Quarter Squares Rule
Quarter Squares Rule
( S T i )'-(4 i )*-*
Quartet
A Set of four, also called a Tetrad.
see also HEXAD, MONAD, QUINTET, TETRAD, TRIAD
Quartic Curve
A general plane quartic curve is a curve of the form
Ax 4 + By 4 + Cx z y + Dx 2 y 2 + Exy 3 + Fx s + Gy 3
+Hx 2 y + Ixy 2 + Jx 2 + Ky 2 + Lxy + Mx + Ny + O = 0.
(1)
The incidence relations of the 28 bitangents of the gen-
eral quartic curve can be put into a One- TO- One cor-
respondence with the vertices of a particular POLYTOPE
in 7-D space (Coxeter 1928, Du Val 1931). This fact is
essentially similar to the discovery by Schoutte (1910)
that the 27 Solomon's Seal Lines on a Cubic Sur-
face can be connected with a POLYTOPE in 6-D space
(Du Val 1931). A similar but less complete relation ex-
ists between the tritangent planes of the canonical curve
of genus 4 and an 8-D POLYTOPE (Du Val 1931).
The maximum number of DOUBLE POINTS for a nonde-
generate quartic curve is three.
A quartic curve of the form
y 2 = (x- a)(x - 0(x - 7)0 - 6) (2)
can be written
(_!^ a = (i-^H 1 _!zi!)( 1 _«z£) l
\x — a/ \ x — a/ V x — a/ \ x — a/
(3)
(4)
(5)
and so is CUBIC in the coordinates
Y =
x — a
y
x — a 2
This transformation is a BlRATIONAL TRANSFORMA-
TION.
(a) . (b)
Quartic Equation 1489
Let P and Q be the Inflection Points and R and 5
the intersections of the line PQ with the curve in Figure
(a) above. Then
A = C
B = 2A
(6)
(7)
In Figure (b), let UV be the double tangent, and T the
point on the curve whose x coordinate is the average of
the x coordinates of U and V. Then UV\\PQ\\RS and
D = F
E = V2D.
(8)
(9)
In Figure (c), the tangent at P intersects the curve at
W. Then
G = SB. (10)
Finally, in Figure (d), the intersections of the tangents
at P and Q are W and X. Then
H = 27B
(11)
(Honsberger 1991).
see also CUBIC SURFACE, PEAR-SHAPED CURVE,
Solomon's Seal Lines
References
Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six
and Seven Dimensions." Proc. Cambridge Phil. Soc. 24,
7-9, 1928.
Du Val, P. "On the Directrices of a Set of Points in a Plane."
Proc. London Math. Soc. Ser. 2 35, 23-74, 1933.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 114-118, 1991.
Schoutte, P. H. "On the Relation Between the Vertices of a
Definite Sixdimensional Polytope and the Lines of a Cubic
Surface." Proc. Roy. Akad. Acad. Amsterdam 13, 375-
383, 1910.
Quartic Equation
A general quartic equation (also called a BIQUADRATIC
EQUATION) is a fourth-order POLYNOMIAL of the form
z 4 + azz 2, + a<iz 2 + a\z + ao = 0. (1)
The ROOTS of this equation satisfy NEWTON'S RELA-
TIONS:
xi + x 2 + x 3 + x 4 = -a 3 (2)
CC1X2 + X1X3 + ^1^4 + #2#3 + #2#4 + X3X4 = «2 (3)
X1X2X3 + CC2#3#4 + XiX2a?4 + X\X$X± — —a\ (4)
x\X2Xzx$ = ao, (5)
where the denominators on the right side are all 04 = 1.
Ferrari was the first to develop an algebraic technique
for solving the general quartic. He applied his technique
1490 Quartic Equation
Quartic Equation
(which was stolen and published by Cardano) to the
equation
x 4 + 6x 2 - 60x + 36 = (6)
(Smith 1994, p. 207).
The x 3 term can be eliminated from the general quartic
(1) by making a substitution of the form
z = x — A,
x 4 + (a 3 - 4A)z 3 + (a 2 - 3a 3 A + 6A 2 )x
+ (ai - 2a 2 A + 3a 3 A - 4A 3 )z
Adding and subtracting x 2 u + u 2 /4 to (10) gives
x 4 + x 2 u + |n 2 - t 2 u - \u 2 +px 2 + qx + r = 1 (14)
which can be rewritten
(x 2 - \u) 2 - [(u - p)x 2 -qx + {\u 2 - r)] = (15)
(Birkhoff and Mac Lane 1965). The first term is a per-
fect square P 2 , and the second term is a perfect square
Q 2 for those u such that
q - 4(u - p)(\u -r).
(16)
This is the resolvent CUBIC, and plugging a solution u\
back in gives
P 2 -Q 2 = (P + Q)(P-Q),
so (15) becomes
(17)
(x 2 + \u 1 +Q){x 2 + \u 1 -Q), (18)
where
Q^Ax-B
A = \/ui — p
R- q
(19)
(20)
(21)
Let y\ be a Real Root of the resolvent Cubic Equa-
tion
y 3 — fl22/ 2 + {o>iQ>z — 4ao)y + (4a 2 ao — ai — a3 clq) = 0.
(22)
The four ROOTS are then given by the ROOTS of the
equation
( 7 ) x 2 + i(a 3 ± v / «3 2 -4a 2 +4j /1 )
+ |(2/i T \/j/i 2 - 4a ) = 0, (23)
which are
+(ao — aiA + a 2 A -
-a 3 A*
+ A 4 ).
(8)
Letting A = a 3 /4 so
z = x - |A
(9)
then gives
x +px + qx + r,
where
(10)
where
— 3 2
p = a 2 - gfl3
(11)
1 i 1 3
5 = ai — 2*^2^3 + g«3
(12)
D= <
1 .1 2
r = ao — ^aia3 -r Ye a2(l3 ~
3 „ 4
" 256 fl 3 •
(13)
4 L
Z3 = — 4 ^3 — j "^ + 2^
^4 = —4^3 — 2^ "" 2^»
ra 3 2 -a 2 +yi
(24)
(25)
(26)
(27)
(28)
' ^fas 2 - R 2 - 2a 2 + 4-(4a 3 a 2 - 8ai - a-j 3 )^ 1
#7^0
^faa 2 -2a 2 + 2 A /yi 2 -4^
tf =
(29)
' v/faa 2 - # 2 - 2a 2 - |(4a 3 a 2 - 8a! - as 3 )/*- 1
R^0
E= <
A/fas 2 - 2a 2 - 2^/2/i 2 - 4a
^ R = 0.
(30)
Another approach to solving the quartic (10) defines
a = (xi + £ 2 )(£3 + Z4) = -(a?i + x 2 ) 2 (31)
/3 = (a* + z 3 )(z 2 + x 4 ) = -(a* + x 3 ) 2 (32)
7 = Od + 2:4) (a?2 4- ^3) = -(^2 + x 3 ) 2 , (33)
where use has been made of
xi + X2 + xs + £4 = (34)
(which follows since a 3 = 0), and
h{x) = (x - a) (a; - /?)(a; - 7) (35)
= x 3 - (a + /3 + 7)x 2 + (a/3 + c*7 + 0*y)x - a/37.
(36)
Quartic Reciprocity Theorem
Comparing with
P(x) = x + px + qx + r
(37)
— (x - xi)(x — X2)(X — X3)(X - X4) (38)
= x* +
II XiXj
>i*3
+ (a?i + x 2 )(a;i + x 3 )(a; 2 + £3)^
- XxX2X^(x\ +x 2 + £3), (39)
gives
h(x) = x — 2px
(p 2 -r)z + g 2 . (40)
Solving this CUBIC EQUATION gives a, j3> and 7, which
can then be solved for the roots of the quartic Xi
(Faucette 1996).
see also Cubic Equation, Discriminant (Polynom-
ial), Quintic Equation
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 17-18, 1972.
Berger, M. §16.4.1-16.4.11.1 in Geometry J. New York:
Springer- Verlag, 1987.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 12, 1987.
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra,
3rd ed. New York: Macmillan, pp. 107-108, 1965.
Ehrlich, G. §4.16 in Fundamental Concepts of Abstract Alge-
bra, Boston, MA: PWS-Kent, 1991.
Faucette, W. M. "A Geometric Interpretation of the Solution
of the General Quartic Polynomial." Amer. Math. Monthly
103, 51-57, 1996.
Smith, D. E. A Source Book in Mathematics. New York:
Dover, 1994.
van der Waerden, B. L. §64 in Algebra, Vol. 1. New York:
Springer- Verlag, 1993.
Quartic Reciprocity Theorem
Gauss stated the case n = 4 using the Gaussian Inte-
gers.
see also RECIPROCITY THEOREM
References
Ireland, K. and Rosen, M. "Cubic and Biquadratic Reci-
procity." Ch. 9 in A Classical Introduction to Modern
Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 108-137, 1990.
Quartic Surface
An Algebraic Surface of Order 4. Unlike Cubic
Surfaces, quartic surfaces have not been fully classi-
fied.
see also Bohemian Dome, Burkhardt Quartic,
Cassini Surface, Cushion, Cyclide, Desmic Sur-
face, Kummer Surface, Miter Surface, Piri-
form, Roman Surface, Symmetroid, Tetrahe-
droid, Tooth Surface
Quasiamicable Pair 1491
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 9, 1986.
Fischer, G. (Ed.). Plates 40-41, 45-49, and 52-56
in Mathematische Modelle/ Mathematical Models, Bild-
band/ Photograph Volume. Braunschweig, Germany:
Vieweg, pp. 40-41, 45-49, and 52-56, 1986.
Hunt, B. "Some Quartic Surfaces." Appendix B,5 in The Ge-
ometry of Some Special Arithmetic Quotients. New York:
Springer- Verlag, pp. 310-319, 1996.
Jessop, C Quartic Surfaces with Singular Points. Cam-
bridge, England: Cambridge University Press, 1916.
Quartile
One of the four divisions of observations which have
been grouped into four equal-sized sets based on their
Rank. The quartile including the top RANKED mem-
bers is called the first quartile and denoted Q\. The
other quartiles are similarly denoted Q2, Q3, and Qa>
For N data points with N of the form An + 5 (for n = 0,
1, . . . ), the HINGES are identical to the first and third
quartiles.
see also HlNGE, INTERQUARTILE RANGE, QUARTILE
Deviation, Quartile Variation Coefficient
Quartile Deviation
QD=§(Q 3
■Qi),
where Qi and Q 2 are INTERQUARTILE RANGES.
see also QUARTILE VARIATION COEFFICIENT
Quartile Range
see Interquartile Range
Quartile Skewness Coefficient
see Bowley Skewness
Quartile Variation Coefficient
Q 3 + Qi
where Qi and Q 2 are INTERQUARTILE RANGES.
Quasiamicable Pair
Let a(m) be the DIVISOR FUNCTION of m. Then two
numbers m and n are a quasiamicable pair if
cr(m) = cr(n) = m + n + 1.
The first few are (48, 75), (140, 195), (1050, 1925),
(1575, 1648), ... (Sloane's A005276). Quasiamicable
numbers are sometimes called Betrothed Numbers
or Reduced Amicable Pairs.
see also Amicable Pair
References
Beck, W. E. and Najar, R. M. "More Reduced Amicable
Pairs." Fib. Quart. 15, 331-332, 1977.
1492 Quasiconformal Map
Quasiregular Polyhedron
Guy, R. K. "Quasi- Amicable or Betrothed Numbers." §B5 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 59-60, 1994.
Hagis, P. and Lord, G. "Quasi-Amicable Numbers." Math.
Comput. 31, 608-611, 1977.
Sloane, N. J. A. Sequence A005276/M5291 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Quasiconformal Map
A generalized Con formal Map.
see also BELTRAMI DIFFERENTIAL EQUATION
References
Iyanaga, S. and Kawada, Y. (Eds.). "Quasiconformal Map-
pings." §347 in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 1086-1088, 1980.
Quasigroup
A GROUPOID S such that for all a, b e S y there exist
unique x,y £ S such that
ax = b
ya = b.
Quasiperiodic Motion
The type of motion executed by a DYNAMICAL SYSTEM
containing two incommensurate frequencies.
Quasirandom Sequence
A sequence of n-tuples that fills n-space more uniformly
than uncorrelated random points. Such a sequence is
extremely useful in computational problems where num-
bers are computed on a grid, but it is not known in ad-
vance how fine the grid must be to obtain accurate re-
sults. Using a quasirandom sequence allows stopping at
any point where convergence is observed, whereas the
usual approach of halving the interval between subse-
quent computations requires a huge number of compu-
tations between stopping points.
see also Pseudorandom Number, Random Number
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Quasi- (that is, Sub-) Random Sequences."
§7.7 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 299-306, 1992.
No other restrictions are applied; thus a quasigroup need
not have an Identity Element, not be associative, etc.
Quasigroups are precisely GROUPOIDS whose multiplica-
tion tables are LATIN SQUARES. A quasigroup can be
empty.
see also Binary OPERATOR, GROUPOID, Latin
Square, Loop (Algebra), Monoid, Semigroup
References
van Lint, J. H. and Wilson, R. M. A Course in Combina-
torics. New York: Cambridge University Press, 1992.
Quasiperfect Number
A least Abundant Number, i.e., one such that
a(n) = 2n + 1.
Quasiperfect numbers are therefore the sum of their non-
trivial DIVISORS. No quasiperfect numbers are known,
although if any exist, they must be greater than 10 35
and have seven or more DIVISORS. Singh (1997) called
quasiperfect numbers SLIGHTLY EXCESSIVE NUMBERS.
see also Abundant Number, Almost Perfect Num-
ber, Perfect Number
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.
Singh, S. FermaVs Enigma: The Epic Quest to Solve
the World's Greatest Mathematical Problem. New York:
Walker, p. 13, 1997.
Quasiperiodic Function
see WeierstraB Sigma Function, WeierstraB
Zeta Function
Quasiregular Polyhedron
A quasiregular polyhedron is the solid region inte-
rior to two DUAL regular polyhedra with Schlafli
SYMBOLS{p, q} and {<?,£>}. Quasiregular polyhedra are
denoted using a Schlafli Symbol of the form {£},
with
a)
CM:}-
Quasiregular polyhedra have two kinds of regular faces
with each entirely surrounded by faces of the other kind,
equal sides, and equal dihedral angles. They must sat-
isfy the Diophantine inequality
1 1 1 ,
- + - + -> 1.
p q r
(2)
But p, q > 3, so r must be 2. This means that the possi-
ble quasiregular polyhedra have symbols { 3 j, 1 4 }, and
{*}. Now
{sH 3 ' 4 > < s >
is the Octahedron, which is a regular Platonic Solid
and not considered quasiregular. This leaves only two
convex quasiregular polyhedra: the CUBOCTAHEDRON
{*} and the ICOSIDODECAHEDRON {j?}.
If nonconvex polyhedra are allowed, then additional
quasiregular polyhedra are the GREAT DODECAHEDRON
{5, §} and the GREAT ICOSIDODECAHEDRON {3, §}
(Hart).
For faces to be equatorial {/i},
h = V4iVi + 1 -1.
(4)
Quasirhombicosidodecahedron
Quaternion 1493
The EDGES of quasiregular polyhedra form a system
of Great Circles: the Octahedron forms three
Squares, the Cuboctahedron four Hexagons, and
the Icosidodecahedron six Decagons. The Ver-
tex Figures of quasiregular polyhedra are Rhombuses
(Hart). The EDGES are also all equivalent, a prop-
erty shared only with the completely regular PLATONIC
Solids.
see also Cuboctahedron, Great Dodecahedron,
Great Icosidodecahedron, Icosidodecahedron,
Platonic Solid
References
Coxeter, H. S. M. "Quasi-Regular Polyhedra." §2-3 in Regu-
lar Polytopes, 3rd ed. New York: Dover, pp. 17-20, 1973.
Hart, G. W. "Quasi- Regular Polyhedra." http://www.li.
net / - george / virtual - polyhedra / quasi - regular -
info.html.
Quasirhombicosidodecahedron
see Great Rhombicosidodecahedron (Uniform)
Quasirhombicuboctahedron
see Great Rhombicuboctahedron (Uniform)
Quasisimple Group
A Finite Group L is quasisimple if L = [L, L] and
L/Z(L) is a Simple Group.
see also Component, Finite Group, Simple Group
Quasithin Theorem
In the classical quasithin case of the Quasi-Unipotent
Problem, if G does not have a "strongly embedded"
Subgroup, then G is a Group of Lie-Type in charac-
teristic 2 of Lie Rank 2 generated by a pair of parabolic
Subgroups Pi and P2, or G is one of a short list of
exceptions.
see also Lie-Type Group, Quasi-Unipotent Prob-
lem
Quasitruncated Cuboctahedron
see Great Truncated Cuboctahedron
Quasitruncated Dodecadocahedron
see Truncated Dodecadodecahedron
Quasitruncated Dodecahedron
see Truncated Dodecahedron
Quasitruncated Great Stellated
Dodecahedron
see Great Stellated Truncated Dodecahedron
Quasitruncated Hexahedron
see Stellated Truncated Hexahedron
Quasitruncated Small Stellated
Dodecahedron
see Small Stellated Truncated Dodecahedron
Quasi-Unipotent Group
A GROUP G is quasi-unipotent if every element of G of
order p is Unipotent for all Primes p such that G has
2>-Rank > 3.
Quasi-Unipotent Problem
see Quasithin Theorem
Quaternary
The BASE 4 method of counting in which only the DIG-
ITS 0, 1, 2, and 3 are used. These DIGITS have the
following multiplication table.
X
1
2
3
1
1
2
3
2
2
10
12
3
3
12
21
see also BASE (NUMBER), BINARY, DECIMAL, HEXA-
DECIMAL, Octal, Ternary
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 9-10,
1991.
$$ Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Bases.m.
Quaternary Tree
see Quadtree
Quaternion
A member of a noncommutative Division ALGEBRA
first invented by William Rowan Hamilton. The quater-
nions are sometimes also known as HYPERCOMPLEX
Numbers and denoted EL While the quaternions are
not commutative, they are associative.
The quaternions can be represented using complex 2x2
Matrices
H
z
-w*
w
z*
a + ib c + id
— c + id a — ib
(i)
where z and w are COMPLEX Numbers, a, 6, c, and
d are Real, and z* is the Complex Conjugate of
z. By analogy with the Complex Numbers being rep-
resentable as a sum of REAL and IMAGINARY PARTS,
a • 1 + bi, a quaternion can also be written as a linear
combination
H = all + 61 + cJ + dK
(2)
1494 Quaternion
of the four matrices
U =
1 =
Js
K =
"l
o"
1
i
—i
"
1
-1
~0
i
i
(3)
(4)
(5)
(6)
(Note that here, U is used to denote the Identity Ma-
trix, not I.) The matrices are closely related to the
Pauli Spin Matrices cr x , <r yj <r z , combined with the
Identity Matrix. From the above definitions, it fol-
lows that
i 2 = -u
J 2 = -U
K 2 = -U.
(7)
(8)
(9)
Therefore I, J, and K are three essentially different so-
lutions of the matrix equation
-u,
(10)
which could be considered the square roots of the nega-
tive identity matrix.
In M 4 , the basis of the quaternions can be given by
(ii)
i =
10
-10
1
0-1
0-
-1
0-1
10
1
0-1
1
1
0-10
10 0"
10
10
1_
(12)
(13)
(14)
The quaternions satisfy the following identities, some-
times known as HAMILTON'S RULES,
(15)
A 1 ~ 2
I =j
= k 2
= -
ij =
-ji =
--k
jk =
-kj
— i
(16)
(17)
ki — —ik = j.
They have the following multiplication table.
Quaternion
(18)
1
i
3
k
1
1
i
3
k
i
i
-1
k
-3
3
3
-k
-1
i
k
k
3
—i
-1
The quaternions ±1, ±i, ±7, and ±k form a non-Abelian
GROUP of order eight (with multiplication as the group
operation) known as Q%.
The quaternions can be written in the form
a = a\ + a 2 i + dzj + a^k. (19)
The conjugate quaternion is given by
a* = ai — a 2 i — dzj — 0,4k. (20)
The sum of two quaternions is then
a+b = (a 1 + bi) + (a 2 +b 2 )i + {a 3 + b 3 )j + (a4+b 4 )k, (21)
and the product of two quaternions is
ab = (ai&i — a 2 b 2 — G3&3 — 04^4)
+ (ai&2 + ^2&1 + a3^4 — (1463)*
+ (ai& 3 — a 2 6 4 + a3 ^i + a 4^)j
+ (ai&4 + a2&3 — &3&2 + »46i)A;,
so the norm is
(22)
n(a) = Vaa* = Va*a = ya\ 2 + a 2 2 4- Q>z 2 + ck 2 .
(23)
In this notation, the quaternions are closely related to
Four- Vectors.
Quaternions can be interpreted as a SCALAR plus a VEC-
TOR by writing
a = a\ + 0,21 + azj + 0,4k — (ai,a),
(24)
where a = [02 «3 0,4]. In this notation, quaternion mul-
tiplication has the particularly simple form
qiqt = (si,v x ) • (S2,v 2 )
= {s\s 2 - vi - v 2 , siv 2 + S2V1 tvix v 2 ). (25)
Division is uniquely defined (except by zero) , so quater-
nions form a Division Algebra. The inverse of a
quaternion is given by
z," 1 °
aa*
and the norm is multiplicative
n(ab) = n(a)n(b).
(26)
(27)
Quaternion
Queens Problem 1495
In fact , the product of two quaternion norms immedi-
ately gives the Euler Four-Square Identity.
A rotation about the Unit VECTOR n by an angle 9 can
be computing using the quaternion
g=( 5 ,v) = (cos(§0),nsin(§0)) (28)
(Arvo 1994 } Hearn and Baker 1996). The components of
this quaternion are called EULER PARAMETERS. After
rotation, a point p = (0, p) is then given by
V =<m
qpq
(29)
since n(q) = 1. A concatenation of two rotations, first
qi and then 52, can be computed using the identity
Julstrom, B. A. "Using Real Quaternions to Represent Ro-
tations in Three Dimensions." UMAP Modules in Under-
graduate Mathematics and Its Applications, Module 652.
Lexington, MA: COMAP, Inc., 1992.
Kelland, P. and Tait, P. G. Introduction to Quaternions, 3rd
ed. London: Macmillan, 1904.
Nicholson, W. K. Introduction to Abstract Algebra. Boston,
MA: PWS-Kent, 1993.
Tait, P. G. An Elementary Treatise on Quaternions , 3rd ed.,
enl Cambridge, England: Cambridge University Press,
1890.
Tait, P. G. "Quaternions." Encyclopedia Britannica, 9th
ed. ca, 1886. ftp://ftp.netcom.com/pub/hb/hbaker/
quaternion/tait/Encyc-Brit . ps . gz.
Quattuordecillion
In the American system, 10 45 .
see also LARGE NUMBER
£
o.
_£
2_
2
o_
o.
I I I lol 1 I
1 1 |oi 1 1 r
2
°.
o_ :
a
-0.
£_
1 1 1 k> i 1 1:
| 1 1 1 1 1 -
0.
Q
£
0.
o__
_£
0.
_£
0_
0.
2
P_
£
o__
I I I lot II
I I I I I ToT
_£
o_
£
o_
£
Q
_ 1 1 | pl I r :
~ p i 1 1 1 1 ~
2
&
2
2
o_
£_
lol II
Q2(qipql)q2 = (qzqifyiqiqZ) = (q2qi)p(q2qi)* (30) Queens Problem
(Goldstein 1980).
see also Biquaternion, Cayley-Klein Parameters,
Complex Number, Division Algebra, Euler Pa-
rameters, Four- Vector, Octonion
References
Altmann, S. L. Rotations, Quaternions, and Double Groups.
Oxford, England: Clarendon Press, 1986.
Arvo, J. Graphics Gems 2. New York: Academic Press,
pp. 351-354 and 377-380, 1994.
Baker, A. L. Quaternions as the Result of Algebraic Opera-
tions. New York: Van Nostrand, 1911.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Item 107, Feb. 1972.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 230-234, 1996.
Crowe, M. J. A History of Vector Analysis: The Evolution
of the Idea of a Vectorial System. New York: Dover, 1994.
Dickson, L. E. Algebras and Their Arithmetics. New York:
Dover, 1960.
Du Val, P. Homographies, Quaternions, and Rotations. Ox-
ford, England: Oxford University Press, 1964.
Ebbinghaus, H. D.; Hirzebruch, F.; Hermes, H.; Prestel, A;
Koecher, M.; Mainzer, M.; and Remmert, R. Numbers.
New York: Springer- Verlag, 1990.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA:
Addison- Wesley, p. 151, 1980.
Hamilton, W. R. Lectures on Quaternions: Containing a
Systematic Statement of a New Mathematical Method.
Dublin: Hodges and Smith, 1853.
Hamilton, W. R. Elements of Quaternions. London: Long-
mans, Green, 1866.
Hamilton, W. R. The Mathematical Papers of Sir William
Rowan Hamilton. Cambridge, England: Cambridge Uni-
versity Press, 1967.
Hardy, A. S. Elements of Quaternions. Boston, MA: Ginn,
Heath, & Co., 1881.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Cambridge, England: Clarendon
Press, 1965.
Hearn, D. and Baker, M. P. Computer Graphics: C Version,
2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 419-420
and 617-618, 1996.
Joly, C. J. A Manual of Quaternions. London: Macmillan,
1905.
0.
a
2_
_a
P_
p.
£
I I Moll I
i°i 1 1 1 I r
0_
Q_
a_
a
Q
0.
I II lol 1
I I I I I P I Z
0.
0.
Q
0.
_£
£_
I I I I IpI III
Q
P
p
Q
9
0.
p
Q
p
p
Q
Q
9
P
p
Q
p
2
P
P
What is the maximum number of queens which can be
placed on an n x n CHESSBOARD such that no two attack
one another? The answer is n queens, which gives eight
queens for the usual 8x8 board (Madachy 1979). The
number of different ways the n queens can be placed on
an n x n chessboard so that no two queens may attack
each other for the first few n are 1, 0, 0, 2, 10, 4, 40, 92,
. . . (Sloane's A000170, Madachy 1979). The number of
rotationally and reflectively distinct solutions are 1, 0,
0, 1, 2, 1, 6, 12, 46, 92, . . . (Sloane's A002562; Dudeney
1970; p. 96). The 12 distinct solutions for n = 8 are
illustrated above, and the remaining 80 are generated
by Rotation and Reflection (Madachy 1979).
Q
Q__
Q
Q
1496 Queens Problem
Queue
The minimum number of queens needed to occupy or
attack all squares of an 8 x 8 board is 5. Dudeney (1970,
pp. 95-96) gave the following results for the number of
distinct arrangements N p (k,n) of A; queens attacking or
occupying every square of an n x n board for which every
queen is attacked ("protected") by at least one other.
k Queens
n x n
N v (k,n)
2
4
3
3
5
37
3
6
1
4
7
5
Dudeney (1970, pp. 95-96) also gave the following re-
sults for the number of distinct arrangements N u (k,n)
of k queens attacking or occupying every square of an
n x n board for which no two queens attack one another
(they are "not protected").
k Queens
n x n
N u (k,n)
1
2
1
1
3
1
3
4
2
3
5
2
4
6
17
4
7
1
5
8
91
Vardi (1991) generalizes the problem from a square
chessboard to one with the topology of the TORUS. The
number of solutions for n queens with n Odd are 1, 0,
10, 28, 0, 88, . . . (Sloane's A007705). Vardi (1991) also
considers the toroidal "semiqueens" problem, in which
a semiqueen can move like a rook or bishop, but only on
Positive broken diagonals. The number of solutions to
this problem for n queens with n Odd are 1, 3, 15, 133,
2025, 37851, . . . (Sloane's A006717), and for EVEN n.
Chow and Velucchi give the solution to the question,
"How many different arrangements of k queens are pos-
sible on an order n chessboard?" as l/8th of the COEF-
FICIENT of a k b n2 ~ k in the POLYNOMIAL
p(a,6, n)
' (a + b) n2 + 2(o + b) n (a 2 + b 2 ) {n2 ~ n)/2
+3(a 2 +6 2 ) n2/2 + 2(a 4 +6 4 ) n2 / 4
n even
(a + b) n2 + 2(o + 6)(a 4 + fc 4 )^ 2 " 1 )/ 4
+(a + 6)(a 2 + 6 2 )^ 2 - 1 ^ 2
+4(a + b) n (a 2 + b 2 ) {n2 ~ n)/2 n odd.
Velucchi also considers the nondominating queens prob-
lem, which consists of placing n queens on an order
n chessboard to leave a maximum number U(n) of
unat tacked vacant cells. The first few values are 0, 0, 0,
1, 3, 5, 7, 11, 18, 22, 30, 36, 47, 56, 72, 82, . . . (Sloane's
A001366). The results can be generalized to k queens
on an n x n board.
see also Bishops Problem, Chess, Kings Problem,
Knights Problem, Knight's Tour, Rooks Prob-
lem
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Reeve-
ations and Essays, 13th ed. New York: Dover, pp. 166-
169, 1987.
Campbell, P. J. "Gauss and the 8-Queens Problem: A Study-
in the Propagation of Historical Error." Historia Math. 4,
397-404, 1977.
Chow, T. and Velucchi, M. "Different Dispositions in the
Chessboard." http : //www. cli . di . unipi . it /-velucchi/
diff .txt.
Dudeney, H. E. "The Eight Queens." §300 in Amusements
in Mathematics. New York: Dover, p. 89, 1970.
Erbas, C. and Tanik, M. M. "Generating Solutions to the
TV-Queens Problem Using 2-Circulants." Math. Mag. 68,
343-356, 1995.
Erbas, C.; Tanik, M, M.; and Aliyzaicioglu, Z. "Linear Con-
gruence Equations for the Solutions of the TV-Queens Prob-
lem." Inform. Proc. Let. 41, 301-306, 1992.
Ginsburg, J. "Gauss's Arithmetization of the Problem of n
Queens." Scripta Math. 5, 63—66, 1939.
Guy, R. K. "The n Queens Problem." §C18 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 133-135, 1994.
Kraitchik, M. "The Problem of the Queens" and "Domina-
tion of the Chessboard." §10.3 and 10.4 in Mathematical
Recreations. New York: W. W. Norton, pp. 247-256, 1942.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 34-36, 1979.
Polya, G. "Uber die 'doppelt-periodischen' Losungen des n-
Damen-Problems." In Mathematische Unterhaltungen und
Spiele (Ed. W. Ahrens). 1918.
Riven, I.; Vardi, I.; and Zimmerman, P. "The n-Queens Prob-
lem." Amer. Math. Monthly 101, 629-639, 1994.
Riven, I. and Zabih, R. "An Algebraic Approach to Con-
straint Satisfaction Problems." In Proc. Eleventh Internat.
Joint Conference on Artificial Intelligence, Vol. 1, August
20-25, 1989. Detroit, MI: IJCAII, pp. 284-289, 1989.
Ruskey, F. "Information on the n Queens Problem."
http : //sue . esc .uvic . ca/ -cos /inf /mis c /Queen. html.
Sloane, N. J. A. Sequences A001366, A000170/M1958,
A006717/M3005, A007705/M4691, and A002562/M0180
in "An On-Line Version of the Encyclopedia of Integer Se-
quences." Sloane, N. J, A. and Plouffe, S. Extended entry
for M0180 in The Encyclopedia of Integer Sequences. San
Diego: Academic Press, 1995.
Vardi, I. "The n-Queens Problems." Ch. 6 in Computational
Recreations in Mathematica. Redwood City, CA: Addison-
Wesley, pp. 107-125, 1991.
Velucchi, M. "Non-Dominating Queens Problem." http://
www. cli . di .unipi . it/-velucchi/queens . txt.
Queue
A queue is a special kind of LIST in which elements
may only be removed from the bottom by a POP action
or added to the top using a PUSH action. Examples
of queues include people waiting in line, and submitted
jobs waiting to be printed on a printer. The study of
queues is called QUEUING THEORY.
see also List, Queuing Theory, Stack
Queuing Theory
Queuing Theory
The study of the waiting times, lengths, and other prop-
erties of Queues.
References
Allen, A. O. Probability, Statistics, and Queueing Theory
with Computer Science Applications. Orlando, FL: Aca-
demic Press, 1978.
Quicksort
The fastest known SORTING ALGORITHM (on average,
and for a large number of elements), requiring 0(n\gn)
steps. Quicksort is a recursive algorithm which first
partitions an array {ai}7-i according to several rules
(Sedgewick 1978):
1. Some key v is in its final position in the array (i.e.,
if it is the jth smallest, it is in position clj).
2. All the elements to the left of a 3 - are less than or equal
to a,j. The elements oi, a2, . . . , clj-i are called the
"left subfile."
3. All the elements to the right of a 3 - are greater than
or equal to clj. The elements a^+i, . . . , a n are called
the "right subfile."
Quicksort was invented by Hoare (1961, 1962), has
undergone extensive analysis and scrutiny (Sedgewick
1975, 1977, 1978), and is known to be about twice as
fast as the next fastest SORTING algorithm. In the worst
case, however, quicksort is a slow n 2 algorithm (and for
quicksort, "worst case" corresponds to already sorted).
see also HEAPSORT, SORTING
References
Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Struc-
tures and Algorithms. Reading, MA: Addis on- Wesley,
pp. 260-270, 1987.
Hoare, C. A. R. "Partition: Algorithm 63," "Quicksort: Al-
gorithm 64," and "Find: Algorithm 65." Comm. ACM A,
321-322, 1961.
Hoare, C. A. R. "Quicksort." Computer J. 5, 10-15, 1962.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
t erring, W. T. "Quicksort." §8.2 in Numerical Recipes
in FORTRAN: The Art of Scientific Computing, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 323-327, 1992.
Sedgewick, R. Quicksort. Ph.D. thesis. Stanford Computer
Science Report STAN-CS-75-492. Stanford, CA: Stanford
University, May 1975.
Sedgewick, R. "The Analysis of Quicksort Programs." Acta
Informatica 7, 327-355, 1977.
Sedgewick, R. "Implementing Quicksort Programs." Comm.
A CM 21, 847-857, 1978.
Quillen-Lichtenbaum Conjecture
A technical CONJECTURE which connects algebraic k-
Theory to Etale cohomology. The conjecture was made
more precise by Dwyer and Friedlander (1982). Thoma-
son (1985) established the first half of this conjecture,
but the entire conjecture has not yet been established.
Quintic Equation 1497
References
Dwyer, W. and Friedlander, E. "Etale .KT-Theory and Arith-
metic." Bull Amer. Math. Soc. 6, 453-455, 1982.
Thomason, R. W. "Algebraic JC-Theory and Etale Cohomol-
ogy." Ann. Sci. Ecole Norm. Sup. 18, 437-552, 1985.
Weibel, C. A. "The Mathematical Enterprises of Robert
Thomason." Bull. Amer. Math. Soc. 34, 1-13, 1996.
Quincunx
The pattern V of dots on the "5" side of a 6-sided DIE.
The word derives from the Latin words for both one and
five.
see also Dice
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 9 and 22, 1996.
Quindecillion
In the American system, 10 48 .
see also LARGE NUMBER
Quintet
A Set of five.
see also Hexad, Monad, Quartet, Tetrad, Triad
Quintic Equation
A general quintic cannot be solved algebraically in terms
of finite additions, multiplications, and root extractions,
as rigorously demonstrated by Abel and Galois.
Euler reduced the general quintic to
x 5 - 10qx 2 -p = 0.
(1)
A quintic also can be algebraically reduced to PRINCIPAL
Quintic Form
x 5 + a2# 2 + aix + ao = 0.
(2)
By solving a quartic, a quintic can be algebraically re-
duced to the Bring Quintic Form
x — x — a — 0,
as was first done by Jerrard.
Consider the quintic
4
(3)
Y[[x - (w j m + w 4j u 2 )] = 0,
(4)
3 =
where w = e 2 ™ /5 and m and u 2 are COMPLEX NUM-
BERS. This is called DE MoiVRE'S QUINTIC. Generalize
it to
4
Y[[x - (w j U! + w 2j u 2 + co 3j u 3 + uj 4j u 4 )] = 0. (5)
J=0
1498 Quintic Equation
Quintic Equation
Expanding,
(u; j ui + J*iu2 + v 3j U3 + w 4j U4) 5
—5U(u; j ui + u; 2j u 2 + v Sj U3 + uf 4j U4) 4
-5V (w j mi + w 2j n 2 + to 3j u 3 + uj aj u a ) 2
+hW{u j u x + u> 2i u 2 + u 3j u 3 + w 4j 'u 4 )
+[5(JC-y)-Z]=0 ) (6)
where
U = tiit/4 -(- u2ti 3 (7)
V = U\U2 2 + ^2«4 2 + ^3^1 + ^4^3 (8)
W = Ui 2 U4 2 + U2 2, U3 — Wl V>2 — Ui U± — Us U\
— 1A4 3 ^3 — U1U2U3U4 (9)
X = lii 3 W3^4 + U 2 S UiUs + ti 3 3 ^2U4 + Ua U1U2 (10)
00 22 22 22
y = U1W3 ^4 + 1*2^1 W3 + ^3^2 W4 + ^4^1 ^2
where
Z = ni 5 + u 2 5 -\r us B + n 4 5 .
The ms satisfy
U1U4 + ^2^3 =
^1^2 2 + 1t2^4 + U3U1 + U4U3 =
(11)
'(12)
(13)
(14)
22223 3 3 3
U\ U4 + V>2 U3 — U\ U2 — U2 U4 ~ Us U\ — U4 U3
(15)
5[(W1 3 W3^4 + U2 3 U±U3 + U3 3 UsU4 + U4 U1U2)
00 22 22 2 2 \ i
- (U1U3 t*4 + U2U1 U3 + ^3^2 U4 H" ^4^1 U2 )J
- (Ux 5 + i/ 2 5 + U3 5 + ^4 5 ) = 6. (16)
Spearman and Williams (1994) show that an irreducible
quintic
x + ax + b —
(17)
with Rational Coefficients is solvable by radicals
IFF there exist rational numbers e = ±1, c > 0, and
e ^ such that
5e 4 (3-4ec)
c 2 + l
-4e 5 (lle + 2c)
(18)
(19)
c 2 + l
The Roots are then
Xj = e{^u\ + u? 2j U2 + u; 3j U3 + o> ^4), (20)
ui =
W2 =
^3 =
li4
vi 2 v
\ 1/5
D 2
2 \ !/ 5
V 3 V 4 \
D 2 J
V2 2 V
D 2
\ 1/5
2 \ 1/5
V4 V 2 \
D 2 J
Vl = Vd + \lD-t\Tb
V2 = -^Td - Vd + cVd
V3
VD- VD-eVD
V4,
D = c + 1.
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
The general quintic can be solved in terms of THETA
FUNCTIONS, as was first done by Hermite in 1858. Kron-
ecker subsequently obtained the same solution more sim-
ply, and Brioshi also derived the equation. To do so,
reduce the general quintic
asx 5 + a^x + asx + a-ix + a\x + ao =
into Bring Quintic Form
x + p = 0.
Then define
k = tan
1 sin- 1 ( 16 V
5Sm V25V/5P 2 /.
= f-sgn(3[p]) forK[p]=0
- \ sgn(SR[p]) for X\p] /
r , ^(fc 2 ) 1/8
2-5»/ 4 > /*(l-* 2 )
(30)
(31)
(32)
(33)
(34)
(35)
where k is the Modulus, m = k 2 is the Parameter,
and q is the NOME. Solving
q(m) = B *"*'<»0/*<»»>
for m gives the inverse parameter
^2 4 (g)
m(q) =
*3 4 («)
(36)
(37)
Quintic Equation
Quintic Equation 1499
The ROOTS are then given by
Xl = (-lf^b{[m(e- 2jri/5 q 1/5 )] l/8
+i[m(e 2 ' H/ y /5 )] 1/8 }
x{[m(c- 4 * i/ V /5 )] 1/8 + [m{e 4 ™ /& q 1/5 )] 1/8 }
x{[rn(q 1/5 )} 1/S + <W)- 1/ 'W)] 1/ "} (38)
x 2 = 6{-[m(g 1/5 )] 1/8 + e a ' i/ *[m(e" i/6 q 1/ ' i )] 1/B }
x{ e - 3 " </4 [m(e- 2 " i/5 g 1/5 )] 1/8 +i[m( e 4 ' ri/5 9 1/5 )] 1/8 }
x{i[m(e- 4iri/5 g 1/5 )] 1/8 + q 5/8 (q 5 )- 1/8 [m(q b )] 1/s }
(39)
x 3 = fe{e- 3,ri/4 [m( e - 27ri/5 g 1/5 )] 1/8
-i[m( e -«" < 'V /B )] 1/8 )}{-M9 1/5 )] 1/8
-i[m(e 4,r</5 ? 1/5 )] 1/8 }
x{e 3 ' ri / 4 [m(e 2 ' ri/5 g 1 / 5 )] 1/8 + g 5/8 (g 5 )- 1/8 [m(g 5 )] 1/8 }
(40)
x 4 = 6{[™(9 1/5 )] 1/8 - iMe-^/y' 5 )] 1 ' 8 )}
X { _ e 3-/4 [m(e 2xV 5g I/5 )] l/ 8 _ <[m(c 4^/5 g l/5 )] l/8j.
x{e- 3,ri/4 [m( e - 2 " /5 g 1/5 )] 1/8
+9 5/ Vr x "W)] 1/8 } (41)
x 5 = 6{[m(g 1/5 )] 1/8 - e- 3 " i/4 [m(e- 2 " i/5 q 1/5 )} 1/8 }
x{ _ e 3-/4 [m(e 2^/5 g l/5 )] l/8 + i[m(e -4-/5 g l/ 5)] l/8 }
xU-ilmle^q 1 ' 6 )] 1 ' 8 + q 5/8 (q 5 r 1/8 {m(q 5 )] 1/a }.
(42)
Felix Klein used a TSCHIRNHAUSEN TRANSFORMATION
to reduce the general quintic to the form
z 5 + haz 2 + 5bz + c = 0.
(43)
He then solved the related ICOSAHEDRAL EQUATION
I(z,l,Z) = z 5 (-l + llz 5 + z 10 ) 5
-[1 + z 30 - 10005O 10 + z 20 ) + 522(-* 6 + z 25 )] 2 Z = 0,
(44)
where Z is a function of radicals of a, 6, and c. The
solution of this equation can be given in terms of Hy-
PERGEOMETRIC FUNCTIONS as
Z-V 6 ° 2 F 1 (--L,f,f,1728Z)
Z"/"^", 41,1,17282) '
(45)
Another possible approach uses a series expansion,
which gives one root (the first one in the list below)
of
t-t- p.
(46)
All five roots can be derived using differential equations
(Cockle 1860, Harley 1862). Let
F 1 (p) = F 2 (p)
ir(„\— zp (1 2 3 4.1 3 5. 3125 ,4^
p ( n \ — p (JL 13 17 21.3 5 3. 3125 4 x
r $\P) — 4-^3^20' 20' 20' 20' 4 » 4' 2> 256 " '
jp ( n \ _ p (JL I 11 13.5 3 7. 3125 n 4x
J"4V^; — 4J"3Vin) in' 10' 10' 4' 2' 4' 256 " />
(47)
(48)
(49)
(50)
then the Roots are
* — „ C fl 2 3 4. 1 3 5. 3125 .4^
*1 - -P4.P3I5, 5, 5, 5, 21 4' 4' "256"^ J
h = -Fi{p) + \pF2(p) + £ P 2 F 3 (p) + ^p 3 F,(p)
(51)
(52)
h - -F!(p) + ipF a (p) - £p 2 F 3 (p) + £p 3 F 4 (p)
(53)
U = -iFi(p) + \ P F 2 (p) - ±ip 2 F,(p) - ±p a Fi(p)
(54)
is = iFi(p) + \pF 2 {p) + &p 2 F 3 (p) - ip 8 F 4 (p).
(55)
This technique gives closed form solutions in terms of
Hypergeometric Functions in one variable for any
POLYNOMIAL equation which can be written in the form
x p + bx q + c.
(56)
Cadenhad, Young, and Runge showed in 1885 that all
irreducible solvable quintics with COEFFICIENTS of x 4 ,
a; 3 , and x 2 missing have the following form
a . + 5^(4 y + 3) x+ V(a, + l)(4 y + 3) =
V 2 + 1 I/ 2 + 1
where /i and v are RATIONAL.
5ee also Bring Quintic Form, Bring- Jerrard Quin-
tic Form, Cubic Equation, de Moivre's Quin-
tic, Principal Quintic Form, Quadratic Equa-
tion, Quartic Equation, Sextic Equation
References
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra,
3rd ed. New York: Macmillan, pp. 418-421, 1965.
Chowla, S. "On Quintic Equations Soluble by Radicals."
Math. Student 13, 84, 1945.
Cockle, J. "Sketch of a Theory of Transcendental Roots."
Phil. Mag. 20, 145-148, 1860.
Cockle, J. " On Transcendental and Algebraic Solution —
Supplemental Paper." Phil. Mag. 13, 135-139, 1862.
Davis, H. T. Introduction to Nonlinear Differential and In-
tegral Equations. New York: Dover, p. 172, 1960.
Dummit, D. S. "Solving Solvable Quintics." Math. Comput.
57, 387-401, 1991.
Glashan, J. C. "Notes on the Quintic." Amer. J. Math. 8,
178-179, 1885.
Harley, R. "On the Solution of the Transcendental Solution
of Algebraic Equations." Quart. J. Pure Appl. Math. 5,
337-361, 1862.
Hermite, C. "Sulla risoluzione delle equazioni del quinto
grado." Annali di math, pura ed appl. 1, 256-259, 1858.
1500
Quintic Surface
Quintuplet
King, R. B. Beyond the Quartic Equation. Boston, MA:
Birkhauser, 1996.
King, R. B. and Cranfield, E. R. "An Algorithm for Calcu-
lating the Roots of a General Quintic Equation from Its
Coefficients." J. Math. Phys. 32, 823-825, 1991.
Rosen, M. I. "Niels Hendrik Abel and Equations of the Fifth
Degree." Amer. Math. Monthly 102, 495-505, 1995.
Shurman, J. Geometry of the Quintic. New York: Wiley,
1997.
Spearman, B. K, and Williams, K. S. "Characterization of
Solvable Quintics x 5 + aa3 + 6." Amer. Math. Monthly 101,
986-992, 1994.
Wolfram Research. "Solving the Quintic." Poster. Cham-
paign, IL: Wolfram Research, 1995. http://www.wolf ram.
com/posters /quintic.
Young, G. P. "Solution of Solvable Irreducible Quintic Equa-
tions, Without the Aid of a Resolvent Sextic." Amer. J.
Math. 7, 170-177, 1885.
Quintic Surface
A quintic surface is an ALGEBRAIC SURFACE of degree
5. Togliatti (1940, 1949) showed that quintic surfaces
having 31 ORDINARY Double Points exist, although
he did not explicitly derive equations for such surfaces.
Beauville (1978) subsequently proved that 31 double
points was the maximum possible, and quintic surfaces
having 31 ORDINARY Double Points are therefore
sometimes called TOGLIATTI SURFACES, van Straten
(1993) subsequently constructed a 3-D family of solu-
tions and in 1994, Barth derived the example known as
the Dervish.
see also Algebraic Surface, Dervish, Kiss Sur-
face, Ordinary Double Point
References
Beauville, A. "Surfaces algebriques complexes." Asterisque
54, 1-172, 1978.
Endraft, S. "Togliatti Surfaces." http://www . mathematik .
uni - mainz . de / Algebraische Geometrie / docs /
Etogliatti . shtml.
Hunt, B. "Algebraic Surfaces." http: //www. mathematik.
uni-kl . de/-wwwagag/Galerie . html.
Togliatti, E. G. "Una notevole superficie de 5° ordine con
soli punti doppi isolati." Vierteljschr. Naturforsch. Ges.
Zurich 85, 127-132, 1940.
Togliatti, E. "Sulle superficie monoidi col massimo numero di
punti doppi." Ann. Mat. Pura AppL 30, 201-209, 1949.
van Straten, D. "A Quintic Hypersurface in F with 130
Nodes." Topology 32, 857-864, 1993.
Quintillion
In the American system, 10 18 .
see also Large Number
Quintuple Product Identity
A.k.a. the Watson Quintuple Product Identity.
oo
]1(1 - <7 n )U - zq n )(l - z-V^Xl - A 2 "" 1 )
X(l-z- 2 q 2n - 1 )= J2 (z 3 ™ - z- 3 "*- 1 ),"* 2 "*- 1 )/ 3 .
(1)
It can also be written
rj(i-9 2n )(i-? 2n -^)(i-? 2n -v i )
n=l
w /i 4n — 3 2w-, An — 4 -2\
X(l-q z ){l~q z )
oo
E3n 2 -2nr/ 3n , — 3n\ / 3n — 2 , — (3n-2)\i / n \
q [{z +z )-(z +z K } )\ (2)
71= — OO
or
oo
J2 (-i) fc g (3fe2 - fc)/ V fc (i + z9 fc )
k= — oo
oo
3 = 1
x(i + z"V J )(i+*<r )- (3)
Using the Notation of the Ramanujan Theta Func-
tion (Berndt, p. 83),
f(B 3 /q,q 5 /B 3 ) - B 2 f(q/B 3 ,B 3 q 5 )
ft 2, H-B 2 ,-q 2 /B*)
f{ ~ q) f(Bq, q /B) ■ < 4)
see also Jacobi Triple Product, Ramanujan Theta
Functions
References
Berndt, B. C. Ramanujan's Notebooks, Part III. New York:
Springer- Verlag, 1985.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 306-309, 1987.
Gasper, G. and Rahman, M. Basic Hyper geometric Series.
Cambridge, England: Cambridge University Press, 1990.
Quintuple
A group of five elements, also called a Quintuplet or
Pentad.
see also Monad, Pair, Pentad, Quadruple, Quad-
ruplet, Quintuplet, Tetrad, Triad, Triplet,
Twins
Quintuplet
A group of five elements, also called a QUINTUPLE or
Pentad.
see also Monad, Pair, Pentad, Quadruple, Quad-
ruplet, Quintuplet, Tetrad, Triad, Triplet,
Twins
Quota Rule
Quotient-Difference Table 1501
Quota Rule
A Recurrence Relation between the function Q aris-
ing in Quota Systems,
Q(n,r) = Q(n - l,r - 1) + Q(n - l,r).
References
Young, S. C; Taylor, A. D.; and Z wicker, W. S. "Count-
ing Quota Systems: A Combinatorial Question from Social
Choice Theory." Math. Mag. 68, 331-342, 1995.
Quota System
A generalization of simple majority voting in which a list
of quotas {^o, . . . , q n } specifies, according to the number
of votes, how many votes an alternative needs to win
(Taylor 1995). The quota system declares a tie unless
for some &, there are exactly k tie votes in the profile
and one of the alternatives has at least qu votes, in which
case the alternative is the choice.
Let Q(n) be the number of quota systems for n voters
and Q(n, r) the number of quota systems for which go =
r + 1, so
Q(n)= J2 Q( n > r )
r=[n/2\
UfA)'
where [x\ is the FLOOR FUNCTION. This produces the
sequence of CENTRAL BINOMIAL COEFFICIENTS 1, 2, 3,
6, 10, 20, 35, 70, 126, ... (Sloane's A001405). It may
be defined recursively by Q(0) = 1 and
Q(n + 1)
J 2Q(n) for n even
1 2Q(n) - C( n+ i)/ 2 for n odd,
where C k is a CATALAN NUMBER (Young et al. 1995).
The function Q(n,r) satisfies
Q(n } r)
+ 1/ v r + 2 /
for r > n/2 — 1 (Young et al. 1995). Q{n,r) satisfies the
Quota Rule.
see also BINOMIAL COEFFICIENT, CENTRAL BINOMIAL
Coefficient
References
Sloane, N. J. A. Sequence A001405/M0769 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Taylor, A. Mathematics and Politics: Strategy, Voting,
Power, and Proof. New York: Springer- Verlag, 1995.
Young, S. C; Taylor, A. D.; and Zwicker, W. S. "Count-
ing Quota Systems: A Combinatorial Question from Social
Choice Theory." Math. Mag. 68, 331-342, 1995.
Quotient
The ratio q = r/s of two quantities r and s, where s ^ 0.
see also Division, Quotient Group, Quotient Ring,
Quotient Space
Quotient-Difference Algorithm
The ALGORITHM of constructing and interpreting a
Quotient-Difference Table which allows intercon-
version of Continued Fractions, Power Series, and
Rational Functions approximations.
see also Quotient-Difference Table
Quotient-Difference Table
^J|l § _ X 2 -EW_1M2)(1) ,
N
1
Hk 1 21 3 .5'"
i £ -
u -i
_i i
-i—i-
-l_l
2 3
A quotient-difference table is a triangular Array of
numbers constructed by drawing a sequence of n num-
bers in a horizontal row and placing a 1 above each. An
additional "1" is then placed at the beginning and end
of the row of Is, and the value -of rows underneath the
original row is then determined by looking at groups of
adjacent numbers
N
W X E
S
and computing
X 2 -EW
N
for the elements falling within a triangle formed by the
diagonals extended from the first and last "1," as illus-
trated above.
0s in quotient-difference tables form square "windows"
which are bordered by GEOMETRIC PROGRESSIONS.
Quotient-difference tables eventually yield a row of 0s
IFF the starting sequence is defined by a linear RECUR-
RENCE RELATION. For example, continuing the above
example generated by the FIBONACCI Numbers
1111111
112 3 5
-1 1 -1
1 1
1
1 1
1 2
-1 1
1 1 1
1 1 2
-1 1
1
3
-1
1 1
3 5
-1 1
111
5 8
1
1 1
8 13
-1
1502 Quotient Group
Quotient Space
1
1
1
1
1
1
1
1
1
1
1
1
2
3
5
8
13
21
-1
1
-1
1
-1
1
and it can be seen that a row of Os emerges (and fur-
thermore that an attempt to extend the table will result
in division by zero). This verifies that the FIBONACCI
Numbers satisfy a linear recurrence, which is in fact
given by the well-known formula
F n = Fn-l + F n -2*
However, construction of a quotient-difference table for
the Catalan Numbers, Motzkin Numbers, etc., does
not lead to a row of zeros, suggesting that these numbers
cannot be generated using a linear recurrence.
see also DIFFERENCE TABLE, FINITE DIFFERENCE
References
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, pp. 85-89, 1996.
Quotient Group
The quotient group of G with respect to a SUBGROUP H
is denoted G/H and is read "G modulo H." The slash
Notation conflicts with that for a Field Extension,
but the meaning can be determined based on context.
see also ABHYANKAR'S CONJECTURE, FIELD EXTEN-
SION, Outer Automorphism Group, Subgroup
Quotient Ring
The quotient ring of R with respect to a RING modulo
some Integer n is denoted R/nR and is read "the ring
R modulo n." If n is a Prime p, then Z/_pZ is the
Finite Field F p . For Composite
Quotient Space
The quotient space X/~ of a TOPOLOGICAL Space X
and an EQUIVALENCE RELATION ~ on X is the set
of EQUIVALENCE CLASSES of points in X (under the
Equivalence Relation ~) together with the topol-
ogy given by a SUBSET U of X/~. U of X/~ is open
Iff UaGUd is open in X.
This can be stated in terms of MAPS as follows: if q :
X — > Xf~ denotes the MAP that sends each point to
its Equivalence Class in X/~ y the topology on X/~
can be specified by prescribing that a subset of X/~ is
open Iff g _1 [the set] is open.
In general, quotient spaces are not well behaved, and lit-
tle is known about them. However, it is known that any
compact metrizable space is a quotient of the CANTOR
SET, any compact connected n-dimensional MANIFOLD
for n > is a quotient of any other, and a function out
of a quotient space / : X/~— > Y is continuous Iff the
function / o q : X — > Y is continuous.
Let D n be the closed n-D DISK and S n_1
ary, the (n - 1)-D sphere. Then B n /S n ~
homeomorphic to S n ), provides an example of a. quo-
tient space. Here, D n /S n ~ is interpreted as the space
obtained when the boundary of the n-DlSK is collapsed
to a point, and is formally the "quotient space by the
equivalence relation generated by the relations that all
points in § n ~ are equivalent."
see also Equivalence Relation, Topological
Space
References
Munkres, J. R. Topology: A First Course. Englewood Cliffs,
NJ: Prentice-Hall, 1975.
its bound-
(which is
n*
with distinct p*, Z/pZ is ISOMORPHIC to the DIRECT
Sum
Z/ P Z = F P1 <g>F P2 <g)...®F Pfe .
see also FINITE FIELD, RING
Quotient Rule
The Derivative rule
_d_
dx
g(x)f'(x)-f(x)g'(x)
[9{x)Y
see also Chain Rule, Derivative, Power Rule,
Product Rule
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 11, 1972.
R
The Field of Real Numbers.
see also C, C*, I, N, Q, R", IR + , Z
R"
The Real Negative numbers.
see also R, M +
R +
The Real Positive numbers.
see also R, R~
rfc(n)
The number of representations of n by A; squares is de-
noted rfc(n). The Mathematical (Wolfram Research,
Champaign, IL) function NumberTheory'NumberTheory
Functions 'SumQfSquaresRCk.n] gives Vk{n).
r2{n) is often simply written r(n). Jacobi solved the
problem for k = 2, 4, 6, and 8. The first cases k —
2, 4, and 6 were found by equating COEFFICIENTS of
the Theta Function tf 3 (», #3 2 (z), and i9 3 4 (z). The
solutions for k = 10 and 12 were found by Liouville and
Eisenstein, and Glaisher (1907) gives a table of rfe(n) for
k = 2s = 18. rs(n) was found as a finite sum involving
quadratic reciprocity symbols by Dirichlet. rs(n) and
r7(n) were found by Eisenstein, Smith, and Minkowski.
7*(n) = T2{n) is whenever n has a PRIME divisor of the
form 4fc+3 to an Odd Power; it doubles upon reaching
a new Prime of the form 4k + 1. It is given explicitly
by
"(«)
E (-D
(d-l)/2 _
= 4[di(n)-d 3 (n)], (1)
d=l,3,...|n
where dfc(n) is the number of DIVISORS of n of the form
Am + fc. The first few values are 4, 4, 0, 4, 8, 0, 0, 4,
4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0,
0, . . . (Sloane's A004018). The first few values of the
summatory function
V,
R(n) = 2^r(n)
(2)
where are 0, 4, 8, 8, 12, 20, 20, 20, 24, 28, 36,
... (Sloane's A014198). Shanks (1993) defines instead
R'(n) = 1 + R(n), with R'(0) = 1. A Lambert Series
for r(n) is
\n + l n
71 = 1 Tl=l
(Hardy and Wright 1979).
(3)
r k (n) 1503
1000 2000 3000 4000 5000
Asymptotic results include
500 1000 1500 2000
^TV 2 (&) = 7rn + 0(Vn)
fc=i
E
ra(fc)
= K + tt In n + 0(n /,! ),
(4)
(5)
where K is a constant known as the SlERPlNSKl CON-
STANT. The left plot above
!><*)
— 7T71,
(6)
with ± yfn illustrated by the dashed curve, and the right
plot shows
r 2 (k)
E
7rlnn,
(7)
with the value of K indicated as the solid horizontal line.
The number of solutions of
(8)
2,2,2
x +y + z = n
for a given n without restriction on the signs or rela-
tive sizes of a, ?/, and z is given by rs(n). If n > 4 is
Squarefree, then Gauss proved that
{24h(-n) for n = 3 (mod 8)
12/i(-4n) for n = 1, 2, 5, 6 (mod 8) (9)
forn = 7 (mod 8)
(Arno 1992), where h(x) is the CLASS NUMBER of x.
Additional higher-order identities are given by
r 4 (n) = 8^cZ = 8<j(n) (10)
d\n
= 24 ]P d = 24a (n) (11)
d=l,3,...|Ti
no(n) = $[E 4 (n) + 16JSi(n) + 8 X4 (n)] (12)
r 2 4(n) = p 2 4(n)
+ gfK-ir'^Mn) - 512r(±n)] f (13)
where
E<(n)= J2 (-I) 1 '- 1 "'* (14)
d=l,3,...|n
E' 4 (n)= ^ (-l) (d, " 1)/a d* (15)
d' = l,3,...|n
X4(n) = J £ (a + 6i) 4 , (16)
a 2 +fe 2 =n
1504
R-Estimate
Rabbit Constant
d! = n/dy dk{n) is the number of divisors of n of the
form 4m + &, p24(n) is a SINGULAR Series, cr(n) is the
Divisor Function, cr (n) is the Divisor Function of
order (i.e., the number of DIVISORS), and r is the TAU
Function.
Similar expressions exist for larger EVEN k, but they
quickly become extremely complicated and can be writ-
ten simply only in terms of expansions of modular func-
tions.
see also Class Number, Landau-Ramanujan Con-
stant, Prime Factors, Sierpinski Constant, Tau
Function
References
Arno, S. "The Imaginary Quadratic Fields of Class Number
4." Acta Arith. 60, 321-334, 1992.
Boulyguine. Comptes Rendus Paris 161, 28-30, 1915.
New York: Chelsea, p. 317, 1952.
Glaisher, J. W. L. "On the Numbers of a Representation of
a Number as a Sum of It Squares, where 2r Does Not
Exceed 18." Proc. London Math. Soc. 5, 479-490, 1907.
Grosswald, E. Representations of Integers as Sums of
Squares. New York: Springer-Verlag, 1985.
Hardy, G. H. "The Representation of Numbers as Sums of
Squares." Ch. 9 in Ramanujan: Twelve Lectures on Sub-
jects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, 1959.
Hardy, G. H. and Wright, E. M. "The Function r(n)," "Proof
of the Formula for r(n)" "The Generating Function of
r(n)," and "The Order of r(n)." §16.9, 16.10, 17.9 , and
18.7 in An Introduction to the Theory of Numbers, 5th ed.
Oxford, England: Clarendon Press, pp. 241-243, 256-258,
and 270-271, 1979.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 162-153, 1993.
Sloane, N. J. A. Sequence A004018/M3218 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
i2-Estimate
A Robust Estimation based on Rank tests. Ex-
amples include the statistic of the KOLMOGOROV-
Smirnov Test, Spearman Rank Correlation, and
Wilcoxon Signed Rank Test.
see also L-Estimate, M-Estimate, Robust Estima-
tion
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Robust Estimation." §15.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 694-700, 1992.
Raabe's Test
Given a Series of Positive terms m and a Sequence
of Positive constants {a»}, use Rummer's Test
p = lim a n fln+i ) •
n-»oo y Un + 1 J
with a n = n, giving
p = lim
= lim
n— ►oo
U n
U n +l
-(n + 1)
\Un+l J
Defining
p = p + 1 = lim
\ Un+1 J .
then gives Raabe's test:
1. If p > 1, the Series Converges.
2. If p < 1, the Series Diverges.
3. If p = 1, the Series may Converge or Diverge.
see also CONVERGENT SERIES, CONVERGENCE TESTS,
Divergent Series, Rummer's Test
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 286-287, 1985.
Bromwich, T. J. I'a and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, p. 39, 1991.
Rabbit Constant
The limiting RABBIT SEQUENCE written as a BINARY
FRACTION 0.1011010110110... 2 (Sloane's A005614),
where 62 denotes a Binary number (a number in base-
2). The Decimal value is
R = 0.7098034428612913146 . . .
(Sloane's A014565).
Amazingly, the rabbit constant is also given by the CON-
TINUED Fraction [0, 2 F °, 2 Fl , 2 F2 , 2 F3 , . . .], where F n
are FIBONACCI Numbers with F Q taken as (Gard-
ner 1989, Schroeder 1991). Another amazing connec-
tion was discovered by S. Plouffe. Define the Beatty
Sequence {a*} by
a; = \i(j>\ ,
where [zj is the Floor Function and <j) is the Golden
Ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ...
(Sloane's A000201). Then
*=5>-.
see also RABBIT SEQUENCE, THUE CONSTANT, THUE-
Morse Constant
References
Finch, S. "Favorite Mathematical Constants." http;//www.
mathsof t . com/asolve/constant/cntf rc/cntf re .html.
Rabbit-Duck Illusion
Racah V -Coefficient 1505
Gardner, M. Penrose Tiles and Trapdoor Ciphers. . . and the
Return of Dr. Matrix, reissue ed. New York: W. H. Free-
man, pp. 21-22, 1989.
Plouffe, S. "The Rabbit Constant to 330 Digits." http://
lacim . uqam . ca/piDATA/rabbit . txt .
Schroeder, M. Fractals, Chaos, Power Laws: Minutes from
an Infinite Paradise. New York: W. H. Freeman, p. 55,
1991.
Sloane, N. J. A. Sequences A005614, A014565, and A000201/
M2322 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Rabbit-Duck Illusion
A perception ILLUSION in which the brain switches be-
tween seeing a rabbit and a duck.
see also Young Girl-Old Woman Illusion
Rabbit Sequence
A SEQUENCE which arises in the hypothetical repro-
duction of a population of rabbits. Let the Substitu-
tion Map — > 1 correspond to young rabbits grow-
ing old, and 1 -> 10 correspond to old rabbits produc-
ing young rabbits. Starting with and iterating using
String Rewriting gives the terms 1, 10, 101, 10110,
10110101, 1011010110110, .... The limiting sequence
written as a BINARY FRACTION 0.1011010110110. . . 2
(Sloane's A005614), where 62 denotes a BINARY number
(a number in base-2) is called the RABBIT CONSTANT.
see also Rabbit Constant, Thue-Morse Sequence
References
Schroeder, M. Fractals, Chaos, Power Laws: Minutes from
an Infinite Paradise. New York: W. H. Freeman, p. 55,
1991.
Sloane, N. J. A. Sequence A005614 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Rabdology
see Napier's Bones
Rabin-Miller Strong Pseudoprime Test
A PRIMALITY TEST which provides an efficient proba-
bilistic ALGORITHM for determining if a given number is
Prime. It is based on the properties of STRONG PSEU-
doprimes. Given an Odd Integer n, let n — 2 r s + 1
with s Odd. Then choose a random integer a with
1 < a < n-1. If a 5 = 1 (mod n) or a 2 ' ' 5 = -1 (mod n)
for some < j < r — 1, then n passes the test. A Prime
will pass the test for all a.
The test is very fast and requires no more than (1 +
o(l)) lg n multiplications (mod n), where Lg is the LOG-
ARITHM base 2. Unfortunately, a number which passes
the test is not necessarily PRIME. Monier (1980) and
Rabin (1980) have shown that a COMPOSITE NUMBER
passes the test for at most 1/4 of the possible bases a.
The Rabin-Miller test (combined with a LUCAS PSEU-
doprime test) is the Primality Test used by
Mathematical versions 2.2 and later (Wolfram Re-
search, Champaign, IL). As of 1991, the combined test
had been proven correct for all n < 2.5 x 10 10 , but not
beyond. The test potentially could therefore incorrectly
identify a large Composite Number as Prime (but not
vice versa). STRONG PSEUDOPRIME tests have been sub-
sequently proved valid for every number up to 3.4 x 10 14 .
see also LUCAS-LEHMER TEST, MILLER'S PRIMALITY
Test, Pseudoprime, Strong Pseudoprime
References
Arnault, F. "Rabin-Miller Primality Test: Composite Num-
bers Which Pass It." Math. Comput. 64, 355-361, 1995.
Miller, G. "Riemann's Hypothesis and Tests for Primality."
J. Comp. Syst. Set. 13, 300-317, 1976.
Monier, L. "Evaluation and Comparison of Two Efficient
Probabilistic Primality Testing Algorithms." Theor. Corn-
put. Sci. 12, 97-108, 1980.
Rabin, M. O. "Probabilistic Algorithm for Testing Primal-
, ity." J. Number Th. 12, 128-138, 1980.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 15-17, 1991.
Rabinovich-Fabrikant Equation
The 3-D Map
x = y(z — 1 -j- x ) + jx
y = x(3z + 1 - x 2 ) -j- 71/
i = — 2z(a -f xy)
(Rabinovich and Fabrikant 1979). The parameters are
most commonly taken as 7 = 0.87 and a = 1.1. It has
a Correlation Exponent of 2.19 ± 0.01.
References
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.
Rabinovich, M. I. and Fabrikant, A. L. Sov. Phys. JETP 50,
311-317, 1979.
Racah ^-Coefficient
The Racah V-COEFFICIENTS are written
V(jiJ2J]mim 2 m)
(i)
and are sometimes expressed using the related
Clebsch-Gordon Coefficients
Ch im2 = (JiJ2m 1 7n 2 \jiJ2Jm),
(2)
1506 Racah W-CoefEcient
Radau Quadrature
or WlGNER 3J-SYMBOLS. Connections among the three
are
(jiJ2mim 2 \jiJ2m)
(-l)- J ' 1+J ' 2 -"V2i + l ( jl h j 1
v ' v I mi m2 —my
(3)
{jiJ2mim2 \jiJ2Jm)
= (-l) i+m y/2j + lVtiiJ2j; mim 2 - m) (4)
V{j 1 j 2 j;m 1 m 2 m) = (_l)-*+'»-W ( Jl j2 Jl
w J •" ' v ' V m2 mi m2
(5)
see also Clebsch-Gordon Coefficient, Racah
PF-COEFFICIENT, WlGNER 3j-SYMBOL, WlGNER 6j-
SYMBOL, WlGNER 9J-SYMBOL
References
Sobel'man, I. I. "Angular Momenta." Ch. 4 in Atomic Spec-
tra and Radiative Transitions, 2nd ed. Berlin: Springer-
Verlag, 1992.
Racah VT-Coefficient
Related to the Clebsch-Gordon Coefficients by
(JiHJ'WJuhMJ"])
= y/{2J' + 1)(2J" + 1) W{JiJ 2 JJs; J' J")
and
(JlJ 2 [J / ]J3|J r i^[J / V2)
= y/(2J' + 1)(2J" + 1) W{J[ J 3 J 2 J"; JJ X ).
see also Clebsch-Gordon Coefficient, Racah
^-Coefficient, Wigner 3j-Symbol, Wigner 6j-
Symbol, Wigner 9j-Symbol
References
Messiah, A. "Racah Coefficients and '6j' Symbols." Ap-
pendix C.H in Quantum Mechanics, Vol. 2. Amsterdam,
Netherlands: North-Holland, pp. 1061-1066, 1962.
Sobel'man, I. I. "Angular Momenta." Ch. 4 in Atomic Spec-
tra and Radiative Transitions, 2nd ed. Berlin: Springer-
Verlag, 1992.
Radau Quadrature
A Gaussian QuADRATURE-like formula for numerical
estimation of integrals. It requires m + 1 points and
fits all Polynomials to degree 2m, so it effectively fits
exactly all POLYNOMIALS of degree 2m - 1. It uses a
Weighting Function W(x) = 1 in which the end-
point — 1 in the interval [—1,1] is included in a total
of n ABSCISSAS, giving r = n — 1 free abscissas. The
general formula is
/l "
f{x) dx = wif(-l) + ]TV/(xi).
1 ;_o
(1)
The free abscissas x» for i = 2, . . . , n are the roots of
the Polynomial
P n -i(x) + P n (x)
l + x
(2)
where P(x) is a LEGENDRE Polynomial. The weights
of the free abscissas are
Wi =
1 — Xi
, (3)
(4)
n'[P n -i{xi)] 2 (l-*0[^-i(*0J a
and of the endpoint
2
wi = — .
n 2
The error term is given by
2 2n ~ 1 n[(n-l)!] 4 (2Tt -i )r ^ ( .
E ~ [(2n-l)!]3 f K) ' (5)
for £e (-1,1).
n
Xi
Wi
2
-1
0.5
0.333333
1.5
3
-1
0.222222
-0.289898
1.02497
0.689898
0.752806
4
-1
0.125
-0.575319
0.657689
0.181066
0.776387
0.822824
0.440924
5
-1
0.08
-0.72048
0.446208
-0.167181
0.623653
0.446314
0.562712
0.885792
0.287427
SAS and weights can be compu
all n.
n Xi
Wi
§(l-x/6) £(16 + ^/6)
|(lW6) ±(16-y/E)
see also Chebyshev Quadrature, Lobatto Quad-
rature
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 888, 1972.
Chandrasekhar, S. Radiative Transfer. New York: Dover,
p. 61, 1960.
Hildebrand, F. B. Introduction to Numerical Analysis. New
York: McGraw-Hill, pp. 338-343, 1956.
Rademacher Function
Radical Integer 1507
Rademacher Function
see Square Wave
Radial Curve
Let C be a curve and let O be a fixed point. Let P be
on C and let Q be the Curvature Center at P. Let
Pi be the point with P\0 a line segment PARALLEL and
of equal length to PQ. Then the curve traced by Pi is
the radial curve of C. It was studied by Robert Tucker
in 1864. The parametric equations of a curve (/, g) with
Radial Point (xo 9 yo) are
X = Xq —
y = 2/0 +
g'(/ ,a +g ,a )
f'9" ~ f"9'
/'(/' a +g' 2 )
Curve
Radial Curve
astroid
catenary
cycloid
deltoid
logarithmic spiral
tractrix
quadrifolium
kampyle of Eudoxus
circle
trifolium
logarithmic spiral
kappa curve
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 40 and 202, 1972.
Yates, R. C. "Radial Curves." A Handbook on Curves and
Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 172-
174, 1952.
Radial Point
The point with respect to which a Radial Curve is
computed.
see also RADIANT POINT
Radian
A unit of angular measure in which the Angle of an
entire CIRCLE is 27r radians. There are therefore 360°
per 27r radians, equal to 180/7T or 57.29577951°/radian.
A Right Angle is n/2 radians.
see also ANGLE, ARC MINUTE, ARC SECOND, DEGREE,
Gradian, Steradian
Radiant Point
The point of illumination for a CAUSTIC.
see also Caustic, Radial Point
Radical
The symbol Vfx used to indicate a root is called a radi-
cal. The expression Vfx is therefore read u x radical n,"
or "the nth Root of x" n = 2 is written ^/x and is
called the SQUARE ROOT of x. n = 3 corresponds to
the Cube Root. The quantity under the root is called
the Radicand.
Some interesting radical identities are due to Ramanu-
jan, and include the equivalent forms
(2 1/3 + l)(2 1/3 -l) 1/3 = 3 1/3
and
(2l /3_ 1) l/3 = ( i )1 /3_ ( | ) l/3 + ( | ) l/3
Another such identity is
(5 l/3 _ 4 l/ 3) l/2 = | (2 l/3 + 20 l/8 _ 25 l/ 3)>
see also Cube Root, Nested Radical, Power, Rad-
ical Integer, Radicand, Root (Radical), Square
Root, Vinculum
Radical Axis
see Radical Line
Radical Center
The Radical Lines of three Circles are Concurrent
in a point known as the radical center (also called the
POWER Center). This theorem was originally demon-
strated by Monge (Dorrie 1965, p. 153).
see also Apollonius' Problem, Concurrent,
Monge's Problem, Radical Line
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, 1965.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 32, 1929.
Radical Integer
A radical integer is a number obtained by closing the In-
tegers under Addition, Division, Multiplication,
Subtraction, and Root extraction. An example of
such a number is y/7 + \/^2
^3+ v / l + \ / 2. The
radical integers are a subring of the ALGEBRAIC INTE-
GERS. If there are ALGEBRAIC INTEGERS which are not
radical integers, they must at least be cubic.
see also ALGEBRAIC INTEGER, ALGEBRAIC NUMBER,
Euclidean Number
1508 Radical Line
Radical Line
The Locus of points of equal Power with respect to
two nonconcentric Circles which is Perpendicular
to the line of centers (the CHORDAL THEOREM; Dorrie
1965). Let the circles have RADII T\ and ti and their
centers be separated by a distance d. If the Circles
intersect in two points, then the radical line is the line
passing through the points of intersection. If not, then
draw any two Circles which cut each original Circle
twice. Draw lines through each pair of points of inter-
section of each CIRCLE. The line connecting their two
points of intersection is then the radical line.
The radical line is located at distances
di =
d H- ri — Ti
<fe = -
2d
d 2 +r 2 2
■ n
2d
along the line of centers from C\ and C2, respectively,
where
d = di — di.
The radical line of any two POLAR CIRCLES is the AL-
TITUDE from the third vertex.
see also Chordal Theorem, Coaxal Circles, In-
verse Points, Inversion, Power (Circle), Radi-
cal Center
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC; Math. Assoc. Amer., pp. 31-34, 1967.
Dixon, R. Mathographics. New York: Dover, p. 68, 1991.
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 153,
1965.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle, Boston,
MA: Houghton Mifflin, pp. 28-34 and 176-177, 1929.
Radicand
The quantity under a RADICAL sign.
see also RADICAL, VINCULUM
Radius
€S^
Radius of Gyration
The distance from the center of a CIRCLE to its PERI-
METER, or from the center of a SPHERE to its surface.
The radius is equal to half the DIAMETER.
see also Bertrand's Problem, Circle, Circumfer-
ence, Diameter, Extent, Inversion Radius, Kin-
ney's Set, Pi, Radius of Convergence, Radius of
Curvature, Radius (Graph), Radius of Gyration,
Radius of Torsion, Radius Vector, Sphere
Radius of Convergence
The RADIUS (or 1-D distance in the 1-D case) over which
series expansion CONVERGES.
Radius of Curvature
The radius of curvature is given by
1
R =
(1)
where k is the CURVATURE. At a given point on a curve,
R is the radius of the OSCULATING CIRCLE. The symbol
p is sometimes used instead of R to denote the radius of
curvature.
(1)
Let x and y be given parametrically by
x — x(t)
(2)
(2)
y = y(*)i
(3)
ely,
then
R (x' 2 + y' 2 ) 3/2
x'y" — y'x"
(4)
where x f = dx/dt and y — dy/dt. Similarly, if the
curve is written in the form y = /(#), then the radius
of curvature is given by
fi + <» ,,M
R =
d^5
(5)
see also Bend (Curvature), Curvature, Osculat-
ing Circle, Torsion (Differential Geometry)
References
Kreyszig, E. Differential Geometry. New York: Dover, p. 34,
1991.
Radius (Graph)
The minimum ECCENTRICITY of any Vertex of a
Graph.
Radius of Gyration
A function quantifying the spatial extent of the structure
of a curve. It is defined by
Rn
2/ °°p(r)<fr
where p(r) is the LENGTH DISTRIBUTION FUNCTION.
Small compact patterns have small R g .
References
Pickover, C. A. Keys to Infinity, New York: W. H. Freeman,
pp. 204-206, 1995.
Radius of Torsion
Radius of Torsion
1
<j = -,
r
where r is the TORSION. The symbol (p is also sometimes
used instead of cr.
see also Torsion (Differential Geometry)
References
Kreyszig, E. Differential Geometry. New York: Dover, p. 39,
1991.
Radius Vector
The VECTOR r from the ORIGIN to the current position.
It is also called the POSITION VECTOR. The derivative
of r satisfies
dv Id, s 1 d , 2n dr
r d-t = 3dt {vT)= 2dt {r) = r Tt =rV '
where v is the magnitude of the VELOCITY (i.e., the
Speed).
Radix
The Base of a number system, i.e., 2 for Binary, 8
for Octal, 10 for Decimal, and 16 for Hexadecimal.
The radix is sometimes called the Base or SCALE.
see also BASE (Number)
Rado's Sigma Function
see Busy Beaver
Radon-Nikodym Theorem
A Theorem which gives Necessary and Sufficient
conditions for a countably additive function of sets can
be expressed as an integral over the set.
References
Doob, J. L. "The Development of Rigor in Mathematical
Probability (1900-1950)." Amer. Math. Monthly 103,
586-595, 1996.
Radon Transform
An Integral Transform whose inverse is used to re-
construct images from medical CT scans. A technique
for using Radon transforms to reconstruct a map of a
planet's polar regions using a spacecraft in a polar orbit
has also been devised (Roulston and Muhleman 1997).
The Radon transform can be denned by
/oo
f(x,r + px)dx
-oo
/oo /»oo
/ f{x,y)8[y - (r +px)] dy dx = C/(p,r), (1)
-oo J — oo
where p is the Slope of a line and r is its intercept. The
inverse Radon transform is
f{x ' y) = hj_ ijHMPiV-P^dp, (2)
Radon Transform 1509
where if is a HlLBERT TRANSFORM. The transform can
also be denned by
R'(r,a)[f(x,y)}
/oo /»oo
/ f (x,y)5(r — xcosa — y sin a) dxdy y (3)
■ oo J — c
-co */ —CO
where r is the Perpendicular distance from a line to
the origin and a is the Angle formed by the distance
Vector.
Using the identity
T[R[f (w, a)]] = T 2 [/(«,»)],
(4)
where T is the Fourier Transform, gives the inver-
sion formula
f(x,y) =
/*7T />00
c / ^[fl[/(w,a)]]|w|e <w(lcos<:,+l ' sinQ) dwdQ.
i/O J -oo
(5)
The Fourier Transform can be eliminated by writing
f{x,y)
/»7T /»0
JO J-c
R[f(r, a)]W(r, a, x, y) dr da, (6)
where W is a WEIGHTING FUNCTION such as
W(r, a.)X,y) = h(xcosa + y sin a — r) = T~ [|tu|]. (7)
Nievergelt (1986) uses the inverse formula
f(x,y) = - lim
7T c^-0
Jo J -c
R[f(r + a; cos a + ysina, a)]G c (r) dr da, (8)
where
l
7TC 2
for |r| < c
G ' w = [M 1 -^) "" "•'><•
(9)
Ludwig's Inversion Formula expresses a function in
terms of its Radon transform. R'(r,a) and R(p,r) are
related by
p = cot a r = r esc a
r — a = cot -1 p.
1+p 2
(10)
(11)
The Radon transform satisfies superposition
R(p,r)[Mx,y) + Mx,y)] = U 1 (p,T) + U 2 (p,T), (12)
(13)
1510 Radon Transform
linearity
R(p>T)[af(x,y)] = all far),
scaling
^^(H)]-' '" ('?•*)• (14)
Rotation, with R$ Rotation by Angle <f>
1
R(p,T)[R<t>f{x,y)] =
U
| coscj) + ps'm<f>\
p — tan <f>
1+p tan <f> ' cos <p -f
i ) t
\- p sin <j> J
(15)
and skewing
R(P> r)[f(ax + by, ex + dy)]
1
|a + 6p|
U
c + dp d — b(c + bd)
T
a + bp J a-\-bp
(16)
(Durrani and Bisset 1984).
The line integral along p, r is
I=y/l+p*U(p t T). (17)
The analog of the 1-D CONVOLUTION Theorem is
R(l>,T)[f(x,v)*g(y)] = U(p,T)*g(r) y (18)
the analog of Plancherel's Theorem is
/OO /*'
U(p,r)dT= /
-cx> ** —
oo /»oo
U{p,r)dT= / / f(x,y)dxdy, (19)
» J —oo J — OO
and the analog of Parseval's Theorem is
J — C
B(P,T)[/(lB,»)] a dT
■£/
f 2 (x,y)dxdy.
oo */ — OO
(20)
If / is a continuous function on C, integrable with re-
spect to a plane Lebesgue Measure, and
/
fds =
(21)
for every (doubly) infinite line / where s is the length
measure, then / must be identically zero. However, if
the global integrability condition is removed, this result
fails (Zalcman 1982, Goldstein 1993).
see also TOMOGRAPHY
References
Anger, B. and Portenier, C. Radon Integrals. Boston, MA:
Birkhauser, 1992.
Armitage, D. H. and Goldstein, M. "Nonuniqueness for the
Radon Transform." Proc. Amer. Math. Soc. 117, ITS-
ITS, 1993.
Deans, S. R. The Radon Transform and Some of Its Appli-
cations. New York: Wiley, 1983.
Radon Transform — Cylinder
Durrani, T. S. and Bisset, D. "The Radon Transform and its
Properties." Geophys. 49, 1180-1187, 1984.
Esser, P. D. (Ed.). Emission Computed Tomography: Cur-
rent Trends. New York: Society of Nuclear Medicine, 1983.
Gindikin, S. (Ed.). Applied Problems of Radon Transform.
Providence, RI: 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, 1979.
Helgason, S. The Radon Transform. Boston, MA:
Birkhauser, 1980,
Kunyansky, L. A. "Generalized and Attenuated Radon
Transforms: Restorative Approach to the Numerical In-
version." Inverse Problems 8, 809-819, 1992.
Nievergelt, Y. "Elementary Inversion of Radon's Transform."
SIAM Rev. 28, 79-84, 1986.
Rann, A. G. and Katsevich, A. I. The Radon Transform and
Local Tomography. Boca Raton, FL: CRC Press, 1996.
Robinson, E. A. "Spectral Approach to Geophysical Inversion
Problems by Lorentz, Fourier, and Radon Transforms."
Proc. Inst Electr. Electron. Eng. 70, 1039-1053, 1982.
Roulston, M. S. and Muhleman, D. O. "Synthesizing Radar
Maps of Polar Regions with a Doppler-Only Method."
Appl. Opt. 36, 3912-3919, 1997.
Shepp, L. A. and Kruskal, J. B. "Computerized Tomogra-
phy: The New Medical X-Ray Technology." Amer. Math.
Monthly 85, 420-439, 1978.
Strichartz, R. S. "Radon Inversion — Variation on a Theme."
Amer. Math. Monthly 89, 377-384 and 420-423, 1982.
Zalcman, L. "Uniqueness and Nonuniqueness for the Radon
Transform." Bull. London Math. Soc. 14, 241-245, 1982.
Radon Transform — Cylinder
K(p,i)
Let the 2-D cylinder function be defined by
«-»>■{; iiti «"
Then the Radon transform is given by
/oo /»oo
/ f{x,y)6[y-(T+px)]dydx, (2)
■oo J — OO
where to
5(x) = ^j e~ ik * (3)
J — OO
is the Delta Function.
^ /»2tt />R />oo
s'm0—pr cos 9)
r dr dO dk
- /»00 /*27T pR
— / ^ / / e- ikr(3ind ~ pcos0) rdrdOdk.
2n y_oo Jo Jo
(4)
Radon Transform — Cylinder
Now write
sin0-pcos0 = \/l+p 2 cos(<9 + 0) = y/l+p 2 cosfl',
(5)
with <fi a phase shift. Then
1_
2n
— iky/ 1+p 2 r cos & f jr\t
- /»00 /»J2 / />27T
^i-.o 6 Jo \Jo
^ poo pR
= _L / e * fcr / 2<rrJ {k^l+p 2 r)rdrdk
27r J -co Jo
/oo pR
e ikT / Jo^v^^^^-
-oo Jo
Then use
/ t n+1 J„(t)dt = z n+1 J„ + i(z),
Jo
which, with n — 0, becomes
/ tJ (t)dt = 2Ji(z).
Jo
Define
rdr =
so the inner integral is
f » J R v /l4-p 2
t = ky/l+pPr
dt = k^/TTp 2 dr
tdt
k 2 {l+p 2 )
»a\»
(6)
(7)
(8)
(9)
(10)
(11)
/
Jo
Jo(t)
tdt
k 2 {l+p 2 )
-L—^/l + P 2 J 1 (kR^l+p 2 )
_ JiikRy/l+p*)
A; 2 (1+p'
and the Radon transform becomes
R, (12)
^'jwL- — s
2fl r°° cosjk^JijkR^l+p 2 )
1+p 2 Jo k
dk
v-
dk
^/R 2 {\+p 2 )-T 2 forr 2 <R 2 {l+p 2 )
P 2 )-
(13)
\0 forr 2 >fl 2 (l+p 2 )
Radon Transform — Gaussian 1511
Converting to R l using p = cot a,
#'(r,a) =
\/(l + cot 2 a)i£ 2 — r 2 esc 2 a
a/I + cot 2 a
2
y esc 2 a.R 2 — r 2 esc 2 a
= 2yfR?
dd \ rdr dk which could have been derived more simply by
ry/lP-T*
J2'(r,a) = / dy.
s/R?-T*
(14)
(15)
Radon Transform — Delta Function
For a Delta Function at (xo,yo),
/oo /»oo
/ S(x-x )5(y-yo)S[y~(r-\-px)] dydx
-oo J — oo
1 /»oo /»oo /»oo
= ^/ / 7 e " lfcIw_(T+ '" >1 '( a5 - a 'o)*(y-i»)
J —oo J —oo J —oo
x dk dy dx
/oo
e ikpx 8(x-x )dx dk
-oo
= — f e ik T e - ik vo e Mp*o dk
2tt /
«/ — oo
= _L f°° e^+w-^dk^Sir + pxo-yo).
27T J-oo
Radon Transform — Gaussian
= — L- [ e- [l2+(T -
0V2W-OO
3x„ 2 W5/, 2
(z 2 +y 3 )/2<r'
x <% — ( r + p x )] ^y dx
+px) 2 ]/2cr 2 i
v / i+^ 2
] dx
-t 2 /[2(l+p 2 )^ 2 ]
1512 Radon Transform — Square
Radon Transform — Square
R(p, t)
/oo /»oo
/ f(x, y)S[y - (r + px)] dydx, (1)
-OO v — <
where
-OO «/ — OO
/(x,y) = {J
for x,y 6 [—a, a]
otherwise
and
*(*)
2?r /
«/ — OO
(2)
(3)
is the Delta Function.
1 /*a /»a /»oo
1 /*oo r /»a /»a
2?r
dfc
1 ifcr -I- r — iky
27r — ik tkp
c ™._^ [c -**»]a 1 [ c **P«]- odjfe
i Z 100 1
= Jl. / e zfer -^[-2isin(A;a)][2*sin(A;pa)]dA;
2*" J,^ fc 2 p
_ 2_ [°° sin(fca) sin(fepa)e ifer rffc
" *TP 7-oo * 3
4 Z" 00 sin(A;a) sin(A;pa) cos(fcr)
= A r S in[fc(r + a)]~sin[fe(r-a)] ^
Wo ^
_ 2 f /*°° sin[A;(r + a)] sin(fepa)
Z 100 sin[fe(r - a)] sin(fepa) rffc \ ^ ^
From Gradshteyn and Ryzhik (1979, equation 3.741.3),
' sin (ax) sin (6a;) , x , ,. . n , ,, IN ,_,
— i — ; - v , y cte = f7rsgn(a6)mm(|a|,|6|), (5)
Jo
R(P> t) = - {sgn[(r + a)pa] min(|r + o|, \pa\)
V
- sgn[(r - a)pa] min(|r - a|, |po|)} . (6)
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, 1979.
Railroad Track Problem
Railroad Track Problem
(l+M)
Given a straight segment of track of length Z, add a small
segment Al so that the track bows into a circular ARC.
Find the maximum displacement d of the bowed track.
The Pythagorean Theorem gives
r* = x* + {\l)\
But R is simply x + d, so
R 2 = {x + df = x 2 + 2xd + d 2
Solving (1) and (2) for x gives
P - d 2
2d
(1)
(2)
(3)
Expressing the length of the Arc in terms of the central
angle,
Ul + Al) = 6(d+x) = 0[d +
\l 2 -d?
2d
-p£^)-m-
But 6 is given by
tan „ 11 = I'M
dl
1/2 _ d 2 lp _ d 2>
so plugging 6 in gives
j2 , 1 ;2
2d
tan
dl
\P - d?
d(l + Al) = (d 2 + \l 2 ) tan" 1 (ip^d?) •
(4)
(5)
(6)
(7)
For I > d,
dZ _ 4d / 4d 2
t' 2 (i-£)~ ' v P
Therefore,
d(J + Al)
4d
/\ 4rf \
(8)
(d 2 + \l 2 )
Ad 16d
Z +
?-H?)'(' +J £)]-
(9)
Ramanujan 6-10-8 Identity
Ramanujan Cos/Cosh Identity 1513
Keeping only terms to order (d/l) 3 ,
tf + A^-HH+^-i^ (10)
A ,«( 8 _ ¥) * Hizi«* = f* (1 i)
! 3 / 3 I
d 2 = llAl
and
dK \^J\iKi = \V6iKi.
(12)
(13)
If we take 2 = 1 mile = 5280 feet and A/ = 1 foot, then
d « 44.450 feet.
Ramanujan 6-10-8 Identity
Let ad = 6c, then
64[(a + 6 + c) 6 + (6 + c + d) 6 - (c + d + a) 6
-(d + a + 6) 6 + (a - d) 6 - (6 - c) 6 ]
x[(a + & + c) 10 + (& + c + d) 10 -(c + d + a) 10
-(d + a + 6) 10 + (a - d) 10 - (6 - c) 10 ]
= 45[(a + 6 + c) 8 + (6 + c + d) 8 - (c + d + a) 8
~(d + a + 6) 8 + (a-d) 8 -(6-c) 8 ] 2 . (1)
This can also be expressed by denning
F 2m (a,M,rf) = (a + b + c) 2m + (6 + c + d) 2m
-{c+d+af™ ~{d+a+bf m + {a~d) 2rn ~{b-c) 2m (2)
f2m(x,y) = (l + z + y) 2m + (x+y + xy) 2m -(y + 2y + l) 2m
-(xy + 1 + x) 2m + (1 - *</) 2m - (x- y) 2m . (3)
Then
F2m{a, b,c,d) = a m /2m(#,y), (4)
and identity (1) can then be written
Mf6{x,y)fio{x,y) =45/ 8 2 (x jy ). (5)
Incidentally,
Mx,y) = Q
U{x,y) = 0.
(6)
(7)
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 3 and 102-106, 1994.
Berndt, B. C. and Bhargava, S. "A Remarkable Identity
Found in Ramanujan's Third Notebook." Glasgow Math.
J. 34, 341-345, 1992.
Berndt, B. C. and Bhargava, S. "Ramanujan — For Low-
brows." Amer. Math. Monthly 100, 644-656, 1993.
Bhargava, S. "On a Family of Ramanujan's Formulas for
Sums of Fourth Powers." Ganita 43, 63-67, 1992.
Hirschhorn, M. D. "Two or Three Identities of Ramanujan."
Amer. Math, Monthly 105, 52-55, 1998.
Nanjundiah, T. S. "A Note on an Identity of Ramanujan."
Amer. Math. Monthly 100, 485-487, 1993.
Ramanujan, S. Notebooks. New York: Springer- Verlag,
pp. 385-386, 1987.
Ramanujan Constant
The Irrational constant
R = e Wl63 = 262537412640768743.99999999999925. ..
which is very close to an Integer. Numbers such as the
Ramanujan constant can be found using the theory of
Modular Functions. A few rather spectacular exam-
ples are given by Ramanujan (1913-14), including the
one above, and can be generated using some amazing
properties of the j-FUNCTlON.
M. Gardner (Apr. 1975) played an April Fool's joke on
the readers of Scientific American by claiming that this
number was exactly an INTEGER. He admitted the hoax
a few months later (Gardner, July 1975).
see also Almost Integer, Class Number, j-
Function
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays , 13th ed. New York: Dover, p. 387,
1987.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988.
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.
Good, I. J. "What is the Most Amazing Approximate Integer
in the Universe?" Pi Mu Epsilon J. 5, 314-315, 1972.
Plouffe, S. "e^v^, the Ramanujan Number." http://
lacim . uqara . ca/piDATA/ramanu j an . txt .
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart. J. Pure Appl. Math. 45, 350-372, 1913-1914.
Wolfram, S. The Mathematica Book, 3rd ed. New York;
Cambridge University Press, p. 52, 1996.
Ramanujan Continued Fraction
Let /(a, 6) be a Ramanujan Theta Function. Then
f(~q,-q 4 ) _ 1 q q 2 q 3
/(-<? 2 ,-4 3 ) 1+1+1+1 + ...'
where the quantity on the right is a Continued Frac-
tion.
see also Ramanujan Theta Functions
Ramanujan Cos/Cosh Identity
■+»E
cos(n9)
cosh(ri7r)
+
! + 2 E
cosh(n#)
cosh(n7r)
2r 4 (f)
where Y(z) is the GAMMA FUNCTION.
1514 Ramanujan-Eisenstein Series
Ramanujan-Eisenstein Series
Let t be a discriminant,
q= -e
rry/t
(1)
then
* ( „=L(,)=.-,.f;e£W"
n2fc+l
-(")' <'-«■)
00 j 3 2fc
4 ( g ) = M(g) = 1 + 240^^
& = 1
'22P
2fc
= (?) (1 - fc2fc,2)
(2)
(3)
5 rt 2fc
fc = l
= (^) 6 (l-2fc 2 )(l+ifcV 2 ). (4)
see a/so Klein's Absolute Invariant, Pi
References
Borwein, J. M. and Borwein, P. B. "Class Number Three
Ramanujan Type Series for l/7r." J. Comput. Appl. Math.
46, 281-290, 1993.
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart J. Pure Appl Math. 45, 350-372, 1913-1914.
Ramanujan Function
n
^ n ' El+2 Ep)b
fc=i
00 1
0(a) = lim 0(a, n) = 1 + 2 V
afc
The values of 0(n) for n = 2, 3, ... are
4>(2) = 2 In 2
0(3) = ln3
0(4) -fin 2
0(6) = |ln3+|ln4.
Ramanujan g- and G- Functions
Following Ramanujan (1913-14), write
J[ (1 + e - kn ^) = 2 1/ V ,rv/s/24 G n
fc = l,3,5,...
CO
TJ (1 - e -<-^) = 2 1/4 e-"^ /24 3 „.
fc=l,3,5,...
Ramanujan g- and G-Functions
These satisfy the equalities
£ 4n = 2 g n G n
G n = Gt/ n
Qn = 54/n
4 — (gnGn) (Gn — <?n )•
(3)
(4)
(5)
(6)
G n and g n can be derived using the theory of Modular
FUNCTIONS and can always be expressed as roots of al-
gebraic equations when n is RATIONAL. For simplicity,
Ramanujan tabulated g n for n Even and G n for n Odd.
However, (6) allows G n and g n to be solved for in terms
of g n and <3 n , giving
<?n - § (^ 8 + VGn 16 -G n ~*y /8 (7)
/ \l/8
G n = I ^ n 8 + V5n 16 + Gp n - 8 J . (8)
Using (3) and the above two equations allows g* n to be
computed in terms of g n or G n
_ J for n ev(
2 1 / 8 G n (G n S + ^n^-Gn" 8 ) 1
#4n =
even
/8
(9)
for n odd.
In terms of the Parameter k and complementary Pa-
rameter k\
Gn = (2fcxr i/i2 (10)
/ , 2 X 1/12
Here,
fcn=A*(n) (12)
is the Elliptic Lambda Function, which gives the
value of k for which
K'(k)
y/n. (13)
K(k)
Solving for A*(n) gives
A'(n) = 5n 6 [\/s« 12 + S»- 12 - Sn 6 ]-
(14)
(15)
Analytic values for small values of n can be found in Ra-
manujan (1913-1914) and Borwein and Borwein (1987),
and have been compiled in Weisstein (1996). Ramanu-
jan (1913-1914) contains a typographical error labeling
C?465 aS C?265.
see also G-Function
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
M\ Analytic Number Theory and Computational Complexity.
V J New York: Wiley, pp. 139 and 298, 1987.
Ramanujan, S. "Modular Equations and Approximations to
7T." Quart J. Pure. Appl Math. 45, 350-372, 1913-1914.
# Weisstein, E. W. "Elliptic Singular Values." http://www.
(2) astro . Virginia. edu/-ewv6n/math/notebooks/Elliptic
Singular.
Ramanujan's Hypergeometric Identity
Ramanujan's Hypergeometric Identity
Ramanujan Psi Sum 1515
Ramanujan's Interpolation Formula
4' 4 . _1
1 ' X )
lF, *
r 2 (f)
where 2 i<i(a,6; c; a) is a HYPERGEOMETRIC FUNCTION,
zF 2 (a, 6, c; d; e; x) is a GENERALIZED HYPERGEOMETRIC
Function, and T(z) is a Gamma Function.
References
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 106, 1959.
Ramanujan's Hypothesis
see Tau Conjecture
Ramanujan's Identity
4>*(x)
where
^(x)= jj(l-x m )
and P(n) is the Partition Function P.
see also RAMANUJAN'S SUM IDENTITY
Ramanujan's Integral
J — C
2 cos (it)
x 2 e -it/2 + y 2 e it/2
(M + ^)/2
X Jfl+V
^2 cos (ft) (a: 2 e-"/2 +y 2 e «/2)
it(v-ix){2
where J n (^) is a Bessel Function of the First
Kind.
References
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.
r^- i E(-i)v^(ib)d a! =2^ a)
J a ^ sm(s7r)
J- \%) poo o" fc
r»(f)r»(|)' / x- 1 J](-i) fc | r A(*)dx = r(.)A(-«), (2)
v0 1 n
where A(z) is the Dirichlet LAMBDA FUNCTION and
T(z) is the GAMMA FUNCTION. Equation (2) is obtained
from (1) by defining
4>(U):
A(«)
r(i + w)'
(3)
These formulas give valid results only for certain classes
of functions.
References
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
pp. 15 and 186-195, 1959.
Ramanujan's Master Theorem
Suppose that in some NEIGHBORHOOD of x = 0,
~ <P(k)(-x) k
w = E^
Then
F
Jo
x n ~ 1 F(x)dx = r(n)<f>(-n).
References
Berndt, B. C. Ramanujan's Notebooks: Part L New York:
Springer- Verlag, p. 298, 1985.
Ramanujan-Petersson Conjecture
A Conjecture for the Eigenvalues of modular forms
under HECKE OPERATORS.
Ramanujan Psi Sum
A sum which includes both the Jacobi Triple Prod-
uct and the g-BlNOMIAL THEOREM as special cases.
Ramanujan's sum is
£
(a)n n _ (ax) 00 (q/ax) o(q) 00 (b/a) 00
(b) n (x) <X) (b/ax) 00 (b) 00 (q/a)oo
where the Notation (q)k denotes ^-Series. For b — q,
this becomes the gr-BlNOMIAL THEOREM.
see also JACOBI TRIPLE PRODUCT, g-BlNOMIAL THEO-
REM, ^-Series
1516 Ramanujan's Square Equation
Ramanujan Theta Functions
Ramanujan's Square Equation where
It has been proved that the only solutions to the DlO-
phantine Equation
2 n - 7 = x 2
are n = 3, 4, 5, 7, and 15 (Beeler et al. 1972, Item 31).
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Ramanujan's Sum
The sum
c q (m) = Yl ^ ihrnl \ (1)
where h runs through the residues RELATIVELY PRIME
to q, which is important in the representation of numbers
by the sums of squares. If (g, q') = 1 (i.e., q and <?' are
Relatively Prime), then
c qq '{m) = c q (m)c q '(m).
(2)
For argument 1,
Cb(l) =/*(*)> (3)
where y, is the MOBIUS FUNCTION, and for general m,
4>(b)
Cb(m)
""{(b,m))
^((^))
(4)
see also MOBIUS FUNCTION, WEYL'S CRITERION
References
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, p. 254, 1991.
Ramanujan's Sum Identity
If
1-f 53x + 9x 2
1
- 82a;-
82a; 2 + x 3
.2- 26a
; - 12x 2
1
- 82a; -
82x 2 + x 3
2 + 8a;
- 10a; 2
= y a n x n
71 = 1
oo
= 22 b n x n
(1)
(2)
1 - 82z - 82x 2 + x 3
22cnX n y (3)
n~0
then
(4)
a n ' + b n ' = c n ' + (-l) n .
Hirschhorn (1995) showed that
a n = i[(64 + 8V85)a n + (64 - 8^85)/?" - 43(-l) B ]
(5)
b n = i[(77 + 7V85)a n + (77 - 7^85 )/T + 16(-l) n ]
(6)
c n = i[(93 + 9\/85)a n + (93 - 9^85 )/T - 16(-1)"],
(7)
a= |(83 + 9>/85)
0= i(83-9\/85).
(8)
(9)
Hirschhorn (1996) showed that checking the first seven
cases n = to 6 is sufficient to prove the result.
References
Hirschhorn, M. D. "An Amazing Identity of Ramanujan."
Math. Mag. 68, 199-201, 1995.
Hirschhorn, M. D. "A Proof in the Spirit of Zeilberger of an
Amazing Identity of Ramanujan." Math. Mag. 69, 267-
269, 1996.
Ramanujan's Tau-Dirichlet Series
see TAU-DIRICHLET SERIES
Ramanujan's Tau Function
see Tau Function
Ramanujan Theta Functions
Ramanujan's one- variable theta function is defined by
¥>( x ) = ^
(1)
m= — oo
It is equal to the function in the Jacobi Triple Prod-
uct with z = 1,
oo
v(x) = G(i) = n^ + * 2 "~ x ) 2 (i - * 2n )
71 = 1
OO oo
= Y, x™ 2 =l + 2]Tx m2 . (2)
m= — oo m=0
Special values include
V(%\ /nil
(3)
(4)
w^_r(f) /r(|)
r(|) V 2 1 /4;r
^" W2 )-W5
*° ~ r(I)
Ae ) = —j— m (5)
Ramanujan's two-variable theta function is defined by
oo
f{a,b)= J2 fl -(-+i)/a 6 -(-D/a (6)
Tl— — OO
for \ab\ < 1. It satisfies
/(-l,o) = (7)
f{a,b) = f(b,a) = (-a; ab) 00 (-6;a6) 00 (a&;a6) 00 (8)
l)/2
Ramp Function
oo oo
= V7-l) fe g fc(2fc_1)/2 + y^(-l) fc ^ fc t 2fc+1 >
fc=0 fc^l
= tea)oo, (9)
where (g)oo are gr-SERIES.
see also JACOBI TRIPLE PRODUCT, SCHROTER'S FOR-
MULA, g-SERIES
Ramp Function
R{x) = xH(x)
H(x)dx
/x
■c
-f
J — c
(1)
(2)
H(x')H(x-x')dx' (3)
= H(x) * H(x), (4)
where H(x) is the Heaviside Step Function and * is
the Convolution. The Derivative is
R'(x) = -H(x).
(5)
The Fourier Transform of the ramp function is given
by
?[R{x)\
f
c R(x) dx = iriS 1 (2nk)
47T 2 k 2 '
(6)
where S(x) is the Delta Function and S'(x) its De-
rivative.
see also Fourier Transform — Ramp Function,
Heaviside Step Function, Rectangle Function,
Sgn, Square Wave
Ramphoid Cusp
Ramsey Number 1517
A type of CUSP as illustrated above for the curve x +
x 2 y 2 - 2x 2 y - xy 2 + y 2 = 0.
see also Cusp
References
Walker, R. J. Algebraic Curves. New York: Springer- Verlag,
pp. 57-58, 1978.
Ramsey Number
The Ramsey number R(m> n) gives the solution to the
Party Problem, which asks the minimum number of
guests R(m, n) that must be invited so that at least m
will know each other (i.e., there exists a CLIQUE of order
m) or at least n will not know each other (i.e., there
exists an independent set of order n). By symmetry, it
is true that
R{m,n) = R(n>m). (1)
It also must be true that
#(2,m) =m. (2)
A generalized Ramsey number is written
i2(mi,...,mfc;n) (3)
and is the smallest Integer R such that, no matter
how each n-element SUBSET of an r-element Set are
colored with k colors, there exists an i such that there is
a Subset of size mi^ all of whose n-element Subsets are
color i. The usual Ramsey numbers are then equivalent
to R(m, n) = R(m, n; 2).
Bounds are given by
R(k 9 l) < <
R(k-l,l) + R{k,l-l)- 1
for R(k- 1,1) and
R(k, I — 1) even
R(k- l,Z) + i2(M-l)
otherwise
and
R(k,k) <4R(k-2,k) + 2
(4)
(5)
(Chung and Grinstead 1983). Erdos proved that for
diagonal Ramsey numbers R(k,k) }
k2 k ' 2
eV2
< R(k,k).
(6)
This result was subsequently improved by a factor of 2
by Spencer (1975). i?(3, k) was known since 1980 to be
bounded from above by C2& 2 /lnA:, and Griggs (1983)
showed that ci — 5/12 was an acceptable limit. J.-H.
Kim (Cipra 1995) subsequently bounded R(3, k) by a
similar expression from below, so
Cl g- k < Ri 3, k) < C2 £- k .
(7)
1518 Ramsey Number
Ramsey Number
Burr (1983) gives Ramsey numbers for all 113 graphs
with no more than 6 Edges and no isolated points.
A summary of known results up to 1983 for R(m, n)
is given in Chung and Grinstead (1983). Radziszowski
maintains an up-to-date list of the best current bounds,
reproduced in part in the following table for R(m, n; 2).
m
n
R{m^n)
3
3
6
3
4
9
3
5
14
3
6
18
3
7
23
3
8
28
3
9
36
3
10
[40, 43]
3
11
[46, 51]
3
12
[52, 60]
3
13
[60, 69]
3
14
[m, 78]
3
15
[73, 89]
3
16
[79, oo]
3
17
[92, oo]
3
18
[98, oo]
3
19
[106, oo]
3
20
[109, oo]
3
21
[122, oo]
3
22
[125, oo]
3
23
[136, oo]
m
n
R(m,n)
4
4
18
4
5
25
4
6
[35, 41]
4
7
[49, 62]
4
8
[55, 85]
4
9
[69, 116]
4
10
[80, 151]
4
11
[93, 191]
4
12
[98, 238]
4
13
[112, 291]
4
14
[119, 349]
4
15
[128, 417]
771
n
R{m,n)
5
5
[43, 49]
5
6
[58, 87]
5
7
[80, 143]
5
8
[95, 216]
5
9
[116, 371]
5
10
[1, 445]
m
n
R(m,n)
6
6
6
6
6
6
7
8
9
10
[102, 165]
[109, 300]
[122, 497]
[153, 784]
[167, 1180]
m
n
R(m,n)
7
7
7
7
7
8
9
10
[205, 545]
[1, 1035]
[1, 1724]
[1, 2842]
m
n
R(m,n)
8
8
8
8
9
10
[282, 1874]
[1, 3597]
[1, 6116]
m
n
R(m,n)
9
9
9
10
[565, 6680]
[1, 12795]
m
n
R(m, n)
10
10
[798, 23981]
m
n
R(m^n)
11
11
[522, oo]
Known values for generalized Ramsey numbers are given
in the following table.
