |>/3s a .
(3)
Therefore, for a regular hexagon,
R /tt\ 2
— = sec 1 — 1 = — = ,
r \6J y/3'
(4)
so
A * -( R Y - 4
A T \r) 3 '
(5)
A Plane Perpendicular to a C 3 axis of a Cube,
Dodecahedron, or Icosahedron cuts the solid in
a regular HEXAGONAL CROSS-SECTION (Holden 1991,
pp. 22-23 and 27). For the CUBE, the PLANE passes
through the Midpoints of opposite sides (Steinhaus
1983, p. 170; Cundy and Rollett 1989, p. 157; Holden
1991, pp. 22-23). Since there are four such axes for the
Cube and Octahedron, there are four possibly hexag-
onal cross- sections. Since there are four such axes in
each case, there are also four possibly hexagonal cross-
sections.
Take seven CIRCLES and close-pack them together in a
hexagonal arrangement. The PERIMETER obtained by
wrapping a band around the CIRCLE then consists of
six straight segments of length d (where d is the DIAME-
TER) and 6 arcs with total length 1/6 of a CIRCLE. The
Perimeter is therefore
p = (12 + 27r)r = 2(6 + n)r.
(6)
see also Cube, Cyclic Hexagon, Dissection, Do-
decahedron, Graham's Biggest Little Hexagon,
Hexagon Polyiamond, Hexagram, Magic Hexa-
gon, Octahedron, Pappus's Hexagon Theorem,
Pascal's Theorem, Talisman Hexagon
References
Cundy, H. and Rollett, A. "Hexagonal Section of a Cube."
§3.15.1 in Mathematical Models, 3rd ed. Stradbroke, Eng-
land: Tarquin Pub., p. 157, 1989.
Dixon, R. Mathographics. New York: Dover, p. 16, 1991.
Holden, A. Shapes, Space, and Symmetry. New York; Dover,
1991.
Pappas, T. "Hexagons in Nature." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 74-75, 1989.
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, 1983.
Hexagon Polyiamond
Hexagon Polyiamond
A 6-POLYIAMOND.
see also Hexagon
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Hexagonal Number
A Figurate Number and 6-Polygonal Number of
the form n(2n - 1). The first few are 1, 6, 15, 28, 45,
... (Sloane's A000384). The Generating Function
of the hexagonal numbers
x{3x + }} = x + 6x 2 + 15z 3 + 28x 4 + . . . .
(1 - x) 6
Every hexagonal number is a TRIANGULAR NUMBER
since
r(2r-l) = £(2r-l)[(2r-l) + l].
In 1830, Legendre (1979) proved that every number
larger than 1791 is a sum of four hexagonal numbers,
and Duke and Schulze-Pillot (1990) improved this to
three hexagonal numbers for every sufficiently large in-
teger. The numbers 11 and 26 can only be represented
as a sum using the maximum possible of six hexagonal
numbers:
11 = 1 + 1 + 1 + 1 + 1 + 6
26= 1 + 1 + 6 + 6 + 6 + 6.
see also Figurate Number, Hex Number, Triangu-
lar Number
References
Duke, W. and Schulze-Pillot, R. "Representations of Integers
by Positive Ternary Quadratic Forms and Equidistribution
of Lattice Points on Ellipsoids." Invent. Math. 99, 49-57,
1990.
Guy, R. K. "Sums of Squares." §C20 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 136-138, 1994.
Legendre, A.-M. Theorie des nombres, J^th ed., 2 vols. Paris:
A. Blanchard, 1979.
Sloane, N. J. A. Sequence A000384/M4108 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hexahedron 837
Hexagonal Pyramidal Number
A Pyramidal Number of the form n(n+l)(4n- l)/6,
The first few are 1, 7, 22, 50, 95, . . . (Sloane's A002412).
The Generating Function of the hexagonal pyrami-
dal numbers is
X ^ X + }} = x + 7x 2 + 22z 3 + 50x 4 + . . . .
(x - l) 4
References
Sloane, N. J. A. Sequence A002412/M4374 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hexagonal Scalenohedron
An irregular DODECAHEDRON which is also a TRAPE-
ZOHEDRON.
see also DODECAHEDRON, TRAPEZOHEDRON
References
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 63, 1990.
Hexagonal Tiling
see Tiling
Hexagram
The Star Polygon {6, 2}, also known as the Star of
David.
see also DISSECTION, PENTAGRAM, SOLOMON'S SEAL
Knot, Star Figure, Star of Lakshmi
Hexagrammum Mysticum Theorem
see Pascal's Theorem
Hexahedron
A hexahedron is a six-sided POLYHEDRON. The regu-
lar hexahedron is the CUBE, although there are seven
topologically different CONVEX hexahedra (Guy 1994).
see also CUBE
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 189, 1994.
838
Hexahemioctahedron
HighLife
Hexahemioctahedron
The Dual Polyhedron of the Cubohemioctahe-
dron.
Hexakaidecagon
see HEXADECAGON
Hexakis Icosahedron
see DlSDYAKIS TRIACONTAHEDRON
Hexakis Octahedron
see DlSDYAKIS DODECAHEDRON
Hexlet
Also called Soddy's Hexlet. Consider three mutually
tangent Spheres A, B, and C. Then construct a chain
of Spheres tangent to each of A, £?, and C threading
and interlocking with the A — B — C ring. Surprisingly,
every chain closes into a "necklace" after six Spheres
regardless of where the first Sphere is placed. This is
a special case of Kollros' Theorem. The centers of
a Soddy hexlet always lie on an Ellipse (Ogilvy 1990,
p. 63).
see also Coxeter's Loxodromic Sequence of Tan-
gent Circles, Kollros' Theorem, Steiner Chain
References
Coxeter, H. S. M. "Interlocking Rings of Spheres." Scripta
Math. 18, 113-121, 1952.
Gosset, T. "The Hexlet." Nature 139, 251-252, 1937.
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 49-50, 1976.
Morley, F. "The Hexlet." Nature 139, 72-73, 1937.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 60-72, 1990.
Soddy, F. "The Bowl of Integers and the Hexlet." Nature
139, 77-79, 1937.
Soddy, F. "The Hexlet." Nature 139, 154 and 252, 1937.
HexLife
An alternative Life game similar to Conway's, which
is played on a hexagonal grid. No set of rules has yet
emerged as uniquely interesting.
see also HighLife
References
Hensel, A. "A Brief Illustrated Glossary of Terms in Con-
way's Game of Life." http://www.cs.jhu.edu/-callahan/
glossary.html.
Hexomino
One of the 35 6-Polyominoes.
References
Pappas, T. "Triangular, Square & Pentagonal Numbers."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, p. 214, 1989.
Heyting Algebra
An Algebra which is a special case of a Logos.
see also LOGOS, TOPOS
Hh Function
Let
Z{x) ee -^e~* 2/2
Q(x)
dt,
(1)
(2)
where Z and Q are closely related to the NORMAL DIS-
TRIBUTION Function, then
Hh_ n (x) = (-l) n - 1 v / 2^z ( "- 1) (x) (3)
HM«) = ^Hh_ l( «)£
Q(x)
Z(x)
(4)
see also Normal Distribution Function, Tetra-
choric Function
Hi-Q
A triangular version of PEG SOLITAIRE with 15 holes
and 14 pegs. Numbering hole 1 at the apex of the tri-
angle and thereafter from left to right on the next lower
row, etc., the following table gives possible ending holes
for a single peg removed (Beeler et al. 1972, Item 76).
Because of symmetry, only the first five pegs need be
considered. Also because of symmetry, removing peg 2
is equivalent to removing peg 3 and flipping the board
horizontally.
remove possible ending pegs
1,7 = 10, 13
2, 6, 11, 14
3 = 12, 4, 9, 15
13
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. Item 75 in
HAKMEM. Cambridge, MA: MIT Artificial Intelligence
Laboratory, Memo AIM-239, Feb. 1972.
Higher Arithmetic
An archaic term for Number THEORY.
Highest Weight Theorem
A theorem proved by E. Cartan (1913) which classifies
the irreducible representations of COMPLEX semisimple
Lie Algebras.
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
HighLife
An alternate set of LIFE rules similar to Conway's, but
with the additional rule that six neighbors generate a
birth. Most of the interest in this variant is due to the
presence of a so-called replicator.
see also HEXLlFE, LIFE
References
Hensel, A. "A Brief Illustrated Glossary of Terms in Con-
way's Game of Life." http://www.cs.jhu.edu/-callahan/
glossary.html.
Highly Abundant Number
Hilbert Basis Theorem
839
Highly Abundant Number
see Highly Composite Number
Highly Composite Number
A Composite Number (also called a Superabundant
Number) is a number n which has more FACTORS than
any other number less than n. In other words, o~(n)/n
exceeds a(k)/k for all k < n, where cr(n) is the Dl VISOR
FUNCTION. They were called highly composite numbers
by Ramanujan, who found the first 100 or so, and su-
perabundant by Alaoglu and Erdos (1944).
There are an infinite numbers of highly composite num-
bers, and the first few are 2, 4, 6, 12, 24, 36, 48, 60,
120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, . . .
(Sloane's A002182). Ramanujan (1915) listed 102 up
to 6746328388800 (but omitted 293, 318, 625, 600, and
29331862500). Robin (1983) gives the first 5000 highly
composite numbers, and a comprehensive survey is given
by Nicholas (1988).
If
jV = 2 a2 3 a3 -"p ap
(1)
is the prime decomposition of a highly composite num-
ber, then
1. The PRIMES 2, 3, . . . , p form a string of consecutive
Primes,
2. The exponents are nonincr easing, so a2 > a 3 > . . . >
a p , and
3. The final exponent a p is always 1, except for the two
cases N = 4 = 2 2 and N = 36 = 2 2 • 3 2 , where it is
2.
Let Q(x) be the number of highly composite numbers
< x. Ramanujan (1915) showed that
Km Qi - X) m
nm = oo.
x-^oo ma;
(2)
Erdos (1944) showed that there exists a constant a >
such that
Q(x) > (lnx) 1+ci (3)
Nicholas proved that there exists a constant ci > such
that
Q(x)«(lnx) C2 . (4)
see also Abundant Number
References
Alaoglu, L. and Erdos, P. "On Highly Composite and Similar
Numbers." Trans. Amer. Math. Soc. 56, 448-469, 1944.
Andree, R. V. "Ramanujan's Highly Composite Numbers."
Abacus 3, 61-62, 1986.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, p. 53, 1994.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, p. 323,
1952.
Flammenkamp, A. http://www.minet .uni-jena.de/-achim/
highly.html.
Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., p. 112, 1973.
Honsberger, R. "An Introduction to Ramanujan's Highly
Composite Numbers." Ch. 14 in Mathematical Gems III.
Washington, DC: Math. Assoc. Amer., pp. 193-207, 1985.
Kanigel, R. The Man Who Knew Infinity: A Life of the
Genius Ramanujan. New York: Washington Square Press,
p. 232, 1991.
Nicholas, J.-L. "On Highly Composite Numbers." In Ra-
manujan Revisited: Proceedings of the Centenary Confer-
ence (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin).
Boston, MA: Academic Press, pp. 215-244, 1988.
Ramanujan, S. "Highly Composite Numbers." Proc. London
Math. Soc. 14, 347-409, 1915.
Ramanujan, S. Collected Papers. New York: Chelsea, 1962.
Robin, G. "Methodes d'optimalisation pour un probleme de
theories des nombres." RAIRO Inform. Theor. 17, 239-
247, 1983.
Sloane, N. J. A. Sequence A002182/M1025 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Wells, D. The Penguin Dictionary of Curious and Interesting
Numbers. New York: Penguin Books, p. 128, 1986.
Higman-Sims Group
The Sporadic Group HS.
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat.bham.ac.uk/atlas/HS.html.
Hilbert's Axioms
The 21 assumptions which underlie the GEOMETRY pub-
lished in Hilbert's classic text Grundlagen der Geome-
tric The eight Incidence Axioms concern collinear-
ity and intersection and include the first of EUCLID'S
Postulates, The four Ordering Axioms concern the
arrangement of points, the five CONGRUENCE AXIOMS
concern geometric equivalence, and the three Continu-
ity AXIOMS concern continuity. There is also a single
parallel axiom equivalent to Euclid's PARALLEL POSTU-
LATE.
see also CONGRUENCE AXIOMS, CONTINUITY AXIOMS,
Incidence Axioms, Ordering 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.
Hilbert Basis Theorem
If R is a NOETHERIAN RING, then 5 = R[X] is also a
Noetherian Ring.
see also Algebraic Variety, Fundamental System,
Syzygy
References
Hilbert, D. "Uber die Theorie der algebraischen Formen."
Math. Ann. 36, 473-534, 1890.
840
Hubert's Constants
Hilbert Hotel
Hilbert's Constants
>N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Extend Hilbert's Inequality by letting p, q > 1 and
so that
P Q
0<A = 2----<1.
P Q
(1)
(2)
Levin (1937) and Steckin (1949) showed that
V^ V^ a mh n < f 7r(q - 1) 1
l^i jLt (m + n) x - 1 * CSC Xq J
m=l n=l V V L J J
poo \ i/p / /»oo \ i/q
J [f(x)} p dxj (jf [g(x)Ydxj (3)
and
c)fl(y)
/°° [°° {^^ dxdy <tt esc
Jo Jo ( x + v) x
"•(q - 1)
(/>oo \ 1/P / />oo \ 1/q
y [/(x)]'dxj l J [g(x)]"dx\ . (4)
Mitrinovic et al. (1991) indicate that this constant is the
best possible.
see also Hilbert's Inequality
References
Finch, S. "Favorite Mathematical Constants." http: //www.
mathsof t . com/asolve/constant /hilbert /hilbert .html.
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities
Involving Functions and Their Integrals and Derivatives.
Dordrecht, Netherlands: Kluwer, 1991.
Steckin, S. B. "On the Degree of Best Approximation to Con-
tinuous Functions." Dokl. Akad. Nauk SSSR 65, 135-137,
1949.
Hilbert Curve
A Lindenmayer System invented by Hilbert (1891)
whose limit is a Plane-Filling Curve which fills
a square. Traversing the VERTICES of an n-D Hy-
PERCUBE in GRAY CODE order produces a genera-
tor for the n-D Hilbert curve (Goetz). The Hilbert
curve can be simply encoded with initial string
H L", String Rewriting rules "L" -> "+RF-LFL-FR+",
H R"->"-LF+RFR+FL-", and angle 90° (Peitgen and Saupe
1988, p. 278).
A related curve is the Hilbert II curve, shown
above (Peitgen and Saupe 1988, p. 284). It is
also a Lindenmayer System and the curve can be
encoded with initial string "X", STRING REWRIT-
ING rules "X" -> "XFYFX+F+YFXFY-F-XFYFX", "Y" ->
"YFXFY-F-XFYFX+F+YFXFY", and angle 90°.
see also Lindenmayer System, Peano Curve,
Plane-Filling Curve, Sierpinski Curve, Space-
Filling Curve
References
Bogomolny, A. "Plane Filling Curves." http://www.cut-
the-knot.com/do_you_know/hilbert.html.
Dickau, R. M. "Two-Dimensional L-Systems." http://
forum.swarthmore.edu/advanced/robertd/lsys2d.html.
Dickau, R. M. "Three-Dimensional L-Systems." http://
forum.swarthmore.edu/advanced/robertd/lsys3d.html.
Goetz, P. "Phil's Good Enough Complexity Dictionary."
http : //www . cs . buffalo . edu/~goetz/dict .html.
Hilbert, D. "Uber die stetige Abbildung einer Linie auf ein
Flachenstiick." Math. Ann. 38, 459-460, 1891.
Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Frac-
tal Images. New York: Springer- Verlag, pp. 278 and 284,
1988.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 198-206, 1991.
Hilbert Function
Let F — {pi, . . . ,p m } C P 2 be a collection of m distinct
points. Then the number of conditions imposed by V
on forms of degree d is called the Hilbert function hr of
T. If curves X\ and X2 of degrees d and e meet in a
collection r of d • e points, then for any fc, the number
hr(k) of conditions imposed by T on forms of degree k
is independent of X\ and X2 and is given by
h r (k)
-Cr)-('-r)
-e-r) + e~v +2 >
where the BINOMIAL COEFFICIENT (£) is taken as if
a< 2 (Cayley 1843).
References
Eisenbud, D.; Green, M.; and Harris, J. "Cayley-Bacharach
Theorems and Conjectures." Bull. Amer. Math. Soc. 33,
295-324, 1996.
Hilbert Hotel
Let a hotel have a Denumerable set of rooms num-
bered 1, 2, 3, Then any finite number n of
guests can be accommodated without evicting the cur-
rent guests by moving the current guests from room i
to room i + n. Furthermore, a DENUMERABLE number
Hilbert's Inequality
Hilbert's Problems
841
of guests can be similarly accommodated by moving the
existing guests from i to 2i, freeing up a D ENUMERABLE
number of rooms 2% — 1.
References
Lauwerier, H. "Hilbert Hotel." In Fractals: Endlessly Re-
peated Geometric Figures, Princeton, NJ: Princeton Uni-
versity Press, p. 22, 1991.
Pappas, T. "Hotel Infinity." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, p. 37, 1989,
Hilbert's Inequality
Given a Positive Sequence {a n },
£
Y--
j -n
n= — oo
<7T
\ n= — oo
where the a n s are REAL and "square summable."
Another INEQUALITY known as Hilbert's applies to
NONNEGATIVE sequences {a n } and {& n },
oo oo
EE
OimOm,
m + n
< 7TCSC
unless all a n or all b n are 0. If f(x) and g(x) are NON-
NEGATIVE integrable functions, then the integral form
is
-a
[f{x)Ydx
[g{x)] q dx
l/Q
The constant 7T csc(7r/P) is the best possible, in the sense
that counterexamples can be constructed for any smaller
value.
References
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 308-309, 1988.
Hilbert Matrix
A MATRIX H with elements
Hij^ii + j-l)- 1
for i, j = 1, 2, . . . , n. Although the Matrix Inverse is
given analytically by
(f 1 );
(-l) i+j (n + i-l)\(n + j-l)\
i + i - 1 [(i - l)!(j - 1)!] 2 (^ - i)l(n - j)\ '
Hilbert matrices are difficult to invert numerically. The
Determinants for the first few values of H n are given
in the following table.
n det(H n )
2 8.33333 x 10"
-4
1-7
3 4.62963 x 10
4 1.65344 x 10"
5 3.74930 x 10 -12
6 5.36730 x 10~ 18
Hilbert's Nullstellansatz
Let K be an algebraically closed field and let I be an
Ideal in K{x), where x = (a?i,#2, . • . yX n ) is a finite set
of indeterminates. Let p e K(x) be such that for any
(ci, . . . , c n ) in K n , if every element of I vanishes when
evaluated if we set each {xi = c,), then p also vanishes.
Then p J lies in J for some j. Colloquially, the theory of
algebraically closed fields is a complete model.
Hilbert Number
see Gelfond-Schneider Constant
Hilbert Polynomial
Let T be an Algebraic Curve in a projective space of
Dimension n, and let p be the Prime Ideal defining T,
and let x(Pi m ) De tne number of linearly independent
forms of degree m modulo p. For large m, x(p> m ) 1S a
Polynomial known as the Hilbert polynomial.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 36, 1980.
Hilbert's Problems
A set of (originally) unsolved problems in mathematics
proposed by Hilbert. Of the 23 total, ten were presented
at the Second International Congress in Paris in 1900.
These problems were designed to serve as examples for
the kinds of problems whose solutions would lead to the
furthering of disciplines in mathematics.
la. Is there a transfinite number between that of a
Denumerable Set and the numbers of the CON-
TINUUM? This question was answered by Godel
and Cohen to the effect that the answer depends
on the particular version of Set Theory as-
sumed.
lb. Can the Continuum of numbers be considered a
Well-Ordered Set? This question is related
to Zermelo's AXIOM OF CHOICE. In 1963, the
Axiom of Choice was demonstrated to be inde-
pendent of all other Axioms in Set Theory, so
there appears to be no universally valid solution
to this question either.
2. Can it be proven that the AXIOMS of logic are con-
sistent? Godel's Incompleteness Theorem
indicated that the answer is "no," in the sense
842
Hilbert's Problems
Hilberfs Problems
that any formal system interesting enough to for-
mulate its own consistency can prove its own con-
sistency Iff it is inconsistent.
3. Give two TETRAHEDRA which cannot be de-
composed into congruent TETRAHEDRA directly
or by adjoining congruent TETRAHEDRA. Max
Dehn showed this could not be done in 1902.
W. F. Kagon obtained the same result indepen-
dently in 1903.
4. Find GEOMETRIES whose AXIOMS are closest to
those of Euclidean Geometry if the Ordering
and Incidence Axioms are retained, the CON-
GRUENCE AXIOMS weakened, and the equivalent
of the Parallel Postulate omitted. This prob-
lem was solved by G. Hamel.
5 . Can the assumption of differentiability for
functions denning a continuous transformation
GROUP be avoided? (This is a generalization of
the Cauchy Functional Equation.) Solved by
John von Neumann in 1930 for bicompact groups.
Also solved for the Abelian case, and for the solv-
able case in 1952 with complementary results by
Montgomery and Zipin (subsequently combined
by Yamabe in 1953). Andrew Glean showed in
1952 that the answer is also "yes" for all locally
bicompact groups.
6. Can physics be axiomized?
7. Let a ^ 1 ^ be Algebraic and (3 Irrational.
Is a p then TRANSCENDENTAL? Proved true in
1934 by Aleksander Gelfond (Gelfond's THEO-
REM; Courant and Robins 1996).
8. Prove the Riemann Hypothesis. The Conjec-
ture has still been neither proved nor disproved.
9. Construct generalizations of the RECIPROCITY
Theorem of Number Theory.
10. Does there exist a universal algorithm for solving
Diophantine Equations? The impossibility of
obtaining a general solution was proven by Ju-
lia Robinson and Martin Davis in 1970, following
proof of the result that the equation n = i*2m
(where F 2m is a FIBONACCI NUMBER) is Dio-
phantine by Yuri Matijasevich (Matijasevic 1970,
Davis 1973, Davis and Hersh 1973, Matijasevic
1993).
11. Extend the results obtained for quadratic fields to
arbitrary INTEGER algebraic fields.
12. Extend a theorem of Kronecker to arbitrary alge-
braic fields by explicitly constructing Hilbert class
fields using special values. This calls for the con-
struction of Holomorphic FUNCTIONS in several
variables which have properties analogous to the
exponential function and elliptic modular func-
tions (Holtzapfel 1995).
13. Show the impossibility of solving the general sev-
enth degree equation by functions of two variables.
14. Show the finiteness of systems of relatively inte-
gral functions.
15. Justify Schubert's Enumerative Geometry
(Bell 1945).
16. Develop a topology of Real algebraic curves and
surfaces. The SHIMURA-TANIYAMA CONJECTURE
postulates just this connection. See Ilyashenko
and Yakovenko (1995) and Gudkov and Utkin
(1978).
17. Find a representation of definite form by
Squares.
18. Build spaces with congruent POLYHEDRA.
19. Analyze the analytic character of solutions to vari-
ational problems.
20. Solve general BOUNDARY VALUE PROBLEMS.
21. Solve differential equations given a MONODROMY
Group. More technically, prove that there always
exists a FuCHSIAN SYSTEM with given singular-
ities and a given MONODROMY GROUP. Several
special cases had been solved, but a NEGATIVE so-
lution was found in 1989 by B. Bolibruch (Anasov
and Bolibruch 1994).
22. Uniformization.
23. Extend the methods of Calculus of Varia-
tions.
References
Anasov, D. V. and Bolibruch, A. A. The Riemann- Hilbert
Problem. Braunschweig, Germany: Vieweg, 1994.
Bell, E. T. The Development of Mathematics, 2nd ed. New
York: McGraw-Hill, p. 340, 1945.
Borowski, E. J. and Borwein, J. M. (Eds.). "Hilbert Prob-
lems." Appendix 3 in The Harper Collins Dictionary of
Mathematics. New York: Harper- Collins, p. 659, 1991.
Boyer, C and Merzbach, U. "The Hilbert Problems." His-
tory of Mathematics, 2nd ed. New York: Wiley, pp. 610-
614, 1991.
Browder, Felix E. (Ed.). Mathematical Developments Aris-
ing from Hilbert Problems. Providence, RI: Amer. Math.
Soc, 1976.
Courant, R. and Robbins, H. What is Mathematics? : An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 107, 1996.
Davis, M. "Hilbert's Tenth Problem is Unsolvable." Amer.
Math. Monthly 80, 233-269, 1973.
Davis, M. and Hersh, R. "Hilbert's 10th Problem." Sci.
Amer., pp. 84-91, Nov. 1973.
Gudkov, D. and Utkin, G. A. Nine Papers on Hilbert's 16th
Problem. Providence, RI: Amer. Math. Soc, 1978.
Holtzapfel, R.-P. The Ball and Some Hilbert Problems.
Boston, MA: Birkhauser, 1995.
Ilyashenko, Yu. and Yakovenko, S. (Eds.). Concerning the
Hilbert 16th Problem. Providence, RI: Amer. Math. Soc,
1995.
Matijasevic, Yu. V. "Solution to of the Tenth Problem of
Hilbert." Mat. Lapok 21, 83-87, 1970.
Matijasevich, Yu. V. Hilbert's Tenth Problem. Cambridge,
MA: MIT Press, 1993.
Hilbert-Schmidt Norm
Hilbert-Schmidt Norm
The Hilbert-Schmidt norm of a MATRIX A is denned as
|A|a = A E ay '
Hilbert-Schmidt Theory
The study of linear integral equations of the Predholm
type with symmetric kernels
K(x,t) =K(t,x).
References
Arfken, G. "Hilbert-Schmidt Theory." §16.4 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 890-897, 1985.
Hill Determinant 843
Hilbert's Theorem
Every MODULAR SYSTEM has a MODULAR SYSTEM
Basis consisting of a finite number of Polynomials.
Stated another way, for every order n there exists a non-
singular curve with the maximum number of circuits and
the maximum number for any one nest.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 61, 1959.
Hilbert Transform
g(y) = -
f{x) = i
7T
f(x) dx
x-y
g{y) dy
y- x
Hilbert Space
A Hilbert space is Vector Space H with an INNER
Product {/, g) such that the NORM defined by
turns H into a COMPLETE METRIC SPACE. If the INNER
PRODUCT does not so define a NORM, it is instead known
as an Inner Product Space.
Examples of FlNITE-dimensional Hilbert spaces include
1. The Real Numbers W 1 with (v,u) the vector Dot
Product of v and u.
2. The Complex Numbers C n with (v, it) the vector
Dot Product oft; and the Complex Conjugate
of it.
An example of an iNFlNITE-dimensional Hilbert space is
L 2 , the Set of all Functions / : R -> K such that the
Integral of f 2 over the whole Real Line is Finite.
In this case, the INNER PRODUCT is
</,
\9) = Jf(x.
)g(x) dx.
see also Titchmarsh Theorem
References
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, pp. 267-272, 1965.
Hill Determinant
A Determinant which arises in the solution of the
second-order Ordinary Differential Equation
2 d 2 tp dtp
+ x^+|±/ l V + i/i 2 -& +
1L
dx 2 ' ~dx ' V 4 '" ~ ' 2 " ' ' 4z 2
Writing the solution as a POWER SERIES
^ = 0.
(1)
^ ~ zl anxS
+2n
(2)
n~ — oo
gives a RECURRENCE RELATION
/i 2 a n+ i + [2h 2 - 46 + 16(n + \s) 2 ]a n + h a n _i = 0. (3)
The value of s can be computed using the Hill determi-
nant
A Hilbert space is always a BANACH SPACE, but the
converse need not hold.
see also Banach Space, L 2 -Norm, L 2 -Space, Liou-
ville Space, Parallelogram Law, Vector Space
References
Sansone, G. "Elementary Notions of Hilbert Space." §1.3 in
Orthogonal Functions, rev. English ed. New York: Dover,
pp. 5-10, 1991.
Stone, M. H. Linear Transformations in Hilbert Space and
Their Applications Analysis. Providence, RI: Amer. Math.
Soc, 1932.
A(-) =
where
t"+2)-
2
4-a*
«a a 2
a 2 = i6-K
o
_2i
■ (4)
(5)
(6)
(7)
844 Hill's Differential Equation
and <j is the variable to solve for. The determinant can
be given explicitly by the amazing formula
Hippopede
A(s) = A(0) -
sin 2 (7rs/2)
sin 2 (±7r v /& Z l*?)
(8)
where
A(0)
64+2h 3 -4b
144+2h 2 -4b
1
16 + 2fc 2 -4b
64+2h 2 -4b
1
2h 2 -4b
16 + 2h 2 ~4b
1
16 + 2fc 2 -4b
(9)
leading to the implicit equation for s,
sin 2 (±7rs) = A(0)sin 2 {\<xyjb- \h? j . (10)
see also HILL'S DIFFERENTIAL EQUATION
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 555-562, 1953.
Hill's Differential Equation
dt 2
<j)(t)x,
where <j> is periodic. It can be written as
CO
00 + 2 ^ n cos(2nz)
dx 2
= 0,
where 6 n are known constants. A solution can be given
by taking the "DETERMINANT" of an infinite MATRIX.
see also HlLL DETERMINANT
Hillam's Theorem
If / : [a, 6] — > [a, b] (where [a, b] denotes the CLOSED
Interval from a to b on the Real Line) satisfies a
Lipschitz Condition with constant K, i.e., if
\f(x)-f{y)\<K\x-y\
for all x, y € [a, 6], then the iteration scheme
Xn + l = (1 - X)x n + A/(x n ),
where A = l/(if + 1), converges to a FIXED POINT of /.
References
Falkowski, B.-J. "On the Convergence of Hillam's Iteration
Scheme." Math. Mag. 69, 299-303, 1996.
Geist, R.; Reynolds, R.; and Suggs, D. "A Markovian Frame-
work for Digital Halftoning." ACM Trans. Graphics 12,
136-159, 1993.
Hillam, B. P. "A Generalization of Krasnoselski's Theorem
on the Real Line." Math. Mag. 48, 167-168, 1975.
Krasnoselski, M. A. "Two Remarks on the Method of Suc-
cessive Approximations." Uspehi Math. Nauk (N. S.) 10,
123-127, 1955.
Hindu Check
see Casting Out Nines
Hinge
1 M 4n+5
150 895 1895
250 895 1099 1775
688 895 1166 1699
795 795 1333 1693
795 1499
The upper and lower hinges are descriptive statistics of
a set of N data values, where N is of the form N —
4n + 5 with n — 0, 1, 2, . . . . The hinges are obtained by
ordering the data in increasing order oi, . . . , a;v, and
writing them out in the shape of a "w" as illustrated
above. The values at the bottom legs are called the
hinges Hi and B.% (and the central peak is the Median).
In this ordering,
Hi = a n +2 = a(iv+3)/4
M = a2n+3 = 0(JV+l)/2
H 2 = a 3 n+4 ~ a(3JV+l)/4-
For N of the form An + 5, the hinges are identical to
the Quartiles. The difference H 2 - Ht is called the
H-SPREAD.
see also H-Spread, Haberdasher's Problem, Me-
dian (Statistics), Order Statistic, Quartile,
Trimean
References
Tukey, J. YV\ Explanatory Data Analysis.
Addison-Wesley, pp. 32-34, 1977.
Hippias' Quadratrix
see Quadratrix of Hippias
Reading, MA:
Hippopede
Histogram
Hodge's Theorem 845
A curve also known as a Horse Fetter and given by
the polar equation
r 2 ^4b(a-bsin6).
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 144-146, 1972.
Histogram
123456789 10
The grouping of data into bins (spaced apart by the so-
called Class Interval) plotting the number of mem-
bers in each bin versus the bin number. The above his-
togram shows the number of variates in bins with CLASS
Interval 1 for a sample of 100 real variates with a Un-
iform Distribution from and 10. Therefore, bin 1
gives the number of variates in the range 0-1, bin 2 gives
the number of variates in the range 1-2, etc.
see also OGIVE
Hitch
A KNOT that secures a rope to a post, ring, another
rope, etc., but does not keep its shape by itself.
see also Clove Hitch, Knot, Link, Loop (Knot)
References
Owen, P. Knots. Philadelphia, PA: Courage, p. 17, 1993.
Hitting Set
Let S be a collection S of subsets of a finite set X. The
smallest subset Y of X that meets every member of S
is called the hitting set or Vertex Cover. Finding the
hitting set is an NP-COMPLETE PROBLEM.
Hjelmslev's Theorem
When all the points P on one line are related by an
ISOMETRY to all points P f on another, the MIDPOINTS
of the segments PP' are either distinct and collinear or
coincident.
HJLS Algorithm
An algorithm for finding INTEGER RELATIONS whose
running time is bounded by a polynomial in the num-
ber of real variables. It is much faster than other algo-
rithms such as the Ferguson-Forcade Algorithm,
LLL Algorithm, and PSOS Algorithm.
Unfortunately, it is numerically unstable and therefore
requires extremely high precision. The cause of this in-
stability is not known (Ferguson and Bailey 1992), but is
believed to derive from its reliance on Gram-Schmidt
ORTHONORMALIZATION, which is know to be numeri-
cally unstable (Golub and van Loan 1989).
see also Ferguson-Forcade Algorithm, Integer
Relation, LLL Algorithm, PSLQ Algorithm,
PSOS Algorithm
References
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.
Golub, G. H. and van Loan, C. F. Matrix Computations, 3rd
ed. Baltimore, MD: Johns Hopkins, 1996.
Hastad, J.; Just, B.; Lagarias, J. C; and Schnorr, C, P.
"Polynomial Time Algorithms for Finding Integer Rela-
tions Among Real Numbers." SI AM J. Comput. 18, 859-
881, 1988.
HK Integral
Named after Henstock and KurzweiL Every LEBESGUE
INTEGRABLE function is HK integrable with the same
value.
References
Shenitzer, A. and Steprans, J. "The Evolution of Integra-
tion." Amer. Math. Monthly 101, 66-72, 1994.
Hodge Star
On an oriented n-D RiEMANNlAN MANIFOLD, the Hodge
star is a linear FUNCTION which converts alternating
Differential /c-Forms to alternating (n — &)-forms.
If w is an alternating fc-FORM, its Hodge star is given
by
w{v u . . . ,Vk) = {*w)(v k +i,-. .,v n )
when v±, . . . , v n is an oriented orthonormal basis.
see also Stokes' Theorem
Hodge's Theorem
On a compact oriented FlNSLER MANIFOLD without
boundary, every COHOMOLOGY class has a UNIQUE har-
monic representative. The Dimension of the Space of
all harmonic forms of degree p is the pth Betti Number
of the Manifold.
see also Betti Number, Cohomology, Dimension,
Finsler Manifold
References
Chern, S.-S. "Finsler Geometry is Just Riemannian Geome-
try without the Quadratic Restriction." Not. Amer. Math.
Soc. 43, 959-963, 1996.
846
Hoehn's Theorem
Hofstadter-Conway $10,000 Sequence
Hoehn's Theorem
A geometric theorem related to the Pentagram and
also called the Pratt-Kasapi Theorem.
\V 1 W 1 \ \V 2 W 2 \ \V 3 Ws\ \V 4 W 4 \ \V 5 W 5 \ _ 1
\W 2 V 3 \ \W S V 4 \ \W 4 V 5 \ \W 5 Vi\ |WiV 2 |
|ViW 2 | \V 2 W 3 \ \V 3 W 4 \ \V 4 W 5 \ \Vf>Wi\
= 1.
\WiV 3 \ \w 2 v 4 \ \W 3 V 5 \ |W 4 Vi| \W 5 V 2 \
In general, it is also true that
\V{Wi\ _ IViVj+iVi+41 IViVi+iVi+aVi+sl
\W i+1 Vi
i+lVi+2\
\ViV i+1 V i+2 V i+4 \ IVi+aVi+sVi+il
This type of identity was generalized to other figures in
the plane and their duals by Pinkernell (1996).
References
Chou, S. C. Mechanical Geometry Theorem Proving. Dor-
drecht, Netherlands: Reidel, 1987.
Griinbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the
Area Principle." Math. Mag. 68, 254-268, 1995.
Hoehn, L. "A Menelaus-Type Theorem for the Pentagram."
Math. Mag. 68, 254-268, 1995.
Pinkernell, G. M. "Identities on Point-Line Figures in the
Euclidean Plane." Math. Mag. 69, 377-383, 1996.
Hoffman's Minimal Surface
A minimal embedded surface discovered in 1992 con-
sisting of a helicoid with a Hole and Handle (Science
News 1992). It has the same topology as a punctured
sphere with a handle, and is only the second complete
embedded minimal surface of finite topology and infi-
nite total curvature discovered (the HELICOID being the
first).
A three-ended minimal surface GENUS 1 is sometimes
also called Hoffman's minimal surface (Peterson 1988).
see also Helicoid
References
Peterson, I. Mathematical Tourist: Snapshots of Modern
Mathematics. New York: W. H. Freeman, pp, 57-59, 1988.
"Putting a Handle on a Minimal Helicoid." Sci. News 142,
276, Oct. 24, 1992.
Hoffman-Singleton Graph
The only GRAPH of DIAMETER 2, GlRTH 5, and VA-
LENCY 7. It contains many copies of the PETERSEN
Graph.
References
Hoffman, A. J. and Singleton, R. R. "On Moore Graphs of
Diameter Two and Three." IBM J. Res. Develop. 4, 497-
504, 1960.
Hofstadter-Conway $10,000 Sequence
The Integer Sequence defined by the Recurrence
Relation
a(n) = a(a(n — 1)) + a(n — a(n — 1)),
with a(l) = a(2) = 1. The first few values are 1, 1,
2, 2, 3, 4, 4, 4, 5, 6, ... (Sloane's A004001). Plotting
a(n)/n against n gives the Batrachion plotted below.
Conway (1988) showed that lim n _>. 00 a(n)/n = 1/2 and
offered a prize of $10,000 to the discoverer of a value of n
for which \a(i)/i — 1/2 1 < 1/20 for i > n. The prize was
subsequently claimed by Mallows, after adjustment to
Conway's "intended" prize of $1,000 (Schroeder 1991),
who found n = 1489.
a(n)/n takes a value of 1/2 for n of the form 2 k with
k = 1, 2, Pickover (1996) gives a table of analogous
values of n corresponding to different values of \a(n)/n —
1/2| < e.
1000
see also Blancmange Function, Hofstadter's Q-
Sequence, Mallow's Sequence
References
Conolly, B. W. "Meta-Fibonacci Sequences." In Fibonacci
and Lucas Numbers, and the Golden Section (Ed. S. Va-
jda). New York: Haistead Press, pp. 127-138, 1989.
Conway, J. "Some Crazy Sequences." Lecture at AT&T Bell
Labs, July 15, 1988.
Guy, R. K. "Three Sequences of Hofstadter." §E31 in Un-
solved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 231-232, 1994.
Kubo, T. and Vakil, R. "On Conway's Recursive Sequence."
Disc. Math. 152, 225-252, 1996.
Mallows, C. "Conway's Challenging Sequence." Amer. Math.
Monthly 98, 5-20, 1991.
Pickover, C. A. "The Drums of Ulupu." In Mazes for
the Mind: Computers and the Unexpected. New York:
St. Martin's Press, 1993.
Pickover, C. A. "The Crying of Fractal Batrachion 1,489."
Ch. 25 in Keys to Infinity. New York: W. H. Freeman,
pp. 183-191, 1995.
Schroeder, M. "John Horton Conway's 'Death Bet.'" Frac-
tals, Chaos, Power Laws. New York: W. H. Freeman,
pp. 57-59, 1991.
Sloane, N. J. A. Sequence A004001/M0276 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hofstadter Figure-Figure Sequence
Hofstadter's Q -Sequence 847
Hofstadter Figure-Figure Sequence
Define F(l) = 1 and 5(1) = 2 and write
F(n) = F(n- 1) + S(n - 1),
where the sequence {S(n)} consists of those integers
not already contained in {F(n)}. For example, F(2) =
F(l) + 5(1) = 3, so the next term of S(n) is 5(2) = 4,
giving F(3) = F(2) 4- 5(2) = 7. The next integer is 5,
so 5(3) = 5 and F(4) = F(3) + 5(3) = 12. Continuing
in this manner gives the "figure" sequence F(n) as 1, 3,
7, 12, 18, 26, 35, 45, 56, . . . (Sloane's A005228) and the
"space" sequence as 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, ...
(Sloane's A030124).
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 73, 1989.
Sloane, N. J. A. Sequences A030124 and A005288/M2629 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Hofstadter G-Sequence
The sequence defined by G(0) = and
G(n) = n - G?(G(n - 1)).
The first few terms are 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8,
9, 9, ... (Sloane's A005206).
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 137, 1989.
Sloane, N. J. A. Sequence A005206/M0436 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hofstadter ^-Sequence
The sequence defined by H(0) = and
H(n)=n-H(H(H(n-l))).
The first few terms are 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9,
10, 10, 11, 12, 13, 13, 14, . . . (Sloane's A005374).
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 137, 1989.
Sloane, N. J. A. Sequence A005374/M0449 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hofstadter Male- Female Sequences
The pair of sequences defined by F(0) = 1, M(0) = 0,
and
F{n) = n-M{F(n-l))
M(n) = n - F(M(n - 1)).
The first few terms of the "male" sequence M{n) are
0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, . . . (Sloane's
A005379), and the first few terms of the "female" se-
quence F(n) are 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9,
... (Sloane's A005378).
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 137, 1989.
Sloane, N. J. A. Sequences A005378/M0263 and A005379/
M0278 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Hofstadter Point
The r-HOFSTADTER TRIANGLE of a given TRIANGLE
AABC is perspective to AABC, and the Perspective
Center is called the Hofstadter point. The Triangle
Center Function is
sm(rA)
a — ——
sm(r — vA)
As r ->- 0, the TRIANGLE CENTER FUNCTION ap-
proaches
A
oc = — ,
a
and as r -¥ 1, the Triangle Center Function ap-
proaches
a
see also Hofstadter Triangle
References
Kimberling, C. "Hofstadter Points." Nieuw Arch. Wiskunder
12, 109-114, 1994.
Kimberling, C. "Major Centers of Triangles." Amer. Math.
Monthly 104, 431-438, 1997.
Kimberling, C. "Hofstadter Points." http://www.
evansville.edu/-ck6/tcenters/recent/hofstad.html.
Hofstadter's Q-Sequence
The Integer Sequence given by
Q(n) = Q(n - Q(n - 1)) + Q{n - Q(n - 2)),
with Q(l) = Q(2) = 1. The first few values are 1, 1, 2, 3,
3, 4, 5, 5, 6, 6, . . . (Sloane's A005185; illustrated above).
These numbers are sometimes called Q-Numbers.
see also Hofstadter-Conway $10,000 Sequence,
Mallow's Sequence
References
Conolly, B. W. "Meta-Fibonacci Sequences." In Fibonacci
and Lucas Numbers, and the Golden Section (Ed. S. Va-
jda). New York: Halstead Press, pp. 127-138, 1989.
848 Hofstadter Sequences
Holder Sum Inequality
Guy, R. "Some Suspiciously Simple Sequences." Amer.
Math. Monthly 93, 186-191, 1986.
Hofstadter, D. R. Godel, Escher Bach: An Eternal Golden
Braid. New York: Vintage Books, pp. 137-138, 1980.
Pickover, C. A. "The Crying of Fractal Batrachion 1,489."
Ch. 25 in Keys to Infinity. New York: W. H. Freeman,
pp. 183-191, 1995.
Sloane, N. J. A. Sequence A005185/M0438 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hofstadter Sequences
Let 6i = 1 and 62 = 2 and for n > 3, let b n be the least
Integer > 6 n -i which can be expressed as the Sum of
two or more consecutive terms. The resulting sequence
is 1, 2, 3, 5, 6, 8, 10, 11, 14, 16, . . . (Sloane's A005243).
Let ci = 2 and C2 = 3, form all possible expressions of
the form dCj - 1 for 1 < i < j < n, and append them.
The resulting sequence is 2, 3, 5, 9, 14, 16, 17, 18, ...
(Sloane's A05244).
see also Hofstadter-Conway $10,000 Sequence,
Hofstadter's CJ-Sequence
References
Guy, R. K. "Three Sequences of Hofstadter." §E31 in Un-
solved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 231-232, 1994.
Sloane, N. J. A. Sequences A005243/M0623 and A00524/
M0705 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Hofstadter Triangle
For a Nonzero Real Number r and a Triangle
AABC, swing Line Segment BC about the vertex B
towards vertex A through an Angle rB. Call the line
along the rotated segment L. Construct a second line l!
by rotating LINE SEGMENT BC about vertex C through
an Angle rC. Now denote the point of intersection of L
and V by A(r). Similarly, construct B(r) and C(r). The
Triangle having these points as vertices is called the
Hofstadter r- triangle. Kimberling (1994) showed that
the Hofstadter triangle is perspective to AABC, and
calls Perspective Center the Hofstadter Point.
see also HOFSTADTER POINT
References
Kimberling, C. "Hofstadter Points." Nieuw Arch. Wiskunde
12, 109-114, 1994.
Kimberling, C. "Hofstadter Points." http : //www .
evansville.edu/-ck6/tcenters/recent/hofstad.html.
Holder Condition
A function (j>{t) satisfies the Holder condition on two
points h and £2 on an arc L when
wt 2 )-0(ti)i<;iit2-tir,
with A and 11 Positive Real constants.
Holder Integral Inequality
If
1 1
- + -
p q
with p, q > 1, then
J a
\f(x)g(x)\dx
I
\f(x)\"dx
1/p
jf
\g(x)\ q dx
1/9
with equality when
i ff (x)i = C |/(*)r\
If p = q — 2, this inequality becomes SCHWARZ'S IN-
EQUALITY.
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,
Holder, O. "Uber einen Mittelwertsatz." Gottingen Nachr.,
44, 1889.
Riesz, F. "Untersuchungen iiber Systeme integrierbarer
Funktionen." Math. Ann. 69, 456, 1910.
Riesz, F. "Su alcune disuguaglianze." Boll. Un. Mat. It. 7,
77-79, 1928.
Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, pp. 32-33, 1991.
Holder Sum Inequality
p q.
with p, q > 1, then
J2\a k b k \< I ^|o fc |-
i/p
1/9
Ei**
, fc=i
with equality when \bk\ = c\a k \ v *■ If p — q = 2, this
becomes the Cauchy 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. 1092, 1979.
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 10-15, 1988.
Hole
Homeomorphic 849
Hole
A hole in a mathematical object is a TOPOLOGICAL
structure which prevents the object from being contin-
uously shrunk to a point. When dealing with TOPO-
LOGICAL Spaces, a Disconnectivity is interpreted as
a hole in the space. Examples of holes are things like
the hole in the "center" of a Sphere or a Circle and
the hole produced in Euclidean Space cutting a Knot
out from it.
Singular Homology Groups form a Measure of the
hole structure of a Space, but they are one particu-
lar measure and they don't always pick up everything.
Homotopy Groups of a Space are another measure
of holes in a Space, as well as Bordism Groups, k-
Theory, Cohomotopy Groups, and so on.
There are many ways to measure holes in a space.
Some holes are picked up by Homotopy Groups that
are not picked up by Homology Groups, and some
holes are picked up by HOMOLOGY GROUPS that are
not picked up by HOMOTOPY GROUPS, (For example,
in the TORUS, HOMOTOPY GROUPS "miss" the two-
dimensional hole that is given by the TORUS itself, but
the second Homology Group picks that hole up.) In
addition, Homology Groups don't pick up the vary-
ing hole structures of the complement of KNOTS in 3-
space, but the first HOMOTOPY Group (the fundamen-
tal group) does.
see also Branch Cut, Branch Point, Cork Plug,
Cross-Cap, Genus (Surface), Singular Point
(Function), Spherical Ring, Torus
Holomorphic Function
A synonym for ANALYTIC FUNCTION.
see also Analytic Function, Homeomorphic
Holonomic Constant
A limiting value of a Holonomic Function near a Sin-
gular POINT. Holonomic constants include Apery's
Constant, Catalan's Constant, Polya's Random
Walk Constants for d > 2, and Pi.
Holonomic Function
A solution of a linear homogeneous ORDINARY DIFFER-
ENTIAL Equation with Polynomial Coefficients.
see also HOLONOMIC CONSTANT
References
Zeilberger, D. "A Holonomic Systems Approach to Special
Function Identities." J. Comput. Appl. Math. 32, 321-
348, 1990.
Holonomy
A general concept in CATEGORY THEORY involving the
globalization of topological or differential structures.
see also MONODROMY
Home Plate
8.5
8.5
Home plate in the game of BASEBALL is an irregular
PENTAGON. However, the Little League rulebook's spec-
ification of the shape of home plate (Kreutzer and Ker-
ley 1990), illustrated above, is not physically realizable,
since it requires the existence of a (12, 12, 17) RIGHT
TRIANGLE, whereas
12 2 + 12 2 = 288 ^ 289 =
17 J
(Bradley 1996).
see also BASEBALL COVER
References
Bradley, M. J. "Building Home Plate: Field of Dreams or
Reality?" Math. Mag. 69, 44-45, 1996.
Kreutzer, P. and Kerley, T. Little League's Official How-to-
Play Baseball Book. New York: Doubleday, 1990.
Homeoid
A shell bounded by two similar ELLIPSOIDS having a
constant ratio of axes. Given a Chord passing through
a homeoid, the distance between inner and outer inter-
sections is equal on both sides. Since a spherical shell
is a symmetric case of a homeoid, this theorem is also
true for spherical shells (CONCENTRIC CIRCLES in the
PLANE), for which it is easily proved by symmetry ar-
guments.
see also CHORD, ELLIPSOID
Homeomorphic
There are two possible definitions:
1. Possessing similarity of form,
2. Continuous, One-TO-One, Onto, and having a con-
tinuous inverse.
The most common meaning is possessing intrinsic topo-
logical equivalence. Two objects are homeomorphic if
they can be deformed into each other by a continuous,
invertible mapping. Homeomorphism ignores the space
in which surfaces are embedded, so the deformation can
be completed in a higher dimensional space than the
surface was originally embedded. MIRROR IMAGES are
homeomorphic, as are MOBIUS BANDS with an Even
number of half twists, and MOBIUS BANDS with an ODD
number of twists.
In Category Theory terms, homeomorphisms are
Isomorphisms in the Category of Topological
Spaces and continuous maps.
see also HOMOMORPHIC, POLISH SPACE
850 Homeomorphic Group
HOMFLY Polynomial
Homeomorphic Group
If the Elements of two Groups are n to 1 and the
correspondences satisfy the same GROUP multiplication
table, the GROUPS are said to be homeomorphic.
see also Isomorphic Groups
Homeomorphic Type
The following three pieces of information completely de-
termine the homeomorphic type of the surface (Massey
1967):
1. Orientability,
2. Number of boundary components,
3. Euler Characteristic.
see also ALGEBRAIC TOPOLOGY, EULER CHARACTER-
ISTIC
References
Massey, W. S. Algebraic Topology: An Introduction. New-
York: Springer- Verlag, 1996.
Homeomorphism
see Homeomorphic, Homeomorphic Group, Home-
omorphic Type, Topologically Conjugate
HOMFLY Polynomial
A 2- variable oriented KNOT POLYNOMIAL P L {a,z) mo-
tivated by the JONES POLYNOMIAL (Preyd et al. 1985).
Its name is an acronym for the last names of its co-
discoverers: Hoste, Ocneanu, Millett, Preyd, Lickorish,
and Yetter (Freyd et al 1985). Independent work re-
lated to the HOMPLY polynomial was also carried out
by Prztycki and Traczyk (1987). HOMFLY polynomial
is defined by the Skein Relationship
a~ 1 P L+ (a, z) - aP L _ (a, z) = zP Lo (a, z) (1)
(Doll and Hoste 1991), where v is sometimes written in-
stead of a (Kanenobu and Sumi 1993) or, with a slightly
different relationship, as
aP L+ (a, z) - oc'^Pl. (a, z) = zP Lo (a, z) (2)
(Kauffman 1991). It is also defined as P L {t,m) in terms
of Skein Relationship
£Pl+ + r x P L _ + mP Lo =
(3)
(Lickorish and Millett 1988). It can be regarded as a
nonhomogeneous Polynomial in two variables or a ho-
mogeneous POLYNOMIAL in three variables. In three
variables the SKEIN RELATIONSHIP is written
xP L+ (x, y, z) + yP L . (x, y, z) + zP Lo (x, t/, z) = 0. (4)
It is normalized so that Punknot = 1. Also, for n unlinked
unknotted components,
Pl(x,v,z)
(_«±*y
(5)
This POLYNOMIAL usually detects CHIRALITY but does
not detect the distinct ENANTIOMERS of the KNOTS
09 42, 10o48, lOon, 10o9i, IO104, and IO125 (Jones 1987).
The HOMFLY polynomial of an oriented KNOT is the
same if the orientation is reversed. It is a generalization
of the JONES POLYNOMIAL V(t), satisfying
V(t)=P(a = t,z = t 1/2 -t- 1/2 ) (6)
V(t) = P{i = it-\m - i(t~ 1/2 - t 1/2 )). (7)
It is also a generalization of the Alexander Polynom-
ial V(z), satisfying
A(z) = P(a = M = t 1/a -r 1/3 ).
(8)
The HOMFLY Polynomial of the Mirror Image K*
of a Knot K is given by
Pjc*(/ ) m) = P if (r 1 ,m) )
(9)
so P usually but not always detects Chirality.
A split union of two links (i.e., bringing two links to-
gether without intertwining them) has HOMFLY poly-
nomial
P{L X UL 2 ) = -(£ + r 1 )m _1 P(Li)P(L a ). (10)
Also, the composition of two links
P(Li#L a ) = P(Li)P(L a ),
(11)
so the Polynomial of a Composite Knot factors into
Polynomials of its constituent knots (Adams 1994).
Mutants have the same HOMFLY polynomials. In
fact, there are infinitely many distinct KNOTS with
the same HOMFLY POLYNOMIAL (Kanenobu 1986).
Examples include (05ooi, IO132), (O8008, IO129) (O8016,
lOise), and (IO025, lOose) (Jones 1987). Incidentally,
these also have the same Jones Polynomial.
M. B. Thistlethwaite has tabulated the HOMFLY poly-
nomial for Knots up to 13 crossings.
see also Alexander Polynomial, Jones Polynom-
ial, Knot Polynomial
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 171-172, 1994.
Doll, H. and Hoste, J. "A Tabulation of Oriented Links."
Math. Comput. 57, 747-761, 1991.
Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, W. B. R.; Millett,
K.; and Oceanu, A. "A New Polynomial Invariant of Knots
and Links." Bull. Amer. Math. Soc. 12, 239-246, 1985.
Homoclinic Point
Homography 851
Jones, V. "Hecke Algebra Representations of Braid Groups
and Link Polynomials." Ann. Math. 126, 335-388, 1987.
Kanenobu, T. "Infinitely Many Knots with the Same Poly-
nomial." Proc. Amer. Math. Soc. 97, 158-161, 1986.
Kanenobu, T. and Sumi, T. "Polynomial Invariants of 2-
Bridge Knots through 22 Crossings." Math. Comput. 60,
771-778 and S17-S28, 1993.
Kauffman, L. H. Knots and Physics. Singapore: World Sci-
entific, p. 52, 1991.
Lickorish, W. B. R. and Millett, B. R. "The New Polynomial
Invariants of Knots and Links." Math. Mag. 61, 1-23,
1988.
Morton, H. R. and Short, H. B. "Calculating the 2-Variable
Polynomial for Knots Presented as Closed Braids." J. Al-
gorithms 11, 117-131, 1990.
Przytycki, J. and Traczyk, P. "Conway Algebras and Skein
Equivalence of Links." Proc. Amer. Math. Soc. 100, 744—
748, 1987.
Stoimenow, A. "Jones Polynomials." http://www.
informatik.hu-berlin.de/-stoimeno/ptab/jlO.html.
^ Weisstein, E. W. "Knots and Links." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/Knots.m.
A small DISK centered near a homoclinic point in-
cludes infinitely many periodic points of different pe-
riods. Poincare showed that if there is a single homo-
clinic point, there are an infinite number. More specifi-
cally, there are infinitely many homoclinic points in each
small disk (Nusse and Yorke 1996).
see also Heteroclinic Point
References
Nusse, H. E. and Yorke, J. A. "Basins of Attraction." Science
271, 1376-1380, 1996.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, p. 145, 1989.
Homogeneous Coordinates
see TRILINEAR COORDINATES
Homogeneous Function
A function which satisfies
Homoclinic Point
A point where a stable and an unstable separatrix (in-
variant manifold) from the same fixed point or same
family intersect. Therefore, the limits
lim f k (X)
and
exist and are equal.
lim
fc— J- — oo
f k (X)
X"\
J^TX
H-
Refer to the above figure. Let X be the point of in-
tersection, with X\ ahead of X on one Manifold and
X-i ahead of X of the other. The mapping of each of
these points TX\ and TX2 must be ahead of the map-
ping of X, TX. The only way this can happen is if the
Manifold loops back and crosses itself at a new homo-
clinic point. Another loop must be formed, with T X
another homoclinic point. Since T 2 X is closer to the hy-
perbolic point than TX, the distance between T 2 X and
TX is less than that between X and TX. Area preser-
vation requires the Area to remain the same, so each
new curve (which is closer than the previous one) must
extend further. In effect, the loops become longer and
thinner. The network of curves leading to a dense Area
of homoclinic points is known as a homoclinic tangle or
tendril. Homoclinic points appear where CHAOTIC re-
gions touch in a hyperbolic FIXED POINT.
f(tx,ty) = t n f(x,y)
for a fixed n. Means, the WeierstraB Elliptic
Function, and Triangle Center Functions are ho-
mogeneous functions. A transformation of the variables
of a TENSOR changes the TENSOR into another whose
components are linear homogeneous functions of the
components of the original TENSOR.
see also Euler's Homogeneous Function Theorem
Homogeneous Numbers
Two numbers are homogeneous if they have identical
Prime Factors. An example of a homogeneous pair is
(6, 36), both of which share Prime Factors 2 and 3:
6 = 2-3
36
2 2 -3 2 .
see also Heterogeneous Numbers, Prime Factors,
Prime Number
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 146, 1983.
Homogeneous Polynomial
A multivariate polynomial (i.e., a POLYNOMIAL in more
than one variable) with all terms having the same de-
gree. For example, x 3 + xyz + y 2 z + z 3 is a homogeneous
polynomial of degree three.
see also POLYNOMIAL
Homographic
see MOBIUS TRANSFORMATION
Homography
A ClRCLE-preserving transformation composed of an
Even number of inversions.
see also Antihomography
852 Homological Algebra
Homothetic
Homological Algebra
An abstract ALGEBRA concerned with results valid for
many different kinds of SPACES.
References
Hilton, P. and Stammbach, U. A Course in Homological Al-
gebra, 2nd ed. New York: Springer- Verlag, 1997.
Weibel, C. A. An Introduction to Homological Algebra. New-
York: Cambridge University Press, 1994.
Homologous Points
The extremities of Parallel RADII of two Circles are
called homologous with respect to the Similitude Cen-
ter collinear with them.
see also Antihomologous Points
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 19, 1929.
Homolographic Equal Area Projection
see Mollweide Projection
Homology (Geometry)
A Perspective Collineation in which the center and
axis are not incident.
see also ELATION, HARMONIC HOMOLOGY, PERSPEC-
TIVE Collineation
Homology Group
The term "homology group" usually means a singular
homology group, which is an ABELIAN GROUP which
partially counts the number of HOLES in a TOPOLOG-
ICAL SPACE. In particular, singular homology groups
form a Measure of the HOLE structure of a Space, but
they are one particular measure and they don't always
pick up everything.
In addition, there are "generalized homology groups"
which are not singular homology groups.
Homology (Topology)
Historically, the term "homology" was first used in a
topological sense by Poincare. To him, it meant pretty
much what is now called a COBORDISM, meaning that
a homology was thought of as a relation between MAN-
IFOLDS mapped into a Manifold. Such Manifolds
form a homology when they form the boundary of a
higher-dimensional MANIFOLD inside the MANIFOLD in
question.
To simplify the definition of homology, Poincare sim-
plified the spaces he dealt with. He assumed that all
the spaces he dealt with had a triangulation (i.e., they
were "SlMPLlClAL COMPLEXES"). Then instead of talk-
ing about general "objects" in these spaces, he restricted
himself to subcomplexes, i.e., objects in the space made
up only on the simplices in the TRIANGULATION of the
space. Eventually, Poincare's version of homology was
dispensed with and replaced by the more general SINGU-
LAR Homology. Singular Homology is the concept
mathematicians mean when they say "homology."
In modern usage, however, the word homology is used to
mean HOMOLOGY GROUP. For example, if someone says
"X did Y by computing the homology of Z," they mean
"X did Y by computing the HOMOLOGY GROUPS of Z."
But sometimes homology is used more loosely in the
context of a "homology in a SPACE," which corresponds
to singular homology groups.
Singular homology groups of a SPACE measure the ex-
tent to which there are finite (compact) boundaryless
Gadgets in that Space, such that these Gadgets are
not the boundary of other finite (compact) GADGETS in
that Space.
A generalized homology or cohomology theory must sat-
isfy all of the Eilenberg-Steenrod Axioms with the
exception of the DIMENSION Axiom.
see also COHOMOLOGY, DIMENSION Axiom, Eilen-
berg-Steenrod Axioms, Gadget, Homological
Algebra, Homology Group, Simplicial Complex,
Simplicial Homology, Singular Homology
Homomorphic
Related to one another by a HOMOMORPHISM.
Homomorphism
A term used in Category Theory to mean a general
MORPHISM.
see also HOMEOMORPHISM, MORPHISM
Homoscedastic
A set of Statistical Distributions having the same
Variance.
see also Heteroscedastic
Homothecy
see Dilation
Homothetic
Two figures are homothetic if they are related by a DILA-
TION (a dilation is also known as a HOMOTHECY). This
means that they lie in the same plane and correspond-
ing sides are Parallel; such figures have connectors
of corresponding points which are CONCURRENT at a
point known as the HOMOTHETIC CENTER. The HO-
MOTHETIC Center divides each connector in the same
ratio k, known as the SIMILITUDE RATIO. For figures
which are similar but do not have Parallel sides, a
Similitude Center exists.
see also Dilation, Homothetic Center, Perspec-
tive, Similitude Ratio
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
nomothetic Center
Homothetic Center
The meeting point of lines that connect corresponding
points from HOMOTHETIC figures. In the above figure,
O is the homothetic center of the HOMOTHETIC figures'
ABCDE and A'B'C'D'E'. For figures which are similar
but do not have Parallel sides, a Similitude Center
exists (Johnson 1929, pp. 16-20).
Given two nonconcentric CIRCLES, draw RADII PARAL-
LEL and in the same direction. Then the line joining the
extremities of the Radii passes through a fixed point
on the line of centers which divides that line externally
in the ratio of RADII. This point is called the exter-
nal homothetic center, or external center of similitude
(Johnson 1929, pp. 19-20 and 41).
If Radii are drawn Parallel but instead in opposite
directions, the extremities of the Radii pass through a
fixed point on the line of centers which divides that line
internally in the ratio of RADII (Johnson 1929, pp. 19-
20 and 41). This point is called the internal homothetic
center, or internal center of similitude (Johnson 1929,
pp. 19-20 and 41).
The position of the homothetic centers for two circles of
radii r», centers (xi,yi), and segment angle are given
by solving tha simultaneous equations
y-V2
yt
V2 -yi
X2 — Xi
(x - X 2 )
vt
vt
x 2
(x - X 2 )
for {x,y), where
xf = Xi + ( — l)Vj cos#
yt =y* + (-l)Visin0,
Homothetic Position 853
and the plus signs give the external homothetic center,
while the minus signs give the internal homothetic cen-
ter.
As the above diagrams show, as the angles of the paral-
lel segments are varied, the positions of the homothetic
centers remain the same. This fact provides a (slotted)
LINKAGE for converting circular motion with one radius
to circular motion with another.
The six homothetic centers of three circles lie three by
three on four lines (Johnson 1929, p. 120), which "en-
close" the smallest circle.
The homothetic center of triangles is the PERSPECTIVE
Center of Homothetic Triangles. It is also called
the Similitude Center (Johnson 1929, pp. 16-17).
see also Apollonius' Problem, Perspective, Simil-
itude Center
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
$ Weisstein, E. W. "Plane Geometry." http: //www. astro.
Virginia . edu/ ~eww6n/math/notebooks/PlaneGeometry . m.
Homothetic Position
Two similar figures with Parallel homologous Lines
and connectors of HOMOLOGOUS POINTS CONCURRENT
at the Homothetic Center are said to be in homo-
thetic position. If two Similar figures are in the same
plane but the corresponding sides are not Parallel,
there exists a self-HOMOLOGOUS Point which occupies
the same homologous position with respect to the two
figures.