*(...; 2)
Bounds
#(3, 3, 3; 2)
17
£(3,3,4; 2)
[30, 32]
#(3, 3, 5; 2)
[45, 59]
^(3, 4, 4; 2)
[55, 81]
#(3, 4, 5; 2)
> 80
^(4, 4, 4; 2)
[128, 242]
#(3, 3, 3, 3; 2)
[51, 64]
R(3, 3, 3, 4; 2)
[87, 159]
#(3,3,3,3, 3; 2)
[162, 317]
R(3, 3, 3, 3, 3, 3; 2)
[1, 500]
fl(...;3)
Bounds
fl(4,4;3)
[14, 15]
see also Clique, Complete Graph, Extremal
Graph, Irredundant Ramsey Number, Schur
Number
References
Burr, S. A. "Generalized Ramsey Theory for Graphs — A Sur-
vey." In Graphs and Combinatorics (Ed. R. A. Bari and
F. Harary). New York: Springer-Verlag, pp. 52-75, 1964.
Burr, S. A. "Diagonal Ramsey Numbers for Small Graphs."
J. Graph Th. 7, 57-69, 1983.
Chartrand, G. "The Problem of the Eccentric Hosts: An
Introduction to Ramsey Numbers." §5.1 in Introductory
Graph Theory. New York: Dover, pp. 108-115, 1985.
Chung, F. R. K. "On the Ramsey Numbers JV(3, 3, . . . , 3; 2)."
Discrete Math. 5, 317-321, 1973.
Chung, F. and Grinstead, C. G. "A Survey of Bounds for
Classical Ramsey Numbers." J. Graph. Th. 7, 25-37,
1983.
Cipra, B. "A Visit to Asymptopia Yields Insights into Set
Structures." Science 267, 964-965, 1995.
Exoo, G. "On Two Classical Ramsey Numbers of the Form
fi(3,n). M SIAM J. Discrete Math. 2, 488-490, 1989.
Exoo, G. "Announcement: On the Ramsey Numbers i?(4,6),
fl(5,6) and 71(3,12)." Ars Combin. 35, 85, 1993.
Ramsey's Theorem
Random Distribution 1519
Exoo, G. "Some New Ramsey Colorings." Electronic J.
Combinatorics 5, No. 1, R29, 1-5, 1998. http://www.
combinatorics . org/Volume J>/v5iltoc .html.
Folkmann, J. "Notes on the Ramsey Number iV(3,3,3,3)."
J. Combinat. Theory. Ser. A 16, 371-379, 1974.
Gardner, M. "Mathematical Games: In Which Joining Sets of
Points by Lines Leads into Diverse (and Diverting) Paths."
Sci. Amer. 237, 18-28, 1977.
Gardner, M. Penrose Tiles and Trapdoor Ciphers. . . and the
Return of Dr. Matrix, reissue ed. New York: W. H. Free-
man, pp. 240-241, 1989.
Giraud, G. "Une minoration du nombre de quadrangles uni-
colores et son application a la majoration des nombres de
Ramsey binaires bicolors." C. R. Acad. Sci. Paris A 276,
1173-1175, 1973.
Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey
Theory, 2nd ed. New York: Wiley, 1990.
Graver, J. E. and Yackel, J. "Some Graph Theoretic Results
Associated with Ramsey's Theorem." J. Combin. Th. 4,
125-175, 1968.
Greenwood, R. E. and Gleason, A. M. "Combinatorial Rela-
tions and Chromatic Graphs." Canad. J. Math. 7, 1-7,
1955.
Griggs, J. R. "An Upper Bound on the Ramsey Numbers
fl(3,fc)." J. Comb. Th. A 35, 145-153, 1983.
Grinstead, C. M. and Roberts, S. M. "On the Ramsey Num-
bers #(3,8) and #(3,9)." J. Combinat Th. Ser. B 33,
27-51, 1982.
Guldan, F. and Tomasta, P. "New Lower Bounds of Some
Diagonal Ramsey Numbers." J. Graph. Th. 7, 149-151,
1983.
Hanson, D. "Sum-Free Sets and Ramsey Numbers." Discrete
Math. 14, 57-61, 1976.
Harary, F. "Recent Results on Generalized Ramsey Theory
for Graphs." Graph Theory and Applications (Ed. Y. Alai,
D. R. Lick, and A. T. White). New York: Springer- Verlag,
pp. 125-138, 1972.
Hill, R. and Irving, R. W. "On Group Partitions Associated
with Lower Bounds for Symmetric Ramsey Numbers." Eu-
ropean J. Combin. 3, 35—50, 1982.
Kalbfleisch, J. G. Chromatic Graphs and Ramsey's Theorem.
Ph.D. thesis, University of Waterloo, January 1966.
McKay, B. D. and Min, Z. K. "The Value of the Ramsey
Number #(3,8)." J. Graph Th. 16, 99-105, 1992.
McKay, B. D. and Radziszowski, S. P. "#(4,5) = 25." J.
Graph. Th 19, 309-322, 1995.
Piwakowski, K. "Applying Tabu Search to Determine New
Ramsey Numbers." Electronic J. Combinatorics 3, R6,
1-4, 1996. http: //www. combinatorics. org/Volume^3/
volume3.html#R6.
Radziszowski, S. P. "Small Ramsey Numbers." Electronic J.
Combin. 1, DSl 1-29, Rev. Mar. 25, 1996. http://ejc.
math . gatech . edu : 8080/ Journal/Surveys/ds 1 . ps.
Radziszowski, S. and Kreher, D. L, "Upper Bounds for Some
Ramsey Numbers #(3, A;)." J. Combinat. Math. Combin.
Comput. 4, 207-212, 1988.
Spencer, J. H. "Ramsey's Theorem — A New Lower Bound."
J. Combinat. Theory Ser. A 18, 108-115, 1975.
Wang, Q. and Wang, G. "New Lower Bounds for the Ramsey
Numbers #(3,$)." Beijing Daxue Xuebao 25, 117-121,
1989.
Whitehead, E. G. "The Ramsey Number N(3, 3,3, 3; 2)."
Discrete Math. 4, 389-396, 1973.
will contain a green SUBGRAPH Km or a red subgroup
K n . Furthermore,
#(m, n) < R(m — 1, n) + #(m, n - 1)
if m y n > 3. The theorem can be equivalently stated
that, for all € N, there exists an n G N such that any
complete DIGRAPH on n VERTICES contains a complete
transitive SUBGRAPH of m VERTICES. Ramsey's theo-
rem is a generalization of the PIGEONHOLE PRINCIPLE
since
tf(2,2,...,2) =t + l.
see also DlLWORTH'S LEMMA, NATURAL INDEPEN-
DENCE Phenomenon, Pigeonhole Principle, Ram-
sey Number
References
Graham, R. L.; Rothschild, B. L.; and Spencer, J. H. Ramsey
Theory, 2nd ed. New York: Wiley, 1990.
Spencer, J. "Large Numbers and Unprovable Theorems."
Amer. Math. Monthly 90, 669-675, 1983.
Randelbrot Set
The FRACTAL-like figure obtained by performing the
same iteration as for the MANDELBROT Set, but adding
a random component i?,
z n+ i — z n 2 +c + R.
In the above plot, R = R x + iR y , where R X} R y €
[-0.05,0.05].
see also Mandelbrot Set
References
Dickau, R. M. "Randelbrot Set." http: //forum. svarthmore
. edu/advanced/robertd/randelbrot . html.
Ramsey's Theorem
A generalization of Dilworth's Lemma. For each
m, n e N with m, n > 2, there exists a least INTEGER
R(m,n) (the Ramsey Number) such that no matter
how the Complete Graph K R ( mtn ) is two-colored, it
Random Distribution
A Distribution in which the variates occur with Prob-
abilities asymptotically matching their "true" under-
lying Distribution is said to be random.
see also Distribution, Random Number
1520 Random Dot Stereogram
Random Walk
Random Dot Stereogram
see Stereogram
Random Graph
A random graph is a GRAPH in which properties such
as the number of NODES, EDGES, and connections be-
tween them are determined in some random way. Erdos
and Renyi showed that for many monotone-increasing
properties of random graphs, graphs of a size slightly
less than a certain threshold are very unlikely to have
the property, whereas graphs with a few more Edges
are almost certain to have it. This is known as a PHASE
Transition.
see also Graph (Graph Theory), Graph Theory
References
Bollobas, B. Random Graphs. London: Academic Press,
1985.
Steele, J. M. "Gibbs' Measures on Combinatorial Objects and
the Central Limit Theorem for an Exponential Family of
Random Trees." Prob. Eng. Inform. Sci. 1, 47-59, 1987.
Random Matrix
A random matrix is a MATRIX of given type and size
whose entries consist of random numbers from some
specified distribution.
see also MATRIX
Random Number
Computer-generated random numbers are sometimes
called Pseudorandom Numbers, while the term "ran-
dom" is reserved for the output of unpredictable physi-
cal processes. It is impossible to produce an arbitrarily
long string of random digits and prove it is random.
Strangely, it is very difficult for humans to produce a
string of random digits, and computer programs can be
written which, on average, actually predict some of the
digits humans will write down based on previous ones.
The Linear Congruence Method is one algorithm
for generating Pseudorandom Numbers. The initial
number used as the starting point in a random number
generating algorithm is known as the SEED. The good-
ness of random numbers generated by a given Algo-
rithm can be analyzed by examining its Noise Sphere.
see also Bays' Shuffle, Cliff Random Number
Generator, Quasirandom Sequence, Schrage's
Algorithm, Stochastic
References
Bassein, S. "A Sampler of Randomness." Amer. Math.
Monthly 103, 483-490, 1996.
Bratley, P.; Fox, B. L.; and Schrage, E. L. A Guide to Sim-
ulation, 2nd ed. New York: Springer- Verlag, 1996.
Dahlquist, G. and Bjorck, A. Ch. 11 in Numerical Methods.
Englewood Cliffs, NJ: Prentice-Hall, 1974.
Deak, I. Random Number Generators and Simulation. New
York: State Mutual Book & Periodical Service, 1990.
Forsythe, G. E.; Malcolm, M. A,; and Moler, C. B. Ch. 10 in
Computer Methods for Mathematical Computations. En-
glewood Cliffs, NJ: Prentice-Hall, 1977.
Gardner, M. "Random Numbers." Ch. 13 in Mathematical
Carnival: A New Round-Up of Tantalizers and Puzzles
from Scientific American. New York: Vintage, 1977.
James, F. "A Review of Pseudorandom Number Generators."
Computer Physics Comm. 60, 329-344, 1990.
Kac, M. "What is Random?" Amer. Sci. 71, 405-406, 1983.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 200-201
and 205-207, 1962.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 151-154,
1951.
Knuth, D. E. Ch. 3 in The Art of Computer Programming,
Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA:
Addison-Wesley, 1981.
Marsaglia, G. "A Current View of Random Number Genera-
tors." In Computer Science and Statistics: Proceedings of
the Symposium on the Interface, 16th, Atlanta, Georgia,
March 1984 (Ed. L. Billard). New York: Elsevier, 1985.
Park, S. and Miller, K. "Random Number Generators: Good
Ones are Hard to Find." Comm. ACM 31, 1192-1201,
1988.
Peterson, I. The Jungles of Randomness: A Mathematical
Safari. New York: Wiley, 1997.
Pickover, C. A. "Computers, Randomness, Mind, and In-
finity." Ch. 31 in Keys to Infinity. New York: W. H.
Freeman, pp. 233-247, 1995.
Press, W. H.; Flannery, B. R; Teukolsky, S. A.; and Vet-
terling, W. T. "Random Numbers." Ch. 7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 266-306, 1992.
Schrage, L. "A More Portable Fortran Random Number Gen-
erator." ACM Trans. Math. Software 5, 132-138, 1979.
Schroeder, M. "Random Number Generators." In Number
Theory in Science and Communication, with Applications
in Cryptography, Physics, Digital Information, Computing
and Self- Similarity, 3rd ed. New York: Springer- Verlag,
pp. 289-295, 1990.
Random Percolation
see Percolation Theory
Random Polynomial
A Polynomial having random Coefficients.
see also KAC FORMULA
Random Variable
A random variable is a measurable function from a
Probability Space (£,§, P) into a Measurable
Space (S',§') known as the State Space.
see also Probability Space, Random Distribution,
Random Number, State Space, Variate
References
Gikhman, I. I. and Skorokhod, A. V. Introduction to the The-
ory of Random Processes. New York: Dover, 1997.
Random Walk
A random process consisting of a sequence of discrete
steps of fixed length. The random thermal perturba-
tions in a liquid are responsible for a random walk phe-
nomenon known as Brownian motion, and the collisions
of molecules in a gas are a random walk responsible for
diffusion. Random walks have interesting mathematical
Random Walk—l-D
properties that vary greatly depending on the dimension
in which the walk occurs and whether it is confined to
a lattice.
see also Random Walk — 1-D, Random Walk — 2-D,
Random Walk — 3-D, Self-Avoiding Walk
References
Barber, M. N. and Ninham, B. W. Random and Restricted
Walks: Theory and Applications. New York: Gordon and
Breach, 1970.
Chandrasekhar, S. In Selected Papers on Noise and Stochas-
tic Processes (Ed. N. Wax). New York: Dover, 1954.
Doyle, P. G. and Snell, J. L. Random Walks and Electric
Networks. Washington, DC: Math. Assoc. Amer, 1984.
Dykin, E. B. and Uspenskii, V. A. Random Walks. New
York: Heath, 1963.
Feller, W. An Introduction to Probability Theory and Its Ap-
plications, Vol 1, 3rd ed. New York: Wiley, 1968.
Gardner, M. "Random Walks." Ch. 6-7 in Mathematical Cir-
cus: More Puzzles, Games, Paradoxes, and Other Math-
ematical Entertainments. Washington, DC: Math. Assoc.
Amer., 1992.
Hughes, B. D. Random Walks and Random Environments,
Vol. 1: Random Walks. New York: Oxford University
Press, 1995.
Hughes, B. D. Random Walks and Random Environments,
Vol. 2: Random Environments. New York: Oxford Uni-
versity Press, 1996.
Lawler, G. F. Intersections of Random Walks. Boston, MA:
Birkhauser, 1996.
Spitzer, F. Principles of Random Walk, 2nd ed. New York:
Springer- Verlag, 1976.
Random Walk — 1-D
Let N steps of equal length be taken along a Line. Let
p be the probability of taking a step to the right, q the
probability of taking a step to the left, m the number
of steps taken to the right, and n 2 the number of steps
taken to the left. The quantities p, q, rii, Ti2, and N are
related by
P + flf = l (1)
and
m+ri2 = N.
(2)
Now examine the probability of taking exactly m steps
out of N to the right. There are (*) = ( ni + n2 ) ways
of taking m steps to the right and n^ to the left, where
(£) is a Binomial Coefficient. The probability of
taking a particular ordered sequence of m and m steps
is p ni q n2 . Therefore,
(m+n2)!
N\
ni!ri2' ni!(iv — m)\
ni N—ni
P q
(3)
where n\ is a Factorial. This is a Binomial Distri-
bution and satisfies
N
Y l P(m) = (p + q) N = l N = l.
(4)
Random Walk—l-D 1521
The Mean number of steps n\ to the right is then
but
)-V iv:
n ni N- ni
,XJ ^ ni UN-
ni=0
-n x )! P q *'
(5)
niP = P ~dp P '
(6)
<m> - E
N\
m\(N -m)\
d ni \ N-
% p ) q
= P
d_ y- NI -ni-JV-ni
dp £-^ 7l\\(N - n±)
ni—0
= p-^(p + q) N =pN(p + q) N - 1 = P N. (7)
From the Binomial Theorem,
<n 2 ) =N- (m) = N(l -p) = qN. (8)
The Variance is given by
<r ni 2 — {m 2 ) - (ni) 2 . (9)
But
C«i
! > = E
ni N — ni 2 / 1 r\\
P l q l ni , (10)
ni!(iV- ni)!
2 711
m p = tii
P
2
p V) pni ={ p i)
^ ni\(N-m)l ydp) p q
= ( jlY v" m "i *— i
\ p dp I 2^ n 1 \{N-n 1 )\ P q
v ' ni=0
= {p^) (p + v) N = -^\pM(p + <i)N-i]
= pWp + g)"" 1 + P N(N - l)(p + qf- 2 }
= p[N + pN(N-l)]
= pN[l +pN -p] = (Np) 2 + Npq
= (m) 2 + Npq. (11)
Therefore,
* ni 2 = (m 2 ) - (m) 2 = Npq, (12)
and the Root-Mean-Square deviation is
ffm = \/Npq. (13)
1522
Random Walk—l-D
For a large number of total steps AT, the BINOMIAL DIS-
TRIBUTION characterizing the distribution approaches a
Gaussian Distribution.
Consider now the distribution of the distances d n trav-
eled after a given number of steps,
djv = m — ri2 = 2ni - JV,
(14)
as opposed to the number of steps in a given direction.
The above plots show cLn(p) for N — 200 and three val-
ues p = 0.1, p = 0.5, and p = 0.9, respectively. Clearly,
weighting the steps toward one direction or the other in-
fluences the overall trend, but there is still a great deal of
random scatter, as emphasized by the plot below, which
shows three random walks all with p = 0.5.
Surprisingly, the most probable number of sign changes
in a walk is 0, followed by 1, then 2, etc.
For a random walk with p = 1/2, the probability Pjv(d)
of traveling a given distance d after N steps is given in
the following table.
steps
-5
_4
-3
-2
-1
1
2
3
4
5
1
1
1
2
_1
2
2
1
4
2
4
1
4
3
i
8
3
8
3
8
i
8
4
JL
16
4
16
6
16
16
_1_
16
5
i
o
5
o
10
10
5
n
j_
32
32
32
32
32
32
In this table, subsequent rows are found by adding HALF
of each cell in a given row to each of the two cells diago-
nally below it. In fact, it is simply PASCAL'S TRIANGLE
padded with intervening zeros and with each row multi-
plied by an additional factor of 1/2. The COEFFICIENTS
in this triangle are given by
P N {d)
2-(^}
(15)
Random Walk—l-D
The expectation value of the distance after N steps is
therefore
{d N )= Yl ww*)
d=-iV,-(iV-2),...
N
ON Z^
\d\N\
2 n L^ (M±£\\{*Lz£Y
d=-iV,-(JV-2),... V 2 )• \ 2 }'
(16)
This sum can be done symbolically by separately con-
sidering the cases N Even and N Odd. First, consider
Even N so that N = 2 J. Then
(d2j)
m
2 N
2-^ ( 2J+d \\ f 2J-d \i
-2(J-1),...
. V- l<*l ■ V- \d\
Z^ ( 2J+d \i ( 2J-d \\ ~*~ Z^t ( W+d \\
-m'm
(^)!(^)!
2 N
E
\2d\
_^ ( 2J-\-2d \\ ( 2J-2d \\
(J-l),... \ 2 )•{ 2 J'
\2d\
2^f ( 2J+2d \\ ( 2J-2d \\
d=l,2„.. V 2 J' V 2 )'
2 n
'E;
2d
(J + d)\(J-d)\
N\ x^ d
d=i
j
7^2 Zs
2»-*^ i {J + d)\{J-d)V
But this sum can be evaluated analytically as
J
(17)
(J + d)\(J-d)\
d~i
(j + d)\(j-d)\ 2r 2 (i + jy
(18)
which, when combined with N = 2 J and plugged back
in, gives
r(2j + i)j r(2j)
{ 2J> ~ 2 2J - 1 r 2 (i + J) 2 2J - 2 r 2 (j) '
But the Legendre Duplication Formula gives
2 2J - 1/2 r(J)r(J+i)
r(2j)
V2tt
(19)
(20)
% _x. 23J -i/ ar (j)r(j+i) 2 r(j+f)
(21)
2 2J - 2 r 2 (j)
Random Walk— 1-D
Now consider AT Odd, so N = 2 J - 1. Then
(d 2 j-i) =
iV!
2 N
\d\
2J-1
+ E
Z-< / 2J-l+d \i ( 2J-l-d \i
d=-( 2 j-i), v 2 r v 2 ;*
-(2J+1),...
Ml
Z^f [ 2J-l+d \; f 2J-l-d \\
iV!
: 2*" 1
AT!
: 2 iV - 1
r(2J)
2 2J " 2
: T(2 J)
2J-1
E
[=1,3,
13
E
^ ( 2J-l+d \l ( 2J-l-d \t
d-1
( 2J-2+d \\ / 2J-d \;
2d-l
d=2,4
" J
E
(J + d-l)!(J-d)!
l + J-2Fi(l,-J;J;l) 1
2 2J-2 F ( j)r(! + j) ^ r( 2 j)
™r(j)r(j + i/2)
-[l + J-afi(l,-J;J;-l)] + l
2 2J-2 r 2(J)J
[l + J- 2 Fi(l,-J;J ; -l)] + l. (22)
2 T(J+i)
VS Jr(j)
But the Hypergeometric Function 2 -Fi has the spe-
cial value
2Fl (l,-J,J,-l)-—^j-y )+ l,
(daj-i)
2 r(j+i)
0F r(j) •
Summarizing the EVEN and Odd solutions,
{d N ) =
2 r(j+i)
Vi r(j) '
where
J = |AT for iV even
J=I(AT+1) for AT odd.
(23)
(24)
(25)
(26)
Written explicitly in terms of N,
{d N ) = <
( 2 rc^jv+l)
-4= — ^-t — — tor iV even
v^ r(^iV)
2£(^) forArodd .
(27)
Random Walk— 1-D
The first few values of (djv) are then
(do)=0
<di> = <<fe> = 1
<d 3 ) = (d.) =
(*> = <*) =
(d 7 ) = (d s ) =
(dg) = {dio) =
<dii> = <dia> =
1523
3
2
15
8
35
16
315
128
693
256
(dl3> = {^14} = fo24 *
Now, examine the asymptotic behavior of (d;sr). The
asymptotic expansion of the Gamma Function ratio is
r(j)
^(-B-
128J 2
+ .
•) (28)
(Graham ei al. 1994), so plugging in the expression for
{(In) gives the asymptotic series
(d N ) =
S(
1 1
^ 4iV + 327V 2
128iV 3
21
20487V 4
...),
(29)
where the top signs are taken for N Even and the bot-
tom signs for N Odd. Therefore, for large N,
(d N )
(30)
which is also shown in Mosteller et al. (1961, p. 14).
see also Binomial Distribution, Catalan Number,
P-Good Path, Polya's Random Walk Constants,
Random Walk — 2-D, Random Walk — 3-D, Self-
Avoiding Walk
References
Chandrasekhar, S. "Stochastic Problems in Physics and As-
tronomy." Rev. Modern Phys. 15, 1-89, 1943. Reprinted
in Noise and Stochastic Processes (Ed. N. Wax). New
York: Dover, pp. 3-91, 1954.
Feller, W. Ch. 3 in An Introduction to Probability Theory
and Its Applications, Vol. 1, 3rd ed. } rev. printing. New
York: Wiley, 1968.
Gardner, M. Chs. 6-7 in Mathematical Carnival: A New
Round-Up of Tantalizers and Puzzles from Scientific
American. New York: Vintage Books, 1977.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Answer
to problem 9.60 in Concrete Mathematics: A Foundation
for Computer Science, 2nd ed. Reading, MA: Addison-
Wesley, 1994.
Hersh, R. and Griego, R. J. "Brownian Motion and Potential
Theory." Sci. Amer. 220, 67-74, 1969.
Kac, M. "Random Walk and the Theory of Brownian Mo-
tion." Amer. Math. Monthly 54, 369-391, 1947. Reprinted
in Noise and Stochastic Processes (Ed. N. Wax). New
York: Dover, pp. 295-317, 1954.
Mosteller, F.; Rourke, R. E. K.; and Thomas, G. B. Proba-
bility and Statistics. Reading, MA: Addison- Wesley, 1961.
1524 Random Walk— 2-D
Random Walk — 2-D
-15 -10
In a PLANE, consider a sum of N 2-D VECTORS with
random orientations. Use PHASOR notation, and let the
phase of each VECTOR be RANDOM. Assume N unit
steps are taken in an arbitrary direction (i.e., with the
angle 6 uniformly distributed in [0,27r) and not on a
Lattice), as illustrated above. The position z in the
Complex Plane after N steps is then given by
£^'
(i)
which has ABSOLUTE SQUARE
N N
j=l fc=l
j=l fc=l
(2)
Therefore,
<w a >="+( i> <( *'
-»k)
(3)
Each step is likely to be in any direction, so both 9j
and Ok are RANDOM VARIABLES with identical MEANS
of zero, and their difference is also a random variable.
Averaging over this distribution, which has equally likely
Positive and Negative values yields an expectation
value of 0, so
<W 2 > = N. (4)
The root- mean-square distance after TV unit steps is
therefore
VN, (5)
j^lrms
so with a step size of /, this becomes
d Tms = iVn.
In order to travel a distance d
<d
TV;
(!)'
(6)
(7)
Range (Image)
steps are therefore required.
-10
"^E:
*HL
a, -fro
i 15
+H
10
^
Amazingly, it has been proven that on a 2-D Lattice,
a random walk has unity probability of reaching any
point (including the starting point) as the number of
steps approaches Infinity.
see also POLYA'S RANDOM WALK CONSTANTS, RAN-
DOM Walk — 1-D, Random Walk — 3-D
Random Walk — 3-D
68
4
On a 3-D LATTICE, a random walk has less than unity
probability of reaching any point (including the start-
ing point) as the number of steps approaches infinity.
The probability of reaching the starting point again
is 0.3405373296. . . . This is one of POLYA'S RANDOM
Walk Constants.
see also POLYA'S RANDOM WALK CONSTANTS, RAN-
DOM Walk — 1-D, Random Walk — 2-D
Range (Image)
If T is Map over a Domain D, then the range of T is
defined as
Range(T) = T(D) = {T(X) : X e D}.
The range T(D) is also called the Image of D under T.
see also Domain, Map
Range (Line Segment)
Ranunculoid
1525
Range (Line Segment)
The set of all points on a Line Segment, also called a
Pencil.
see also Perspectivity, Section (Pencil)
References
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, p. 8,
1961.
Range (Statistics)
R = max(xi) — min(^t).
(i)
For small samples, the range is a good estimator of the
population STANDARD Deviation (Kenney and Keep-
ing 1962, pp. 213-214). For a continuous Uniform Dis-
tribution
P(a
for < x < C
for \x\ < C,
the distribution of the range is given by
D(R) = N(*)"- l -iN-l)(*) K .
(2)
(3)
Given two samples with sizes m and n and ranges Ri
and i?2, let u = R1/R2. Then
D(u) = <
m(m — l)n(n — 1)
(m + Ti)(ro+n — l)(m+n — 2)
x[(m + n)u m - 2 - (m + ra-2)u m_1 ]
for < u < 1
m(m— l)n(n — 1)
(m+n)(m-\-n — l)(m+n — 2)
x [(m + n)u~ n -(m + n~ 2)u~ n ~ 1 ]
for 1 < u < oo.
(4)
The Mean is
1*>U
(m — l)n
(m+l)(n-2) }
and the MODE is
(m-2)(m+n)
(m-l)(m+n-2)
(n+l)(m+n-2)
n(m+7i)
for 7n — n < 2
for m — n > 2.
(5)
(6)
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 213-214,
1962.
Rank
In a total generality, the "rank" of a mathematical ob-
ject is defined whenever that object is Free. In gen-
eral, the rank of a Free object is the Cardinality of
the Free generating Subset G. The word "rank" also
refers to several unrelated concepts in mathematics in-
volving groups, quadratic forms, sequences, statistics,
and tensors.
see also RANK (GROUP), RANK (QUADRATIC FORM),
Rank (Sequence), Rank (Statistics), Rank (Ten-
sor)
Rank (Group)
For an arbitrary finitely generated Abelian GROUP G,
the rank of G is defined to be the rank of the Free
generating SUBSET G modulo its TORSION SUBGROUP.
For a finitely generated GROUP, the rank is defined to
be the rank of its "Abelianization."
see also ABELIAN GROUP, BETTI NUMBER, BURNSIDE
Problem, Quasithin Theorem, Quasi-Unipotent
Group, Torsion (Group Theory)
Rank (Quadratic Form)
For a Quadratic Form Q in the canonical form
Q — yi 2 + y2 2 + . . . + y P 2 - y P +\ 2 - y v +2 2 - ... - y 2 ,
the rank is the total number r of square terms (both
Positive and Negative).
see also Signature (Quadratic Form)
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1105, 1979.
Rank (Sequence)
The position of a Rational Number in the Sequence
1, §, 2, |, 3, \, |, f, 4, |, ..., ordered in terms of
increasing NUMERATOR+DENOMINATOR.
see also ENCODING, FAREY SERIES
Rank (Statistics)
The Ordinal Number of a value in a list arranged in
a specified order (usually decreasing).
see also Spearman Rank Correlation, Wilcoxon
Rank Sum Test, Wilcoxon Signed Rank Test,
Zipf's Law
Rank (Tensor)
The total number of CONTRAVARIANT and COVARIANT
indices of a TENSOR. The rank of a TENSOR is indepen-
dent of the number of DIMENSIONS of the SPACE.
Rank Object
1
> 2
scalar
vector
tensor
see also Contravariant Tensor, Covariant Ten-
sor, Scalar, Tensor, Vector
Ranunculoid
1 /
N \
_v
\ \
Sf
\
/ /
1 I
\ J
1
\ \
i \
/ ]
f \
/ /
1526
RAT-Free Set
Rational Approximation
An Epicycloid with n = 5 cusps, named after the but-
tercup genus Ranunculus (Madachy 1979).
see also EPICYCLOID.
References
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 223, 1979.
Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 79—
80, 1995.
RAT-Free Set
A RAT-free set is a set of points, no three of which
determine a RIGHT TRIANGLE. Let f(n) be the smallest
RAT-free subset guaranteed to be contained in a planar
set of n points, then the function f(n) is bounded by
y/n < f(n) < 2y/n.
References
Abbott, H. L. "On a Conjecture of Erdos and Silverman in
Combinatorial Geometry." J. Combin. Th. A 29, 380-381,
1980.
Chan, W. K. "On the Largest RAT-FREE Subset of a Finite
Set of Points." Pi Mu Epsilon, Spring 1987.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 250-251, 1991.
Seidenberg, A. "A Simple Proof of a Theorem of Erdos and
Szekeres." J. London Math. Soc. 34, 352, 1959.
Ratio
The ratio of two numbers r and s is written r/s, where
r is the NUMERATOR and s is the DENOMINATOR. The
ratio of r to s is equivalent to the QUOTIENT r/s. Bet-
ting Odds written as r : s correspond to s/(r + s). A
number which can be expressed as a ratio of Integers
is called a RATIONAL NUMBER.
see also DENOMINATOR, FRACTION, NUMERATOR,
Odds, Quotient, Rational Number
Ratio Distribution
Given two distributions Y and X with joint probabil-
ity density function f(x,y), let U = Y/X be the ratio
distribution. Then the distribution function of u is
D{u) = P(U < u)
= P(Y < uX\X > 0) + P(Y > uX\X < 0)
/»co pux pO />0
= / / f{x,y)dydx + I I f(x,y)dydx.
JO Jo J — oo J ux
(i)
The probability function is then
/»oo /»0
P(u) = D'(u)= I xf(x,ux)dx — / xf(x,ux)dx
Jo J — oo
/oo
\x\f(x,ux) dx. (2
■oo
For variates with a standard NORMAL DISTRIBUTION,
the ratio distribution is a CAUCHY DISTRIBUTION. For
a Uniform Distribution
f(x,y)
(1 for x, y G [0,1]
10
otherwise,
(3)
u <
for < u < 1
— ri^. 2 i 1 / u - *
2u 2
for u > 1.
(4)
see also CAUCHY DISTRIBUTION
Ratio Test
Let Uk be a SERIES with POSITIVE terms and suppose
p = hm .
k— J-oo Uk
Then
1. If p < 1, the Series Converges.
2. If p > 1 or p = oo, the SERIES DIVERGES.
3. If p = 1, the Series may Converge or Diverge.
The test is also called the Cauchy Ratio Test or
d'Alembert Ratio Test.
see also Convergence Tests
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 282-283, 1985.
Bromwich, T. J. Pa. and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, p. 28, 1991.
Rational Approximation
If r is any number and n is any INTEGER, then there is
a Rational Number m/n for which
rt ^ ml
< r < -.
n n
(1)
If r is Irrational and k is any Whole Number, there
is a FRACTION m/n with n < k and for which
< r < — .
n nk
(2)
Furthermore, there are an infinite number of FRACTIONS
m/n for which
m 1
— < "2
n rr
<r- — < -^. (3)
Hurwitz has shown that for an Irrational Number (
<-z\<Z1Z> ( 4 )
there are infinitely RATIONAL NUMBERS h/k if < c <
\/5, but if c > \/5, there are some £ for which this
approximation holds for only finitely many h/k.
Rational Canonical Form
Rational Integer 1527
Rational Canonical Form
There is an invertible matrix Q such that
Q-'TQ = diag[L(Vi), L(iM, • • • , L(tf.)]>
where L(/) is the companion MATRIX for any MONIC
Polynomial
/(A) = / + /iA + ... + /nA n
with f n = 1. The POLYNOMIALS ipi are called the "in-
variant factors" of T, and satisfy ipi+i\ipi for i = s — 1,
..., 1 (Hartwig 1996).
References
Gantmacher, F. R. The Theory of Matrices, Vol. 1. New
York: Chelsea, 1960.
Hartwig, R. E. "Roth's Removal Rule and the Rational
Canonical Form.' 1 Amer. Math. Monthly 103, 332-335,
1996.
Herstein, I. N. Topics in Algebra, 2nd ed. New York:
Springer- Verlag, p. 162, 1975.
Hoffman, K. and Kunze, K. Linear Algebra, 3rd ed. Engle-
wood Cliffs, NJ: Prentice-Hall, 1996.
Lancaster, P. and Tismenetsky, M. The Theory of Matrices,
2nd ed. New York: Academic Press, 1985.
Turnbull, H. W. and Aitken, A. C. An Introduction to the
Theory of Canonical Matrices, 2nd impression. New York:
Blackie and Sons, 1945.
Rational Cuboid
see Euler Brick
Rational Distances
It is possible to find six points in the PLANE, no three on
a Line and no four on a CIRCLE (i.e., none of which are
COLLINEAR or CONCYCLIC), such that all the mutual
distances are RATIONAL. An example is illustrated by
Guy (1994, p. 185).
It is not known if a TRIANGLE with INTEGER sides, ME-
DIANS, and AREA exists (although there are incorrect
PROOFS of the impossibility in the literature). How-
ever, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz
have showed that there are infinitely many triangles with
Rational sides (Heronian Triangles) with two Ra-
tional Medians (Guy 1994, p. 188).
see also COLLINEAR, CONCYCLIC, CYCLIC QUADRILAT-
ERAL, Equilateral Triangle, Euler Brick, Hero-
nian Triangle, Rational Quadrilateral, Ratio-
nal Triangle, Square, Triangle
References
Guy, R. K. "Six General Points at Rational Distances" and
"Triangles with Integer Sides, Medians, and Area." §D20
and D21 in Unsolved Problems in Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 185-190 and 188-190,
1994.
Rational Domain
see Field
Rational Double Point
There are nine possible types of ISOLATED SINGULARI-
TIES on a Cubic Surface, eight of them rational double
points. Each type of ISOLATED SINGULARITY has an as-
sociated normal form and COXETER-DYNKIN DIAGRAM
(Ai, A 2 , A 3 , A 4 , A 5 , £>4, A>, E 6 and E 6 ).
The eight types of rational double points (the Eq type
being the one excluded) can occur in only 20 combi-
nations on a Cubic Surface (of which Fischer 1986
gives 19): A u 2A U $A U 4A U A 2 , (A 2 ,Ai), 2A 2 ,
(2A 2 ,Ai), 3A 2 , A s , (A s ,Ai) y (A 3 ,2Ai), A 4 , (A 4 ,Ai),
Ab, (As,Ai), At, -Ds, and E& (Looijenga 1978, Bruce
and Wall 1979, Fischer 1986).
In particular, on a CUBIC SURFACE, precisely those con-
figurations of rational double points occur for which the
disjoint union of the COXETER-DYNKIN DIAGRAM is
a Subgraph of the Coxeter-Dynkin Diagram E g .
Also, a surface specializes to a more complicated one
precisely when its graph is contained in the graph of the
other one (Fischer 1986).
see also Coxeter-Dynkin Diagram, Cubic Surface,
Isolated Singularity
References
Bruce, J. and Wall, C. T. C. "On the Classification of Cubic
Surfaces." J. London Math. Soc. 19, 245-256, 1979.
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 13, 1986.
Fischer, G. (Ed.). Plates 14-31 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 17-31, 1986.
Looijenga, E. "On the Semi-Universal Deformation of a Sim-
ple Elliptic Hypersurface Singularity. Part II: The Dis-
criminant." Topology 17, 23-40, 1978.
Rodenberg, C "Modelle von Flachen dritter Ordnung." In
Mathematische Abhandlungen aus dem Verlage Mathema-
tischer Modelle von Martin Schilling. Halle a. S., 1904.
Rational Function
A quotient of two polynomials P(z) and Q(z),
R(z)
_-P(*)
-Q(z)'
is called a rational function. More generally, if P and Q
are POLYNOMIALS in multiple variables, their quotient
is a rational function.
see also ABEL'S CURVE THEOREM, CLOSED FORM,
Fundamental Theorem of Symmetric Functions,
Quotient-Difference Algorithm, Rational Inte-
ger, Rational Number, Riemann Curve Theorem
Rational Integer
A synonym for Integer. The word "rational" is some-
times used for emphasis to distinguish it from other
types of "integers" such as Cyclotomic Integers,
Eisenstein Integers, and Gaussian Integers.
see also Cyclotomic Integer, Eisenstein Integer,
Gaussian Integer, Integer, Rational Number
1528
Rational Number
Ray
References
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, p. 1, 1979.
Rational Number
A number that can be expressed as a FRACTION p/q
where p and q are INTEGERS, is called a rational num-
ber with Numerator p and Denominator q. Num-
bers which are not rational are called Irrational Num-
bers. Any rational number is trivially also an ALGE-
BRAIC Number.
For a, 6, and c any different rational numbers, then
multiple of the other. Here, (0, 0, 0) is not considered
to be a valid point. The triples (a, 6, 1) correspond
to the ordinary points (a, 6), and the triples (a, 6, 0)
correspond to the POINTS AT INFINITY, usually called
the Line at Infinity.
The rational points on ELLIPTIC CURVES over the GA-
LOIS Field GF(q) are 5, 7, 9, 10, 13, 14, 16, . . . (Sloane's
A005523).
see also Elliptic Curve, Line at Infinity, Point at
Infinity
References
Sloane, N. J. A. Sequence A005523/M3757 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
(a-6) 2 (6-c) 2 (c-a) 2
is the SQUARE of a rational number (Honsberger 1991).
The probability that a random rational number has an
Even Denominator is 1/3 (Beeler et al. 1972, Item
54).
see also ALGEBRAIC INTEGER, ALGEBRAIC NUMBER,
Anomalous Cancellation, Denominator, Dirich-
let Function, Fraction, Integer, Irrational
Number, Numerator, Quotient, Transcendental
Number
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Courant, R. and Robbins, H. "The Rational Numbers." §2.1
in What is Mathematics?: An Elementary Approach to
Ideas and Methods, 2nd ed. Oxford, England: Oxford Uni-
versity Press, pp. 52-58,, 1996.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 52-53, 1991.
Rational Point
A K-T&tional point is a point (X, Y) on an ALGEBRAIC
Curve, where X and Y are in a Field K.
The rational point may also be a Point at Infinity.
For example, take the ELLIPTIC CURVE
Y 2 = X 3 + X + 42
and homogenize it by introducing a third variable Z so
that each term has degree 3 as follows:
ZY Z
X s + XZ 2 + 42Z 3 .
Now, find the points at infinity by setting Z = 0, ob-
taining
Q = X 3 .
Solving gives X — 0, Y equal to any value, and (by
-definition) Z = 0. Despite freedom in the choice of Y,
there is only a single Point at Infinity because the
two triples (Xi, Yi, Zi), (X2, Y2, Z 2 ) are considered
to be equivalent (or identified) only if one is a scalar
Rational Quadrilateral
A rational quadrilateral is a Quadrilateral for which
the sides, DIAGONALS, and AREA are RATIONAL, The
simplest case has sides a = 52, b = 25, c = 39, and
d — 60 and DIAGONALS of length p = 63 and q = 56.
see also Area, Diagonal (Polygon), Rational
Quadrilateral
Rational Triangle
A rational triangle is a TRIANGLE all of whose sides are
Rational Numbers and all of whose Angles are Ra-
tional numbers of DEGREES. The only such triangle is
the Equilateral Triangle (Conway and Guy 1996).
see also Equilateral Triangle, Fermat's Right
Triangle Theorem, Right Triangle
References
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.
RATS Sequence
A sequence produced by the instructions "reverse, add,
then sort the digits," where zeros are suppressed. For
example, after 668 we get
668 + 866 = 1534,
so the next term is 1345. Applied to 1, the sequence
gives 1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444,
55778, . . . (Sloane's A004000)
see also 196-Algorithm, Kaprekar Routine, Re-
versal, Sort-Then-Add Sequence
References
Sloane, N. J. A. Sequence A004000/M1137 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Ray
A B
A VECTOR AB from a point A to a point B. In GEOM-
ETRY, a ray is usually taken as a half-infinite LINE with
one of the two points A and B taken to be at INFINITY.
see also Line, Vector
Rayleigh Distribution
Rayleigh Distribution
The distribution with Probability Function
P(r) =
-r 2 /2s 2
(1)
Rayleigh-Ritz Variational Technique
Rayleigh Differential Equation
y" -»{l-\y 2 )y+y = 0,
where \i > 0. Differentiating and setting y
the van der Pol Equation.
see also VAN DER POL EQUATION
Rayleigh's Formulas
The formulas
1529
for r € [0,oo). The MOMENTS about are given by
H' m = r m P(r)dr = s- 2 / r m+1 e~ T /2a dr
Jo Jo
= s- 2 I m+ i (^) , (2)
where I(x) is a Gaussian Integral. The first few of
these are
h(a~
) = 3«
) = ioVOTT
) = §a -s/air
h(a *) = a ,
so
^ = S " 2 i(2 S 2 ) = l
(3)
(4)
(5)
(6)
(7)
(8)
fJ ,[ = S - 2 \(2s 2 )V2^= \ S V2^ = Syf^ (9)
M ' 2 = S - 2 |(2 S 2 ) 2 = 2 S 2 (10)
/ii = S - 2 |(2 S 2 ) 2 ^/2^ : = f s 3 ^ = 3s 3 ^1 (11)
/ii = S " 2 (2s 2 ) 3 = 8s 4 . (12)
The Moments about the Mean are
M2 = M2 - (^i) 2 = — y-s 2 ( 13 )
M3 = ^3 - 3/4A«i + 2(Mi) 3 = Tf (t - 3)« 3 (14)
^4 = ^4- 4/X3M1 + 6a»2(/*i) 2 - 3(m - l')
32-3tt 2 4
■S
(15)
4 — 7T
so the Mean, Variance, Skewness, and Kurtosis are
I* (16)
(17)
(18)
(19)
a = fi 2 =
fi 3 _ 2(tt - 3)VtF
(4 - tt) 3 / 2
71 = ^3 =
72
^4
■3 =
2(-3tt 2 + 12?r - 8)
(tt-4)*
- y gives
n f 1 <i\ n sinz
dzj z
1/ „( z) = - z »(-i£)
1 d\ n cosz
for n = 0, 1, 2, ... , where j n (z) is a Spherical Bessel
Function of the First Kind and y n {z) is a Spheri-
cal Bessel Function of the Second Kind.
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 439, 1972.
Rayleigh-Ritz Variational Technique
A technique for computing ElGENFUNCTlONS and
EIGENVALUES. It proceeds by requiring
■■/
J= I \p(x)y x 2 - q(x)y 2 ]dx
(1)
to have a STATIONARY VALUE subject to the normaliza-
tion condition
/
(2)
y w(x) dx = 1
and the boundary conditions
py»y\ b a - 0. (3)
This leads to the Sturm-Liouville Equation
d ( dy
which gives the stationary values of
J jrw dx
as
F[y„(x)] = A„,
(4)
(5)
(6)
where X n are the Eigenvalues corresponding to the
ElGENFUNCTION y n .
References
Arfken, G. "Rayleigh-Ritz Variational Technique." §17.8 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 957-961, 1985.
1530 Rayleigh's Theorem
Real Part
Rayleigh's Theorem
see Parseval's Theorem
Re-Entrant Circuit
A Cycle in a Graph which terminates at the starting
point.
see also Cycle (Graph), Eulerian Circuit, Hamil-
tonian Cycle
Real Analysis
That portion of mathematics dealing with functions of
real variables. While this includes some portions of TO-
POLOGY, it is most commonly used to distinguish that
portion of CALCULUS dealing with real as opposed to
Complex Numbers.
Real Axis
see Real Line
Real Function
A Function whose Range is in the Real Numbers is
said to be a real function.
see also COMPLEX FUNCTION, SCALAR FUNCTION,
Vector Function
Pick two real numbers x and y at random in (0, 1) with
a Uniform Distribution. What is the Probability
-Peven that [x/y], where [r] denotes Nint, the nearest
Integer to r, is Even? The answer may be found as
follows (Putnam Exam).
r. / x / h \ f p ( ay < x<b v)
a < — < o ] = J
y
\P(i<v<z)
foJ*dxdy=±(b-a) for0<a<6<l
!ll^ d V dx =h~k for Ka< 6
(1)
^P(0<^<|) + £p( 2 n-l<^<2n + l)
n—l
= * ( *~ 0) + £[2(2n-ir2(2n+i).
OO
= 4 + ^ Un- 1 + 4n-l)
= i + (!-t + 7-§ + ---) = i + (l-tan- 1 l)
= \ ~ \ = |(5 " *) » 46.460%. (2)
Real Line
-1.72 { V2
-+44
ULU
H ►
-10 12 3 4
A Line with a fixed scale so that every Real Number
corresponds to a unique POINT on the LINE. The gen-
eralization of the real line to 2-D is called the Complex
Plane.
see also Abscissa, Complex Plane
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. 57, 1996.
Real Matrix
A Matrix whose elements consist entirely of Real
Numbers.
Real Number
The set of all RATIONAL and IRRATIONAL numbers is
called the real numbers, or simply the "reals," and de-
noted M. The set of real numbers is also called the
Continuum, denoted C.
The real numbers can be extended with the addition of
the Imaginary Number i y equal to y/^1. Numbers of
the form x + iy, where x and y are both real, are then
called Complex Numbers. Another extension which
includes both the real numbers and the infinite ORDINAL
Numbers of Georg Cantor is the Surreal Numbers.
Plouffe's "Inverse Symbolic Calculator" includes a huge
database of 54 million real numbers which are algebraic-
ally related to fundamental mathematical constants and
functions.
see also Complex Number, Continuum, i t Imagi-
nary Number, Integer Relation, Rational Num-
ber, Real Part, Surreal Number
References
Plouffe, S. "Inverse Symbolic Calculator." http://www.cecm.
sfu. ca/projects/ISC/.
Plouffe, S. "Plouffe's Inverter." http://www.lacim.uqam.ca/
P i/.
Putnam Exam. Problem B-3 in the 54th Putnam Exam.
Real Part
The real part 3ft of a Complex NUMBER z = x + iy is
the REAL Number not multiplying i, so R[x + iy] = x.
In terms of z itself,
»[*] = *(*+*•),
where z* is the Complex Conjugate of z.
see also ABSOLUTE SQUARE, COMPLEX CONJUGATE,
Imaginary 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.
Real Polynomial
Reciprocation 1531
Real Polynomial
A Polynomial having only Real Numbers as Coef-
ficients.
see also POLYNOMIAL
Real Projective Plane
The closed topological MANIFOLD, denoted MP 2 , which
is obtained by projecting the points of a plane E from
a fixed point P (not on the plane), with the addition
of the Line at Infinity, is called the real projective
plane. There is then a one-to-one correspondence be-
tween points in E and lines through P. Since each line
through P intersects the sphere S centered at P and
tangent to E in two Antipodal Points, MP 2 can be
described as a Quotient Space of S 2 by identifying any
two such points. The real projective plane is a NONORI-
ENTABLE SURFACE.
The Boy Surface, Cross-Cap, and Roman Surface
are all homeomorphic to the real projective plane and,
because MP 2 is nonorientable, these surfaces contain
self-intersections (Kuiper 1961, Pinkall 1986).
set also Boy Surface, Cross-Cap, Nonorientable
Surface, Projective Plane, Roman Surface
References
Geometry Center. "The Projective Plane." http://www.
geom.umn.edu/zoo/toptype/pplane/.
Gray, A. "Realizations of the Real Projective Plane." §12.5
in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 241-245, 1993.
Klein, F. §1.2 in Vorlesungen ilber nicht-euklidische Geome-
trie. Berlin, 1928.
Kuiper, N. H. "Convex Immersion of Closed Surfaces in i? 3 ."
Comment. Math. Helv. 35, 85-92, 1961.
Pinkall, U. Mathematical Models from the Collections of Uni-
versities and Museums (Ed. G. Fischer). Braunschweig,
Germany: Vieweg, pp. 64-65, 1986.
Real Quadratic Field
A Quadratic Field Q(Vd) with D > 0.
see also Quadratic Field
Realizer
A Set of R of Linear Extensions of a Poset P =
(X, <) is a realizer of P (and is said to realize P) pro-
vided that for all x, y £ X, x < y Iff x is below y in
every member of R.
see also Dominance, Linear Extension, Partially
Ordered Set, Poset Dimension
of n distinct elements (I, a, 6, c, . . . , n), the set of prod-
ucts (a/, a 2 , a&, ac, . . . , an) reproduces the n original dis-
tinct elements in a new order.
see also GROUP
Reciprocal
The reciprocal of a REAL or Complex NUMBER z is
its Multiplicative Inverse 1/z. The reciprocal of a
Complex Number z — x + iy is given by
1 x — iy _ x y
x + ;
x 2 -\-y 2 x 2 + y 2 x 2 + y 2
Reciprocal Difference
The reciprocal differences are closely related to the DI-
VIDED Difference. The first few are explicitly given
by
X - X! , ,
p(x 0i xi) = — — (1)
P2(XQ,XI,X2) =
P3(X0,Xi,X2 y Xz)
fo-fi
Xq — X2
p(XO)Xi) — p(xi,X2)
XQ — Xs
+ /i (2)
pn V^O) 2^1 , . . . , X n )
p2(xo i Xi 1 X2) — p2(xi,X2,X$)
+p(xx,x 2 ) (3)
•EO X-n
pn-l(^0, • • • ,#n-l) — pn-l(&l, • • • j^n)
+pn-2(xu...,X n -l)- (4)
see also BACKWARD DIFFERENCE, CENTRAL DIFFER-
ENCE, Divided Difference, Finite Difference,
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. 878, 1972.
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 443, 1987.
Reciprocal Polyhedron
see Dual Polyhedron
Reciprocating Sphere
see MlDSPHERE
Rearrangement Theorem
Each row and each column in the Group multiplication
table lists each of the GROUP elements once and only
once. From this, it follows that no two elements may
be in the identical location in two rows or two columns.
Thus, each row and each column is a rearranged list of
the GROUP elements. Stated otherwise, given a GROUP
Reciprocation
An incidence-preserving transformation in which points
and lines are transformed into their poles and polars.
A Projective GEOMETRY-like Duality Principle
holds for reciprocation.
References
Coxeter, H. S. M. and Greitzer, S. L. "Reciprocation." §6.1
in Geometry Revisited. Washington, DC: Math. Assoc.
Amer., pp. 132-136, 1967.
1532 Reciprocity Theorem
Rectangle Squaring
Reciprocity Theorem
If there exists a Rational Integer x such that, when
n, p, and q are Positive Integers,
x n = q (mod p) ,
then q is the n-adic reside of p, i.e., q is an n-adic residue
of p IFF x n = q (mod p) is solvable for x.
The first case to be considered was n = 2 (the QUADRA-
TIC Reciprocity Theorem), of which Gauss gave the
first correct proof. Gauss also solved the case n = 3
(Cubic Reciprocity Theorem) using Integers of
the form a + 6p, when p is a root if x 2 -f x + 1 =
and a, 6 are rational Integers. Gauss stated the case
n — 4 (Quartic Reciprocity Theorem) using the
Gaussian Integers.
Proof of n-adic reciprocity for PRIME n was given by
Eisenstein in 1844-50 and by Kummer in 1850-61.
In the 1920s, Artin formulated Artin's RECIPROCITY
Theorem, a general reciprocity law for all orders.
see also Artin Reciprocity, Cubic Reciprocity
Theorem, Langlands Reciprocity, Quadratic
Reciprocity Theorem, Quartic Reciprocity The-
orem, Rook Reciprocity Theorem
Rectangle
A closed planar QUADRILATERAL with opposite sides of
equal lengths a and 6, and with four RIGHT ANGLES.
The AREA of the rectangle is
A — a&,
and its DIAGONALS are of length
p, q = V a 2 + b 2 .
A Square is a degenerate rectangle with a — 6.
see also Golden Rectangle, Perfect Rectangle,
Square
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 122, 1987.
Eppstein, D. "Rectilinear Geometry." http://www.ics.uci.
edu/ -eppstein/ junky ard/rect .html.
Rectangle Function
The rectangle function II(;c) is a function which is
outside the interval [—1,1] and unity inside it. It is
also called the Gate Function, Pulse Function, or
Window Function, and is defined by
U(x).
o
for
for
for
x >
x\<
(1)
The function f(x) = hU((x — c)/b) has height h, center
c, and full-width b. Identities satisfied by the rectangle
function include
Il{x) = H{x+\)-H(x-\) (2)
= H(\+x) + H{\-x)-l (3)
= H(\-x 2 ) (4)
= i[sgn(a:+i)-s g ii(x-i)] > (5)
where H(x) is the Heaviside Step Function. The
Fourier Transform of the rectangle function is given
by
^P(*)]
J — c
e- 2 * ikx Tl{x) dx = sinc(Trfc), (6)
where sinc(z) is the SlNC FUNCTION.
see also Fourier Transform — Rectangle Func-
tion, Heaviside Step Function, Ramp Function
Rectangle Squaring
Given a Rectangle U3BCDE, draw EF = DE on an
extension of BE. Bisect BF and call the MIDPOINT G.
Now draw a SEMICIRCLE centered at G, and construct
the extension of ED which passes through the SEMI-
CIRCLE at H. Then OEKLH has the same Area as
nZlBCDE. This can be shown as follows:
A{UJBCDE) = BEED = BE-EF
= (a + 6)(a-6) = a 2 - b 2 = c 2 .
References
Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1
in Journey Through Genius: The Great Theorems of
Mathematics. New York: Wiley, pp. 13-14, 1990.
Rectangular Coordinates
Rectifying Latitude 1533
Rectangular Coordinates
see Cartesian Coordinates
Rectangular Projection
see Equirectangular Projection
Rectangular Distribution
see Uniform Distribution
Rectangular Hyperbola
A Right Hyperbola of the special form
xy = afc,
so that the Asymptotes are the lines x = and y = 0.
The rectangular hyperbola is sometimes also called an
Equilateral Hyperbola.
see also Hyperbola, Right Hyperbola
References
Courant, R. and Robbins, H. What is Mathematics?: An El-
ementary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 76-77, 1996.
Rectangular Parallelepiped
J-
V
A closed box composed of 3 pairs of rectangular faces
placed opposite each other and joined at RIGHT AN-
GLES to each other. This PARALLELEPIPED therefore
corresponds to a rectangular "box." If the lengths of
the sides are denoted a, 6, and c, then the VOLUME is
V = abc,
the total Surface Area is
A- 2(a6 + 6c + ca),
and the length of the "space" DIAGONAL is
dabc = ya 2 + b 2 + c 2 .
a)
(2)
(3)
If a — b = c, then the rectangular parallelepiped is a
Cube.
see also Cube, Euler Brick, Parallelepiped
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 127, 1987.
Rectifiable Current
The space of currents arising from rectifiable sets by in-
tegrating a differential form is called the space of 2-D
rectifiable currents. For C a closed bounded rectifiable
curve of a number of components in R , C bounds a rec-
tifiable current of least Area. The theory of rectifiable
currents generalizes to ra-D surfaces in K n .
see also INTEGRAL CURRENT, REGULARITY THEOREM
References
Morgan, F. "What is a Surface?" Amer. Math. Monthly 103,
369-376, 1996.
Rectifiable Set
The rectifiable sets include the image of any Lipschitz
FUNCTION / from planar domains into R 3 . The full set
is obtained by allowing arbitrary measurable subsets of
countable unions of such images of Lipschitz functions as
long as the total AREA remains finite. Rectifiable sets
have an "approximate" tangent plane at almost every
point.
References
Morgan, F. "What is a Surface?" Amer. Math. Monthly 103,
369-376, 1996.
Rectification
Rectification is the determination of the length of a
curve.
see also QUADRABLE, SQUARING
Rectifying Latitude
An Auxiliary Latitude which gives a sphere having
correct distances along the meridians. It is denoted /z
(or w) and is given by
7rM
2M "
(1)
M p is evaluated for M at the north pole (4> = 90°), and
M is given by
M
= a(l-e 2 ) /
Jo
(1 - e 2 sin 2 0) 3 /2
e 2 sin <j) d<p ■
e 2 sin <j) cos
\J 1 — e 2 sin 2 <j>
(2)
A series for M is
M = a[(l - he 2 - -he
3 A 5 „6 \ .
i- 64 e ~ 256 e -•••)<?
-(|e 2 + ie 4 + I H s e 6 + ...) S in(2,A)
-(J^e 6 + ...)sm(6<t>) + ...],
(3)
1534 Rectifying Plane
Recurrence Sequence
and a series for fj, is
H = <P-(%e 1 -±e 1 3 + ...)sm(24>)
+ (lfe 1 2 -Me 1 4 + ...)sin(44>)
- (§ ei 3 - . . .) S in(6<« + (fie! 4 - . . .) sin(8tf>) + . . . ,
(4)
where
ei =
_i-vT
1 + vT
The inverse formula is
» = \i + (fei - f| ei 3 + . . .) sin(2/i)
+ (fie 1 2 -§fe 1 4 + ...)sin(4 M )
+ (We 1 3 -...)sin(6 M )
+ (^Z ei 4 -...)sin(8 M ) + ....
(5)
(6)
Recurrence Sequence
A sequence of numbers generated by a RECURRENCE
Relation is called a recurrence sequence. Perhaps
the most famous recurrence sequence is the FIBONACCI
Numbers.
If a sequence {x n } with x\ = x 2 — 1 is described by a
two-term linear recurrence relation of the form
X n — A-Xn — 1 ~T -DXn
(i)
for n > 3 and A and B constants, then the closed form
for x n is given by
a 71 ~/3 n
a- (3
(2)
where a and j3 are the Roots of the Quadratic Equa-
tion
x - Ax ~ B = 0,
(3)
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, pp. 125-
128, 1921.
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 16-17, 1987.
Rectifying Plane
The PLANE spanned by the TANGENT VECTOR T and
Binormal Vector B .
see also BINORMAL VECTOR, TANGENT VECTOR
Recurrence Relation
A mathematical relationship expressing f n as some com-
bination of fi with i < n. The solutions to linear recur-
rence can be computed straightforwardly, but QUAD-
RATIC Recurrences are not so well understood. The
sequence generated by a recurrence relation is called a
Recurrence Sequence. Perhaps the most famous ex-
ample of a recurrence relation is the one denning the
Fibonacci Numbers,
F n — F n -2 + F n ~i
for n > 3 and with Fi = F 2 = 1.
see also Argument Addition Relation, Argu-
ment Multiplication Relation, Clenshaw Recur-
rence Formula, Quadratic Recurrence, Recur-
rence Sequence, Reflection Relation, Transla-
tion Relation
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Recurrence Relations and Clenshaw's Recur-
rence Formula." §5.5 in Numerical Recipes in FORTRAN:
The Art of Scientific Computing, 2nd ed. Cambridge, Eng-
land: Cambridge University Press, pp. 172-178, 1992.
a=\{A+>/A*+AB)
{3= ±(A-y/A*+4B).
The general second-order linear recurrence
X n — AXn-1 + BXn-2
(4)
(5)
(6)
for constants A and B with arbitrary x\ and X2 has
terms
X2 = X2
x$ — Bx\ + Ax2
X4 = Bx 2 + ABxi + A 2 x 2
x$ = B 2 xi + 2ABx 2 + A 2 Bx 1 + A 3 x 2
x 6 = B 2 x 2 + 2AB 2 X! + 3A 2 Bx2 + A 3 Bx x + A 4 x 2 .
Dropping xi, x 2 , and A, this can be written
1
1
B
1
B
B
1
B 2
2B
B
1
B 2
2B 2
3B
B 1,
which is simply PASCAL'S TRIANGLE on its side. An
arbitrary term can therefore be written as
[i(n + fc-2)J\ fc . (n
gm(n + *"2)J\
A K B l
-fc-l)/2j
XZi [n+fc (m ° d 2 )l a ; 2 [ Tl + fe + 1 ( mod 2 )1. (7)
-(Ax, - x 2 ) J2 A 2k -^ 2 B~ k+n - 2 ' k
k=0
n-1
n-k-2
Recurrence Sequence
Recurrence Sequence 1535
The general linear third-order recurrence
X n = AXn-l + BXn-2 + Cx n -3 (9)
has solution
( a~ n . 0~ n
Xn - XI y A + 2aB + 3q2c + A + 2j3B+ 3/3 2 C
+ -Ll )
A + 2<yB + 3>y 2 Cj
^—
A + 2aB +
-(Axi - x 2 )
P 1
3a 2 C A + 2J3B + 3I3 2 B
-(Bxi -h Ax2 - x 3 )
p 2- n
(
\A + 2aB +
A + 2 7 C + 37 2 C
3a 2 C
-,2-n
+ "
+
0-
(10)
A + 2/3£ + 3/3 2 C A + 2 7 £ + 3 7 2 C
where a, j3, and 7 are the roots of the polynomial
Cx 3 + Bx 2 + Ax = 1. (11)
A QUOTIENT-DIFFERENCE Table eventually yields a
line of Os Iff the starting sequence is defined by a linear
recurrence relation.
A linear second-order recurrence
/ n+ l = Xfn + yfn-1 (12)
can be solved rapidly using a "rate doubling,"
fn+2 = (x 2 + 2y)f n - y 2 f n -2, (13)
"rate tripling"
/n+3 = (x z + 3xy)f n + y 3 /n~3, (14)
or in general, "rate fc-tupling" formula
fn + k — Pkfn + qkfn-k, (15)
where
Po - 2 (16)
Pi=x (17)
p* = 2(-y) fc / 2 !T fc (x/(2tVy)) (18)
p k+1 = xp k + ypfc-i (19)
(here, T k (x) is a Chebyshev Polynomial OF THE
First Kind) and
qo = -1
qi = y
qk = -(-y) fc
5fc+i = ~yqk
(20)
(21)
(22)
(23)
(Beeler et al. 1972, Item 14).
Let
s(X) = |}(1 - a,X) n < = 1 - aiX - . . . - a B , (24)
where the generalized POWER sum a(h) for h — 0, 1, . . .
is given by
a{h) = Y^Mh)<*i k ,
(25)
with distinct NONZERO roots a», COEFFICIENTS i4»(/i)
which are Polynomials of degree m - 1 for Positive
INTEGERS n*, and i € [l,Tra]. Then the sequence {a/i}
with a^ = a(ft) satisfies the Recurrence Relation
cth+n = Sidh+Ti-i + * - 4- s n a h (26)
(Meyerson and van der Poorten 1995).
The terms in a general recurrence sequence belong to a
finitely generated Ring over the INTEGERS, so it is im-
possible for every RATIONAL NUMBER to occur in any
finitely generated recurrence sequence. If a recurrence
sequence vanishes infinitely often, then it vanishes on
an arithmetic progression with a common difference 1
that depends only on the roots. The number of values
that a recurrence sequence can take on infinitely often
is bounded by some INTEGER I that depends only on
the roots. There is no recurrence sequence in which
each INTEGER occurs infinitely often, or in which ev-
ery Gaussian Integer occurs (Myerson and van der
Poorten 1995).
Let /x(n) be a bound so that a nondegenerate INTEGER
recurrence sequence of order n takes the value zero at
least fi(n) times. Then /a(2) = 1, /x(3) = 6, and //(4) > 9
(Myerson and van der Poorten 1995). The maximal case
for fi(3) is
a n+3 = 2a n+2 - 4a n +i + 4a„ (27)
with
a = ai = (28)
a 2 = 1. (29)
The zeros are
^0 = a l — a 4 = 0>6 = &13 = &52 = (30)
(Beukers 1991).
see also Binet Forms, Binet's Formula, Fast Fi-
bonacci Transform, Fibonacci Sequence, Lucas
Sequence, Quotient-Difference Table, Skolem-
Mahler-Lerch Theorem
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Beukers, F. "The Zero-Multiplicity of Ternary Recurrences."
Composito Math. 77, 165-177, 1991.
Myerson, G. and van der Poorten, A. J. "Some Problems
Concerning Recurrence Sequences." Amer. Math. Monthly
10? 698-705, 1995.
1536 Recurring Digital Invariant
Red-Black Tree
Recurring Digital Invariant
To define a recurring digital invariant of order fc, com-
pute the sum of the fcth powers of the digits of a number
n. If this number n' is equal to the original number n,
then n = ri is called a /c-Narcissistic NUMBER. If
not, compute the sums of the fcth powers of the digits
of n', and so on. If this process eventually leads back
to the original number n, the smallest number in the se-
quence {n, n\ n", . . .} is said to be a fc-recurring digital
invariant. For example,
55 : 5 3 + 5 3 = 250
250 : 2 3 + 5 3 + 3 = 133
133 : I 3 + 3 3 + 3 3
55,
so 55 is an order 3 recurring digital invariant. The fol-
lowing table gives recurring digital invariants of orders
2 to 10 (Madachy 1979).
Order RDIs
Cycle Lengths
2 4 8
3 55, 136, 160, 919 3, 2, 3, 2
4 1138, 2178 7, 2
5 244, 8294, 8299, 9044, 9045, 28, 10, 6, 10, 22,
10933,24584, 58618, 89883 4, 12, 2, 2
6 17148, 63804, 93531, 239459, 30, 2, 4, 10, 3
282595
7 80441, 86874, 253074, 376762, 92, 56, 27, 30, 14, 21
922428, 982108, five more
8 6822, 7973187, 8616804
9 322219, 2274831, 20700388,
eleven more
10 20818070, five more
see also 196-Algorithm, Additive Persistence,
Digital Root, Digitadition, Happy Number,
Kaprekar Number, Narcissistic Number, Vam-
pire Number
References
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 163-165, 1979.
Recursion
A recursive process is one in which objects are defined in
terms of other objects of the same type. Using some sort
of Recurrence Relation, the entire class of objects
can then be built up from a few initial values and a small
number of rules. The FIBONACCI NUMBERS are most
commonly denned recursively. Care, however, must be
taken to avoid Self-Recursion, in which an object is
defined in terms of itself, leading to an infinite nesting.
see also Ackermann Function, Primitive Recur-
sive Function, Recurrence Relation, Recur-
rence Sequence, Richardson's Theorem, Self-
Recursion, Self-Similarity, TAK Function
References
Buck, R. C. "Mathematical Induction and Recursive Defini-
tions." Amer. Math. Monthly 70, 128-135, 1963.
Knuth, D. E. "Textbook Examples of Recursion." In Ar-
tificial Intelligence and Mathematical Theory of Compu-
tation, Papers in Honor of John McCarthy (Ed. V. Lif-
schitz). Boston, MA: Academic Press, pp. 207-229, 1991.
Peter, R. Rekursive Funktionen. Budapest: Akad. Kiado,
1951.
Recursive Function
A recursive function is a function generated by (1) ADDI-
TION, (2) Multiplication, (3) selection of an element
from a list, and (4) determination of the truth or fal-
sity of the Inequality a < b according to the technical
rules:
1. If F and the sequence of functions Gi, . . . , G n are
recursive, then so is F(Gi t . . . , G n ).
2. If F is a recursive function such that there is an x
for each a with if (a, x) — 0, then the smallest x can
be obtained recursively.
A Turing Machine is capable of computing recursive
functions.
see also TURING MACHINE
References
Kleene, S. C. Introduction to Metamathematics. Princeton,
NJ: Van Nostrand, 1952.
Recursive Monotone Stable Quadrature
A Quadrature (Numerical Integration) algorithm
which has a number of desirable properties.
References
Favati, P.; Lotti, G.; and Romani, F. "Interpolary Integration
Formulas for Optimal Composition." ACM Trans. Math.
Software 17, 207-217, 1991.
Favati, P.; Lotti, G.; and Romani, F. "Algorithm 691: Im-
proving QUADPACK Automatic Integration Routines."
ACM Trans. Math. Software 17, 218-232, 1991.
Red-Black Tree
An extended Binary Tree satisfying the following con-
ditions:
1. Every node has two Children, each colored either
red or black.
2. Every LEAF node is colored black.
3. Every red node has both of its CHILDREN colored
black.
4. Every path from the ROOT to a Leaf contains the
same number (the "black- height" ) of black nodes.
Let n be the number of internal nodes of a red-black
tree. Then the number of red-black trees for n = 1,
2, ... is 2, 2, 3, 8, 14, 20, 35, 64, 122, ... (Sloane's
A001131). The number of trees with black roots and
red roots are given by Sloane's A001137 and Sloane's
A001138, respectively.
Let T h be the Generating Function for the number of
red-black trees of black-height h indexed by the number
of Leaves. Then
T h + 1 (x) = [T h (x)] 2 + [T h (x)] 4
(1)
Red Net
Reeb Foliation
1537
where 2i(x) = x+x 2 . UT(x) is the GENERATING FUNC-
TION for the number of red-black trees, then
T(x) = x 4- x 2 + T(x 2 (l + x) 2 )
(2)
(Ruskey). Let rb(n) be the number of red-black trees
with n LEAVES, r(n) the number of red-rooted trees,
and b(n) the number of black-rooted trees. All three of
the quantities satisfy the RECURRENCE RELATION
R(n)
Ef 2m \
ln-2mj
n/4<n<n/2 v 7
R(m),
(3)
where (£) is a BINOMIAL COEFFICIENT, rb(l) = 1,
rb(2) = 2 for R{n) = rb(n), r(l) = r(3) = 0, r(2) = 1
for #(n) = r(n), and 6(1) = 1 for R(n) = b(n) (Ruskey).
References
Beyer, R. "Symmetric Binary B-Trees: Data Structures and
Maintenance Algorithms." Acta Informal. 1, 290-306,
1972.
Rivest, R. L.; Leiserson, C. E.; and Cormen, R. H. Introduc-
tion to Algorithms. New York: McGraw-Hill, 1990.
Ruskey, F. "Information on Red-Black Trees." http://sue.
esc . uvic . ca/-cos/inf /tree/RedBlackTree .html.
Sloane, N. J. A. Sequences A001131, A001137, and A001138
in "An On-Line Version of the Encyclopedia of Integer Se-
quences."
Red Net
The coloring red of two COMPLETE SUBGRAPHS of n/2
points (for EVEN n) in order to generate a BLUE-EMPTY
Graph.
see also Blue-Empty Graph, Complete Graph
Reduced Amicable Pair
see Quasiamicable Pair
Reduced Fraction
A FRACTION a/b written in lowest terms, i.e., by divid-
ing Numerator and Denominator through by their
Greatest Common Divisor (a, b). For example, 2/3
is the reduced fraction of 8/12.
see also Fraction, Proper Fraction
Reduced Latitude
see Parametric Latitude
Reducible Crossing
A crossing in a LINK projection which can be removed
by rotating part of the Link, also called Removable
Crossing.
see also Alternating Knot
Reducible Representation
see Irreducible Representation
Reducible Matrix
A SQUARE n x n matrix A = aij is called reducible if
the indices 1, 2, . . . , n can be divided into two disjoint
nonempty sets ii, £2, . •
fi + v = n) such that
z M and ji, 72,
j u (with
for a = 1, 2, . . . , /x and /? = 1, 2, . . . , 1/. A SQUARE MA-
TRIX which is not reducible is said to be Irreducible.
see also Square Matrix
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.
Reduction of Order
see Ordinary Differential Equation-
Order
-Second-
Reduction Theorem
If a fixed point is added to each group of a special com-
plete series, then the resulting series is complete.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 253, 1959.
Redundancy
n
R(X U . . .X n ) = Y, H ( X i) ~ H(Xu- * • . x ")i
i-i
where H(xi) is the ENTROPY and H(X l7 * * -,X n ) is the
joint ENTROPY. Linear redundancy is defined as
n
L(Xi,...,X„) = -i5^1n<r 4 ,
t=i
where ai are EIGENVALUES of the correlation matrix.
see also PREDICTABILITY
References
Eraser, A. M. "Reconstructing Attractors from Scalar Time
Series: A Comparison of Singular System and Redundancy
Criteria." Phys. D 34, 391-404, 1989.
Palus, M. "Identifying and Quantifying Chaos by Using
Information-Theoretic Functional. " In Time Series Pre-
diction: Forecasting the Future and Understanding the
Past (Ed. A. S. Weigend and N. A. Gerschenfeld). Proc.
NATO Advanced Research Workshop on Comparative
Time Series Analysis held in Sante Fe, NM, May 14-17,
1992. Reading, MA: Addison- Wesley, pp. 387-413, 1994.
Reeb Foliation
The Reeb foliation of the Hypersphere S 3 is a Folia-
tion constructed as the UNION of two solid Tori with
common boundary.
see also FOLIATION
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 287-288, 1976.
1538
Reef Knot
Reflection Property
Reef Knot
see Square Knot
Refinement
A refinement X of a COVER Y is a COVER such that
every element x € X is a Subset of an element y €Y.
see also Cover
Reflection
The operation of exchanging all points of a mathemati-
cal object with their Mirror IMAGES (i.e., reflections in
a mirror). Objects which do not change HANDEDNESS
under reflection are said to be Amphichiral; those that
do are said to be Chiral.
If the Plane of reflection is taken as the yz-PLANE,
the reflection in 2- or 3-D SPACE consists of making the
transformation x —> —x for each point. Consider an ar-
bitrary point xq and a Plane specified by the equation
ax + by + xz + d = 0.
This Plane has Normal Vector
and the POINT-PLANE DISTANCE is
\ax + by + cz -f d\
D =
vV + b 2 + c 2
(1)
(2)
(3)
The position of the point reflected in the given plane is
therefore given by
Xq = xo — 2Dn
' Xo
_2o
2\ax + by + czq + d\
(4)
see also Amphichiral, Chiral, Dilation, Enan-
tiomer, Expansion, Glide, Handedness, Improper
Rotation, Inversion Operation, Mirror Image,
Projection, Reflection Property, Reflection
Relation, Reflexible, Rotation, Rotoinversion,
Translation
Reflection Property
In the plane, the reflection property can be stated as
three theorems (Ogilvy 1990, pp. 73-77):
1. The LOCUS of the center of a variable CIRCLE, tan-
gent to a fixed Circle and passing through a fixed
point inside that CIRCLE, is an ELLIPSE.
2. If a variable Circle is tangent to a fixed CIRCLE
and also passes through a fixed point outside the
CIRCLE, then the LOCUS of its moving center is a
Hyperbola.
3. If a variable CIRCLE is tangent to a fixed straight line
and also passes through a fixed point not on the line,
then the LOCUS of its moving center is a PARABOLA.
Let a : / -> I 2 be a smooth regular parameterized
curve in M 2 defined on an Open Interval /, and let
F\ and F% be points in P \a{I), where P n is an n-D
Projective Space. Then a has a reflection property
with FOCI F\ and F 2 if, for each point P 6 a(I),
1. Any vector normal to th e cu rve a at P lies in the
Span of the vectors F\P and F 2 A
2. The line normal to a at P bisects one of the pairs of
opposite Angles formed by the intersection of the
lines joining Fi and F 2 to P.
A smooth connected plane curve has a reflection
property Iff it is part of an ELLIPSE, HYPERBOLA,
Parabola, Circle, or straight Line.
Foci
Sign Both foci finite
One focus Both foci
finite oo
distinct + confocal ellipses confocal || lines
parabolas
distinct — confocal hyperbola confocal || lines
and _L bisector parabolas
of interfoci line
segment
equal concentric circles 1 1 lines
Let S € M 3 be a smooth connected surface, and let F\
and F 2 be points in P 3 \S, where P n is an n-D PRO-
JECTIVE SPACE. Then S has a reflection property with
Foci F x and F 2 if, for each point P e S,
1. Any vec tor n orm al to S at P lies in the SPAN of the
vectors F\P and F 2 A
2. The line normal to S at P bisects one of the pairs
of opposite angles formed by the intersection of the
lines joining F\ and F 2 to P.
A smooth connected surface has a reflection property
Iff it is part of an ELLIPSOID of revolution, a Hyper-
BOLOID of revolution, a PARABOLOID of revolution, a
Sphere, or Plane.
Foci
Sign
Both foci finite
One focus
finite
Both foci
oo
distinct
+
confocal ellipsoids
confocal
paraboloids
1 1 planes
distinct
confocal hyperboloids
and plane _L bisector
of interfoci line
segment
confocal
paraboloids
1 1 planes
equal
concentric spheres
|j planes
see also Billiards
References
Drucker, D. "Euclidean Hypersurfaces with Reflective Prop-
erties." Geometrica Dedicata 33, 325-329, 1990.
Drucker, D. "Reflective Euclidean Hypersurfaces." Geomet-
rica Dedicata 39, 361-362, 1991.
Reflection Relation
Regular Isotopy 1539
Drucker, D. "Reflection Properties of Curves and Surfaces."
Math. Mag. 65, 147-157, 1992.
Drucker, D. and Locke, P. "A Natural Classification of Curves
and Surfaces with Reflection Properties." Math. Mag. 69,
249-256, 1996.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 73-77, 1990.
Wegner, B. "Comment on 'Euclidean Hypersurfaces with Re-
flective Properties'." Geometrica Dedicata 39, 357—359,
1991.
Reflection Relation
A mathematical relationship relating f(—x) to f{x).
see also Argument Addition Relation, Argument
Multiplication Relation, Recurrence Relation,
Translation Relation
Reflexible
An object is reflexible if it is superposable with its image
in a plane mirror. Also called AMPHICHIRAL.
see also Amphichiral, Chiral, Enantiomer, Hand-
edness, Mirror Image, Reflection
References
Ball, W. W. R. and Coxeter, H. S. M. "Polyhedra." Ch. 5 in
Mathematical Recreations and Essays, 13th ed. New York:
Dover, p. 130, 1987.
Reflexible Map
An AUTOMORPHISM which interchanges the two vertices
of a regular map at each edge without interchanging the
vertices.
see also EDMONDS' Map
Reflexive Closure
The reflexive closure of a binary RELATION R on a Set
X is the minimal REFLEXIVE RELATION R f on X that
contains R. Thus aR'a for every element a of X and
aR'b for distinct elements a and 6, provided that aRb.
see also Reflexive Reduction, Reflexive Rela-
tion, Relation, Transitive Closure
Reflexive Graph
see Directed Graph
Reflexive Reduction
The reflexive reduction of a binary RELATION R on a
SET X is the minimum relation R' on X with the same
Reflexive Closure as R. Thus aR'b for any elements
a and b of X, provided that a and b are distinct and
aRb.
see also REFLEXIVE CLOSURE, RELATION, TRANSITIVE
Reduction
Reflexive Relation
A Relation R on a Set S is reflexive provided that
xRx for every x in S.
see also RELATION
Reflexivity
A Reflexive Relation.
Region
An open connected set is called a region (sometimes also
called a Domain).
Regression
A method for fitting a curve (not necessarily a straight
line) through a set of points using some goodness-of-
fit criterion. The most common type of regression is
Linear Regression.
see also Least Squares Fitting, Linear Regres-
sion, Multiple Regression, Nonlinear Least
Squares Fitting, Regression Coefficient
References
Kleinbaum, D. G. and Kupper, L. L. Applied Regression
Analysis and Other Multivariable Methods. North Scit-
uate, MA: Duxbury Press, 1978.
Regression Coefficient
The slope b of a line obtained using linear Least
Squares Fitting is called the regression coefficient.
see also Correlation Coefficient, Least Squares
Fitting
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, p. 254, 1951.
Regula Falsi
see False Position Method
Regular Function
see Holomorphic Function
Regular Graph
A Graph is said to be regular of degree r if all Local
Degrees are the same number r. Then
E ■
where E is the number of EDGES. The connected 3-
regular graphs have been determined by G. Brinkman
up to 24 Vertices.
see also COMPLETE GRAPH, COMPLETELY REGULAR
Graph, Local Degree, Superregular Graph
References
Chartrand, G. Introductory Graph Theory.
Dover, p. 29, 1985.
New York:
Regular Isotopy
The equivalence of MANIFOLDS under continuous defor-
mation within the embedding space. KNOTS of opposite
Chirality have Ambient Isotopy, but not regular
isotopy.
see also Ambient Isotopy
1540 Regular Isotopy Invariant
Regular Polyhedron
Regular Isotopy Invariant
see Bracket Polynomial
Regular Local Ring
A regular local ring is a LOCAL RING R with MAXIMAL
IDEAL m so that m can be generated with exactly d ele-
ments where d is the Krull Dimension of the Ring R.
Equivalently, R is regular if the VECTOR SPACE m/m 2
has dimension d,
see also Krull Dimension, Local Ring, Regular
Ring, Ring
References
Eisenbud, D. Commutative Algebra with a View Toward Al-
gebraic Geometry. New York: Springer- Verlag, p. 242,
1995.
Regular Number
A number which has a finite DECIMAL expansion. A
number which is not regular is said to be nonregular.
see also Decimal Expansion, Repeating Decimal
Regular Parameterization
A parameterization of a Surface x(u y v) in u and v is
regular if the Tangent Vectors
— and
ou
<9x
dv
are always Linearly Independent.
Regular Patch
A regular patch is a PATCH x : U — > W 1 for which
the JACOBIAN J(-x)(u 7 v) has rank 2 for all (u t v) € U.
A Patch is said to be regular at a point (uq,vo) 6 U
providing that its JACOBIAN has rank 2 at (uo^vo). For
example, the points at <f> = ±7r/2 in the standard param-
eterization of the SPHERE (cos0sin<£, sin ^ sin 0, cos<f>)
are not regular.
An example of a PATCH which is regular but not IN-
JECTIVE is the CYLINDER defined parametrically by
(cos k, sin it, v) with u € (— oo, oo) and v € (—2, 2). How-
ever, if x : U — > W 1 is an injective regular patch, then x
maps U diffeomorphically onto x(?7).
see also INJECTIVE PATCH, PATCH, REGULAR SURFACE
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 187, 1993.
Regular Point
see Ordinary Point
Regular Polygon
An n-sided POLYGON in which the sides are all the same
length and are symmetrically placed about a common
center. The sum of PERPENDICULARS from any point
to the sides of a regular polygon of n sides is n times
the APOTHEM. Only certain regular polygons are "CON-
structible" with Ruler and Straightedge.
n
Regular Polygon
3
equilateral triangle
4
square
5
pentagon
6
hexagon
7
heptagon
8
octagon
9
nonagon
10
decagon
12
dodecagon
15
pentadecagon
16
hexadecagon
17
heptadecagon
18
octadecagon
20
icosagon
30
triacontagon
see also Constructible Polygon, Geometrogra-
phy, Heptadecagon, Polygon
References
Bishop, W. "How to Construct a Regular Polygon."
Math. Monthly 85, 186-188, 1978.
Amer.
Regular Polyhedron
A polyhedron is said to be regular if its FACES and Ver-
tex Figures are Regular (not necessarily Convex)
polygons (Coxeter 1973, p. 16). Using this definition,
there are a total of nine regular polyhedra, five being
the Convex Platonic Solids and four being the Con-
cave (stellated) Kepler-Poinsot Solids. However,
the term "regular polyhedra" is sometimes used to refer
exclusively to the Convex Platonic Solids.
It can be proven that only nine regular solids (in the
Coxeter sense) exist by noting that a possible regular
polyhedron must satisfy
cos 2 (jj+cos 2 ^j+cos 2 (^)=l.
Gordon showed that the only solutions to
1 + cos <(>i + cos c(>2 + cos <f>z =
of the form 0, = irrrii/ni are the permutations of
(§71-, §71-, |7r) and (§7r, |7r, |7r). This gives three per-
mutations of (3, 3, 4) and six of (3, 5, |) as possible
solutions to the first equation. Plugging back in gives
the SCHLAFLI SYMBOLS of possible regular polyhedra as
{3,3}, {3,4}, {4,3}, {3,5}, {5,3}, {3, f}, {§ ,3}, {5, §},
and {§,5} (Coxeter 1973, pp. 107-109). The first five of
Regular Prime
Regular Surface 1541
these are the PLATONIC SOLIDS and the remaining four
the Kepler-Poinsot Solids.
Every regular polyhedron has e + 1 axes of symmetry,
where e is the number of EDGES, and 3h/2 PLANES of
symmetry, where h is the number of sides of the corre-
sponding Petrie Polygon.
see also Convex Polyhedron, Kepler-Poinsot
Solid, Petrie Polygon, Platonic Solid, Poly-
hedron, Polyhedron Compound, Sponge, Vertex
Figure
References
Coxeter, H. S. M. "Regular and Semi-Regular Polytopes I."
Math. Z. 46, 380-407, 1940.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, pp. 1-17, 93, and 107-112, 1973.
Cromwell, P. R. Polyhedra. New York: Cambridge University
Press, pp. 85-86, 1997.
Regular Prime
A Prime which does not Divide the Class Number
h(p) of the CYCLOTOMIC Field obtained by adjoining
a Primitive pra Root of unity to the rational Field.
A Prime p is regular Iff p does not divide the Nu-
merators of the Bernoulli Numbers B 10 , Bi 2 , . . . ,
i?2 P -2- A Prime which is not regular is said to be an
Irregular Prime.
In 1915, Jensen proved that there are infinitely many
Irregular Primes. It has not yet been proven that
there are an Infinite number of regular primes (Guy
1994, p. 145). Ofthe 283,145 Primes <4x10 6 , 171,548
(or 60.59%) are regular (the conjectured FRACTION is
e _1/2 « 60.65%). The first few are 3, 5, 7, 11, 13, 17,
19, 23, 29, 31, 41, 43, 47, . . . (Sloane's A007703).
see also BERNOULLI NUMBER, FERMAT'S THEOREM, IR-
REGULAR Prime
References
Buhler, J.; Oandall, R. Ernvall, R.; and Metsankyla, T. "Ir-
regular Primes and Cyclotomic Invariants to Four Million."
Math. Comput. 61, 151-153, 1993.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 145, 1994.
Ribenboim, P. "Regular Primes." §5.1 in The New Book
of Prime Number Records. New York: Springer- Verlag,
pp. 323-329, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 153, 1993.
Sloane, N. J. A. Sequence A007703/M2411 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Regular Ring
In the sense of von Neumann, a regular ring is a RING
R such that for all a € R, there exists a b 6 R satisfying
a = aba.
see also Regular Local Ring, Ring
References
Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H,
Freeman, p. 196, 1989.
Regular Sequence
Let there be two PARTICULARLY WELL-BEHAVED
Functions F(x) and p T (x). If the limit
/oo
Pt(x
-oo
)F(x)dx
exists, then p T (x) is a regular sequence of PARTICU-
LARLY Well-Behaved Functions.
Regular Singular Point
Consider a second-order ORDINARY DIFFERENTIAL
Equation
y" + P(x)y'+Q(x)y = 0.
If P{x) and Q(x) remain FINITE at x = Xo, then xo
is called an ORDINARY POINT. If either P(x) or Q(x)
diverges as x — > xo> then xo is called a singular point. If
either P{x) or Q(x) diverges as x — > xo but (x — xo)P(x)
and (x — xo) 2 Q(x) remain Finite as x — > xo, then x =
Xo is called a regular singular point (or NONESSENTIAL
Singularity).
see also IRREGULAR SINGULARITY, 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.
Regular Singularity
see Regular Singular Point
Regular Surface
A Subset M C W 1 is called a regular surface if for each
point p € M, there exists a NEIGHBORHOOD V of p in
W 1 and a Map x : U -> W 1 of a Open Set U C R 2
onto V n M such that
1. x is differentiate,
2. x :U ^V r\M is & HOMEOMORPHISM,
3. Each map x : U -> M is a REGULAR PATCH.
Any open subset of a regular surface is also a regular
surface.
see also REGULAR PATCH
References
Gray, A. "The De0nition of a Regular Surface in R n ." §10.4
in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 195-200, 1993.
1542 Regular Triangle Center
Relation
Regular Triangle Center
A Triangle Center is regular Iff there is a Triangle
Center Function which is a Polynomial in A, a, 6,
and c (where A is the AREA of the TRIANGLE) such that
the Trilinear Coordinates of the center are
/(a,6,c) : /(6,c,a) : /(c,a,6).
The IsOGONAL Conjugate of a regular center is a regu-
lar center. Furthermore, given two regular centers, any
two of their HARMONIC CONJUGATE POINTS are also
regular centers.
see also ISOGONAL CONJUGATE, TRIANGLE CENTER,
Triangle Center Function
Regularity Theorem
An AREA-minimizing surface (Rectifiable Current)
bounded by a smooth curve in R 3 is a smooth subman-
ifold with boundary.
see also Minimal Surface, Rectifiable Current
References
Morgan, F. "What is a Surface?" Amer. Math. Monthly 103,
369-376, 1996.
Regularized Beta Function
The regularized beta function is defined by
I(z;a,b)
B{z;a,b)
B{a,b) '
where B(z;a,b) is the incomplete Beta Function and
B(a,b) is the complete Beta Function.
see also Beta Function, Regularized Gamma
Function
Regularized Gamma Function
The regularized gamma functions are defined by
P(a,z) = l-Q(a,z) =
7(<*i*)
T(a)
and
g(a,z) = l-P(a,z) = ^i) >
where 7(0, z) and F(a, z) are incomplete Gamma Func-
tions and T(a) is a complete GAMMA Function. Their
derivatives are
A.
dz
A
dz
P(a i z) = e~ z z a - 1
Q{a,z)
— z a — 1
e z
see also Gamma Function, Regularized Beta
Function
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 160—161, 1992.
Regulus
The locus of lines meeting three given Skew LINES.
( "Regulus" is also the name of the brightest star in the
constellation Leo.)
Reidemeister Moves
twist
untwist
ii.
m.
poke
unpoke
slide -^ |
In the 1930s, Reidemeister first rigorously proved that
Knots exist which are distinct from the UNKNOT. He
did this by showing that all KNOT deformations can be
reduced to a sequence of three types of "moves," called
the (I) Twist Move, (II) Poke Move, and (III) Slide
Move.
Reidemeister'S Theorem guarantees that moves I, II,
and III correspond to Ambient Isotopy (moves II and
III alone correspond to REGULAR ISOTOPY). He then
defined the concept of COLORABILITY, which is invariant
under Reidemeister moves.
see also Ambient Isotopy, Colorable, Markov
Moves, Regular Isotopy, Unknot
Reidemeister's Theorem
Two Links can be continuously deformed into each
other Iff any diagram of one can be transformed into
a diagram of the other by a sequence of REIDEMEISTER
Moves.
see also Reidemeister Moves
Reinhardt Domain
A Reinhardt domain with center c is a DOMAIN D in
C n such that whenever D contains Zo, the DOMAIN D
also contains the closed POLYDISK.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 101, 1980.
Relation
A relation is any Subset of a Cartesian Product.
For instance, a SUBSET of A x B, called a (binary) "re-
lation from A to £," is a collection of Ordered Pairs
(a, b) with first components from A and second compo-
nents from £, and, in particular, a Subset of A x A is
called a "relation on A" For a binary relation R, one
often writes aRb to mean that (a, b) is in R.
Relative Error
Remainder Theorem 1543
see also Adjacency Relation, Antisymmetric Re-
lation, Argument Addition Relation, Argu-
ment Multiplication Relation, Cover Relation,
Equivalence Relation, Irreflexive, Partial Or-
der, Recurrence Relation, Reflection Rela-
tion, Reflexive Relation, Symmetric Relation,
Transitive, Translation Relation
Relative Error
Let the true value of a quantity be x and the measured
or inferred value xo. Then the relative error is defined
/\X Xq — X Xq
Sx =
1,
where Ax is the ABSOLUTE ERROR. The relative error
of the Quotient or Product of a number of quantities
is less than or equal to the SUM of their relative errors.
The Percentage Error is 100% times the relative
error.
see also Absolute Error, Error Propagation,
Percentage 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.
Relative Extremum
A Relative Maximum or Relative Minimum, also
called a Local EXTREMUM.
see also EXTREMUM, GLOBAL EXTREMUM, RELATIVE
Maximum, Relative Minimum
Relative Maximum
A Maximum within some Neighborhood which need
not be a GLOBAL Maximum.
see also Global Maximum, Maximum, Relative
Minimum
Relative Minimum
A Minimum within some Neighborhood which need
not be a Global Minimum.
see also GLOBAL MINIMUM, MINIMUM, RELATIVE MAX-
IMUM
Relatively Prime
Two integers are relatively prime if they share no com-
mon factors (divisors) except 1. Using the notation
(m,n) to denote the Greatest Common Divisor,
two integers m and n are relatively prime if (m,n) =
1. Relatively prime integers are sometimes also called
STRANGERS or COPRIME and are denoted m In.
The probability that two Integers picked at random
are relatively prime is [C(2)] _1 — 6/71- 2 , where C,{z) is
the Riemann Zeta Function. This result is related
to the fact that the Greatest Common Divisor of m
and n, (m,n) = A;, can be interpreted as the number of
Lattice Points in the Plane which lie on the straight
LINE connecting the VECTORS (0,0) and (m,n) (exclud-
ing (m,n) itself). In fact Q/ir 2 the fractional number of
Lattice Points Visible from the Origin (Castellanos
1988, pp. 155-156).
Given three INTEGERS chosen at random, the probabil-
ity that no common factor will divide them all is
[C(3)] _1 wl.202" 1 =0.832...,
where £(3) is Apery's CONSTANT. This generalizes to
k random integers (Schoenfeld 1976).
see also DIVISOR, GREATEST COMMON DIVISOR, VISI-
BILITY
References
Castellanos, D. "The Ubiquitous Pi." Math. Mag. 61, 67-98,
1988.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, pp. 3-4, 1994.
Schoenfeld, L. "Sharper Bounds for the Chebyshev Functions
0(x) and $(x) t II." Math. Comput. 30, 337-360, 1976.
Relaxation Methods
Methods of solving an Ordinary Differential Equa-
tion by replacing it with a FINITE DIFFERENCE equa-
tion on a regular grid spanning the domain of interest.
The finite difference equations are then solved using an
n-D Newton's Method or other similar algorithm.
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Richardson Extrapolation and the Bulirsch-
Stoer Method." §17.3 in Numerical Recipes in FORTRAN:
The Art of Scientific Computing, 2nd ed. Cambridge, Eng-
land: Cambridge University Press, pp. 753-763, 1992.
Remainder
In general, a remainder is a quantity "left over" after
performing a particular algorithm. The term is most
commonly used to refer to the number left over when two
integers are divided by each other in Integer Division.
For example, 55\7 = 7, with a remainder of 6, Of course
in real division, there is no such thing as a remainder
since, for example, 55/7 = 7 4- 6/7.
The term remainder is also sometimes to the Residue
of a Congruence.
see also DIVISION, INTEGER DIVISION, RESIDUE (CON-
GRUENCE)
Remainder Theorem
see Polynomial Remainder Theorem
1544
Rembs 7 Surfaces
Renyi's Parking Constants
Rembs' Surfaces
A special class of Enneper's Surfaces which can be
given parametrically by
x = a(U cosu — U sinu)
y = —a(U sin u -f- U' cos u)
z = v — aV ,
where
U =
v =
cosh(u\/C)
7c
cos(vVC + 1 )
VcTi
2V
(c + i)(u 2 -v 2 y
The value of v is restricted to
|v| < Vo =
2VC+T
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(Reckziegel 1986), and the values v - ±Vo correspond
to the ends of the cleft in the surface.
see also Enneper's Surfaces, Kuen Surface, Siev-
ert's Surface
References
Fischer, G. (Ed.). Plate 88 in Mathematische Mod-
elle/ 'Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 84, 1986.
Reckziegel, H. "Sievert's Surface." §3.4.4.3 in Mathemati-
cal Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 39-40, 1986.
Rembs, E. "Enneper'sche Flachen konstanter posi-
tiver Krummung und Hazzidakissche Transformationen."
Jahrber. DMV 39, 278-283, 1930.
Removable Crossing
see Reducible Crossing
Removable Singularity
A Singular Point z of a Function f(z) for which
it is possible to assign a Complex Number in such a
way that f(z) becomes Analytic. A more precise way
of defining a removable singularity is as a Singularity
zo of a function f(z) about which the function f(z) is
bounded. For example, the point xo = is a removable
singularity in the Sinc Function sine a: = sinx/x, since
this function satisfies sincO = 1.
Rencontres Number
see Derangement, Subfactorial
Rendezvous Values
see Magic Geometric Constants
Renyi's Parking Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Given the Closed Interval [0, x] with x > 1, let 1-D
"cars" of unit length be parked randomly on the interval.
The MEAN number M(x) of cars which can fit (without
overlapping!) satisfies
M(i)= {^/r
for < x < 1
M{y)dy for x > 1.
The mean density of the cars for large x is
M(x)
(1)
m = lim
x— )-oo X
0.7475979203
,oo / /■* ! _ e -v \
I exp —2 / dy I dx
Jo \ Jo y J
Furthermore,
M(x) = mx + m - 1 + Q(x~ n )
(2)
(3)
for all n (Renyi 1958), which was strengthened by
Dvoretzky and Robbins (1964) to
M(x) = mx + m-\ +
2e\ x ~ 3/2
(I)
(4)
Dvoretzky and Robbins (1964) also proved that
inf «±1<™< sup m+1. (5 )
c<t<x+l t+1 ~ ~ x <t<x + l t+1
Let V(x) be the variance of the number of cars, then
Dvoretzky and Robbins (1964) and Mannion (1964)
showed that
_ ,. V(x)
v = Um
x— too X
f{-jf
J e— '^(y)* 1
x exp
e-**R % (y)dy + i
.Jo J
dx = 0.038156 ... , (6)
where
Rx (x) = M(x) - mx -m+1 (7)
f (1 - m - mx) 2 for < x < 1
4(1 -m) 2 for x = 1
, +/ :c " 1 ^i(y)^i(^-y-i)rf2/] J
(8)
Rep-Tile
Repunit 1545
and the numerical value is due to Blaisdell and Solomon
(1970). Dvoretzky and Robbins (1964) also proved that
. r V(t) <, <, V(t)
inf : < v < sup
x<t<x+l t+1
x<t<x + l
t + V
and that
V(x) = vx + v + G
(-
V x
4e\ a
(9)
(10)
Palasti (1960) conjectured that in 2-D,
M(x,y) 2
lim = m ,
x,y^oo Xy
(ii)
but this has not yet been proven or disproven (Finch).
References
Blaisdell, B. E. and Solomon, H. "On Random Sequential
Packing in the Plane and a Conjecture of Palasti." J. AppL
Prob. 7, 667-698, 1970.
Dvoretzky, A. and Robbins, H. "On the Parking Problem."
Publ. Math. Inst. Hung. Acad. Sci. 9, 209-224, 1964.
Finch, S. "Favorite Mathematical Constants." http://vwv.
mathsof t . com/asolve/constant/renyi/renyi . html.
Mannion, D. "Random Space-Filling in One Dimension."
Publ. Math. Inst. Hung. Acad. Sci. 9, 143-154, 1964.
Palasti, I. "On Some Random Space Filling Problems." Publ.
Math. Inst. Hung. Acad. Sci. 5, 353-359, 1960.
Renyi, A. "On a One-Dimensional Problem Concerning Ran-
dom Space- Filling." Publ. Math. Inst. Hung. Acad. Sci.
3, 109-127, 1958.
Solomon, H. and Weiner, H. J. "A Review of the Packing
Problem." Comm. Statist. Th. Meth. 15, 2571-2607,
1986.
Rep-Tile
A POLYGON which can be divided into smaller copies of
itself.
see also DISSECTION
References
Gardner, M. Ch. 19 in The Unexpected Hanging and Other
Mathematical Diversions. Chicago, IL: Chicago University
Press, 1991.
Repartition
see Adele
Repeating Decimal
A number whose decimal representation eventually be-
comes periodic (i.e., the same sequence of digits repeats
indefinitely) is called a repeating decimal. Numbers
such as 0.5 can be regarded as repeating decimals since
0.5 = 0.5000 . . . = 0.4999 . . .. All RATIONAL NUMBERS
have repeating decimals, e.g., 1/11 = 0.09. However,
Transcendental Numbers, such as n — 3.141592 . . .
do not.
see also CYCLIC NUMBER, DECIMAL EXPANSION, FULL
Reptend Prime, Irrational Number, Midy's The-
orem, Rational Number, Regular Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 53-54,
1987.
Courant, R. and Robbins, H. "Rational Numbers and Peri-
odic Decimals." §2.2.4 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 66-68, 1996.
Repfigit Number
see Keith Number
Replicate
One out of a set of identical observations in a given
experiment under identical conditions.
Reptend Prime
see Full Reptend Prime
Representation
The representation of a Group G on a Complex Vec-
tor Space V is a group action of G on V by linear
transformations. Two finite dimensional representations
7r on V and n on V' are equivalent if there is an invert-
ible linear map E : V *-> V r such that Tv'{g)E = En(g)
for all g £ G. 7T is said to be irreducible if it has no
proper NONZERO invariant SUBSPACES.
see also CHARACTER (MULTIPLICATIVE), PETER- WEYL
Theorem, Primary Representation, Schur's
Lemma
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Repdigit
A number composed of a single digit is called a repdigit.
If the digits are all Is, the repdigit is called a Repunit.
The Beast Number 666 is a repdigit.
see also Keith Number, Repunit
Repunit
A (generalized) repunit to the base 6 is a number of the
form
71 b - 1 "
The term "repunit" was coined by Beiler (1966), who
also gave the first tabulation of known factors. Repunit s
M n = Ml = 2 n - 1 with b = 2 are called MERSENNE
1546 Repunit
Residual vs. Predictor Plot
Numbers. If b = 10, the number is called a repunit
(since the digits are all Is). A number of the form
Rn —
10 n - 1
10- 1
Rn
10 n - 1
2
000225
3
003462
4
002450
5
003463
6
003464
7
023000
8
023001
9
002452
10
002275
11
016123
12
016125
is therefore a (decimal) repunit of order n.
b Sloane 6-Repunits
1.3, 7, 15, 31,63, 127, ...
1.4, 13,40, 121,364, ...
1, 5, 21,85, 341, 1365, ...
1, 6, 31, 156, 781,3906, ...
1, 7, 43, 259, 1555, 9331, ...
1, 8, 57, 400, 2801, 19608, . . .
1, 9, 73, 585, 4681, 37449, ...
1, 10, 91, 820, 7381,66430, ...
1, 11, 111, 1111, 11111, ...
1, 12, 133, 1464, 16105, 177156, ...
1, 13, 157, 1885, 22621, 271453, ...
Williams and Seah (1979) factored generalized repunits
for 3 < b < 12 and 2 < n < 1000. A (base-10) re-
punit can be PRIME only if n is PRIME, since other-
wise 10 ab - 1 is a Binomial Number which can be fac-
tored algebraically. In fact, if n = 2a is Even, then
10 2a - 1 = (10 a - l)(10 a + 1). The only base-10 repunit
PRIMES R n for n < 16,500 are n = 2, 19, 23, 317, and
1031 (Sloane's A004023; Madachy 1979, Williams and
Dubner 1986, Ball and Coxeter 1987). The number of
factors for the base-10 repunits for n = 1, 2, . . . are 1,
1, 2, 2, 2, 5, 2, 4, 4, 4, 2, 7, 3, . . , (Sloane's A046053).
b Sloane n of Prime b- Repunits
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, . . .
3, 7, 13, 71, 103, 541, 1091, 1367, ...
3, 7, 11, 13, 47, 127, 149, 181, 619, . . .
2, 3, 7, 29, 71, 127, 271, 509, 1049, . . .
5, 13, 131, 149, 1699, ...
2, 19, 23, 317, 1031, ...
17, 19, 73, 139, 907, 1907, 2029, 4801, ...
2, 3, 5, 19, 97, 109, 317, 353, 701, . . .
A table of the factors not obtainable algebraically
for generalized repunits (a continuously updated ver-
sion of Brillhart et al. 1988) is maintained on-
line. These tables include factors for 10 n — 1
(with n < 209 odd) and 10 n + 1 (for n < 210
Even and Odd) in the files ftp://sable.ox.ac.uk/
pub /math/ Cunningham/ 10- and ftp://sable.ox.ac.
uk/pub /math/ Cunningham/ 10+. After algebraically fac-
toring i? n , these are sufficient for complete factoriza-
tions. Yates (1982) published all the repunit factors for
n < 1000, a portion of which are reproduced in the
Mathematical® notebook by Weisstein.
A Smith Number can be constructed from every fac-
tored repunit.
2
000043
3
028491
5
004061
6
004062
7
004063
10
004023
11
005808
12
004064
see also Cunningham Number, Fermat Number,
Mersenne Number, Repdigit, Smith Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 66, 1987.
Beiler, A. H. "11111. .. 111." Ch. 11 in Recreations in the
Theory of Numbers: The Queen of Mathematics Enter-
tains. New York: Dover, 1966.
Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.;
and Tuckerman, B. Factorizations of b n ± 1, 6 = 2,
3,5,6,7,10,11,12 Up to High Powers, rev. ed. Provi-
dence, RI: Amer. Math. Soc, 1988. Updates are avail-
able electronically from ftp://sable.ox.ac.uk/pub/math/
Cunningham.
Dubner, H. "Generalized Repunit Primes." Math. Comput.
61, 927-930, 1993.
Guy, R. K. "Mersenne Primes. Repunits. Fermat Numbers.
Primes of Shape k • 2 n + 2." §A3 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 8-13, 1994.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 152-153, 1979.
Ribenboim, P. "Repunits and Similar Numbers." §5.5 in The
New Booh of Prime Number Records. New York: Springer-
Verlag, pp. 350-354, 1996.
Snyder, W. M. "Factoring Repunits." Am. Math. Monthly
89, 462-466, 1982.
Sorli, R. "Factorization Tables." http://www.maths.uts,
edu.au/staff/ron/fact/fact.html.
# Weisstein, E. W. "Repunits." http : //www . astro .
virginia.edu/-eww6n/math/notebooks/Repunit .m.
Williams, H. C. and Dubner, H. "The Primality of #1031."
Math. Comput. 47, 703-711, 1986.
Williams, H. C. and Seah, E. "Some Primes of the Form
(a n - l)/(a - 1). Math. Comput. 33, 1337-1342, 1979.
Yates, S. "Prime Divisors of Repunits." J. Recr. Math. 8,
33-38, 1975.
Yates, S. "The Mystique of Repunits." Math. Mag. 51, 22-
28, 1978.
Yates, S. Repunits and Reptends. Delray Beach, FL: S. Yates,
1982.
Residual
The residual is the sum of deviations from a best-fit
curve of arbitrary form.
R = ^pi ~ f(xi,a u ...,a n )] 2 .
The residual should not be confused with the CORRE-
LATION Coefficient.
Residual vs. Predictor Plot
A plot of yi vs. e* = & — yi. Random scatter indicates
the model is probably good. A pattern indicates a prob-
lem with the model. If the spread in e* increases as yi
increases, the errors are called HETEROSCEDASTIC.
Residue Class
Residue Class
The residue classes of a function f(x) mod n are all pos-
sible values of the RESIDUE f(x) (mod n). For example,
the residue classes of x 2 (mod 6) are {0, 1, 3, 4}, since
2 == (mod 6)
l 2 = 1 (mod 6)
3 2 == 3 (mod 6)
4 2 = 4 (mod 6)
are all the possible residues. A Complete Residue
System is a set of integers containing one element from
each class, so in this case, {0, 1,9,4} would be a Com-
plete Residue System.
The 0(m) residue classes prime to m form a GROUP un-
der the binary multiplication operation (mod ra), where
<j>(m) is the TOTIENT FUNCTION (Shanks 1993) and the
Group is classed a Modulo Multiplication Group.
see also Complete Residue System, Congruence,
Cubic Number, Quadratic Reciprocity Theo-
rem, Quadratic Residue, Residue (Congruence),
Square Number
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 56 and 59-63, 1993.
Residue (Complex Analysis)
The constant a_i in the Laurent Series
Residue Theorem (Complex Analysis) 1547
37 i3 ^ 37 i+4+s _ 3 (_4)(_ 1 ) = 12 ( mod 17).
see also COMMON RESIDUE, CONGRUENCE, MINIMAL
Residue
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 55-56, 1993.
Residue Index
p - 1 divided by the Haupt-Exponent of a base b mod
p for a given Prime p.
see also Haupt-Exponent
Residue Theorem (Complex Analysis)
Given a complex function /(z), consider the Laurent
Series
f(z) = ^ &n(z- Z ) n
of f(z) is called the residue of f(z). The residue is a
very important property of a complex function and ap-
pears in the amazing Residue Theorem of Contour
Integration.
see also Contour Integration, Laurent Series,
Residue Theorem
References
Arfken, G. "Calculus of Residues." §7.2 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 400-421, 1985.