854 nomothetic Triangles
Hopf Link
Homothetic Triangles
Nonconcurrent TRIANGLES with PARALLEL sides are al-
ways Homothetic. Homothetic triangles are always
Perspective Triangles. Their Perspective Cen-
ter is called their HOMOTHETIC CENTER.
Homotopy
A continuous transformation from one FUNCTION to an-
other. A homotopy between two functions / and g
from a SPACE X to a SPACE Y is a continuous MAP
G from X G [0, 1] >-> Y such that G(x,0) = f(x) and
G(Xj 1) = g(x). Another way of saying this is that a
homotopy is a path in the mapping SPACE Map(X, Y)
from the first FUNCTION to the second.
see also /i-COBORDlSM
Homotopy Axiom
One of the Eilenberg-Steenrod Axioms which states
that, if / : (X, A) -+ (V, B) is homotopic to g : (X, A) ->■
{Y,B), then their INDUCED MAPS /* : H n (X,A) ->
H n (Y,B) and g* : H n (X,A) -» H n (Y,B) are the same.
Homotopy Group
A Group related to the Homotopy classes of Maps
from Spheres S n into a Space X.
see also COHOMOTOPY GROUP
Hook
A 6-Polyiamond.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Hook Length Formula
A Formula for the number of Young Tableaux
associated with a given YOUNG DIAGRAM. In each
box, write the sum of one plus the number of boxes
horizontally to the right and vertically below the
box (the "hook length"). The number of tableaux
is then n! divided by the product of all "hook
lengths". The Combinatorica f NumberOf Tableaux func-
tion in Mathematical implements the hook length for-
mula.
see also YOUNG DIAGRAM, YOUNG TABLEAU
References
Jones, V. "Hecke Algebra Representations of Braid Groups
and Link Polynomials." Ann. Math. 126, 335-388, 1987.
Skiena, S. Implementing Discrete Mathematics: Combina-
torics and Graph Theory with Mathematica. Reading,
MA: Addison-Wesley, 1990.
Homotopy Theory
The branch of Algebraic Topology which deals with
Homotopy Groups.
References
Aubry, M. Homotopy Theory and Models. Boston, MA: Birk-
hauser, 1995.
Honeycomb
A TESSELLATION in ra-D, for n > 3. The only regular
honeycomb in 3-D is {4,3,4}, which consists of eight
cubes meeting at each Vertex. The only quasiregular
honeycomb (with regular cells and semiregular VERTEX
Figures) has each Vertex surrounded by eight Tet-
3,4
rahedra and six Octahedra and is denoted
There are many semiregular honeycombs, such as
in which each Vertex consists of two Octa-
{?»}.
hedra {3,4} and four Cuboctahedra
see also SPONGE, TESSELLATION
W
References
Bulatov, V. "Infinite Regular Polyhedra." http : //www .
physics. orst.edu/~bulatov/polyhedr a/ infinite/.
Hoof
see Cylindrical Wedge
Hopf Algebra
Let a graded module A have a multiplication <f> and a
co- multiplication ip. Then if <j> and ip have the unity of
k as unity and ip : (A, <f>) — >■ (A y (f>) <g> (A, <p) is an algebra
homomorphism, then (A,<f>,tp) is called a Hopf algebra.
Hopf Bifurcation
The Bifurcation of a Fixed Point to a Limit Cycle
(Tabor 1989).
References
Guckenheimer, J. and Holmes, P. Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields, 3rd
ed. New York: Springer- Verlag, pp. 150-154, 1997.
Marsden, J. and McCracken, M. Hopf Bifurcation and Its
Applications. New York: Springer- Verlag, 1976.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, p. 197, 1989.
Hopf Circle
see Hopf Map
Hopf Link
OD
The Link 2? which has Jones Polynomial
v(t) = -t-r 1
Hop f Map
and HOMFLY POLYNOMIAL
P(z,cx) = z~ (a~ -
It has Braid Word <ti 2 .
5 )+;
Hopf Map
The first example discovered of a Map from a higher-
dimensional Sphere to a lower-dimensional Sphere
which is not null-HOMOTOPlC. Its discovery was a shock
to the mathematical community, since it was believed at
the time that all such maps were null-HOMOTOPlC, by
analogy with HOMOLOGY GROUPS. The Hopf map takes
points (Xi, X2, X3, X4) on a 3-sphere to points on a
2-sphere (#1, £2, £3)
xi = 2{X 1 X 2 + X3X4)
X2 — 2(XiX4 — X2X3)
X3 = (Xi 2 + X 3 2 ) - (X 2 2 + x 4 2 ).
Every point on the two SPHERES corresponds to a CIR-
CLE called the HOPF CIRCLE on the 3-SPHERE.
Hopf 's Theorem
A Necessary and Sufficient condition for a Mea-
sure which is quasi-invariant under a transformation to
be equivalent to an invariant PROBABILITY MEASURE is
that the transformation cannot (in a measure theoretic
sense) compress the Space.
Horizontal
Oriented in position PERPENDICULAR to up-down, and
therefore PARALLEL to a flat surface.
see also VERTICAL
Horizontal- Vertical Illusion
see Vertical-Horizontal Illusion
Horn Angle
The configuration formed by two curves starting at a
point, called the VERTEX V, in a common direction.
They are concrete illustrations of non- Archimedean ge-
ometries.
References
Kasner, E. "The Recent Theory of the Horn Angle." Scripta
Math 11, 263-267, 1945.
Horn Cyclide
Horner's Method 855
The inversion of a HORN TORUS. If the inversion center
lies on the torus, then the horn cyclide degenerates to a
Parabolic Horn Cyclide.
see also Cyclide, Horn Torus, Parabolic Cyclide,
Ring Cyclide, Spindle Cyclide, Torus
Horn Torus
One of the three STANDARD TORI given by the para-
metric equations
x = (c + a cos v) cos u
y — (c 4- a cos v) sin u
z = a sin v
(i)
(2)
(3)
with a — c. The inversion of a horn torus is a HORN
Cyclide (or Parabolic Horn Cyclide), The above
left figure shows a horn torus, the middle a cutaway,
and the right figure shows a CROSS-SECTION of the horn
torus through the zz-plane.
see also Cyclide, Horn Cyclide, Ring Torus, Spin-
dle Torus, Standard Tori, Torus
References
Gray, A. "Tori." §11.4 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 218-220, 1993.
Pinkall, U. "Cyclides of Dupin." §3.3 in Mathematical Models
from the Collections of Universities and Museums (Ed.
G. Fischer). Braunschweig, Germany: Vieweg, pp. 28-30,
1986.
Horned Sphere
see Alexander's Horned Sphere, Antoine's
Horned Sphere
Horner's Method
Let
P(x) = a n x n -h . . . -h ao
and b n = a n . If we then define
bk = a>k + bk-iXo
(i)
(2)
for k = n — 1, n — 2,
therefore follows that
0, we obtain bo = P(xq). It
P(x) = (x-x )Q{x) + b , (3)
where
Q(x) = bnx 71 ' 1 + 6 n -ix n 2 + . . . + b 2 x + 6i.
(4)
In addition,
P'(x) = Q{x) + {x-x )Q'(x) (5)
P'(xo) = Q(x ). (6)
856
Horner's Rule
Hundred
Horner's Rule
A rule for Polynomial computation which both re-
duces the number of necessary multiplications and re-
sults in less numerical instability due to potential sub-
traction of one large number from another. The rule
simply factors out POWERS of x, giving
a n x n + a n -\x n ~ + . . . + ao = ((a n x + a n -i)x + . . .)x + clq.
References
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 9, 1991.
Horocycle
The LOCUS of a point which is derived from a fixed point
Q by continuous parallel displacement.
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 300, 1969.
Horse Fetter
see Hippopede
Horseshoe Map
see Smale Horseshoe Map
Hough Transform
A technique used to detect boundaries in digital images.
Householder's Method
A RoOT-finding algorithm based on the iteration for-
mula
x n+1 -x n f , M y. mxn)]2 j.
This method, like Newton's Method, has poor con-
vergence properties near any point where the Deriva-
tive f'(x) = 0.
see also Newton's Method
References
Householder, A. S. The Numerical Treatment of a Single
Nonlinear Equation. New York: McGraw-Hill, 1970.
Howell Design
Let S be a set of n + 1 symbols, then a Howell design
H(s,2n) on symbol set S is an s x s array H such that
1. Every cell of H is either empty or contains an un-
ordered pair of symbols from 5,
2. Every symbol of S occurs once in each row and col-
umn of H, and
3. Every unordered pair of symbols occurs in at most
one cell of H.
References
Colbourn, C. J. and Dinitz, J. H. (Eds.) "Howell Designs."
Ch. 26 in CRC Handbook of Combinatorial Designs. Boca
Raton, FL: CRC Press, pp. 381-385, 1996.
Hub
The central point in a Wheel Graph W n . The hub has
Degree n - 1.
see also Wheel Graph
Huffman Coding
A lossless data compression algorithm which uses a small
number of bits to encode common characters. Huffman
coding approximates the probability for each character
as a POWER of 1/2 to avoid complications associated
with using a nonintegral number of bits to encode char-
acters using their actual probabilities.
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Huffman Coding and Compression of Data."
Ch. 20.4 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 896-901, 1992.
Hull
see AFFINE Hull, Convex Hull
Humbert's Theorem
The Necessary and Sufficient condition that an al-
gebraic curve has an algebraic INVOLUTE is that the ARC
Length is a two- valued algebraic function of the coor-
dinates of the extremities. Furthermore, this function
is a Root of a Quadratic Equation whose Coeffi-
cients are rational functions of x and y.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 195, 1959.
Hundkurve
see Tractrix
Hundred
100 = 10 2 . Madachy (1979) gives a number of algebraic
equations using the digits 1 to 9 which evaluate to 100,
such as
(7 - 5) 2 + 96 + 8 - 4 - 3 - 1 = 100
3 2 +91 + 7 + 8-6-5-4 = 100
V9 - 6 + 72 - (1)(3!) - 8 + 45 = 100
123 - 45 - 67 + 89 = 100,
and so on.
see also 10, BILLION, HUNDRED, LARGE NUMBER, MIL-
LION, Thousand
References
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 156-159, 1979.
Hunt's Surface
Hunt's Surface
An Algebraic SURFACE given by the implicit equation
4{x 2 + y 2 + z 2 - 13) 3 + 27(3x 2 + y 2 - 4z 2 - 12) 2 = 0.
References
Hunt, B. "Algebraic Surfaces." http: //wwv.mathematik.
uni~kl.de/~wwwagag/ Galerie.html.
Nordstrand, T. "Hunt's Surface." http://www.uib.no/
people/nf ytn/hunttxt .htm.
Huntington Equation
An equation proposed by Huntington (1933) as part of
his definition of a BOOLEAN ALGEBRA,
n(n(x) + y) + n(n(x) + n(y)) = x.
see also Robbins Algebra, Robbins Equation
References
Huntington, E. V. "New Sets of Independent Postulates for
the Algebra of Logic, with Special Reference to White-
head and Russell's Principia Mathematical Trans. Amer.
Math. Soc. 35, 274-304, 1933.
Huntington, E. V. "Boolean Algebra. A Correction." Trans.
Amer. Math. Soc. 35, 557-558, 1933.
Hurwitz Equation
The Diophantine Equation
2 . 2 , , 2
ax\X2 ■
which has no INTEGER solutions for a > n.
see also LAGRANGE NUMBER (DIOPHANTINE EQUA-
TION)
References
Guy, R. K. "Markoff Numbers." §D12 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 166-168, 1994.
Hurwitz Polynomial 857
Hurwitz's Irrational Number Theorem
As Lagrange showed, any IRRATIONAL NUMBER a has
an infinity of rational approximations p/q which satisfy
v^
(1)
Similarly, if a ^ f (1 + a/5),
<
and if a ^ f(l + \/5 ) ^ a/2,
<
x/SV'
5 1
/22lq 2
(2)
In general, even tighter bounds of the form
V
q ^ L n q 2
(3)
(4)
can be obtained for the best rational approximation pos-
sible for an arbitrary irrational number a, where the L n
are called LAGRANGE NUMBERS and get steadily larger
for each "bad" set of irrational numbers which is ex-
cluded.
see also Hurwitz's Irrational Number Theo-
rem, Liouville's Rational Approximation Theo-
rem, Liouville-Roth Constant, Markov Number,
Roth's Theorem, Segre's Theorem, Thue-Siegel-
Roth Theorem
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 40, 1987.
Chandrasekharan, K. An Introduction to Analytic Number
Theory. Berlin: Springer- Verlag, p. 23, 1968.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 187-189, 1996.
Hurwitz Number
A number with a continued fraction whose terms are the
values of one or more POLYNOMIALS evaluated on con-
secutive Integers and then interleaved. This property
is preserved by MOBIUS TRANSFORMATIONS (Beeler et
al. 1972, p. 44).
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Hurwitz Polynomial
A Polynomial with Real Positive Coefficients
and ROOTS which are either NEGATIVE or pairwise con-
jugate with Negative Real Parts.
858 Hurwitz-Radon Theorem
Hurwitz- Radon Theorem
Determined the- possible values of r and n for which
there is an Identity of the form
/ 2 , , 2\ / 2 . , 2\ 2 . , 2
(Xi + . . . + X r )(yi + . • . + Vt ) = Z\ + • . • + Z n -
Hurwitz's Root Theorem
Let {/(as)} be a SEQUENCE of ANALYTIC FUNCTIONS
REGULAR in a region (7, and let this sequence be UNI-
FORMLY Convergent in every Closed Subset of G.
If the Analytic Function
lim fn(x) = f(x)
n — ► oo
does not vanish identically, then if x = a is a zero of
f(x) of order fc, a Neighborhood \x - a\ < 5 of x = a
and a number N exist such that if n > N, f n {x) has
exactly k zeros in \x — a\ < S.
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 22, 1975.
Hurwitz Zeta Function
A generalization of the Riemann Zeta Function with
a Formula
^• a ) = E(fcTa
(fc + a) s
(1)
where any term with k + a = is excluded. The Hur-
witz zeta function can also be given by the functional
equation
CU
= ar(i-.)(2^r 1 5:™(T + ?2 f)f( 1 -'?)
(2)
(Apostol 1976, Miller and Adamchik), or the integral
((s,a) = ±a +
5-1
+2| C °(a 2 + y 2 )- s/2 {sin[ 5 tan- 1 (^)]}
e 2 *y - 1 '
(3)
If $t[z] < 0, then
2r(i - z)
{(*,*) =
(27T) 1
(?)
cos(27ran)
n-l
sin(27ran)
fnz\ v^ sin(27r
(4)
Hurwitz Zeta Function
The Hurwitz zeta function satisfies
C(0,o) = i-o (5)
£C(0,a) = ln[T(a)]-fln(2ir)
£c(0,0) = iln(27r),
(6)
(7)
where T(z) is the Gamma Function. The Polygamma
FUNCTION ip m {z) can be expressed in terms of the Hur-
witz zeta function by
ifmiz) = (-ir +1 m!C(l + m, Z ).
For Positive integers k 7 p y and q > p }
(8)
C' I -2fc + 1
■;)
[iP(2k) -ln(27rq)]B 2 k(p/q)
2k
ty{2k) ~ ln(27r)]B 2fc
q 2k 2k
(-l) fc+1 7r ^^ (2wpn
(27rq) 2k
n=l ^ ' ^ '
+
C'(-2fe + l)
(9)
where B n is a Bernoulli Number, B n (x) a Ber-
noulli Polynomial, i/> n (z) is a Polygamma Func-
tion, and C(z) is a Riemann Zeta Function (Mil-
ler and Adamchik). Miller and Adamchik also give the
closed-form expressions
f(-2* + l,±) = -
£ 2fc ln2 (2 2fc ~ 1 - l)C / (-2fc + l)
4 k k
2 2fe-
• H +i 4)
(10)
= =F-
(9*-l)B 2 *7r B 2fc ln3
3/ V3(3 2k - 1 - l)8fc (3 2fc -!)4fc
.(-l)"i>2k-i(l) (3 ^-'-l)C'(-2fc + l)
2\/3(67r) 2fc -
2(3 2fc " 1 )
(ID
.(4 fc + l)B 2fc 7r , (4 fc - 1 -l)B 2fc ln2
4 fc +!fc
23*-ifc
. (-l) fc ^ 2t -i(l) (2 2fc - 1 -l)C'(-2fc + l)
4(87r) 2fc - 1 2 4fc -!
(9* - l)(2 2fc " 1 + l)B 2k n
(12)
C' -2fc + 1
6 /
v^ 2 *" 1 )^
B 2fc (3 2fc ~ 1 -l)ln2 £F 2fe (2 2fc - 1 -l)ln3
(6 2fc - 1 )4A; (e 2 *- 1 )^
(-1)*(2"- 1 + l)^ 3fc -i(|)
2v/3(127r) 2fe - 1
(2 2fe-i _ 1)(3 2fe-i _ i)^(-2A; + 1)
+
2(6 2 *" 1 )
(13)
Hutton's Formula
Hyperbola 859
see also KHINTCHINE'S CONSTANT, POLYGAMMA FUNC-
TION, Psi Function, Riemann Zeta Function, Zeta
Function
References
Apostol, T. M. Introduction to Analytic Number Theory.
New York: Springer- Verlag, 1995.
Elizalde, E.; Odintsov, A. D.; and Romeo, A. Zeta Regu-
larization Techniques with Applications. River Edge, NJ:
World Scientific, 1994.
Knopfmacher, J. "Generalised Euler Constants." Proc. Ed-
inburgh Math. Soc. 21, 25-32, 1978.
Magnus, W. and Oberhettinger, F. Formulas and Theorems
for the Special Functions of Mathematical Physics, 3rd ed.
New York: Springer- Verlag, 1966.
Miller, J. and Adamchik, V. "Derivatives of the Hurwitz
Zeta Function for Rational Arguments." Submitted to
J. Symb. Comput. http://www.wolfram.com/-victor/
articles/hurwitz.html.
Spanier, J. and Oldham, K. B. "The Hurwitz Function
£(i/;u)." Ch. 62 in An Atlas of Functions. Washington,
DC: Hemisphere, pp. 653-664, 1987.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, ^th ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 268-269, 1950.
Hutton's Formula
The Machin-Like Formula
i7r = 2tan- 1 (|) + tan- 1 (i).
The other two-term Machin-Like FORMULAS are Eu-
ler's Machin-Like Formula, Hermann's Formula,
and Machines Formula.
Hutton's Method
see Lambert's Method
Hyperbola
In general, a hyperbola is defined as the LOCUS of all
points in the PLANE the difference of whose distance
from two fixed points (the Foci Fi and F 2 ) separated
by a distance 2c, where
c = yja?+b 2 ,
(1)
is a given Positive constant. By analogy with the defi-
nition of the Ellipse, the equation for a hyperbola with
Semimajor Axis a parallel to the x-Axis and Semimi-
NOR Axis b parallel to the y-AxiS is given by
(x -xo) 2 (y ~yo)
Unlike the ELLIPSE, no points of the hyperbola actually
lie on the SEMIMINOR Axis, but rather the ratio b/a
determined the vertical scaling of the hyperbola. The
Eccentricity of the hyperbola is defined as
a V or
(3)
In the standard equation of the hyperbola, the center is
located at (a:o,t/o), tne FOCI are at (#o ± c, yo), and the
vertices are at (xq ± a, yo). The so-called Asymptotes
(shown as the dashed lines in the above figures) can be
found by substituting for the 1 on the right side of the
general equation (2),
y = ±-{x - xo) + j/o,
and therefore have SLOPES dbb/a.
(4)
The special case a = b (the left diagram above) is known
as a Right Hyperbola because the Asymptotes are
Perpendicular.
In Polar Coordinates, the equation of a hyperbola
centered at the ORIGIN (i.e., with x = yo = 0) is
a 2 b 2
b 2 cos 2 9- a 2 sin 2 0'
In Polar Coordinates centered at a Focus,
a(e 2 - 1)
T = .
1 — e cos 9
(5)
(6)
The two-center BIPOLAR COORDINATES equation with
origin at a FOCUS is
fi — T2 = zt2a.
The parametric equations for the hyperbola are
x = ±a cosh t
y = bsinht.
The Curvature and Tangential Angle are
4>(t)
[cosh(2i)]" 3/2
- tan -1 (tanh£).
(7)
(8)
(9)
(10)
(11)
b 2
(2)
The special case of the Right Hyperbola was first
studied by Menaechmus. Euclid and Aristaeus wrote
about the general hyperbola, but only studied one
branch of it. The hyperbola was given its present name
by Apollonius, who was the first to study both branches.
The Focus and Directrix were considered by Pappus
(MacTutor Archive). The hyperbola is the shape of an
orbit of a body on an escape trajectory (i.e., a body
860 Hyperbola Evolute
Hyperbolic Automorphism
with positive energy), such as some comets, about a
fixed mass, such as the sun.
The LOCUS of the apex of a variable Cone containing
an Ellipse fixed in 3-space is a hyperbola through the
Foci of the Ellipse. In addition, the Locus of the
apex of a Cone containing that hyperbola is the origi-
nal Ellipse. Furthermore, the Eccentricities of the
ELLIPSE and hyperbola are reciprocals.
see also Conic Section, Ellipse, Hyperboloid,
Jerabek's Hyperbola, Kiepert's Hyperbola,
Parabola, Quadratic Curve, Rectangular Hy-
perbola, Reflection Property, Right Hyper-
bola
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 199-200, 1987.
Casey, J. "The Hyperbola." Ch. 7 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. 250-284, 1893.
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 75—76, 1996.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 79-82, 1972.
Lee, X. "Hyperbola." http://www.best.com/-xah/Special
PlaneCurves_dir/Hyperbola_dir/hyperbola.html.
Lockwood, E. H. "The Hyperbola." Ch. 3 in i Book
of Curves. Cambridge, England: Cambridge University
Press, pp. 24-33, 1967.
MacTutor History of Mathematics Archive. "Hyperbola."
http : //www-groups . dcs . st-and. ac .uk/ -history /Curves
/Hyperbola. html.
Hyperbola Evolute
The Evolute of a Rectangular Hyperbola is the
Lame Curve
(ax) 2/3 -(by) 2/3 = (a + b) 2/3 .
From a point between the two branches of the EVOLUTE,
two NORMALS can be drawn to the HYPERBOLA. How-
ever, from a point beyond the EVOLUTE, four NORMALS
can be drawn.
Hyperbola Inverse Curve
■oe
\
\
\
1
r~^
s~\
/
1
\^J
^J
i
/
/
*■--
--'
\
For a Hyperbola with a = b with Inversion Center
at the center, the Inverse Curve
2k cost
a[3-cos(2t)]
ksin(2t)
a[3 - cos(2£)]
(1)
(2)
For an Inversion Center at the Vertex, the Inverse
Curve
x = a-\-
y = a +
4k cost sin 2 (|i)
a[5 - 4 cos t + cos(2£) - 2 sin(2t)]
fc(tan£ — 1)
a[(secr.-l) 2 + (tanr.-l) 2 ]
(3)
(4)
is a Right Strophoid.
For an Inversion Center at the Focus, the Inverse
Curve
kcost(l — ecost)
a(cost — e) 2
y
Ve^lksin(2t)
2a(cost — e) 2
is a LlMAgON, where e is the ECCENTRICITY.
-ee-
~<0
(5)
(6)
For a Hyperbola with a = \^3b and Inversion Cen-
ter at the Vertex, the Inverse Curve
x = 6 +
y = b +
2k cos £(\/3 — cos t)
6[9 - 4\/3 cos t + cos(2i) - 2 sin(2t)]
fe(tant — 1)
b[(y/3aect - l) 2 + (tan* - l) 2 ]
(7)
(8)
is a Maclaurin Trisectrix.
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, p. 203, 1972.
Hyperbola Pedal Curve
The Pedal Curve of a Hyperbola with the Pedal
Point at the Focus is a Circle. The Pedal Curve
of a Rectangular Hyperbola with Pedal Point at
the center is a Lemniscate.
Hyperbolic Automorphism
see ANOSOV AUTOMORPHISM
is a Lemniscate.
Hyperbolic Cosecant
Hyperbolic Cotangent 861
Hyperbolic Cosecant
-2
-20
The hyperbolic cosecant is denned as
1 2
csch x = —
sinhx e x — e~ x
see also Bernoulli Number, Bipolar Coordinates,
Bipolar Cylindrical Coordinates, Cosecant,
Helmholtz Differential Equation — Toroidal
Coordinates, Hyperbolic Sine, Poinsot's Spirals,
Surface of Revolution, Toroidal Function
References
Abramowitz, M. and Stegun, C. A. (Eds.)- "Hyperbolic
Functions." §4.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 83-86, 1972.
Spanier, J. and Oldham, K. B. "The Hyperbolic Secant
sech(a:) and Cosecant csch(x) Functions." Ch. 29 in An At-
las of Functions. Washington, DC: Hemisphere, pp. 273-
278, 1987.
Hyperbolic Cosine
140
120
100
80
60
40
20
-6
-4
-2
2
4
6
| Cosh z|
The hyperbolic cosine is defined as
coshx = \{e x +e x ).
This function describes the shape of a hanging cable,
known as the CATENARY.
see also BIPOLAR COORDINATES, BIPOLAR CYLIN-
DRICAL Coordinates, Bispherical Coordinates,
Catenary, Catenoid, Chi, Conical Function,
Correlation Coefficient — Gaussian Bivariate
Distribution, Cosine, Cubic Equation, de Moiv-
re's Identity, Elliptic Cylindrical Coordi-
nates, Elsasser Function, Fibonacci Hyper-
bolic Cosine, Fibonacci Hyperbolic Sine, Hyper-
bolic Geometry, Hyperbolic Lemniscate Func-
tion, Hyperbolic Sine, Hyperbolic Secant,
Hyperbolic Tangent, Inversive Distance, La-
place's Equation — Bipolar Coordinates, La-
place's Equation — Bispherical Coordinates, La-
place's Equation — Toroidal Coordinates, Lem-
niscate Function, Lorentz Group, Mathieu Dif-
ferential Equation, Mehler's Bessel Function
Formula, Mercator Projection, Modified Bes-
sel Function of the First Kind, Oblate Spher-
oidal Coordinates, Prolate Spheroidal Coordi-
nates, Pseudosphere, Ramanujan Cos/Cosh Iden-
tity, Sine-Gordon Equation, Surface of Revolu-
tion, Toroidal Coordinates
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic
Functions." §4.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 83-86, 1972.
Spanier, J. and Oldham, K. B. "The Hyperbolic Sine sinh(z)
and Cosine cosh(a;) Functions." Ch. 28 in An Atlas of
Functions. Washington, DC: Hemisphere, pp. 263-271,
1987.
Hyperbolic Cotangent
=J -->■ 2 4
The hyperbolic cotangent is defined as
e x +e~ x e 2x + l
coth x =
e x - e~
862 Hyperbolic Cube
Hyperbolic Fixed Point (Map)
Its Laurent Series is
A Quadratic Surface given by the equation
cotha? — — h |# — jjtx
see also BERNOULLI NUMBER, BIPOLAR COORDINATES,
Bipolar Cylindrical Coordinates, Cotangent,
Fibonacci Hyperbolic Cotangent, Hyperbolic
Tangent, Laplace's Equation — Toroidal Coor-
dinates, Lebesgue Constants (Fourier Series),
Prolate Spheroidal Coordinates, Surface of
Revolution, Toroidal Coordinates, Toroidal
Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic
Functions." §4.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 83-86, 1972.
Spanier, J. and Oldham, K. B. "The Hyperbolic Tangent
tanh(:r) and Cotangent coth(a;) Functions." Ch. 30 in
An Atlas of Functions. Washington, DC: Hemisphere,
pp. 279-284, 1987.
Hyperbolic Cube
A hyperbolic version of the Euclidean CUBE.
see also Hyperbolic Dodecahedron, Hyperbolic
Octahedron, Hyperbolic Tetrahedron
References
Rivin, I. "Hyperbolic Polyhedron Graphics." http://www .
mathsource . com/ cgi -bin /Math Source / Applications /
Graphics/3D/0201-788.
Hyperbolic Cylinder
2 2
x y
a 2 b 2
see also Elliptic Paraboloid, Paraboloid
Hyperbolic Dodecahedron
A hyperbolic version of the Euclidean DODECAHEDRON.
see also HYPERBOLIC CUBE, HYPERBOLIC OCTAHE-
DRON, Hyperbolic Tetrahedron
References
Rivin, I. "Hyperbolic Polyhedron Graphics." http://www -
mathsource , com/ cgi -bin /Math Source /Applications /
Graphics/3D/0201-788.
Hyperbolic Fixed Point (Differential
Equations)
A Fixed Point for which the Stability Matrix has
Eigenvalues Ai < < A 2 , also called a Saddle
Point.
see also Elliptic Fixed Point (Differential Equa-
tions), Fixed Point, Stable Improper Node, Sta-
ble Spiral Point, Stable Star, Unstable Im-
proper Node, Unstable Node, Unstable Spiral
Point, Unstable Star
References
Tabor, M. "Classification of Fixed Points." §1.4.b in Chaos
and Integrability in Nonlinear Dynamics: An Introduc-
tion. New York: Wiley, pp. 22-25, 1989.
Hyperbolic Fixed Point (Map)
A Fixed Point of a Linear Transformation (Map)
for which the rescaled variables satisfy
(S - a) 2 + 4/? 7 > 0.
see also Elliptic Fixed Point (Map), Linear
Transformation, Parabolic Fixed Point
Hyperbolic Functions
Hyperbolic Functions
The hyperbolic functions sinh, cosh, tanh, csch, sech,
coth (Hyperbolic Sine, Hyperbolic Cosine, etc.)
share many properties with the corresponding CIRCU-
LAR Functions. The hyperbolic functions arise in
many problems of mathematics a nd math ematical phys-
ics in which integrals involving y/l + x 2 arise (whereas
the Circular Functions involve y/1 - x 2 ).
For instance, the HYPERBOLIC Sine arises in the grav-
itational potential of a cylinder and the calculation of
the Roche limit. The HYPERBOLIC Cosine function is
the shape of a hanging cable (the so-called CATENARY).
The Hyperbolic Tangent arises in the calculation of
magnetic moment and rapidity of special relativity. All
three appear in the Schwarzschild metric using exter-
nal isotropic Kruskal coordinates in general relativity.
The Hyperbolic Secant arises in the profile of a lam-
inar jet. The HYPERBOLIC COTANGENT arises in the
Langevin function for magnetic polarization.
The hyperbolic functions are defined by
6 — 6
sinhz = = — sinh(— z)
cosh z =
tanhz =
csch z =
sech z =
coth z ~
2
e z + e~
= cosh(— z)
e — e e — 1
e z + e~ z ~ e 2z + 1
2
e z - e~ z
2
e z + e"
e z + e~
e^ + 1
e 2z - 1'
For purely IMAGINARY arguments,
sinh(iz) = isinz
cosh(iz) = cosz.
(i)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
The hyperbolic functions satisfy many identities anoma-
lous to the trigonometric identities (which can be in-
ferred using Osborne's Rule) such as
cosh 2 x — sinh 2 x = 1
cosh x H- sinh x = e x
cosh x — sinh x = e~
(9)
(10)
(11)
See also Beyer (1987, p. 168). Some half-angle FORMU-
LAS are
tanh
coth
(z \ sinh x + i sin y
2 / cosh x + cos y
(z \ sinh x — i sin y
2 / cosh x — cos y
(12)
(13)
Hyperbolic Functions 863
Some double-angle FORMULAS are
sinh(2x) = 2 sinh x cosh x (14)
cosh(2cc) = 2 cosh 2 x - 1 = 1 + 2 sinh 2 x. (15)
Identities for Complex arguments include
sinh(x -I- iy) = sinh x cos y -f i cosh x sin y (16)
cosh(# + iy) = cosh x cosy + £ sinh x sin y. (17)
The Absolute Squares for Complex arguments are
(18)
| sinh(z)| 2 = sinh 2 x + sin 2 y
|cosh(z)| = sinh x -f cos y.
Integrals involving hyperbolic functions include
dx
(19)
/
x\/a + bx
= ln
= ln
= ln
y/a + bx — y/a
y/a + bx + y/a
(y/a + bx — y/a) 2
(a + bx) — a
(a + bx) - 2y/i
i{a + bx) + a
bx
(20)
If b > 0, then
/
xy/a~+bx
-In
2a + 6a; — 2ya(a + 6x)
6x
-|(= + o-Vs(s + o
(21)
Let z = 2a/bx + 1, and a/bx — (z — l)/2 and
dec
/
xy/a + 6x
In [2 - ^/(z-l)(z + l)]
( z — yz 2 — 1 1 = cosh _1 (z)
'(' + =)
V^fs ) ' (22)
In
cosh
2 tanh
see a/so Hyperbolic Cosecant, Hyperbolic Co-
sine, Hyperbolic Cotangent, Generalized Hy-
perbolic Functions, Hyperbolic Inverse Func-
tions, Hyperbolic Secant, Hyperbolic Sine, Hy-
perbolic Tangent, Hyperbolic Inverse Func-
tions, Osborne's Rule
864 Hyperbolic Geometry
Hyperbolic Knot
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic
Functions." §4.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 83-86, 1972.
Beyer, W. H. "Hyperbolic Function." CRC Standard Math-
ematical Tables, 28th ed. Boca Raton, FL: CRC Press,
pp. 168-186, 1987.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 126-131, 1967.
Yates, R. C. "Hyperbolic Functions." A Handbook on Curves
and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 113-118, 1952.
Hyperbolic Geometry
A Non-Euclidean Geometry, also called Lobachev-
sky-Bolyai- Gauss Geometry, having constant Sec-
tional Curvature — 1. This Geometry satisfies all
of Euclid's Postulates except the Parallel Postu-
late, which is modified to read: For any infinite straight
Line L and any Point P not on it, there are many other
infinitely extending straight LINES that pass through P
and which do not intersect L.
In hyperbolic geometry, the sum of Angles of a Tri-
angle is less than 180°, and TRIANGLES with the same
angles have the same areas. Furthermore, not all TRI-
ANGLES have the same Angle sum (c.f. the AAA The-
orem for Triangles in Euclidean 2-space). The best-
known example of a hyperbolic space are SPHERES in
Lorentzian 4-space. The PoiNCARE Hyperbolic Disk
is a hyperbolic 2-space. Hyperbolic geometry is well un-
derstood in 2-D, but not in 3-D.
Geometric models of hyperbolic geometry include the
Klein-Beltrami Model, which consists of an Open
Disk in the Euclidean plane whose open chords corre-
spond to hyperbolic lines. A 2-D model is the POINCARE
Hyperbolic Disk. Felix Klein constructed an analytic
hyperbolic geometry in 1870 in which a POINT is repre-
sented by a pair of Real Numbers (#1,3:2) with
x^+x* 2 < 1
(i.e., points of an Open Disk in the Complex Plane)
and the distance between two points is given by
d(xyX) = a cosh
1 — X1X1 — X2X2
V 7 ! - xi 2 - x 2 2 V 1 " x i 2 ~ x 2 2
The geometry generated by this formula satisfies all of
Euclid's Postulates except the fifth. The Metric of
this geometry is given by the CAYLEY-KLEIN-HlLBERT
Metric,
9ii
912
922
a 2 (l-x 2 2 )
(1-zi 2 -X2 2 ) 2
a 2 x\X2
(1-Xi 2 ~X 2 2 ) 2
a 2 {l- Xl 2 )
{l-Xi 2 -X2 2 ) 2 '
Hilbert extended the definition to general bounded sets
in a Euclidean Space.
see also Elliptic Geometry, Euclidean Geome-
try, Hyperbolic Metric, Klein-Beltrami Model,
Non-Euclidean Geometry, Schwarz-Pick Lemma
References
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 57-60, 1990.
Eppstein, D. "Hyperbolic Geometry." http://www.ics.uci.
edu/-eppstein/ junkyard/hyper. html.
Stillwell, J. Sources of Hyperbolic Geometry. Providence, RI:
Amer. Math. Soc, 1996.
Hyperbolic Inverse Functions
sinh" 1 ^) = In (a + ^o? + 6 2 )
cosh -1 z = In ( z db y z 2 — 1 J
csch" 1 z = ln(l± y/l + z 2 )
sech -1 z = In ( I
V z J
coth- 1 ^=iln(|3^V
(1)
(2)
(3)
(4)
(5)
(6)
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic
Functions." §4.6 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 86-89, 1972.
Hyperbolic Knot
A hyperbolic knot is a KNOT that has a complement that
can be given a metric of constant curvature — 1. The
only KNOTS which are not hyperbolic are TORUS KNOTS
and Satellite Knots (including Composite Knots),
as proved by Thurston in 1978. Therefore, all but six of
the PRIME KNOTS with 10 or fewer crossings are hyper-
bolic. The exceptions with nine or fewer crossings are
03ooi (the(3,2)-TORUS Knot), 05ooi, 07ooi, O8019 (the
(4,3)-Torus Knot), and 09 oi.
Almost all hyperbolic knots can be distinguished by
their hyperbolic volumes (exceptions being 05 002 and a
certain 12-crossing knot; see Adams 1994, p. 124). It has
been conjectured that the smallest hyperbolic volume is
2.0298. . . , that of the Figure-of-Eight Knot.
MUTANT Knots have the same hyperbolic knot volume.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 119-127, 1994.
Adams, C; Hildebrand, M.; and Weeks, J. "Hyperbolic In-
variants of Knots and Links." Trans. Amer. Math. Soc.
326, 1-56, 1991.
^ Weisstein, E. W. "Knots and Links." http: //www. astro.
Virginia. edu/~eww6n/math/notebooks/Knots.m.
Hyperbolic Lemniscate Function
Hyperbolic Paraboloid 865
Hyperbolic Lemniscate Function
By analogy with the Lemniscate Functions, hyper-
bolic lemniscate functions can also be defined
px
arcsinhlemn x = (1 + i 4 ) 1 ' 2 dt
Jo
arccoshlemn x = / (1 + t ) ' dt.
J X
Let < 9 < tt/2 and < v < 1, and write
6jj l= { v dt
2 _ J VT+P'
(1)
(2)
(3)
where fi is the constant obtained by setting 6 = 7r/2 and
v = l. Then
<"M>
(4)
where if(fe) is a complete ELLIPTIC INTEGRAL OF THE
FIRST Kind, and Ramanujan showed
2 tan 1 v = 6 + y2
sin(2nfl)
n cosh(n7r) '
8 2 v ; ^ (2n + 1
(-l) n cos[(2n + l)0]
(2n + l)cosh[i(2n+l)7rJ
(5)
(6)
and
ln (r^) =ln[tan( ^ + 2^
A (-irsin[(2n+l)^]
Z-, ( 2n + l)[eC^+ 1 ) 7r -l]
(7)
(Berndt 1994).
see a/50 LEMNISCATE FUNCTION
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 255-258, 1994.
Hyperbolic Map
A linear MAP W 1 is hyperbolic if none of its Eigenval-
ues have modulus 1. This means that IR n can be written
as a direct sum of two A-invariant SUBSPACES E 3 and
E u (where s stands for stable and u for unstable). This
means that there exist constants C > and < A < 1
such that
||A n v|| <CA n |M| if v€E s
\\A- n v\\ <CA n |M| if veE u
for n = 0, 1,
see also Pesin Theory
Hyperbolic Metric
The Metric for the Poincare Hyperbolic Disk, a
model for HYPERBOLIC GEOMETRY. The hyperbolic
metric is invariant under conformal maps of the disk
onto itself.
see also HYPERBOLIC GEOMETRY, POINCARE HYPER-
BOLIC Disk
References
Bear, H. S. "Part Metric and Hyperbolic Metric." Amer.
Math. Monthly 98, 109-123, 1991.
Hyperbolic Octahedron
A hyperbolic version of the Euclidean Octahedron,
which is a special case of the ASTROIDAL ELLIPSOID
with a = 6 = c = 1. It is given by the parametric
equations
x = (cos u cost;) 3
y — (siniicosv) 3
• 3
z = sin v
for u £ [— 7r/2,7r/2] and v £ [— 7r, tt].
see also Astroidal Ellipsoid, Hyperbolic Cube,
Hyperbolic Dodecahedron, Hyperbolic Tetra-
hedron
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 305-306, 1993.
Nordstrand, T. "Astroidal Ellipsoid." http://www.uib.no/
people/nf ytn/asttxt .htm.
Rivin, I. "Hyperbolic Polyhedron Graphics." http://www .
mathsource . com/ cgi -bin /Math Source /Applications /
Graphics/3D/0201-788.
Hyperbolic Paraboloid
A Quadratic Surface given by the Cartesian equation
b 2
(1)
866 Hyperbolic Partial Differential Equation
(left figure). This form has parametric equations
Hyperbolic Secant
x(u,v) = a(u + v)
(2)
y(u,v) = ±bv
(3)
z(u,v) = u + 2uv
(4)
(Gray 1993, p. 336). An alternative form is
z-xy
(5)
(right figure; Fischer 1986), which has parametric equa-
tions
x(u, v) = u
y(u,v) = v
z(U) v) = uv.
(6)
(?)
(8)
see also Elliptic Paraboloid, Paraboloid, Ruled
Surface
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 3-4, 1986.
Fischer, G. (Ed.). Plates 7-9 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 8-10, 1986.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 211-212 and 336,
1993.
Meyer, W. "Spezielle algebraische Flachen." Encylopadie der
Math. Wiss. Ill, 22B, 1439-1779.
Salmon, G. Analytic Geometry of Three Dimensions. New-
York: Chelsea, 1979.
Hyperbolic Partial Differential Equation
A Partial Differential Equation of second-order,
i.e., one of the form
Au xx + 2Bu xy + Cu y y + Du x + Eu v + F = 0,
is called hyperbolic if the MATRIX
Z =
(1)
(2)
satisfies det(Z) < 0. The Wave Equation is an exam-
ple of a hyperbolic partial differential equation. Initial-
boundary conditions are used to give
«(ar, y, t) = g(x, y, t) for x e dQ y t > (3)
u(x,2/,0) =va{x,y) in fl (4)
u t (x,y,0) = vi(x,y) in f>, (5)
u xy = f{u x ,u u x,y) (6)
where
holds in Q.
see also Elliptic Partial Differential Equation,
Parabolic Partial Differential Equation, Par-
tial Differential Equation
Hyperbolic Plane
In the hyperbolic plane H 2 , a pair of LINES can be Par-
allel (diverging from one another in one direction and
intersecting at an IDEAL Point at infinity in the other),
can intersect, or can be Hyperparallel (diverge from
each other in both directions).
see also EUCLIDEAN PLANE, RIGID MOTION
Hyperbolic Point
A point p on a REGULAR SURFACE M £ R 3 is said to
be hyperbolic if the GAUSSIAN CURVATURE K(p) <
or equivalently, the PRINCIPAL CURVATURES «i and k 2 ,
have opposite signs.
see also Anticlastic, Elliptic Point, Gaussian
Curvature, Hyperbolic Fixed Point (Differen-
tial Equations), Hyperbolic Fixed Point (Map),
Parabolic Point, Planar Point, Synclastic
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 280, 1993.
Hyperbolic Polyhedron
A Polyhedron in a Hyperbolic Geometry.
see Hyperbolic Cube, Hyperbolic Dodecahedron,
Hyperbolic Octahedron, Hyperbolic Tetrahe-
dron
Hyperbolic Rotation
Also known as the LORENTZ TRANSFORMATION Or PRO-
CRUSTIAN Stretch. Leaves each branch of the HYPER-
BOLA x'y' = xy invariant and transforms Circles into
Ellipses with the same Area.
/ -l
x = ii x
V = W-
Hyperbolic Rotation (Crossed)
Exchanges branches of the Hyperbola x'y = xy.
i -l
x = fl x
y - -m-
Hyperbolic Secant
Hyperbolic Sine
Hyperbolic Spiral 867
The hyperbolic secant is defined as
1 2
secha; =
coshx e x + e~ x
It has a MAXIMUM at x = and inflection points at
x = ±sech~ 1 (l/V2) « 0.881374.
see also BENSON'S FORMULA, CATENARY, CATENOID,
Euler Number, Hyperbolic Cosine, Oblate
Spheroidal Coordinates, Pseudosphere, Secant,
Surface of Revolution, Tractrix, Tractroid
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic
Functions." §4.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 83-86, 1972.
Spanier, J. and Oldham, K. B. "The Hyperbolic Secant
sech(x) and Cosecant csch(x) Functions." Ch. 29 in An At-
las of Functions. Washington, DC: Hemisphere, pp. 273-
278, 1987.
Hyperbolic Sine
150
100
50
-6 -i
|^-~- — -2
-50
-100
-150
2
4
6
|Sinh z |
The hyperbolic sine is defined as
sinha; = \{e x — e x ).
see also Beta Function (Exponential), Bipo-
lar Coordinates, Bipolar Cylindrical Coor-
dinates, Bispherical Coordinates, Catenary,
Catenoid, Conical Function, Cubic Equation, de
Moivre's Identity, Dixon-Ferrar Formula, El-
liptic Cylindrical Coordinates, Elsasser Func-
tion, Fibonacci Hyperbolic Cosine, Fibonacci
Hyperbolic Sine, Gudermannian Function, He-
licoid, helmholtz differential equation —
Elliptic Cylindrical Coordinates, Hyperbolic
Cosecant, Laplace's Equation — Bispherical Co-
ordinates, Laplace's Equation — Toroidal Co-
ordinates, Lebesgue Constants (Fourier Se-
ries), Lorentz Group, Mercator Projection,
Miller Cylindrical Projection, Modified Bes-
sel Function of the Second Kind, Modified
Spherical Bessel Function, Modified Struve
Function, Nicholson's Formula, Oblate Spher-
oidal Coordinates, Parabola Involute, Parti-
tion Function P, Poinsot's Spirals, Prolate
Spheroidal Coordinates, Ramanujan's Tau Func-
tion, Schlafli's Formula, Shi, Sine, Sine-Gordon
Equation, Surface of Revolution, Toroidal Co-
ordinates, Toroidal Function, Tractrix, Wat-
son's Formula
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic
Functions." §4.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 83-86, 1972.
Spanier, J. and Oldham, K. B. "The Hyperbolic Sine sinh(x)
and Cosine cosh(a:) Functions." Ch. 28 in An Atlas of
Functions. Washington, DC: Hemisphere, pp. 263-271,
1987.
Hyperbolic Space
see Hyperbolic Geometry
Hyperbolic Spiral
An Archimedean Spiral with Polar equation
6*
The hyperbolic spiral originated with Pierre Varignon
in 1704 and was studied by Johann Bernoulli between
1710 and 1713, as well as by Cotes in 1722 (MacTutor
Archive) .
see also Archimedean Spiral, Spiral
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. "Hyperbolic Spi-
ral." http: //www-groups . dcs . st-and.ac . uk/ -history/
Curves/Hyperbolic. html.
868 Hyperbolic Spiral Inverse Curve
Hyperboloid
Hyperbolic Spiral Inverse Curve
Taking the pole as the Inversion Center, the Hyper-
bolic Spiral inverts to Archimedes' Spiral
Hyperbolic Spiral Roulette
The Roulette of the pole of a Hyperbolic Spiral
rolling on a straight line is a Tractrix.
Hyperbolic Substitution
A substitution which can be used to transform integrals
involving square roots into a more tractable form.
Form
Substitution
x = a sinh u
x = a cosh u
yjx 2 + a 2
y/x 2 — a 2
see also Trigonometric Substitution
Hyperbolic Tangent
By way of analogy with the usual TANGENT
sin a?
tan x = ,
cos a;
the hyperbolic tangent is defined as
sinhx e x — e~ x
tanh x =
coshx e x + e~ x e 2x + 1 '
where sinh a; is the Hyperbolic Sine and cosh a: is the
Hyperbolic Cosine. The hyperbolic tangent can be
written using a CONTINUED FRACTION as
tanhx = ^ .
1+-
3 +
5 + ...
see also Bernoulli Number, Catenary, Correla-
tion Coefficient — Gaussian Bivariate Distribu-
tion, Fibonacci Hyperbolic Tangent, Fisher's z f -
Transformation, Hyperbolic Cotangent, Lor-
entz Group, Mercator Projection, Oblate
Spheroidal Coordinates, Pseudosphere, Surface
of Revolution, Tangent, Tractrix, Tractroid
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Hyperbolic
Functions." §4.5 in Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables, 9th
printing. New York: Dover, pp. 83-86, 1972.
Spanier, J. and Oldham, K. B. "The Hyperbolic Tangent
tanh(a;) and Cotangent coth(x) Functions." Ch. 30 in
An Atlas of Functions. Washington, DC: Hemisphere,
pp. 279-284, 1987.
Hyperbolic Tetrahedron
A hyperbolic version of the Euclidean Tetrahedron.
see also HYPERBOLIC CUBE, HYPERBOLIC DODECAHE-
DRON, Hyperbolic Octahedron
References
Rivin, I. "Hyperbolic Polyhedron Graphics." http://vwv .
mathsource . com/ cgi -bin /Math Source /Applications/
Graphics/3D/0201-788.
Hyperbolic Umbilic Catastrophe
A Catastrophe which can occur for three control fac-
tors and two behavior axes.
see also Elliptic Umbilic Catastrophe
Hyperboloid
A Quadratic Surface which may be one- or two-
sheeted.
The one-sheeted circular hyperboloid is a doubly RULED
Surface. When oriented along the z- Axis, the one-
sheeted circular hyperboloid has CARTESIAN COORDI-
NATES equation
+
2 2
V- - fl - i
(1)
Hyperboloid
and parametric equation
x = ay 1 -f- u 2 cosv
y = ay 1 + u 2 sinv
(2)
(3)
(4)
for v E [0, 27r) (left figure). Other parameterizations
include
x(U)V) = a(cosu =F usinu)
t/(n, v) = a(sin u±v cos tt)
z(u, t;) = ±cu,
(middle figure), or
x(ujv) = acoshvcosii
j/(u, v) = a cosh v sin it
z(tz,v) = csinhv
(5)
(6)
(7)
(8)
(9)
(10)
(right figure). An obvious generalization gives the one-
sheeted Elliptic Hyperboloid.
A two-sheeted circular hyperboloid oriented along the
z-Axis has Cartesian Coordinates equation
— + V- - £L
a 2 a? c 2
The parametric equations are
-1.
(11)
x = a sinh u cos t>
(12)
y ~ a sinh t/ sin v
(13)
z = iccoshw
(14)
for v E [0, 2tt). Note that the plus and minus signs in
z correspond to the upper and lower sheets. The two-
sheeted circular hyperboloid oriented along the a- Axis
has Cartesian equation
y
= 1
(15)
Hyperboloid Embedding
and parametric equations
x — ±a cosh u cosh v
y = a sinh u cosh v
z = c sinh v
869
(16)
(17)
(18)
(Gray 1993, p. 313). Again, an obvious generalization
gives the two-sheeted Elliptic Hyperboloid.
The Support Function of the hyperboloid of one sheet
2 2 2
a 2 6 2 c 2
(2 2 2\ _1 / 2
and the Gaussian Curvature is
/£" = -
a 2 6 2 c 2 '
(19)
(20)
(21)
The Support Function of the hyperboloid of two
sheets
2 2 2
fl _ y_ _ £_
a 2 6 2 c 2
= 1
-1/2
and the GAUSSIAN CURVATURE is
K =
a 2 b 2 c 2
(22)
(23)
(24)
(Gray 1993, pp. 296-297).
see also CATENOID, ELLIPSOID, ELLIPTIC HYPER-
BOLOID, Hyperboloid Embedding, Paraboloid,
Ruled Surface
References
Fischer, G. (Ed.). Plates 67 and 69 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 62 and 64, 1986.
Gray, A. "The Hyperboloid of Revolution." §18.5 in Modern
Differential Geometry of Curves and Surfaces. Boca Ra-
ton, FL: CRC Press, pp. 296-297, 311-314, and 369-370,
1993.
Hyperboloid Embedding
A 4-Hyperboloid has Negative Curvature, with
R"
2,2.2
■ x +y + z
rt dx rt dy n dz
2x h 2y— + 2z —
dw dw dw
2w = 0.
Since
dw =
r = xx + yy + zz,
xdx + ydy + zdz _ r • dr
w
Vr 2 - R 2 '
(1)
(2)
(3)
(4)
870 Hypercomplex Number
To stay on the surface of the Hyperboloid,
ds 2 = dx 2 + dy 2 + dz 2 - dw 2
2 2 2 v dr
— dx + dy -h dz - =r-
r 2 — it 2
= dr +r 2 dQ 2 +
dr 2
i * 2 '
(5)
Hypercomplex Number
A number having properties departing from those of
the Real and Complex Numbers. The most com-
mon examples are BlQUATERNIONS, EXTERIOR ALGE-
BRAS, Group algebras, Matrices, Octonions, and
Quaternions.
References
van der Waerden, B. L. A History of Algebra from al~
Khwarizmi to Emmy Noether. New York: Springer- Verlag,
pp. 177-217, 1985.
Hypercube
The generalization of a 3-Cube to n-D, also called a
Measure Polytope. It is a regular Polytope with
mutually Perpendicular sides, and is therefore an Or-
THOTOPE. It is denoted j n and has SCHLAFLI SYMBOL
{4, 3,3 }. The number of fc-cubes contained in an n-
Hyperelliptic Function
Gardner, M. "Hypercubes." Ch. 4 in Mathematical Carni-
val: A New Round- Up of Tantalizers and Puzzles from
Scientific American. New York: Vintage Books, 1977.
Geometry Center. "The Tesseract (or Hypercube)." http://
www.geom.umn.edu/docs/outreach/4-cube/.
Pappas, T. "How Many Dimensions are There?" The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 204-205, 1989.
Hyperdeterminant
A technically defined extension of the ordinary DE-
TERMINANT to "higher dimensional" HYPERMATRICES.
Cayley (1845) originally coined the term, but subse-
quently used it to refer to an Algebraic Invariant of
a multilinear form. The hyperdeterminant of the 2x2x2
HYPERMATRIX A = aijk (for i,j, k = 0, 1) is given by
det(A) = (a oo 2 ani + a 001 On + a 010 a 101 + a n aioo )
— 2(a oo a ooi a iioOni + aooo a oioOioitiiii + a oo a oiiaioo a in
+ a oi0 io a iox a no + aooi&on a no a ioo + cioio a oii a ioi a ioo)
+ 4(aooo a on a ioi^no + aooi a oiottioo a xii)'
The above hyperdeterminant vanishes Iff the following
system of equations in six unknowns has a nontrivial
solution,
aoooZoyo 4- aoio#o2/i + aioo^iyo + aiio^i^i =
aooi^oyo + aoiizoyi + ^oi^iS/o + amxiyi =
=
aooo^o^o + aooi^o^x + ciiooXiZo + aioiEi^i — „
aoio^o^o + clquXqZi + cluqXizq -f omxizi =
^ n aoooyoZo + aooij/o^i + aoio2/i^o + aonj/i^i =
n — 2
cube can be found from the COEFFICIENTS of (2k + l) n . a 100 yozo 4- aioiyoZi + ano2/i2o + amS/i^i = 0.
i4
^
The 1-hypercube is a Line Segment, the 2-hypercube
is the Square, and the 3-hypercube is the Cube. The
hypercube in M 4 , called a TESSERACT, has the SCHLAFLI
Symbol {4,3,3} and Vertices (±1,±1,±1,±1). The
above figures show two visualizations of the TESSERACT.
The figure on the left is a projection of the TESSERACT
in 3-space (Gardner 1977), and the figure on the right is
the Graph of the Tesseract symmetrically projected
into the PLANE (Coxeter 1973). A TESSERACT has 16
Vertices, 32 Edges, four Squares, and eight Cubes.
see also Cross Polytope, Cube, Hypersphere,
Orthotope, Parallelepiped, Polytope, Simplex,
Tesseract
see also DETERMINANT, HYPERMATRIX
References
Cayley, A. "On the Theory of Linear Transformations."
Cambridge Math. J. 4, 193-209, 1845.
GePfand, I. M.; Kapranov, M. M.; and Zelevinsky, A. V.
"Hyperdeterminants." Adv. Math. 96, 226-263, 1992.
Schlafli, L. "Uber die Resultante eine Systemes mehrerer
algebraischer Gleichungen." Denkschr. Kaiserl. Akad.
Wiss.j Math.-Naturwiss. Klasse 4, 1852.
Hyperellipse
71/771 .
y +c
n/m.
-c = 0,
References
Coxeter, H. S. M. Regular Polytopes,
Dover, p. 123, 1973.
3rd ed. New York:
with n/m > 2. If n/m < 2, the curve is a HYPOELLIPSE.
see also Ellipse, Hypoellipse, Superellipse
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 82, 1993.
Hyperelliptic Function
see Abelian Function
Hyperelliptic Integral
Hyperelliptic Integral
see Abelian Integral
Hyperfactorial
The function defined by
H{n) = K(n + 1) = l^ 3 • • • n n ,
where K is the iC-FUNCTION and the first few val-
ues for n = 1, 2, ... are 1, 4, 108, 27648, 86400000,
4031078400000, 3319766398771200000, ... (Sloane's
A002109), and these numbers are called hyperfactorials
by Sloane and Plouffe (1995).
see also G-FUNCTION, GLAISHER-KlNKELIN CON-
STANT, if-FUNCTION
References
Sloane, N. J. A. Sequence A002109/M3706 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Hypergeometric Differential Equation
x(x-l)^ + [(l + a + P)x-y]^+a/3y = 0.
It has Regular Singular Points at 0, 1, and oo.
Every Ordinary Differential Equation of second-
order with at most three REGULAR SINGULAR POINTS
can be transformed into the hypergeometric differential
equation.
see also Confluent Hypergeometric Differential
Equation, Confluent Hypergeometric Function,
Hypergeometric Function
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 542-543, 1953.
Hypergeometric Distribution
Let there be n ways for a successful and m ways for an
unsuccessful trial out of a total of n + m possibilities.
Take N samples and let Xi equal 1 if selection i is suc-
cessful and if it is not. Let x be the total number of
successful selections,
= 5>-
(i)
The probability of i successful selections is then
P(x = i) =
[# ways for i successes] [# ways for N — i unsuccesses]
[total number of ways to select]
/n\ / 77i \ n\ m\
_ \i)\N-i) _ i!(n-i!) (m+i-N)\(N-i)l
(n + 7n)l
(T)
jVl(JV-n-m)!
n\m\Nl(N — m — n)\
i\(n - i)\{m + i - N)\(N - i)l(n + m)
• (2)
Hypergeometric Distribution 871
The ith selection has an equal likelihood of being in any
trial, so the fraction of acceptable selections p is
P =
n
n-\- m
P(Xi = 1) =
n
n + m
(3)
(4)
The expectation value of x is
N \ N
l i=l
i=X
N
^ n
The Variance is
N
nN
-\-m n-\-m
= Np.
(5)
var(#) = ^ var(^) + ^ ^P cov(xi i Xj). (6)
i=i
=1 j=i
Since Xi is a Bernoulli variable,
var(x i )=p(l-p) = ^(l-^)
n + m V n + m/
n /n + ra — n\ _ nm , .
~n + 7n\ n-\-m J (n + m) 2 '
(8)
ENnm
var(a^) = 7 — ■ rx .
(n + my-
i=l
For i < j, the COVARIANCE is
cov(a;i,a!j-) = {xtXj) - {xi}{xj}. (9)
The probability that both i and j are successful for i ^ j
P(xi = 1,xj = 1) = P(xi = l)P(xj = l\xi - 1)
n n — 1
n + mn + m — 1
n(n — 1)
(n -f m)(n + m — 1) '
(10)
But since Xi and Xj are random BERNOULLI variables
(each or 1), their product is also a BERNOULLI variable.
In order for XiXj to be 1, both x» and Xj must be 1,
(XiXj) - P(xiXj = 1) = P(Xi = l,Xj - 1)
n n — 1
n+mn+m— 1
n(n — 1)
(n + m)(rz + m — 1) *
(ii)
872 Hypergeometric Distribution
Combining (11) with
(Xi) (Xj) =
gives
COv(Xi,Xj)
and the KURTOSIS
(12)
n H- ?n n -f- m (n + ra) 2 '
(n + m)(n 2 — n) — n 2 (n + m — 1)
(n -f ra) 2 (n + m — 1)
n 3 + mn 2 — n 2 — mn — n 3 — n 2 m + n 2
72
Hypergeometric Function
F{m,n,N)
mnN(-3 + m + n)(-2 + m + ra)(-m - ra + JV) '
(21)
(n + 77i) 2 (n + m — 1)
mn
(n + m) 2 (n + m — 1) '
(13)
where
F(m, n, TV) = m 3 — m 5 + 3m 2 n — 6m 3 n + m 4 n + 3mn 2
- 12m V + 8m 3 n 2 + n 3 - 6mn 3 + 8m V
+ mn 4 - n 5 - 6m 3 iV + 6m 4 N + 18m 2 niV
- 6m 3 niV + 18mn 2 iV - 24m 2 n 2 iV - 6n 3 N
- 6mn 3 N + 6n 4 A^ + 6m 2 N 2 - QrnN 2
There are a total of AT 2 terms in a double summation
over N. However, i = j for N of these, so there are a
total of N 2 - N = N(N - 1) terms in the Covariance
summation
- 24mniV 2 + 12m 2 nN 2 + 6n 2 iV 2
+ 12mn 2 iV 2 - 6n 3 N 2 .
(22)
The Generating Function is
N N
N(N - l)mn
fc)
EE^^- (n ;;; ( ;;r_ ir ^ M-p*^-*,^-**!;**), (23)
1=1 .7=1 \ N J
1=1 j-1
Combining equations (6), (8), (11), and (14) gives the
Variance
var(#) =
Nmn
N(N - \)mn
(n + m) 2 (n + m) 2 (n + m — 1)
_ Nmn / _ N - 1 \
(m + n) 2 \ n + m — 1/
_ Nmn / N + m- 1 -JV + 1 N
(n + m) 2 V n + m — 1 /
_ Nmn(n + m — N)
(n + m) 2 (n + m — 1) '
so the final result is
(x) — Np
and, since
and
we have
np(l - p) =
m
n + m
mn
(n -f- m) 2 '
(15)
(16)
(17)
(18)
— var(z) = Np(l - p) [1 )
_ mnN(m + n- iV)
(m + n) 2 (m -f n — 1) *
The Skewness is
(19)
7i =
q-p J N- 1 / iV-2n \
\ N-m V JV-2 /
f
/npq
(m - n)(m + n - 2N)
m + n — 2
m + n — 1
mnN(m -f n — A/") '
(20)
where 2 Fi(a, 6;c; 2) is the HYPERGEOMETRIC FUNC-
TION.
If the hypergeometric distribution is written
h n (x,s)= yx ')r x \ (24)
then
2_\h n (x,s)u x = A 2^1 (-5, -np\nq - s + l;w). (25)
References
Beyer, W. H. Ci?C Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 532-533, 1987.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, pp. 113-114, 1992.
Hypergeometric Function
A Generalized Hypergeometric Function
v F q (a\, . . . , a v \ 61, . . . , b q ] x) is a function which can be
defined in the form of a HYPERGEOMETRIC SERIES, i.e.,
a series for which the ratio of successive terms can be
written
afc+i _ P(k) _ (k + ai)(k + q 2 ) • • • (fc + a P )
a k ~ Q(k) ~ (k + b l )(k + b 2 )--{k + b q )(k + l) X '
(1)
(The factor of k + 1 in the DENOMINATOR is present
for historical reasons of notation.) The function
2 Fi(a, b\ c; x) corresponding to p = 2, qr = 1 is the first
hypergeometric function to be studied (and, in general,
arises the most frequently in physical problems), and so
is frequently known as "the" hypergeometric equation.
Hypergeometric Function
Hypergeometric Function 873
To confuse matters even more, the term "hypergeomet-
ric function" is less commonly used to mean CLOSED
Form.
The hypergeometric functions are solutions to the HY-
PERGEOMETRIC Differential Equation, which has a
Regular Singular Point at the Origin. To derive
the hypergeometric function based on the HYPERGEO-
METRIC Differential Equation, plug
y = Y2 AnZn
n=0
oo
y = y^nA n z n ~ l
71 =
OO
(2)
(3)
(4)
into
z(l - z)y" + [c - (a + b + l)a]y - aby = (5)
to obtain
oo oo
^2 n(n - ^Anz 71 - 1 -Y^n(n~ l)A n z n
n—0 n=0
oo oo
+c ^ nA ^ n ~ 1 + (a + b + 1) ]P nA n z n
n = n=r0
oo
-ab^2 A nZ n = (6)
Yl n{n - ^Anz 71 ' 1 -Y2n{n- l)A n z n
n~2 n=0
oo oo
+ cY^nAnZ 71 ' 1 - (a + b + l)Y]nA n z n
n— 1 n=l
oo
-a6^^ n ^ n = (7)
71 =
oo oo
y Jjt + l)nA n +iz n — N^ n(n — l)A n z n
71 = 71 =
OO oo
+ C J2( n + l)^n+l2" - (a + & + 1) 5Z nA " 2 "
n=0 7i=0
oo
-ab^AnZ 71 = (8)
71 =
OO
^[n(n + 1)j4„+i - n(n - l)A n + c(n + l)A„_i
71 =
-(a + b + l)nA n - a6A n ]2 n = (9)
^{(n + l)(n + c),4n+i
71 =
-[n{n -l + a + 6+l) + ab]A n } z n = (10)
OO
^{(n+l)(n + c)A n+ i
n=0
-[n 2 + (a + 6)n + ab]A„}z n = 0, (11)
so
and
y = A
(n + a)(n + 6)
A " +1_ (n+l)(n + cr n
a6 0(0 + 1)6(6 + 1) j
1+ l!c* + 2!c(c+l) Z + -
(12)
(13)
This is the regular solution and is denoted
2 F 1 (a,6; C; z) = l+— Z + 2 , c(c + 1) * +■■
-E
(a)n(6)n >?"