Residue (Congruence)
The number b in the CONGRUENCE a = b (mod m) is
called the residue of a (mod m). The residue of large
numbers can be computed quickly using CONGRUENCES.
For example, to find 37 13 (mod 17), note that
37 = 3
37 2 = 3 2 = 9 = -8
37 4 = 81 = -4
37 8 = 16 = -1,
/(*)= J2 m* -*>) n -
(1)
Integrate term by term using a closed contour 7 encir-
cling Zq,
/ f(z) dz = 2_J a n I (z - z ) n dz
= 2_\ a n (z ~ Zo) n dz
n= — oo "*
+a_i / -~ + ]T a n f(z - zo) n dz. (2)
J-r Z Z ° n=0 A
The Cauchy Integral Theorem requires that the
first and last terms vanish, so we have
/ f(z)dz = a-i /
J *y J y
dz
zq
(3)
But we can evaluate this function (which has a POLE at
20) using the Cauchy Integral Formula,
/(*>)
2?ri /
f(z) dz
Zo
(4)
This equation must also hold for the constant function
f(z) — 1, in which case it is also true that f(zo) = 1, so
2-iri J z — zq '
/ = 27T2,
(5)
(6)
1548 Residue Theorem (Complex Analysis)
Resolution
and (3) becomes
/
f(z) dz = 27rza_i.
(7)
The quantity a_i is known as the RESIDUE of f(z) at zq.
Generalizing to a curve passing through multiple poles,
(7) becomes
/
poles in 7
f(z)dz = 2iri ]T nfr.^V-'i.
(8)
where n is the WINDING NUMBER and the (i) superscript
denotes the quantity corresponding to Pole i.
If the path does not completely encircle the RESIDUE,
take the CAUCHY PRINCIPAL VALUE to obtain
/•
f(z)dz = (<9 2 -<9i)m_i.
If / has only ISOLATED SINGULARITIES, then
4 ;) eC*
(9)
(10)
The residues may be found without explicitly expanding
into a Laurent Series as follows.
f(z)= J2 «»(*-*>)"•
(11)
If f(z) has a POLE of order m at Zo, then a n = for
n < — m and a_ m ^ 0. Therefore,
oo oo
/(*)= y. an{z-zo) n = Y, a - m +"( z - z °y m+n
(z - z ) m f(z) = 2ja_ m+n (z - z ) n
(12)
(13)
-[(z - z ) m f(z)} = £na_ m+n (z - zoT~ l
n=0
oo
= 2_j na ~ rn+n ^ z ~ z °) n ~ 1
n-l
oo
= ^(n + l)a_ m+ri+1 (z - z ) n (14)
n=0
-£;[(* ~ zorf(z)} = J>(n + l)a_ m+n+1 (z - zo)^ 1
n=0
oo
= ^2n(n + l)a_ m+n+ i(^-^o) n ~ 1
n-l
oo
= ]P(n + l)(n + 2)a_ m+n+2 (z - z ) n . (15)
Iterating,
jm-l
oo
= J^(n + l)(n + 2)(n + m - l)a n -i(* - 2o) n
71 =
= (m — l)!a_i
oo
+ ]P(n + l)(n 4- 2)(n + m - l)a„_i(* - zo)"' 1 . (16)
So
J7TI — 1
lim —
z— ►zo dz'
jKz-zorm]
= lim (m - l)!a_i + = (m - l)!a_i, (17)
Z— J-ZQ
and the RESIDUE is
1
a-i
(m-iy.dz 1
jm-l
-[(z- Z0 ) m /W], =Z0 . (18)
This amazing theorem says that the value of a CONTOUR
Integral in the Complex Plane depends only on the
properties of a few special points inside the contour.
see also Cauchy Integral Formula, Cauchy Inte-
gral Theorem, Contour Integral, Laurent Se-
ries, Pole, Residue (Complex Analysis)
Residue Theorem (Group)
If two groups are residual to a third, every group residual
to one is residual to the other. The Gambier extension of
this theorem states that if two groups are pseudoresidual
to a third, then every group pseudoresidual to the first
with an excess greater than or equal to the excess of the
first minus the excess of the second is pseudoresidual to
the second, with an excess > 0.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, pp. 30-31, 1959.
Resolution
Resolution is a widely used word with many different
meanings. It can refer to resolution of equations, reso-
lution of singularities (in ALGEBRAIC Geometry), reso-
lution of modules or more sophisticated structures, etc.
In a Block Design, a Partition R of a BIBD's set
of blocks B into PARALLEL CLASSES, each of which in
turn partitions the set V, is called a resolution (Abel
and Furino 1996).
A resolution of the Module M over the Ring R is a
complex of .R-modules d and morphisms di and a MOR-
PHISM e such that
Ci-l
Co -> e M ->
Resolution Class
Resonance Overlap Method 1549
satisfying the following conditions:
1. The composition of any two consecutive morphisms
is the zero map,
2. For all i, (ker di) / (im di+i) = 0,
3. C /(kere)-M,
where ker is the kernel and im is the image. Here, the
quotient
(ker di)
(imdi+i)
is the ith HOMOLOGY GROUP.
If all modules C\ are projective (free), then the resolu-
tion is called projective (free). There is a similar concept
for resolutions "to the right" of M, which are called in-
jective resolutions.
see also HOMOLOGY GROUP, MODULE, MORPHISM,
Ring
References
Abel, R. J. R. and Purino, S. C. "Resolvable and Near Re-
solvable Designs." §1.6 in The CRC Handbook of Combi-
natorial Designs (Ed. C. J. Colbourn and J. H. Dinitz).
Boca Raton, FL: CRC Press, p. 4 and 87-94, 1996.
Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H.
Freeman, p. 339, 1989.
Resolution Class
see Parallel Class
Resolution Modulus
The least Positive Integer m* with the property that
x{y) = 1 whenever y = 1 (mod m*) and (j/, m) = 1.
Resolvable
A balanced incomplete Block Design (B y V) is called
resolvable if there exists a Partition R of its set of
blocks B into PARALLEL CLASSES, each of which in turn
partitions the set V. The partition R is called a Reso-
lution.
see also Block Design, Parallel Class
References
Abel, R. J. R. and Purino, S. C. "Resolvable and Near Re-
solvable Designs." §1.6 in The CRC Handbook of Combi-
natorial Designs (Ed. C. J. Colbourn and J. H. Dinitz).
Boca Raton, FL: CRC Press, p. 4 and 87-94, 1996.
Resolving Tree
A tree of LINKS obtained by repeatedly choosing a cross-
ing, applying the SKEIN RELATIONSHIP to obtain two
simpler Links, and repeating the process. The Depth
of a resolving tree is the number of levels of links, not in-
cluding the top. The Depth of the Link is the minimal
depth for any resolving tree of that Link.
Resonance Overlap
Isolated resonances in a Dynamical System can
cause considerable distortion of preserved TORI in their
Neighborhood, but they do not introduce any Chaos
into a system. However, when two or more resonances
are simultaneously present, they will render a system
nonintegrable. Furthermore, if they are sufficiently
"close" to each other, they will result in the appearance
of widespread (large-scale) CHAOS.
To investigate this problem, Walker and Ford (1969)
took the integrable Hamiltonian
H (I U I 2 ) = h + h - /? - 3/i/ 2 + h 2
and investigated the effect of adding a 2:2 resonance and
a 3:2 resonance
H(I,0) = Ho (I) + a/1/2 cos(26'i - 2<9 2 )
4-/?/i 3/2 / 2 cos(20i-30 2 ).
At low energies, the resonant zones are well-separated.
As the energy increases, the zones overlap and a "macro-
scopic zone of instability" appears. When the overlap
starts, many higher-order resonances are also involved
so fairly large areas of PHASE SPACE have their TORI
destroyed and the ensuing CHAOS is "widespread" since
trajectories are now free to wander between regions that
previously were separated by nonresonant TORI.
Walker and Ford (1969) were able to numerically pre-
dict the energy at which the overlap of the resonances
first occurred. They plotted the 02-axis intercepts of
the inner 2:2 and the outer 2:3 separatrices as a func-
tion of total energy. The energy at which they crossed
was found to be identical to that at which 2:2 and 2:3
resonance zones began to overlap.
see also Chaos, Resonance Overlap Method
References
Walker, G. H. and Ford, J. "Amplitude Instability and Er-
godic Behavior for Conservative Nonlinear Oscillator Sys-
tems." Phys. Rev. 188, 416-432, 1969.
Resonance Overlap Method
A method for predicting the onset of widespread CHAOS.
see also GREENE'S METHOD
References
Chirikov, B. V. "A Universal Instability of Many-
Dimensional Oscillator Systems." Phys. Rep, 52, 264-379,
1979.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, pp. 154-163, 1989.
1550 Restricted Divisor Function
Restricted Divisor Function
50 100 150 200 250 300
The sum of the Aliquot Divisors of n, given by
s(n) = o~(n) — n,
where a(n) is the Divisor Function. The first few
values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... (Sloane's
A001065).
see also DIVISOR FUNCTION
References
Sloane, N. J. A. Sequence A001065/M2226 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Restricted Growth Function
see Restricted Growth String
Restricted Growth String
For a Set Partition of n elements, the n-character
string aia2...a n in which each character gives the
BLOCK (Bo, Bi, ...) into which the corresponding el-
ement belongs is called the restricted growth string (or
sometimes the RESTRICTED GROWTH FUNCTION). For
example, for the Set Partition {{1},{2},{3,4}}, the
restricted growth string would be 0122. If the BLOCKS
are "sorted" so that a\ = 0, then the restricted growth
string satisfies the INEQUALITY
flt+i < 1 + max{ai, a2, . . . , a;}
for t = l, 2, ...,n — 1 .
References
Ruskey, F. "Info About Set Partitions." http://sue . esc .
uvic.ca/~cos/inf/setp/SetPartitions.html.
Resultant
Given a Polynomial p(x) of degree n with roots ai,
i = 1, . . . , n and a Polynomial q(x) of degree m with
roots /3j, j = 1, . . . , m, the resultant is defined by
R{p,q) = YlY[{Pi-ai).
t=l j=l
There exists an ALGORITHM similar to the Euclid-
ean Algorithm for computing resultants (Pohst and
Zassenhaus 1989). The resultant is the DETERMINANT
Reuleaux Triangle
of the corresponding Sylvester Matrix. Given p and
q, then
h{x) = R(q(t),p{x-t))
is a Polynomial of degree mn ) having as its roots all
sums of the form oci + (3j .
see also DISCRIMINANT (POLYNOMIAL), SYLVESTER
Matrix
References
Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Num-
ber Theory. Cambridge, England: Cambridge University
Press, 1989.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 348, 1991.
Retardance
A shift in PHASE.
see also PHASE
Reuleaux Polygon
A curvilinear polygon built up of circular ARCS. The
Reuleaux polygon is a generalization of the REULEAUX
Triangle.
see also Curve of Constant Width, Reuleaux Tri-
angle
References
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 52-54, 1991.
Reuleaux Triangle
A Curve of Constant Width constructed by drawing
arcs from each Vertex of an Equilateral Triangle
between the other two VERTICES. It is the basis for the
Harry Watt square drill bit. It has the smallest AREA
for a given width of any Curve of Constant Width.
The Area of each meniscus-shaped portion is
A = §7rr
VS \ (it v/3\ 2 m
where we have subtracted the AREA of the wedge from
that of the Equilateral Triangle. The total Area
is then
^- 3I 6 4
V3
2 , VO 2
r -\ — r =
4
TV- V^
(2)
When rotated in a square, the fractional AREA covered
is
^covered = 2^3 + |tt = 0.9877700392 .... (3)
Reversal
Rhombic Dodecahedral Number 1551
The center does not stay fixed as the Triangle is ro-
tated, but moves along a curve composed of four arcs of
an Ellipse (Wagon 1991).
see also Curve of Constant Width, Flower of
Life, Piecewise Circular Curve, Reuleaux Poly-
gon
References
Bogomolny, A. "Shapes of Constant Width." http://www.
cut-the-knot . com/do _you_know/cwidth . html.
Eppstein, D. "Reuleaux Triangles." http://www.ics.uci.
e du/ - epps t e in/ junky ard/r euleaux.html.
Reuleaux, F. The Kinematics of Machinery. New York:
Dover, 1963.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 52-54 and 381-383, 1991.
Yaglom, I. M. and Boltyansky, B. G. Convex Shapes, Mos-
cow: Nauka, 1951.
Reversal
The reversal of a decimal number abc • * • is • ■ ■ cba.
Ball and Coxeter (1987) consider numbers whose re-
versals are integral multiples of themselves. PALIN-
DROMIC NUMBER and numbers ending with a Zero
are trivial examples. The first few nontrivial examples
are 8712, 9801, 87912, 98901, 879912, 989901, 8799912,
9899901, 87128712, 87999912, 98019801, 98999901,
... (Sloane's A031877). The pattern continues for
large numbers, with numbers of the form 879- • -912
equal to 4 times their reversals and numbers of the
form 989- ■ -901 equal to 9 times their reversals. In
addition, runs of numbers of either of these forms
can be concatenated to yield numbers of the form
87 9 ■ ■ • 9 12 • • • 87 9 ■ ■ • 9 12, equal to 4 times their rever-
sals, and 989---901---989---901, equal to 9 times
their reversals.
The product of a 2-digit number and its reversal is never
a SQUARE NUMBER except when the digits are the same
(Ogilvy 1988). Numbers whose product is the reversal
of the products of their reversals include
312 x 221 = 68952
213 x 122 = 25986
(Ball and Coxeter 1987, p. 14).
see also RATS SEQUENCE
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 14-15,
1987.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, pp. 88-89, 1988.
Sloane, N. J. A. Sequence A031877 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Reversion of Series
see Series Reversion
Reverse-Then- Add Sequence
An integer sequence produced by the 196- ALGORITHM.
see also 196-Algorithm, Sort-Then-Add Sequence
Reznik's Identity
For P and Q POLYNOMIALS in n variables,
\P-Q\2 2
= £
\P h i " ) (Di,...,Dn)Q(xi,...,x n )\ 2 2
i\,,..,i n >0
111 -'-Inl
where A = dfdxi, \X\ 2 is the Bombieri NORM, and
p(«i,-,in) = jyh. ...£)j»p.
Bombieri's Inequality follows from this identity.
see also BEAUZAMY AND DEGOT'S IDENTITY
Rhodonea
see Rose
Rhomb
see Rhombus
Rhombic Dodecahedral Number
A FlGURATE NUMBER which is constructed as a cen-
tered Cube with a Square Pyramid appended to each
face,
RhoDodn = CCub n + 6P n -i = (2n - l)(2n 2 - 2n + 1),
where CCub n is a CENTERED CUBE NUMBER and P n is
a Pyramidal Number. The first few are 1, 15, 65, 175,
369, 671, ... (Sloane's A005917). The GENERATING
FUNCTION of the rhombic dodecahedral numbers is
;c(lH-llg + lla: a + a: 3 )
{x-iy
= 3 + 15ar + 65ar + 175x +....
see also Octahedral Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 53-54, 1996.
Sloane, N. J. A. Sequence A005917/M4968 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Reverse Greedy Algorithm
An algorithm for computing a Unit Fraction.
see also Greedy Algorithm, Unit Fraction
1552 Rhombic Dodecahedron
Rhombic Dodecahedron
The Dual Polyhedron of the Cuboctahedron, also
sometimes called the RHOMBOIDAL DODECAHEDRON
(Cotton 1990). Its 14 vertices are joined by 12 RHOM-
BUSES, and one possible way to construct it is known as
the BAUSPIEL. The rhombic dodecahedron is a ZONO-
HEDRON and a SPACE-FILLING Polyhedron. The ver-
tices are given by (±1, ±1, ±1), (±2, 0, 0), (0, ±2, 0),
(0, 0, ±2).
The edges of the CuBE-OCTAHEDRON COMPOUND in-
tersecting in the points plotted above are the diagonals
of Rhombuses, and the 12 Rhombuses form a rhombic
dodecahedron (Ball and Coxeter 1987).
see also BAUSPIEL, CUBE-OCTAHEDRON COMPOUND,
Dodecahedron, Pyritohedron, Rhombic Tria-
CONTAHEDRON, RHOMBUS, TRIGONAL DODECAHE-
DRON, ZONOHEDRON
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 137,
1987.
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 62, 1990.
Rhombic Icosahedron
A ZONOHEDRON which can be derived from the TRIA-
CONTAHEDRON by removing any one of the zones and
bringing together the two pieces into which the remain-
der of the surface is thereby divided.
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 143,
1987.
Bilinski, S. "Uber die Rhomb enisoeder." Glasnik Mat.-Fiz.
Astron. Drustro Mat. Fix. Hrvatske Ser. II 15, 251-263,
1960.
Rhombicosacron
Rhombic Polyhedron
A Polyhedron with extra square faces, given by the
Schlafli Symbol r{^}.
see also RHOMBIC DODECAHEDRON, RHOMBIC ICOSA-
HEDRON, Rhombic Triacontahedron, Snub Poly-
hedron, Truncated Polyhedron
Rhombic Triacontahedron
A Zonohedron which is the Dual Polyhedron of
the Icosidodecahedron. It is composed of 30 Rhom-
buses joined at 32 vertices. Ede (1958) enumerates
13 basic series of stellations of the rhombic triaconta-
hedron, the total number of which is extremely large.
Messer (1995) describes 226 stellations. The intersect-
ing edges of the Dodecahedron-Icosahedron Com-
pound form the diagonals of 30 RHOMBUSES which com-
prise the Triacontahedron. The Cube 5-Compound
has the 30 facial planes of the rhombic triacontahedron
(Ball and Coxeter 1987).
see also Cube 5-Compound, Dodecahedron-Icosa-
hedron Compound, Icosidodecahedron, Rhombic
Dodecahedron, Rhombus, Zonohedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 137,
1987.
Bulatov, V.v "Stellations of Rhombic Triacontahedron."
http://www . physics . orst . edu/ -bulatov /polyhedra/
rtc/.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub,, p. 127, 1989.
Ede, J. D. "Rhombic TViacontahedra." Math. Gazette 42,
98-100, 1958,
Messer, P. W, "Les etoilements du rhombitricontaedre et
plus." Structural Topology No. 21, 25-46, 1995.
Rhombicosacron
The Dual Polyhedron of the Rhombicosahedron.
Rhombicosahedron
Rhombicosahedron
Rhombus
1553
The Uniform Polyhedron C/" 56 whose Dual Poly-
hedron is the Rhombicosacron. It has Wythoff
5
Symbol 2 3 f . Its faces are 20{6} + 30{4}. The Cir-
2
CUMRADIUS for unit edge length is
R=±V7.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 149-150, 1971.
Rhombicosidodecahedron
see Bigyrate Diminished Rhombicosidodec-
ahedron, Diminished Rhombicosidodecahedron,
Great Rhombicosidodecahedron (Archimedean),
Great Rhombicosidodecahedron (Uniform), Gy-
rate Bidiminished Rhombicosidodecahedron,
Gyrate Rhombicosidodecahedron, Metabidimin-
ished Rhombicosidodecahedron, Metabigyrate
Rhombicosidodecahedron, Metagyrate Dimin-
ished Rhombicosidodecahedron, Parabidimin-
ished Rhombicosidodecahedron, Parabigyrate
Rhombicosidodecahedron, Paragyrate Dimin-
ished Rhombicosidodecahedron, Small Rhomb-
icosidodecahedron, Tridiminished Rhombicosi-
dodecahedron, Trigyrate Rhombicosidodecahe-
dron
Rhombicuboctahedron
see Great Rhombicuboctahedron (Archimedean),
Great Rhombicuboctahedron (Uniform), Small
Rhombicuboctahedron
Rhombidodecadodecahedron
It has Schlafli Symbol r
{!}
and Wythoff Sym-
bol f 5 | 2. Its faces are 12{|} + 30{4} + 12{5}. The
CIRCUMRADIUS for unit edge length is
R=\y/l.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 116-117, 1989.
Rhomb ihexacr on
see Great Rhombihexacron, Small Rhombihex-
ACRON
Rhombihexahedron
see Great Rhombihexahedron, Small Rhombihex-
ahedron
Rhombitruncated Cuboctahedron
see Great Rhombicuboctahedron (Archimedean)
Rhombitruncated Icosidodecahedron
see Great Rhombicosidodecahedron (Archimed-
ean)
Rhombohedron
A Parallelepiped bounded by six congruent Rhombs.
see also Parallelepiped, Rhomb
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 142 and
161, 1987.
Rhomboid
A Parallelogram in which angles are oblique and ad-
jacent sides are of unequal length.
see also Diamond, Lozenge, Parallelogram,
Quadrilateral, Rhombus, Skew Quadrilateral,
Trapezium, Trapezoid
Rhomboidal Dodecahedron
see Rhombic Dodecahedron
Rhombus
The Uniform Polyhedron L7 38 whose Dual Polyhe-
dron is the Medial Deltoidal Hexecontahedron.
A Quadrilateral with both pairs of opposite sides
Parallel and all sides the same length, i.e., an equilat-
eral Parallelogram. The word Rhomb is sometimes
1554 Rhumb Line
used instead of rhombus. The DIAGONALS p and q of a
rhombus satisfy
p 2 + g a =4a a ,
and the AREA is
A = |pg.
A rhombus whose ACUTE ANGLES are 45° is called a
Lozenge.
see also Diamond, Lozenge, Parallelogram,
Quadrilateral, Rhombic Dodecahedron, Rhom-
bic Icosahedron, Rhombic Triacontahedron,
Rhomboid, Skew Quadrilateral, Trapezium,
Trapezoid
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.
Rhumb Line
see Loxodrome
Ribbon Knot
If the KNOT K is the boundary K = /(S 1 ) of a singular
disk / : O ->■ § 3 which has the property that each self-
intersecting component is an arc A C /(P ) for which
f 1 (A) consists of two arcs in D 2 , one of which is inte-
rior, then K is said to be a ribbon knot. Every ribbon
knot is a SLICE KNOT, and it is conjectured that every
Slice Knot is a ribbon knot.
see also Slice KNOT
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 225, 1976.
Ribet's Theorem
If the Taniyama-Shimura Conjecture holds for all
semistable Elliptic Curves, then Fermat's Last
Theorem is true. Before its proof by Ribet in 1986,
the theorem had been called the epsilon conjecture. It
had its roots in a surprising result of G. Prey.
see also Elliptic Curve, Fermat's . .st Theorem,
Modular Form, Modular Function, Taniyama-
Shimura Conjecture
Riccati-Bessel Functions
S n (z) = Zj n (z) = yj —Jn+1/2(Z)
C n {z) = -zn n {z)
N n+1/2 {z),
where j n (z) and n nK z) are SPHERICAL BESSEL FUNC-
TIONS of the First and Second Kind.
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Riccati-Bessel
Functions." §10.3 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, p, 445, 1972.
Rice Distribution
Riccati Differential Equation
y' =P(z) + Q(z)y + R(z)y 2 , (1)
where y* = dyjdz. The transformation
(2)
leads to the second-order linear homogeneous equation
R(z)y" - [R'(z) + Q(z)R(z)]y' + [R(z)] 2 P(z)y = 0. (3)
Another equation sometimes called the Riccati differen-
tial equation is
z 2 w" + [z 2 - n(n + l)]w = 0, (4)
which has solutions
w = Azj n (z) + Bzy n (z). (5)
Yet another form of "the" Riccati differential equation
is
az n +by\
d v __» , ^ (6)
dz
which is solvable by algebraic, exponential, and logarith-
mic functions only when n = — 4m/(2m ± 1), for m = 0,
1,2,....
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Riccati-Bessel
Functions." §10.3 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, p. 445, 1972.
Glaisher, J. W. L. "On Riccati's Equation." Quart J. Pure
AppL Math. 11, 267-273, 1871.
Ricci Curvature
The mathematical object which controls the growth rate
of the volume of metric balls in a MANIFOLD.
see also BISHOP'S INEQUALITY, MlLNOR'S THEOREM
Ricci Tensor
where R X »\ K is the RiEMANN Tensor.
see also Curvature Scalar, Riemann Tensor
Rice Distribution
^M-^)'°(^)'
where I (z) is a MODIFIED BESSEL FUNCTION OF THE
First Kind and Z > 0. For a derivation, see Papoulis
(1962). For |V| = 0, this reduces to the RAYLEIGH DIS-
TRIBUTION.
see also RAYLEIGH DISTRIBUTION
References
Papoulis, A. The Fourier Integral and Its Applications. New
York: McGraw-Hill, 1962.
Richard's Paradox
Riemann Function 1555
Richard's Paradox
It is possible to describe a set of Positive Integers
that cannot be listed in a book containing a set of count-
ing numbers on each consecutively numbered page.
Richardson Extrapolation
The consideration of the result of a numerical calculation
as a function of an adjustable parameter (usually the
step size). The function can then be fitted and evaluated
at ft = to yield very accurate results. Press et ai.
(1992) describe this process as turning lead into gold.
Richardson extrapolation is one of the key ideas used in
the popular and robust BULIRSCH-STOER ALGORITHM
of solving Ordinary Differential Equations.
see also Bulirsch-Stoer Algorithm
References
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., p. 106, 1990.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Richardson Extrapolation and the Bulirsch-
Stoer Method." §16.4 in Numerical Recipes in FORTRAN:
The Art of Scientific Computing, 2nd ed. Cambridge, Eng-
land: Cambridge University Press, pp. 718-725, 1992.
Richardson's Theorem
Let R be the class of expressions generated by
1. The Rational Numbers and the two Real Num-
bers 7r and In 2,
2. The variable x,
3. The operations of ADDITION, MULTIPLICATION, and
composition, and
4. The Sine, Exponential, and Absolute Value
functions.
Then if E € R, the predicate "E = 0" is recursively
Undecidable.
see also Recursion, Undecidable
References
Caviness, B. F. "On Canonical Forms and Simplification." J.
Assoc. Comp. Mach. 17, 385-396, 1970.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, 1996.
Richardson, D. "Some Unsolvable Problems Involving Ele-
mentary Functions of a Real Variable." J. Symbolic Logic
33, 514-520, 1968.
Ridders' Method
A variation of the False Position Method for find-
ing ROOTS which fits the function in question with an
exponential.
see also False Position Method
References
Ostrowski, A. M. Ch. 12 in Solutions of Equations and Sys-
tems of Equations, 2nd ed. New York: Academic Press,
1966.
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.
Ralston, A. and Rabinowitz, P. §8.3 in A First Course in
Numerical Analysis, 2nd ed. New York: McGraw-Hill,
1978.
Ridders, C. F. J. "A New Algorithm for Computing a Sin-
gle Root of a Real Continuous Function." IEEE Trans.
Circuits Systems 26, 979-980, 1979.
Ridge
An (n - 2)-D FACE of an n-D POLYTOPE.
see also POLYTOPE
Riemann- Christoffel Tensor
see Riemann Tensor
Riemann Curve Theorem
If two algebraic plane curves with only ordinary singular
points and CUSPS are related such that the coordinates
of a point on either are Rational Functions of a cor-
responding point on the other, then the curves have the
same Genus (Curve). This can be stated equivalent ly
as the Genus of a curve is unaltered by a BlRATlONAL
Transformation.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 120, 1959.
Riemann Differential Equation
see Riemann P-Differential Equation
Riemann's Formula
J(x) = Li(aO - ^ Li 0O + ln 2 /
J X
dt
r.(t 2 -l)ln*'
where Li (as) is the LOGARITHMIC INTEGRAL, the sum is
taken over all nontrivial zeros p (i.e., those other than
-2, -4, . . . ) of the Riemann Zeta Function C(s), and
J(x) is Riemann Weighted Prime-Power Count-
ing Function.
see also LOGARITHMIC INTEGRAL, PRIME NUM-
BER Theorem, Riemann Weighted Prime-Power
Counting Function, Riemann Zeta Function
Riemann Function
The function obtained by approximating the RlEMANN
Weighted Prime-Power Counting Function J 2 in
fi(n)
( \ V^ /* n 7 { l/n\
(i)
1556 Riemann Hypothesis
Riemann Hypothesis
by the LOGARITHMIC Integral Li(z). This gives
R(n) = 1 + J2
(lnn) k
k((k + 1) fc!
— -Li(n ),
(2)
(3)
where £(z) is the Riemann Zeta Function, fi(n) is
the Mobius Function, and Li(x) is the Logarithmic
Integral. Then
n{x) = R(x)-^2R(x P ),
(4)
where w is the PRIME COUNTING FUNCTION. Ramanu-
jan independently derived the formula for R(n), but
nonrigorously (Berndt 1994, p. 123).
see also Mangoldt Function, Prime Number The-
orem, Riemann-Mangoldt Function, Riemann
Zeta Function
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, 1994.
Conway, J. H. and Guy, R. K. The Booh of Numbers. New
York: Springer- Verlag, pp. 144-145, 1996.
Riesel, H. and Gohl, G. "Some Calculations Related to Rie-
mann's Prime Number Formula." Math. Comput. 24,
969-983, 1970.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 28-29 and 362-372, 1991.
Riemann Hypothesis
First published in Riemann (1859), the Riemann hy-
pothesis states that the nontrivial ROOTS of the RIE-
MANN Zeta Function
oo
(1)
where seC (the COMPLEX NUMBERS), all lie on the
"Critical Line" &[$] = 1/2, where R[z] denotes the
Real Part of z. The Riemann hypothesis is also known
as Artin's Conjecture.
In 1914, Hardy proved that an Infinite number of val-
ues for s can be found for which £(s) = and R[s] = 1/2.
However, it is not known if all nontrivial roots s satisfy
3R[s] = 1/2, so the conjecture remains open. Andre Weil
proved the Riemann hypothesis to be true for field func-
tions (Weil 1948, Eichler 1966, Ball and Coxeter 1987).
In 1974, Levin showed that at least 1/3 of the ROOTS
must lie on the CRITICAL LINE (Le Lionnais 1983), a
result which has since been sharpened to 40% (Vardi
1991, p. 142). It is known that the zeros are symmetri-
cal placed about the line $s[s] = 0.
The Riemann hypothesis is equivalent to A < 0, where
A is the de Bruijn-Newman Constant (Csordas et
al. 1994). It is also equivalent to the assertion that for
some constant c,
| Li(a;) — 7v(x)\ < cy/x lnx,
(2)
where Li(x) is the LOGARITHMIC INTEGRAL and it is the
Prime Counting Function (Wagon 1991).
The hypothesis was computationally tested and found to
be true for the first 2 x 10 8 zeros by Brent et al. (1979),
a limit subsequently extended to the first 1.5 x 10 9 + 1
zeros by Brent et al. (1979). Brent's calculation covered
zeros a + it in the region < t < 81, 702, 130.19.
There is also a finite analog of the Riemann hypothe-
sis concerning the location of zeros for function fields
defined by equations such as
ay
1 + fcz m + c = 0.
(3)
This hypothesis, developed by Weil, is analogous to the
usual Riemann hypothesis. The number of solutions for
the particular cases (/, m) = (2, 2), (3,3), (4,4), and (2,4)
were known to Gauss.
see also Critical Line, Extended Riemann Hy-
pothesis, Gronwall's Theorem, Mertens Conjec-
ture, Mills' Constant, Riemann Zeta Function
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 75, 1987.
Brent, R. P.; Vandelune, J.; te Riele, H. J. J.; and Winter,
D. T. "On the Zeros of the Riemann Zeta Function in the
Critical Strip. I." Math. Comput. 33, 1361-1372, 1979.
Brent, R. P.; Vandelune, J.; te Riele, H. J. J.; and Winter,
D. T. "On the Zeros of the Riemann Zeta Function in the
Critical Strip. II." Math. Comput. 39, 681-688, 1982. Ab-
stract available at ftp://nimbus.anu.edu.au/pub/Brent/
rpb070a.avi. Z.
Csordas, G.; Smith, W.; and Varga, R. S. "Lehmer Pairs of
Zeros, the de Bruijn-Newman Constant and the Riemann
Hypothesis." Constr. Approx. 10, 107-129, 1994.
Eichler, M. Introduction to the Theory of Algebraic Numbers
and Functions. New York: Academic Press, 1966.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 25, 1983.
Odlyzko, A. "The 10 20 th Zero of the Riemann Zeta Function
and 70 Million of Its Neighbors."
Riemann, B. "Uber die Anzahl der Primzahlen unter einer
gegebenen Grosse," Mon. Not. Berlin Akad. } pp. 671-680,
Nov. 1859.
Sloane, N. J. A. Sequence A002410/M4924 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Vandelune, J. and te Riele, H. J. J. "On The Zeros of the
Riemann Zeta-Function in the Critical Strip. III." Math.
Comput. 41, 759-767, 1983.
Vandelune, J.; te Riele, H. J. J.; and Winter, D. T. "On the
Zeros of the Riemann Zeta Function in the Critical Strip.
IV." Math. Comput. 46, 667-681, 1986.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 33, 1991.
Weil, A. Sur les courbes algebriques et les varieVes qui s'en
deduisent. Paris, 1948.
Riexnann Integral
Riemann-Mangoldt Function 1557
Riemann Integral
The Riemann integral is the INTEGRAL normally en-
countered in CALCULUS texts and used by physicists and
engineers. Other types of integrals exist (e.g., the Leb-
ESGUE INTEGRAL), but are unlikely to be encountered
outside the confines of advanced mathematics texts.
The Riemann integral is based on the JORDAN MEA-
SURE, and defined by taking a limit of a Riemann Sum,
pa n
/ f{x)dx= lim y"f(xt)Ax k (1)
/, max Ax*.— J-0 z — '
n
f(x,y)dA= lim J2f{xl,yl)AA k (2)
SO
fe=l
//
[f[f(x,y,z)dV= lim S"f(x h ,yl,zl)AV k ,
JJJ maxAVfO^
(3)
where a < x <b and x£, y%, and z* k are arbitrary points
in the intervals Axk, Aj/fc, and Azk, respectively. The
value max Axk is called the Mesh Size of a partition of
the interval [a, 6] into subintervals Axk-
As an example of the application of the Riemann integral
definition, find the AREA under the curve y — x r from
to a. Divide (a, b) into n segments, so Axk — ^^ = h,
then
f( Xl ) = /(0) =
f{x a ) = f(Ax k ) = h T
f(x 3 ) = f(2Ax k ) = (2h) r .
(4)
(5)
(6)
By induction
f(x k ) = f([k - l]Ax fc ) = [(* - l)h] r = h r {k - l) r , (7)
so
f(x k )Ax k = h r+1 (k-l) r
n n
J2f(^)Ax k =h r+1 Y^(k-l) r -
k-1 k=l
For example, take r = 2.
n n
^/(x fe )A^=/i 3 ^-l) 2
fc=i fe=i
/ n n n \
n(n + l)(2n+l) n(n + l) ,
— -± — 2 — - + n
(10)
/= lim S^ f(x k *)Axk = lim y^/(xfe)Aaj fc
fc=l
fc=i
= lim /i
n— >-oo
"n(n + l)(2n+l) rt n(n+l)
[ 6 2 2 +n J
= a lim
n— j-oo
= ia 3 .
"n(n + l)(2n + 1) n(n + 1) ( n
6n 3 n 3 n 3
3 ^
see also INTEGRAL, Riemann Sum
References
(11)
Kestelman, H. "Riemann Integration." Ch. 2 in Modern
Theories of Integration, 2nd rev. ed. New York: Dover,
pp. 33-66, 1960.
Riemann's Integral Theorem
Associated with an irreducible curve of GENUS (Curve)
p, there are p Linearly Independent integrals of the
first sort. The ROOTS of the integrands are groups of
the canonical series, and every such group will give rise
to exactly one integral of the first sort.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 274, 1959.
Riemann-Lebesgue Lemma
Sometimes also called MERCER'S THEOREM.
lim /
J o
K(\,z)Csm(nz)dz =
for arbitrarily large C and "nice" K(\,z). Gradshteyn
and Ryzhik (1979) state the lemma as follows. If f(x)
is integrable on [7r, 7r], then
r
lim / f(x) sin(tx) dx — >
(8)
and
(9)
References
lim
t—yoo
J
J —It
F
J — 7T
f{x) cos(fcc) dx —¥ 0.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1101, 1979.
Riemann-Mangoldt Function
£—j n
n>l
= Li(x) — 2_, ei(plnar) — In 2
nontrivial p
C(p)=o
r dt
J x t{t'-l)\nt'
+
(1)
1558 Riemann Mapping Theorem
Riemann-Roch Theorem
where $(z) is the Riemann Zeta Function, Li(x) is
the Logarithmic Integral and ei(x) is the Exponen-
tial Integral. The Mangoldt Function is given by
where
A(r») = {
Inp if n — p m for (m > 1) and p prime
otherwise
C(x) _ ^ A(n
CW
n s
for »[a] > 1.
A(n)
'<■> = ££?•
(2)
(3)
(4)
The SUMMATORY Riemann- Mangoldt function is denned
by
^(x) = ^A(n)=0(x) + 0(x 1/2 ) + .... (5)
n<x
see also Prime Number Theorem, Riemann Func-
tion
References
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 364-365, 1991.
Riemann Mapping Theorem
Let zq be a point in a simply connected region R ^ C.
Then there is a unique Analytic FUNCTION w = f(z)
mapping R one-to-one onto the DISK \w\ < 1 such that
/(z ) = and f'(z Q ) — 0. The COROLLARY guarantees
that any two simply connected regions except R can be
mapped CONFORMALLY onto each other.
Riemann's Moduli Problem
Find an ANALYTIC parameterization of the compact
Riemann Surfaces in a fixed Homomorphism class.
The Ahlfors-Bers Theorem proved that Riemann's
Moduli Space gives the solution.
see also Ahlfors-Bers Theorem, Riemann's Mod-
uli Space
Riemann's Moduli Space
Riemann's moduli space R p is the space of ANALYTIC
Equivalence Classes of Riemann Surfaces of fixed
Genus p.
see also Ahlfors-Bers Theorem, Riemann's Mod-
uli Problem, Riemann Surface
Riemann P-Differential Equation
dz 2
+
1 - a - a' 1-/3-/3' 1 - 7 - 1
z-b
du
dz
aa'(a-6)(q-c) 00'(b- c)(b- a)
z — a z — b
+
77 / (c-a)(c-fr)
a + a'+jS + jS' + T + V^l.
(z — a)(z — b)(z — c)
-0,
Solutions are Riemann P-Series (Abramowitz and Ste-
gun 1972, pp. 564-565).
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Riemann's Dif-
ferential Equation." §15.6 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, pp. 564-565, 1972.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 541-543, 1953.
Riemann P- Series
The solutions to the Riemann P-Differential Equa-
tion
z = P
Solutions are given in terms of the HYPERGEOMETRIC
Function by
x 2 Fi (a + (3 + 7, a + 0' + 7; 1 + a - a ; A)
x 3 Fi (a + + 7, oc' + + 7; 1 + a! - a; A)
/z-a\ a (z-c\i'
U * = (^-b) [J^b)
x 2 Fi(a + + 7', ol + 0' + 7'; 1 + ol - a; A)
/z~a\ a ' (z-cyt 1
u *={7^b) [z-^b)
x 2 F 1 (a + + 7, a + 0' + 7; 1 + a - a; A),
where
A= (*-")(c-i>)
(z — b)(c — a) '
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Riemann's Dif-
ferential Equation." §15.6 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, pp. 564-565, 1972,
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 541-543, 1953.
Whit taker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 283-284, 1990.
Riemann-Roch Theorem
The dimension of a complete series is equal to the sum
of the order and index of specialization of any group,
less the GENUS of the base curve
r = N + i + p.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 261, 1959.
Riemann Series Theorem
Riemann Series Theorem
By a suitable rearrangement of terms, a conditionally
convergent SERIES may be made to converge to any de-
sired value, or to DIVERGE.
References
Bromwich, T. J. I'a. and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, p. 74, 1991.
Riemann-Siegel Functions
Re[RiemannSiegelZ z] Im[RiemannSiegelZ z] |RiemannSiegelZ z |
15-10
10-10
For a Real Positive £, the Riemann-Siegel Z function
is defined by
Z(t) = e w(t) C(|+it).
The top plot superposes Z{t) (thick line) on \C{\ +^)l>
where C(z) is the Riemann Zeta Function.
10
-10
-5
-10
Re[RiemannSiegelTheta z] Im[RiemannSiegelTheta zj
30
|RiemannSiegelTheta z |
The Riemann-Siegel theta function appearing above is
defined by
= 5[lnr(£ + fit)- \t\niv]
= arg[T(iH- \%t)]- \thx<x.
Riemann Surface 1559
These functions are implemented in Mathematica®
(Wolfram Research, Champaign, IL) as RiemannSiegelZ
[z] and RiemannSiegelThetaCz] , illustrated above.
see also Riemann Zeta Function
References
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 143, 1991.
Riemann Space
see Metric Space
Riemann Sphere
A 1-D Complex Manifold C*, which is the one-point
compactification of the Complex numbers C U {oo},
together with two charts. For all points in the COM-
PLEX Plane, the chart is the Identity Map from
the SPHERE (with infinity removed) to the COMPLEX
PLANE. For the point at infinity, the chart neighbor-
hood is the sphere (with the ORIGIN removed), and the
chart is given by sending infinity to and all other points
z to 1/z.
Riemann- Stieltjes Integral
see Stieltjes Integral
Riemann Sum
Let a CLOSED Interval [a, b] be partitioned by points
a < xi < x 2 < .. . < Xn-i < 6, the lengths of the
resulting intervals between the points are denoted Aasi,
Ax2, . . . , Ax n . Then the quantity
y^j(x* k )Ax k
k=i
is called a Riemann sum for a given function f(x) and
partition. The value maxAx^ is called the MESH SIZE
of the partition. If the LIMIT max Ax k — > exists, this
limit is known as the Riemann INTEGRAL of f(x) over
the interval [a, 6]. The shaded areas in the above plots
show the Lower and Upper Sums for a constant Mesh
Size.
see also Lower Sum, Riemann Integral, Upper Sum
Riemann Surface
The Riemann surface S of the ALGEBRAIC FUNCTION
FIELD K is the set of nontrivial discrete valuations on
K. Here, the set S corresponds to the IDEALS of the
Ring A of Integers of K over C(^). (A consists of the
elements of K that are ROOTS of MONIC POLYNOMIALS
over C [-?].)
1560
Riemann Tensor
Riemann Zeta Function
see also ALGEBRAIC FUNCTION FIELD, IDEAL, RING
References
Fischer, G. (Ed.). Plates 123-126 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 120-123, 1986.
Riemann Tensor
A TENSOR sometimes known as the RiEMANN-
Christoffel Tensor. Let
*-£-e{y}.
{s u\ .
i j"
) Kind. Th<
iw.-JJ-lYj-fclv}.
(i)
where the quantity inside the { " , " Ms a Christof-
fel Symbol of the Second Kind. Then
(2)
Broken down into its simplest decomposition in iV-D,
1
Riemann Theta Function
Let the Imaginary Part of a g x g Matrix F be Pos-
itive DEFINITE, and m — (mi, . . . ,m 9 ) be a row VEC-
TOR with coefficients in Z. Then the Riemann theta
function is defined by
&(u) = ^exp[27ri(m T u+ |F T m)].
see also Ramanujan Theta Functions, Theta
Function
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 9, 1980.
Riemann Weighted Prime-Power Counting
Function
The Riemann weighted prime-power counting function
is defined by
RxpvK — jy _ 2
R
(gxuRuK — gxnRfiu ~ g^vRxn + g^^Rxv)
(gx^g^K — gx^g^u) + C\^ VK . (3)
Mx)
{N-l){N-2) y
Here, R^ u is the RlCCl TENSOR, R is the CURVATURE
Scalar, and Cx^ K is the Weyl Tensor. In terms of
the Jacobi Tensor J^^p,
R^au(3 = 3(J„ a /3 ~ J f3av)- ( 4 )
The Riemann tensor is the only tensor that can be con-
structed from the METRIC TENSOR and its first and
second derivatives,
for p™ with p a prime/ ]\
otherwise
l ™ J 2 -
r 2+iT ^
= lim — ^ I — lnC(s)ds.
*-oo27Tt J 2 _ iT S
(2)
The Prime Counting Function is given in terms of
Jiix) by
n
^ — ' 77,
(3)
R a W = r^ 5j7 - T^s + r« 7 rg, - r^ - r^,*, (5) The function also satisfies the identity
where T are CONNECTION COEFFICIENTS and c are
Commutation Coefficients. The number of inde-
pendent coordinates in n-D is
^M= r Mx)x-'- 1 dx.
(4)
_ 1 2/ 2
n'(n* - 1),
(6)
and the number of Scalars which can be constructed
from Rx^vk and g^ v is
S "=[±n(n~l)(n-2)(n
for n = 2
+ 3) for ra = l,n > 2.
(7)
see also MANGOLDT FUNCTION, PRIME COUNTING
Function, Riemann's Formula
Riemann Xi Function
see Xi Function
Riemann Zeta Function
In 1-D, Aim = 0.
n
c n
S n
1
2
1
1
3
6
3
4
20
14
see also BlANCHI IDENTITIES, CHRISTOFFEL SYM-
BOL of the Second Kind, Commutation Co-
efficient, Connection Coefficient, Curvature
Scalar, Gaussian Curvature, Jacobi Tensor,
Petrov Notation, Ricci Tensor, Weyl Tensor
Riemann Zeta Function
Riemann Zeta Function
1561
The Riemann zeta function can be defined by the inte-
gral
where x > 1. If x is an Integer n, then
,,n — 1 „ — u„,ti — 1
W e 1i _ u n _i V - ^ -ku n-1
= = e u > e u
u - 1 1 — e _u ^— '
/•oo n _i _°°_ A
Jo e"- 1 £^io
e-^u" -1 du.
(2)
(3)
Let y = /en, then dy = kdu and
oo /»oo
— fcu n— 1 j
e u du
if
e y y n dy,
(4)
where T(n) is the GAMMA Function. Integrating the fi-
nal expression in (4) gives T(n), which cancels the factor
l/r(n) and gives the most common form of the Riemann
zeta function,
«») = Ei-
(5)
At n = 1, the zeta function reduces to the HARMONIC
SERIES (which diverges), and therefore has a singularity.
In the Complex Plane, trivial zeros occur at —2, —4,
—6, . . . , and nontrivial zeros at
s = ex + it
(6)
for < (7 < 1. The figures below show the structure of
C(z) by plotting |C(z)| and l/|C(z)|.
10 20 30 40 50 60
The Riemann Hypothesis asserts that the nontrivial
Roots of C(s) all have Real Part <t = R[s] = 1/2, a
line called the "CRITICAL STRIP." This is known to be
true for the first 1.5 x 10 12 roots (Brent et al 1979). The
above plot shows | £(1/2 -hit) | for t between and 60. As
can be seen, the first few nontrivial zeros occur at t —
14.134725, 21.022040, 25.010858, 30.424876, 32.935062,
37.586178, . . . (Wagon 1991, pp. 361-362 and 367-368).
The Riemann zeta function can also be defined in terms
of Multiple Integrals by
The Riemann zeta function can be split up into
Ca+fl) = *(*)e- W(t) > («)
where z{t) and 0(t) are the RlEMANN-SlEGEL Func-
tions. An additional identity is
lim C(«) - — r
s — ► ! S — 1
where 7 is the EULER-MASCHERONI CONSTANT.
(9)
The Riemann zeta function is related to the Dirichlet
Lambda Function X(u) and Dirichlet Eta Func-
tion r](u) by
and
C(") = AM = vM
2 V 2" — 1 2" — 2
<i{u) + r i {v) = 2\{v)
(10)
(11)
(Spanier and Oldham 1987). It is related to the LlOU-
ville Function \{n) by
C(2f)
C(»)
n
\{n)
(12)
(Lehman 1960, Hardy and Wright 1979). Furthermore,
C 2 (s) _ f^ 2"<">
t
1562
Riemann Zeta Function
where u)(n) — o"o(n) is the number of different prime
factors of n (Hardy and Wright 1979).
A generalized Riemann zeta function C(s,a) known as
the Hurwitz Zeta Function can also be defined such
that
C(s) = C(s,0). (14)
The Riemann zeta function may be computed analyti-
cally for EVEN n using either CONTOUR INTEGRATION
or PARSEVAL'S THEOREM with the appropriate FOUR-
IER SERIES. An interesting formula involving the prod-
uct of Primes was first discovered by Euler in 1737,
C (,)(l-2-)= (!+£ + £ + ...) (l-£)
C(x)(l- 2-)(l -3-*)
- (l — — — }- (— — — "\
V + 3 1 5* 7* 7 V3* + 9 X + 15* + " 7
(16)
C(x)(l -2-X1-3-) -..(I- p- x )---
OO
=c(x)n( i -p _x )= i - ( i7 )
n=2
Here, each subsequent multiplication by the next Prime
p leaves only terms which are POWERS of p~ x . There-
fore,
C(x) =
U(i-p- x )
p=2
(18)
where p runs over all PRIMES. Euler's product formula
can also be written
cw=(i-2- s )- 1 n (wr 1 n c 1 -'-"*) -1 -
(19)
(20)
9 = 1
(mod 4)
(mod
i
4)
A few
sum
identities involving £(n) are
oo
-i] = i
£(-l)"[C(n) " 1] = I-
(21)
The Riemann zeta function is related to the GAMMA
Function T(z) by
r(f)ir-"cw = r(i=i)*-< l ->/'c(i-.). (22)
£(n) was proved to be transce ndental for all even n by
Euler. Apery (1979) proved c(3J to^e IRRATIONAL with
Riemann Zeta Function
the aid of the k~ 3 sum formula below. As a result, £(3)
is sometimes called Apery's Constant.
oo
oo . \fc — 1
oo
c(4, ^S^j (25)
(Guy 1994, p. 257). A relation of the form
« B > = Z »El^ (26)
has been searched for with Z$ a RATIONAL or ALGE-
BRAIC Number, but if Z 5 is a Root of a Polynomial
of degree 25 or less, then the Euclidean norm of the co-
efficients must be larger than 2 x 10 37 (Bailey, Bailey
and Plouffe). Therefore, no such sums are known for
((n) are known for n > 5.
The zeta function is defined for R[s] > 1, but can be
analytically continued to R[s] > as follows
53(-i)»„-+5] n - = 2 J2 n ~ s
n=l n=l n=2,4,.,.
oo oo
= 2 ^2(2ky s = 2 1 - 3 ]T k" (27)
fc=i fe=i
oo
Y,(-l) n n- s +«s) = 2 1 -°«s) (28)
71 = 1
OO
71=1
The DERIVATIVE of the Riemann zeta function is defined
by
C'( S ) = - S ^fc- s lnfc = -^^. (30)
fc = l
As s -> 0,
C'(0) = -iln(2T).
For Even n = 2k,
n.
(31)
(32)
where B n is a BERNOULLI NUMBER. Another intimate
connection with the BERNOULLI NUMBERS is provided
by
B„ = (-l)" +1 nC(l-n). (33)
Riemann Zeta Function
Riemann Zeta Function
1563
No analytic form for £(n) is known for Odd n = 2/c + l,
but (,(2k 4- 1) can be expressed as the sum limit
C(2&+1)
t2fc + l
lim OI , , >
i=l
Him)
2fc + l
(34)
(Stark 1974). The values for the first few integral argu-
ments are
C(o) = -i
C(l) = oo
C(3) = 1.2020569032...
^ = To
C(5) = 1.0369277551...
C(7) = 1.0083492774...
C(9) = 1.0020083928...
10
<™ = 93*55-
Euler gave C(2) to £(26) for Even n, and Stieltjes (1993)
determined the values of C(2), • - . , C(^0) to 30 digits of
accuracy in 1887. The denominators of C(^ n ) f° r n —
1, 2, ... are 6, 90, 945, 9450, 93555, 638512875, ...
(Sloane's A002432).
Using the LLL ALGORITHM, Plouffe (inspired by Zucker
1979, Zucker 1984, and Berndt 1988) has found some
beautiful infinite sums for £(n) with Odd n. Let
5± ^£pr
h n {e 2 * h ±iy
then
C(3) = I | 5 ^ 3 -2S_(3)
C(5)=3k 7 r 5 -i5_(5)-^S + (5)
294'
19
C(7)
C(9) = 5T5jf75^-ii5-(9)-45sS+(9)
C(H) =
C(13) =
•25_ (7)
r n -25-(ll)
(35)
(36)
(37)
(38)
(39)
(40)
13
16512
8255
5_(13)-
8^5 5 +( 13 )
C(15) =
C(17) = T
13687
390769879500
397549
12024529867250
2
tt 15 -2S-(15)
17 261632 {
(41)
(42)
71 130815 ^-
.(17)
5+ (17)
C(19)
C(21) =
130815
7708537
21438612514068750
68529640373
tt 19 -25_(19)
(43)
(44)
1881063815762259253125 '
2
|S_(21)
lS+(21)
(45)
2 4 6 8 10
The inverse of the Riemann Zeta Function 1/C(p) is
the asymptotic density of pth-powerfree numbers (i.e.,
Squarefree numbers, Cubefree numbers, etc.). The
following table gives the number Q p (n) of pth-powerfree
numbers < n for several values of n.
p 1/C(p) 10 100 10 3
10*
10 &
10°
2 0.607927 7 61 608 6083 60794 607926
3 0.831907 9 85 833 8319 83190 831910
4 0.923938 10 93 925 9240 92395 923939
5 0.964387 10 97 965 9645 96440 964388
6 0.982953 10 99 984 9831 98297 982954
The value for £(2) can be found using a number of dif-
ferent techniques (Apostol 1983, Choe 1987, Giesy 1972,
Holme 1970, Kimble 1987, Knopp and Schur 1918, Kor-
tram 1996, Matsuoka 1961, Papadimitriou 1973, Sim-
mons 1992, Stark 1969, Stark 1970, Yaglom and Yaglom
1987). The problem of finding this value analytically
is sometimes known as the Basler Problem (Castel-
lanos 1988). Yaglom and Yaglom (1987), Holme (1970),
and Papadimitrou (1973) all derive the result from DE
MoiVRE's Identity or related identities.
Consider the FOURIER SERIES of f{x) = x 2n
oo oo
f(x) = |a + \J a m cos(mx) + Vj b m sin (ma), (46)
which has coefficients given by
a = —
7T
f(x) dx
7T
.2TI+1
TV \2n + 1
" * Jo
= 27T 2n
" 2n + l
x 2n dx
* J*
2 J"
* Jo
7T /
J — 7T
x n cos(mx) dx
x n cos (ma) dx
sin(ma) dx = 0,
(47)
(48)
(49)
where the latter is true since the integrand is ODD.
Therefore, the FOURIER SERIES is given explicitly by
2n +
— + y a-m cos(mx).
(50)
1564 Riemann Zeta Function
Now, a m is given by the COSINE INTEGRAL
= -(-l) n+1 (2n)!
sm(mx) y.
(-1)*
+ cos
h)]
(-1)
A;=0
fc + 1
(2Jfc)!m 2n - 2fc + 1
jfc=i
(2k-3)\m 2n - 2k + 2 *
(51)
But cos(mTr) = ( — l) m , and sin(m7r) = sinO = 0, so
o m = |(-l)- +l (2n)!(-l) m £;
(-1>
Jfe + 1
fc=l
{2k - 3)!m 2Tl - 2fc + 2
M-ir + - 2 (2n)!|: (2fc j^; n _ 2t+2 7r — .
Now, if n = 1,
= 4(-ir
(52)
fc=i
(2A; - 3)!m 4
(-l)!m 2 m 2
(53)
so the Fourier Series is
' 2 — (-l) m cos(mx)
Ji n
-y^E
m=l
m z
Letting x = 7t gives cos(m7r) = ( — l) m , so
2 7T
7T = h '
3 ^ ^ m'
771 — 1
OO
and we have
« 2 )-Ei
(54)
(55)
(56)
Higher values of n can be obtained by finding a m and
proceeding as above.
The value £(2) can also be found simply using the ROOT
Linear Coefficient Theorem. Consider the equa-
tion sin z = and expand sin in a Maclaurin Series
smz = z-- + - + ... =
(57)
where w = z 2 . But the zeros of sin(z) occur at 7r, 27r, 37r,
. . . , so the zeros of sinw — sin yfz occur at tt 2 , (27r) 2 ,
Therefore, the sum of the roots equals the COEFFICIENT
of the leading term
- + — + — + -i-i (59)
tt 2 ^ (2tt) 2 + (3tt 2 ) + 3! ~ 6' ( '
Riemann Zeta Function
which can be rearranged to yield
« 2 > = T-
(60)
Yet another derivation (Simmons 1992) evaluates the
integral using the integral
= f [{x+\x 2 y+\x i y 2 + ...)}ldy
Jo
Jo
dy
y 2 y 3
V+& + & + -
\dy
l + ^ + ^+---- (61)
To evaluate the integral, rotate the coordinate system
by tt/4 so
x = u cos 6 — v sin = | y/2 (u — v) (62)
y = usin0 + t> cos0 = |>/2(u + t;) (63)
and
zy=f(u 2 -u 2 ) (64)
l-xy=\{2-u 2 +v 2 ). (65)
Then
fV2/2 /*u
/=4 /" ' r _dudv_
Jo Jo 2-u*+v*
V2 fV2-<u
+4
Jy/2/2 Jo
dudv
2-u 2 +v 2
Now compute the integrals Ii and J2.
^v/2/2
S/1 + /2. (66)
Jo [Jo Z-u 2 +v
= 4 / "[ — J^ tan- 1
= 4 / * tan" 1 ( " 1 d«. (67)
y/2-u 2
Make the substitution
u = v2 sin
y/2 - v? = y/2cos9
du = \f2 cos 6 d8 y
(68)
(69)
(70)
Riemann Zeta Function
Riemann Zeta Function 1565
so
tan"
+ (- F ±J)=*n-*( ' / * aa9 )=9 (71)
\V2-u 2 J \V2cos0j
and
/i = 4 / -t=4 0V2 cosOde = 2[0 2 ]* /6 = ^-.
J V 2 cos 6 18
(72)
72 can also be computed analytically,
dv
2-u 2 +v 2
Jy/2/2 Jo
v^-w
c?n
(73)
But
tan'
\V2-u 2 J V V^costf y
( 1 - sinfl \ __ _i /_cos_0\
I cosfl ;~ tan U + sin^
tan
= tan
= tan
sin(§7r-0)
1 + cos(|tt- 0)_
1 f 2sm[I(I ff -g)]cos[I(^-g)] |
\ 2cos2[I(± 7 r-0)] /.
= 1(^-0),
(74)
/2 = 4 / V-^(^-^)^ cos ^
A/6 V2cos0
<£6>
=mh-k];;:
^ 8 16/ V24 144 /J 9 ' [ }
Combining I\ and I2 gives
C(2) = / 1 +/ 2 = I? + T = 6
(76)
see also Abel's Functional Equation, Debye
Functions, Dirichlet Beta Function, Dirich-
let Eta Function, Dirichlet Lambda Func-
tion, Harmonic Series, Hurwitz Zeta Func-
tion, Khintchine's Constant, Lehmer's Phenome-
non, Psi Function, Riemann Hypothesis, Riemann
P-Series, Riemann-Siegel Functions, Stieltjes
Constants, Xi Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Riemann Zeta
Function and Other Sums of Reciprocal Powers." §23.2
in Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing. New-
York: Dover, pp. 807-808, 1972.
Apery, R. "Irrationalite de £(2) et C(3)." Asterisque 61, 11-
13, 1979.
Apostol, T. M. "A Proof that Euler Missed: Evaluating C(2)
the Easy Way." Math, Intel 5, 59-60, 1983.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 332-335, 1985.
Ayoub, R. "Euler and the Zeta Function." Amer. Math.
Monthly 71, 1067-1086, 1974.
Bailey, D. H. "Multiprecision Translation and Execution of
Fortran Programs." ACM Trans. Math, Software. To ap-
pear.
Bailey, D. and Plouffe, S. "Recognizing Numerical
Constants." http : //www . cecm . sf u . ca/organics/papers/
bailey.
Berndt, B. C. Ch. 14 in Ramanujan's Notebooks, Part II.
New York: Springer-Verlag, 1988.
Borwein, D. and Borwein, J. "On an Intriguing Integral and
Some Series Related to C(4)." Proc. Amer. Math. Soc.
123, 1191-1198, 1995.
Brent, R. P.; van der Lune, J.; te Riele, H. J. J.; and Winter,
D. T. "On the Zeros of the Riemann Zeta Function in the
Critical Strip 1." Math. Comput. 33, 1361-1372, 1979.
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag.
61, 67-98, 1988.
Choe, B. R. "An Elementary Proof of £)~ =1 ± = ^."
Amer. Math. Monthly 94, 662-663, 1987. "
Davenport, H. Multiplicative Number Theory, 2nd ed. New
York: Springer-Verlag, 1980.
Edwards, H. M. Riemann's Zeta Function. New York: Aca-
demic Press, 1974.
Farmer, D. W. "Counting Distinct Zeros of the Riemann
Zeta-Function." Electronic J, Combinatorics 2, Rl,
1-5, 1995. http : //www . combinatorics . org/Volume_2/
volume2 .html#Rl.
Giesy, D. P. "Still Another Proof that J] 1/k 2 = tt 2 /6."
Math. Mag. 45, 148-149, 1972.
Guy, R. K. "Series Associated with the ^-Function." §F17 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer-Verlag, pp. 257-258, 1994.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, p. 255, 1979.
Holme, F. "Ein enkel beregning av 5^^ p"»" Nordisk Mat.
Tidskr. 18, 91-92 and 120, 1970.
Ivic, A. A. The Riemann Zeta-Function. New York: Wiley,
1985.
Ivic, A. A. Lectures on Mean Values of the Riemann Zeta
Function. Berlin: Springer-Verlag, 1991.
Karat suba, A. A. and Voronin, S. M. The Riemann Zeta-
Function. Hawthorne, NY: De Gruyter, 1992.
Katayama, K. "On Ramanujan's Formula for Values of Rie-
mann Zeta-Function at Positive Odd Integers." Acta
Math. 22, 149-155, 1973.
Kimble, G. "Euler's Other Proof." Math. Mag. 60, 282,
1987.
Knopp, K. and Schur, I. "Uber die Herleitug der Gleichung
V 00 \ — ^-." Archiv der Mathematik u. Physik 27,
m-176, 1918.
Kortram, R A. "Simple Proofs for J^Hi P" ~ ^T and
sinx = zlir=i ( 1_ Pp)'" Math - Ma 9' 69 ' 122 - 125 >
1996.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 35, 1983.
1566 Riemannian Geometry
Riesel Number
Lehman, R. S. "On Liouville's Function." Math. Comput.
14, 311-320, 1960.
Matsuoka, Y. "An Elementary Proof of the Formula
V°° JL = sl» Amer. Math. Monthly 68, 486-487,
mr. 1 k 6
Papadimitriou, I. "A Simple Proof of the Formula
$3~ Jl. = uly Amer. Math. Monthly 80, 424-425,
1973. 1
Patterson, S. J. An Introduction to the Theory of the Rie-
mann Z eta- Function. New York: Cambridge University-
Press, 1988.
Plouffe, S. "Identities Inspired from Ramanujan Notebooks."
http : //www . lacim. uqam . ca/plouf f e/identit ies .html.
Simmons, G. F. "Euler's Formula ^^T V™ 2 — 7t ' 2 /6 by Dou-
ble Integration." Ch. B. 24 in Calculus Gems: Brief Lives
and Memorable Mathematics. New York: McGraw-Hill,
1992.
Sloane, N. J. A. Sequence A002432/M4283 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Spanier, J. and Oldham, K. B. "The Zeta Numbers and Re-
lated Functions." Ch. 3 in An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 25-33, 1987.
Stark, E. L. "Another Proof of the Formula J27=i ~£* = IT""
Amer. Math. Monthly 76, 552-553, 1969.
Stark, E. L. "1 - \ + J - ^ + . . . = f|." Praxis Math. 12,
1-3, 1970.
Stark, E. L. "The Series ^Hi k ~ S s = 2 > 3 > 4 ' ' * ' » 0nce
More." Math. Mag. 47, 197-202, 1974.
Stieltjes, T. J. Oeuvres Completes, Vol. 2 (Ed. G. van Dijk.)
New York: Springer- Verlag, p. 100, 1993.
Titchmarsh, E. C. The Zeta-Function of Riemann. New
York: Stechert-Hafher Service Agency, 1964.
Titchmarsh, E. C. and Heath-Brown, D. R. The Theory of
the Riemann Zeta-Function, 2nd ed, Oxford, England:
Oxford University Press, 1986.
Vardi, I. "The Riemann Zeta Function." Ch. 8 in Com-
putational Recreations in Mathematica. Reading, MA:
Addison- Wesley, pp. 141-174, 1991.
Wagon, S. "The Evidence: Where Are the Zeros of Zeta of
5?" Math. Intel 8, 57-62, 1986.
Wagon, S. "The Riemann Zeta Function." §10.6 in Mathe-
matica in Action. New York: W. H. Freeman, pp. 353-362,
1991.
Yaglom, A. M. and Yaglom, I. M. Problem 145 in Challenging
Mathematical Problems with Elementary Solutions, Vol. 2.
New York: Dover, 1987.
Zucker, I. J. "The Summation of Series of Hyperbolic Func-
tions." SIAM J. Math. Anal 10, 192-206, 1979.
Zucker, I. J. "Some Infinite Series of Exponential and Hy-
perbolic Functions." SIAM J. Math. Anal. 15, 406-413,
1984.
Riemannian Geometry
The study of MANIFOLDS having a complete RIEMAN-
NIAN METRIC. Riemannian geometry is a general space
based on the Line Element
ds = F(x , . . . , x ; dx , . . . , dx ) ,
with F(x y y) > for y ^ a function on the TANGENT
BUNDLE TM. In addition, F is homogeneous of degree
1 in y and of the form
F 2 — gij(x) dx 1 dx j
(Chern 1996). If this restriction is dropped, the resulting
geometry is called FlNSLER GEOMETRY.
References
Besson, G.; Lohkamp, J.; Pansu, P.; and Petersen, P. Rie-
mannian Geometry. Providence, RI: Amer. Math. Soc,
1996.
Buser, P. Geometry and Spectra of Compact Riemann Sur-
faces. Boston, MA: Birkhauser, 1992.
Chavel, I. Eigenvalues in Riemannian Geometry. New York:
Academic Press, 1984.
Chavel, I. Riemannian Geometry: A Modern Introduction.
New York: Cambridge University Press, 1994.
Chern, S.-S. "Finsler Geometry is Just Riemannian Geome-
try without the Quadratic Restriction." Not. Amer. Math.
Soc. 43, 959-963, 1996.
do Carmo, M. P. Riemannian Geometry. Boston, MA: Birk-
hauser, 1992.
Riemannian Geometry (Non-Euclidean)
see Elliptic Geometry
Riemannian Manifold
A Manifold possessing a Metric Tensor. For a com-
plete Riemannian manifold, the METRIC d(x^y) is de-
fined as the length of the shortest curve (GEODESIC)
between x and y.
see also Bishop's Inequality, Cheeger's Finiteness
Theorem
Riemannian Metric
Suppose for every point i in a COMPACT MANIFOLD
M, an Inner Product {-, -) x is defined on a Tangent
Space T x M of M at x. Then the collection of all these
INNER PRODUCTS is called the Riemannian metric. In
1870, ChristofFel and Lipschitz showed how to decide
when two Riemannian metrics differ by only a coordi-
nate transformation.
see also Compact Manifold, Line Element, Metric
Tensor
Riesel Number
There exist infinitely many Odd Integers k such that
k-2 n -l is Composite for every n > 1. Numbers k with
this property are called RlESEL NUMBERS, and anal-
ogous numbers with the minus sign replaced by a plus
are called SlERPINSKI NUMBERS OF THE SECOND KIND.
The smallest known Riesel number is k = 509,203, but
there remain 963 smaller candidates (the smallest of
which is 659) which generate only composite numbers for
all n which have been checked (Ribenboim 1996, p. 358).
Let a(k) be smallest n for which (2A:-l)-2 n -l is PRIME,
then the first few values are 2, 0, 2, 1, 1, 2, 3, 1, 2, 1, 1,
4, 3, 1, 4, 1, 2, 2, 1, 3, 2, 7, . . . (Sloane's A046069), and
second smallest n are 3, 1, 4, 5, 3, 26, 7, 2, 4, 3, 2, 6, 9,
2, 16, 5, 3, 6, 2553, . . . (Sloane's A046070).
see also CUNNINGHAM NUMBER, MERSENNE NUMBER,
Sierpinski's Composite Number Theorem, Sier-
pinski Number of the Second Kind
References
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, p. 357, 1996.
Riesz-Fischer Theorem
Rigby Points 1567
Riesel, H. "Nagra stora primtal." Elementa 39, 258-260,
1956.
Sloane, N. J. A. Sequence A046068 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Riesz-Fischer Theorem
A function is L 2 - (square-) integrable Iff its Fourier
Series is L2 -convergent. The application of this theo-
rem requires use of the Lebesgue Integral.
see also Lebesgue Integral
Riesz Representation Theorem
Let / be a bounded linear FUNCTIONAL on a HlLBERT
SPACE H. Then there exists exactly one xo G H such
that f(x) = (x,x ) for all x 6 H . Also, ||/|| = ||x ||.
see also Functional, Hilbert Space
References
Debnath, L. and Mikusiriski, P. Introduction to Hilbert
Spaces with Applications. San Diego, CA: Academic Press,
1990.
Riesz's Theorem
Every continuous linear functional U[f] for / 6 C[a, b]
can be expressed as a Stieltjes Integral
u[f]
f
J a
f(x) dw(x)j
where w(x) is determined by U and is of bounded vari-
ation on [a, b] .
see also Stieltjes Integral
References
Kestelman, H. "Riesz's Theorem." §11.5 in Modern Theories
of Integration, 2nd rev, ed. New York: Dover, pp. 265-269,
1960.
Riffle Shuffle
A Shuffle, also called a Faro Shuffle, in which a
deck of 2n cards is divided into two Halves which are
then alternatively interleaved from the left and right
hands (an "in-shume" ) or from the right and left hands
(an "out-shuffle"). Using an "in-shuffle," a deck origi-
nally arranged as 12345678 would become 5 16 2 7
3 8 4. Using an "out-shuffle," the deck order would be-
come 15263748. Riffle shuffles are used in card tricks
(Mario 1958ab, Adler 1973), and also in the theory of
parallel processing (Stone 1971, Chen et al 1981).
In general, card k moves to the position originally oc-
cupied by the 2kth card (mod 2n + 1). Therefore, in-
shuffling 2n cards 2n times (where 2n + 1 is Prime) re-
sults in the original card order. Similarly, out-shuffling
2n cards 2n - 2 times (where 2n - 1 is Prime) results
in the original order (Diaconis et al 1983, Conway and
Guy 1996). Amazingly, this means that an ordinary
deck of 52 cards is returned to its original order after 8
out-shuffles.
Morris (1994) further discusses aspects of the perfect
riffle shuffle (in which the deck is cut exactly in half
and cards are perfectly interlaced). Ramnath and Scully
(1996) give an algorithm for the shortest sequence of in-
and out-shuffles to move a card from arbitrary position
i to position j. This algorithm works for any deck with
an EVEN number of cards and is O(logn).
see also CARDS, SHUFFLE
References
Adler, I. "Make Up Your Own Card Tricks." J. Recr. Math.
6, 87-91, 1973.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 323-
325, 1987.
Chen, P. Y.; Lawrie, D. H.; Yew, P.-C; and Padua, D. A.
"Interconnection Networks Using Shuffles." Computer 33,
55-64, Dec. 1981.
Conway, J. H. and Guy, R. K. "Fractions Cycle into Deci-
mals." In The Book of Numbers. New York: Springer-
Verlag, pp. 163-165, 1996.
Diaconis, P.; Graham, R. L.; and Kantor, W. M. "The Math-
ematics of Perfect Shuffles." Adv. Appl. Math. 4, 175-196,
1983.
Gardner, M. Mathematical Carnival: A New Round- Up of
Tantalizers and Puzzles from Scientific American. Wash-
ington, DC: Math. Assoc. Amer., 1989.
Herstein, I. N. and Kaplansky, I. Matters Mathematical. New
York: Harper & Row, 1974.
Mann, B. "How Many Times Should You Shuffle a Deck of
Cards." UMAP J. 15, 303-332, 1994.
Mario, E. Faro Notes. Chicago, IL: Ireland Magic Co., 1958a.
Mario, E. Faro Shuffle. Chicago, IL: Ireland Magic Co.,
1958b.
Medvedoff, S. and Morrison, K. "Groups of Perfect Shuffles."
Math. Mag. 60, 3-14, 1987.
Morris, S. B. and Hartwig, R. E. "The Generalized Faro Shuf-
fle." Discrete Math. 15, 333-346, 1976.
Peterson, I. Islands of Truth: A Mathematical Mystery
Cruise. New York: W. H. Freeman, pp. 240-244, 1990.
Ramnath, S. and Scully, D. "Moving Card i to Position j
with Perfect Shuffles." Math. Mag. 69, 361-365, 1996.
Stone, H. S. "Parallel Processing with the Perfect Shuffle."
IEEE Trans. Comput. 2, 153-161, 1971.
Rigby Points
The Perspective Centers of the Tangential and
Contact Triangles of the inner and outer Soddy
Points. The Rigby points are given by
Ri = l+%Ge
R% =1- §Ge,
where / is the INCENTER and Ge is the Gergonne
Point.
see also Contact Triangle, Gergonne Point,
Griffiths Points, Incenter, Oldknow Points,
Soddy Points, Tangential Triangle
References
Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
1568 Right Angle
Right Strophoid
Right Angle
An Angle equal to half the Angle from one end of a
line segment to the other. A right angle is 7r/2 radians
or 90°. A Triangle containing a right angle is called a
Right Triangle. However, a Triangle cannot con-
tain more than one right angle, since the sum of the two
right angles plus the third angle would exceed the 180°
total possessed by a TRIANGLE.
see also Acute Angle, Oblique Angle, Obtuse An-
gle, Right Triangle, Semicircle, Straight An-
gle, Thales' Theorem
Right Conoid
A Ruled Surface is called a right conoid if it can be
generated by moving a straight LINE intersecting a fixed
straight LINE such that the LINES are always PERPEN-
DICULAR (Kreyszig 1991, p. 87). Taking the PERPEN-
DICULAR plane as the xy-pl&ne and the line to be the
cc-AxiS gives the parametric equations
x(u, v) = i>cosi9(u)
y(u, v) = vsin$(u)
z(U)V) = h(u)
(Gray 1993). Taking h(u) = 2u and d(u) = u gives the
Helicoid.
see also Helicoid, Plucker's Conoid, Wallis's
Conical Edge
References
Dixon, R. Mathographics. New York: Dover, p. 20, 1991.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 351-352, 1993.
Kreyszig, E. Differential Geometry. New York: Dover, 1991.
Right Hyperbola
A Hyperbola for which the Asymptotes are Per-
pendicular. This occurs when the Semimajor and
SEMIMINOR AXES are equal. Taking a = b in the equa-
tion of a Hyperbola with Semimajor Axis parallel to
the x-Axis and Semiminor Axis parallel to the y-AxiS
(i.e., vertical Directrix),
Right Strophoid
(x-xq) 2 _ (y - y ) 2
a 2 b 2
= 1
therefore gives
(x — xq) 2
(y-yo) 2 = a 2 '
A special type of right hyperbola is the so-called RECT-
ANGULAR Hyperbola, which has equation xy = ab.
see also HYPERBOLA, RECTANGULAR HYPERBOLA
Right Line
see Line
The Strophoid of a line L with pole O not on L and
fixed point O' being the point where the PERPENDICU-
LAR from O to L cuts L is called a right strophoid. It is
therefore a general STROPHOID with a = n/2.
The right strophoid is given by the Cartesian equation
2 c — x 2
y = — r~ x >
c + x
or the polar equation
r = ccos(20)sec0.
The parametric form of the strophoid is
*(*) =
»(*) =
t(t 2 - i)
t 2 + i
The right strophoid has CURVATURE
4(1 + 3i 2 )
K(t) = --
(l + 6t 2 +t 4 ) 3 / 2
and Tangential Angle
<£(t) = -2 tan -1 1- tan" 1 (- -^ J
(1)
(2)
(3)
(4)
(5)
(6)
The right strophoid first appears in work by Isaac Bar-
row in 1670, although Torricelli describes the curve in
his letters around 1645 and Roberval found it as the LO-
CUS of the focus of the conic obtained when the plane
cutting the Cone rotates about the tangent at its vertex
(MacTutor Archive). The Area of the loop is
Aioop = fc 2 (4-7r)
(7)
(MacTutor Archive).
Let C be the CIRCLE with center at the point where
the right strophoid crosses the x-axis and radius the
distance of that point from the origin. Then the right
strophoid is invariant under inversion in the CIRCLE C
and is therefore an Anallagmatic Curve.
see also Strophoid, Trisectrix
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 71, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 100-104, 1972.
Lockwood, E. H. "The Right Strophoid." Ch. 10 in A Book
of Curves. Cambridge, England: Cambridge University
Press, pp. 90-97, 1967.
MacTutor History of Mathematics Archive. "Right Stro-
phoid." http : //www-groups . dcs . st-and . ac . uk/-history
/Curves/Right .html.
Right Strophoid Inverse Curve
Right Strophoid Inverse Curve
Rigid 1569
The INVERSE Curve of a right strophoid is the same
strophoid.
Right Triangle
A Triangle with an Angle of 90° (tt/2 radians). The
sides a, 6, and c of such a TRIANGLE satisfy the PY-
THAGOREAN THEOREM. The largest side is convention-
ally denoted c and is called the HYPOTENUSE.
For any three similar shapes on the sides of a right tri-
angle,
Ai + A 2 = As, (1)
which is equivalent to the PYTHAGOREAN THEOREM.
For a right triangle with sides a, 6, and HYPOTENUSE c,
let r be the INRADIUS. Then
\ab = \ra + \rb + \rc = \r{a + 6 + c).
Solving for r gives
ab
r — ; .
a + b + c
But any PYTHAGOREAN TRIPLE can be written
2 2
a — m — n
b = 2mn
so (5) becomes
(m — n )2mn
(2)
(3)
(4)
(5)
(6)
= n(m — n), (7)
m 2 — n 2 + 2mn + m 2 + n 2
which is an INTEGER when m and n are integers.
C
Given a right triangle AABC, draw the Altitude AH
from the Right ANGLE A. Then the triangles AAHC
and ABHA are similar.
In a right triangle, the MIDPOINT of the HYPOTENUSE
is equidistant from the three VERTICES (Dunham 1990).
This can be proved as follows. Given AABC, let M
be the MIDPOINT of AB (so that AM = BM). Draw
DM\\CA, then since ABDM is similar to ABC A, it
follows that BD = DC. Since both ABDM and
ACDM are right triangles and the corresponding legs
are equal, the HYPOTENUSES are also equal, so we have
AM — BM = CM and the theorem is proved.
see also Acute Triangle, Archimedes' Midpoint
Theorem, Brocard Midpoint, Circle-Point Mid-
point Theorem, Fermat's Right Triangle Theo-
rem, Isosceles Triangle, Malfatti's Right Tri-
angle Problem, Obtuse Triangle, Pythagorean
Triple, Quadrilateral, RAT-Free Set, Triangle
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 121, 1987.
Dunham, W. Journey Through Genius: The Great Theorems
of Mathematics. New York: Wiley, pp. 120-121, 1990.
Rigid
A Framework is rigid Iff continuous motion of the
points of the configuration maintaining the bar con-
straints comes from a family of motions of all EUCLID-
EAN Space which are distance-preserving. A Graph G
is (generically) d-rigid if, for almost all (i.e., an open
dense set of) CONFIGURATIONS of p, the FRAMEWORK
G(p) is rigid in R d .
One of the first results in rigidity theory was the RIGID-
ITY THEOREM by Cauchy in 1813. Although rigidity
problems were of immense interest to engineers, inten-
sive mathematical study of these types of problems has
occurred only relatively recently (Connelly 1993, Graver
et al. 1993).
see also Bar (Edge), Flexible Polyhedron, Frame-
work, Laman's Theorem, Liebmann's Theorem,
Rigidity Theorem
References
Connelly, R. "Rigidity." Ch. 1.7 in Handbook of Convex Ge-
ometry, Vol. A (Ed. P. M. Gruber and J. M. Wills). Am-
sterdam, Netherlands: North-Holland, pp. 223-271, 1993.
Crapo, H. and Whiteley, W. "Statics of Frameworks and Mo-
tions of Panel Structures, A Projective Geometry Introduc-
tion." Structural Topology 6, 43-82, 1982.
1570 Rigid Motion
Ring Cyclide
Graver, J.; Servatius, B.; and Servatius, H. Combinatorial
Rigidity. Providence, RI: Amer. Math. Soc, 1993.
Rigid Motion
A transformation consisting of ROTATIONS and TRANS-
LATIONS which leaves a given arrangement unchanged.
see also Euclidean Motion, Plane, Rotation
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. 141, 1996.
Rigidity Theorem
If the faces of a convex POLYHEDRON were made of
metal plates and the EDGES were replaced by hinges,
the Polyhedron would be Rigid. The theorem was
stated by Cauchy (1813), although a mistake in this pa-
per went unnoticed for more than 50 years. An example
of a concave "FLEXIBLE POLYHEDRON" (with 18 trian-
gular faces) for which this is not true was given by Con-
nelly (1978), and a Flexible Polyhedron with only
14 triangular faces was subsequently found by StefFen
(Mackenzie 1998).
see also FLEXIBLE POLYHEDRON, RIGID
References
Cauchy, A. L. "Sur les polygons et le polyheders." XVIe
Cahier\yi, 87-89, 1813.
Connelly, R. "A Flexible Sphere." Math. Intel 1, 130-131,
1978.
Graver, J.; Servatius, B.; and Servatius, H. Combinatorial
Rigidity. Providence, RI: Amer. Math. Soc, 1993.
Mackenzie, D. "Polyhedra Can Bend But Not Breathe." Sci-
ence 279, 1637, 1998.
Ring
A ring is a set together with two Binary Operators
£(+, *) satisfying the following conditions:
1. Additive associativity: For all a,b,c 6 5, (a+6) + c =
a+(b + c),
2. Additive commutativity: For all a, b £ S, a + b =
6 + a,
3. Additive identity: There exists an element £ S
such that for all a E 5, + a = a + = a,
4. Additive inverse: For every a 6 S there exists —ainS
such that a + (—a) = (—a) + a = 0,
5. Multiplicative associativity: For all a,b,c £ 5, (a*
b) * c = a * (6 * c),
6. Left and right distributivity: For all a,b,c G 5, a *
(6+c) = (a*b) + (a*c) and (6+c)*a = (6*a) + (c*a).
A ring is therefore an Abelian GROUP under addition
and a SEMIGROUP under multiplication. A ring must
contain at least one element, but need not contain a
multiplicative identity or be commutative. The number
of finite rings of n elements for n = 1, 2, ... , are 1, 2, 2,
11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, . . . (Sloane's A027623
and A037234; Fletcher 1980). In general, the number of
rings of order p z for p an ODD PRIME is 3p + 50 and 52
for p = 2 (Ballieu 1947, Gilmer and Mott 1973).
A ring with a multiplicative identity is sometimes called
a Unit Ring. Fraenkel (1914) gave the first abstract
definition of the ring, although this work did not have
much impact.
A ring that is COMMUTATIVE under multiplication, has
a unit element, and has no divisors of zero is called an
Integral Domain. A ring which is also a Commuta-
tive multiplication group is called a Field. The sim-
plest rings are the INTEGERS Z, POLYNOMIALS R[x] and
R[x, y] in one and two variables, and Square nxn REAL
Matrices.
Rings which have been investigated and found to be of
interest are usually named after one or more of their in-
vestigators. This practice unfortunately leads to names
which give very little insight into the relevant properties
of the associated rings.
see also Abelian Group, Artinian Ring, Chow
Ring, Dedekind Ring, Division Algebra, Field,
Gorenstein Ring, Group, Group Ring, Ideal,
Integral Domain, Module, Nilpotent Element,
Noetherian Ring, Number Field, Prime Ring,
Prufer Ring, Quotient Ring, Regular Ring,
Ringoid, Semiprime Ring, Semiring, Semisimple
Ring, Simple Ring, Unit Ring, Zero Divisor
References
Ballieu, R. "Anneaux finis; systemes hypercomplexes de rang
trois sur un corps commutatif." Ann. Soc. Sci. Bruxelles.
Ser. 7 61, 222-227, 1947.
Fletcher, C. R. "Rings of Small Order." Math. Gaz. 64,
9-22, 1980. mm
Fraenkel, A. "Uber die Teiler der Null und die Zerlegung von
Ringen." J. Reine Angew. Math. 145, 139-176, 1914.
Gilmer, R. and Mott, J. "Associative Rings of Order p 3 ."
Proc. Japan Acad. 49, 795-799, 1973.
Kleiner, I. "The Genesis of the Abstract Ring Concept."
Amer. Math. Monthly 103, 417-424, 1996.
Sloane, N. J. A. Sequences A027623 and A037234 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
van der Waerden, B. L. A History of Algebra. New York:
Springer- Verlag, 1985.
Ring Cyclide
The inversion of a Ring Torus, If the inversion center
lies on the torus, then the ring cyclide degenerates to a
Parabolic Ring Cyclide.
Ring Function
Robbins Equation 1571
see also Cyclide, Parabolic Cyclide, Ring Cy-
clide, Ring Torus, Spindle Cyclide, Torus
Ring Function
see Toroidal Function
Rising Factorial
see POCHHAMMER SYMBOL
Rivest- Shamir- Adleman Number
see RSA Number
Ring Torus
RMS
see Root-Mean-Square
Robbin Constant
One of the three STANDARD TORI given by the para-
metric equations
x — (c + a cos v) cos u
y — (c + a cos v) sin u
z = a sin v
with c> a. This is the TORUS which is generally meant
when the term "torus" is used without qualification.
The inversion of a ring torus is a RING CYCLIDE if the
Inversion Center does not lie on the torus and a Par-
abolic Ring Cyclide if it does. The above left figure
shows a ring torus, the middle a cutaway, and the right
figure shows a CROSS-SECTION of the ring torus through
the #2-plane.
see also Cyclide, Horn Torus, Parabolic Ring Cy-
clide, Ring Cyclide, Spindle 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.
Ringoid
A ringoid R is a set (fi, +, x) with two binary operators,
conventionally denoted addition (+) and multiplication
(x), where x distributes over + left and right:
a(b + c) = ab + acand(b + c)a — ba -j- ca.
A ringoid can be empty.
see also Binary Operator, Ring, Semiring
References
Rosenfeld, A. An Introduction to Algebraic Structures. New
York: Holden-Day, 1968,
Risch Algorithm
An Algorithm for indefinite integration.
see also INDEFINITE INTEGRAL
R:
4
105
+ T75fV2-
17
105
V3+f ln(l + \/2)
+ f ln(2 + y/Z) - j^tt = 0.661707182 .
see also TRANSFINITE DIAMETER
References
Plouffe, S. "The Robbin Constant." http://lacim.uqam.ca/
piDATA/robbin.txt.
Robbin's Inequality
If the fourth MOMENT fi 4 ^ 0, then
P{\x-fi 4 \ > A) <
where a 2 is the VARIANCE.
^4+3(iV- 1)<T 4
AT 3 A 4
Robbins Algebra
Building on work of Huntington (1933), Robbins con-
jectured that the equations for a Robbins algebra, com-
mutivity, associativity, and the ROBBINS EQUATION
n(n(x + y) + n(x + n(y))) = x,
imply those for a BOOLEAN ALGEBRA. The conjecture
was finally proven using a computer (McCune 1997).
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.
McCune, W. "Solution of the Robbins Problem." J. Au-
tomat. Reason. 19, 263-276, 1997.
McCune, W. "Robbins Algebras are Boolean." http://www.
mcs.anl.gov/-mccune/papers/robbins/.
Nelson, E. "Automated Reasoning." http : //www . math .
pr inc et on . e du/ ~ne 1 s on/ ar . html .
Robbins Equation
n(n(x 4- y) + n(x + n(y))) = x.
see also ROBBINS ALGEBRA
1572
Robertson Condition
Rogers-Ramanujan Continued Fraction
Robertson Condition
For the Helmholtz Differential Equation to be
Separable in a coordinate system, the SCALE FACTORS
hi in the Laplacian
^ hj
1 d (h x h 2 hz d
h 2 hz dm \ hi 2 dui
(i)
and the functions fi(ui) and $^ defined by
^^(fn^)+(kl 2 $ n l+k 2 2 $n2+ks 2 <S> n 3)X n =0
fn OUn \ OUn J
(2)
must be of the form of a Stackel Determinant
5=|* n
$11 $12 $13
$21 $22 $23
$31 $32 $33
h\h2h$
fl(ui)f 2 (U2)h(us)'
(3)
see also HELMHOLTZ DIFFERENTIAL EQUATION, LA-
place's Equation, Separation of Variables,
Stackel Determinant
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, p. 510, 1953.
Robertson Conjecture
A conjecture due to M. S. Robertson (1936) which treats
a Univalent Power Series containing only Odd pow-
ers within the UNIT Disk. This conjecture IMPLIES the
BlEBERBACH CONJECTURE and follows in turn from the
Milin Conjecture, de Branges' proof of the Bieber-
bach Conjecture proceeded by proving the Milin
Conjecture, thus establishing the Robertson conjec-
ture and hence implying the truth of the BlEBERBACH
Conjecture.
see also BlEBERBACH CONJECTURE, MlLIN CONJEC-
TURE
References
Stewart, I. From Here to Infinity: A Guide to Today's
Mathematics. Oxford, England: Oxford University Press,
p. 165, 1996.
Robertson- Seymour Theorem
A generalization of the KURATOWSKI REDUCTION THE-
OREM by Robertson and Seymour, which states that the
collection of finite graphs is well-quasi-ordered by minor
embeddability, from which it follows that Kuratowski's
"forbidden minor" embedding obstruction generalizes to
higher genus surfaces.
Formally, for a fixed INTEGER g > 0, there is a finite
list of graphs L(g) with the property that a graph C
embeds on a surface of genus g Iff it does not contain,
as a minor, any of the graphs on the list L.
References
Fellows, M. R. "The Robertson-Seymour Theorems: A Sur-
vey of Applications." Comtemp. Math. 89, 1-18, 1987.
Robin Boundary Conditions
Partial Differential Equation Boundary Condi-
tions which, for an elliptic partial differential equation
in a region fB, specify that the sum of an and the normal
derivative of u = / at all points of the boundary of fi,
a and / being prescribed.
Robin's Constant
see Transfinite Diameter
Robinson Projection
A PSEUDOCYLINDRICAL MAP PROJECTION which dis-
torts shape, AREA, scale, and distance to create attrac-
tion average projection properties.
References
Dana, P. H. "Map Projections." http://www.utexas.edu/
depts/grg/gcraft/notes/mapproj/mapproj .html.
Robust Estimation
An estimation technique which is insensitive to small
departures from the idealized assumptions which have
been used to optimize the algorithm. Classes of
such techniques include M-ESTIMATES (which fol-
low from maximum likelihood considerations), L-
ESTIMATES (which are linear combinations of Order
Statistics), and ^-Estimates (based on Rank tests).
see also L-Estimate, M-Estimate, .R-Estimate
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Robust Estimation." §15.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 694-700, 1992.
Rodrigues's Curvature Formula
dN + Ki dx = 0,
where N is the unit NORMAL VECTOR and k» is one of
the two Principal Curvatures.
see also Normal Vector, Principal Curvatures
Rodrigues Formula
An operator definition of a function. A Rodrigues for-
mula may be converted into a Schlafli INTEGRAL.
see also SCHLAFLI INTEGRAL
Rogers-Ramanujan Continued Fraction
see Ramanujan Continued Fraction
Rogers-Ramanujan Identities
Roman Coefficient 1573
Rogers-Ramanujan Identities
For \q\ < 1 and using the NOTATION of the RAMANUJAN
THETA FUNCTION, the Rogers-Ramanujan identities are
1L, [ a \u
/(~g 5 )
H-q 2 ,-Q 3 )
/(-g s ) = y
*(fc+l)
(i)
(2)
where (q)k are ^-SERIES. Written out explicitly (Hardy
1959, p. 13),
1 +
1 +
l-<? (l-g)(l-<7 2 ) (l-<?)(l-S 2 )(l-<7 3 )
1
(1 -g)(l- « fl ) ■■■(!- ^Kl-flf 9 )-
- + :
: + :
1-9 (i-9)(i-* 2 ) (i-<?)(i-g 2 )(i-<? 3 )
1
(3)
+ ...
(i-g 2 )(i-<? 7 )---(i-s 3 )(i-</ 8 )--
The identities can also be written succinctly as
k 2 +ak
■ (4)
1 +
Ea
(l_ g )(l-g*). ..(!-«*)
=n n
j=0
(l-g5j + a+l)( 1 _ g 5i-a+4)»
(5)
where a = 0, 1.
Other forms of the Rogers-Ramanujan identities include
E
, (r,q)k(q;q)n-k z -^ (q;q) n -k(q;q)n+k
k k
and
?(«;
2g fc
;g)fe (?; <?)«-«:
= E
(-l) fc (l + g fc )^ 5fc2 - fc)/2
(9;g)n-*(<?;<?)n+fc
(7)
(Petkovsek et al. 1996).
see also ANDREWS-SCHUR IDENTITY
References
Andrews, G. E. The Theory of Partitions. Cambridge, Eng-
land: Cambridge University Press, 1985.
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. 17-20, 1986.
Andrews, G. E. and Baxter, R. J. "A Motivated Proof of the
Rogers-Ramanujan Identities." Amer. Math. Monthly 96,
401-409, 1989.
Bressoud, D. M. Analytic and Combinatorial Generaliza-
tions of the Rogers-Ramanujan Identities. Providence, RI:
Amer. Math. Soc, 1980.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 13, 1959.
Paule, P "Short and Easy Computer Proofs of the Rogers-
Ramanujan Identities and of Identities of Similar Type."
Electronic J. Combinatorics 1, R10, 1-9, 1994. http://
www. comb inatorics.org/Volume_l/volumel. html#R10.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, p. 117, 1996.
Robinson, R. M. "Comment to: 'A Motivated Proof of the
Rogers-Ramanujan Identities.'" Amer. Math. Monthly 97,
214-215, 1990.
Rogers, L. J. "Second Memoir on the Expansion of Certain
Infinite Products." Proc. London Math. Soc. 25, 318-343,
1894.
Sloane, N. J. A. Sequence A006141/M0260 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Rolle's Theorem
Let / be differentiate on (a, b) and continuous on [a, b].
If /(a) = /(b) = 0, then there is at least one point
c € (a, b) where /'(c) = 0.
see also FIXED POINT THEOREM, MEAN- VALUE THEO-
REM
Roman Coefficient
A generalization of the Binomial Coefficient whose
Notation was suggested by Knuth,
[n}\
[k]\[n-k]V
(1)
The above expression is read "Roman n choose &."
Whenever the BINOMIAL COEFFICIENT is defined (i.e.,
n > fc > or fc > > n), the Roman coefficient agrees
with it. However, the Roman coefficients are defined for
values for which the BINOMIAL COEFFICIENTS are not,
e.g.,
where
n
-1
k
n <
1
={i
L» + 11
(_l) fc +( fc >°)
[*1 '
for n <
for n > 0.
(2)
(3)
(4)
The Roman coefficients also satisfy properties like those
of the Binomial Coefficient,
(5)
n
n
k
n — k
n
k
n
n
— r
k
r
r
k
— r
an analog of PASCAL'S FORMULA
n
k
=
n- l"
k
+
n- r
k-1
(6)
(7)
1574 Roman Factorial
and a curious rotation/reflection law due to Knuth
(-i)
fc+(fc>0)
—n
fc-1
= (-l)"
n+(n>0)
-A:
n-1
(8)
(Roman 1992).
see also Binomial Coefficient, Roman Factorial
References
Roman, S. "The Logarithmic Binomial Formula." Amer.
Math. Monthly 99, 641-648, 1992.
Roman Factorial
(n\
[n]\=\ (-D — 1
I (-n-l)!
for n >
for n < 0.
(i)
The Roman factorial arises in the definition of the HAR-
MONIC Logarithm and Roman Coefficient. It obeys
the identities
[nl!=lnlln-l"|! (2)
Lnl!
T = [nl [n - 11 • • • [n - k + 11 (3)
(4)
[n-k]
Lnl!L-n-ll! = (-l)" +( " <0)
where
and
w-{J
for n #
for n =
n < 0= |
1 for n <
for n > 0.
(5)
(6)
see a/so HARMONIC LOGARITHM, HARMONIC NUMBER,
Roman Coefficient
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.
Roman Numeral
A system of numerical notations used by the Romans. It
is an additive (and subtractive) system in which letters
are used to denote certain "base" numbers, and arbi-
trary numbers are then denoted using combinations of
symbols.
Character Numerical Value
I
V
X
L
C
D
M
1
5
10
50
100
500
1000
For example, the number 1732 would be denoted MD-
CCXXXII. One additional rule states that, instead of
using four symbols to represent a 4, 40, 9, 90, etc., such
numbers are instead denoted by preceding the symbol
Roman Surface
for 5, 50, 10, 100, etc., with a symbol indicating subtrac-
tion. For example, 4 is denoted IV, 9 as IX, 40 as XL,
etc. However, this rule is generally not followed on the
faces of clocks, where IIII is usually encountered instead
of IV.
Roman numerals are encountered in the release year for
movies and occasionally on the numerals on the faces of
watches and clocks, but in few other modern instances.
They do have the advantage that Addition can be done
"symbolically" (and without worrying about the "place"
of a given Digit) by simply combining all the symbols
together, grouping, writing groups of 5 Is as V, groups
of 2 Vs as X, etc.
Roman Surface
A Quartic Nonorientable Surface, also known as
the Steiner Surface. The Roman surface is one of
the three possible surfaces obtained by sewing a MOBIUS
Strip to the edge of a Disk. The other two are the Boy
Surface and Cross-Cap, all of which are homeomor-
phic to the Real Projective Plane (Pinkall 1986).
The center point of the Roman surface is an ordi-
nary Triple Point with (±1,0,0) = (0,±1,0) =
(0,0, ±1), and the six endpoints of the three lines of
self-intersection are singular PINCH POINTS, also known
as Whitney Singularities. The Roman surface is es-
sentially six Cross-Caps stuck together and contains a
double Infinity of Conics.
The Roman surface can given by the equation
(x 2 +y 2 + z 2 -k 2 ) 2 = [(^-fc) 2 -2 a : 2 ][(^ + fc) 2 -2y 2 ]. (1)
Solving for z gives the pair of equations
k(y 2 - x 2 ) ± (x 2 - y 2 )y/V -x*-y*
2(x 2 + y 2 )
(2)
If the surface is rotated by 45° about the 2- Axis via the
Rotation Matrix
Rz(45°)
x/2
1
1 0]
-1
1
1.
to give
= R*(45°)
(3)
(4)
Roman Surface
Rook Number
1575
then the simple equation
x 2 y 2 + x 2 z 2 + y 2 z 2 + 2kxyz = (5)
results. The Roman surface can also be generated us-
ing the general method for NONORIENTABLE SURFACES
using the polynomial function
f(x,y,z) = (xy,yz,zx)
(Pinkall 1986). Setting
x = cos u sin v
y = sin u sin v
z = cos V
(6)
(7)
(8)
(9)
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, p. 19, 1986.
Fischer, G. (Ed.). Plates 42™44 and 108-114 in Mathematis-
che Modelle/ Mathematical Models, Bildband/ Photograph
Volume. Braunschweig, Germany: Vieweg, pp. 42-44 and
108-109, 1986,
Geometry Center. "The Roman Surface." http://www.geom.
umn.edu/zoo/toptype/pplane/roman/.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 242-243, 1993.
Nordstrand, T. "Steiner's Roman Surface." http://www.
uib.no/people/nfytn/steintxt.htm.
Pinkall, U. Mathematical Models from the Collections of Uni-
versities and Museums (Ed. G. Fischer). Braunschweig,
Germany: Vieweg, p. 64, 1986.
Wang, P. "Renderings." http://www.ugcs.caltech.edu/
-pet erw/portf olio/renderings/.
in the former gives
Roman Symbol
x(u, v)
\(2u) sir
u, v) = |sinucos(2u)
z(UjV) = | costtsin(2^)
(10)
(11)
(12)
for u G [0,27r) and v G [— 7r/2,7r/2]. Flipping sinu
and qosv and multiplying by 2 gives the form shown
by Wang.
A HOMOTOPY (smooth deformation) between the Ro-
man surface and BOY SURFACE is given by the equa-
tions
x(u,v) =
y(u y v) =
z(u,v) =
\/2cos(2ti) cos 2 v + cosusin(2i;)
2 - aV2sin(3u)sin(2t;)
V / 2sin(2n) cos 2 v — sinwsin(2i;)
2 — ay2 sin(3w) sin(2u)
3 cos 2 v
2 -av / 2sin(3u)sin(2i;)
(13)
(14)
(15)
for u G [— 7r/2,7r/2] and v G [0, 7r] as a varies from to
1. a = corresponds to the Roman surface and a = 1
to the Boy Surface (Wang).
see also Boy Surface, Cross-Cap, Heptahedron,
Mobius Strip, Nonorientable Surface, Quartic
Surface, Steiner Surface
w-{;
n for n/0
see also Roman Factorial, Harmonic Logarithm
References
Roman, S. "The Logarithmic Binomial Formula." Amer.
Math. Monthly 99, 641-648, 1992.
Romberg Integration
A powerful NUMERICAL INTEGRATION technique which
uses k refinements of the extended Trapezoidal Rule
to remove error terms less than order D{N~ 2k ). The
routine advocated by Press et ai. (1992) makes use of
Neville's Algorithm.
References
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., pp. 106-107, 1990.
Dahlquist, G. and Bjorck, A. §7.4.1-7.4.2 in Numerical Meth-
ods. Englewood Cliffs, NJ: Prentice-Hall, 1974.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Romberg Integration." §4,3 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 134-135, 1992.
Ralston, A. and Rabinowitz, P. §4.10 in A First Course in
Numerical Analysis, 2nd ed. New York: McGraw-Hill,
1978.
Stoer, J.; and Burlisch, R. §3.4-3.5 in Introduction to Nu-
merical Analysis. New York: Springer- Verlag, 1980.
Rook Number
The rook numbers r% of an n x n BOARD B are the
number of subsets of size n such that no two elements
have the same first or second coordinate. In other word,
it is the number of ways of placing n rooks on B such
that none attack each other. The rook numbers of a
board determine the rook numbers of the complemen-
tary board £?, defined to be d x d\B. This is known
as the Rook Reciprocity Theorem. The first few
rook numbers are 1, 2, 7, 23, 115, 694, 5282, 46066, . . .
(Sloane's A000903). For an n x n board, each n x n
1576 Rook Reciprocity Theorem
Room Square
Permutation Matrix corresponds to an allowed con-
figuration of rooks.
see also ROOK RECIPROCITY THEOREM
References
Sloane, N. J. A. Sequence A000903/M1761 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Rook Reciprocity Theorem
£rj?(d -*)!**
£(-
l) k ri(d-
k)\x k (x + l) d -
References
Chow, T. Y. "The Path-Cycle Symmetric Function of a Di-
graph." Adv. Math, 118, 71-98, 1996.
Chow, T. "A Short Proof of the Rook Reciprocity Theorem."
Electronic J. Combinatorics 3, RIO, 1-2, 1996. http://
www . combinatorics . org/Volume J/volume3 .html#R10.
Goldman, J. R.; Joichi, J. T.; and White, D. E. "Rook The-
ory I. Rook Equivalence of Ferrers Boards." Proc. Amcr.
Math. Soc. 52, 485-492, 1975.
Riordan, J. An Introduction to Combinatorial Analysis. New
York: Wiley, 1958.
Rooks Problem
R
R
R
R
R
R
R
R
The rook is a CHESS piece which may move any num-
ber of spaces either horizontally or vertically per move.
The maximum number of nonattacking rooks which may
be placed on an n x n CHESSBOARD is n. This arrange-
ment is achieved by placing the rooks along the diagonal
(Madachy 1979). The total number of ways of placing
n nonattacking rooks on an n x n board is n! (Madachy
1979, p. 47). The number of rotationally and reflectively
inequivalent ways of placing n nonattacking rooks on
an n x n board are 1, 2, 7, 23, 115, 694, . . . (Sloane's
A000903; Dudeney 1970, p. 96; Madachy 1979, pp. 46-
54).
R
R
R
R
R
R
R
R
The minimum number of rooks needed to occupy or at-
tack all spaced on an 8 x 8 Chessboard is 8, illustrated
above (Madachy 1979).
Consider an n x n chessboard with the restriction that,
for every subset of {1, . . . , n}, a rook may not be put
in column s + j (mod n) when on row j, where the rows
are numbered 0, 1, ..., n — 1. Vardi (1991) denotes
the number of rook solutions so restricted as rook(s,n).
rook({l},n) is simply the number of DERANGEMENTS
on n symbols, known as a SUBFACTORIAL. The first few
values are 1, 2, 9, 44, 265, 1854, . . . (Sloane's A000166).
rook({l,2},n) is a solution to the Married Couples
Problem, sometimes known as MENAGE NUMBERS.
The first few MENAGE NUMBERS are -1, 1, 0, 2, 13,
80, 579, ... (Sloane's A000179).
Although simple formulas are not known for general {1,
. . . , p}, Recurrence Relations can be used to com-
pute rook({l, . . . ,p},n) in polynomial time for p = 3,
. . . , 6 (Metropolis et ah 1969, Mine 1978, Vardi 1991).
see also Chess, Menage Number, Rook Number,
Rook Reciprocity Theorem
References
Dudeney, H. E. "The Eight Rooks." §295 in Amusements in
Mathematics. New York: Dover, p. 88, 1970.
Kraitchik, M. "The Problem of the Rooks" and "Domina-
tion of the Chessboard." §10.2 and 10.4 in Mathematical
Recreations. New York: W. W. Norton, pp. 240-247 and
255-256, 1942.
Madachy, J. S. Madachy 's Mathematical Recreations. New
York: Dover, pp. 36-37, 1979.
Metropolis, M.; Stein, M. L.; and Stein, P. R. "Permanents
of Cyclic (0, 1) Matrices." J. Combin. Th. 7, 291-321,
1969.
Mine, H. §3.1 in Permanents. Reading, MA: Addison- Wesley,
1978.
Riordan, J. Chs. 7-8 in An Introduction to Combinatorial
Analysis. Princeton, NJ: Princeton University Press, 1978.
Sloane, N. J. A. Sequences A000903/M1761, A000166/
M1937, and A000179/M2062 in "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, Extended entry for M2062 in
The Encyclopedia of Integer Sequences. San Diego: Aca-
demic Press, 1995.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, pp. 123-124, 1991.
Room Square
A Room square (named after T. G. Room) of order n
(for n Odd) is an arrangement in an n x n SQUARE
Matrix of n + 1 objects such that each cell is either
empty or holds exactly two different objects. Further-
more, each object appears once in each row and column
and each unordered pair occupies exactly one cell. The
Room square of order 2 is shown below.
1,2
The Room square of order 8 is
Root
Root-Mean-Square 1577
1,8
5,7
3,4
2,6
3,7
2,8
6,1
4,5
5,6
4,1
3,8
7,2
6,7
5,2
4,8
1,3
2,4
7,1
6, 3
5 ,8
3, 5
1,2
7,4
6,8
4,6
2, 3
1,5
7,8
References
Dinitz, J. H. and Stinson, D. R. In Contemporary Design
Theory: A Collection of Surveys (Ed. J. H. Dinitz and
D. R. Stinson). New York: Wiley, 1992.
Gardner, M. Time Travel and Other Mathematical Bewil-
derments. New York: W. H. Freeman, pp. 146-147 and
151-152, 1988.
Mullin, R. C. and Nemeth, E. "On Furnishing Room
Squares." J. Combin. Th. 7, 266-272, 1969.
Mullin, R. D. and Wallis, W. D. "The Existence of Room
Squares." Aequationes Math. 13, 1-7, 1975.
O'Shaughnessy, C. D. "On Room Squares of Order 6m + 2."
J. Combin. Th. 13, 306-314, 1972.
Room, T. G. "A New Type of Magic Square" (Note 2569).
Math. Gaz. 39, 307, 1955.
Wallis, W. D. "Solution of the Room Square Existence Prob-
lem." J. Combin. Th. 17, 379-383, 1974.
Root
The roots of an equation
f(x) =
(1)
are the values of x for which the equation is satisfied.
The Fundamental Theorem of Algebra states that
every POLYNOMIAL equation of degree n has exactly n
roots, where some roots may have a multiplicity greater
than 1 (in which case they are said to be degenerate).
To find the nth roots of a Complex Number, solve the
equation z n = w. Then
z n = \z\ n [cos(n6) + ism(n9)] = \w\ (cos <j> -\- i sin <(>) , (2)
Laguerre's Method, Lambert's Method, Lehmer-
Schur Method, Lin's Method, Maehly's Proce-
dure, Muller's Method, Newton's Method, Rid-
ders' Method, Root Dragging Theorem, Schro-
der's Method, Polynomial, Secant Method,
Sturm Function, Sturm Theorem, Tangent Hy-
perbolas Method, WeierstraB Approximation
Theorem
References
Arfken, G. "Appendix 1: Real Zeros of a Function." Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 963-967, 1985.
Boyer, C. B. A History of Mathematics. New York: Wiley,
1968.
Householder, A. S. The Numerical Treatment of a Single
Nonlinear Equation. New York: McGraw-Hill, 1970.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Roots of Polynomials." §9.5 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 362-372, 1992.
Root Dragging Theorem
If any of the ROOTS of a POLYNOMIAL are increased,
then all of the critical points increase.
References
Anderson, B. "Polynomial Root Dragging." Amer. Math.
Monthly 100, 864-866, 1993.
Root Linear Coefficient Theorem
The sum of the reciprocals of ROOTS of an equation
equals the NEGATIVE COEFFICIENT of the linear term
in the MACLAURIN SERIES.
see also Newton's Relations
Root-Mean- Square
The root-mean-square (RMS) of a variate Z, sometimes
called the Quadratic Mean, is the SQUARE ROOT of
the mean squared value of x:
and
\z\ = |H 1/n
arg(z)
(3)
(4)
Rolle proved that any number has n nth roots (Boyer
1968, p. 476). Householder (1970) gives an algorithm for
constructing root-finding algorithms with an arbitrary
order of convergence. Special root-finding techniques
can often be applied when the function in question is a
Polynomial.
see also Bailey's Method, Bisection Procedure,
Brent's Method, Crout's Method, Descartes'
Sign Rule, False Position Method, Fundamen-
tal Theorem of Symmetric Functions, Graeffe's
Method, Halley's Irrational Formula, Hal-
ley's Method, Halley's Rational Formula,
Horner's Method, Householder's Method, Hut-
ton's Method, Isograph, Jenkins-Traub Method,
(i/e;
= <
fP(x)* 2
(1)
for a discrete distribution
(2)
for a continuous distribution.
f P(x)dx
Hoehn and Niven (1985) show that
R(ai +c,a 2 + c,... t a n + c) < c + #(ai,a 2 ,. . . ,a„)
for any Positive constant c.
Physical scientists often use the term root-mean-square
as a synonym for STANDARD DEVIATION when they refer
to the SQUARE Root of the mean squared deviation of
a signal from a given baseline or fit.
1578 Root (Radical)
see also Arithmetic-Geometric Mean, Arith-
metic-Harmonic Mean, Generalized Mean, Ge-
ometric Mean, Harmonic Mean, Harmonic-
Geometric Mean, Mean, Median (Statistics),
Standard Deviation, Variance
References
Hoehn, L. and Niven, I. "Averages on the Move." Math.
Mag. 58, 151-156, 1985,
Root (Radical)
The nth root (or "RADICAL") of a quantity z is a value
r such that z = r n , and therefore is the INVERSE FUNC-
TION to the taking of a POWER. The nth root is de-
noted r = Vfz or, using POWER notation, r — z 1 ^ 71 .
The special case of the SQUARE ROOT is denoted y/z.
The quantities for which a general FUNCTION equals
are also called ROOTS, or sometimes ZEROS.
see also Cube Root, Root, Square Root, Vinculum
Root Test
Let uk be a Series with Positive terms, and let
p = lim u k 1/k .
k—^oo
1. If p < 1, the Series Converges.
2. If p > 1 or p = oo, the Series Diverges.
3. If p = 1, the Series may Converge or Diverge.
This test is also called the Cauchy Root Test.
see also Convergence Tests
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 281-282, 1985.
Bromwich, T. J. Pa and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, pp. 31-39, 1991.
Root (Tree)
A special node which is designated to turn a Tree into
a Rooted Tree. The root is sometimes also called
"Eve," and each of the nodes which is one Edge fur-
ther away from a given Edge is called a Child. Nodes
connected to the same node are then called SIBLINGS.
see also Child, Rooted Tree, Sibling, Tree
Root of Unity
The nth Roots of Unity are Roots ( k = e 2nik/p of
the Cyclotomic Equation
x p = 1,
which are known as the DE M0IVRE NUMBERS.
see also CYCLOTOMIC EQUATION, DE MOIVRE'S IDEN-
TITY, de Moivre Number, Unity
References
Courant, R. and Robbins, H. "De Moivre's Formula and the
Roots of Unity." §5.3 in What is Mathematics?: An Ele-
mentary Approach to Ideas and Methods, 2nd ed. Oxford,
England: Oxford University Press, pp. 98-100, 1996.
Rooted Tree
Rooted Tree
1 •
2 I
V
v *
O = root
A Tree with a special node called the "Root" or
"EVE." Denote the number of rooted trees with n nodes
by T n , then the GENERATING FUNCTION is
oo
T(x) = ^ T n x n = x + x 2 + 2x 3 + 4x 4 + 9x 5 + 20z 6
n=0
+48x 7 + 115a; 8 + 286x 9 + 719z 10 + . . . (1)
(Sloane's A000081). This Power Series satisfies
T(x) = xexp
£j*v)
t(x) = T(x)-±[T 2 (x)-T(x%
(2)
(3)
where t(x) is the GENERATING FUNCTION for unrooted
Trees. A Generating Function for T n can be writ-
ten using a product involving the sequence itself ss
oo oo
(1-X n )
(4)
The number of rooted trees can also be calculated from
the Recurrence Relation
r «+i = 7E(£ <n, ') T <-
•j+i)
(5)
3 = 1
d\j
with To = and T\ = 1, where the second sum is over
all d which Divide j (Finch).
see also Ordered Tree, Red-Black Tree, Weakly
Binary Tree
References
Finch, S. "Favorite Mathematical Constants." http://vvv.
mathsoft.com/asolve/constant/otter/otter.html.
Ruskey, F. "Information on Rooted Trees." http://sue.csc
.uvic.ca/-cos/inf/tree/RootedTree.html.
Sloane, N. J. A. Sequence A000081/M1180 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
RosattVs Theorem
Rotation Formula
1579
Rosatti's Theorem
There is a one-to-one correspondence between the sets
of equivalent correspondences (not of value 0) on an ir-
reducible curve of GENUS (Curve) p, and the rational
COLLINEATIONS of a projective space of 2p — 1 dimen-
sions which leave invariant a space of p — 1 dimensions.
The number of linearly independent correspondences
will be that of linearly independent COLLINEATIONS.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 339, 1959.
Rose
A curve which has the shape of a petalled flower. This
curve was named RHODONEA by the Italian mathemati-
cian Guido Grandi between 1723 and 1728 because it
resembles a rose (MacTutor Archive). The polar equa-
tion of the rose is
r = asin(n#),
or
r = acos(n#).
If n is Odd, the rose is n-petalled. If n is Even, the
rose is 2n-petalled. If n is IRRATIONAL, then there are
an infinite number of petals.
The QUADRIFOLIUM is the rose with n = 2. The rose is
the Radial Curve of the Epicycloid.
see also DAISY, MAURER ROSE, STARR ROSE
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 175-177, 1972.
Lee, X. "Rose." http://www.best.com/-xah/SpecialPlane
Curves_dir/Rose_dir/rose . html.
MacTutor History of Mathematics Archive. "Rhodonea
Curves." http: // www - groups . dcs . st - and .ac.uk/
-hist ory/Curves/Rhodonea. html.
Wagon, S. "Roses." §4.1 in Mathematica in Action. New
York: W. H. Freeman, pp. 96-102, 1991.
Rosenbrock Methods
A generalization of the Runge-Kutta METHOD for so-
lution of Ordinary Differential Equations, also
called Kaps-Rentrop Methods.
see also Runge-Kutta Method
References
Press, W, H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 730-735, 1992.
Rossler Model
The nonlinear 3-D MAP
X = -(Y + Z)
Y = X +0.2Y
Z - 0.2 + XZ - cZ.
see also Lorenz System
References
Dickau, R. M. "Rossler Attractor." http://www.prairienet.
org/ -pops/rossler. html.
Peitgen, H.-O.; Jiirgens, H.; and Saupe, D. §12.3 in Chaos
and Fractals: New Frontiers of Science, New York:
Springer- Verlag, pp. 686-696, 1992.
Rotation
The turning of an object or coordinate system by an AN-
GLE about a fixed point. A rotation is an Orientation-
Preserving Orthogonal Transformation. Eu-
ler's Rotation Theorem states that an arbitrary ro-
tation can be parameterized using three parameters.
These parameters are commonly taken as the EULER
Angles. Rotations can be implemented using Rota-
tion Matrices.
The rotation Symmetry Operation for rotation by
360°/n is denoted "ra." For periodic arrangements of
points ("crystals"), the CRYSTALLOGRAPHY RESTRIC-
TION gives the only allowable rotations as 1, 2, 3, 4, and
6.
see also Dilation, Euclidean Group, Euler's Rota-
tion Theorem, Expansion, Improper Rotation, In-
finitesimal Rotation, Inversion Operation, Mir-
ror Plane, Orientation-Preserving, Orthogo-
nal Transformation, Reflection, Rotation For-
mula, Rotation Group, Rotation Matrix, Rota-
tion Operator, Rotoinversion, Shift, Transla-
tion
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 211, 1987.
Yates, R. C. "Instantaneous Center of Rotation and the Con-
struction of Some Tangents." A Handbook on Curves and
Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 119-
122, 1952.
Rotation Formula
1580 Rotation Group
Roth's Removal Rule
A formula which relates the VECTOR r' to the ANGLE
$ in the above figure (Goldstein 1980). Referring to the
figure,
r' = OI$ + W$ + V$
= n(fi • r) + [r — n(n • r)] cos $ + (r x n) sin $
= rcos$ + n(n * r)(l — cos<I>) + (r x n) sin<&.
The Angle $ and unit normal n may also be expressed
as Euler Angles. In terms of the Euler PARAME-
TERS,
r' =r(e 2 -ei 2 -e 2 2 -e 3 2 ) + 2e(e-r) + 2(rx n)sin$.
see also Euler Angles, Euler Parameters
References
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA:
Addison-Wesley, 1980.
Rotation Group
There are three representations of the rotation groups,
corresponding to Expansion/Dilation, Rotation,
and Shear.
Rotation Matrix
When discussing a Rotation, there are two possible
conventions: rotation of the axes and rotation of the
object relative to fixed axes.
curve rotated by angle
In M 2 , let a curve be rotated by a clockwise Angle 0, so
that the original axes of the curve are x and y, and the
new axes of the curve are x' and y ' . The MATRIX trans-
forming the original curve to the rotated curve, referred
to the original x and y axes, is
i.e.,
cos 9 sin 9
— sin 9 cos 9
Re
(1)
(2)
% y
axes rotated by angle 9
On the other hand, let the axes with respect to which
a curve is measured be rotated by a clockwise Angle
0, so that the original axes are xo and yo, and the new
axes are x and y. Then the MATRIX transforming the
coordinates of the curve with respect to x and y is given
by the Matrix Transpose of the above matrix:
Ri
cos 8 — sin 9
sin 9 cos 9
R^x .
(3)
(4)
In R 3 , rotations of the x-, y-, and z-axes give the ma-
trices
ri o o "
Rx(a) =
cos a sin a
_ — sin a cos a _
" cos f3 — sin f3 '
(5)
Rv(/?) =
1
_ sin /3 cos
cos 7 sin 7 0"
(6)
R.(7) =
— sin 7 cos 7
1.
(7)
see also Euler Angles, Euler's Rotation Theo-
rem, Rotation
Rotation Number
The period for a QUASIPERIODIC trajectory to pass
through the same point in a SURFACE OF SECTION. If
the rotation number is IRRATIONAL, the trajectory will
densely fill out a curve in the SURFACE OF SECTION. If
the rotation number is RATIONAL, it is called the WIND-
ING NUMBER, and only a finite number of points in the
Surface of Section will be visited by the trajectory.
see also Quasiperiodic Function, Surface of Sec-
tion, Winding Number (Map)
Rotation Operator
The rotation operator can be derived from examining
an Infinitesimal Rotation
\ Qit / space
\dt)
body
+ o;x,
where d/dt is the time derivative, u? is the ANGULAR
Velocity, and x is the Cross Product operator.
see also Acceleration, Angular Acceleration, In-
finitesimal Rotation
Roth's Removal Rule
If the matrices A, X, B, and C satisfy
AX - XB = C,
then
1 X"
1
"A C"
B
1 -X"
1
=
A 0"
B
Roth's Theorem
Roulette 1581
where I is the Identity Matrix.
References
Roth, W. E. "The Equations AX-YB = C and AX-XB =
C in Matrices." Proc. Amer. Math, Soc. 3, 392-396, 1952.
Turnbull, H. W. and Aitken, A. C. An Introduction to the
Theory of Canonical Matrices. New York: Dover, p. 422,
1961.
Roth's Theorem
For Algebraic a
p
a
<
7 2 + €'
with e > 0, has finitely many solutions. Klaus Roth
received a Fields Medal for this result.
see also Hurwitz Equation, Hurwitz's Irrational
Number Theorem, Lagrange Number (Ratio-
nal Approximation), Liouville's Rational Ap-
proximation Theorem, Liouville-Roth Constant,
Markov Number, Segre's Theorem, Thue-Siegel-
Roth Theorem
References
Davenport, H. and Roth, K. F. "Rational Approximations to
Algebraic Numbers." Mathematika 2, 160-167, 1955.
Roth, K. F. "Rational Approximations to Algebraic Num-
bers." Mathematika 2, 1-20, 1955.
Roth, K. F. "Corrigendum to 'Rational Approximations to
Algebraic Numbers'." Mathematika 2, 168, 1955.
Rotkiewicz Theorem
If n > 19, there exists a base-2 PSEUDOPRIME between
n and n 2 . The theorem was proved in 1965.
see also PSEUDOPRIME
References
Rotkiewicz, A. "Les intervalles contenants les nombres pseu-
doprimiers." Rend. Circ. Mat. Palermo Ser. 2 14, 278-
280, 1965.
Rotkiewicz, A. "Sur les nombres de Mersenne depourvus de
diviseurs carres er sur les nombres naturels n, tel que n 2 —
2" - 2." Mat. Vesnik 2 (17), 78-80, 1965.
Rotkiewicz, A. "Sur les nombres pseudoprimiers carres."
Elem. Math. 20, 39-40, 1965.
Rotoinversion
see IMPROPER ROTATION
Rotor
A convex figure that can be rotated inside a POLY-
GON (or Polyhedron) while always touching every side
(or face). The least Area rotor in a Square is the
Reuleaux Triangle. The least Area rotor in an
Equilateral Triangle is a Lens with two 60° Arcs
of Circles and Radius equal to the Triangle Alti-
tude.
There exist nonspherical rotors for the TETRAHEDRON,
Octahedron, and Cube, but not for the Dodecahe-
dron and ICOSAHEDRON.
see also LENS, REULEAUX TRIANGLE
References
Gardner, M. The Unexpected Hanging and Other Mathemat-
ical Diversions. Chicago, IL: Chicago University Press,
p. 219, 1991.
Rotunda
A class of solids whose only true member is the PEN-
TAGONAL Rotunda.
see also Elongated Rotunda, Gyroelongated Ro-
tunda, Pentagonal Rotunda, Triangular Hebe-
sphenorotunda
References
Johnson, N. W. "Convex Polyhedra with Regular Faces."
Canad. J. Math. 18, 169-200, 1966.
Rouche's Theorem
Given two functions / and g ANALYTIC in A with 7
a simple loop HOMOTOPIC to a point in A, if \g(z)\ <
|/(;z)| for all z on 7, then / and / + g have the same
number of ROOTS inside 7.
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 22, 1975.
Roulette
The curve traced by a fixed point on a closed convex
curve as that curve rolls without slipping along a sec-
ond curve. The roulettes described by the FOCI of CON-
IC S when rolled upon a line are sections of MINIMAL
Surfaces (i.e., they yield Minimal Surfaces when re-
volved about the line) known as UNDULOIDS.
Curve 1
Curve 2
Pole
Roulette
circle
exterior
circle
on c.
epicycloid
circle
interior
circle
on c.
hypocycloid
circle
line
on c.
cycloid
circle
same circle
any point
rose
circle
line
center
parabola
involute
cycloid
line
center
ellipse
ellipse
line
focus
elliptic catenary
hyperbola
line
focus
hyperbolic catenary
hyperbolic
line
origin
tractrix
spiral
line
any curve
on line
involute of curve
logarithmic
line
any point
line
spiral
parabola
equal
parabola
vertex
cissoid of Diodes
parabola
line
focus
catenary
see also Glissette, Unduloid
References
Besant, W. H. Notes on Roulettes and Glissettes, 2nd enl.
ed. Cambridge, England: Deighton, Bell & Co., 1890.
1582
Round
RSA Encryption
Cundy, H. and Rollett, A. "Roulettes and Involutes." §2.6 in
Mathematical Models, 3rd ed. Stradbroke, England: Tar-
quin Pub., pp. 46-55, 1989.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 56-58 and 206, 1972.
Lockwood, E. H. "Roulettes." Ch. 17 in A Book of
Curves. Cambridge, England: Cambridge University
Press, pp. 138-151, 1967.
Yates, R. C. "Roulettes." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 175-185,
1952.
Zwillinger, D. (Ed.). "Roulettes (Spirograph Curves)." §8.2
in CRC Standard Mathematical Tables and Formulae, 3rd
ed. Boca Raton, FL: CRC Press, 1996. http://www.geom.
umn.edu/docs/reference/CRC-formulas/node34.html.
Round
see Nint
Rounding
The process of approximating a quantity, usually done
for convenience or, in the case of numerical computa-
tions, of necessity. If rounding is performed on each of
a series of numbers in a long computation, round-off er-
rors can become important, especially if division by a
small number ever occurs.
see also SHADOWING THEOREM
References
Wilkinson, J. H. Rounding Errors in Algebraic Processes.
New York: Dover, 1994.
Routh's Theorem
If the sides of a Triangle are divided in the ratios A : 1,
ji : 1, and v : 1, the CEVIANS form a central TRIANGLE
whose Area is
A =
(XfJLU - l) 2
(Xy, + A + l)(ixu + ii + l)(i/A + v + 1)
A, (1)
where A is the Area of the original TRIANGLE. For
A = fi = v = n,
n 2 + n + 1
(2)
For n = 2, 3, 4, 5, the areas are y, |, and ~. The
Area of the Triangle formed by connecting the divi-
sion points on each side is
A' =
\[IV
(A + 1)(ai + 1)(i/ + 1)
A.
(3)
Routh's theorem gives Ceva'S Theorem and Mene-
laus' Theorem as special cases.
see also Ceva's Theorem, Cevian, Menelaus' The-
orem
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 211-212, 1969.
Routh-Hurwitz Theorem
Consider the Characteristic Equation
|AI - A| = A n + hX 71 - 1 + . . . + 6 n _iA + b n =
determining the n Eigenvalues A of a Real n x n
Matrix A, where I is the Identity Matrix. Then the
Eigenvalues A all have Negative Real Parts if
Ai > 0,A 2 > 0,...,A n > 0,
where
A fc =
bi
1
••
6,
b 2
61
1
•■
6 5
b 4
b 3
b 2
61
•■
°2fc-l « 2 fe_2 W 2fe-
b 2 k-4 b
'2k-5 D 2k-6
b k
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1119, 1979.
RSA Encryption
A Public-Key Cryptography Algorithm which
uses Prime Factorization as the Trapdoor Func-
tion. Define
n = pq (1)
for p and q PRIMES. Also define a private key d and a
public key e such that
de = 1 (mod <f>(n))
(e,0(n)) = 1,
where <j>(n) is the TOTIENT FUNCTION.
(2)
(3)
Let the message be converted to a number M. The
sender then makes n and e public and sends
E = M e (mod n). (4)
To decode, the receiver (who knows d) computes
E d = (M e ) d = M ed = M N<p(n)+1 = M (mod n), (5)
since N is an Integer. In order to crack the code, d
must be found. But this requires factorization of n since
0(n) = (p-l)(g-l).
(6)
RSA Number
Rubber-Sheet Geometry 1583
Both p and q should be picked so that p ± 1 and q ± 1
are divisible by large Primes, since otherwise the POL-
LARD p-1 Factorization Method or Williams p+l
Factorization Method potentially factor n easily. It
is also desirable to have (f>(<j>(pq)) large and divisible by
large PRIMES.
It is possible to break the cryptosystem by repeated en-
cryption if a unit of Z/<£(n)Z has small ORDER (Sim-
mons and Norris 1977, Meijer 1996), where Z/sZ is the
Ring of Integers between and s — 1 under addition
and multiplication (mod s). Meijer (1996) shows that
"almost" every encryption exponent e is safe from break-
ing using repeated encryption for factors of the form
where
p = 2pi + 1
q = 2 qi + 1,
Pi = 2p 2 + 1
9i = 2^2 + 1,
(7)
(8)
(9)
(10)
and p, pi, £>2, q, qi, and q 2 are all PRIMES. In this case,
<f>(n) = 4piqfi
4>{<t>{n)) = Sp 2 q 2 .
(11)
(12)
Meijer (1996) also suggests that p 2 and q 2 should be of
order 10 75 .
Using the RSA system, the identity of the sender can be
identified as genuine without revealing his private code.
see also Public-Key Cryptography
References
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 166-173, 1985.
Meijer, A. R. "Groups, Factoring, and Cryptography." Math.
Mag. 69, 103-109, 1996.
Rivest, R. L. "Remarks on a Proposed Cryptanalytic Attack
on the MIT Public-Key Cryptosystem." Cryptologia 2,
62-65, 1978.
Rivest, R.; Shamir, A.; and Adleman, L. "A Method for Ob-
taining Digital Signatures and Public Key Cryptosystems."
Comm. ACM 21, 120-126, 1978.
RSA Data Security.® A Security Dynamics Company.
http : //www .rsa. com.
Simmons, G. J. and Norris, M. J. "Preliminary Comments
on the MIT Public-Key Cryptosystem." Cryptologia 1,
406-414, 1977.
RSA Number
Numbers contained in the "factoring challenge" of RSA
Data Security, Inc. An additional number which is not
part of the actual challenge is the RSA-129 number. The
RSA numbers which have been factored are RSA-100,
RSA-110, RSA-120, RSA-129, and RSA-130 (Cowie et
al 1996).
RSA-129 is a 129-digit number used to encrypt one
of the first public-key messages. This message was
published by R. Rivest, A. Shamir, and L. Adleman
(Gardner 1977), along with the number and a $100
reward for its decryption. Despite belief that the
message encoded by RSA-129 "would take millions of
years of break," RSA-129 was factored in 1994 using
a distributed computation which harnessed networked
computers spread around the globe performing a mul-
tiple polynomial QUADRATIC SIEVE FACTORIZATION
Method. The effort was coordinated by P. Leylad,
D. Atkins, and M. Graff. They received 112,011 full fac-
torizations, 1,431,337 single partial factorizations, and
8,881,138 double partial factorizations out of a factor
base of 524,339 Primes. The final Matrix obtained
was 188,346 x 188,346 square.
The text of the message was "The magic words are
squeamish ossifrage" (an ossifrage is a rare, predatory
vulture found in the mountains of Europe), and the FAC-
TORIZATION (into a 64-DlGlT number and a 65-DlGlT
number) is
114381625757888867669235779976146612010218296- • ■
■ ■ ■ 7212423625625618429357069352457338978305971 • • •
• • • 23563958705058989075147599290026879543541
= 3490529510847650949147849619903898133417764* • •
• • • 638493387843990820577 ■ 3276913299326 ■ • •
■ ■ • 6709549961988190834461413177642967992 ■ ■ -
• • • 942539798288533
(Leutwyler 1994, Cipra 1995).
References
Cipra, B. "The Secret Life of Large Numbers." What's Hap-
pening in the Mathematical Sciences, 1995-1996, Vol. 3.
Providence, RI: Amer. Math. Soc, pp. 90-99, 1996.
Cowie, J.; Dodson, B.; Elkenbracht-Huizing, R. M.; Lenstra,
A. K.; Montgomery, P. L.; Zayer, J. A. "World Wide Num-
ber Field Sieve Factoring Record: On to 512 Bits." In Ad-
vances in Cryptology—ASIACRYPT '96 (Kyongju) (Ed.
K. Kim and T. Matsumoto.) New York: Springer-Verlag,
pp. 382-394, 1996.
Gardner, M. "Mathematical Games: A New Kind of Cipher
that Would Take Millions of Years to Break." Sci. Amer.
237, 120-124, Aug. 1977.
Klee, V. and Wagon, S. Old and New Unsolved Problems in
Plane Geometry and Number Theory, rev. ed. Washing-
ton, DC: Math. Assoc. Amer., p. 223, 1991.
Leutwyler, K. "Superhack: Forty Quadrillion Years Early, a
129-Digit Code is Broken." Sci. Amer. 271, 17-20, 1994.
Leyland, P. ftp://sable.ox.ac.uk/pub/math/rsal29.
RSA Data Security.® A Security Dynamics Company,
http : //www . rsa. com.
Taubes, G. "Small Army of Code-breakers Conquers a 129-
Digit Giant." Science 264, 776-777, 1994.
$ Weisstein, E. W. "RSA Numbers." http: //www. astro.
Virginia . edu/ -eww6n/math/notebooks/RSAWumber s . m.
Rubber-Sheet Geometry
see Algebraic Topology
1584
Rubik's Clock
Rudvalis Group
Rubik's Clock
A puzzle consisting of 18 small clocks. There are 12 18
possible configurations, although not all are realizable.
see also Rubik's Cube
References
Denes, J. and Mullen, G. L. "Rubik's Clock and Its Solution."
Math. Mag. 68, 378-381, 1995.
Zeilberger, D. "Doron Zeilberger's Maple Packages and
Programs: RubikClock." http : //www . math . temple . edu/
-zeilberg/programs . html.
Rubik's Cube
A 3 x 3 x 3 Cube in which the 26 subcubes on the outside
are internally hinged in such a way that rotation (by a
quarter turn in either direction or a half turn) is possible
in any plane of cubes. Each of the six sides is painted
a distinct color, and the goal of the puzzle is to return
the cube to a state in which each side has a single color
after it has been randomized by repeated rotations. The
Puzzle was invented in the 1970s by the Hungarian
Erno Rubik and sold millions of copies worldwide over
the next decade.
The number of possible positions of Rubik's cube is
8!12!3 8 2 12
23-2
= 43,252,003,274,489,856,000
(Turner and Gold 1985). Hoey showed using the POLYA-
BURNSIDE LEMMA that there are 901,083,404,981,813,-
616 positions up to conjugacy by whole-cube symme-
tries.
Algorithms exist for solving a cube from an arbitrary ini-
tial position, but they are not necessarily optimal (i.e.,
requiring a minimum number of turns). The maximum
number of turns required for an arbitrary starting po-
sition is still not known, although it is bounded from
above. Michael Reid (1995) produced the best proven
bound of 29 turns (or 42 "quarter-turns"). The proof
involves large tables of "subroutines" generated by com-
puter.
However, Dik Winter has produced a program based on
work by Kociemba which has solved each of millions of
cubes in at most 21 turns. Recently, Richard Korf (1997)
has produced a different algorithm which is practical
for cubes up to 18 moves away from solved. Out of 10
randomly generated cubes, one was solved in 16 moves,
three required 17 moves, and six required 18 moves.
see also RUBIK'S CLOCK
References
Hofstadter, D. R. "Metamagical Themas: The Magic Cube's
Cubies are Twiddled by Cubists and Solved by Cubemeis-
ters." Sci. Amer. 244, 20-39, Mar. 1981.
Larson, M. E. "Rubik's Revenge: The Group Theoretical
Solution." Amer. Math. Monthly 92, 381-390, 1985.
Miller, D. L. W. "Solving Rubik's Cube Using the 'Best-
fast' Search Algorithm and 'Profile' Tables." http: //www.
sunyit.edu/-millerdl/RUBIK.HTM.
Schubart, M. "Rubik's Cube Resource List." http: //www.
best.com/-schubart/rc/resources.html.
Singmaster, D. Notes on Rubik's 'Magic Cube. 7 Hillside, NJ:
Enslow Pub., 1981.
Taylor, D. Mastering Rubik's Cube. New York: Holt, Rine-
hart, and Winston, 1981.
Taylor, D. and Rylands, L. Cube Games: 92 Puzzles & So-
lutions New York: Holt, Rinehart, and Winston, 1981.
Turner, E. C. and Gold, K. F. "Rubik's Groups." Amer.
Math. Monthly 92, 617-629, 1985.
Rudin-Shapiro Sequence
The sequence of numbers given by
where n is written in binary
n = eie 2 . . . efc.
(i)
(2)
It is therefore the parity of the number of pairs of consec-
utive Is in the BINARY expansion of n. The SUMMATORY
sequence is
IZ a "
3=0
which gives
f 2 fc/2 + 1 if k is even
| 2 (*-l)/2 + 1 if k igodd
(3)
(4)
(Blecksmith and Laud 1995).
References
Blecksmith, R. and Laud, P. W. "Some Exact Number The-
ory Computations via Probability Mechanisms." Amer.
Math. Monthly 102, 893-903, 1995.
Brillhart, J.; Erdos, P.; and Morton, P. "On the Sums of the
Rudin-Shapiro Coefficients II." Pac. J. Math. 107, 39-69,
1983.
Brillhart, J. and Morton, P. "Uber Summen von Rudin-
Shapiroschen Koeffizienten." Ill J. Math. 22, 126-148,
1978.
Prance, M. M. and van der Poorten, A. J. "Arithmetic and
Analytic Properties of Paper Folding Sequences." Bull.
Austral. Math. Soc. 24, 123-131, 1981.
Rudvalis Group
The Sporadic Group Ru.
see also SPORADIC GROUP
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat.bham.ac.uk/atlas/Ru.html.
Rule
Ruler 1585
Rule
A usually simple ALGORITHM or IDENTITY. The term is
frequently applied to specific orders of Newton-Cotes
Formulas.
see also Algorithm, BAC-CAB Rule, Bode's Rule,
Chain Rule, Cramer's Rule, Descartes' Sign
Rule, Durand's Rule, Estimator, Euler's Rule,
Euler's Totient Rule, Golden Rule, Hardy's
Rule, Horner's Rule, Identity, L'Hospital's
Rule, Leibniz Integral Rule, Method, Osborne's
Rule, Pascal's Rule, Power Rule, Product Rule,
Quarter Squares Rule, Quota Rule, Quotient
Rule, Roth's Removal Rule, Rule of 72, Simp-
son's Rule, Slide Rule, Sum Rule, Trapezoidal
Rule, Weddle's Rule, Zeuthen's Rule
Rule of 72
5 10 15 20
^actual -Frule 72 (%)
The time required for a given PRINCIPAL to double (as-
suming n = 1 Conversion Period) for Compound
Interest is given by solving
2P = P(l + r)\
or
t =
In 2
ln(l + r)'
(1)
(2)
where Ln is the NATURAL LOGARITHM. This function
can be approximated by the so-called "rule of 72" :
t',
0.72
r
(3)
The above plots show the actual doubling time t (left
plot) and difference between actual and time calculated
using the rule of 72 (right plot) as a function of the
interest rate r.
see also Compound Interest, Interest
References
Avanzini, J. F. Rapid Debt-Reduction Strategies. Fort Worth,
TX: HIS Pub., 1990.
the Gaussian Curvature on a ruled Regular Sur-
face is everywhere Nonpositive.
Examples of ruled surfaces include the elliptic Hyper-
BOLOID of one sheet (a doubly ruled surface)
a(cosu =p v sin u)
b(sinu ± cosu)
±cv
a cos u
bsinu
±v
— a cos u
bsinu
c
(2)
the Hyperbolic Paraboloid (a doubly ruled surface)
(3)
a(u -j- v)
±bv
, u + 2uv
Plucker's Conoid
rcosv
r sin 6
2 cos 9 sin
" au~
a
=
+ v
±b
[u 2 \
_2u_
2 cos sin
+ r
COS0
sin0
(4)
and the Mobius Strip
cosu-\- vcos(^u) cos it
sinu + i;cos(|u) sinti
vsin(~u)
cosu
sin it
+ av
COs(|ia)cOS14
cos ( I u ) sin u
sin(fu)
(5)
(Gray 1993).
The only ruled MINIMAL SURFACES are the PLANE and
HELICOID (Catalan 1842, do Carmo 1986).
see also Asymptotic Curve, Cayley's Ruled Sur-
face, Developable Surface, Director Curve,
Directrix (Ruled Surface), Generalized Cone,
Generalized Cylinder, Helicoid, Noncylindri-
cal Ruled Surface, Plane, Right Conoid, Ruling
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.). Plates 32-33 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 32-33, 1986.
Gray, A. "Ruled Surfaces." Ch. 17 in Modern Differential
Geometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 333-355, 1993.
Ruled Surface
A SURFACE which can be swept out by a moving LINE in
space and therefore has a parameterization of the form
x(u, v) = b(u) + v5(u),
(i)
where b is called the Directrix (also called the BASE
Curve) and S is the Director Curve. The straight
lines themselves are called RULINGS. The rulings of a
ruled surface are ASYMPTOTIC CURVES. Furthermore,
Ruler
A Straightedge with markings to indicate distances.
Although Geometric Constructions are sometimes
said to be performed with a ruler and COMPASS, the
term STRAIGHTEDGE is preferable to ruler since mark-
ings are not allowed by the classical Greek rules.
see also COASTLINE PARADOX, COMPASS, GEOMETRIC
Construction, Geometrography, Golomb Ruler,
Perfect Ruler, Simplicity, Slide Rule, Straight-
edge
1586
Ruler Function
Run
Ruler Function
The exponent of the largest POWER of 2 which DIVIDES
a given number k. The values of the ruler function are
1, 2, 1, 3, 1, 2, 1, 4, 1, 2, . . . (Sloane's A001511).
References
Guy, R. K. "Cycles and Sequences Containing All Permu-
tations as Subsequences." §E22 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
p. 224 1994.
Sloane, N. J. A, Sequence A001511/M0127 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Ruling
One of the straight lines sweeping out a RULED SUR-
FACE. The rulings on a ruled surface are ASYMPTOTIC
Curves.
see also Asymptotic Curve, Director Curve, Di-
rectrix (Ruled Surface), Ruled Surface
Run
A run is a sequence of more than one consecutive iden-
tical outcomes, also known as a CLUMP. Given n BER-
NOULLI Trials (say, in the form of Coin Tossings),
the probability Pt (n) of a run of t consecutive heads or
tails is given by the RECURRENCE RELATION
P t (n) = P t (n - 1) + 2~ f [1 - P t (n - *)], (1)
where P t (n) = for n < t and P t {t) = 2 1_ * (Bloom
1996).
Let Ct(m 7 k) denote the number of sequences of m indis-
tinguishable objects of type A and k indistinguishable
objects of type B in which no t-ruu occurs. The proba-
bility that a t-run does occur is then given by
Pt(m,k) = 1
C t (m,k)
CT) '
(2)
where (*) is a BINOMIAL COEFFICIENT. Bloom (1996)
gives the following recurrence sequence for C t {m,k),
t-i
Ct(m,ft) = ^C t (m-l,fc-i)
i=0
t-i
- ^T C t (m -t,k-i) + e t (m, k), (3)
where
{1 if m = and < k < t
-1 if m-t and < k < t (4)
otherwise.
Another recurrence which has only a fixed number of
terms is given by
Ct{m 1 k) = Ct(m-l J k) + Ct{m,k-l)-Ct(m-t t k-l)
-C t {m -l,k-t) + C t {m-t,k-t)+ e*(m, k), (5)
where
f 1 if (m,fc) = (0,0) or (t,t)
e* t {m,k) = I -1 if ( m ,jfe) = (0,t) or (t, 0) (6)
1 otherwise
(Goulden and Jackson 1983, Bloom 1996). These formu-
las disprove the assertion of Gardner (1982) that "there
will almost always be a clump of six or seven CARDS
of the same color" in a normal deck of cards by giving
P 6 (26,26) = 0.46424.
Given n Bernoulli TRIALS with a probability of suc-
cess (heads) p, the expected number of tails is n(l — p),
so the expected number of tail runs > 1 is « n(l — p)p.
Continuing,
N H = n(l -p)p
R
(7)
is the expected number of runs > R, The longest ex-
pected run is therefore given by
£ = logi/ p [n(l-p)]
(8)
(Gordon et al. 1986, Schilling 1990). Given m 0s and n
Is, the number of possible arrangements with u runs is
afc^c;:;)
f ""t(r--x 1 )(::i
u = 2k
) + (r 2 i )(r 1 1 ) «=2fc+i
(9)
for k an Integer, where (£) is a Binomial Coeffi-
cient. Then
/«
u ~2 \ rn J
(10)
Bloom (1996) gives the expected number of noncontigu-
ous t-runs in a sequence of m 0s and n Is as
Ffr, m A (rn+l)(n)t + (n+l)(m) t . ,
E(n, m, t) = ^^ , (11)
where (o) n is the POCHHAMMER SYMBOL. For m > 10,
u has an approximately Normal Distribution with
Mean and Variance
2mn
\lu = H ; —
2 _ 2mn(2mn — m — n)
(m -f n) 2 (m + n — 1) '
(12)
(13)
see also COIN TOSSING, EULERIAN NUMBER, PERMU-
TATION, S-RUN
References
Bloom, D. M. "Probabilities of Clumps in a Binary Sequence
(and How to Evaluate Them Without Knowing a Lot)."
Math. Mag. 69, 366-372, 1996.
Gardner, M. Aha! Gotcha: Paradoxes to Puzzle and Delight.
New York: W. H. Freeman, p. 124, 1982.
Runge-Kutta Method
Russell's Paradox
1587
Godbole, A. P. "On Hypergeometric and Related Distribu-
tions of Order Je." Commun. Stat.: Th. and Meth. 19,
1291-1301, 1990.
Godbole, A. P. and Papastavnidis, G. (Eds.). Runs and Pat-
terns in Probability: Selected Papers. New York: Kluwer,
1994.
Gordon, L.; Schilling, M. F.; and Waterman, M. S. "An Ex-
treme Value Theory for Long Head Runs." Prob. Th. and
Related Fields 72, 279-287, 1986.
Goulden, I. P. and Jackson, D. M. Combinatorial Enumera-
tion. New York: Wiley, 1983.
Mood, A. M. "The Distribution Theory of Runs." Ann.
Math. Statistics 11, 367-392, 1940.
Philippou, A. N. and Makri, F. S. "Successes, Runs, and
Longest Runs." Stat. Prob. Let. 4, 211-215, 1986.
Schilling, M. F. "The Longest Run of Heads." Coll. Math.
J. 21, 196-207, 1990.
Schuster, E. F. In Runs and Patterns in Probability: Selected
Papers (Ed. A. P. Godbole and S. Papastavridis). Boston,
MA: Kluwer, pp. 91-111, 1994.
Runge-Kutta Method
A method of integrating ORDINARY DIFFERENTIAL
Equations by using a trial step at the midpoint of
an interval to cancel out lower-order error terms. The
second-order formula is
Size Control for Runge-Kutta." §16.1 and 16.2 in Numeri-
cal Recipes in FORTRAN: The Art of Scientific Comput-
ing, 2nd ed. Cambridge, England: Cambridge University
Press, pp. 704-716, 1992.
Runge- Walsh Theorem
Let f(x) be an Analytic Function which is Regular
in the interior of a JORDAN CURVE C and continuous in
the closed Domain bounded by C Then f(x) can be
approximated with an arbitrary accuracy by POLYNO-
MIALS.
see also Analytic Function, Jordan Curve
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 7, 1975.
Running Knot
A Knot which tightens around an object when strained
but slackens when the strain is removed. Running knots
are sometimes also known as slip knots or nooses.
References
Owen, P. Knots. Philadelphia, PA: Courage, p. 60, 1993.
fci = hf(x ni yn)
k 2 = hf(x n + \ h y y n + §A;i)
2M+i =yn + k 2 + £>(h s ),
and the fourth-order formula is
fa = hf(x n ,y n )
k 2 = hf(x n + \h,y n + \fa)
ks = hf(x n + \h,y n + \k 2 )
k 4 = hf(x n + h,y n + k 3 )
2/n+i = y n + §&i + |fa + 5** + 5*4 + 0(h 5 ).
(Press et ah 1992). This method is reasonably simple
and robust and is a good general candidate for numerical
solution of differential equations when combined with an
intelligent adaptive step-size routine.
see also Adams' Method, Gill's Method, Milne's
Method, Ordinary Differential Equation,
rosenbrock methods
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 896-897, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 492-493, 1985.
Cartwright, J. H. E. and Piro, O. "The Dynamics of Runge-
Kutta Methods." Int. J. Bifurcations Chaos 2, 427-449,
1992. http://formentor.uib.es/-julyan/TeX/rkpaper/
root/root. html.
Lambert, J. D. and Lambert, D. Ch. 5 in Numerical Meth-
ods for Ordinary Differential Systems: The Initial Value
Problem. New York: Wiley, 1991.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Runge-Kutta Method" and "Adaptive Step
Russell's Antinomy
Let R be the set of all sets which are not members of
themselves. Then R is neither a member of itself nor not
a member of itself. Symbolically, let R = {x : x $ x}.
Then R e R Iff R £ R.
Bertrand Russell discovered this Paradox and sent it
in a letter to G. Prege just as Prege was completing
Grundlagen der Arithmetik. This invalidated much of
the rigor of the work, and Prege was forced to add a note
at the end stating, "A scientist can hardly meet with
anything more undesirable than to have the foundation
give way just as the work is finished. I was put in this
position by a letter from Mr. Bertrand Russell when the
work was nearly through the press."
see also Grelling's Paradox
References
Courant, R. and Robbins, H. "The Paradoxes of the Infinite."
§2.4.5 in What is Mathematics?: An Elementary Approach
to Ideas and Methods, 2nd ed. Oxford, England: Oxford
University Press, p. 78, 1996.
Frege, G. Foundations of Arithmetic. Evanston, IL: North-
western University Press, 1968.
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, pp. 20-21, 1989.
Russell's Paradox
see Russell's Antinomy
1588 Russian Multiplication
Ryser Formula
Russian Multiplication
Also called Ethiopian Multiplication. To multiply
two numbers a and 6, write do = a and 60 = 6 in
two columns. Under ao, write |_ao/2_|, where [x\ is the
Floor Function, and under 60, write 2&o- Continue
until a* = 1. Then cross out any entries in the b column
which are opposite an Even Number in the a column
and add the b column. The result is the desired product.
For example, for a = 27, b = 35
27 35
13 70
6440
3 280*
1 560
945
Puzzles & Diversions. New York: Simon and Schuster,
1959.
Knuth, D. E. The Art of Computer Programming, Vol. 2:
Seminumerical Algorithms, 2nd ed. Reading, MA:
Addison-Wesley, p. 497, 1981.
Nijenhuis, A. and Wilf, H. Chs. 7-8 in Combinatorial Algo-
rithms. New York: Academic Press, 1975.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, p. Ill, 1991.
Russian Roulette
Russian roulette is a GAME of chance in which one or
more of the six chambers of a gun are filled with bullets,
the magazine is rotated at random, and the gun is shot.
The shooter bets on whether the chamber which rotates
into place will be loaded. If it is, he loses not only his
bet but his life.
A modified version is considered by Blom et al. (1996)
and Blom (1989). In this variant, the revolver is loaded
with a single bullet, and two duelists alternately spin the
chamber and fire at themselves until one is killed. The
probability that the first duelist is killed is then 6/11.
References
Blom, G. Probabilities and Statistics: Theory and Applica-
tions. New York: Springer- Verlag, p. 32, 1989.
Blom, G.; Englund, J,.-E.; and Sandell, D. "General Russian
Roulette." Math. Mag. 69, 293-297, 1996.
Ryser Formula
A formula for the PERMANENT of a MATRIX
perm(a^) = (-1)- £) (-l) N IIE a ^
sC{l,...,n} i=l j£s
where the SUM is over all SUBSETS of {1, . . . , n}, and
\s\ is the number of elements in s. The formula can be
optimized by picking the SUBSETS so that only a single
element is changed at a time (which is precisely a GRAY
Code), reducing the number of additions from n 2 to n.
It turns out that the number of disks moved after the
fcth step in the TOWERS OF Hanoi is the same as the
element which needs to be added or deleted in the kth
Addend of the Ryser Formula (Gardner 1988, Vardi
1991, p. Ill)
see also Determinant, Gray Code, Permanent,
Towers of Hanoi
References
Gardner, M. "The Icosian Game and the Tower of Hanoi."
Ch. 6 in The Scientific American Book of Mathematical
s-Additive Sequence
Saalschiitz's Theorem
1589
s- Additive Sequence
A generalization of an ULAM SEQUENCE in which each
term is the Sum of two earlier terms in exactly s ways.
(s,i)-additive sequences are a further generalization in
which each term has exactly s representations as the
Sum oft distinct earlier numbers. It is conjectured that
O-additive sequences ultimately have periodic differences
of consecutive terms (Guy 1994, p. 233).
see also Greedy Algorithm, Stohr Sequence,
Ulam Sequence
References
Finch, S. R. "Conjectures about s-Additive Sequences." Fib.
Quart 29, 209-214, 1991.
Finch, S. R. "Are O-Additive Sequences Always Regular?"
Amer. Math. Monthly 99, 671-673, 1992.
Finch, S. R. "On the Regularity of Certain 1-Additive Se-
quences." J. Combin. Th. Ser. A. 60, 123-130, 1992.
Finch, S. R. "Patterns in 1-Additive Sequences." Experi-
ment Math. 1, 57-63, 1992.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer-Verlag, pp. 110 and 233, 1994.
Ulam, S. M. Problems in Modern Mathematics. New York:
Interscience, p. ix, 1964.
s- Cluster
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let an n x n MATRIX have entries which are either 1
(with probability p) or (with probability q = 1 - p) .
An s-cluster is an isolated group of s adjacent (i.e., hori-
zontally or vertically connected) Is. Let C n be the total
number of "Site" clusters. Then the value
Ks(p) — lim
(C n )
(1)
called the Mean Cluster Count Per Site or Mean
Cluster Density, exists. Numerically, it is found that
K s {l/2) « 0.065770 . . . (Ziff et al. 1997).
Considering instead "BOND" clusters (where numbers
are assigned to the edges of a grid) and letting C n be
the total number of bond clusters, then
Kb( P )
lim
n— >-oo
(Cn)
(2)
exists. The analytic value is known for p = 1/2,
K B {\) = \y/l-% (3)
(Ziff et al 1997).
see also Bond Percolation, Percolation Theory,
s-Run, Site Percolation
References
Finch, S. "Favorite Mathematical Constants." http://vww.
mathsoft.com/asolve/constant/rndprc/rndprc.html.
Temperley, H. N. V. and Lieb, E. H. "Relations Between the
'Percolation' and 'Colouring' Problem and Other Graph-
Theoretical Problems Associated with Regular Planar Lat-
tices; Some Exact Results for the 'Percolation' Problem."
Proc. Roy. Soc. London A 322, 251-280, 1971.
Ziff, R.; Finch, S.; and Adamchik, V. "Universality of Finite-
Sized Corrections to the Number of Percolation Clusters."
Phys. Rev. Let To appear, 1998.
s-Run
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let v be a n- VECTOR whose entries are each 1 (with
probability p) or (with probability q — 1 — p) . An
s-run is an isolated group of s consecutive Is. Ignoring
the boundaries, the total number of runs R n satisfies
*„=<*»>= (i _ P f yy = P (i - P )(i - P n ),
K(p) = lim Jf B =p(l-p),
n— >-oo
which is called the Mean Run Count Per Site or
Mean Run Density in Percolation Theory.
see also PERCOLATION THEORY, 5-CLUSTER
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/rndprc/rndprc.html.
S-Signature
see Signature (Recurrence Relation)
Saalschiitzian
For a Generalized Hypergeometric Function
P +ir p
ai,a2,
i a P+i . ,
the Saalschiitzian S is defined if
E" = £ a+1 -
see also Generalized Hypergeometric Function
Saalschiitz's Theorem
-x, -y, -z 1 = r(n + l)r(s + 2/ + n + 1)
n + l,-x-y-z ~ T(x + n + l)T(y + n + 1)
r(y + z + n + l)r(z + x + n + 1)
X T(z + n + l)r(x + y + z + n+l)'
where 3 F 2 (a,6,c; d, e; z) is a GENERALIZED HYPERGEO-
METRIC Function and Y(z) is the Gamma Function.
1590
Saddle
Saint Andrew's Cross
It can be derived from the DOUGALL-RAMANUJAN
Identity and written in the symmetric form
3 F 2 (a, 6, c;d, e; 1) =
(d- a)| c |(d-fc)| c |
d\ c \(d — a - 6)| c |
for d+e = a-f-6 + c+l with c a negative integer and (a) n
the POCHHAMMER Symbol (Petkovsek et at. 1996).
see also Dougall-Ramanujan Identity, General-
ized Hypergeometric Function
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Welles-
ley, MA: A. K. Peters, pp. 43 and 126, 1996.
Saddle
A Surface possessing a Saddle Point.
see also HYPERBOLIC PARABOLOID, MONKEY SADDLE,
Saddle Point (Function)
Saddle-Node Bifurcation
see Fold Bifurcation
Saddle Point (Fixed Point)
see Hyperbolic Fixed Point (Differential Equa-
tions), Hyperbolic Fixed Point (Map)
Saddle Point (Function)
A Point of a Function or Surface which is a Sta-
tionary POINT but not an EXTREMUM. An example
of a 1-D Function with a saddle point is f(x) = a; 3 ,
which has
f(x) = Sx
/"Or) = 6x
/'"(*) = 6.
This function has a saddle point at Xq = by the Ex-
TREMUM TEST since f"(x ) = and f" r (x ) = 6^0.
An example of a Surface with a saddle point is the
Monkey Saddle.
Saddle Point (Game)
For a general two-player ZERO-SUM Game,
min min a»j < min max aij .
i<m j<n j<n i<m
A Necessary and Sufficient condition for a saddle
point to exist is the presence of a Payoff Matrix ele-
ment which is both a minimum of its row and a maxi-
mum of its column. A GAME may have more than one
saddle point, but all must have the same VALUE.
see also Game, Payoff Matrix, Value
References
Dresher, M. "Saddle Points." §1.5 in The Mathematics of
Games of Strategy: Theory and Applications. New York:
Dover, pp. 12-14, 1981.
Llewellyn, D. C; Tovey, C; and Trick, M. "Finding Saddle-
points of Two-Person, Zero Sum Games." Amer. Math.
Monthly 95, 912-918, 1988.
Safarevich Conjecture
see Shafarevich Conjecture
Safe
A position in a Game is safe if the person who plays
next will lose.
see also GAME, UNSAFE
Sagitta
The PERPENDICULAR distance s from an Arc's MID-
POINT to the CHORD across it, equal to the RADIUS r
minus the APOTHEM a,
s = r — a.
(i)
For a regular POLYGON of side length a,
« = fl-r=Ia[c8c(£)-cat(£)]
= iatan(£)
= rtan(£)tan(£)
(2)
(3)
(4)
If the two are equal, then write
where R is the CIRCUMRADIUS, r the INRADIUS, a is the
side length, and n is the number of sides.
see also APOTHEM, CHORD, SECTOR, SEGMENT
min min at j = min max a^- = v,
i<.m j<n j<.n i<m
where v is called the VALUE of the GAME. In this case,
there exist optimal strategies for the first and second
players.
Saint Andrew's Cross
Saint Anthony's Cross
Salem Constants 1591
A GREEK CROSS rotated by 45°, also called the crux
decussata. The MULTIPLICATION Sign x is based on
Saint Andrew's cross (Bergamini 1969)
A
see also Cross, Greek Cross, Multiplication Sign
References
Bergamini, D. Mathematics. New York: Time-Life Books,
p. 11, 1969.
Saint Anthony's Cross
n:
:□
A CROSS also called the tau cross or crux commissa.
see also CROSS
Saint Petersburg Paradox
Consider a game in which a player bets on whether a
given TOSS of a COIN will turn up heads or tails. If he
bets $1 that heads will turn up on the first throw, $2
that heads will turn up on the second throw (if it did
not turn up on the first), $4 that heads will turn up on
the third throw, etc., his expected payoff is
i(l)+i(2)+i(4) + ... = i + i + i + ... = oo.
Apparently, the first player can be in the hole by any
amount of money and still come out ahead in the end.
This PARADOX was first proposed by Daniel Bernoulli.
The paradox arises as a result of muddling the distinc-
tion between the amount of the final payoff and the net
amount won in the game. It is misleading to consider
the payoff without taking into account the amount lost
on previous bets, as can be shown as follows. At the
time the player first wins (say, on the nth toss), he will
have lost
Gardner, M. The Scientific American Book of Mathematical
Puzzles & Diversions. New York: Simon and Schuster,
pp. 51-52, 1959.
Kamke, E. Einfuhrung in die Wahrscheinlichkeitstheorie.
Leipzig, Germany, pp. 82-89, 1932.
Keynes, X. M. K. "The Application of Probability to Con-
duct." In The World of Mathematics, Vol. 2 (Ed. K. New-
man). Redmond, WA: Microsoft Press, 1988.
Kraitchik, M. "The Saint Petersburg Paradox." §6.18 in
Mathematical Recreations. New York: W. W. Norton,
pp. 138-139, 1942.
Todhunter, IL. §391 in History of the Mathematical Theory of
Probability, New York: Chelsea, p. 221, 1949.
Sal
see Walsh Function
Salamin Formula
see Brent-Salamin Formula
Salem Constants
Each point of the PlSOT-VlJAYARAGHAVAN CONSTANTS
5 is a Limit Point from both sides of a set T known as
the Salem constants (Salem 1945). The Salem constants
are algebraic Integers > 1 in which one or more of the
conjugates is on the Unit Circle with the others inside
(Le Lionnais 1983, p. 150). The smallest known Salem
number was foundlbylLfehm^ri: (1933) as the largest Real
Root of
X + X
x 3 +x + l = 0,
which is
<n = 1.176280818.
(Le Lionnais 1983, p. 35). Boyd (1977) found the fol-
lowing table of small Salem numbers, and suggested that
ci, (72, 0"3> and a a are the smallest Salem numbers. The
Notation 110-1-1-1 is short for 1 1 -1 -1 -1
-1-10 11, the coefficients of the above polynomial.
D
= 2 n
dollars. In this toss, however, he wins 2 n_1 dollars. This
means that the net gain for the player is a whopping $1,
no matter how many tosses it takes to finally win. As
expected, the large payoff after a long run of tails is
exactly balanced by the large amount that the player
has to invest.
In fact, by noting that the probability of winning on
the nth toss is l/2 n , it can be seen that the probability
distribution for the number of tosses needed to win is
simply a Geometric Distribution with p = 1/2.
see also Coin Tossing, Gambler's Ruin, Geometric
Distribution, Martingale
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 44-45,
1987.
1592
Salesman Problem
Sample Proportion
&k
polynomial
1.1762808183
1.1883681475
1.2000265240
1.2026167437
1.2163916611
1.2197208590
1.2303914344
1.2326135486
1.2356645804
1.2363179318
1.2375048212
12 1.2407264237
13 1.2527759374
1.2533306502
1.2550935168
1.2562211544
1.2601035404
1.2602842369
1.2612309611
20 1.2630381399
21 1.2672964425
1.2806381563
1.2816913715
10
11
14
15
16
17
18
19
22
23
24 1.2824955606
25 1.2846165509
26 1.2847468215
27 1.2850993637
29 1.2851856708
30 1.2851967268
31 1.2851991792
32 1.2852354362
33 1.2854090648
34 1.2863959668
35 1.2867301820
36 1.2917414257
37 1.2920391602
38 1.2934859531
39 1.2956753719
10
18
14
14
10
18
10
20
22
16
26
12
18
20
14
18
24
22
10
26
14
8
26
20
18
26
30
28 1.2851215202 30
30
26
44
30
34
18
26
24
20
10
18
110-1-1-
1-11-10
10 0-1-10
10-10000
10 0-1-1
1-10
10 0-10-1
1-10 0-1
10-1-100
1-10
10-100-1
1-11-10
10 0-1
10-100-1
10-1-101
1-10 0-11
1-10 0-11
1-10-110
10-100-1
1-10 0-
1-10 0-
10 0-1-1
10 0-1
-1 -1 -1 -1
1-22-22-
10 0-10-
1-211-21
10 0-1-
1
1-22-210
10-11-11
1-10
10-1-100
1-10
00000001
10-100-1
1-10 0-11
10-11-10
1-22-22-
1-10 0-11
1-10 0-
10-100-1
10-1-101
1-10 0-11
-11-11
1
-1
0-11
10 0-11
110-1-1
-1
00-101001
-1
__1 _1 _1
-1
0-1
0-11-101-1
0-11-11
10 1
1 1
-1
-1 -1
2 10-11-1
1 -1 -1
0-110-11-1
1-1-1-1-10
-1 2 -2
-1
0-1000-100-1
10-1-1011
-10 0-1
1
-100010010-1
-10 1-1
1 -1
2 2-33-3
-10 1-110-11
10
0-101
-10 1-1
see also PlSOT-VlJAYARAGHAVAN CONSTANTS
References
Boyd, D. W. "Small Salem Numbers." Duke Math. J. 44,
315-328, 1977.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
1983.
Lehmer, D. H. "Factorization of Certain Cyclotomic Func-
tions." Ann. Math., Ser. 2 34, 461-479, 1933.
Salem, R. "Power Series with Integral Coefficients." Duke
Math. J. 12, 153-172, 1945.
Stewart, C. L. "Algebraic Integers whose Conjugates Lie Near
the Unit Circle." Bull Soc. Math. France 106, 169-176,
1978.
Salesman Problem
see Traveling Salesman Problem
Salient Point
A point at which two noncrossing branches of a curve
meet with different tangents.
see also CUSP
Salinon
The above figure formed from four connected Semicir-
cles. The word salinon is Greek for salt cellar, which
the figure resembles.
see also Arbelos, Piecewise Circular Curve, Semi-
circle
Salmon's Theorem
Given a track bounded by two confocal ELLIPSES, if a
ball is rolled so that its trajectory is tangent to the in-
ner Ellipse, the ball's trajectory will be tangent to the
inner ELLIPSE following all subsequent caroms as well.
References
Salmon, G. A Treatise on Conic Sections.
Chelsea, p. 182, 1954.
New York:
Saltus
The word saltus has two different meanings: either a
jump or an oscillation of a function.
Sample Proportion
Let there be x successes out of n BERNOULLI TRIALS.
The sample proportion is the fraction of samples which
were successes, so
(1)
x
p= -•
n
For large n, p has an approximately Normal Distri-
bution. Let RE be the Relative Error and SE the
Standard Error, then
{p} =P
SE(p) = a(p) =
p(l-p)
RE(p) =
2p(l-p)
erf _1 (CI),
(2)
(3)
(4)
where CI is the Confidence Interval and erf 2 is the
Erf function. The number of tries needed to determine
p with Relative Error RE and Confidence Inter-
val CI is
_2[err 1 (CI)] 2 2Kl-p)
(RE) 2
(5)
Sample Space
Sandwich Theorem
1593
Sample Space
Informally, the sample space for a given set of events
is the set of all possible values the events may assume.
Formally, the set of possible events for a given variate
forms a SlGMA ALGEBRA, and sample space is denned
as the largest set in the Sigma Algebra.
See also PROBABILITY SPACE, RANDOM VARIABLE,
Sigma Algebra, State Space
Sample Variance
To estimate the population VARIANCE from a sample
of TV elements with a priori unknown MEAN (i.e., the
MEAN is estimated from the sample itself), we need an
unbiased Estimator for a. This is the k- STATISTIC fc 2}
where
TV
and 77i2 = s 2 is the sample variance
N
- ivX^-*) 3
Note that some authors prefer the definition
N
^D**-*)'
(1)
(2)
(3)
since this makes the sample variance an Unbiased Es-
timator for the population variance.
see also fc-STATISTIC, VARIANCE
Sampling
For infinite precision sampling of a band-limited signal
at the Nyquist Frequency, the signal-to-noise ratio
after N q samples is
SNR:
<roo) =
pa
-1/2
\A + p 2 x/T+p 2 "
(1)
where p is the normalized cross-correlation COEFFI-
CIENT
V<* a (')><2/ 3 (t)>
(2)
For p < 1,
SNR « py/N q . (3)
The identical result is obtained for oversampling. For
undersampling, the SNR decreases (Thompson et al
1986),
see also Nyquist Sampling, Oversampling, Quanti-
zation Efficiency, Sampling Function, Shannon
Sampling Theorem, Sinc Function
References
Feuer, A. Sampling in Digital Signal Processing and Control.
Boston, MA: Birkhauser, 1996.
Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr.
Interferometry and Synthesis in Radio Astronomy. New
York: Wiley, pp. 214-216, 1986.
Sampling Function
The 1-D sampling function is given by
S(x) = V^ S(x-nAx)j
where S is the DlRAC DELTA FUNCTION. The 2-D ver-
sion is
S(Uj V) = VJ 6(U - U n ,V — V n )j
which can be weighted to
S(U,V) = ^Ji^TxT^Dn^l^ ~ « n ,t) —V n ),
where R n is a reliability weight, D n is a density weight
(Weighting Function), and T n is a taper.
see also Shah Function, Sinc Function
Sampling Theorem
In order for a band-limited (i.e., one with a zero POWER
Spectrum for frequencies f > B) baseband (/ > 0)
signal to be reconstructed fully, it must be sampled at a
rate / > 2B. A signal sampled at / = 2B is said to be
Nyquist Sampled, and / = 2B is called the Nyquist
Frequency. No information is lost if a signal is sam-
pled at the Nyquist Frequency, and no additional
information is gained by sampling faster than this rate.
see also Aliasing, Nyquist Frequency, Nyquist
Sampling, Oversampling
San Marco Fractal
The Fractal J(-3/4,0), where J is the Julia Set. It
slightly resembles the MANDELBROT Set.
see also DOUADY'S RABBIT FRACTAL, JULIA SET,
Mandelbrot Set
References
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 173, 1991.
Sandwich Theorem
see Ham Sandwich Theorem, Squeezing Theorem
1594
Sard's Theorem
SAS Theorem
Sard's Theorem
The set of "critical values" of a Map u : R n -> W 1 of
Class C 1 has Lebesgue Measure in R n .
see also Class (Map), Lebesgue Measure
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 682, 1980.
Sarkovskii's Theorem
Order the NATURAL NUMBERS as follows:
^2-9-C...-<2-2-3-<2-2-5-<2-2-7
-< 2 • 2 • 9 ^ . . . -< 2 - 2 • 2 • 3 -< . . .
-< 2 5 -< 2 4 -« 2 3 -< 2 2 -< 2 -< 1.
Now let F be a CONTINUOUS FUNCTION from the REALS
to the REALS and suppose p < q in the above ordering.
Then if F has a point of Least Period p, then F also
has a point of Least Period q.
A special case of this general result, also known as Sar-
kovskii's theorem, states that if a Continuous Real
function has a PERIODIC POINT with period 3, then
there is a Periodic Point of period n for every In-
teger n.
A converse to Sarkovskii's theorem says that if p -< q
in the above ordering, then we can find a CONTINUOUS
Function which has a point of Least Period <z, but
does not have any points of LEAST PERIOD p (Elaydi
1996). For example, there is a CONTINUOUS FUNCTION
with no points of LEAST PERIOD 3 but having points of
all other Least Periods.
see also Least Period
References
Conway, J. H. and Guy, R. K. "Periodic Points." In The
Book of Numbers. New York: Springer- Verlag, pp. 207-
208, 1996.
Devaney, R. L. An Introduction to Chaotic Dynamical Sys-
tems, 2nd ed. Reading, MA: Addison- Wesley, 1989.
Elaydi, S. "On a Converse of Sharkovsky's Theorem." Amer.
Math. Monthly 103, 386-392, 1996.
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, p. 49, 1993.
Sharkovsky, A. N. "Co-Existence of Cycles of a Continuous
Mapping of a Line onto Itself." Ukranian Math. Z. 16,
61-71, 1964.
Stefan, P. "A Theorem of Sharkovsky on the Existence of
Periodic Orbits of Continuous Endomorphisms of the Real
Line." Comm. Math. Phys. 54, 237-248, 1977.
Sarkozy's Theorem
A partial solution to the Erdos Squarefree Con-
jecture which states that the BINOMIAL COEFFICIENT
( 2 ™) is never SQUAREFREE for all sufficiently large n >
no. Sarkozy (1985) showed that if s(n) is the square
part of the Binomial Coefficient ( 2 ™) , then
where C,{z) is the RlEMANN Zeta Function. An upper
bound on n of 2 8 ' 000 has been obtained.
see also Binomial Coefficient, Erdos Squarefree
Conjecture
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, 1980.
Sander, J. W. "A Story of Binomial Coefficients and Primes."
Amer. Math. Monthly 102, 802-807, 1995.
Sarkozy, A. "On the 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.
Sarrus Linkage
A LINKAGE which converts circular to linear motion us-
ing a hinged square.
see also Hart's Inversor, Linkage, Peaucellier IN-
VERSOR
Sarrus Number
see Poulet Number
SAS Theorem
Specifying two sides and the ANGLE between them
uniquely determines a TRIANGLE. Let b be the base
length and h be the height. Then the AREA is
K = \ch = \acs\nB.
a)
The length of the third side is given by the Law of
Cosines,
b 2 = a 2 + c 2 — 2accosI?,
so
b = yo 2 + c 2 — 2accosi?.
Using the Law OF SiNES
a b __ c
sin A sin B sin C
then gives the two other ANGLES as
a sin B
(2)
(3)
A = sin
C — sin
Va 2 + c 2 — 2ac cos B
_i / csini?
yja 2 + c 2 — 2ac cos B
) (4)
.) (5)
ln5(n)-(\/2-2)C(|)v / ^,
see also AAA Theorem, AAS Theorem, ASA The-
orem, ASS Theorem, SSS Theorem, Triangle
Satellite Knot
Scalar Triple Product 1595
Satellite Knot
Let K± be a knot inside a TORUS. Now knot the TORUS
in the shape of a second knot (called the COMPAN-
ION KNOT) K2. Then the new knot resulting from
K\ is called the satellite knot K3. COMPOSITE KNOTS
are special cases of satellite knots. The only Knots
which are not HYPERBOLIC KNOTS are TORUS KNOTS
and satellite knots (including COMPOSITE Knots). No
satellite knot is an Almost Alternating Knot.
see also Almost Alternating Knot, Companion
Knot, Composite Knot, Hyperbolic Knot, Torus
Knot
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 115-118, 1994.
Satisfiability Problem
Deciding whether a given Boolean formula in conjunc-
tive normal form has an assignment that makes the for-
mula "true." In 1971, Cook showed that the problem is
NP-COMPLETE.
see also BOOLEAN ALGEBRA
References
Cook, S. A. and Mitchell, D. G. "Finding Hard Instances
of the Satisfiability Problem: A Survey." In Satisfiability
problem: theory and applications (Piscataway f NJ, 1996).
Theoret. Comput. Sci., Vol. 35. Providence, RI: Amer.
Math. Soc., pp. 1-17, 1997.
Sausage Conjecture
In n-D for n > 5 the arrangement of HYPERSPHERES
whose CONVEX Hull has minimal Content is always
a "sausage" (a set of HYPERSPHERES arranged with
centers along a line), independent of the number of n-
spheres. The CONJECTURE was proposed by Fejes Toth,
and solved for dimensions > 42 by Betke et al. (1994)
and Betke and Henk (1998).
see also Content, Convex Hull, Hypersphere, Hy-
persphere Packing, Sphere Packing
References
Betke, U.; Henk, M.; and Wills, J. M. "Finite and Infinite
Packings." J. Reine Angew. Math. 453, 165-191, 1994.
Betke, U. and Henk, M. "Finite Packings of Spheres." Dis-
crete Comput. Geom. 19, 197-227, 1998.
Croft, H. T.; Falconer, K. J.; and Guy, R. K. Problem D9
in Unsolved Problems in Geometry. New York: Springer-
Verlag, 1991.
Fejes Toth, L. "Research Problems." Periodica Methematica
Hungarica 6, 197-199, 1975.
Savitzky-Golay Filter
A low-pass filter which is useful for smoothing data.
see also Filter
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 183 and 644-645, 1992.
Savoy Knot
see Figure-of-Eight Knot
Scalar
A one-component quantity which is invariant under RO-
TATIONS of the coordinate system.
see also PSEUDOSCALAR, SCALAR FIELD, SCALAR
Function, Scalar Potential, Scalar Triple
Product, Tensor, Vector
Scalar Curvature
see Curvature Scalar
Scalar Field
A MAP / : E n «->■ R which assigns each x a SCALAR
Function /(x).
see also Vector Field
References
Morse, P. M. and Feshbach, H. "Scalar Fields." §1.1 in Meth-
ods of Theoretical Physics, Part I. New York: McGraw-
Hill, pp. 4-8, 1953.
Scalar Function
A function /(#!, . . . , x n ) of one or more variables whose
RANGE is one- dimensional, as compared to a VECTOR
Function, whose Range is three-dimensional (or, in
general, n-dimensional) .
see also Complex Function, Real Function, Vec-
tor Function
Scalar Potential
A conservative VECTOR FIELD (for which the CURL V X
F = 0) may be assigned a scalar potential
<Kx,y,z)- 0(0,0,0)
p(x,0,0)
L
F ds
/ Fi(t,0,0)dt+ / F 2 (x,t 1 Q)dt
</(0,0,0) */(x,0,0)
+
/ F z (x,y,t)dt,
J(x,y,0)
where f F • ds is a Line Integral.
see also POTENTIAL FUNCTION, VECTOR POTENTIAL
Scalar Triple Product
The Vector product
[A, B, C] = A * (B x C) = B • (C x A)
Ai A 2 As
B\ B2 B3
Ci C2 C3
= C ■ (A x B)
which yields a SCALAR (actually, a PSEUDOSCALAR)
1596 Scale
Schauder Fixed Point Theorem
The Volume of a Parallelepiped whose sides are
given by the vectors A, B, and C is
^parallelepiped = |A ■ (B X C)|.
see also Cross Product, Dot Product, Parallel-
epiped, Vector Triple Product
References
Arfken, G. "Triple Scalar Product, Triple Vector Product."
§1.5 in Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 26-33, 1985.
Scale
see Base (Number)
Scale Factor
For a diagonal Metric Tensor gij — guSij, where 5ij
is the Kronecker Delta, the scale factor is defined by
hi = y/gU.
(i)
The Line Element (first Fundamental Form) is then
given by
ds = #ii dxn + 522 dx22 + #33 dx^s
(2)
hi dxn + /l2 dX22 + h>3 dxs3 . (3)
The scale factor appears in vector derivatives of coordi-
nates in Curvilinear Coordinates.
see also Curvilinear Coordinates, Fundamental
Forms, Line Element
Scalene Triangle
A Triangle with three unequal sides.
see also Acute Triangle, Equilateral Triangle,
Isosceles Triangle, Obtuse Triangle, Triangle
Scaling
Increasing a plane figure's linear dimensions by a scale
factor s increases the Perimeter p' — > sp and the Area
A' -► s 2 A.
see also Dilation, Expansion, Fractal, Self-
Similarity
Scattering Operator
An Operator relating the past asymptotic state of a
Dynamical System governed by the Schrodinger equa-
tion
if t m = Hi>{t)
to its future asymptotic state.
see also Wave Operator
Scattering Theory
The mathematical study of the Scattering Operator
and Schrodinger equation.
see also SCATTERING OPERATOR
References
Yafaev, D. R. Mathematical Scattering Theory: General The-
ory. Providence, RI: Amer. Math. Soc, 1996.
Schaar's Identity
A generalization of the GAUSSIAN SUM. For p and q
of opposite Parity (i.e., one is Even and the other is
Odd), Schaar's identity states
v r=0
e -iri/4 V~\
c X ^ ^mt q/p
Vp
■E«
see also Gaussian Sum
References
Evans, R. and Berndt, B. "The Determination of Gauss
Sums." Bull Amer. Math. Soc. 5, 107-129, 1981.
SchanuePs Conjecture
Let Ai, ..., A n 6 C be linearly independent over the
RATIONALS Q, then
Q(Ai,.
,e A ")
has Transcendence degree at least n over Q.
Schanuel's conjecture is a generalization of the
LlNDEMANN-WEIERSTRAfi THEOREM. If the conjecture
is true, then it follows that e and 7v are algebraically
independent. Mcintyre (1991) proved that the truth of
Schanuel's conjecture also guarantees that there are no
unexpected exponential-algebraic relations on the INTE-
GERS Z (Marker 1996).
see also Constant Problem
References
Macintyre, A. "Schanuel's Conjecture and Free Exponential
Rings." Ann. Pure Appl. Logic 51, 241-246, 1991.
Marker, D. "Model Theory and Exponentiation." Not.
Amer. Math. Soc. 43, 753-759, 1996.
Schauder Fixed Point Theorem
Let A be a closed convex subset of a BANACH SPACE
and assume there exists a continuous MAP T sending A
to a countably compact subset T(A) of A. Then T has
fixed points.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 543, 1980.
Schauder, J. "Der Fixpunktsatz in Funktionalraumen." Stu-
dia Math. 2, 171-180, 1930.
Zeidler, E. Applied Functional Analysis: Applications to
Mathematical Physics. New York: Springer- Verlag, 1995.
Scheme
Schinzel Circle 1597
Scheme
A local-ringed Space which is locally isomorphic to an
Affine Scheme.
see also Affine Scheme
References
Iyanaga, S. and Kawada, Y. (Eds.)- "Schemes." §18E in En-
cyclopedic Dictionary of Mathematics. Cambridge, MA:
MIT Press, p. 69, 1980.
Schensted Correspondence
A correspondence between a PERMUTATION and a pair
of Young Tableaux.
see also PERMUTATION, YOUNG TABLEAU
References
Knuth, D. E. The Art of Computer Programming, Vol. 3:
Sorting and Searching, 2nd ed. Reading, MA: Addison-
Wesley, 1973.
Stanton, D. W. and White, D. E. §3.6 in Constructive Com-
binatorics. New York: Springer- Verlag, pp. 85-87, 1986.
Scherk's Minimal Surfaces
A class of Minimal Surfaces discovered by Scherk
(1834) which were the first new surfaces discovered since
Meusnier in 1776. Scherk's first surface is doubly peri-
odic. Scherk's second surface, illustrated above, can be
written parametrically as
x = 2SR[ln(l + re ie ) - ln(l - re i$ )]
y = R[4it<m- 1 (re id )]
z = ft {2i(- ln[l - r 2 e 2ie ) + ln[l + r 2 e 2i6 ])}
for € [0,27r), and r £ (0,1). Scherk's first surface
has been observed to form in layers of block copolymers
(Peterson 1988).
von Seggern (1993) calls
cln
cos(27ry)
cos(27rx)
"Scherk's surface." Beautiful images of wood sculptures
of Scherk surfaces are illustrated by Sequin.
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.
Meusnier, J. B. "Memoire sur la courbure des surfaces."
Mem. des savans etrangers 10 (lu 1776), 477—510, 1785.
Peterson, I. "Geometry for Segregating Polymers." , Sci.
News, 151, Sep. 3, 1988.
Scherk, H. F. "Bemerkung liber der kleinste Flache innerhalb
gegebener Grenzen." J. Reine. angew. Math. 13, 185-208,
1834.
Thomas, E. L.; Anderson, D. M.; Henkee, C. S.; and
Hoffman, D. "Periodic Area- Minimizing Surfaces in Block
Copolymers/' Nature 334, 598-601, 1988.
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 304, 1993.
Wolfram Research "Mathematica Version 2.0 Graphics
Gallery." http : // www . mathsource . com / cgi - bin / Math
Source/Applications/Graphics/3D/0207-155.
Schiffler Point
The Concurrence S of the Euler Lines E n of
the Triangles AXBC, AXCA, AXAB, and AABC
where X is the INCENTER. The TRIANGLE CENTER
Function is
1 b + c — a
cos B + cos C
b + c
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Schiffler Point." http: //www. evansville.
edu/ -ck6/t center s/recent/schiff .html.
Schiffler, K.; Veldkamp, G. R.; and van der Spek, W. A.
"Problem 1018 and Solution." Crux Math. 12, 176-179,
1986.
Schinzel Circle
A CIRCLE having a given number of LATTICE POINTS
on its Circumference. The Schinzel circle halving n
lattice points is given by the equation
(x
(x-
l) 2 + y
2 _ lcfc-1
2 _ i 5 2*
for n = 2k even
for n = 2k + 1 odd.
Note that these solutions do not necessarily have the
smallest possible Radius. For example, while the
Schinzel circle centered at (1/3, 0) and with radius 625/3
1598 Schinzel's Hypothesis
SchlaHi Function
has nine lattice points on its circumference, so does the
circle centered at (1/3, 0) with radius 65/3.
see also Circle, Circle Lattice Points, Ku-
likowski's Theorem, Lattice Point, Schinzel's
Theorem, Sphere
References
Honsberger, R. "Circles, Squares, and Lattice Points."
Ch. 11 in Mathematical Gems I. Washington, DC: Math.
Assoc. Amer., pp. 117-127, 1973.
Kulikowski, T. "Sur l'existence d'une sphere passant par un
nombre donne aux coordonnees entieres." L'Enseignement
Math. Ser. 2 5, 89-90, 1959.
Schinzel, A. "Sur l'existence d'un cercle passant par un
nombre donne de points aux coordonnees entieres."
L'Enseignement Math. Ser. 2 4, 71-72, 1958.
Sierpihski, W. "Sur quelques problemes concernant les points
aux coordonnees entieres." L'Enseignement Math. Ser. 2
4, 25-31, 1958.
Sierpinski, W. "Sur un probleme de H. Steinhaus concernant
les ensembles de points sur le plan." Fund. Math. 46,
191-194, 1959.
Sierpinski, W. A Selection of Problems in the Theory of
Numbers. New York: Pergamon Press, 1964.
Schinzel's Hypothesis
If fi(x), ..., f s {x) are irreducible POLYNOMIALS with
Integer Coefficients such that no Integer n > 1
divides fi(x), . . . , f s (x) for all INTEGERS x, then there
should exist infinitely many x such that fi(x), . . . , f s (x)
are simultaneous PRIME.
References
Schinzel, A. and Sierpinski, W. "Sur certaines hypotheses
concernant les nombres premiers. Remarque." Acta
Arithm. 4, 185-208, 1958.
Schinzel's Theorem
For every POSITIVE INTEGER n, there exists a CIRCLE
in the plane having exactly n LATTICE POINTS on its
Circumference. The theorem is based on the number
r(n) of integral solutions (x,y) to the equation
given by
.2 , 2
x + y =n,
r(n) = 4(di - d 3 ),
(1)
(2)
where d\ is the number of divisors of n of the form 4/c + l
and dz is the number of divisors of the form 4fe + 3. It
explicitly identifies such circles (the SCHINZEL Circles)
as
References
Honsberger, R. "Circles, Squares, and Lattice Points."
Ch. 11 in Mathematical Gems I. Washington, DC: Math.
Assoc. Amer., pp. 117-127, 1973,
Kulikowski, T. "Sur l'existence d'une sphere passant par un
nombre donne aux coordonnees entieres." L'Enseignement
Math. Ser. 2 5, 89-90, 1959.
Schinzel, A. "Sur l'existence d'un cercle passant par un
nombre donne de points aux coordonnees entieres."
L'Enseignement Math. Ser. 2 4, 71-72, 1958.
Sierpinski, W. "Sur quelques problemes concernant les points
aux coordonnees entieres." L'Enseignement Math. Ser. 2
4, 25-31, 1958.
Sierpinski, W. "Sur un probleme de H. Steinhaus concernant
les ensembles de points sur le plan." Fund. Math. 46,
191-194, 1959.
Sierpinski, W. A Selection .of Problems in the Theory of
Numbers. New York: Pergamon Press, 1964.
Schisma
The musical interval by which eight fifths and a major
third exceed five octaves,
(f) 8 (!) _ 3 8 -5 _ 32805
= 1.00112915..
2 5 2 15 32768
see also COMMA OF DlDYMUS, COMMA OF PYTHAGO-
RAS, Diesis
Schlafli Double Six
see Double Sixes
Schlafli's Formula
For R[z] > 0,
i r /2
Jv{z) — — I cos(zsin£ — ut) dt
* Jo
sin(i/7r) f°
* Jo
e e at,
where J u (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.
Schlafli Function
The function giving the VOLUME of the spherical
quadrectangular TETRAHEDRON:
8 \p q r
(x-
-tf+V t = & k - 1
for n :
= 2k
(3)
where
(x-
-|) 2 +2/ 2 = |5 2fc
for n -
= 2fe + l.
7T 2 ,/
Note, however, that these solutions do not necessarily
have the smallest possible radius.
see also Browkin's Theorem, Kulikowski's Theo-
rem, Schinzel Circle
7r / 7r 7r \ __ v-^ / D — sin x sin z \ m
T / U ~ X,V ' 2 ~ V = 2-J V.D + sinxsin^J
m=l
cos(2mx) — cos(2my) + cos(2?n^) — 1 2 2 2
x ^— x -y - z ,
and
D~^
cos 2 x cos 2 z — cos 2 y.
see also TETRAHEDRON
Schlafli Integral
Schnirelmann's Theorem
1599
Schlafli Integral
A definition of a function using a CONTOUR INTEGRAL.
Schlafli integrals may be converted into RODRIGUES
Formulas.
see also Rodrigues Formula
Schlafli's Modular Form
The Modular Equation of degree 5 can be written
\v J \uj V u 2 v 2 ) '
see also MODULAR EQUATION
Schlafli Polynomial
A polynomial given in terms of the Neumann Polyno-
mials On(x) by
S n (x) =
2xO n (x) - 2cos 2 (|n7r)
see also Neumann Polynomial
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1477,
1980.
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 196, 1993.
Schlafli Symbol
The symbol {p, q} is used to denote a TESSELLATION
of regular p-gons, with q of them surrounding each
Vertex. The Schlafli symbol can be used to de-
scribe Platonic Solids, and a generalized version de-
scribes QUASIREGULAR POLYHEDRA and ARCHIMED-
EAN Solids.
see also Archimedean Solid, Platonic Solid,
QUASIREGULAR POLYHEDRON, TESSELLATION
Schlegel Graph
A GRAPH corresponding to POLYHEDRA skeletons. The
POLYHEDRAL Graphs are special cases.
References
Gardner, M. Wheels t Life, and Other Mathematical Amuse-
ments. New York: W. H. Freeman, p. 158, 1983.
Schlomilch's Function
/*oo /»oo
5(1/,*)=/ {l + t)-"e- zt dt = z ,/ - 1 e z u- u e- u du
= Z e W- u/2 ,(l-v)/2(z),
where W k ,m(z) is the WHITTAKER FUNCTION.
Schlomilch's Series
A FOURIER SERIES-like expansion of a twice continu-
ously differentiable function
oo
f(x) ~ ~a +y j a n Jo{nx)
n=l
for < x < 7r, where Jo(x) is a zeroth order BESSEL
FUNCTION OF THE FIRST KlND and
ao
= 2/(0) + - / du / /'(usin0)d<£
* Jo Jo
_ 2 r r /2 ,
a n = — I du I uf (usin(j))cos(n7r)d<f).
* Jo Jo
A special case gives the amazing identity
oo oo
1 = J Q {Z) + 2 ^ hn{z) = [Jo(z)] 2 + 2 Y^i J n(z)} 2 -
see also Bessel Function of the First Kind, Bes-
sel Function Fourier Expansion, Fourier Series
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1473,
1980.
Schmitt-Conway Biprism
A Convex Polyhedron which is Space-Filling, but
only aperiodically, was found by Conway in 1993.
see also CONVEX POLYHEDRON, SPACE-FlLLING POLY-
HEDRON
Schnirelmann Constant
The constant s in SCHNIRELMANN'S THEOREM.
see also Schnirelmann's Theorem
Schnirelmann Density
The Schnirelmann density of a sequence of natural num-
bers is the greatest lower bound of the fractions A(n)/n
where A(n) is the number of terms in the sequence < n.
References
Khinchin, A. Y. "The Landau-Schnirelmann Hypothesis and
Mann's Theorem." Ch. 2 in Three Pearls of Number The-
ory. New York: Dover, pp. 18-36, 1998.
Schnirelmann's Theorem
There exists a POSITIVE INTEGER s such that every suf-
ficiently large Integer is the sum of at most $ Primes.
It follows that there exists a POSITIVE INTEGER s > s
such that every INTEGER > 1 is a sum of at most so
Primes, where s is the Schnirelmann Constant.
The best current estimate is so = 19.
see also Prime Number, Schnirelmann Density,
Waring's Problem
References
Khinchin, A. Y. "The Landau-Schnirelmann Hypothesis and
Mann's Theorem." Ch. 2 in Three Pearls of Number The-
ory. New York: Dover, pp. 18-36, 1998.
1600
Schoenemann's Theorem
Schrage's Algorithm
Schoenemann's Theorem
If the integral COEFFICIENTS C , Ci, . . . , Cjv-i of the
Polynomial
f(x) = Co + dx + C 2 x 2 + . . . + C N ^x N - 1 + x N
are divisible by a Prime Number p, while the free term
Co is not divisible by p 2 , then f(x) is irreducible in the
natural rationality domain.
see also Abel's Irreducibility Theorem, Abel's
Lemma, Gauss's Polynomial Theorem, Kron-
ecker's Polynomial Theorem
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 118,
1965.
Scholz Conjecture
Let the minimal length of an ADDITION CHAIN for a
number n be denoted l(n). Then the Scholz conjecture
states that
/(2 n -l) <n-l + /(n).
The conjecture has been proven for a variety of special
cases but not in general.
see also ADDITION CHAIN
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. Ill, 1994.
Schonflies Symbol
One of the set of symbols d y C 3 , Ci, C2, C3, C4, C5,
C$, CV, C$, Cih, Cshj Cih, Csh, Cehi Civ, C% v , C± v ,
Csv, Cq v , Cqov, £>2j &31 &4, D& y Dq, £>2/i, £>3/i, £>4/i,
£>5/ij £>6hj -C>8h, Dooh, Did, £>3d, £>4d, £>5d, ^6dj ^, Ih }
O, Ofc, 54, Se, Sa, T, Td, and T* used to identify crys-
tallographic symmetry GROUPS.
Cotton (1990), gives a table showing the translations
between Schonflies symbols and HERMANN-MAUGUIN
Symbols. Some of the Schonflies symbols denote dif-
ferent sets of symmetry operations but correspond to
the same abstract GROUP and so have the same CHAR-
ACTER Table.
see also Character Table, Hermann-Mauguin
Symbol, Point Groups, Space Groups, Symmetry
Operation
References
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 379, 1990.
Schonflies Theorem
If J is a simple closed curve in R , the closure of one
of the components of M 2 — J is HOMEOMORPHIC with
the unit 2-BALL. This theorem may be proved using the
Riemann Mapping Theorem, but the easiest proof is
via Morse Theory.
The generalization to n-D is called Mazur's Theo-
rem. It follows from the Schonflies theorem that any
two Knots of S 1 in § 2 or R 2 are equivalent.
see also JORDAN CURVE THEOREM, MAZUR'S THEO-
REM, Riemann Mapping Theorem
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 9, 1976.
Thomassen, C. "The Jordan- Schonflies Theorem and the
Classification of Surfaces." Amer. Math. Monthly 99, 116—
130, 1992.
Schoolgirl Problem
see KlRKMAN'S SCHOOLGIRL PROBLEM
Schoute Coaxal System
The Circumcircle, Brocard Circle, Lemoine
Line, and Isodynamic Points belong to a Coaxal
System orthogonal to the the Apollonius Circles,
called the Schoute coaxal system. In general, there are
12 points whose PEDAL TRIANGLES with regard to a
given Triangle have a given form. They lie six by six
on two Circles of the Schoute coaxal system.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 297-299, 1929.
Schoute's Theorem
In any Triangle, the Locus of a point whose Pedal
Triangle has a constant Brocard Angle and is de-
scribed in a given direction is a CIRCLE of the SCHOUTE
Coaxal System.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 297-299, 1929.
Schoute, P. H. Proc. Amsterdam Acad., 39-62, 1887-1888.
Schrage's Algorithm
An algorithm for multiplying two 32-bit integers modulo
a 32-bit constant without using any intermediates larger
than 32 bits. It is also useful in certain types of Random
Number generators.
References
Bratley, P.; Fox, B. L.; and Schrage, E. L. A Guide to Sim-
ulation, 2nd ed. New York: Springer- Verlag, 1996.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Random Numbers." Ch. 7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
p. 269, 1992.
Schrage, L. "A More Portable Fortran Random Number Gen-
erator." ACM Trans. Math. Software 5, 132-138, 1979.
Schroder-Bernstein Theorem
Schroder-Bernstein Theorem
The Schroder-Bernstein theorem for numbers states that
if
n < m < 71,
then m = n. For Sets, the theorem states that if there
are Injections of the Set A into the Set B and of
B into A, then there is a BlJECTIVE correspondence
between A and B (i.e., they are Equipollent).
see also Bijection, Equipollent, Injection
Schroder's Equation
f(\z) = R{z),
where R(z) = Ax + a 2 x 2 + . . ., A = fl'(O), |A| = 1, and
A n ^ 1 for all n £ N.
Schroder's Method
Two families of equations used to find roots of nonlin-
ear functions of a single variable. The "B" family is
more robust and can be used in the neighborhood of
degenerate multiple roots while still providing a guar-
anteed convergence rate. Almost all other root-finding
methods can be considered as special cases of Schroder's
method. Householder humorously claimed that papers
on root-finding could be evaluated quickly by looking
for a citation of Schroder's paper; if the reference were
missing, the paper probably consisted of a rediscovery
of a result due to Schroder (Stewart 1993).
One version of the "A" method is obtained by applying
Newton's Method to ///',
X n -\-± — X n
f(Xn)f'(x n )
[f'(Xn)] 2 -f(x n )f"(x n )
(Scavo and Thoo 1995).
see also Newton's Method
References
Householder, A. S. The Numerical Treatment of a Single
Nonlinear Equation. New York: McGraw-Hill, 1970.
Scavo, T. R. and Thoo, J. B. "On the Geometry of Halley's
Method." Amer. Math. Monthly 102, 417-426, 1995.
Schroder, E. "Uber unendlich viele Algorithmen zur
Auflosung der Gleichungen." Math. Ann. 2, 317-365,
1870.
Stewart, G. W. "On Infinitely Many Algorithms for Solv-
ing Equations." English translation of Schroder's orig-
inal paper. College Park, MD: University of Maryland,
Institute for Advanced Computer Studies, Department of
Computer Science, 1993. ftp://thales.cs.umd.edu/pub/
reports/imase.ps.
Schroter's Formula
Schroder Number
1601
V
1 [ "~;?
! ~
sfflffiaz
!"'! L
I l / /
-' )£ /
? \A
vr \ V
i/i !
...dl^Li :
zzp.zn t±_
r-r~r
1/ ,\
~y\ ~EL_qz
~zt rM
■-/ -f 4-
n2
\\AA yx
_2t ^±.^_
zrrv.r ~m
~7- -^ - ->-'
/ t
' a /r
Y. \ZZ -SLl.
Jm.
^1e __
±- -V z _
1 z&zna
~^y
<j_ -,^i-
_,z_
/
/H /
ZLJ
The Schroder number S n is the number of LATTICE
PATHS in the Cartesian plane that start at (0, 0), end at
(n,n), contain no points above the line y = x, and are
composed only of steps (0, 1), (1, 0), and (1, 1), i.e., ->,
t, and /\ The diagrams illustrating the paths generat-
ing Si, 52, and S3 are illustrated above. The numbers
S n are given by the RECURRENCE RELATION
n-l
S n — S n -1 + / ^ SkSn-1-kj
fc=0
where So = 1, and the first few are 2, 6, 22, 90, ...
(Sloane's A006318). The Schroder Numbers bear the
same relation to the DELANNOY NUMBERS as the CATA-
LAN Numbers do to the Binomial Coefficients.
see also Binomial Coefficient, Catalan Number,
Delannoy Number, Lattice Path, Motzkin Num-
ber, p-Good Path
References
Sloane, N. J. A. Sequence A006318/M1659 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Schroeder Stairs
see Penrose Stairway
Schroter's Formula
Let a general Theta FUNCTION be defined as
T(x,q)= ]T] x n q n ,
then
T(x,q a )T(x,q b ) =
°]T y k q bk2 T(xyq 2b \q a+b )T(y'x-\ 2ab \q ab(1+b) ).
see also Blecksmith-Brillhart-Gerst Theorem,
Jacobi Triple Product, Ramanujan Theta Func-
tions
1602 Schur Algebra
Schur Number
References
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in
Analytic Number Theory and Computational Complexity.
New York: Wiley, p. Ill, 1987.
Tannery, J. and Molk, J. Elements de la Theorie des Fonc-
tions Elliptiques, 4 vols. Paris: Gauthier-Villars et fils,
1893-1902.
Schur Matrix
The pxp Square Matrix formed by setting s^- = C j ,
where £ is an pth ROOT OF Unity. The Schur matrix
has a particularly simple DETERMINANT given by
detS
rf'\
Schur Algebra
An Auslander algebra which connects the representation
theories of the symmetric group of PERMUTATIONS and
the General Linear Group GL(n,C). Schur algebras
are "quasihereditary."
References
Martin, S. Schur Algebras and Representation Theory. New
York: Cambridge University Press, 1993.
Schur Functor
A FUNCTOR which defines an equivalence of module
Categories.
References
Martin, S. Schur Algebras and Representation Theory. New
York: Cambridge University Press, 1993.
Schur's Inequalities
Let A = a,ij be an n x n Matrix with Complex (or
Real) entries and Eigenvalues A x , A 2 , . - . , A n , then
where p is an Odd Prime and
£>i| 2 < £|ay|
i , j — 1
£>[A;]| 2 <]T
aij + a* j{
&ij Gji
ij = l
References
Gradshteyn, I. S, and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1120, 1979.
Schur's Lemma
For each k 6 N there exists a largest Integer s(k)
(known as the Schur Number) such that no matter
how the set of INTEGERS less than \n\e\ (where [as J
is the Floor Function) is partitioned into k classes,
one class must contain INTEGERS as, y, z such that
x + y = z, where x and y are not necessarily distinct.
The upper bound has since been slightly improved to
[n!(e- 1/24)J.
see also COMBINATORICS, SCHUR NUMBER, SCHUR'S
Theorem
References
Guy, R. K. "Schur's Problem. Partitioning Integers into
Sum-Free Classes" and "The Modular Version of Schur's
Problem." §E11 and E12 in Unsolved Problems in Number
Theory, 2nd ed. New York: Springer- Verlag, pp. 209-212,
1994.
(l if p= 1
\ i if p = 3
= 1 (mod 4)
3 (mod 4).
This determinant has been used to prove the QUADRA-
TIC Reciprocity Law (Landau 1958, Vardi 1991). The
Absolute Values of the Permanents of the Schur
matrix of order 2p + 1 are given by 1, 3, 5, 105, 81,
6765, . . . (Sloane's A003112, Vardi 1991).
Denote the Schur matrix S p with the first row and first
row column omitted by S p . Then
permSp = ppermS^,
where perm denoted the PERMANENT (Vardi 1991).
References
Graham, R. L. and Lehmer, D. H. "On the Permanent of
Schur's Matrix." J. Austral Math. Soc. 21, 487-497,
1976.
Landau, E. Elementary Number Theory. New York: Chelsea,
1958.
Sloane, N. J. A. Sequence A003112/M2509 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 119-122 and 124, 1991.
Schur Multiplier
A property of FINITE SIMPLE GROUPS which is known
for all such GROUPS.
see also Finite Group, Simple Group
Schur Number
The Schur numbers are the numbers in the partition-
ing of a set which are guaranteed to exist by Schur's
Lemma. Schur numbers satisfy the inequality
s(k) > c(315) fc/5
for k > 5 and some constant c. Schur's Theorem also
shows that
s(n) < R(n),
where R(n) is a Ramsey Number. The first few
Schur numbers are 1, 4, 13, 44, (> 157), ... (Sloane's
A045652).
see also RAMSEY NUMBER, RAMSEY'S THEOREM,
Schur's Lemma, Schur's Theorem
References
Frederickson, H. "Schur Numbers and the Ramsey Numbers
7V(3,3,...,3;2)." J. Combin. Theory Ser. A 27, 376-377,
1979.
Schur's Problem
Guy, R. K. "Schur's Problem. Partitioning Integers into
Sum-Free Classes" and "The Modular Version of Schur's
Problem." §E11 and E12 in Unsolved Problems in Number
Theory, 2nd ed. New York: Springer- Verlag, pp. 209-212,
1994.
Sloane, N. J. A. Sequence A045652 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Schur's Problem
see Schur's Lemma
Schur's Representation Lemma
If 7r on V and n' on V f are irreducible representations
and E : V n-> V is a linear map such that ir' (g)E =
Eir(g) for all g £ and group G, then E — or E is
invertible. Furthermore, if V = V' , then E is a Scalar.
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Schur's Theorem
As shown by Schur in 1916, the SCHUR NUMBER s(n)
satisfies
s(n) < R(n)
for n — 1, 2, ... , where R(n) is a RAMSEY NUMBER.
see also Ramsey Number, Schur's Lemma, Schur
Number
Schwarz's Inequality
l<V>ihk}| 2 <Wiltfi>(iMte>.
Written out explicitly
(i)
i
b 1 2 />& fb
ipi(x)ip2(x) dx
j < j [Mx)] 2 dxf
[Mx)l dx I [M*)] dx,
(2)
with equality Iff g(x) = otf{x) with a a constant. To
derive, let ip(x) be a Complex function and A a Com-
plex constant such that ift(x) = f(x) + Xg(x) for some
/ and g. Then
/ ip*ipdx = / f*fdx + A / f* gdx + A* / g* f dx
+AA* g*gdx > 0, (3)
with equality when ip(x) = 0. Now, note that A and A*
are LINEARLY Independent (they are Orthogonal),
so differentiate with respect to one of them (say A*) and
set to zero to minimize J ip*ipdx.
I tp*ipdx — / g* f dx + A / g*gdx = Q
J g* gdx'
(4)
(5)
Schwarz-Pick Lemma
which means that
Jf'gdx
A* = -
f g'gdx
1603
(6)
Plugging back in,
j i>*4>dx= j f'fdx- y/^ I f'gdx
J f*g dx
*gdx T „
*gdx J
fdx +
St
J g'fdx / f*gdx
{Jg'gdxY
/'
gdx > 0.
(7)
Multiplying through by J g* gdx gives
/ f f dx j g*gdx - / g* f dx f*gdx
- / fgdx g*fdx+ g*fdx f*gdx>0 (8)
g*fdx / f*gdx< if f dx / g*gdx (9)
g*fdx\ = / f*gdx\ < I ffdx j g*gdx (10)
or
if\g) I 2 < (f\f) (g\g) .
(ii)
Bessel's Inequality can be derived from this.
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, gth printing. New York: Dover,
p. 11, 1972.
.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 527-529, 1985.
Schwarz-Pick Lemma
If / is an analytic map of the DISK D into B> and / pre-
serves the hyperbolic distance between any two points,
then / is a disk map and preserves all distance.
References
Busemann, H. The Geometry of Geodesies. New York: Aca-
demic Press, p. 41, 1955.
1604 Schwarz Reflection Principle
Scientific Notation
Schwarz Reflection Principle
Let _
, n f (n) (z )
g(z) ^^(z-zo)"
nl
(1)
then
g'{*) =
]P(z - ZoY
/ (n) (*>)
f (n) (zp')
E, * .\ / K 2
(2)
If zo is pure real, then zo = zo*, so
9(z'). (3)
Therefore, if a function f(z) is ANALYTIC over some
region including the Real Line and f(z) is REAL when
z is real, then f*(z) = f(z*).
Schwarz Triangle
The Schwarz triangles are Spherical Triangles
which, by repeated reflection in their indices, lead to
a set of congruent SPHERICAL TRIANGLES covering the
Sphere a finite number of times.
Schwarz triangles are specified by triples of numbers
(p>q, r )' There are four "families" of Schwarz triangles,
and the largest triangles from each of these families are
(22n),(2 2 2/»V2 3 3'>(-4 4 4/'
The others can be derived from
(p q r) = (p x n) + (x q r 2 ),
where
and
cos(-)=-cos(-)
_ co S (f)sin(^-)-co S (f)sin(^)
Ml)
see also Colunar Triangle, Spherical Triangle
References
Coxeter, H. S. M. Regular Poly topes, 3rd ed. New York:
Dover, pp. 112-113 and 296, 1973.
Schwarz, H. A. "Zur Theorie der hypergeometrischen Reihe."
J. reine angew. Math. 75, 292-335, 1873.
Schwarz's Triangle Problem
see Fagnano's Problem
Schwarzian Derivative
The Schwarzian derivative is defined by
Di
Schwarzian —
The Feigenbaum Constant is universal for 1-D MAPS
if its Schwarzian derivative is NEGATIVE in the bounded
interval (Tabor 1989, p. 220).
see also FEIGENBAUM CONSTANT
References
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, 1989.
Schwenk's Formula
Let R + B be the number of MONOCHROMATIC FORCED
Triangles (where R and B are the number of red and
blue Triangles) in an Extremal Graph. Then
where (£) is a Binomial Coefficient and [a; J is the
Floor Function (Schwenk 1972).
see also Extremal Graph, Monochromatic
Forced Triangle
References
Schwenk, A. J. "Acquaintance Party Problem." Amer. Math.
Monthly 79, 1113-1117, 1972.
Scientific Notation
Scientific notation is the expression of a number n in the
form a X 10 p , where
V = L lo gio \ n \\
is the Floor of the base-10 Logarithm of n (the "order
of magnitude" ) , and
n
Top
is a Real Number satisfying 1 < \a\ < 10. For exam-
ple, in scientific notation, the number n = 101,325 has
order of magnitude
p = [log 10 101,325J = L 5 - 00572 J = 5 >
so n would be written 1.01325 x 10 5 . The special case
of does not have a unique representation in scientific
notation, i.e., = x 10° = x 10 1 = . . ..
see also Characteristic (Real Number), Figures,
Mantissa, Significant Figures
Score Sequence
Secant 1605
Score Sequence
The score sequence of a TOURNAMENT is a monotonic
nondecreasing sequence of the Outdegrees of the Ver-
tices. The score sequences for n = 1, 2, . . . are 1, 1,
2,4, 9, 22, 59, 167, ... (Sloane's A000571).
see also TOURNAMENT
References
Ruskey, F. "Information on Score Sequences." http://sue.
esc .uvic . ca/-cos/inf /nump/ScoreSequence .html.
Ruskey, F.; Cohen, R.; Fades, R; and Scott, A. "Alley CATs
in Search of Good Homes." Congres. Numer. 102, 97-110,
1994.
Sloane, N. J. A. Sequence A000571/M1189 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Searching
Searching refers to locating a given element or an el-
ement satisfying certain conditions from some (usually
ordered or partially ordered) table, list, Tree, etc.
see also Sorting, Tabu Search, Tree Searching
References
Knuth, D. E. The Art of Computer Programming, 2nd ed,
Vol. 3: Sorting and Searching. Reading, MA: Addison-
Wesley, 1973.
Press, W. H.; Flannery, B. R; Teukolsky, S. A.; and Vet-
terling, W. T. "How to Search an Ordered Table." §3.4
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 110-113, 1992.
Screw
A Translation along a straight line L and a Rotation
about L such that the angle of ROTATION is proportional
to the TRANSLATION at each instant. Also known as a
Twist.
see also DlNl'S SURFACE, HELICOID, ROTATION, SCREW
Theorem, Seashell, Translation
Screw Theorem
Any motion of a rigid body in space at every instant is
a SCREW motion. This theorem was proved by Mozzi
and Cauchy.
see also SCREW
Scruple
An archaic UNIT FRACTION variously defined as 1/200
(of an hour), 1/10 or 1/12 (of an inch), 1/12 (of a ce-
lestial body's angular diameter), or 1/60 (of an hour or
Degree).
see also Calcus, Uncia
Sea Horse Valley
A portion of the Mandelbrot Set centered around
-1.25 + 0.047z with width approximately 0.009 + 0.0052.
see also Mandelbrot Set
Search Tree
see Tree Searching
Seashell
see Conical Spiral
Secant
The function defined by sec a: = l/cosz, where cos a; is
the Cosine. The Maclaurin Series of the secant is
( — l) n E 2 n 2n
sec a: = — . , , — x
(2n)!
1 + W + £
x* + -^-x 6 + 2Z-x s 4-
**' ~ 720'*' ' ftnfi4^ ~ • • ' '
where E 2n is an Euler NUMBER.
see also Alternating Permutation, Cosecant, Co-
sine, Euler Number, Exsecant, Inverse Secant
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func-
tions." §4.3 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972.
Spanier, J. and Oldham, K. B. "The Secant sec(a;) and Cose-
cant csc(x) Functions." Ch. 33 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 311-318, 1987.
1606
Secant Line
Second Fundamental Tensor
Secant Line
tangent line
secant line
A line joining two points of a curve. In abstract math-
ematics, the points which a secant line connects can be
either Real or Complex Conjugate Imaginary.
see also BlTANGENT, TANGENT LINE, TRANSVERSAL
Line
Secant Method
A RoOT-finding algorithm which assumes a function to
be approximately linear in the region of interest. Each
improvement is taken as the point where the approxi-
mating line crosses the axis. The secant method retains
only the most recent estimate, so the root does not nec-
essarily remain bracketed. When the Algorithm does
converge, its order of convergence is
Urn |e fc+1 |«C|e|*,
k— )-oo
where C is a constant and <j> is the Golden Mean.
f(xn-i) - f(x n -2)
/'(x n -i) _
Xn — 1 X n — 2
f(x n ) ~ f(Xn-l) + f'(Xn)(x n - X n -l) =
^ , f(x n -l) ~ f(x n -2)
(1)
(2)
(3)
/(Z n -l) +
SO
Xn — X n ~ 1
Xn— 1 X n —2
f{x n -l)(x n -i — X-n-l)
(x n ~ SC„-l) = 0, (4)
(5)
f(x n -l) ~ f{x n -l)
see also False Position Method
References
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.
Secant Number
A number, more commonly called an EULER NUMBER,
giving the number of Odd Alternating Permuta-
tions. The term ZAG NUMBER is sometimes also used.
see also ALTERNATING PERMUTATION, EULER NUM-
BER, Euler Zigzag Number, Tangent Number
Sech
see Hyperbolic Secant
Second
see Arc Second
Second Curvature
see Torsion (Differential Geometry)
Second Derivative Test
Suppose f(x) is a Function of x which is twice Dif-
ferentiable at a Stationary Point x .
1. If f"(x ) > 0, then / has a Relative Minimum at
2. If f"(xo) < 0, then / has a Relative MAXIMUM at
The Extremum TEST gives slightly more general con-
ditions under which functions with /"(xo) = 0.
If f(x,y) is a 2-D FUNCTION which has a RELATIVE
Extremum at a point (xo,z/o) and has Continuous
Partial Derivatives at this point, then f x {xo,yo) =
and f y (xo,yo) = 0. The second Partial Derivatives
test classifies the point as a Maximum or Minimum.
Define the DISCRIMINANT as
■LS — fxxjyy — Jxyjyx = Jxxjyy — Jxy •
1. If D > 0, f xv (x ,y ) > and /xx(x 0) yo) +
f yy (xo,yo) > 0, the point is a Relative Minimum.
2. If D > 0, fxx(x ,yo) < 0, and f X x(x 0i yo) +
fyy(xo,yo) < 0, the point is a Relative MAXIMUM.
3. If D < 0, the point is a SADDLE POINT.
4. If D = 0, higher order tests must be used.
see also Discriminant (Second Derivative Test),
Extremum, Extremum Test, First Derivative
Test, Global Maximum, Global Minimum, Hes-
sian Determinant, Maximum, Minimum, Rela-
tive Maximum, Relative Minimum, Saddle Point
(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. 14, 1972.
Second Fundamental Tensor
see Weingarten Map
Section (Graph)
Seek Time 1607
Section (Graph)
A section of a GRAPH is obtained by finding its inter-
section with a Plane.
Section (Pencil)
The lines of a PENCIL joining the points of a Range to
another POINT.
see also Pencil, Range (Line Segment)
Section (Tangent Bundle)
A Vector Field is a section of its Tangent Bundle,
meaning that to every point a; in a MANIFOLD M, a
VECTOR X(x) € T X M is associated, where T x is the
Tangent Space.
see also Tangent Bundle, Tangent Space
Sectional Curvature
The mathematical object k which controls the rate of
geodesic deviation.
see also Bishop's Inequality, Cheeger's Finiteness
Theorem, Geodesic
Sector
The Area of the sector is
A WEDGE obtained by taking a portion of a CIRCLE
with Central Angle 6 < n radians (180°), illustrated
above as the shaded region. A sector of 7r radians would
be a Semicircle. Let R be the radius of the Circle,
c the Chord length, s the Arc Length, h the height
of the arced portion, and d the height of the triangular
portion. Then
R = h + d
s = R0
d = Rcos{\0)
= §ccot(§0)
c = 2Rsin{\6)
= 2dtan(f0)
= 2^R 2 -d?
= 2yJh{2R-h),
The Angle 9 obeys the relationships
•-5 ---(I) -'--(a)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
A = \Rs = \R 2 e
(11)
(Beyer 1987).
see also ClRCLE-ClRCLE INTERSECTION, LENS, OBTUSE
Triangle, Segment
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 125, 1987.
Sectorial Harmonic
A Spherical Harmonic of the form
sin(m<9)iC(cos0)
or
cos(m<9)P^(cos^).
see also SPHERICAL HARMONIC
Secular Equation
see Characteristic Equation
Seed
The initial number used as the starting point in a RAN-
DOM Number generating Algorithm.
Seed 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 6-fold symmetry, forming
a mesmerizing pattern of CIRCLES and LENSES.
see also Circle, Five Disks Problem, Flower of
Life, Venn Diagram
References
Rawles, B. Sacred Geometry Design Sourcebook: Universal
Dimensional Patterns. Nevada City, CA: Elysian Pub.,
p. 15, 1997.
$F Weisstein, E. W. "Flower of Life." http: //www. astro.
Virginia. edu/-eww6n/math /notebooks /Flower Of Life .m.