(c)„ n! '
(14)
where (a) n are POCHHAMMER SYMBOLS. The hyperge-
ometric series is convergent for REAL — 1 < z < 1, and
for z = ±1 if c > a + 6. The complete solution to the
Hypergeometric Differential Equation is
y = ^2^1 (a, 6; c; ^)+Bz 1 " c 2 F 1 (a+l-c, 6+1 -c; 2-c; z).
(15)
Derivatives are given by
d,2Fi(a, 6;c; z) _ a&
dz c
2 Fi(a + l,6+l;c + l;z)(16)
rf 2 2 Fi(a,6;c;z) __ a(a + 1)6(6+1)
dz 2 ~~ c(c+l)
x 2 F 1 (a + 2,6 + 2;c + 2;z) (17)
(Magnus and Oberhettinger 1949, p. 8). An integral
giving the hypergeometric function is
2 F 1 (a i b;c;z)
r(c)
r(6)r(c-
as shown by Euler in 1748.
6)y a-**)-
(18)
A hypergeometric function can be written using Eu-
ler's Hypergeometric Transformations
t-*t
(19)
t-> 1-i
(20)
t-> (1-z-tz) -1
(21)
1-t
(22)
874 Hypergeometric Function
in any one of four equivalent forms
2 Fi(a,b;c;z) = (1 - z)~ a 2 F 1 (a,c-b;c;z/(z - 1))
(23)
= (1 - z)~ b 2 Fi(c - a, b; c; z/{z - 1))
(24)
= (1 - z) c - a - b 2 F 1 (c - a, c - b; c; z).
(25)
It can also be written as a linear combination
2 Fi(a,6;c;^)
_ r(c)r(c - a - 6)
T(c - a)r(c - b)
T(c)r(a + 6 - c)
+ ■
2 Fi(a, 6;a + 6+1 — c; 1 — z)
(1 - z) c
i»r(&)
x 2 i ? i(c - a, c - b; 1 + c - a - fe; 1 - z). (26)
Kummer found all six solutions (not necessarily regular
at the origin) to the HYPERGEOMETRIC DIFFERENTIAL
Equation,
u x (x) = 2 F 1 (a,b;c;z)
ui(x) = 2 i*i(a, 6;a + 6+1- c; 1 — z)
t/ 3 (x) = z~ a 2 Fi(a,a+ 1 — c;a+ 1 - 6; 1/z)
u 4 (x) = z~ b 2 F X {b + 1 - c, 6; 6 + 1 - a; 1/z)
us («) = 2 1-c 2^1 {b + 1 - c, a + 1 - c; 2 - c; z)
u 6 (a:) = (1 - z) c ~ a ~ b 2 Fi{c - a, c - 6; c + 1 - a - 6; 1 - z).
Hypergeometric Function
u[ l) (x) = z 1_c 2 Fi (fa + 1 - c, a + 1 - c; 2 - c; z)
u< 2) (x) = z'-^l - z)^- 1 a Fi(6 + 1 - c, 1 - a; 2 - c; «/(z - 1))
< 2) (x) = z 1_c (l - z) c - a_1 2^(1 - fa, a + 1 - c; 2 - c; z/(* - 1))
ti^ 4) (x) = z^ c (l - z) c " a - b a Fi(l - 6, 1 - a; 2 - c; z)
u< 1} (x) = (1 -z) c ~ a - b 2 F 1 (c-a,c-6;c+l - a - 6; 1 - z)
u < 2 >(x) = z- c (i-z;r a - b
X 2 Fi(c -a, 1 — a;c+l— a — b;l — 1/z)
x 2 Fi(l - 6, c - 6;c+ 1 -a- 6; 1 - 1/z)
(4) / \ c — a — b/-. \ c — a — b
«i ( X ) = Z I 1 ' Z )
x 2 Fi(l-b, l-a;c+l-a-6;l -z).
Goursat (1881) gives many hypergeometric transforma-
tion FORMULAS, including several cubic transformation
Formulas.
Many functions of mathematical physics can be ex-
pressed as special cases of the hypergeometric functions.
For example,
2 F 1 {-l,l + l,l;(l-z)/2)=Pi(z), (27)
where Pi(z) is a LEGENDRE POLYNOMIAL.
(l + zr = 2 Fi(-n, &;&;-*) (28)
\n(l + z) = z 2 F 1 (l t l;2;-z) (29)
Complete ELLIPTIC INTEGRALS and the RiEMANN P-
Series can also be expressed in terms of 2 i*i(a, 6; c\z).
Special values include
Applying EULER'S HYPERGEOMETRIC TRANSFORMA-
TIONS to the Kummer solutions then gives all 24 possi-
ble forms which are solutions to the HYPERGEOMETRIC
DIFFERENTIAL EQUATION
= 2 F 1 (a,6; c; z)
- (1 - z)"%F 1 (a, c - 6; c; z/(z - 1))
= (1 - zy b 2 F l {c-a,b\c*,z/{z- 1))
= (1 - z) c ~ a - b 2 Fi (c - a, c - fa; c; z)
= 2 F 1 (a,6;a + fc+ 1 - c; 1 - z)
= z~ a 2 F 1 (a,a + l-c;a + fa+l — c;l — 1/z)
- z~ b 2 F±{b + 1-c, 6;a + b+l-c;l- 1/z)
= z 1_c 2 F 1 (6+ 1 - c,a + 1 - c; a + fc + 1 - c; 1 - z)
= z~° 2 Fi (a, a + 1 - c; a + 1 - fa; 1/z)
= z _a (l- l/lz) _a 2 F 1 (a,c-6;a + l - fa; 1/(1 - z))
= z-(l-l/*) c — f
X 2 F!(1 - b,o + l - c;a+ 1 - fa; 1/(1 - z))
= z"°(l - l/z) c -°- b ,^(1 - fa, c - fa; a+ 1 - 6; 1/z)
= ^-^F^b + 1 - c, fa; b + 1 - a; 1/z)
= z _b (l- 1/z)'- 6 " 1
X zF^bi -c, 1 -a; 6 + 1 - a; 1/(1 - z))
= z" 6 (l - l/z) _b 2 F 1 {c - a, fa; fa + 1 - a; 1/(1 - z))
= z _b (l - l/z) c - a_b iF^c - a, 1 - a; 6 + 1 - a; 1/z)
2 Fi(a,6;a-6+l;-l)
i"
x)
(»
'x)
<°>
^x)
c.)
^x)
4"
'x)
<•>
'x)
4"
x)
4 1 '
x)
i 1 '
x)
4"
x)
•i"
'x)
4"
'x)
4 1 *
x)
.?'
[x)
.?'
[x)
4 4)
[x)
- 9- r= T(l + a + b)
2 Fi(a, 6; c; |) - 2 a 2 Fi(a, c - 6; c; -1)
„, . w ^^^ M r(i)r[(|(i + a + 6)] /oox
2*1 (a, 6; |(a + 6 + l); |) = ^ rl ^ ; J xlT , rl/ , , ^, (33)
(31)
(32)
27 r[i(i + a)]r[i(i + &)r
2 Fi(a, 1 - a; c; f ) = —y -^
2 Fi(a,6;c;l) =
r[|(o + c)]r[|(i + c -a)]
r(c)r(c - a - 6)
r(c - a)r(c - b) '
Rummer's First Formula gives
2 Fi(| + m - fc, -n; 2m + 1; 1)
(34)
(35)
T(2m + l)r(m + ~ + k + n)
r(m + I + fc)r(2m + 1 + n)
, (36)
where m 7^ —1/2, — 1, —3/2, Many additional
identities are given by Abramowitz and Stegun (1972,
p. 557).
Hypergeometric Function
Hypergeometric Series 875
Hypergeometric functions can be generalized to GENER-
ALIZED Hypergeometric Functions
n i 71 m (ai, . . . , a n ] bi, - - • , bm] z)*
(37)
A function of the form \F\{a\b\z) is called a CONFLU-
ENT Hypergeometric Function, and a function of
the form o^iOM) is called a Confluent Hypergeo-
metric Limit Function.
see also APPELL HYPERGEOMETRIC FUNCTION,
Barnes' Lemma, Bradley's Theorem, Cayley's
Hypergeometric Function Theorem, Clausen
Formula, Closed Form, Confluent Hypergeo-
metric Function, Confluent Hypergeometric
Limit Function, Contiguous Function, Darling's
Products, Generalized Hypergeometric Func-
tion, Gosper's Algorithm, Hypergeometric Iden-
tity, Hypergeometric Series, Jacobi Polynom-
ial, Rummer's Formulas, Rummer's Quadratic
Transformation, Rummer's Relation, Orr's The-
orem, Ramanujan's Hypergeometric Identity,
Saalschutzian, Sister Celine's Method, Zeilber-
ger's Algorithm
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Hypergeometric
Functions." Ch. 15 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 555-566, 1972.
Arfken, G. "Hypergeometric Functions." §13.5 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 748-752, 1985.
Fine, N. J. Basic Hypergeometric Series and Applications.
Providence, RI: Amer. Math. Soc, 1988.
Gasper, G. and Rahman, M. Basic Hypergeometric Series.
Cambridge, England: Cambridge University Press, 1990.
Gauss, C. F. "Disquisitiones Generales Circa Seriem Infini-
tam[^], + [^±^^]x 2
r a(a + l)(q + 2)/3(g + l)(g + 2) i 3+ fitc> p ars p rior> » Q Qm _
X L 1-2.3. -y(T + l)(7 + 2) J
mentationes Societiones Regiae Scientiarum Gottingensis
Recentiores, Vol. II. 1813.
Gessel, I. and Stanton, D. "Strange Evaluations of Hyperge-
ometric Series." SIAM J. Math. Anal. 13, 295-308, 1982.
Gosper, R. W. "Decision Procedures for Indefinite Hyper-
geometric Summation." Proc. Nat. Acad. Sci. USA 75,
40-42, 1978.
Goursat, M. E. "Sur 1 'equation different ielle lineaire qui ad-
met pour integrate la serie hypergeometrique." Ann. Sci.
Ecole Norm. Super. Sup. 10, S3-S142, 1881.
Iyanaga, S. and Kawada, Y. (Eds.). "Hypergeometric Func-
tions and Spherical Functions." Appendix A, Table 18
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, pp. 1460-1468, 1980.
Kummer, E. E. "Uber die Hypergeometrische Reihe." J.
fur die Reine Angew. Mathematik 15, 39-83 and 127-172,
1837.
Magnus, W. and Oberhettinger, F. Formulas and Theorems
for the Special Functions of Mathematical Physics. New-
York: Chelsea, 1949.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 541-547, 1953.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, 1996.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Hypergeometric Functions." §6.12 in Numer-
ical Recipes in FORTRAN: The Art of Scientific Comput-
ing, 2nd ed. Cambridge, England: Cambridge University
Press, pp. 263-265, 1992.
Seaborn, J. B. Hypergeometric Functions and Their Appli-
cations. New York: Springer- Verlag, 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 Gauss Function
F(a, 6;c;x)." Ch. 60 in An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 599-607, 1987.
Hypergeometric Identity
A relation expressing a sum potentially involving Bino-
mial Coefficients, Factorials, Rational Func-
tions, and power functions in terms of a simple re-
sult. Thanks to results by Fasenmyer, Gosper, Zeil-
berger, Wilf, and Petkovsek, the problem of determin-
ing whether a given hypergeometric sum is expressible
in simple closed form and, if so, finding the form, is now
(subject to a mild restriction) completely solved. The al-
gorithm which does so has been implemented in several
computer algebra packages and is called ZEILBERGER'S
Algorithm.
see also GENERALIZED HYPERGEOMETRIC FUNCTION,
Gosper's Algorithm, Hypergeometric Series,
Sister Celine's Method, Wilf-Zeilberger Pair,
Zeilberger's Algorithm
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 18, 1996.
Hypergeometric Polynomial
see Jacobi Polynomial
Hypergeometric Series
A hypergeometric series Ylk ak * s a ser ^ es ^ or wn i cn
ao = 1 and the ratio of consecutive terms is a RATIONAL
Function of the summation index fc, i.e., one for which
Qfc+i
dk
Q(ky
with P(k) and Q(k) POLYNOMIALS. The functions gen-
erated by hypergeometric series are called HYPERGEO-
METRIC Functions or, more generally, Generalized
Hypergeometric Functions. If the polynomials are
completely factored, the ratio of successive terms can be
written
flfc+i
P(k)
(fc + ai)(fc + Q2) • (fe + a P )
a k Q(k) (k + 6i)(t + b 2 ) • • • (k + b q )(k + 1)
x,
where the factor of A; + 1 in the DENOMINATOR is present
for historical reasons of notation, and the resulting GEN-
ERALIZED Hypergeometric Function is written
P F q
a\ a2
bi 62
= y^afcs fc
876 Hypergroup
Hypersphere
If p = 2 and q = 1, the function becomes a traditional
Hypergeometric Function 2^1 (a, b; c; a).
Many sums can be written as GENERALIZED HYPER-
GEOMETRIC FUNCTIONS by inspections of the ratios of
consecutive terms in the generating hypergeometric se-
ries.
see also GENERALIZED HYPERGEOMETRIC FUNCTION,
Geometric Series, Hypergeometric Function,
HYPERGEOMETRIC IDENTITY
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Hyperge-
ometric Series," "How to Identify a Series as Hyperge-
ometric," and "Software That Identifies Hypergeometric
Series." §3.2-3.4 in A=B. Wellesley, MA: A. K. Peters,
pp. 34-42, 1996.
Hypergroup
A Measure Algebra which has many properties as-
sociated with the convolution MEASURE ALGEBRA of a
GROUP, but no algebraic structure is assumed for the
underlying SPACE.
References
Bloom, W. R.; and Heyer, H. The Harmonic Analysis of
Probability Measures on Hypergroups. Berlin: de Gruyter,
1995.
Jewett, R. I. "Spaces with an Abstract Convolution of Mea-
sures." Adv. Math. 18, 1-101, 1975.
Hypermatrix
A generalization of the Matrix to an m x 712 x ■ ■ • array
of numbers.
see also Hyperdeterminant
References
Gel'fand, I. M.; Kapranov, M. M.; and Zelevinsky, A. V.
"Hyperdeterminants." Adv. Math. 96, 226-263, 1992.
Hyper parallel
Two lines in Hyperbolic Geometry which diverge
from each other in both directions.
see also Antiparallel, Ideal Point, Parallel
Hyperperfect Number
A number n is called fc-hyperperfect if
References
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- Verlag, pp. 45-53, 1994.
Sloane, N. J. A. Sequences A007592/M5113 and A007593/
M5121 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Hyperplane
Let ai, a2, . . . , a n be SCALARS not all equal to 0. Then
the Set 5 consisting of all Vectors
X =
Xl
X2
n — 1 + fc^Jrfj,
in W 1 such that
a 1 x 1 + 0,2X2 + . . . + a n x n —
is a SUBSPACE of W 1 called a hyperplane. More gen-
erally, a hyperplane is any co-dimension 1 vector SUB-
SPACE of a Vector Space. Equivalently, a hyperplane
V in a Vector Space W is any Subspace such that
W/V is 1-dimensional. Equivalently, a hyperplane is the
Kernel of any Nonzero linear Map from the Vector
Space to the underlying Field.
Hyperreal Number
Hyperreal numbers are an extension of the REAL NUM-
BERS to include certain classes of infinite and infinites-
imal numbers. A hyperreal number is said to be finite
Iff |x| < n for some Integer n. x is said to be in-
finitesimal IFF |x| < 1/n for all INTEGERS n.
see also Ax-Kochen Isomorphism Theorem, Non-
standard Analysis
References
Apps, P. "The Hyperreal Line." http://www.math.wisc.
edu/-apps/line .html.
Keisler, H. J. "The Hyperreal Line." In Real Numbers, Gen-
eralizations of the Reals, and Theories of Continua (Ed.
P. Ehrlich). Norwell, MA: Kluwer, 1994.
Hyperspace
A Space having Dimension n > 3.
where the summation is over the Proper Divisors with
1 < di < n, giving
ka(n) = (& + l)n + A: + 1,
where a(n) is the Divisor Function. The first few
hyperperfect numbers are 21, 301, 325, 697, 1333, . . .
(Sloane's A007592). 2-hyperperfect numbers include 21,
2133, 19521, 176661, ... (Sloane's A007593), and the
first 3-hyperperfect number is 325.
Hypersphere
The n-hypersphere (often simply called the n-sphere)
is a generalization of the CIRCLE (n = 2) and SPHERE
(n = 3) to dimensions n > 4. It is therefore defined as
the set of n-tuples of points (xi, #2, • • • > x n ) such that
xi 2 + x 2 2 + . . . + x n 2 = R 2 ,
(i)
Hypersphere
where R is the RADIUS of the hypersphere. The CON-
TENT (i.e., n-D VOLUME) of an n- hypersphere of RADIUS
R is given by
Jo
S n r- 1 dr=^^,
(2)
where S n is the hyper-SuRFACE Area of an n-sphere of
unit radius. But, for a unit hypersphere, it must be true
that
S n f
Jo
1 dr
-(»X 3 + - + »n 3 ) ^ ^
But the Gamma Function can be defined by
dr,
f°° 2
T(m) = 2 e" r r 2 ™' 1 ,
Jo
i5 n r(in) = [r(l)]" - (7r 1/2 ) n
h n
2tt
n/2
r(|n)*
This gives the Recurrence Relation
c __ 27r5 n
Jn+2 — ■
n
Using T(n + 1) = nT(n) then gives
£ n iT 7r n/2 J2 n 7r n/2 R n
V n
(in)r(fn) r(l + |n)
(Conway and Sloane 1993).
(3)
(4)
(5)
(6)
(7)
(8)
5
|3
1
'"*>
_:i3V-r±t"
f Jul
\^
35
rt 30
J 25
« 20
J 15
3 10
M 5
^
5 10 15 20
Dimension
5 10 15 20
Dimension
Strangely enough, the hyper-SURFACE Area and CON-
TENT reach MAXIMA and then decrease towards as n
increases. The point of Maximal hyper- Surface Area
satisfies
dS n 7r"/ 2 [ln7r-Vo(in)]
dn
ran)
o,
(9)
Hypersphere 877
where tpo{x) = V(x) is the DlGAMMA FUNCTION. The
point of Maximal Content satisfies
dV n __ 7r n/2 [ln7r-^o(l + |n)] __
dn
2r(l + |n)
= 0.
(10)
Neither can be solved analytically for n, but the numer-
ical solutions are n = 7.25695 ... for hyper- SURFACE
Area and n = 5.25695 ... for Content. As a result,
the 7-D and 5-D hyperspheres have MAXIMAL hyper-
Surface Area and Content, respectively (Le Lion-
nais 1983).
n
v n
Vn/Vn -cube
5„
1
1
1
2
1
2
2
7T
1*
2tt
3
*»
1*
47T
4
i- 2
1 ^r 2
32*
27T 2
5
8 -rr 2
15*
1 -TT 2
60*
3*
6
6*
1 _3
384 *
TT 3
7
16 „3
105*
1 _3
840 7l
16^.3
15*
8
1 -TT 4
24*
1 _4
6144 "
1 * 4
3*
9
32 ^4
945*
1 4
15120^
32 ^.4
105*
10
120 "
1 TT 5
1 -TT 5
12*
122880"
In 4-D, the generalization of SPHERICAL COORDINATES
is defined by
xi = R sin ij) sin cos
X2 — R sin ^ sin <j> sin
X3 = i? sin V 7 cos </>
X4 = Rcosip.
The equation for a 4-sphere is
Xi 2 + X2 2 + X3 2 + X4 2 = i? 2 ,
(11)
(12)
(13)
(14)
(15)
and the LINE ELEMENT is
2 D 2fJ /2 , -2
<te 2 = JE 2 ^ + sin 2 ^(d<£ 2 + sin 2 0<*0 2 )]. (16)
By defining r = Rsinip, the Line ELEMENT can be
rewritten
dr 2
+ r 2 (# 2 + sin 2 <f>d$ 2 ). (17)
The hyper-SURFACE Area is therefore given by
/*7T PIT /»27T
S4 = I Rdtj) I R sin ip d<fr I Rsinip sin <j> d6
Jo Jo Jo
2tt 2 R 3 .
(18)
878 Hypersphere Packing
Hypocycloid
see also Circle, Hypercube, Hypersphere Packing,
Mazur's Theorem, Sphere, Tesseract
References
Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices,
and Groups, 2nd ed. New York: Springer- Verlag, p. 9,
1993.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 58, 1983.
Peterson, I. The Mathematical Tourist: Snapshots of Modern
Mathematics. New York: W. H. Freeman, pp. 96-101,
1988.
Hypersphere Packing
Draw unit n-spheres in an n-D space centered at all ±1
coordinates. Then place an additional Hypersphere at
the origin tangent to the other HYPERSPHERES. Then
the central HYPERSPHERE is contained with the HY-
PERSPHERE with VERTICES at the center of the other
spheres for n between 2 and 8. However, for n — 9, the
central Hypersphere just touches the bounding Hy-
persphere, and for n > 9, the Hypersphere is par-
tially outside the hypercube. This can be seen by finding
the distance from the origin to the center of one of the
HYPERSPHERES
y/(±l)* + ... + (±l)* = yfii.
The radius of the central sphere is therefore y/n— 1. The
distance from the origin to the center of the bounding
hypercube is always 2 (two radii), so the center HYPER-
SPHERE is tangent when y/n — 1 == 2, or n = 9, and
outside for n > 9.
The analog of face-centered cubic packing is the densest
lattice in 4- and 5-D. In 8-D, the densest lattice packing
is made up of two copies of face-centered cubic. In 6- and
7-D, the densest lattice packings are cross-sections of the
8-D case. In 24-D, the densest packing appears to be
the Leech Lattice. For high dimensions (~ 1000-D),
the densest known packings are nonlattice. The densest
lattice packings in n-D have been rigorously proved to
have Packing Density 1, tt/(2V3), n/(3V2), n 2 /16,
tt 2 /(15\/2), tt 3 /(48V5), tt 3 /105, and tt 4 /384 (Finch).
The largest number of unit Circles which can touch
another is six. For Spheres, the maximum number is
12. Newton considered this question long before a proof
was published in 1874. The maximum number of hyper-
spheres that can touch another in n-D is the so-called
Kissing Number.
see also KISSING NUMBER, LEECH LATTICE, SPHERE
Packing
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/hermit/hermit.html.
Gardner, M. Martin Gardner's New Mathematical Diver-
sions from Scientific American. New York: Simon and
Schuster, pp. 89-90, 1966.
Hypervolume
see Content
Hypocycloid
The curve produced by a small CIRCLE of RADIUS b
rolling around the inside of a large CIRCLE of RADIUS
a > b. A hypocycloid is a HYPOTROCHOID with h =
b. To derive the equations of the hypocycloid, call the
Angle by which a point on the small Circle rotates
about its center $, and the ANGLE from the center of
the large CIRCLE to that of the small CIRCLE <j>. Then
(a - b)<f> = 6#,
a — b
<t>.
(i)
(2)
Call p = a — 26. If x(0) = p, then the first point is
at minimum radius, and the Cartesian parametric equa-
tions of the hypocycloid are
x = (a — b) cos (j> — b cos #
= (a - b)cos<j) - fecos ( — - — </>) (3)
y = (a — b) sin <f> + b sin $
= (a — &)sin<£ + b sin f — - — <f>\ . (4)
If x(0) = a instead so the first point is at maximum ra-
dius (on the Circle), then the equations of the hypocy-
cloid are
x = (a — b) cos0 + 6 cos I — - — <j>\ (5)
y = (a — b) sin<£ — 6 sin ( — - — (j>) . (6)
An n-cusped non-self-intersecting hypocycloid has
a/b = n. A 2-cusped hypocycloid is a Line Segment,
as can be seen by setting a = b in equations (3) and (4)
and noting that the equations simplify to
x — asm<
(7)
(8)
Hypocycloid
Hypocycloid 879
A 3-cusped hypocycloid is called a DELTOID or TRICUS-
POID, and a 4-cusped hypocycloid is called an ASTROID.
If a/b is rational, the curve closes on itself and has b
cusps. If a/b is IRRATIONAL, the curve never closes and
fills the entire interior of the CIRCLE.
n-hypocycloids can also be constructed by beginning
with the Diameter of a CIRCLE, offsetting one end by
a series of steps while at the same time offsetting the
other end by steps n times as large in the opposite di-
rection and extending beyond the edge of the CIRCLE.
After traveling around the CIRCLE once, an n-cusped
hypocycloid is produced, as illustrated above (Madachy
1979).
Let r be the radial distance from a fixed point. For RA-
DIUS of Torsion p and Arc Length s, a hypocycloid
can given by the equation
s 2 +p 2 ^ 16r 2 (9)
(Kreyszig 1991, pp. 63-64). A hypocycloid also satisfies
sin ip ■
where
r i = tan *
(10)
(11)
and ip is the Angle between the Radius Vector and
the Tangent to the curve.
The Arc Length of the hypocycloid can be computed
as follows
f a ~ b
x — —(a — b) sin0 — (a — 6) sin ( — - — <f>\
— (a — b) sin <fi + sin ( — - — <f> j
y = (a — b) coscp — (a — 6) cos ( <j)\
= (a — b) cos — cos f — - — <p )
(12)
(13)
x' a + y ,2 = (a-&)
, . * /a - &
+ sin
' <fi + 2 sin <j> sin ( — - — <j>\
in 2 I — - — <f>\ 4- cos 2 <j> — 2 cos cos ( — - — <f>\
+cos2 (nr )]
= (a - bf 1 2 + 2 [sin sin (^^t)
— cos<£cos I — <f>] >
= 2(o - bf [l - cos U + ^J^A]
= 4(a-&) 2 i[l-cos(^)]^4(a-6) 2 sin 2 (g),
(14)
so
ds = <s/x 12 + y f2 d<j> = 2 (a - b) sin (°^\ d<j> (15)
for (j) < (b/2a)n. Integrating,
•(*)=jf*='(-')[-f«-(S)].'
_*^fl [_„(.,) +l]
8b(a — b) . 2 / a
— * sin ' —
The length of a single cusp is then
K)
8&(q - b) . 2 /*r\ _ 86(a - 6)
sin
(D=« <">
If n = a/b is rational, then the curve closes on itself
without intersecting after n cusps. For n = a/b and
with x(0) = a, the equations of the hypocycloid become
x = — [(n — 1) cos0 — cos[(n — l)<j)]a, (18)
V — ~[( n ~ l)sin0 + sin[(n - l)0]a, (19)
and
8&(&n-&) ot/ . Sa(n-1)
s n = n — - — = - = 8b(n - 1) = — i ^. (20)
no n
Compute
xy — ya;' = (a — 6) cos<£ + 6cos ( 0) (b — a)
x sin <p + sin I — - — <p\
— (a — b) sin (j> — b sin I — - — <j> J (a — 6)
cos — cos I — - — J
a0>
2(a 2 -3a6 + 26 2 )sin 2 gV (21)
880 Hypocycloid
Hypocycloid
The AREA of one cusp is then
/>27r6/a
A= \ I (xy -yx')d<t>
Jo
= (a 2 -3a6 + 26 2 )
= (a 2 -3a6 + 26 2 )
__ 6(a 2 - 3a6 + 26 2 )
a£-6sin(f )
2a
> ( 2 ^)
2-irb/a
2a
(22)
If n = a/6 is rational, then after n cusps,
b(a 2 - 3ab + 2b 2 ) « (° 2 " 3 < + 2 £)
r i — — TITT 1 —
A n — 717T-
n 2 -3n + 2 2 _ (n - l)(n - 2) 2
r 7ra = r 7ra . (26)
a — p
nt
a'
then
r 2 = i(a 2 +p 2 )-i(a
p 2 ) cos
(!♦)
= §(a 2 + p 2 )- |(a 2 -p 2 )cos(2nt).
The Polar Angle is
_ y (a-6)sin<£ + 6sin(^0)
tan# — — = — - — -,
x (a- b)coscp-bcos (^</>)
But
b=\{a-p)
a - b — \{a + p)
a — b _ a + p
b a — p^
(29)
(30)
(31)
(32)
(33)
(34)
The equation of the hypocycloid can be put in a form
which is useful in the solution of CALCULUS OF VARI-
ATIONS problems with radial symmetry. Consider the
case x(0) = p, then
2 2,2
r = x + y
= (a — b) 2 cos 2 <j> — 2(a — 6)6 cos <f> cos [ — - — <fi J
+ 6 w(^V)
+ (a — 6) 2 sin 2 <j> + 2(a — 6)6sin<£sin I — - — j
rf *"(H-'*)]
= {(a-6) 2 +6 2 -2(a-6)6
x cos^cos f — - — <j)\ — sin <p sin ( — - — <f>) ?
= (a - 6) 2 + 6 2 - 2(a - 6)6cos (j-</>\ . (24)
But p = a — 26, so 6 = (a — p)/2, which gives
(a _ h f + 6 2 = [„ - I( a - p)] 2 + [i(o - p)f
= [|(« + P)] 2 + [|(a-p)] 2
= l(a 2 + 2ap + p 2 + a 2 - 2ap + p 2 )
2(a-6)6=2[a-i(a-p)]i(a-p)
(25)
= i(a + p)(a-p) = i(a 2 -p 2 ). (26)
Now let
2Qt =y(t>,
6
a — p
Qt
(27)
(28)
f (a + p) sin0 + i(o - p) sin (f±f <*)
§(a + p)cosc£- ±(a-p)cos (fzf^)
(a + p) sin (^fit) + (a - p) sin (^fit)
(a + p) cos (.^fit) - (a - p) cos (^Ht)
a [sin (^fit) +sin(^± £ nt)]
+p[sin(^at) -gjn(a±£nt)]
a [cos (i=£fit) - cos (^nt)]
+p [cos (^nt) + cos (^fit)]
2asin(Ot)cos (ffii) - 2pcos(f2i)sin (f fit)
2asin(fit)sin (ffit) + 2pcos(fit)cos (f fit)
atan(fit) - ptan (ffit)
atan(fit)tan (ffit) +p
(35)
Computing
[atan(fit) - ptan (f fit) + tan (f Qt)]
/ p \ X [a tan(fit) tan (jfit) + pi
V a / [atan(fii)tan(ffii) + p]
- [atan(m) - ptan (*^*)1 tan (f fit)
atan(fif) [l + tan 2 (f fit)]
p[l + tan 2 (ffit)]
= - tan(fit),
P
then gives
tan
tan(ftt)
-Qt
(36)
(37)
Hypocycloid — 3- Cusped
Finally, plugging back in gives
= tan
tan
-tan I cj)
P \a~ P
~ tan [ <f>
P \a- P
p a
a a — p
a — p
(38)
This form is useful in the solution of the Sphere with
Tunnel problem, which is the generalization of the
Brachistochrone Problem, to find the shape of a
tunnel drilled through a SPHERE (with gravity varying
according to Gauss's law for gravitation) such that the
travel time between two points on the surface of the
Sphere under the force of gravity is minimized.
see also CYCLOID, EPICYCLOID
References
Bogomolny, A. "Cycloids." http://www.cut-the-knot.com/
pythagoras/cycloids . html.
Kreyszig, E. Differential Geometry. New York: Dover, 1991.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 171-173, 1972.
Lee, X. "Epicycloid and Hypocycloid." http://www.best.
com/-xah/SpecialPlaneCurvesjdir/EpiHypocycloid^dir/
epiHypocycloid.html.
MacTutor History of Mathematics Archive. "Hypocycloid."
http: //www-groups .dcs.st-and.ac.uk/-history/Curves
/Hypocycloid. html.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 225-231, 1979.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 50-52, 1991.
Yates, R. C. "Epi- and Hypo- Cycloids." A Handbook on
Curves and Their Properties. Ann Arbor, MI: J. W. Ed-
wards, pp. 81-85, 1952.
Hypocycloid — 3-Cusped
see Deltoid
Hypocycloid — 4-Cusped
see Astroid
Hypocycloid Evolute
For x(0) = a,
x = — (a — b) cos0 — b cos ( — - — <f>)
y= ^26[ (a - 6)sin ^ + 6sin (^*).
Hypocycloid Pedal Curve 881
If a/b — n, then
1
y
n-2
1
n-2
[(n — 1) cos <j) — cos[(n - l)<f>]a
[(n - 1) sin <f> + sin[(n - l)<j>]a.
This is just the original HYPOCYCLOID scaled by the
factor (n — 2)/n and rotated by l/(2n) of a turn.
Hypocycloid Involute
The Hypocycloid
a
V
a -2b
a
a -2b
has Involute
a -2b
\(a — b) cos <j> — b cos ( — - — <j> )
(a — b) sin<j) + 6 sin I — - — <f>\
y
a
a -2b
(a — b)cos(f> + 6 cos
(a — b) sin <f> — b sin
which is another HYPOCYCLOID.
Hypocycloid Pedal Curve
The Pedal Curve for a Pedal Point at the center is
a Rose.
882 Hypoellipse
Hyzer's Illusion
Hypoellipse
y n/m + c
n/m
-c = 0,
with n/m < 2. If n/m > 2, the curve is a Hyperel-
LIPSE.
see also Ellipse, Hyperellipse, Superellipse
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 82, 1993.
Hypotenuse
The longest LEG of a RIGHT TRIANGLE (which is the
side opposite the Right Angle).
Hypothesis
A proposition that is consistent with known data, but
has been neither verified nor shown to be false. It is
synonymous with CONJECTURE.
see also BOURGET'S HYPOTHESIS, CHINESE HYPOTH-
ESIS, Continuum Hypothesis, Hypothesis Test-
ing, Nested Hypothesis, Null Hypothesis, Postu-
late, Ramanujan's Hypothesis, Riemann Hypoth-
esis, Schinzel's Hypothesis, Souslin's Hypothesis
Hypothesis Testing
The use of statistics to determine the probability that a
given hypothesis is true.
see also BONFERRONI CORRECTION, ESTIMATE, FlSHER
Sign Test, Paired *-Test, Statistical Test, Type
I Error, Type II Error, Wilcoxon Signed Rank
Test
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. 52-110, 1971.
Iyanaga, S. and Kawada, Y. (Eds.). "Statistical Estimation
and Statistical Hypothesis Testing." Appendix A, Table 23
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, pp. 1486-1489, 1980.
Shaffer, J. P. "Multiple Hypothesis Testing." Ann. Rev.
Psych. 46, 561-584, 1995.
Hypotrochoid
The Roulette traced by a point P attached to a Cir-
cle of radius b rolling around the inside of a fixed CIR-
CLE of radius a. The parametric equations for a hy-
potrochoid are
x = n cos t + i
• ft cos f — t J
y = n sin t — h sin ( — t ) ,
(i)
(2)
where n = a — 6 and h is the distance from P to the
center of the rolling CIRCLE. Special cases include the
HYPOCYCLOID with h = 6, the ELLIPSE with a = 26,
and the ROSE with
6 =
2nh
n + 1
_ (n - l)h
n + 1
(3)
(4)
see also EPITROCHOID, HYPOCYCLOID, SPIROGRAPH
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 165-168, 1972.
Lee, X. "Hypotrochoid." http:// www . best . com / - xah /
Special Plane Curves _ dir / Hypotrochoid _ dir /
hypotrochoid.html.
Lee, X. "Epitrochoid and Hypotrochoid Movie Gallery."
http://www.best . com/ -xah/ Special Plane Curves.dir/
EpiHypoTMovieGalleryjdir/epiHypoTMovieGallery.html.
MacTutor History of Mathematics Archive. "Hypotrochoid."
http : //www-groups . dcs , st-and . ac . uk/ -hi story /Curves
/Hypotrochoid . html.
Hypotrochoid Evolute
x — 1
1 7
\ 1
1 /
\ /
\ (
\
/ \
/'T
/ I
1 V
v — '^
The Evolute of the Hypotrochoid is illustrated
above.
Hyzer's Illusion
see Freemish Crate
Icosahedral Equation 883
Ice Fractal
%
The Imaginary Number i is defined as i = V-T- How-
ever, for some reason engineers and physicists prefer the
symbol j to i. Numbers of the form z = x + iy where
x and y are REAL NUMBERS are called COMPLEX NUM-
BERS, and when z is used to denote a Complex Num-
ber, it is sometimes (in older texts) called an "AFFIX."
The Square Root of i is
/t = ±
i + 1
V2 '
V2
(*+l)
f(i 2 + 2i + l).
(1)
(2)
This can be immediately derived from the EULER FOR-
MULA with x = 7r/2,
„**/2
(3)
Vi = vW 2 = e W4 = cos(±7r)+isin(±7r) =
1 + i
V2 *
(4)
The Principal Value of i % is
(j*/y
i 2 n/2
e~ n/2 = 0.207879.... (5)
see also Complex Number, Imaginary Identity,
Imaginary Number, Real Number, Surreal Num-
ber
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. 89, 1996.
I
see Z
7V
A /*. A "*'y»^*
A A AAA A A ' 'W'A'A'A'H'A'A 1
W
W"
/ v
v¥ V
l+l
+
-h
'+'
1 _
-f-
1"
_ 1
H-
"I
-■£
.+£.+.
TT^TF
■**Aj
*-!f.'+*+'f'+
U' + 'A' + U ^'++1+*^
j~ t -+ f- ¥ ij
j ~|" | | "Hl 1+ifi+i i + ifi+ r
1 + l T l + n K+ffi+ffi+,fi+ r
A FRACTAL (square, triangle, etc.) based on a simple
generating motif. The above plots show the ice triangle,
antitriangle, square, and antisquare. The base curves
and motifs for the fractals illustrated above are shown
below.
see also FRACTAL
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, p. 44,
1991.
^ Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
I-Signature
see Signature (Recurrence Relation)
Iamond
see POLYIAMOND
Icosagon
A 20-sided Polygon. The Swastika is an irregular
icosagon.
see also SWASTIKA
Icosahedral Equation
Hunt (1996) gives the "dehomogenized" icosahedral
equation as
10\3
[{z™ + 1) - 228(z 15 - z b ) + 494z lu )
+I72&uz*{z 10 + lLz 5 - l) 5 = 0.
884 Icosahedral Graph
Icosahedron
Other forms include
T ( r7\ 5 5/ 10 , -i -, 5 5 10\5
I(u, v,Z) = u v (u -\-lluv -v )
r 30 , 30 innAC / 20 10 , 10 20 \
— [u + v — 10005(u v -{• u v )
+522(u 2 5z; 5 - uV5)] 2 Z =
and
5 , 10\5
I(z,l, Z) = z 5 (-l + llz b + z lu y
_[1 + z ™ _ I0005(z 10 + z 20 ) + 522(-z 5 + z 25 )] 2 Z = 0.
References
Hunt, B. TVie Geometry of Some Special Arithmetic Quo-
tients. New York: Springer- Verlag, p. 146, 1996.
Icosahedral Graph
A Polyhedral Graph.
see also Cubical Graph, Dodecahedral Graph,
Octahedral Graph, Tetrahedral Graph
Icosahedral Group
The GROUP Ih of symmetries of the ICOSAHEDRON and
Dodecahedron. The icosahedral group consists of the
symmetry operations E, 12Cs, 12C 5 , 2OC3, 15C2, z,
125i , 125? , 205 6 , and 15cr (Cotton 1990).
see also Dodecahedron, Icosahedron, Octahedral
Group, Tetrahedral Group
References
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 48-50, 1990.
Lomont, J. S. "Icosahedral Group." §3.10.E in Applications
of Finite Groups. New York: Dover, p. 82, 1987.
Icosahedron
A Platonic Solid (P 5 ) with 12 Vertices, 30 Edges,
and 20 equivalent EQUILATERAL TRIANGLE faces 20{3}.
It is described by the SCHLAFLI SYMBOL {3,5}. It is
also Uniform Polyhedron U22 and has Wythoff
Symbol 5 | 23. The icosahedron has the Icosahedral
GROUP Ih of symmetries.
A plane Perpendicular to a C$ axis of an icosahedron
cuts the solid in a regular DECAGONAL CROSS-SECTION
(Holden 1991, pp. 24-25).
A construction for an icosahedron with side length a =
V 50 — 10^5/5 places the end vertices at (0, 0, ±1) and
the central vertices around two staggered Circles of
RADII |\/5 and heights ±|\/5, giving coordinates
± (|>/5cos(§«r), f^/5sin(!«r), ±Vb) (1)
for i = 0, 1, . . . , 4, where all the plus signs or minus
signs are taken together. Explicitly, these coordinates
xo*
±(1^,0,1^5) (2)
xf - ±(£(5 - V5), ^50 + 10^5, iVb) (3)
x± = ±(-^(^ + 5)^^50 -IOa/5, \yfe) (4)
x 3 ± = ±{-U^> - 5 )> -ft ^50 - 10 A \y/E) (5)
xj = ±(^(5 - v^), -i\/50TW5, |x/5). (6)
By a suitable rotation, the VERTICES of an icosahe-
dron of side length 2 can also be placed at (0,±<£, ±1),
(±1,0, ±0), and (±0, ±1,0), where <p is the GOLDEN
RATIO. These points divide the EDGES of an OCTAHE-
DRON into segments with lengths in the ratio 0:1.
The Dual Polyhedron of the icosahedron is the Do-
decahedron. There are 59 distinct icosahedra when
each TRIANGLE is colored differently (Coxeter 1969).
To derive the VOLUME of an icosahedron having edge
length a, consider the orientation so that two VERTICES
are oriented on top and bottom. The vertical distance
between the top and bottom PENTAGONAL DlPYRAMlDS
is then given by
z = y/P-x 2 , (7)
where
i= Iv/3a
(8)
Icosahedron
Icosahedron Stellations 885
is the height of an ISOSCELES TRIANGLE, and the
SAGITTA x = R' — r of the pentagon is
giving
= ^a^V^S-Kh/Sa,
c 2 = ^V / 5-2v / 5a 2 .
(9)
(10)
Plugging (8) and (10) into (7) gives
v»-*c-'^>=V ""''»' V5 '
/l0 + 2\/5 , /i0 + 2a/5
iV50 + 10V5a,
(11)
which is identical to the radius of a Pentagon of side
a. The CIRCUMRADIUS is then
R = h+ \z,
where
h= ^ ^50- 10a/5 o
(12)
(13)
is the height of a PENTAGONAL DlPYRAMID. Therefore,
^ = (/l+ I z) 2
= (j^V / 50-10v / 5+ ^VsO + WsjV
Taking the square root gives the ClRCUMRADIUS
R.
^±(b + VE)a= \>/lO + 2>/EattQM10ba.
(15)
The INRADIUS is
r = ^(3^+ v / 15)a w 0.75576a. (16)
The square of the INTERRADIUS is
2 /l \2 ,2
P = (2 Z ) + X <
= [(J)( l5o)( 50 + 10 ^) + m ( 25 + lOv^ )]a 2
= |(3 + V5)a a > (17)
so
^ ^ V a ( 3 + ^) a = \( l + ^ ) a ~ 0-80901a. (18)
The Area of one face is the Area of an Equilateral
Triangle
A= \a 2 y/?>. (19)
The volume can be computed by taking 20 pyramids of
height r
V = 20[(±A)r] - 20|iv / 3a 2 ^(3v / 3 + \/l5)a
= i(3 + A/5)a 3 . (20)
Apollonius showed that
•^icosahedron -^-icosahedron
•^dodecahedron
^dodecahedron
(21)
where V is the volume and A the surface area.
see also AUGMENTED TRIDIMINISHED ICOSAHEDRON,
Decagon, Dodecahedron, Great Icosahedron,
Icosahedron
Stellations, Metabidiminished Icosahedron, Tri-
diminished Icosahedron, Trigonometry Values —
7T/5
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, 1969.
Davie, T. "The Icosahedron." http://www.dcs.st-and.ac,
uk/~ad/mathrecs/polyhedra/icosahedron.html.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Klein, F. Lectures on the Icosahedron. New York: Dover,
1956.
Pappas, T. "The Icosahedron & the Golden Rectangle." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 115, 1989.
Icosahedron Stellations
Applying the Stellation process to the ICOSAHEDRON
gives
20 + 30 + 60 + 20 + 60 + 120 + 12 + 30 + 60 + 60
cells of ten different shapes and sizes in addition to the
ICOSAHEDRON itself. After application of five restric-
tions due to J. C. P. Miller to define which forms should
be considered distinct, 59 stellations are found to be
possible. Miller's restrictions are
1. The faces must lie in the twenty bounding planes of
the icosahedron.
2. The parts of the faces in the twenty planes must be
congruent, but those parts lying in one place may be
disconnected.
3. The parts lying in one plane must have threefold
rotational symmetry with or without reflections.
4. All parts must be accessible, i.e., lie on the outside
of the solid.
5. Compounds are excluded that can be divided into
two sets, each of which has the full symmetry of the
whole.
Of these, 32 have full icosahedral symmetry and 27 are
Enantiomeric forms. Four are Polyhedron Com-
pounds, one is a Kepler-Poinsot SOLID, and one is
the Dual Polyhedron of an Archimedean Solid.
886
Icosahedron Stellations
Icosahedron Stellations
The only STELLATIONS of PLATONIC SOLIDS which are
Uniform Polyhedra are the three Dodecahedron
Stellations the Great Icosahedron (stellation #
ii).
n
name
1
icosahedron
2
triakisicosahedron
3
octahedron 5-compound
4
echidnahedron
11
great icosahedron
18
tetrahedron 10-compound
20
deltahedron-60
36
tetrahedron 5-compound
v
«
03
04
05
06
>k-
21
l\
24
P A SI>
27
30
m
#
>jfe,
■v
33
.<»%
36
Icosahedron Stellations
40
41
42
1\ A V*
43
44
45
^ ^ ^4
46
47
^ ^ >4f
<!&>>-&
A A"
58
59
see a/so Archimedean Solid Stellation, Dodeca-
hedron Stellations, Stellation
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 146-
147, 1987.
Icosidodecadodecahedron 887
Bulatov, V. "Stellations of Icosahedron." http: //www.
physics.orst.edu/-bulatov/polyhedra/icosahedron/.
Coxeter, H. S. M. The Fifty-Nine Icosahedra. New York:
Springer- Verlag, 1982.
Hart, G. W. "59 Stellations of the Icosahedron." http://
www.li.net/-george/virtual-polyhedra/stellations-
icosahedron-index.html.
Maeder, R. E. Icosahedra. m notebook, http://www.inf.
ethz . ch/department/TI/rm/programs .html.
Maeder, R. E. "The Stellated Icosahedra." Mathematica in
Education 3, 1994. ftp://ftp.inf.ethz.ch/doc/papers/
ti/scs/icosahedra94.ps.gz.
Maeder, R. E. "Stellated Icosahedra." http://wwv.
mathconsult . ch/showroom/icosahedra/.
Wang, P. "Polyhedra." http : //www .ugcs . caltech.edu/
-peterw/portfolio/polyhedra/.
Wenninger, M. J. Polyhedron Models. New York: Cambridge
University Press, pp. 41-65, 1989.
Wheeler, A. H. "Certain Forms of the Icosahedron and a
Method for Deriving and Designating Higher Polyhedra."
Proc. Internal Math. Congress 1, 701-708, 1924.
Icosian Game
The problem of finding a HAMILTONIAN CIRCUIT along
the edges of an ICOSAHEDRON, i.e., a path such that
every vertex is visited a single time, no edge is visited
twice, and the ending point is the same as the starting
point.
see also Hamiltonian Circuit, Icosahedron
References
Herschel, A. S. "Sir Wm. Hamilton's Icosian Game." Quart.
J. Pure Applied Math. 5, 305, 1862.
Icosidodecadodecahedron
The Uniform Polyhedron ?7 4 4 whose Dual Poly-
hedron is the Medial Icosacronic Hexecontahe-
dron. It has Wythoff Symbol | 5 | 3. Its faces are
20{6} + 12{|} + 12{5}. Its CIRCUMRADIUS for unit edge
length is
References
Wenninger, M. J, Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 128-129, 1989.
888 Icosidodecahedron
Icosidodecahedron
Ideal
An Archimedean Solid whose Dual Polyhedron is
the Rhombic Triacontahedron. It is one of the two
convex QUASIREGULAR Polyhedra and has SCHLAFLI
Symbol {;?}. It is also Uniform Polyhedron U 2 a
and has Wythoff Symbol 2 | 3 5. Its faces are 20{3} +
12{5}. The Vertices of an icosidodecahedron of
Edge length 20 _1 are (±2,0,0), (0,±2,0), (0,0, ±2),
(±1,±0-\±1), (±l,±^±<t>- 1 ), (±<f>-\±h±<f>). The
30 Vertices of an Octahedron 5-Compound form an
icosidodecahedron (Ball and Coxeter 1987). FACETED
versions include the SMALL ICOSIHEMIDODECAHEDRON
and Small Dodecahemidodecahedron.
The faces of the icosidodecahedron consist of 20 trian-
gles and 12 pentagons. Furthermore, its 60 edges are bi-
sected perpendicularly by those of the reciprocal RHOM-
BIC Triacontahedron (Ball and Coxeter 1987).
The Inradius, Midradius,
unit edge length are
and ClRCUMRADlUS for
r= |(5 + 3\/5) « 1.46353
p=\ v / eT+^/5 « 1.53884
R= 1(1 + ^5) =<^w 1.61803.
see also ARCHIMEDEAN SOLID, GREAT ICOSIDODECA-
HEDRON, QUASIREGULAR POLYHEDRON, SMALL ICOSI-
HEMIDODECAHEDRON, Small Dodecahemidodeca-
hedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 137,
1987.
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 73, 1989.
Icosidodecahedron Stellation
The first stellation is a Dodecahedron-Icosahedron
Compound.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 73-96, 1989.
Icosidodecatruncated Icosidodecahedron
see ICOSITRUNCATED DODECADODECAHEDRON
Icositruncated Dodecadodecahedron
The Uniform Polyhedron U45 also called the
ICOSIDODECATRUNCATED ICOSIDODECAHEDRON whose
Dual Polyhedron is the Tridyakis Icosahedron.
It has Wythoff Symbol 3 | 5 |. Its faces are 20{6} +
12{10}+12{~}. Its ClRCUMRADlUS for unit edge length
is
R = 2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 130-131, 1989.
Ida Surface
A 3-D shadow of a 4-D Klein Bottle.
see also Klein Bottle
References
Peterson, I. Islands of Truth: A Mathematical Mystery
Cruise. New York: W. H. Freeman, pp. 44-45, 1990.
Ideal
A subset / of elements in a Ring R which forms an
additive Group and has the property that, whenever x
belongs to R and y belongs to /, then xy and yx belong
to i". For example, the set of Even Integers is an ideal
in the RING of INTEGERS. Given an ideal /, it is possible
to define a FACTOR RING R/I.
An ideal may be viewed as a lattice and specified as the
finite list of algebraic integers that form a basis for the
lattice. Any two bases for the same lattice are equiva-
lent. Ideals have multiplication, and this is basically the
Kronecker product of the two bases.
For any ideal i", there is an ideal U such that
iii
z,
where z is a Principal Ideal, (i.e., an ideal of rank
1). Moreover there is a finite list of ideals U such that
this equation may be satisfied for every I. The size of
this list is known as the Class Number. In effect, the
above relation imposes an EQUIVALENCE RELATION on
ideals, and the number of ideals modulo this relation
is the class number. When the Class Number is 1,
the corresponding number RING has unique factoriza-
tion and, in a sense, the class number is a measure of
the failure of unique factorization in the original number
ring.
Ideal Number
Identity Function 889
Dedekind (1871) showed that every Nonzero ideal in
the domain of INTEGERS of a FIELD is a unique product
of Prime Ideals.
see also CLASS NUMBER, DIVISOR THEORY, IDEAL
Number, Maximal Ideal, Prime Ideal, Principal
Ideal
References
Malgrange, B. Ideals of Differentiable Functions,
Oxford University Press, 1966.
London:
Ideal Number
A type of number involving the ROOTS OF Unity which
was developed by Kummer while trying to solve Fer-
mat'S Last Theorem. Although factorization over the
Integers is unique (the Fundamental Theorem of
Algebra), factorization is not unique over the Com-
plex NUMBERS. Over the ideal numbers, however, fac-
torization in terms of the COMPLEX NUMBERS becomes
unique. Ideal numbers were so powerful that they were
generalized by Dedekind into the more abstract IDEALS
in general Rings which are a key part of modern ab-
stract Algebra.
see also Divisor Theory, Fermat's Last Theorem,
Ideal
Ideal (Partial Order)
An ideal J of a Partial Order P is a subset of the
elements of P which satisfy the property that if y G /
and x <y, then x € I. For k disjoint chains in which the
ith chain contains Ui elements, there are (1 + ^i)(l +
712) ■••(! + rik) ideals. The number of ideals of a n-
element Fence Poset is the Fibonacci Number F n .
References
Ruskey, F. "Information on Ideals of Partially Ordered
Sets." http : // sue . esc . uvic . ca / * cos / inf / pose /
Ideals.html,
Steiner, G. "An Algorithm to Generate the Ideals of a Partial
Order." Operat. Res. Let 5, 317-320, 1986.
Ideal Point
A type of Point at Infinity in which parallel lines
in the HYPERBOLIC PLANE intersect at infinity in one
direction, while diverging from one another in the other.
see also Hyperparallel
Idele
The multiplicative subgroup of all elements in the prod-
uct of the multiplicative groups k* whose absolute value
is 1 at all but finitely many u t where k is a number Field
and v a PLACE.
see also Adele
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Idemfactor
see Dyadic
Idempotent
An Operator A such that A 2 = A or an element of an
Algebra x such that x 2 = x.
see also Automorphic Number, Boolean Algebra,
Group, Semigroup
Identity
An identity is a mathematical relationship equating one
quantity to another (which may initially appear to be
different).
see also Abel's Identity, Andrews-Schur Iden-
tity, BAC-CAB Identity, Beauzamy and De-
got's Identity, Beltrami Identity, Bianchi Iden-
tities, Bochner Identity, Brahmagupta Iden-
tity, Cassini's Identity, Cauchy-Lagrange Iden-
tity, Christoffel-Darboux Identity, Chu-Van-
dermonde Identity, de Moivre's Identity, Dou-
gall-Ramanujan Identity, Euler Four-Square
Identity, Euler Identity, Euler Polynomial
Identity, Ferrari's Identity, Fibonacci Identity,
Frobenius Triangle Identities, Green's Identi-
ties, Hypergeometric Identity, Imaginary Iden-
tity, Jackson's Identity, Jacobi Identities, Ja-
cobi's Determinant Identity, Lagrange's Iden-
tity, Le Cam's Identity, Leibniz Identity, Liou-
ville Polynomial Identity, Matrix Polynomial
Identity, Morgado Identity, Newton's Identi-
ties, Quintuple Product Identity, Ramanujan
6-10-8 Identity, Ramanujan Cos/Cosh Identity,
Ramanujan's Identity, Ramanujan's Sum Iden-
tity, Reznik's Identity, Rogers-Ramanujan Iden-
tities, Schaar's Identity, Strehl Identity, Syl-
vester's Determinant Identity, Trinomial Iden-
tity, Visible Point Vector Identity, Watson
Quintuple Product Identity, Worpitzky's Iden-
tity
Identity Element
The identity element i" (also denoted E, e, or 1) of
a GROUP or related mathematical structure S is the
unique elements such that I A = AI = J for every ele-
ment A £ S. The symbol "E" derives from the German
word for unity, "Einheit."
see also BINARY OPERATOR, GROUP, INVOLUTION
(Group), Monoid
Identity Function
890 Identity Map
Illusion
Im[Ident z]
The function f{x) — x which assigns every REAL Num-
ber x to the same Real Number x. It is identical to
the Identity Map.
Identity Map
The Map which assigns every Real Number to the
same Real Number id^. It is identical to the Iden-
tity Function.
Identity Matrix
The identity matrix is defined as the Matrix 1 (or I)
such that
l(X) = X
for all VECTORS X. The identity matrix is
lij — dij
for ijj = 1,2, ..., n, where Sij is the Kronecker
Delta. Written explicitly,
1 =
1
1
Identity Operator
The Operator I which takes a Real Number to the
same Real Number Ir = r.
see also Identity Function, Identity Map
Idoneal Number
A Positive value of D for which the fact that a number
is a MONOMORPH (i.e., the number is expressible in only
one way as x 2 -\-Dy 2 or x 2 — Dy 2 where x 2 is Relatively
Prime to Dy 2 ) guarantees it to be a Prime, Power
of a Prime, or twice one of these. The numbers are
also called Euler's Idoneal Numbers, or Suitable
Numbers.
The 65 idoneal numbers found by Gauss and Euler and
conjectured to be the only such numbers (Shanks 1969)
are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21,
22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70,
72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168,
177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345,
357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848
(Sloane's A000926).
References
Shanks, D. "On Gauss's Class Number Problems." Math.
Comput 23, 151-163, 1969.
Sloane, N. J. A. Sequence A000926/M0476 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Iff
If and only if (i.e., NECESSARY and SUFFICIENT). The
terms "Just If" or "EXACTLY When" are sometimes
used instead. A iff B is written symbolically as A H- B.
A iff B is also equivalent to A ^ B, together with B ^
A, where the symbol => denotes "Implies."
J. H. Conway believes that the word originated with
P. Halmos and was transmitted through Kelley (1975).
Halmos has stated, "To the best of my knowledge, I
DID invent the silly thing, but I wouldn't swear to it in
a court of law. So there — give me credit for it anyway"
(Asimov 1997).
see also EQUIVALENT, EXACTLY ONE, IMPLIES, NECES-
SARY, Sufficient
References
Asimov, D. "Iff." math-fun@cs.arizona.edu posting, Sept. 19,
1997.
Kelley, J. L. General Topology. New York: Springer- Verlag,
1975.
Ill-Conditioned
A system is ill-conditioned if the CONDITION NUMBER
is too large (and singular if it is Infinite).
see also CONDITION NUMBER
Illumination Problem
In the early 1950s, Ernst Straus asked
1. Is every POLYGONAL region illuminable from every
point in the region?
2. Is every POLYGONAL region illuminable from at least
one point in the region?
Here, illuminable means that there is a path from every
point to every other by repeated reflections. Tokarsky
(1995) showed that unilluminable rooms exist in the
plane and 3-D, but question (2) remains open. The
smallest known counterexample to (1) in the PLANE has
26 sides.
see also Art Gallery Theorem
References
Klee, V. "Is Every Polygonal Region Illuminable from Some
Point?" Math. Mag. 52, 180, 1969.
Tokarsky, G. W. "Polygonal Rooms Not Illuminable from
Every Point." Amer. Math. Monthly 102, 867-879, 1995.
Illusion
An object or drawing which appears to have properties
which are physically impossible, deceptive, or counter-
intuitive.
see also Benham's Wheel, Freemish Crate, Gob-
let Illusion, Hermann Grid Illusion, Hermann-
Hering Illusion, Hyzer's Illusion, Impossible
Figure, Irradiation Illusion, Kanizsa Trian-
gle, Muller-Lyer Illusion, Necker Cube, Orbi-
son's Illusion, Parallelogram Illusion, Penrose
Image
Immersion 891
Stairway, Poggendorff Illusion, Ponzo's Illu-
sion, Rabbit-Duck Illusion, Tribar, Vertical-
Horizontal Illusion, Young Girl-Old Woman Il-
lusion, Zollner's Illusion
References
Ausbourne, B. "A Sensory Adventure." http: //www. lainet .
com/illusions/.
Ausbourne, B. "Optical Illusions: A Collection." http://
www. lainet . coro/~ausbourn/.
Ernst, B. Optical Illusions. New York: Taschen, 1996.
Fineman, M. The Nature of Visual Illusion. New York:
Dover, 1996.
Gardner, M. "Optical Illusions." Ch. 1 in Mathematical Cir-
cus: More Puzzles, Games, Paradoxes and Other Math-
ematical Entertainments from Scientific American. New
York: Knopf, 1979.
Gregory, R. L. Eye and Brain, 5th ed. Princeton, NJ: Prince-
ton University Press, 1997.
"Illusions: Central Station." http: //www. heureka.f i/i/
Illusions_ctrl_station.html. en.
Landrigad, D. "Gallery of Illusions." http://valley.uml.
edu/psychology/illusion.html.
Luckiesh, M. Visual Illusions: Their Causes, Characteris-
tics, and Applications. New York: Dover, 1965.
Pappas, T. "History of Optical Illusions." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 172-173, 1989.
Tolansky, S. Optical Illusions. New York: Pergamon Press,
1964.
Imaginary Point
A pair of values x and y one or both of which is Com-
plex.
References
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, p. 2,
1961.
Imaginary Quadratic Field
A Quadratic Field Q(Vd) with D < 0.
see also Quadratic Field
Immanant
For annxn matrix, let 5 denote any permutation ei, ei,
. . . , e n of the set of numbers 1, 2, . . . , n, and let x^ {$)
be the character of the symmetric group corresponding
to the partition (A). Then the immanant |a mn |^ is
defined as
|a m „p = £x <A> (S)P S
where the summation is over the n! permutations of the
Symmetric Group and
Pa ~ &le 1 0,2e 2 ' ' ' °"n,e ri •
Image
see Range (Image)
Imaginary Identity
Imaginary Number
A Complex Number which has zero Real Part, so
that it can be written as a REAL NUMBER multiplied by
the "imaginary unit" i (equal to %/— 1).
see also Complex Number, Galois Imaginary,
Gaussian Integer, i, Real Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 211-216, 1996.
Imaginary Part
The imaginary part £y of a Complex Number z = x+iy
is the Real Number multiplying i, so ^[ai + iy] = y. In
terms of z itself,
2i '
where z* is the COMPLEX CONJUGATE of z.
see also ABSOLUTE SQUARE, COMPLEX CONJUGATE,
Real Part
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.
see also Determinant, Permanent
References
Littlewood, D. E. and Richardson, A. R. "Group Characters
and Algebra." Philos. Trans. Roy. Soc. London A 233,
99-141, 1934.
Littlewood, D. E. and Richardson, A. R. "Immanants of
Some Special Matrices." Quart. J. Math. (Oxford) 5, 269-
282, 1934.
Wybourne, B. G. "Immanants of Matrices." §2.19 in Symme-
try Principles and Atomic Spectroscopy. New York: Wiley,
pp. 12-13, 1970.
Immersed Minimal Surface
see Enneper's Surfaces
Immersion
A special nonsingular Map from one MANIFOLD to an-
other such that at every point in the domain of the map,
the Derivative is an injective linear map. This is equiv-
alent to saying that every point in the DOMAIN has a
Neighborhood such that, up to Diffeomorphisms of
the Tangent Space, the map looks like the inclusion
map from a lower-dimensional EUCLIDEAN SPACE to a
higher-dimensional EUCLIDEAN SPACE.
see also Boy Surface, Eversion, Smale-Hirsch
Theorem
References
Boy, W. "Uber die Curvatura integra und die Topologie
geschlossener Flachen." Math. Ann 57, 151-184, 1903.
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.
892 Impartial Game
Improper Integral
Impartial Game
A Game in which the possible moves are the same for
each player in any position. All positions in all impartial
Games form an additive Abelian Group. For impar-
tial games in which the last player wins (normal form
games), the nim- value of the sum of two Games is the
nim-sum of their nim- values. If the last player loses, the
Game is said to be in misere form and the analysis is
much more difficult.
see also Fair Game, Game, Partisan Game
Implicit Function
A function which is not defined explicitly, but rather is
defined in terms of an algebraic relationship (which can
not, in general, be "solved" for the function in question).
For example, the Eccentric Anomaly E of a body
orbiting on an ELLIPSE with ECCENTRICITY e is defined
implicitly in terms of the mean anomaly M by KEPLER'S
Equation
M = E - e sin E.
Implicit Function Theorem
Given
F 1 (x,y,z,u,v,w) =
F 2 (x,y,z,u,v J w) =
F z (x,y,z,u,v,w) - 0,
if the JACOBIAN
JF(u,V)iv)
d(F u F2,F s )
d(u,v,w)
#o,
then u 7 v 7 and w can be solved for in terms of x 1 y y and
z and Partial Derivatives of u, u, w with respect to
x, y, and z can be found by differentiating implicitly.
More generally, let A be an Open Set in R n+k and let
/ : A -> R n be a C r Function. Write / in the form
f( x i y)-> where x and y are elements of R and W 1 . Sup-
pose that (a, b) is a point in A such that /(a, 6) = and
the Determinant of the n x n Matrix whose elements
are the DERIVATIVES of the n component FUNCTIONS of
/ with respect to the n variables, written as y, evalu-
ated at (a, 6), is not equal to zero. The latter may be
rewritten as
rank(D/(a, b)) = n.