Seek Time
see Point-Point Distance — 1-D
(10)
1608
Segment
Seidel-Entringer-Arnold Triangle
Segment
A portion of a CIRCLE whose upper boundary is a circu-
lar ARC and whose lower boundary is a Chord making
a Central Angle < n radians (180°), illustrated
above as the shaded region. Let R be the radius of the
Circle, c the Chord length, s the Arc Length, h
the height of the arced portion, and d the height of the
triangular portion. Then
R = h + d
(1)
s = R6
(2)
d=Rcos(±0)
(3)
= §ccot(i0)
(4)
= \y/lR?-c>
(5)
c = 2i?sin(i0)
(6)
= 2dtan(i6»)
(7)
= 2-y/-R 2 - d 2
(8)
= 2y/h{2R - h) .
(9)
The Angle obeys the relationships
9= — = 2 cos
R
'(i)—- 1 ®
= 2sin "H^)-
\2Rj
The AREA of the segment is then
(10)
A — ^sector -^-isosceles triangle
:>2
= \R 2 {e-smd)
= \{Rs- cd)
= R 2 cos" 1 (|) - d^R? - d?
1 (^ir) ~( R - h)V 2 Rh-h 2 , (15)
(11)
(12)
(13)
(14)
= R cos
where the formula for the Isosceles Triangle in terms
of the VERTEX angle has been used (Beyer 1987).
see also CHORD, ClRCLE-ClRCLE INTERSECTION, CYL-
INDRICAL Segment, Lens, Parabolic Segment,
Sagitta, Sector, Spherical Segment
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 125, 1987.
Segmented Number
see Prime Number of Measurement
Segner's Recurrence Formula
The recurrence FORMULA
E n = E2E n -l + i?3-En-2 + • ■ - + E n ~\E2
which gives the solution to EULER'S POLYGON DIVISION
Problem.
see also Catalan Number, Euler's Polygon Divi-
sion Problem
Segre's Theorem
For any Real Number r > 0, an IRRATIONAL number
a can be approximated by infinitely many RATIONAL
fractions p/q in such a way that
P
VT+~4rg 2 q
< - -a <
y/T+4rq 2
If r = 1, this becomes HURWITZ'S IRRATIONAL NUMBER
Theorem.
see also Hurwitz's Irrational Number Theorem
Seiberg-Witten Equations
*2 = -t(iM),
where r is the sesquilinear map r : W + x W + — > A + <g>C.
see also WlTTEN'S EQUATIONS
References
Donaldson, S. K. "The Seiberg-Witten Equations and 4-
Manifold Topology." Bull. Amer. Math. Soc. 33, 45-70,
1996.
Morgan, J. W. The Seiberg-Witten Equations and Applica-
tions to the Topology of Smooth Four- Manifolds. Prince-
ton, NJ: Princeton University Press, 1996.
Seiberg-Witten Invariants
see Witten's Equations
Seidel-Entringer-Arnold Triangle
The Number Triangle consisting of the Entringer
Numbers E n ^ arranged in "ox-plowing" order,
£ao — > En
E22 4— E21 4 — E20
E30 — > E31 — >■ E32 - ^ Ess
E44 <— E43 <— E42 <— E41 4— E40
giving
1
0-> 1
14-14-0
0-> 1 -*2-»2
54-54-44-24-0
Seifert Circle
Seifert's Spherical Spiral 1609
see also Bell Number, Boustrophedon Trans-
form, Clark's Triangle, Entringer Number, Eu-
ler's Triangle, Leibniz Harmonic Triangle, Num-
ber Triangle, Pascal's Triangle
References
Arnold, V. I. "Bernoulli-Euler Updown Numbers Associ-
ated with Function Singularities, Their Combinatorics, and
Arithmetics." Duke Math. J. 63, 537-555, 1991.
Arnold, V. I. "Snake Calculus and Combinatorics of Ber-
noulli, Euler, and Springer Numbers for Coxeter Groups."
Russian Math. Surveys 47, 3-45, 1992.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, 1996.
Dumont, D. "Further Triangles of Seidel-Arnold Type and
Continued Fractions Related to Euler and Springer Num-
bers." Adv. Appl. Math. 16, 275-296, 1995.
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,
Seidel, I. "Uber eine einfache Entstehungsweise der
Bernoullischen Zahlen und einiger verwandten Reihen,"
Sitzungsber. Munch. Akad. 4, 157-187, 1877.
Seifert Circle
Eliminate each knot crossing by connecting each of the
strands coming into the crossing to the adjacent strand
leaving the crossing. The resulting strands no longer
cross but form instead a set of nonintersecting CIRCLES
called Seifert circles.
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, p. 96, 1994.
Seifert Conjecture
Every smooth NONZERO VECTOR FIELD on the 3-
SPHERE has at least one closed orbit. The conjecture
was proposed in 1950, proved true for Hopf fibrations,
but proved false in general by Kuperberg (1994).
References
Kuperberg, G. "A Volume-Preserving Counterexample to the
Seifert Conjecture." Comment. Math. Helv. 71, 70-97,
1996.
Kuperberg, G. and Kuperberg, K. "Generalized counterex-
amples to the Seifert Conjecture." Ann. Math. 143, 547-
576, 1996.
Kuperberg, G. and Kuperberg, K. "Generalized Counterex-
amples to the Seifert Conjecture." Ann. Math. 144, 239-
268, 1996.
Kuperberg, K. "A Smooth Counterexample to the Seifert
Conjecture." Ann. Math. 140, 723-732, 1994,
Seifert Form
For K a given KNOT in S 3 , choose a SEIFERT SURFACE
M 2 in § 3 for K and a bicollar M x [-1, 1] in S 3 - K.
If x G H\(M) is represented by a 1-cycle in M, let x +
denote the homology cycle carried by x x 1 in the bi-
collar. Similarly, let x~ denote x x — 1. The function
/ : ffi(M) x ffi(M) -> Z defined by
f(x,y) = lk(x,y + ) y
where Ik denotes the LINKING NUMBER, is called a
Seifert form for K.
see also Seifert Matrix
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 200-201, 1976.
Seifert Matrix
Given a Seifert Form f(x,y), choose a basis ei,
. . . , e 2g for H\{M) as a Z-module so every element is
uniquely expressible as
n\€\ H- . . . + Tl2ge2g
with rii integer, define the Seifert matrix V as the 2g x 2g
integral MATRIX with entries
Vij = lk(ei,et).
The right-hand TREFOIL KNOT has Seifert matrix
V =
-1 1
-1
A Seifert matrix is not a knot invariant, but it can be
used to distinguish between different SEIFERT SURFACES
for a given knot.
see also Alexander Matrix
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 200-203, 1976.
Seifert's Spherical Spiral
Is given by the CYLINDRICAL COORDINATES parametric
equation
r = sn(s)
= ks
z = cn(s),
1610
Seifert Surface
Self-Adjoint Operator
where k is a POSITIVE constant and sn(s) and cn(s) are
Jacobi Elliptic Functions (Whittaker and Watson
1990, pp. 527-528).
References
Bowman, F. Introduction to Elliptic Functions, with Appli-
cations. New York: Dover, p. 34, 1961.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Seifert Surface
An orientable surface with one boundary component
such that the boundary component of the surface is a
given Knot K, In 1934, Seifert proved that such a sur-
face can be constructed for any Knot. The process of
generating this surface is known as Seifert's algorithm.
Applying Seifert's algorithm to an alternating projection
of an alternating knot yields a Seifert surface of minimal
Genus.
There are KNOTS for which the minimal genus Seifert
surface cannot be obtained by applying Seifert's algo-
rithm to any projection of that Knot, as proved by
Morton in 1986 (Adams 1994, p. 105).
see also Genus (Knot), Seifert Matrix
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 95-106, 1994.
Seifert, H. "Uber das Geschlecht von Knotten." Math. Ann.
110, 571-592, 1934.
Self- Adjoint Matrix
A Matrix A for which
A* == (A T )* = A,
where the ADJOINT OPERATOR is denoted A f , A T is
the Matrix Transpose, and * is the Complex Con-
jugate. If a Matrix is self- adjoint, it is said to be
Hermitian.
see also Adjoint Operator, Hermitian Matrix,
Matrix Transpose
In order for the operator to be self-adjoint, i.e.,
C = C\ (4)
the second terms in (1) and (3) must be equal, so
po'(x) =pi(x). (5)
This also guarantees that the third terms are equal, since
Po'(x) = pi(x) => Po"(x) = pi'(x),
so (3) becomes
r rl d 2 ,du
Cu = L ■ u = p —^ 4- p — + P2U
= i( Po ^) +P2U=0 -
(6)
(7)
(8)
The LEGENDRE DIFFERENTIAL EQUATION and the
equation of SIMPLE HARMONIC MOTION are self-adjoint,
but the Laguerre Differential Equation and Her-
mite Differential Equation are not.
A nonself-adjoint second-order linear differential oper-
ator can always be transformed into a self-adjoint one
using Sturm-Liouville Theory. In the special case
p 2 (x) = 0, (8) gives
d_
dx
, ,du~\
i \ du n
du
" c I
dx
Po(x)
dx
Poix) 1
where C is a constant of integration.
(9)
(10)
(11)
(12)
A self-adjoint operator which satisfies the BOUNDARY
Conditions
Self-Adjoint Operator
Given a differential equation
% t \ du du , x
Cu(x) = po — + pi — + p 2 u, (1)
where pi = Pi{x) and u = u(x), the Adjoint Opera-
tor & is defined by
&U = -Tripoli) ~ -fdPlu) +P2U
(2)
= P0 dx 2 ~ + ( 2po ' ~ Pl ^dx + ( p °" -Pi'+^K ( 3 )
V*pU'\ x = a = V*pU*\ x = b
(13)
is automatically a HERMITIAN OPERATOR.
see also ADJOINT OPERATOR, HERMITIAN OPERATOR,
Sturm-Liouville Theory
References
Arfken, G. "Self-Adjoint Differential Equations." §9.1 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 497-509, 1985.
Self-Avoiding Walk
Self Number
1611
Self- Avoiding Walk
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let the number of Random Walks on a d-D lattice
starting at the Origin which never land on the same
lattice point twice in n steps be denoted c(n). The first
few values are
c d (0) = 1
c d (l) = 2d
c d (2) = 2d(2d-l).
The connective constant
fi d = lim [cd{n)] l/n
(1)
(2)
(3)
(4)
is known to exist and be FINITE. The best ranges for
these constants are
/x 2 e [2.62002,2.6939]
fi 3 € [4.572140,4.7476]
,U4 G [6.742945,6.8179]
/x 5 e [8.828529,8.8602]
^ 6 € [10.874038, 10.8886]
(5)
(6)
(7)
(8)
(9)
(Finch).
For the triangular lattice in the plane, fi < 4.278 (Aim
1993), and for the hexagonal planar lattice, it is conjec-
tured that
fi = \/2 + V2 (10)
(Madras and Slade 1993).
The following limits are also believed to exist and to be
Finite:
c(n)
H"n^
c(n)
f j, n nl- 1 (lnn) 1 /^
for d ^ 4
for d = 4,
(ii)
where the critical exponent 7 = 1 for d > 4 (Madras
and Slade 1993) and it has been conjectured that
for d = 2
for d = 3
for d = 4.
(12)
Define the mean square displacement over all n-step self-
avoiding walks uj as
s(n) = (Kn)| 2 ) = ^^|o;(n)| 2 . (13)
The following limits are believed to exist and be FINITE:
lim„_
linin-
s ( n )
f °° n 2 "(lnn)V4
for d ^ 4
for d = 4,
(14)
where the critical exponent v = 1/2 for d > 4 (Madras
and Slade 1993), and it has been conjectured that
for d = 2
for d = 3
for d = 4.
(15)
see also Random Walk
References
Aim, S. E. "Upper Bounds for the Connective Constant of
Self- Avoiding Walks." Combin. Prob. Comput. 2, 115-
136, 1993.
Finch, S. "Favorite Mathematical Constants." http://www.
maths oft . c om/as olve / c ons t ant /cnntv/ cimtv.html.
Madras, N. and Slade, G. The Self- Avoiding Walk. Boston,
MA: Birkhauser, 1993.
Self-Conjugate Subgroup
see Invariant Subgroup
Self-Descriptive Number
A 10-DlGIT number satisfying the following property.
Number the DIGITS to 9, and let DIGIT n be the num-
ber of ns in the number. There is exactly one such
number: 6210001000.
References
Pickover, C. A. "Chaos in Ontario." Ch. 28 in Keys to In-
finity. New York: W. H. Freeman, pp. 217-219, 1995.
Self-Homologous Point
see Similitude Center
Self Number
A number (usually base 10 unless specified otherwise)
which has no GENERATOR. Such numbers were origi-
nally called Columbian Numbers (S. 1974). There are
infinitely many such numbers, since an infinite sequence
of self numbers can be generated from the RECURRENCE
Relation
^•lO^+Cfc-i + S,
(1)
for k — 2, 3, . . . , where C\ — 9. The first few self
numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97,
... (Sloane's A003052).
An infinite number of 2-self numbers (i.e., base-2 self
numbers) can be generated by the sequence
C k = 2 j + C k -i + 1
(2)
for k = 1, 2, ... , where C\ = 1 and j is the number
of digits in Ck-\> An infinite number of n-self numbers
can be generated from the sequence
C* = (n - 2)n*" 1 + C^ + (n - 2)
(3)
1612 Self-Reciprocating Property
Selmer Group
for k = 2, 3, . . . , and
Ci =
fn-1
ln-2
for n even
for n odd.
(4)
Joshi (1973) proved that if k is ODD, then m is a fc-self
number IFF m is ODD. Patel (1991) proved that 2fc,
Ak + 2, and k 2 + 2A; + 1 are fc-self numbers in every EVEN
base /c > 4.
see a/so DIGITADITION
References
Cai, T. "On fc-Self Numbers and Universal Generated Num-
bers." Fib. Quart 34, 144-146, 1996.
Gardner, M. Time Travel and Other Mathematical Bewil-
derments. New York: W. H. Freeman, pp. 115-117, 122,
1988.
Joshi, V. S. Ph.D. dissertation. Gujarat University, Ahmad-
abad, 1973.
Kaprekar, D. R. The Mathematics of New Self- Numbers. De-
vaiali, pp. 19-20, 1963.
Patel, R. B. "Some Tests for fc-Self Numbers." Math. Student
56, 206-210, 1991.
S., B. R. Solution to Problem E 2048. Amer. Math. Monthly
81, 407, 1974.
Sloane, N. J. A. Sequence A003052/M2404 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Self- Reciprocating Property
Let h be the number of sides of certain skew POLYGONS
(Coxeter 1973, p. 15). Then
h =
2{p + q + 2)
10 -p-q '
References
Coxeter, H. S. M. Regular Poly topes, 3rd ed. New York:
Dover, 1973.
Self- Recursion
Self-recursion is a RECURSION which is defined in terms
of itself, resulting in an ill-defined infinite regress.
see Self-Recursion
Self- Similarity
An object is said to be self-similar if it looks "roughly"
the same on any scale. FRACTALS are a particularly
interesting class of self-similar objects.
see also Fractal
References
Hutchinson, J. "Fractals and Self- Similarity." Indiana Univ.
J. Math. 30, 713-747, 1981.
Self-Transversality Theorem
Let 7, r, and s be distinct INTEGERS (mod n), and let
Wi be the point of intersection of the side or diagonal
ViVi+j of the n-gon P = [Vi, . . . , V n ] with the transversal
V i+r Vi+ s . Then a NECESSARY and SUFFICIENT condi-
tion for
" ViWi
n
where AB\\CD and
WiVi
i+j
= (-!)",
AB1
CD] '
is the ratio of the lengths [A, B] and [C, D] with a plus or
minus sign depending on whether these segments have
the same or opposite direction, is that
1. n — 2m is EVEN with j = m (mod n) and s =
r + m (mod n),
2. n is arbitrary and either s = 2r and j = 3r, or
3. r = 2s (mod n) and j = 3s (mod n).
References
Griinbaum, B. and Shepard, G. C. "Ceva, Menelaus, and the
Area Principle." Math. Mag. 68, 254-268, 1995.
Selfridge's Conjecture
There exist infinitely many n > with p n 2 > p n -ip n+i
for all i < n. Also, there exist infinitely many n >
such that 2p n < p n -i + Pn-i for all i < n.
Selfridge-Hurwitz Residue
Let the Residue from Pepin's Theorem be
R n = 3 (ir "" 1)/2 (modF n ),
where F n is a Fermat Number. Selfridge and Hurwitz
use
tf n (mod2 35 -l,2 36 ,2 36 -l).
A nonvanishing R n (mod 2 36 ) indicates that F n is COM-
POSITE for n > 5.
see also Fermat Number, Pepin's Theorem
References
Crandall, R.; Doenias, J.; Norrie, C; and Young, J. "The
Twenty-Second Fermat Number is Composite." Math.
Comput. 64, 863-868, 1995.
Selmer Group
A GROUP which is related to the Taniyama-Shimura
Conjecture.
see also Taniyama-Shimura Conjecture
Semi-Integral
Semicubical Parabola 1613
2c
Semi-Integral
An Integral of order 1/2. The semi-integral of the
Constant Function f(x) — c is
dx- 1 / 2
see also Semiderivative
References
Spanier, J. and Oldham, K. B. An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 8 and 14, 1987.
Semialgebraic Number
A subset of M 71 which is a finite Boolean combination
of sets of the form {x = (asi, . . . , x m ) : f(x) > 0} and
{x:g{x) = 0}, where /,^R[Xi,...,X n ].
References
Bierstone, E. and Milman, P. "Semialgebraic and Subanalytic
Sets." IHES Pub. Math. 67, 5-42, 1988.
Marker, D. "Model Theory and Exponentiation." Not.
Amer. Math. Soc. 43, 753-759, 1996.
Semianalytic
X C M. 71 is semianalytic if, for all x € M n , there is an
open neighborhood U of x such that X n U is a finite
Boolean combination of sets {x £ U : f(x) = 0} and
{x €U : g(x) > 0}, where f,g : U -> R are ANALYTIC.
see also ANALYTIC FUNCTION, PSEUDOANALYTIC
Function, Subanalytic
References
Marker, D. "Model Theory and Exponentiation."
Amer. Math. Soc. 43, 753-759, 1996.
Semicircle
Not
Half a Circle. The Perimeter of the semicircle of
Radius r is
L = 27-4-Trr = r(2 + 7r),
and the Area is
A = 2 I vV - y 2 dy
! / V r2 - y 2
Jo
1 2
2 7rr •
h 3 -
(1)
(2)
(3)
The weighted mean of y is
pr
(y) = 2 / yy^ 2 -V 2 d V =
Jo
The Centroid is then given by
(y) 4r
The semicircle is the CROSS-SECTION of a HEMISPHERE
for any PLANE through the z-AxiS.
see also Arbelos, Arc, Circle, Disk, Hemisphere,
Lens, Right Angle, Salinon, Thales' Theorem,
Yin- Yang
(4)
Semicolon Derivative
see COVARIANT DERIVATIVE
Semiconvergent Series
see Asymptotic Series
Semicubical Parabola
A PARABOLA-like curve with Cartesian equation
y = ax 3 ' 2 , (1)
parametric equations
y = at,
and Polar Coordinates,
tan 2 sec
(2)
(3)
(4)
The semicubical parabola is the curve along which a par-
ticle descending under gravity describes equal vertical
spacings within equal times, making it an ISOCHRONOUS
Curve. The problem of finding the curve having this
property was posed by Leibniz in 1687 and solved by
Huygens (MacTutor Archive).
The Arc Length, Curvature, and Tangential An-
gle are
(4 + 9r) ' -A
(5)
(6)
(7)
«(*) =
K(t) = t(4 + 9* 2 ) 3 / 2
^(t) = tan- 1 (ft).
see also NEILE'S PARABOLA, PARABOLA INVOLUTE
References
Gray, A. "The Semicubical Parabola." §1.7 in Modern Dif-
ferential Geometry of Curves and Surfaces. Boca Raton,
FL: CRC Press, pp. 15-16, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 85-87, 1972.
Lee, X. "Semicubic Parabola." http://www.best.com/-xah/
Special Plane Curves _ dir / Semicubic Parabola _ dir /
semicubicParabola.html.
MacTutor History of Mathematics Archive. "Neile's
Parabola." http : // www - groups . dcs . st - and .ac.uk/
-history/Curves/Neiles .html.
Yates, R. C. "Semi-Cubic Parabola." A Handbook on Curves
and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 186-187, 1952.
1614
Semiderivative
Semiperfect Number
Semiderivative
A Derivative of order 1/2. The semiderivative of the
Constant Function f(x) = c is
d^c c
dx 1 / 2
\^7TX
see also Derivative, Semi-Integral
References
Spanier, J. and Oldham, K. B. An Atlas of Functions. Wash-
ington, DC: Hemisphere, pp. 8 and 14, 1987.
Semidirect Product
The "split" extension G of GROUPS N and F which
contains a SUBGROUP F isomorphic to F with G = FN
and F D N = {e}.
References
lyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 613, 1980.
Semiflow
An Action with G — R + .
see also Flow
Semigroup
A mathematical object defined for a set and a BI-
NARY OPERATOR in which the multiplication operation
is ASSOCIATIVE. No other restrictions are placed on a
semigroup; thus a semigroup need not have an IDEN-
TITY Element and its elements need not have inverses
within the semigroup. A semigroup is an ASSOCIATIVE
Groupoid.
A semigroup can be empty. The total number of semi-
groups of order n are 1, 4, 18, 126, 1160, 15973, 836021,
... (Sloane's A001423). The number of semigroups of
order n with one IDEMPOTENT are 1, 2, 5, 19, 132, 3107,
623615, ... (Sloane's A002786), and with two IDEM-
POTENTS are 2, 7, 37, 216, 1780, 32652, ... (Sloane's
A002787). The number a(n) of semigroups having n
Idempotents are 1, 2, 6, 26, 135, 875, . . . (Sloane's
A002788).
see also ASSOCIATIVE, BINARY OPERATOR, FREE SEMI-
GROUP, Groupoid, Inverse Semigroup, Monoid,
Quasigroup
References
Clifford, A. H. and Preston, G. B. The Algebraic Theory of
Semigroups. Providence, Rl: Amer. Math. Soc, 1961.
Sloane, N. J. A. Sequences A001423/M3550, A002786/
M1522, A002787/M1802, and A002788/M1679 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Semilatus Rectum
Given an ELLIPSE, the semilatus rectum is defined as
the distance L measured from a FOCUS such that
1 = 1(1. 2_\
L - 2 \r+ + r-J'
(1)
where r+ = a(l + e) and r_ = a(l — e) are the APOAPSIS
and Periapsis, and e is the Ellipse's Eccentricity.
Plugging in for r+ and r_ then gives
- - — ( 1 l A _
L ~ 2a Vl-e + 1 + e/ ~
1 (l + e ) + (l- e )
2a
1-e 2
a 1 - e 2 :
so
L = a(l-e 2 ).
(2)
(3)
see also Eccentricity, Ellipse, Focus, Latus Rec-
tum, Semimajor Axis, Semiminor Axis
Semimagic Square
A square that fails to be a Magic Square only because
one or both of the main diagonal sums do not equal the
Magic Constant is called a Semimagic Square.
see also MAGIC SQUARE
Semimajor Axis
Half the distance across an Ellipse along its long prin-
cipal axis.
see also Ellipse, Semiminor Axis
Semiminor Axis
Half the distance across an ELLIPSE along its short prin-
cipal axis.
see also ELLIPSE, SEMIMAJOR AXIS
Semiperfect Magic Cube
A semiperfect magic cube, also called an Andrews
Cube, is a MAGIC Cube for which the cross-section di-
agonals do not sum to the MAGIC CONSTANT.
see also Magic Cube, Perfect Magic Cube
References
Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time
Travel and Other Mathematical Bewilderments. New
York: W. H. Freeman, pp. 213-225, 1988.
Semiperfect Number
A number such as 20 = 1 + 4 + 5 + 10 which is the Sum
of some (or all) its PROPER DIVISORS. A semiperfect
number which is the SUM of all its PROPER DIVISORS is
called a Perfect Number. The first few semiperfect
numbers are 6, 12, 18, 20, 24, 28, 30, 36, 40, . . . (Sloane's
A005835). Every multiple of a semiperfect number is
semiperfect, as are all numbers 2 m p for m > 1 and p a
Prime between 2 m and 2 m+1 (Guy 1994, p. 47).
Semiperimeter
Semiregular Polyhedron 1615
A semiperfect number cannot be DEFICIENT. Rare
Abundant Numbers which are not semiperfect are
called Weird Numbers. Semiperfect numbers are
sometimes also called Pseudoperfect Numbers.
see also Abundant Number, Deficient Number,
Perfect Number, Primitive Semiperfect Num-
ber, Weird Number
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. Sequence A005835/M4094 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Zachariou, A. and Zachariou, E. "Perfect, Semi-Perfect and
Ore Numbers." Bull Soc. Math. Grece (New Ser.) 13,
12-22, 1972.
Semiperimeter
The semiperimeter on a figure is defined as
s= \p,
(1)
where p is the PERIMETER. The semiperimeter of POLY-
GONS appears in unexpected ways in the computation of
their Areas. The most notable cases are in the ALTI-
TUDE, Exradius, and Inradius of a Triangle, the
Soddy Circles, Heron's Formula for the Area of a
Triangle in terms of the legs a, 6, and c
Aa = ys(s — a)(s — b)(s — c),
(2)
and Brahmagupta's Formula for the Area of a
Quadrilateral
-^-quadrilateral —
\ (s — a)(s — b)(s — c)(s — d) — abcdcos 2 I — - — j .
(3)
The semiperimeter also appears in the beautiful
L'Huilier's Theorem about Spherical Triangles.
For a Triangle, the following identities hold,
s — a — |(— a + 6 + c)
s — b — ~(a 4- b — c)
s — c = |(a + 6 — c).
(4)
(5)
(6)
Now consider the above figure. Let I be the INCENTER
of the Triangle AABC, with D, E y and F the tan-
gent points of the INCIRCLE. Extend the line BA with
GA = CE. Note that the pairs of triangles {AD I, API),
(BDI,BEI), (CFI,CEI) are congruent. Then
BG = BD + AD + AG = BD + AD + CE
= \{2BD + 2AD + 2CE)
= \ [(BD + BE) + (AD + AF) + (CE + CF)]
- \ [(BD + AD) + (BE + CE) + (AF + CF)]
= \(AB + BC + AC) = \(a + b + c)^s. (7)
Furthermore,
s-a^ BG ~ BC
= (BD + AD + AG) - (BE + CE)
= (BD + AD + CE) - (BD + CE) = AC (8)
s-b = BG-AC
= (BD + AD + AG) - (AF + CF)
= (BD + AD + CE) - (AD + CE) = BD (9)
s~c = BG-AB = AG (10)
(Dunham 1990). These equations are some of the build-
ing blocks of Heron's derivation of HERON'S FORMULA.
References
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.
Semiprime
A COMPOSITE number which is the PRODUCT of two
Primes (possibly equal). They correspond to the 2-
Almost Primes. The first few are 4, 6, 9, 10, 14, 15,
21, 22, . . . (Sloane's A001358).
see also Almost Prime, Chen's Theorem, Compos-
ite Number, Prime Number
References
Sloane, N. J. A. Sequence A001358/M3274 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Semiprime Ring
Given an Ideal A, a semiprime ring is one for which
A n = Implies A = for any Positive n. Every
Prime Ring is semiprime.
see also PRIME Ring
Semiregular Polyhedron
A Polyhedron or plane Tessellation is called
semiregular if its faces are all Regular Polygons and
its corners are alike (Walsh 1972; Coxeter 1973, pp. 4
and 58; Holden 1991, p. 41). The usual name for a
semiregular polyhedron is an ARCHIMEDEAN SOLIDS, of
which there are exactly 13.
1616 Semiring
Separating Family
see also ARCHIMEDEAN SOLID, POLYHEDRON, TESSEL-
LATION
References
Coxeter, H. S. M. "Regular and Semi- Regular Poly topes I."
Math. Z. 46, 380-407, 1940.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, 1973.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
1991.
Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods
of Semi- Regular Polyhedra." Geometriae Dedicata 1, 117-
123, 1972.
Semiring
A semiring is a set together with two BINARY OPERA-
TORS 5(+, *) satisfying the following conditions:
1. Additive associativity: For all a, 6, c 6 5, (a-\-b)+c =
a + (fc + c),
2. Additive commutativity: For all a, 6 6 5, a + b =
6 + a,
3. Multiplicative associativity: For all a,b,c 6 S, (a*
fr) * c — a * (6 * c),
4. Left and right distributivity: For all a,b,c e 5, a *
(fc+c) = (a*6) + (a*c) and (6-hc)*a = (fc*a) + (c*a).
Thus a semiring is therefore a commutative SEMIGROUP
under addition and a SEMIGROUP under multiplication.
A semiring can be empty.
see also Binary Operator, Ring, Ringoid, Semi-
group
References
Rosenfeld, A. An Introduction to Algebraic Structures. New
York: Holden-Day, 1968.
Semisecant
see Transversal Line
Semisimple
A p-ELEMENT x of a Group G is semisimple if
E(Cg(x)) 7^ 1, where E(H) is the commuting product
of all components of H and Cq{x) is the CENTRALIZER
ofG.
see also CENTRALIZER, p-ELEMENT
Semisimple Algebra
An ALGEBRA with no nontrivial nilpotent IDEALS. In
the 1890s, Cartan, Frobenius, and Molien independently
proved that any finite-dimensional semisimple algebra
over the REAL or COMPLEX numbers is a finite and
unique DIRECT Sum of Simple Algebras. This re-
sult was then extended to algebras over arbitrary fields
by Wedderburn in 1907 (Kleiner 1996).
see also IDEAL, NlLPOTENT ELEMENT, SIMPLE ALGE-
BRA
References
Kleiner, I. "The Genesis of the Abstract Ring Concept."
Amer. Math. Monthly 103, 417-424, 1996.
Semisimple Lie Group
A Lie GROUP which has a simply connected covering
group HOMEOMORPHIC to M. n . The prototype is any
connected closed subgroup of upper TRIANGULAR COM-
PLEX Matrices. The Heisenberg Group is such a
group.
see also Heisenberg Group, Lie Group
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Semisimple Ring
A Semiprime Ring which is also an Artinian Ring.
see also Artinian Ring
Semistable
When a Prime I divides the Discriminant of a El-
liptic Curve E } two or all three roots of E become
congruent mod /. An ELLIPTIC CURVE is semistable if,
for all such PRIMES /, only two roots become CONGRU-
ENT mod / (with more complicated definitions for p = 2
or 3).
see also Discriminant (Elliptic Curve), Elliptic
Curve
Sensitivity
The probability that a STATISTICAL TEST will be posi-
tive for a true statistic.
see also SPECIFICITY, STATISTICAL TEST, TYPE I ER-
ROR, Type II Error
Sentence
A Logic Formula with no Free variables.
Separating Edge
An EDGE of a GRAPH is separating if a path from a point
A to a point B must pass over it. Separating EDGES can
therefore be viewed as either bridges or dead ends.
see also EDGE (Graph)
Separating Family
A Separating Family is a Set of Subsets in which
each pair of adjacent elements are found separated, each
in one of two disjoint subsets. The 26 letters of the
alphabet can be separated by a family of 9,
(abcdefghi) (jklmnopqr) (stuvwxyz)
(abcjklstu) (defmnovwx) (ghipqryz) .
(adgjmpsvy) (behknqtwz) (cfilorux)
The minimal size of the separating family for an n-set is
0, 2, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, . . . (Sloane's A007600).
see also Katona's Problem
References
Honsberger, R. "Cai Mao-Cheng's Solution to Katona's
Problem on Families of Separating Subsets." Ch. 18 in
Mathematical Gems HI. Washington, DC: Math. Assoc.
Amer., pp. 224-239, 1985.
Sloane, N. J. A. Sequence A007600/M0456 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Separation
Sequential Graph 1617
Separation
Two distinct point pairs AC and BD separate each other
if A, B, C, and D lie on a CIRCLE (or line) in such order
that either of the arcs (or the line segment AC) contains
one but not both of B and D. In addition, the point
pairs separate each other if every CIRCLE through A and
C intersects (or coincides with) every CIRCLE through
B and D> If the point pairs separate each other, then
the symbol AC/ /BD is used.
Separation of Variables
A method of solving partial differential equations in a
function <3> and variables #, t/, . . . by making a substi-
tution of the form
^(x,y,...) = X(x)Y(y)---,
breaking the resulting equation into a set of independent
ordinary differential equations, solving these for X(x),
Y(y), . . . , and then plugging them back into the original
equation.
This technique works because if the product of functions
of independent variables is a constant, each function
must separately be a constant. Success requires choice
of an appropriate coordinate system and may not be at-
tainable at all depending on the equation. Separation of
variables was first used by L'Hospital in 1750. It is espe-
cially useful in solving equations arising in mathematical
physics, such as Laplace's Equation, the Helmholtz
Differential Equation, and the Schrodinger equa-
tion.
see also HELMHOLTZ DIFFERENTIAL EQUATION, LA-
place's Equation
References
Arfken, G. "Separation of Variables" and "Separation of
Variables — Ordinary Differential Equations." §2.6 and
§8.3 in Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 111-117 and 448-451,
1985.
Morse, P. M. and Feshbach, H. "Separable Coordinates" and
"Table of Separable Coordinates in Three Dimensions."
§5.1 in Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 464-523 and 655-666, 1953.
Separation Theorem
There exist numbers yi < 2/2 < • . ■ < #n-i, a < y n -i,
y n -i < 6, such that
\ u = a(y„) - a(y„_i),
where v — 1, 2, . . . , n, yo = a and y n = 6. Furthermore,
the zeros xi, ..., x n , arranged in increasing order, al-
ternate with the numbers yi, . . .y n -i, so
x v <y v < x v +i.
More precisely,
a(x v + e) — a(a) < a{y u ) - a(a)
= Ai + . . . + \ u < a(x v +i — e) — a(a)
for v = 1, . . . , n — 1.
see also POINCARE SEPARATION THEOREM, STURMIAN
Separation Theorem
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 50, 1975.
Separatrix
A phase curve (invariant MANIFOLD) which meets a HY-
PERBOLIC Fixed Point (intersection of a stable and an
unstable invariant Manifold). A separatrix marks a
boundary between phase curves with different proper-
ties. For example, the separatrix in the equation of mo-
tion for the pendulum occurs at the angular momentum
where oscillation gives way to rotation.
Septendecillion
In the American system, 10 .
see also LARGE NUMBER
Septillion
In the American system, 10 24 .
see also Large Number
Sequence
A sequence is an ordered set of mathematical objects
which is denoted using braces. For example, the symbol
{2n}^ =1 denotes the infinite sequence of Even Num-
bers {2, 4, ..., 2n, ...}.
see also 196-ALGORITHM, jI-SeQUENCE, ALCUIN'S SE-
QUENCE, 52-Sequence, Beatty Sequence, Car-
michael Sequence, Cauchy Sequence, Conver-
gent Sequence, Degree Sequence, Density (Se-
quence), Fractal Sequence, Giuga Sequence, In-
finitive Sequence, Integer Sequence, Iteration
Sequence, List, Nonaveraging Sequence, Prim-
itive Sequence, Reverse-Then- Add Sequence,
Score Sequence, Series, Signature Sequence,
Sort-Then-Add Sequence, Ulam Sequence
Sequency
The sequency k of a WALSH FUNCTION is defined as half
the number of zero crossings in the time base.
see also WALSH FUNCTION
Sequency Function
see WALSH FUNCTION
Sequential Graph
A Connected Graph having e Edges is said to be
sequential if it is possible to label the nodes i with dis-
tinct INTEGERS fi in{0, 1, 2, . . . , e — 1} such that when
EDGE ij is labeled fi + /?, the set of EDGE labels is
a block of e consecutive integers (Grace 1983, Gallian
1990). No Harmonious Graph is known which cannot
also be labeled sequentially.
1618
Series
Series Multisection
see also Connected Graph, Harmonious Graph
References
Gallian, J. A. "Open Problems in Grid Labeling." Amer.
Math. Monthly 97, 133-135, 1990.
Grace, T. "On Sequential Labelings of Graphs." J. Graph
Th. 7, 195-201, 1983.
Series
A series is a sum of terms specified by some rule. If each
term increases by a constant amount, it is said to be an
Arithmetic Series. If each term equals the previous
multiplied by a constant, it is said to be a Geomet-
ric SERIES. A series usually has an INFINITE' number
of terms, but the phrase INFINITE SERIES is sometimes
used for emphasis or clarity.
If the sum of partial sequences comprising the first few
terms of the series does not converge to a Limit (e.g.,
it oscillates or approaches ±oo), it is said to diverge.
An example of a convergent series is the GEOMETRIC
Series
Bs>
and an example of a divergent series is the HARMONIC
Series
E— = oo.
n
A number of methods known as CONVERGENCE Tests
can be used to determine whether a given series con-
verges. Although terms of a series can have either sign,
convergence properties can often be computed in the
"worst case" of all terms being POSITIVE, and then ap-
plied to the particular series at hand. A series of terms
u n is said to be Absolutely Convergent if the series
formed by taking the absolute values of the u n ,
^M
converges.
An especially strong type of convergence is called Un-
iform CONVERGENCE, and series which are uniformly
convergent have particularly "nice" properties. For ex-
ample, the sum of a Uniformly Convergent series
of continuous functions is continuous. A CONVERGENT
Series can be Differentiated term by term, provided
that the functions of the series have continuous deriva-
tives and that the series of DERIVATIVES is UNIFORMLY
Convergent. Finally, a Uniformly Convergent se-
ries of continuous functions can be INTEGRATED term by
term.
For a table listing the Coefficients for various series
operations, see Abramowitz and Stegun (1972, p. 15).
While it can be difficult to calculate analytical expres-
sions for arbitrary convergent infinite series, many al-
gorithms can handle a variety of common series types.
The program Mathematica® (Wolfram Research, Cham-
paign, IL) implements many of these algorithms. Gen-
eral techniques also exist for computing the numerical
values to any but the most pathological series (Braden
1992).
see also ALTERNATING SERIES, ARITHMETIC SERIES,
Artistic Series, Asymptotic Series, Bias (Series),
Convergence Improvement, Convergence Tests,
Euler-Maclaurin Integration Formulas, Geo-
metric Series, Harmonic Series, Infinite Series,
Melodic Series, ^-Series, Riemann Series Theo-
rem, Sequence, Series Expansion, Series Rever-
sion
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Infinite Series."
§3.6 in Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables, 9th printing. New
York: Dover, p. 14, 1972.
Arfken, G. "Infinite Series." Ch. 5 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 277-351, 1985.
Boas, R. P. Jr. "Partial Sums of Infinite Series, and How
They Grow." Amer. Math. Monthly 84, 237-258, 1977.
Boas, R. P. Jr. "Estimating Remainders." Math. Mag. 51,
83-89, 1978.
Borwein, J. M. and Borwein, P. B. "Strange Series and High
Precision Fraud." Amer. Math. Monthly 99, 622-640,
1992.
Braden, B. "Calculating Sums of Infinite Series." Amer.
Math. Monthly 99, 649-655, 1992.
Bromwich, T. J. Pa. and MacRobert, T. M. An Introduc-
tion to the Theory of Infinite Series, 3rd ed. New York:
Chelsea, 1991.
Hansen, E. R. A Table of Series and Products. Englewood
Cliffs, NJ: Prentice-Hall, 1975.
Hardy, G. H. Divergent Series. Oxford, England: Clarendon
Press, 1949.
Jolley, L. B. W. Summation of Series, 2nd rev. ed. New
York: Dover, 1961.
Knopp, K. Theory and Application of Infinite Series. New
York: Dover, 1990.
Mangulis, V. Handbook of Series for Scientists and Engi-
neers. New York: Academic Press, 1965.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Series and Their Convergence." §5.1 in
Numerical Recipes in FORTRAN: The Art of Scientific
Computing, 2nd ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 159-163, 1992.
Rainville, E. D. Infinite Series. New York: Macmillan, 1967.
Series Expansion
see Laurent Series, Maclaurin Series, Power Se-
ries, Series Reversion, Taylor Series
Series Inversion
see Series Reversion
Series Multisection
If
f(x) = /0 + flX + J2X 2 + . . . + f n X n + . . . ,
then
S(n,j) = fjx> + f J+n x }+n + f j+ 2nx 3 ^ n + .
Series Reversion
is given by
n-l
S(nJ) = -^ w'^fiwx),
n *• — '
t=o
where w = e 2 ™ /r \
see a/50 Series Reversion
References
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 210-214, 1985.
Series Reversion
Series reversion is the computation of the COEFFICIENTS
of the inverse function given those of the forward func-
tion. For a function expressed in a series as
y = a\x + CL2X 2 + asx 3 + . . . , (1)
the series expansion of the inverse series is given by
x = A iy + A 2 y 2 + A 3 y 3 + . . . . (2)
By plugging (2) into (1), the following equation is ob-
tained
y — aiAiy + (a 2 A x 2 + aiA 2 )y 2
+ (a 3 Ai 3 + 2a 2 AiA 2 + a 1 A 3 )y 3
+(3a 3 Ai 2 A 2 + a 2 A 2 2 + a 2 A 1 A 3 ) + . . . . (3)
Equating COEFFICIENTS then gives
Ai = ai" 1 (4)
a 2 . 2 -3 /r\
A2 = Ai = — ai 02 (5)
ai
A 3 =ai" 5 (2a 2 2 -aia 3 ) (6)
Aa = ai~ 7 (5aia2a3 — ai 2 <X4 — 5a2 3 ) (7)
A5 = a\~ (6ai a2a4 + 3ai a 2 a 3 + 14a2 — ai as
- 21aia 2 2 a 3 ) (8)
^6 = d\~ (7ai a2ds + 7ai a 3 a4 + 84aia2 a 3
— a\ a& — 28ai a2a 3 — 42a 2 — 28ai a 2 0,4)
(9)
A7 = a\~ (8ai a 2 a6 + 8ai a 3 as + 4a 1 04
+ 120ai 2 a 2 3 a 4 + 180ai 2 a 2 2 a 3 2 + 132a 2 6
— ai a*? — 36ai a2 as — 72ai a2a 3 a4
- 12ai 3 a 3 3 - 330aia 2 4 a 3 ) (10)
(Dwight 1961, Abramowitz and Stegun 1972, p. 16). A
derivation of the explicit formula for the nth term is
given by Morse and Feshbach (1953),
Serpentine Curve 1619
1
noi r
£ (- 1 )"
n(n + l)---(n-l + s + t + u+../
1 sHlul---
©'©'•■•
where
s + 2t + Su + . .
n- 1.
(12)
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 316-317, 1985.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 297, 1987.
Dwight, H. B. Table of Integrals and Other Mathematical
Data, 4th ed. New York: Macmillan, 1961.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 411-413, 1953.
Serpentine Curve
A curve named and studied by Newton in 1701 and con-
tained in his classification of CUBIC CURVES. It had
been studied earlier by L'Hospital and Huygens in 1692
(MacTutor Archive).
The curve is given by the CARTESIAN equation
3/0*0
and parametric equations
abx
x 2 + a 2
x(t) = a cot t
y(t) = b sin t cost.
(i)
(2)
(3)
The curve has a MAXIMUM at x = a and a MINIMUM at
x = —a, where
and inflection points at x = ±v3a, where
V «( T \- ^x{x 2 -2>a 2 ) _
The Curvature is given by
2abx(x 2 -3a 2 )
k(x) =
(z 2 + a 2 ) 3
' . (a 3 b-afcx 2 ) 2 l
1+ (*2+a2)4 J
3/2
K(t)
4a/2 ab[2 cos(2t) - 1] cot t esc 2 t
{6 2 [1 + cos(4t)] + 2a 2 esc 4 f} 3/2 '
(4)
(5)
(6)
(7)
(11)
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 111-112, 1972.
MacTutor History of Mathematics Archive. "Serpentine."
http : //www-groups . des . st-and . ac . uk/ -history/Curves
/Serpentine. html.
1620
Serret-Frenet Formulas
Set Theory
Serret-Frenet Formulas
see FRENET FORMULAS
Set
A set is a FINITE or INFINITE collection of objects. Older
words for set include AGGREGATE and CLASS. Russell
also uses the term MANIFOLD to refer to a set. The
study of sets and their properties is the object of Set
THEORY. Symbols used to operate on sets include A
(which denotes the Empty Set 0), v = (which denotes
the Power Set of a set), n (which means "and" or
Intersection), and U (which means "or" or Union).
The Notation A b , where A and B are arbitrary sets,
is used to denote the set of MAPS from B to A. For
example, an element of X would be a MAP from the
Natural Numbers N to the set X. Call such a func-
tion /, then /(l), / (2), etc., are elements of X, so call
them si, 32, etc. This now looks like a SEQUENCE of el-
ements of X, so sequences are really just functions from
N to X. This NOTATION is standard in mathematics
and is frequently used in symbolic dynamics to denote
sequence spaces.
Let E, F, and G be sets. Then operation on these sets
using the n and U operators is COMMUTATIVE
EnF = FHE (1)
EUF = FUE, (2)
(EnF)nG = ED(FnG) (3)
An (U Si ) =\J( A nBi) (4)
(£UF)UG = £U(FUG), (5)
and Distributive
(E n F) U G = (E U G) n (F U G) (6)
(E u F) n G = (e n G) u (F n G). (r)
The proofs follow trivially using Venn Diagrams.
p (i)A = Y, p{Ai) - (8)
The table below gives symbols for some common sets in
mathematics.
Associative
Symbol
Set
l n
n-ball
c
complex numbers
c n ,
C (n)
n-differentiable functions
w
n-disk
e
quaternions
i
integers
N
natural numbers
Q
rational numbers
R n
real numbers in n-D
S n
n-sphere
z
integers
z n
integers (mod n)
ir
negative integers
z +
positive integers
z*
nonnegative integers
see also Aggregate, Analytic Set, Borel Set, C,
Class (Set), Coanalytic Set, Definable Set, De-
rived Set, Double-Free Set, Extension, Ground
Set, I, Intension, Intersection, Kinney's Set,
Manifold, N, Perfect Set, Poset, Q, R, Set Dif-
ference, Set Theory, Triple-Free Set, Union,
Venn Diagram, Well-Ordered Set, Z, Z~, Z +
References
Courant, R. and Robbins, H. "The Algebra of Sets." Supple-
ment to Ch. 2 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 108-116, 1996.
Set Difference
The set difference A\B is denned by
A\B = {x : x e A and x B}.
The same symbol is also used for Quotient GROUPS.
Set Partition
A set partition of a Set 5 is a collection of disjoint
Subsets B , Bi, ... of 5 whose Union is £, where
each Bi is called a BLOCK. The number of partitions of
the Set {k}% =1 is called a Bell Number.
see also Bell Number, Block, Restricted Growth
String, Stirling Number of the Second Kind
References
Ruskey, F. "Info About Set Partitions." http://sue . esc .
uvic . ca/~cos/inf /setp/SetPartitions .html.
Set Theory
The mathematical theory of SETS. Set theory is closely
associated with the branch of mathematics known as
Logic.
There are a number of different versions of set the-
ory, each with its own rules and AXIOMS. In or-
der of increasing CONSISTENCY STRENGTH, several ver-
sions of set theory include PEANO ARITHMETIC (or-
dinary Algebra), second-order arithmetic (Analy-
sis), Zermelo-Fraenkel Set Theory, Mahlo, weakly
Sexagesimal
Sexy Primes 1621
compact, hyper-Mahlo, ineffable, measurable, Ramsey,
super compact, huge, and n-huge set theory.
Given a set of REAL NUMBERS, there are 14 versions of
set theory which can be obtained using only closure and
complement (Beeler et al. 1972, Item 105).
see also Axiomatic Set Theory, Consistency
Strength, Continuum Hypothesis, Descriptive
Set Theory, Impredicative, Naive Set Theory,
Peano Arithmetic, Set, Zermelo-Fraenkel Set
Theory
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, pp. 36-44, Feb. 1972.
Brown, K. S. "Set Theory and Foundations." http: //www.
seanet . com/-ksbrown/if oundat .htm.
Courant, R. and Robbins, H. "The Algebra of Sets." Supple-
ment to Ch. 2 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 108-116, 1996.
Devlin, K. The Joy of Sets: Fundamentals of Contemporary
Set Theory, 2nd ed. New York: Springer- Verlag, 1993.
Halmos, P. R. Naive Set Theory. New York: Springer- Verlag,
1974.
MacTutor History of Mathematics Archive. "The Beginnings
of Set Theory." http://www-groups.dcs.st-and.ac.uk/-
history / HistTopics / Beginnings _ of _ set .theory . html.
Stewart, I. The Problems of Mathematics, 2nd ed. Oxford:
Oxford University Press, p. 96, 1987.
Sexagesimal
The base-60 notational system for representing REAL
Numbers. A base-60 number system was used by the
Babylonians and is preserved in the modern measure-
ment of time (hours, minutes, and seconds) and ANGLES
(Degrees, Arc Minutes, and Arc Seconds).
see also Base (Number), Binary, Decimal, Hexa-
decimal, Octal, Quaternary, Scruple, Ternary,
Vigesimal
References
Bergamini, D. Mathematics. New York: Time-Life Books,
pp. 16-17, 1969.
$ Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/~eww6n/math/notebooks/Bases.m.
Sexdecillion
In the American system, 10 51 .
see also LARGE NUMBER
Sextic Equation
The general sextic polynomial equation
x 6 + a$x + a±x + a$x + Q>2X + a\x + ao —
can be solved in terms of HYPERGEOMETRIC FUNCTIONS
in one variable using Klein's approach to solving the
Quintic Equation.
see also Cubic Equation, Quadratic Equation,
Quartic Equation, Quintic Equation
References
Coble, A. B. "The Reduction of the Sextic Equation to the
Valentiner Form— Problem." Math. Ann. 70, 337-350,
1911a.
Coble, A. B. "An Application of Moore's Cross-ratio Group
to the Solution of the Sextic Equation." Trans. Amer.
Math. Soc. 12, 311-325, 1911b.
Cole, F. N. "A Contribution to the Theory of the General
Equation of the Sixth Degree." Amer. J. Math. 8, 265-
286, 1886.
Sextic Surface
An Algebraic Surface which can be represented im-
plicitly by a polynomial of degree six in x, y, and z.
Examples are the BARTH Sextic and Boy SURFACE.
see also Algebraic Surface, Barth Sextic, Boy
Surface, Cubic Surface, Decic Surface, Quadra-
tic Surface, Quartic Surface
References
Catanese, F. and Ceresa, G. "Constructing Sextic Surfaces
with a Given Number of Nodes." J. Pure Appl Algebra
23, 1-12, 1982.
Hunt, B. "Algebraic Surfaces." http: //www. mathematik.
uni-kl . de/-wwwagag/Galerie . html.
Sextillion
In the American system, 10 21 .
see also Large Number
Sexy Primes
Since a Prime Number cannot be divisible by 2 or 3,
it must be true that, for a Prime p, p = 6 (mod 1,5).
This motivates the definition of sexy primes as a pair
of primes (p, q) such that p — q = 6 ( "sexy" since "sex"
is the Latin word for "six."). The first few sexy prime
pairs are (5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23,
29), (31, 37), (37, 43), (41, 47), (47, 53), ... (Sloane's
A023201 and A046117).
Sexy constellations also exist. The first few sexy triplets
(i.e., numbers such that each of (p,p + 6,p + 12) is
Prime but p+ 18 is not Prime) are (7, 13, 19), (17, 23,
29), (31, 37, 43), (47, 53, 59), ... (Sloane's A046118,
A046119, and A046120). The first few sexy quadruplets
are (11, 17, 23, 29), (41, 47, 53, 59), (61, 67, 73, 79),
(251, 257, 263, 269), ... (Sloane's A046121, A046122,
A046123, A046124). Sexy quadruplets can only begin
with a Prime ending in a "1." There is only a sin-
gle sexy quintuplet, (5, 11, 17, 23, 29), since every fifth
number of the form 6n±l is divisible by 5, and therefore
cannot be PRIME.
see also PRIME CONSTELLATION, PRIME QUADRUPLET,
Twin Primes
References
Sloane, N. J. A. Sequences A023201, A046117, A046118,
A046119, A046120, A046121, A046122, A046123, and
A046124 in "An On-Line Version of the Encyclopedia of
Integer Sequences."
Trotter, T. "Sexy Primes." http://www.geocities.com/
CapeCanaveral/Launchpad/8202/sexyprim.html.
1622 Seydewitz's Theorem
Seydewitz's Theorem
If a Triangle is inscribed in a Conic Section, any
line conjugate to one side meets the other two sides in
conjugate points.
see also CONIC SECTION, TRIANGLE
Sgn
Also called SlGNUM. It can be defined as
sgn
i-1 x <0
x =
1 x >
sgn(x) = 2H(x) - 1,
(1)
(2)
where H(x) is the Heaviside Step Function. For
x ^ 0, this can be written
sgn(z)
for x / 0.
(3)
see also Heaviside Step Function, Ramp Function
Shadow
The Surface corresponding to the region of obscuration
when a solid is illuminated from a point light source (lo-
cated at the Radiant Point). A Disk is the Shadow
of a Sphere on a PLANE perpendicular to the Sphere-
Radiant Point line. If the Plane is tilted, the shadow
can be the interior of an Ellipse or a Parabola.
see also Projective Geometry
Shadowing Theorem
Although a numerically computed CHAOTIC trajectory-
diverges exponentially from the true trajectory with the
same initial coordinates, there exists an errorless trajec-
tory with a slightly different initial condition that stays
near ("shadows") the numerically computed one. There-
fore, the Fractal structure of chaotic trajectories seen
in computer maps is real.
References
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, pp. 18-19, 1993.
Shafarevich Conjecture
A conjecture which implies the MORDELL CONJECTURE,
as proved in 1968 by A. N. Parshin.
see also MORDELL CONJECTURE
References
Stewart, I. The Problems of Mathematics, 2nd ed. Oxford,
England: Oxford University Press, p. 45, 1987.
Shallit Constant
Shah Function
oo
ID(x)= ]P S(x-n)
(1)
n—~ oo
where S(x) is the Delta Function, so III (x) = for
a;^Z (i.e., x not an INTEGER). The shah function obeys
the identities
m(o,) = if;*(.-2)
7l= — oo
III (-a) = III (x)
UI(x + n) = m(x),
for 2nGZ (i.e., n a half-integer).
It is normalized so that
r n+l/2
/"
J n-
HI (x) dx = 1.
(2)
(3)
(4)
(5)
fn-1/2
The "sampling property" is
oo
III (*)/(*)= Y, f(n)S{x-n) (6)
n= — oo
and the "replicating property" is
oo
III (*) * /(*) = Y, /(*-»). ( 7 )
n= — oo
where * denotes CONVOLUTION.
see also CONVOLUTION, DELTA FUNCTION, IMPULSE
Pair
Shah- Wilson Constant
see Twin Primes Constant
Shallit Constant
Define /(xi,X2, . . . ,x n ) with Xi POSITIVE as
n k
i=l l<i<A;<Ti j = i
Then
min / = 3n — C + o(l)
as n increases, where the Shallit constant is
C = 1.369451403937...
(Shallit 1995). In their solution, Grosjean and De Meyer
(quoted in Shallit 1995) reduced the complexity of the
problem.
References
MacLeod, A. http://www.mathsoft .com/asolve/constant/
shapiro/macleod.html.
Shallit, J. Solution by C. C. Grosjean and H. E. De Meyer. "A
Minimization Problem." Problem 94-15 in SIAM Review
37, 451-458, 1995.
Shallow Diagonal
Shapiro's Cyclic Sum Constant 1623
Shallow Diagonal
see Pascal's Triangle
Shanks' Algorithm
An Algo rithm whi ch finds the least Nonnegative
value of ya (mod p) for given a and Prime p.
Shanks' Conjecture
Let p(g) be the first PRIME which follows a PRIME Gap
of g between consecutive PRIMES. Shanks' conjecture
holds that
to[p(p)] ~ Vs-
see also PRIME DIFFERENCE FUNCTION, PRIME GAPS
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 21, 1994.
Rivera, C. "Problems & Puzzles (Conjectures): Shanks'
Conjecture." http: //www. sci .net .mx/~crivera/ppp/
conj_009.htm.
Shanks, D. "On Maximal Gaps Between Successive Primes."
Math. Comput. 18, 646-651, 1964.
Shannon Entropy
see Entropy
Shannon Sampling Theorem
see Sampling Theorem
Shape Operator
The negative derivative
S(v)
-£» V N
(1)
of the unit normal N vector field of a Surface is called
the shape operator (or Weingarten Map or Second
Fundamental Tensor). The shape operator S is
an Extrinsic Curvature, and the Gaussian Curva-
ture is given by the DETERMINANT of S. If x : U -> R 3
is a Regular Patch, then
5(x u ) - -N tt (2)
5(x v ) = -N„. (3)
At each point p on a Regular Surface Mel 3 , the
shape operator is a linear map
S : M p -► M p . (4)
The shape operator for a surface is given by the WEIN-
GARTEN Equations.
see also CURVATURE, FUNDAMENTAL FORMS, WEIN-
GARTEN Equations
References
Gray, A. "The Shape Operator," "Calculation of the Shape
Operator," and "The Eigenvalues of the Shape Opera-
tor." §14.1, 14.3, and 14.4 in Modern Differential Geome-
try of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 268-269, 274-279, 1993.
Reckziegel, H. In Mathematical Models from the Collections
of Universities and Museums (Ed. G. Fischer). Braun-
schweig, Germany: Vieweg, p. 30, 1986.
Shapiro's Cyclic Sum Constant
N. B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Consider the sum
fn(XlyX 2 ,. .. ,X n ) =
Xl
+
x 2
X2 + Xs Xs + X4
+ ..
+ ^L_ +
.a.'.w (1)
where the XjS are NONNEGATIVE and the DENOMINA-
TORS are Positive, Shapiro (1954) asked if
fn(xux 2 ,...jX n ) > \n (2)
for all n. It turns out (Mitrinovic et al. 1993) that this
Inequality is true for all Even n < 12 and Odd n <
23. Ranikin (1958) proved that for
(3)
f(n) = inf / n (xi,X2, ■• .,£„),
a:>0
lim
/(«)
n>l n
7 x 10"
(4)
A can be computed by letting <j>(x) be the Convex Hull
of the functions
3/i = e
3/2 =
e x _|_ e x/2 •
Then
A= |0(O) = 0.4945668...
(Drinfeljd 1971).
A modified sum was considered by Elbert (1973):
gn(XljXi y ... y X n )
(5)
(6)
(7)
Xl + Xz X2 + #4
X± + X 2 X 2 +#3
X x -1 + Xi X n + X 2
Consider
where
jjl = lim
n— ► oo n
Xn—1 "T X n
9(n)
Xji ~T~ X\
g(n) = inf g n (xi,X2,- • -,a; n ),
x>0
and let %p{x) be the Convex Hull of
y? =
Then
1 + e*
1 + e*/ 2 '
fi = il>{0) = 0.978012..
see also Convex Hull
(8)
(9)
(10)
(11)
(12)
(13)
References
Drinfeljd, V. G. "A Cyclic Inequality." Math. Notes. Acad.
Sci. USSR 9, 68-71, 1971.
Elbert, A. "On a Cyclic Inequality." Period. Math. Hungar.
4, 163-168, 1973.
Finch, S. "Favorite Mathematical Constants." http: //www.
mathsof t . c om/ as olve/ const ant /shapiro/shapiro .html.
Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classi-
cal and New Inequalities in Analysis. New York: Kluwer,
1993.
1624 Sharing Problem
Sheaf (Topology)
Sharing Problem
A problem also known as the POINTS PROBLEM or UN-
FINISHED GAME. Consider a tournament involving k
players playing the same game repetitively. Each game
has a single winner, and denote the number of games
won by player i at some juncture Wi. The games are in-
dependent, and the probability of the ith player winning
a game is pi. The tournament is specified to continue
until one player has won n games. If the tournament is
discontinued before any player has won n games so that
Wi < n for i — 1, . . . , fc, how should the prize money
be shared in order to distribute it proportionally to the
players' chances of winning?
For player i, call the number of games left to win n =
n — Wi > the "quota." For two players, let p~p\ and
q = p2 = 1 — P be the probabilities of winning a single
game, and a = ri = n — w± and b = r^ = n — W2 be
the number of games needed for each player to win the
tournament. Then the stakes should be divided in the
ratio m : n, where
m = p
n = q
1+ a g+ o(a_M) g2
a(q + l)---(a + 6-2) b _ ±
~ h ""~ h (6-1)! q
6 6(6+1) 2
1+ lP+— 2i~ *
6(6+1). -.(6 + a-2) ,
+ . . . H — p
(a-1)
(1)
(2)
(Kraitchik 1942).
If i players have equal probability of winning ( "cell prob-
ability"), then the chance of player i winning for quotas
ri, ..., r k is
Wi = Dj B " 1 (ri,... ) ri_i,rt+i,...,r fc ;ri), (3)
where D is the DlRICHLET INTEGRAL of type 2D. Simi-
larly, the chance of player i losing is
Li = Cf" 1 (ri,...,ri-i,rt + i,...,r fe ;ri), (4)
where C is the DlRICHLET INTEGRAL of type 2C. If the
cell quotas are not equal, the general Dirichlet integral
£) a must be used, where
For h = 4 with quota vector r = (n, r2, r3, r±) and A =
Pi +P3 +P4,
^)='f"fC;;iti,r)(?)'*(5)'(sr
x ^ /a (n , ra + 1 + j)D™ (r 4 - j, r 3 - i). (7)
Pi
i-5X">
(5)
An expression for k — 5 is given by Sobel and Frankow-
ski (1994, p. 838).
see also Dirichlet Integrals
References
Kraitchik, M. "The Unfinished Game." §6.1 in Mathematical
Recreations. New York: W. W. Norton, pp. 117-118, 1942.
Sobel, M. and Frankowski, K. "The 500th Anniversary of the
Sharing Problem (The Oldest Problem in the Theory of
Probability)." Amer. Math. Monthly 101, 833-847, 1994.
Sharkovsky's Theorem
see SARKOVSKII'S THEOREM
Sharpe's Differential Equation
A generalization of the BESSEL DIFFERENTIAL EQUA-
TION for functions of order 0, given by
zy" + y + (z + A)y = 0.
Solutions are
y = c ±< *iFi(f =Ffti4;l;=F2w),
where 1 F 1 (a\b;x) is a Confluent Hypergeometric
Function.
see also Bessel Differential Equation, Conflu-
ent Hypergeometric Function
Sharpe Ratio
A risk-adjusted financial measure developed by Nobel
Laureate William Sharpe. It uses a fund's standard de-
viation and excess return to determine the reward per
unit of risk. The higher a fund's Sharpe ratio, the better
the fund's "risk-adjusted" performance.
see also ALPHA, BETA
Sheaf (Geometry)
The set of all Planes through a Line.
see also Line, Pencil, Plane
References
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, p. 12,
1961.
If ri = r and ai = 1, then Wi and Li reduce to 1/k
as they must. Let P(ri, . . . , r k ) be the joint probability
that the players would be RANKED in the order of the
ns in the argument list if the contest were completed.
For k = 3,
P(ri,r 2 ,r 3 ) = C^ 1 ' 1) (r 1 ,r 2 ,r 3 ).
(6)
Sheaf (Topology)
A topological GADGET related to families of ABELIAN
Groups and Maps.
References
Iyanaga, S. and Kawada, Y. (Eds.). "Sheaves." §377 in En-
cyclopedic Dictionary of Mathematics, Cambridge, MA:
MIT Press, p. 1171-1174, 1980.
Shear
Shear
Shi
1625
A transformation in which all points along a given Line
L remain fixed while other points are shifted parallel to
L by a distance proportional to their Perpendicular
distance from L. Shearing a plane figure does not change
its AREA. The shear can also be generalized to 3-D, in
which Planes are translated instead of lines.
Shear Matrix
The shear matrix
ej,- is obtained from the IDENTITY
MATRIX by inserting s at (i,i), e.g.,
ri
5
0]
1
.0
1.
see also Elementary Matrix
Shephard's Problem
Measurements of a centered convex body in Euclidean
n-space (for n > 3) show that its brightness function
(the volume of each projection) is smaller than that of
another such body. Is it true that its VOLUME is also
smaller? C. M. Petty and R. Schneider showed in 1967
that the answer is yes if the body with the larger bright-
ness function is a projection body, but no in general for
every n.
References
Gardner, R. J. "Geometric Tomography." Not. Amer. Math.
Soc. 42, 422-429, 1995.
Sheppard's Correction
A correction which must be applied to the MOMENTS
computed from Normally Distributed data which
have been binned. The corrected versions of the second,
third, and fourth moments are
M 2 =M2 <0) -£C 2
(1)
M3=M3 (0)
(2)
(0) 1 (0) , 7 2
HA = H4 K ~ 2M2' + ^C ,
(3)
where c is the CLASS INTERVAL. If d r is the rth Cu-
MULANT of an ungrouped distribution and n r the rth
CUMULANT of the grouped distribution with CLASS IN-
TERVAL c, the corrected cumulants (under rather restric-
tive conditions) are
Bjl,
for r odd
for r even,
(4)
where B r is the rth BERNOULLI NUMBER, giving
«! = K\
(5)
' 1 2
K 2 — K2 - ^ C
(6)
t
K>3 — ^3
(7)
K4 = K4 + Y20C
(8)
K5 — /^5
(9)
' 1 6
Kq — K 6 — 252 C •
(10)
For a proof, see Kendall et al. (1987).
References
Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall's Ad-
vanced Theory of Statistics, Vol. 1: Distribution Theory,
6th ed. New York: Oxford University Press, 1987.
Kenney, J. F. and Keeping, E. S. "Sheppard's Correction."
§4.12 in Mathematics of Statistics, Pt. 2, 2nd ed. Prince-
ton, NJ: Van Nostrand, pp. 80-82, 1951.
Sherman-Morrison Formula
A formula which allows the new MATRIX to be computed
for a small change to a MATRIX A. If the change can be
written in the form
u(g) v
for two vectors u and v, then the Sherman- Morrison
formula is
1 + A
where
A = v.A _1 u.
see also WOODBURY FORMULA
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Sherman-Morrison Formula." In Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 65-67, 1992.
Shi
1626
Shift
Sibling
Shi(z)
f z sinht
Jo ^ r
dt.
The function is given by the Mathematical (Wolfram
Research, Champaign, IL) command SinhlntegralCz].
see also Chi, Cosine Integral, Sine Integral
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Co-
sine Integrals." §5.2 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 231-233, 1972.
Shift
A Translation without Rotation or distortion.
see also DILATION, EXPANSION, ROTATION, TRANSLA-
TION, Twirl
Shift Property
see Delta Function
Shimura-Taniyama Conjecture
see Taniyama-Shimura Conjecture
Shimura-Taniyama- Weil Conjecture
see Taniyama-Shimura Conjecture
Shoe Surface
A surface given by the parametric equations
y(u,v) = v
z(u,v) = iu 3
1 2
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 634, 1993.
Shortening
A KNOT used to shorten a long rope.
see also Bend (Knot)
References
Owen, P. Knots. Philadelphia, PA: Courage, p. 65, 1993.
Shuffle
The randomization of a deck of Cards by repeated
interleaving. More generally, a shuffle is a rearrange-
ment of the elements in an ordered list. Shuffling by
exactly interleaving two halves of a deck is called a Rif-
fle SHUFFLE. Normal shuffling leaves gaps of different
lengths between the two layers of cards and so random-
izes the order of the cards.
A deck of 52 Cards must be shuffled seven times for it
to be randomized (Aldous and Diaconis 1986, Bayer and
Diaconis 1992). This is intermediate between too few
shuffles and the decreasing effectiveness of many shuf-
fles. One of Bayer and Diaconis's randomness Crite-
ria, however, gives 31gfc/2 shuffles for a A;-card deck,
yielding 11-12 shuffles for 52 CARDS. Keller (1995)
shows that roughly In k shuffles are needed just to ran-
domize the bottom card.
see also Bays' Shuffle, Cards, Faro Shuffle,
Monge's Shuffle, Riffle Shuffle
References
Aldous, D. and Diaconis, P. "Shuffling Cards and Stopping
Times." Amer. Math. Monthly 93, 333-348, 1986.
Bayer, D. and Diaconis, P. "Trailing the Dovetail Shuffle to
Its Lair." Ann. Appl. Probability 2, 294-313, 1992.
Keller, J. B. "How Many Shuffles to Mix a Deck?" SIAM
Review 37, 88-89, 1995.
Morris, S. B. "Practitioner's Commentary: Card Shuffling."
UMAP J. 15, 333-338, 1994.
Rosenthal, J. W. "Card Shuffling." Math. Mag. 54, 64-67,
1981.
Siamese Dodecahedron
see Snub Disphenoid
Siamese Method
A method for constructing MAGIC SQUARES of Odd or-
der, also called DE LA LOUBERE'S METHOD.
see also MAGIC SQUARE
Sibling
Two nodes connected to the same node in a ROOTED
Tree are called siblings.
see also CHILD, ROOTED TREE
Shoemaker's Knife
see Arbelos
Sicherman Dice
Sierpinski Arrowhead Curve 1627
Sicherman Dice
2
6
4
2
1
3
8
4
1
5
3
3
A pair of DICE which have the same Odds for throwing
every number as a normal pair of 6-sided Dice. They
are the only such alternate arrangement.
see also Dice, Efron's Dice
Sici Spiral
The spiral
x = c ci t
y = c(sit- |tt),
where ci(t) and si(i) are the Cosine Integral and Sine
Integral and c is a constant.
see also Cosine Integral, Sine Integral, Spiral
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, pp. 204 and 270, 1993.
Side
The edge of a Polygon and face of a Polyhedron are
sometimes called sides.
Sidon Sequence
see B2-SEQUENCE
Siegel Disk Fractal
see also DOUADY'S RABBIT FRACTAL, JULIA SET,
Mandelbrot Set, San Marco Fractal
References
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 176, 1991.
Siegel Modular Function
A r n -invariant meromorphic function on the space of
all n x n complex symmetric matrices with POSITIVE
Imaginary Part. In 1984, H. Umemura expressed the
ROOTS of an arbitrary POLYNOMIAL in terms of elliptic
Siegel functions.
References
Iyanaga, S. and Kawada, Y. (Eds.). "Siegel Modular Func-
tions." §34F in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 131-132, 1980.
Siegel's Paradox
If a fixed Fraction x of a given amount of money P is
lost, and then the same FRACTION x of the remaining
amount is gained, the result is less than the original and
equal to the final amount if a Fraction x is first gained,
then lost. This can easily be seen from the fact that
[P(l - a;)](l + x) = P(l - x 2 ) < P
[P(l + x)]{l -x) = P(l - x 2 ) < P.
Siegel's Theorem
An Elliptic Curve can have only a finite number of
points with Integer coordinates.
see also ELLIPTIC CURVE
References
Davenport, H. "Siegel's Theorem." Ch. 21 in Multiplica-
tive Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 126-125, 1980.
Sierpinski Arrowhead Curve
A Julia Set with c = -0.390541 - 0. 586788*. The
Fractal somewhat resembles the better known Man-
delbrot Set.
A Fractal which can be written as a Lindenmayer
System with initial string "YF", String Rewriting
rules "X" -> "YF+XF+Y", "Y" -> "XF-YF-X", and an-
gle 60°.
see also DRAGON CURVE, HlLBERT CURVE, KOCH
Snowflake, Lindenmayer System, Peano Curve,
Peano-Gosper Curve, Sierpinski Curve, Sierpin-
ski Sieve
References
Dickau, R. M. "Two-Dimensional L-Systems." http://
forum . swarthmore . edu/ advanced/robe rtd/lsys2d . html.
1628 Sierpinski Carpet
Sierpiriski Carpet
m
"ml
A Fractal which is constructed analogously to the
Sierpinski Sieve, but using squares instead of trian-
gles. Let N n be the number of black boxes, L n the
length of a side of a white box, and A n the fractional
Area of black boxes after the nth iteration. Then
N n
L n
(i) n = 3- n
A n = L n 2 N n = (l) n .
The Capacity Dimension is therefore
r lniV n ln(8 n )
ctcap = - lim - — — - = - hm
n-^oo In Lin n~¥oo
3 In 2
(1)
(2)
(3)
ln8
— lim , . . —
n-+oo ln(3" n ) ln3
ln3
1.892789261.
(4)
see also Menger Sponge, Sierpinski Sieve
References
Dickau, R, M. "The Sierpinski Carpet." http:// forum .
swarthmore . edu/advanced/robertd/carpet .html.
Peitgen, H.-O.; Jurgens, H.; and Saupe, D. Chaos and Frac-
tals: New Frontiers of Science. New York: Springer-
Verlag, pp. 112-121, 1992.
^ Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
Sierpiriski's Composite Number Theorem
There exist infinitely many Odd Integers k such that
&-2 n + l is Composite for every n > 1. Numbers k with
this property are called Sierpinski Numbers of the
Second Kind, and analogous numbers with the plus
sign replaced by a minus are called RlESEL NUMBERS.
It is conjectured that the smallest Sierpinski Number
OF the Second Kind is k = 78,557 and the smallest
Riesel Number is k — 509,203.
see also Cunningham Number, Sierpinski Number
of the Second Kind
References
Buell, D. A. and Young, J. "Some Large Primes and the Sier-
pinski Problem." SRC Tech. Rep. 88004, Supercomputing
Research Center, Lanham, MD, 1988.
Jaeschke, G. "On the Smallest k such that k - 2^ + 1 are
Composite." Math. Comput. 40, 381-384, 1983.
Jaeschke, G. Corrigendum to "On the Smallest k such that
k • 2^ + 1 are Composite." Math. Comput. 45, 637, 1985,
Keller, W. "Factors of Fermat Numbers and Large Primes of
the Form k ■ 2 n + 1." Math. Comput. 41, 661-673, 1983.
Keller, W. "Factors of Fermat Numbers and Large Primes of
the Form k • 2 n + 1, II." In prep.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 357-359, 1996.
Riesel, H. "Nagra stora primtal." Elementa 39, 258-260,
1956.
Sierpinski Curve
Sierpinski, W. "Sur un probleme concernant les nombres k •
2 n + 1." Elem. d. Math. 15, 73-74, 1960.
see also COMPOSITE NUMBER, SIERPINSKI NUMBERS
of the Second Kind, Sierpinski's Prime Sequence
Theorem
Sierpiriski Constant
2,6
2.58
2.56
1500
2000
Let rk(n) denote the number of representations of n by
k squares, then the SUMMATORY FUNCTION of r 2 {k)/k
has the ASYMPTOTIC expansion
^Z^) =ir + 7rlnn + 0(n -v 2))
where K = 2.5849817596 is the Sierpinski constant. The
above plot shows
E
r*(k)
— 7rlnn,
with the value of K indicated as the solid horizontal line.
see also rk(n)
References
Sierpiriski, W. Oeuvres Choiseies, Tome 1. Editions Scien-
tifiques de Pologne, 1974.
Sierpiriski Curve
There are several FRACTAL curves associated with Sier-
piriski. The above curve is one example, and the SIER-
PINSKI Arrowhead Curve is another. The limit of the
curve illustrated above has Area
A - -5-
A — 12'
The Area for a related curve illustrated by Cundy and
Rollett (1989) is
A=|(7-4V2).
Sierpinski Gasket
Sierpinski Sieve 1629
see also Exterior Snowflake, Gosper Island,
Hilbert Curve, Koch Antisnowflake, Koch
Snowflake, Peano Curve, Peano-Gosper Curve,
Sierpinski Arrowhead Curve
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 67-68, 1989.
Dickau, R. M. "Two- Dimensional L-Systems." http://
forum . swarthmore . edu/advanced/robertd/lsys2d . html.
Gardner, M. Penrose Tiles and Trapdoor Ciphers. . . and the
Return of Dr. Matrix, reissue ed. New York: W. H. Free-
man, p. 34, 1989.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 207, 1991.
Sierpinski Gasket
see Sierpinski Sieve
Sierpinski-Menger Sponge
see Menger Sponge
The smallest odd k such that k + 2 n is COMPOSITE for
all n < k are 773, 2131, 2491, 4471, 5101, ....
see also MERSENNE NUMBER, RlESEL NUMBER, SlER-
pinski's Composite Number Theorem
References
Buell, D. A. and Young, J. "Some Large Primes and the Sier-
pinski Problem." SRC Tech. Rep. 88004, Supercomputing
Research Center, Lanham, MD, 1988.
Jaeschke, G. "On the Smallest k such that k • 2 N + 1 are
Composite." Math. Comput. 40, 381-384, 1983.
Jaeschke, G. Corrigendum to "On the Smallest k such that
k ■ 2 N + 1 are Composite." Math. Comput. 45, 637, 1985.
Keller, W. "Factors of Fermat Numbers and Large Primes of
the Form k • 2 n + 1." Math. Comput. 41, 661-673, 1983.
Keller, W. "Factors of Fermat Numbers and Large Primes of
the Form k • 2 n + 1, II." In prep.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 357-359, 1996.
Sierpinski, W. "Sur un probleme concernant les nombres k ■
2 n + 1." Elem. d. Math. 15, 73-74, 1960.
Sloane, N. J. A. Sequence A046067 in "An On-Line Version
of the Encyclopedia of Integer Sequences."046068
Sierpinski Number of the First Kind
Numbers of the form S n = n n + 1. The first few are 2,
5, 28, 257, 3126, 46657, 823544, 16777217, . . . (Sloane's
A014566). Sierpinski proved that if S n is PRIME with
n > 2, then S n = F m+2 ™, where F m is a FERMAT NUM-
BER with m > 0. The first few such numbers are F\ = 5,
F 3 = 257, F 6 , JFii, F 20 , and F 37 . Of these, 5 and 257 are
PRIME, and the first unknown case is F37 > 10 3x10 .
see also Cullen Number, Cunningham Number,
Fermat Number, Woodall Number
References
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 155, 1979.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, p. 74, 1989.
Sloane, N. J. A. Sequence A014566 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Sierpinski's Prime Sequence Theorem
For any M, there exists a t' such that the sequence
2 , ,/
n + t
where n ~ 1, 2, ... contains at least M PRIMES.
see also Dirichlet's Theorem, Fermat An + 1 The-
orem, Sierpinski's Composite Number Theorem
References
Abel, U. and Siebert, H. "Sequences with Large Numbers of
Prime Values." Amer. Math. Monthly 100, 167-169, 1993.
Ageev, A. A. "Sierpinski's Theorem is Deducible from Euler
and Dirichlet." Amer. Math. Monthly 101, 659-660, 1994.
Forman, R. "Sequences with Many Primes." Amer. Math.
Monthly 99, 548-557, 1992.
Garrison, B. "Polynomials with Large Numbers of Prime Val-
ues." Amer. Math. Monthly 97, 316-317, 1990.
Sierpinski, W. "Les binomes x 2 +n et les nombres premiers."
Bull. Soc. Roy. Sci. Liege 33, 259-260, 1964.
Sierpinski Number of the Second Kind
A number k satisfying Sierpinski's Composite Num-
ber Theorem, i.e., such that k ■ 2" + 1 is Composite
for every n > 1. The smallest known is k = 78,557,
but there remain 35 smaller candidates (the smallest of
which is 4847) which are known to generate only com-
posite numbers for n < 18, 000 or more (Ribenboim
1996, p. 358).
Let a(k) be smallest n for which (2k — 1) ■ 2 n + 1 is
Prime, then the first few values are 0, 1, 1, 2, 1, 1, 2, 1,
3, 6, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 1, 2, 2, 583, . . . (Sloane's
A046067). The second smallest n are given by 1, 2, 3,
4, 2, 3, 8, 2, 15, 10, 4, 9, 4, 4, 3, 60, 6, 3, 4, 2, 11, 6,
9, 1483, ... (Sloane's A046068). Quite large n can be
required to obtain the first prime even for small k. For
example, the smallest prime of the form 383 • 2 n -f 1 is
383 * 2 6393 + 1. There are an infinite number of Sierpinski
numbers which are PRIME.
Sierpinski Sieve
A Fractal described by Sierpinski in 1915. It
is also called the SIERPINSKI GASKET or Sier-
pinski Triangle. The curve can be written
as a LlNDENMAYER SYSTEM with initial string
"FXF— FF— FF", String Rewriting rules "F" ->
"FF", "X"->"--FXF++FXF++FXF-- M , and angle 60°.
Let N n be the number of black triangles after iteration
n, L n the length of a side of a triangle, and A n the
fractional Area which is black after the nth iteration.
Then
N„=3 n
(1)
L n = (§)" = 2-
(2)
A n = L n 2 N n = (|) n .
(3)
1630 Sierpinski Sponge
Sieve of Eratosthenes
The Capacity Dimension is therefore
hm — —
n— >oo in Lin
lim
ln(3 n )
In 3
^ ""*_ i„ r n™»ln(2-") In 2
= 1.584962501.... (4)
In Pascal's Triangle, coloring all Odd numbers black
and EVEN numbers white produces a Sierpinski sieve.
see also Lindenmayer System, Sierpinski Arrow-
head Curve, Sierpinski Carpet, Tetrix
References
Crownover, R. M. Introduction to Fractals and Chaos. Sud-
bury, MA: Jones & B art let t, 1995.
Dickau, R. M. "Two- Dimensional L-Systems." http://
forum . swarthmore . edu/advanced/robertd/lsys2d . html.
Dickau, R. M. "Typeset Fractals." Mathematica J. 7, 15,
1997.
Dickau, R. "Sierpinski-Menger Sponge Code and Graphic."
http : // www . mathsource . com / cgi - bin / Math Source /
Applications/Graphics/0206-110.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 13-
14, 1991,
Peitgen, H.-O.; Jiirgens, H.; and Saupe, D. Chaos and Frac-
tals: New Frontiers of Science. New York: Springer-
Verlag, pp. 78-88, 1992.
Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal
Images. New York: Springer- Verlag, p. 282, 1988.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 108 and 151-153, 1991.
Wang, P. "Renderings." http : //www . ugcs . caltech . edu/
-peterw/portf olio/renderings/.
# Weisstein, E. W. "Fractals." http://www. astro. Virginia.
edu/~eww6n/math/notebooks/Fractal.m.
Sierpinski Sponge
see Tetrix
Sierpinski Tetrahedron
see Tetrix
Sierpiriski's Theorem
see SlERPINSKl'S COMPOSITE NUMBER THEOREM,
Sierpinski's Prime Sequence Theorem
Sierpinski Triangle
see Sierpinski Sieve
Sieve
A process of successively crossing out members of a list
according to a set of rules such that only some remain.
The best known sieve is the Eratosthenes Sieve for
generating Prime Numbers. In fact, numbers gener-
ated by sieves seem to share a surprisingly large number
of properties with the Prime Numbers.
see also Happy Number, Number Field Sieve Fac-
torization Method, Prime Number, Quadratic
Sieve Factorization Method, Sierpinski Sieve,
Sieve of Eratosthenes, Wallis Sieve
References
Halberstam, H. and Richert, H.-E. Sieve Methods. New York:
Academic Press, 1974.
Pomerance, C. "A Tale of Two Sieves." Not Amer. Math.
Soc. 43, 1473-1485, 1996.
Sieve of Eratosthenes
123i5i7i9l|0
1 2 3 J 5 1 7 H *
11 lb 13 lk 15 1J6 17 lis 19 2T0
11 ife 13 lk 1I5 lie 17 ife 19 2I0
21 2I2 23 2k 25 2?6 27 2J8 29 3J0
2|l 2J2 23 U 25 2?6 2l7 2J8 29 ib
31 3J2 33 3J4 35 3J6 37 3J8 39 4°
31 3b 3J3 3k 35 it 37 3?8 3J9 4J0
41 4J2 43 4J4 45 4J6 47 4J8 49 5?0
41 ifc 43 4k 4J5 4J6 47 ih 49 5?0
1!3 ! i I'!l!!
1 2 M s S M S 1
11 J| 13 ^ g $ 17 JJ 19 g
11 a i3 » a * » a i9 a
f f « g 2j 5 £ % $ 2 9 g
3} f 2 MH 5 f * i* 29 Ml
3i fff f \i " f * #
31 f f * a a " * * a
41 4| 43 4J4 4| 4J6 47 ife 49 (b
41 $ « t a t « a * it
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\ , 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
therefore crosses out PRIMES up to I \/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.
Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, pp. 100-101, 1989.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 20-21, 1996.
Sievert Integral
Sigmoid Function 1631
Sievert Integral
The integral
/
Jo
-x sec 4>
d<j>.
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Sievert Inte-
gral." §27.4 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 1000-1001, 1972.
Sievert's Surface
A special case of Enneper'S Surfaces which can be
given parametrically by
x — r cos c
y = r sin q>
_ ln[tan(|v)] + a(C+l)cosv
where
v / cTT
+ tan -1 (tanWC + l)
C + 1 — C sin 2 v cos 2 u
_ a^J{C + 1)(1 + Csin 2 u) sin^
= 7d '
(i)
(2)
(3)
(4)
(5)
(6)
with |u| < 7r/2 and < v < tv (Reckziegel 1986).
see also Enneper's Surfaces, Kuen Surface,
Rembs' Surfaces
References
Fischer, G. (Ed.). Plate 87 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 83, 1986.
Reckziegel, H. "Sievert's Surface." §3.4.4.3 in Mathemati-
cal Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 38-39, 1986.
Sievert, H. Uber die Zentralfldchen der Enneperschen
Flachen konstanten Krummungsmafies. Dissertation,
Tubingen, 1886.
Sifting Property
The property
/
/(yWx-y)dy = /(x)
obeyed by the DELTA FUNCTION <5(x).
see also Delta Function
Sigma Algebra
Let X be a Set. Then a cr- algebra F is a nonempty
collection of SUBSETS of X such that the following hold:
1. The Empty Set is in F.
2. If A is in F, then so is the complement of A.
3. If A n is a Sequence of elements of F, then the
Union of the A n s is in F.
If S is any collection of subsets of X, then we can always
find a cr-algebra containing 5, namely the POWER Set
of X. By taking the Intersection of all <r-algebras
containing 5, we obtain the smallest such a-algebra. We
call the smallest <x-algebra containing S the a-algebra
generated by S.
see also Borel Sigma Algebra, Borel Space, Mea-
surable Set, Measurable Space, Measure Alge-
bra, Standard Space
Sigma Function
see Divisor Function
Sigmoid Curve
see Sigmoid Function
Sigmoid Function
The function
y
1 + e~ x
which is the solution to the Ordinary Differential
Equation
dy
dx
= y(i-y)-
It has an inflection point at x = 0, where
see also Exponential Function, Exponential
Ramp
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 124, 1993.
1632 Sign
Sign
The sign of a number, also called Sgn, is —1 for a NEG-
ATIVE number (i.e., one with a MINUS SIGN "-"), for
the number Zero, or +1 for a Positive number (i.e.,
one with a Plus Sign "+").
see also Absolute Value, Minus Sign, Negative,
Plus Sign, Positive, Sgn, Zero
Signalizer Functor Theorem
0(G; A) = (0(a) : a <= A - 1)
is an ^4-invariant solvable p'-subgroup of G.
Signature (Knot)
The signature s(K) of a Knot K can be defined using
the Skein Relationship
s (unknot) =
s(K + ) - s(K-) e {0,2},
and
4\s(K)<r>V(K)(2i) >0,
where V(K) is the Alexander-Conway Polynomial
and V(K)(2i) is an Odd Number.
Many UNKNOTTING NUMBERS can be determined using
a knot's signature.
see also Skein Relationship, Unknotting Number
References
Gordon, C. McA.; Litherland, R. A.; and Murasugi, K. "Sig-
natures of Covering Links." Canad. J. Math. 33,381-394,
1981.
Murasugi, K. "On the Signature of Links." Topology 9, 283-
298, 1970.
Murasugi, K. "Signatures and Alexander Polynomials of
Two-Bridge Knots." C. R. Math. Rep. Acad. Sci. Canada
5, 133-136, 1983.
Murasugi, K. "On the Signature of a Graph." C. R. Math.
Rep. Acad. Sci. Canada 10, 107-111, 1988.
Murasugi, K. "On Invariants of Graphs with Applications to
Knot Theory." Trans. Amer. Math. Soc. 314, 1-49, 1989.
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, 1976.
Stoimenow, A. "Signatures." http://www.informatik.hu-
berlin.de/~stoimeno/ptab/siglO.html.
Signature (Quadratic Form)
The signature of the QUADRATIC FORM
Q = y 2 + y* + . . . + y P 2 - y P + 2 - y P +2 2 - . . . - y 2
is the number s of POSITIVE squared terms in the re-
duced form. (The signature is sometimes defined as
2s -r.)
see also p-Signature, Rank (Quadratic Form),
Sylvester's Inertia Law, Sylvester's Signature
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1105, 1979.
Signed Deviation
Signature (Recurrence Relation)
Let a sequence be defined by
A-i = s
A = 3
A\ — r
A n = rA n -i - sA n -2 + A n -3-
Also define the associated Polynomial
f(x) = x s — rx 2 + sx + 1,
and let A be its discriminant. The Perrin Sequence
is a special case corresponding to A n (0, — 1). The sig-
nature mod m of an Integer n with respect to the
sequence Ak(r, s) is then defined as the 6-tuple (A- n -x,
A-n, A- n+1 , A n -\, A n , A n +i) (mod m).
1. An Integer n has an S-signature if its signature
(mod n) is (A_ 2 , A_i, Ao, Ai, A 2 ).
2. An INTEGER n has a Q-signature if its signature
(mod n) is CONGRUENT to (A,s,i?,i?,r, C) where,
for some INTEGER a with f(a) = (mod n), A =
a" 2 + 2a, B = -m 2 + (r 2 - s)a, and C = a 2 + 2a~ l .
3. An Integer n has an I-signature if its signature
(mod n) is CONGRUENT to (r, s, £>', 2?, r, s), where
D' + D = rs - 3 and (£>' - D) 2 = A.
see also Perrin Pseudoprime
References
Adams, W. and Shanks, D. "Strong Primality Tests that Are
Not Sufficient." Math. Comput. 39, 255-300, 1982.
Grantham, J. "Frobenius Pseudoprimes." http://www.
dark, net /pub/grantham/pseudo/pseudo.ps
Signature Sequence
Let be an Irrational Number, define S(0) = {c +
dO : c,d € N}, and let c n (0) + d n 9(6) be the sequence
obtained by arranging the elements of S(0) in increasing
order. A sequence x is said to be a signature sequence if
there Exists a Positive Irrational Number 6 such
that x = {c n (0)}, and x is called the signature of 9.
The signature of an Irrational Number is a Fractal
Sequence. Also, if a: is a signature sequence, then the
Lower-Trimmed Subsequence is V(x) = x.
References
Kimberling, C. "Fractal Sequences and Interspersions." Ars
Combin. 45, 157-168, 1997.
Signed Deviation
The signed deviation is defined by
Am = (ui — u),
so the average deviation is
Ui
Ui — u
0.
see also ABSOLUTE DEVIATION, DEVIATION, DISPER-
SION (Statistics), Mean Deviation, Quartile De-
viation, Standard Deviation, Variance
Significance
Silverman Constant
1633
Significance
Let 5 = z < Observed. A value < a < 1 such
that P(S) < a is considered "significant" (i.e., is not
simply due to chance) is known as an Alpha Value.
The Probability that a variate would assume a value
greater than or equal to the observed value strictly by
chance, P(5) y is known as a P- Value.
Depending on the type of data and conventional prac-
tices of a given field of study, a variety of different alpha
values may be used. One commonly used terminology
takes P(S) > 5% as "not significant," 1% < P(S) < 5%,
as "significant" (sometimes denoted *), and P(S) < 1%
as "highly significant" (sometimes denoted **). Some
authors use the term "almost significant" to refer to
5% < P(S) < 10%, although this practice is not rec-
ommended.
see also Alpha Value, Confidence Interval, P-
Value, Probable Error, Significance Test, Sta-
tistical Test
Significance Test
A test for determining the probability that a given result
could not have occurred by chance (its Significance).
see also Significance, Statistical Test
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 491-492, 1987.
Significant Digits
When a number is expressed in SCIENTIFIC NOTATION,
the number of significant figures is the number of DIG-
ITS needed to express the number to within the uncer-
tainty of measurement. For example, if a quantity had
been measured to be 1.234 ± 0.002, four figures would
be significant. No more figures should be given than
are allowed by the uncertainty. For example, a quantity
written as 1.234 ± 0.1 is incorrect; it should really be
written as 1.2 ± 0.1.
The number of significant figures of a MULTIPLICATION
or DIVISION of two or more quantities is equal to the
smallest number of significant figures for the quantities
involved. For ADDITION or MULTIPLICATION, the num-
ber of significant figures is determined with the smallest
significant figure of all the quantities involved. For ex-
ample, the sum 10.234 + 5.2 + 100.3234 is 115.7574, but
should be written 115.8 (with rounding), since the quan-
tity 5.2 is significant only to ±0.1.
see also Nint, Round, Truncate
Significant Figures
see Significant Digits
Signpost
A 6-POLYIAMOND.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Signum
see Sgn
Silver Constant
The Real Root of the equation
x 3 — 5a; + 6x — 1 = 0,
which is 3.2469 It is the seventh BERAHA CON-
STANT.
see also Beraha Constants
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
pp. 51 and 143, 1983.
Silver Mean
see Silver Ratio
Silver Ratio
The quantity denned by the Continued Fraction
*s = [2,2,2,...] = 2+-
1
2+-
2 + -
It follows that
so
(Ss ~ I) 2 = 2,
5s = v^+1 = 2.41421....
see also GOLDEN RATIO, GOLDEN RATIO CONJUGATE
Silverman Constant
^ <f>(n)<r(n) = 11 ( 1 + 2^p2fe_ p fc-i 1
n=l p prime \ fe=l /
= 1.786576459...,
where <f>(n) is the Totient FUNCTION and cr(n) is the
Divisor Function.
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/totient/totient .html.
Zimmerman, P. http : // www . mathsof t . com / asolve /
constant/tot ient/zimmermn. html.
1634
Silverman's Sequence
Similitude Ratio
Silverman's Sequence
Let /(l) = 1, and let f(n) be the number of occurrences
of n in a nondecr easing sequence of INTEGERS. Then
the first few values of f(n) are 1, 2, 2, 3, 3, 4, 4, 4, 5, 5,
5, . . . (Sloane's A001462). The asymptotic value of the
nth term is <f> 2 ~ <i> n <i> ~ 1 , where <j> is the GOLDEN RATIO.
References
Guy, R. K. "Silverman's Sequences." §E25 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 225-226, 1994.
Sloane, N. J. A. Sequence A001462/M0257 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Similar
directly similar inversely similar
Two figures are said to be similar when all corresponding
Angles are equal. Two figures are Directly Similar
when all corresponding ANGLES are equal and described
in the same rotational sense. This relationship is written
A ~ B. (The symbol ~ is also used to mean "is the same
order of magnitude as" and "is Asymptotic to.") Two
figures are Inversely Similar when all corresponding
ANGLES are equal and described in the opposite rota-
tional sense.
see also Directly Similar, Inversely Similar, Sim-
ilarity Transformation
References
Project Mathematics! Similarity. Videotape (27 minutes).
California Institute of Technology. Available from the
Math. Assoc. Amer.
Similarity Axis
see d'Alembert's Theorem
Similarity Dimension
To multiply the size of a d-D object by a factor a, c = a d
copies are required, and the quantity
j _ mc
In a
is called the similarity dimension.
Similarity Point
External (or positive) and internal (or negative) simi-
larity points of two Circles with centers C and C' and
Radii r and r' are the points E and / on the lines CC
such that
CE __ r_
C'E ~ r"
CI
CI
r
"r 7
Similarity Transformation
An ANGLE-preserving transformation. A similarity
transformation has a transformation MATRIX A' of the
form
A' -BAB" 1 .
If A is an Antisymmetric Matrix (a^ = -a,-*) and B
is an Orthogonal Matrix, then
(bab* 1 )^ = bikdkib^ 1 = — &ifcOjk&j~. = -b kiQ>ik{b )~ jt
= —b kidkibji — bjiaikb ki
-(bab~
Similarity transformations and the concept of SELF-
SlMlLARITY are important foundations of FRACTALS
and Iterated Function Systems.
see also CONFORMAL TRANSFORMATION
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 83-
103, 1991.
Similitude Center
Also called a Self-Homologous Point. If two Sim-
ilar figures lie in the plane but do not have parallel
sides (they are not HOMOTHETIC), there exists a cen-
ter of similitude which occupies the same homologous
position with respect to the two figures. The LOCUS of
similitude centers of two nonconcentric circles is another
circle having the line joining the two nomothetic centers
as its DIAMETER.
There are a number of interesting theorems regarding
three CIRCLES (Johnson 1929, pp. 151-152).
1. The external similitude centers of three circles are
COLLINEAR.
2. Any two internal similitude centers are COLLINEAR
with the third external one.
3. If the center of each circle is connected with the in-
ternal similitude center of the other three [sic], the
connectors are CONCURRENT.
4. If one center is connected with the internal simil-
itude center of the other two, the others with the
corresponding external centers, the connectors are
Concurrent.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 19-27 and 151-153, 1929.
Similitude Ratio
Two figures are HOMOTHETIC if they are related by a
Dilation (a dilation is also known as a Homothecy).
This means that the connectors of corresponding points
are CONCURRENT at a point which divides each connec-
tor in the same ratio k } known as the similitude ratio.
see also CONCURRENT, DILATION, HOMOTHECY, HO-
MOTHETIC
Simple Algebra
Simple Harmonic Motion 1635
Simple Algebra
An Algebra with no nontrivial Ideals.
see also Algebra, Ideal, Semisimple Algebra
Simple Continued Fraction
A Continued Fraction
o- = bo +
a\
61 +
a 2
(1)
b 2 +
a 3
63 + . . .
in which the biS are all unity, leaving a continued fraction
of the form
a = clq +
(2)
ai +
a 2 +
a 3 + . . .
A simple continued fraction can be written in a compact
abbreviated Notation as
cr = [ao,ai,a 2 ,a 3 ,...].
(3)
Bach and Shallit (1996) show how to compute the JA-
COBI SYMBOL in terms of the simple continued fraction
of a Rational Number a/6.
see also CONTINUED FRACTION
References
Bach, E. and Shallit, J. Algorithmic Number Theory,
Vol 1: Efficient Algorithms. Cambridge, MA: MIT Press,
pp. 343-344, 1996.
Simple Curve
A curve is simple closed if it does not cross itself.
see also Jordan Curve
Simple Graph
A Graph for which at most one Edge connects any two
nodes.
see also Adjacency Matrix, Edge (Graph)
Simple Group
A simple group is a GROUP whose NORMAL SUBGROUPS
(Invariant Subgroups) are Order one or the whole
of the original Group. Simple groups include Alter-
nating Groups, Cyclic Groups, Lie-Type Groups
(five varieties), and SPORADIC GROUPS (26 varieties,
including the MONSTER GROUP). The CLASSIFICATION
Theorem of finite simple groups states that such groups
can be classified completely into the three types:
1. Cyclic Groups of Prime Order,
2. Alternating Groups of degree at least five
3. Lie-Type Chevalley Groups,
4. Lie-Type (Twisted Chevalley Groups or the
Tits Group), and
5. Sporadic Groups.
Burnside's Conjecture states that every non-
Abelian Simple Group has Even Order.
see also ALTERNATING GROUP, BURNSIDE'S CONJEC-
TURE, Chevalley Groups, Classification Theo-
rem, Cyclic Group, Feit-Thompson Theorem, Fi-
nite Group, Group, Invariant Subgroup, Lie-
Type Group, Monster Group, Schur Multiplier,
Sporadic Group, Tits Group, Twisted Chevalley
Groups
Simple Harmonic Motion
Simple harmonic motion refers to the periodic sinusoidal
oscillation of an object or quantity. Simple harmonic
motion is executed by any quantity obeying the DIF-
FERENTIAL Equation
x + ujq x = 0,
(i)
where x denotes the second DERIVATIVE of x with re-
spect to t, and uo is the angular frequency of oscillation.
This Ordinary Differential Equation has an irreg-
ular Singularity at 00. The general solution is
x = Asin((jj t) + Bcos(iOot)
= C cos(u>ot 4- 0),
(2)
(3)
where the two constants A and B (or C and <j>) are
determined from the initial conditions.
Many physical systems undergoing small displacements,
including any objects obeying Hooke's law, exhibit sim-
ple harmonic motion. This equation arises, for example,
in the analysis of the flow of current in an electronic
CL circuit (which contains a capacitor and an induc-
tor). If a damping force such as Friction is present, an
additional term /3x must be added to the DIFFERENTIAL
Equation and motion dies out over time.
Adding a damping force proportional to x, the first de-
rivative of x with respect to time, the equation of motion
for damped simple harmonic motion is
x + f3x + uo x — 0,
(4)
where f3 is the damping constant. This equation arises,
for example, in the analysis of the flow of current in
an electronic CLR circuit, (which contains a capacitor,
an inductor, and a resistor). This ORDINARY DIFFER-
ENTIAL Equation can be solved by looking for trial
solutions of the form x = e rt . Plugging this into (4)
gives
(r 2 +^r + u;o 2 )e r *-0 (5)
r 2 +/3r + u;o 2
:0.
(6)
1636 Simple Harmonic Motion
This is a QUADRATIC EQUATION with solutions
r=|(-/3±V/3 2 -W). (7)
There are therefore three solution regimes depending on
the SIGN of the quantity inside the SQUARE ROOT,
a = /3 2 - W- (8)
The three regimes are
1. a > is POSITIVE: overdamped,
2. a = is Zero: critically damped,
3. a < is Negative: underdamped.
If a periodic (sinusoidal) forcing term is added at angular
frequency a;, the same three solution regimes are again
obtained. Surprisingly, the resulting motion is still pe-
riodic (after an initial transient response, corresponding
to the solution to the unforced case, has died out), but it
has an amplitude different from the forcing amplitude.
The "particular" solution x p (t) to the forced second-
order nonhomogeneous ORDINARY DIFFERENTIAL
Equation
x + p(t)x + q(t)x = A cos(u;£)
due to forcing is given by the equation
(9)
, m „ tt\ f x *(t)9(t) ,. , „ m / xi(t)g(t)
x p (t) = -xi(t) J w{t) dt + x 2 (t)J w ^ dt,
(10)
where x\ and Xi are the homogeneous solutions to the
unforced equation
x + p(t)x + q(t)x =
(ii)
Simple Harmonic Motion
= 2w .
(13)
The above plot shows an underdamped simple harmonic
oscillator with w — 0.3, j3 = 0.15. The solid curve is for
(A, B) = (1, 0), the dot-dashed for (0, 1), and the dotted
for (1/2, 1/2). In this case, a = so the solutions of the
form x = e rt satisfy
r± = \{-(3) = -\0 = -wo-
One of the solutions is therefore
x\ = e .
(14)
(15)
In order to find the other linearly independent solution,
we can make use of the identity
x 2 (t) = Xi(t)
I
e - Jp(t)dt
M*)] 2
dt.
(16)
Since we have p(t) = 2o; , e J * simplifies to e 2wot .
Equation (16) therefore becomes
/-2u> t f
-UJ t
(17)
The general solution is therefore
x = {A + Bt)e-" Qt . (18)
In terms of the constants A and B, the initial values are
3(0) = A
x(0) =B~Au,
(19)
(20)
and W{t) is the WRONSKIAN of these two functions.
Once the sinusoidal case of forcing is solved, it can be
generalized to any periodic function by expressing the
periodic function in a FOURIER SERIES.
A = x(0)
B = aj'(0) + woa!(0).
(21)
(22)
2.5r
0.5
5 10 15 20
Critical damping is a special case of damped simple har-
monic motion in which
For sinusoidally forced simple harmonic motion with
critical damping, the equation of motion is
x + 2ujox + ujq x — A cos(u;£), (23)
and the WRONSKIAN is
W(t) = X\X2 — X\X2
a -"0t/ -u t
UQt. -wot
(e- w °'-a;ote- wot )+woe- wot te
:e- 2t " ot (l-wot + wot) = e- 2tl ' ot .
(24)
a = f3 2 - 4m 2 = 0,
(12)
Simple Harmonic Motion
Simple Harmonic Motion 1637
Plugging this into the equation for the particular solu-
tion gives
X p (t) :
Ot)
+ te
■■Ae
a(u>t)
)t f te-" Qt Acos(u
I e -2u> t
I e -2w t
'* - / te^ * cas(vt) dt + t / e w °*cos(wt)dt
= Ae-** (- a < f"" a , 3 [(a; a +*u; a fa>o-^o a +too a )
L (w 2 +w 2 ) 2
x cos(wi) + u;(tu) 2 — 2u; + tu; 2 ) sin(wt)]
e w o* ^
+ 1 ~— [uj cos(u;i) + u> sin(u>i) >
a> 2 + a> 2 J
(u> 2 +u> 2 ) 2
[(wo 2 — w 2 ) cos(u>i) + 2a;u; sin(a>£)].
(25)
In order to put this in the desired form, note that we
want to equate
C cos + S sin 6 = Q cos(<9 + (5)
= Q (cos cos J - sin sin J) . (26)
This means
C = Q cos 5 = u;o — w
(27)
5 = — Qsin<5 = 2ojo;o,
(28)
so
Q = y/C* + S 2
(29)
* = ta„-i(-§).
(30)
Plugging in,
Q= VW - 2u; 2 to 2 + u; 4 + 4u/ 2 u> 2
= yW + 2o; 2 cc; 2 + a; 4 = u; 2 + w 2 .
J = tan -i(_i^\
\a> 2 - wo 2 /
The solution in the requested form is therefore
A
(31)
(32)
P (w a +u;o a ) 2
(u>o + u; ) cos(o;t + 5)
u; 2 + ^o 2
where J is defined by (32).
cos(u;£ + <5),
(33)
0.6
0.4
0.2
5 10 15 20
Overdamped simple harmonic motion occurs when
2 - W > 0,
a = (3 2 - 4a> 2 > 0.
(34)
(35)
The above plot shows an overdamped simple harmonic
oscillator with uj = 0.3, /3 = 0.075. The solid curve is
for (A,B) = (1,0), the dot-dashed for (0, 1), and the
dotted for (1/2, 1/2). The solutions are
xi — e "
x 2 = e r +',
where
r± = |(-/3 ± V/3 2 - 4w 2 :
The general solution is therefore
x = Ae r - t + Be , '+ t ,
(36)
(37)
(38)
(39)
where A and B are constants. The initial values are
x(0) = A + B
x'(0) = Ar_ +Br + ,
i4 = z(O) +
r+x(0) -x f (0)
B = -
T- — r+
r + a;(0)-a; / (0)
r_ — 7*4-
(40)
(41)
(42)
(43)
For a cosinusoidally forced overdamped oscillator with
forcing function g(t) = Ccos(u>£), the particular solu-
tions are
where
n = i(-/3+ v / /3 2 -4wo 2 )
r 2 = \{-0 - y/0* - 4o> 2 ).
(44)
(45)
(46)
(47)
1638 Simple Harmonic Motion
These give the identities
r\ + r 2 = -0
t\ — V2 = \//3 2 - 4a;o 2
and
(48)
(49)
U>0
= W - (^ - ^H = J[(n + r 2 ) 2 - (r x - r 2 y\
= \[2nr2 + 2rir 2 ] = nrj.
The Wronskian is
(50)
W(t) = yiy 2 - yij/2 = e rit r 2 e r2 ' - r ie ri V 2t
= (ra - ri)e (ri+r » ) *. (51)
The particular solution is
where
Vi9{t) _ C u; sin(u;£) — r 2 cos(u;£)
vi =
V 2
~J W(t) r 2 -ri e p a*(r 2 2 +a; 2 )
_ /" V2g(t) _ C u;sin(u>£) — ri cos(wt)
(52)
(53)
W(t) r-2-ri e r i t (r 2 2 +w 2 )
(54)
Therefore,
Vp = C
c
cos{ujt){r\r2 - w 2 ) - sin(a;t)a;(ri + r 2 )
(n 2 +a; 2 )(r2 2 + cc; 2 )
(ivo 2 — a; 2 ) cos(a;t) + 0oj sin(a;t)
a; 2 /3 2 + (a; 2 -a;o 2 )
V(w a -wo 2 ) 2 +/? a " a
cos(a;£ + J), (55)
" tx; 2 ^ 2 + (^ 2 -a;o 2 ) 2
x cos(u>£ + S)
C
^0W + (v 2 -u;o 2 )
where
Vo; 2 — wo /
(56)
Simple Harmonic Motion
Underdamped simple harmonic motion occurs when
f3 2 -4wo 2 <0, (57)
a = 2 - 4u> 2 < 0.
(58)
The above plot shows an underdamped simple harmonic
oscillator with u = 0.3, = 0.4. The solid curve is for
(A, B) = (1, 0), the dot-dashed for (0, 1), and the dotted
for (1/2, 1/2). Define
7 = V^=! v /4u;o 2 -/? 2 ,
then solutions satisfy
where
r± = |(-)9 ± V^ - 4wo a ),
and are of the form
x = e -(P/ 2±i -i)\
Using the Euler Formula
e ,x = cos x + i sin a;,
this can be rewritten
x = e - (0/2)t [cos (ft) ± isin (ft)] .
(59)
(60)
(61)
(62)
(63)
(64)
We are interested in the real solutions. Since we are deal-
ing here with a linear homogeneous ODE, linear sums
of Linearly Independent solutions are also solutions.
Since we have a sum of such solutions in (64), it follows
that the Imaginary and Real Parts separately satisfy
the ODE and are therefore the solutions we seek. The
constant in front of the sine term is arbitrary, so we can
identify the solutions as
x 1 = e" ( ^ /2)t cos( 7 t)
x 2 -e- ( ^ /2)t sin( 7 t),
so the general solution is
x = e- W2)t [Acos(ft) + Bsin(ft)].
The initial values are
x(0) = A
x'(O) = -±0A + B,f
(65)
(66)
(67)
(68)
(69)
so A and B can be expressed in terms of the initial
conditions by
A = x(0)
B
0x(O) + x'(0)
27
(70)
(71)
Simple Harmonic Motion
Simple Harmonic Motion 1639
For a cosinusoidally forced underdamped oscillator with
forcing function g(t) = Ccos(ujt) y use
(72)
(73)
where
7 =
a =
■P
to obtain
4tt;o'
! -/3 2 :
2
= 47 2
= 2a.
The particular
solutions are
yi(t) = e at cos(7t)
W(«) = e _a " sin( 7 i).
(74)
(75)
(76)
(77)
(78)
The Wronskian is
= e~ at cos(7t)[-ae~ at sin( 7 t) + e" at 7cos(7t)]
- e~ at sin(7t)[-ae~ a * cos(7*) - e~ ai 7sin(7i)]
= e~ 2at {a[- sin(7*) cos(7*) + sin(7t) cos(7*)]
+ 7 [cos 2 (7i)+sin 2 (7t)]}
= je' 2a \ (79)
The particular solution is given by
Vv = -S/i^i +2/2^2,
(80)
where
t>i
«2
" / ^W = 7 / ^ C ° S(7t) C ° S(U;i) * (81)
= / ^ = f / eat cos(7£ ) cos(c ^ *■ (82)
Using computer algebra to perform the algebra, the par-
ticular solution is
y P (t) = C
= C
= c
= C
(a 2 + 7 2 - a; 2 ) cos(a;£) + 2aa; sin(a;£)
[a 2 + (7-a;) 2 ][a 2 + (7 + ^) 2 ]
(wo 2 - c^ 2 ) cos(a;t) + /3u>sin(a;£)
(a 2 +7 2 +^ 2 ) 2 -4 7 2 u> 2
(a;o 2 — uj 2 ) cos(a;t) + f3u sin(a;t)
(a; 2 +a; 2 ) 2 -4i(4a;o 2 -/3 2 V 2
(ujq 2 — a; 2 ) cos(a;t) 4- f3u; sin(u;£)
(wo 2 -u; 2 ) 2 -a; 2 (4a;o 2 -/? 2 )
(u; 2 -w 2 ) 2 -u; 2 (W-/? 2 )
cos(a;£ -f J)
= C- — t^ —* — n ^ cos(ojt + d),
(a; 2 -^ 2 ) 2 -a; 2 (4a;o 2 -/3 2 )
5 = tan x ( — 5 r- J .
(84)
If the forcing function is sinusoidal instead of cosinu-
soidal, then
5' = S - \iz = tan" 1 x - |tt = tan" 1 ( ) , (85)
(86)
5' = tan
2 2
-1 / wo — C^
0a;
Simple Harmonic Motion Quadratic
Perturbation
Given a simple harmonic oscillator with a quadratic per-
turbation ex 2 ,
x + ujq 2 x — aex = 0,
(i)
find the first-order solution using a perturbation
method. Write
x — xq + ex\ + . . . ,
(2)
(3)
X = Xo + €Xi -f . . . .
Plugging (2) and (3) back into (1) gives
(x + exi) + (ojo 2 ico+^o 2 ea:i)-ae(xo + 2xoxie + ..0 = 0.
(4)
Keeping only terms of order e and lower and grouping,
we obtain
(x 4- u; 2 x ) + {xi 4- oj 2 xi - ax 2 )e = 0. (5)
Since this equation must hold for all POWERS of €, we
can separate it into the two differential equations
Xq + O>0 xo =
-.2 2
Xl 4" ^0 %1 = OCXq .
(6)
(7)
The solution to (6) is just
xo = Acos(uJot 4- 4>). (8)
Setting our clock so that <£ — gives
xo = Acos(ojo^). (9)
Plugging this into (7) then gives
xi + a;o 2 xi = olA? cos 2 (o>o£). (10)
The two homogeneous solutions to (10) are
(83)
xi = cos(ojoi)
X2 = sin(o;o£).
(ii)
(12)
1640 Simple Harmonic Motion
Simplex
The particular solution to (10) is therefore given by
•/
X2(t)g(t)
X v {t) = -Xi (t) / ""^r'^' dt + X^t)
where
/Xl
1
(t)9(t)
W{t)
dt,
(13)
(14)
g(t) = aA 2 cos 2 (u;ot),
and the WRONSKIAN is
W = X1X2 — X1X2
= cos(^ot)wo cos(aJoi) — [— wo sin(u;ot)] sin(u;ot)
- wo. (15)
Plugging everything into (13),
x p = ocA
r-
sin(u;ot)cos 2 (wot)
wo
2 ' cos(wot) / ""^ WUL ^ """ v~^v ^
+ sin(«, t) J C °^ UJ0t)
Wo
eft
= < sin(wot) / [1 - sin 2 (u;ot)] cos(uJot) dt
wo [ J
,/si
— cos(wot) / sin(u;ot) cos (wot) dt
}■
Now let
u = sin(u;ot)
du = ujq cos(a;ot) dt
v = cos(ujot)
dv — —wo sin(u;ot) dt.
(16)
(17)
(18)
(19)
(20)
Then
aA 2
Xp ~ u,o 2
aA 2
sin(wo^)
/<.-.■
) dw + cos(wot) v dv
I"
= — j [sin(w t)(l - |u 3 ) + cos(w t)|i> 3 ]
ocA 2
Wo 2
{sin(wot)[l - § sin 3 (wot)]
-f I cos (u; t) cos 3 (wot)}
j2
= — - {|[cos 4 (aj t) - sin 4 (w t)] + sm 2 (u> t)}
= — - {§ [cos 2 (w t) - sin 2 (w t)] + sin 2 (uj t)}
— — ^-|[cos 2 (a;ot) + 2sin 2 (u;ot)]
Wo'
aA 2
3w 2
aA 2
6a; 2
[2 -cos 2 (w t)] :
[3-cos(2a;ot)].
aA 2
3a; 2
{2- i[l + cos(2w t)]}
(21)
Plugging cco(t) and (21) into (2), we obtain the solution
aA 2
x{t) = Acos(u> t) - - — -e[cos(2^ t) - 3]. (22)
OCJo
Simple Harmonic Oscillator
see Simple Harmonic Motion
Simple Interest
Interest which is paid only on the Principal and not
on the additional amount generated by previous INTER-
EST payments. A formula for computing simple interest
is
a(t) = o(0)(l + rt),
where a(t) is the sum of Principal and INTEREST at
time t for a constant interest rate r.
see also Compound Interest, Interest
References
Kellison, S. G. Theory of Interest, 2nd ed. Burr Ridge, IL:
Richard D. Irwin, 1991.
Simple Polygon
A Polygon P is said to be simple (or Jordan) if the
only points of the plane belonging to two EDGES of P are
the Vertices of P. Such a polygon has a well-defined
interior and exterior.
see also POLYGON, REGULAR POLYGON, TWO-EARS
Theorem
References
Toussaint, G. "Anthropomorphic Polygons." Amer. Math.
Monthly 122, 31-35, 1991.
Simple Ring
A Nonzero Ring S whose only (two-sided) Ideals are
S itself and zero. Every commutative simple ring is a
Field. Every simple ring is a Prime Ring.
see also Field, Ideal, Prime Ring, Ring
Simplex
The generalization of a tetrahedral region of space to
n-D. The boundary of a fc-simplex has k + 1 0-faces
(Vertices), k(k + l)/2 1-faces (Edges), and (J+*) i-
faces, where (£) is a Binomial Coefficient.
The simplex in 4-D is a regular TETRAHEDRON ABCD
in which a point E along the fourth dimension through
the center of ABCD is chosen so that EA = EB —
EC — ED = AB. The 4-D simplex has SCHLAFLI SYM-
BOL {3,3,3}.
Simplex
point
line segment
equilateral triangular plane region
tetrahedral region
4-simplex
The only irreducible spherical simplexes generated by
reflection are A n (n > 1), B n {n > 4), C n (n > 2),
D v 2 (p > 5), E 6y E 7 , E&, F 4 , £3, and G 4 . The only
irreducible Euclidean simplexes generated by reflection
Simplex Method
Simpson's Rule 1641
are W 2} Pm {m > 3), Q m {m > 5), R m {m > 3), S m
(m > 4), V 3i T 7 , T 8 , T 9 , and l/ 5 .
The regular simplex in n-D with n > 5 is denoted a n
and has Schlafli Symbol { 3, . . . , 3 }.
see also COMPLEX, CROSS POLYTOPE, EQUILATERAL
Triangle, Line Segment, Measure Polytope,
Nerve, Point, Simplex Method, Tetrahedron
References
Eppstein, D. "Triangles and Simplices." http://www.ics.
uci.edu/-eppstein/junkyard/triangulation.html.
Simplex Method
A method for solving problems in LINEAR PROGRAM-
MING. This method, invented by G. B. Dantzig in 1947,
runs along Edges of the visualization Solid to find the
best answer. In 1970, Klee and Minty constructed ex-
amples in which the simplex method required an expo-
nential number of steps, but such cases seem never to
be encountered in practical applications.
A much more efficient (POLYNOMiAL-time) Algorithm
was found in 1984 by N. Karmarkar. This method goes
through the middle of the Solid and then transforms
and warps. It offers many advantages over the simplex
method (Nemirovsky and Yudin 1994).
see also LINEAR PROGRAMMING
References
Nemirovsky, A. and Yudin, N. Interior- Point Polynom-
ial Methods in Convex Programming. Philadelphia, PA:
SIAM, 1994.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Downhill Simplex Method in Multidi-
mensions" and "Linear Programming and the Simplex
Method." §10.4 and 10.8 in Numerical Recipes in FOR-
TRAN: The Art of Scientific Computing, 2nd ed. Cam-
bridge, England: Cambridge University Press, pp. 402-406
and 423-436, 1992.
Tokhomirov, V. M. "The Evolution of Methods of Convex
Optimization." Amer. Math. Monthly 103, 65-71, 1996.
Simplicial Complex
A simplicial complex is a Space with a Triangula-
tion. Objects in the space made up of only the sim-
plices in the triangulation of the space are called sim-
plicial subcomplexes. When only simplicial complexes
and subcomplexes are considered, defining HOMOLOGY
is particularly easy (and, in fact, combinatorial because
of its finite/counting nature). This kind of homology is
called Simplicial Homology.
see also HOMOLOGY (TOPOLOGY), NERVE, SIMPLICIAL
Homology, Space, Triangulation
Simplicial Homology
The type of HOMOLOGY which results when the spaces
being studied are restricted to Simplicial Complexes
and subcomplexes.
see also SIMPLICIAL COMPLEX
Simplicity
The number of operations needed to effect a Geomet-
ric Construction as determined in Geometrogra-
PHY. If the number of operations of the five GEOMET-
ROGRAPHIC types are denoted mi, rri2, ru, 712, and 713,
respectively, then the simplicity is mi +7712+^1 +^2 +^3
and the symbol m\S\ + rriiSi + n\C\ + 712C2 + n 3 C%.
It is apparently an unsolved problem to determine if a
given Geometric Construction is of smallest possi-
ble simplicity.
see also GEOMETRIC CONSTRUCTION, GEOMETROGRA-
PHY
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, 1976.
Simply Connected
A Connected Domain is said to be simply connected
(also called 1-connected) if any simple closed curve can
be shrunk to a point continuously in the set. If the
domain is CONNECTED but not simply, it is said to be
Multiply Connected.
A SPACE S is simply connected if it is 0-connected and
if every MAP from the 1-SPHERE to S extends continu-
ously to a MAP from the 2-DlSK. In other words, every
loop in the SPACE is contractible.
see also CONNECTED SPACE, MULTIPLY CONNECTED
Simpson's Paradox
It is not necessarily true that averaging the averages of
different populations gives the average of the combined
population.
References
Paulos, J. A. A Mathematician Reads the Newspaper. New
York: BasicBooks, p. 135, 1995.
Simpson's Rule
Let h = (b — a)/n, and assume a function f(x) is defined
at points f(a + kh) = yk for k — 0, . . . , n. Then
/
f(x) dx = ~h(yi + 4y 2 + 2y 3 + 4y 4 + . . .
+2y n -2 + 4j/ n -l + Vn) ~ Rn,
where the remainder is
i? n = ^(6-a) 4 / (4 V)
for some x* G [a, b].
see also Bode's RULE, NEWTON-COTES FORMULAS,
Simpson's 3/8 Rule, Trapezoidal Rule
References
Abramowitz, M. and Stegun, C. A. (Eds,). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 886, 1972.
1642 Simpson's 3/8 Rule
Simpson's 3/8 Rule
Sine Function
f
f(x) dx = |M/i + 3/2 + 3/ 3 + U) - &ft 6 / (4) (0-
see also Bode's Rule, Newton-Cotes Formulas,
Simpson's Rule
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 886, 1972.
Simson Line
The Simson line is the Line containing the feet of the
perpendiculars from a point on the ClRCUMClRCLE of
a TRIANGLE to the sides (or their extensions) of the
Triangle. The Simson line is sometimes known as the
WALLACE-SlMSON LINE, since it does not appear in any
work of Simson (Johnson 1929, p. 137).
The ANGLE between the Simson lines of two points P
and P f is half the ANGLE of the arc PP f . The Simson
line of any Vertex is the Altitude through that Ver-
tex. The Simson line of a point opposite a Vertex is
the corresponding side. If T1T2T3 is the Simson line of a
point T of the ClRCUMClRCLE, then the triangles TT X T 2
and TA2A1 are directly similar.
see also ClRCUMClRCLE
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 40-41 and 43-
45, 1967.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 137-139, 1929.
Sine Function
78
Of. 6
D.4
/0.2
-15 N
s^/0
\-5
-0.2
\ 5 / ^
%^y 15
A function also called the SAMPLING FUNCTION and de-
fined by
( 1 for x =
snic(x) = J ^ otherwisej (1)
where sin a? is the Sine function. Let U(x) be the Rect-
angle Function, then the Fourier Transform of
II(a;) is the sine function
-F[n(x)] = sinc(?rfc).
(2)
The sine function therefore frequently arises in physical
applications such as Fourier transform spectroscopy as
the so-called Instrument Function, which gives the
instrumental response to a DELTA FUNCTION input. Re-
moving the instrument functions from the final spectrum
requires use of some sort of DECONVOLUTION algorithm.
The sine function can be written as a complex INTEGRAL
by noting that
sinc(nir)
__ sin(nx)
nx
1 e tnx -e~
nx 2i
= X \e itx ] n = — /
2inx [ J_n 2n I
J —n
e ixt dt.
(3)
The sine function can also be written as the INFINITE
Product
sin a:
x
00
n-(j)-
(4)
k=l
Definite integrals involving the sine function include
Jo
F
Jo
F
Jo
F
Jo
F
Jo
sinc(x) dx = \k
sine 2 (x) dx
sinc 3 (x) dx = ~tt
sinc 4 (x)dx — \ir
sin.c*(x)dx — ||| 7r.
(5)
(6)
(7)
(8)
(9)
Sine Function
These are all special cases of the amazing general result
.l-c(_ 1 }l(a-b)/2\
f
Jo
Sin X , 77
: — ax =
2 a ~ c (b- 1)!
X E (-l) fe (^)(«-2A:) 6 - 1 [ln(a-2fc)r, (10)
x°
[a/2j-c
where a and b are POSITIVE integers such that a > b > c,
c = a -b ( m od 2), [x\ is the FLOOR FUNCTION, and 0°
is taken to be equal to 1 (Kogan). This spectacular for-
mula simplifies in the special case when n is a POSITIVE
EVEN integer to
J x 2n 2(2n-l)! \ n-1 /' V }
where (£) is an Eulerian Number (Kogan). The so-
lution of the integral can also be written in terms of the
Recurrence Relation for the coefficients
c(a, b) = <
( * ( a_1 "l
for b = 1 or b = 2
(F3lf ( or2)[(a-lMa-2,6-2)
—a • c(a, b — 2)] otherwise
(12)
(Zimmerman).
*i R :
The half-infinite integral of sinc(cc) can be derived using
Contour Integration. In the above figure, consider
the path 7 = 71+712+72+721. Now write z = Re 10 . On
an arc, dz = iRe %e dO and on the x-AxiS, dz = e %e dR.
Write
/oo . p % z
^dx = <5 ^-dx, (13)
-oo J 7
where $ denotes the IMAGINARY POINT. Now define
= lim F ^^p iOR^dO
r x ^o J^ R\e ie
PR2 e iR
+ lim lim / — - dR
Hi-yOH2->oo J R R
+ i im r??vM dx + i im / 1 ^(„^),
R2^OoJ Z R 1^°Jr 2 -R
(14)
Sine Function 1643
where the second and fourth terms use the identities
1 and e 17r = — 1. Simplifying,
,0
lim / <
1= lim / exp{iR 1 e ie )i0d0+ / — dfl
+ lim
R2— yoo
f
Jo
exp(iz)
r 0+ ~-iR
dz +
«/ oo
0~ iH
f-x poo iR no iR
= -/ i0dJ9+ / V djR + + / -^^
Jo Jo+ K J-00 K
(15)
where the third term vanishes by JORDAN'S LEMMA.
Performing the integration of the first term and com-
bining the others yield
I = —iix +
Rearranging gives
f
J — c
■dz = 0.
r
J — O
/oo
S1I1Z
-00
■ dz = 27T,
dz = 7T.
(16)
(17)
(18)
The same result is arrived at using the method of
Residues by noting
/ = 0+§27riRes[/(z)] a= o
(2-0)-
: in [e \z=o
%7X,
(19)
so
3(7) = 7T. (20)
Since the integrand is symmetric, we therefore have
Jo
■ dx = |7T,
(21)
giving the Sine INTEGRAL evaluated at as
si(0) = - n^dx = -±7r. (22)
Jo x
1644 Sinclair's Soap Film Problem
Sine
An interesting property of sinc(:c) is that the set of LO-
CAL EXTREMA of sinc(a;) corresponds to its intersections
with the COSINE function cos(a;), as illustrated above.
see also Fourier Transform, Fourier Trans-
form — Rectangle Function, Instrument Func-
tion, Jinc Function, Sine, Sine Integral
References
Kogan, S. "A Note on Definite Integrals Involving Trigono-
metric Functions." http://www.mathsoft.coia/asolve/
constant /pi/sin/sin. html.
Morrison, K. E. "Cosine Products, Fourier Transforms, and
Random Sums." Amer. Math. Monthly 102, 716-724,
1995.
Sinclair's Soap Film Problem
Find the shape of a soap film (i.e., Minimal Surface)
which will fill two inverted conical FUNNELS facing each
other is known as Sinclair's soap film problem (Bliss
1925, p. 121). The soap film will assume the shape of a
Catenoid.
see also Catenoid, Funnel, Minimal Surface
References
Bliss, G. A. Calculus of Variations. Chicago, IL: Open
Court, pp. 121-122, 1925.
Isenberg, C. The Science of Soap Films and Soap Bubbles.
New York: Dover, p. 81, 1992.
Sinclair, M. E. "On the Minimum Surface of Revolution in
the Case of One Variable End Point." Ann. Math. 8,
177-188, 1907.
Sine
sin 6
Let 6 be an Angle measured counterclockwise from the
a;- Axis along the arc of the UNIT CIRCLE. Then sin0 is
the vertical coordinate of the arc endpoint. As a result of
this definition, the sine function is periodic with period
27r. By the PYTHAGOREAN THEOREM, sin also obeys
the identity
(i)
sin + cos = 1.
Sin z|
The sine function can be defined algebraically by the
infinite sum
(_l)-i
E l-lJ 2n-l
(2n-l)\
(2)
and Infinite Product
sin x = x
n('-^)-
n=l V 7
(3)
It is also given by the Imaginary Part of the complex
exponential
sinx = 3[e ix ]. (4)
The multiplicative inverse of the sine function is the
Cosecant, defined as
1
smx
(5)
Using the results from the EXPONENTIAL SUM FORMU-
LAS
> sin(nx) = S
_n=0
sm(±Nx) i{N _ 1)x/2
sin(§a:)
^^sin^iV-1)]. (6)
Similarly,
\ p n sin(nx) = S
p e
1 —pe~
psmx
1 — 2p cos x + p 2
(7)
1 — 2p cos x + p 2
Other identities include
sin(n0) = 2 cos sin[(n - 1)0] - sin[(n - 2)0] (8)
- / \ f n \ n-l • fn\ n-3 • 3
S1 n(nx) = ycos ™ - ^ cos *s,n *
+ (^Jcos n - 5 xsin 5 a:-... ) (9)
Sine
Sine-Gordon Equation 1645
where (£) is a Binomial Coefficient.
Cvijovic and Klinowski (1995) show that the sum
S„(a) = Y,
sin(2fc + l)a
(2k + 1)"
(10)
has closed form for i/ = 2n + 1,
&n+i(ot)
(-I)" 2n+l P (OL
*.£), (id
4(2n)!
where £?„(a;) is an EULER POLYNOMIAL.
A Continued Fraction representation of sin a; is
X
1 + -
(2-3-x 2 ) +
2-3X 2
(4-5-x 2 ) +
4-5x 2
(6-7-z 2 ) + .
(12)
The value of sin(27r/n) is IRRATIONAL for all n except 4
and 12, for which sin(7r/2) = 1 and sin(?r/6) = 1/2.
The FOURIER Transform of sin(27r/e ;r) is given by
/oo
e' 27rik ° x sm(27Tk x) dx
-oo
= ^i[5{k + ko)-8(k-k )]. (13)
Definite integrals involving sin a; include
f
Jo
/»oo
/ si
/
/■
Jo
sin(;c ) dx = \v2tt
sin( a: 3 )dx=|r(i)
sin(x 4 )da; = - cos(§7r)r(|)
sin(x 5 )dx=i(V5-l)r(|),
(14)
(15)
(16)
(17)
where T(x) is the Gamma Function.
see also ANDREW'S SINE, COSECANT, COSINE, FOURIER
Transform — Sine, Hyperbolic Sine, Sinc Func-
tion, Tangent, Trigonometry
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func-
tions." §4.3 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972.
Cvijovic, D. and Klinowski, J. "Closed-Form Summation of
Some Trigonometric Series." Math, Comput 64, 205-210,
1995.
Hansen, E. R. A Table of Series and Products. Englewood
Cliffs, NJ: Prentice-Hall, 1975.
Project Mathematics! Sines and Cosines, Parts I-III. Video-
tapes (28, 30, and 30 minutes). California Institute of
Technology. Available from the Math. Assoc. Amer.
Spanier, J. and Oldham, K. B. "The Sine sin(x) and Co-
sine cos(x) Functions." Ch. 32 in An Atlas of Functions.
Washington, DC: Hemisphere, pp. 295-310, 1987.
Sine- Gordon Equation
A Partial Differential Equation which appears in
differential geometry and relativistic field theory. Its
name is a pun on its similar form to the KLEIN- GORDON
Equation. The sine-Gordon equation is
vtt — v xx -h sin?; = 0,
(i)
where v u and v xx are PARTIAL DERIVATIVES. The equa-
tion can be transformed by defining
giving
v^ v — sin v.
Traveling wave analysis gives
: - Z0 = \/c 2 -l /
df
v /2[rf-2sin 2 (i/)]
For d = 0,
z-z = ±y/l-c 2 ln[±tan(i/)]
Letting z = £77 then gives
zf" + /' = sin/,
Letting g = e 1 * gives
» 9'
12 2g' - g 2 + 1
+
2z
= 0,
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
which is the third PAINLEVE TRANSCENDENT. Look for
a solution of the form
v(x,t) = 4 tan l
4>(x)
Taking the partial derivatives gives
4>xx = — k <f> -\- m <f> + n
i/, tt = fcV + (m 2 - l)i(> 2 - n 2 ,
(10)
(11)
(12)
which can be solved in terms of Elliptic Functions.
A single SOLITON solution exists with k = n = 0, m > 1:
where
v = 4 tan
=
1
exp 1
' ±x-
-Bt
yi-
-0 2
y/m 2
-1
(13)
(14)
1646 Sine Integral
A two-SOLiTON solution exists with k — 0, m > 1:
sinh({3m,x)
v = 4 tan
/3 cosh(/3mt)
(15)
A SOLITON-antisoliton solution exists with k ^ 0, n = 0,
ttt/ > 1:
tj = —4 tan"
A "breather" solution is
sinh(f3mx)
cosh(mt)
v = —4 tan
m sin(\/l — 7n 2 i)
Vl - m 2 cosh(ma;)
(16)
(17)
References
Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and
Chaos. Cambridge, England: Cambridge University Press,
pp. 199-200, 1990.
Sine Integral
1.5
1
0.5
-10
-5
-0.6
Al
A. 5
5
10
Re [Sinlntegral z
Sinlntegral z|
There are two types of "sine integrals" commonly de-
fined,
f
Jo
si(s)= r^dt
and
si(cc)
/*°° sint
/ ~
J x
dt
= ^N^) -ei(-ix)}
= ^M*^) - e i("^)l
= Si(z) - |tt,
where ei(x) is the EXPONENTIAL INTEGRAL and
ei(a;) = — ei(— x).
(i)
(2)
(3)
(4)
(5)
Sine Integral
Si(x) is the function returned by the Mathematica®
(Wolfram Research, Champaign, IL) command Sin
Integral [x] and displayed above. The half-infinite in-
tegral of the Sinc Function is given by
si(0)
/»oo
I sin x . i
= — / dX = — ~7T.
Jo x
(6)
To compute the integral of a sine function times a power
/
1=1 x 2ti sin (ma?) dx,
use Integration by Parts. Let
u = x 2n dv = sin(rax) dx
du = 2nx n dx v = cos(mz),
771
(7)
(8)
(9)
I = x n cos(mx) +
2n / ,„_!
m J
cos(raa?) da;. (10)
Using Integration by Parts again,
u = x 2n ~ x dv = cos(mx) dx
1
(11)
/
du = (2n - l)a; 2 dx u = — sin(roa?) (12)
m
x 2n sin(raa;) dx = a; n cos(roa;)
2n r 1
m
2n
[ 1 2n-l / \
— x cosimx)
Lro
- / x 2n ~ 2 s'm(mx) dx
-x Zn s'm(mx) H -x 2n x sin(mx)
m ra
2n-2
(2w)(2n-
n-1) /
> 2 J
x n sin(ma:) dx
-x 2n cos(mx) H ^-a? 2n 1 sin(ma?) + . . .
m m^
+
|n)! f o
1 _2n
m
(2n)!
sin(roa;) da?
(2n)!
= — — aT" 1 cos(mas) H ^a? n_ sinfraa?) + . .
2n 2n-l
-r
rr?/
771
2n+l
cos(ma:)
= cos(mz)V(-l) fc+1 - \ 7 7, afc+1 s aw -
v y Z_^ v y (2n - 2fc)!m 2fc+1
fc=0
,2n-2fc+l
(13)
Sine-Tangent Theorem
Letting k' = n — fc, so
/x-sinKx),,
n
= cos(mx)X)(-l) B - fc+1 (2 ifc)!maB _ aik+1
(2n)! 2fe
X
k=0
n-1
(2n)! 2fc+I
+ sin(mx) Vf-D""^ 1 ^ x
fc=0
:(-l) n+1 (2n)!
n
+ sin(mx) >
fc =
(-l) fe+1
(2fc)!m :
(- 1 ) „2*
1^271-2^ + 1*
(2& - 3)!m 2 "- 2fc + 2
General integrals of the form
I(k.
J)= f°
Jo
dx
(14)
(15)
are related to the Sinc Function and can be computed
analytically.
see also Chi, Cosine Integral, Exponential In-
tegral, Nielsen's Spiral, Shi, Sici Spiral, Sinc
Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Sine and Co-
sine Integrals." §5.2 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 231-233, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 342-343, 1985.
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 Cosine and Sine Inte-
grals." Ch. 38 in An Atlas of Functions. Washington, DC:
Hemisphere, pp. 361-372, 1987.
Sine- Tangent Theorem
If
sin a m
Singular Point (Differential Equation) 1647
Singly Even Number
An Even Number of the form 4n + 2 (i.e., an Integer
which is DIVISIBLE by 2 but not by 4). The first few
for n = 0, 1, 2, ... are 2, 6, 10, 14, 18, ... (Sloane's
A016825)
see also Doubly Even Number, Even Number, Odd
Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 30, 1996.
Sloane, N. J. A. Sequence A016825 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Singular Homology
The general type of HOMOLOGY which is what mathe-
maticians generally mean when they say "homology."
Singular homology is a more general version than
Poincare's original SlMPLlClAL HOMOLOGY.
see also HOMOLOGY (TOPOLOGY), SlMPLICIAL HOMO-
LOGY
Singular Point (Algebraic Curve)
A singular point of an ALGEBRAIC CURVE is a point
where the curve has "nasty" behavior such as a CUSP
or a point of self-intersection (when the underlying field
K is taken as the REALS). More formally, a point (a, 6)
on a curve f(x,y) = is singular if the x and y Par-
tial Derivatives of / are both zero at the point (a, 6).
(If the field K is not the Reals or Complex Numbers,
then the PARTIAL DERIVATIVE is computed formally us-
ing the usual rules of CALCULUS.)
Consider the following two examples. For the curve
then
sin/3 n
tan[§(a-/3)] __ m -n
tan[|(a + /5)] ~ m + n'
Sines Law
see Law of Sines
0,
the CUSP at (0, 0) is a singular point. For the curve
x 2 + y 2 = -1,
(0, i) is a nonsingular point and this curve is nonsingular.
see also ALGEBRAIC CURVE, CUSP
Singular Point (Differential Equation)
Consider a second-order ORDINARY Differential
Equation
|/" + P(x) ? /' + 0(x)y = 0.
If P{x) and Q(x) remain FINITE at x = #o, then xq
is called an ORDINARY POINT. If either P[x) or Q(x)
diverges as x — > xo, then xq is called a singular point.
Singular points are further classified as follows:
1. If either P{x) or Q{x) diverges as x — > xq but (x —
xo)P(x) and (x — xo) 2 Q{x) remain FINITE as x — ► #o,
then x = xq is called a REGULAR SINGULAR POINT
(or NONESSENTIAL SINGULARITY).
1648 Singular Point (Function)
Singular Value Decomposition
2. If P(x) diverges more quickly than l/(x — xq), so
(x — xq)P(x) approaches Infinity as x -* xo, or
Q(x) diverges more quickly than l/(x — Xq) 2 Q so
that (x - x ) 2 Q(x) goes to Infinity as x -► x ,
then x is called an IRREGULAR SINGULARITY (or
Essential Singularity).
see also Irregular Singularity, Regular Singu-
lar Point, Singularity
References
Arfken, G. "Singular Points." §8.4 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 451-454, 1985.
Singular Point (Function)
Singular points (also simply called "singularities") are
points z in the Domain of a Function / where /
fails to be Analytic. Isolated Singularities may
be classified as Essential Singularities, Poles, or
Removable Singularities.
Essential Singularities are Poles of Infinite or-
der.
A Pole of order n is a singularity zq of f(z) for which
the function (z — Zo) n f(z) is nonsingular and for which
(z — zo) k f(z) is singular for k = 0, 1, . . . , n — 1.
Removable Singularities are singularities for which
it is possible to assign a Complex Number in such a
way that f(z) becomes ANALYTIC. For example, the
function f(z) — z 2 /z has a Removable Singularity
at 0, since f(z) = z everywhere but 0, and f(z) can be
set equal to at z = 0. REMOVABLE SINGULARITIES are
not POLES.
The function f(z) = csc(l/z) has POLES at 2 =
l/(27m), and a nonisolated singularity at 0.
see also ESSENTIAL SINGULARITY, IRREGULAR SINGU-
LARITY, Ordinary Point, Pole, Regular Singular
Point, Removable Singularity, Singular Point
(Differential Equation)
References
Arfken, G. "Singularities." §7.1 in Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 396-
400, 1985.
Singular Series
p,q
where S v , q is a GAUSSIAN Sum, and T(s) is the GAMMA
Function.
Singular System
A system is singular if the CONDITION NUMBER is IN-
FINITE and ILL-CONDITIONED if it is too large.
see also Condition Number, Ill-Conditioned
Singular Value
A Modulus k r such that
K'(k r )
K(k r )
= V?,
where K(k) is a complete ELLIPTIC INTEGRAL OF THE
First Kind, and K'(k r ) = K(y/\ - k r 2 ). The Ellip-
tic Lambda Function A*(r) gives the value of k r .
Abel (quoted in Whittaker and Watson 1990, p. 525)
proved that if r is an INTEGER, or more generally when-
ever
K'(k) _ a + by/K
K(k) ~ c + dv^'
where a, 6, c, d, and n are INTEGERS, then the MODULUS
k is the Root of an algebraic equation with Integer
Coefficients.
see also ELLIPTIC INTEGRAL SINGULAR VALUE, ELLIP-
TIC Integral of the First Kind, Elliptic Lambda
Function, Modulus (Elliptic Integral)
References
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.
Singular Value Decomposition
An expansion of a Real M x N Matrix by Orthog-
onal Outer Products according to
*-£
SfcUfcVfc,
where s\ > S2 > • • • > 0,
and
K = min{M, N}
ujufc' = vjv k ' = 6 k k>
(1)
(2)
(3)
Here Sij is the Kronecker Delta and A T is the Ma-
trix Transpose.
see also CHOLESKY DECOMPOSITION, LU DECOMPOSI-
TION, QR Decomposition
References
Nash, J. C. "The Singular- Value Decomposition and Its Use
to Solve Least-Squares Problems." Ch. 3 in Compact
Numerical Methods for Computers: Linear Algebra and
Function Minimisation, 2nd ed. Bristol, England: Adam
Hilger, pp. 30-48, 1990.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Singular Value Decomposition." §2.6 in
Numerical Recipes in FORTRAN: The Art of Scientific
Computing, 2nd ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 51-63, 1992.
Singularity
Sinusoidal Spiral Pedal Curve 1649
Singularity
In general, a point at which an equation, surface, etc.,
blows up or becomes DEGENERATE.
see also Essential Singularity, Isolated Singu-
larity, Singular Point (Algebraic Curve), Sin-
gular Point (Differential Equation), Singular
Point (Function), Whitney Singularity
Sinh
see Hyperbolic Sine
Sink (Directed Graph)
Sinusoidal Spiral
A curve of the form
sink
m
A vertex of a Directed Graph with no exiting edges,
also called a Terminal.
see also Directed Graph, Network, Source
Sink (Map)
A stable fixed point of a MAP which, in a dissipative
Dynamical System, is an Attractor.
see also Attractor, Dynamical System
Sinusoidal Projection
An equal AREA Map PROJECTION.
x = (A — Ao)cos0
V = <f>-
The inverse FORMULAS are
<j> = y
A = A +
COS(j>
(1)
(2)
(3)
(4)
References
Snyder, J, P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 243-248, 1987.
: a 71 cos(n$)
with n Rational, which is not a true Spiral. Sinu-
soidal spirals were first studied by Maclaurin. Special
cases are given in the following table.
n
Curve
-2
hyperbola
-1
line
i
2
parabola
1
3
Tschirnhausen cubic
logarithmic spiral
1
3
Cayley sextic
1
2
cardioid
1
circle
2
Bernoulli lemniscate
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, p. 184, 1972.
Lee, X. "Sinusoid." http : //www . best . com/ ~xah/ Special
PlaneCurves_dir/Sinusoid_dir/sinusoid.html.
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 175, 1967.
MacTutor History of Mathematics Archive. "Sinusoidal Spi-
rals." http: //www-groups . dcs . st-and.ac.uk/-history/
Curves/Sinusoidal. html.
Sinusoidal Spiral Inverse Curve
The Inverse Curve of a Sinusoidal Spiral
r = a (l/n) [cos(nt)] 1/n
with Inversion Center at the origin and inversion ra-
dius k is another SINUSOIDAL SPIRAL
■ ka {1/n) [cos(nt)] 1/n .
Sinusoidal Spiral Pedal Curve
The Pedal Curve of a Sinusoidal Spiral
r = ar /r ^[cos(n£)] /n
with Pedal Point at the center is another Sinusoidal
Spiral
x = cos 1+1/n (nt) cos[(n + l)t]
y = cos 1+1/n (nt) sin[(n + 1)<].
1650
Sister Celine 7 s Method
Six-Color Theorem
Sister Celine's Method
A method for finding RECURRENCE RELATIONS for hy-
pergeometric polynomials directly from the series ex-
pansions of the polynomials. The method is effec-
tive and easily implemented, but usually slower than
Zeilberger's Algorithm. Given a sum /(n) =
^2 k F(n 1 k) } the method operates by finding a recur-
rence of the form
^2 5Z aij ( n ) F ( n - j> * - *) = o
i=0 j=0
Site Percolation
site percolation bond percolation
A Percolation which considers the lattice vertices as
the relevant entities (left figure).
see also Bond Percolation, Percolation Theory
by proceeding as follows (Petkovsek et al. 1996, p. 59):
1. Fix trial values of J and J.
2. Assume a recurrence formula of the above form
where a>ij(n) are to be solved for.
3. Divide each term of the assumed recurrence by
F(n,k) and reduce every ratio F(n—j,k — i)/F(n,k)
by simplifying the ratios of its constituent factorials
so that only RATIONAL FUNCTIONS in n and k re-
main.
4. Put the resulting expression over a common DENOM-
INATOR, then collect the numerator as a POLYNOM-
IAL in k.
5. Solve the system of linear equations that results af-
ter setting the coefficients of each power of k in the
Numerator to for the unknown coefficients a^.
6. If no solution results, start again with larger J or J.
Under suitable hypotheses, a "fundamental theorem"
(Verbaten 1974, Wilf and Zeilberger 1992, Petkovsek et
al 1996) guarantees that this algorithm always succeeds
for large enough / and J (which can be estimated in ad-
vance). The theorem also generalizes to multivariate
sums and to q- and multi-g-sums (Wilf and Zeilberger
1992, Petkovsek et al. 1996).
see also Generalized Hypergeometric Function,
Gosper's Algorithm, Hypergeometric Identity,
Hypergeometric Series, Zeilberger's Algorithm
References
Fasenmyer, Sister M. C. Some Generalized Hypergeometric
Polynomials. Ph.D. thesis. University of Michigan, Nov.
1945.
Fasenmyer, Sister M. C. "Some Generalized Hypergeometric
Polynomials." Bull. Amer. Math. Soc. 53, 806-812, 1947.
Fasenmyer, Sister M. C. "A Note on Pure Recurrence Rela-
tions." Amer. Math. Monthly 56, 14-17, 1949.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Sister Celine's
Method." Ch. 4 in A=B. Wellesley, MA: A. K. Peters,
pp. 55-72, 1996.
Rainville, E. D. Chs. 14 and 18 in Special Functions. New
York: Chelsea, 1971.
Verbaten, P. "The Automatic Construction of Pure Recur-
rence Relations." Proc. EUROSAM '74, ACM-SIGSAM
Bull 8, 96-98, 1974.
Wilf, H. S. and Zeilberger, D. "An Algorithmic Proof Theory
for Hypergeometric (Ordinary and "<j") Multisum/lntegral
Identities." Invent. Math. 108, 575-633, 1992.
Siteswap
A siteswap is a sequence encountered in JUGGLING in
which each term is a POSITIVE integer, encoded in BI-
NARY. The transition rule from one term to the next
consists of changing some to 1, subtracting 1, and then
dividing by 2, with the constraint that the DIVISION by
two must be exact. Therefore, if a term is EVEN, the bit
to be changed must be the units bit. In siteswaps, the
number of 1-bits is a constant.
Each transition is characterized by the bit position of
the toggled bit (denoted here by the numeral on top of
the arrow). For example,
111-^10011-^1011-^10101-^1011-^111
•A 100011-^10101 -^1110-^111-^1011 . . .
The second term is given from the first as follows:
000111 with bit 5 flipped becomes 100111, or 39. Sub-
tract 1 to obtain 38 and divide by two to obtain 19,
which is 10011.
see also JUGGLING
References
Juggling Information Service. "Siteswaps." http://www.
juggling.org/help/siteswap.
Six-Color Theorem
To color any map on the SPHERE or the PLANE requires
at most six-colors. This number can be easily be reduced
to five, and the Four-Color Theorem demonstrates
that the NECESSARY number is, in fact, four.
see also FOUR-COLOR THEOREM, HEAWOOD CONJEC-
TURE, Map Coloring
References
Franklin, P. "A Six Colour Problem." J. Math. Phys. 13,
363-369, 1934.
Hoffman, I. and Soifer, A. "Another Six-Coloring of the
Plane." Disc. Math. 150, 427-429, 1996.
Saaty, T. L. and Kainen, P. C. The Four- Color Problem:
Assaults and Conquest. New York: Dover, 1986.
Skein Relationship
Skew Symmetric Matrix 1651
Skein Relationship
A relationship between Knot Polynomials for links
in different orientations (denoted below as L+, Lo, and
L-). J. H. Conway was the first to realize that the
Alexander Polynomial could be denned by a rela-
tionship of this type.
N,
\
)(
/
.X
K
-o
see also Alexander Polynomial, HOMFLY Poly-
nomial, Signature (Knot)
Skeleton
The GRAPH obtained by collapsing a POLYHEDRON into
the PLANE. The number of topologically distinct skele-
tons N(n) with n Vertices is given in the following
table.
n
N(n)
4
5
6
1
2
7
References
Gardner, M. Martin Gardner's New Mathematical Diver-
sions from Scientific American. New York: Simon and
Schuster, p. 233, 1966.
Skeleton Division
A LONG Division in which most or all of the digits
are replaced by a symbol (usually asterisks) to form a
Cryptarithm.
see also CRYPTARITHM
Skew Conic
Also known as a Gauche Conic, Space Conic,
Twisted Conic, or Cubical Conic Section. A
third-order Space Curve having up to three points in
common with a plane and having three points in com-
mon with the plane at infinity. A skew cubic is deter-
mined by six points, with no four of them COPLANAR.
A line is met by up to four tangents to a skew cubic.
A line joining two points of a skew cubic (REAL or con-
jugate imaginary) is called a SECANT of the curve, and
a line having one point in common with the curve is
called a SEMISECANT or TRANSVERSAL. Depending on
the nature of the roots, the skew conic is classified as
follows:
1. The three Roots are Real and distinct (CUBICAL
Hyperbola).
2. One root is Real and the other two are COMPLEX
Conjugates (Cubical Ellipse).
3. Two of the Roots coincide (Cubical Parabolic
Hyperbola).
4. All three Roots coincide (Cubical Parabola).
See also CONIC SECTION, CUBICAL ELLIPSE, CUBI-
CAL Hyperbola, Cubical Parabola, Cubical Par-
abolic Hyperbola
Skew Field
A Field in which the commutativity of multiplication
is not required, more commonly called a DIVISION AL-
GEBRA.
see also Division Algebra, Field
Skew Lines
Two or more Lines which have no intersections but are
not Parallel, also called Agonic Lines. Since two
Lines in the Plane must intersect or be Parallel,
skew lines can exist only in three or more DIMENSIONS.
see also Gallucci's Theorem, Regulus
Skew Polyomino
see also L-POLYOMINO, SQUARE
Straight Polyomino, T-Polyomino
Polyomino,
Skew Quadrilateral
A four-sided QUADRILATERAL not contained in a plane.
The problem of finding the minimum bounding surface
of a skew quadrilateral was solved by Schwarz (1890) in
terms of ABELIAN INTEGRALS and has the shape of a
Saddle. It is given by solving
(1 + /y 2 )/« - 2/x/y/xy + (1 + f^fyy = 0.
see also QUADRILATERAL
References
Isenberg, C. The Science of Soap Films and Soap Bubbles.
New York: Dover, p. 81, 1992.
Forsyth, A. R. Calculus of Variations. New York: Dover,
p. 503, 1960.
Schwarz, H. A. Gesammelte Mathematische Abhandlungen,
2nd ed. New York: Chelsea.
Skew Symmetric Matrix
A Matrix A where
A T - -A,
with A T denoting the MATRIX TRANSPOSE.
see also MATRIX TRANSPOSE, SYMMETRIC MATRIX
1652
Skewes Number
Skewes Number
The Skewes number (or first Skewes number) is the num-
ber Ski above which n(n) < Li(n) must fail (assuming
that the RlEMANN HYPOTHESIS is true), where ir(n) is
the Prime Counting Function and Li(n) is the Log-
arithmic Integral.
Ski = e e
10 1
,27/4
The Skewes number has since been reduced to e
8.185 x 10 370 by te Riele (1987), although Conway and
Guy (1996) claim that the best current limit is 10 1167 .
In 1914, Littlewood proved that the inequality must, in
fact, fail infinitely often.
The second Skewes number Sk2 is the number above
which 7r(n) < Li(n) must fail (assuming that the Rie-
MANN Hypothesis is false). It is much larger than the
Skewes number Ski ,
Sk 2 = 10 1
Sklar's Theorem
The Pearson Mode Skewness is defined by
(4)
Pearson's Skewness Coefficients are defined by
3 [mean] — [mode]
[mean] — [mode]
and
3 [mean] — [median]
(5)
(6)
The Bowley Skewness (also known as Quartile
Skewness Coefficient) is defined by
(Qa - Qa) - (Qa - Qi) _Qi~ 2Q 2 + Qs
Q9-Q1
Qa-Qi
, (7)
where the Qs denote the Interquartile Ranges. The
Momental Skewness is
2<r 3 '
(8)
see also Graham's Number, Riemann Hypothesis
References
Asimov, L "Skewered!" Of Matters Great and Small. New-
York: Ace Books, 1976. Originally published in Magazine
of Fantasy and Science Fiction, Nov. 1974.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 63, 1987.
Boas, R. P. "The Skewes Number." In Mathematical Plums
(Ed. R. Honsberger). Washington, DC: Math. Assoc.
Amer., 1979.
Conway, J. H. and Guy, R. K. The Book of Numbers, New
York: Springer- Verlag, p. 61, 1996.
Lehman, R. S. "On the Difference n(x) - \i(x)" Acta Arith.
11, 397-410, 1966.
te Riele, H. J. J. "On the Sign of the Difference tt(x) -li(z)."
Math. Comput. 48, 323-328, 1987.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, p. 30, 1991.
Skewness
The degree of asymmetry of a distribution. If the distri-
bution has a longer tail less than the maximum, the
function has NEGATIVE skewness. Otherwise, it has
POSITIVE skewness. Several types of skewness are de-
fined. The Fisher Skewness is defined by
An Estimator for the Fisher Skewness 71 is
7i
M3 _ M3
M2 3 / 2 cr 3 '
(1)
where fi3 is the third Moment, and ^2 1 ^ 2 = & is the
Standard Deviation. The Pearson Skewness is
defined by
ft -(£)•-*•. w
The Momental Skewness is defined by
J™) = I
7i.
(3)
9i
k 2 ^ 2 '
(9)
where the ks are /c-Statistics. The Standard Devi-
ation of g\ is
(10)
6
N
see also Bowley Skewness, Fisher Skewness,
Gamma Statistic, Kurtosis, Mean, Momental
Skewness, Pearson Skewness, Standard Devia-
tion
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 928, 1972.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Moments of a Distribution: Mean, Vari-
ance, Skewness, and So Forth." §14.1 in Numerical Recipes
in FORTRAN: The Art of Scientific Computing, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 604-609, 1992.
Sklar's Theorem
Let if be a 2-D distribution function with marginal dis-
tribution functions F and G. Then there exists a COP-
ULA C such that
H(x,y) = C(F(x),G(y)).
Conversely, for any univariate distribution functions F
and G and any COPULA C, the function H is a two-
dimensional distribution function with marginals F and
G. Furthermore, if F and G are continuous, then C is
unique.
Skolem-Mahler-Lerch Theorem
Slutzky-Yule Effect 1653
Skolem-Mahler-Lerch Theorem
If {a ,ai,...} is a Recurrence Sequence, then the
set of all k such that au = is the union of a finite
(possibly Empty) set and a finite number (possibly zero)
of full arithmetical progressions, where a full arithmetic
progression is a set of the form {r, r + d, r 4- 2d, . . .} with
re [0,d).
References
Myerson, G. and van der Poorten, A. J. "Some Problems
Concerning Recurrence Sequences." Amer. Math. Monthly
102, 698-705, 1995.
Skolem Paradox
Even though ARITHMETIC is uncountable, it possesses
a countable "model."
Skolem Sequence
A Skolem sequence of order n is a sequence S =
{si, S2, . . - , S2n} of 2n integers such that
1. For every A; £ {1,2,. ..,n}, there exist exactly two
elements Si,Sj G S such that Si — Sj = k, and
2. If St = Sj = k with i < j, then j — i = k.
References
Colbourn, C. J. and Dinitz, J. H. (Eds.) "Skolem Sequences."
Ch. 43 in CRC Handbook of Combinatorial Designs. Boca
Raton, FL: CRC Press, pp. 457-461, 1996.
Slide Rule
A mechanical device consisting of a sliding portion and a
fixed case, each marked with logarithmic axes. By lining
up the ticks, it is possible to do MULTIPLICATION by tak-
ing advantage of the additive property of LOGARITHMS.
More complicated slide rules also allow the extraction of
roots and computation of trigonometric functions. The
development of the desk calculator (and subsequently
pocket calculator) rendered slide rules largely obsolete
beginning in the 1960s.
see also ABACUS, RULER, STRAIGHTEDGE
References
Electronic Teaching Laboratories. Simplify Math: Learn to
Use the Slide Rule. New Augusta, IN: Editors and Engi-
neers, 1966.
Saffold, R. The Slide Rule. Garden City, NY: Doubleday,
1962.
Slightly Defective Number
see Almost Perfect Number
Slightly Excessive Number
see Quasiperfect Number
Slip Knot
see Running Knot
Slant Height
The height of an object (such as a Cone) measured
along a side from the edge of the base to the apex.
Slice Knot
A Knot K in
a Disk A 2 in 1
= dD is a slice knot if it bounds
which has a TUBULAR NEIGHBOR-
HOOD A 2 x© whose intersection with § is a Tubular
Neighborhood K xB 2 for K.
Every Ribbon KNOT is a slice knot, and it is conjectured
that every slice knot is a Ribbon KNOT.
see also Ribbon Knot, Tubular Neighborhood
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 218, 1976.
Slide Move
slide -"^ |
The Reidemeister Move of type III.
see also REIDEMEISTER MOVES
Slope
A quantity which gives the inclination of a curve or line
with respect to another curve or line. For a LINE in the
Plane making an Angle with the x-Axis, the Slope
m is a constant given by
Ay
m ~ — — = tantf,
Ax
where Ax and Ay are changes in the two coordinates
over some distance. It is meaningless to talk about the
slope in 3-D unless the slope with respect to what is spec-
ified.
Slothouber-Graatsma Puzzle
Assemble six 1 x 2 x 2 blocks and three lxlxl blocks
into a 3 x 3 x 3 CUBE.
see also Box-Packing Theorem, Conway Puzzle,
Cube Dissection, de Bruijn's Theorem, Klarner's
Theorem, Polycube
References
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 75-77, 1976.
Slutzky-Yule Effect
A Moving Average may generate an irregular oscilla-
tion even if none exists in the original data.
see also Moving Average
1654
Sluze Pearls
Small Dodecahemicosacron
Sluze Pearls
see Pearls of Sluze
Smale-Hirsch Theorem
The Space of Immersions of a Manifold in another
Manifold is Homotopically equivalent to the space
of bundle injections from the Tangent Space of the
first to the TANGENT BUNDLE of the second.
see also HOMOTOPY, IMMERSION, MANIFOLD, TAN-
GENT Bundle, Tangent Space
Smale Horseshoe Map
The basic topological operations for constructing an At-
TRACTOR consist of stretching (which gives sensitivity to
initial conditions) and folding (which gives the attrac-
tion). Since trajectories in Phase Space cannot cross,
the repeated stretching and folding operations result in
an object of great topological complexity.
The Smale horseshoe map consists of a sequence of op-
erations on the unit square. First, stretch by a factor of
2 in the x direction, then compress by 2a in the y direc-
tion. Then, fold the rectangle and fit it back into the
square. Repeating this generates the horseshoe at trac-
tor. If one looks at a cross-section of the final structure,
it is seen to correspond to a Cantor Set.
see also Attractor, Cantor Set
References
Gleick, J. Chaos: Making a New Science. New York: Pen-
guin, pp. 50-51, 1988.
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, p. 77, 1990.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, 1989.
Small Cubicuboctahedron
Small Ditrigonal Dodecacronic
Hexecontahedron
The Dual Polyhedron of the Small Ditrigonal
Dodecicosidodecahedron.
Small Ditrigonal Dodecicosidodecahedron
The Uniform Polyhedron Uaz whose Dual Polyhe-
dron is the Small Ditrigonal Dodecacronic Hex-
econtahedron. It has Wythoff Symbol 3 § | 5. Its
faces are 20{3} + 12{|} + 12{10}. Its CIRCUMRADIUS
with a = 1 is
R= |\/34 + 6V5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 126-127, 1971.
Small Ditrigonal Icosidodecahedron
The Uniform Polyhedron U30 whose Dual Polyhe-
dron is the Small Triambic Icosahedron. It has
Wythoff Symbol 3 | 3 f . Its faces are 20{3} + 12{§ }.
A Faceted version is the Ditrigonal Dodecadodec-
AHEDRON. Its CIRCUMRADIUS with a — 1 is
R=±yfi.
Uniform Polyhedron Ui$ whose Dual Polyhedron
is the Small Hexacronic Icositetrahedron. It has
Wythoff Symbol § 4 1 4. Its faces are 8{3} + 6{4} +
6{8}. The CIRCUMRADIUS for the solid with unit edge
length is
R = fv / 5 + 2v / 2.
FACETED versions include the GREAT RHOMBICUB-
octahedron (Uniform) and Small Rhombihexahe-
dron.
References
Wenninger, M. J, Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 104-105, 1971.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 106-107, 1971.
Small Dodecacronic Hexecontahedron
The Dual Polyhedron of the Small Dodecicosido-
decahedron.
Small Dodecahemicosacron
The Dual Polyhedron of the Small Dodecahemi-
cosahedron.
Small Dodecahemicosahedron
Small Dodecahemicosahedron
Small Hexagrammic Hexecontahedron 1655
Small Dodecicosahedron
The Uniform Polyhedron C/ 62 whose Dual Poly-
hedron is the Small Dodecahemicosacron. It has
Wythoff Symbol § § | 3. Its faces are 10{6} + 12{§}.
It is a Faceted version of the Icosidodecahedron.
Its ClRCUMRADIUS with unit edge length is
J2= 1.
The Uniform Polyhedron Ubo whose Dual Polyhe-
dron is the Small Dodecicosacron. It has Wyth-
3
. Its faces are 20{6} + 12{10}. Its
ClRCUMRADIUS with a = 1 is
R= Wte + 6y/E.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 155, 1971.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 141-142, 1971.
Small Dodecahemidodecacron
The Dual Polyhedron of the Small Dodecahemi-
dodecahedron.
Small Dodecahemidodecahedron
Small Dodecicosidodecahedron
The Uniform Polyhedron U51 whose Dual Polyhe-
dron is the Small Dodecahemidodecacron. It has
3
Wythoff Symbol 25 f . Its faces are 30{4} + 12{10}.
2
Its ClRCUMRADIUS with a = 1 is
The Uniform Polyhedron U 33 whose Dual Poly-
hedron is the Small Dodecacronic Hexecontahe-
dron. It has Wythoff Symbol § 5 | 5. Its faces are
20{3} + 12{5} + 12{10}. It is a FACETED version of
the Small Rhombicosidodecahedron. Its Circum-
radius with a = 1 is
J2= i\/ll + 4v / 5.
R= i\/ll+4v / 5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 113-114, 1971.
Small Dodecicosacron
The Dual Polyhedron of the Small Dodecicosa-
hedron.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 110-111, 1971.
Small Hexacronic Icositetrahedron
The Dual Polyhedron of the Small Cubicubocta-
hedron.
Small Hexagonal Hexecontahedron
The Dual Polyhedron of the Small Snub Icosicosi-
dodecahedron.
Small Hexagrammic Hexecontahedron
The Dual Polyhedron of the Small Retrosnub
Icosicosidodecahedron.
1656
Small Icosacronic Hexecontahedron
Small Rhombicosidodecahedron
Small Icosacronic Hexecontahedron
The Dual Polyhedron of the Small Icosicosido-
DECAHEDRON.
Small Icosicosidodecahedron
The Uniform Polyhedron U31 whose Dual Poly-
hedron is the Small Icosacronic Hexecontahe-
dron. It has Wythoff Symbol f 5|5. Its faces are
12{5} 4- 6{10}. Its Circumradius with a = 1 is
R = <j> = 1(1 + V5),
where </> is the GOLDEN RATIO.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 143, 1971.
Small Icosihemidodecacron
The Dual Polyhedron of the Small Icosihemido-
decahedron.
Small Icosihemidodecahedron
The Uniform Polyhedron U^ 9 whose Dual Poly-
hedron is the Small Icosihemidodecacron. It has
Wythoff Symbol § 3 | 5. Its faces are 20{3} + 6{10}.
It is a Faceted version of the Icosidodecahedron.
Its Circumradius with a = 1 is
R^4>= £(1 + V5).
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p, 140, 1971.
Small Inverted Retrosnub
Icosicosidodecahedron
see Small Retrosnub Icosicosidodecahedron
Small Multiple Method
An algorithm for computing a UNIT FRACTION.
Small Number
Guy's "Strong Law of Small Numbers" states that
there aren't enough small numbers to meet the many-
demands made of them. Guy (1988) also gives several
interesting and misleading facts about small numbers:
1. 10% of the first 100 numbers are SQUARE NUMBERS,
2. A Quarter of the numbers < 100 are Primes.
3. All numbers less than 10, except for 6, are PRIME
Powers.
4. Half the numbers less than 10 are FIBONACCI NUM-
BERS.
see also Large Number, Strong Law of Small
Numbers
References
Guy, R. K. "The Strong Law of Small Numbers." Amer.
Math. Monthly 95, 697-712, 1988.
Small Retrosnub Icosicosidodecahedron
The Uniform Polyhedron £/ 72 also called the
Small Inverted Retrosnub Icosicosidodecahe-
dron whose Dual Polyhedron is the Small Hexa-
grammic Hexecontahedron. It has Wythoff Sym-
bol I § § §. Its faces are 100(3} + 12{f }. It has CIR-
CUMRADIUS with a = 1
R= 1^/13 + 3^/5- \/l02 + 46 V5
« 0.580694800133921.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 194-199, 1971.
Small Rhombicosidodecahedron
Small Rhombicuboctahedron
Small Rhombihexahedron 1657
An Archimedean Solid whose Dual Polyhedron is
the Deltoidal Hexecontahedron. It has Schlafli
Symbol r{j?}. It is also Uniform Polyhedron
C/27 with Wythoff Symbol 35 1 2. Its faces are
20{3} + 30{4} + 12{5}. The Small Dodecicosido-
decahedron and Small Rhombidodecahedron are
FACETED versions. The Inradius, Midradius, and
Circumradius for a = 1 are
r = £(15 + 2^5)^11 + 4^/5 = 2.12099 . . .
p= ^10 + 4^ = 2.17625...
R=\ V / H + 4v / 5 = 2.23295 ....
see also Great Rhombicosidodecahedron (Archi-
medean), Great Rhombicosidodecahedron (Uni-
form)
Small Rhombicuboctahedron
A
A
A
A
V
V
V
V
An Archimedean Solid also (inappropriately) called
the Truncated Icosidodecahedron. This name is
inappropriate since truncation would yield rectangu-
lar instead of square faces. Its DUAL POLYHEDRON
is the Deltoidal Icositetrahedron, also called the
Trapezoidal Icositetrahedron. It has Schlafli
Symbol r{^}. It is also Uniform Polyhedron
U10 and has Wythoff Symbol 3 4 1 2. Its Inradius,
Midradius, and Circumradius for a — 1 are
r= ^(6 + \/2)\/5 + 2\/2 = 1.22026...
p = \ ^4 + 2^2 = 1.30656 . . .
R=\ a/5 + 2\/2 = 1.39897 ....
A version in which the top and bottom halves are rotated
with respect to each other is known as the ELONGATED
Square Gyrobicupola.
see also Elongated Square Gyrobicupola, Great
Rhombicuboctahedron (Archimedean), Great
Rhombicuboctahedron (Uniform)
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 137-
138, 1987.
Small Rhombidodecacron
The Dual Polyhedron of the Small Rhombidodec-
ahedron.
Small Rhombidodecahedron
The Uniform Polyhedron U39 whose Dual Poly-
hedron is the Small Rhombidodecacron. It has
3
Wythoff Symbol 2 5 § . Its faces are 30{4} + 12{10}.
2
It is a Faceted version of the Small Rhombicosido-
decahedron. Its Circumradius with a = 1 is
R= |\/ll + 4\/5.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 113-114, 1971.
Small Rhombihexacron
The Dual Polyhedron of the Small Rhombihexa-
hedron.
Small Rhombihexahedron
The Uniform Polyhedron ?7 18 whose Dual Polyhe-
dron is the Small Rhombihexacron. It has Wyth-
3
f . Its faces are 12{4} + 6{8}. It is
2
1658
Small Snub Icosicosidodecahedron
Small Stellated Truncated Dodecahedron
a Faceted version of the Small Rhombicuboctahe-
DRON. Its ClRCUMRADIUS with a = 1 is
R= |V5 + 2V2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 134, 1971.
Small Snub Icosicosidodecahedron
The Uniform Polyhedron C/32 whose Dual Polyhe-
dron is the Small Hexagonal Hexecontahedron.
It has Wythoff Symbol | 33 f (Har'El 1993 gives the
symbol as , ^
ClRCUMRADIUS for a
§33.) Its faces are 100{3} + 12{§}. Its
1 is
R
Y 13 + 3^5 + \/l02 + 46^
= 1.4581903307387.
References
Har'El, Z. "Uniform Solution for Uniform Polyhedra." Ge-
ometriae Dedicata 47, 57-110, 1993.
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 172-173, 1971.
Small Stellapentakis Dodecahedron
The Dual Polyhedron of the Truncated Great
Dodecahedron.
Small Stellated Dodecahedron
One of the Kepler-Poinsot Solids whose Dual
Polyhedron is the Great Dodecahedron. Its
Schlafli Symbol is {§,5}. It is also Uniform Poly-
hedron U34 and has Wythoff Symbol 5 1 2 f . It was
originally called the URCHIN by Kepler. It is composed
of 12 Pentagrammic faces. Its faces are 12{§}. The
easiest way to construct it is to build twelve pentagonal
Pyramids
and attach them to the faces of a DODECAHEDRON.
The ClRCUMRADIUS of the small stellated dodecahedron
with a = 1 is
R = |5 1/4 0" 1/2 = i5 1/4 ^2(>/5-l).
see also GREAT DODECAHEDRON, GREAT ICOSAHE-
dron, Great Stellated Dodecahedron, Kepler-
Poinsot Solid
References
Fischer, G. (Ed.). Plate 103 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 102, 1986.
Rawles, B. Sacred Geometry Design Sourcebook: Universal
Dimensional Patterns. Nevada City, CA: Elysian Pub.,
p. 219, 1997.
Small Stellated Triacontahedron
see Medial Rhombic Triacontahedron
Small Stellated Truncated Dodecahedron
The Uniform Polyhedron C/ 5 8 also called the
Quasitruncated Small Stellated Dodecahedron
whose Dual Polyhedron is the Great Pentakis Do-
decahedron. It has Schlafli Symbol t'{§,5} and
Wythoff Symbol 25 | §. Its faces are 12{5} + 12{^}.
Its ClRCUMRADIUS with a — 1 is
r= Wm-ioVE-
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 151, 1971.
Small Triakis Octahedron
Small Triakis Octahedron
Smarandache Constants
1659
The Dual Polyhedron of the Truncated Cube.
see also GREAT TRIAKIS OCTAHEDRON
Small Triambic Icosahedron
The Dual Polyhedron of the Small Ditrigonal
ICOSIDODECAHEDRON.
Small World Problem
The small world problem asks for the probability that
two people picked at random have at least one acquain-
tance in common.
see also Birthday Problem
Smarandache Ceil Function
A SMARANDACHE-like function which is defined where
Sk(n) is denned as the smallest integer for which
n\Sk(n) k , The Smarandache Sk(n) function can there-
fore be obtained by replacing any factors which are fcth
powers in n by their k roots. The functions Sk{n) for
k = 2, 3, ..., 6 for values such that Sk(n) ^ n are
tabulated by Begay (1997).
Si(n) = n, so the first few values of Si(n) are 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, . . .
(Sloane's A000027). The first few values of S 2 {n) are 1,
2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19,
10, . . . (Sloane's A019554) The first few values of 5 3 (n)
are 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 4, 17,
6, 19, 10, . . . (Sloane's A019555) The first few values of
S 4 (n) are 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2,
17, 6, 19, 10, . . . (Sloane's A007947).
see also PSEUDOSMARANDACHE FUNCTION, SMARAN-
DACHE Function, Smarandache-Kurepa Func-
tion, SMARANDACHE NEAR-TO-PRIMORIAL FUNC-
TION, Smarandache Sequences, Smarandache-
Wagstaff Function, Smarandache Function
References
Begay, A. "Smarandache Ceil Functions," Bull Pure Appl
Sci. 16E, 227-229, 1997.
"Functions in Number Theory." http://www.gallup.unm.
edu/~smarandache/FUNCTl . TXT.
Sloane, N. J. A. Sequences A007947, A019554, A019555, and
A0472/M000027 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
Smarandache, F. Collected Papers, Vol. 2. Kishinev,
Moldova: Kishinev University Press, 1997.
Smarandache, F. Only Problems, Not Solutions!, J^th ed.
Phoenix, AZ: Xiquan, 1993.
Smarandache Constants
The first Smarandache constant is denned as
* - E jsm > 1 - 093111 '
71 = 2
where S(n) is the SMARANDACHE FUNCTION. Cojo-
caru and Cojocaru (1996a) prove that Si exists and is
bounded by 0.717 < Si < 1.253. The lower limit given
above is obtained by taking 40,000 terms of the sum.
Cojocaru and Cojocaru (1996b) prove that the second
Smarandache constant
oo . .
S 2 = \^ -^ ~ 1.71400629359162
n=2
is an Irrational Number.
Cojocaru and Cojocaru (1996c) prove that the series
E
m 2 s(i)
0.719960700043708
converges to a number 0.71 < S3 < 1.01, and that
54(a) "5n£^
converges for a fixed REAL Number a > 1. The values
for small a are
S 4 (l) » 1.72875760530223
S 4 (2) w 4.50251200619297
S 4 (3) « 13.0111441949445
5 4 (4) « 42.4818449849626
S 4 (5) sa 158.105463729329.
Sandor (1997) shows that the series
~ (-lr-'Sjn)
s 5 = E
1660
Smarandache Constants
Smarandache Function
converges to an IRRATIONAL. Burton (1995) and Du-
mitrescu and Seleacu (1996) show that the series
= V-
S(n)
(n+l)\
converges. Dumitrescu and Seleacu (1996) show that
the series
S(n)
and
07 ~ Z^( n + r )\
2-^ (n — r)!
converge for r a natural number (which must be nonzero
in the latter case). Dumitrescu and Seleacu (1996) show
that
59 ~ Z^ V-n 5(i)
n=2 £-^i=2 i\
converges. Burton (1995) and Dumitrescu and Seleacu
(1996) show that the series
and
*ȣ
f? 2 [S(n)]"y/[S{n) + l]\
converge for a > 1.
see also Smarandache Function
References
Burton, E. "On Some Series Involving the Smarandache
Function." Smarandache Notions J. 6, 13—15, 1995.
Burton, E. "On Some Convergent Series." Smarandache No-
tions J. 7, 7-9, 1996.
Cojocaru, I. and Cojocaru, S. "The First Constant of
Smarandache." Smarandache Notions J. 7, 116—118,
1996a.
Cojocaru, I. and Cojocaru, S. "The Second Constant of
Smarandache." Smarandache Notions J. 7, 119-120,
1996b.
Cojocaru, I. and Cojocaru, S. "The Third and Fourth Con-
stants of Smarandache." Smarandache Notions J. 7, 121—
126, 1996c.
"Constants Involving the Smarandache Function." http://
www . gallup .unm.edu/-smarandache/CONSTANT . TXT.
Dumitrescu, C. and Seleacu, V. "Numerical Series Involving
the Function 5." The Smarandache Function. Vail: Erhus
University Press, pp. 48-61, 1996.
Ibstedt, H. Surfing on the Ocean of Numbers — A Few
Smarandache Notions and Similar Topics. Lupton, AZ:
Erhus University Press, pp. 27-30, 1997.
Sandor, J. 'On The Irrationality Of Certain Alternative
Smarandache Series." Smarandache Notions J. 8, 143—
144, 1997.
Smarandache, F. Collected Papers, Vol, 1. Bucharest, Ro-
mania: Tempus, 1996.
Smarandache, F. Collected Papers, Vol. 2. Kishinev,
Moldova: Kishinev University Press, 1997.
Smarandache Function
500
400
200
100
200
300
400
500
The smallest value S(n) for a given n for which n\S(n)\
(n divides S(n) Factorial). For example, the number
8 does not divide 1!, 2!, 3!, but does divide 4! = 4-3-2-1 =
8 • 3, so 5(8) = 4. For a PRIME p, S(p) = p, and for an
Even Perfect Number r, S(r) is Prime (Ashbacher
1997).
The Smarandache numbers for n = 1, 2, . . . are 1, 2, 3,
4, 5, 3, 7, 4, 6, 5, 11, ... (Sloane's A002034). Letting
a(n) denote the smallest value of n for which S(n) = 1,
2, ..., then a(n) is given by 1, 2, 3, 4, 5, 9, 7, 32,
27, 25, 11, 243, ... (Sloane's A046021). Some values
of S(n) first occur only for very large n, for example,
5(59,049) = 24, 5(177,147) = 27, 5(134,217,728) =
30, 5(43,046,721) = 36, and 5(9,765,625) = 45.
D. Wilson points out that if we let
I(n,p)
•V(n,p)
p-1
be the power of the Prime p in n!, where E(n,p) is the
sum of the base-p digits of n, then it follows that
a(n) = minp J(n - 1 ' p)+1 ,
where the minimum is taken over the PRIMES p dividing
n. This minimum appears to always be achieved when
p is the Greatest Prime Factor of n.
The incrementally largest values of S(n) are 1, 2, 3, 4, 5,
7, 11, 13, 17, 19, 23, 29, ... (Sloane's A046022), which
occur for n = 1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... ,
i.e., the values where S(n) = n.
Tutescu (1996) conjectures that the DlOPHANTlNE
Equation S(n) — S(n + 1) has no solution.
see also FACTORIAL, GREATEST PRIME FACTOR, PSEU-
dosmarandache Function, Smarandache Ceil
Function, Smarandache Constants, Smaran-
dache-Kurepa Function, Smarandache Near-
to-Primorial Function, Smarandache- Wagstaff
Function
References
Ashbacher, C. An Introduction to the Smarandache Func-
tion. Cedar Rapids, IA: Decisionmark, 1995.
Ashbacher, C. "Problem 4616." School Set Math. 97, 221,
1997.
Smarandache-Kurepa Function
Smarandache Sequences 1661
Begay, A. "Smarandache Ceil Functions." Bulletin Pure
Appl. Sci. India 16E, 227-229, 1997.
Dumitrescu, C. and Seleacu, V. The Smarandache Function.
Vail, AZ: Erhus University Press, 1996.
"Functions in Number Theory." http://www.gallup.unm.
edu/~smarandache/FUNCTl . TXT.
Ibstedt, H. Surfing on the Ocean of Numbers — A Few
Smarandache Notions and Similar Topics. Lupton, AZ:
Erhus University Press, pp. 27-30, 1997.
Sandor, J. "On Certain Inequalities Involving the Smaran-
dache Function." Abstracts of Papers Presented to the
Amer. Math. Soc. 17, 583, 1996.
Sloane, N. J. A. Sequences A046021, A046022, A046023, and
A002034/M0453 in "An On-Line Version of the Encyclo-
pedia of Integer Sequences."
Smarandache, F. Collected Papers, Vol 1. Bucharest, Ro-
mania: Tempus, 1996.
Smarandache, F. Collected Papers, Vol. 2. Kishinev,
Moldova: Kishinev University Press, 1997.
Tutescu, L. "On a Conjecture Concerning the Smarandache
Function." Abstracts of Papers Presented to the Amer.
Math. Soc. 17, 583, 1996.
Smarandache-Kurepa Function
Given the sum-of-factorials function
E(n) = ^fc!,
SK(p) is the smallest integer for p Prime such that 1 -f
E[SK(p— 1)] is divisible by p. The first few known values
of SK(p) are 2, 4, 6, 6, 5, 7, 7, 12, 22, 16, 55, 54, 42 ;
24, . . . for p = 2, 5, 7, 11, 17, 19, 23, 31, 37, 41, 61, 71,
73, 89, ... . The values for p = 3, 13, 29, 43, 47, 53, 67,
79, 83, . . . , if they are finite, must be very large (e.g.,
SK(3) > 100,000).
see also PSEUDOSMARANDACHE FUNCTION, SMARAN-
DACHE Ceil Function, Smarandache Function,
Smarandache- Wagstaff Function, Smarandache
Function
References
Ashbacher, C. "Some Properties of the Smarandache-Kurepa
and Smarandache- Wagstaff Functions." Math. Informatics
Quart 7, 114-116, 1997.
Mudge, M. "Introducing the Smarandache-Kurepa and
Smarandache- Wagstaff Functions." Smarandache Notions
J. 7, 52-53, 1996.
Mudge, M. "Introducing the Smarandache-Kurepa and
Smarandache-Wagstaff Functions." Abstracts of Papers
Presented to the Amer. Math. Soc. 17, 583, 1996.
Smarandache Near-to-Primorial Function
SNTP(n) is the smallest Prime such that p# - 1, p#,
or p# + 1 is divisible by n, where p# is the PRIMORIAL
of p. Ashbacher (1996) shows that SNTP(n) only exists
1. If there are no square or higher powers in the factor-
ization of n, or
2. If there exists a PRIME q < p such that n\{q# ±
1), where p is the smallest power contained in the
factorization of n.
Therefore, SNTP{n) does not exist for the SQUAREFUL
numbers n = 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, . . .
(Sloane's A002997) The first few values of SNTP{n),
where defined, are 2, 2, 2, 3, 3, 3, 5, 7, . . . (Sloane's
A046026).
see also PRIMORIAL, SMARANDACHE FUNCTION
References
Ashbacher, C. "A Note on the Smarandache Near- To-
Primordial Function." Smarandache Notions J. 7, 46-49,
1996.
Mudge, M. R. "The Smarandache Near-To-Primorial Func-
tion." Abstracts of Papers Presented to the Amer. Math.
Soc. 17, 585, 1996.
Sloane, N. J. A. Sequence A002997 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Smarandache Paradox
Let A be some attribute (e.g., possible, present, per-
fect, etc.). If all is A, then the non-^4 must also be A
For example, "All is possible, the impossible too," and
"Nothing is perfect, not even the perfect."
References
Le, C. T. "The Smarandache Class of Paradoxes." Bull.
Transylvania Univ. Brasov 36, 7-8, 1994.
Le, C. T. "The Smarandache Class of Paradoxes." Bull. Pure
Appl. Sci. 14E, 109-110, 1995.
Le, C. T. "The Smarandache Class of Paradoxes." J. Indian
Acad. Math. 18, 53-55, 1996.
Mitroiescu, I. The Smarandache Class of Paradoxes. Glen-
dale, AZ: Erhus University Press, 1994.
Mitroiescu, I. "The Smarandache's Class of Paradoxes Ap-
plied in Computer Science." Abstracts of Papers Presented
to the Amer. Math. Soc. 16, 651, 1995.
Smarandache Sequences
Smarandache sequences are any of a number of simply
generated Integer Sequences resembling those con-
sidered in published works by Smarandache such as the
Consecutive Number Sequences and Euclid Num-
bers (Iacobescu 1997). Other Smarandache- type se-
quences are given below.
1. The concatenation of n copies of the Integer n:
1, 22, 333, 4444, 55555, ... (Sloane's A000461;
Marimutha 1997),
2. The concatenation of the first n FIBONACCI NUM-
BERS: 1, 11, 112, 1123, 11235, . . . (Sloane's A019523;
Marimutha 1997),
3. The smallest number that is the sum of squares of
two distinct earlier terms: 1, 2, 5, 26, 29, 677, ...
(Sloane's A008318, Bencze 1997),
4. The smallest number that is the sum of squares of
any number of distinct earlier terms: 1, 1, 2, 4, 5, 6,
16, 17, ... (Sloane's A008319, Bencze 1997),
5. The smallest number that is not the sum of squares
of two distinct earlier terms: 1, 2, 3, 4, 6, 7, 8, 9, 11,
. . . (Sloane's A008320, Bencze 1997),
6. The smallest number that is not the sum of squares
of any number of distinct earlier terms: 1, 2, 3, 6, 7,
8, 11, . . . (Sloane's A008321, Bencze 1997),
1662
Smarandache Sequences
Smarandache Sequences
7. The smallest number that is a sum of cubes of two
distinct earlier terms: 1, 2, 9, 730, 737, . . . (Sloane's
A008322, Bencze 1997),
8. The smallest number that is a sum of cubes of any
number of distinct earlier terms: 1, 1, 2, 8, 9, 512,
513, 514, . . . (Sloane's A008323, Bencze 1997),
9. The smallest number that is not a sum of cubes of
two of distinct earlier terms: 1, 2, 3, 4, 5, 6, 7, 8, 10,
. . . (Sloane's A008380, Bencze 1997),
10. The smallest number that is not a sum of cubes of
any number of distinct earlier terms: 1, 2, 3, 4, 5, 6,
7, 10, 11, . . . (Sloane's A008381, Bencze 1997),
11. The number of PARTITIONS of a number n — 1, 2,
. . . into Square Numbers: l, 1, l, l, 2, 2, 2, 2, 3,
4, 4, 4, 5, 6, 6, 6, 8, 9, 10, 10, 12, 13, ... (Sloane's
A001156, Iacobescu 1997),
12. The number of PARTITIONS of a number n = 1, 2,
... into Cubic Numbers: 1, 1, 1, 1, 1, 1, 1, 1, 2,
2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, . . . (Sloane's
A003108, Iacobescu 1997),
13. Two copies ofthe first n POSITIVE integers: 11,1212,
123123, 12341234, . . . (Sloane's A019524, Iacobescu
1997),
14. Numbers written in base of triangular numbers: 1,
2, 10, 11, 12, 100, 101, 102, 110, 1000, 1001, 1002,
. . . (Sloane's A000462, Iacobescu 1997),
15. Numbers written in base of double factorial numbers:
1, 10, 100, 101, 110, 200, 201, 1000, 1001, 1010, ...
(Sloane's A019513, Iacobescu 1997),
16. Sequences starting with terms {a\ , ai } which contain
no three-term arithmetic progressions starting with
{1,2}: 1, 2, 4, 5, 10, 11, 13, 14, 28, ... (Sloane's
A033155, Iacobescu 1997, Mudge 1997, Weisstein),
17. Numbers of the form (n!) 2 + 1: 2, 5, 37, 577, 14401,
518401, 25401601, 1625702401, 131681894401, ...
(Sloane's A020549, Iacobescu 1997),
18. Numbers of the form (n!) 3 + 1: 2, 9, 217, 13825,
1728001, 373248001, 128024064001, ... (Sloane's
A019514, Iacobescu 1997),
19. Numbers of the form 1 -j- l!2!3! ■ ■ ■ nl: 2, 3, 13, 289,
34561, 24883201, 125411328001, 5056584744960001,
. . . (Sloane's A019515, Iacobescu 1997),
20. Sequences starting with terms {a\ , 0,2} which contain
no three-term geometric progressions starting with
{1,2}: 1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, . . .
(Sloane's A000452, Iacobescu 1997),
21. Numbers repeating the digit 1 p n times, where p n is
the nth prime: 11,111,11111,1111111,... (Sloane's
A031974, Iacobescu 1997). These are a subset ofthe
REPUNITS,
22. Integers with all 2s, 3s, 5s, and 7s (prime digits)
removed: 1, 4, 6, 8, 9, 10, 11, 1, 1, 14, 1, 16, 1, 18,
19, 0, . . . (Sloane's A019516, Iacobescu 1997),
23. Integers with all 0s, Is, 4s, and 9s (square digits)
removed: 2, 3, 5, 6, 7, 8, 2, 3, 5, 6, 7, 8, 2, 2, 22, 23,
. . . (Sloane's A031976, Iacobescu 1997).
24. (Smarandache-Fibonacci triples) Integers n such
that S{n) = S{n - 1) + S(n - 2), where S(k) is the
Smarandache Function: 3, 11, 121, 4902, 26245,
. . . (Sloane's A015047; Aschbacher and Mudge 1995;
Ibstedt 1997, pp. 19-23; Begay 1997). The largest
known is 19,448,047,080,036,
25. (Smarandache-Radu triplets) Integers n such that
there are no primes between the smaller and larger
of S(n) and S(n + 1): 224, 2057, 265225, ...
(Sloane's A015048; Radu 1994/1995, Begay 1997, Ib-
stedt 1997). The largest known is 270,329,975,921,
205,253,634,707,051,822,848,570,391,313,
26. (Smarandache crescendo sequence): Integers ob-
tained by concatenating strings ofthe first n-f 1 inte-
gers for n = 0, 1, 2, ... : 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, . . .
(Sloane's A002260; Brown 1997, Brown and Castillo
1997). The nth term is given by n-m(m+l)/2 + l,
where m = [(y/Sn + 1 - l)/2j , with [x\ the FLOOR
Function (Hamel 1997),
27. (Smarandache descrescendo sequence): Integers ob-
tained by concatenating strings of the first n inte-
gers for n = ..., 2, 1: 1, 2, 1, 3, 2, 1, 4, 3, 2, 1,
. . . (Sloane's A004736; Smarandache 1997, Brown
1997),
28. (Smarandache crescendo pyramidal sequence): Inte-
gers obtained by concatenating strings of rising and
falling integers: 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3,
2, 1, . . . (Sloane's A004737; Brown 1997, Brown and
Castillo 1997, Smarandache 1997),
29. (Smarandache descrescendo pyramidal sequence):
Integers obtained by concatenating strings of falling
and rising integers: 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1,
2, 3, 4, ... (Brown 1997),
30. (Smarandache crescendo symmetric sequence): 1, 1,
1, 2, 2, 1, 1, 2, 3, 3, 2, 1, ... (Sloane's A004739,
Brown 1997, Smarandache 1997),
31. (Smarandache descrescendo symmetric sequence): 1,
1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, ... (Sloane's A004740;
Brown 1997, Smarandache 1997),
32. (Smarandache permutation sequence): Numbers ob-
tained by concatenating sequences of increasing
length of increasing ODD NUMBERS and decreasing
Even Numbers: 1, 2, 1, 3, 4, 2, 1, 3, 5, 6, 4, 2, . . .
(Sloane's A004741; Brown 1997, Brown and Castillo
1997),
33. (Smarandache pierced chain sequence): Numbers of
the form c(n) = 1010101 for n = 0, 1, . . . : 101,
1010101, 10101010101, ... (Sloane's A031982; Ash-
bacher 1997). In addition, c(n)/101 contains no
Primes (Ashbacher 1997),
Smarandache Sequences
Smith Conjecture 1663
34. (Smarandache symmetric sequence): 1, 11, 121,
1221, 12321, 123321, . . . (Sloane's A007907; Smaran-
dache 1993, Dumitrescu and Seleacu 1994, sequence
3; Mudge 1995),
35 . (Smarandache square- digital sequence) : square
numbers all of whose digits are also squares: 1, 4, 9,
49, 100, 144, ... (Sloane's A019544; Mudge 1997),
36. (Square-digits): numbers composed of digits which
are squares: 1, 4, 9, 10, 14, 19, 40, 41, 44, 49, ...
(Sloane's A066030),
37. (Smarandache square-digital sequence): square-digit
numbers which are themselves squares: 1, 4, 9, 49,
100, 144, ... (Sloane's A019544; Mudge 1997),
38. (Cube-digits): numbers composed of digits which are
cubes: 1, 4, 10, 11, 14, 40, 41, 44, 100, 101, ...
(Sloane's A046031),
39. (Smarandache cube-digital sequence) : cube-digit
numbers which are themselves cubes: 1, 8, 1000,
8000, 1000000, . . . (Sloane's A019545; Mudge 1997),
40. (Prime-digits): numbers composed of digits which
are primes: 2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, . . .
(Sloane's A046034),
41. (Smarandache prime-digital sequence): prime-digit
numbers which are themselves prime: 2, 3, 5, 7, 23,
37, 53, . . . (Smith 1996, Mudge 1997).
see also ADDITION CHAIN, CONSECUTIVE NUMBER SE-
QUENCES, Cubic Number, Euclid Number, Even
Number, Fibonacci Number, Integer Sequence,
Odd Number, Partition, Smarandache Function,
Square Number
References
Aschbacher, C. Collection of Problems On Smarandache No-
tions. Vail, AZ: Erhus University Press, 1996.
Aschbacher, C. and Mudge, M. Personal Computer World.
pp. 302, Oct. 1995.
Begay, A. "Smarandache Ceil Functions." Bull Pure Appl
Sci. 16E, 227-229, 1997.
Bencze, M. "Smarandache Recurrence Type Sequences."
Bull Pure Appl Sci. 16E, 231-236, 1997.
Bencze, M. and Tutescu, L. (Eds.). Some Notions and Ques-
tions in Number Theory, Vol 2. http://www.gallup.unm.
edu/~smarandache/SNAQINT2.TXT.
Brown, J. "Crescendo &; Descrescendo." In Richard Henry
Wilde: An Anthology in Memoriam (1789-1847) (Ed.
M. Myers). Bristol, IN: Bristol Banner Books, p. 19, 1997.
Brown, J. and Castillo, J. "Problem 4619." School Sci. Math.
97, 221-222, 1997.
Dumitrescu, C. and Seleacu, V. (Ed.). Some Notions and
Questions in Number Theory, J^th ed. Glendale, AZ: Er-
hus University Press, 1994. http://www.gallup.unm.edu/
* smarandache/ SNAQINT.TXT.
Dumitrescu, C. and Seleacu, V. (Ed.). Proceedings of the
First International Conference on Smarandache Type No-
tions in Number Theory. Lupton, AZ: American Research
Press, 1997.
Hamel, E. Solution to Problem 4619. School Sci. Math. 97,
221-222, 1997.
Iacobescu, F. "Smarandache Partition Type and Other Se-
quences." Bull. Pure Appl. Sci. 16E, 237-240, 1997.
Ibstedt, H. Surfing on the Ocean of Numbers — A Few
Smarandache Notions and Similar Topics. Lupton, AZ:
Erhus University Press, 1997.
Kashihara, K. Comments and Topics on Smarandache No-
tions and Problems.ail, AZ: Erhus University Press, 1996.
Mudge, M. "Top of the Class." Personal Computer World,
674-675, June 1995.
Mudge, M. "Not Numerology but Numeralogy!" Personal
Computer World, 279-280, 1997.
Programs and the Abstracts of the First International Con-
ference on Smarandache Notions in Number Theory.
Craiova, Romania, Aug. 21-23, 1997.
Radu, I. M. Mathematical Spectrum 27, 43, 1994/1995.
Sloane, N. J. A. Sequences A001156/M0221 and A003108/
M0209 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Smarandache, F. "Properties of the Numbers." Tempe, AZ:
Arizona State University Special Collection, 1975.
Smarandache, F. Only Problems, Not Solutions!, J^th ed.
Phoenix, AZ: Xiquan, 1993.
Smarandache, F. Collected Papers, Vol. 2. Kishinev,
Moldova: Kishinev University Press, 1997.
Smith, S. "A Set of Conjectures on Smarandache Sequences."
Bull Pure Appl. Sci. 15E, 101-107, 1996.
Smarandache- WagstafF Function
Given the sum-of- Facto rials function
E(n) = J>,
SW(p) is the smallest integer for p PRIME such that
E[SW(p)] is divisible by p. The first few known values
are 2, 4, 5, 12, 19, 24, 32, 19, 20, 20, 20, 7, 57, 6, . . . for
p = 3, 11, 17, 23, 29, 37, 41, 43, 53, 67, 73, 79, 97, ... .
The values for 5, 7, 13, 31, . . . , if they are finite, must
be very large.
see also FACTORIAL, SMARANDACHE FUNCTION
References
Ashbacher, C. "Some Properties of the Smarandache-Kurepa
and Smarandache- Wags taff Functions." Math. Informatics
Quart. 7, 114-116, 1997.
"Functions in Number Theory." http://www.gallup.unm.
edu/-smarandache/FUNCTl .TXT.
Mudge, M. "Introducing the Smarandache-Kurepa and
Smarandache- Wagstaff Functions." Smarandache Notions
J. 7, 52-53, 1996.
Mudge, M. "Introducing the Smarandache-Kurepa and
Smarandache-Wagstaff Functions." Abstracts of Papers
Presented to the Amer. Math. Soc. 17, 583, 1996.
Smith Brothers
Consecutive Smith Numbers. The first two brothers
are (728, 729) and (2964, 2965).
see also Smith Number
Smith Conjecture
The set of fixed points which do not move as a knot is
transformed into itself is not a KNOT. The conjecture
was proved in 1978.
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 350-351, 1976.
1664 Smith's Markov Process Theorem
Smooth Number
Smith's Markov Process Theorem
Consider
P2(yi,t\y s ,t 3 )
P2(yi,t 1 \y2,ti)Ps(yuti;y2 1 t2\ys,t3)dy2. (1)
/
If the probability distribution is governed by a MARKOV
Process, then
Ps{yuti;y2 i t2\ys,ts) = P2O/2, t 2 \ ys,t 3 )
= P2(y2\ys,t3-t 2 ). (2)
Assuming no time dependence, so t\ = 0,
ft(yi|y3,*3)= P2{yi\y2,t 2 )P2(y2\y3,t3-t2)dy 2 . (3)
see also Markov Process
Smith's Network Theorem
In a Network with three EDGES at each VERTEX, the
number of Hamiltonian Circuits through a specified
Edge is or Even.
see also Edge (Graph), Hamiltonian Circuit, Net-
work
Smith Normal Form
A form for Integer matrices.
Smith Number
A Composite Number the Sum of whose Digits is
the sum of the DIGITS of its PRIME factors (excluding
1). (The PRIMES are excluded since they trivially satisfy
this condition). One example of a Smith number is the
Beast Number
666 = 2 ■ 3 • 3 - 37,
6 + 6 + 6 = 2 + 3 + 3 + (3 + 7) = 18.
Another Smith number is
4937775 = 3 • 5 ■ 5 • 65837,
since
4+9 + 3+7+7+7+5 = 3 + 5 + 5 + (6 + 5+8 + 3 + 7) = 42.
The first few Smith numbers are 4, 22, 27, 58, 85,
94, 121, 166, 202, 265, 274, 319, 346, ... (Sloane's
A006753). There are 360 Smith numbers less than 10 4
and 29,928 < 10 6 . McDaniel (1987a) showed that an
infinite number exist.
A generalized fc-Smith number can also be defined as
a number m satisfying S p (m) = kS(m), where S p is
the sum of prime factors and S is the sum of digits.
There are 47 1-Smith numbers, 21 2-Smith numbers,
three 3-S.mith numbers, and one 7-Smith, 9-Smith, and
14-Smith number < 1000.
A Smith number can be constructed from every factored
Repunit R n , The largest known Smith number is
9xiW(10 4594 + 3xl0 2297
+ 1)
1476 -,^3913210
see also MONICA SET, PERFECT NUMBER, REPUNIT,
Smith Brothers, Suzanne Set
References
Gardner, M. Penrose Tiles and Trapdoor Ciphers. . . and the
Return of Dr. Matrix, reissue ed. New York: W. H. Free-
man, pp. 99-300, 1989.
Guy, R. K. "Smith Numbers." §B49 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer-Verlag,
pp. 103-104, 1994.
McDaniel, W. L. "The Existence of Infinitely Many fc-Smith
Numbers." Fib. Quart, 25, 76-80, 1987a.
McDaniel, W. L. "Powerful K-Smith Numbers." Fib. Quart
25, 225-228, 1987b.
Oltikar, S. and Weiland, K. "Construction of Smith Num-
bers." Math. Mag. 56, 36-37, 1983.
Sloane, N. J. A. Sequence A006753/M3582 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Wilansky, A. "Smith Numbers." Two-Year College Math. J.
13, 21, 1982.
Yates, S. "Special Sets of Smith Numbers." Math. Mag. 59,
293-296, 1986.
Yates, S. "Smith Numbers Congruent to 4 (mod 9)." J. Recr.
Math. 19, 139-141, 1987.
Smooth Manifold
Another word for a C°° (infinitely differentiate) MAN-
IFOLD. A smooth manifold is a TOPOLOGICAL MANI-
FOLD together with its "functional structure" (Bredon
1995) and so differs from a TOPOLOGICAL MANIFOLD
because the notion of differentiability exists on it. Every
smooth manifold is a TOPOLOGICAL MANIFOLD, but not
necessarily vice versa. (The first nonsmooth TOPOLOG-
ICAL Manifold occurs in 4-D.) In 1959, Milnor showed
that a 7-D HYPERSPHERE can be made into a smooth
manifold in 28 ways.
see also DlFFERENTIABLE MANIFOLD, HYPERSPHERE,
Manifold, Topological Manifold
References
Bredon, G. E. Topology & Geometry. New York: Springer-
Verlag, p. 69, 1995.
Smooth Number
An Integer is A;-smooth if it has no Prime Factors
> k. The probability that a random POSITIVE INTEGER
< n is fc-smooth is ip(n, k)/n t where ifi(n, k) is the num-
ber of fc-smooth numbers < n. This fact is important in
Smooth Surface
Snake Polyiamond 1665
application of Kraitchik's extension of Fermat's Fac-
torization Method because it is related to the num-
ber of random numbers which must be examined to find
a suitable subset whose product is a square.
Since about 7r(fc) fc-smooth numbers must be found
(where 7r(fc) is the Prime Counting Function), the
number of random numbers which must be examined
is about 7r(k)n/ip(n,k). But because it takes about
7r(k) steps to determine if a number is fc-smooth using
Trial Division, the expected number of steps needed
to find a subset of numbers whose product is a square
is ~ [7r(A;)] 2 n/V J (n, k) (Pomerance 1996). Canfield et al.
(1983) showed that this function is minimized when
k ~ exp(| Vlnnlnlnn)
and that the minimum value is about
exp(2 Vln n In In n ) .
In the Continued Fraction Factorization Algo-
rithm, n can be taken as 2*Jn, but in Fermat's FAC-
TORIZATION METHOD, it is n x / 2+c . k is an estimate
for the largest PRIME in the FACTOR BASE (Pomerance
1996).
References
Canfield, E. FL; Erdos, P.; and Pomerance, C. "On a Problem
of Oppenheim Concerning 'Factorisation Numerorum.'" J.
Number Th. 17, 1-28, 1983.
Pomerance, C. "On the Role of Smooth Numbers in Number
Theoretic Algorithms." In Proc. Internat. Congr. Math.,
Zurich, Switzerland, 1994, Vol 1 (Ed. S. D. Chatterji).
Basel: Birkhauser, pp. 411-422, 1995.
Pomerance, C. "A Tale of Two Sieves." Not. Amer. Math.
Soc. 43, 1473-1485, 1996.
Smooth Surface
A surface PARAMETERIZED in variables u and v is called
smooth if the TANGENT VECTORS in the u and v direc-
tions satisfy
T u xT v / 0,
where A x B is a CROSS PRODUCT.
Snake
A simple circuit in the d-HYPERCUBE which has no
chords (i.e., for which all snake edges are edges of the
HYPERCUBE). Klee (1970) asked for the maximum
length s(d) of a d-snake. Klee (1970) gave the bounds
s(d) 1
12-
4(d-l) - 2 d 2 7d(d-l) 2 + 2
(1)
for d > 6 (Danzer and Klee 1967, Douglas 1969), as well
as numerous references. Abbott and Katchalski (1988)
show
and Snevily (1994) showed that
^ 2n " I 1 -20^11) (3)
for n < 12, and conjectured
s(d) < 3 • 2 n ' 3 + 2 (4)
for n < 5. The first few values for s(d) for d = 1, 2, . . . ,
are 2, 4, 6, 8, 14, 26, . . . (Sloane's A000937).
see also HYPERCUBE
References
Abbott, H. L. and Katchalski, M. "On the Snake in the Box
Problem." J. Combin. Th. Ser. B 44, 12-24, 1988.
Danzer, L. and Klee, V. "Length of Snakes in Boxes." J.
Combin. Th. 2, 258-265, 1967.
Douglas, R. J. "Some Results on the Maximum Length of
Circuits of Spread k in the d-Cube." J. Combin. Th. 6,
323-339, 1969.
Evdokimov, A. A. "Maximal Length of a Chain in a Unit
n-Dimensional Cube." Mat. Zametki 6, 309-319, 1969.
Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903-909, 1989.
Kautz, W. H. "Unit-Distance Error- Checking Codes." IRE
Trans. Elect Comput. 7, 177-180, 1958.
Klee, V. "What is the Maximum Length of a d-Dimensional
Snake?" Amer. Math. Monthly 77, 63-65, 1970.
Sloane, N. J. A. Sequence A000937/M0995 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Snevily, H. S. "The Snake- in- the- Box Problem: A New Upper
Bound." Disc. Math. 133, 307-314, 1994.
Snake Eyes
A roll of two Is (the lowest roll possible) on a pair of
six-sided DICE. The probability of rolling snake eyes is
1/36, or 2.777. . . %.
see also Boxcars
Snake Oil Method
The expansion of the two sides of a sum equality in terms
of Polynomials in x™ and y k , followed by closed form
summation in terms of x and y. For an example of the
technique, see Bloom (1995).
References
Bloom, D. M. "A Semi-Unfriendly Identity." Problem 10206.
Solution by R. J. Chapman. Amer. Math. Monthly 102,
657-658, 1995.
Wilf, H. S. Generatingfunctionology, 2nd ed. New York:
Academic Press, 1993.
Snake Polyiamond
s{d) > 77 • 2 d
(2)
A 6-POLYIAMOND.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
1666
Snedecor's F -Distribution
Snub Cube
Snedecor's F-Distribution
If a random variable X has a Chi-Squared DISTRIBU-
TION with m degrees of freedom (xm 2 ) and a random
variable Y has a Chi-Squared Distribution with n
degrees of freedom (xn 2 )> and X and Y are independent,
then
Letting
Y/n
(1)
is distributed as Snedecor's F-distribution with m and
n degrees of freedom
/(F(m,n)) =
p / m+n \ (rn\ m / 2 jp(m-2)/2
(m+n)/2
r(?)r(?)(i + £F)
for < F < oo. The MOMENTS about are
A*i
M2 =
Ms =
M4
n-2
n 2 (m + 2)
m(n- 2)(n-4)
n 3 (m + 2)(m-f-4)
m 2 (n-2)(n-4)(n-6)
n 4 (m + 2)(m-h4)(m + 6)
m 3 (n - 2)(n - 4)(n - 6)(n - 8) '
so the Moments about the Mean are given by
(2)
(3)
(4)
(5)
(6)
^2 =
M3
M4
_ 2n 2 (m + n-2)
m(n-2) 2 (n-4)
8n 3 (m + n - 2)(2m + n - 2)
m 2 (n-2) 3 (n-4)(n-6)
12n 4 (m + n-2)
m 3 (n - 2) 4 (n - 4)(n - 6)(n - 8)
(7)
(8)
g{m,n), (9)
where
g(m, n) = mn + 4n + m n + 8mn — 16n
+ 10m 2 -20m + 16, (10)
and the Mean, VARIANCE, Skewness, and Kurtosis
are
n
n — £t
2n 2 (m + n-2)
<r 2 =
m(n-2) 2 (n-4)
72
T 3
M4
2(n-4) 2m + 1
7i - — = 2 A
a 3 V m(m + n — 2) n — 6
12/i(m, n)
m(m + n — 2)(n — 6)(n — 8) '
(ii)
(12)
(13)
(14)
where
/i(m,n) = n 3 + 5mn 2 — 8n 2 + 5m 2 n — 32ran
+20n - 22m 2 + 44m - 16. (15)
mF
V) =
1+-
(16)
gives a Beta Distribution.
see also Beta Distribution, Chi-Squared Distribu-
tion, Student's ^-Distribution
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 536, 1987.
Snellius-Pothenot Problem
A Surveying Problem which asks: Determine the po-
sition of an unknown accessible point P by its bearings
from three inaccessible known points A, £, and C.
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.
Snowflake
see Exterior Snowflake, Koch Antisnowflake,
Koch Snowflake, Pentaflake
Snub Cube
An Archimedean Solid also called the Snub Cub-
octahedron whose VERTICES are the 24 points on the
surface of a SPHERE for which the smallest distance be-
tween any two is as great as possible. It has two ENAN-
TIOMERS, and its DUAL POLYHEDRON is the PENTAG-
ONAL Icositetrahedron. It has Schlafli Symbol
s{^}. It is also Uniform Polyhedron U X2 and has
Wythoff Symbol | 2 34. Its faces are 32{3} + 6{4}.
Snub Cuboctahedron
Snub Dodecadodecahedron 1667
The INRADIUS, MlDRADIUS, and ClRCUMRADIUS for
a = 1 are
r = 1.157661791...
p = 1.247223168...
R = \\i \ ! 3? + t = 1.3437133737446. . . ,
2 V # 2 - 5# + 4
where
x = (19 + 3\/33) 1/3 ,
and the exact expressions for r and p can be computed
using
R 2 - \a 2
R
yj& ~ W-
see a/50 Snub Dodecahedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th erf. New York: Dover, p. 139,
1987.
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.
Snub Cuboctahedron
see Snub Cube
The coordinates of the VERTICES may be found by solv-
ing the set of four equations
i 2 . 2
L + X 2 + Zl
= 4
{X2
-I) 2
+ {zz - Zif
= 4
2
X2
+ (zs - z 2 ) 2
= 4
x 2 2
-\-X 2 2
+ (Z2 - Zi) 2
= 4
lkno
wns cC2i ^l) 22) and
23-
X2
= 1.28917
Zl
= 1.15674
Z2
= 1.97898
z$
= 3.13572.
The analytic solution requires solving the CUBIC EQUA-
TION and gives
a 2 = 1 - 7 ■ 2" 2/3 (l - iVs)a~' - § • 2~ 1/3 (1 + iy/s)a
zi = | .2" 1/2 [-48 + 6/?(l + i\/3)+/? 2 (l-z\/3)
+ 147/3 7 (>/3 - i) + 42/? 2 7 (v / 3 + i)} l/ \
where
a= (12iV237-54) 1/3
/3 = 3 1/3 (2i\/237-9) 1/3
7 = (9i + 2v / 237)~ 1 -
Snub Disphenoid
One of the convex DELTAHEDRA also known as the
Siamese Dodecahedron. It is Johnson Solid J&4-
(0,1, z 3 )
v (0,-l,z 3 )
(-^0,z 2 )
(* 2 > 0, z 2 )
(0, -x 2 , z 2 )
(-1,0,0)
(1,0,0)
Snub Dodecadodecahedron
The Uniform Polyhedron L/40 whose Dual Poly-
hedron is the Medial Pentagonal Hexecontahe-
dron. It has Wythoff Symbol | 2 § 5. Its faces are
12{ § } + 60{3} + 12{5}. It has ClRCUMRADIUS for a = 1
of
#=1.27443994.
see also Snub Cube
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th erf. New York: Dover, p. 139,
1987.
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.
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 174-176, 1971.
1668 Snub Dodecahedron
Snub Dodecahedron
An Archimedean Solid, also called the Snub Icos-
IDODECAHEDRON, whose DUAL POLYHEDRON is the
Pentagonal Hexecontahedron. It has Schlafli
Symbol s{;!}. It is also Uniform Polyhedron U 2 q
and has Wythoff Symbol | 235- Its faces are 80{3} +
12{5}. For a = 1, it has INRADIUS, Midradius, and
ClRCUMRADIUS
r = 2.039873155...
p = 2.097053835...
R
1 / 8-2 2 / 3 - 16x + 2 1 / 3 x 2
2V8-2 2 /3 _ I0x + 2 1 / 3 z 2
2.15583737511564...,
where
x~ (49 + 27\/5 + 3^6 ^93 + 49 V5 ]
and the exact expressions for r and p can be computed
using
R 2 - W
R
i
R?
i«".
References
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.
Snub Icosidodecadodecahedron
Sociable Numbers
The Uniform Polyhedron Ua& whose Dual Poly-
hedron is the Medial Hexagonal Hexecontahe-
dron. It has Wythoff Symbol | 3 § 5. Its faces are
12{|} + 80{3} + 12{5}. It has ClRCUMRADIUS for a = 1
of
ff-i
2 V 2 4 / 3
24/3 _ Ux + 2 2 / s x 2
where
Sx + 2 2 / 3 x 2
1.12689791279994...,
c = (25 + 3V^9) 1/3 .
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 177-178, 1971.
Snub Icosidodecahedron
see Snub Dodecahedron
Snub Polyhedron
A polyhedron with extra triangular faces, given by the
Schlafli Symbol s{ p }.
see also Rhombic Polyhedron, Truncated Polyhe-
dron
Snub Square Antiprism
see Johnson Solid
Soap Bubble
see Bubble
Soccer Ball
see Truncated Icosahedron
Sociable Numbers
Numbers which result in a periodic Aliquot SE-
QUENCE. If the period is 1, the number is called a PER-
FECT NUMBER. If the period is 2, the two numbers are
called an Amicable Pair. If the period is t > 3, the
number is called sociable of order t. Only two sociable
numbers were known prior to 1970, the sets of orders
5 and 28 discovered by Poulet (1918). In 1970, Cohen
discovered nine groups of order 4.
The table below summarizes the number of sociable cy-
cles known as given in the compilation by Moews (1995).
order
known
3
4
38
5
1
6
2
8
2
9
1
28
1
Social Choice Theory
Soddy Circles 1669
see also ALIQUOT SEQUENCE, PERFECT NUMBER, UNI-
TARY Sociable Numbers
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM : 239, Item 61, Feb. 1972.
Borho, W. "Uber die Fixpunkte der fc-fach iterierten Teil-
erersummenfunktion." Mitt. Math. Gesellsch. Hamburg 9,
34-48, 1969.
Cohen, H. "On Amicable and Sociable Numbers." Math.
Comput. 24, 423-429, 1970.
Devitt, J. S.; Guy, R. K.; and Selfridge, J. L. Third Report on
Aliquot Sequences, Congr. Numer. XVIII, Proc. 6th Man-
itoba Conf. Numerical Math, pp. 177-204, 1976.
Flammenkamp, A. "New Sociable Numbers." Math. Com-
put. 56, 871-873, 1991.
Gardner, M. "Perfect, Amicable, Sociable." Ch. 12 in Math-
ematical Magic Show: More Puzzles, Games, Diversions,
Illusions and Other Mathematical Sleight- of- Mind from
Scientific American. New York: Vintage, pp. 160-171,
1978.
Guy, R. K. "Aliquot Cycles or Sociable Numbers." §B7 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 62-63, 1994.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 145-146, 1979.
Moews, D. and Moews, P. C. "A Search for Aliquot Cycles
Below 10 10 ." Math. Comput. 57, 849-855, 1991.
Moews, D. and Moews, P. C. "A Search for Aliquot Cycles
and Amicable Pairs." Math. Comput. 61, 935-938, 1993.
Moews, D. "A List of Aliquot Cycles of Length Greater than
2." Rev. Dec. 18, 1995. http://xraysgi.ims.uconn.edu:
8080/sociable . txt.
Poulet, P. Question 4865. L'intermed. des Math. 25, 100-
101, 1918.
te Riele, H. J. J. "Perfect Numbers and Aliquot Sequences."
In Computational Methods in Number Theory, Part I.
(Eds. H. W. Lenstra Jr. and R. Tijdeman). Amsterdam,
Netherlands: Mathematisch Centrum, pp. 141-157, 1982.
# Weisstein, E. W. "Sociable and Amicable Num-
bers." http : //www . astro . Virginia . edu/ -eww6n/math/
notebooks/Sociable. m.
Social Choice Theory
The theory of analyzing a decision between a collection
of alternatives made by a collection of n voters with sep-
arate opinions. Any choice for the entire group should
reflect the desires of the individual voters to the extent
possible.
Fair choice procedures usually satisfy ANONYMITY (in-
variance under permutation of voters), DUALITY (each
alternative receives equal weight for a single vote), and
MONOTONICITY (a change favorable for X does not hurt
X). Simple majority vote is anonymous, dual, and
monotone. MAY'S THEOREM states a stronger result.
see also Anonymous, Dual Voting, May's Theo-
rem, Monotonic Voting, Voting
References
Taylor, A. Mathematics and Politics: Strategy, Voting,
Power, and Proof. New York: Springer- Verlag, 1995.
Young, S. C; Taylor, A. D.; and Zwicker, W. S. "Count-
ing Quota Systems: A Combinatorial Question from Social
Choice Theory." Math. Mag. 68, 331-342, 1995.
Socrates' Paradox
Socrates is reported to have stated: "One thing I know-
is that I know nothing."
see also Liar's Paradox
References
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
p. 134, 1995.
Soddy Circles
Given three distinct points A, B, and C, let three CIR-
CLES be drawn, one centered about each point and each
one tangent to the other two. Call the RADII n (r 3 = a',
n = 6', r 2 = c'). Then the CIRCLES satisfy
a + b' = c
a + c — o
+ c — a,
as shown in the diagram below.
A
(1)
(2)
(3)
Solving for the RADII then gives
a! - \{b + c-a)
b' = |(o + c-6)
c' = i(o + 6-c).