Then there exists a NEIGHBORHOOD B of a in R k and
a unique C r FUNCTION g : B -)• W 1 such that g(a) = b
and f(x,g(x)) = for all x £ B.
see also CHANGE OF VARIABLES THEOREM, JACOBIAN
References
Munkres, J. R. Analysis on Manifolds. Reading, MA:
Addison- Wesley, 1991.
Implies
The symbol => means "implies" in the mathematical
sense. Let A be true. If this implies that B is also true,
then the statement is written symbolically as A => B,
or sometimes A C B. If A =>- B and B => A (i.e, A =>
BAB^A), then A and B are said to be EQUIVALENT,
a relationship which is written symbolically as A <£> B
or A ^ B,
see also EQUIVALENT
Impossible Figure
A class of Illusion in which an object which is physi-
cally unrealizable is apparently depicted.
see also Freemish Crate, Home Plate, Illusion,
Necker Cube, Penrose Stairway, Tribar
References
Cowan, T. M. "The Theory of Braids and the Analysis of
Impossible Figures." J. Math. Psych. 11, 190-212, 1974.
Cowan, T. M. "Supplementary Report: Braids, Side Seg-
ments, and Impossible Figures." J. Math. Psych. 16,
254-260, 1977.
Cowan, T. M. "Organizing the Properties of Impossible Fig-
ures." Perception 6, 41-56, 1977.
Cowan, T. M. and Pringle, R. "An Investigation of the
Cues Responsible for Figure Impossibility." J. Exper.
Psych. Human Perception Performance 4, 112-120, 1978.
Ernst, B. Adventures with Impossible Figures. Stradbroke,
England: Tarquin, 1987.
Harris, W. F. "Perceptual Singularities in Impossible Pic-
tures Represent Screw Dislocations." South African J. Sci.
69, 10-13, 1973.
Fineman, M. The Nature of Visual Illusion. New York:
Dover, pp. 119-122, 1996.
Jablan, S. "Impossible Figures." http: //members. tripod.
com/-modularity/impos.htm and "Are Impossible Figures
Possible?" http: //members .tripod, com/ -modularity/
kulpa . htm.
Kulpa, Z. "Are Impossible Figures Possible?" Signal Pro-
cessing 5, 201-220, 1983.
Kulpa, Z. "Putting Order in the Impossible." Perception 16,
201-214, 1987.
Sugihara, K. "Classification of Impossible Objects." Percep-
tion 11, 65-74, 1982.
Terouanne, E. "Impossible Figures and Interpretations of
Polyhedral Figures." J. Math. Psych. 27, 370-405, 1983.
Terouanne, E. "On a Class of ^Impossible' Figures: A New
Language for a New Analysis." J. Math. Psych. 22, 24-47,
1983.
Thro, E. B. "Distinguishing Two Classes of Impossible Ob-
jects." Perception 12, 733-751, 1983.
Wilson, R. "Stamp Corner: Impossible Figures." Math. In-
tel! 13, 80, 1991.
Impredicative
Definitions about a Set which depend on the entire Set.
Improper Integral
An INTEGRAL which has either or both limits INFINITE
or which has an Integrand which approaches Infinity
at one or more points in the range of integration.
see also Definite Integral, Indefinite Integral,
Integral, Proper Integral
Improper Node
Incenter
893
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Improper Integrals." §4.4 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 135-140, 1992.
Improper Node
A Fixed Point for which the Stability Matrix has
equal nonzero EIGENVECTORS.
see also Stable Improper Node, Unstable Im-
proper Node
Improper Rotation
The Symmetry Operation corresponding to a a Ro-
tation followed by an INVERSION OPERATION, also
called a ROTOINVERSION. This operation is denoted n
for an improper rotation by 360° /n, so the CRYSTAL-
LOGRAPHY Restriction gives only 1, 2, 3, 4, 6 for
crystals. The MIRROR PLANE symmetry operation is
(a;, j/, z) -» (as, y, — z)> etc., which is equivalent to 2.
Impulse Pair
-1/2 1/2
The even impulse pair is the FOURIER TRANSFORM of
cos(7rfc),
ii(z) = §«*(* + !) + §*(*-§)•
It satisfies
IIC*) */(*)= \J{X+ I) + 1/(3, -I),
where * denotes CONVOLUTION, and
II(x)dx = 1.
(1)
(2)
(3)
1/2
-1/2
The odd impulse pair is the FOURIER TRANSFORM of
isin(7rs),
Impulse Symbol
Bracewell's term for the Delta Function.
see also IMPULSE PAIR
References
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, 1965.
In-and-Out Curve
etc.
A curve created by starting with a circle, dividing it into
six arcs, and flipping three alternating arcs. The process
is then repeated an infinite number of times.
Inaccessible Cardinal
An inaccessible cardinal is a Cardinal Number which
cannot be expressed in terms of a smaller number of
smaller cardinals.
Inaccessible Cardinals Axiom
see also Lebesgue Measurability Problem
Inadmissible
A word or string which is not ADMISSIBLE.
Incenter
The center I of a TRIANGLE'S INCIRCLE. It can be found
as the intersection of ANGLE BISECTORS, and it is the
interior point for which distances to the sidelines are
equal. Its Trilinear Coordinates are 1:1:1. The
distance between the incenter and ClRCUMCENTER is
y/R(R-2r).
The incenter lies on the EULER LINE only for an ISOS-
CELES Triangle. It does, however, lie on the Soddy
LINE. For an EQUILATERAL TRIANGLE, the ClRCUM-
CENTER O, Centroid G, Nine-Point Center F, Or-
THOCENTER H, and DE LONGCHAMPS POINT Z all co-
incide with I.
The incenter and Excenters of a Triangle are an
Orthocentric System. The Power of the incenter
with respect to the ClRCUMCIRCLE is
p:
ai<22<23
a>i + o>2 + as
I^ziiti+i)-^-!).
(4)
(Johnson 1929, p. 190). If the incenters of the TRIAN-
GLES A,4i# 2 #3, ^A 2 H Z A 1 , and AA 3 Fitf 2 are X x , X 2 ,
and X3, then X2X3 is equal and parallel to J2/3, where
Hi are the FEET of the ALTITUDES and U are the in-
centers of the TRIANGLES. Furthermore, Xi, X2, X3,
are the reflections of / with respect to the sides of the
Triangle Ahhh (Johnson 1929, p. 193).
894
Incenter-Excenter Circle
Incident
If four points are on a CIRCLE (i.e., they are CON-
CYCLIC), the incenters of the four TRIANGLES form a
Rectangle whose sides are parallel to the lines con-
necting the middle points of opposite arcs. Furthermore,
the connectors pass through the center of the RECTAN-
GLE (Fuhrmann 1890, p. 50; Johnson 1929, pp. 254-
255). More generally, the 16 incenters and excenters of
the Triangles whose Vertices are four points on a
CIRCLE, are the intersections of two sets of four Paral-
lel lines which are mutually PERPENDICULAR (Johnson
1929, p. 255).
see also Centroid (Orthocentric System), Cir-
CUMCENTER, EXCENTER, GERGONNE POINT, INCIR-
CLE, Inradius, Orthocenter
References
Carr, G. S. Formulas and Theorems in Pure Mathematics,
2nd ed. New York: Chelsea, p. 622, 1970.
Dixon, R. Mathographics. New York: Dover, p. 58, 1991.
Fuhrmann, W. Synthetische Beweise Planimetrischer Sdtze.
Berlin, 1890.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 182-194, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Incenter." http : //www . evansville . edu/
~ck6/tcenters/class/incenter.html.
Incenter-Excenter Circle
Given a triangle AAi^^, the points Ai, J, and J\ lie
on a line, where J is the INCENTER and J\ is the EX-
CENTER corresponding to A\. Furthermore, the CIRCLE
with JJi as the DIAMETER has P as its center, where P
is the intersection of A\ 3\ with the ClRCUMClRCLE of
AA1A2A3, and passes through A 2 and A3. This CIRCLE
has Radius
r = \a\ sec(|ai) = 2Rsin(\cx\).
It arises because JJi J2 J3 forms an ORTHOCENTRIC Sys-
tem.
see also ClRCUMClRCLE, EXCENTER, EXCENTER-
Excenter Circle, Incenter, Orthocentric Sys-
tem
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 185, 1929.
Incidence Axioms
The eight of Hilbert's Axioms which concern
collinearity and intersection; they include the first four
of Euclid's Postulates.
see also Absolute Geometry, Congruence Ax-
ioms, Continuity Axioms, Euclid's Postulates,
Hilbert's Axioms, Ordering 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.
Incidence Matrix
For a A;-D POLYTOPE life, the incidence matrix is defined
by
fc
Vij
{i
if 11^. _ x belongs to Il£
if nj. _ ± does not belong to II fc .
The zth row shows which II^s surround II fc .
jth column shows which IIfc_is bound II fc .
!, and the
Incidence
matrices are also used to specify PROJECTIVE PLANES.
The incidence matrices for a Tetrahedron ABCD are
v 1
AD
BD
CD
BC
AC
AB
A
1
1
1
B
1
1
1
C
1
1
1
D
1
1
1
r?
BCD
ACD
ABD
ABC
AD
1
1
BD
1
1
CD
1
1
BC
1
1
AC
1
1
AB
1
1
~,3
V
ABCD
BCD
1
ACD
1
ABD
1
ABC
1
see also ADJACENCY MATRIX, fc-CHAIN, /c-ClRCUIT
Incident
Two objects which touch each other are said to be inci-
dent.
see also INCIDENCE MATRIX
Incircle
Incircle
The Inscribed Circle of a Triangle AABC. The
center i" is called the INCENTER and the RADIUS r the
INRADIUS. The points of intersection of the incircle with
T are the Vertices of the Pedal Triangle of T with
the Incenter as the Pedal Point (c.f. Tangential
Triangle). This Triangle is called the Contact
Triangle.
The Area K of the Triangle AABC is given by
K = AAIC + AC IB + AAIB
= \br + \ar + \cr = |(a + 6 + c)r = sr,
where s is the Semiperimeter.
Using the incircle of a TRIANGLE as the INVERSION CEN-
TER, the sides of the TRIANGLE and its C IRC UM CIRCLE
are carried into four equal CIRCLES (Honsberger 1976,
p. 21). Pedoe (1995, p. xiv) gives a Geometric Con-
struction for the incircle.
see also ClRCUMCIRCLE, CONGRUENT INCIRCLES
Point, Contact Triangle, Inradius, Triangle
transformation principle
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 11-13, 1967.
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., 1976.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 182-194, 1929.
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., 1995.
Inclusion- Exclusion Principle
If Ai, . . . , Ak are finite sets, then
O
i=l
£(-D i+1 £,
where & is the sum of the CARDINALITIES of the inter-
sections of the sets taken i at a time.
Inclusion Map
Given a Subset B of a Set A, the Injection / : B -> A
defined by f(b) = b for all b € B is called the inclusion
map.
see also Long Exact Sequence of a Pair Axiom
Indefinite Integral 895
Incommensurate
Two lengths are called incommensurate or incommensu-
rable if their ratio cannot be expressed as a ratio of whole
numbers. Irrational Numbers and Transcenden-
tal Numbers are incommensurate with the integers.
see also IRRATIONAL NUMBER, PYTHAGORAS'S CON-
STANT, Transcendental Number
Incomplete Gamma Function
see Gamma Function
Incompleteness
A formal theory is said to be incomplete if it contains
fewer theorems than would be possible while still retain-
ing Consistency.
see also CONSISTENCY, GODEL'S INCOMPLETENESS
Theorem
References
Chaitin, G. J. "G. J. Chaitin's Home Page." http://www.
cs . auckland . ac . nz/CDMTCS/chaitin.
Increasing Function
A function f{x) increases on an INTERVAL J if /(&) >
f{a) for all b > a, where a, b € I. Conversely, a function
f(x) decreases on an Interval I if /(&) < f(a) for all
b > a with a, b 6 /.
If the Derivative f'(x) of a Continuous Function
f(x) satisfies f(x) > on an Open INTERVAL (a, 6),
then f(x) is increasing on (a, &). However, a function
may increase on an interval without having a derivative
defined at all points. For example, the function x x ' z
is increasing everywhere, including the origin x = 0,
despite the fact that the DERIVATIVE is not defined at
that point.
see also DECREASING FUNCTION, DERIVATIVE, NONDE-
creasing Function, Nonincreasing Function
Increasing Sequence
For a Sequence {a n }, if a n +i-a n > for n > x, then a
is increasing for n > x. Conversely, if a n +i — a n < for
n > a?, then a is DECREASING for n > x. If a n +i/a n > 1
for all n > x, then a is increasing for n > x. Conversely,
if a n+ i/a n < 1 for all n > x, then a is decreasing for
n > x.
Indefinite Integral
An Integral
/
f(x) dx
without upper and lower limits, also called an An-
tiderivative. The first Fundamental Theorem of
Calculus allows Definite Integrals to be computed
896 Indefinite Quadratic Form
Index
in terms of indefinite integrals. If F is the indefinite in-
tegral for /(z), then
/'
f(x)dx = F(b)-F(a).
see also ANTIDERIVATIVE, CALCULUS, DEFINITE INTE-
GRAL, Fundamental Theorems of Calculus, Inte-
gral
Indefinite Quadratic Form
A QUADRATIC FORM Q(x) is indefinite if it is less than
for some values and greater than for others. The
QUADRATIC FORM, written in the form (x, Ax), is in-
definite if Eigenvalues of the Matrix A are of both
signs.
see also Positive Definite Quadratic Form, Posi-
tive Semidefinite Quadratic Form
References
Gradshteyn, L S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1106, 1979.
Indegree
The number of inward directed EDGES from a given
Vertex in a Directed Graph.
see also Local Degree, Outdegree
Independence Axiom
A rational choice between two alternatives should de-
pend only on how they differ.
Independence Complement Theorem
If sets E and F are Independent, then so are E and
F\ where F' is the complement of F (i.e., the set of all
possible outcomes not contained in F). Let U denote
"or" and n denote "and." Then
P(E) = P(EFUEF f ) (1)
= P(EF) + P(EF') - P(EF n EF ! ), (2)
where AB is an abbreviation for Af) B. But E and F
are independent, so
P{EF) = P(E)P(F).
(3)
Also, since F and F 1 are complements, they contain no
common elements, which means that
P(EFnEF f ) = (4)
for any E. Plugging (4) and (3) into (2) then gives
P(E) = P(E)P{F) + P{EF'). (5)
Rearranging,
P(EF') = P(E)[1 - P(F)] = P(E)P(F') J (6)
Q.E.D.
see also INDEPENDENT STATISTICS
Independence Number
The number
a(G) = max(|C/| : U C V independent)
for a Graph G. The independence number of the DE
Bruun Graph of order n is given by 1, 2, 3, 7, 13, 28,
... (Sloane's A006946).
References
Sloane, N. J. A. Sequence A006946/M0834 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Independent Equations
see Linearly Independent
Independent Sequence
see Strongly Independent, Weakly Independent
Independent Statistics
Two variates A and B are statistically independent Iff
the Conditional Probability P(A\B) of A given B
satisfies
P(A\B) = P(A), (1)
in which case the probability of A and B is just
P(AB) = P(A n B) = P(A)P(B). (2)
Similarly, n events Ai, j42, • . . , A n are independent Iff
WrM=n^)-
(3)
K i=l / i=l
Statistically independent variables are always UN COR-
RELATED, but the converse is not necessarily true.
see also BAYES' FORMULA, CONDITIONAL PROBABIL-
ITY, INDEPENDENCE COMPLEMENT THEOREM, UNCOR-
RELATED
Independent Vertices
A set of Vertices A of a Graph with Edges V is
independent if it contains no EDGES.
see also INDEPENDENCE NUMBER
Indeterminate Problems
see Diophantine Equation — Linear
Index
A statistic which assigns a single number to several in-
dividual statistics in order to quantify trends. The best-
known index in the United States is the consumer price
index, which gives a sort of "average" value for infla-
tion based on the price changes for a group of selected
products.
Index Set
Induction Axiom
897
Let p n be the price per unit in period n, q n be the quan-
tity produced in period n, and v n = p n q n be the value of
the n units. Let q a be the estimated relative importance
of a product. There are several types of indices defined,
among them those listed in the following table.
Index
Abbr. Formula
Bowley index Pb
Fisher index Pf
Geometric mean index Pq
Harmonic mean index Ph
Lasp eyres' s index Pl
Marshall-Edgeworth index Pme
Mitchell index Pm
Paasche's index Pp
Walsh index Pw
see also Bowley Index, Fisher Index, Geometric
Mean Index, Harmonic Mean Index, Laspeyres'
Index, Marshall-Edgeworth Index, Mitchell In-
dex, Paasche's Index, Residue Index, Walsh In-
dex
References
Fisher, I. The Making of Index Numbers: A Study of Their
Varieties, Tests and Reliability, 3rd ed. New York: Au-
gustus M. Kelly, 1967.
Kenney, J. F. and Keeping, E. S. "Index Numbers." Ch. 5
in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ:
Van Nostrand, pp. 64-74, 1962.
Mudgett, B. D. Index Numbers. New York: Wiley, 1951.
Index Set
A Stochastic Process is a family of Random Vari-
ables {#(£,•),£ e J} from some Probability Space
(S,S,P) into a State Space (S',§')> where J is the
index set of the process.
References
Doob, J. L. "The Development of Rigor in Mathematical
Probability (1900-1950)." Amer. Math. Monthly 103,
586-595, 1996.
Index Theory
A branch of TOPOLOGY dealing with topological invari-
ants of Manifolds.
References
Roe, J. Index Theory, Coarse Geometry, and Topology of
Manifolds. Providence, RI: Amer. Math. Soc, 1996.
Upmeier, H. Toeplitz Operators and Index Theory in Several
Complex Variables. Boston, MA: Birkhauser, 1996.
Indicatrix
A spherical image of a curve. The most common indi-
catrix is Dupin's Indicatrix.
see also DUPIN'S INDICATRIX
Indicial Equation
The RECURRENCE RELATION obtained during applica-
tion of the Frobenius Method of solving a second-
order ordinary differential equation. The indicial equa-
tion (also called the Characteristic Equation) is
obtained by noting that, by definition, the lowest or-
der term x k (that corresponding to n — 0) must have a
Coefficient of zero. For an example of the construc-
tion of an indicial equation, see BESSEL DIFFERENTIAL
Equation.
1. If the two ROOTS are equal, only one solution can be
obtained.
2. If the two ROOTS differ by a noninteger, two solu-
tions can be obtained.
3. If the two ROOTS differ by an Integer, the larger
will yield a solution. The smaller may or may not.
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 532-534, 1953.
Indifference Principle
see Insufficient Reason Principle
Induced Map
If / : (X,A) -> (Y,B) is homotopic to g : (X,A) ->
(r,£), then f m : H n {X,A) -> H n (Y 7 B) and g* :
H n (Xj A) -> H n (Y i B) are said to be the induced maps.
see also Eilenberg-Steenrod Axioms
Induced Norm
see Natural Norm
Induction
The use of the INDUCTION PRINCIPLE in a PROOF. In-
duction used in mathematics is often called MATHEMAT-
ICAL Induction.
References
Buck, R. C. "Mathematical Induction and Recursive Defini-
tions." Amer. Math. Monthly 70, 128-135, 1963.
Induction Axiom
The fifth of PEANO'S AXIOMS, which states: If a SET 5
of numbers contains zero and also the successor of every
number in 5, then every number is in S.
see also Peano's Axioms
898 Induction Principle
Infimum Limit
Induction Principle
The truth of an INFINITE sequence of propositions Pi for
i — 1, . . . , oo is established if (1) P± is true, and (2) Pk
Implies P k +i for all k.
References
Courant, R. and Robbins, H. "The Principle of Mathematical
Induction" and "Further Remarks on Mathematical Induc-
tion." §1.2.1 and 1.7 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 9-11 and 18-20,
1996.
Inequality
A mathematical statement that one quantity is greater
than or less than another, "a is less than 6" is denoted
a < 6, and "a is greater than 6" is denoted a > b. "a
is less than or equal to 6" is denoted a < 6, and "a
is greater than or equal to &" is denoted a > b. The
symbols a < b and a > b are used to denote "a is much
less than 6" and "a is much greater than 6," respectively.
Solutions to the inequality \x — a\ < b consist of the set
{x : — b < x — a < 6}, or equivalently {x : a — b < x <
a + b}. Solutions to the inequality |cc — a| > b consist of
the set {x : x — a > 6} U {x : x — a < — &}. If a and b
are both Positive or both Negative and a < 6, then
1/a > 1/6.
see also ABC CONJECTURE, ARITHMETIC-LOGARITH-
mic-Geometric Mean Inequality, Bernoulli In-
equality, Bernstein's Inequality, Berry-Osseen
Inequality, Bienayme-Chebyshev Inequal-
ity, Bishop's Inequality, Bogomolov-Miyaoka-
Yau Inequality, Bombieri's Inequality, Bonfer-
roni's Inequality, Boole's Inequality, Carle-
man's Inequality, Cauchy Inequality, Cheby-
shev Inequality, Chi Inequality, Copson's In-
equality, Erdos-Mordell Theorem, Exponen-
tial Inequality, Fisher's Block Design Inequal-
ity, Fisher's Estimator Inequality, Garding's In-
equality, Gauss's Inequality, Gram's Inequal-
ity, Hadamard's Inequality, Hardy's Inequal-
ity, Harnack's Inequality, Holder Integral In-
equality, Holder's Sum Inequality, Isoperimet-
ric Inequality, Jarnick's Inequality, Jensen's In-
equality, Jordan's Inequality, Kantrovich In-
equality, Markov's Inequality, Minkowski In-
tegral Inequality, Minkowski Sum Inequality,
Morse Inequalities, Napier's Inequality, No-
sarzewska's Inequality, Ostrowski's Inequal-
ity, Ptolemy Inequality, Robbin's Inequality,
Schroder-Bernstein Theorem, Schur's Inequal-
ities, Schwarz's Inequality, Square Root In-
equality, Steffensen's Inequality, Stolarsky's
Inequality, Strong Subadditivity Inequality,
Triangle Inequality, Turan's Inequalities, Wei-
erstraB Product Inequality, Wirtinger's In-
equality, Young 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. 16, 1972.
Beckenbach, E. F. and Bellman, Richard E. An Introduction
to Inequalities. New York: Random House, 1961.
Beckenbach, E. F. and Bellman, Richard E. Inequalities, 2nd
rev. print. Berlin: Springer- Verlag, 1965.
Hardy, G. H.; Littlewood, J. E.; and Polya, G, Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
1952.
Kazarinoff, N. D. Geometric Inequalities. New York: Ran-
dom House, 1961.
Mitrinovic, D. S. Analytic Inequalities. New York: Springer-
Verlag, 1970.
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical
& New Inequalities in Analysis. Dordrecht, Netherlands:
Kluwer, 1993.
Mitrinovic, D. S.; Pecaric, J. E.; Fink, A. M. Inequalities In-
volving Functions & Their Integrals & Derivatives, Dor-
drecht, Netherlands: Kluwer, 1991.
Mitrinovic, D. S.; Pecaric, J. E.; and Volenec, V. Recent Ad-
vances in Geometric Inequalities. Dordrecht, Netherlands:
Kluwer, 1989.
Inexact Differential
An infinitesimal which is not the differential of an actual
function and which cannot be expressed as
dz
-(*),* + (g).*
the way an EXACT DIFFERENTIAL can. Inexact differ-
entials are denoted with a bar through the d. The most
common example of an inexact differential is the change
in heat dQ encountered in thermodynamics.
see also EXACT DIFFERENTIAL, PFAFFIAN FORM
References
Zemansky, M. W. Heat and Thermodynamics, 5th ed. New
York: McGraw-Hill, p. 38, 1968.
Inf
see Infimum, Infimum Limit
Infimum
The greatest lower bound of a set. It is denoted
inf.
5
see also Infimum Limit, Supremum
Infimum Limit
The limit infimum of a set is the greatest lower bound
of the Closure of a set. It is denoted
lim inf .
s
see also Infimum, Supremum
Infinary Divisor
Infinary Divisor
p x is an infinary divisor of p y (with y > 0) if p x \ y -ip y >
This generalizes the concept of the fc-ARY DIVISOR.
see also INFINARY PERFECT NUMBER, fc-ARY DIVISOR
References
Cohen, G. L. "On an Integer's Infinary Divisors." Math.
Comput. 54, 395-411, 1990.
Cohen, G. and Hagis, P. "Arithmetic Functions Associated
with the Infinary Divisors of an Integer." Internat. J.
Math. Math. Sci. 16, 373-383, 1993.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 54, 1994.
Infinary Multiperfect Number
Let croo(n) be the Sum of the Infinary Divisors of
a number n. An infinary fc-multiperfect number is a
number n such that croo(n) = kn. Cohen (1990) found
13 infinary 3-multiperfects, seven 4-multiperfects, and
two 5-multiperfects.
see also INFINARY PERFECT NUMBER
References
Cohen, G. L. "On an Integer's Infinary Divisors." Math.
Comput. 54, 395-411, 1990.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 54, 1994.
Infinary Perfect Number
Let <Too(n) be the Sum of the INFINARY DIVISORS of
a number n. An infinary perfect number is a number
n such that <7oo(n) - 2n. Cohen (1990) found 14 such
numbers. The first few are 6, 60, 90, 36720, . . . (Sloane's
A007257).
see also Infinary Multiperfect Number
References
Cohen, G. L. "On an Integer's Infinary Divisors." Math.
Comput. 54, 395-411, 1990.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 54, 1994.
Sloane, N. J. A. Sequence A007257/M4267 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Infinite
Greater than any assignable quantity of the sort in ques-
tion. In mathematics, the concept of the infinite is made
more precise through the notion of an INFINITE Set.
see also COUNTABLE SET, COUNTABLY INFINITE SET,
Finite, Infinite Set, Infinitesimal, Infinity
Infinite Product
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
A Product involving an Infinite number of terms.
Such products can converge. In fact, for POSITIVE a n ,
the Product n^Li an conver g es to a Nonzero num-
ber IFF ^ j °°_ 1 \na n converges.
Infinite Product 899
Infinite products can be used to define the COSINE
cos a;
=n
4x J
7r 2 (2n-l) 2
(1)
Gamma Function
T(z) =
■n(^f)«- /r
(2)
Sine, and Sinc Function. They also appear in the
Polygon Circumscribing Constant
°° i
(5)'
(3)
An interesting infinite product formula due to Euler
which relates 7r and the nth PRIME p n is
(4)
(5)
n°°
J. xi=n
Pn
rr
A. ±z=n
l + i
.i)(pn-i)/a "|
Pn J
(Blatner 1997).
The product
fi('+i)
(6)
has closed form expressions for small Positive integral
P>2,
n( 1+ i)-^ m
Tl = l
n( i+ ^) = r° sh( ' 7rv/5) (8)
n=l
n/ 1 , 1 \ cosh(7ry / 2)-cos(7rV2)
l 1+ ^J = 2^ ^
n=l
oo
J] (l+ -L) = |r[ e xp(|,ri)]r[ex P (f^)]|- 2 . (10)
The d- Analog expression
[00!] d =
T> — 3 V /
(11)
900 Infinite Series
also has closed form expressions,
n('-£K
71 = 3
n/_ _8_\ sinh(7ry/3)
V n 3 ) 42tt\/3
n=B
n(i-5)- stah(2 "
1207T
(12)
(13)
(14)
A (l- j^f) = |r[exp(|7ri)]r[2exp(|7rz)]|- 2 .(15)
see also COSINE, DlRICHLET ETA FUNCTION, EU-
ler Identity, Gamma Function, Iterated Ex-
ponential Constants, Polygon Circumscribing
Constant, Polygon Inscribing Constant, Q-
Function, Sine
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 75, 1972.
Arfken, G. "Infinite Products." §5.11 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 346-351, 1985.
Blatner, D. The Joy of Pi. New York: Walker, p. 119, 1997.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/infprd/infprd.html.
Hansen, E. R. A Table of Series and Products. Englewood
Cliffs, NJ: Prentice-Hall, 1975.
Whittaker, E. T. and Watson, G. N. §7.5 and 7.6 in A Course
in Modern Analysis, J^th ed. Cambridge, England: Cam-
bridge University Press, 1990.
Infinite Series
A Series with an Infinite number of terms.
see also SERIES
Infinite Set
A Set of 5 elements is said to be infinite if the ele-
ments of a Proper Subset S f can be put into One-
TO-One correspondence with the elements of S. An
infinite set whose elements can be put into a One-TO-
One correspondence with the set of Integers is said
to be Countably Infinite; otherwise, it is called Un-
countably Infinite.
see also Aleph-0, Aleph-1, Cardinal Number,
Countably Infinite Set, Continuum, Finite, In-
finite, Infinity, Ordinal Number, Transfinite
Number, Uncountably Infinite Set
References
Courant, FL and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, p. 77, 1996.
Infinitesimal Rotation
Infinitesimal
A quantity which yields after the application of some
Limiting process. The understanding of infinitesimals
was a major roadblock to the acceptance of CALCULUS
and its placement on a firm mathematical foundation.
see also Infinite, Infinity, Nonstandard Analysis
Infinitesimal Analysis
An archaic term for CALCULUS.
Infinitesimal Matrix Change
Let B, A, and e be square matrices with e small, and
define
B = A(l+e), (1)
where I is the IDENTITY MATRIX. Then the inverse of
B is approximately
B- 1 = (|-e)A- x .
This can be seen by multiplying
(2)
BB" 1 = (A + Ae)(A~ 1 -eA- 1 )
= AA" 1 - AeA" 1 + AeA" 1 - Ae 2 A _1
= | _ Ae'A" 1 « 1. (3)
Note that if we instead let B' = A + e, and look for an
inverse of the form B' _1 = A -1 + C, we obtain
B'B'" 1 = (A + e)(A" 1 + C) = AA" 1 + AC + eA" 1 + eC
= l + AC + e(C- r -A- 1 ) = l. (4)
In order to eliminate the e term, we require C = -A -1 .
However, then AC = —I, so BB -1 = so there can be
no inverse of this form.
The exact inverse of B can be found as follows.
B = A(l + e) = A(l + A- 1 e), (5)
so
B-^AO + A^e)]- 1 . (6)
Using a general MATRIX INVERSE identity then gives
B- 1 = (l + A _1 e)- 1 A- 1 .
(7)
Infinitesimal Rotation
An infinitesimal transformation of a VECTOR r is given
by
r' = (l + e)r, (1)
where the Matrix e is infinitesimal and I is the IDEN-
TITY MATRIX. (Note that the infinitesimal transforma-
tion may not correspond to an inversion, since inversion
Infinitesimal Rotation
Infinity 901
is a discontinuous process.) The COMMUTATIVITY of in-
finitesimal transformations ei and £% is established by
the equivalence of
(i + ei)(l + e 2 ) = l 2 + eil + le 2 + eie 2 w l + ei+e 2 (2)
(l+e 2 )(l + ei) = l 2 + e 2 l + lei + e 2 ei « l + e 2 +ei. (3)
Now let
A=l+e. (4)
The inverse A" is then I — e, since
AA"
(l + e)(l-e) = l 2 -e 2 ^L
(5)
Since we are defining our infinitesimal transformation to
be a rotation, Orthogonality of Rotation Matri-
ces requires that
A T = A-\ (6)
but
A^-l-e
(l + e) T = r + e T = l + e T ,
(8)
so e = — e T and the infinitesimal rotation is Antisym-
metric. It must therefore have a Matrix of the form
(9)
The differential change in a vector r upon application of
the Rotation Matrix is then
dQ 3
-dQ 2
-dQ 3
dfii
dQ 2
-dfti
dr = r' — r = (I + e)r — r = er.
Writing in MATRIX form,
(10)
dv =
~ x~
y
_z _
dQ 3 -dQ 2 '
-dQ 3 dfii
dQ 2 -dtoi
=
" y dQ 3 — z dQ,2
zdQi — xdQ 3
xd&2 — ydfli _
= (ydQ 3 — zdQ,2)yL + (zdQi — xdQ 3 )y
+ (xd£l2 — ydQi)z
— r x dQ.
Therefore,
where
/dr\ dQ
I — J = rx — =rxu>,
\ dt / rotation, body at
d<j>
(11)
(12)
(13)
ws * =n *- (14)
The total rotation observed in the stationary frame will
be a sum of the rotational velocity and the velocity in the
rotating frame. However, note that an observer in the
stationary frame will see a velocity opposite in direction
to that of the observer in the frame of the rotating body,
SO
(iL.-(iL+-'- <I5 >
This can be written as an operator equation, known as
the Rotation Operator, defined as
\ OX / space V
d ) +ux. (16)
dt)
body
see also Acceleration, Euler Angles, Rotation,
Rotation Matrix, Rotation Operator
Infinitive Sequence
A sequence {x n } is called an infinitive sequence if, for
every i, x n = i for infinitely many n. Write a(i,j) for
the jth index n for which x n = i. Then as i and j range
through iV, the array A = a(i,j), called the associative
array of #, ranges through all of N.
see also FRACTAL SEQUENCE
References
Kimberling, C. "Fractal Sequences and Interspersions." Ars
Combin. 45, 157-168, 1997.
Infinitude of Primes
see Euclid's Theorems
Infinity
An unbounded number greater than every Real Num-
ber, most often denoted as oo. The symbol oo had been
used as an alternative to M (1,000) in Roman Numer-
als until 1655, when John Wallis suggested it be used
instead for infinity.
Infinity is a very tricky concept to work with, as ev-
idenced by some of the counterintuitive results which
follow from Georg Cantor's treatment of INFINITE Sets.
Informally, l/oo = 0, a statement which can be made
rigorous using the LIMIT concept,
Similarly,
lim - = 0.
x— >-oo X
lim — = oo,
s-K)+ X
where the notation + indicates that the LIMIT is taken
from the POSITIVE side of the REAL LINE.
see also Aleph, Aleph-0, Aleph-1, Cardinal Num-
ber, Continuum, Continuum Hypothesis, Hilbert
Hotel, Infinite, Infinite Set, Infinitesimal, Line
at Infinity, L'Hospital's Rule, Point at Infinity,
Transfinite Number, Uncountably Infinite Set,
Zero
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer-Verlag, p. 19, 1996.
902
Inflection Point
Injection
Courant, R. and Robbins, H. "The Mathematical Analysis of
Infinity." §2.4 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 77-88, 1996.
Hardy, G. H. Orders of Infinity, the 'infinitarcalcul' of Paul
Du Bois-Reymond, 2nd ed. Cambridge, England: Cam-
bridge University Press, 1924.
Lavine, S. Understanding the Infinite. Cambridge, MA: Har-
vard University Press, 1994.
Maor, E. To Infinity and Beyond: A Cultural History of the
Infinite. Boston, MA: Birkhauser, 1987.
Moore, A. W. The Infinite. New York: Routledge, 1991.
Morris, R. Achilles in the Quantum Universe: The Definitive
History of Infinity. New York: Henry Holt, 1997.
Peter, R. Playing with Infinity. New York: Dover, 1976.
Smail, L. L. Elements of the Theory of Infinite Processes.
New York: McGraw-Hill, 1923.
Vilenskin, N. Ya. In Search of Infinity. Boston, MA:
Birkhauser, 1995.
Wilson, A. M. The Infinite in the Finite. New York: Oxford
University Press, 1996.
Zippin, L. Uses of Infinity. New York: Random House, 1962.
Inflection Point
A point on a curve at which the SIGN of the CURVATURE
(i.e., the concavity) changes. The FIRST DERIVATIVE
Test can sometimes distinguish inflection points from
EXTREMA for DlFFERENTIABLE functions f(x).
see also CURVATURE, DlFFERENTIABLE, EXTREMUM,
First Derivative Test, Stationary Point
Information Dimension
Define the "information function" to be
N
I=-Y,Pi(e)HP<(<)], (1)
where P»(e) is the NATURAL MEASURE, or probability
that element i is populated, normalized such that
and
j>i(e) = l.
(2)
The information dimension is then defined by
/
d inf = - lim
o+ ln(e)
lim y
fi(e)ln[fi(c)]
ln(6) *
(3)
If every element is equally likely to be visited, then P»(e)
is independent of i, and
^P i (e) = iVP i (e) = l,
(4)
£*■»(*)
dinf = lim
= lim
o+ lne
C _K)+ In e
= — lim
InJV
e -K)+ ln(e)
(6)
where d cap is the CAPACITY DIMENSION.
see also CORRELATION EXPONENT
References
Farmer, J. D. "Chaotic Attractors of an Infinite- dimensional
Dynamical System." Physica D 4, 366-393, 1982.
Nayfeh, A. H. and Balachandran, B. Applied Nonlinear
Dynamics: Analytical, Computational, and Experimental
Methods. New York: Wiley, pp. 545-547, 1995.
Information Entropy
see Entropy
Information Theory
The branch of mathematics dealing with the efficient
and accurate storage, transmission, and representation
of information.
see also Coding Theory, Entropy
References
Goldman, S. Information Theory. New York: Dover, 1953.
Lee, Y. W. Statistical Theory of Communication. New York:
Wiley, 1960.
Pierce, J. R. An Introduction to Information Theory. New
York: Dover, 1980.
Reza, F. M. An Introduction to Information Theory. New
York: Dover, 1994.
Singh, J. Great Ideas in Information Theory, Language and
Cybernetics. New York: Dover, 1966.
Zayed, A. I. Advances in Shannon's Sampling Theory. Boca
Raton, FL: CRC Press, 1993.
Initial Value Problem
An initial value problem is a problem that has its condi-
tions specified at some time t = to. Usually, the problem
is an Ordinary Differential Equation or a Par-
tial Differential Equation. For example,
V 2 u = f
u — U\
in ft
t = t
on dft,
where dft denotes the boundary of ft, is an initial value
problem.
see also BOUNDARY CONDITIONS, BOUNDARY VALUE
Problem, Partial Differential Equation
References
Eriksson, K.; Estep, D.; Hansbo, P.; and Johnson, C. Compu-
tational Differential Equations. Lund, Sweden: Studentlit-
teratur, 1996.
Pi&
N J
(5)
Injection
see One-to-One
Injective
Inject ive
A Map is injective when it is One-to-One, i.e., / is
injective when x ^ y IMPLIES f(x) / f(y).
see also ONE-TO-ONE, SURJECTIVE
Injective Patch
An injective patch is a PATCH such that x(^i,ui) =
x(ii2,f2) implies that U\ — u<i and v\ — V2. An example
of a PATCH which is injective but not REGULAR is the
function defined by (u s ,v 3 ,uv) for u,v G ( — 1,1). How-
ever, if x : U — y M n is an injective regular patch, then x
maps U diffeomorphically onto x(t/).
see also PATCH, REGULAR PATCH
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 187, 1993.
Inner Automorphism Group
A particular type of AUTOMORPHISM GROUP which ex-
ists only for GROUPS. For a GROUP G, the inner auto-
morphism group is defined by
Inn(G) = {a a : a G G} C Aut(G)
where a a is an AUTOMORPHISM of G defined by
cr a {x) = axa~ .
see also AUTOMORPHISM, AUTOMORPHISM GROUP
Inner Product
see DOT PRODUCT
Inner Product Space
An inner product space is a Vector Space which has
an Inner Product. If the Inner Product defines a
NORM, then the inner product space is called a HlLBERT
Space.
see also Hilbert Space, Inner Product, Norm
Inradius
The radius of a Triangle's Incircle or of a Polyhe-
dron's INSPHERE, denoted r. For a TRIANGLE,
Inradius
903
_ 1 (b-rc- a)(c + a - b)(a + b - c) = A
T " 2V a+b+c ~ s U
= 4Rsin(|ai) sin(|a2) sin(^a 3 ),
(2)
where A is the AREA of the TRIANGLE, a, 6, and c are
the side lengths, s is the Semiperimeter, and R is the
ClRCUMRADlUS (Johnson 1929, p. 189).
Equation (1) can be derived easily using TRILINEAR CO-
ORDINATES. Since the INCENTER is equally spaced from
all three sides, its trilinear coordinates are 1:1:1, and its
exact trilinear coordinates are r : r : r. The ratio k of
the exact trilinears to the homogeneous coordinates is
given by
k =
2A
a + b + c s
But since k = r in this case,
s
Q. E. D.
Other equations involving the inradius include
abc
Rr
45
A = rr±r2Tz
(3)
(4)
(5)
(6)
cos A + cos B + cos C — 1 + — (7)
R
r = 2R cos A cos B cos C (8)
a 2 +b 2 + c 2 =4rR + 8R 2 , (9)
where n are the EXRADII (Johnson 1929, pp. 189-191).
As shown in RIGHT TRIANGLE, the inradius of a RIGHT
TRIANGLE of integral side lengths x, y, and z is also
integral, and is given by
r = xy (1Q)
x + y + z
where z is the HYPOTENUSE. Let d be the distance be-
tween inradius r and ClRCUMRADlUS R, d = rR. Then
R 2 -d 2 = 2Rr
1 1
+
1
(11)
(12)
R-d R+d r v '
(Mackay 1886-87). These and many other identities are
given in Johnson (1929, pp. 186-190).
Expressing the MlDRADIUS p and ClRCUMRADlUS R in
terms of the midradius gives
(13)
(14)
Vp 2 + l a2
r 2 - y
R
for an ARCHIMEDEAN SOLID.
see also CARNOT'S THEOREM, ClRCUMRADlUS, MlDRA-
DIUS
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Mackay, J. S. "Historical Notes on a Geometrical Theorem
and its Developments [18th Century]." Proc. Edinburgh
Math. Soc. 5, 62-78, 1886-1887.
Mackay, J. S. "Formulas Connected with the Radii of the In-
circle and Excircles of a Triangle." Proc. Edinburgh Math.
Soc. 12, 86-105.
Mackay, J. S. "Formulas Connected with the Radii of the In-
circle and Excircles of a Triangle." Proc. Edinburgh Math.
Soc. 13, 103-104.
904
Inscribed
Integer-Matrix Form
Inscribed
A geometric figure which touches only the sides (or in-
terior) of another figure.
see also CIRCUMSCRIBED, INCENTER, INCIRCLE, INRA-
DIUS
Inscribed Angle
The Angle with Vertex on a Circle's Circumfer-
ence formed by two points on a Circle's CIRCUMFER-
ENCE. For Angles with the same endpoints,
9 C = 2#i,
where C is the CENTRAL ANGLE.
see also Central Angle
References
Pedoe, D. Circles: A Mathematical View, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., pp. xxi~xxii, 1995.
Inside-Outside Theorem
Let P{z) and Q(z) be POLYNOMIALS with COMPLEX
arguments and deg(Q) > deg(P *f 2). Then
2 «Ein 5 ide 7 ReS /W
/—/outside 7
Res/(^)
inside 7
outside 7,
where Res are the RESIDUES.
Insphere
A Sphere Inscribed in a given solid.
see also ClRCUMSPHERE, MlDSPHERE
Int
see Integer Part
Integer
One of the numbers . . . , -2, —1, 0, 1, 2, The Set
of Integers forms a Ring which is denoted Z. A given
Integer n may be Negative (n e Z"), Nonnegative
(n € Z*), ZERO (n = 0), or Positive (neZ + = N).
The Ring Z has Cardinality of N . The Generating
Function for the Positive Integers is
/(*) =
(l-^) :
= x + 2x 2 + 3x 3 + Ax A + . .
There are several symbols used to perform operations
having to do with conversion between REAL NUMBERS
and integers. The symbol |xj ("FLOOR x n ) means "the
largest integer not greater than x," i.e., int(x) in com-
puter parlance. The symbol [x] means "the nearest in-
teger to x n (Nint), i.e., nint(x) in computer parlance.
The symbol \x] ("Ceiling x") means "the smallest in-
teger not smaller x," or -int(-x), where int(x) is the
Integer Part of x.
see also ALGEBRAIC INTEGER, ALMOST INTEGER,
Complex Number, Counting Number, Cyclo-
tomic Integer, Eisenstein Integer, Gaussian In-
teger, N, Natural Number, Negative, Positive,
Radical Integer, Real Number, Whole Number,
Z,Z", Z + ,Z*, Zero
Integer Division
Division in which the fractional part (remainder) is dis-
carded is called integer division and is sometimes de-
noted \. Integer division can be defined as a\b = [a/b\,
where "/" denotes normal division and [x\ is the FLOOR
Function. For example,
Instrument Function
The finite FOURIER COSINE TRANSFORM of an APO-
dization Function, also known as an Apparatus
Function. The instrument function I(k) correspond-
ing to a given ApODIZATION FUNCTION A(x) is then
given by
m
/a
■a
cos(27rA;x)^4(a;) dx.
see also Apodization Function,
Transform
Fourier Cosine
Insufficient Reason Principle
A principle also called the Indifference Principle
which was first enunciated by Johann Bernoulli. The
insufficient reason principle states that, if we are igno-
rant of the ways an event can occur and therefore have
no reason to believe that one way will occur preferen-
tially to another, it will occur equally likely in any way.
10/3 = 3+1/3
10\3 = 3.
Integer Factorization
see Prime Factorization
Integer-Matrix Form
Let Q(x) = Q(x) = Q{xi,X2, . . . ,CEn) be an integer-
valued n-ary QUADRATIC FORM, i.e., a POLYNOMIAL
with integer COEFFICIENTS which satisfies Q(x) > for
Real x ^ 0. Then Q(x) can be represented by
where
Q(x) = x T Ax,
2 dxidxj
Integer Module
Integer Sequence 905
is a Positive symmetric matrix (Duke 1997). If A has
POSITIVE entries, then Q(x) is called an integer matrix
form. Conway et at. (1997) have proven that, if a POS-
ITIVE integer matrix quadratic form represents each of
1, 2, 3, 5, 6, 7, 10, 14, and 15, then it represents all
Positive Integers.
see also FIFTEEN THEOREM
References
Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane,
N. J. A. "The Primary Pretenders." Acta Arith. 78, SOT-
SIS, 1997.
Duke, W. "Some Old Problems and New Results about Quad-
ratic Forms." Not. Amer. Math. Soc. 44, 190-196, 1997.
Integer Module
see Abelian Group
Integer Part
The function int(x) gives the INTEGER PART of x.
In many computer languages, the function is denoted
int (x) , but in mathematics, it is usually called the
FLOOR Function and denoted \_x\.
see also Ceiling Function, Floor Function, Nint
Integer Relation
A set of Real Numbers zci, . . . , x n is said to possess
an integer relation if there exist integers ai such that
aiXi -f aixi + . . . + a n x n = 0,
with not all ai = 0. An interesting example of such
a relation is the 17-VECTOR (1, x, x ,
5 ) with
x = 3 1 / 4 — 2 1 / 4 , which has an integer relation (1, 0, 0,
0, -3860, 0, 0, 0, -666, 0, 0, 0, -20, 0, 0, 0, 1), i.e.,
1 - 3860x - 666x°
20a: 12 + x 16 - 0.
This is a special case of finding the polynomial of degree
n = rs satisfied by x = 3 1/V — 2 1 / 3 .
Algorithms for finding integer relations include the
Ferguson-Forcade Algorithm, HJLS Algorithm,
LLL Algorithm, PSLQ Algorithm, PSOS Algo-
rithm, and the algorithm of Lagarias and Odlyzko
(1985). Perhaps the simplest (and unfortunately most
inefficient) such algorithm is the Greedy Algorithm.
Plouffe's "Inverse Symbolic Calculator" site includes a
huge 54 million database of REAL NUMBERS which are
algebraically related to fundamental mathematical con-
stants.
see also CONSTANT PROBLEM, FERGUSON-FORCADE
Algorithm, Greedy Algorithm, Hermite-Linde-
mann Theorem, HJLS Algorithm, Lattice Reduc-
tion, LLL Algorithm, PSLQ Algorithm, PSOS
Algorithm, Real Number, Lindemann-Weier-
straB Theorem
References
Bailey, D. and Plouffe, S. "Recognizing Numerical
Constants." http : //www . cecm . sf u . c a/ organics /papers/
bailey.
Lagarias, J. C. and Odlyzko, A. M. "Solving Low-Density
Subset Sum Problems." J. ACM 32, 229-246, 1985.
Plouffe, S. "Inverse Symbolic Calculator." http://www.cecm.
sfu.ca/projects/ISC/.
Integer Sequence
A Sequence whose terms are Integers. The most
complete printed references for such sequences are
Sloane (1973) and its update, Sloane and Plouffe (1995).
Sloane also maintains the sequences from both works to-
gether with many additional sequences in an on-line list-
ing. In this listing, sequences are identified by a unique
6-DlGIT A-number. Sequences appearing in Sloane and
Plouffe (1995) are ordered lexicographically and identi-
fied with a 4-DlGlT M-number, and those appearing in
Sloane (1973) are identified with a 4-Digit N-number.
Sloane's huge (and enjoyable) database is accessible by
either e-mail or web browser. To look up sequences by
e-mail, send a message to either sequencesQresearch.
att . com or superseekerQresearch. att . com containing
lines of the form lookup 5 14 42 132 To use the
browser version, point to http://www.research.att.
com/ -njas/sequences/eisonline. html.
see also Aronson's Sequence, Combinatorics, Con-
secutive Number Sequences, Conway Sequence,
Eban Number, Hofstadter-Conway $10,000 Se-
quence, Hofstadter's Q-Sequence, Levine-O'Sul-
livan Sequence, Look and Say Sequence, Mal-
low's Sequence, Mian-Chowla Sequence, Morse-
Thue Sequence, Newman-Conway Sequence,
Number, Padovan Sequence, Perrin Sequence,
RATS Sequence, Sequence, Smarandache Se-
quences
References
Aho, A. V. and Sloane, N. J. A. "Some Doubly Exponential
Sequences." Fib. Quart. 11, 429-437, 1973.
Bernstein, M. and Sloane, N. J. A. "Some Canonical Se-
quences of Integers." Linear Algebra and Its Applications
226-228, 57-72, 1995.
Erdos, P.; Sarkozy, E.; and Szemeredi, E. "On Divisibility
Properties of Sequences of Integers." In Number The-
ory, Colloq. Math. Soc. Jdnos Bolyai, Vol. 2. Amsterdam,
Netherlands: North-Holland, pp. 35-49, 1970.
Guy, R. K. "Sequences of Integers." Ch. E in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 199-239, 1994.
Krattenthaler, C. "RATE: A Mathematica Guessing Ma-
chine." http : //radon . mat . univie . ac . at/People/kratt /
rate/rate .html.
Ostman, H. Additive Zahlentheorie I, II. Heidelberg, Ger-
many: Springer- Verlag, 1956.
Pomerance, C. and Sarkozy, A. "Combinatorial Number The-
ory." In Handbook of Combinatorics (Ed. R. Graham,
M. Grotschel, and L. Lovasz). Amsterdam, Netherlands:
North-Holland, 1994.
Ruskey, F. "The (Combinatorial) Object Server." http://
sue. csc.uvic.ca/-cos.
Sloane, N. J. A. A Handbook of Integer Sequences. Boston,
MA: Academic Press, 1973.
906 Integrable
Integral
Sloane, N. J. A. "Find the Next Term." J. Recr. Math. 7,
146, 1974.
Sloane, N. J. A. "An On-Line Version of the Encyclo-
pedia of Integer Sequences." Elec. J. Combin. 1,
Fl 1-5, 1994. http://www.combinatorics.org/Volume_l/
volumel .html#Fl.
Sloane, N. J. A. "Some Important Integer Sequences." In
CRC Standard Mathematical Tables and Formulae (Ed.
D. Zwillinger). Boca Raton, FL: CRC Press, 1995.
Sloane, N. J. A. "An On-Line Version of the Encyclopedia
of Integer Sequences." http://www.research.att.com/
-njas/sequences/eisonline .html.
Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer
Sequences. San Diego, CA: Academic Press, 1995.
Stohr, A. "Geloste und ungeloste Fragen iiber Basen der
naturlichen Zahlenreihe I, II." J. reine angew. Math. 194,
40-65 and 111-140, 1955.
Turan, P. (Ed.). Number Theory and Analysis: A Collection
of Papers in Honor of Edmund Landau (1877-1938). New-
York: Plenum Press, 1969.
^ Weisstein, E. W. "Integer Sequences." http: //www. astro.
Virginia . edu / - eww6n / math / notebooks / Integer
Sequences. m.
Integrable
A function for which the INTEGRAL can be computed is
said to be integrable.
see also DlFFERENTIABLE, INTEGRAL, INTEGRATION
Integral
An integral is a mathematical object which can be in-
terpreted as an AREA or a generalization of AREA. Inte-
grals, together with Derivatives, are the fundamental
objects of CALCULUS. Other words for integral include
Antiderivative and Primitive. The Riemann In-
tegral is the simplest integral definition and the only
one usually encountered in elementary CALCULUS. The
Riemann Integral of the function f(x) over x from a
to b is written
/
f(x)dx.
(i)
Every definition of an integral is based on a particu-
lar Measure. For instance, the Riemann Integral is
based on Jordan Measure, and the Lebesgue Inte-
gral is based on LEBESGUE MEASURE. The process of
computing an integral is called INTEGRATION (a more
archaic term for INTEGRATION is QUADRATURE), and
the approximate computation of an integral is termed
Numerical Integration.
There are two classes of (Riemann) integrals: Definite
Integrals
/
f(x) dx,
(2)
which have upper and lower limits, and INDEFINITE IN-
TEGRALS, which are written without limits. The first
Fundamental Theorem of Calculus allows Defi-
nite Integrals to be computed in terms of Indefinite
Integrals, since if F is the Indefinite Integral for
/(#), then
«/ a
f(x) dx = F(b) - F(a).
(3)
Wolfram Research (http://www.integrals.com) main-
tains a web site which will integrate many common (and
not so common) functions. However, it cannot solve
some simple integrals such as
/ — (x vsin x) dx
fix COS X i \
= / I — t + V sin x I dx (4)
J \2Vsinx /
J[^L 2 {x\nx)]dx
__ /* |"(ma; + l)ln(l-a;ln:c)"
£ln:c
dx, (5)
where L^ is the DiLOGARlTHM. Furthermore, it gives
an incorrect answer of 7r 1_2 ^ 3 /(v / 3 • 4^ 3 ) to
i.ir/2
r(v / 3)=/ -
Jo 1
dx
+ (tanx)
V3 ~ 2
(6)
This integral and, in fact, the generalized integral for
arbitrary a
n
/.tt/2
dx
+ (tana)"'
(7)
have a "trick" solution which takes advantage of the
trigonometric identity
tan(|7r — x) = cot a;
Letting z = (tanz) a ,
(8)
r 1 dx r 1 dx
= r /4 — f w/ * dx
-f:\^^)-i>
= **•
(9)
Integral
Integral 907
Here is a list of common INDEFINITE INTEGRALS:
„r+l
\ + c
+ c
sin x dx = — cos £ + C
cos x dx = sin a; + C
secx| + C
esc x dx = In | esc x — cot x\ + C
J r + 1
a dx = - —
J lna
/
/
/ tan x dx — In
/
= In [tan(§x)] + C
1, / 1 — cos a; \ ~
= - In ( + C
2 V 1 + cos a; /
/ sec x dx = In | sec x + tan x| + C
= gd- 1 (x) + C
I cotxdx = In | sinx| + C i
/ sec xdx — tan x + C '
/ sec x tan x dx = sec x + C i
f «.->** = ,„»-> x -VT=* + c
/sin-^^sin-x+V^ + C
/ tan -1 xdx — xtan -1 x — | ln(l + x 2 ) + C
/• d^ =:=sin _ 1 / £ N
i Va 2 - x 2 \aJ
J Va 2 - x 2 \a)
J a 2 +x 2 a \a/
J a z + x* a V a /
/dx 1 _i /x\ _,
— = - sec (-) +C
xy/x 2 — a 2 a Va/
/ dx - 1 -- 1 H i -
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
/
X 1
sin 2 (ax) dx = - - — sin(2ax) + C (33)
/
/
/
/
snudu — k 1 ln(dnu — kcnu) + C (34)
u - E(u)
sn udu —
k 2
+ C
(35)
cnudu = A; sin (fcsnu) + C (36)
dn u du = sin" 1 (sn u) + C, (37)
where sinx is the SlNE; cosx is the COSINE; tanx is the
Tangent; esc x is the Cosecant; sec x is the Secant;
cotx is the COTANGENT; cos^x is the INVERSE Co-
sine; sin -1 x is the INVERSE SINE; tan -1 is the INVERSE
Tangent; sn u, en it, and dntt are Jacobi Elliptic
Functions; E(u) is a complete Elliptic Integral of
the Second Kind; and gd(x) is the Gudermannian
Function.
To derive (15), let u = cosx, so du = — sin xdx and
du
/f sinx , f du
tan x = I dx — — / —
J cosx J u
= -ln|u| + C = -ln|cosx| + C
= In | cosx| _1 + C = In | sec x| + C. (38)
To derive (18), let u = cscx — cotx, so du —
( — esc x cot x + esc 2 x) dx and
/ esc x dx = /
■/
esc x — cot X
CSC x ; — ax
CSC x — cot X
esc 2 x + cot X CSC X
csc x + cot x
dx
/
du
= / — =ln|u| + C
— In | csc x — cot x| + C
To derive (19), let
so
and
u = secx + tanx,
du = (sec x tan x + sec x) dx
(39)
(40)
(41)
/ sec x dx = /
■/
sec x + tan x .
sec x ax
sec x + tan x
sec 2 x + sec x + tan x
sec x + tan x
dx
/
dn
= / — = In Ittl + C
= In | sec x + tan a;| + C.
(42)
908 Integral
To derive (20), let u = sinx, so du = cosxdx and
/[ cos x _ /* du
cot xdx — / — — dx = —
J sinx 7 «
= In |u| + C = In | sinz| + C. (43)
Differentiating integrals leads to some useful and pow-
erful identities, for instance
_d_
dx
I f(x) dx —
J a
/(*).
(44)
which is the first Fundamental Theorem of Calcu-
lus.
) dx = —f{x)
dx
/ f(x,t)dt= /
J a J a
f
dx
f(x,t)dt
(45)
(46)
j /*X /*X r\
f(x,t)dt = f(x,t)+ — f(x,t)dt. (47)
dx J '^-'-'- ^~'-' ■ / Qx
J a «/ a
If f(x,t) is singular or Infinite, then
d
dx
«/ a
f(x,t) dx
'-[ [(*-«)£ + (*-«)?? + /
x — a I _ L' d#
The Leibniz Identity is
U (»)
ft
di, (48)
/•u(x) ~
+ / ^f(x,t)dt. (49)
Other integral identities include
f(t)dtdx= / (x~t)f(t)dt (50)
/>x /»a:
*/ a «/ a
/•x /»t n /»*3 />t 2
/ dt n / dtn-x*- / di 2 / /(^l)di
Jo Jo Jo Jo
i r
(»-l)'io
{x-tr^mdt (5i)
-— (x, J fc ) = rf jfc J fc + xj — J k = J + rV ■ J (52)
OXk OXk
L""L& M -L"-"'
-X
rV-J(fr.
(53)
Integrals of the form
f fix
)dx
Integral
(54)
with one INFINITE LIMIT and the other NONZERO may
be expressed as finite integrals over transformed func-
tions. If f(x) decreases at least as fast as 1/x 2 , then
let
x
x z
dx = —x dt
**•
(55)
(56)
(57)
and
fib
/«■>*-£>(!)*-/ '?>G)*
1/a
If f(x) diverges as (x — a) y for 7 € [0, 1], let
(58)
(59)
z = t VCi-7) +a
dx = _J_ t d/i-7)-i df = _J_ t [i-(i-r)l/(i-r) dt
1 — 7 1 — 7
,7/(1-7)
7-1
t = («-a) 1 - 7 ,
dt
(60)
(61)
and
b
d /' , ' (a!) / , /
— / f(x,t)dt = v'(x)f(x,v(x)) -uf(x,u(x)) I
dx Ju(x) Ja
f(x) dx ■
-I
1-7
(6-a)!-7
t w-'>f(t ini -"+a)dt. (62)
If f(x) diverges as (x + b) y for 7 6 [0, 1], let
x = b — t
1/(1-7)
dx
1
7-1
t=(b-x) 1 - y ,
t^^dt
(63)
(64)
(65)
and
/ f(x) dx = —
= / C ' r /(1 - 7) /(6-* 1/(1 " 7> )*. (66)
■7
.(b-a) 1 "^
If the integral diverges exponentially, then let
. —x
t = e
(67)
dt = — e drr
(68)
a; = — lni,
(69)
Integral
and
J°°f( X )dx = J e /(-lnt)f.
(70)
Integrals with rational exponents can often be solved
by making the substitution u = z 1/n , where n is the
Least Common Multiple of the Denominator of the
exponents.
Integration rules include
f
f(x) dx =
/>& pa
/ f(x)dx = - I f(x)dx.
J a Jb
(71)
(72)
For c G (a, 6),
/*0 /»C /»'
/ f{x)dx= / /(x)dx + /
t/a </ a •/ c
f(x)dx= / /(x)dx + / /(a;) da. (73)
If g' is continuous on [a, 6] and / is continuous and has
an antiderivative on an INTERVAL containing the values
of g(x) for a < x < 6, then
J a
f(g(x))g'(x)dx
P9(b)
) du. (74)
Liouville showed that the integrals
sinx
dx
x dx I ES (75)
/2 /* p X f giri oj /*
e~ x dx I — dx I dx I
J x J x J
cannot be expressed as terms of a finite number of ele-
mentary functions. Other irreducibles include
I x x dx I x~ x dx I \/sinxdx. (76)
Chebyshev proved that if U, V, and W are RATIONAL
Numbers, then
/
x u (A + Bx v ) w dx
(77)
is integrable in terms of elementary functions IFF (U +
l)/V, W,otW±(U+ 1)/V is an INTEGER (Ritt 1948,
Shanks 1993).
There are a wide range of methods available for NUMERI-
CAL INTEGRATION. A good source for such techniques is
Press et al. (1992). The most straightforward numerical
integration technique uses the NEWTON-COTES FORMU-
LAS (also called QUADRATURE FORMULAS), which ap-
proximate a function tabulated at a sequence of regu-
larly spaced INTERVALS by various degree POLYNOMI-
ALS. If the endpoints are tabulated, then the 2- and 3-
point formulas are called the TRAPEZOIDAL Rule and
Integral 909
Simpson's Rule, respectively. The 5-point formula is
called BODE'S RULE. A generalization of the TRAPE-
ZOIDAL Rule is Romberg Integration, which can
yield accurate results for many fewer function evalua-
tions.
If the analytic form of a function is known (instead
of its values merely being tabulated at a fixed number
of points), the best numerical method of integration is
called Gaussian Quadrature. By picking the optimal
ABSCISSAS at which to compute the function, Gaussian
quadrature produces the most accurate approximations
possible. However, given the speed of modern comput-
ers, the additional complication of the Gaussian Quad-
rature formalism often makes it less desirable than
the brute-force method of simply repeatedly calculat-
ing twice as many points on a regular grid until conver-
gence is obtained. An excellent reference for GAUSSIAN
Quadrature is Hildebrand (1956).
see also A-lNTEGRABLE, ABELIAN INTEGRAL, CAL-
CULUS, Chebyshev-Gauss Quadrature, Cheby-
shev Quadrature, Darboux Integral, Definite
Integral, Denjoy Integral, Derivative, Dou-
ble Exponential Integration, Euler Integral,
Fundamental Theorem of Gaussian Quadra-
ture, Gauss-Jacobi Mechanical Quadrature,
Gaussian Quadrature, Haar Integral, Hermite-
Gauss Quadrature, Hermite Quadrature, HK
Integral, Indefinite Integral, Integration,
Jacobi-Gauss Quadrature, Jacobi Quadrature,
Laguerre-Gauss Quadrature, Laguerre Quad-
rature, Lebesgue Integral, Lebesgue-Stieltjes
Integral, Legendre-Gauss Quadrature, Legen-
dre Quadrature, Lobatto Quadrature, Me-
chanical Quadrature, Mehler Quadrature,
Newton-Cotes Formulas, Numerical Integra-
tion, Peron Integral, Quadrature, Radau Quad-
rature, Recursive Monotone Stable Quadra-
ture, Riemann-Stieltjes Integral, Romberg In-
tegration, Riemann Integral, Stieltjes Inte-
gral
References
Beyer, W. H. "Integrals." CRC Standard Mathematical Ta-
bles, 28th ed. Boca Raton, FL: CRC Press, pp. 233-296,
1987.
Bronstein, M. Symbolic Integration I: Transcendental Func-
tions. New York: Springer- Verlag, 1996.
Gordon, R. A. The Integrals of Lebesgue, Denjoy, Perron,
and Henstock. Providence, Rl: Amer. Math. Soc, 1994.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1993.
Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 319-323, 1956.
Piessens, R.; de Doncker, E.; Uberhuber, C. W.; and Ka-
haner, D. K. QUADPACK: A Subroutine Package for Au-
tomatic Integration. New York: Springer- Verlag, 1983.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Integration of Functions." Ch. 4 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
910 Integral Brick
Integral Equation
2nd ed. Cambridge, England: Cambridge University Press,
pp. 123-158, 1992.