(4)
(5)
(6)
The above TRIANGLE has sides a, 6, and c, and
Semiperimeter
s= |(a + 6 + c). (7)
Plugging in,
2a = (a' + &') + («' + c / ) + (6 / + c , ) = 2(a'+6' + c), (8)
1670 Soddy Circles
Soddy's Hexlet
giving
o! + b* + c = s. (9)
In addition,
a = b + c = a -\- b + c — a — s — a . (10)
Switching a and a' to opposite sides of the equation and
noting that the above argument applies equally well to
b' and c' then gives
Solving for « n +i gives
a = s — a
b' =s-b
c = s — c.
(11)
(12)
(13)
As can be seen from the first figure, there exist exactly
two nonintersecting Circles which are Tangent to all
three CIRCLES. These are called the inner and outer
Soddy circles (S and S' , respectively), and their centers
are called the inner and outer SODDY POINTS.
The inner Soddy circle is the solution to the FOUR
COINS PROBLEM. The center S of the inner Soddy cir-
cle is the Equal Detour Point, and the center of
the outer Soddy circle S f is the ISOPERIMETRIC POINT
(Kimberling 1994).
Frederick Soddy (1936) gave the FORMULA for finding
the Radii of the Soddy circles (r*) given the RADII n
(i — 1, 2, 3) of the other three. The relationship is
rj/ 2 , 2 . 2 , 2\
2(ei +e 2 +€ 3 +e 4 )
(ei+e 2 +€3 + e4) 2 , (14)
where e* = ±m ~ ±l/n are the so-called Bends, de-
fined as the signed Curvatures of the Circles. If the
contacts are all external, the signs are all taken as POS-
ITIVE, whereas if one circle surrounds the other three,
the sign of this circle is taken as NEGATIVE (Coxeter
1969). Using the QUADRATIC FORMULA to solve for e 4 ,
expressing in terms of radii instead of curvatures, and
simplifying gives
rt =
T1V2V3
T2T3 + 7*i(r2 4- 7*3) ± 2y rir2Vs(ri + r 2 + rs)
(15)
Here, the Negative solution corresponds to the outer
Soddy circle and the POSITIVE one to the inner Soddy
circle.
This Formula is called the Descartes Circle The-
orem since it was known to Descartes. However, Soddy
also extended it to SPHERES. Gosper has further ex-
tended the result to n + 2 mutually tangent n-D Hy-
PERSPHERES, whose CURVATURES satisfy
Kn + l =
v^\/( EL Ki Y - ( n - x ) ELo Ki2 + EL Ki
n- 1
(17)
For (at least) n = 2 and 3, the Radical equals
/(n)V«o«i ***«n, (18)
where V is the Content of the Simplex whose vertices
are the centers of the n-f-1 independent HYPERSPHERES.
The RADICAND can also become NEGATIVE, yielding an
Imaginary « n +i. For n = 3, this corresponds to a
sphere touching three large bowling balls and a small
BB, all mutually tangent, which is an impossibility.
Bellew has derived a generalization applicable to a CIR-
CLE surrounded by n CIRCLES which are, in turn, cir-
cumscribed by another CIRCLE. The relationship is
n
[n(c n - l) 2 + 1] ]T Ki 2 +n(3nc n 2 -2n- 6)c n 2 (c n - 1) 2 =
[n(c- 1 l) + l]' x{n(cte - 1) ' + 11 S>
-\-nc n (c n - l)(nc n 2 + (3 - n)c n - 4])},
where
:„ = esc (-)
C n —
(19)
(20)
For n = 3, this simplifies to the Soddy formula.
see also APOLLONIUS CIRCLES, APOLLONIUS' PROB-
LEM, Arbelos, Bend (Curvature), Circumcircle,
Descartes Circle Theorem, Four Coins Prob-
lem, Hart's Theorem, Pappus Chain, Sphere
Packing, Steiner Chain
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New-
York: Wiley, pp. 13-14, 1969.
Elides, N. D. and Fukuta, J. "Problem E3236 and Solution."
Amer. Math. Monthly 97, 529-531, 1990.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, p. 181, 1994.
"The Kiss Precise." Nature 139, 62, 1937,
Soddy, F. "The Kiss Precise." Nature 137, 1021, 1936.
Vandeghen, A. "Soddy's Circles and the De Longchamps
Point of a Triangle." Amer. Math. Monthly 71, 176-179,
1964.
Soddy's Hexlet
see Hexlet
n+l
\k» — n) Kj 2 = 0.
(16)
Soddy Line
Solenoidal Field 1671
Soddy Line
A Line on which the Incenter /, Gergonne Point
Ge, and inner and outer Soddy Points S and S' lie
(the latter two of which are the Equal Detour Point
and the ISOPERIMETRIC Point). The Soddy line can be
given parametrically by
J + \Ge,
where A is a parameter. It is also given by
£(/-e)a = 0,
where cyclic permutations of d, e, and / are taken and
the sum is over TRILINEAR COORDINATES a, /?, and 7.
A Center
—4 outer Griffiths point Gr'
—2 outer Oldknow point OV
— I outer Rigby point Ri'
— 1 outer Soddy center S f
incenter I
1 inner Soddy center 5
1 inner Rigby point Ri
2 inner Oldknow point 01
4 inner Griffiths point Gr
00 Gergonne point
S\ i", 5, and Ge are Collinear and form a Harmonic
Range (Vandeghen 1964, Oldknow 1996). There are a
total of 22 Harmonic Ranges for sets of four points
out of these 10 (Oldknow 1996).
The Soddy line intersects the Euler Line in the DE
Longchamps Point, and the Gergonne Line in the
Fletcher Point.
see also DE LONGCHAMPS POINT, EULER LlNE,
Fletcher Point, Gergonne Point, Griffiths
Points, Harmonic Range, Incenter, Oldknow
Points, Rigby Points, Soddy Points
References
Oldknow, A. "The Euler- Gergonne- Soddy Triangle of a Tri-
angle." Amer. Math. Monthly 103, 319-329, 1996.
Vandeghen, A. "Soddy's Circles and the De Longchamps
Point of a Triangle." Amer. Math. Monthly 71, 176-179,
1964.
Soddy Points
Given three mutually tangent CIRCLES, there exist ex-
actly two nonintersecting CIRCLES TANGENT to all three
CIRCLES. These are called the inner and outer SODDY
CIRCLES, and their centers are called the inner and outer
Soddy points. The outer Soddy circle is the solution to
the Four Coins Problem. The center S of the inner
Soddy circle is the EQUAL DETOUR Point, and the cen-
ter of the outer Soddy circle S' is the ISOPERIMETRIC
POINT (Kimberling 1994).
see also EQUAL DETOUR POINT, ISOPERIMETRIC
Point, Soddy Circles
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, p. 181, 1994.
Sofa Constant
see Moving Sofa Constant
Sol Geometry
The Geometry of the Lie Group R Semidirect
Product with R 2 , where R acts on R 2 by (*, (as, y)) -»
(e t x,e~ t y).
see also THURSTON'S GEOMETRIZATION CONJECTURE
Soldner's Constant
Consider the following formulation of the Prime Num-
ber THEOREM,
•w-E^jTe-
where fi(m) is the MOBIUS FUNCTION and c (some-
times also denoted fj.) is Soldner's constant. Ramanujan
found c = 1.45136380 . . . (Hardy 1969, Le Lionnais 1983,
Berndt 1994). Soldner (cited in Nielsen 1965) derived
the correct value of c as 1.4513692346. . . , where c is the
root of
(Le Lionnais 1983).
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 123-124, 1994.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 45, 1959.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 39, 1983.
Nielsen, N. Theorie des Integrallogarithms. New York:
Chelsea, p. SS y 1965.
Solenoidal Field
A solenoidal Vector Field satisfies
V-B =
(1)
for every Vector B, where V-B is the Divergence. If
this condition is satisfied, there exists a vector A, known
as the Vector Potential, such that
B = V x A,
(2)
where V x A is the CURL. This follows from the vector
identity
V-B = V-(V x A) = 0. (3)
If A is an Irrotational Field, then
A x r (4)
is solenoidal. If u and v are irrotational, then
u x v (5)
1672
Solid
Solid of Revolution
is solenoidal. The quantity
(V-u) x (Vt>),
(6)
where Vu is the GRADIENT, is always solenoidal. For a
function <p satisfying LAPLACE'S EQUATION
V> = 0,
(7)
it follows that V0 is solenoidal (and also IRROTA-
tional).
see also BELTRAMI FIELD, CURL, DIVERGENCE, DlVER-
genceless field, gradient, irrotational field,
Laplace's Equation, Vector Field
References
Gradshteyn, I. S* and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1084, 1980.
Solid
A closed 3-D figure (which may, according to some ter-
minology conventions, be self-intersecting). Among the
simplest solids are the Sphere, Cube, Cone, Cylin-
der, and more generally, the POLYHEDRA.
see also Apple, Archimedean Solid, Catalan
Solid, Cone, Cork Plug, Cube, Cuboctahe-
dron, Cylinder, Cylindrical Hoof, Cylindrical
Wedge, Dodecahedron, Geodesic Dome, Great
Dodecahedron, Great Icosahedron, Great
Rhombicosidodecahedron (Archimedean), Great
Rhombicuboctahedron (Archimedean), Great
Stellated Dodecahedron, Icosahedron, Icosi-
dodecahedron, Johnson Solid, Kepler-Poinsot
Solid, Lemon, Mobius Strip, Octahedron, Pla-
tonic Solid, Polyhedron, Pseudosphere,
Rhombicosidodecahedron, Rhombicuboctahe-
dron, Small Stellated Dodecahedron, Snub
Cube, Snub Dodecahedron, Solid of Revolution,
Sphere, Steinmetz Solid, Stella Octangula,
Surface, Tetrahedron, Torus, Truncated Cube,
Truncated Dodecahedron, Truncated Icosahe-
dron, Truncated Octahedron, Truncated Tet-
rahedron, Uniform Polyhedron, Wulff Shape
Solid Angle
Defined as the SURFACE AREA Vt of a UNIT SPHERE
which is subtended by a given object S. Writing the
Spherical Coordinates as for the Colatitude
(angle from the pole) and 8 for the LONGITUDE (az-
imuth),
Q = A
projected
//.-
sin <p dO dcj).
Solid angle is measured in Steradians, and the solid
angle corresponding to all of space being subtended is
47r Steradians.
see also Sphere, Steradian
Solid Geometry
That portion of GEOMETRY dealing with SOLIDS, as op-
posed to Plane Geometry. Solid geometry is con-
cerned with POLYHEDRA, SPHERES, 3-D SOLIDS, lines
in 3-space, PLANES, and so on.
see also Geometry, Plane Geometry, Spherical
Geometry
References
Alt shiller- Court, N. Modern Pure Solid Geometry. New
York: Chelsea, 1979.
Bell, R. J. T. An Elementary Treatise on Coordinate Geom-
etry of Three Dimensions. London: Macmillan, 1926.
Conn, P. M. Solid Geometry. New York: Routledge, 1968.
Frost, P. Solid Geometry, 3rd ed. London: Macmillan, 1886.
Lines, L. Solid Geometry. New York: Dover, 1965.
Salmon, G. Treatise on the Analytic Geometry of Three Di-
mensions, 6th ed. London: Longmans Green, 1914.
Shute, W. G.; Shirk, W. W.; and Porter, G. F. Solid Geom-
etry. New York: American Book Co., 1960.
Wentworth, G. A. and Smith, D. E. Solid Geometry. Boston,
MA: Ginn and Company, 1913.
Solid Partition
Solid partitions are generalizations of PLANE PARTI-
TIONS. MacMohan (1960) conjectured the GENERATING
Function for the number of solid partitions was
/(*)
1
(l-z)(l-z 2 ) 3 (l-z 3 ) 6 (l-^ 4 ) 1
but this was subsequently shown to disagree at n = 6
(Atkin et al. 1967). Knuth (1970) extended the tabula-
tion of values, but was unable to find a correct generat-
ing function. The first few values are 1, 4, 10, 26, 59,
140, . . . (Sloane's A000293).
References
Atkin, A. O. L.; Bratley, P.; MacDonald, I. G.; and McKay,
J. K. S. "Some Computations for m-Dimensional Parti-
tions." Proc. Cambridge Philos. Soc. 63, 1097-1100, 1967.
Knuth, D. E. "A Note on Solid Partitions." Math. Comput.
24, 955-961, 1970.
MacMahon, P. A. "Memoir on the Theory of the Partitions
of Numbers. VI: Partitions in Two-Dimensional Space, to
which is Added an Adumbration of the Theory of Parti-
tions in Three-Dimensional Space." Phil. Trans. Roy. Soc.
London Ser. A 211, 345-373, 1912b.
MacMahon, P. A. Combinatory Analysis, Vol. 2. New York:
Chelsea, pp. 75-176, 1960.
Sloane, N. J. A. Sequence A3392/M000293 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Solid of Revolution
To find the VOLUME of a solid of rotation by adding up
a sequence of thin cylindrical shells, consider a region
bounded above by y = /(#), below by y = g(x), on the
left by the LINE x = a, and on the right by the LINE
x — b. When the region is rotated about the t/-AxiS,
the resulting VOLUME is given by
V
[
2tt / x[f{x)
- g(x)] dx.
Solidus
To find the volume of a solid of rotation by adding up
a sequence of thin flat disks, consider a region bounded
above by y = /(#), below by y = g{x), on the left by the
Line x = a, and on the right by the Line x = b. When
the region is rotated about the cc-AxiS, the resulting
Volume is
Solvable Group 1673
Solomon's Seal Knot
V
-jf
i[f(x)} 2 -[g(x)} 2 }dx.
see also Surface of Revolution, Volume
Solidus
The diagonal slash "/" used to denote DIVISION for in-
line equations such as a/6, l/(x — l) 2 , etc. The solidus
is also called a DIAGONAL.
see also DIVISION, OBELUS
Solitary Number
A number which does not have any FRIENDS. Solitary
numbers include all PRIMES and POWERS of PRIMES.
More generally, numbers for which (n, <x(n)) = 1 are
solitary, where (a, b) is the Greatest Common Divi-
sor of a and b and a(n) is the DIVISOR FUNCTION. The
first few solitary numbers are 1, 2, 3, 4, 5, 7, 8, 9, 11,
13, 16, 17, 19, 21, ... (Sloane's A014567).
see also FRIEND
References
Anderson, C. W. and Hickerson, D. Problem 6020. "Friendly
Integers." Amer. Math. Monthly 84, 65-66, 1977.
Sloane, N. J. A. Sequence A014567 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Soliton
A stable isolated (i.e., solitary) traveling wave solution
to a set of equations.
see also Lax Pair, Sine-Gordon Equation
References
Bullough, R. K. and Caudrey, P. J. (Eds.). Solitons. Berlin:
Springer- Verlag, 1980.
Dodd, R. K. Solitons and Nonlinear Equations. London:
Academic Press, 1984.
Drazin, P. G. and Johnson, R. S, Solitons: An Introduction.
Cambridge, England: Cambridge University Press, 1988.
Filippov, A. The Versatile Solitons. Boston, MA:
Birkhauser, 1996.
Gu, C. H. Soliton Theory and Its Applications. New York:
Springer- Verlag, 1995.
Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and
Chaos. Cambridge, England: Cambridge University Press,
1990.
Lamb, G. L. Jr. Elements of Soliton Theory. New York:
Wiley, 1980.
Makhankov, V. G.; Fedyann, V. K,; and Pashaev, O. K.
(Eds.). Solitons and Applications. Singapore: World Sci-
entific, 1990.
Newell, A. C. Solitons in Mathematics and Physics.
Philadelphia, PA: SIAM, 1985.
Olver, P. J. and Sattinger, D. H. (Eds.). Solitons in Physics,
Mathematics, and Nonlinear Optics. New York: Springer-
Verlag, 1990.
Remoissent, M. Waves Called Solitons, 2nd ed. New York:
Springer- Verlag, 1996.
The (5,2) Torus Knot 05 oi with Braid Word <ti 5 .
Solomon's Seal Lines
The 27 Real or Imaginary straight Lines which lie
on the general CUBIC Surface and the 45 triple tan-
gent Planes to the surface. All are related to the 28
Bitangents of the general Quartic Curve.
Schoutte (1910) showed that the 27 lines can be put into
a One- TO- One correspondence with the vertices of a
particular POLYTOPE in 6-D space in such a manner that
all incidence relations between the lines are mirrored in
the connectivity of the POLYTOPE and conversely (Du
Val 1931). A similar correspondence can be made be-
tween the 28 bitangents and a 7-D POLYTOPE (Coxeter
1928) and between the tritangent planes of the canoni-
cal curve of genus four and an 8-D POLYTOPE (Du Val
1933).
see also BRIANCHON'S THEOREM, CUBIC SURFACE,
Double Sixes, Pascal's Theorem, Quartic Sur-
face, Steiner Set
References
Bell, E. T. The Development of Mathematics, 2nd ed. New
York: McGraw-Hill, pp. 322-325, 1945.
Coxeter, H. S. M. "The Pure Archimedean Polytopes in Six
and Seven Dimensions." Proc. Cambridge Phil. Soc. 24,
7-9, 1928.
Du Val, P. "On the Directrices of a Set of Points in a Plane."
Proc. London Math. Soc. Ser. 2 35, 23-74, 1933.
Schoutte, P. H. "On the Relation Between the Vertices of a
Definite Sixdimensional Polytope and the Lines of a Cubic
Surface." Proc. Roy. Akad. Acad. Amsterdam 13, 375-
383, 1910.
Solomon's Seal Polygon
see Hexagram
Solvable Congruence
A Congruence that has a solution.
Solvable Group
A solvable group is a group whose composition indices
are all PRIME NUMBERS. Equivalently, a solvable is a
GROUP having a "normal series" such that each "nor-
mal factor" is ABELIAN. The term solvable derives from
this type of group's relationship to Galois's Theorem,
namely that the SYMMETRIC GROUP S n is insoluble for
n > 5 while it is solvable for n = 1, 2, 3, and 4. As a
result, the POLYNOMIAL equations of degree > 5 are not
solvable using finite additions, multiplications, divisions,
and root extractions.
1674 Solvable Lie Group
Somos Sequence
Every FINITE GROUP of order < 60, every ABELIAN
Group, and every Subgroup of a solvable group is solv-
able.
see also Abelian Group, Composition Series, Ga-
lois's Theorem, Symmetric Group
References
Lomont, J. S. Applications of Finite Groups. New York:
Dover, p. 26, 1993.
Berlekamp, E. R.; Conway, J. H.; and Guy, R. K. Ch. 24
in Winning Ways, For Your Mathematical Plays, Vol. 2:
Games in Particular. London: Academic Press, 1982.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 203-205, 1989.
Gardner, M. Ch. 6 in The Second Scientific American Book
of Mathematical Puzzles & Diversions: A New Selection.
New York: Simon and Schuster, pp. 65-77, 1961.
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, pp. 168-169, 1983.
Solvable Lie Group
The connected closed SUBGROUPS (up to an ISOMOR-
PHISM) of Complex Matrices that are closed under
conjugate transpose and have a discrete finite center.
Examples include SPECIAL Linear GROUPS, Symplec-
tic Groups, and certain isometry groups of Quadra-
tic Forms.
see also LIE GROUP
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Soma Cube
A solid Dissection puzzle invented by Piet Hein during
a lecture on Quantum Mechanics by Werner Heisenberg.
There are seven soma pieces composed of all the irregular
face-joined cubes (POLYCUBES) with < 4 cubes. The
object is to assemble the pieces into a CUBE. There are
240 essentially distinct ways of doing so (Beeler et al.
1972, Berlekamp et al. 1982), as first enumerated one
rainy afternoon in 1961 by J. H. Conway and Mike Guy.
A commercial version of the cube colors the pieces black,
green, orange, white, red, and blue. When the 48 sym-
metries of the cube, three ways of assembling the black
piece, and 2 5 ways of assembling the green, orange,
white, red, and blue pieces are counted, the total num-
ber of solutions rises to 1,105,920.
see also Cube Dissection, Polycube
References
Albers, D. J. and Alexanderson, G. L. (Eds.). Mathematical
People: Profiles and Interviews. Boston, MA: Birkhauser,
p. 43, 1985.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 112-
113, 1987.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Item 112, Feb. 1972.
Somer-Lucas Pseudoprime
An Odd Composite Number N is called a Somer-
Lucas d-pseudoprime (with d > 1) if there EXISTS a
nondegenerate LUCAS SEQUENCE U(P,Q) with U = 0,
Ui = 1, D = P 2 - 4Q, such that (N y D) = 1 and
the rank appearance of N in the sequence U(P, Q) is
(l/a)(N - {D/N)), where (D/N) denotes the Jacobi
Symbol.
see also Lucas Sequence, Pseudoprime
References
Ribenboim, P. "Somer-Lucas Pseudoprimes." §2.X.D in The
New Book of Prime Number Records, 3rd ed. New York:
Springer- Verlag, pp. 131-132, 1996.
Sommerfeld's Formula
There are (at least) two equations known as Sommer-
feld's formula. The first is
Mz)
2?r / , .
J — 7] + %'
2-7T — 77 + 100
e izcost e iu(t-n/2) d ^
where J v (z) is a BeSSEL FUNCTION OF THE FIRST
Kind. The second states that under appropriate re-
strictions,
I Jo{Tr)e 7^=W = -,
t*V T2 + fc2
VrM^
see also WEYRICH'S FORMULA
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, pp. 1472 and
1474, 1980.
Somos Sequence
The Somos sequences are a set of related symmetrical
Recurrence Relations which, surprisingly, always
give integers. The Somos sequence of order k is defined
by
Elk/2]
■ =1 an-jan-(fc-i)
a n = — ,
dn-k
Sondat's Theorem
Sophie Germain Prime 1675
where [x\ is the FLOOR FUNCTION and clj = 1 for j = 0,
. . . , fc — 1. The 2- and 3-Somos sequences consist entirely
of Is. The A;-Somos sequences for k = 4, 5, 6, and 7 are
i 2
aTi-ian-3 + a n _2
ln-4
a n _ia n -4 + a n _2an-3
On-5
a n _6
1
a n -7 '
[a„_ia n -5 + a Tl _2tt n -4 + &n-3 ]
giving 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, ... (Sloane's
A006720), 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, ...
(Sloane's A006721), 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421,
1103, ... (Sloane's A006722), 1, 1, 1, 1, 1, 1, 3, 5, 9,
17, 41, 137, 769, ... (Sloane's A006723). Gale (1991)
gives simple proofs of the integer-only property of the
4-Somos and 5-Somos sequences. Hickerson proved 6-
Somos generates only integers using computer algebra,
and empirical evidence suggests 7-Somos is also integer-
only.
However, the A> Somes sequences for k > 8 do not give
integers. The values of n for which a n first becomes
nonintegral for the fc-Somos sequence for k = 8, 9, ...
are 17, 19, 20, 22, 24, 27, 28, 30, 33, 34, 36, 39, 41, 42,
44, 46, 48, 51, 52, 55, 56, 58, 60, . . . (Sloane's A030127).
see also Gobel's Sequence, Heronian Triangle
References
Buchholz, R. H. and Rathbun, R. L. "An Infinite Set of Heron
Triangles with Two Rational Medians." Amer. Math.
Monthly 104, 107-115, 1997.
Gale, D. "Mathematical Entertainments: The Strange and
Surprising Saga of the Somos Sequences." Math. Intel.
13, 40-42, 1991.
Sloane, N. J. A. Sequences A030127, A006720/M0857,
A006721/M0735, A006722/M2457, and A006723/M2456
in "An On- Line Version of the Encyclopedia of Integer Se-
quences."
Sondat's Theorem
The Perspective Axis bisects the line joining the two
Orthocenters.
see also Orthocenter, Perspective Axis
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 259, 1929.
Sonine's Integral
Jm{x)
2x"
2 m - 7 T(m
i p 1
Jnix^e^ii-t 2 ) 771 - 71 - 1 ^,
where J m {x) is a BESSEL FUNCTION OF THE FIRST
Kind and T(x) is the Gamma Function.
see also HANKEL'S INTEGRAL, POISSON INTEGRAL
Sonine Polynomial
A polynomial which differs from the associated La-
GUERRE Polynomial by only a normalization constant,
r (x) = — e x -r- e x ) = 7 — — ,
T ^r+.(*)
X
x~
s!(r + s)!0! (s - l)!(r + s - l)!l!
x r ~ 2
+ (r-2)!(r + s-2)!2! "'
1 -(r + l)/2 x/2ttt f \
= 8\(r + s)\ X e W s+r/2+1/2 , r/2 (x),
where W k} m(z) is a Whittaker Function.
see also LAGUERRE POLYNOMIAL, WHITTAKER FUNC-
TION
Sonine-Schafheitlin Formula
f
Jo
J^{at)J v {bt)t- x dt
aT[(/* + i/-A + l)/2]
2 x b*- x + 1 r[(- f i + v + A + l)/2]r(Ax 4- 1)
x 2 Fi((/x + i/-A+l)/2,(/ i -i/-A-r-l)/2;^+l;a76 2 ),
where *% + v - A + 1] > 0, 3fc[A] > -1, < a <
6, Jv{x) is a Bessel Function of the First Kind,
T(x) is the GAMMA Function, and 2 F 1 (a,b;c]x) is a
Hypergeometric Function.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1474,
1980.
Sophie Germain Prime
A Prime p is said to be a Sophie Germain prime if both
p and 2p + 1 are Prime. The first few Sophie Germain
primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131,
... (Sloane's A005384).
Around 1825, Sophie Germain proved that the first case
of Fermat's Last Theorem is true for such primes,
i.e., if p is a Sophie Germain prime, there do not exist
Integers x, y, and z different from and not multiples
of p such that
x p + y p = z p .
Sophie Germain primes p of the form p = k - 2 n —
1 (which makes 2p + 1 a PRIME) are COMPOSITE
Mersenne Numbers. Since the largest known Com-
posite Mersenne Number is M p with p = 39051 x
2 6001 — 1, p is the largest known Sophie Germain prime.
see also CUNNINGHAM CHAIN, FERMAT'S LAST THEO-
REM, Mersenne Number, Twin Primes
1676
Sorites Paradox
Space
References
Dubner, H. "Large Sophie Germain Primes." Math. Comput.
65, 393-396, 1996.
Ribenboim, P. "Sophie Germane Primes." §5.2 in The New
Book of Prime Number Records. New York: Springer-
Verlag, pp. 329-332, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 154-157, 1993,
Sloane, N. J. A. Sequence A005384 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Sorites Paradox
Sorites paradoxes are a class of paradoxical arguments
also known as little-by-little arguments. The name
"sorites" derives from the Greek word soros, meaning
"pile" or "heap". Sorites paradoxes are exemplified by
the problem that a single grain of wheat does not com-
prise a heap, nor do two grains of wheat, three grains
of wheat, etc. However, at some point, the collection
of grains becomes large enough to be called a heap, but
there is apparently no definite point where this occurs.
see also Unexpected Hanging Paradox
Sort-Then-Add Sequence
A sequence produced by sorting the digits of a number
and adding them to the previous number. The algorithm
terminates when a sorted number is obtained. For n =
1, 2, ... , the algorithm terminates on 1, 2, 3, 4, 5, 6, 7,
8, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 33, ...
(Sloane's A033862). The first few numbers not known
to terminate are 316, 452, 697, 1376, 2743, 5090, . . .
(Sloane's A033861). The least numbers of sort-then-add
persistence n = 1, 2, . . . , are 1, 10, 65, 64, 175, 98, 240,
325, 302, 387, 198, 180, 550, . . . (Sloane's A033863).
see also 196-Algorithm, RATS Sequence
References
Sloane, N. J. A. Sequences A033861, A033862, and A033863
in "An On-Line Version of the Encyclopedia of Integer Se-
quences."
Sorting
Sorting is the rearrangement of numbers (or other or-
derable objects) in a list into their correct lexographic
order. Alphabetization is therefore a form of sorting.
Because of the extreme importance of sorting in almost
all database applications, a great deal of effort has been
expended in the creation and analysis of efficient sorting
algorithms.
see also HEAPSORT, ORDERING, QUICKSORT
References
Knuth, D. E. The Art of Computer Programming, Vol 3:
Sorting and Searching, 2nd ed. Reading, MA: Addison-
Wesley, 1973.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Sorting." Ch. 8 in Numerical Recipes in FOR-
TRAN: The Art of Scientific Computing, 2nd ed. Cam-
bridge, England: Cambridge University Press, pp. 320-
339, 1992.
Source
sink
A vertex of a Directed Graph with no entering edges.
see also Directed Graph, Network, Sink (Di-
rected Graph)
Sous-Double
A Multiperfect Number P 3 . Six sous-doubles are
known, and these are believed to comprise all sous-
doubles.
see also Multiperfect Number, Sous-Triple
Souslin's Hypothesis
Every dense linear order complete set without endpoints
having at most u; disjoint intervals is order isomorphic
to the Continuum of Real Numbers, where u> is the
set of Natural Numbers.
References
Iyanaga, S. and Kawada, Y. (Eds.). "Souslin's Hypothe-
sis." §35E.4 in in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, p. 137, 1980.
Souslin Set
The continuous image of a Polish Space, also called
an Analytic Set.
see also Analytic Set, Polish Space
Sous- Triple
A Multiperfect Number P 4 - 36 sous-triples are
known, and these are believed to comprise all sous-
triples.
see also Multiperfect Number, Sous-Double
Space
The concept of a space is an extremely general and im-
portant mathematical construct. Members of the space
obey certain addition properties. Spaces which have
been investigated and found to be of interest are usually
named after one or more of their investigators. This
practice unfortunately leads to names which give very
little insight into the relevant properties of a given space.
One of the most general type of mathematical spaces is
the Topological Space.
see also Affine Space, Baire Space, Banach
Space, Base Space, Bergman Space, Besov Space,
Borel Space, Calabi-Yau Space, Cellular Space,
Chu Space, Dodecahedral Space, Drinfeld's
Symmetric Space, Eilenberg-Mac Lane Space,
Euclidean Space, Fiber Space, Finsler Space,
Space of Closed Paths
Space-Filling Function 1677
First-Countable Space, Frechet Space, Func-
tion Space, G-Space, Green Space, Hausdorff
Space, Heisenberg Space, Hilbert Space, Hy-
perbolic Space, Inner Product Space, L 2 -Space,
Lens Space, Line Space, Linear Space, Liou-
ville Space, Locally Convex Space, Locally Fi-
nite Space, Loop Space, Mapping Space, Measure
Space, Metric Space, Minkowski Space, Muntz
Space, Non-Euclidean Geometry, Normed Space,
Paracompact Space, Planar Space, Polish Space,
Probability Space, Projective Space, Quotient
Space, Riemann's Moduli Space, Riemann Space,
Sample Space, Standard Space, State Space,
Stone Space, Teichmuller Space, Tensor Space,
Topological Space, Topological Vector Space,
Total Space, Vector Space
Space of Closed Paths
see Loop Space
Space Conic
see Skew Conic
Space Curve
A curve which may pass through any region of 3-D space,
as contrasted to a Plane Curve which must lie in a
single PLANE. Von Staudt (1847) classified space curves
geometrically by considering the curve
0: /->:
(1)
at to = and assuming that the parametric functions
<pi(t) for i — 1, 2, 3 are given by POWER SERIES which
converge for small t. If the curve is contained in no
PLANE for small £, then a coordinate transformation
puts the parametric equations in the normal form
h(t)=t 1+kl +...
» 3 (£)=i 3+fcl+fc3 + fe3 +.
(2)
(3)
(4)
for integers &i, fo, k$ > 0, called the local numerical
invariants.
see also Curve, Cyclide, Fundamental Theorem
of Space Curves, Helix, Plane Curve, Seifert's
Spherical Spiral, Skew Conic, Space-Filling
Function, Spherical Spiral, Surface, Viviani's
Curve
References
do Carmo, M.; Fischer, G.; Pinkall, U.; and Reckziegel,
fL "Singularities of Space Curves." §3.1 in Mathemati-
cal Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 24-25, 1986.
Fine, H. B. "On the Singularities of Curves of Double Cur-
vature." Amer. J. Math. 8, 156-177, 1886.
Fischer, G. (Ed.). Plates 57-64 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 58-59, 1986.
Gray, A. "Curves in M. n " and "Curves in Space." §1.2 and
Ch. 7 in Modern Differential Geometry of Curves and Sur-
faces, Boca Raton, FL: CRC Press, pp. 4-6 and 123-151,
1993.
Griffiths, P. and Harris, J. Principles of Algebraic Geometry.
New York: Wiley, 1978.
Saurel, P. "On the Singularities of Tortuous Curves." Ann.
Math. 7, 3-9, 1905.
Staudt, C von. Geometrie der Lage. Nurnberg, Germany,
1847.
Wiener, C "Die Abhangigkeit der Ruckkehrelemente der
Projektion einer unebenen Curve von deren der Curve
selbst." Z. Math. & Phys. 25, 95-97, 1880.
Space Diagonal
The Line Segment connecting opposite Vertices (i.e.,
two VERTICES which do not share a common face) in a
Parallelepiped or other similar solid.
see also Diagonal (Polygon), Diagonal (Polyhe-
dron), Euler Brick
Space Distance
The maximum distance in 3-D can occur no more than
2 n — 2 times. Also, there exists a fixed number c such
that no distance determined by a set of n points in 3-
D space occurs more than cn 5 ^ 3 times. The maximum
distance can occur no more than [|^ 2 J times in 4-D,
where [x\ is the Floor Function.
References
Honsberger, FL Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., pp. 122-123, 1976.
Space Division
The number of regions into which space can be divided
by n Spheres is
N= §n(n 2 -3n + 8),
giving 2, 4, 8, 16, 30, 52, 84, . . . (Sloane's A046127).
see also Plane Division
References
Sloane, N. J. A. Sequence A046127 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Space-Filling Curve
see Space-Filling Function
Space-Filling Function
A "Curve" (i.e., a continuous map of a 1-D Interval)
into a 2-D area (a PLANE-FILLING Function) or a 3-D
volume.
see also Hilbert Curve, Peano Curve, Peano-
Gosper Curve, Plane-Filling Curve, Sierpinski
Curve, Space-Filling Polyhedron
1678 Space-Filling Polyhedron
Sparse Matrix
References
Pappas, T. "Paradoxical Curve-Space- Filling Curve." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 208, 1989.
Platzman, L. K. and Bartholdi, J. J. "Spacefilling Curves
and the Planar Travelling Salesman Problem." J. Assoc.
Comput. Mach. 46, 719-737, 1989.
Wagon, S. "A Spacefilling Curve." §6.3 in Mathematica in
Action. New York: W. H. Freeman, pp. 196-209, 1991.
Space-Filling Polyhedron
A space-filling polyhedron is a POLYHEDRON which can
be used to generate a TESSELLATION of space. There
exists one 16-sided space-filling POLYHEDRON, but it
is unknown if this is the unique 16-sided space-filler.
The Cube, Rhombic Dodecahedron, and Trun-
cated Octahedron are space-fillers, as are the Elon-
gated Dodecahedron and hexagonal Prism. These
five solids are all "primary" PARALLELOHEDRA (Coxeter
1973).
P. Schmitt discovered a nonconvex aperiodic polyhedral
space-filler around 1990, and a convex POLYHEDRON
known as the Schmitt-Conway BlPRISM which fills
space only aperiodically was found by J. H. Conway in
1993 (Eppstein).
see also Cube, Elongated Dodecahedron,
Keller's Conjecture, Parallelohedron, Prism,
Rhombic Dodecahedron, Schmitt-Conway Bi-
prism, Tessellation, Tiling, Truncated Octahe-
dron
References
Coxeter, H. S. M. Regular Poly topes, 3rd ed. New York:
Dover, pp. 29-30, 1973.
Critchlow, K. Order in Space: A Design Source Book. New
York: Viking Press, 1970.
Devlin, K. J. "An Aperiodic Convex Space-filler is Discov-
ered." Focus: The Newsletter of the Math. Assoc. Amer.
13, 1, Dec. 1993.
Eppstein, D. "Re: Aperiodic Space-Filling Tile?." http://
www.ics.uci.edu/-eppstein/junkyaxd/biprism.html.
Holden, A. Shapes, Space, and Symmetry. New York: Dover,
pp. 154-163, 1991.
Thompson, D'A. W. On Growth and Form, 2nd ed., compl.
rev. ed. New York: Cambridge University Press, 1992,
Tutton, A. E. H. Crystallography and Practical Crystal Mea-
surement, 2nd ed. London: Lubrecht 8z Cramer, pp. 567
and 723, 1964.
Space Groups
The space groups in 2-D are called Wallpaper
GROUPS. In 3-D, the space groups are the symmetry
GROUPS possible in a crystal lattice with the translation
symmetry element. There are 230 space groups in M 3 ,
although 11 are MIRROR IMAGES of each other. They
are listed by Hermann-Mauguin Symbol in Cotton
(1990).
see also Hermann-Mauguin Symbol, Lattice
Groups, Point Groups, Wallpaper Groups
References
Arfken, G. "Crystallographic Point and Space Groups."
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, p. 248-249, 1985.
Buerger, M. J. Elementary Crystallography. New York: Wi-
ley, 1956.
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, pp. 250-251, 1990.
Span (Geometry)
The largest possible distance between two points for a
finite set of points.
see also Jung's Theorem
Span (Link)
The span of an unoriented LINK diagram (also called
the Spread) is the difference between the highest and
lowest degrees of its Bracket Polynomial. The span
is a topological invariant of a knot. If a Knot K has a
reduced alternating projection of n crossings, then the
span of K is 4n.
see also Link
Span (Polynomial)
The difference between the highest and lowest degrees
of a Polynomial.
Span (Set)
For a Set 5, the span is defined by max 5— min 5, where
max is the Maximum and min is the MINIMUM.
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 207, 1994.
Span (Vectors)
The span of Subspace generated by VECTORS vi and
v 2 e V is
Span(vi,v 2 ) = {rvi + sv 2 :r,s6K}.
Sparse Matrix
A Matrix which has only a small number of Nonzero
elements.
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Sparse Linear Systems." §2.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 63-82, 1992.
Spearman Rank Correlation
Special Unitary Group 1679
Spearman Rank Correlation
The Spearman rank correlation is defined by
X>3/
VE* 2 £s/ 2
= i-«E
N{N 2 -1)'
(1)
The Variance, Kurtosis, and higher order Moments
are
2
a =
72 = -
73 = 75
AT-1
114
257V 5iV 2
. = 0.
(2)
(3)
(4)
Student was the first to obtain the VARIANCE. The
Spearman rank correlation is an i£-ESTlMATE.
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 634-637, 1992.
Special Curve
see Plane Curve, Space Curve
Special Function
see Function
Special Linear Group
The special linear group SL n (q) is the Matrix Group
corresponding to the set of n x n COMPLEX MATRI-
CES having DETERMINANT +1. It is a SUBGROUP of
the General Linear Group GL n (q) and is also a Lie
Group.
see also GENERAL LINEAR GROUP, SPECIAL ORTHOG-
ONAL Group, Special Unitary Group
References
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.;
and Wilson, R A. "The Groups GL n (q) t SL n {q), PGL n (q),
and PSL n (q) = L n (q)." §2.1 in Atlas of Finite Groups:
Maximal Subgroups and Ordinary Characters for Simple
Groups. Oxford, England: Clarendon Press, p. x, 1985.
Special Matrix
A matrix whose entries satisfy
if j > i + 1
+ 1
{0 if j > i
-1 ifj = z-
or 1 if j < i.
There are 2 n_1 special Minimal Matrices of size nxn.
References
Knuth, D. E. "Problem 10470." Amer. Math. Monthly 102,
655, 1995.
Special Orthogonal Group
The special orthogonal group SO n (q) is the SUBGROUP
of the elements of GENERAL ORTHOGONAL GROUP
GO n {q) with DETERMINANT 1.
see also General Orthogonal Group, Special Lin-
ear Group, Special Unitary Group
References
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker,
R. A.; and Wilson, R. A. "The Groups GO n (q), SO n (q),
PGO n (q), and PSO n (q), and O n (q)- n §2.4 in Atlas of
Finite Groups: Maximal Subgroups and Ordinary Char-
acters for Simple Groups. Oxford, England: Clarendon
Press, pp. xi-xii, 1985.
Special Point
A POINT which does not lie on at least one ORDINARY
Line.
see also ORDINARY POINT
References
Guy, R. K. "Unsolved Problems Come of Age."
Monthly 96, 903-909, 1989.
Amer. Math.
Special Series Theorem
If the difference between the order and the dimension of
a series is less than the GENUS (Curve), then the series
is special.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 253, 1959.
Special Unitary Group
The special unitary group SU n (q) is the set ofnxn
Unitary Matrices with Determinant +1 (having
n 2 - 1 independent parameters). SU(2) is HOMEOMOR-
phic with the Orthogonal Group 0^(2). It is also
called the Unitary Unimodular Group and is a Lie
GROUP. The special unitary group can be represented
by the Matrix
U(a,b)
a b
-b* a*
(1)
where a* a + 6*6 = 1 and a, 6 are the Cayley-Klein
Parameters. The special unitary group may also be
represented by the MATRIX
U&riX)
or the matrices
u*{U) =
e ,£ cos rj e 1 ^ sin jj
-e -tl " sin T] e~ 1 ^ cos T]
cos(|</>) isin(^4>)
isin(|</>) cos(|</>)
cos(i/3) sin(|/3)
-sin(i/3) cos(|/3)
(2)
(3)
(4)
(5)
1680
Species
Spectral Rigidity
The order 2j + 1 representation is
EW W W.7)
= £
(-1)"-'- V(j + p)!(j - p)!(j + g)!(j - <?)!
(j — p — Tn)\(j + q — m)\(m + p — q)\m\
xe i9Q cos 2;,+9 - p - 2m (i
(i/3)sin=
p+2m-q/lm ip7
(|0)«
(6)
The summation is terminated by putting l/(—N)\ = 0.
The Character is given by
x u) (
' \ — j 1 + 2 cos a + . . . + 2 cos(ja)
' a ' ~ | 2[cos(|a) + cos(fa) + . . . + cos(ja)
, ,i , for j =0,1,2,...
sin( jOj
'"■K'+gM for 7 = i *
(7)
see a/50 ORTHOGONAL GROUP, SPECIAL LINEAR
Group, Special Orthogonal Group
References
Arfken, G. "Special Unitary Group, SU(2) and SU(2)-0£
Homomorphism." Mathematical Methods for Physicists,
3rd ed. Orlando, FL: Academic Press,- pp. 253-259, 1985.
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker,
R. A.; and Wilson, R. A. "The Groups GU n (q)> 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.
Species
A species of structures is a rule F which
1. Produces, for each finite set 27, a finite set F[U],
2. Produces, for each bijection cr : U — Y V, a function
F[a] :F[U] -► F[V}.
The functions F[a] should further satisfy the following
functorial properties:
1. For all bijections a : U — ► V and r :V — ^ W,
F[roa] =F[r]oF[cr] y
2. For the Identity Map Idiy :[/->[/,
F[Id] = Id .
V F[U]
Specificity
The probability that a STATISTICAL TEST will be nega-
tive for a negative statistic,
see also SENSITIVITY, STATISTICAL TEST, TYPE I ER-
ROR, Type II Error
Spectral Norm
The Natural Norm induced by the L 2 -Norm. Let
A f be the Adjoint of the Square Matrix A, so that
A = a^, then
||A||2 = (maximum eigenvalue of A' A) 1 ' 2
||Ax|| a
= max ", , , .
11*1(2^0 ||x|| 2
see also L 2 -Norm, Matrix Norm
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1115, 1979.
Strang, G. §6.2 and 7.2 in Linear Algebra and Its Applica-
tions, J^th ed. New York: Academic Press, 1980.
Spectral Power Density
P y {v)= lim -
T->oo 1
L
r/2
T/2
[y(t)-y]e- 2 ' M dt
/.T/2
\ P y {v)dv^ lim - / [y{t)-yfdt
JO T ^°° T J -T/2
= {{y-yf) = <7y 2 -
see also POWER SPECTRUM
Spectral Radius
Let A be an n x n MATRIX with COMPLEX or Real ele-
ments with Eigenvalues Ai, . . . , A„. Then the spectral
radius p(A) of A is
p(A) = max |A;|.
Kx<n
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, pp. 1115-1116, 1979.
An element cr G F[U] is called an F-structure on U (or
a structure of species F on U). The function F[a] is
called the transport of F-structures along a.
References
Bergeron, F.; Labelle, G.; and Leroux, P, Combinatorial
Species and Tree-Like Structures. Cambridge, England:
Cambridge University Press, p. 5, 1998.
Spectral Rigidity
The mean square deviation of the best local fit straight
line to a staircase cumulative spectral density over a
normalized energy scale.
References
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, p. 341, 1993.
Spectral Theorem
Sphere 1681
Spectral Theorem
Let H be a Hilbert Space, B(H) the set of Bounded
linear operators from H to itself, and cr{T) the SPEC-
TRUM of T. Then if T e B(H) and T is normal, there
exists a unique resolution of the identity E on the Borel
subsets of <t(T) which satisfies
T= int XdE(X).
*{T)
Furthermore, every projection E{uj) Commutes with
every S e B(H) which Commutes with T.
References
Rudin, W. Theorem 12.23 in Functional Analysis, 2nd ed.
New York: McGraw-Hill, 1991.
Spectrum (Operator)
Let T be an Operator on a Hilbert Space. The
spectrum cr(T) of T is the set of A such that (T — XI)
is not invertible on all of the HILBERT Space, where
the As are Complex Numbers and / is the Identity
Operator. The definition can also be stated in terms
of the resolvent of an operator
p(T) = {X : (T - XI) is invertible},
and then the spectrum is defined to be the complement
of p(T) in the Complex Plane. The reason for doing
this is that it is easy to demonstrate that p(T) is an
Open Set, which shows that the spectrum is closed.
see also Hilbert Space
Spectrum Sequence
A spectrum sequence is a SEQUENCE formed by succes-
sive multiples of a Real Number a rounded down to
the nearest INTEGER s n — [na\ . If a is IRRATIONAL,
the spectrum is called a Beatty Sequence.
see also Beatty Sequence, Lagrange Spectrum,
Markov Spectrum
Speed
The Scalar |v| equal to the magnitude of the Veloc-
ity v.
see also ANGULAR VELOCITY, VELOCITY
Spence's Function
see also Spence's Integral
References
Berestetskii, V. B.; Lifschitz, E. M.; and Ditaevskii, L. P.
Quantum Electrodynamics, 2nd ed. Oxford, England:
Pergamon Press, p. 596, 1982.
Spence's Integral
F(x) = U 2 {l-x)
where Li 2 (z) is the DlLOGARlTHM.
see also SPENCE'S FUNCTION
Spencer's 15-Point Moving Average
A Moving Average using 15 points having weights —3,
-6, -5, 3, 21, 46, 67, 74, 67, 46, 21, 3, -5, -6, and -3.
It is sometimes used by actuaries.
see also MOVING Average
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 223, 1962.
Sperner's Theorem
The Maximum Cardinality of a collection of Subsets
of a ^-element Set T, none of which contains another,
is the Binomial Coefficient (. J 2 i), where \x\ is the
Floor Function.
see also CARDINALITY
Sphenocorona
see Johnson Solid
Sphenoid
see DlSPHENOlD
Sphenomegacorona
see Johnson Solid
Sphere
F(I)s j-^i±n d ,
1682 Sphere
Sphere
A sphere is defined as the set of all points in M which
are a distance r (the "RADIUS") from a given point (the
"Center"). Twice the Radius is called the Diameter,
and pairs of points on opposite sides of a DIAMETER are
called Antipodes. The term "sphere" technically refers
to the outer surface of a "BUBBLE," which is denoted § 2 .
However, in common usage, the word sphere is also used
to mean the UNION of a sphere and its INTERIOR (a
"solid sphere"), where the INTERIOR is called a BALL.
The Surface Area of the sphere and Volume of the
Ball of Radius r are given by
5 = 4nr
V = f 7rr 3
(1)
(2)
Radius r, then a solid Ball is obtained. Converting
to "standard" parametric variables a = p, u — 8, and
v = <j> gives the first FUNDAMENTAL FORMS
E = a sin v
F =
G = a,
second Fundamental Forms
e = a 2 sin 2 v
g = a
(10)
(11)
(12)
(13)
(14)
(15)
(Beyer 1987, p. 130). In On the Sphere and Cylinder
(ca. 225 BC), Archimedes became the first to derive
these equations (although he expressed 7r in terms of
the sphere's circular cross-section). The fact that
^sphere
•'circumscribed cylinder ^sphere
was also known to Archimedes.
2 (3)
Any cross-section through a sphere is a CIRCLE (or, in
the degenerate case where the slicing PLANE is tangent
to the sphere, a point). The size of the Circle is maxi-
mized when the PLANE defining the cross-section passes
through a Diameter.
The equation of a sphere of RADIUS r is given in CARTE-
SIAN Coordinates by
2,2,2
x + y +z
which is a special case of the ELLIPSOID
and Spheroid
2 2 2
a* V c 2
2,2 2
x +V , * =1
a 2 "*" c 2
(4)
(5)
(6)
A sphere may also be specified in SPHERICAL COORDI-
NATES by
x = p cos 9 sin 4>
y = p sin 9 sin <f>
Z ~ p COS 0,
(7)
(8)
(9)
where 9 is an azimuthal coordinate running from to 2n
(LONGITUDE), is a polar coordinate running from to
7r (Colatitude), and p is the Radius. Note that there
are several other notations sometimes used in which the
symbols for 9 and <j> are interchanged or where r is used
instead of p. If p is allowed to run from to a given
Area Element
dA = asinu,
Gaussian Curvature
a 2
and Mean Curvature
H= 1 -.
(16)
(17)
(18)
A sphere may also be represented parametrically by let-
ting u = r cos 0, so
x = y r 2 — u 2 cos 9
y = v r 2 — v? sin#
(19)
(20)
(21)
where 9 runs from to 27r and u runs from — r to r.
Given two points on a sphere, the shortest path on
the surface of the sphere which connects them (the
Sphere Geodesic) is an Arc of a Circle known as a
Great Circle. The equation of the sphere with points
(xi ,yijZi) and (#2 , yi , ^2 ) lying on a DIAMETER is given
by
(x-xi){x-x 2 ) + {y-yi)(y-y2) + (z-z 1 )(z-z 2 ) = 0.
(22)
Four points are sufficient to uniquely define a sphere.
Given the points {xi>yi,Zi) with i = 1, 2, 3, and 4, the
sphere containing them is given by the beautiful DE-
TERMINANT equation
= (23)
x 2 + y 2 + z 2
X
y
z 1
xi 2 + yi 2 +zi 2
Xi
yi
z x 1
2 1 2 1 2
X2 +2/2 +2 2
x 2
yi
z 2 1
2 , 2 , 2
X3 + ys + z z
X 3
ys
Zz 1
2 1 2 1 2
XA +2/4 + 24
£4
2/4
24 1
(Beyer 1987, p. 210).
Sphere
The generalization of a sphere in n dimensions is called a
Hypersphere. An n-D Hypersphere can be specified
by the equation
xi 2 +x 2 2 + ... + £ n 2 =r 2 .
(24)
The distribution of ANGLES for random rotation of a
sphere is
(25)
P(0)^sm 2 (l0),
giving a MEAN of 7r/2 + 2/ir.
To pick a random point on the surface of a sphere, let u
and v be random variates on [0, 1]. Then
9 = 2ttu
= cos _1 (2v- 1).
This works since the Solid Angle is
dQ = sin 4>d9 d<j> = dO d(cos <f>) .
(26)
(27)
(28)
Another easy way to pick a random point on a Sphere
is to generate three gaussian random variables x, y, and
z. Then the distribution of the vectors
1
^/ X 2 +y*+ Z <> [ z _
(29)
is uniform over the surface S 2 . Another method is to
pick z from a UNIFORM DISTRIBUTION over [-1,1] and
9 from a UNIFORM DISTRIBUTION over [0,27r). Then
the points
Vl-z 2 cos0~
VT
■ z 2 sin 9
z
(30)
are uniformly distributed over § .
Pick four points on a sphere. What is the probability
that the TETRAHEDRON having these points as VER-
TICES contains the Center of the sphere? In the 1-D
case, the probability that a second point is on the oppo-
site side of 1/2 is 1/2. In the 2-D case, pick two points.
In order for the third to form a TRIANGLE containing
the CENTER, it must lie in the quadrant bisected by a
Line Segment passing through the center of the Cir-
cle and the bisector of the two points. This happens
for one QUADRANT, so the probability is 1/4. Similarly,
for a sphere the probability is one Octant, or 1/8.
Pick two points at random on a unit sphere. The first
one can be assigned the coordinate (0, 0, 1) without
loss of generality. The second point can be given the
coordinates (sin</>, 0cos</>) with 9 = since all points
with the same (f> are rotationally identical. The distance
between the two points is then
r = y/sin 2 <f>+ (1 - cos0) 2 = y/2 - cos <f) = 2sin(§0).
(31)
Sphere 1683
Because the surface Area element is
dQ = sin <f)d9d<j), (32)
the probability that two points are a distance r apart is
P*(r) =
J™ S(<f> — r) sin 4> d<j>
Jq sin <fi d<f>
= f / 6[r- 2 sin(±<t>)] sin 4>d<j).
Jo
The Delta Function contributes when
|r = sin(i^)
= 2sin- 1 (H,
(33)
(34)
(35)
P^r) = \ sin[2sin" 1 (|r)] = sin[sin l {\ r)} cos[sin x {\r)]
r*. (36)
However, we need
P r (r)dr = Pt(r)^dr,
ar
(37)
and
dr= §cos(§0)^ = ±^l-sin 2 (±0)d
d4 _ 2_
dr ~ ,/4~^'
and
Pr(r) = \r ^4 - r 2
v 7 ^
(38)
(39)
(40)
for r £ [0,2]. Somewhat surprisingly, the largest dis-
tances are the most common, contrary to most people's
intuition. A plot of 15 random lines is shown below.
1684 Sphere- Cylinder Intersection
Sphere Eversion
The Moments about zero are
H f n = (r n )= / r n dr
I
Jo
->n + l
2 + n'
giving the first few as
' 4
Ml = 3
M2 = 2
#*' — 16
M3 - -g-
// — 16
M4 - t-
Moments about the MEAN are
2 2
M2 = tr = 5
^3 = "lis
^ 4 — 135 '
so the Skewness and Kurtosis are
71 = ^/2
72 = -f.
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
(50)
(51)
see also Ball, Bing's Theorem, Bubble, Cir-
cle, Dandelin Spheres, Diameter, Ellipsoid,
Exotic Sphere, Fejes Toth's Problem, Hy-
persphere, Liebmann's Theorem, Liouville's
Sphere-Preserving Theorem, Mikusinski's Prob-
lem, Noise Sphere, Oblate Spheroid, Osculat-
ing Sphere, Parallelizable, Prolate Spheroid,
Radius, Space Division, Sphere Packing, Tennis
Ball Theorem
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, 1987.
Eppstein, D. "Circles and Spheres." http://vww . ics . uci .
edu/~eppstein/ junkyard/sphere. html.
Geometry Center. "The Sphere." http://www.geom.umn,
edu/zoo/toptype/sphere/.
Sphere- Cylinder Intersection
see Cylinder-Sphere Intersection
Sphere Embedding
A 4-sphere has Positive Curvature, with
Since
dw —
R 2 = x 1 + y 2 + z 2 + w 2
n dx rt dy n dz
2x-~ + 2y-z- + 2z~— + 2w = 0.
aw aw dw
r~x-k + yy + zz,
xdx + ydy -f- zdz _ r ■ dr
VR 2 - t 2 '
(1)
(2)
(3)
(4)
To stay on the surface of the sphere,
ds 2 - dx 2 + dy 2 + dz 2 + dw 2
dx z + dy* + dz A +
m 2 j 2
2 r dr
, 2 . 2 , 2 , dr
dr + r dfl +
R 2 -r 2
2
fl2
-1
dr 2 I 1 + -5^— + r 2 da 2
= dr
^-1
+ r 2 dQ 2
1 -
^+r 2 <*0 2 .
(5)
W
With the addition of the so-called expansion parameter,
this is the Robertson- Walker line element.
Sphere Eversion
Smale (1958) proved that it is mathematically possible
to turn a SPHERE inside-out without introducing a sharp
crease at any point. This means there is a regular homo-
topy from the standard embedding of the 2-SPHERE in
Euclidean 3-space to the mirror-reflection embedding
such that at every stage in the homotopy, the sphere is
being Immersed in Euclidean Space. This result is
so counterintuitive and the proof so technical that the
result remained controversial for a number of years.
In 1961, Arnold Shapiro devised an explicit eversion but
did not publicize it. Phillips (1966) heard of the result
and, in trying to reproduce it, actually devised an inde-
pendent method of his own. Yet another eversion was
devised by Morin, which became the basis for the movie
by Max (1977). Morin's eversion also produced explicit
algebraic equations describing the process. The origi-
nal method of Shapiro was subsequently published by
Francis and Morin (1979).
see also Eversion, Sphere
References
Francis, G. K. Ch. 6 in A Topological Picturebook. New York:
Springer- Verlag, 1987.
Francis, G. K. and Morin, B. "Arnold Shapiro's Eversion of
the Sphere." Math. Intell. 2, 200-203, 1979.
Levy, S. Making Waves: A Guide to the Ideas Behind Out-
side In. Wellesley, MA: A. K. Peters, 1995.
Levy, S. "A Brief History of Sphere Eversions." http : //www .
geom .umn . edu/docs/outreach/oi/history . html.
Levy, S.; Maxwell, D.; and Munzner, T. Outside-In.
22 minute videotape. http://www.geom.umn.edu/docs/
outreach/oi/.
Max, N. "Turning a Sphere Inside Out." Videotape.
Chicago, IL: International Film Bureau, 1977.
Peterson, I. Islands of Truth: A Mathematical Mystery
Cruise. New York: W. H. Freeman, pp. 240-244, 1990.
Petersen, I. "Forging Links Between Mathematics and Art."
Science News 141, 404-405, June 20, 1992.
Phillips, A. "Turning a Surface Inside Out." Set. Amer. 214,
112-120, Jan. 1966.
Smale, S. "A Classification of Immersions of the Two-
Sphere." Trans. Amer. Math. Soc. 90, 281-290, 1958.
Sphere Geodesic
Sphere Packing 1685
Sphere Geodesic
see Great Circle
Sphere Packing
Let T) denote the Packing Density, which is the frac-
tion of a Volume filled by identical packed Spheres.
In 2-D (CIRCLE PACKING), there are two periodic pack-
ings for identical CIRCLES: square lattice and hexagonal
lattice. Fejes Toth (1940) proved that the hexagonal lat-
tice is indeed the densest of all possible plane packings
(Conway and Sloane 1993, pp. 8-9).
In 3-D, there are three periodic packings for identical
spheres: cubic lattice, face-centered cubic lattice, and
hexagonal lattice. It was hypothesized by Kepler in 1611
that close packing (cubic or hexagonal) is the densest
possible (has the greatest 77), and this assertion is known
as the Kepler Conjecture. The problem of finding
the densest packing of spheres (not necessarily periodic)
is therefore known as the KEPLER Problem. The Ke-
pler Conjecture is intuitively obvious, but the proof
still remains elusive. However, Gauss (1831) did prove
that the face-centered cubic is the densest lattice pack-
ing in 3-D (Conway and Sloane 1993, p. 9). This result
has since been extended to Hypersphere Packing.
In 3-D, face-centered cubic close packing and hexagonal
close packing (which is distinct from hexagonal lattice),
both give
V =
3\/2
74.048%.
(1)
For packings in 3-D, C. A. Rogers (1958) showed that
77 < Vl8 (cos
H
77.96355700%
(2)
(Le Lionnais 1983). This was subsequently improved to
77.844% (Lindsey 1986), then 77.836% (Muder 1988).
However, Rogers (1958) remarks that "many mathe-
maticians believe, and all physicists know" that the ac-
tual answer is 74.05% (Conway and Sloane 1993, p. 3).
"Random" close packing in 3-D gives only 77 w 64%
(Jaeger and Nagel 1992).
The Packing Densities for several packing types are
summarized in the following table.
Packing
77 (exact)
77 (approx.)
square lattice (2-D)
7T
4
0.7854
hexagonal lattice (2-D)
7T
0.9069
cubic lattice
7T
6
0.5236
hexagonal lattice
7T
3V3
0.6046
face-centered cubic lattice
TV
3v/2
0.7405
random
—
0.6400
For cubic close packing, pack six SPHERES together in
the shape of an EQUILATERAL TRIANGLE and place an-
other Sphere on top to create a Triangular Pyra-
mid. Now create another such grouping of seven
Spheres and place the two PYRAMIDS together facing
in opposite directions. A CUBE emerges. Consider a
face of the Cube, illustrated below.
The "unit cell" cube contains eight 1/8-spheres (one at
each Vertex) and six Hemispheres. The total Vol-
ume of Spheres in the unit cell is
'spheres in unit cell
(8-1+6- |)fr 3
8
47T
16 ,3
(3)
The diagonal of the face is 4r, so each side is 2y/2r. The
Volume of the unit cell is therefore
(2^2r) 3 ^lev^r 3 .
The Packing Density is therefore
?7ccp
16_„2
16V2r 3
7T
3\/2
(4)
(5)
(Conway and Sloane 1993, p. 2).
Hexagonal close packing must give the same values, since
sliding one sheet of SPHERES cannot affect the volume
they occupy. To verify this, construct a 3-D diagram
containing a hexagonal unit cell with three layers. Both
the top and the bottom contain six 1/6-SPHERES and
one HEMISPHERE. The total number of spheres in these
two rows is therefore
2(6± + l|) = 3.
(6)
The VOLUME of Spheres in the middle row cannot be
simply computed using geometry. However, symmetry
requires that the piece of the SPHERE which is cut off
is exactly balanced by an extra piece on the other side.
There are therefore three SPHERES in the middle layer,
for a total of six, and a total VOLUME
''spheres in unit cell
6-fr 3 (3 + 3)
: 87rr
(7)
1686 Sphere Packing
Sphere-Sphere Intersection
The base of the HEXAGON is made up of 6 Equilateral
TRIANGLES with side lengths 2r. The unit cell base
Area is therefore
Amit cell - 6[|(2r)(v / 3r)] = 6\/3r 2 .
(8)
The height is the same as that of two Tetrahedra
length 2r on a side, so
'■unit cell
giving
rjHCP
= 2 2r
8^7"*
(6^) (4rvT) 3v ^
(9)
(10)
(Conway and Sloane 1993, pp. 7 and 9).
If we had actually wanted to compute the VOLUME of
Sphere inside and outside the Hexagonal Prism, we
could use the Spherical Cap equation to obtain
(3-^)=&*r 8 (9-V3)
1 3
— 7IT
9
(11)
Vd = *r 3 [| - ^(9 - VS)] = ^7rr 3 (36 - 9 + Vs)
= i7rr 3 (27+v/3). (12)
The rigid packing with lowest density known has 77 m
0.0555 (Gardner 1966). To be Rigid, each SPHERE must
touch at least four others, and the four contact points
cannot be in a single HEMISPHERE or all on one equator.
If spheres packed in a cubic lattice, face-centered cu-
bic lattice, and hexagonal lattice are allowed to expand,
they form cubes, hexagonal prisms, and rhombic dodec-
ahedra. Compressing a random packing gives polyhedra
with an average of 13.3 faces (Coxeter 1958, 1961).
For sphere packing inside a Cube, see Goldberg (1971)
and Schaer (1966).
see also Cannonball Problem, Circle Pack-
ing, DODECAHEDRAL CONJECTURE, HEMISPHERE,
Hermite Constants, Hypersphere, Hypersphere
Packing, Kepler Conjecture, Kepler Problem,
Kissing Number, Local Density, Local Density
Conjecture, Sphere
References
Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices,
and Groups, 2nd ed. New York: Springer- Verlag, 1993.
Coxeter, H, S. M. "Close-Packing and so Forth." Illinois J.
Math. 2, 746-758, 1958.
Coxeter, H, S. M. "Close Packing of Equal Spheres." Sec-
tion 22.4 in Introduction to Geometry, 2nd ed. New York:
Wiley, pp. 405-411, 1961.
Coxeter, H. S. M. "The Problem of Packing a Number of
Equal Nonoverlapping Circles on a Sphere." Trans. New
York Acad. ScL 24, 320-331, 1962.
Critchlow, K. Order in Space: A Design Source Book. New
York: Viking Press, 1970.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 195-197, 1989.
Eppstein, D. "Covering and Packing." http://www.ics.uci
.edu/-eppstein/ junkyard/cover. html.
Fejes Toth, G. "Uber einen geometrischen Satz." Math. Z.
46, 78-83, 1940.
Fejes Toth, G. Lagerungen in der Ebene, auf der Kugel und
in Raum, 2nd ed. Berlin: Springer- Verlag, 1972.
Gardner, M. "Packing Spheres." Ch. 7 in Martin Gardner's
New Mathematical Diversions from Scientific American.
New York: Simon and Schuster, 1966.
Gauss, C. F. "Besprechung des Buchs von L. A. See-
ber: Intersuchungen iiber die Eigenschaften der posi-
tiven ternaren quadratischen Formen usw." Gottingsche
Gelehrte Anzeigen (1831, July 9) 2, 188-196, 1876.
Goldberg, M. "On the Densest Packing of Equal Spheres in
a Cube." Math. Mag. 44, 199-208, 1971.
Hales, T. C. "The Sphere Packing Problem." J. Comput.
Appl. Math 44, 41-76, 1992.
Jaeger, H. M. and Nagel, S. R. "Physics of Granular States,"
Science 255, 1524, 1992.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 31, 1983.
Lindsey, J. H. II. "Sphere Packing in M 3 ." Math. 33, 137-
147, 1986.
Muder, D. J. "Putting the the Best Face of a Voronoi Poly-
hedron." Proc. London Math. Soc. 56, 329-348, 1988.
Rogers, C. A. "The Packing of Equal Spheres." Proc. London
Math. Soc. 8, 609-620, 1958.
Rogers, C. A. Packing and Covering. Cambridge, England:
Cambridge University Press, 1964.
Schaer, J. "On the Densest Packing of Spheres in a Cube."
Can. Math. Bui 9, 265-270, 1966.
Sloane, N. J. A. "The Packing of Spheres." Sci. Amer. 250,
116-125, 1984.
Stewart, I. The Problems of Mathematics, 2nd ed. Oxford,
England: Oxford University Press, pp. 69-82, 1987.
Thompson, T. M. From Error- Correcting Codes Through
Sphere Packings to Simple Groups. Washington, DC:
Math. Assoc. Amer., 1984.
Sphere Point Picking
see Fejes Toth's Problem
Sphere-Sphere Intersection
Let two spheres of RADII R and r be located along the x-
Axis centered at (0,0,0) and (d, 0,0), respectively. Not
surprisingly, the analysis is very similar to the case of
Sphere-Sphere Intersection
Spherical Bessel Differential Equation 1687
the Circle-Circle Intersection. The equations of
the two Spheres are
2 . 2
■y +z
(x-d) 2 +y 2 +z 2 =r :
Combining (1) and (2) gives
(x-d) 2 + {R 2 -x 2 ) = r 2 .
Multiplying through and rearranging give
x 2 - 2dx + d 2
R 2 .
Solving for x gives
d 2 - r 2 + R 2
2d
(1)
(2)
(3)
(4)
(5)
The intersection of the Spheres is therefore a curve
lying in a PLANE parallel to the yz-plane at a single
^-coordinate. Plugging this back into (1) gives
2 , 2
y + z
R A -x
4d 2 R 2
= R 2
(d 2 -
/ d 2„ r 2 +jR2 y
+ R 2 ) 2
Ad?
which is a CIRCLE with RADIUS
(6)
a= WjV^R 2 ~(d 2 -r 2 +R 2 ) 2
1_
2d
{(-d + r - R)(-d - r + R)
x [(-d + r + R)(d + r + R)] 1/2 .
(7)
The Volume of the 3-D Lens common to the two
spheres can be found by adding the two SPHERICAL
Caps. The distances from the SPHERES' centers to the
bases of the caps are
di
d 2
■ x
d — x,
so the heights of the caps are
(r-R + d)(r + R-d)
hi = R — d\
d 2
2d
(R-r + d)(R + r-d)
2d
(8)
(9)
(10)
(11)
The Volume of a Spherical Cap of height ti for a
Sphere of Radius R' is
V(R',h')= l-Kh' 2 {3R' -ti).
(12)
Letting R\ = R and R2 — r and summing the two caps
gives
V = V{Ri,h!) +V(R 2 ,h 2 )
it{R + r- d) 2 (d 2 + 2dr - 3r 2 + 2dR + 6rR - 3R 2 )
12d
(13)
This expression gives V = for d = r + R as it must.
In the special case r = R, the VOLUME simplifies to
V = ^7r(4i? + d)(2#-d) 2 .
(14)
see also APPLE, ClRCLE-ClRCLE INTERSECTION, DOU-
BLE Bubble, Lens, Sphere
Sphere with Tunnel
Find the tunnel between two points A and Bona grav-
itating SPHERE which gives the shortest transit time
under the force of gravity. Assume the SPHERE to be
nonrotating, of RADIUS a, and with uniform density p.
Then the standard form EULER-LAGRANGE DIFFEREN-
TIAL Equation in polar coordinates is
r^(r 3 - ra 2 ) + r> 2 (2a 2 - r 2 ) + aV = 0, (1)
along with the boundary conditions r(<f> = 0) = ro,
r^(0 = 0) = 0, r(<j> = <j>a) = «, and r((j> = <j> B ) = a.
Integrating once gives
2
r> =
r
r 2 '
(2)
But this is the equation of a HYPOCYCLOID generated by
a Circle of Radius \(a - r ) rolling inside the Circle
of RADIUS a, so the tunnel is shaped like an arc of a
HYPOCYCLOID. The transit time from point A to point
B is
where
GM
rt 2
ag
\izpGa
(3)
(4)
is the surface gravity with G the universal gravitational
constant.
Spherical Bessel Differential Equation
Take the Helmholtz Differential Equation
V 2 F + A; 2 F =
(1)
in Spherical Coordinates. This is just Laplace's
Equation in Spherical Coordinates with an addi-
tional term,
d 2 e
d 2 R^^ 2dR
dr 2 r dr r 2 sin 2 <j> d0 2
$R
cos <f> d$
1 d 2 $
r z sin <p d(p v* d(f> 2
1688 Spherical Bessel Differential Equation
Spherical Bessel Function
Multiply through by r 2 /R$G,
r 2 d 2 R 2rdR 2 2 1 d 2 B
R dr 2 + R dr + T + G sin 2 <j> d6 2
cos 6 d§ 1 d 2 $
_l r 1 — o.
$ sin <j) d(j) <£ dcp 2
:x^=o. (3)
This equation is separable in R. Call the separation
constant n(n -f 1),
££+£?♦""*•+»■
Now multiply through by i?,
dR
r 2 $? + 2r^ + [*V - n(n + l)]fl = 0.
dr 2
dr
(4)
(5)
This is the SPHERICAL BESSEL DIFFERENTIAL EQUA-
TION. It can be transformed by letting x = kr, then
dRlr) , dR(r)
r — : — = kr-
dr
kdr
kr
dx
Similarly,
i d 2 R(r) _
r — r- r — — x
dR{r) _^ dR(r)
d(kr) ~ "
2 d 2 R(r)
dr 2
so the equation becomes
dx 2
i d 2 R
dx 2
dR
dx
x 2 ^ + 2x^ + [x 2 - n(n + 1)]R = 0.
(6)
(7)
(8)
Now look for a solution of the form R(r) = Z(x)x 1 ' 2 1
denoting a derivative with respect to a: by a prime,
R' = Z'x~ 1/2 - \Zx~ 3/2
R" = Z"x~ 1/2 - \Z'x- z/2 - \Z'x~ 3/2
- \{-\)Zx-^ 2
-- Z"x~ 1/2 - Z'x
*.-»/* + \Zx~ b ' 2 ,
(9)
(10)
x 2 {Z"x- 1 ' 2 - Z'x~ 3 ' 2 + \Zx- 5 ' 2 )
+2x{Z'x- 1/2 - \Zx~ 3/2 ) + [x 2 - n(n+ l)]Zx~ 1/2 =
(11)
x 2 (Z" - Z'x' 1 + \Zx~ 2 ) + 2x(Z' - \Zx~ x )
+[x 2 - n(n + \)]Z = (12)
x 2 Z" + (-x + 2x)Z' + [f-l + x 2 -n(n + l)}Z = (13)
But the solutions to this equation are BESSEL FUNC-
TIONS of half integral order, so the normalized solutions
to the original equation are
R(r) = A
Jn+i/2(kr)
ifkr
+ B
'n+l/2
(kr)
yfkr
(15)
which are known as SPHERICAL BESSEL FUNCTIONS.
The two types of solutions are denoted j n (x) (SPHERI-
CAL Bessel Function of the First Kind) or n n (x)
(Spherical Bessel Function of the Second Kind),
and the general solution is written
where
R(r) = A'j n (kr) + B'n n {kr),
n n (z)
_ /7T in+1/2
(z)
V~z
(16)
(17)
(18)
see also Spherical Bessel Function, Spherical
Bessel Function of the First Kind, Spherical
Bessel Function of the Second Kind
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 437, 1972.
Spherical Bessel Function
A solution to the Spherical Bessel Differential
Equation. The two types of solutions are denoted j n (x)
(Spherical Bessel Function of the First Kind) or
n n (x) (Spherical Bessel Function of the Second
Kind).
see also SPHERICAL BESSEL FUNCTION OF THE FIRST
Kind, Spherical Bessel Function of the Second
Kind
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bes-
sel Functions." §10.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 437-442, 1972.
Arfken, G. "Spherical Bessel Functions." §11.7 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 622-636, 1985.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Bessel Functions of Fractional Order, Airy
Functions, Spherical Bessel Functions." §6.7 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 234-245, 1992.
Z" + xZ' + [x 2 -(n 2 +n+ \)}Z =
x 2 Z" + xZ' + [x 2 - (n + \f\Z -
(14)
Spherical Bessel Function
Spherical Bessel Function of the First Kind
Spherical Cap 1689
M x )
2 n x 7]
(2n+l)!!
J n +i/2(x)
2-^ s!(2s + 2n+l)!
s=0
1-
l!(2n + 3)
+ 7
(I- 2 ) 5
+ .
2!(2n + 3)(2n + 5)
= ( .i)v(4) B -.
V x dx } x
The first few functions are
. , . sin x
3o(x) =
x
. . x sinx cosx
k( x ) = ~2 ~
x* x
• ( \ ( 3 x \ ■ 3
see also Poisson Integral Representation, Ray-
leigh's Formulas
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bes-
sel Functions." §10.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 437-442, 1972.
Arfken, G. "Spherical Bessel Functions." §11.7 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 622-636, 1985.
Spherical Bessel Function of the Second
Kind
Un(x) = ^—Y n+1/2 {x)
(-1)
Tl + 1
r£
2 n x"
71 =
(2n-l)J!
x n+i
(-l)'(s -n)! a ,
s\(2s-2n)\
±x 2
2 X
2!(l-2n)(3-2n)
/ 7T
2x
1!(1 - 2n)
+
= (-ir\/^j-n- 1/2 w.
The first few functions are
cos a;
71q(x) = — -
m(x) = -
X
cos x sin x
n2( * ) = ~GI4)
_3_
x 2
see also Rayleigh's Formulas
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bes-
sel Functions." §10.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 437-442, 1972.
Arfken, G. "Spherical Bessel Functions." §11.7 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 622-636, 1985.
Spherical Bessel Function of the Third Kind
see Spherical Hankel Function of the First
Kind, Spherical Hankel Function of the Second
Kind
Spherical Cap
A spherical cap is the region of a Sphere which lies
above (or below) a given PLANE. If the PLANE passes
through the Center of the Sphere, the cap is a Hemi-
sphere. Let the Sphere have Radius R, then the Vol-
ume of a spherical cap of height h and base RADIUS a is
1690 Spherical Cap
Spherical Coordinates
given by the equation of a SPHERICAL SEGMENT (which
is a spherical cut by a second Plane)
^spherical segment = g7r/l(3a +36 + h )
with 6 = 0, giving
Vcap = |7r/i(3a -f h ).
Using the PYTHAGOREAN THEOREM gives
(R~h) 2 + a =i* 2 ,
which can be solved for a 2 as
a =2itt-/i 2 ,
and plugging this in gives the equivalent formula
Ka P = l*h 2 (3R-h).
(1)
(2)
(3)
(4)
(5)
In terms of the so-called Contact Angle (the angle
between the normal to the sphere at the bottom of the
cap and the base plane)
R - h = Rsint
aEsin
(V).
(6)
(7)
Vcap = |tt^ 3 (2 - 3 sin a + sin 3 a). (8)
Consider a cylindrical box enclosing the cap so that the
top of the box is tangent to the top of the Sphere. Then
the enclosing box has VOLUME
Vbox = 7ra 2 h — 7v(Rcosa)[R(l — sin a)]
= 7rR 3 (l - sina — sin 2 a -f sin 3 a), (9)
so the hollow volume between the cap and box is given
by
Vbox - Kap = f 7Ti? 3 (l - 3sin 2 Q + 2 SUl 3 a). (10)
If a second Plane cuts the cap, the resulting Spher-
ical Frustum is called a Spherical Segment. The
Surface Area of the spherical cap is given by the same
equation as for a general ZONE:
'S'cap = %7vRh.
(11)
see also CONTACT ANGLE, DOME, FRUSTUM, HEMI-
SPHERE, Solid of Revolution, Sphere, Spherical
Segment, Torispherical Dome, Zone
Spherical Coordinates
z
A system of CURVILINEAR COORDINATES which is nat-
ural for describing positions on a SPHERE or SPHEROID.
Define to be the azimuthal ANGLE in the rcy-PLANE
from the z-AxiS with < 6 < 2n (denoted A when re-
ferred to as the LONGITUDE), <j> to be the polar ANGLE
from the z- AXIS with < <f> < it (COLATITUDE, equal
to <j> — 90° - 5 where S is the LATITUDE), and r to be
distance (Radius) from a point to the ORIGIN.
Unfortunately, the convention in which the symbols 9
and <f> are reversed is frequently used, especially in phys-
ics, leading to unnecessary confusion. The symbol p is
sometimes also used in place of r. Arfken (1985) uses
(r, </>, 0), whereas Beyer (1987) uses (p, 0, <j>). Be very
careful when consulting the literature.
In this work, the symbols for the azimuthal, polar, and
radial coordinates are taken as 0, </>, and r, respectively.
Note that this definition provides a logical extension of
the usual Polar Coordinates notation, with 6 re-
maining the ANGLE in the ay-PLANE and <j> becoming
the Angle out of the PLANE.
■ \fx 2 J r y 2 +z 2
--' (i)
■'(;)•
(l)
(2)
(3)
where r £ [0, oo), 6 [0, 27r), and <f> 6 [0,7r]. In terms of
Cartesian Coordinates,
x = r cos sin <j>
y = r sin sin <j>
z = r cos 0.
The Scale Factors are
h r = 1
h$ = r sin
h,j> = r,
(4)
(5)
(6)
(7)
(8)
(9)
Spherical Coordinates
Spherical Coordinates 1691
so the Metric Coefficients are
g rr = 1
gee = r 2 sin 2
9<t>4> = r .
The Line Element is
ds = drr + r d<j> </> + r sin dO 0,
the AREA element
da. = r 2 sin d0 d0 f ,
and the VOLUME ELEMENT
The Gradient is
dV = r 2 sin dO d<j> dr.
The JACOBIAN is
S(r,M)
: r I sin 0| .
The Position Vector is
" r cos sin
r = r sin sin
r cos
so the Unit Vectors are
f =
dr
1 — 1 _
1 dr |
" cos 6 sin
sin 9 sin
COS0
*1
4>
e =
dr
d$ _
1 dr \
" — sin 9 '
cos 9
\d9\
=
dr
d<j> _
\dr\ ~
\ d<p \
" cos 9 CO
sin co
— sin <
50"
3
Derivatives of the Unit Vectors are
dr
Dr
Or
00
86
do
dO
dv
dO
d<f>
d<f>
— sin 9 sin
cos 9 sin
— cos#
— sin#
— sin 9 cos
cos 9 cos
cos#
sin 9 cos
— sin
= sin
cos — sin f
= COS
=
— cos 9 sin
— sin 9 sin
— cos
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
„ -9 1 x d 1 h d
or r oq) rsin0 oO
V r f =
V r = O
v r <£ = 6
sin 00
V«r :
= id
r sin r
(30)
(31)
(32)
(33)
(34)
Vfl g = _cos0</> + sin0f = _cot0^_l. (35)
r sin r r
„ A , COS 00 1 , . A
r sin r
(36)
Now, since the Connection COEFFICIENTS are given
by T) k = *i ■ (VfcX,),
(37)
(38)
(39)
p<P =
1 oi
T
COt <f> r\
COt <j> r\
.
1
-ij
The Divergence is
V • F = A% + r%A 3
= [A T <T + (T r rr A r + T r g T A 6 + r; r A*]
+[A% + (T e Te A r + r e ee A e + T%A*)]
g T dr + g e 86 * g* dcj> +l + + '
+ (l A r + + S2^ A *) + (l A r +0 + 0)
\r r ) \r /
1
= JLa v + -A r +
dr r r sin 30
or, in Vector notation,
8 A » + 10 A *+!*± A * t
r oq> r
\r dr J
-(r 2 F r ) +
„ , 1 d cot
r a0 r
r y sin
(40)
3F*
30
dr
r sin 30
(sm^Ji-r^r^-. (41)
rsin0 d9
The Covariant Derivatives are given by
gkk oxk
(42)
1692 Spherical Coordinates
Spherical Coordinates
so
or or
A r ,e =
1 0A r
■tU
1 8A r
r sin <f> dd ~ ro r sin 06
1 0A r _ A^
rsin0 d(j) r
T r9 A 9
Ar:d
I dA r
r d(f>
V i A. _ 1 Mr _ ^ ,
r 50
= K^"^)
^^ = ~~E A OrAi — — —
ar ar
(43)
9
(44)
(45)
(46)
Ae-.e =
1 &4
- TieAi
r sin #0
—^-OAedO - T^A* - rj,j4 r
r sin
1 <9A# cot
rsin0 06
+
H
r r
_ldA e ri , dA 9
e " t '~r~dF~ * r ** d<j>
A.fc r —
dAj
Or
1 d>r-^i —
dA*
(47)
(48)
(49)
1 dA<t>
r sin 06
TLAi
i oa
rsin0 06
<f> t^Q
- 4>9
1 OA<f> cot
r sin 06
ldA<t>
A e
r 0(j) r
i _ 1 dA<t> r
(50)
(51)
The Commutation Coefficients are given by
c a/3^ = [e tt ,e/3] = V a e/3 - V,ge a (52)
[*,*] = [0,0] = [£,0 = 0, (53)
so Crr = c% 9 = c%& = 0, where a = r, 0, 0.
[f,0] = -[0,r] = V.0 - V*r = - -6 = --0, (54)
so c rd = -c er = --, c^ = c^ = 0.
M] = -0,*] = O-±0=-±£, (55)
[0, 0] = -[0, 0] = - cot 00 - = - cot 00, (56)
izing,
ro
01
c r =
.0
0.
(58)
ro
_ 1
1
c $ =
i
T
±COt0
(59)
Lo
-±cot0 J
ro
o -\]
c* =
(60)
i
c
)
Time derivatives of the POSITION VECTOR are
" cos 6 sin r — r sin sin 00 + r cos 6 cos '
sin sin r + r cos sin + r sin cos
cos <f>r — r sin
" cos 6 sin "I r — sin 6
sin sin r + r sin cos £
COS0 J L
" COS COS "
+ r sin 6 cos
— sin
— f r + r sin00 + r 00. (61)
The Speed is therefore given by
v = |r| = \/r 2 +r 2 sin 2 00 2 +r 2 2 . (62)
The Acceleration is
£ — ( — sin 6 sin 00r + cos 6 cos 0r0 + cos 6 sin 0r)
— (sin 6 sin 0f + r cos 6 sin 00 2 + r sin cos 000
+ r sin sin 00) + (cos 6 cos 0r0 — r sin 6 cos 00
— r cos sin 00 + r cos cos 00)
= —2 sin sin 00r + 2 cos cos 0r0 — 2r sin cos 000
+ cos sin (j>r — r sin sin 00 + r cos cos 00
-rcos0sin0(0 2 + 2 ) (63)
j/ = (sin sin 0f + r cos sin 00 + r cos sin 00)
+ (cos sin 0r0 — r sin sin 00 2 + r cos cos 000
+ r cos sin 00) + (sin cos 0r0 + r cos cos
— r sin sin 00 2 + r sin cos 00)
— 2 cos sin 00r + 2 sin cos 0r0 + 2r cos cos
+ sin sin 0f + r cos sin 00 + r sin cos 00
-rsin0sin0(0 2 +0 2 ) (64)
z = (cos 0f — sin 0r0) — (f sin 00 + r cos 00 2 4- r sin 00)
-r cos 00 + cos 0r — 2 sin 00r — r sin 00.
(65)
o o 1 , i
C^^ = -C^ = - COt 0.
(57)
Spherical Coordinates
Plugging these in gives
" cos sin '
r = (r — r<j> )
sin sin
COS
" — sin "
+ (2r cos (j>0(j) + r sin 00)
cos0
" COS COS "
'cos 6
+ (2r0 + r0)
sin cos ^
— r sin 00 2
sin0
— sin
L o
(66)
but
sin 0r + COS 0</> =
cos sin 2 + cos cos 2
sin sin 2 + sin cos 2
cos
sin0
(67)
r = (r — 7*0 )r + (2r cos 000 + 2 sin 00r + r sin 00)0
+ (2r0 + r<j))<p — r sin 00 (sin 0r + cos (fxj))
= (f — r0 — r sin 00 )r
+ (2 sin 00r + 2r cos 000 + r sin 00)0
+(2r0 + rij> - r sin cos 00 2 )</>. (68)
Time DERIVATIVES of the UNIT VECTORS are
— sin sin 00 + cos cos
cos sin 00 + sin cos
— sin
0:
0:
-cos 001
* cos "
-sin 00
= -0
sin0
— sin cos — cos sin
cos cos — sin sin
— cos
sin 00 + 00
(69)
= — 0(sin 0f + cos 00)
(70)
= — 0r + cos 000.
(71)
The Curl is
V x
f- x
£<-*»> -^
r sin0
1
+ -
r
i dF r
sin0 #0
!<•*»
r
' d
di
0.
(72)
Spherical Coordinates
The Laplacian is
l d / 2 d\ . l a 2
1693
v 2 =
+
r 2 dr
1 3
r 2 sin 50
( r2 ^)
r 2 sin 2 S0 2
( Sin0 ^)
r 2 \
r . 2 _^ +2r A ) +
5r 2 dr
r 2 sin 2 50 2
r 2 sin
COS0 — +Sm0^-j
50
1
<90 2
5 2
d 2 2d_
dr 2 r dr r 2 sin 2 50 2
cos0 d Id 2
_l r 1 .
r 2 sin 50 r 2 50 2
The vector LAPLACIAN is
(73)
V 2 V:
1 d 2 (rv r )
+ ;
2 Qv$
ae 2
r 2 sin 2 8 d<f> 2
2 9 v d> 2i
1 g 2 (rv g ) , 1 fi 2 t>
r fir 2 "•" r 2 a^ 2
r 2 sin 9 0<j>
2 2 cot dv <f>
9 2 v e
r 2 sin 2 6 d<p 2
r* r* sin t
+
2 0«
Sr 2 ~T r2 902 "T r 2 sin 2 <j 5 02
50
+
2 cot fl 3^0
10 0<£
2 cot 9 .
r 2 v »
cot 6v&
r 2 ee
v
r 2 sin 2 6
i cot 9l V
^ T> 2 fi0
*0
(74)
To express Partial Derivatives with respect to Carte-
sian axes in terms of Partial Derivatives of the spher-
ical coordinates,
r cos sin 0]
r sin sin (75)
r cos
cos sin (f>dr — r sin sin dO + r cos cos d0
sin sin dr + r sin cos dO -\- r sin cos d(f>
cos <f>dr — r sin d0
"x"
y
=
. z .
dx~
dy
=
dz\
=
cos sin — rsin0sin0 rcos0cos0
sin sin r sin cos r sin cos
cos0 — r sin0
dr
d6
d(f>
(76)
Upon inversion, the result is
dr
dO
d<t>
cos 9 sin sin sin cos
sin cos 9
r sin
cos 6 cos <^>
r sin <f>
sin 6 cos ^
sin 4>
dx
dy
dz
(77)
1694 Spherical Design
Spherical Hankel Function of the Second Kind
The Cartesian Partial DERIVATIVES in spherical coor-
dinates are therefore
_d__cfrd_ dl_d_ d$d_
dx dx dr dx dO dx d<p
= cos 6 sin <f>-
d
dr r sin <j>
sin d cos cos .<p d
d(f>
(78)
dy dy dr dy dO dy dcj)
, _ . . d cos 9 d sin cos <b d
= sin0sm0— H ^-?^ H ^r
or r sin at/ r a<p
5 5r 9
dz
<90_d d±^
dz dr + dzd9 + dz d(j>
COS0
d sin</> d
dr r d<p
(79)
(80)
(Gasiorowicz 1974, pp. 167-168).
The Helmholtz Differential Equation is separable
in spherical coordinates.
see also Colatitude, Great Circle, Helmholtz
Differential Equation — Spherical Coordinates,
Latitude, Longitude, Oblate Spheroidal Coor-
dinates, Prolate Spheroidal Coordinates
References
Arfken, G. "Spherical Polar Coordinates." §2.5 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, pp. 102-111, 1985.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 212, 1987.
Gasiorowicz, S. Quantum Physics. New York: Wiley, 1974.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, p. 658, 1953.
Spherical Design
X is a spherical i-design in E Iff it is possible to exactly
determine the average value on E of any Polynomial
/ of degree at most t by sampling / at the points of X.
In other words,
1
/ /(0de =
I E
1
volume E t
1*1
References
£/(*)■
xex
Colbourn, C. J. and Dinitz, J. H. (Eds.) "Spherical t-
Designs." Ch. 44 in CRC Handbook of Combinatorial De-
signs. Boca Raton, FL: CRC Press, pp. 462-466, 1996.
Spherical Excess
The difference between the sum of the angles of a
Spherical Triangle and 180°.
see also ANGULAR DEFECT, DESCARTES TOTAL ANGU-
LAR Defect, Girard's Spherical Excess Formula,
L'Huilier's Theorem, Spherical Triangle
Spherical Frustum
see Spherical Segment
Spherical Geometry
The study of figures on the surface of a Sphere (such as
the Spherical Triangle and Spherical Polygon),
as opposed to the type of geometry studied in PLANE
Geometry or Solid Geometry.
see also Plane Geometry, Solid Geometry, Spher-
ical Trigonometry, Thurston's Geometrization
Conjecture
Spherical Hankel Function of the First Kind
h£\x) == ^H^ 1/2 (x) = j n (x) + in„(s),
where H^(x) is the Hankel Function of the First
Kind and j n {x) and n n {x) are the SPHERICAL Bessel
Functions of the First and Second Kinds. Explic-
itly, the first few are
%
■ — t
x
Hq *(x) = —(sin a? — icosx) — — — e 1,
M"w=r«(-i-i)
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Spherical Bes-
sel Functions." §10.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 437-442, 1972.
Spherical Hankel Function of the Second
Kind
hg\x) = iJ^H™ 1/2 (x) = j n (x) - tn n (x),
where H {2) (x) is the HANKEL FUNCTION OF THE SEC-
OND Kind and j n (x) and n n (x) are the SPHERICAL BES-
SEL Functions of the First and Second Kinds. Ex-
plicitly, the first is
/in 2 (x) — — (sinx + icosx) = -e
X X
References
Abramowitz, M. and Stegun, C A. (Eds.). "Spherical Bes-
sel Functions." §10.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 437-442, 1972.
Spherical Harmonic
Spherical Harmonic 1695
Spherical Harmonic
The spherical harmonics Yj m (0,0) are the angular por-
tion of the solution to Laplace's Equation in Spher-
ical Coordinates where azimuthal symmetry is not
present. Some care must be taken in identifying the no-
tational convention being used. In the below equations,
9 is taken as the azimuthal (longitudinal) coordinate,
and as the polar (latitudinal) coordinate (opposite the
notation of Arfken 1985).
y ™W^(^ Pr(cos * )e< " (1)
where m = — Z, —1 + 1, .
tion is chosen such that
-,o,
/ and the normaliza-
p 2-K /»7T
Jo Jo
Y^Y™ sin <t>d(j)de
/»27T />1
Jo J -i
Y t m YF d(cos<f>)de = 6 mrn ,5 llf , (2)
where Smn is the Kronecker Delta. Sometimes, the
Condon-Shortley Phase (-l) m is prepended to the
definition of the spherical harmonics.
Integrals of the spherical harmonics are given by
J YiT y iT y iT dU = y(Mi + l)(2h + l)(2/,Tj):
x (h h h\(h h h \ (3)
\0 0y yrm m 2 m 3 J' w
where f 2 3 is a WlGNER 3J-SYMBOL
V 771 1 TU2 TTls J
(which is related to the CLEBSCH-GORDON COEFFI-
CIENTS). The spherical harmonics obey
yr
i
(21 + 1)! . / , -
- — - — — sm 0e
47T
2/ + 1
Pi (cos 0)
47T
t — (-1) Y x ,
where P t (x) is a Legendre Polynomial.
(4)
(5)
(6)
Y§(e, &
Y?(6, <p) Yi(6,
Y°2(B, <
Yl(B, <p)
Yi(8, (p)
i »*.
*>»
I *c # *
The above illustrations show |y™(0,0)| (top) and
R[Yr(0,<p)] and S[y, m (M)] (bottom). The first few
spherical harmonics are
1 1
20r~
t'-IVs*^
*-W§
COS0
v i 1 / 3 . . «
- -_2 1 / 15 . 2 , -2i0
i 1 / 15 . , , _i^
Yo = -\ — sin © cos 6 e
2 2 A/ 2tt r
^2
y, 1
■o_ 1 /5
2 "IV*
(3 cos 2 0-1)
1 / 15 • A A i*
~~\l Sin $ C0S $ e
2 V 27T
4 V 27T
3 1 /35 3 , -3t«
I, = — a / — sin © e
3 8 V 7T
y, = t
1 /105 .
4 V 2?r
sin 0cos0e
-2%e
Y 3 l — - \ — sin 0(5 cos — l)e
y 3 ° = - J - (5 cos 3 - 3 cos 0)
Y£ = -iJ— sin0(5cos 2 0-l)e^
2 1 /105 . 2 j , lid
y 3 = -a —— sm 0cos0e
4 V 27r
1 /35 . 3 , 3ifl
y/(ft <
7/(0, KiC ft Yi(0, .
1696 Spherical Harmonic
Written in terms of CARTESIAN COORDINATES,
ie _ x + iy
Y "
Y?
1 1
1 /3
2 V *■ ^Jx 2 +y 2 + z 2
1/3 x + iy
2 V 25r y/x 2 + y 2 +z 2
r o_l[5( 3z 2 A
2 4 V 7T ^a; 2 +j/ 2 + z 2 y
<-i _ ! / 15 z(x + iy)
2\ 2wx 2 +y 2 + z 2
2 _ 1 /lj" (x + iy) 2
4 V 2tt a; 2 + y 2 + z 2 '
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
These can be separated into their Real and IMAGINARY
Parts
Yr\6,<f>) = ^(cos^sin^) (16)
Y( mc (M) = Pr (cos <f>)cos{mO).
(17)
The Zonal Harmonics are defined to be those of the
form
P?(cos9). (18)
The Tesseral Harmonics are those of the form
sm(m<t>)Pn(cosO) (19)
cos(m<t>)P™(cosQ) (20)
for n^m. The SECTORIAL Harmonics are of the form
sin(m<p)P™(cose) (21)
cos(m<p)P™(cosO). (22)
The spherical harmonics form a COMPLETE Orthonor-
MAL Basis, so an arbitrary Real function f(6><f>) can
be expanded in terms of COMPLEX spherical harmonics
f(9,4>) = Y^ £ ATY t m (e,4>),
(23)
Spherical Harmonic Addition Theorem
or Real spherical harmonics
oo I
1=0 m~Q
(24)
see also CORRELATION COEFFICIENT, SPHERICAL HAR-
MONIC Addition Theorem, Spherical Harmonic
Closure Relations, Spherical Vector Harmonic
References
Arfken, G. "Spherical Harmonics." §12.6 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 680-685, 1985.
Ferrers, N. M. An Elementary Treatise on Spherical Harmon-
ics and Subjects Connected with Them. London: Macmil-
lan, 1877.
Groemer, H. Geometric Applications of Fourier Series and
Spherical Harmonics. New York: Cambridge University
Press, 1996.
Hobson, E. W. The Theory of Spherical and Ellipsoidal Har-
monics. New York: Chelsea, 1955.
MacRobert, T. M. and Sneddon, I. N. Spherical Harmonics:
An Elementary Treatise on Harmonic Functions, with Ap-
plications, 3rd ed. rev. Oxford, England: Pergamon Press,
1967.
Press, W. H,; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Spherical Harmonics." §6.8 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 246-248, 1992.
Sansone, G. "Harmonic Polynomials and Spherical Harmon-
ics," "Integral Properties of Spherical Harmonics and the
Addition Theorem for Legendre Polynomials," and "Com-
pleteness of Spherical Harmonics with Respect to Square
Integrable Functions." §3.18-3.20 in Orthogonal Func-
tions, rev. English ed. New York: Dover, pp. 253-272,
1991.
Sternberg, W. and Smith, T. L. The Theory of Potential
and Spherical Harmonics, 2nd ed. Toronto: University of
Toronto Press, 1946.
Spherical Harmonic Addition Theorem
A Formula also known as the Legendre Addition
Theorem which is derived by finding Green's Func-
tions for the Spherical Harmonic expansion and
equating them to the generating function for LEGEN-
DRE Polynomials. When 7 is defined by
cos 7 = cos $1 cos 82 + sin #1 sin 62 cos <fii — <f> 2 ,
p " (coS7) = drr £ (-irc(«i^r m («^)
47T
2n +
T J2 Y£(0u<t>i)Y™(h><h)
1=0 ro=-l
= F 7l (cos<9i)P n (cos6> 2 )
(n - m)\ nn
+2 J2 ^^T^m(cos^)P-(cos^)cos[m(^-^ 2 )].
Spherical Harmonic Closure Relations
Spherical Ring 1697
References
Arfken, G. "The Addition Theorem for Spherical Harmon-
ics." §12.8 in Mathematical Methods for Physicists, 3rd
ed. Orlando, FL: Academic Press, pp. 693-695, 1985.
Spherical Harmonic Closure Relations
The sum of the absolute squares of the SPHERICAL HAR-
MONICS y; m (i9,0) over all values of m is
Y, |iT(M)| a =
2 2/ + 1
47T
The double sum over m and / is given by
oo I
1=0 m~-l
J2 Yr(eu<t>i)Yr*{62,<h)
i
sin#i
5{9i - 62)5(<j> 1 - <f> 2 )
= <5(COS01 — COS 02)^(01 ~ 02),
where S(x) is the Delta Function.
Spherical Harmonic Tensor
A tensor defined in terms of the TENSORS which satisfy
the Double Contraction Relation.
see also Double Contraction Relation, Spherical
Harmonic
Spherical Helix
The Tangent Indicatrix of a Curve of Constant
PRECESSION is a spherical helix. The equation of a
spherical helix on a Sphere with Radius r making an
Angle 6 with the z-axis is
x(ip) = |r(l -f cos 0) cos i[)
-§r(l-cos0)cos(i±^) (1)
y(ip) = |r(l -f- cos 0) sin ip
-Hl-co^)sin(i±^^) (2)
, ,n . ^ / cos 9 ,\ ,.
z{ip) = rsmd cos [- -i/)) . (3)
\ J. COS O /
The projection on the ay-plane is an Epicycloid with
Radii
a = r cos 6 (4)
b = rsin 2 (±0). (5)
see also HELIX, LOXODROME, SPHERICAL SPIRAL
References
Scofield, P. D, "Curves of Constant Precession." Amer.
Math. Monthly 102, 531-537, 1995.
Spherical Point System
How can n points be distributed on a Sphere such that
they maximize the minimum distance between any pair
of points? This is Fejes Toth's Problem.
see also Fejes Toth's Problem
Spherical Polygon
A closed geometric figure on the surface of a Sphere
which is formed by the ARCS of Great CIRCLES. The
spherical polygon is a generalization of the SPHERICAL
Triangle. If is the sum of the Radian Angles of
a spherical polygon on a Sphere of Radius r, then the
Area is
S = [0 - (n - 2)n]r 2 .
see also GREAT CIRCLE, SPHERICAL TRIANGLE
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 131, 1987.
Spherical Ring
A Sphere with a Cylindrical Hole cut so that the
centers of the Cylinder and Sphere coincide, also
called a Napkin Ring.
The volume of the entire Cylinder is
V C y\ = ttLR ,
and the VOLUME of the upper segment is
V seg = \<Kh{m 2 + h 2 ),
where
R=fr
\V
(1)
(2)
(3)
(4)
so the Volume removed upon drilling of a CYLINDRICAL
hole is
Kern = V cy l + 2V seg = 7T[LR 2 + \h{?>R 2 + h*)]
- tt(LR 2 + hR 2 + f/i 3 )
= .[L{r 2 ~\L 2 ) + {r-\L){r 2 -\L 2 )
\L,
+ \{r-\Lf]
12
v 2 L - \RL 2 + iL 3 )
3 V 2 J
= 4Lr 2 - ii 3 + (r 3 2 , „ 4 _ , gJ
+ |(r 3 -|r 2 L+|rL 2 -|L 3 )]
= ^[|r 3 + (l-i-i)r 2 L + (-i + l)M 2
4 3 1 r 3
§7r(8r 3 - L 3 ),
(5)
1698 Spherical Sector
Spherical Spiral
so
Vieft = Sphere " ^rem = l^ 3 - (f TTr* - |ttL 3 ) = \irL\
(6)
Spherical Sector
The Volume of a spherical sector, depicted above, is
given by
V = fTrRX
where h is the vertical height of the upper and lower
curves.
see also CYLINDRICAL SEGMENT, SPHERE, SPHERICAL
Cap, Spherical Segment, Zone
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 131, 1987.
Spherical Segment
A spherical segment is the solid defined by cutting a
Sphere with a pair of Parallel Planes, It can be
thought of as a Spherical Cap with the top truncated,
and so it corresponds to a Spherical Frustum. The
surface of the spherical segment (excluding the bases) is
called a ZONE.
Call the Radius of the Sphere R and the height of
the segment (the distance from the plane to the top of
Sphere) h. Let the Radii of the lower and upper bases
be denoted a and 6, respectively. Call the distance from
the center to the start of the segment d, and the height
from the bottom to the top of the segment h. Call the
Radius parallel to the segment r, and the height above
the center y. Then r 2 = R 2 — y 2 ,
pd+h
pd-\-h pd
V = 7rr 2 dy = 7r [R z - y z ) dy
J d J d
1„, 3 1 d+h
id
= n [R 2 y - |y 3 ] ? h = n{R 2 h - i [(d + hf - d*]}
= iz[R 2 h - §(d 3 + Sd 2 h + 3h 2 d + h s - d 3 )]
= ir{R 2 h-d 2 h-h 2 d-\h z )
= nh(R 2 -d 2 -hd- |/i 2 ).
Using
a 2 =R 2 - d 2
(1)
(2)
b 2 = R 2 - (d+h) 2 =:R 2 -d 2 - 2dh - h 2 , (3)
gives
a + b 2 = 2iT - 2d 2 - 2dh - h
R 2 -d 2 ~dh= ±(a 2 +6 2 + /i 2 ),
(4)
(5)
V = nh[l(a 2 + b 2 + h 2 ) - \h 2 ) = ith{\a 2 + \b 2 + \h 2 )
= |7r/i(3a 2 +36 2 + ^ 2 ).
(6)
The surface area of the ZONE (which excludes the top
and bottom bases) is given by
S = 27rRh.
(7)
see also ARCHIMEDES' PROBLEM, FRUSTUM, HEMI-
SPHERE, Sphere, Spherical Cap, Spherical Sec-
tor, Surface of Revolution, Zone
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 130, 1987.
Spherical Shell
A generalization of an Annulus to 3-D. A spherical shell
is the intersection of two concentric BALLS of differing
Radii.
see also Annulus, Ball, Chord, Sphere, Spherical
Helix
Spherical Spiral
Spherical Symmetry
Spherical Trigonometry 1699
The path taken by a ship which travels from the south
pole to the north pole of a SPHERE while keeping a fixed
(but not Right) Angle with respect to the meridians.
The curve has an infinite number of loops since the
separation of consecutive revolutions gets smaller and
smaller near the poles. It is given by the parametric
equations
x = cos t cos c
y = sin t cos c
z = — sin c,
where
c = tan l (at)
and a is a constant.
see also Mercator Projection, Seifert's Spheri-
cal Spiral
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 162, 1993.
Lauwerier, H, "Spherical Spiral." In Fractals: Endlessly Re-
peated Geometric Figures. Princeton, NJ: Princeton Uni-
versity Press, pp. 64-66, 1991.
Spherical Symmetry
Let A and B be constant VECTORS. Define
Q = 3(A.r)(B.f)-A.B.
Then the average of Q over a spherically symmetric sur-
face or volume is
(Q) = (3cos 2 0-l)(A.B) = O,
since /3cos 2 — l\ = over the sphere.
Spherical Tessellation
see Triangular Symmetry Group
Spherical Triangle
A spherical triangle is a figure formed on the surface of a
sphere by three great circular arcs intersecting pairwise
in three vertices. The spherical triangle is the spherical
analog of the planar TRIANGLE. Let a spherical triangle
have Angles a, /3, and 7 and RADIUS r. Then the
Area of the spherical triangle is
*f = r 2 [(a + /? + 7 )-7r].
The sum of the angles of a spherical triangle is between
180° and 540°. The amount by which it exceeds 180° is
called the SPHERICAL Excess and is denoted E or A.
The study of angles and distances of figures on a sphere
is known as SPHERICAL TRIGONOMETRY.
see also Colunar Triangle, Girard's Spherical
Excess Formula, L'Huilier's Theorem, Spherical
Polygon, Spherical Trigonometry
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 79, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 131 and 147-150, 1987.
Spherical Trigonometry
Define a SPHERICAL TRIANGLE on the surface of a unit
Sphere, centered at a point O, with vertices A, B }
and C. Define Angles a = LBOC, b = LCOA, and
c = /.AOB. Let the Angle between Planes AOB and
AOC be a, the Angle between Planes BOC and AOB
be 0, and the Angle between Planes BOC and AOC
be 7. Define the VECTORS
EL=Ot
h = oi
c = od.
(1)
(2)
(3)
Then
(a x b) • (a x c) = (|a| |b| sinc)(|a| |c| sin6) cosa
= sin 6 sin c cos a . (4)
Equivalently,
(a x b) • (a x c) = a • [b x (a x c)]
= a • [a(b • c) — c(a • b)]
= (b-c)-(a.c)(a-b)
= cos a — cos c cos b. (5)
Since these two expressions are equal, we obtain the
identity
cos a = cos b cos c + sin b sin c cos a
(6)
The identity
I (a x b) x (a x c)|
sin a = j -^ — * —
|a x b||a x c|
_ [a,b,c]
sin b sin c '
]a[b,a,c]+b[a,a,c][
sin b sin c
(7)
1700 Spherical Trigonometry
Spherical Trigonometry
where [a,b,c] is the Scalar Triple Product, gives a
spherical analog of the LAW OF SlNES,
sin a _ sin/3 _ sin 7 __ QVol(OABC)
sin a sin b sin c sin a sin b sin c '
(8)
where Vo\(OABC) is the Volume of the Tetrahe-
dron. From (7) and (8), it follows that
sin a cos j3 = cos b sin c — sin b cos c cos a (9)
cos a cos 7 = sin a cot 6 — sin 7 cot /3. (10)
These are the fundamental equalities of spherical
trigonometry.
There are also spherical analogs of the Law of COSINES
for the sides of a spherical triangle,
cos a = cos 6 cose + sin b sin c cos A (11)
cos& — cosccosa + sin c sin a cos I? (12)
cose = cos a cos b + sin a sin b cos C, (13)
and the angles of a spherical triangle,
cos A=— cos B cos C + sin B sin C cos a (14)
cos B — — cos C cos ^4 + sin C sin .4 cos b (15)
cos C = — cos ^4 cos B + sin A sin i? cos c (16)
(Beyer 1987), as well as the Law OF TANGENTS
tan[i(fl-fc)]_tan[l(A-B)]
tan[±(a + 6)] tan[±(i4 + B)]'
Let
5~ |(a + 6 + c)
S=f(A + £ + C),
then the half-angle formulas are
J sm(s - a)
h
tW 1 R) —
J sin(s — 6)
rW 1 ^- *
1 sm(s - c)
where
b 2 -
sin(s — a) sin(s — 6) sin(s — c)
sms
and the half-side formulas are
(17)
(18)
(19)
(20)
(21)
(22)
= tan 2 r, (23)
tan(ia) = K cos(S - A)
tan(i&) = K cos(S - B)
tan(ic) = Kcos(S-C),
(24)
(25)
(26)
where
K z
cos S
(cos(S - A) cos(5 - B) cos(5 - C)
= tan R,
(27)
where R is the RADIUS of the SPHERE on which, the
spherical triangle lies.