Ritt, J. F. Integration in Finite Terms. New York: Columbia
University Press, p. 37, 1948.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 145, 1993.
Wolfram Research. "The Integrator." http://www.
integrals . com
Integral Brick
see Euler Brick
Integral Calculus
That portion of "the" Calculus dealing with Inte-
grals.
see also CALCULUS, DIFFERENTIAL CALCULUS, INTE-
GRAL
Integral Cuboid
see Euler Brick
Integral Current
A Rectifiable Current whose boundary is also a
Rectifiable Current.
Integral Curvature
Given a GEODESIC TRIANGLE (a triangle formed by the
arcs of three GEODESICS on a smooth surface),
/
J ABC
Kda = A + B + C-n.
Given the EULER CHARACTERISTIC x,
K da — 27tx,
//■
so the integral curvature of a closed surface is not altered
by a topological transformation.
see also Gauss-Bonnet Formula, Geodesic Trian-
gle
Integral Domain
A Ring that is Commutative under multiplication, has
a unit element, and has no divisors of 0. The Integers
form an integral domain.
see also Field, Ring
Integral Drawing
A Graph drawn such that the Edges have only In-
teger lengths. It is conjectured that every PLANAR
GRAPH has an integral drawing.
References
Harborth, H. and Moller, M. "Minimum Integral Drawings
of the Platonic Graphs." Math. Mag. 67, 355-358, 1994.
Integral Equation
If the limits are fixed, an integral equation is called a
Fredholm integral equation. If one limit is variable, it
is called a Volterra integral equation. If the unknown
function is only under the integral sign, the equation is
said to be of the "first kind." If the function is both
inside and outside, the equation is called of the "second
kind." A Fredholm equation of the first kind is of the
form
/(*)= f K(x,t)4>(t)dt. (1)
J a
A Fredholm equation of the second kind is of the form
<f>{x) = f{x) + X f K(x, t)(f>(t) dt. (2)
J a
A Volterra equation of the first kind is of the form
f(x) = I K(x y t)<f>(t)dt. (3)
J a
A Volterra equation of the second kind is of the form
4>( x ) = f( x )+ f K{x,t)<j>{t)dt, (4)
J a
where the functions K(x,t) are known as Kernels. In-
tegral equations may be solved directly if they are Sep-
arable. Otherwise, a NEUMANN SERIES must be used.
A Kernel is separable if
n
K(x,t) = \JT l M j (x)N i (t).
(5)
This condition is satisfied by all Polynomials and
many TRANSCENDENTAL FUNCTIONS. A FREDHOLM
Integral Equation of the Second Kind with sep-
arable Kernel may be solved as follows:
t/ o
<j>(x) = f(x) + / K(x, t)<f>(t) dt
= f{x) + \^M j {x) J NjitMQdt
3 = 1 ^ a
n
= /(*) + A ^cjAf^s), (6)
3 = 1
where
»/ a
Cj = I Nj(t)(j)(t)dt.
(7)
Now multiply both sides of (7) by N{(x) and integrate
over dx.
f
<f>(x)Ni(x)dx
/ f(x)Ni(x)dx + \ 1 ^c j / Mj(x)Ni(x)dx.
J a -J Jo,
(8)
Integral of Motion
Integrand 911
By (7), the first term is just c». Now define
(x)f(x)dx
b t = f Ni
J a
dij = / Ni(x)Mj(x)dx,
J a
(9)
(10)
so (8) becomes
Ci = bi + A y ciij Cj
(ii)
j=i
Writi
ng this in matrix form,
C = B + AAC,
(12)
so
(I - AA)C = B
(13)
C = (l-AA) _1 B.
(14)
see also Fredholm Integral Equation of the
First Kind, Fredholm Integral Equation of the
Second Kind, Volterra Integral Equation of
the First Kind, Volterra Integral Equation of
the Second Kind
References
Corduneanu, C. Integral Equations and Applications. Cam-
bridge, England: Cambridge University Press, 1991.
Davis, H, T. Introduction to Nonlinear Differential and In-
tegral Equations. New York: Dover, 1962.
Kondo, J. Integral Equations. Oxford, England: Clarendon
Press, 1992.
Lovitt, W. V. Linear Integral Equations. New York: Dover,
1950.
Mikhlin, S. G. Integral Equations and Their Applications
to Certain Problems in Mechanics, Mathematical Phys-
ics and Technology, 2nd rev. ed. New York: Macmillan,
1964.
Mikhlin, S. G. Linear Integral Equations. New York: Gordon
& Breach, 1961.
Pipkin, A. C. A Course on Integral Equations. New York:
Springer- Verlag, 1991.
Porter, D. and Stirling, D. S. G. Integral Equations: A
Practical Treatment, from Spectral Theory to Applications.
Cambridge, England: Cambridge University Press, 1990.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Integral Equations and Inverse Theory."
Ch. 18 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 779-817, 1992.
Tricomi, F. G. Integral Equations. New York: Dover, 1957.
Integral of Motion
A function of the coordinates which is constant along a
trajectory in Phase Space. The number of DEGREES
of Freedom of a Dynamical System such as the
Duffing Differential Equation can be decreased
by one if an integral of motion can be found. In general,
it is very difficult to discover integrals of motion.
Integral Sign
The symbol J used to denote an Integral J f(x) dx.
The symbol was chosen to be a stylized script "S" to
stand for "summation."
see also INTEGRAL
Integral Test
Let Yl u k be- a series with POSITIVE terms and let f(x)
be the function that results when k is replaced by x in
the Formula for u^. If / is decreasing and continuous
for x > 1 and
lim f(x) = 0,
then
and
£
Uk
[
f(x) dx
both converge or diverge, where 1 < t < oo. The test is
also called the CAUCHY INTEGRAL TEST or MACLAURIN
Integral Test.
see also CONVERGENCE TESTS
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 283-284, 1985.
Integral Transform
A general integral transform is denned by
g{a) = f
f(t)K(a,t)dt,
where K(a,t) is called the Kernel of the transform.
see also Fourier Transform, Fourier-Stieltjes
Transform, H-Transform, Hadamard Trans-
form, Hankel Transform, Hartley Transform,
Hough Transform, Operational Mathematics,
Radon Transform, Wavelet Transform, Z-
Transform
References
Arfken, G. "Integral Transforms." Ch. 16 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 794-864, 1985.
Carslaw, H. S. and Jaeger, J. C. Operational Methods in Ap-
plied Mathematics.
Davies, B. Integral Transforms and Their Applications, 2nd
ed. New York: Springer- Verlag, 1985.
Poularikas, A. D. (Ed.). The Transforms and Applications
Handbook. Boca Raton, FL: CRC Press, 1995.
Zayed, A. I. Handbook of Function and Generalized Function
Transformations. Boca Raton, FL: CRC Press, 1996.
Integrand
The quantity being INTEGRATED, also called the Ker-
nel. For example, in J f(x)dx, fix) is the integrand.
see also INTEGRAL, INTEGRATION
912 Integrating Factor
Integrating Factor
A Function by which an Ordinary Differential
EQUATION is multiplied in order to make it integrable.
see also ORDINARY DIFFERENTIAL EQUATION
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 526-529, 1953.
Integration
The process of computing or obtaining an INTEGRAL. A
more archaic term for integration is QUADRATURE.
see also CONTOUR INTEGRATION, INTEGRAL, INTEGRA-
TION by Parts, Measure Theory, Numerical Inte-
gration
Integration Lattice
A discrete subset of IR S which is closed under addition
and subtraction and which contains Z s as a SUBSET.
see also LATTICE
References
Sloan, I. H. andJoe, S. Lattice Methods for Multiple Integra-
tion. New York: Oxford University Press, 1994.
Integration by Parts
A first-order (single) integration by parts uses
d(uv) = u dv + v du (1)
/ d{uv) = uv = / udv 4- / vdu, (2)
udv = uv — vdu (3)
rt> rf(b)
/ udv = [uv]a — I vdu.
Ja J f(a)
so
and
(4)
Now apply this procedure n times to J f( n \x)g(x)dx.
u — g{x) dv = y\x)dx
du = g(x)dx i; = / (n_1) 0).
(5)
(6)
Therefore,
ff^g(x)dx = g(x)f (n - 1 \x)- f f^- 1 \x)g'(x)dx.
(7)
But
f f (n - 1) (x)g'(x)dx
= g'(x)f (n - 2) (x) - f f^- 2) (x)g"{x)dx
(8)
Integration by Parts
J f (n - 2) (x)g"(x)dx
= g"(x)f( n - 3) (x)-Jf (n - s) (x)g (3) (x)dx, (9)
so
f f {n) {x)g{x) dx = g(x)f {n ~ 1) (x) - g'(x)f (n - 2) (x)
+g"(x)f< n - s) (x)-... + (-l) n f f(x)g (n Hx)dx. (10)
Now consider this in the slightly different form
J f{x)g(x) dx. Integrate by parts a first time
u = f(x) dv = g(x)dx (11)
du = f{x) dx v = / g(x) dx, (12)
so
/
f(x)g{x) dx = f(x) I g{x) dx
/■
/ [/«•■
) dx
f'(x)dx. (13)
Now integrate by parts a second time,
u = f'(x) dv= I g(x)(dx) 2 (14)
du = f"(x)dx v= [[gixXdx) 2 , (15)
so
/ f(x)g(x)dx = f(x) / g(x)dx - f(x) II g{x){dx) 2
+ j\jj \(x)(dx) 2 ]f"(x)dx. (16)
Repeating a third time,
/ f(x)g(x)dx = f(x) / g(x)dx - f'(x) II g(x)(dx) 2
f"(x)JJJg(x){dx) a
+f"
I [III
g(x)(dxf
f"'(x)dx. (17)
Intension
Therefore, after n applications,
/ f(x)g(x) dx = f{x) / g{x) dx - f(x) 11 g{x){dxf
+f"(x)JJJg(x)(dxf-...
+(-l)^f^( x )j...Jg( x )(dx) n+1
Interpolation 913
+(
-l)-/
g(x)(dx)
n + l
L n+l
/ (n+1) (z)<iz. (18)
If f (n+1) (x) = (e.g., for an nth degree POLYNOMIAL),
the last term is 0, so the sum terminates after n terms
and
/ f(x)g(x) dx = f(x) / g(x) dx
-f{x) JJ g{x){dxf + /"(*) jJJ g{x){dxf - . . .
+(-i)»+ 1 / (n) («) / ■ ■ • Jg{x){dxr + \ (19)
References
Abramowitz, M. and Stegun, C. A, (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, gth printing. New York: Dover,
p. 12, 1972.
Intension
A definition of a Set by mentioning a defining property.
see also EXTENSION
References
Russell, B. "Definition of Number." Introduction to Mathe-
matical Philosophy. New York: Simon and Schuster, 1971.
Interchange Graph
see Line Graph
Interest
Interest is a fee (or payment) made for the borrowing
(or lending) of money. The two most common types
of interest are Simple Interest, for which interest is
paid only on the initial Principal, and COMPOUND IN-
TEREST, for which interest earned can be re-invested to
generate further interest.
see also Compound Interest, Conversion Period,
Rule of 72, Simple Interest
References
Kellison, S. G. Theory of Interest, 2nd ed. Burr Ridge, IL:
Richard D. Irwin, 1991.
Interior
That portion of a region lying "inside" a specified
boundary. For example, the interior of the SPHERE is a
Ball.
see also Exterior
Interior Angle Bisector
see Angle Bisector
Intermediate Value Theorem
If / is continuous on a Closed Interval [a, b] and c is
any number between f(a) and f(b) inclusive, there is at
least one number x in the Closed Interval such that
f{x) = c.
see also WeierstraB Intermediate Value Theorem
Internal Bisectors Problem
see Steiner-Lehmus Theorem
Internal Knot
One of the knots £ P +i, - . . , £ m -p-i of a B-Spline with
control points Po, . . . , P n and Knot Vector
T — {toj^l) • • • j^m},
where
p = m — n — 1.
see also B-Spline, Knot Vector
Interpolation
The computation of points or values between ones that
are known or tabulated using the surrounding points or
values.
see also AlTKEN INTERPOLATION, BESSEL'S INTER-
POLATION Formula, Everett Interpolation, Ex-
trapolation, Finite Difference, Gauss's In-
terpolation Formula, Hermite Interpolation,
Lagrange Interpolating Polynomial, Newton-
Cotes Formulas, Newton's Divided Difference
Interpolation Formula, Osculating Interpola-
tion, Thiele's Interpolation Formula
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Interpolation."
§25.2 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 878-882, 1972.
Iyanaga, S. and Kawada, Y. (Eds.). "Interpolation." Ap-
pendix A, Table 21 in Encyclopedic Dictionary of Mathe-
matics. Cambridge, MA: MIT Press, pp. 1482-1483, 1980.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Interpolation and Extrapolation." Ch. 3
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 99-122, 1992.
914 Interquartile Range
Intrinsic Curvature
Interquartile Range
Divide a set of data into two groups (high and low) of
equal size at the Median if there is an Even number of
data points, or two groups consisting of points on either
side of the Median itself plus the Median if there is
an Odd number of data points. Find the MEDIANS of
the low and high groups, denoting these first and third
quartiles by Qi and Q$. The interquartile range is then
defined by
IQR=Q 3 -Qi.
see also ^-Spread, Hinge, Median (Statistics)
Inter radius
see MlDRADIUS
Intersection
The intersection of two sets A and B is the set of ele-
ments common to A and B. This is written AnB, and
is pronounced "A intersection B" or "A cap B" The in-
tersection of sets Ai through A n is written f]^ := . 1 Ai. The
intersection of lines AB and CD is written AB O CD.
see also And, Union
Interspersion
An Array A = a^, i 7 j > 1 of Positive Integers is
called an interspersion if
1. The rows of A comprise a PARTITION of the POSI-
TIVE Integers,
2. Every row of A is an increasing sequence,
3. Every column of A is a (possibly FINITE) increasing
sequence,
4. If (uj) and (vj) are distinct rows of A and if p and
q are any indices for which u p < v q < i£p+i, then
If an array A = aij is an interspersion, then it is a DIS-
PERSION. If an array A = a{hj) 1S an interspersion,
then the sequence {x n } given by {x n = i : n = (i 7 j)}
for some j is a FRACTAL SEQUENCE. Examples of in-
terspersion are the Stolarsky Array and Wythoff
Array.
see also Dispersion (Sequence), Fractal Se-
quence, Stolarsky Array
References
Kimberling, C. "Interspersions and Dispersions." Proc.
Amer. Math. Soc. 117, 313-321, 1993.
Kimberling, C. "Fractal Sequences and Interspersions." Ars
Combin. 45, 157-168, 1997.
Intersphere
see Midsphere
Interval
A collection of points on a LINE SEGMENT. If the end-
points a and b are FINITE and are included, the interval
is called Closed and is denoted [a, b]. If one of the end-
points is ±00, then the interval still contains all of its
Limit Points, so [a, 00) and (—00, 6] are also closed in-
tervals. If the endpoints are not included, the interval
is called OPEN and denoted (a, 6). If one endpoint is
included but not the other, the interval is denoted [a, b)
or (a,b] and is called a Half-Closed (or Half-Open)
interval.
see also Closed Interval, Half-Closed Interval,
Limit Point, Open Interval, Pencil
Interval Graph
A GRAPH G = (V,E) is an interval graph if it captures
the Intersection Relation for some set of Intervals
on the Real Line. Formally, P is an interval graph
provided that one can assign to each v 6 V an interval
I v such that I u nl v is nonempty precisely when uv € E.
see also Comparability Graph
References
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.
Fishburn, P. C. Interval Orders and Interval Graphs: A
Study of Partially Ordered Sets. New York: Wiley, 1985.
Gilmore, P. C. and Hoffman, A. J. "A Characterization of
Comparability Graphs and of Interval Graphs." Canad. J.
Math. 16, 539-548, 1964.
Lekkerkerker, C. G. and Boland, J. C. "Representation of a
Finite Graph by a Set of Intervals on the Real Line." Fund.
Math. 51, 45-64, 1962.
Interval Order
A POSET P = (X, <) is an interval order if it is ISO-
MORPHIC to some set of Intervals on the Real Line
ordered by left-to-right precedence. Formally, P is an in-
terval order provided that one can assign to each x £ X
an Interval [xl,xr] such that xr < yL in the Real
Numbers Iff x < y in P.
see also PARTIALLY ORDERED SET
References
Fishburn, P. C. Interval Orders and Interval Graphs: A
Study of Partially Ordered Sets. New York: Wiley, 1985.
Wiener, N. "A Contribution to the Theory of Relative Posi-
tion." Proc. Cambridge Philos. Soc. 17, 441-449, 1914.
Intrinsic Curvature
A Curvature such as Gaussian Curvature which
is detectable to the "inhabitants" of a surface and not
just outside observers. An EXTRINSIC CURVATURE, on
the other hand, is not detectable to someone who can't
study the 3-dimensional space surrounding the surface
on which he resides.
see also CURVATURE, EXTRINSIC CURVATURE, GAUS-
SIAN Curvature
Intrinsic Equation
Inverse Cosecant
915
Intrinsic Equation
An equation which specifies a CURVE in terms of intrin-
sic properties such as Arc Length, Radius of Cur-
vature, and Tangential Angle instead of with ref-
erence to artificial coordinate axes. Intrinsic equations
are also called Natural Equations.
see also Cesaro Equation, Natural Equation,
Whewell Equation
References
Yates, R. C. "Intrinsic Equations." A Handbook on Curves
and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 123-126, 1952.
Intrinsically Linked
A Graph is intrinsically linked if any embedding of it
in 3-D contains a nontrivial Link. A Graph is intrinsi-
cally linked IFF it contains one of the seven PETERSEN
GRAPHS (Robertson et al. 1993).
The Complete Graph K 6 (left) is intrinsically linked
because it contains at least two linked Triangles. The
Complete ^-Partite Graph 1^3,3,1 (right) is also in-
trinsically linked.
see also Complete Graph, Complete A;-Partite
Graph, Petersen Graphs
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 217-221, 1994.
Robertson, N,; Seymour, P. D,; and Thomas, R. "Linkless
Embeddings of Graphs in 3-Space." Bull. Amer. Math.
Soc. 28, 84-89, 1993.
Invariant
A quantity which remains unchanged under certain
classes of transformations. Invariants are extremely use-
ful for classifying mathematical objects because they
usually reflect intrinsic properties of the object of study.
see Adiabatic Invariant, Alexander Invariant,
Algebraic Invariant, Arf Invariant, Integral of
Motion
References
Hunt, B. "Invariants." Appendix B.l in The Geometry of
Some Special Arithmetic Quotients. New York: Springer-
Verlag, pp. 282-290, 1996.
Invariant Density
see Natural Invariant
Invariant (Elliptic Function)
The invariants of a WEIERSTRAfl ELLIPTIC FUNCTION
are defined by
g 2 = 60S Q mn
Here,
g 3 55 140E'n mT T 6 .
Qmn = 2muJi — 2no>2,
where uj\ and U2 are the periods of the Elliptic Func-
tion.
Invariant Manifold
When stable and unstable invariant Manifolds inter-
sect, they do so in a Hyperbolic Fixed Point (Sad-
dle Point). The invariant Manifolds are then called
Separatrices. A Hyperbolic Fixed Point is char-
acterized by two ingoing stable MANIFOLDS and two
outgoing unstable Manifolds. In integrable systems,
incoming W s and outgoing W u MANIFOLDS all join up
smoothly.
A stable invariant MANIFOLD W s of a FIXED POINT Y*
is the set of all points Y such that the trajectory passing
through Yq tends to Y* as j — > 00.
An unstable invariant Manifold W u of a Fixed Point
Y* is the set of all points Yq such that the trajectory
passing through Yb tends to Y* as j — > — 00.
see also Homoclinic Point
Invariant Point
see Fixed Point (Transformation)
Invariant Subgroup
see Normal Subgroup
Inverse Cosecant
1.5 ■
1.25
1
0.75
0.5
0.25
Re[ArcCsc z]
Im[ArcCsc z]
The function esc 1 x, also denoted arccsc(;c), where esc x
is the Cosecant and the Superscript -1 denotes an
916 Inverse Cosine
Inverse Function, not the multiplicative inverse. The
inverse cosecant satisfies
esc as = sec
y x 2 - 1
for Positive or Negative x, and
CSC~ X = 7T + csc~ (—as)
(1)
(2)
for x > 0. The inverse cosecant is given in terms of other
inverse trigonometric functions by
esc = cos
i^p)
(3)
= cot- 1 ( V / » 2 -l) (4)
= §7r — sec" 1 x = -§7r - sec _1 (-x) (5)
= sin-(I) (6)
for x > 0.
see also COSECANT INVERSE SlNE, SlNE
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 142-143, 1987.
Inverse Cosine
The function cos -1 as, also denoted arccos(as), where
cos as is the Cosine and the superscript —1 denotes
an Inverse Function, not the multiplicative inverse.
The MACLAURIN Series for the inverse cosine range
-1 < x < 1 is
X 112^ 1152 "^ V /
cos as = o7r — as— ~as
The inverse cosine satisfies
cos as — 7r — cos (— as J
Inverse Cotangent
for Positive and Negative as, and
cos" 1 = f 7T - cos^a/I-^ 2 ) (3)
for x > 0. The inverse cosine is given in terms of other
inverse trigonometric functions by
"-- 1 — *"'(;^p) (4)
= §7r + sin _1 (-x) = |7r — sin -1 as (5)
= ±7r — tan'
(6)
for Positive or Negative x, and
cos" 1 a; = esc" 1 (-^L=j (7)
= sec" 1 (i) (8)
= siir 1 ( V / l-z 2 ) (9)
= ten -(^Z) (10)
for as > 0.
see also Cosine, Inverse Secant
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circu-
lar Functions." §4.4 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 79-83, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 142-143, 1987.
Inverse Cotangent
(2)
The function cot 1 as, also denoted arccot(as), where
cot as is the Cotangent and the superscript —1 denotes
an Inverse Function and not the multiplicative in-
verse. The Maclaurin SERIES is given by
cot" 1 x = \tz - x + \x z - \x* + \x 7 - |as 9 + . . . , (1)
Inverse Cotangent
and Power Series by
cot a; = a; — 3# +5^ ~ 7 X ~t~ 9 X "•" * *
Euler derived the INFINITE series
(2)
cot x X = X
+
x 2 + l 3(x 2 + l) 2
+ 7
2-4
3 ■ 5(x 2 + l) 3
(Wetherfield 1996).
The inverse cotangent satisfies
cot -1 x — tv — cot~ 1 (— a;)
for Positive and Negative x, and
+ .
(3)
COt X = ^7T — COt
-G)
(4)
(5)
for x > 0. The inverse cotangent is given in terms of
other inverse trigonometric functions by
cot x x — cos
v?n
1 * -1
= |7T — sin
(6)
(7)
= \tv + tan _1 (-x) = |tt — tan -1 a; (8)
for Positive or Negative x, and
cot 1 x = csc 1 (ya; 2 + l)
sec
for x > 0.
A number
v Vz 2 + 1
t x = cot~ x 7
(9)
(10)
(11)
(12)
(13)
where x is an INTEGER or RATIONAL NUMBER, is some-
times called a GREGORY NUMBER. Lehmer (1938a)
showed that cot -1 (a/6) can be expressed as a finite sum
of inverse cotangents of INTEGER arguments
cot" 1 ^) =£(-l)*- 1 cot- 1 n*,
where
(14)
(15)
Inverse Cotangent 917
with [x\ the FLOOR FUNCTION, and
di+i = din + i + bi
&i_i_i = di — mbi,
(16)
(17)
with ao = a and 60 = &, and where the recurrence is
continued until b k+ i — 0. If an INVERSE TANGENT sum
is written as
tan
" 1 n = ^/ fc tan 1 n k + f tan \ (18)
then equation (14) becomes
cot -1 n — 2^ fk cot -1 nk + ccot" 1 1, (19)
jt=i
where
c=2-f-2j2fr
(20)
Inverse cotangent sums can be used to generate
Machin-Like Formulas.
An interesting inverse cotangent identity attributed to
Charles Dodgson (Lewis Carroll) by Lehmer (1938b;
Bromwich 1965, Castellanos 1988ab) is
cot _1 (p + r) + tan" ^p + g) = tarT 1 ^ (21)
where
(22)
1 + p — qr.
Other inverse cotangent identities include
2 cot" 1 (2a:) - cot -1 x = cot -1 (4a; 3 + 3x) (23)
3 cot 1 (3x) — cot 1 x = cot
/ 27s 4 + 18x 2 -l \
I 8. j'
(24)
as well as many others (Bennett 1926, Lehmer 1938b).
see also COTANGENT, INVERSE TANGENT, MACHIN'S
Formula, Machin-Like Formulas, Tangent
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circu-
lar Functions." §4.4 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 79-83, 1972.
Bennett, A. A. "The Four Term Diophantine Arccotangent
Relation." Ann. Math. 27, 21-24, 1926.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 142-143, 1987.
Bromwich, T. J. I. and MacRobert, T. M. An Introduction to
the Theory of Infinite Series, 3rd ed. New York: Chelsea,
1991.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988a.
Castellanos, D. "The Ubiquitous Pi. Part II." Math. Mag.
61, 148-163, 1988b.
Lehmer, D. H. "A Cotangent Analogue of Continued Frac-
tions." Duke Math. J. 4, 323-340, 1938a.
Lehmer, D. H. "On Arccotangent Relations for n. n Amer.
Math. Monthly 45, 657-664, 1938b.
# Weisstein, E. W. "Arccotangent Series." http:// www .
astro . Virginia . edu / - eww6n / math / notebooks / Cot
Series. m.
Wetherfield, M. "The Enhancement of Machin's Formula by
Todd's Process." Math. Gaz., 333-344, July 1996.
918
Inverse Curve
Inverse Hyperbolic Cosine
Inverse Curve
Given a Circle C with Center O and Radius fc, then
two points P and Q are inverse with respect to C if OP *
OQ = k 2 . HP describes a curve Ci, then Q describes
a curve Ci called the inverse of C\ with respect to the
circle C (with Inversion Center 0). If the Polar
equation of C is r(#), then the inverse curve has polar
equation
k 2
If O = (x ,yo) and P =
equations
X = Xq +
y = yo +
(/(*)> 5(*))> tnen tne inverse has
* 2 (/-*o])
(/-x ) 2 + (5-2/o) 2
fe 2 (g-yo)
(/ - xo) 2 + (g - yo)
2 '
Curve
Inversion
Center
Inverse Curve
Archimedean spiral
origin
Archimedean spiral
cardioid
cusp
parabola
circle
any pt.
another circle
cissoid of Diodes
cusp
parabola
cochleoid
origin
quadratrix of Hippias
epispiral
origin
Rose
Fermat's spiral
origin
lituus
hyperbola
center
lemniscate
hyperbola
vertex
right strophoid
hyperbola with
vertex
Maclaurin trisectrix
a - \/3
lemniscate
center
hyperbola
lituus
origin
Fermat spiral
logarithmic spiral
origin
logarithmic spiral
Maclaurin trisectrix
focus
Tschirnhausen's cubic
parabola
focus
cardioid
parabola
vertex
cissoid of Diocles
quadratrix of Hippias
cochleoid
right strophoid
origin
the same right strophoid
sinusoidal spiral
origin
sinusoidal spiral inverse
curve
Tschirnhausen cubic
sinusoidal spiral
see also INVERSION, INVERSION CENTER, INVERSION
Circle
References
Lee, X. "Inversion." http://www.best .com/ ~xah/Special
PlaneCurves_dir/Inversion_dir /inversion, html.
Lee, X. "Inversion Gallery." http://www . best . com/ -xah/
Special Plane Curves _ dir / Inversion Gallery _ dir /
inversionGallery.html.
Yates, R. C. "Inversion." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 127-134,
1952.
Inverse Function
Given a FUNCTION f(x), its inverse f~ 1 (x) is defined by
/(/ _1 (a:)) = x. Therefore, f{x) and f~\x) are reflec-
tions about the line y = x.
Inverse Hyperbolic Cosecant
7
6
5
4
3
2
1
Re[ArcCsch z]
Im[ArcCsch z]
The Inverse Function of the Hyperbolic Cose-
cant, denoted csch -1 x.
see also HYPERBOLIC COSECANT
Inverse Hyperbolic Cosine
3
2.5
2
1.5
0.5
RetArcCosh z]
The Inverse Function of the Hyperbolic Cosine,
denoted cosh -1 x.
see also HYPERBOLIC COSINE
Inverse Filter
A linear Deconvolution Algorithm.
Inverse Hyperbolic Cotangent
Inverse Hyperbolic Cotangent
Inverse Points 919
3
2.5
2
1.5
1
0.5
Re[ArcCoth z]
Im[ArcCoth z]
The Inverse Function of the Hyperbolic Cotan-
gent, denoted coth -1 x.
see also HYPERBOLIC COTANGENT
Inverse Hyperbolic Functions
The Inverse of the Hyperbolic Functions, denoted
cosh" 1 x 1 coth -1 x, csch -1 x, seen -1 cc, sinh - x, and
tanh -1 x,
see also HYPERBOLIC FUNCTIONS
References
Spanier, J. and Oldham, K. B. "The Inverse Hyperbolic Func-
tions." Ch. 31 in An Atlas of Functions, Washington, DC:
Hemisphere, pp. 285-293, 1987.
Inverse Hyperbolic Secant
The Inverse Function of the Hyperbolic Secant,
denoted sech -1 x.
see also HYPERBOLIC SECANT
Inverse Hyperbolic Sine
3
2
1
-10
-5
-2
-3
5
10
Re[ArcSinh z]
Im[ArcSinh z]
The Inverse Function of the Hyperbolic Sine, de-
noted sinh -1 x.
see also Hyperbolic Sine
Inverse Hyperbolic Tangent
The Inverse Function of the Hyperbolic Tangent,
denoted tanh -1 x.
see also Hyperbolic Tangent
Inverse Matrix
see also Matrix Inverse
Inverse Points
Points which are transformed into each other through
Inversion about a given Inversion Circle. The point
P' which is the inverse point of a given point P with re-
spect to an Inversion Circle C may be constructed
geometrically using a COMPASS only (Courant and Rob-
bins 1996).
see also Geometric Construction, Inversion, Po-
lar, Pole (Geometry)
920 Inverse Quadratic Interpolation
References
Courant, R. and Robbins, H. "Geometrical Construction of
Inverse Points." §3.4.3 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 144-145, 1996.
Inverse Quadratic Interpolation
The use of three prior points in a RoOT-finding ALGO-
RITHM to estimate the zero crossing.
Inverse Scattering Method
A method which can be used to solve the initial value
problem for certain classes of nonlinear Partial DIF-
FERENTIAL EQUATIONS. The method reduces the ini-
tial value problem to a linear INTEGRAL EQUATION in
which time appears only implicitly. However, the solu-
tions u(x,t) and various of their derivatives must ap-
proach zero asa;-> ±oo (Infeld and Rowlands 1990).
see also AbLOWTTZ-RAMANI-SeGUR CONJECTURE,
Backlund Transformation
References
Infeld, E. and Rowlands, G. "Inverse Scattering Method."
§7.4 in Nonlinear Waves, Solitons, and Chaos. Cam-
bridge, England: Cambridge University Press, pp. 192-
196, 1990.
Miura, R. M. (Ed.) Backlund Transformations, the Inverse
Scattering Method, Solitons, and Their Applications. New
York: Springer- Verlag, 1974.
Inverse Secant
Re[ArcSec z]
Im[ArcSec z]
The function sec -1 z, where sec a; is the SECANT and the
superscript —1 denotes the INVERSE FUNCTION, not the
multiplicative inverse. The inverse secant satisfies
-l -i
sec x — esc
Vx 2 - 1
for POSITIVE or Negative x, and
sec - x = 7r + sec - (—x)
(1)
(2)
Inverse Sine
for x > 0. The inverse secant is given in terms of other
inverse trigonometric functions by
sec - x — cos - I — ) (3)
—"'(^r) (4)
= ~7V — csc~ x — —\-k — csc~ ( — x) (5)
—-(*S) <„
= tan- 1 (V* 2 -l) (7)
for x > 0.
see also INVERSE COSECANT, SECANT
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 141-143, 1987.
Inverse Semigroup
The abstract counterpart of a PSEUDOGROUP formed by
certain subsets of a Groupoid which admit a MULTI-
PLICATION.
References
Weinstein, A. "Groupoids: Unifying Internal and External
Symmetry." Not Amer. Math. Soc. 43, 744-752, 1996.
Inverse Sine
1.5
1
0.5
-1
-0.5
-0.5
-1
-1.5
0.5
1
RetArcSin z]
|ArcSin zj
The function sin -1 #, where sin a; is the Sine and the
superscript —1 denotes the INVERSE FUNCTION, not the
multiplicative inverse. The inverse sine satisfies
sin x = — sin (—x)
for Positive and Negative #, and
- 1 = ±n-snr l (y/l^tf)
(1)
(2)
Inverse Tangent
for x > 0. The inverse sine is given in terms of other
inverse trigonometric functions by
sin -1 x = cos" 1 (— x) — |7r = \-k — cos -1 x (3)
2" ~ 2'
■ X A
tan
for Positive or Negative x, and
^ x = cos 1 (y 1 — x 2 )
cot
i fVT^:
-(:)
1
vT^i?
(4)
(5)
(6)
(7)
(8)
(9)
for x > 0.
5ee a/50 INVERSE COSINE, SINE
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circu-
lar Functions." §4.4 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 79-83, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 142-143, 1987.
Inverse Tangent
1.5
1
0.5
-10
-5
-0.fi
/-I
-1.5
5
10
Re[ArcTan z
ArcTan z j
The inverse tangent is also called the arctangent and is
denoted either tan" 1 x or arctan x. It has the MAC-
laurin Series
tan
-■-£
(-l)"a 2 " +1
2n + l
• J-/ 'i'^ 1 r; X 7 "T • • • •
(1)
Inverse Tangent 921
A more rapidly converging form due to Euler is given by
2 2n (rc!) 2 x 2n+1
tan
Ix -Z2( 2n
n=Q
(2n + l)! (+x 2 )"+ x
(2)
(Castellanos 1988). The inverse tangent satisfies
tan" 1 x = - tan _1 (-x) (3)
for Positive and Negative x, and
tan = \-k — tan
for x > 0. The inverse tangent is given in terms of other
inverse trigonometric functions by
tan x = \iz — cos
(5)
^ y/x 2 + 1
for Positive or Negative x, and
cot 1 (—x) — \n = \-k — cot 1 x (6)
(7)
(8)
(9)
(10)
(11)
, -i -i/l
tan x = cos
vV + l
i / Vx^Ti
= sec" 1 (v / ^ 2 + l)
for x > 0.
In terms of the Hypergeometric Function,
tan~ 1 x = x 2 Fi(i, |;f;-z 2 )
l + x :
X a*i(l,l;§;- *~
2 'l + x 2
(12)
(13)
(Castellanos 1988). Castellanos (1986, 1988) also gives
some curious formulas in terms of the FIBONACCI NUM-
BERS,
tan
-i _ v~^ ( — l) n i ? 2n+it'
x ~ 2Lt 5"(2n+l
(14)
( z lT^Wit 2ri+1
5"(2n + 1
= 5 y (-i)"'w_ — (15)
^ (2n + 1)(« + Vit 2 + 1 ) 2 " +1
_y. (-l)"5"+ a F aB+1 8
^(2n+l)(t) + v^T5) 2 »+ 1 '
n=0 v 7 v '
922
where
Inverse Tangent
t =
2x
1 +
u=l(l + f^)
and v is the largest Positive Root of
Sxv 4 - lOOu 3 - 450xu 2 + 875i> + 625z = 0.
The inverse tangent satisfies the addition FORMULA
tan" 1 x + tan -1 y = tan' 1 ( ^^- J (20)
as well as the more complicated FORMULAS
tan_1 (^) =tan_1 (D +tan_1 (^n)
(21)
tan " I (;)= 2tM, " , (s)- tm " 1 (i?T3=) (22)
tan
+ tan
i
p 2 +pq+l J '
(23)
the latter of which was known to Euler. The inverse
tangent FORMULAS are connected with many interesting
approximations to Pi
tan _1 (l + x) = \ir + \x - \x 2 + ^z 3 + ±x 5
+ ^ 6 +iT 2 -s 7 + ---- (24)
Euler gave
-i y (1 2-4 2 2-4- 6 3 \ /or ,
tan 1 x=|(-y + — ,* + ^-^ + . . .) , (25)
where
2/ : -
1 + x 2
(26)
The inverse tangent has CONTINUED FRACTION repre-
sentations
tan x •
(27)
1 +
3 +
4z
9x*
5+ T
16x 2
7+-
9 + ...
x
1 +
X
(28)
3 - a: +
9x
5 - 3z +
25aT
7 - 5aT + . . .
Inverse Tangent
To find tan" 1 # numerically, the following ARITHMETIC-
Geometric MEAN-like Algorithm can be used. Let
(29)
(30)
(17)
/-. , 2\-l/2
a = (1 + x ) '
b = 1.
(18)
Then compute
a»+i = \{di + bi)
(19)
bi+i = yai+i6i,
V
and the inverse tangent is given by
tan 1 x = lim — == —
(31)
(32)
(33)
(Acton 1990).
An inverse tangent tan -1 n with integral n is called re-
ducible if it is expressible as a finite sum of the form
tan
-^E/-
'k tan rife ,
(34)
where f k are POSITIVE or NEGATIVE INTEGERS and m
are ilNTEGERS < n. tan" 1 m is reducible IFF all the
PRIME factors of 1 + m 2 occur among the PRIME factors
of 1 4- n 2 for n = 1, . . . , m - 1. A second NECESSARY
and Sufficient condition is that the largest PRIME fac-
tor of 1 + m 2 is less than 2m. Equivalent to the second
condition is the statement that every GREGORY NUM-
BER t x — cot -1 x can be uniquely expressed as a sum
in terms of t m s for which m is a ST0RMER NUMBER
(Conway and Guy 1996). To find this decomposition,
write
arg(l -f in) = arg JJ(1 + n h i) fk , (35)
(36)
so the ratio
n t= i(i+^) /fc
T — ;
1 + in
is a Rational Number. Equation (36) can also be
written
r 2 (l + n 2 ) = IJ(l+n* 2 ) /fc . (37)
k = l
Writing (34) in the form
tan" 1 n = ]P f k tan" 1 n k + / tan" 1 1 (38)
fc=i
allows a direct conversion to a corresponding INVERSE
Cotangent Formula
cot
" 1 n = ^/ fc cot ^fc + ccot x l, (39)
where
c = 2-f-2^f r
k=l
(40)
Inverse Trigonometric Functions
Inversion
923
Todd (1949) gives a table of decompositions of tan -1 n
for n < 342. Conway and Guy (1996) give a similar
table in terms of ST0RMER NUMBERS.
Arndt and Gosper give the remarkable inverse tangent
identity
^2n + l
sin
y ^ tan 1 a,k
(-ir£r=rn-:rh-t-(^)]
2n+ 1
v / rE^
(41)
, 2 + l)
see also Inverse Cotangent, Tangent
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circu-
lar Functions." §4.4 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 79-83, 1972.
Acton, F. S, "The Arctangent." In Numerical Methods
that Work f upd. and rev. Washington, DC: Math. Assoc.
Amer., pp. 6-10, 1990.
Arndt, J. "Completely Useless Formulas." http://www.jjj.
de/hfloat/hf loatpage.html#f ormulas.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Item 137, Feb. 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 142-143, 1987.
Castellanos, D. "Rapidly Converging Expansions with Fi-
bonacci Coefficients." Fib. Quart. 24, 70-82, 1986.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988.
Conway, J. H. and Guy, R. K. "St0rmer's Numbers." The
Book of Numbers. New York: Springer- Verlag, pp. 245-
248, 1996.
Todd, J. "A Problem on Arc Tangent Relations." Amer.
Math. Monthly 56, 517-528, 1949.
Inverse Trigonometric Functions
Inverse Functions of the Trigonometric Func-
tions written cos -1 #, cot -1 z, esc -1 x, sec -1 x, sin -1 x,
and tan -1 x.
see also Inverse Cosecant, Inverse Cosine, In-
verse Cotangent, Inverse Secant, Inverse Sine,
Inverse Tangent
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Inverse Circu-
lar Functions." §4.4 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 79-83, 1972.
Spanier, J. and Oldham, K. B. "Inverse Trigonometric Func-
tions." Ch. 35 in An Atlas of Functions. Washington, DC:
Hemisphere, pp. 331-341, 1987.
Two figures are said to be Similar when all correspond-
ing Angles are equal, and are inversely similar when all
corresponding ANGLES are equal and described in the
opposite rotational sense.
see also DIRECTLY SIMILAR, SIMILAR
Inversion
Inversion is the process of transforming points to their
INVERSE POINTS. This sort of inversion was first sys-
tematically investigated by Jakob Steiner. Two points
are said to be inverses with respect to an Inversion
Circle with Inversion Center O = (x 0j y ) and In-
version Radius k if PT and PS are line segments sym-
metric about OP and tangent to the Circle, and Q is
the intersection of OP and ST. The curve to which a
given curve is transformed under inversion is called its
Inverse Curve.
Note that a point on the Circumference of the In-
version Circle is its own inverse point. The inverse
points obey
k OQ y {l)
OP x OQ,
(2)
where k 2 is called the POWER. The equation for the in-
verse of the point (x,y) relative to the INVERSION CIR-
CLE with Inversion Center (xo,yo) and inversion ra-
dius k is therefore
Xq
yo +
k (x — Xq)
(x - x ) 2 + (y-yo) 2
k 2 (y-yo)
(x - xq) 2 + (y - y ) 2 '
In vector form,
X = x +
fc 2 (x — Xo)
■ Xq
(3)
(4)
(5)
Any Angle inverts to an opposite Angle.
Inversely Similar
inversely similar
924
Inversion
Inversive Distance
Treating Lines as Circles of Infinite Radius, all Cir-
cles invert to CIRCLES. Furthermore, any two nonin-
tersecting circles can be inverted into concentric circles
by taking the INVERSION CENTER at one of the two lim-
iting points (Coxeter 1969), and ORTHOGONAL CIRCLES
invert to ORTHOGONAL CIRCLES (Coxeter 1969).
The inverse of a CIRCLE of RADIUS a with CENTER (x, y)
with respect to an inversion circle with INVERSION CEN-
TER (0, 0) and INVERSION RADIUS k is another CIRCLE
with Center (x f ,y) = (sx,sy) and Radius r' = \s\a,
where
k 2
s = "TX^ a • W
x* -\- y* — a*
The above plot shows a checkerboard centered at (0, 0)
and its inverse about a small circle also centered at (0,
0) (Dixon 1991).
see also Arbelos, Hexlet, Inverse Curve, Inver-
sion Circle, Inversion Operation, Inversion Ra-
dius, Inversive Distance, Inversive Geometry,
Midcircle, Pappus Chain, Peaucellier Inversor,
Polar, Pole (Geometry), Power (Circle), Radi-
cal Line, Steiner Chain, Steiner's Porism
References
Courant, R. and Robbins, H. "Geometrical Transformations.
Inversion." §3.4 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 140-146, 1996.
Coxeter, H. S. M. "Inversion in a Circle" and "Inversion of
Lines and Circles." §6.1 and 6.3 in Introduction to Geom-
etry, 2nd ed. New York: Wiley, p. 77-83, 1969.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 108-114, 1967.
Dixon, R. "Inverse Points and Mid-Circles." §1.6 in Matho-
graphics. New York: Dover, pp. 62-73, 1991.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 43-57, 1929.
Lockwood, E. H. "Inversion." Ch. 23 in A Book of
Curves. Cambridge, England: Cambridge University
Press, pp. 176-181, 1967.
Ogilvy, C S. Excursions in Geometry. New York: Dover,
pp. 25-31, 1990.
# Weisstein, E. W. "Plane Geometry." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/PlaneGeometry.m.
Inversion Center
The point that INVERSION OF A CURVE is performed
with respect to.
see also INVERSE POINTS, INVERSION CIRCLE, INVER-
SION Radius, Inversive Distance, Polar, Pole
(Geometry), Power (Circle)
Inversion Circle
The Circle with respect to which a Inverse Curve
is computed or relative to which INVERSE POINTS are
computed.
see also INVERSE POINTS, INVERSION CENTER, INVER-
SION Radius, Inversive Distance, Midcircle, Po-
lar, Pole (Geometry), Power (Circle)
Inversion Operation
The Symmetry Operation (x t y,z) -» (—x,—y,—z).
When used in conjunction with a ROTATION, it becomes
an Improper Rotation.
Inversion Radius
The Radius used in performing an Inversion with re-
spect to an Inversion Circle.
see also INVERSE POINTS, INVERSION CENTER, IN-
VERSION Circle, Inversive Distance, Polar, Pole
(Geometry), Power (Circle)
Inversive Distance
The inversive distance is the NATURAL LOGARITHM of
the ratio of two concentric circles into which the given
circles can be inverted. Let c be the distance between
the centers of two nonintersecting CIRCLES of Radii a
and b < a. Then the inversive distance is
S = cosh
2 i l2 2
a +b - c
2ab
(Coxeter and Greitzer 1967).
The inversive distance between the SODDY CIRCLES is
given by
S = 2 cosh" 1 2,
Inversive Geometry
Involute 925
and the Circumcircle and Incircle of a Triangle
with CiRCUMRADlUS R and Inradius r are at inversive
distance
5 — 2 sinh"
V2V R)
(Coxeter and Greitzer 1967, pp. 130-131).
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 123-124 and
127-131, 1967.
Inversive Geometry
The Geometry resulting from the application of the
INVERSION operation. It can be especially powerful for
solving apparently difficult problems such as STEINER'S
PORISM and APOLLONIUS' PROBLEM.
see also Hexlet, Inverse Curve, Inversion, Peau-
cellier inversor, polar, pole (geometry),
Power (Circle), Radical Line
References
Ogilvy, C. S. "Inversive Geometry" and "Applications of In-
versive Geometry." Chs. 3 — 4 in Excursions in Geometry.
New York: Dover, pp. 24-55, 1990.
Inverted Funnel
see also Funnel, Sinclair's Soap Film Problem
Inverted Snub Dodecadodecahedron
The Uniform Polyhedron Uq whose Dual Polyhe-
dron is the Medial Inverted Pentagonal Hexe-
CONTAHEDRON. It has WYTHOFF SYMBOL | 2 § 5. Its
faces are 12{§} + 60{3} + 12{5}. It has CiRCUMRADlUS
for unit edge length of
R « 0.8516302.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 180-182, 1989.
Invertible Knot
A knot which can be deformed into itself but with the
orientation reversed. The simplest noninvertible knot is
O8017. No general technique is known for determining
if a Knot is invertible. Burde and Zieschang (1985)
give a tabulation from which it is possible to extract the
invertible knots up to 10 crossings.
see also Amphichiral Knot
References
Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter,
1985.
Involuntary
A Linear Transformation of period two. Since a
Linear Transformation has the form,
A' =
olX +
(1)
7A + (T
applying the transformation a second time gives
„ = a\'+0 = {ct 2 +0 1 )\ + 0{ct + 8)
7 A' + 5 (a + S)jX + 1 + 5 2 ' K }
For an involuntary, A" = A, so
7 (a + 5)X 2 -r (S 2 - a 2 )A - (a + 8)0 = 0. (3)
Since each COEFFICIENT must vanish separately,
cry H- 7<5 =
S 2 - a 2 =
ol(3 + ps = 0.
(4)
(5)
(6)
The first equation gives 6 = ±a. Taking 5 = a would
require 7 = = 0, giving A = A', the identity transfor-
mation. Taking S = —a gives 6 = —a, so
A' =
a\ +
7A — a'
(7)
the general form of an INVOLUTION.
see also Cross-Ratio, Involution (Line)
References
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, pp. 14-
15, 1961.
Involute
Attach a string to a point on a curve. Extend the string
so that it is tangent to the curve at the point of at-
tachment. Then wind the string up, keeping it always
taut. The LOCUS of points traced out by the end of
the string is the involute of the original curve, and the
original curve is called the EvOLUTE of its involute. Al-
though a curve has a unique EvOLUTE, it has infinitely
many involutes corresponding to different choices of ini-
tial point. An involute can also be thought of as any
926
Involute
Irradiation Illusion
curve Orthogonal to all the Tangents to a given
curve.
The equation of the involute is
n = r - sf ,
where T is the TANGENT VECTOR
T =
and s is the Arc Length
dr
dt
I dr I
I dt |
(1)
(2)
(3)
This can be written for a parametrically represented
function (f{t),g(t)) as
x(t) = f-
sf
y(t) = g
Vf' 2 +9' 2
Vf' 2 +9' 2 '
(4)
(5)
Curve
Involute
astroid
cardioid
catenary
circle catacaustic
for a point source
circle
cycloid
deltoid
ellipse
epicycloid
hypocycloid
logarithmic spiral
Neile's parabola
nephroid
nephroid
astroid 1/2 as large
cardioid 3 times as large
tractrix
limacon
circle involute (a spiral)
equal cycloid
deltoid 1/3 as large
ellipse involute
reduced epicycloid
similar hypocycloid
equal logarithmic spiral
parabola
Cayley's sextic
nephroid 2 times as large
see also Evolute, Humbert's Theorem
References
Cundy, H. and Rollett, A. "Roulettes and Involutes." §2.6 in
Mathematical Models, 3rd ed. Stradbroke, England: Tar-
quin Pub., pp. 46-55, 1989.
Dixon, R. "String Drawings." Ch. 2 in Mathographics. New
York: Dover, pp. 75-78, 1991.
Gray, A. "Involutes." §5.4 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 81-85, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 40-42 and 202, 1972.
Lee, X. "Involute." http : //www .best . com/~xah/Special
PlaneCurves_dir/Involute_dir/involute.html.
Lockwood, E. H. "Evolutes and Involutes." Ch. 21 in A Book
of Curves. Cambridge, England: Cambridge University
Press, pp. 166-171, 1967.
Pappas, T. "The Involute." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, p. 187, 1989.
Yates, R. C. "Involutes." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 135-137,
1952.
Involution (Group)
An element of order 2 in a Group (i.e., an element A
of a Group such that A 2 = /, where I is the Identity
Element),
see also Group, Identity Element
Involution (Line)
Pairs of points of a line, the product of whose distances
from a Fixed POINT is a given constant. This is more
concisely denned as a PROJECTIVITY of period two.
see also INVOLUNTARY
Involution (Operator)
An Operator of period 2, i.e., an Operator * which
satisfies ((a)*)* = a.
Involution (Set)
An involution of a Set S is a PERMUTATION of S which
does not contain any cycles of length > 2. The PER-
MUTATION Matrices of an involution are Symmetric.
The number of involutions I(n) of a Set containing the
first n integers is given by the RECURRENCE RELATION
I(n) = I(n - 1) + (n - l)I(n - 2).
For n = 1, 2, . . . , the first few values of I{n) are 1, 2,
4, 10, 26, 76, ... (Sloane's A000085). The number of
involutions on n symbols cannot be expressed as a fixed
number of hypergeometric terms (Petkovsek et al. 1996,
p. 160).
see also PERMUTATION
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, 1996.
Ruskey, F. "Information on Involutions." http: //sue . esc .
uvic. ca/ -cos /inf /perm/ Involutions. html.
Sloane, N. J. A. Sequence A00085/M1221 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Involution (Transformation)
A TRANSFORMATION of period 2.
Irradiation Illusion
The Illusion shown above which was discovered by
Helmholtz in the 19th century. Despite the fact that
the two above figures are identical in size, the white
hole looks bigger than the black one in this ILLUSION.
References
Pappas, T. "Irradiation Optical Illusion. " The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, p. 199,
1989.
Irrational Number
Irreducible Matrix 927
Irrational Number
A number which cannot be expressed as a FRACTION pjq
for any INTEGERS p and q. Every TRANSCENDENTAL
Number is irrational. Numbers of the form n 1 '™ are
irrational unless n is the rath POWER of an INTEGER.
Numbers of the form log n ra, where log is the LOGA-
RITHM, are irrational if m and n are INTEGERS, one of
which has a PRIME factor which the other lacks. e r is
irrational for rational r/0. The irrationality of e was
proven by Lambert in 1761; for the general case, see
Hardy and Wright (1979, p. 46). n n is irrational for
Positive integral n. The irrationality of tv was proven
by Lambert in 1760; for the general case, see Hardy and
Wright (1979, p. 47). Apery's Constant C(3) (where
C(z) is the RlEMANN ZETA FUNCTION) was proved irra-
tional by Apery (Apery 1979, van der Poorten 1979).
Prom GELFOND'S THEOREM, a number of the form a b
is Transcendental (and therefore irrational) if a is
Algebraic ^ 0, 1 and b is irrational and Algebraic.
This establishes the irrationality of e" (since (— l)~ l =
(e* w )"* = e T ), 2^, and eir. Nesterenko (1996) proved
that 7r + e* is irrational. In fact, he proved that it, e n
and r(l/4) are algebraically independent, but it was not
previously known that n + e n was irrational.
Given a POLYNOMIAL equation
X + Crr
-!X
+ . . . + c ,
(1)
where Ci are INTEGERS, the roots Xi are either integral
or irrational. If cos(2#) is irrational, then so are cos#,
sin0, and tan 5.
Irrationality has not yet been established for 2 e , 7r e , 7r ,
or 7 (where 7 is the EULER-MASCHERONI CONSTANT).
Quadratic Surds are irrational numbers which have
periodic CONTINUED FRACTIONS.
Hurwitz's Irrational Number Theorem gives
bounds of the form
P
<
L n q*
(2)
for the best rational approximation possible for an ar-
bitrary irrational number a, where the L n are called
Lagrange Numbers and get steadily larger for each
"bad" set of irrational numbers which is excluded.
The Series
E
(Tk(n)
(3)
where <7 k (n) is the DIVISOR FUNCTION, is irrational for
k — 1 and 2. The series
El ^-^ d(n)
2 n — 1 — ^-^ 2 n
(4)
where d(n) is the number of divisors of n, is also irra-
tional, as are
q n +r
(-i) n
q n +r
(5)
for q an INTEGER other than p, ±1, and r a RATIONAL
NUMBER other than or -q n (Guy 1994).
see also ALGEBRAIC INTEGER, ALGEBRAIC NUMBER,
Almost Integer, Dirichlet Function, Ferguson-
Forcade Algorithm, Gelfond's Theorem, Hur-
witz's Irrational Number Theorem, Near Noble
Number, Noble Number, Pythagoras's Theorem,
Quadratic Irrational Number, Rational Num-
ber, Segre's Theorem, Transcendental Number
References
Apery, R. "Irrationalite de ((2) et C(3)." Asterisque 61, 11-
13, 1979.
Courant, R. and Robbins, H. "Incommensurable Segments,
Irrational Numbers, and the Concept of Limit." §2.2 in
What is Mathematics?: An Elementary Approach to Ideas
and Methods, 2nd ed. Oxford, England: Oxford University
Press, pp. 58-61, 1996.
Guy, R. K. "Some Irrational Series." §B14 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, p. 69, 1994.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.
Manning, H. P. Irrational Numbers and Their Representa-
tion by Sequences and Series. New York: Wiley, 1906.
Nesterenko, Yu. "Modular Functions and Transcendence
Problems." C. R. Acad. Sci. Paris Ser. I Math. 322,
909-914, 1996.
Nesterenko, Yu. V. "Modular Functions and Transcendence
Questions." Mat. Sb. 187, 65-96, 1996.
Niven, I. M. Irrational Numbers. New York: Wiley, 1956.
Niven, I. M. Numbers: Rational and Irrational. New York:
Random House, 1961.
Pappas, T. "Irrational Numbers & the Pythagoras Theorem."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, pp. 98-99, 1989.
van der Poorten, A. "A Proof that Euler Missed. . . Apery's
Proof of the Irrationality of C(3)." Math. Intel. 1,196-203,
1979.
Irrationality Measure
see LlOUVILLE-ROTH CONSTANT
Irrationality Sequence
A sequence of POSITIVE INTEGERS {a n } such that
5^1/(a n 6n) is Irrational for all integer sequences
{&n}. Erdos showed that {2 2 } is an irrationality se-
quence.
References
Guy, R. K. "Irrationality Sequence." §E24 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, p. 225, 1994.
Irreducible Matrix
A Square Matrix which is not Reducible is said to
be irreducible.
928 Irreducible Polynomial
Irregular Pair
Irreducible Polynomial
A Polynomial or polynomial equation is said to be
irreducible if it cannot be factored into polynomials of
lower degree over the same Field.
The number of binary irreducible polynomials of degree
n is equal to the number of n-bead fixed NECKLACES
of two colors: 1, 2, 3, 4, 6, 8, 14, 20, 36, ... (Sloane's
A000031), the first few of which are given in the follow-
ing table.
n Polynomials
1 x
2 x,x+l
3 x, x 2 + x + 1, x + 1
4 x, x 3 + x + 1, x 3 + x 2 + 1, x + 1
see also FIELD, GALOIS FIELD, NECKLACE, POLYNOM-
IAL, Primitive Irreducible Polynomial
References
Sloane, N. J. A. Sequences A000031/M0564 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.
Irreducible Representation
An irreducible representation of a GROUP is a represen-
tation for which there exists no UNITARY TRANSFORMA-
TION which will transform the representation MATRIX
into block diagonal form. The irreducible representa-
tion has a number of remarkable properties.
see also Group, Ito's Theorem, Unitary Transfor-
mation
Irreducible Semiperfect Number
see Primitive Pseudoperfect Number
Irreducible Tensor
Given a general second Rank Tensor Aij and a Met-
ric gij, define
6 = A ij9 ij = A\
u/ = e ijk A jk
(1)
(2)
(3)
where 5ij is the KRONECKER Delta and e ijk is the
Levi-Civita Symbol. Then
mi + \8gij + \tijku
= [i(A y + Aji) - \ gi jA k k ] + \A k k9ij + \u jk [e^ k A^]
= i(A y + A,.) + i(tf *? - 8?8})A Xll
— 2 V^ij * Aji) + 2 [Aij Aji) — Aij,
(4)
Irredundant Ramsey Number
Let Gi, G 2 , . . . , G t be a t-EDGE coloring of the Com-
plete GRAPH K n , where for each i = 1, 2, . . . , t, G» is
the spanning SUBGRAPH of K n consisting of all EDGES
colored with the ith. color. The irredundant Ramsey
number s(<?i, • ■ ■ ,<ft) is the smallest INTEGER n such
that for any t-EDGE coloring of K n , the Complement
Graph Gi has an irredundant set of size qi for at least
one i = 1, . . . , t. Irredundant Ramsey numbers were
introduced by Brewster et al. (1989) and satisfy
s(qu.-.,qt) < Ji(gi,...,gt)-
For a summary, see Mynhardt (1992).
s
Bounds
Reference
*(3,3)
6
Brewster et al 1989
5(3,4)
8
Brewster et al 1989
S (3,5)
12
Brewster et al. 1989
S (3,6)
15
Brewster et al 1990
S (3,7)
18
Chen and Rousseau 1995,
Cockayne et al 1991
5(4,4)
13
Cockayne et al 1992
5(3,3,3)
13
Cockayne and Mynhardt 1994
where 0, u;*, and &ij are Tensors of Rank 0, 1, and 2.
see also TENSOR
References
Brewster, R. C; Cockayne, E. J.; and Mynhardt, C. M. "Irre-
dundant Ramsey Numbers for Graphs." J. Graph Theory
13, 283-290, 1989.
Brewster, R. C; Cockayne, E. J.; and Mynhardt, C. M. "The
Irredundant Ramsey Number 5(3,6)." Quaest. Math. 13,
141-157, 1990.
Chen, G. and Rousseau, C. C. "The Irredundant Ramsey
Number a(3,7).» J. Graph. Th. 19, 263-270, 1995.
Cockayne, E. J.; Exoo, G.; Hattingh, J. H.; and Mynhardt,
C. M. "The Irredundant Ramsey Number 5(4,4)." Util.
Math. 41, 119-128, 1992.
Cockayne, E. J.; Hattingh, J. H.; and Mynhardt, C. M. "The
Irredundant Ramsey Number 5(3,7)." Util Math. 39,
145-160, 1991.
Cockayne, E. J. and Mynhardt, C. M. "The Irredundant
Ramsey Number 5(3,3,3) = 13." J. Graph. Th. 18, 595-
604, 1994.
Hattingh, J. H. "On Irredundant Ramsey Numbers for
Graphs." J. Graph Th. 14, 437-441, 1990.
Mynhardt, C. M. "Irredundant Ramsey Numbers for Graphs:
A Survey." Congres. Numer. 86, 65-79, 1992.
Irreflexive
A Relation R on a Set S is irreflexive provided that
no element is related to itself; in other words, xRx for
no x in S.
see also RELATION
Irregular Pair
If p divides the NUMERATOR of the BERNOULLI NUMBER
£?2fc for < 2k < p — 1, then (p, 2k) is called an irregular
pair. For p < 30000, the irregular pairs of various forms
are p — 16843 for (p,p — 3), p = 37 for (p,p — 5), none
for (p,p - 7), and p = 67, 877 for (p,p- 9).
see also Bernoulli Number, Irregular Prime
References
Johnson, W. "Irregular Primes and Cyclotomic Invariants."
Math. Comput. 29, 113-120, 1975.
Irregular Prime
ISBN 929
Irregular Prime
Primes for which Rummer's theorem on the unsolvabil-
ity of Fermat's Last Theorem does not apply. An
irregular prime p divides the NUMERATOR of one of the
Bernoulli Numbers Bio, B12, . .., B 2p -2, as shown
by Kummer in 1850. The Fermat EQUATION has no
solutions for Regular Primes.
20 40 60 80 100 120
Number of Irregular Primes
An Infinite number of irregular Primes exist, as
proven in 1915 by Jensen. The first few irregular primes
are 37, 59, 67, 101, 103, 131, 149, 157, ... (Sloane's
A000928). Of the 283,145 Primes less than 4 x 10 6 ,
111,597 (or 39.41%) are regular. The conjectured FRAC-
TION is 1 - e _1/2 w 39.35% (Ribenboim 1996, p. 415).
see also Bernoulli Number, Fermat's Last Theo-
rem, Irregular Pair, Regular Prime
References
Buhler, J.; Crandall, R.; Ernvall, R.; and Metsankyla, T. "Ir-
regular Primes and Cyclotomic Invariants to Four Million."
Math. Comput. 60, 151-153, 1993.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, p. 202, 1979.
Johnson, W. "Irregular Primes and Cyclotomic Invariants."
Math. Comput 29, 113-120, 1975.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 325-329 and 414-425,
1996.
Sloane, N. J. A. Sequence A000928/M5260 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Stewart, C. L. "A Note on the Fermat Equation." Mathe-
matika 24, 130-132, 1977.
Irregular Singularity
Consider a second-order Ordinary Differential
Equation
y" + P(x)y' + Q(x)y = 0.
If P(x) and Q(x) remain FINITE at a; = xq, then
x is called an ORDINARY POINT. If either P(x) or
Q(x) diverges as x -» #o, then xq is called a singular
ppint. If P(x) diverges more quickly than l/(x — xo),
so (x — xo)P(x) approaches Infinity as x -> xq, or
Q(x) diverges more quickly than l/(x — Xq) 2 Q so that
(x — xq) 2 Q(x) goes to Infinity as x — > xo, then xo is
called an IRREGULAR SINGULARITY (or ESSENTIAL SIN-
GULARITY).
see also ORDINARY POINT, REGULAR SINGULAR POINT,
Singular Point (Differential Equation)
References
Arfken, G. "Singular Points." §8.4 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 451-453 and 461-463, 1985.
Irrotational Field
A Vector Field v for which the Curl vanishes,
V x v = 0.
see also BELTRAMI FlELD, CONSERVATIVE FIELD,
Solenoidal Field, Vector Field
Isarithm
see Equipotential Curve
ISBN
Publisher
Digits
Addison-Wesley 0201
Amer. Math. Soc. 0821
Cambridge University Press 0521
CRC Press 0849
Dover 0486
McGraw-Hill 0070
Oxford University Press 0198
Springer- Verlag 0387
Wiley 0471
The International Standard Book Number (ISBN) is a
10-digit Code which is used to identify a book uniquely.
The first four digits specify the publisher, the next five
digits the book, and the last digit dio is a check digit
which may be in the range 0-9 or X (where X equals
10). The check digit is computed from the equation
10di + 9d 2 + 8d 3 + • • ■ + 2cfo + dio = (mod 11) .
For example, the number for this book is 0-8493-9640-9,
and
10-0 + 9-8 + 8-4 + 7'9 + 6-3 + 5*9
+4-6 + 3-4 + 2-0 + l-9 = 275 = 25-ll = (mod 11),
as required.
see also Code
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. 894, 1992.
930 Island
Island
Isodynamic Points
If an integrable QUASIPERIODIC system is slightly per-
turbed so that it becomes nonintegrable, only a finite
number of n-CYCLES remain as a result of MODE LOCK-
ING. One will be elliptical and one will be hyperbolic.
Surrounding the Elliptic Fixed Point is a region of
stable Orbits which circle it, as illustrated above in the
Standard Map with K = 1.5. As the map is iteratively
applied, the island is mapped to a similar structure sur-
rounding the next point of the elliptic cycle. The map
thus has a chain of islands, with the Fixed Point alter-
nating between ELLIPTIC (at the center of the islands)
and HYPERBOLIC (between islands). Because the un-
perturbed system goes through an INFINITY of rational
values, the perturbed system must have an Infinite
number of island chains.
see also Mode Locking, Orbit (Map), Quasiperi-
odic Function
Isobaric Polynomial
A Polynomial in which the sum of Subscripts is the
same in each term.
see also Homogeneous Polynomial
Isochronous Curve
see Semicubical Parabola, Tautochrone Prob-
lem
Isoclinal
see Isocline
Isocline
A graphical method of solving an Ordinary Differ-
ential Equation of the form
dy
dx
f(x,y)
by plotting a series of curves f(x,y) = [const], then
drawing a curve Perpendicular to each curve such
that it satisfies the initial condition. This curve is the
solution to the Ordinary Differential Equation.
References
Karman, T. von and Biot, M. A. Mathematical Methods in
Engineering: An Introduction to the Mathematical Treat-
ment of Engineering Problems. New York: McGraw-Hill,
pp. 3 and 7, 1940.
Isoclinic Groups
Two GROUPS G and H are said to be isoclinic if there
are isomorphisms G/Z(G) -> H/Z(H) and G" -> H\
where Z(G) is the Center of the group, which identify
the two commutator maps.
References
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.;
and Wilson, R. A. "Isoclinism." §6.7 in Atlas of Fi-
nite Groups: Maximal Subgroups and Ordinary Characters
for Simple Groups. Oxford, England: Clarendon Press,
pp. xxiii-xxiv, 1985.
Isodynamic Points
The first and second isodynamic points of a TRIANGLE
AABG can be constructed by drawing the triangle's
Angle Bisectors and Exterior Angle Bisectors.
Each pair of bisectors intersects a side of the triangle
(or its extension) in two points Da and Di2, for i — 1,
2, 3. The three CIRCLES having DuD 12 , D21D22, and
D 31 D 32 as Diameters are the Apollonius Circles
Ci, C2, and C3. The points S and S" in which the three
Apollonius Circles intersect are the first and second
isodynamic points, respectively.
S and S' have TRIANGLE CENTER FUNCTIONS
a = sin(^4 ± |7r),
respectively. The Antipedal Triangles of both
points are Equilateral and have Areas
A' = 2A[cotwcot(|7r)],
where w is the Brocard Angle.
The isodynamic points are ISOGONAL CONJUGATES of
the Isogonic Centers. They lie on the Brocard
Axis. The distances from either isodynamic point to
the Vertices are inversely proportional to the sides.
The Pedal Triangle of either isodynamic point is an
Equilateral Triangle. An Inversion with either
Isoenergetic Nondegeneracy
Isogonal Line 931
isodynamic point as the INVERSION CENTER transforms
the triangle into an Equilateral Triangle.
The CIRCLE which passes through both the isodynamic
points and the Centroid of a TRIANGLE is known as
the Parry Circle.
see also Apollonius Circles, Brocard Axis, Cen-
troid (Triangle), Isogonic Centers, Parry Cir-
cle
References
Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.
London: Hodgson, p. 106, 1913.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 295-297, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Isoenergetic Nondegeneracy
The condition for isoenergetic nondegeneracy for a Ham-
iltonian
H = H (I) + eH 1 (1,0)
d 2 H
diidij
dli
dH Q
dli
/0,
which guarantees the EXISTENCE on every energy level
surface of a set of invariant tori whose complement has
a small MEASURE.
References
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, pp. 113-114, 1989.
Isogonal Conjugate
The isogonal conjugate X^ 1 of a point X in the plane of
the Triangle AABC is constructed by reflecting the
lines AX, BX, and CX about the Angle Bisectors
at A, B, and C. The three reflected lines CONCUR at
the isogonal conjugate. The Trilinear Coordinates
of the isogonal conjugate of the point with coordinates
a : P : 7
- 1 :/?- 1 :
7
Sections that Circumscribe the Triangle. The type
of Conic Section is determined by whether the line d
meets the ClRCUMClRCLE C",
1. If d does not intersect <7\ the isogonal transform is
an Ellipse;
2. If d is tangent to C', the transform is a PARABOLA;
3. If d cuts C, the transform is a HYPERBOLA, which
is a Rectangular Hyperbola if the line passes
through the ClRCUMCENTER
(Casey 1893, Vandeghen 1965).
The isogonal conjugate of a point on the ClRCUMClRCLE
is a POINT AT INFINITY (and conversely). The sides of
the Pedal Triangle of a point are Perpendicular to
the connectors of the corresponding VERTICES with the
isogonal conjugate. The isogonal conjugate of a set of
points is the LOCUS of their isogonal conjugate points.
The product of ISO TO MIC and isogonal conjugation is a
Collineation which transforms the sides of a Trian-
gle to themselves (Vandeghen 1965).
see also Antipedal Triangle, Collineation, Iso-
gonal Line, Isotomic Conjugate Point, Line at
Infinity, Symmedian Line
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 rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 153-158, 1929.
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.
Isogonal Line
angle bisector
Isogonal conjugation maps the interior of a Triangle
onto itself. This mapping transforms lines onto Conic
A B
The line L' through a TRIANGLE VERTEX obtained by
reflecting an initial line L (also through a VERTEX)
about the Angle Bisector. If three lines from the
Vertices of a Triangle AABC are Concurrent at
X = L1L2L3, then their isogonal lines are also Con-
current, and the point of concurrence X' — L^L^L^
is called the ISOGONAL CONJUGATE point.
see also ISOGONAL CONJUGATE
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 153-157, 1929.
932 Isogonic Centers
Isometry
Isogonic Centers
The first isogonic center Fi of a TRIANGLE is the Fer-
MAT Point. The second isogonic center F 2 is con-
structed analogously with the first isogonic center ex-
cept that for F 2 , the EQUILATERAL TRIANGLES are con-
structed on the same side of the opposite Vertex. The
second isogonic center has Triangle Center Func-
tion
a = csc(^4 — g7r).
Its Antipedal Triangle is Equilateral and has
Area
2A = [-1 + cotwcot(|7r)],
where u> is the BrOCARD Angle.
The first and second isogonic centers are ISOGONAL
Conjugates of the Isodynamic Points.
see also Brocard Angle, Equilateral Triangle,
Fermat Point, Isodynamic Points, Isogonal Con-
jugate
References
Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.
London: Hodgson, p. 107, 1913.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Isograph
The substitution of re id for z in a POLYNOMIAL p(z).
p(z) is then plotted as a function of 9 for a given r in
the Complex Plane. By varying r so that the curve
passes through the Origin, it is possible to determine a
value for one Root of the Polynomial.
Isohedral Tiling
Let S(T) be the group of symmetries which map a
Monohedral Tiling T onto itself. The Transitiv-
ity Class of a given tile T is then the collection of all
tiles to which T can be mapped by one of the symmetries
of S(T). If T has k Transitivity Classes, then T is
said to be &-isohedral. Berglund (1993) gives examples
of fc-isohedral tilings for k = 1, 2, and 4.
see also ANISOHEDRAL 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.
Isohedron
A convex POLYHEDRON with symmetries acting transi-
tively on its faces. Every isohedron has an Even number
of faces (Griinbaum 1960).
References
Griinbaum, B. "On Polyhedra in £ 3 Having All Faces Con-
gruent." Bull. Research Council Israel 8F, 215-218, 1960.
Griinbaum, B. and Shepard, G. C. "Spherical Tilings with
Transitivity Properties." In The Geometric Vein: The
Coxeter Festschrift (Ed. C. Davis, B. Griinbaum, and
F. Shenk). New York: Springer- Verlag, 1982.
Isolated Point
A point on a curve, also known as an Acnode or Her-
mit Point, which has no other points in its NEIGHBOR-
HOOD.
Isolated Singularity
An isolated singularity is a SINGULARITY for which there
exists a (small) Real NUMBER e such that there are no
other Singularities within a Neighborhood of radius
e centered about the SINGULARITY.
The types of isolated singularities possible for CUBIC
SURFACES have been classified (Schlafli 1864, Cayley
1869, Bruce and Wall 1979) and are summarized in the
following table from Fischer (1986).
Double Pt.
Symbol
Normal Form
Coxeter
Name
Diagram
conic
c 2
2,2,2
x +y +z
Ai
biplanar
B 3
2 , 2,3
x + y + 2
A2
biplanar
B 4
x 2 + y 2 +z 4
A 3
biplanar
B 5
x 2 -r y 2 + z 5
A 4
biplanar
Be
x 2 + y 2 + z 6
A s
uniplanar
U G
x 2 A-z(y 2 +z 2 )
D 4
uniplanar
u 7
x 2 + z(y 2 -r z 3 )
D s
uniplanar
Us
x 2 +y z +z A
E 6
elliptic cone pt.
—
xy 2 - Az z
-g 2 x 2 y + g 3 x s
E&
see also CUBIC SURFACE, RATIONAL DOUBLE POINT,
Singularity
References
Bruce, J. and Wall, C. T. C. "On the Classification of Cubic
Surfaces." J. London Math. Soc. 19, 245-256, 1979.
Cayley, A. "A Memoir on Cubic Surfaces." Phil. Trans. Roy.
Soc. 159, 231-326, 1869.
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 12-13, 1986.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 380-381, 1953.
Schlafli, L. "On the Distribution of Surfaces of Third Order
into Species." Phil. Trans. Roy. Soc. 153, 193-247, 1864.
Isolating Integral
An integral of motion which restricts the Phase SPACE
available to a Dynamical System.
Isometry
A Bijective Map between two Metric Spaces that
preserves distances, i.e.,
d{f(x)J(y)) = d(x,y),
where / is the MAP and d(a,b) is the DISTANCE func-
tion.
An isometry of the Plane is a linear transformation
which preserves length. Isometries include ROTATION,
Translation, Reflection, Glides, and the Iden-
tity Map. If an isometry has more than one FIXED
Isometric Latitude
Isoperimetric Inequality 933
POINT, it must be either the identity transformation or
a reflection. Every isometry of period two (two appli-
cations of the transformation preserving lengths in the
original configuration) is either a reflection or a half turn
rotation. Every isometry in the plane is the product of
at most three reflections (at most two if there is a Fixed
Point). Every finite group of isometries has at least one
Fixed Point.
see also Distance, Euclidean Motion, Hjelmslev's
Theorem, Length (Curve), Reflection, Rota-
tion, Translation
References
Gray, A. "Isometries of Surfaces." §13.2 in Modern Differen-
tial Geometry of Curves and Surfaces. Boca Raton, FL:
CRC Press, pp. 255-258, 1993.
Isometric Latitude
An Auxiliary Latitude which is directly proportional
to the spacing of parallels of Latitude from the equator
on an ellipsoidal Mercator Projection. It is defined
by
e sm <
ip = In
tan(
**+§*)(r
+ e sin 4>
e/2
(1)
where the symbol r is sometimes used instead of iff. The
isometric latitude is related to the Conformal Lati-
tude by
ip = lntan(^7r+ \x).
The inverse is found by iterating
(2)
<j)~2 tan
exp(<0)
( 1 + e sin (j)
y 1 — e sin
e/2
|7T,
with the first trial as
2tan~ 1 (e^)- §tt.
(3)
(4)
see also Latitude
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. 15, 1987.
Isomorphic Groups
Two GROUPS are isomorphic if the correspondence be-
tween them is One-to-One and the "multiplication"
table is preserved. For example, the Point Groups C 2
and D\ are isomorphic GROUPS, written C^ = P>\ or
C 2 ^ Di (Shanks 1993). Note that the symbol .9* is
also used to denote geometric CONGRUENCE.
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, 1993.
Isomorphic Posets
Two POSETS are said to be isomorphic if their "struc-
tures" are entirely analogous. Formally, POSETS P =
(X, <) and Q = (X\ <') are isomorphic if there is a
BlJECTION / from X to X' such that x < x l precisely
when f(x) <' f(x).
Isomorphism
Isomorphism is a very general concept which appears in
several areas of mathematics. Formally, an isomorphism
is BlJECTlVE MORPHISM. Informally, an isomorphism
is a map which preserves sets and relations among ele-
ments.
A space isomorphism is a VECTOR SPACE in which ad-
dition and scalar multiplication are preserved. An iso-
morphism of a Topological Space is called a Home-
OMORPHISM.
Two groups Gx and Gz with binary operators + and x
are isomorphic if there exists a map f : Gi *-^ G2 which
satisfies
f(x + y) = f(x)xf(y).
An isomorphism preserves the identities and inverses of
a GROUP. A GROUP which is isomorphic to itself is
called an Automorphism.
see also AUTOMORPHISM, AX-KOCHEN ISOMORPHISM
Theorem, Homeomorphism, Morphism
Isoperimetric Inequality
Let a PLANE figure have AREA A and PERIMETER p.
Let the Circle of Perimeter p have Radius r. Then
4ttA
<1,
Isomorphic Graphs
Two GRAPHS which contain the same number of Ver-
tices connected in the same way are said to be isomor-
phic. Formally, 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).
References
Chartrand, G. "Isomorphic Graphs." §2.2 in Introductory
Graph Theory. New York: Dover, pp. 32-40, 1985.
where the quantity on the left is known as the ISOPERI-
METRIC Quotient.
934 Isoperimetric Point
Isosceles Tetrahedron
Isoperimetric Point
The point S' which makes the Perimeters of the TRI-
ANGLES ABS'C, ACS' A, and AAS'B equal. The
isoperimetric point exists Iff the largest Angle of the
triangle satisfies
max(A, B, C) < 2sin~ 1 (f) « 1.85459 rad « 106.26°,
or equivalently
a + b + c>4R-\-r,
where a, 6, and c are the side lengths of AABC, r is the
INRADIUS, and R is the ClRCUMRADlUS. The isoperi-
metric point is also the center of the outer SODDY CIR-
CLE of AABC and has Triangle Center Function
ot = 1-
2A
a(b + c~ a)
= sec(^A) cos(±B) cos(±C) - 1.
see also Equal Detour Point, Perimeter, Soddy
Circles
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C "Isoperimetric Point and Equal Detour
Point." http://www.evansville.edu/~ck6/tcenters/
recent/isoper.html.
Kimberling, C and Wagner, R. W. "Problem E 3020 and
Solution." Amer. Math. Monthly 93, 650-652, 1986.
Veldkamp, G. R. "The Isoperimetric Point and the Point(s) of
Equal Detour." Amer. Math. Monthly 92, 546-558, 1985.
Isoperimetric Problem
Find a closed plane curve of a given length which en-
closes the greatest AREA. The solution is a CIRCLE. If
the class of curves to be considered is limited to smooth
curves, the isoperimetric problem can be stated symbol-
ically as follows: find an arc with parametric equations
x = x(t)j y — y(t) for t € [ii,r- 2 ] such that x(ti) — #(£ 2 ),
y(ti) — 2/(^2) (where no further intersections occur) con-
strained by
such that
x' 2 + y' 2 dt
(xy - x'y)dt
is a Maximum.
see also DlDO'S PROBLEM, ISOVOLUME PROBLEM
References
Bogomolny, A. "Isoperimetric Theorem and
Inequality." http : //www . cut-the-knot . com/do _you_know/
isoperimetric.html.
Isenberg, C. Appendix V in The Science of Soap Films and
Soap Bubbles. New York: Dover, 1992.
Isoperimetric Quotient
A quantity defined in the ISOPERIMETRIC INEQUALITY
p 2
see also Isoperimetric Inequality
Isoperimetric Theorem
Of all convex n-gons of a given PERIMETER, the one
which maximizes Area is the regular n-gon.
see also Isoperimetric Inequality, Isoperimetric
Problem
Isopleth
see Equipotential Curve
Isoptic Curve
For a given curve C, consider the locus of the point P
from where the TANGENTS from P to C meet at a fixed
given Angle. This is called an isoptic curve of the given
curve.
Curve
Isoptic
cycloid
epicycloid
hypocycloid
parabola
curtate or prolate cycloid
epitrochoid
hypo trochoid
hyperbola
New
sinusoidal spiral sinusoidal spiral
see also Orthoptic Curve
References
Lawrence, J. D. A Catalog of Special Plane Curves.
York: Dover, pp. 58-59 and 206, 1972.
Yates, R. C. "Isoptic Curves." A Handbook on Curves and
Their Properties. Ann Arbor, Ml: J. W. Edwards, pp. 138-
140, 1952.
Isosceles Tetrahedron
A nonregular Tetrahedron in which each pair of op-
posite EDGES are equal such that all triangular faces are
congruent. A Tetrahedron is isosceles Iff the sum of
the face angles at each VERTEX is 180°, and IFF its In-
SPHERE and ClRCUMSPHERE are concentric.
The only way for all the faces of a TETRAHEDRON to
have the same Perimeter or to have the same AREA is
for them to be fully congruent, in which case the tetra-
hedron is isosceles.
Isosceles Triangle
Isospectral Manifolds 935
see also Circumsphere, Insphere, Isosceles Trian-
gle, Tetrahedron
References
Brown,, B. H. "Theorem of Bang. Isosceles Tetrahedra."
Amer. Math. Monthly 33, 224-226, 1926.
Honsberger, R. "A Theorem of Bang and the Isosceles Tet-
rahedron." Ch. 9 in Mathematical Gems II. Washington,
DC: Math. Assoc. Amer., pp. 90-97, 1976.
Isosceles Triangle
Isoscelizer
A Triangle with two equal sides (and two equal An-
gles). The name derives from the Greek iso (same) and
skelos (Leg). The height of the above isosceles triangle
can be found from the PYTHAGOREAN THEOREM as
The Area is therefore given by
A = \ah = \aJb 2 - \a 2 .
(1)
(2)
There is a surprisingly simple relationship between the
Area and Vertex Angle 9. As shown in the above
diagram, simple TRIGONOMETRY gives
h = Rcos(l9)
a = i2sin(§0),
(3)
(4)
so the Area is
A= \(2a)h = ah = R 2 cos(|0)sin(§0) = ±R 2 sin6.
(5)
No set of n > 6 points in the PLANE can determine only
Isosceles Triangles.
see also Acute Triangle, Equilateral Triangle,
Internal Bisectors Problem, Isosceles Tetrahe-
dron, Isoscelizer, Obtuse Triangle, Point Pick-
ing, Pons Asinorum, Right Triangle, Scalene
Triangle, Steiner-Lehmus Theorem
An isoscelizer of an Angle A in a Triangle A ABC
is a Line Segment IabIac where Iab lies on AB and
I ac on AC such that AAIabIac is an ISOSCELES TRI-
ANGLE.
see also CONGRUENT ISOSCELIZERS POINT, ISOSCELES
Triangle, Yff Center of Congruence
Isospectral Manifolds
\<^t,
DRUMS that sound the same, i.e., have the same eigen-
frequency spectrum. Two drums with differing AREA,
Perimeter, or Genus can always be distinguished.
However, Kac (1966) asked if it was possible to construct
differently shaped drums which have the same eigenfre-
quency spectrum. This question was answered in the
affirmative by Gordon et aL (1992). Two such isospec-
tral manifolds are shown in the right figure above (Cipra
1992).
References
Chapman, S. J. "Drums That Sound the Same." Amer.
Math. Monthly 102, 124-138, 1995.
Cipra, B. "You Can't Hear the Shape of a Drum." Science
255, 1642-1643, 1992.
Gordon, C; Webb, D.; and Wolpert, S. "Isospectral Plane
Domains and Surfaces via Riemannian Orbifolds." Invent.
Math. 110, 1-22, 1992.
Gordon, C; Webb, D.; and Wolpert, S. "You Cannot Hear
the Shape of a Drum." Bull. Amer. Math. Soc. 27, 134-
138, 1992.
Kac, M. "Can One Hear the Shape of a Drum?" Amer. Math.
Monthly 73, 1-23, 1966.
936
Isothermal Parameterization
Isotopy
Isothermal Parameterization
A parameterization is isothermal if, for £ = u + iv and
, ,.v dxk .dx k
the identity
0i 2 (C) + ^2 2 (C) + ^3 2 (O = o
holds.
see a/50 Minimal Surface, Temperature
Isotomic Conjugate Point
The point of concurrence Q of the ISOTOMIC Lines rel-
ative to a point P. The isotomic conjugate a' : $ : 7'
of a point with Trilinear Coordinates a : @ : 7 is
(a 2 a)" 1 : (fc 2 /?)' 1 : (c 2 7 )"
(1)
The isotomic conjugate of a LINE d having trilinear
equation
la + m/3 + U7 (2)
is a Conic Section circumscribed on the Triangle
AABC (Casey 1893, Vandeghen 1965). The isotomic
conjugate of the Line at Infinity having trilinear equa-
tion
aa. + b(3 + C7 = (3)
is Steiner's Ellipse
o f ^ f f ^ ' 'of
a b c
(4)
(Vandeghen 1965). The type of Conic Section to
which d is transformed is determined by whether the
line d meets Steiner's ELLIPSE E.
1. If d does not intersect E, the isotomic transform is
an Ellipse.
2. If d is tangent to £7, the transform is a PARABOLA.
3. If d cuts E, the transform is a Hyperbola, which
is a Rectangular Hyperbola if the line passes
through the isotomic conjugate of the Orthocen-
ter
(Casey 1893, Vandeghen 1965).
There are four points which are isotomically self-
conjugate: the CENTROID M and each of the points
of intersection of lines through the VERTICES PARAL-
LEL to the opposite sides. The isotomic conjugate of the
Euler Line is called Jerabek's Hyperbola (Casey
1893, Vandeghen 1965).
Vandeghen (1965) calls the transformation taking points
to their isotomic conjugate points the CEVIAN TRANS-
FORM. The product of isotomic and ISOGONAL is a
Collineation which transforms the sides of a Trian-
gle to themselves (Vandeghen 1965).
see also Cevian Transform, Gergonne Point, Iso-
gonal Conjugate, Jerabek's Hyperbola, Nagel
Point, Steiner's Ellipse
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 rev. enl. ed. Dublin: Hodges, Figgis, &; Co., 1893.
Eddy, R. H. and Fritsch, R. "The Conies of Ludwig Kiepert:
A Comprehensive Lesson in the Geometry of the Triangle."
Math. Mag. 67, 188-205, 1994.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 157-159, 1929.
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.
Isotomic Lines
A
A \ P3 G 3
Given a point P in the interior of a TRIANGLE
A^4!^4 2 ^3, draw the CEVIANS through P from each
Vertex which meet the opposite sides at Pi, P2, and
P3. Now, mark off point Q\ along side A?Az such that
^.3 Pi = A2 Qij etc., i.e., so that Qi and Pi are equidis-
tance from the MIDPOINT of AjA k . The lines A1Q1,
A2Q2, and ^.3(33 then coincide in a point Q known as
the Isotomic Conjugate Point.
see also CEVIAN, ISOTOMIC CONJUGATE POINT, MID-
POINT
Isotone Map
A MAP which is monotone increasing and therefore
order-preserving .
Isotope
To rearrange without cutting or pasting.
Isotopy
A HOMOTOPY from one embedding of a MANIFOLD M
in N to another such that at every time, it is an embed-
ding. The notion of isotopy is category independent, so
notions of topological, piecewise-linear, smooth, isotopy
(and so on) exist. When no explicit mention is made,
"isotopy" usually means "smooth isotopy."
see also Ambient Isotopy, Regular Isotopy
Isotropic Tensor
Iterated Function System 937
Isotropic Tensor
A TENSOR which has the same components in all rotated
coordinate systems.
rank
isotropic tensors
all
1
none
2
Kronecker delta
3
1
4
3
Isovolume Problem
Find the surface enclosing the maximum volume per
unit surface Area / = V/S. The solution is a Sphere,
which has
_ g7rr _ i
isphere — ^^ — jT.
see also Dido's Problem, Isoperimetric Problem
References
Bogomolny, A. "Isoperimetric Theorem and
Inequality." http : //www . cut— the-knot . com/do_you_know/
isoperimetric.html.
Isenberg, C. Appendix VI in The Science of Soap Films and
Soap Bubbles. New York: Dover, 1992.
Isthmus
see Bridge (Graph)
Iterated Exponential Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Euler (Le Lionnais 1983) and Eisenstein (1844) showed
that the function h(x) = x x , where x x is an ab-
breviation for ar x \ converges only for e~ e < x < e 1 ' 6 ,
that is, 0.0659. . . < x < 1.44466. . . . The value it con-
verges to is the inverse of x 1 ^ , which has a closed form
expression in terms of Lambert's VT-Function,
h{z)
W(-]nz)
— In z
(Corless et al). Knoebel (1981) gives
3 2 (lnz) 2 4 3 (lnz) 3
h(z) = l + lna;+ „, + \, + ■
3!
4!
(1)
(2)
(Vardi 1991). A Continued Fraction due to Khovan-
skii (1963) is
x L " = 1+-
2(x - 1)
x* + 1 ■
(x 2 -l)(x-l) 2
Sx(x -j- 1)
(4x 2 - l)(x - l) 2
5x(ae + l)
(9x 2 - l)(x- l) 2
7x(x + 1) - . . .
The function g{x) = a^ 1 ^ converges only for
e" 1/e < x < e e , that is, 0.692 . . . < x < 15.154 .... The
value it converges to is the inverse of x x .
Some interesting related integrals are
/V,_ vizir 1
Jo £r nT
= 0.7834305107...
pi oo
/ x~ x dx = V— = 1.
Jo tri nn
2912859971...
(4)
(5)
(3)
(Spiegel 1968, Abramowitz and Stegun 1972).
see also Lambert's ^-Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
1972.
Baker, I. N. and Rippon, P. J. "A Note on Complex Itera-
tion." Amer. Math. Monthly 92, 501-504, 1985.
Barrows, D. F. "Infinite Exponentials." Amer. Math.
Monthly 43, 150-160, 1936.
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 . 2.
Creutz, M. and Sternheimer, R. M. "On the Convergence of
Iterated Exponentiation, Part I." Fib. Quart 18, 341-347,
1980.
Creutz, M. and Sternheimer, R. M. "On the Convergence of
Iterated Exponentiation, Part II." Fib. Quart. 19, 326-
335, 1981.
de Villiers, J. M. and Robinson, P. N. "The Interval of
Convergence and Limiting Functions of a Hyperpower Se-
quence." Amer. Math. Monthly 93, 13-23, 1986.
Eisenstein, G. "Entwicklung von a aa ." J. Reine angew.
Math. 28, 49-52, 1844.
Finch, S. "Favorite Mathematical Constants." http: //www.
mathsoft.com/asolve/constant/itrexp/itrexp.html.
Khovanskii, A. N. The Application of Continued Fractions
and Their Generalizations to Problems in Approximation
Theory. Groningen, Netherlands: P. Noordhoff, 1963.
Knoebel, R. A. "Exponentials Reiterated." Amer. Math.
Monthly 88, 235-252, 1981.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
pp. 22 and 39, 1983.
Mauerer, H. "Uber die Funktion x x fur ganzzahliges Ar-
gument (Abundanzen)." Mitt Math. Gesell. Hamburg 4,
33-50, 1901.
Spiegel, M, R. Mathematical Handbook of Formulas and Ta-
bles. New York: McGraw-Hill, 1968.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, p. 12, 1991.
Iterated Function System
A finite set of contraction maps w% for i — 1, 2, ...,
N, each with a contractivity factor s < 1, which map a
compact METRIC Space onto itself. It is the basis for
FRACTAL image compression techniques.
see also Barnsley's Fern, Self-Similarity
938
Iterated Radical
Iwasawa's Theorem
References
Barnsley, M. F. "Fractal Image Compression." Not. Amer.
Math. Soc. 43, 657-662, 1996.
Barnsley, M. Fractals Everywhere, 2nd ed. Boston, MA: Aca-
demic Press, 1993.
Barnsley, M. F. and Demko, S. G. "Iterated Function Systems
and the Global Construction of Fractals." Proc. Roy. Soc.
London, Ser. A 399, 243-275, 1985.
Barnsley, M. F. and Hurd, L. P. Fractal Image Compression.
Wellesley, MA: A. K, Peters, 1993.
Diaconis, P. M. and Shashahani, M. "Products of Random
Matrices and Computer Image Generation." Contemp.
Math. 50, 173-182, 1986.
Fisher, Y. Fractal Image Compression. New York: Springer-
Verlag, 1995.
Hutchinson, J. "Fractals and Self-Similarity." Indiana Univ.
J. Math. 30, 713-747, 1981.
Wagon, S. "Iterated Function Systems." §5.2 in Mathematica
in Action. New York: W. H. Freeman, pp. 149-156, 1991.
Iterated Radical
see Nested Radical
Iteration Sequence
A Sequence {a,j} of Positive Integers is called an
iteration sequence if there EXISTS a strictly increasing
sequence {s k } of Positive Integers such that a± =
si > 2 and Oj = s aj _ x for j = 2, 3, . . . . A Necessary
and SUFFICIENT condition for {a,j} to be an iteration
sequence is
CLj > 2a j -i — CLj-1
for all j > 3.
References
Kimberling, C. "Interspersions and Dispersions." Proc.
Amer. Math. Soc. 117, 313-321, 1993.
Iverson Bracket
Let S be a mathematical statement, then the Iverson
bracket is defined by
«.{;
if S is true
if S is false.
This notation conflicts with the brackets sometimes used
to denote the Floor Function. For this reason, and
because of the elegant symmetry of the FLOOR FUNC-
TION and Ceiling Function symbols [x\ and \x] , the
use of [x] to denote the FLOOR FUNCTION should be
deprecated.
see also Ceiling Function, Floor Function
References
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science. Read-
ing, MA: Addison- Wesley, p. 24, 1990.
Iverson, K. E. A Programming Language. New York: Wiley,
p. 11, 1962.
Iwasawa's Theorem
Every finite-dimensional Lie Algebra of characteristic
p ^ has a faithful finite-dimensional representation.
References
Jacobson, N. Lie Algebras. New York: Dover, pp. 204-205,
1979.
Ito's Lemma
V t -V = / MS u ,T-u)dS u - / fr(S u ,T-u)du
Jo Jo
+ \<T 2 fsj
Jo
fxx(S u ,T — u) du,
where V t = f(S u r) for < r = T - t < T, and / G
C 2 ' 1 ((0,oo)x[0,T]).
References
Price, J. F. "Optional Mathematics is Not Optional." Not
Amer. Math. Soc. 43, 964-971, 1996.
Ito's Theorem
The dimension d of any IRREDUCIBLE REPRESENTATION
of a Group G must be a Divisor of the index of each
maximal normal Abelian SUBGROUP of G.
see also Abelian Group, Irreducible Representa-
tion, Subgroup
References
Lomont, J. S. Applications of Finite Groups. New York:
Dover, p. 55, 1993.
j-Function 939
The symbol used by engineers and some physicists to
denote i, the IMAGINARY NUMBER x/^T.
j- Conductor
see Frey Curve
j-Function
"^07z -0.15 -0.1 -0,05
-1000
-2000
0.01 -0.005
The j-function is defined as
jfa) = 1728J(V5),
where
4 [1-A(g) + A 2 (g)] 3
JW -27 A»(«)[l - Afa)]»
(1)
(2)
is Klein's Absolute Invariant, X(q) the Elliptic
Lambda Function
\(q) = k 2 (q)
(3)
and di a THETA FUNCTION. This function can also be
specified in terms of the Weber FUNCTIONS /, /i, / 2 ,
72 , and 73 as
3(z) =
[/ 24 (z) ~ 16] 3
/ 24 W
[/i 24 (*) + 16] 3
h 2 \z)
\tf\z) + 16] 3
f2 2i (z)
73 2 (z) + 1728
(4)
(5)
(6)
(7)
(8)
(Weber 1902, p. 179; Atkin and Morain 1993).
The j-function is MEROMORPHIC function on the upper
half of the Complex Plane which is invariant with
respect to the SPECIAL LINEAR GROUP 51(2, Z). It has
a Fourier Series
j(g) = 2_/ ° n9 "'
for the Nome
„2irit
q = e
(9)
(10)
with $s[t] > 0. The coefficients in the expansion of the
j-function satisfy:
1. a n = for n < — 1 and a_i = 1,
2. all a n s are INTEGERS with fairly limited growth with
respect to n, and
3. j(q) is an ALGEBRAIC Number, sometimes a Ra-
tional Number, and sometimes even an Integer
at certain very special values of q (or i) .
The latter result is the end result of the massive and
beautiful theory of COMPLEX multiplication and the
first step of Kronecker's so-called "JUGENDTRAUM."
Then all of the COEFFICIENTS in LAURENT SERIES
j(q) = - + 744 + 196884g + 21494760? 2
q
+864299970g 3 +20245856256g 4 +333202640600g 5 + . . .
(ID
(Sloane's A000521) are POSITIVE INTEGERS (Rankin
1977). Let d be a Positive Squarefree Integer,
and define
-\i(i + tV3)
for d = 1 or 2 (mod 4)
for d = 3 (mod 4).
(12)
Then the Nome is
e 2ni(iVd)
27Ti(l + Z\/d)/2
■{:
-nVd
for d = 1 or 2 (mod 4)
for d = 3 (mod 4).
(13)
It then turns out that j(q) is an ALGEBRAIC INTEGER
of degree h(-d), where h(-d) is the Class Number of
the Discriminant -d of the Quadratic Field Q(y/n)
(Silverman 1986). The first term in the Laurent Se-
ries is then g" 1 = e " 27rV ^ or -e _7rv ^, and all the
later terms are POWERS of q' 1 , which are small num-
bers. The larger n, the faster the series converges. If
h(-d) = 1, then j(q) is a Algebraic Integer of de-
gree 1, i.e., just a plain INTEGER. Furthermore, the
Integer is a perfect Cube.
The numbers whose LAURENT SERIES give INTEGERS
are those with CLASS NUMBER 1. But these are precisely
the Heegner Numbers -1, -2, -3, -7, -11, -19,
-43, -67, -163. The greater (in Absolute Value)
the Heegner Number d, the closer to an Integer is
the expression e 71 "^ - ", since the initial term in j(q) is
the largest and subsequent terms are the smallest. The
best approximations with h(—d) = 1 are therefore
e 7rv^3 _ 96(} 3 + 744 _ 2 2 x 1Q -4 ^
e nV ^ w 5280 3 + 744 - 1.3 x 10" 6 (15)
e 7rvT63 _ 640320 3 + 744 _ y g x 10 -13 ^
940 j -Function
Jackson's Identity
The exact values of j(q) corresponding to the Heegner
Numbers are
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
i(-
— 7T\
-e ) =
12 3
j(e~
2tt\/2 \ _
20 3
j(-e"
-7T-s/3 \
o 3
j(-e'
-7T\/7\ _
:-l5 3
j(~e~
ttVTTn _
-32 3
j(-e~
^V^ \ _
-96 3
i(-e
-tt\/43
) = -960 3
j(-e- 7rv/ ^) = -5280 3
j(-e~ 7r>/I ^) = -640320 3 .
(The number 5280 is particularly interesting since it is
also the number of feet in a mile.) The Almost In-
teger generated by the last of these, e 163 (corre-
sponding to the field Q(\/-163) and the IMAGINARY
quadratic field of maximal discriminant), is known as
the Ramanujan Constant.
^v^ e -*^ ^ and e*"^ are also ALMOST INTEGERS.
These correspond to binary quadratic forms with dis-
criminants —88, —148, and —232, all of which have
CLASS NUMBER two and were noted by Ramanujan
(Berndt 1994).
It turns out that the j-function also is important in the
Classification Theorem for finite simple groups, and
that the factors of the orders of the Sporadic Groups,
including the celebrated Monster Group, are also re-
lated.
see also ALMOST INTEGER, KLEIN'S ABSOLUTE INVARI-
ANT, Weber Functions
References
Atkin, A. O. L. and Morain, F. "Elliptic Curves and Primal-
ity Proving." Math. Comput 61, 29-68, 1993.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 90-91, 1994.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 117-118, 1987.
Cohn, H. Introduction to the Construction of Class Fields.
New York: Dover, p. 73, 1994.
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.
Morain, F. "Implementation of the Atkin- Goldwasser-Kilian
Primality Testing Algorithm." Rapport de Recherche 911,
INRIA, Oct. 1988.
Rankin, R. A. Modular Forms. New York: Wiley, 1985.
Rankin, R. A. Modular Forms and Functions. Cambridge,
England: Cambridge University Press, p. 199, 1977.
Serre, J. P. Cours d'arithmetique. Paris: Presses Universi-
taires de France, 1970.
Silverman, J. H. The Arithmetic of Elliptic Curves. New
York: Springer- Verlag, p. 339, 1986.
Sloane, N. J. A. Sequence A000521/M5477 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Weber, H. Lehrbuch der Algebra, Vols. I-II. New York:
Chelsea, 1979.
^ Weisstein, E. W. "j-Function." http: //www. astro.
virginia.edu/-eww6n/math/notebooks/jFunction.in.
^-Invariant
An invariant of an ELLIPTIC CURVE closely related to
the Discriminant and defined by
■(F) = 28 3 3fl3
The determination of j as an ALGEBRAIC INTEGER in
the QUADRATIC Field Q(j) is discussed by Greenhill
(1891), Weber (1902), Berwick (1928), Watson (1938),
Gross and Zaiger (1985), and Dorman (1988). The norm
of j in Q(j) is the Cube of an Integer in Z.
see also DISCRIMINANT (ELLIPTIC CURVE), ELLIPTIC
Curve, Frey Curve
References
Berwick, W. E. H. "Modular Invariants Expressible in Terms
of Quadratic and Cubic Irrationalities." Proc. London
Math. Soc. 28, 53-69, 1928.
Dorman, D. R. "Special Values of the Elliptic Modular Func-
tion and Factorization Formulae." J. reine angew. Math.
383, 207-220, 1988.
Greenhill, A. G. "Table of Complex Multiplication Moduli."
Proc. London Math. Soc. 21, 403-422, 1891.
Gross, B. H. and Zaiger, D. B. "On Singular Moduli." J.
reine angew. Math. 355, 191-220, 1985.
Watson, G. N. "Ramanujans Vermutung iiber Zerfallungsan-
zahlen." J. reine angew. Math. 179, 97-128, 1938.
Weber, H. Lehrbuch der Algebra, Vols. I-II. New York:
Chelsea, 1979.
Jackson's Difference Fan
If, after constructing a Difference Table, no clear
pattern emerges, turn the paper through an Angle of
60° and compute a new table. If necessary, repeat the
process. Each ROTATION reduces POWERS by 1, so the
sequence {k 71 } multiplied by any POLYNOMIAL in n is
reduced to Os by a fc-fold difference fan.
References
Conway, J. H. and Guy, R. K. "Jackson's Difference Fans." In
The Book of Numbers. New York: Springer- Verlag, pp. 84—
85, 1996.
Jackson's Identity
A ^-Series identity involving
(aq)™{aqde)™(adec)™{aqcd)™
where
(aqc)^(aqd)^(aqe)^(aqcde)^ '
(aK = (l-a)(l-aq)...(l-aq n - 1 ).
see also ^-Series
References
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
pp. 109-110, 1959.
Jackson, F. H. "Summation of g-Hypergeometric Series."
Messenger Math. 47, 101-112, 1917.
Jackson's Theorem
Jacobi's Determinant Identity 941
Jackson's Theorem
Jackson's theorem is a statement about the error E n (f)
of the best uniform approximation to a REAL FUNCTION
f(x) on [-1, 1] by Real Polynomials of degree at most
n. Let f(x) be of bounded variation in [—1, 1] and let
M' and V' denote the least upper bound of |/(x)| and
the total variation of f(x) in [— 1, 1], respectively. Given
the function
F(x) = F(-1)+ / f(x)dx
L
then the coefficients
o„ = \{2n + 1) J F{x)P n (x) dx
(1)
(2)
of its LEGENDRE Series, where P n (x) is a Legendre
Polynomial, satisfy the inequalities
o„ <
(^(M' + V')n
\^{M' + V')n
_3/2 for n > 1
3/2 forn>2.
(3)
Moreover, the Legendre Series of F{x) converges uni-
formly and absolutely to F(x) in [—1,1].
Bernstein strengthened Jackson's theorem to
2n£?2n(a) <
4n
7r(2n+l) 7T
0.6366.
(4)
A specific application of Jackson's theorem shows that
if
a(x) = \x\, (5)
then
E n (a) < -■
(°)
see also Legendre Series, Picone's Theorem
References
Cheney, E. W. Introduction to Approximation Theory. New-
York: McGraw-Hill, 1966.
Jackson, D. The Theory of Approximation. New York:
Amer. Math. Soc, p. 76, 1930.
Rivlin, T. J. An Introduction to the Approximation of Func-
tions. New York: Dover, 1981.
Sansone, G. Orthogonal Functions, rev. English ed. New
York: Dover, pp. 205-208, 1991.
Jaco-Shalen-Johannson Torus
Decomposition
Irreducible orientable Compact 3-Manifolds have a
canonical (up to ISOTOPY) minimal collection of dis-
jointly Embedded incompressible Tori such that each
component of the 3-MANIF0LD removed by the TORI is
either "atoroidal" or "Seifert-fibered."
Jacobi Algorithm
A method which can be used to solve a TRIDIAGONAL
MATRIX equation with largest absolute values in each
row and column dominated by the diagonal element.
Each diagonal element is solved for, and an approximate
value plugged in. The process is then iterated until it
converges. This algorithm is a stripped-down version of
the Jacobi Method of matrix diagonalization.
see also Jacobi Method, Tridiagonal Matrix
References
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., pp. 161-163, 1990.
Jacobi-Anger Expansion
oo
c <»co.« = ^2 i n J n (z)e in \
n~ — oo
where J n (z) is a BESSEL FUNCTION OF THE FIRST
KIND. The identity can also be written
= J (z) +2^T i n Jn(z) cos(nfl).
This expansion represents an expansion of plane waves
into a series of cylindrical waves.
see also Bessel Function of the First Kind
Jacobi's Curvature Theorem
The principal normal indicatrix of a closed space curve
with nonvanishing curvature bisects the Area of the
unit sphere if it is embedded.
Jacobi's Determinant Identity
Let
A=[* °] ,„
A"' =[?£]. (2)
where B and W are k x k Matrices. Then
(det Z)(det A) = det B. (3)
The proof follows from equating determinants on the
two sides of the block matrices
(4)
where I is the IDENTITY Matrix and is the zero ma-
trix.
References
Gantmacher, F. R. The Theory of Matrices, Vol. 1. New-
York: Chelsea, p. 21, 1960.
Horn, R. A. and Johnson, C. R. Matrix Analysis. Cambridge,
England: Cambridge University Press, p. 21, 1985.
[B Dl
[1 X]
[B 0]
E C
z
E 1
942 Jacobi Differential Equation
Jacobi Differential Equation
(l-x 2 )y' + \j3-a-(a+l3+2)x]y'+n{n+a+P+l)y =
(1)
or
d_
dx
[(i-*r +i (i+xf + y]
+n{n + a + f3 + 1)(1 - x) a (l + xfy = 0. (2)
The solutions are JACOBI POLYNOMIALS. They can be
transformed to
d 2 u
+
1 1 - a 2 1 1 - f3 2
dx 2 [4(1 -x) 2 4(l + z) 2
+
n(n + a + /? + 1) + f (a + l)(/3 + 1)
u - 0, (3)
1-z 2
where
u - u(x) = (1 - a;) (a+1)/2 (l + ^) w+1)/2 Pi a '' 3) (x) ) (4)
and
d 2 u
dO 2
+
1 ^2 1/32
4 ~ Q | 4 -P
4sin 2 (±0) 4cos 2 (±0)
(-
+ n +
a + /3 + r
u = 0, (5)
where
u - u(0) - sin^^^^Jcos^+'^d^P^^Ccosd).
(6)
Jacobi Differential Equation (Calculus of
Variations)
— to,,* -«,, = -^(f y ^V + fy'y / n') - (fyyV + fyy'V) = 0,
where
fl(s, »?»»?) = 2 (/»»»? + 2 fyy'VV +fy'y'n )•
This equations arises in the CALCULUS OF VARIATIONS.
References
Bliss, G. A. Calculus of Variations. Chicago, IL: Open
Court, pp. 162-163, 1925.
Jacobi Elliptic Functions
Jacobi Elliptic Functions
The Jacobi elliptic functions are standard forms of El-
liptic Functions. The three basic functions are de-
noted cn(u, k), dn(uj fc), and sn(u, k) } where fc is known
as the Modulus. In terms of Theta Functions,
sn(u, k) =
cn(u, fc) =
dn(tx, k) —
^3t9 4 (^3" 2 )
(1)
(2)
(3)
(Whittaker and Watson 1990, p. 492), where tf* = ^(0)
(Whittaker and Watson 1990, p. 464). Ratios of Jacobi
elliptic functions are denoted by combining the first let-
ter of the Numerator elliptic function with the first of
the Denominator elliptic function. The multiplicative
inverses of the elliptic functions are denoted by reversing
the order of the two letters. These combinations give a
total of 12 functions: cd, en, cs, dc, dn, ds, nc, nd, ns,
sc, sd, and sn. The Amplitude <f> is defined in terms of
snu by
y — sin</> = sn(ti, fc). (4)
The k argument is often suppressed for brevity so, for
example, sn(w, k) can be written snu.
The Jacobi elliptic functions are periodic in K(k) and
K'(k) as
sn(u + 2mK + 2niK\ k) = (™l) m sn(u, k) (5)
cn(u + 2mK + 2mA'', k) = (-l) m+n cn(u, k) (6)
dn(u + 2mK + 2niK\ k) = (-l) n dn(u, Jfe), (7)
where K(k) is the complete Elliptic Integral of the
First Kind, K'(k) = K{k f ), and k' = y/l - k 2 (Whit-
taker and Watson 1990, p. 503).
The cnx, dncc, and sn# functions may also be defined
as solutions to the differential equations
§ = -(l + *») tf + 2*V
dx
cfy
dx 2
2 = -(1 - 2k 2 )y - 2k 2 y 3
(2 - k 2 )y - 2y 3 .
(8)
(9)
(10)
The standard Jacobi elliptic functions satisfy the iden-
tities
2 , 2
sn u + en u
1
k sn u + dn u = 1
; 2 2 ,i/2 j 2
« cn u+fc = dn u
2 . t /2 2 i 2
cn u + fc sn u = dn u.
(ii)
(12)
(13)
(14)
Jacobi Elliptic Functions
Special values are
cn(0) = 1
dn(0) = 1
cn{K) =
dn(JK") = ft' = y/l - ft 2 ,
sn(K) = 1,
(15)
(16)
(17)
(18)
(19)
where K = K(k) is a complete Elliptic Integral of
THE First Kind and k' is the complementary MODULUS
(Whittaker and Watson 1990, pp. 498-499).
In terms of integrals,
/»sn u
u= (l-t 2 ) 1 "^!-* 2 * 2 )' 1 ^* (20)
Jo
= / (t 2 - i)- i/2 (t 2 - z 2 r i/2 dt (21)
J nsu
= f 1 (i-t 2 r i/2 {k ,2 +k 2 t 2 r i/2 dt (22)
J cnu
/nc u
(t 2 - iy 1/2 (k ,2 t 2 + k 2 y 1/2 dt (23)
= f 1 (i - t 2 r i/2 (t 2 - k' 2 r i/2 dt (24)
</dnu
(i 2 -l)- 1/2 (l-fc' 2 t 2 )- 1/2 rft (25)
= [ V (l + t 2 )- 1/2 (l + k' 2 t 2 )- 1/2 dt (26)
Jo
/>oo
= / (t 2 + i)- i/2 (t 2 +k ,2 )- i/2 dt (27)
J CS u
= [ Sd "(l-k' 2 t 2 )- 1/2 (l + k 2 t 2 )- 1/2 dt (28)
Jo
= r (t 2 - k' 2 y i/2 (t 2 + k 2 r i/2 dt (29)
t/ds u
= r du (i - * 2 )- i/2 (i - fc 2 t 2 )- i/2 dt (30)
= r (* 2 -i)- i/2 (< 2 -fc 2 )- i/2 dt (31)
t/dc u
(Whittaker and Watson 1990, p. 494).
Jacobi elliptic functions addition formulas include
, . snucnvdn^ + sn^cnwdnw , N
sn(u + v) = —^ — 2 2 ( 32 )
. x en u en v — sn u sn v dn w dn v /onX
cn(u + v) = — — „- = (33)
1 — ft 3 sn 2 wsn 2 i;
, , , dnitdn -u — k 2 snwsn venuenv ._ ,.
1 — ft 2 sn J w sn 2 v
Jacobi Elliptic Functions
Extended to integral periods,
sn(u + K) •
cn(u + K) = - 1
cnu
dnu
k f snu
dnit
dn(u + K) =
k'
dnu
sn(u + 2K) — -snu
cn{u + 2K) — — cnu
dn(u + 2K) = dniA
For COMPLEX arguments,
sn(it, k) dn(t>, ft')
943
(35)
(36)
(37)
(38)
(39)
(40)
sn(u + iv)
cn(u 4- iv) =
dn(u + iv) =
l-dn 2 (u,ft)sn 2 (u,ft')
cn(u, ft) dn(u, ft) si
l-dn 2 (u,ft)
cn(u, ft) cn(u, A:')
i cn(u, A;) dn(u, ft) sn(v, ft') cn(v, ft') , .
+ l-dn 2 (u,ft)sn 2 (v,ft') ( ]
l-dn 2 (w,ft)sn 2 (t;,ft')
i sn(u, ft) dn(u, ft) sn(v, ft') dn(w, ft')
l-dn 2 (u,ft)sn 2 (i;,ft')
dn(ti, ft) cn(t>, ft') dn(v, ft')
l-dn 2 (u,ft)sn 2 (t;,ft')
ift 2 sn(u, ft) cn(u, ft) sn(v, ft')
l-dn 2 (u,ft)sn 2 (v,ft')
(42)
(43)
Derivatives of the Jacobi elliptic functions include
dsnu
du
denu
du
ddnu
du
= en u dn u
(44)
= sn u dn u
(45)
= —ft 2 snu cnu.
(46)
Double-period formulas involving the Jacobi elliptic
functions include
,_ x 2 snu cnu dnu
sn(2u) = — ■ r^ — -.
v J 1 - ft 2 sn 4 u
cn(2u)
1 — 2 sn 2 u + ft 2 sn 4 u
1 - ft 2 sn 4 u
(47)
(48)
, ,„ , 1 — 2ft 2 sn 2 u + ft 2 sn 4 u ,.„.
dn(2w) = l^k^-u • (49)
Half-period formulas involving the Jacobi elliptic func-
tions include
vTTft 7
sn(^) =
dn(|K) = v/fe 7 .
(50)
(51)
(52)
944
Jacobi Function of the First Kind
Squared formulas include
1-
2
sn u ■
cn(2u)
1 4 dn(2u)
2 dn(2u) 4 cn(2ii)
en u — — ; — ' ,\ — -
1 4 dn(2u)
2 _ dn(2u) 4 cn(2u)
1 + cn(2u)
(53)
(54)
(55)
see also Amplitude, Elliptic Function, Jacobi's
Imaginary Transformation, Theta Function,
Weierstrass Elliptic Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Jacobian El-
liptic Functions and Theta Functions." Ch. 16 in Hand-
book of Mathematical Functions with Formulas, Graphs,
and Mathematical Tables, 9th printing. New York: Dover,
pp. 567-581, 1972.
Bellman, R. E. A Brief Introduction to Theta Functions.
New York: Holt, Rinehart and Winston, 1961.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 433, 1953.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Elliptic Integrals and Jacobi Elliptic Func-
tions." §6.11 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 254-263, 1992.
Spanier, J. and Oldham, K. B. "The Jacobian Elliptic Func-
tions." Ch. 63 in An Atlas of Functions. Washington, DC:
Hemisphere, pp. 635-652, 1987.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Jacobi Function of the First Kind
see Jacobi Polynomial
Jacobi Function of the Second Kind
l(«.0)
a /~ _L l\-0
QjT J "(«) = 2— 1 (x^l)- a (x + l)
x I (1 - t) n+c
{X + t) n+li {x-t)- n - 1 dt.
In the exceptional case n = 0,a + /3 + l = 0,a noncon-
stant solution is given by
Q (a) {x) = ln(z + 1) + ir _1 sm(ira)(x - l)~ a (x + l) - "
' \n(l + t)dt.
f 1 (1 -«)"(! + ,
7-i *-*
References
Szego, G. "Jacobi Polynomials." Ch. 4 in Orthogonal Polyno-
mials, J^th ed. Providence, RI: Amer. Math. Soc, pp. 73-
79, 1975.
Jacobi-Gauss Quadrature
Jacobi-Gauss Quadrature
Also called Jacobi Quadrature or Mehler Quad-
rature. A Gaussian Quadrature over the interval
[-1, 1] with Weighting Function W{x) = (l-z) a (l+
x)* 3 . The Abscissas for quadrature order n are given by
the roots of the Jacobi Polynomials pI? ,(5) (x). The
weights are
Wi =
^n+lTn
A-nPn \&i)*, n -\-i y^i)
(1)
where A n is the Coefficient of x n in P^ a,p \x). For
Jacobi Polynomials,
7n
= r(2n + a + ff + l)
71 2™n!r(n + a + /? + l)'
where Y(z) is a GAMMA FUNCTION. Additionally,
1 2 2n+a+ ^ +1 n!
2 2n (n\) 2 2n4a4/?4l
r(n + a + l)r(n + j3 + l)
T(n + a + /3 + l)
2n 4 a 4 4 2 T(n 4- a 4 l)T(n 4 4- 1)
n + OL + + l r(n + a + /3+l)
(2)
, (3)
Wi
VA(xi)V n+1 (xi)
r(n + a + l)r(n + j9 + l) 2 ' 2n+ °' +0+1 nl
T(n + a + + 1) (l-x*)[Vi(xi)]*'
(4)
where
V m = P^(x)
,(«,/3)/ s 2 n n!
(-I)"'
The error term is
r(n 4 a 4 l)r(n 4 4 l)T(n 4 a 4 4 1)
(5)
(6)
E n =
(2n 4 a 4 4 l)[r(2n 4 a 4 4 l)] 2
X (2n)! / l5J
(?)
(Hildebrand 1959).
References
Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 331-334, 1956.
Jacobi Identities
Jacobi Polynomial 945
Jacobi Identities
"The" Jacobi identity is a relationship
[A, [B, C}} + [B, [C, A}} + [C, [A, B}} = 0, (1),
between three elements A, £?, and C, where [A, B] is the
Commutator. The elements of a Lie Group satisfy
this identity.
Relationships between the Q-Functions Qi are also
known as Jacobi identities:
Q1Q2Q3 = 1,
(2)
equivalent to the Jacobi Triple Product (Borwein
and Borwein 1987, p. 65) and
(3)
Q 2 8 = 16gQi 8 + Q 3 8 ,
where
(4)
K = K{k) is the complete Elliptic Integral of the
First Kind, and K'{k) = K{k f ) = K{y/T=W). Using
Weber Functions
r -l/24/-i
(5)
(6)
(7)
* l/2 1/12^
(5) and (6) become
/i/ 2 / = v / 2 (8)
/ 8 = /i 8 + / 2 8 (9)
(Borwein and Borwein 1987, p. 69).
see also Commutator, Jacobi Triple Product, Q-
Function, Weber Functions
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.
Jacobi's Imaginary Transformation
For Jacobi Elliptic Functions snw, cn^, and dnu,
sn(iu, k) = i
cn(iu, k)
dn(m, k)
.sn(ti, fc')
cn(u, &')
1
cn(u, k')
dn(it, k')
cn(u, k')
(Abramowitz and Stegun 1972, Whittaker and Watson
1990).
see also JACOBI ELLIPTIC FUNCTIONS
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 592 and 595, 1972.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, ^th ed. Cambridge, England: Cambridge Uni-
versity Press, p. 505, 1990.
Jacobi Matrix
see Jacobi Rotation Matrix, Jacobian
Jacobi Method
A method of diagonalizing MATRICES using JACOBI
Rotation Matrices. It consists of a sequence of
Orthogonal Similarity Transformations, each of
which eliminates one off-diagonal element.
see also Jacobi Algorithm, Jacobi Rotation Ma-
trix
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Jacobi Transformation of a Symmetric Ma-
trix." §11.1 in Numerical Recipes in FORTRAN: The Art
of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 456-462, 1992.
Jacobi Polynomial
Also known as the Hypergeometric Polynomials,
they occur in the study of ROTATION GROUPS and in
the solution to the equations of motion of the symmetric
top. They are solutions to the JACOBI DIFFERENTIAL
Equation. Plugging
i/=0
(i)
into the differential equation gives the RECURRENCE
Relation
[ 1 -v{v + a+f3+l)]a u -2{v + l)(v + a + \)a l/ + 1 = (2)
for v — 0, 1, . . . , where
7 = n(n + a + /? + l). (3)
Solving the RECURRENCE RELATION gives
:>(*,/9)
(*) =
t^ft-x-
2 n nl
r a -*r a (i+ *r
d n
dx n
x^-[{l-x) a + n {l + xf+ n ] (4)
for a,f3 > — 1. They form a complete orthogonal sys-
tem in the interval [—1,1] with respect to the weighting
function
w n (x) = (l-x) a (l + xf, (5)
and are normalized according to
(6)
where (?) is a BINOMIAL COEFFICIENT. Jacobi polyno-
mials can also be written
(„,„) = T(2n + a + (3 + 1)
nW(n + a + + 1)
xG„(a + /?+l,0 + l,i(x+l)), (7)
946 Jacobi Polynomial
where T(z) is the Gamma Function and
Jacobi polynomials are Orthogonal satisfying
Jacobi Polynomial
(8)
L
P^- 0) Pi a - 0) (l - x) a (l + xf dx
2 a+f>+1 T{n + a + 1)I> + + 1)
2n + a + + l nW(n + a + + 1)
Omri' \v)
The Coefficient of the term x" in pt a ' beta > ( x ) j s given
by
r(2n + a + + l) ()
n ~ 2"nW{n + a + 0+lY K '
They satisfy the RECURRENCE RELATION
2(n + l)(n + a + /3 + l)(2n + a + /3)P^f \x)
= [(2n + a + + l)(a 2 -0 2 ) + (2n + a + 0) 3 x}Pi a ^{x)
-2(n + a)(n + 0)(2n + a + + 2)P^f > (x), (11)
where (m)„ is the RISING FACTORIAL
(m)„ = m(m + l)---(m + n-l) = i-^— -^ (12)
The Derivative is given by
£[P^ ) (*)] = i(n + a + i 8+l)^tt 1,/J+1) (*)- (13)
The Orthogonal Polynomials with Weighting
Function (6 - x) a (x - a)* 3 on the Closed Interval
[a, 6] can be expressed in the form
[const.]^ (2^|-l) (H)
(Szego 1975, p. 58).
Special cases with a = are
p(°.<*)/yi - r(> + a + 1)I> + 1) p (q,-i/2) f2 a _ n
^ w-r(^ + a + i)r(2i/ + i) n l j
_ T{2u + a + l)T{v+l) p (-i/a,a)/ 1 2 _Jx
- (-1) r(i/ + a + i)r(2i/ + i) n l x j
(15)
p(«..«0 M _ rO + a + 2)l> + l) „ p(a .i/2) f 2 2 _ ,
p ^+lw-r(l,+ a +1)1x2^+2)^ ^ 1)
_ r(2^ + a + 2)i>+l) rP (i/2, Q ) ri 2 a >
- (_1) T(v + a + l)T(2u + 2) X ^ U } '
(16)
Further identities are
Pi a+1 ^(x) =
P<^ +1 V)
2n + a + /? + 2
n + a+ljP^-Cn+^P^fW
1-z
2
2n + a + + 2
(n + j9 + l)Pi a > 0) (x) + (n + l)i£ff (a)
l + x
(17)
(18)
A 2i/ T Q + ^+ir(i/ + i)r(i/ + a + /3 + i)
Z^ 2«+/ 3 + 1 i> + a 4- i)r(i/ + /? + 1)
xP^(x)Qi^(y)
_l (y-l)-"(y + l)- (3 2- a -f
2 y - z 2n + a + /3 + 2
r(n + 2)r(rz + a + /3 + 2)
X r(n + a + l)r(n + /3+l)
x-y
(Szego 1975, p. 79).
The Kernel Polynomial is
(19)
2n + a + + 2
T(n + 2)r(n + a + /3 + 2)
X T(n + a + l)r(n + ^ + l)
x-y
(20)
(Szego 1975, p. 71).
The Discriminant is
D W) = a-^"- 1 ) f[ v »-* n +*{ v + a y-\v + py- 1
x(n + ^ + a + ^) n -" (21)
(Szego 1975, p. 143).
For a = = 0, P,! 0,0 ^) reduces to a LEGENDRE POLY-
NOMIAL. The Gegenbauer Polynomial
O.0, * x) = ^±4p<— >(2x - 1) (22)
l(2n + pj
and Chebyshev Polynomial of the First Kind can
also be viewed as special cases of the Jacobi POLYNO-
MIALS. In terms of the HYPERGEOMETRIC FUNCTION,
**"»(*)= ( n ; a )
x 2 F l {~n,n + a + 0;a + l;\{\-x)) (23)
«-)=(r)(x)'
x aJ Fi(-n,-n-/?;a + l;|^y). (24)
Jacobi Polynomial
Jacob! Symbol 947
Let N\ be the number of zeros in x G ( — 1, 1), N2 the
number of zeros in x € (-00, -1), and iV 3 the number
of zeros in x E (1, 00). Define Klein's symbol
r if u <
E(u) — I \u\ if u positive and nonintegral (25)
where \x\ is the FLOOR FUNCTION, and
X(c h p) = E[±(\2n + a + f3 + l\-\a\-\p\ + l)}(26)
y(a,/?) = E[f(-|2n + a + /3 + l| + |a|-|^| + l)]
(27)
Z{a,p) = E[\(-\2n + CL + P±l\-\a\ + \p\ + l)].
(28)
If the cases a = —1, —2, , , , , — n, (3 — — 1, —2, . . . , — n,
and n + a + /3 = — 1, —2, . . . , — n are excluded, then the
number of zeros of Pn in the respective intervals are
N x {a,P):
f2|_i(* + l)J for(-ir("r)(^)>0
\2LiXj+l for(-ir(^)(^)<0
(29)
iV2(a ' /3) -\2L|yJ+l for( 2 " + r /3 )( n : /3 )<0
(30)
for(-^)("r)<0
(31)
(Szego 1975, pp. 144-146).
The first few POLYNOMIALS are
(x) = l
P[^\x) = i[2(a + 1) + (a + + 2)(x - 1)]
^ 2 (a,/3) (*) = I [4(a + 1)2 + 4(a + /? + 3)(a + 2)(s - 1)
+ (a + /? + 2) 2 (z~l) 2 ],
where (m) n is a RISING FACTORIAL. See Abramowitz
and Stegun (1972, pp. 782-793) and Szego (1975, Ch. 4)
for additional identities.
see also CHEBYSHEV POLYNOMIAL OF THE FIRST KIND,
Gegenbauer Polynomial, Jacobi Function of the
Second Kind, Rising Factorial, Zernike Poly-
nomial
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.
Iyanaga, S. and Kawada, Y. (Eds.). "Jacobi Polynomials."
Appendix A, Table 20. V in Encyclopedic Dictionary of
Mathematics. Cambridge, MA: MIT Press, p. 1480, 1980.
Szego, G. "Jacobi Polynomials." Ch. 4 in Orthogonal Poly-
nomials, 4th ed. Providence, RI: Amer. Math. Soc, 1975.
Jacobi Quadrature
see JACOBI-GAUSS QUADRATURE
Jacobi Rotation Matrix
A Matrix used in the Jacobi Transformation
method of diagonalizing Matrices. It contains cos</>
in p rows and columns and sin <j) in q rows and columns,
Ppq —
cos</> •
• •
• sin<j)
• 1 •
sin<^> •
• •
* coscp
see also JACOBI TRANSFORMATION
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Jacobi Transformation of a Symmetric Ma-
trix." §11.1 in Numerical Recipes in FORTRAN: The Art
of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 456—462, 1992.
Jacobi Symbol
The product of LEGENDRE SYMBOLS (n/p») for each of
the PRIME factors pi such that m = JIi^' denoted
(n/m). When misa PRIME, the Jacobi symbol reduces
to the Legendre Symbol. The Jacobi symbol satisfies
the same rules as the Legendre Symbol
(n/m)(n/m) — (n/(mm)) (1)
(n/m)(n jm) = ((nri)/m) (2)
(n 2 /m) = (n/m 2 ) = 1 if (m y n) = 1 (3)
(n/m) = (n/m) if n = n (mod m) (4)
(-l/m) = (-l)<— ^^{^
for ?n = 1 (mod 4)
for m = — 1 (mod 4)
(5)
w-> -(-)«--«• -{l, £::s£33
(n/m)
( (m/n)
\ -(m/n
for m or n = 1 (mod 4)
) for m, n = 3 (mod 4).
(6)
(7)
Written another way, for m and n RELATIVELY Prime
Odd Integers with n > 3,
(m/n) = (-l) (m - 1)(n - 1)/4 (n/m).
(8)
The Jacobi symbol is not denned if m < or m is Even.
Bach and Shallit (1996) show how to compute the Jacobi
symbol in terms of the Simple CONTINUED FRACTION
of a Rational Number a/b.
see also Kronecker Symbol
948
Jacobi Tensor
Jacobi Triple Product
References
Bach, E. and Shallit, J. Algorithmic Number Theory,
Vol 1: Efficient Algorithms. Cambridge, MA: MIT Press,
pp. 343-344, 1996.
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.
Riesel, H. "Jacobi's Symbol." Prime Numbers and Com-
puter Methods for Factorization , 2nd ed. Boston, MA:
Birkhauser, pp. 281-284, 1994.
Jacobi Tensor
where R is the RlEMANN TENSOR.
see also Riemann Tensor
Jacobi's Theorem
Let M r be an r-rowed Minor of the nth order Deter-
minant |A| associated with an n x n Matrix A = a^
in which the rows i\ } Z2, . .., i r are represented with
columns k\ } fe, . . • , k T . Define the complementary mi-
nor to M T as the (n — &)-rowed MINOR obtained from
|A| by deleting all the rows and columns associated with
M r and the signed complementary minor M^ r > to M r to
be
Ti^-(r) __ /_-.yi+i2 + ---+ir + fc 1 +fc 2 + ---+fcr
x [complementary minor toM^].
Let the MATRIX of cofactors be given by
An Ai2 - ■ • Ai n
A21 A22 ' • ' A2n
with Mr and M r the corresponding r-rowed minors of
|A| and A, then it is true that
m; = |A| p - 1 m (p) .
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1109-1100, 1979.
Jacobi Theta Function
see Theta Function
Jacobi Transformation
see Jacobi Method
Jacobi Triple Product
The Jacobi triple product is the beautiful identity
00 / jn-i\
n(i-^)(i+x 2 »-v)^i+^j
00
Em 2 2m / n \
X Z . (1)
m= — 00
In terms of the Q-FUNCTIONS, (1) is written
Q1Q2Q3 = 1,
(2)
which is one of the two Jacobi Identities. For the
special case of z = 1, (1) becomes
00
^)^g(i)=H( i+x2 "" i ) 2 ( i - x2 ")
n = l
00 00
- Y, * m2 =l + 25> m2 , (3)
m = — 00
where tp(x) is the one-variable Ramanujan Theta
Function.
To prove the identity, define the function
00 / *2n-l\
n=l ^ '
Z 2 J
= (l + xz 2 )
x (1 + xV) I 1 +
(4)
Then
F{xz) = (1 + sV) (l + ^j) (1 + sV) (l + J)
(1 + 1V) 1 +
Taking (5) + (4),
F{xz)
F(z)
z 2 J
V 1+ xz 2 ) (l + xz 2 )
(5)
xz 2 + 1 1
xz 2 1 + xz 2 xz 2 '
which yields the fundamental relation
xz 2 F{xz) -F{z).
Now define
G(z)=F(z)l[(l-x 2n )
(6)
(7)
(8)
Jacobi Triple Product
oo
G{xz)=F{xz)~[[(l-x 2n ).
n=l
Using (7), (9) becomes
(9)
G(xz)
F{z)
U(l-x 2n ) =
G(z) = xz 2 G(xz).
G 3. m
(ii)
Expand G in a Laurent Series. Since G is an Even
Function, the Laurent Series contains only even
terms.
G(z)= Y, a - z2m -
m=-oo
Equation (11) then requires that
oo oo
\^ a m z 2m = xz 2 22 a m(xz) 2
(12)
m= — oo
m= — oo
H a "
m— — oo
,^ +1 z 2Tn+2 . (13)
This can be re-indexed with m' = m — 1 on the left side
of (13)
OO CO
E„ 2m \~^ „ 2ro-l 2m /-,.,>>
which provides a RECURRENCE RELATION
__ 2m- 1
(15)
a\ = aox (16)
a2 — aix = aox = a^x = ao£ (17)
a 3
5 5+4 9 3 J / no \
= a?,x = aox = aox = aox . (loj
The exponent grows greater by (2m — 1) for each increase
in m of 1. It is given by
£(2m-
_ m(mtij
2
— m = m .
(19)
n=l
Therefore,
m 2
(20)
This means that
oo
G(z) = a ]T xm2 * 2m ( 21 )
Jacobi Zeta Function 949
The Coefficient ao must be determined by going back
to (4) and (8) and letting z = 1. Then
oo
F(l) = Y[(l + x* n - 1 )(l + x 2n - 1 )
n=l
oo
= ]7(l + x 2n - 1 ) 2 (22)
n=l
oo
G(l) = F(l)[](l-a ; 2 ' 1 )
n=l
oo oo
=n(i+- 2n - i ) 2 ii( i - a;2n )
n=l n=l
CO
= ]T(l + x 2 "- 1 ) 2 (l- a; 2n ), (23)
since multiplication is ASSOCIATIVE. It is clear from this
expression that the ao term must be 1, because all other
terms will contain higher POWERS of x. Therefore,
ao = 1,
so we have the Jacobi triple product,
(24)
g(z) = rja-^xi+z 2 "- 1 * 2 ) ( 1+i V)
71 = 1 ^ '
OO
m 2m
X Z
(25)
m= — oo
see also Euler Identity, Jacobi Identities, Q-
Function, Quintuple Product Identity, Ra-
manujan Psi Sum, Ramanujan Theta Functions,
Schroter's Formula, Theta 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, pp. 63-64, 1986.
Borwein, J. M. and Borwein, P. B. "Jacobi's Triple Prod-
uct and Some Number Theoretic Applications." Ch. 3 in
Pi & the AGM: A Study in Analytic Number Theory and
Computational Complexity. New York: Wiley, pp. 62-101,
1987.
Jacobi, C. G. J. Fundamentia Nova Theoriae Functionum
Ellipticarum. Regiomonti, Sumtibus f rat rum Borntraeger,
p. 90, 1829.
Whit taker, E. T. and Watson, G. N. A Course in Modern
Analysis, ^.th ed. Cambridge, England: Cambridge Uni-
versity Press, p. 470, 1990.
Jacobi Zeta Function
Denoted zn(u, fe) or Z(u).
Z{(j>\m) = E(4>\m) -
E(m)F(4>\m)
K(m)
m—~oo
950
Jacobian
Jacobian Group
where <j> is the AMPLITUDE, m is the PARAMETER, and
F and K are Elliptic Integrals of the First Kind,
and E is an ELLIPTIC INTEGRAL OF THE SECOND KIND.
See Gradshteyn and Ryzhik (1980, p. xxxi) for expres-
sions in terms of THETA FUNCTIONS.
see also Zeta 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. 595, 1972.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.
Jacobian
Given a set y = f(x) of n equations in n variables xi,
. . . , Xnj written explicitly as
y =
7i
h
(i)
or more explicitly as
2/i = /i(a:i, • • • ,Xn)
(2)
the Jacobian matrix, sometimes simply called "the Ja-
cobian'* (Simon and Blume 1994) is denned by
(3)
The Determinant of J is the Jacobian Determi-
nant (confusingly, often called "the Jacobian" as well)
and is denoted
" dyi ,
dxx
dx n
J(#l, #2, £3) —
dy n
dx n J
J =
— I d (yii---iVn)
a(xi,...,x n )
Taking the differential
dy = y x rfx
(4)
(5)
shows that J is the DETERMINANT of the MATRIX y x ,
and therefore gives the ratios of n-D volumes (Con-
tents) in y and x,
dyi • - • dy n =
d(y u ...,y n
d(xx, . . . ,i n )
dx\ * ■ ■ dx n .
(6)
The concept of the Jacobian can also be applied to n
functions in more than n variables. For example, con-
sidering f(u,v,w) and g(U)V,w), the Jacobians
d(f,g)
d(u,v)
d(f,g)
d(u, w)
u
fv
9u
9v
u
/«
9u
9w
(7)
(8)
can be defined (Kaplan 1984, p. 99).
3 variables, the Jacobian takes the
For the case of n
special form
J/(xi,x 2 ,x 3 ) =
dy_
dxi
— x —I
0x2 dxs I
(9)
where a*b is the DOT PRODUCT and b x c is the CROSS
Product, which can be expanded to give
d(yi,y2,ya)
d(x 1 ,X 2 ,X3)
dyi
dyi
dyi
dxi
dx 2
dx 3
i>V2
dy 2
dy 2
dxi
0x2
dxs
Qyz
&V3
®V3.
dxi
dx 2
dxs
(10)
see also CHANGE OF VARIABLES THEOREM, CURVILIN-
EAR Coordinates, Implicit Function Theorem
References
Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA:
Addison-Wesley, pp. 98-99, 123, and 238-245, 1984.
Simon, C. P. and Blume, L. E. Mathematics for Economists.
New York: W. W. Norton, 1994.
Jacobian Conjecture
If det[F f (x)] = 1 for a POLYNOMIAL mapping F (where
det is the Determinant), then F is Bijective with
Polynomial inverse.
Jacobian Curve
The Jacobian of a linear net of curves of order n is a
curve of order 3(n — 1). It passes through all points
common to all curves of the net. It is the LOCUS of
points where the curves of the net touch one another
and of singular points of the curve.
see also Cayleyian Curve, Hessian Covariant,
Steinerian Curve
References
Coolidge, J, L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 149, 1959.
Jacobian Determinant
see Jacobian
Jacobian Group
The Jacobian group of a 1-D linear series is given by in-
tersections of the base curve with the JACOBIAN CURVE
of itself and two curves cutting the series.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 283, 1959.
Jacobsthal-Lucas Number
Jacobsthal-Lucas Number
see Jacobsthal Number
Jacobsthal-Lucas Polynomial
see Jacobsthal Polynomial
Jacobsthal Number
The Jacobsthal numbers are the numbers obtained by
the U n s in the LUCAS SEQUENCE with P = 1 and
Q = — 2 j corresponding to a = 2 and b = — 1. They
and the Jacobsthal-Lucas numbers (the V^s) satisfy the
Recurrence Relation
*Jn — *Jrt — 1 ~P "Jn-
(i)
The Jacobsthal numbers satisfy Jo = and Ji = 1 and
are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... (Sloane's
A001045). The Jacobsthal-Lucas numbers satisfy jo = 2
and j x = 1 and are 2, 1, 5, 7, 17, 31, 65, 127, 257, 511,
1025, ... (Sloane's A014551). The properties of these
numbers are summarized in Horadam (1996). They are
given by the closed form expressions
L(n-1)/2J
r=0 V /
Ln/2j
jn = y^ —
n I n — r
(2)
(3)
where \x\ is the Floor Function and (?) is a Bino-
mial Coefficient. The Binet forms are
J„ = i (a n - 6") = i[2" -(-!)"]
(4)
(5)
j B = a" + & B = 2" + (-l) B .
The Generating Functions are
oo
Y^ Jix'- 1 = (l-x- 2X 2 )- 1 (6)
^2 jt^~ l = (1 + 4x)(l -x- 2x 2 )~
The Simson Formulas are
^n + Wn-l Jn
(-1)"2"
(7)
(8)
jn+ljn-l-j n 2 = 9(-l) n " 1 2 n - 1 = -9(J„+i J„_i- J n 2 ).
(9)
Summation FORMULAS include
2_^ J; — |(J n +2 - 3)
i=2
n
^Ji = |0"n+2 -5).
(10)
(11)
Jacobsthal Polynomial 951
Interrelationships are
jnJn = J2n (12)
jn = Jn+1 + 2 J n _i (13)
9J n = jn + 1 + 2j n _! (14)
jn+1 + jn = 3( Jn+1 + Jn) - 3 • 2 n (15)
jn+1 - jn = 3( Jn+1 - Jn) + 4(-l) n+1 = 2" + 2(-l) n+1
(16)
j n+1 - 2j n = 3(2J„ - J n+1 ) = 3(-l) n+1 (17)
2in+i +j n -i = 3(2J n+1 + J n _x) + 6(-l) n+1 (18)
jn+r + Jn-r = 3( J n +r + Jn-r) + 4(-l)"~ r (19)
= 2 n " r (2 2r + 1) + 2(-l) n - r (20)
Jn + r Jn — r ~ 0\J n + r Jn — r) = 2 \Z — 1J (21)
j„ = 3J n + 2(-l) n (22)
3J n +j n =2 n+1 (23)
Jn + j n = 2 J„+i (24)
jn+2Jn~2 ~ jj = -9(J n+2 J„-2 - J„) 2 = 9(-l) n 2 n - 2
Jrnjn l Jnjm — ^Jm + n
(26)
Jmjn T vJmJn — ^Jm+n
(27)
j„ 2 + 9J„ 2 = 2j2n
(28)
Jmjn Jnjm — (, *■) ^ Jm — n \^)
jmjn - 9Jm Jn = (-l) n 2" +1 j ro _ n (30)
(Horadam 1996)
2 9J„ 2 - (-l) n 2 n+2 (31)
References
Horadam, A. F. "Jacobsthal and Pell Curves." Fib. Quart.
26, 79-83, 1988.
Horadam, A. F. "Jacobsthal Representation Numbers." Fib.
Quart. 34, 40-54, 1996.
Sloane, N. J. A. Sequences A014551 and A001045/M2482 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Jacobsthal Polynomial
The Jacobsthal polynomials are the Polynomials ob-
tained by setting p(x) = 1 and q(x) — 2x in the LUCAS
Polynomial Sequence. The first few Jacobsthal poly-
nomials are
Ji(x) = l
J*(x) = 1
Js(x) = l + 2x
J^z) = 1 + 4x
J 5 (x) = 4x 2 +6x + l,
952 Janko Groups
Jeep Problem
and the first few Jacobs thai-Lucas polynomials are
J2(x) = 4# + 1
jz{x) = 6x + 1
M x )
: 8X + SX + 1
20aT + 10a: + l.
Jacobsthal and Jacobsthal-Lucas polynomials satisfy
Jn(l) = Jn
jn(l) = jn
where J n is a JACOBSTHAL NUMBER and j n is a
Jacobsthal-Lucas Number.
Janko Groups
The Sporadic Groups Ji, J2, ^3 and J 4 . The Janko
group J2 is also known as the Hall-JanKO GROUP.
see also SPORADIC GROUP
References
Wilson, R. A. "ATLAS of Finite Group Represen-
tation." http://for.mat.bham.ac.uk/atlas/Jl.html,
J2.html, J3.html, and J4.html.
Japanese Triangulation Theorem
Let a convex Cyclic Polygon be Triangulated in
any manner, and draw the INCIRCLE to each TRIANGLE
so constructed. Then the sum of the INRADII is a con-
stant independent of the TRIANGULATION chosen. This
theorem can be proved using CARNOT'S THEOREM. It is
also true that if the sum of INRADII does not depend on
the TRIANGULATION of a POLYGON, then the POLYGON
is Cyclic.
see also Carnot's Theorem, Cyclic Polygon, In-
CIRCLE, INRADIUS, TRIANGULATION
References
Honsberger, R. Mathematical Gems III. Washington, DC:
Math, Assoc. Amer., pp. 24-26, 1985.
Lambert, T. "The Delaunay Triangulation Maximizes the
Mean Inradius." Proc. Sixth Canadian Conf. Comput. Ge-
ometry. Saskatoon, Saskatchewan, Canada, pp. 201-206,
August 1994.
Jarnick's Inequality
Given a Convex plane region with Area A and PERI-
METER p, then
\N-A\< Py
where N is the number of enclosed LATTICE POINTS.
see also LATTICE POINT, NOSARZEWSKA'S INEQUALITY
Jeep Problem
Maximize the distance a jeep can penetrate into the
desert using a given quantity of fuel. The jeep is allowed
to go forward, unload some fuel, and then return to its
base using the fuel remaining in its tank. At its base,
it may refuel and set out again. When it reaches fuel it
has previously stored, it may then use it to partially fill
its tank. This problem is also called the Exploration
PROBLEM (Ball and Coxeter 1987).
Given n + / (with < / < 1) drums of fuel at the
edge of the desert and a jeep capable of holding one
drum (and storing fuel in containers along the way),
the maximum one-way distance which can be traveled
(assuming the jeep travels one unit of distance per drum
of fuel expended) is
+ 1 ^ 2z - 1
i=l
2n + T + |[7 + 21n24-^ (|+n)],
2n +
/
where 7 is the EULER-MASCHERONI CONSTANT and
^n(z) the POLYGAMMA FUNCTION.
For example, the farthest a jeep with n = 1 drum can
travel is obviously 1 unit. However, with n = 2 drums of
gas, the maximum distance is achieved by filling up the
jeep's tank with the first drum, traveling 1/3 of a unit,
storing 1/3 of a drum of fuel there, and then returning
to base with the remaining 1/3 of a tank. At the base,
the tank is filled with the second drum. The jeep then
travels 1/3 of a unit (expending 1/3 of a drum of fuel),
refills the tank using the 1/3 of a drum of fuel stored
there, and continues an additional 1 unit of distance on
a full tank, giving a total distance of 4/3. The solutions
for n = 1, 2, ... drums are 1, 4/3, 23/15, 176/105,
563/315, ..., which can also be written as a(n)/b(n),
where
a(n)= Q + i + ...+ ^i- i .)LCM(l J 3 ) 5 l ...,2n-l)
6(n) = LCM(l,3 J 5,...,2n-l)
(Sloane's A025550 and A025547).
see also HARMONIC NUMBER
References
Alway, G. C. "Crossing the Desert." Math. Gaz. 41, 209,
1957.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 32, 1987.
Bellman, R. Exercises 54-55 Dynamic Programming. Prince-
ton, NJ: Princeton University Press, p. 103, 1955.
Fine, N. J. "The Jeep Problem." Amer. Math. Monthly 54,
24-31, 1947.
Gale, D. "The Jeep Once More or Jeeper by the Dozen."
Amer. Math. Monthly 77, 493-501, 1970.
Gardner, M. The Second Scientific American Book of Math-
ematical Puzzles & Diversions: A New Selection. New
York: Simon and Schuster, pp. 152 and 157-159, 1961.
Jenkins-Traub Method
Jerk 953
Haurath, A.; Jackson, B.; Mitchem, J.; and Schmeichel, E.
"Gale's Round- Trip Jeep Problem." Amer. Math. Monthly
102, 299-309, 1995.
Helmer, O. "A Problem in Logistics: The Jeep Problem,"
Project Rand Report No. Ra 15015, Dec. 1947,
Phipps, C. G. "The Jeep Problem, A More General Solution."
Amer, Math, Monthly 54, 458-462, 1947.
Jensen Polynomial
Let f(x) be a real Entire Function of the form
/(*) = $>*!['
k=0
Jenkins- Traub Method
A complicated POLYNOMIAL RoOT-finding algorithm
which is used in the IMSL® (IMSL, Houston, TX) li-
brary and which Press et al. (1992) describe as "prac-
tically a standard in black-box POLYNOMIAL ROOT-
finders."
References
IMSL, Inc. IMSL Math/Library User's Manual. Houston,
TX: IMSL, Inc.
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. 369, 1992.
Ralston, A. and Rabinowitz, P. §8.9-8.13 in A First Course
in Numerical Analysis, 2nd ed. New York: McGraw-Hill,
1978.
Jensen's Formula
/•27T
Jo
ln|;z + e i *|d0 = 27rln :f \ z \,
where
In = max(0, In a;)
and lnx is the NATURAL LOGARITHM.
Jensen's Inequality
For a Real Continuous Concave Function
Y,f( X i) < f (Y. X i
if / is concave down,
E
n ~ \ n J
if / is concave up, and
E/(^) __ f fJ2 x ^
m
IFF xi = X2 — . . . — x n . A special case is
where the 7fcS are Positive and satisfy Turan's In-
equalities
7fc 2 - 7fc-i7fc+i >
for k = 1, 2, The Jensen polynomial g(t) associated
with f(x) is then given by
k=o x /
,k
where (j) is a BINOMIAL COEFFICIENT.
References
Csordas, G.; Varga, R. S.; and Vincze, I. "Jensen Polynomials
with Applications to the Riemann ^-Function." J. Math.
Anal Appl. 153, 112-135, 1990.
Jerabek's Hyperbola
The ISOGONAL CONJUGATE of the EULER LINE. It
passes through the the vertices of a TRIANGLE, the
Orthocenter, Circumcenter, the Lemoine Point,
and the ISOGONAL CONJUGATE points of the NlNE-
Point Center and de Longchamps Point.
see also CIRCUMCENTER, DE LONGCHAMPS POINT, EU-
ler Line, Isogonal Conjugate, Lemoine Point,
Nine-Point Center, Orthocenter
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 rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893.
Pinkernell, G. M. "Cubic Curves in the Triangle Plane." J.
Geom. 55, 141-161, 1996.
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.
Jerk
The jerk j is defined as the time DERIVATIVE of the
Vector Acceleration a,
J = II
da.
dt'
y/x\X 2 " • X n <
see also ACCELERATION, VELOCITY
with equality Iff X\ = X2 = . . . = x n .
see also Concave Function, Mahler's Measure
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.
954 Jinc Function
Jinc Function
The jinc function is defined as
jinc(x) = — — ,
x
where Ji(x) is a Bessel Function of the First
Kind, and satisfies lim x _K) jinc(x) = 1/2. The Deriva-
TIVE of the jinc function is given by
jinc (x) = .
The function is sometimes normalized by multiplying by
a factor of 2 so that jinc(O) = 1 (Siegman 1986, p. 729).
see also Bessel Function of the First Kind, Sinc
Function
References
Siegman, A. E. Lasers. Sausalito, CA: University Science
Books, 1986.
Jitter
A Sampling phenomenon produced when a waveform
is not sampled uniformly at an interval t each time, but
rather at a series of slightly shifted intervals t + At» such
that the average (AU) = 0.
see also Ghost, Sampling
JoachimsthaPs Equation
Using Clebsch-Aronhold Notation,
>n n , >n — 1 >• n — 1 , 1 / 1 \>n-2 a2 n — 2 2 ,
fi a y + fi &a y a x + 2 n \ n ~ -Ufi & a y a x + ...
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves, New
York: Dover, p. 89, 1959.
Johnson Circle
The ClRCUMCIRCLE in JOHNSON'S THEOREM.
see also JOHNSON'S THEOREM
Johnson's Equation
The Partial Differential Equation
d ( , u \ 3a 2
— I U t + UU X + 2 Uxxx + — I + -^ U W —
which arises in the study of water waves.
References
Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and
Chaos. Cambridge, England: Cambridge University Press,
p. 223, 1990.
Johnson Solid
Johnson Solid
The Johnson solids are the CONVEX POLYHEDRA hav-
ing regular faces (with the exception of the completely
regular PLATONIC SOLIDS, the "SEMiREGULAR" AR-
CHIMEDEAN Solids, and the two infinite families of
PRISMS and ANTIPRISMS). There are 28 simple (i.e.,
cannot be dissected into two other regular-faced poly-
hedra by a plane) regular-faced polyhedra in addition
to the Prisms and Antiprisms (Zalgaller 1969), and
Johnson (1966) proposed and Zalgaller (1969) proved
that there exist exactly 92 Johnson solids in all.
A database of solids and VERTEX Nets of these solids is
maintained on the Bell Laboratories Netlib server, but
a few errors exist in several entries. A concatenated and
corrected version of the files is given by Weisstein, to-
gether with Mathematical® (Wolfram Research, Cham-
paign, IL) code to display the solids and nets. The fol-
lowing table summarizes the names of the Johnson solids
and gives their images and nets.
1. Square pyramid
Johnson Solid
6. Pentagonal rotunda
7. Elongated triangular pyramid
8. Elongated square pyramid
^A A A A A
9. Elongated pentagonal pyramid
10. Gyroelongated square pyramid
11. Gyroelongated pentagonal pyramid
Johnson Solid 955
14, Elongated triangular dipyramid
15. Elongated square dipyramid
16. Elongated pentagonal dipyramid
17. Gyroelongated square dipyramid
18. Elongated triangular cupola
19. Elongated square cupola
<a
y
20. Elongated pentagonal cupola
956 Johnson Solid
21. Elongated pentagonal rotunda
Johnson Solid
I I I 1 J [ I I ZD
22. Gyroelongated triangular cupola
23. Gyroelongated square cupola
]A
24. Gyroelongated pentagonal cupola
25. Gyroelongated pentagonal rotunda
27. Triangular orthobicupola
28. Square orthobicupola
<l
A<
A
V
>v
>
<
V
>
30. Pentagonal orthobicupola
31. Pentagonal gyrobicupola
32. Pentagonal orthocupolarontunda
33. Pentagonal gyro cup olarotunda
35. Elongated triangular orthobicupola
Johnson Solid
36. Elongated triangular gyrobicupola
37. Elongated square gyrobicupola
<HA
215
38. Elongated pentagonal orthobicupola
39. Elongated pentagonal gyrobicupola
n
40. Elongated pentagonal orthocupolarotunda
41. Elongated pentagonal gyrocupolarotunda
42. Elongated pentagonal orthobirotunda
Johnson Solid 957
43. Elongated pentagonal gyrobirotunda
44. Gyroelongated triangular bicupola
45. Gyroelongated square bicupola
\AAAA
K.
AAAA
46. Gyroelongated pentagonal bicupola
47. Gyroelongated pentagonal cupolarotunda
48. Gyroelongated pentagonal birotunda
49. Augmented triangular prism
958 Johnson Solid
50. Biaugmented triangular prism
Johnson Solid
51. Triaugmented triangular prism
52. Augmented pentagonal prism
53. Biaugmented pentagonal prism
54. Augmented hexagonal prism
55. Parabiaugmented hexagonal prism
56. Metabiaugmented hexagonal prism
57. Triaugmented hexagonal prism
58. Augmented dodecahedron
59. Parabiaugmented dodecahedron
60. Metabiaugmented dodecahedron
61. Triaugmented dodecahedron
64. Augmented tridiminished icosahedron
Johnson Solid
65. Augmented truncated tetrahedron
66. Augmented truncated cube
67. Biaugmented truncated cube
68. Augmented truncated dodecahedron
69. Parabiaugmented truncated dodecahedron
70. Metabiaugmented truncated dodecahedron
71. Triaugmented truncated dodecahedron
Johnson Solid 959
72. Gyrate rhombicosidodecahedron
73. Parabigyrate rhombicosidodecahedron
74. Metabigyrate rhombicosidodecahedron
75. Trigyrate rhombicosidodecahedron
76. Diminished rhombicosidodecahedron
77. Paragyrate diminished rhombicosidodecahedron
78. Metagyrate diminished rhombicosidodecahedron
79. Bigyrate diminished rhombicosidodecahedron
mhm
f9H8 ~ ; ~ '
^8 ;
960 Johnson Solid
80. Parabidiminished rhombicosidodecahedron
Johnson Solid
81. Metabidiminished rhombicosidodecahedron
82. Gyrate bidiminished rhombicosidodecahedron
83. Tridiminished rhombicosidodecahedron
88. Sphenomegacorona
92. Triangular hebesphenorotunda
The number of constituent n-gons ({n}) for each John-
son solid are given in the following table.
Johnson Solid
Johnson Solid
961
Jn {3} {4} {5} {6} {8} {10}
1 4 1
2 5 1
3 4 3 1
4 4 5 1
5 5 5 1 1
6 10 6 1
7 4 3
8 4 5
9 5 5 1
10 12 1
11 15 1
12 6
13 10
14 6 3
15 8 4
16 10 5
17 16
18 4 9 1
19 4 13 1
20 5 15 1 1
21 10 10 6 1
22 16 3 1
23 20 5 1
24 25 5 1 1
25 30 6 1
26 4 4
27 8 6
28 8 10
29 8 10
30 10 10 2
31 10 10 2
32 15 5 7
33 15 5 7
34 20 12
35 8 12
36 8 12
37 8 18
38 10 20 2
39 10 20 2
40 15 15 7
41 15 15 7
42 20 10 12
43 20 10 12
44 20 6
45 24 10
46 30 10 2
Jn {3} {4} {5} {6} {8} {10}
47 35 5 7
48 40 12
49 6 2
50 10 1
51 14
52 4 4 2
53 8 3 2
54 4 5 2
55 8 4 2
56 8 4 2
57 12 3 2
58 5 11
59 10 10
60 10 10
61 15 9
62 10 2
63 5 3
64 7 3
65 8 3 3
66 12 5 5
67 16 10 4
68 25 5 1 11
69 30 10 2 10
70 30 10 2 10
71 35 15 3 9
72 20 30 12
73 20 30 12
74 20 30 12
75 20 30 12
76 15 25 11 1
77 15 25 11 1
78 15 25 11 1
79 15 25 11 1
80 10 20 10 2
81 10 20 10 2
82 10 20 10 2
83 5 15 9 3
84 12
85 24 2
86 12 2
87 16 1
88 16 2
89 18 3
90 20 4
91 8 2 4
92 13 3 3 1
see also ANTIPRISM, ARCHIMEDEAN SOLID, CONVEX
Polyhedron, Kepler-Poinsot Solid, Polyhedron,
Platonic Solid, Prism, Uniform Polyhedron
References
Bell Laboratories, http://netlib.bell-labs.com/netlib/
polyhedra/.
Bulatov, V. "Johnson Solids." http://www.physics.orst.
edu/ -bulatov/polyhedra/ j ohnson/.
Cromwell, P. R. Polyhedra, New York: Cambridge University
Press, pp. 86-92, 1997.
962
Johnson's Theorem
Hart, G. W. "NetLib Polyhedra DataBase." http://www.li.
net/ *george/virtual-polyhedra/netlib-info. html.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Hume, A. Exact Descriptions of Regular and Semi-Regular
Polyhedra and Their Duals. Computer Science Technical
Report #130. Murray Hill, NJ: AT&T Bell Laboratories,
1986.
Johnson, N. W. "Convex Polyhedra with Regular Faces."
Canad. J. Math. 18, 169-200, 1966.
Pugh, A. "Further Convex Polyhedra with Regular Faces."
Ch. 3 in Polyhedra: A Visual Approach. Berkeley, CA:
University of California Press, pp. 28-35, 1976.
$ Weisstein, E. W. "Johnson Solids." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/JohnsonSolids.m.
# Weisstein, E. W. "Johnson Solid Netlib Database." http://
www . astro . Virginia . edu/-eww6n/raath/notebooks/
JohnsonSolids.dat.
Zalgaller, V. Convex Polyhedra with Regular Faces. New
York: Consultants Bureau, 1969.
Johnson's Theorem
Let three equal Circles with centers Ci, C2, and C3
intersect in a single point O and intersect pairwise in
the points P, Q, and R. Then the ClRCUMCIRCLE J of
APQR (the so-called JOHNSON Circle) is congruent to
the original three.
see also ClRCUMCIRCLE, JOHNSON CIRCLE
References
Emch, A. "Remarks on the Foregoing Circle Theorem."
Amer. Math. Monthly 23, 162-164, 1916.
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 18-21, 1976.
Johnson, R. "A Circle Theorem." Amer. Math. Monthly 23,
161-162, 1916.
Join (Graph)
Let x and y be distinct nodes of G which are not joined
by an EDGE. Then the graph G/xy which is formed by
adding the Edge (cc, y) to G is called a join of G.
Join (Spaces)
Let X and Y be TOPOLOGICAL SPACES. Then their join
is the factor space
Joint Theorem
where ~ is the EQUIVALENCE RELATION
{t — t' — and x = x'
or
t = t* — 1 and y = y.
see also CONE (SPACE), SUSPENSION
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 6, 1976.
Joint Distribution Function
A joint distribution function is a Distribution Func-
tion in two variables defined by
D(x,y)=P(X<x y Y<y) (1)
D x (x) = D(x,oo) (2)
D y (y) = D(oo,y) (3)
so that the joint probability function
P[(*,V) G C)] = , -^ P{x,y)dxdy (4)
(x,y) e C
P(xeA,yeB)= P(x,y)dxdy (5)
J B J A
P{x,y) = P{x E (-oo 9 x],y€ (-00, y]}
/b pa
I P(x,y)dxdy (6)
-00 J — c
-00 </ — 00
P(a < x < a + da,b < y <b + db)
pb+db pa-\-da
= / P(x,y)dxdy^P{a,b)dadb. (7)
Jb J a
A multiple distribution function is of the form
Z?(ai, . . . ,a n ) = P(xi < ai, . . . ,x n < a n ).
see also Distribution Function
Joint Probability Density Function
see Joint Distribution Function
Joint Theorem
see Gaussian Joint Variable Theorem
(8)
x*y = (X x Y x I)/ ~
Jonah Formula
Jonah Formula
A formula for the generalized CATALAN NUMBER p d q i.
The general formula is
n — q
k-1
Y^p d <
n — pi
qz \ k-i
where (£) is a Binomial Coefficient, although
Jonah's original formula corresponded to p = 2, q —
(Hilton and Pederson 1991).
References
Hilton, P. and Pederson, J. "Catalan Numbers, Their Gener-
alization, and Their Uses." Math. Intel. 13, 64-75, 1991.
Jones Polynomial
The second Knot Polynomial discovered. Unlike the
first-discovered ALEXANDER POLYNOMIAL, the Jones
polynomial can sometimes distinguish handedness (as
can its more powerful generalization, the HOMFLY
Polynomial). Jones polynomials are Laurent Poly-
nomials in t assigned to an M 3 KNOT. The Jones poly-
nomials are denoted V L (t) for Links, V K (t) for Knots,
and normalized so that
VJinknot(t) = 1.
(1)
For example, the Jones polynomial of the TREFOIL
Knot is given by
i(t) = t + t 3 - t\
(2)
If a Link has an ODD number of components, then Vl
is a Laurent Polynomial over the Integers; if the
number of components is EVEN, Vl(£) is t 1 ' 2 times a
Laurent Polynomial. The Jones polynomial of a
Knot Sum Li#L 2 satisfies
V Ll #L t = (V tl )(Vt a ).
(3)
X.
\
X
/
y
L + L L_
The Skein Relationship for under- and overcrossings
is
fV L+ - tV L _ = (t 1/2 - t- 1/2 )V Lo . (4)
Combined with the link sum relationship, this allows
Jones polynomials to be built up from simple knots and
links to more complicated ones.
Some interesting identities from Jones (1985) follow. For
any LINK L,
V L (-l) = A i (-l), (5)
Jones Polynomial 963
where Al is the Alexander Polynomial, and
V L (l) = (-2)*-\ (6)
where p is the number of components of L. For any
Knot K,
V K (e 2wi/3 ) = 1 (T)
and
dt
V K (1) = 0.
(8)
Let K" denote the MIRROR IMAGE of a KNOT K. Then
V K '(t) = V K (t- 1 ). (9)
For example, the right-hand and left-hand TREFOIL
Knots have polynomials
Vt«feu(t) = * + t 3 - t* (10)
Vt rrf bu'(*) = r 1 + r s -r 4 . (n)
Jones defined a simplified trace invariant for knots by
1 - V K (t)
Wk ® = (i-W-tY
The Arf Invariant of Wk is given by
Aii(K) = W K {i)
(12)
(13)
(Jones 1985), where i is y/^1. A table of the W poly-
nomials is given by Jones (1985) for knots of up to eight
crossings, and by Jones (1987) for knots of up to 10
crossings. (Note that in these papers, an additional
polynomial which Jones calls V is also tabulated, but
it is not the conventionally defined Jones polynomial.)
Jones polynomials were subsequently generalized to the
two-variable HOMFLY Polynomials, the relationship
being
V(t) ~P(a = t 9 x = t
- f *. - f 1 / 2
-1/2
V(t) = P(£ = it,m = i(i
- ,V+-V2
.1/2
))•
(14)
(15)
They are related to the KAUFFMAN POLYNOMIAL F by
ir(t)=F(-r s/4 ,r 1/4 + t 1/4 ).
(16)
Jones (1987) gives a table of BRAID WORDS and W poly-
nomials for knots up to 10 crossings. Jones polynomi-
als for KNOTS up to nine crossings are given in Adams
(1994) and for oriented links up to nine crossings by
Doll and Hoste (1991). All Prime Knots with 10 or
fewer crossings have distinct Jones polynomials. It is
not known if there is a nontrivial knot with Jones poly-
nomial 1. The Jones polynomial of an (m, n)-TORUS
Knot is
.(m — l)(n-l)/2/-i _ £""1+1 _ £ n + 1 _|_ +™+n\
(17)
964 Jones Polynomial
Jordan-Holder Theorem
Let k be one component of an oriented LINK L. Now
form a new oriented Link L* by reversing the orienta-
tion of k. Then
V L * = t~
V(L),
where V is the Jones polynomial and A is the Linking
NUMBER of k and L — k. No such result is known for
HOMFLY Polynomials (Lickorish and Millett 1988).
Birman and Lin (1993) showed that substituting the
POWER Series for e x as the variable in the Jones poly-
nomial yields a POWER Series whose COEFFICIENTS
are VASSILIEV POLYNOMIALS.
Let L be an oriented connected Link projection of n
crossings, then
n > span V(L), (18)
with equality if L is Alternating and has no Remov-
able Crossing (Lickorish and Millett 1988).
There exist distinct KNOTS with the same Jones poly-
nomial. Examples include (05ooi, IO132), (O8008, IO129),
(O8016, lOise), (IO025, lOose), (IO022, IO035), (10 O4 i,
IO094), (IO043, IO091), (IO059, IO106), (lOoeo, IO083),
(IO071, IO104), (IO073, lOose), (lOosi, IO109), and (IO137,
IO155) (Jones 1987). Incidentally, the first four of these
also have the same HOMFLY POLYNOMIAL.
Witten (1989) gave a heuristic definition in terms of
a topological quantum field theory, and Sawin (1996)
showed that the "quantum group" U q {sl2) gives rise to
the Jones polynomial.
see also Alexander Polynomial, HOMFLY Poly-
nomial, Kauffman Polynomial F, Knot, Link,
Vassiliev Polynomial
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, 1994.
Birman, J. S. and Lin, X.-S. "Knot Polynomials and Vas-
siliev's Invariants." Invent. Math. Ill, 225-270, 1993.
Doll, H. and Hoste, J. "A Tabulation of Oriented Links."
Math. Comput. 57, 747-761, 1991,
Jones, V. "A Polynomial Invariant for Knots via von Neu-
mann Algebras." Bull. Am. Math. Soc. 12, 103-111, 1985.
Jones, V. "Hecke Algebra Representations of Braid Groups
and Link Polynomials." Ann. Math. 126, 335-388, 1987.
Lickorish, W. B. R. and Millett, B. R. "The New Polynomial
Invariants of Knots and Links." Math. Mag. 61, 1-23,
1988.
Murasugi, K. "Jones Polynomials and Classical Conjectures
in Knot Theory." Topology 26, 297-307, 1987.
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.
Sawin, S. "Links, Quantum Groups, and TQFTS." Bull.
Amer. Math. Soc. 33, 413-445, 1996.
Stoimenow, A. "Jones Polynomials." http://www.
informatik.hu-berlin.de/-stoimeno/ptab/jlO.html.
Thistlethwaite, M. "A Spanning Tree Expansion for the Jones
Polynomial." Topology 26, 297-309, 1987.
$$ Weisstein, E. W. "Knots and Links." http: //www. astro.
virginia.edu/-eww6n/math/notebooks /Knot s.m.
Witten, E. "Quantum Field Theory and the Jones Polynom-
ial." Comm. Math. Phys. 121, 351-399, 1989.
Jonquiere's Function
see POLYGAMMA FUNCTION
Jordan Algebra
A nonassociative algebra with the product of elements
A and B defined by the Anticommutator {A,B} =
AB + BA.
see also ANTICOMMUTATOR
Jordan Curve
A Jordan curve is a plane curve which is topologically
equivalent to (a HOMEOMORPHIC image of) the Unit
Circle.
It is not known if every Jordan curve contains all four
Vertices of some Square, but it has been proven
true for "sufficiently smooth" curves and closed convex
curves (Schnirelmann). For every Triangle T and Jor-
dan curve J, J has an INSCRIBED TRIANGLE similar to
T.
see also JORDAN CURVE THEOREM, UNIT CIRCLE
Jordan Curve Theorem
If J is a simple closed curve in M , then R — J has
two components (an "inside" and "outside"), with J the
BOUNDARY of each.
see also JORDAN CURVE, SCHONFLIES THEOREM
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 9, 1976.
Jordan Decomposition Theorem
Let V ^ (0) be a finite dimensional VECTOR Space over
the COMPLEX NUMBERS, and let A be a linear operator
on V. Then V can be expressed as a Direct Sum of
cyclic subspaces.
References
Gohberg, I. and Goldberg, S. "A Simple Proof of the Jor-
dan Decomposition Theorem for Matrices." Amer. Math.
Monthly 103, 157-159, 1996.
Jordan-Holder Theorem
The composition quotient groups belonging to two COM-
POSITION Series of a Finite Group G are, apart from
their sequence, ISOMORPHIC in pairs. In other words, if
I CH S C ...CH 2 CH 1 CG
is one COMPOSITION SERIES and
I CK t C...CK 2 CK 1 cG
Jordan's Inequality
Jordan Polygon 965
is another, then t = s, and corresponding to any compo-
sition quotient group Kj/Kj+x, there is a composition
quotient group Hi/Hi+i such that
Then
K<
Hi
This theorem was proven in 1869-1889.
see also Composition Series, Finite Group, Iso-
morphic Groups
References
Lomont, J. S. Applications of Finite Groups. New York:
Dover, p. 26, 1993.
Jordan's Inequality
1.25 1.5
— x < sinx < x.
7T
References
Yuefeng, F. "Jordan's Inequality." Math. Mag. 69, 126,
1996.
Jordan's Lemma
Jordan's lemma shows the value of the INTEGRAL
■/.
1 = / f(x)e lax dx
(1)
along the Real AXIS is for "nice" functions which
satisfy lim^oo \f(Re i9 )\ = 0. This is established using
a Contour Integral I r which satisfies
lim \I R \ < - lim e = 0.
R— kx> a J2— kx>
To derive the lemma, write
x = Re i9 = R(cos + i sin 0)
dx = iRe i6 dO
and define the CONTOUR INTEGRAL
"-F
Jo
£ f r% i0\ iaRcos 6 ~aR sin 9 • Ty i&
f(Re )e iRe
(2)
(3)
(4)
dO (5)
|/«l
Jo
IHRe'll \e
i9\\ i iaRcos$\ i — aRsin6\
\i\\e t& \dO
R [^ \f{Re i9 )\e' aRsin9 dO
Jo
pn/2
Jo
2R / \f(Re")\e
i0\ I — afisin 9
d6.
(6)
Now, if limfi_>oo \f(Re %6 )\ = 0, choose an e such that
\f(Re ie )\<e 9 so
/.tt/2
\I R \ <2Re / e " ajRsin
Jo
But, for e [0,71-/2],
-6 < sin0,
\I R \<2Re [^ e' 2aRe/7r dO
Jo
'dQ.
(7)
(8)
= 2eR-
-(l-e- aK ).
As long as limj^oo \f{z)\ = 0) Jordan's lemma
lim \I R \ < - lim e =
R^oo a R->oo
(9)
(10)
then follows.
see also CONTOUR INTEGRATION
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 406-408, 1985.
Jordan Measure
Let the set M correspond to a bounded, NONNEGATIVE
function / on an interval < f(x) < c for x G [a, b]. The
Jordan measure, when it exists, is the common value of
the outer and inner Jordan measures of M.
The outer Jordan measure is the greatest lower bound of
the areas of the covering of M, consisting of finite unions
of RECTANGLES. The inner Jordan measure of M is the
difference between the AREA c(a-b) of the RECTANGLE
S with base [a, b] and height c, and the outer measure
of the complement of M in 5.
References
Shenitzer, A. and Steprans, J. "The Evolution of Integra-
tion." Amer. Math. Monthly 101, 66-72, 1994.
Jordan Polygon
see Simple Polygon
966 Josephus Problem
Jugendtraum
Josephus Problem
Given a group of n men arranged in a CIRCLE under the
edict that every mth man will be executed going around
the CIRCLE until only one remains, find the position
L(n, m) in which you should stand in order to be the last
survivor (Ball and Coxeter 1987). The original problem
consisted of a CIRCLE of 41 men with every third man
killed (n = 41, m = 3). In order for the lives of the last
two men to be spared, they must be placed at positions
31 (last) and 16 (second-to-last).
The following array gives the original position of the last
survivor out of a group of n = 1, 2, . . . , if every mth
man is killed:
1
2
1
3
3
2
4
1
1
2
5
3
4
1
2
6
5
1
5
1
4
7
7
4
2
6
3
5
8
1
7
6
3
1
4
4
9
3
1
1
8
7
2
3 8
10
5
4
5
3
3
9
17 8
(Sloane's A032434). The survivor for m = 2 can be
given analytically by
L(n,2) = l + 2n-2 1+LlgnJ ,
where [n\ is the FLOOR FUNCTION and LG is the LOG-
ARITHM to base 2. The first few solutions are therefore
1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 9, 11, 13, 15, 1, . . . (Sloane's
A006257).
Mott-Smith (1954) discusses a card game called "Out
and Under" in which cards at the top of a deck are
alternately discarded and placed at the bottom. This is
a Josephus problem with parameter m = 2, and Mott-
Smith hints at the above closed-form solution.
The original position of the second-to-last survivor is
given in the following table for n = 2, 3, . . . :
1
1
2
1
1
3
1
1
2
4
3
2
1
2
5
1
1
5
1
4
6
3
1
2
1
3
4
7
1
4
6
3
1
3
4
8
3
1
1
2
7
1
3
7
9
5
4
5
3
3
8
1
6
(Sloane's A032435).
Another version of the problem considers a CIRCLE of
two groups (say, "A" and "B") of 15 men each, with ev-
ery ninth man cast overboard. To save all the members
of the "A" group, the men must be placed at positions
1, 2, 3, 4, 10, 11, 13, 14, 15, 17, 20, 21, 25, 28, 29, giving
the ordering
AAAABBBBBAABAAABABBAABBBABBAAB
which can be remembered with the aid of the
Mnemonic "From numbers' aid and art, never will fame
depart." Consider the vowels only, assign a = 1, e = 2,
i = 3 ? o ~ 4, u = 5, and alternately add a number of
letters corresponding to a vowel value, so 4A (o), 5B (u),
2A (e), etc. (Ball and Coxeter 1987).
If every tenth man is instead thrown overboard, the men
from the "A" group must be placed in positions 1, 2, 4,
5, 6, 12, 13, 16, 17, 18, 19, 21, 25, 28, 29, giving the
sequence
AABAAABBBBBAABBAAAABABBBABBAAB
which can be constructed using the MNEMONIC "Rex
paphi cum gente bona dat signa serena" (Ball and Cox-
eter 1987).
see also KlRKMAN'S SCHOOLGIRL PROBLEM, NECK-
LACE
References
Bachet, C. G. Problem 23 in Problemes plaisans et
delectables, 2nd ed. p. 174, 1624.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 32-36,
1987.
Kraitchik, M. "Josephus' Problem." §3.13 in Mathematical
Recreations. New York: W. W. Norton, pp. 93-94, 1942.
Mott-Smith, G. Mathematical Puzzles for Beginners and En-
thusiasts. New York: Dover, 1954.
Sloane, N. J. A. Sequence A006257/M2216 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Jug
see Three Jug Problem
J ugendtraum
Kronecker proved that all the Galois extensions of the
RATIONALS Q with ABELIAN Galois groups are SUB-
FIELDS of cyclotomic fields Q(^n), where //„ is the group
of nth ROOTS OF UNITY. He then sought to find a sim-
ilar function whose division values would generate the
Abelian extensions of an arbitrary Number Field. He
discovered that the j-FUNCTION works for IMAGINARY
quadratic number fields K, but the completion of this
problem, known as Kronecker's Jugendtraum ("dream
of youth"), for other fields remains one of the great un-
solved problems in NUMBER THEORY.
see also ^-FUNCTION
References
Shimura, G. Introduction to the Arithmetic Theory of Auto-
morphic Functions. Princeton, NJ: Princeton University
Press, 1981.
Juggling
Jumping Champion 967
Juggling
The throwing and catching of multiple objects such that
at least one is always in the air. Some aspects of jug-
gling turn out to be quite mathematical. The best ex-
amples are the two-handed asynchronous juggling se-
quences known as "SlTESWAPS."
see also SlTESWAP
References
Buhler, J.; Eisenbud, D.; Graham, R.; and Wright, C. "Jug-
gling Drops and Descents." Amer. Math. Monthly 101,
507-519, 1994.
Donahue, B. "Jugglers Now Juggle Numbers to Compute
New Tricks for Ancient Art." New York Times, pp. B5
and BIO, Apr. 16, 1996.
Juggling Information Service. "Siteswaps." http://wwv.
juggling.org/help/siteswap,
Julia Fractal
see Julia Set
Julia Set
Let R(z) be a rational function
*M - P{Z
Q{zY
(1)
where z e C, C* is the Riemann Sphere Cu{oo}, and
P and Q are POLYNOMIALS without common divisors.
The "filled-in" Julia set Jr is the set of points z which
do not approach infinity after R(z) is repeatedly applied.
The true Julia set is the boundary of the filled-in set
(the set of "exceptional points"). There are two types
of Julia sets: connected sets and Cantor Sets.
For a Julia set J c with c < 1, the CAPACITY DIMENSION
is
l+^2+<?(|c| 3 ). (2)
■^capacity
For small c, J c is also a JORDAN Curve, although its
points are not COMPUTABLE.
Quadratic Julia sets are generated by the quadratic
mapping
Z n +l — Z n +C
(3)
for fixed c. The special case c = -0.123 + 0.745z is
called Douady's Rabbit Fractal, c = -0.75 is called
the San Marco Fractal, and c ~ -0.391 - 0.587i
is the SlEGEL Disk Fractal. For every c, this trans-
formation generates a FRACTAL. It is a CONFORMAL
Transformation, so angles are preserved. Let J be
the Julia Set, then x' \- > x leaves J invariant. If a
point P is on J, then all its iterations are on J. The
transformation has a two- valued inverse. If b = and y
is started at 0, then the map is equivalent to the Logis-
tic Map. The set of all points for which J is connected
is known as the Mandelbrot Set.
see also Dendrite Fractal, Douady's Rabbit
Fractal, Fatou Set, Mandelbrot Set, Newton's
Method, San Marco Fractal, Siegel Disk Frac-
tal
References
Dickau, R. M. "Julia Sets." http://forum.swarthmore.edu/
advanced/robertd/julias .html.
Dickau, R. M. "Another Method for Calculating Julia Sets."
http:// forum . swarthmore . edu / advance&y robertd /
inversejulia.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.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 124-
126, 138-148, and 177-179, 1991.
Peitgen, H.-O. and Saupe, D. (Eds.). "The Julia Set," "Julia
Sets as Basin Boundaries," "Other Julia Sets," and "Ex-
ploring Julia Sets." §3.3.2 to 3.3.5 in The Science of Frac-
tal Images. New York: Springer-Verlag, pp. 152-163, 1988.
Schroeder, M. Fractals, Chaos, Power Laws. New York:
W, H. Freeman, p. 39, 1991.
Wagon, S. "Julia Sets." §5.4 in Mathematica in Action. New
York: W. H. Freeman, pp. 163-178, 1991.
Jump
A point of Discontinuity.
see also Discontinuity, Jump Angle, Jumping
Champion
Jump Angle
A Geodesic Triangle with oriented boundary yields
a curve which is piecewise DlFFERENTlABLE. Further-
more, the Tangent Vector varies continuously at all
but the three corner points, where it changes suddenly.
The angular difference of the tangent vectors at these
corner points are called the jump angles.
see also ANGULAR DEFECT, GAUSS-BONNET FORMULA
Jumping Champion
An integer n is called a JUMPING CHAMPION if n is
the most frequently occurring difference between con-
secutive primes n < N for some N (Odlyzko et at. ).
This term was coined by J. H. Conway in 1993. There
are occasionally several jumping champions in a range.
Odlyzko et al. give a table of jumping champions for
n < 1000, consisting mainly of 2, 4, and 6. 6 is the
jumping champion up to about n « 1.74 x 10 35 , at
which point 30 dominates. At n « 10 425 , 210 becomes
champion, and subsequent PRIMORIALS are conjectured
to take over at larger and larger n. Erdos and Straus
(1980) proved that the jumping champions tend to in-
finity under the assumption of a quantitative form of the
fc-tuples conjecture.
see also Prime Difference Function, Prime Gaps,
Prime Number, Primorial
References
Erdos, P.; and Straus, E. G. "Remarks on the Differences
Between Consecutive Primes." Elem. Math. 35, 115-118,
1980.
968 Jung's Theorem Just One
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer-Verlag, 1994.
Nelson, H. "Problem 654." X Recr. Math, 11, 231, 1978-
1979.
Odlyzko, A.; Rubinstein, M.; and Wolf, M. "Jumping
Champions." http : //www . research . att . com/*amo/doc/
recent.html.
Jung's Theorem
Every finite set of points with Span d has an enclosing
Circle with Radius no greater than \/3d/3.
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 28, 1983.
Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 103-110,
1957.
Just If
see Iff
Just One
see Exactly One
k-ary Divisor
K
fe-ary Divisor
Let a DIVISOR d of n be called a 1-ary divisor if d J_ n/d.
Then d is called a fc-ary divisor of n, written d|kn, if the
Greatest Common (k - l)-ary divisor of d and (n/d)
is 1.
In this notation, d\n is written d\on, and d||n is written
d|in. p x is an Infinary Divisor of p y (with y > 0) if
P*|y-lP V -
see a/50 Divisor, Greatest Common Divisor, Infi-
nary Divisor
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 54, 1994.
k- Chain
Any sum of a selection of IlfcS, where life denotes a fc-D
POLYTOPE.
see also fc-ClRCUlT
fc- Circuit
A fc-CHAlN whose bounding (k - 1)-Chain vanishes.
fc-Coloring
A fc-coloring of a Graph G is an assignment of one of
k possible colors to each vertex of G such that no two
adjacent vertices receive the same color.
see also COLORING, EDGE-COLORING
References
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:
Assaults and Conquest. New York: Dover, p. 13, 1986.
A>Form
see Differential £;-Form
X-Function
K-Function
An extension of the K-function
iiC(nH-l) = l 1 2 2 3 3 -
denned by
K(z)
G(z)
969
(i)
(2)
Here, G{z) is the G-Function defined by
(n!)»
G(n + 1);
if n =
K{n+1) ~ \0!l!2!--(n-l)! if n > 0.
(3)
tt
The if-function is given by the integral
oz-l
K(z) = (27v)- {z - 1)/2 exp
GH
ln(<!) dt
(4)
and the closed-form expression
K{z)=exp[C(-l,z)-C(-l)], (5)
where £(z) is the RlEMANN Zeta FUNCTION, £'(z) its
Derivative, £(a, z) is the Hurwitz Zeta Function,
and
~dC(s,z)
C'(a,z)
ds
(6)
K(z) also has a STIRLING-Iike series
K{z + 1) = {2 l '\ 1 z) 1 ^z^ 1 )
(\ 2 , 1 ^4 Bq
4 • 5 • 6z 4
-•••), (7)
where
*i = mi)} 8
= e -(ln2)/3-12C , (-l)
^2 2/3 7re 7_1 ~ C ' C2)/CC2) ,
(8)
(9)
(10)
and 7 is the EULER-MASCHERONI Constant (Gosper).
The first few values of K(n) for n = 2, 3, ... are 1,
4, 108, 27648, 86400000, 4031078400000, . . . (Sloane's
A002109). These numbers are called HYPERFACTORI-
ALS by Sloane and Plouffe (1995).
see also G-Function, Glaisher-Kinkelin Con-
stant, Hyperfactorial, Stirling's Series
References
Sloane, N. J. A. Sequence A002109/M3706 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
970 K-Graph
K-Gvaph
The GRAPH obtained by dividing a set of VERTICES {1,
. . . , n} into k — 1 pairwise disjoint subsets with VER-
TICES of degree rai, . . . , nfc-i, satisfying
n = ni + . . . + rifc^i,
and with two VERTICES joined IFF they lie in distinct
Vertex sets. Such Graphs are denoted K ni1 ..., nk .
see also Bipartite Graph, Complete Graph, Com-
plete A;-Partite Graph, ^-Partite Graph
fc-Matrix
A fc-matrix is a kind of CUBE ROOT of the IDENTITY
Matrix denned by
k =
— '
i
1
It satisfies
I,
where I is the IDENTITY MATRIX.
see also Cube Root, Quaternion
fc-Partite Graph
A A;-partite graph is a GRAPH whose VERTICES can be
partitioned into k disjoint sets so that no two vertices
within the same set are adjacent.
see also COMPLETE fc-PARTITE GRAPH, K-GRAPH
References
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:
Assaults and Conquest New York: Dover, p. 12, 1986.
fc-Statistic
An Unbiased Estimator of the Cumulants m of
a Distribution. The expectation values of the k-
statistics are therefore given by the corresponding Cu-
MULANTS
(hi) = ki
(k 2 ) — K2
(k 3 ) = K 3
{k 4 } = «4
(1)
(2)
(3)
(4)
(Kenney and Keeping 1951, p. 189). For a sample of
size, N, the first few fc-statistics are given by
fci
k 2
k 3
m\
N
N-l
TTl2
N 2
-m 3
{N-l){N-2)
N 2 [(N + l)m 4 - 3(iV - l)m 2 2 ]
(N - 1){N - 2)(N - 3)
(5)
(6)
(7)
(8)
k-Statistic
where mi is the sample MEAN, 7712 is the sample VARI-
ANCE, and mi is the sample zth MOMENT about the
Mean (Kenney and Keeping 1951, pp. 109-110, 163-
165, and 189; Kenney and Keeping 1962). These statis-
tics are obtained from inverting the relationships
(mi) = (i
i \ N ~ X
(m 2 ) = -Jj—m
(9)
(10)
/ m2 2\ = ( N - *)[(* ~ l )v* + (^ ~ 2N + 3 W] (n)
(-3) = {N -T- 2) »s
N 3
N 2
(12)
(m 4 ) =
(N - 1)[(N 2 - 3N + 3)^X4 + 3(2iV - 3)^2 2 ]
N s
(13)
The first moment (sample Mean) is
N
2 = 1
and the expectation value is
< mi > = (^E Z4 ) =/1 *
(14)
(15)
The second MOMENT (sample STANDARD DEVIATION)
is
m 2 = ((* - m) 2 ) = (x 2 ) - 2n (x) +(i 2 = (a= 2 ) - A* 2
N / N \ 2
i=i
N
1=1
N 2
y 2?i 1 y XiXj
*,i=i
N-l
N 2
N N
/ ; Xi ~ JV2 2-^ XiX ^
(16)
2 = 1
and the expectation value is
N
^) = i V 1 (^E a; ' 2 )-^( E^ ;
i=l
N-l , N(N-l) 2
-M2 -
N *-* N 2 >* >
since there are N(N — 1) terms XiXj, using
(XiXj) = (a*) <Xj) = (xi) 2 ,
(17)
(18)
k-Statistic
k-Statistic 971
and where ja 2 is the MOMENT about 0. Using the iden-
tity
and simplifying then gives
M2 = ^2 + M
(19)
to convert to the MOMENT fi 2 about the MEAN and
simplifying then gives
(m 2 > = —ff-V>2
(20)
The factor (N - 1)/N is known as BESSEL's CORREC-
TION.
The third MOMENT is
m 3 = ((as - /i) 3 ) = (as 3 - 3fix 2 + 3fi 2 x - // 3 )
= (x 3 ) - 3fi (x 2 ) + Sfi 2 (x) - fi 3
= (as 3 ) - 3fi (as 2 ) + 3ju 3 - fi 3
= (as 3 ) -3Ai(as 2 ) + 2/x 3
" TV.
+ 21
+
(E-)
iV3
Now use the identities
-t a j
(21)
(22)
(e^He^e^+E 1
( \Jasj J = \^ast 3 + 3 \jasi 2 asj +6 N^ XiXjX k) (23)
where it is understood that sums over products of vari-
ables exclude equal indices. Plugging in
ms =
+ {-^2+ 3 -^)T, Xi2 ^ +6 -w^ XiXiXk - (24)
The expectation value is then given by
(m 3 > = (jf ~ Jp + J^) N »*
(25)
where \x 2 is tne MOMENT about 0. Plugging in the iden-
tities
(J>2 — V>2 + M
(26)
(27)
(JV-l)(iV-2)
< m 3) = J^ M3
(28)
(Kenney and Keeping 1951, p. 189).
The fourth Moment is
m 4 = ((x - fj,) 4 ) = (x 4 - 4x 3 n + 6xV - 4xp 3 + ft*)
= (x 4 ) - Aft <x 3 ) + 6ft 2 <x 2 ) - 3/x 4
^(E^(E-)-^(E^
Now use the identities
(E *•) 2 (E x * 2 ) = E x<4 + 2 E ^
+2 J^ x 2 x 2 + 2 ^ Xi 2 x i x fe (31)
(yjxij = /Jxi 4 +4^Jxi 3 Xj i +6^Jxi 2 Xj 2
2
+12 \^ XjXfc + 24 \J XjXjXfcXi. (32)
Plugging in,
/ 1 4 6 3\v-^ 4
+ (-£ +2 -£-*-fOE«^
+ ( 2 -^r-«-]^)E a "V
+ ( 2 -^- i2 -J0e*^ x *
o
The expectation value is then given by
/ 12 _ JLJEP
+ Viv3 - ^
+
(HL-ZtL) i
Viv 3 iv 4
72
) iiV(AT - 1)(JV - 2)^ M 2
-—±N(N-1)(N- 2)(N - 3)n 4 , (34)
where ^ are MOMENTS about 0. Using the identities
\i% = M2 + ^ 2 (35)
M3 = A*3 + 3^2M + V? (36)
M4 = ^4 + 4^3// + 6/z 2 ^ 2 + ^ 4 (37)
972 k-Statistic
and simplifying gives
(m 4 )
(N - l)[(iV 2 - 3N + 3)^4 + 3(2AT - 3)^ 2 2
iV 3
(Kenney and Keeping 1951, p. 189).
The square of the second moment is
(38)
m 2 2 = «z 2 ) - m 2 ) 2 = {x 2 Y - 2f (x 2 ) + n*
= (^p.1;=(^-)'(sE'.1
-MX- , )'-*(Z«)'(X'>')
+
^(E^)"
iV 4
Now use the identities
(39)
(J2 xi *) = $>i 4 + 2$>«V (40)
+2 ^ x ? x i j + 2 ^2 x ? x i Xk ( 41 )
2
+4Vn s ij + \2^2xjX k 4- 2\22 XiX i XkXl - ( 42 )
Plugging in,
= ( N 2 N s + N4 )z2 Xi4
+ (- 2 -^+ 4 -^)£^-
24 v— *
+ JjiZ^ XiX i XkXl ( 43 )
The expectation value is then given by
»\-(-L 2
2 4
+ ^)iV M 4
+ (-^ + ^)" JV(iV - 1)(iV - 2)M2M2
94
+ ^£*(* " 1)(^ - ^ - ^ ( 44 )
k-Statistic
where ^ are MOMENTS about 0. Using the identities
A* 2 = V>2 + ^
M3 = M3 + 3^2M + M 3
/i 4 = ^4 + 4^3/i 4- 6^i2M + M
(45)
(46)
(47)
and simplifying gives
7 2 , (N - 1)[(N - 1) M4 + (iV 2 - 27V + 3) M2 2 ]
(m 2 ) =
iV 3
(Kenney and Keeping 1951, p. 189).
The VARIANCE of k 2 is given by
var(fc 2 ) = -r} + ■
N (N- 1)k 2 2 '
so an unbiased estimator of var(&2) is given by
2fc 2 2 AT + (iV-l)A;4
var(fc 2 ) :
N(N + 1)
(48)
(49)
(50)
(Kenney and Keeping 1951, p. 189). The Variance of
kz can be expressed in terms of CUMULANTS by
, . kb 9k 2 k 4 . 9k 3 2 ,
var(fc 3 ) = 1T7 + — 7 + tt 7 + ■
6k 2 *
N N-l N-l N(N-l)(N-2)'
(51)
An Unbiased Estimator for var(fc 3 ) is
var(fc3) =
6k 2 2 N(N - 1)
(N - 2)(N + 1)(N + 3)
(Kenney and Keeping 1951, p. 190).
(52)
Now consider a finite population. Let a sample of N
be taken from a population of M. Then Unbiased Es-
timators M 2 for the population Mean ^, M 2 for the
population VARIANCE ^2, G\ for the population Skew-
NESS 71, and G2 for the population KURTOSIS 72 are
Mi =/i
M ~ N
_ M -2N / M-l
1_ M-2 ]/ N(M-N) 11
(M - 1)(M 2 - 6MiV + M + 6JV 2 ) 72
iV(M-2)(M-3)(M-iV)
_ 6M(MAT + M- AT 2 -1)
iV(M-2)(M-3)(M-iV)
(53)
(54)
(55)
(56)
(Church 1926, p. 357; Carver 1930; Irwin and Kendall
1944; Kenney and Keeping 1951, p. 143), where 71 is
the sample SKEWNESS and 72 is the sample KURTOSIS.
see also GAUSSIAN DISTRIBUTION, KURTOSIS, MEAN,
Moment, Skewness, Variance
k-Subset
k-Tuple Conjecture 973
References
Carver, H. C. (Ed.)- "Fundamentals of the Theory of Sam-
pling." Ann. Math. Stat. 1, 101-121, 1930.
Church, A. E. R. "On the Means and Squared Standard-
Deviations of Small Samples from Any Population."
Biometrika 18, 321-394, 1926.
Irwin, J. O. and Kendall, M. G. "Sampling Moments of Mo-
ments for a Finite Population." Ann. Eugenics 12, 138-
142, 1944.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Kenney, J. F. and Keeping, E. S. "The fc-Statistics." §7.9 in
Mathematics of Statistics , Pt. 1, 3rd ed, Princeton, NJ:
Van Nostrand, pp. 99-100, 1962.
k- Subset
A fc-subset is a Subset containing exactly k elements.
see also SUBSET
fc-Theory
A branch of mathematics which brings together ideas
from algebraic geometry, Linear Algebra, and Num-
ber Theory. In general, there are two main types of
fc-theory: topological and algebraic.
Topological fc-theory is the "true" fc-theory in the sense
that it came first. Topological A;- theory has to do with
Vector Bundles over Topological Spaces. Ele-
ments of a fc-theory are Stable Equivalence classes
of Vector Bundles over a Topological Space. You
can put a Ring structure on the collection of Stably
Equivalent bundles by denning Addition through the
Whitney Sum, and Multiplication through the Ten-
sor Product of Vector Bundles. This defines "the
reduced real topological fc-theory of a space."
"The reduced A;-theory of a space" refers to the same
construction, but instead of REAL VECTOR BUNDLES,
Complex Vector Bundles are used. Topological k-
theory is significant because it forms a generalized Co-
HOMOLOGY theory, and it leads to a solution to the vec-
tor fields on spheres problem, as well as to an under-
standing of the J-homeomorphism of HOMOTOPY THE-
ORY.
Algebraic fc-theory is somewhat more involved. Swan
(1962) noticed that there is a correspondence between
the CATEGORY of suitably nice TOPOLOGICAL SPACES
(something like regular HAUSDORFF SPACES) and C*-
ALGEBRAS. The idea is to associate to every SPACE the
C*-Algebra of Continuous Maps from that Space
to the Reals.
A Vector Bundle over a Space has sections, and
these sections can be multiplied by CONTINUOUS Func-
tions to the Reals. Under Swan's correspondence,
Vector Bundles correspond to modules over the C*-
Algebra of Continuous Functions, the Modules
being the modules of sections of the VECTOR BUNDLE.
This study of MODULES over C*-ALGEBRA is the start-
ing point of algebraic fc-theory.
The QuiLLEN-LlCHTENBAUM CONJECTURE connects al-
gebraic fc-theory to Etale cohomology.
see also C*-ALGEBRA
References
Srinivas, V. Algebraic k-Theory, 2nd ed. Boston, MA:
Birkhauser, 1995.
Swan, R. G. "Vector Bundles and Projective Modules."
Trans. Amer. Math. Soc. 105, 264-277, 1962.
A;- Tuple Conjecture
The first of the HARDY-LlTTLEWOOD CONJECTURES.
The fc-tuple conjecture states that the asymptotic num-
ber of Prime Constellations can be computed ex-
plicitly. In particular, unless there is a trivial divisi-
bility condition that stops p, p + ai, . . . , p + a^ from
consisting of Primes infinitely often, then such Prime
Constellations will occur with an asymptotic den-
sity which is computable in terms of ai, ..., a*,. Let
< mi < 77i2 < . • . < 77ifc, then the fc-tuple conjecture
predicts that the number of PRIMES p < x such that
p + 2mi, p + 2m 2 , • • . , p + 2m k are all Prime is
P(a:;rai,m2, . . . ,mfc) ~ C(m 1 ,m2, - . - ,mfc)
f
dt
ln fc+1 *'
where
C(mi,m 2 ,...,mjfe) = 2 fc |J
1-
w{q\m\ ,m2,...,rofe)
(1-5)
fc+1
the product is over Odd Primes 5, and
w (<?; 7771,7712,... ,m fc )
(1)
(2)
(3)
denotes the number of distinct residues of 0, mi, . . . ,
mk (mod q) (Halberstam and Richert 1974, Odlyzko).
If k = 1, then this becomes
<*->-»n?te§m5i
(1 - 1)'
q\m
(4)
This conjecture is generally believed to be true, but has
not been proven (Odlyzko et al. ). The following spe-
cial case of the conjecture is sometimes known as the
Prime Patterns Conjecture. Let S be a Finite
set of Integers. Then it is conjectured that there ex-
ist infinitely many k for which {k + s : s € S} are all
Prime Iff 5 does not include all the Residues of any
Prime. The Twin Prime Conjecture is a special
case of the prime patterns conjecture with 3 = {0,2}.
This conjecture also implies that there are arbitrarily
long Arithmetic Progressions of Primes.
see also Arithmetic Progression, Dirichlet's
Theorem, Hardy-Littlewood Conjectures, A;-
Tuple Conjecture, Prime Arithmetic Progres-
sion, Prime Constellation, Prime Quadruplet,
974 Kabon Triangles
Kakeya Needle Problem
Prime Patterns Conjecture, Twin Prime Con-
jecture, Twin Primes
References
Brent, R. P. "The Distribution of Small Gaps Between Suc-
cessive Primes." Math. Comput. 28, 315-324, 1974.
Brent, R. P. "Irregularities in the Distribution of Primes and
Twin Primes." Math. Comput. 29, 43-56, 1975.
Halberstam, E. and Richert, H.-E. Sieve Methods. New York:
Academic Press, 1974.
Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Par-
titio Numerorum.' III. On the Expression of a Number as
a Sum of Primes." Acta Math. 44, 1-70, 1922.
Odlyzko, A.; Rubinstein, M.; and Wolf, M. "Jumping Cham-
pions."
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, pp. 66-68,
1994.
Kabon Triangles
The largest number N(n) of nonoverlapping TRIANGLES
which can be produced by n straight LINE SEGMENTS.
The first few terms are 1, 2, 5, 7, 11, 15, 21, . . . (Sloane's
A006066).
References
Sloane, N. J. A. Sequence A006066/M1334 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Kac Formula
The expected number of Real zeros E n of a Random
Polynomial of degree n is
As n ■
where
Ci = -
7T
^ I «
o — o
- 4 -f
* Jo
CO,
9
E n •
In 2
(n + l) 2 £ 2 ~
(t 2 - l) 2 (t 2
I) 2
dt
1 _ (n + l) 2 t 2 "
(1-i 2 ) 2 (l-t 2n + 2 ) 2
dt.
- In n + d + h 0(n~
7r nn
(1)
(2)
(3)
4e -2x
(1
X+l
dx
-2x\2
= 0.6257358072.... (4)
The initial term was derived by Kac (1943).
References
Edelman, A. and Kostlan, E. "How Many Zeros of a Random
Polynomial are Real?" Bull. Amer. Math. Soc. 32, 1-37,
1995.
Kac, M. "On the Average Number of Real Roots of a Random
Algebraic Equation." Bull. Amer. Math. Soc. 49, 314-
320, 1943.
Kac, M. "A Correction to 'On the Average Number of Real
Roots of a Random Algebraic Equation'." Bull. Amer.
Math. Soc. 49, 938, 1943.
Kac Matrix
The (n + 1) x (n + 1) Tridiagonal Matrix (also called
the Clement Matrix) defined by
r0
n
On
1
n —
1
2
n-2
n- 1
1
Lo
n
oJ
The Eigenvalues are 2k - n for k = 0, 1, . . . , n.
Kahler Manifold
A manifold for which the Exterior Derivative of the
Fundamental Form Q associated with the given Her-
mitian metric vanishes, so dQ = 0.
References
Amoros, J. Fundamental Groups of Compact Kahler Mani-
folds. Providence, RI: Amer. Math. Soc, 1996.
Iyanaga, S. and Kawada, Y. (Eds.). "Kahler Manifolds."
§232 in Encyclopedic Dictionary of Mathematics. Cam-
bridge, MA: MIT Press, pp. 732-734, 1980.
Kakeya Needle Problem
What is the plane figure of least Area in which a line
segment of width 1 can be freely rotated (where transla-
tion of the segment is also allowed)? Besicovitch (1928)
proved that there is no Minimum Area. This can be
seen by rotating a line segment inside a DELTOID, star-
shaped 5-oid, star-shaped 7-oid, etc. When the figure
is restricted to be convex, Cunningham and Schoenberg
(1965) found there is still no minimum AREA. How-
ever, the smallest simple convex domain in which one
can put a segment of length 1 which will coincide with
itself when rotated by 180° is
^(5-2^)^ = 0.284258...
(Le Lionnais 1983).
see also Curve of Constant Width, Lebesgue Min-
imal Problem, Reuleaux Polygon, Reuleaux Tri-
angle
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 99-101,
1987.
Besicovitch, A. S. "On Kakeya's Problem and a Similar One."
Math. Z. 27, 312-320, 1928.
Besicovitch, A. S. "The Kakeya Problem." Amer. Math.
Monthly 70, 697-706, 1963.
Cunningham, F. Jr. and Schoenberg, I. J. "On the Kakeya
Constant." Canad. J. Math. 17, 946-956, 1965.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 24, 1983.
Ogilvy, C. S. A Calculus Notebook. Boston: Prindle, Weber,
& Schmidt, 1968.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 147-153, 1990.
KakutanVs Fixed Point Theorem
Pal, J. "Ein Minimumproblem fur Ovale." Math. Ann. 88,
311-319, 1921.
Plouffe, S. "Kakeya Constant." http://lacim.uqam.ca/
piDATA/kakeya . txt.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 50-52, 1991.
KakutanPs Fixed Point Theorem
Every correspondence that maps a compact convex sub-
set of a locally convex space into itself with a closed
graph and convex nonempty images has a fixed point.
see also FIXED POINT THEOREM
Kakutani's Problem
see Collatz Problem
Kalman Filter
An Algorithm in Control Theory introduced by
R. Kalman in 1960 and refined by Kalman and R. Bucy.
It is an Algorithm which makes optimal use of im-
precise data on a linear (or nearly linear) system with
Gaussian errors to continuously update the best esti-
mate of the system's current state.
see also WIENER FILTER
References
Chui, C. K. and Chen, G. Kalman Filtering: With Real-Time
Applications, 2nd ed. Berlin: Springer- Verlag, 1991.
Grewal, M. S. Kalman Filtering: Theory & Practice. Engle-
wood Cliffs, NJ: Prentice-Hall, 1993.
KAM Theorem
see Kolmogorov-Arnold-Moser Theorem
Kaplan- Yorke Conjecture
Kanizsa Triangle
975
Kampyle of Eudoxus
A curve studied by Eudoxus in relation to the classical
problem of Cube Duplication. It is given by the polar
equation
a,
and the parametric equations
x = a sec t
y — a tan t sec t
with t e [-7r/2,7r/2].
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 141-143, 1972.
MacTutor History of Mathematics Archive. "Kampyle of Eu-
doxus." http: //www-groups . dcs . st-and. ac .uk/ -history
/Curves/Kampyle . html.
V ~7
An optical ILLUSION, illustrated above, in which the
eye perceives a white upright Equilateral Triangle
where none is actually drawn.
see also Illusion
References
Bradley, D. R. and Petry, H. M. "Organizational Determi-
nants of Subjective Contour." Amer. J. Psychology 90,
253-262, 1977.
Fineman, M. The Nature of Visual Illusion. New York:
Dover, pp. 26, 137, and 156, 1996.
Kantrovich Inequality
Suppose xi < X2 < . . . < x n are given POSITIVE num-
bers. Let Ai, . . . , A n > and ^ Xj — 1. Then
(E A ^)(E A ^ _1 )<^G- a ,
where
A— |(zi +x n )
G = y/xiXn
are the Arithmetic and Geometric Mean, respec-
tively, of the first and last numbers.
References
Ptak, V. "The Kantrovich Inequality." Amer. Math. Monthly
102, 820-821, 1995.
Kaplan- Yorke Conjecture
There are several versions of the Kaplan- Yorke con-
jecture, with many of the higher dimensional ones re-
maining unsettled. The original Kaplan- Yorke conjec-
ture (Kaplan and Yorke 1979) proposed that, for a
two-dimensional mapping, the CAPACITY DIMENSION D
equals the Kaplan- Yorke Dimension D K y,
D = D K y = d Ly a = l + — ,
0~2
where <j\ and <r 2 are the Lyapunov Characteristic
Exponents. This was subsequently proven to be true in
1982. A later conjecture held that the KAPLAN- YORKE
Dimension is generically equal to a probabilistic dimen-
sion which appears to be identical to the INFORMATION
DIMENSION (Prederickson et al. 1983). This conjecture
is partially verified by Ledrappier (1981). For invertible
2-D maps, v = a — D, where v is the CORRELATION
Exponent, a is the Information Dimension, and D
is the Capacity Dimension (Young 1984).
976 Kaplan-Yorke Dimension
Kaprekar Routine
see also CAPACITY DIMENSION, KAPLAN- YORKE DI-
MENSION, Lyapunov Characteristic Exponent,
Lyapunov Dimension
References
Chen, Z. M. "A Note on Kaplan-Yorke- Type Estimates on
the Fractal Dimension of Chaotic Attractors." Chaos, Soli-
tons, and Fractals 3, 575-582, 1994.
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.
Kaplan, J. L. and Yorke, J. A. In Functional Differen-
tial Equations and Approximations of Fixed Points (Ed.
H.-O. Peitgen and H.-O. Walther). Berlin: Springer-
Verlag, p. 204, 1979.
Ledrappier, F. "Some Relations Between Dimension and Lya-
punov Exponents." Commun. Math. Phys. 81, 229—238,
1981.
Worzbusekros, A. "Remark on a Conjecture of Kaplan and
Yorke." Proc. Amer. Math. Soc. 85, 381-382, 1982.
Young, L. S. "Dimension, Entropy, and Lyapunov Exponents
in Differentiable Dynamical Systems." Phys. A 124, 639-
645, 1984
Kaplan-Yorke Dimension
<Tl + . . . + CTj
Kappa Curve
D K y = j 4-
Wj+i\
where <n < a n are Lyapunov Characteristic Expo-
nents and j is the largest Integer for which
Ai + . . . + A,- > 0.
If v = a = D, where v is the CORRELATION EX-
PONENT, a the Information Dimension, and D the
Hausdorff Dimension, then
D<D K y
(Grassberger and Procaccia 1983).
References
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.
Kaplan-Yorke Map
A curve also known as GUTSCHOVEN's CURVE which
was first studied by G. van Gutschoven around 1662
(MacTutor Archive). It was also studied by Newton
and, some years later, by Johann Bernoulli. It is given
by the Cartesian equation
/ 2 , 2\ 2 2 2
(x + y )y = a x ,
by the polar equation
v — a cot 0,
and the parametric equations
x — a cos t cot t
y ~ a COS t.
(1)
(2)
(3)
(4)
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York; Dover, pp. 136 and 139-141, 1972.
MacTutor History of Mathematics Archive. "Kappa Curve."
http: //www-groups . dcs . st-and.ac .uk/ -history /Curves
/Kappa. html.
Kaprekar Number
Consider an rc-digit number k. Square it and add the
right n digits to the left n or n — 1 digits. If the resultant
sum is k, then k is called a Kaprekar number. The first
few are 1, 9, 45, 55, 99, 297, 703, . . . (Sloane's A006886).
9^ =81
8 + 1 = 9
Xn-\-l — ^Xn
y n+1 - ay n + cos(47rz n ),
where x nj y n are computed mod 1. (Kaplan and Yorke
1979). The Kaplan-Yorke map with a = 0.2 has COR-
RELATION Exponent 1.42 ±0.02 (Grassberger Procac-
cia 1983) and CAPACITY DIMENSION 1.43 (Russell et al.
1980).
References
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.
Kaplan, J. L. and Yorke, J. A. In Functional Differen-
tial Equations and Approximations of Fixed Points (Ed.
H.-O. Peitgen and H.-O. Walther). Berlin: Springer-
Verlag, p. 204, 1979.
Russell, D. A.; Hanson, J. D.; and Ott, E. "Dimension of
Strange Attractors." Phys. Rev. Let. 45, 1175-1178, 1980.
297^ = 88,209 88 + 209 = 297.
see also Digital Root, Digitadition, Happy Num-
ber, Kaprekar Routine, Narcissistic Number,
Recurring Digital Invariant
References
Sloane, N. J. A. Sequence A006886/M4625 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Kaprekar Routine
A routine discovered in 1949 by D. R. Kaprekar for 4-
digit numbers, but which can be generalized to fc-digit
numbers. To apply the Kaprekar routine to a number
n, arrange the digits in descending (n f ) and ascending
(n") order. Now compute K{n) = n — n" and iterate.
The algorithm reaches (a degenerate case), a constant,
Kaps-Rentrop Methods
Karatsuba Multiplication 977
or a cycle, depending on the number of digits in k and
the value of n.
For a 3-digit number n in base 10, the Kaprekar routine
reaches the number 495 in at most six iterations. In
baser, there is a unique number ((r-2)/2,r-l,r/2) r to
which n converges in at most (r + 2)/2 iterations IFF r is
EVEN. For any 4-digit number n in base-10, the routine
terminates on the number 6174 after seven or fewer steps
(where it enters the 1-cycle K(Q174) = 6174).
2. 0,0, 9, 21, {(45), (49)}, ...,
3. 0, 0, (32, 52), 184, (320, 580, 484),
4. 0, 30, {201, (126, 138)}, (570, 765), {(2550), (3369),
(3873)},...,
5. 8, (48, 72), 392, (1992, 2616, 2856, 2232), (7488,
10712, 9992, 13736, 11432),
6. 0, 105, (430, 890, 920, 675, 860, 705), {5600, (4305,
5180)}, {(27195), (33860), (42925), (16840, 42745,
35510)}, ...,
7. 0, (144, 192), (1068, 1752, 1836), (9936, 15072,
13680, 13008, 10608), (55500, 89112, 91800, 72012,
91212, 77388), ...,
8. 21, 252, {(1589, 3178, 2723), (1022, 3122, 3290,
2044, 2212)}, {(17892, 20475), (21483, 25578, 26586,
21987)},...,
9. (16, 48), (320, 400), {(2256, 5312, 3856), (3712,
5168, 5456)}, {41520, (34960, 40080, 55360, 49520,
42240)}, ... ;
10. 0, 495, 6174, {(53955, 59994), (61974, 82962, 75933,
63954), (62964, 71973, 83952, 74943)}, ...,
see also 196- Algorithm, Kaprekar Number, RATS
Sequence
References
Eldridge, K. E. and Sagong, S. "The Determination of
Kaprekar Convergence and Loop Convergence of All 3-
Digit Numbers." Amer. Math. Monthly 95, 105-112, 1988.
Kaprekar, D. R. "An Interesting Property of the Number
6174." Scripta Math. 15, 244-245, 1955.
Trigg, C W. "All Three-Digit Integers Lead to..." The
Math. Teacher, 67, 41-45, 1974.
Young, A. L. "A Variation on the 2-digit Kaprekar Routine."
Fibonacci Quart 31, 138-145, 1993.
Kaps-Rentrop Methods
A generalization of the Runge-Kutta Method for so-
lution of Ordinary Differential Equations, also
called Rosenbrock Methods.
see also Runge-Kutta Method
References
Press, W. H.; Flannery, B. R; 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. 730-735, 1992.
Kapteyn Series
A series of the form
y^a n ^+n[0 + n)z] t
a=0
where J n {z) is a Bessel Function of the First
Kind. Examples include Kapteyn's original series
and
oo
-J— = l + 2Vj n (nz)
1 — z ^-^
2(T^) = $>" (2nz) -
see also Bessel Function of the First Kind, Neu-
mann Series (Bessel Function)
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1473,
1980.
Karatsuba Multiplication
It is possible to perform MULTIPLICATION of LARGE
NUMBERS in (many) fewer operations than the usual
brute-force technique of "long multiplication." As dis-
covered by Karatsuba and Ofman (1962), Multiplica-
tion of two n-DiGiT numbers can be done with a Bit
Complexity of less than n 2 using identities of the form
(a + &.10 n )(c + cM0 n )
= ac+ [{a + b)(c + d)-ac- bd]l0 n + bd • 10 2n . (1)
Proceeding recursively then gives BlT COMPLEXITY
0(n lg3 ), where lg3 = 1.58... < 2 (Borwein et al.
1989). The best known bound is O(nlgnlglgn) steps
for n » 1 (Schonhage and Strassen 1971, Knuth 1981).
However, this ALGORITHM is difficult to implement, but
a procedure based on the Fast FOURIER TRANSFORM is
straightforward to implement and gives Bit COMPLEX-
ITY 0((\gn) 2+e n) (Brigham 1974, Borodin and Munro
1975, Knuth 1981, Borwein et al. 1989).
As a concrete example, consider MULTIPLICATION of two
numbers each just two "digits" long in base w,
Ni = ao + a\W
iV 2 = b + biw,
(2)
(3)
then their PRODUCT is
P = iViiV 2
= a bo + (ao&i + aibo)w + a\b\W
= P0 + PlW + P2W 2 •
(4)
978 Karatsuba Multiplication
Katona's Problem
Instead of evaluating products of individual digits, now
write
qo = o,obo
qi = (a + ai)(6 + h)
q 2 = aibi.
(5)
(6)
(7)
The key term is qi, which can be expanded, regrouped,
and written in terms of the pj as
^1 — Pi + PO + P2 •
(8)
However, since po = <?o, and p2 = #2, it immediately
follows that
Po = qo
(9)
Pi = qi - qo - 92
(10)
P2 =92,
(11)
so the three "digits" of p have been evaluated using three
multiplications rather than four. The technique can be
generalized to multidigit numbers, with the trade-off be-
ing that more additions and subtractions are required.
Now consider four- "digit" numbers
Ni — ao + a\W + a^vj + a^w ,
(12)
which can be written as a two- "digit" number repre-
sented in the base w 2 ,
Ni = {do + a\w) + (a 2 + a 3 w ) * i
The "digits" in the new base are now
a ~ ao ~j- aiw
a\ = a,2 + azw,
(13)
References
Borodin, A. and Munro, I. The Computational Complexity
of Algebraic and Numeric Problems. New York: American
Elsevier, 1975.
Borwein, J. M.; B or we in, P. B.; and Bailey, D. H. "Ramanu-
jan, Modular Equations, and Approximations to Pi, or
How to Compute One Billion Digits of Pi." Amer. Math.
Monthly 96, 201-219, 1989.
Brigham, E. O. The Fast Fourier Transform. Englewood
Cliffs, NJ: Prentice-Hall, 1974.
Brigham, E. O. Fast Fourier Transform and Applications.
Englewood Cliffs, NJ: Prentice-Hall, 1988.
Cook, S. A. On the Minimum Computation Time of Func-
tions. Ph.D. Thesis. Cambridge, MA: Harvard University,
pp. 51-77, 1966.
Hollerbach, U. "Fast Multiplication Sc Division of Very Large
Numbers." sci. math. research posting, Jan. 23, 1996.
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.
Knuth, D. E. The Art of Computing, Vol. 2: Seminumer-
ical Algorithms, 2nd ed. Reading, MA: Addison- Wesley,
pp. 278-286, 1981.
Schonhage, A. and Strassen, V. "Schnelle Multiplication
Grosser Zahlen." Computing 7, 281-292, 1971.
Toom, A. L. "The Complexity of a Scheme of Functional
Elements Simulating the Multiplication of Integers." Dokl.
Akad. Nauk SSSR 150, 496-498, 1963. English translation
in Soviet Mathematics 3, 714-716, 1963.
Zuras, D. "More on Squaring and Multiplying Large Inte-
gers." IEEE Trans. Comput. 43, 899-908, 1994.
Katona's Problem
Find the minimum number f(n) of subsets in a SEPA-
RATING Family for a Set of n elements, where a Sepa-
rating Family is a Set of Subsets in which each pair
of adjacent elements is found separated, each in one of
two disjoint subsets. For example, the 26 letters of the
alphabet can be separated by a family of nine:
(abcdefghi)
(jklmnopqr)
(stuvwxyz)
(14)
(abcjklstu)
(defmnovwx)
(ghipqryz)
(15)
(adgjmpsvy)
(behknqtwz)
(cfilorux)
and the Karatsuba algorithm can be applied to Ni and
7V 2 in this form. Therefore, the Karatsuba algorithm
is not restricted to multiplying two-digit numbers, but
more generally expresses the multiplication of two num-
bers in terms of multiplications of numbers of half the
size. The asymptotic speed the algorithm obtains by re-
cursive application to the smaller required subproducts
isC?(n lg3 ) (Knuth 1981).
When this technique is recursively applied to multidigit
numbers, a point is reached in the recursion when the
overhead of additions and subtractions makes it more
efficient to use the usual 0(n 2 ) Multiplication algo-
rithm to evaluate the partial products. The most effi-
cient overall method therefore relies on a combination
of Karatsuba and conventional multiplication.
see also Complex Multiplication, Multiplication,
Strassen Formulas
The problem was posed by Katona (1973) and solved by
C. Mao-Cheng in 1982,
f{n) = min J2p + 3 flog 3 (g) j : p = 0, 1, 2} ,
where \x] is the Ceiling Function. f(n) is nonde-
creasing, and the values for n = 1, 2, ... are 0, 2, 3,
4, 5, 5, 6, 6, 6, 7, . . . (Sloane's A07600). The values at
which f(n) increases are 1, 2, 3, 4, 5, 7, 10, 13, 19, 28,
37, ... (Sloane's A007601), so /(26) = 9, as illustrated
in the preceding example.
see also SEPARATING FAMILY
References
Honsberger, R. "Cai Mao-Cheng's Solution to Katona's
Problem on Families of Separating Subsets." Ch. 18 in
Mathematical Gems III. Washington, DC: Math. Assoc.
Amer., pp. 224-239, 1985.
Kauffman Polynomial F
Keith Number 979
Katona, G. O. H. "Combinatorial Search Problem." In A
Survey of Combinatorial Theory (Ed. J. N. Srivasta et
al.). Amsterdam, Netherlands: North- Holland, pp. 285—
308, 1973.
Sloane, N. J. A. Sequences A007600/M0456 and A007601/
M0525 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Kauffman Polynomial F
A semi-oriented 2-variable KNOT POLYNOMIAL defined
by
F L {a,z) = a- wW (\L\), (1)
where L is an oriented LINK DIAGRAM, w(L) is the
WRITHE of L, \L\ is the unoriented diagram correspond-
ing to L, and (L) is the BRACKET POLYNOMIAL. It
was developed by Kauffman by extending the BLM/Ho
Polynomial Q to two variables, and satisfies
F(l,x) =Q(x).
(2)
The Kauffman POLYNOMIAL is a generalization of the
Jones Polynomial V(t) since it satisfies
v(t) = F(-r s/4 ,r 1/4
+* 1/4 ),
(3)
but its relationship to the HOMFLY POLYNOMIAL is
not well understood. In general, it has more terms than
the HOMFLY Polynomial, and is therefore more pow-
erful for discriminating KNOTS. It is a semi-oriented
Polynomial because changing the orientation only
changes F by a POWER of a. In particular, suppose
L* is obtained from L by reversing the orientation of
component fc, then
F L . = a 4X F L ,
(4)
where A is the LINKING NUMBER of k with L - k (Lick-
orish and Millett 1988). F is unchanged by MUTATION.
F Li +f L9 =F(L 1 )F(L 2 )
F Li ul 2 = [(a 1 +a)x
1]Fl 1 Fl 2 -
(5)
(6)
M. B. Thistlethwaite has tabulated the Kauffman 2-
variable POLYNOMIAL for KNOTS up to 13 crossings.
References
Lickorish, W. B. R. and Millett, B. R. "The New Polynomial
Invariants of Knots and Links." Math. Mag. 61, 1—23,
1988.
Stoimenow, A. "Kauffman Polynomials." http://www.
informatik.hu-berlin.de/-stoimeno/ptab/klO.html.
# Weisstein, E. W. "Knots and Links." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/Knots.m.
Kauffman Polynomial X
A 1-variable Knot Polynomial denoted X or C.
C L {A) = {-A*y w ^{L),
(1)
where (L) is the BRACKET POLYNOMIAL and w(L) is
the Writhe of L. This Polynomial is invariant under
Ambient Isotopy, and relates Mirror Images by
C L * =Cl{A- x ).
(2)
It is identical to the Jones Polynomial with the
change of variable
£(r *'•) = V(t).
(3)
The X Polynomial of the Mirror Image K* is the
same as for K but with A replaced by A~ l .
References
Kauffman, L. H. Knots and Physics. Singapore: World Sci-
entific, p. 33, 1991.
Kei
The Imaginary Part of
e-'^Kvixe™' 4 ) = ker„(a;) +ikei„(x).
see also Bei, Ber, Ker, Kelvin Functions
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func-
tions." §9.9 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 379-381, 1972.
Keith Number
A Keith number is an n-digit Integer N such that if
a Fibonacci- like sequence (in which each term in the
sequence is the sum of the n previous terms) is formed
with the first n terms taken as the decimal digits of
the number iV, then AT itself occurs as a term in the
sequence. For example, 197 is a Keith number since
it generates the sequence 1, 9, 7, 17, 33, 57, 107, 197,
... (Keith). Keith numbers are also called REPFIGIT
Numbers.
There is no known general technique for finding Keith
numbers except by exhaustive search. Keith numbers
are much rarer than the PRIMES, with only 52 Keith
numbers with < 15 digits: 14, 19, 28, 47, 61, 75, 197,
742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647,
7909, . . . (Sloane's A007629). In addition, three 15-digit
Keith numbers are known (Keith 1994). It is not known
if there are an INFINITE number of Keith numbers.
References
Esche, H. A. "Non-Decimal Replicating Fibonacci Digits." J.
Recr. Math. 26, 193-194, 1994.
Heleen, B. "Finding Repfigits — A New Approach." J. Recr.
Math. 26, 184-187, 1994.
980 Keller's Conjecture
Kepler Conjecture
Keith, M. "Repfigit Numbers." J. Recr. Math. 19, 41-42,
1987.
Keith, M. "All Repfigit Numbers Less than 100 Billion
(10 11 )." J. Recr. Math. 26, 181-184, 1994.
Keith, M. "Keith Numbers." http : //users . aol . com/
s6sj7gt/mikekeit .htm.
Robinson, N. M. "All Known Replicating Fibonacci Digits
Less than One Thousand Billion (10 12 )." J. Recr. Math.
26, 188-191, 1994.
Shirriff, K. "Computing Replicating Fibonacci Digits." J.
Recr. Math. 26, 191-193, 1994.
Sloane, N. J. A. Sequence A007629/M4922 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
"Table: Repfigit Numbers (Base 10* ) Less than 10 15 ." J.
Recr. Math. 26, 195, 1994.
Keller's Conjecture
Keller conjectured that tiling an n-D space with n-D
HYPERCUBES of equal size yields an arrangement in
which at least two hypercubes have an entire (n — 1)-D
"side" in common. The CONJECTURE has been proven
true for n = 1 to 6, but disproven for n > 10.
References
Cipra, B. "If You Can't See It, Don't Believe It." Science
259, 26-27, 1993.
Cipra, B. WhaVs Happening in the Mathematical Sciences,
Vol 1. Providence, RI: Amer. Math. Soc, pp. 24, 1993.
Kelvin's Conjecture
What space-filling arrangement of similar polyhedral
cells of equal volume has minimal surface AREA?
Kelvin proposed the 14-sided TRUNCATED OCTAHE-
DRON. Wearie and Phelan (1994) discovered another
14-sided Polyhedron that has 3% less Surface Area.
References
Gray, J. "Parsimonious Polyhedra." Nature 367, 598-599,
1994.
Wearie, D. and Phelan, R. "A Count er-Example to Kelvin's
Conjecture on Minimal Surfaces." Philos. Mag. Let. 69,
107-110, 1994.
Kelvin Functions
Kelvin defined the Kelvin functions BEI and BER ac-
cording to
J u (xe 37Ti/4 ) = ber„(x)+ibei,,(x), (1)
where J u (s) is a Bessel Function OF THE First
Kind, and the functions Kei and Ker by
e -vK%i j{ i/ ^ xe ' K ' t / ) __ ker I/ (x) -h ikei u (x)^ (2)
where K v (x) is a Modified Bessel Function of the
Second KIND. For the special case v = 0,
Jo(iVix) = J Q (±V2(i- l)x) = bev(x)-\-ibei(x). (3)
see also Bei, Ber, Kei, Ker
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func-
tions." §9.9 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 379-381, 1972.
Spanier, J. and Oldham, K. B. "The Kelvin Functions."
Ch. 55 in An Atlas of Functions. Washington, DC: Hemi-
sphere, pp. 543-554, 1987.
Kelvin Transformation
The transformation
, / ,, (a\ n - 2 (a 2 x[ a 2 x' n \
v(x u ...,x n )= {-) ^ — .....—j,
where
/2 / 2 / 2
r = x x + . . . + x n .
Ifu(xi,. . . , x n ) is a Harmonic Function on a Domain
D of W 1 (with n > 3), then v(x' l ,... ,x f n ) is HARMONIC
onD'.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 623, 1980.
Kempe Linkage
A double rhomboid LINKAGE which gives rectilinear mo-
tion from circular without an inversion.
References
Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 126—127,
1957.
Kepler Conjecture
In 1611, Kepler proposed that close packing (cubic or
hexagonal) is the densest possible Sphere Packing
(has the greatest «), and this assertion is known as the
Kepler conjecture. Finding the densest (not necessarily
periodic) packing of spheres is known as the KEPLER
Problem.
A putative proof of the Kepler conjecture was put for-
ward by W.-Y. Hsiang (Hsiang 1992, Cipra 1993), but
was subsequently determined to be flawed (Conway et
al. 1994, Hales 1994). According to J. H. Conway, no-
body who has read Hsiang's proof has any doubts about
its validity: it is nonsense.
see also Dodecahedral Conjecture, Kepler Prob-
lem
References
Cipra, B. "Gaps in a Sphere Packing Proof?" Science 259,
895, 1993.
Conway, J. H.; Hales, T. C; Muder, D. J.; and Sloane,
N. J. A. "On the Kepler Conjecture." Math. Intel 16,
5, Spring 1994.
Eppstein, D. "Sphere Packing and Kissing Numbers."
http:// www . ics . uci . edu / - eppstein / junkyard /
spher epack . html .
Hales, T. C. "The Sphere Packing Problem." J. Comput.
Appl. Math. 44, 41-76, 1992.
Hales, T. C. "Remarks on the Density of Sphere Packings in
3 Dimensions." Combinatori 13, 181-197, 1993.
Hales, T. C. "The Status of the Kepler Conjecture." Math.
Intel 16, 47-58, Summer 1994.
Hales, T. C. 'The Kepler Conjecture." http://www.math.
lsa.umich.edu/~hales/kepler.html.
Hsiang, W.-Y. "On Soap Bubbles and Isoperimetric Regions
in Noncompact Symmetrical Spaces. 1." Tohoku Math. J.
44, 151-175, 1992.
Hsiang, W.-Y. "A Rejoinder to Hales's Article." Math. Intel
17, 35-42, Winter 1995.
Kepler's Equation
Kepler's Equation 981
Kepler's Equation
Let M be the mean anomaly and E the ECCENTRIC
Anomaly of a body orbiting on an Ellipse with Ec-
centricity e, then
M = E - e sin E.
(i)
For M not a multiple of 7r, Kepler's equation has a
unique solution, but is a TRANSCENDENTAL EQUATION
and so cannot be inverted and solved directly for E given
an arbitrary M. However, many algorithms have been
derived for solving the equation as a result of its impor-
tance in celestial mechanics.
Writing a E as a Power Series in e gives
CO
£ = M + ^a n e n , (2)
where the coefficients are given by the Lagrange In-
version Theorem as
Ln/2j . ,
°« = ^j E (- 1 )* ( I) ( n - 2 *) n ~ 1 sin t( n - 2 *) M i
(3)
(Wintner 1941, Moulton 1970, Henrici 1974, Finch).
Surprisingly, this series diverges for
e> 0.6627434193...,
(4)
a value known as the LAPLACE LIMIT. In fact, E con-
verges as a Geometric Series with ratio
1 + vT+lr
: exp(\/l + e 2 )
(5)
(Finch).
There is also a series solution in BESSEL FUNCTIONS OF
the First Kind,
E — M + y^ -J n (ne) sin(nM).
*- — ' n
(6)
n-\
This series converges for all e < 1 like a GEOMETRIC
Series with ratio
i + vT^7
:exp(\/l - e 2 ).
(7)
The equation can also be solved by letting tp be the
Angle between the planet's motion and the direction
Perpendicular to the Radius Vector. Then
tan-0 :
esinE
(8)
Alternatively, we can define e in terms of an intermedi
ate variable <j>
e = sin 0,
then
sin[|(u — E)] = */- sm(~<j))smv
sin[|(v + E)] = J - cos(\4>)sinv.
(9)
(10)
(11)
Iterative methods such as the simple
E i+1 = M + esin£7i (12)
with E = work well, as does NEWTON'S METHOD,
M + esinEi - Ei
Ei+i — Ei + ■
1 — e cos Ei
(13)
In solving Kepler's equation, Stieltjes required the solu-
tion to
e x (x-l) = e- x (x + l), (14)
which is 1.1996678640257734... (Goursat 1959, Le Li-
onnais 1983).
see also Eccentric Anomaly
References
Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed.,
rev. ed. Richmond, VA: Willmann-Bell, 1988.
Dorrie, H. "The Kepler Equation." §81 in 100 Great Prob-
lems of Elementary Mathematics: Their History and So-
lutions. New York: Dover, pp. 330-334, 1965.
Finch, S. "Favorite Mathematical Constants." http://wvv.
mathsoft.com/asolve/constant/lpc/lpc.html.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA:
Addison- Wesley, pp. 101-102 and 123-124, 1980.
Goursat, E. A Course in Mathematical Analysis, Vol. 2. New
York: Dover, p. 120, 1959.
Henrici, P. Applied and Computational Complex Analysis,
Vol. 1: Power Series-Integration-Conformal Mapping-
Location of Zeros. New York: Wiley, 1974.
Ioakimids, N. I. and Papadakis, K. E. "A New Simple Method
for the Analytical Solution of Kepler's Equation." Celest.
Mech. 35, 305-316, 1985.
Ioakimids, N. I. and Papadakis, K. E. "A New Class of Quite
Elementary Closed-Form Integrals Formulae for Roots of
Nonlinear Systems." Appl. Math. Comput. 29, 185-196,
1989.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 36, 1983.
Marion, J. B. and Thornton, S. T. "Kepler's Equations." §7.8
in Classical Dynamics of Particles & Systems, 3rd ed. San
Diego, CA: Harcourt Brace Jovanovich, pp. 261-266, 1988.
Moulton, F. R. An Introduction to Celestial Mechanics, 2nd
rev. ed. New York: Dover, pp. 159-169, 1970.
Siewert, C. E. and Burniston, E. E. "An Exact Analytical
Solution of Kepler's Equation." Celest. Mech. 6, 294-304,
1972.
Wintner, A. The Analytic Foundations of Celestial Mechan-
ics. Princeton, NJ: Princeton University Press, 1941.
982 Kepler's Folium
Kepler's Folium
The curve with implicit equation
[(a: - b) 2 + y 2 ][x(x - b) + y 2 ] - 4a(x - b)y 2 .
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 71-72, 1993.
Kepler-Poinsot Solid
The Kepler-Poinsot solids are the four regular CONCAVE
POLYHEDRA with intersecting facial planes. They are
composed of regular Concave Polygons and were un-
known to the ancients. Kepler discovered two of them
about 1619. These two were subsequently rediscovered
by Poinsot, who also discovered the other two, in 1809.
As shown by Cauchy, they are stellated forms of the
Dodecahedron and Icosahedron.
The Kepler-Poinsot solids, illustrated above, are
known as the Great Dodecahedron, Great Icos-
ahedron, Great Stellated Dodecahedron, and
Small Stellated Dodecahedron. Cauchy (1813)
proved that these four exhaust all possibilities for regu-
lar star polyhedra (Ball and Coxeter 1987).
A table listing these solids, their DUALS, and COM-
POUNDS is given below.
Polyhedron
Dual
great dodecahedron
great Icosahedron
great stellated dodec.
small stellated dodec.
great stellated dodec.
great icosahedron
small stellated dodec. great dodecahedron
Polyhedron
Compound
great dodecahedron
great icosahedron
great stellated dodec.
small stellated dodec.
great dodecahedron-
small stellated dodec.
great icosahedron-
great stellated dodec.
great icosahedron-
great stellated dodec.
great dodecahedron-
small stellated dodec.
The polyhedra { § , 5} and {5, § } fail to satisfy the Poly-
Ker
where V is the number of faces, E the number of edges,
and F the number of faces, despite the fact that formula
holds for all ordinary polyhedra (Ball and Coxeter 1987).
This unexpected result led none less than Schlafli (1860)
to conclude that they could not exist.
In 4-D, there are 10 Kepler-Poinsot solids, and in n-
D with n > 5, there are none. In 4-D, nine of the
solids have the same Vertices as {3,3,5}, and the
tenth has the same as {5,3,3}. Their Schlafli Sym-
BOLSare{§,5,3}, {3,5, f}, {5, §,5}, {§, 3, 5}, {5,3, f },
{f, 5, §}, {5, |,3}, {3, f,5}, {§,3,3}, and {3, 3, f}.
Coxeter et al. (1954) have investigated star "Archimed-
ean" polyhedra.
see also ARCHIMEDEAN SOLID, DELTAHEDRON, JOHN-
SON Solid, Platonic Solid, Polyhedron Com-
pound, Uniform Polyhedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 144-
146, 1987.
Cauchy, A. L. "Recherches sur les polyedres." J. de VEcole
Polytechnique 9, 68-86, 1813.
Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller,
J. C. P. "Uniform Polyhedra." Phil Trans. Roy. Soc. Lon-
don Ser. A 246, 401-450, 1954.
Pappas, T. "The Kepler-Poinsot Solids." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, p. 113,
1989.
Quaisser, E. "Regular Star-Polyhedra." Ch. 5 in Mathemat-
ical Models from the Collections of Universities and Mu-
seums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 56-62, 1986.
Schlafli. Quart J. Math. 3, 66-67, 1860.
Kepler Problem
Finding the densest not necessarily periodic Sphere
Packing.
see also Kepler Conjecture, Sphere Packing
Kepler Solid
see Kepler-Poinsot Solid
Ker
The Real Part of
e' vni/2 K v (xe vi/4 ) = ker„(x)+ikii v (x),
where K v (x) is a MODIFIED BESSEL FUNCTION OF THE
Second Kind.
see also Bei, Ber, Kei, Kelvin Functions
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Kelvin Func-
tions." §9.9 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 379-381, 1972.
hedral Formula
V -E + F = 2,
Keratoid Cusp
Keratoid Cusp
Khintchine's Constant
983
The Plane Curve given by the Cartesian equation
2 2,5
y = x y + x .
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.
Kernel (Integral)
The function K(a,t) in an Integral or Integral
Transform
</(
J a
f{t)K(a,t)dt.
see also BERGMAN KERNEL, POISSON KERNEL
Kernel (Linear Algebra)
see Nullspace
Kernel Polynomial
The function
K n (x Ql x) = K n (x,x ) = K n (x,x )
which is useful in the study of many POLYNOMIALS.
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, 1975.
Kervaire's Characterization Theorem
Let G he a GROUP, then there exists a piecewise linear
Knot K n ~ 2 in § n forn > 5 with G = 7n(S n - K) Iff
G satisfies
1. G is finitely presentable,
2. The Abelianization of G is infinite cyclic,
3. The normal closure of some single element is all of
G,
4. H2(G) = 0; the second homology of the group is
trivial.
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 350-351, 1976.
Ket
A CONTRAVARIANT VECTOR, denoted \ifi). The ket is
Dual to the Covariant Bra 1- Vector {ip\. Taken
together, the Bra and ket form an Angle Bracket
(bra+ket = bracket) {ip\ip). The ket is commonly en-
countered in quantum mechanics.
see also Angle Bracket, Bra, Bracket Product,
Contravariant Vector, Covariant Vector, Dif-
ferential £:-Form, One-Form
Khinchin Constant
see Khintchine's Constant
Khintchine's Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
2.4
Let
£ = [qo j <?i » ■ • •] = Qo +
(1)
<7i +
qi +
$3 + • • •
be the Simple Continued Fraction of a Real Num-
ber x, where the numbers qi are called Partial Quo-
tients. Khintchine (1934) considered the limit of the
Geometric Mean
G n (x) — (q!q 2 •••q n ) 1/n
(2)
as n — > oo. Amazingly enough, this limit is a constant
independent of x — except if x belongs to a set of Mea-
sure 0-given by
^ = 2.685452001.
(3)
(Sloane's A002210), as proved in Kac (1959). The values
G n {x) are plotted above for n = 1 to 500 and x = 7r,
l/7r, sinl, the Euler-Mascheroni Constant 7, and
the Copeland-Erdos Constant. Real Numbers x
for which limn-^oo G n (x) ^ K include x = e, v2, V^,
and the Golden Ratio <£, all of which have periodic
Partial Quotients, plotted below.
984 Khintchine's Constant
100
300
400
500
The Continued Fraction for K is [2, 1, 2, 5, 1, 1, 2,
1, 1, ...] (Sloane's A002211). It is not known if K is
Irrational, let alone Transcendental. Bailey et al.
(1995) have computed K to 7350 DIGITS.
Explicit expressions for K include
*=n
1 +
n(n + 2)
Inn/ In 2
\n2\xiK= ^7T + §(ln2) 2 + ' v ' "
F
Jo
In
oo
hm-1
In 2 ^— ' m
771 = 1
K(2m) - 1],
(4)
(5)
(6)
where C,{z) is the Riemann Zeta Function and
fcm = 5^-
(7)
i=i
(Shanks and Wrench 1959). Gosper gave
In 2 ^ ? '
(8)
J=2
where £'(jz) is the DERIVATIVE of the RlEMANN ZETA
Function. An extremely rapidly converging sum also
due to Gosper is
\nK = j^ X)| ~ ln(* + l)pn(fc + 3)
fc=0 I
-21n(fc + 2)+ln(fc + l)]
(_l)^(2-2 fc+2 )
fc + 2
+ ln(fc + 1)
Jfe+2
£
.s=l
ln(fc + l) ,
(fc + 1)*+=" Cl* + 2 >* + 2 J
(-l)*(2-2 5 )
(fc+l) a s
(9)
where £(s,a) is the Hurwitz Zeta Function.
Khintchine's Constant
Khintchine's constant is also given by the integral
ttx(1 — x 2 )
In21n(fi0= / * v ln
2 Jo ^(l + z)
sin(Trz)
dx. (10)
If Pn/Qn is the nth CONVERGENT of the CONTINUED
Fraction of x, then
\im(Q n ) 1/n = lim f^) Vn =e- 2/(121n2) « 3.27582
n— )-oo n— >oo \ X /
(ii)
for almost all REAL x (Levy 1936, Finch). This num-
ber is sometimes called the LEVY CONSTANT, and the
argument of the exponential is sometimes called the
Khintchine-Levy Constant.
Define the following quantity in terms of the fcth partial
quotient qk,
M(s f n,x)= [~J2
l/s
Qk
Then
lim M(l,n, x) = oo
n— ► oo
(12)
(13)
for almost all real x (Khintchine, Knuth 1981, Finch),
and
Af(l,n,a;) ~0(lnn). (14)
Furthermore, for s < 1, the limiting value
lim M(s,n,x) = K(s)
(15)
exists and is a constant K(s) with probability 1 (Rockett
and Szusz 1992, Khintchine 1997).
see also CONTINUED FRACTION, CONVERGENT,
Khintchine-Levy Constant, Levy Constant, Par-
tial Quotient, Simple Continued Fraction
References
Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the
Khintchine Constant." Math. Comput. 66, 417-431, 1997.
Finch, S. "Favorite Mathematical Constants." http://vvv.
mathsof t , com/asolve/constant/khntchn/khntchn.html.
Kac, M. Statistical Independence and Probability, Analysts
and Number Theory. Providence, Rl: Math. Assoc. Amer.,
1959,
Khinchin, A. Ya. Continued Fractions. New York: Dover,
1997.
Knuth, D. E. Exercise 24 in The Art of Computer Program-
ming, Vol. 2: Seminumerical Algorithms, 2nd ed. Read-
ing, MA: Addison- Wesley, p. 604, 1981.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 46, 1983.
Lehmer, D. H. "Note on an Absolute Constant of Khint-
chine." Amer. Math. Monthly 46, 148-152, 1939.
Phillipp, W. "Some Metrical Theorems in Number Theory."
Pacific J. Math. 20, 109-127, 1967.
PloufFe, S. "Plouffe's Inverter: Table of Current Records for
the Computation of Constants." http://lacim.uqam.ca/
pi/records .html.
Khintchine-Levy Constant
Rockett, A. M. and Sziisz, P. Continued Fractions. Singa-
pore: World Scientific, 1992.
Shanks, D. and Wrench, J. W. "Khintchine's Constant."
Amer. Math. Monthly 66, 148-152, 1959.
Sloane, N. J. A. Sequences A002210/M1564 and A002211/
M0118 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Vardi, I. "Khinchin's Constant." §8.4 in Computational
Recreations in Mathematica. Reading, MA: Addison-
Wesley, pp. 163-171, 1991.
Wrench, J. W. "Further Evaluation of Khintchine's Con-
stant." Math. Comput. 14, 370-371, 1960.
Khintchine-Levy Constant
A constant related to Khintchine's Constant defined
by
KL.
= 1.1865691104....
12 In 2
see also Khintchine's Constant, Levy Constant
References
Plouffe, S. "Khintchine-Levy Constant." http://lacim,
uqam . ca/piDATA/klevy . txt .
Khovanski's Theorem
If /i,...,/ m : M 71 — > R. are exponential polynomials,
then {x E M 71 : fi(x) = ■ ■ ■ f n {x) = 0} has finitely many
connected components.
References
Marker, D. "Model Theory and Exponentiation." Not
Amer. Math. Soc. 43, 753-759, 1996.
Kiepert's Conies
see Kiepert's Hyperbola, Kiepert's Parabola
Kiepert's Hyperbola
A curve which is related to the solution of LEMOINE's
Problem and its generalization to Isosceles Trian-
gles constructed on the sides of a given TRIANGLE. The
Vertices of the constructed Triangles are
A! - - sin $ : sin(C + <j)) : sin(S + <j>) (1)
B' = sin(C + 0) : - sin</» : sin(^4 + <p) (2)
C' ~ sin(J3 + <f>) : sm(A + <f>) : - sin <f>, (3)
where <f> is the base Angle of the Isosceles Triangle.
Kiepert showed that the lines connecting the Vertices
of the given TRIANGLE and the corresponding peaks of
the Isosceles Triangles Concur. The Trilinear
Coordinates of the point of concurrence are
sin(B + <j>) sin(C + (f>) : sin(C + <f>) sin(A + </>) :
sin(^4 + <(>) sm(B + 0). (4)
The locus of this point as the base ANGLE varies is given
by the curve
sin(B - C) sin(C - A) sm(A - B)
a 7
= 6c(c 2 -c 2 ) i ca(c 2 -a 2 ) | ab(a* - b 2 ) _ ^
Kiepert's Hyperbola 985
Writing the Trilinear Coordinates as
cti = diSi, (6)
where di is the distance to the side opposite a* of length
Si and using the POINT-LlNE DISTANCE FORMULA with
(#0,2/0) written as (:c,y),
di =
I (2/1+2 -yi+i)(x-xi+i)
Si
(xj+2 -a?t+i)(y - 2/i+i)
, (7)
where 2/4 = 2/1 and 2/5 = 2/2 gives the FORMULA
3
2^Si + iSi + 2(s i+1 - S i+2 )
1=1
X
Si
3
(2/1+2 - Vi+i){x - ffi»+i) - (x i+2 - Xi+i)(y - 2/j+i)
= (8)
„2 2
(s i+ i ~ s i+2 )
r-f (y*+2 - 2/i+i)(z - z i+1 ) - (xi+2 - x i+1 )(y - y i+ i)
= 0. (9)
Bringing this equation over a common DENOMINATOR
then gives a quadratic in x and y, which is a CONIC
Section (in fact, a Hyperbola). The curve can also
be written as csc(A -f t) : csc(£? -f t) : csc(C -f i), as t
varies over [— 7r/4, 7r/4].
Kiepert's hyperbola passes through the triangle's CEN-
troid M (0 = 0), Orthocenter H {(j> = tt/2), Ver-
tices A (<j> = -a if a < 7r/2 and = 7r-a:ifa:> 7r/2),
B (<f> = -/3), C (<f> = -7), Fermat Point F x (<f> = tt/3),
second ISOGONIC CENTER F 2 (</> = -tt/3), ISOGONAL
Conjugate of the Brocard Midpoint (<j> - u>), and
Brocard's Third Point Z z (<p = a;), where a; is the
Brocard Angle (Eddy and Fritsch 1994, p. 193).
The Asymptotes of Kiepert's hyperbola are the Sim-
son Lines of the intersections of the Brocard Axis
with the ClRCUMCiRCLE. Kiepert's hyperbola is a
Rectangular Hyperbola. In fact, all nondegenerate
conies through the VERTICES and Ortho CENTER of a
Triangle are Rectangular Hyperbolas the centers
986
Kiepert's Parabola
Killing Vectors
of which lie halfway between the Isogonic Centers
and on the Nine-Point CIRCLE. The LOCUS of centers
of these HYPERBOLAS is the NlNE-POINT CIRCLE,
The ISOGONAL CONJUGATE curve of Kiepert's hyper-
bola is the Brocard Axis. The center of the Incircle
of the Triangle constructed from the Midpoints of
the sides of a given TRIANGLE lies on Kiepert's hyper-
bola of the original TRIANGLE.
see also Brocard Angle, Brocard Axis, Brocard
Points, Centroid (Triangle), Circumcircle, Iso-
gonal Conjugate, Isogonic Centers, Isosceles
Triangle, Lemoine's Problem, Nine-Point Cir-
cle, Orthocenter, Simson Line
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 rev. enl. ed. Dublin: Hodges, Figgis, &; Co., 1893.
Eddy, R. H. and Fritsch, R. "The Conies of Ludwig Kiepert:
A Comprehensive Lesson in the Geometry of the Triangle."
Math. Mag. 67, 188-205, 1994.
Kelly, P. J. and Merriell, D. "Concentric Polygons." Amer.
Math. Monthly 71, 37-41, 1964.
Mineuer, A. "Sur les asymptotes de l'hyperbole de Kiepert."
Mathesis 49, 30-33, 1935.
Rigby, J. F. "A Concentrated Dose of O Id-Fashioned Geom-
etry." Math. Gaz. 57, 296-298, 1953.
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.
Kiepert's Parabola
Let three similar Isosceles Triangles AA'BC,
AAB'C, and AABC f be constructed on the sides of a
Triangle AABC. Then the Envelope of the axis
of the Triangles AABC and AA'B'C is Kiepert's
parabola, given by
smA(sin 2 B - sin 2 C) sin B (sin 2 C - sin 2 A)
u v
sin C(sin 2 A — sin 2 B) _
w
a(b 2 ~ c 2 ) + b(c 2 - a 2 ) + c(a 2 - 6 2 ) =
U V w
where [u, v, w] are the TRILINEAR COORDINATES for a
line tangent to the parabola. It is tangent to the sides
of the TRIANGLE, the line at infinity, and the Lemoine
Line. The Focus has Triangle Center Function
a = csc(B - C).
(3)
The Euler Line of a triangle is the Directrix of
Kiepert's parabola. In fact, the DIRECTRICES of all
parabolas inscribed in a TRIANGLE pass through the
Orthocenter. The Brianchon Point for Kiepert's
parabola is the Steiner Point.
see also Brianchon Point, Envelope, Euler
Line, Isosceles Triangle, Lemoine Line, Steiner
Points
Kieroid
Let the center B of a CIRCLE of Radius a move along
a line BA. Let O be a fixed point located a distance c
away from AB. Draw a SECANT LINE through O and
D, the Midpoint of the chord cut from the line DE
(which is parallel to AB) and a distance b away. Then
the LOCUS of the points of intersection of OD and the
CIRCLE Pi and P2 is called a kieroid.
Special Case
Curve
6 =
b = a
b = a — — c
conchoid of Nicomedes
cissoid plus asymptote
strophoid plus asymptote
References
Yates, R. C. "Kieroid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 141-142,
1952,
Killing's Equation
The equation defining KILLING Vectors.
£xgab = X a; b + X b;a = 2X( a;6 ) = 0,
where C is the Lie Deftivative.
see also KILLING VECTORS
Killing Vectors
If any set of points is displaced by X z dxi where all dis-
tance relationships are unchanged (i.e., there is an ISOM-
etry), then the Vector field is called a Killing vector.
9ab
dx fC dx' d
dx a dx b
■9cd(x)j
so let
to, a , a
x = x + ex
dx ta
dx b
= o b + ex >b
(1)
(2)
9ab(x) = (SI + €x c , a ) (S$ + ex* , 6 ) g c d(x € + eX e )
= (SI + ez c iQ ) (S% + ex d jb ) [g c d(x) + eX e g cd (x), e + . . .]
= gab(x) + e[g ad X d jb + g bd X d , a + X e g ab , e ] + G(e 2 )
= Cxgab, (3)
where C is the Lie Derivative. An ordinary deriva-
tive can be replaced with a covariant derivative in a Lie
Derivative, so we can take as the definition
gab-c =
be cc
gabg = o a ,
(4)
(5)
which gives KILLING'S EQUATION
C-X9ab — X a ;b + Xb-a = 2X( a; (,) = 0. (6)
Kimberling Sequence
A Killing vector X b satisfies
9 c X c -ab - RabX —
X a ;bc = -flabcd-si
X a;b ;b + R a c X c = 0,
where R a b is the RlCCI TENSOR and R abc d is the RlE-
mann Tensor.
Kings Problem 987
A 2-sphere with METRIC
ds 2 = d6 2 + sin 2 6d(j) 2
(10)
has three Killing vectors, given by the angular momen-
tum operators
L x — - cos </>— + cot#sin<£— (11)
ott o<p
Ly = sin <f>— + cot cos 0— - (12)
C7C7 C/0
L *-^'
(13)
vectors in Euclidean 3-space are
x 1 = —
dx
(14)
2 8
x = n~
(15)
.3 5
(16)
4 9 a
x =y d~z- z ^
(17)
s_ d d
X — Z X
OX OZ
(18)
x e =x d__ JL
dy dx'
(19)
In Minkowski Space, there are 10 Killing vectors
X? = af for 2 = 1,2,3,4 (20)
X° k = (21)
X l k = e lkrn x m for k = 1,2,3 (22)
[0a:fc] for fc = l,2,3. (23)
^ = ^m
The first group is TRANSLATION, the second ROTATION,
and the final corresponds to a "boost."
Kimberling Sequence
A sequence generated by beginning with the Positive
integers, then iteratively applying the following algo-
rithm:
1. In iteration z, discard the zth element,
2. Alternately write the i + k and i — kih elements until
k = i,
3. Write the remaining elements in order.
The first few iterations are therefore
(7)
nn
2
3
4
5
6
7
8
9
10
11
2
[31
4
5
6
7
8
9
10
11
12
(8)
4
2
m
6
7
8
9
10
11
12
13
(9)
6
2
7
4
8
9
10
11
12
13
14
.IE-
8
7
9
2
10
6
11
12
13
14
15
(_> I & JJ 1U V AJ. 4-~ J.V J.T: J-W
The diagonal elements form the sequence 1, 3, 5, 4, 10,
7, 15, ... (Sloane's A007063).
References
Guy, R. K. "The Kimberling Shuffle." §E35 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 235-236, 1994.
Kimberling, C. "Problem 1615." Crux Math. 17, 44, 1991.
Sloane, N. J. A. Sequence A007063/M2387 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Kimberling Shuffle
see also Kimberling Sequence
Kings Problem
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
The problem of determining how many nonattacking
kings can be placed on an n x n CHESSBOARD. For
n = 8, the solution is 16, as illustrated above (Madachy
1979). In general, the solutions are
K(r,
i\n 2 n
\I(n + l) 2 n
even
odd
(1)
(Madachy 1979), giving the sequence of doubled squares
1, 1, 4, 4, 9, 9, 16, 16, ... (Sloane's A008794). This
sequence has Generating Function
1 + aT
(l~x*) 2 (l-x)
= 1 + x + 4a; 2 + 4a; 3 + 9a: 4 + 9a; 5 4- . . . .
(2)
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
Kg
988 King Walk
Kirkman Triple System
The minimum number of kings needed to attack or oc-
cupy all squares on an 8 x 8 CHESSBOARD is nine, illus-
trated above (Madachy 1979).
see also BISHOPS PROBLEM, CHESS, HARD HEXAGON
Entropy Constant, Knights Problem, Queens
Problem, Rooks Problem
References
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 39, 1979.
King Walk
see Delannoy Number
Kinney's Set
A set of plane Measure that contains a CIRCLE of
every RADIUS.
References
Falconer, K. J. The Geometry of Fractal Sets. New York:
Cambridge University Press, 1985.
Fejzic, H. "On Thin Sets of Circles." Amer. Math. Monthly
103, 582-585, 1996.
Kinney, J. R. "A Thin Set of Circles." Amer. Math. Monthly
75, 1077-1081, 1968.
Kinoshita-Terasaka Knot
The Knot with Braid Word
3 2 -1-2 -1 -1 -1
0~\ &3 &2 &z 0~\ <72 <7l <T3 0~2
Its Jones Polynomial is
r 4 (-l + 2t - 2t 2 + 2i 3 + t 6 - 2t 7 + 2t 8 - 2t 9 + t 10 ),
the same as for CONWAY'S KNOT. It has the same AL-
EXANDER Polynomial as the Unknot.
References
Kinoshita, S. and Terasaka, H. "On Unions of Knots." Osaka
Math. J. 9, 131-153, 1959.
Kinoshita-Terasaka Mutants
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 49-50, 1994.
Kirby Calculus
The manipulation of Dehn Surgery descriptions by a
certain set of operations.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, p. 263, 1994.
Kirby's List
A list of problems in low- dimensional TOPOLOGY main-
tained by R. C. Kirby. The list currently runs about 380
pages.
References
Kirby, R. "Problems in Low-Dimensional Topology."
http : //math . berkeley . edu/ -kirby/.
Kirkman's Schoolgirl Problem
In a boarding school there are fifteen schoolgirls who al-
ways take their daily walks in rows of threes. How can
it be arranged so that each schoolgirl walks in the same
row with every other schoolgirl exactly once a week?
Solution of this problem is equivalent to constructing a
KIRKMAN Triple System of order n = 2. The follow-
ing table gives one of the 7 distinct (up to permutations
of letters) solutions to the problem.
Sun Mon Tue Wed Thu Fri
Sat
ABC ADE AFG AHI AJK ALM ANO
DHL BIK BHJ BEG CDF BEF BDG
EJN CMO CLN BMN CLO CIJ CHK
FIO FHN DIM DJO EHM DKN EIL
GKM GJL EKO FKL GIN GHO FJM
(The table of Dorrie 1965 contains a misprint in which
the a\ = B and a-z = C entries for Wednesday and
Thursday are written simply as a.)
see also JOSEPHUS PROBLEM, KlRKMAN TRIPLE SYS-
TEM, 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.
Dorrie, H. §5 in 100 Great Problems of Elementary Mathe-
matics: Their History and Solutions. New York: Dover,
pp. 14-18, 1965.
Frost, A. "General Solution and Extension of the Problem
of the 15 Schoolgirls." Quart. J. Pure Applied Math. 11,
1871.
Kirkman, T. P. "On a Problem in Combinatories." Cam-
bridge and Dublin Math. J. 2, 191-204, 1847.
Kirkman, T. P. Lady's and Gentleman's Diary. 1850.
Kraitchik, M. §9.3.1 in Mathematical Recreations. New York:
W. W. Norton, pp. 226-227, 1942.
Peirce, B. "Cyclic Solutions of the School-Girl Puzzle." As-
tron. J. 6, 169-174, 1859-1861.
Ryser, H. J. Combinatorial Mathematics. Buffalo, NY:
Math. Assoc. Amer., pp. 101-102, 1963.
Kirkman Triple System
A Kirkman triple system of order v = 6n + 3 is a
Steiner Triple System with parallelism (Ball and
Coxeter 1987), i.e., one with the following additional
stipulation: the set of b = (2n + l)(3n + 1) triples is
partitioned into 3n + 1 components such that each com-
ponent is a (2n + l)-subset of triples and each of the v
elements appears exactly once in each component. The
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