Additional formulas include the HAVERSINE formulas
hava = hav(6 — c) + sin6sincsin(s — c) (28)
sin(s — b) sin(s — c)
hav A =
(29)
• a • ( 3 °)
sin b sin c
hav[?r- (£ + <?)] + sin B sin C hava, (31)
sin b sin c
hava — hav(6 — c)
Gauss's Formulas
sin[i(a-fe)] sin[i(A-B)]
sin(|c)
sin[|(a + 6)]
sin(|c)
cos[|(a — 6)]
cos(|c)
cos(|C)
cos[±(A-B)]
sin(fC)
sin[^(A + g)]
cos(|C)
cos[|(a + 6)] cos[±(A + B)]
cos(|c)
and Napier's Analogies
sin(iC)
sin[±(A-B)] _ tan[|(a-6)]
sin[i(A + S)]
cos[*(A-B)]
cos[i(A + B)]
sin[f(a-fr)]
sin[|(a + fe)]
cos[|(a-6)]
tan(fc)
tan[|(a + &)]
tan(±c)
tan[|(A-B)]
cot(IC)
tan[i(A + S)]
cos[i(a + fe)] cot(fC)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(Beyer 1987).
see also Angular Defect, Descartes Total Angu-
lar Defect, Gauss's Formulas, Girard's Spher-
ical Excess Formula, Law of Cosines, Law of
Sines, Law of Tangents, L'Huilier's Theorem,
Napier's Analogies, Spherical Excess, Spherical
Geometry, Spherical Polygon, Spherical Trian-
gle
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 131 and 147-150, 1987.
Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed.,
rev. ed. Richmond, VA: Willmann-Bell, 1988.
Smart, W. M. Text- Book on Spherical Astronomy, 6th ed.
Cambridge, England: Cambridge University Press, 1960.
Spherical Vector Harmonic
Spider and Fly Problem 1701
Spherical Vector Harmonic
see Vector Spherical Harmonic
Spheroid
A spheroid is an ELLIPSOID
2 2 /i ■ 2 j 2-2/i'2jl 2 2 ±
r cos 6 sin <b r sin t/ sin d> r cos © H ,„ .
5 " + 5 ~ + 2-^ = 1 (!)
or o J c z
with two Semimajor Axes equal. Orient the Ellipse
so that the a and b axes are equal, then
r 2 cos 2 9 sin 2 <j> r 2 sin 2 sin 2 r 2 cos 2 <p
+
1,
1 (2)
(3)
a z c^
where a is the equatorial Radius and c is the polar
Radius. Here <f> is the colatitude, so take 5 = 7r/2 — <fi
to express in terms of latitude.
«2 rtrt „2 c 2 -2 p
r cos o r sin o
(4)
Spheroidal Wavefunction
Whittaker and Watson (1990, p. 403) define the internal
and external spheroidal wavefunctions as
5 ™ = 2ir (n + m)\ Pn {XV)Pn {COS0) sin ( ™ 0)
S™ = 27r ^~^; Q-(ir)Q-(cos^)^ n S (m0).
see a/so ELLIPSOIDAL HARMONIC, OBLATE SPHEROIDAL
Wave Function, Prolate Spheroidal Wave Func-
tion, Spherical Harmonic
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Spheroidal Wave
Functions." Ch. 21 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 751-759, 1972.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 642-644, 1953.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, 4th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Rewriting cos 2 5 = 1 — sin 2 5 gives
?+■'*'•'(?-?)-'
2 / i , 2 . 2 ^ a? - <?
r 1 + a sin d — ^-^—
(5)
= r 2 (l + sin 2 ^r^]=a 2 , (6)
Sphinx
A 6-Polyiamond named for its resemblance to the
Great Sphinx of Egypt.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
SO
a (l
2 2\ - 1 / 2
r = a ( 1 + sin o ~ —
Spider and Fly Problem
(7)
If a > c, the spheroid is Oblate. If a < c, the spheroid
is Prolate. If a — c, the spheroid degenerates to a
Sphere.
see also DARWIN-DE SlTTER SPHEROID, ELLIPSOID,
Oblate Spheroid, Prolate Spheroid
Spheroidal Harmonic
A spheroidal harmonic is a special case of the ELLIP-
SOIDAL Harmonic which satisfies the differential equa-
tion
_d_
dx
["-'>£]♦(*■
, c 2 x 2_rn Q
1-x 2
on the interval — 1 < x < 1.
see also ELLIPSOIDAL HARMONIC
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "A Worked Example: Spheroidal Harmon-
ics." §17.4 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 764-773, 1992.
p
WyS
! 24
fly *
32
In a rectangular room (a Cuboid) with dimensions 30' x
12' x 12', a spider is located in the middle of one 12' x 12'
wall one foot away from the ceiling. A fly is in the middle
of the opposite wall one foot away from the floor. If the
fly remains stationary, what is the shortest distance the
spider must crawl to capture the fly? The answer, 40',
can be obtained by "flattening" the walls as illustrated
above.
References
Pappas, T. "The Spider & the Fly Problem." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 218 and 233, 1989.
1702 Spider Lines
Spindle Torus
Spider Lines
see Epitrochoid
Spiegeldrieck
see FUHRMANN TRIANGLE
Spieker Center
The center of the Spieker Circle. It is the Centroid
of the Perimeter of the original Triangle. The third
Brocard Point is COLLINEAR with the Spieker center
and the ISOTOMIC CONJUGATE Point of its Incenter.
see also Brocard Points, Centroid (Triangle), In-
center, Isotomic Conjugate Point, Perimeter,
Spieker Circle, Taylor Center
References
Casey, J. A Treatise on the Analytical Geometry of the Point,
Line, Circle, and Conic Sections, Containing an Account
of Its Most Recent Extensions, with Numerous Examples,
2nd ed., rev, enl. Dublin: Hodges, Figgis, & Co., p. 81,
1893.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 226-229 and 249, 1929.
Kimberling, C "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Spieker Circle
The Incircle of the Medial Triangle. The center of
the Spieker circle is called the Spieker Center.
see also INCIRCLE, MEDIAL TRIANGLE, SPIEKER CEN-
TER
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 226-228, 1929.
Spigot Algorithm
An Algorithm which generates digits of a quantity one
at a time without using or requiring previously com-
puted digits. Amazingly, spigot ALGORITHMS are known
for both Pi and e.
Spijker's Lemma
The image on the RlEMANN SPHERE of any CIRCLE
under a COMPLEX rational mapping with NUMERATOR
and Denominator having degrees no more than n has
length no longer than 2mr.
References
Edelman, A. and Kostlan, E. "How Many Zeros of a Random
Polynomial are Real?" Bull. Amer. Math. Soc. 32, 1-37,
1995.
Spindle Cyclide
The inversion of a SPINDLE TORUS. If the inversion cen-
ter lies on the torus, then the spindle cyclide degenerates
to a Parabolic Spindle Cyclide.
see also Cyclide, Horn Cyclide, Parabolic Cy-
clide, Ring Cyclide, Spindle Torus, Torus
Spindle Torus
One of the three STANDARD TORI given by the para-
metric equations
x = (c + a cos v) cos u
y = (c + a cos v) sin u
z = a sin i?
with c < a. The exterior surface is called an Apple
and the interior surface a Lemon. The above left figure
shows a spindle torus, the middle a cutaway, and the
right figure shows a cross-section of the spindle torus
through the x^-plane.
see also Apple, Cyclide, Horn Torus, Lemon, Par-
abolic Spindle Cyclide, Ring Torus, Spindle Cy-
clide, 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.
Spinode
Spirograph 1703
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.
Spinode
see also Acnode, Crunode, Cusp, Tacnode
Spinor
A two-component COMPLEX column VECTOR. Spinors
are used in physics to represent particles with half-
integral spin (i.e., Fermions).
References
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.
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.
Spira Mirabilis
see Logarithmic Spiral
Spiral
In general, a spiral is a curve with t(s)/k(s) equal to a
constant for all s, where r is the TORSION and k, is the
Curvature.
see also ARCHIMEDES' SPIRAL, CIRCLE INVOLUTE,
Conical Spiral, Cornu Spiral, Cotes' Spi-
ral, Daisy, Epispiral, Fermat's Spiral, Hyper-
bolic Spiral, Logarithmic Spiral, Mice Problem,
Nielsen's Spiral, Phyllotaxis, Poinsot's Spirals,
Polygonal Spiral, Spherical Spiral
Spiric Section
References
uci . edu /
Eppstein, D. "Spirals." http:// www . ics
eppstein/ junkyard/ spiral. html.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 54-
66, 1991.
Lockwood, E. H. "Spirals." Ch. 22 in A Book of
Curves. Cambridge, England: Cambridge University
Press, pp. 172-175, 1967.
Yates, R. C "Spirals." A Handbook on Curves and Their
Properties. Ann Arbor, Ml: J. W. Edwards, pp. 206-216,
1952,
Spiral Point
A Fixed Point for which the Eigenvalues are Com-
plex Conjugates.
see also Stable Spiral Point, Unstable Spiral
Point
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.
A curve with Cartesian equation
(r 2 - a 2 + c 4- x 2 + y 2 ) = 4r 2 (z 2 + c 2 ).
Around 150 BC, Menaechmus constructed Conic Sec-
tions by cutting a CONE by a PLANE. Two hundred
years later, the Greek mathematician Perseus investi-
gated the curves obtained by cutting a TORUS by a
Plane which is Parallel to the line through the center
of the Hole of the Torus (MacTutor).
In the FORMULA of the curve given above, the TORUS
is formed from a CIRCLE of RADIUS a whose center is
rotated along a CIRCLE of RADIUS r. The value of c
gives the distance of the cutting PLANE from the center
of the Torus.
When c = 0, the curve consists of two CIRCLES of
RADIUS a whose centers are at (r, 0) and (— r, 0). If
c = r + a, the curve consists of one point (the origin),
while if c > r + a, no point lies on the curve. The above
curves have (a,6,r) = (3,4,2), (3, 1, 2) (3, 0.8, 2), (3,
1, 4), (3, 1, 4.5), and (3, 0, 4.5).
References
MacTutor History of Mathematics Archive. "Spiric Sec-
tions." http : //www-groups . dcs . st-and . ac . uk/~history/
Curves/Spiric . html.
Spirograph
A HYPOTROCHOID generated by a fixed point on a CIR-
CLE rolling inside a fixed CIRCLE. It has parametric
equations,
x = (R + r) cos - (r + p) cos (^-^o) (1)
y = (R + r)sin<9 - (r + p)sin (~^) , (2)
where R is the radius of the fixed circle, r is the radius
of the rotating circle, and p is the offset of the edge of
the rotating circle. The figure closes only if i?, r, and p
are Rational. The equations can also be written
x = xq [mcost 4- acos(nt)] — yo [m sin t — asin(nt)]
(3)
y = 2/o [tu cos t + a cos(nt)] + xq [m sin t — a sin(nt)] ,
(4)
1 704 Spirograph
Spirolateral
where the outer wheel has radius 1, the inner wheel a
radius p/g, the pen is placed a units from the center,
the beginning is at 9 radians above the x-axis, and
<1~P
Q
q-p
P
Xo = COS
yo = sin 0.
The following curves are for a = z/10, with i = 1, 2,
10, and = 0.
(5)
(6)
(7)
(8)
(?,<?) = (2,5)
(P,9) = (2,7)
(p,«) = (l,3)
(P,*) = (M)
(P,g) = (3,7)
Additional attractive designs such as the following can
also be made by superposing individual spirographs.
(P,9) = (l,5)
see also EPITROCHOID,
Rose, Spirolateral
Hypotrochoid, Maurer
Spirolateral
A figure formed by taking a series of steps of length 1,2,
. , . , n, with an angle turn after each step. The symbol
for a spirolateral is ai ''"' afc n©, where the a*s indicate
that turns are in the —0 direction for these steps.
Spirolateral
Sponge 1705
r
n
■u
j
V
A
References
Gardner, M. "Worm Paths." Ch. 17 in Knotted Dough-
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, 1986.
Odds, F. C. "Spirolaterals." Math. Teacher 66, 121-124,
1973.
Spline
An interpolating POLYNOMIAL which uses information
from neighboring points to obtain a degree of global
smoothness.
see also B-Spline, Bezier Spline, Cubic Spline,
NURBS Curve
References
Bartels, R. H.; Beatty, J. C; and Barsky, B. A. An Introduc-
tion to Splines for Use in Computer Graphics and Geo-
metric Modelling. San Francisco, CA: Morgan Kaufmann,
1987.
de Boor, C. A Practical Guide to Splines. New York:
Springer- Verlag, 1978.
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.
Spath, H. One Dimensional Spline Interpolation Algorithms.
Wellesley, MA: A. K. Peters, 1995.
Splitting
A type
B type
Splitting Algorithm
A method for computing a UNIT FRACTION,
method always terminates (Beeckmans 1993).
This
References
Beeckmans, L. "The Splitting Algorithm for Egyptian Frac-
tions." J. Number Th. 43, 173-185, 1993.
Sponge
A sponge is a solid which can be parameterized by IN-
TEGERS p, g, and n which satisfy the equation
2 sin
i) =cos (i)-
see also MAURER ROSE, SPIROGRAPH
The possible sponges are {p, q\k} = {6,6|3}, {6,4|4},
{4,6|4}, {3,6|6}, and {4,4|oo} (Ball and Coxeter 1987).
see also Honeycomb, Menger Sponge, Sierpinski
Sponge, Tetrix
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 152,
1987.
Cromwell, P. R. Polyhedra. New York: Cambridge University
Press, p. 79, 1997.
1706 Sporadic Group
Square
Sporadic Group
One of the 26 finite SIMPLE GROUPS. The most com-
plicated is the Monster Group. A summary, as given
by Conway et al. (1985), is given below.
Sym
Name
Order
M A
Afu
Mathieu
M 12
Mathieu
M 22
Mathieu
M 23
Mathieu
M 24
Mathieu
J 2 = HJ Janko
Suz
Suzuki
HS
Higman-Sims
McL
McLaughlin
Co z
Conway
Co 2
Conway
CO!
Conway
He
Held
Fi 22
Fischer
Fi 23
Fischer
**«
Fischer
HN
Harada-Norton
Th
Thompson
B
Baby Monster
M
Monster
Ji
Janko
O'N
O'Nan
Jz
Janko
Ly
Lyons
Ru
Rudvalis
J 4
Janko
2 4 -3 2 -5 ■ 11 11
2 6 ■ 3 3 -5 ■ 11 2 2
2 7 - 3 2 -5- 7 * 11 12 2
2 7 ■ 3 2 ■ 5- 7 • 11 • 23 11
2 10 - 3 3 ■ 5 • 7- 11 - 23 11
2 7 • 3 3 ■ 5 2 • 7 2 2
2 13 . 3 7 • 5 2 • 7 • 11 • 13 6 2
2 9 . 3 2 -5 3 • 7 • 11 2 2
2 7 - 3 6 ■ 5 3 - 7- 11 3 2
2 10 • 3 7 • 5 3 • 7 • 11 • 23 11
2 1S - 3 6 - 5 3 • 7 • 11 ■ 23 11
2 21 • 3 9 -5 4 • 7 2 - 11 - 13- 23 2 1
2 10 • 3 3 • 5 2 • 7 3 • 17 12
2 17 -3 9 -5 2 ■ 7- 11 ■ 13 6 2
2 18 ■ 3 13 ■ 5 2 • 7- 11 • 13 • 17 • 23 1 1
2 21 • 3 16 • 5 2 • 7 3 ■ 11 ■ 13 • 17 3 2
■23 • 29
2 14 ■ 3 6 • 5 6 • 7 • 11 • 19 12
2 15 ■ 3 10 -5 3 • 7 2 - 13- 19-31 1 1
2 41 • 3 13 -5 6 -7 2 - 11 - 13- 17- 19 2 1
•23 -31-47
2 46 , 320 , 5 9 , ? 6 , 1X 2 . 13 3 . 17 . 19 i i
•23 • -29 -31 -41 -47-59- 71
2 3 - 3 - 5 - 7 • 11 • 19 11
2 9 • 3 4 • 7 3 -5 • 11 - 19 ■ 31 3 2
2 7 - 3 5 -5 ■ 17 ■ 19 3 2
2 8 -3 7 -5 6 • 7 • 11 ■ 31 -37-67 1 1
2 14 -3 3 • 5 3 - 7- 13- 29 2 1
2 21 -3 3 • 5- 7- ll 3 • 23-29 -31 1 1
•37-43
see also Baby Monster Group, Conway Groups,
Fischer Groups, Harada-Norton Group, Held
Group, Higman-Sims Group, Janko Groups, Lyons
Group, Mathieu Groups, McLaughlin Group,
Monster Group, O'Nan Group, Rudvalis Group,
Suzuki Group, Thompson Group
References
Aschbacher, M. Sporadic Groups. New York: Cambridge
University Press, 1994.
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.;
and Wilson, R. A. Atlas of Finite Groups: Maximal Sub-
groups and Ordinary Characters for Simple Groups. Ox-
ford, England: Clarendon Press, p. viii, 1985.
Math. Intell. Cover of volume 2, 1980.
Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat .bham.ac .uk/atlas#spo.
Sports
see also Baseball, Bowling, Checkers, Chess, Go
Spr ague- Grundy Function
see Nim- Value
Sprague-Grundy Number
see Nim- Value
Sprague-Grundy Value
see Nim- Value
Spread (Link)
see Span (Link)
Spread (Tree)
A Tree having an infinite number of branches and
whose nodes are sequences generated by a set of rules.
see also FAN
Spun Knot
A 3-D Knot spun about a plane in 4-D. Unlike SUS-
PENDED KNOTS, spun knots are smoothly embedded at
the poles.
see also Suspended Knot, Twist-Spun Knot
Squarable
An object which can be constructed by SQUARING is
called squarable.
Square
The term square is sometimes used to mean SQUARE
NUMBER. When used in reference to a geometric figure,
however, it means a convex QUADRILATERAL with four
equal sides at Right Angles to each other, illustrated
above.
The Perimeter of a square with side length a is
L = Aa
and the AREA is
(i)
(2)
The INRADIUS r, Circumradius R, and Area A can
be computed directly from the formulas for a general
regular POLYGON with side length a and n = 4 sides,
7 a cot
(;)-ws.
A — \na 2 cot I — J = a 2 .
R = ^acsc
(3)
(4)
(5)
The length of the Diagonal of the Unit SQUARE is y/2,
sometimes known as PYTHAGORAS'S CONSTANT.
Square
Square Bracket Polynomial 1707
The Area of a square inscribed inside a Unit SQUARE
as shown in the above diagram can be found as follows.
Label x and y as shown, then
x 2 + y 2 = r 2
(^/l + ri-xf+y 2 = 1.
Plugging (6) into (7) gives
(^/l + r 2 - x) 2 + (r 2 - x 2 ) = 1.
(6)
(7)
(8)
Expanding
- 2zV 1 + r 2 + 1 + r 2 4- r 2 - x 2 = 1 (9)
and solving for x gives
Plugging in for y yields
x/l + r 2
y = yr 2 — x 2 =
vm
The area of the shaded square is then
A=(^T7^-x-y) 2 = { ± T ^
(Detemple and Harold 1996).
(10)
(11)
(12)
The Straightedge and Compass construction of the
square is simple. Draw the line OPq and construct a
circle having OPq as a radius. Then construct the per-
pendicular OB through O. Bisect P OB and PqOB to
locate Pi and ft, where Pq is opposite ft- Similarly,
construct Pz and Pa on the other SEMICIRCLE. Con-
necting P1P2P3P4 then gives a square.
As shown by Schnirelmann, a square can be INSCRIBED
in any closed convex planar curve (Steinhaus 1983). A
square can also be CIRCUMSCRIBED about any closed
curve (Steinhaus 1983).
An infinity of points in the interior of a square are known
whose distances from three of the corners of a square are
Rational Numbers. Calling the distances a, 6, and c
where s is the side length of the square, these solutions
satisfy
(s 2 +b>
a 2 ) 2 + (s 2 +b 2 -c 2 ) 2 = (2bs) 2
(13)
(Guy 1994). In this problem, one of a, 6, c, and s is
DIVISIBLE by 3, one by 4, and one by 5. It is not known
if there are points having distances from all four corners
RATIONAL, but such a solution requires the additional
condition
a 2 +c z = b 2 + <r.
(14)
In this problem, s is Divisible by 4 and a, 6, c, and d
are ODD. If s is not DIVISIBLE by 3 (5), then two of a,
b, c, and d are DIVISIBLE by 3 (5) (Guy 1994).
see also BROWKIN'S THEOREM, DISSECTION, DOUGLAS-
Neumann Theorem, Finsler-Hadwiger Theorem,
Lozenge, Perfect Square Dissection, Pythago-
ras's Constant, Pythagorean Square Puz-
zle, Rectangle, Square Cutting, Square Num-
ber, Square Packing, Square Quadrants, Unit
Square, von Aubel's Theorem
References
Detemple, D. and Harold, S. "A Round-Up of Square Prob-
lems." Math. Mag. 69, 15-27, 1996.
Dixon, R. Mathographics. New York: Dover, p, 16, 1991.
Eppstein, D. "Rectilinear Geometry." http://www.ics.uci.
edu/-eppstein/junkyard/rect .html.
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.
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, p. 104, 1983.
Square Bracket Polynomial
A Polynomial which is not necessarily an invariant of
a Link. It is related to the Dichroic Polynomial. It
is defined by the Skein Relationship
and satisfies
and
B L+ =q l/2 vB Lo +B Loo ,
-^unknot = Q
J^LUunknot =3 &L'
(1)
(2)
(3)
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 235-241, 1994.
1708 Square Cupola
Square Cupola
Square Number
Johnson Solid J4. The bottom eight Vertices are
(±i(l + v/2),±i,0),(±i,±i(l + >/2),0),
and the top four VERTICES are
^•^•(^Tl)-
Square Curve
see SlERPINSKI CURVE
Square Cutting
The average number of regions into which N lines divide
a Square is
±N(N-1)tt + N + 1
(Santalo 1976).
see also Circle Cutting
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ asolve/constant/geom/ geom.html.
Santalo, L. A. Integral Geometry and Geometric Probability.
Reading, MA: Addis on- Wesley, 1976.
Square- Free
see Squarefree
Square Gyrobicupola
see Johnson Solid
Square Integrable
A function f(x) is said to be square integrable if
J — C
\f{x)\ 2 dx
is finite.
see also Integrable, L 2 -Norm, Titchmarsh Theo-
rem
References
Sansone, G. "Square Integrable Functions." §1.1 in Orthogo-
nal Functions, rev. English ed. New York: Dover, pp. 1-2,
1991.
Square Knot
A composite Knot of six crossings consisting of a KNOT
Sum of a Trefoil Knot and its Mirror Image. The
Granny Knot has the same Alexander Polynomial
(x 2 — x + 1) 2 as the square knot. The square knot is also
called the Reef Knot.
see also Granny Knot, Mirror Image, Trefoil
Knot
References
Owen, P. Knots. Philadelphia, PA: Courage, p. 50, 1993.
Square Matrix
A Matrix for which horizontal and vertical dimensions
are the same (i.e., an n x n Matrix).
see also MATRIX
Square Number
A Figurate Number of the form m — n r, where n
is an INTEGER. A square number is also called a PER-
FECT Square. The first few square numbers are 1, 4,
9, 25, 36, 49, . . . (Sloane's A000290). The Generating
Function giving the square numbers is
x(x + 1)
(1-x) 3
= x + 4x 2 + 9x 3 + 16x 4 + . . . . (1)
The kth nonsquare number an is given by
= n + [\ + v^J ,
(2)
where |^J is the FLOOR FUNCTION, and the first few
are 2, 3, 5, 6, 7, 8, 10, 11, . . . (Sloane's A000037).
The only numbers which are simultaneously square and
Pyramidal (the Cannonball Problem) are Pi = 1
and P24 = 4900, corresponding to Si = 1 and S70 =
4900 (Dickson 1952, p. 25; Ball and Coxeter 1987, p. 59;
Ogilvy 1988), as conjectured by Lucas (1875, 1876) and
proved by Watson (1918). The Cannonball Problem
is equivalent to solving the DlOPHANTlNE EQUATION
|a;(x + l)(2a; + l)
(3)
Square Number
Square Number 1709
(Guy 1994, p. 147).
The only numbers which are square and Tetrahedral
are Tei = 1, Te 2 = 4, and Te 48 = 19600 (giving Si = 1,
S 2 = 4, and Si 4 o = 19600), as proved by Meyl (1878;
cited in Dickson 1952, p. 25; Guy 1994, p. 147). In
general, proving that only certain numbers are simulta-
neously figurate in two different ways is far from elemen-
tary.
To find the possible last digits for a square number, write
n = 10a+6 for the number written in decimal NOTATION
as abio (a, 6 = 0, 1, ... , 9). Then
n = 100a 2 + 20a& + 6 2 ,
(4)
so the last digit of n 2 is the same as the last digit of 6 2 .
The following table gives the last digit of b 2 for 6 = 0,
1, . . . , 9. As can be seen, the last digit can be only 0,
1, 4, 5, 6, or 9.
1
2
3
4
5
6
7
8
9
1
4
9
_6
.5
_6
_9
_4
_1
We can similarly examine the allowable last two digits
by writing abcio as
n = 100a + 106 + c,
(5)
n = (100a + 106 + cf
= 10V + 2(1000a6 + lOOac + 106c) + 1006 2 + c
2
(6)
(10 V + 2000a6 + lOOac + 1006 2 ) + 206c + c 2 ,
so the last two digits are given by 206c + c 2 = c(206 + c).
But since the last digit must be 0, 1, 4, 5, 6, or 9, the
following table exhausts all possible last two digits.
c
h
1
2
3
4
5
6
7
8
9
1
01
21
41
61
81
-01
_21
-41
_61
.81
4
16
96
_76
_56
_36
-16
_96
_76
_56
_36
5
25
_25
J25
_25
-25
,25
.25
.25
-25
_25
6
36
.56
_76
_96
_16
_36
_56
,76
_96
.16
9
81
_61
.41
.21
_01
_81
_61
-41
_21
_01
The only possibilities are 00, 01, 04, 09, 16, 21, 24, 25,
29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, and 96,
which can be summarized succinctly as 00, el, e4, 25,
06, and e9, where e stands for an Even Number and o
for an Odd Number. Additionally, unless the sum of
the digits of a number is 1, 4, 7, or 9, it cannot be a
square number.
The following table gives the possible residues mod n
for square numbers for n = 1 to 20. The quantity s(n)
gives the number of distinct residues for a given n.
n
s{n)
X*
' (mod
»)
2
2
o,
1
3
2
o,
1
4
2
o,
1
5
3
o,
1,
4
6
4
o,
1,
3,4
7
4
o,
1,
2,4
8
3
o,
1,
4
9
4
o,
1,
4,7
10
6
o,
1,
4,5,
6,
9
11
6
o,
1,
3,4,
5,
9
12
4
o,
1,
4,9
13
7
o,
1,
3,4,
9,
10, 12
14
8
o,
1,
2,4,
7,
8, 9, 11
15
6
o,
1,
4,6,
9,
10
16
4
o,
1,
4,9
17
9
o,
1,
2,4,
8,
9, 13, 15, 16
18
8
o,
1,
4,7,
9,
10, 13, 16
19
10
o,
1,
4,5,
6,
7, 9, 11, 16, 17
20
6
o,
1,
4,5,
9,
16
In general, the Odd squares are congruent to 1 (mod 8)
(Conway and Guy 1996). Stangl (1996) gives an explicit
formula by which the number of squares s(n) in Z n (i.e.,
mod n) can be calculated. Let p be an Odd Prime.
Then s(n) is the MULTIPLICATIVE FUNCTION given by
5(2) = 2
*(p) = ±(p+l)
(P*2)
*(p 2 ) = £(p a -p + 2)
5(2")
s{p n )
f|(2- 1
J 2(p
I 2(i
+ 4)
+ 5)
(p+i)
+ 2p+l
2(p+l)
for n even
for n odd
for n > 3 even
for n > 3 odd.
(7)
(8)
(9)
(10)
(11)
s(n) is related to the number q(n) of QUADRATIC
Residues in Z n by
q(p n ) = s(p n ) - s(p n ' 2 )
(12)
for n > 3 (Stangl 1996).
For a perfect square n, {n/p) = or 1 for all Odd
PRIMES p < n where (n/p) is the Legendre SYMBOL.
A number n which is not a perfect square but which
satisfies this relationship is called a PSEUDOSQUARE.
The minimum number of squares needed to represent
the numbers 1, 2, 3, . . . are 1, 2, 3, 1, 2, 3, 4, 2, 1, 2, . . .
(Sloane's A002828), and the number of distinct ways to
represent the numbers 1, 2, 3, ... in terms of squares
are 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, . . . (Sloane's A001156).
A brute-force algorithm for enumerating the square per-
mutations of n is repeated application of the GREEDY
Algorithm. However, this approach rapidly becomes
impractical since the number of representations grows
extremely rapidly with n, as shown in the following ta-
ble.
1710 Square Number
Square Number
n
Square Partitions
10
4
50
104
100
1116
150
6521
200
27482
S W Sloane Numbers
Every POSITIVE integer is expressible as a SUM of (at
most) g{2) = 4 square numbers (WARING's PROBLEM).
(Actually, the basis set is {0, 1, 4, 9, 16, 25, 36, 64, 81,
100, . . . }, so 49 need never be used.) Furthermore, an
infinite number of n require four squares to represent
them, so the related quantity G{2) (the least Integer
n such that every POSITIVE INTEGER beyond a certain
point requires G(2) squares) is given by (3(2) = 4.
Numbers expressible as the sum of two squares are those
whose Prime Factors are of the form 4k — 1 taken to
an EVEN Power. Numbers expressible as the sum of
three squares are those not of the form 4 k (8l 4- 7) for
fc, / > 0. The following table gives the first few numbers
which require N — 1, 2, 3, and 4 squares to represent
them as a sum.
N Sloane Numbers
1 000290 1, 4, 9, 16, 25, 36, 49, 64, 81, . . .
2 000415 2, 5, 8, 10, 13, 17, 18, 20, 26, 29, . . .
3 000419 3, 6, 11, 12, 14, 19, 21, 22, 24, 27, . . .
4 004215 7, 15, 23, 28, 31, 39, 47, 55, 60, 63, . . .
The Fermat 4n + 1 Theorem guarantees that every
PRIME of the form An 4 1 is a sum of two SQUARE NUM-
BERS in only one way.
There are only 31 numbers which cannot be expressed
as the sum of distinct squares: 2, 3, 6, 7, 8, 11, 12, 15,
18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60,
67, 72, 76, 92, 96, 108, 112, 128 (Sloane's A001422; Guy
1994). All numbers > 188 can be expressed as the sum
of at most five distinct squares, and only
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . .
2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29,
50, 65, 85, 125, 130, 145, 170, 185, . .
3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22,
27, 33, 38, 41, 51, 57, 59, 62, 69, 74, .
54, 66, 81, 86, 89, 99, 101, 110, 114, .
129, 134, 146, 153, 161, 171, 189, ...
4, 7, 10, 12, 13, 15, 16, 18, 19, 20, . . .
31, 34, 36, 37, 39, 43, 45, 47, 49, . . .
28, 42, 55, 60, 66, 67, 73, 75, 78, . . .
52, 58, 63, 70, 76, 84,87, 91,93, ...
The number of INTEGERS < x which are squares or sums
of two squares is
1
1
000290
2
1
025284
2
2
025285
3
1
025321
3
2
025322
3
3
025323
3
4
025324
4
1
025357
4
2
025358
4
3
025359
4
4
025360
N(x) ~kx(\nx)- l/2 >
where
k =
N
n a -»■-»)■
(16)
(17)
r=4n+3
r prime
(Landau 1908; Le Lionnais 1983, p. 31). The product
of four distinct Nonzero Integers in Arithmetic
Progression is square only for (—3, —1, 1, 3), giv-
ing (-3)(-l)(l)(3) = 9 (Le Lionnais 1983, p. 53). It
is possible to have three squares in Arithmetic Pro-
gression, but not four (Dickson 1952, pp. 435-440). If
these numbers are r 2 , s 2 , and £ 2 , there are POSITIVE
Integers p and q such that
r = \p 2 -2pq-q 2 \
s=p 2 +q 2
t =p 2 + 2pq-q 2 ,
(18)
(19)
(20)
124 = 1 + 4 4 9 + 25 + 36 4 49
(13)
where (j>, q) = 1 and one of r, s, or t is EVEN (Dick-
son 1952, pp. 437-438). Every three-term progression of
squares can be associated with a PYTHAGOREAN Triple
(X,F,Z)by
and
188 = 1 + 4 + 9 + 25 + 49 + 100
(14)
require six distinct squares (Bohman et at. 1979; Guy
1994, p. 136). In fact, 188 can also be represented using
seven distinct squares:
188 = 1 4- 4 4 9 + 25 + 36 + 49 4 64.
(15)
The following table gives the numbers which can be rep-
resented in W different ways as a sum of 5 squares. For
example,
50 = l 2 + 7 2 = 5 2 + 5 2
can be represented in two ways (W — 2) by two squares
(5 = 2).
Y=l(t-r)
Z = s
(21)
(22)
(23)
(Robertson 1996).
Catalan's Conjecture states that 8 and 9 (2 3 and
3 2 ) are the only consecutive POWERS (excluding and
1), i.e., the only solution to Catalan's Diophantine
Problem. This Conjecture has not yet been proved
or refuted, although R. Tijdeman has proved that there
can be only a finite number of exceptions should the
Conjecture not hold. It is also known that 8 and 9
are the only consecutive CUBIC and square numbers (in
either order).
Square Number
Square Number 1711
A square number can be the concatenation of two
squares, as in the case 16 = 4 2 and 9 = 3 2 giving
169= 13 2 .
It is conjectured that, other than 10 2n , 4 x 10 2ti and
9 x 10 2n , there are only a FINITE number of squares
n 2 having exactly two distinct NONZERO DIGITS (Guy
1994, p. 262). The first few such n are 4, 5, 6, 7, 8, 9,
11, 12, 15, 21, . . . (Sloane's A016070), corresponding to
n 2 of 16, 25, 36, 49, 64, 81, 121, . . . (Sloane's A016069).
The following table gives the first few numbers which,
when squared, give numbers composed of only certain
digits. The only known square number composed only
of the digits 7, 8, and 9 is 9. Vardi (1991) considers
numbers composed only of the square digits: 1, 4, and
9.
Digits Sloane n, n
1, 2, 3 030175 1, 11, 111, 36361, 363639, ...
030174 1, 121, 12321, 1322122321, . . .
1, 4, 6 027677 1, 2, 4, 8, 12, 31, 38, 108, . . .
027676 1, 4, 16, 64, 144, 441, 1444, . . .
1, 4, 9 027675 1, 2, 3, 7, 12, 21, 38, 107, . . .
006716 1, 4, 9, 49, 144, 441, 1444, 11449, . . .
2, 4, 8 027679 2, 22, 168, 478, 2878, 210912978, . . .
027678 4, 484, 28224, 228484, 8282884, . . .
4, 5, 6 030177 2, 8, 216, 238, 258, 738, 6742, . . .
030176 4, 64, 46656, 56644, 66564, . . .
Brown Numbers are pairs (m,n) of Integers satis-
fying the condition of Brocard's PROBLEM, i.e., such
that
999, 390, 432 2 = 998, 781, 235, 573, 146, 624,
(30)
n\ + 1 = m ,
(24)
where n! is a FACTORIAL. Only three such numbers are
known: (5,4), (11,5), (71,7). Erdos conjectured that
these are the only three such pairs.
Either 5x 2 + 4 = y 2 or 5a; 2 — 4 = y 2 has a solution in
Positive Integers Iff, for some n, (x,y) = (F n ,L n ),
where F n is a FIBONACCI Number and L n is a Lucas
Number (Honsberger 1985, pp. 114-118).
The smallest and largest square numbers containing the
digits 1 to 9 are
11,826 2 = 139,854,276,
30,384" =923,187,456.
(25)
(26)
The smallest and largest square numbers containing the
digits to 9 are
32,043 2 = 1,026,753,849,
99,066 2 = 9,814,072,356
(27)
(28)
(Madachy 1979, p. 159). The smallest and largest square
numbers containing the digits 1 to 9 twice each are
335, 180, 136 2 = 112, 345, 723, 568, 978, 496 (29)
and the smallest and largest containing 1 to 9 three
times are
10, 546, 200, 195, 312 2
= 111, 222, 338, 559, 598, 866, 946, 777, 344 (31)
31,621,017,808, 182 2
= 999, 888, 767, 225, 363, 175, 346, 145, 124 (32)
(Madachy 1979, p. 159).
Madachy (1979, p. 165) also considers number which are
equal to the sum of the squares of their two "halves"
such as
1233 = 12 2 + 33 2
8833 = 88 2 + 33 2
10100 = 10 2 + 100 2
5882353 = 588 2 + 2353 2 ,
(33)
(34)
(35)
(36)
in addition to a number of others.
see also Antisquare Number, Biquadratic Num-
ber, Brocard's Problem, Brown Numbers, Can-
nonball Problem, Catalan's Conjecture, Cen-
tered Square Number, Clark's Triangle, Cubic
Number, Diophantine Equation, Fermat 4n + 1
Theorem, Greedy Algorithm, Gross, Lagrange's
Four-Square Theorem, Landau-Ramanujan Con-
stant, Pseudosquare, Pyramidal Number, r k {n),
Squarefree, Square Triangular Number, War-
ing's Problem
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 59, 1987.
Bohman, J.; Froberg, C.-E.; and Riesel, H. "Partitions in
Squares." BIT 19, 297-301, 1979.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 30-32, 1996.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, 1952.
Grosswald, E. Representations of Integers as Sums of
Squares. New York: Springer- Verlag, 1985.
Guy, R. K. "Sums of Squares" and "Squares with Just Two
Different Decimal Digits." §C20 and F24 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 136-138 and 262, 1994.
Honsberger, R. "A Second Look at the Fibonacci and Lucas
Numbers." Ch. 8 in Mathematical Gems III. Washington,
DC: Math. Assoc. Amer., 1985.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
1983. t
Lucas, E. Question 1180. Nouv. Ann. Math. Ser. 2 14, 336,
1875.,
Lucas, E. Solution de Question 1180. Nouv. Ann. Math. Ser.
£15, 429-432, 1876.
Madachy, J. S. Madachy y s Mathematical Recreations. New
York: Dover, pp. 159 and 165, 1979.
Meyl, A.-J.-J. Solution de Question 1194. Nouv. Ann. Math.
17, 464-467, 1878.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, pp. 77 and 152, 1988.
1712 Square Orthobicupola
Square Pyramid
Pappas, T. "Triangular, Square & Pentagonal Numbers."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, p. 214, 1989.
Pietenpol, J. L. "Square Triangular Numbers." Amer. Math.
Monthly 69, 168-169, 1962.
Robertson, J. P. "Magic Squares of Squares." Math. Mag.
69, 289-293, 1996.
Stangl, W. D. "Counting Squares in Z„." Math. Mag. 69,
285-289, 1996.
Taussky-Todd, O. "Sums of Squares." Amer. Math. Monthly
77, 805-830, 1970.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 20 and 234-237, 1991.
Watson, G.N. "The Problem of the Square Pyramid." Mes-
senger. Math. 48, 1-22, 1918.
Square Orthobicupola
see Johnson Solid
Square Packing
Find the minimum size SQUARE capable of bounding n
equal SQUARES arranged in any configuration. The only
packings which have been proven optimal are 2, 3, 5, and
Square Numbers (4, 9, . . . ). If n = a 2 - a for some
a, it is Conjectured that the size of the minimum
bounding square is a for small n. The smallest n for
which the CONJECTURE is known to be violated is 1560.
The size is known to scale as fc 6 , where
|(3-V3)<6<|.
The best packing of a SQUARE inside a PENTAGON, il-
lustrated above, is 1.0673. . . .
References
Erdos, P. and Graham, R. L. "On Packing Squares with
Equal Squares." J. Combin. Th. Ser. A 19, 119-123, 1975.
Friedman, E. "Packing Unit Squares in Squares." Elec.
J. Combin. DS7, 1-24, Mar. 5, 1998. http://vw.
combinatorics . org/Surveys/.
Gardner, M. "Packing Squares." Ch. 20 in Fractal Music,
HyperCards, and More Mathematical Recreations from Sci-
entific American Magazine. New York: W. H. Freeman,
1992.
Gobel, F. "Geometrical Packing and Covering Problems."
In Packing and Covering in Combinatorics (Ed. A. Schri-
jver). Amsterdam: Tweede Boerhaavestraat, 1979.
Roth, L. F. and Vaughan, K. C. "Inefficiency in Packing
Squares with Unit Squares." J. Combin. Th. Ser. A 24,
170-186, 1978.
Square Polyomino
□
see also L-Polyomino, Skew Polyomino, Straight
Polyomino, T-Polyomino
n
Exact
Decimal
1
1
1
2
2
2
3
2
2
4
2
2
5
2+fV2
2.707. . .
6
3
3
7
3
3
8
3
3
9
3
3
10
3+fx/2
3.707. . .
11
3.877. . .
12
4
4
13
4
4
14
4
4
15
4
4
16
4
4
17
4+|V2
4.707. . .
18
2(7 + ^)
4.822. . .
19
3+fx/2
4.885...
20
5
5
21
5
5
22
5
5
23
5
5
24
5
5
25
5
5
26
5.650...
Square Pyramid
A square pyramid is a PYRAMID with a SQUARE base.
If the top of the pyramid is cut off by a PLANE, a square
Pyramidal Frustum is obtained. If the four Tri-
angles of the square pyramid are EQUILATERAL, the
square pyramid is the "regular" POLYHEDRON known as
Johnson Solid Ji and, for side length a, has height
h= \V2a.
(1)
Using the equation for a general Pyramid, the Volume
of the "regular" is therefore
V= \hA b = \V2a z .
(2)
If the apex of the pyramid does not lie atop the center
of the base, then the Slant Height is given by
^h 2 + \a\
(3)
where h is the height and a is the length of a side of the
base.
Square Pyramid
a ahfl ail
(a) (b) (c)
Consider a HEMISPHERE placed on the base of a square
pyramid (having side lengths a and height h). Further,
let the hemisphere be tangent to the four apex edges.
Then what is the volume of the Hemisphere which is
interior the pyramid (Cipra 1993)?
From Fig. (a), the ClRCUMRADIUS of the base is a/y/2.
Now find h in terms of r and a. Fig. (b) shows a CROSS-
SECTION cut by the plane through the pyramid's apex,
one of the base's vertices, and the base center. This
figure gives
1=v /|7^
so the Slant Height is
(4)
(5)
s= ^Jh 2 + \a 2 =b + c = ^\a 2 -r 2 + ^h 2 -r 2 . (6)
Solving for h gives
/i =
ra
yja? - 2r 2
(7)
We know, however, that the HEMISPHERE must be tan-
gent to the sides, so r = a/ 2, and
yfd-
2 "fa 2
i
2
a= |v^a.
(8)
Fig. (c) shows a CROSS-SECTION through the center,
apex, and midpoints of opposite sides. The PYTHAGO-
REAN THEOREM once again gives
We now need to find x and y.
(9)
J\a 2 -x 2 + d = /.
(10)
But we know I and h, and d is given by
d = yjh 2 - x 2 ,
(11)
so
J\a 2 -x 2 + J\a 2 - x 2 = |V3a.
(12)
Solving gives
Square Pyramid 1713
x=|\/6a, (13)
so
y
= v ^r^ =v /TTI = y3_2 0= _^.
(14)
We can now find the Area of the Spherical Cap as
V cap = ±irH(3A 2 +H 2 ), (15)
where
A = y =
2x/3
(16)
H = r — x
»-;*-• (!-£)• (ir)
V c
cap — 6
iTra 3
= X
= |™ 3
3 (i2) + (2 vej (2 vej
_4 + ^4 + 6 Vg)\\2 V6J
2 _
3 ^6
1 1
i)
3 2^6 3^
-»«-a-^
6^6/
Therefore, the volume within the pyramid is
'l
(18)
T7 2 3 i T / 2 13 2 3
Vinside = 5 at - 4Vca P = g^ga - nira
3 "~ 1 2 ev'ej
2 3/1 1
S™U 2
7 \
m ^ ~ = + 6^ J = f *"'
V6V6 8^
\9^ 4
(19)
This problem appeared in the Japanese scholastic apti-
tude test (Cipra 1993).
see also SQUARE PYRAMIDAL NUMBER
References
Cipra, B. "An Awesome Look at Japan Math SAT." Science
259, 22, 1993.
1714 Square Pyramidal Number
Square Root
Square Pyramidal Number
A Figurate Number of the form
P„ = £n(n + l)(2n + l),
(1)
corresponding to a configuration of points which form
a Square Pyramid, is called a square pyramidal num-
ber (or sometimes, simply a Pyramidal Number). The
first few are 1, 5, 14, 30, 55, 91, 140, 204, . . . (Sloane's
A000330). They are sums of consecutive pairs of Tet-
rahedral Numbers and satisfy
P n = f(2n + l)T n ,
where T n is the nth Triangular Number.
(2)
The only numbers which are simultaneously SQUARE
and pyramidal (the Cannonball Problem) are Pi = 1
and P24 = 4900, corresponding to Si = 1 and S70 —
4900 (Dickson 1952, p. 25; Ball and Coxeter 1987, p. 59;
Ogilvy 1988), as conjectured by Lucas (1875, 1876) and
proved by Watson (1918). The proof is far from ele-
mentary, and is equivalent to solving the DlOPHANTlNE
Equation
y 2 = ±x(x + l)(2x + l)
(3)
(Guy 1994, p. 147). However, an elementary proof has
also been given by a number of authors.
Numbers which are simultaneously TRIANGULAR and
square pyramidal satisfy the DlOPHANTlNE EQUATION
3(2y + l) 2 = 8x 3 + 12z 2 + 4z + 3.
(4)
The only solutions are x — —1, 0, 1, 5, 6, and 85 (Guy
1994, p. 147). Beukers (1988) has studied the problem
of finding numbers which are simultaneously TETRAHE-
DRAL and square pyramidal via INTEGER points on an
Elliptic Curve. He finds that the only solution is the
trivial Ta = P x = 1.
see also Tetrahedral Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 59, 1987.
Beukers, F. "On Oranges and Integral Points on Certain
Plane Cubic Curves." Nieuw Arch. Wish. 6, 203-210,
1988.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 47-50, 1996.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, 1952.
Guy, R. K. "Figurate Numbers." §D3 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 147-150, 1994.
Lucas, E. Question 1180. Nouvelles Ann. Math. Ser. 2 14,
336, 1875.
Lucas, E. Solution de Question 1180. Nouvelles Ann. Math.,
Ser. 5 15,429-432, 1876.
Ogilvy, C. S. and Anderson, J. T. Excursions in Number
Theory. New York: Dover, pp. 77 and 152, 1988.
Sloane, N. J. A. Sequence A000330/M3844 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Watson, G. N. "The Problem of the Square Pyramid." Mes-
senger. Math. 48, 1-22, 1918.
Square Quadrants
P S
^c^
I B
/T^
B
cV
A
\C
I B
^C^
B
Q R Q R
The areas of the regions illustrated above can be found
from the equations
A + 4£ + 4C = 1
A + 3B + 2C = \-k.
(1)
(2)
Since we want to solve for three variables, we need a
third equation. This can be taken as
A + 2B + C = 2E + D,
(3)
where
(4)
(5)
D + E=\tz,
leading to
A+2B + C = D + 2E = 2{D + E)-D= |tt-|V3. (6)
Combining the equations (1), (2), and (6) gives the ma-
trix equation
(7)
1 4 41
r A i
r x 1
1 3 2
B
_
\«
1 2 1.
C _
Utt-I^J
which can be inverted to yield
1
12'
B = -1 + |V3+ ^7T
C=l-iV3+j7T.
(8)
(9)
(10)
References
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 67-69, 1991.
Square Root
Square Root
Im[Sqrt z]
|Sqrt z|
A square root of x is a number r such that r 2 = x. This
is written r = x 1/2 (x to the 1/2 POWER) or r = y/x.
The square root function f(x) = yfx is the Inverse
Function of f(x) = x 2 . Square roots are also called
Radicals or Surds. A general Complex Number z
has two square roots. For example, for the real POSITIVE
number x = 9, the two square roots are \/9 = ±3, since
3 2 = (-3) 2 = 9. Similarly, for the real NEGATIVE num-
ber x = — 9, the two square roots are v 7 — 9 = ±3i, where
i is the Imaginary Number defined by i 2 = -1. In
common usage, unless otherwise specified, "the" square
root is generally taken to mean the POSITIVE square
root.
The square root of 2 is the IRRATIONAL NUMBER y/2 ftj
1.41421356 (Sloane's A002193), which has the simple
periodic CONTINUED FRACTION 1, 2, 2, 2, 2, 2,
The square root of 3 is the IRRATIONAL NUMBER y/3 «
1.73205081 (Sloane's A002194), which has the simple
periodic Continued Fraction 1, 1, 2, 1, 2, 1, 2, —
In general, the CONTINUED Fractions of the square
roots of all POSITIVE integers are periodic.
The square roots of a COMPLEX NUMBER are given by
y/ x + iy = ±y/x 2 -\-y 2 | cos -tan" 1 f- j
+2£
As can be seen in the above figure, the Imaginary Part
of the complex square root function has a Branch Cut
along the NEGATIVE real axis.
A Nested Radical of the form y/a±by/c can some-
times be simplified into a simple square root by equating
V a ± by/c = Vd ± y/e .
(2)
Square Root 1715
A sequence of approximations a/6 to y/n can be derived
by factoring
a 2 - nb 2 = ±1 (7)
(where — 1 is possible only if —1 is a Quadratic
Residue of n). Then
{a + by/n){a-by/n) = ±1 (8)
(a + by/K) k (a - b^) k = (±l) k = ±1, (9)
and
(l + v^) 1 = i + \M (io)
(l + ^) 2 = (l + n) + 2v^ (11)
(l + y/n)(a + by/n) = (a + bn) + y/n(a + b). (12)
Therefore, a and b are given by the RECURRENCE RE-
LATIONS
a,i = a%—i + bi-in
bi = di-i + 6i_i
(13)
(14)
with a\ — b\ = 1. The error obtained using this method
is
a /-
<
1
(15)
6(a + V^) 26 2 '
The first few approximants to y/n are therefore given by
.2
MU + n),
1 + 3n 1 + 6n + rT 1 + lOn + 5rr
3 + n ' 4(ra+l) ' 5 + lOn + n 2
(16)
This Algorithm is sometimes known as the Bhaska-
ra-Brouckner Algorithm. For the case n = 2, this
gives the convergents to y/2 as 1, 3/2, 7/5, 17/12, 41/29,
99/70, ....
Another general technique for deriving this sequence,
known as Newton's Iteration, is obtained by letting
x — y/n. Then x = n/x, so the SEQUENCE
1 / n
Xk = - I Xk-i H
2 V Xfc-i
(17)
Squaring gives
converges quadratically to the root. The first few ap-
proximants to y/n are therefore given by
a ± by/c = d + e ± 2v / 5e ,
a = d + e
6 2 c = 4<fe.
Solving for d and e gives
d, e =
a ± Va 2 — 6 2 c
(3)
(4)
(5)
(6)
1,§(1 + n),
1 + 6n + rf
4(n+l) '
1 + 26n + 70n 2 + 28n 3 + n 4
8(l + n)(l + 6n + n 2 )
(18)
For \/2, this gives the convergents 1, 3/2, 17/12,
577/408, 665857/470832, ....
see also Continued Square Root, Cube Root,
Nested Radical, Newton's Iteration, Quadratic
Surd, Root of Unity, Square Number, Square
Triangular Number, Surd
1716 Square Root Inequality
Square Triangular Number
References
Sloane, N. J. A. Sequences A002193/M3195 and A002194/
M4326 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Spanier, J. and Oldham, K. B. "The Squa re- Root Func-
tion y/bx + c and Its Reciprocal," "Th e by/a 2 - x 2 Func-
tion and Its Reciprocal," and "The by/x 2 + a Function."
Chs. 12, 14, and 15 in An Atlas of Functions. Washing-
ton, DC: Hemisphere, pp. 91-99, 107-115, and 115-122,
1987.
Williams, H. C. "A Numerical Investigation into the Length
of the Period of the Continued Fraction Expansion of y/D."
Math. Comp. 36, 593-601, 1981.
Square Root Inequality
1
2y/n + 1 - 2y/n< -4= < 2\fn - 2y/n - 1 .
\/n
Square Root Method
The square root method is an algorithm which solves
the Matrix Equation
Au = g
(1)
for u, with Aapxp Symmetric Matrix and g a
given Vector. Convert A to a Triangular Matrix
such that
T T T = A,
where T T is the Matrix Transpose. Then
T T k = g
Tu = k,
(2)
(3)
(4)
giving T from A. Now solve for k in terms of the SijS
and g,
suki — gi
S12&1 + S22&2 = 92
sijki + s 2 jh 2 + . . . + Sjjkj = 0j, (8)
which gives
fc 2 =
9i
511
92 — Si2k\
522
Qj — Sijki — S2jk 2 — ... — Sj-i,jfcj-i
• (9)
Finally, find u from the s^s and k,
snui + S12U2 . . . + S\pU p = k\
S22U2 + . . • 4- S2pU p — k 2
Sppltp ~ = - Kp ,
giving the desired solution,
U p -1 =
(10)
Kp — 1 Sp — i^pUp
Sp— l,p — 1
fej ~ gj.j+l^' + l ~ S 3,3+2Uj + 2 - ... - SjpUp
571
(11)
T =
SIX 512
$22
giving the equations
(5)
511
= an
511S12
= ai2
„ 2 1 » 2
512 + 5 2 2
= ^22
s
2 1 „
lj + 52j
2 + ... + s,j 3
= ajj
Slj + S2j5 2 fe + . . . + SjjSjk
= Gjfe.
These j
?ive
sn
= y/au.
512
_ ai2
511
522
= V fl 22
- 512 2
S 3J
= V a ij
_ Q-.2 _
52> 2 --..-5
• 1 - 2
_ djk -
SljSlfc -
S2jS2k — . . . ~
" S J — l.J 5 J — l,fc
(6)
,(7)
see a/so LU DECOMPOSITION
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 298-300,
1951.
Square Triangular Number
A number which is simultaneously SQUARE and TRIAN-
GULAR. The first few are 1, 36, 1225, 41616, 1413721,
48024900, ... (Sloane's A001110), corresponding to
Ti — Si, Tg = Sq, T49 = £35, T288 = ^204) Tiqsi =
5n89, ... (Pietenpol 1962), but there are an infinite
number, as first shown by Euler in 1730 (Dickson 1952).
The general FORMULA for a square triangular number
ST n is 6 2 c 2 , where b/c is the nth convergent to the CON-
TINUED Fraction of y/2 (Ball and Coxeter 1987, p. 59;
Conway and Guy 1996). The first few are
1 3 7 17 41 99 239
1' 2' 5' 12' 29' 70' 169'
(1)
The Numerators and Denominators give solutions
to the Pell Equation
x 2 - 2y 2 = ±1,
(2)
Square Triangular Number
Squared 1717
but can also be obtained by doubling the previous Frac-
tion and adding to the FRACTION before that. The con-
nection with the Pell Equation can be seen by letting
N denote the JVth TRIANGULAR NUMBER and M the
Mth Square Number, then
Defining
\N{N + l) =M 2
x = 2N + l
y = 2M
then gives the equation
x 2 - 2y = 1
(3)
(4)
(5)
(6)
(Conway and Guy 1996). Numbers which are simul-
taneously Triangular and Square Pyramidal also
satisfy the DlOPHANTINE EQUATION
3(2y + l) 2 = 8x 3 + 12z 2 + Ax 4- 3. (7)
The only solutions are x = -1, 0, 1, 5, 6, and 85 (Guy
1994, p. 147).
A general FORMULA for square triangular numbers is
ST n =
(8)
(l + y/2) 2n -(l- 72) 2
4v^
= ^[(17+12v / 2) n + (17-12v / 2) n -2]. (9)
The square triangular numbers also satisfy the RECUR-
RENCE Relation
ST n =34ST„_i-ST n _ 2 + 2
(10)
Un+2 = 6^n+l — U n ,
(11)
0, u\ = 1, where ST n = u n 2 .
A curious
product formula for ST n is given by
ST„ = 2 a - B Il[3 + cos(^)]. (12)
fc = l
An amazing Generating Function is
1 + x
f(x) = - w „ nA rr =' 1 + 36x + 1225a;' + . . .
JK } (1 -x)(l -34z + ;r 2 )
(13)
(Sloane and Plouffe 1995).
see also Square Number, Square Root, Triangu-
lar Number
References
Allen, B. M. "Squares as Triangular Numbers." Scripta
Math. 20, 213-214, 1954.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, 1987.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 203-205, 1996.
Dickson, L. E. A History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, pp. 10, 16, and
27, 1952.
Guy, R. K. "Sums of Squares" and "Figurate Numbers."
§C20 and §D3 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 136-138 and 147-
150, 1994,
Khatri, M. N. "Triangular Numbers Which are Also
Squares." Math. Student 27, 55-56, 1959.
Pietenpol, J. L. "Square Triangular Numbers." Problem E
1473. Amer. Math. Monthly 69, 168-169, 1962.
Sierpinski, W. Teoria Liczb, 3rd ed. Warsaw, Poland: Mono-
grafie Mate maty czne t* 19, p. 517, 1950.
Sierpinski, W. "Sur les nombres triangulaires carres." Pub.
Faculte d'Electrotechnique VUniversite Belgrade, No. 65,
1-4, 1961.
Sierpinski, W. "Sur les nombres triangulaires carres." Bull
Soc. Royale Sciences Liege, 30 ann., 189-194, 1961.
Sloane, N. J. A. Sequence A001110/M5259 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Walker, G. W. "Triangular Squares." Problem E 954. Amer.
Math. Monthly 58, 568, 1951.
Square Wave
The square wave is a periodic waveform consisting of
instantaneous transitions between two levels which can
be denoted ±1. The square wave is sometimes also called
the RADEMACHER FUNCTION. Let the square wave have
period 2L. The square wave function is ODD, so the
Fourier Series has a = a„ = and
bn=lf o sin (^) (fa
4.2/1 \ 4 f n even
— — sin Un7r) = — <
mr z Tin I 1 n odd.
The Fourier SERIES for the square wave is therefore
n=l,3,5,.
1 . / nirx \
n sm [-r)-
see also Hadamard Matrix, Walsh Function
References
Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr.
Inter ferometry and Synthesis in Radio Astronomy. New
York: Wiley, p. 203, 1986.
Squared
A number to the Power 2 is said to be squared, so that
x 2 is called "as squared."
see also CUBED, SQUARE ROOT
1718 Squared Square
Square ful
Squared Square
see Perfect Square Dissection
Squarefree
60 ■
50
40
30
20
10
20 40 60 80 100
A number is said to be squarefree (or sometimes
QUADRATFREI; Shanks 1993) if its PRIME decomposi-
tion contains no repeated factors. All PRIMES are there-
fore trivially squarefree. The squarefree numbers are 1,
2, 3, 5, 6, 7, 10, 11, 13, 14, 15, . . . (Sloane's A005117).
The SQUAREFUL numbers (i.e., those that contain at
least one square) are 4, 8, 9, 12, 16, 18, 20, 24, 25, ...
(Sloane's A013929).
The asymptotic number Q(n) of squarefree numbers < n
is given by
fir?
Q(n)=^ + 0{V^) (1)
(Hardy and Wright 1979, pp. 269-270). Q(n) for n =
10, 100, 1000, ... are 7, 61, 608, 6083, 60794, 607926,
. . . , while the asymptotic density is 1/C(2) = 6/7T 2 «
0.607927, where £(n) is the Riemann Zeta Function.
The Mobius Function is given by
if n has one or more repeated prime factors
if n = 1
if n is product of k distinct primes,
(2)
so p,{n) ^ indicates that n is squarefree. The asymp-
totic formula for Q(x) is equivalent to the formula
£l«(»)l
6x
+ 0{y^c)
(3)
(Hardy and Wright 1979, p. 270)
There is no known polynomial-time algorithm for recog-
nizing squarefree INTEGERS or for computing the square-
free part of an Integer. In fact, this problem may
be no easier than the general problem of integer fac-
torization (obviously, if an integer n can be factored
completely, n is squarefree Iff it contains no dupli-
cated factors). This problem is an important unsolved
problem in NUMBER THEORY because computing the
Ring of integers of an algebraic number field is re-
ducible to computing the squarefree part of an In-
teger (Lenstra 1992, Pohst and Zassenhaus 1997).
The Mathematica® (Wolfram Research, Champaign,
IL) function NumberTheory 'NumberTheoryFunctions '
SquareFreeQ [n] determines whether a number is
squarefree.
The largest known SQUAREFUL FIBONACCI NUMBER
is F 336 , and no Squareful Fibonacci Numbers F p
are known with p Prime. All numbers less than
2.5 x 10 15 in Sylvester's Sequence are squarefree,
and no SQUAREFUL numbers in this sequence are known
(Vardi 1991). Every Carmichael Number is square-
free. The Binomial Coefficients ( 2n ~ 1 ) are square-
free only for n = 2, 3, 4, 6, 9, 10, 12, 36, . . . , with no
others less than n = 1500. The CENTRAL BINOMIAL
Coefficients are Squarefree only for n = 1, 2, 3, 4,
5, 7, 8, 11, 17, 19, 23, 71, ... (Sloane's A046098), with
no others less than 1500.
see also BINOMIAL COEFFICIENT, BlQUADRATEFREE,
Composite Number, Cubefree, Erdos Squarefree
Conjecture, Fibonacci Number, Korselt's Crite-
rion, Mobius Function, Prime Number, Riemann
Zeta Function, Sarkozy's Theorem, Square Num-
ber, Squareful, Sylvester's Sequence
References
Bellman, R, and Shapiro, H. N. "The Distribution of Square-
free Integers in Small Intervals." Duke Math. J. 21, 629-
637, 1954.
Hardy, G. H. and Wright, E. M. "The Number of Square-
free Numbers." §18.6 in An Introduction to the Theory
of Numbers, 5th ed. Oxford, England: Clarendon Press,
pp. 269-270, 1979.
Lenstra, H. W. Jr. "Algorithms in Algebraic Number The-
ory." Bull Amer. Math. Soc. 26, 211-244, 1992.
Pohst, M. and Zassenhaus, H. Algorithmic Algebraic Num-
ber Theory. Cambridge, England: Cambridge University
Press, p. 429, 1997.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 114, 1993.
Sloane, N. J. A. Sequences A013929 and A005117/M0617 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Vardi, L "Are All Euclid Numbers Squarefree?" §5.1 in Com-
putational Recreations in Mathematica. Reading, MA:
Addison-Wesley, pp. 7-8, 82-85, and 223-224, 1991.
Squareful
A number is squareful, also called NONSQUAREFREE, if it
contains at least one SQUARE in its prime factorization.
Such a number is also called Squareful. The first few-
are 4, 8, 9, 12, 16, 18, 20, 24, 25, . . . (Sloane's A013929).
The greatest multiple prime factors for the squareful
integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 3, . . . (Sloane's
A046028). The least multiple prime factors for squareful
integers are 2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 2, 2, 2, . . . (Sloane's
A046027).
see also GREATEST PRIME FACTOR, LEAST PRIME FAC-
TOR, Smarandache Near-to-Primorial Function,
Squarefree
Squaring
Stability Matrix 1719
References
Sloane, N. J. A. Sequences A013929, A046027, and A046028
in "An On-Line Version of the Encyclopedia of Integer Se-
quences."
Squaring
Squaring is the GEOMETRIC CONSTRUCTION, using only
Compass and Straightedge, of a Square which has
the same area as a given geometric figure. Squaring
is also called QUADRATURE. An object which can be
constructed by squaring is called SQU ARABLE.
see also CIRCLE SQUARING, COMPASS, CONSTRUCTIBLE
Number, Geometric Construction, Rectangle
Squaring, Straightedge, Triangle Squaring
Squeezing Theorem
Using the LAW OF COSINES
Let there be two functions f-(x) and f+{x) such that
f(x) is "squeezed" between the two,
If
/-(a) </(*)< /+(*).
r = lim f-(x) = lim /+(#),
then lim x _).a f{x) = r. In the above diagram the func-
tions f-{x) = -x 2 and /+(#) = x 2 "squeeze" x 2 sin(c;c)
at 0, so lim x _> £ 2 sin(cx) = 0. The squeezing theorem
is also called the SANDWICH THEOREM.
SSS Theorem
Specifying three sides uniquely determines a Triangle
whose AREA is given by Heron's FORMULA,
where
A = y/s(s — a)(s — b)(s — c),
s= Ua + b + c)
(1)
(2)
is the Semiperimeter of the Triangle. Let R be the
Circumradius, then
A =
abc
b 2 +c 2 -
- 2bc cos A
(4)
a 2 +c 2
— 2accosB
(5)
a 2 + b 2
— 2ab cos C
(6)
gives the three ANGLES as
A = cos
B = cos"
C = cos"
f b 2 +c 2 -b 2 \
\ lac J
( a 2 +b 2 -c 2 \
\ 2ab J '
(7)
(8)
(9)
see also AAA Theorem, AAS Theorem, ASA The-
orem, ASS Theorem, Heron's Formula, SAS The-
orem, Semiperimeter, Triangle
Stability
The robustness of a given outcome to small changes in
initial conditions or small random fluctuations. CHAOS
is an example of a process which is not stable.
see also STABILITY MATRIX
Stability Matrix
Given a system of two ordinary differential equations
x = f{x,y)
(1)
(2)
let xq and yo denote Fixed Points with x = y = 0, so
f{xo y yo) =
0(3o,yo) = 0.
Then expand about (#0,2/0) so
Sx = f x (x 0i yo)5x + f y (x ,yo)5y
+ fxy(xo J yo)Sx6y-\- ...
Sy = g x (x ,yo)Sx + g y (x ,yo)Sy
+ g xy (xo, yo)5xSy+
To first-order, this gives
d_
dt
Sx
Sy
fx{xo,yo) f y (xo,yo)
g x (xo,yo) g y (xo,yo)
Sx
Sy
(3)
(4)
(5)
(6)
(7)
(3)
where the 2x2 Matrix, or its generalization to higher
dimension, is called the stability matrix. Analysis of
the Eigenvalues (and Eigenvectors) of the stability
matrix characterizes the type of Fixed Point.
see also Elliptic Fixed Point (Differential Equa-
tions), Fixed Point, Hyperbolic Fixed Point
1720
Stabilization
Stack
(Differential Equations), Linear Stability, Sta-
ble Improper Node, Stable Node, Stable Spiral
Point, Stable Star, Unstable Improper Node,
Unstable Node, Unstable Spiral Point, Unsta-
ble Star
References
Tabor, M. "Linear Stability Analysis." §1,4 in Chaos and In-
tegrability in Nonlinear Dynamics: An Introduction. New
York: Wiley, pp. 20-31, 1989.
Stabilization
n-\
A type II Markov Move.
see also MARKOV MOVES
Stable Equivalence
Two Vector Bundles are stably equivalent Iff Iso-
morphic Vector Bundles are obtained upon Whit-
ney Summing each Vector Bundle with a trivial
Vector Bundle.
see also Vector Bundle, Whitney Sum
Stable Spiral Point
A Fixed Point for which the Stability Matrix has
Eigenvalues of the form A± = -a±ij3 (with a,/? > 0).
see also Elliptic Fixed Point (Differential
Equations), Fixed Point, Hyperbolic Fixed
Point (Differential Equations), Stable Im-
proper Node, Stable Node, Stable Star, Unsta-
ble Improper 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.
Stable Star
A Fixed Point for which the Stability Matrix has
one zero EIGENVECTOR with NEGATIVE EIGENVALUE
A <0.
see also Elliptic Fixed Point (Differential
Equations), Fixed Point, Hyperbolic Fixed
Point (Differential Equations), Stable Im-
proper Node, Stable Node, Stable Spiral Point,
Unstable Improper Node, Unstable Node, Unsta-
ble Spiral Point, Unstable Star
References
Tabor, M. "Classification of Fixed Points." §l,4.b in Chaos
and Integrability in Nonlinear Dynamics: An Introduc-
tion. New York: Wiley, pp. 22-25, 1989.
Stable Improper Node
A Fixed Point for which the Stability Matrix has
equal NEGATIVE EIGENVALUES.
see also Elliptic Fixed Point (Differential Equa-
tions), Fixed Point, Hyperbolic Fixed Point
(Differential Equations), Stable Node, Stable
Spiral Point, Unstable Improper Node, Unsta-
ble 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.
Stable Node
A Fixed Point for which the Stability Matrix has
both Eigenvalues Negative, so Ai < A 2 < 0.
see also Elliptic Fixed Point (Differential
Equations), Fixed Point, Hyperbolic Fixed
Point (Differential Equations), Stable Im-
proper 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.
Stable Type
A Polynomial equation whose Roots all have Nega-
tive Real Parts. For a Real Quadratic Equation
z 2 + Bz + C = 0,
the stability conditions are £?, C > 0. For a REAL CUBIC
Equation
z z + Az 2 + Bz + C - 0,
the stability conditions are A, £?, C > and AB > C.
References
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra,
3rd ed. New York: Macmillan, pp. 108-109, 1965.
Stack
A Data Structure which is a special kind of List in
which elements may be added to or removed from the
top only. These actions are called a Push or a POP,
respectively. Actions may be taken by popping one or
more values, operating on them, and then pushing the
result back onto the stack.
Stacks are used as the basis for computer languages such
as FORTH, PostScript® (Adobe Systems), and the RPN
language used in Hewlett-Packard® programmable cal-
culators.
see also LIST, POP, Push, Queue
Stackel Determinant
Standard Deviation 1721
Stackel Determinant
A DETERMINANT used to determine in which coordinate
systems the Helmholtz Differential Equation is
separable (Morse and Feshbach 1953). A determinant
S=\$ n
$11 $12 $13
$21 $22 $23
$31 $32 $33
(1)
in which $ n j are functions of ui alone is called a Stackel
determinant. A coordinate system is separable if it
obeys the ROBERTSON CONDITION, namely that the
Scale Factors hi in the Laplacian
V 2 = V^ 1 d f hih 2 h z d \ ( .
2L^ tnhihsdm I hi 2 du x ) y }
can be rewritten in terms of functions f%{ui) defined by
1 d ( /11/12/13 \
d
h\h,2hz dui \ hi 2 dui
g(ui+i,Ui+ 2 ) d
h\h,2hz du
= hSTid^i \ fi d^J (3)
such that S can be written
hih2h,3
5 =
fl{Ui)f2{u 2 )f3{u 3 )'
(4)
When this is true, the separated equations are of the
form
^^-f/n^)+(^l 2 $nl+^ 2 $n2+fc 3 2 $ Tl 3)X n =0
fn OU n \ OU n J
(5)
The $ijS obey the minor equations
Mi = $22$33 - $23$32
M 2 = $13$31 - $12$33
M 3 — $12$23 - $13$22
s_
h\
hV
which are equivalent to
Mi $n + M 2 $2i + M 3 $3i = S
Ml $12 + M2$22 + M3$32 =
Ml$13 + M 2 $23 + M 3 $33 = 0.
(6)
(7)
(8)
(9)
(10)
(11)
This gives a total of four equations in nine unknowns.
Morse and Feshbach (1953, pp. 655-666) give not only
the Stackel determinants for common coordinate sys-
tems, but also the elements of the determinant (although
it is not clear how these are derived).
see also Helmholtz Differential Equation, La-
place's Equation, Poisson's Equation, Robert-
son Condition, Separation of Variables
References
Morse, P. M. and Feshbach, H. "Tables of Separable Coordi-
nates in Three Dimensions." Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 509-511 and 655-
666, 1953.
Stamp Folding
The number of ways of folding a strip of stamps has
several possible variants. Considering only positions of
the hinges for unlabeled stamps without regard to orien-
tation of the stamps, the number of foldings is denoted
U(n). If the stamps are labelled and orientation is taken
into account, the number of foldings is denoted N(n).
Finally, the number of symmetric foldings is denoted
S(n). The following table summarizes these values for
the first n.
n
S{n)
U(n)
N(n)
1
1
1
1
2
1
1
1
3
2
2
6
4
4
5
16
5
6
14
50
6
8
39
144
7
18
120
462
8
20
358
1392
9
56
1176
4536
10
3572
see also MAP FOLDING
References
Gardner, M. "The Combinatorics of Paper-Folding." In
Wheels, Life, and Other Mathematical Amusements. New-
York: W. H. Freeman, pp. 60-73, 1983.
Ruskey, F. "Information of Stamp Folding." http:// sue .
csc.uvic.ca/-cos/inf /perm/ StampFolding.html.
Sloane, N. J. A. A Handbook of Integer Sequences. Boston,
MA: Academic Press, p. 22, 1973.
Standard Deviation
The standard deviation is defined as the Square Root
of the Variance,
a - y/{x*) - (x) 2 = VmT 1 ^
(1)
where /i = (x) is the Mean and fi' 2 = (#/ is the sec-
ond Moment about 0. The variance a 2 is equal to the
second Moment about the Mean,
: /x 2 .
(2)
The square root of the SAMPLE VARIANCE is the "sam-
ple" standard deviation,
sn
\
- ^(Xi - xY
(3)
1722
Standard Error
Standard Map
It is a Biased Estimator of the population standard
deviation. As unbiased ESTIMATOR is given by
sjv-i
N
\ N-l ^ }
(4)
Physical scientists often use the term ROOT-MEAN-
SQUARE as a synonym for standard deviation when they
refer to the SQUARE ROOT of the mean squared devia-
tion of a signal from a given baseline or fit.
see also Mean, Moment, Root-Mean-Square, Sam-
ple Variance, Standard Error, Variance
Standard Error
The square root of the ESTIMATED VARIANCE of a quan-
tity. The standard error is also sometimes used to mean
z=l i=l
see also Standard Deviation
Standard Map
:i'iv ;:
~\. <£.$-■ ' . l .
' )%;/ : '
..' ; ;<??'■•.
'.-...■"■'■' ''■■
. ;: . »
. ■ v :
.;■ .'•-'■
■'"..^■> v ;
: r::
A 2-D Map, also called the Taylor-Greene-Chirikov
Map in some of the older literature.
/n+l = In + K Sin n (1)
0n+l = n + J„+l = i» + 6> n + tfsin0 ni (2)
where 7 and are computed mod 27T and K is a POSI-
TIVE constant. An analytic estimate of the width of the
Chaotic zone (Chirikov 1979) finds
51 = Be
-AK- 1 ' 2
(3)
Numerical experiments give A w 5.26 and B ss 240.
The value of K at which global Chaos occurs has been
bounded by various authors. GREENE'S Method is the
most accurate method so far devised.
Author
Bound
Fraction
Decimal
Hermann
>
34
0.029411764
Italians
>
0.65
Greene
tt
-
0.971635406
MacKay and Pearson
<
63
64
0.984375000
Mather
<
4
3
1.333333333
Fixed Points are found by requiring that
J»+l = /n (4)
0n+l = 0n- (5)
The first gives Ksin0 n = 0, so sin0 n = and
0n = O,7T. (6)
The second requirement gives
J n + irsin<9 n = / n = 0. (7)
The Fixed Points are therefore (1,0) = (0,0) and
(0,7r). In order to perform a Linear Stability analy-
sis, take differentials of the variables
dln+l = dl n + K COS n dO n
dOn+i = dI n + (l + K cos n ) d0 n
In Matrix form,
SIn+l
SQn+l
1 K COS n
1 1 + K COS n
5I n
50 n
(8)
(9)
(10)
The EIGENVALUES are found by solving the CHARAC-
TERISTIC Equation
1 — A K cos n
1 l + ifcosfln-A
= 0,
A 2 -A(*rcos0 n -f2) + l = O
(11)
(12)
A± = \[KcosO n + 2± v/(*:cos0 n + 2) 2 -4]. (13)
For the FIXED POINT (0,7r),
4° ,7r) = \[2-K± y/(2-K)*-4]
(2-K±y/K 2 -4K)
(14)
The Fixed Point will be stable if |»(A (0,,r) )| < 2. Here,
that means
\\2-K\<l
(15)
\2-K\<2
(16)
-2 < 2-K <2
(17)
-4 < -K <
(18)
Standard Normal Distribution
Stanley's Theorem 1723
so if e [0,4). For the Fixed Point (0, 0), the Eigen-
values are
A ( ± °.°) = i [ 2 + K ± y/(K + iY-i]
= \{2 + K±^K*+AK).
(19)
If the map is unstable for the larger EIGENVALUE, it is
unstable. Therefore, examine A^ ' . We have
<1,
2 + K + ^K 2 + AK
-2<2 + K+ ^K 2 + AK < 2
-4 - K < y/K 2 + 4K < -K.
(20)
(21)
(22)
But K > 0, so the second part of the inequality cannot
be true. Therefore, the map is unstable at the FIXED
Point (0, 0).
References
Chirikov, B. V. "A Universal Instability of Many-
Dimensional Oscillator Systems." Phys. Rep. 52, 264-379,
1979.
Standard Normal Distribution
A Normal Distribution with zero Mean (fj, = 0) and
unity Standard Deviation (a 2 = 1).
see also NORMAL DISTRIBUTION
Standard Space
A Space which is Isomorphic to a Borel Subset B
of a Polish Space equipped with its Sigma Algebra
of Borel Sets.
see also Borel Set, Polish Space, Sigma Algebra
Standard Tori
full view
cutaway
cross-section
ring
torus
horn
torus
spindle
torus
One of the three classes of TORI illustrated above and
given by the parametric equations
The three different classes of standard tori arise from the
three possible relative sizes of a and c. c> a corresponds
to the Ring TORUS shown above, c = a corresponds to
a HORN Torus which touches itself at the point (0, 0,
0), and c < a corresponds to a self-intersecting SPIN-
DLE TORUS (Pinkall 1986). If no specification is made,
"torus" is taken to mean RING TORUS.
The standard tori and their inversions are CYCLIDES.
see also Apple, Cyclide, Horn Torus, Lemon, Ring
Torus, Spindle Torus, Torus
References
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.
Standardized Moment
Denned for samples Xi, i = 1, . . . , N by
„ 1 V^ r Mr
where
Zi =
The first few are
Oil
=
a 2
—
1
cxz
=
M3
5 3
OJ 4
=
M4
s 4
(1)
(2)
(3)
(4)
(5)
(6)
see also Kurtosis, Moment, Skewness
Standardized Score
see z-Score
Stanley's Theorem
The total number of Is that occur among all unordered
Partitions of a Positive Integer is equal to the sum
of the numbers of distinct parts of (i.e., numbers in)
those Partitions.
see also ELDER'S THEOREM, PARTITION
References
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer, pp. 6-8, 1985,
x = (c + a cos v) cos u
y = (c + a cos v) sin u
z = asinv.
(i)
(2)
(3)
1724
Star
Star Number
Star
In formal geometry, a star is a set of 2n VECTORS =bai ,
. . . , =ba n which form a fixed center in Euclidean 3-
SPACE. In common usage, a star is a STAR POLYGON
(i.e., regular convex polygon) such as the PENTAGRAM
or Hexagram
see also Cross, Eutactic Star, Star of Goliath,
Star Polygon
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 72-
77, 1991.
^ Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
Star of Goliath
see NONAGRAM
Star of David
see HEXAGRAM
Star Figure
A Star PoLYGON-like figure | E } for which p and q are
not Relatively Prime.
see also Star Polygon
Star (Fixed Point)
A Fixed Point which has one zero Eigenvector.
see Stable Star, Unstable Star
Star Fractal
A Fractal composed of repeated copies of a Penta-
gram or other polygon.
The above figure shows a generalization to different off-
sets from the center.
Star Graph
The fc-star graph is a Tree on k + 1 nodes with one
node having valency k and the others having valency 1.
Star graphs S n are always GRACEFUL.
Star of Lakshmi
The Star Figure {8/2}, which is used by Hindus to
symbolize Ashtalakshmi, the eight forms of wealth. This
symbol appears prominently in the Lugash national mu-
seum portrayed in the fictional film Return of the Pink
Panther.
see also Dissection, Hexagram, Pentagram, Star
Figure, Star Polygon
References
Savio, D. Y. and Suryanaroyan, E. R. "Chebyshev Polyno-
mials and Regular Polygons." Amer. Math. Monthly 100,
657-661, 1993.
Star Number
The number of cells in a generalized Chinese checkers
board (or "centered" Hexagram).
S n = 6n(n + l) + l = S n -! + 12(n - 1). (1)
The first few are 1, 13, 37, 73, 121, ... (Sloane's
A003154). Every star number has Digital Root 1 or
4, and the final digits must be one of: 01, 21, 41, 61, 81,
13, 33, 53, 73, 93, or 37.
The first Triangular star numbers are 1, 253, 49141,
9533161, . . . (Sloane's A006060), and can be computed
using
TS n =
3[(7 + 4x/3) 2n ~ 1 + (7-4 v / 3) 2n - 1 ] - 10
32
194r£ n _i +60- TS n -2.
(2)
The first few SQUARE star numbers are 1, 121, 11881,
1164241,114083761,... (Sloane's A006061). SQUARE
star numbers are obtained by solving the DlOPHANTINE
Equation
2x 2 + 1 = 3y 2 (3)
Star Polygon
Stationary Point 1725
and can be computed using
[(5 + 2>/6) w (V6 - 2) - (5 - 2y/6) n (V6 + 2)f
SS n =
(4)
see also Hex Number, Square Number, Triangular
Number
References
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 A003154/M4893, A006060/
M5425, and A006061/M5385 in "An On-Line Version of
the Encyclopedia of Integer Sequences."
Star Polygon
A star polygon {p/q}, with p,q POSITIVE INTEGERS, is
a figure formed by connecting with straight lines every
qth point out of p regularly spaced points lying on a
Circumference. The number q is called the Density
of the star polygon. Without loss of generality, take
q < p/2.
The usual definition (Coxeter 1969) requires p and q to
be Relatively Prime. However, the star polygon can
also be generalized to the Star Figure (or "improper"
star polygon) when p and q share a common divisor
(Savio and Suryanaroyan 1993). For such a figure, if
all points are not connected after the first pass, i.e., if
(p,q) ^ 1, then start with the first unconnected point
and repeat the procedure. Repeat until all points are
connected. For (p, q) ^ 1, the {p/q} symbol can be
factored as
{;}--{*}■
where
/ _ P
p = r
n
„' - q
q = «'
n
to give n {p 1 /q} figures, each rotated by 2tt/p radians,
or 360%.
If q = 1, a Regular Polygon {p} is obtained. Spe-
cial cases of {p/q} include {5/2} (the Pentagram),
{6/2} (the Hexagram, or Star of David), {8/2} (the
Star of Lakshmi), {8/3} (the Octagram), {10/3}
(the Decagram), and {12/5} (the Dodecagram).
The star polygons were first systematically studied by
Thomas Bradwardine.
see also DECAGRAM, HEXAGRAM, NONAGRAM, OCTA-
gram, Pentagram, Regular Polygon, Star of
Lakshmi, Stellated Polyhedron
References
Coxeter, H. S. M. "Star Polygons." §2.8 in Introduction to
Geometry, 2nd ed. New York: Wiley, pp. 36-38, 1969.
Frederickson, G. "Stardom." Ch. 16 in Dissections: Plane
and Fancy. New York: Cambridge University Press,
pp. 172-186, 1997.
Savio, D. Y. and Suryanaroyan, E. R. "Chebyshev Polyno-
mials and Regular Polygons." Amer. Math. Monthly 100,
657-661, 1993.
Star Polyhedron
see Kepler-Poinsot Solid
Starr Rose
a = 8, b = 16, c = 16 a = 6, b = 18, c = 18
see also Maurer Rose
References
Wagon, S. "Variations of Circular Motion." §4.5 in Mathe-
matica in Action. New York: W. H. Freeman, pp. 137-140,
1991.
State Space
The measurable space (S',§') into which a RANDOM
Variable from a Probability Space is a measurable
function.
see also PROBABILITY SPACE, RANDOM VARIABLE
Stationary Point
(1)
f'M < 0, i
f"(x)>0\
/'U)<0\ f'(x)>0
fix) =
A
V
f\x) =
fix) > / \fXx) <
(2)
stationary point
minimum
maximum
(3)
1726 Stationary Tangent
Statistics
A point xo at which the Derivative of a Function
f(x) vanishes,
f(x ) = 0.
A stationary point may be a MINIMUM, MAXIMUM, or
Inflection Point.
see also CRITICAL POINT, DERIVATIVE, EXTREMUM,
First Derivative Test, Inflection Point, Maxi-
mum, Minimum, Second Derivative Test
Stationary Tangent
see Inflection Point
Stationary Value
The value at a STATIONARY POINT.
Statistic
A function of one or more random variables,
see also Anderson-Darling Statistic, Kuiper
Statistic, Variate
Statistical Test
A test used to determine the statistical SIGNIFICANCE
of an observation. Two main types of error can occur:
1. A Type I Error occurs when a false negative result
is obtained in terms of the Null HYPOTHESIS by
obtaining a false positive measurement.
2. A Type II Error occurs when a false positive result
is obtained in terms of the Null Hypothesis by
obtaining a false negative measurement.
The probability that a statistical test will be positive for
a true statistic is sometimes called the test's SENSITIV-
ITY, and the probability that a test will be negative for
a negative statistic is sometimes called the SPECIFICITY.
The following table summarizes the names given to the
various combinations of the actual state of affairs and
observed test results.
result
name
true positive result
false negative result
true negative result
false positive result
sensitivity
1 — sensitivity
specificity
1 — specificity
Multiple-comparison corrections to statistical tests are
used when several statistical tests are being performed
simultaneously. For example, let's suppose you were
measuring leg length in eight different lizard species and
wanted to see whether the MEANS of any pair were dif-
ferent. Now, there are 8!/2!6! = 28 pairwise comparisons
possible, so even if all of the population means are equal,
it's quite likely that at least one pair of sample means
would differ significantly at the 5% level. An ALPHA
Value of 0.05 is therefore appropriate for each individ-
ual comparison, but not for the set of all comparisons.
In order to avoid a lot of spurious positives, the ALPHA
Value therefore needs to be lowered to account for the
number of comparisons being performed. This is a cor-
rection for multiple comparisons. There are many differ-
ent ways to do this. The simplest, and the most conser-
vative, is the Bonferroni Correction. In practice,
more people are more willing to accept false positives
(false rejection of Null HYPOTHESIS) than false neg-
atives (false acceptance of NULL HYPOTHESIS), so less
conservative comparisons are usually used.
see also ANOVA, Bonferroni Correction, Chi-
Squared Test, Fisher's Exact Test, Fisher
Sign Test, Kolmogorov-Smirnov Test, Likeli-
hood Ratio, Log Likelihood Procedure, Nega-
tive Likelihood Ratio, Paired £-Test, Paramet-
ric Test, Predictive Value, Sensitivity, Signif-
icance Test, Specificity, Type I Error, Type
II Error, Wilcoxon Rank Sum Test, Wilcoxon
Signed Rank Test
Statistics
The mathematical study of the Likelihood and Prob-
ability of events occurring based on known informa-
tion and inferred by taking a limited number of sam-
ples. Statistics plays an extremely important role in
many aspects of economics and science, allowing edu-
cated guesses to be made with a minimum of expensive
or difficult-to-obtain data.
see also Box-and- Whisker Plot, Buffon-Laplace
Needle Problem, Buffon's Needle Problem,
Chernoff Face, Coin Flipping, de Mere's Prob-
lem, Dice, Distribution, Gambler's Ruin, Index,
Likelihood, Moving Average, P- Value, Popula-
tion Comparison, Power (Statistics), Probabil-
ity, Residual vs. Predictor Plot, Run, Sharing
Problem, Statistical Test, Tail Probability
References
Brown, K. S. "Probability." http://www.seanet.com/
-ksbrown/ iprobabi , htm.
Babu, G. and Feigelson, E. Astro statistics. New York: Chap-
man & Hall, 1996.
Dixon, W. J. and Massey, F. J. Introduction to Statistical
Analysis, 4th ed. New York: McGraw-Hill, 1983.
Doob, J. L. Stochastic Processes. New York: Wiley, 1953.
Feller, W. An Introduction to Probability Theory and Its Ap-
plications, Vol. 1, 3rd ed. New York: Wiley, 1968.
Feller, W. An Introduction to Probability Theory and Its Ap-
plications, Vol. 2, 2nd ed. New York: Wiley, 1968.
Fisher, N. I.; Lewis, T.; and Embleton, B. J. J. Statistical
Analysis of Spherical Data. Cambridge, England: Cam-
bridge University Press, 1987.
Fisher, R. A. and Prance, G. T. The Design of Experiments,
9th ed. rev. New York: Hafner, 1974.
Fisher, R. A. Statistical Methods for Research Workers, 14th
ed,, rev. and enl. Darien, CO: Hafner, 1970.
Goldberg, S. Probability: An Introduction. New York:
Dover, 1986.
Gonick, L. and Smith, W. The Cartoon Guide to Statistics.
New York: Harper Perennial, 1993.
Goulden, C. H. Methods of Statistical Analysis, 2nd ed. New
York: Wiley, 1956.
Hoel, P. G.; Port, S. C; and Stone, C. J. Introduction to
Statistical Theory. New York: Houghton Mifflin, 1971.
Statistics
Steenrod Algebra 1727
Hogg, R. V. and Tanis, E. A. Probability and Statistical In-
ference, 3rd ed. New York: Macrnillan, 1988,
Keeping, E. S. Introduction to Statistical Inference. New
York: Dover, 1995.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.
Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall's Ad-
vanced Theory of Statistics, Vol. 1: Distribution Theory,
6th ed.0340614307 New York: Oxford University Press,
1987.
Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall's Ad-
vanced Theory of Statistics, Vol. 2A: 5th ed. New York:
Oxford University Press, 1987.
Kendall, M. G.; Stuart, A.; and Ord, J. K. Kendall's Ad-
vanced Theory of Statistics, Vol. 2B: Bayesian Inference.
New York: Oxford University Press, 1987.
Keynes, J. M. A Treatise on Probability. London: Macrnil-
lan, 1921.
Mises, R. von Mathematical Theory of Probability and Statis-
tics. New York: Academic Press, 1964.
Mises, R. von Probability, Statistics, and Truth, 2nd rev.
English ed. New York: Dover, 1981.
Mood, A. M. Introduction to the Theory of Statistics. New
York: McGraw-Hill, 1950.
Mostelier, F. Fifty Challenging Problems in Probability with
Solutions. New York: Dover, 1987.
Mostelier, F.; Rourke, R. E. K.; and Thomas, G. B. Prob-
ability: A First Course, 2nd ed. Reading, MA: Addison-
Wesley, 1970.
Neyman, J. First Course in Probability and Statistics. New
York: Holt, 1950.
Ostle, B. Statistics in Research: Basic Concepts and Tech-
niques for Research Workers, 4 th ed. Ames, I A: Iowa State
University Press, 1988.
Papoulis, A. Probability, Random Variables, and Stochastic
Processes, 2nd ed. New York: McGraw-Hill, 1984,
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Statistical Description of Data." Ch. 14
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 603-649, 1992.
Pugh, E. M. and Winslow, G. H. The Analysis of Physical
Measurements. Reading, MA: Addison- Wesley, 1966.
Renyi, A. Foundations of Probability. San Francisco, CA:
Holden-Day, 1970.
Robbins, H. and van Ryzin, J, Introduction to Statistics.
Chicago, IL: Science Research Associates, 1975.
Ross, S. M. A First Course in Probability. New York:
Macrnillan, 1976.
Ross, S. M, Introduction to Probability and Statistics for En-
gineers and Scientists. New York: Wiley, 1987.
Ross, S. M. Applied Probability Models with Optimization
Applications. New York: Dover, 1992.
Ross, S. M. Introduction to Probability Models, 5th ed. New
York: Academic Press, 1993.
Snedecor, G. W. Statistical Methods Applied to Experiments
in Agriculture and Biology, 5th ed. Ames, IA: State Col-
lege Press, 1956.
Tippett, L, H. C. The Methods of Statistics: An Introduc-
tion Mainly for Experimentalists, 3rd rev. ed. London:
Williams and Norgate, 1941.
Todhunter, I. A History of the Mathematical Theory of Prob-
ability from the Time of Pascal to that of Laplace. New
York: Chelsea, 1949.
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, 1977.
Uspensky, J. V. Introduction to Mathematical Probability.
New York: McGraw-Hill, 1937.
Weaver, W. Lady Luck: The Theory of Probability. New
York: Dover, 1963.
Whittaker, E. T. and Robinson, G. The Calculus of Observa-
tions: A Treatise on Numerical Mathematics, 4th ed. New
York: Dover, 1967.
Young, H. D. Statistical Treatment of Experimental Data.
New York: McGraw-Hill, 1962.
Yule, G. U. and Kendall, M. G. An Introduction to the The-
ory of Statistics, 14th ed., rev. and enl. New York: Hafner,
1950.
Staudt- Clausen Theorem
see VON STAUDT-CLAUSEN THEOREM
Steenrod Algebra
The Steenrod algebra has to do with the COHOMOL-
OGY operations in singular COHOMOLOGY with INTE-
GER mod 2 Coefficients. For every n e Z and
i £ {0,1,2,3,...} there are natural transformations of
Functors
Sq i :H n (^Z 2 )->H n+i (^Z 2 )
satisfying:
1. Sq i = for i > n.
2. Sq n (x) = x — x for all x e H n {X,A;Z 2 ) and all
pairs (X, A).
3. Sq°^id Hn ^ Z2y
4. The Sq l maps commute with the coboundary maps
in the long exact sequence of a pair. In other words,
Sq* :ir>;Z 2 )->ir + >;Z 2 )
is a degree i transformation of cohomology theories.
5. (Cartan Relation)
Sq'ix w j,) = H^ k ^Sq J {x) - Sq h {y).
6. (Adem Relations) For i < 2j,
Sq* o S<f(x) = EW, ( j . * 2k 1 ) Sj +i - k o Sq k (x).
7. Sq l o £ = S o Sq l where E is the cohomology suspen-
sion isomorphism.
The existence of these cohomology operations endows
the cohomology ring with the structure of a MODULE
over the Steenrod algebra A, defined to be T{F'j {Sq % :
i 6 {0,1,2,3, ...}})/#, where F% (•) is the free mod-
ule functor that takes any set and sends it to the free
Z2 module over that set. We think of F% {Sq 1 : i €
{0, 1,2,.. .}} as being a graded Z 2 module, where the
i-th gradation is given by Z 2 • Sq 1 , This makes the
tensor algebra T(F% {Sq { : i € {0,1,2,3,...}}) into a
Graded Algebra over Z 2 . R is the Ideal generated
by the elements Sq'Sq 3 + ^Lo{^-2h) s ^~ ks ^ and
1728 Steenrod-Eilenberg Axioms
Steinbach Screw
1 + Sq° for < i < 2j. This makes A into a graded Z2
algebra.
By the definition of the Steenrod algebra, for any Space
(X, A), H*(X, A] Z 2 ) is a MODULE over the Steenrod al-
gebra A, with multiplication induced by Sq t -x = Sq l (x).
With the above definitions, cohomology with COEFFI-
CIENTS in the RING Z 2 , H*(m;Z 2 ) is a FUNCTOR from
the category of pairs of TOPOLOGICAL SPACES to graded
modules over A.
see also Adem Relations, Cartan Relation, Coho-
mology, Graded Algebra, Ideal, Module, Topo-
logical Space
Steenrod-Eilenberg Axioms
see Eilenberg-Steenrod Axioms
Steenrod's Realization Problem
When can homology classes be realized as the image
of fundamental classes of MANIFOLDS? The answer is
known, and singular BORDISM GROUPS provide insight
into this problem.
see also BORDISM GROUP, MANIFOLD
Steepest Descent Method
An Algorithm for calculating the Gradient V/(P)
of a function at an n-D point P. The steepest descent
method starts at a point Po and, as many times as
needed, moves from P, to P;+i by minimizing along
the line extending from P^ in the direction of — V/(Pi),
the local downhill gradient. This method has the severe
drawback of requiring a great many iterations for func-
tions which have long, narrow valley structures. In such
cases, a Conjugate Gradient Method is preferable.
see also CONJUGATE GRADIENT METHOD, GRADIENT
References
Arfken, G. "The Method of Steepest Descents." §7.4 in Math-
ematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, pp. 428-436, 1985.
Menzel, D. (Ed.). Fundamental Formulas of Physics, Vol. 2,
2nd ed. New York: Dover, p. 80, 1960.
Morse, P. M. and Feshbach, H. "Asymptotic Series; Method
of Steepest Descent." §4.6 in Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 434-443, 1953.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 414, 1992.
Steffenson's Formula
U = /o + \piP + l)*i/a " W - l)pS-i/2
+(5 3 4- S 4 )6$ /2 + (S3 " Sa)SU/2 + ■ ■ • , (1)
for p e [-f , §], where S is the CENTRAL DIFFERENCE
and
S2n+1 = 2{2n+l)
(2)
s — p (p +n \
(3)
bin+1 ~ 2n + 2\2n+l)
<$2ti+1
fp + n+l\
~ 52n+2= V 2n + 2 )
(4)
£2n+l
- S 2 n+2 = " ( £ " ) ,
(5)
where (£) is a Binomial COEFFICIENT.
see also Central Difference, Stirling's Finite
Difference Formula
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 433, 1987.
Steffensen's Inequality
Let f(x) be a NONNEGATIVE and monotonic decreasing
function in [a, b] and g(x) satisfy such that < g(x) < 1
in [a, 6], then
/»& nb i*a-\-k
I f(x)dx< / f(x)g(x)dx < / f{x)dx,
J b — k J a J a
where
References
/ 9(v.
J a
) dx.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1099, 1979.
Steinbach Screw
A SURFACE generated by the parametric equations
x(u,v) = ucosv
y(u,v) = iisint>
z(u, v) = v cosu.
The above image uses u € [—4,4] and v (E [0,6.25].
References
Naylor, B. "Steinbach Screw 1." http://www.garlic.com/
-bnaylor/rtsteinl .html.
Pickover, C. A. Mazes for the Mind: Computers and the
Unexpected. New York: St. Martin's Press, 1992.
Wang, P. "Renderings." http://www.ugcs.caltech.edu/
*peterw/portf olio/renderings/.
Steiner Chain
Steiner Chain
Given two nonconcentric CIRCLES with one interior to
the other, if small TANGENT CIRCLES can be inscribed
around the region between the two Circles such that
the final CIRCLE is TANGENT to the first, the CIRCLES
form a Steiner chain.
The simplest way to construct a Steiner chain is to per-
form an INVERSION on a symmetrical arrangement on n
circles packed between a central circle of radius b and an
outer concentric circle of radius a. In this arrangement,
sin I — =
Vn/ a +
a — b
(1)
so the ratio of the radii for the small and large circles is
■GO
a l + sin(^)-
(2)
To transform the symmetrical arrangement into a
Steiner chain, find an Inversion Center which trans-
forms two centers initially offset by a fixed distance c to
the same point. This can be done by equating
k 2 x
k 2 (x-c)
x 2 — a 2 (x — c) 2 — b 2 '
(3)
giving the offset of the inversion center from the large
circle's center as
a 2 - b 2 + c 2 ± y/(a? - b 2 + c 2
4a 2 c
2c
(4)
Plugging in a fixed value of a fixes b, which therefore
determines x for a given c. Equivalently, a Steiner chain
results whenever the Inversive Distance between the
two original circles is given by
* = 2b. [«*(£)+ tang)] (5)
= 2b MS + £)] (6)
(Coxeter and Greitzer 1967). The centers of the circles
in a Steiner chain lie on an ELLIPSE (Ogilvy 1990, p. 57).
Steiner'S PORISM states that if a Steiner chain is
formed from one starting circle, then a Steiner chain
is also formed from any other starting circle.
Steiner's Ellipse 1729
see also Arbelos, Coxeter's Loxodromic Sequence
of Tangent Circles, Hexlet, Pappus Chain,
Steiner's Porism
References
Coxeter, H. S. M. "Interlocking Rings of Spheres." Scripta
Math. 18, 113-121, 1952.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 87, 1969.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 124-126, 1967.
Forder, H. G. Geometry, 2nd ed. London: Hutchinson's Uni-
versity Library, p. 23, 1960.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Sci. Amer.
240, 18-28, Jan. 1979.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 113-115, 1929.
Ogilvy, C S. Excursions in Geometry. New York: Dover,
pp. 51-54, 1990.
# Weisstein, E. W. "Plane Geometry." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/PlaneGeometry.m.
Steiner Construction
A construction done using only a STRAIGHTEDGE. The
Poncelet-Steiner Theorem proves that all construc-
tions possible using a COMPASS and STRAIGHTEDGE are
possible using a STRAIGHTEDGE alone, as long as a fixed
Circle and its center, two intersecting CIRCLES with-
out their centers, or three nonintersecting CIRCLES, are
drawn beforehand.
see also Geometric Construction, Mascher-
oni Construction, Poncelet-Steiner Theorem,
Straightedge
References
Dorrie, H. "Steiner's Straight-Edge Problem." §34 in 100
Great Problems of Elementary Mathematics: Their His-
tory and Solutions. New York: Dover, pp. 165-170, 1965.
Steiner, J. Geometric Constructions with a Ruler, Given a
Fixed Circle with Its Center. Translated from the first Ger-
man ed. (1833). New York: Scripta Mathematica, 1950.
Steiner's Ellipse
Let a' : 0' : 7' be the ISOTOMIC CONJUGATE POINT
of a point with TRILINEAR COORDINATES a : (3 : 7.
The isotomic conjugate of the Line at Infinity having
trilinear equation
act + b/3 + C7 =
0W 7V a'0'
a c
0,
known as Steiner's ellipse (Vandeghen 1965).
see also ISOTOMIC CONJUGATE POINT, LINE AT INFIN-
ITY
References
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.
1730 Steiner's Hypocycloid
Steiner's Porism
Steiner's Hypocycloid
see Deltoid
Steiner-Lehmus Theorem
Any Triangle that has two equal Angle Bisec-
tors (each measured from a Vertex to the opposite
sides) is an ISOSCELES TRIANGLE. This theorem is
also called the INTERNAL BISECTORS PROBLEM and
Lehmus' Theorem.
see also ISOSCELES TRIANGLE
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, p. 72, 1952.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New-
York: Wiley, p. 9, 1969.
Coxeter, H. S, M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 14-16, 1967.
Gardner, M. Martin Gardner's New Mathematical Diver-
sions from Scientific American. New York: Simon and
Schuster, pp. 198-199 and 206-207, 1966.
Henderson, A. "The Lehmus-Steiner-Terquem Problem in
Global Survey." Scripta Math. 21, 223-232 and 309-312,
1955.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 72-73, 1975.
Steiner Points
There are two different types of points known as Steiner
points.
The point of CONCURRENCE of the three lines drawn
through the VERTICES of a TRIANGLE PARALLEL to the
corresponding sides of the first Brocard Triangle. It
lies on the Circumcircle opposite the Tarry Point
and has Triangle Center Function
bc(a 2 ~b 2 ){a
c 2 )-
The Brianchon Point for Kiepert's Parabola is the
Steiner point. The Lemoine Point K is the Steiner
point of the first Brocard Triangle. The Simson
Line of the Steiner point is Parallel to the line OK,
when O is the Circumcenter and K is the Lemoine
Point.
If triplets of opposites sides on a Conic Section in
PASCAL'S THEOREM are extended for all permutations
of Vertices, 60 Pascal Lines are produced. The 20
points of their 3 by 3 intersections are called Steiner
points,
see also Brianchon Point, Brocard Trian-
gles, Circumcircle, Conic Section, Kiepert's
Parabola, Lemoine Point, Pascal Line, Pascal's
Theorem, Steiner Set, Steiner Triple System,
Tarry Point
References
Casey, J. A Treatise on the Analytical Geometry of the Point,
Line, Circle, and Conic Sections, Containing an Account
of Its Most Recent Extensions, with Numerous Examples,
2nd ed., rev. enl. Dublin: Hodges, Figgis, &; Co., p. 66,
1893.
Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.
London: Hodgson, p. 102, 1913.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 281-282, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Steiner's Porism
If a STEINER Chain is formed from one starting cir-
cle, then a STEINER CHAIN is formed from any other
starting circle. In other words, given two nonconcen-
tric Circles, draw Circles successively touching them
and each other. If the last touches the first, this will
also happen for any position of the first CIRCLE.
see also HEXLET, STEINER CHAIN
References
Coxeter, H. S. M. "Interlocking Rings of Spheres." Scripta
Math. 18, 113-121, 1952.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 87, 1969.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 124-126, 1967.
Forder, H. G. Geometry, 2nd ed. London: Hutchinson's Uni-
versity Library, p. 23, 1960.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Sci. Amer.
240, 18-28, Jan. 1979.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 113-115, 1929.
Ogilvy, C. S. Excursions in Geometry. New York: Dover,
pp. 53-54, 1990.
Steiner 7 s Problem
Steiner Set 1731
Steiner's Problem
1.4
2 4 6 8 10
For what value of x is f(x) = x 1/x a MAXIMUM? The
maximum occurs at x = e, where
/'(x)-aT 2+1/l (l-lnz) = 0,
which gives a maximum of
i/fi
1.444667861..
The function has an inflection point at x = 0.581933 . . .,
where
f"{x) = x" 4+1/a: [l - 3x + (lnx)(2x - 2 + lnx)] = 0.
see also Fermat's Problem
Steiner Quadruple System
A Steiner quadruple system is a Steiner System S(t =
3, A; = 4, v), where 5 is a v-set and B is a collection of
fc-sets of 5 such that every t-subset of 5 is contained
in exactly one member of B. Barrau (1908) established
the uniqueness of 5(3,4,8),
1
2
4
8
3 5
6
7
2
3
5
8
1 4
6
7
3
4
6
8
1 2
5
7
4
5
7
8
1 2
3
6
1
5
6
8
2 3
4
7
2
6
7
8
1 3
4
5
1
3
7
8
2 4
5
6
and 5(3,4
10)
1
2
4
5
1
2
3
7
1
3
5
8
2
3
5
6
2
3
4
8
2
4
6
9
3
4
6
7
3
4
5
9
3
5
7
4
5
7
8
4
5
6
1
4
6
8
5
6
8
9
1
5
6
7
2
5
7
9
6
7
9
2
6
7
8
3
6
8
1
7
8
3
7
8
9
1
4
7
9
1
2
8
9
4
8
9
2
5
8
2
3
9
1
5
9
1
3
6
9
1
3
4
1
2
6
2
4
7
(1935) showed the existence of at least one 5(3,4,14).
Hanani (1960) proved that a NECESSARY and SUFFI-
CIENT condition for the existence of an 5(3, 4, v) is that
v = 2 or 4 (mod 6).
The number of nonisomorphic steiner quadruple systems
of orders 8, 10, 14, and 16 are 1, 1, 4 (Mendelsohn
and Hung 1972), and at least 31,021 (Lindner and Rosa
1976).
see also Steiner System, Steiner Triple System
References
Barrau, J. A. "On the Combinatory Problem of Steiner."
K. Akad. Wet. Amsterdam Proc. Sect. Sci. 11, 352-360,
1908.
Bays, S. and de Week, E. "Sur les syst ernes de quadruples."
Comment. Math. Helv. 7, 222-241, 1935.
Fitting, F. "Zyklische Losungen des Steiner'schen Problems."
Nieuw. Arch. Wisk. 11, 140-148, 1915.
Hanani, M. "On Quadruple Systems." Canad. J. Math. 12,
145-157, 1960.
Lindner, C. L. and Rosa, A. "There are at Least 31,021 Non-
isomorphic Steiner Quadruple Systems of Order 16." UtiU
itas Math. 10, 61-64, 1976.
Lindner, C. L. and Rosa, A. "Steiner Quadruple Systems — A
Survey." Disc. Math. 22, 147-181, 1978.
Mendelsohn, N. S. and Hung, S. H. Y. "On the Steiner Sys-
tems 5(3,4,14) and 5(4,5,15)." Utilitas Math. 1, 5-95,
1972.
Steiner's Segment Problem
Given n points, find the line segments with the shortest
possible total length which connect the points. The seg-
ments need not necessarily be straight from one point
to another.
For three points, if all Angles are less than 120°, then
the line segments are those connecting the three points
to a central point P which makes the ANGLES (A) PB i
(B) PC, and (C) PA all 120°. If one Angle is greater
that 120°, then P coincides with the offending ANGLE.
For four points, P is the intersection of the two diago-
nals, but the required minimum segments are not nec-
essarily these diagonals.
A modified version of the problem is, given two points,
to find the segments with the shortest total length con-
necting the points such that each branch point may be
connected to only three segments. There is no general
solution to this version of the problem.
Steiner Set
Three sets of three LINES such that each line is incident
with two from both other sets.
see also Solomon's Seal Lines, Steiner Points,
Steiner Triple System
Fitting (1915) subsequently constructed the cyclic sys-
tems 5(3,4,26) and 5(3,4,34), and Bays and de Week
1732
Steiner Surface
Steiner Triple System
Steiner Surface
A projection of the VERONESE SURFACE into 3-D (which
must contain singularities) is called a Steiner surface.
A classification of Steiner surfaces allowing complex
parameters and projective transformations was accom-
plished in the 19th century. The surfaces obtained by
restricting to real parameters and transformations were
classified into 10 types by Coffman et at. (1996). Ex-
amples of Steiner surfaces include the ROMAN SURFACE
(Coffman type 1) and Cross-Cap (type 3).
The Steiner surface of type 2 is given by the implicit
equation
2 2
x y
2 2,22 n
x z + y z — xyz — 0,
and can be transformed into the ROMAN SURFACE or
CROSS- Cap by a complex projective change of coordi-
nates (but not by a real transformation). It has two
pinch points and three double lines and, unlike the RO-
MAN Surface or Cross-Cap, is not compact in any
affine neighborhood.
The Steiner surface of type 4 has the implicit equation
y
2 2 2 2 2 2 4
2xy ~ xz + x y + x z — z
0,
and two of the three double lines of surface 2 coincide
along a line where the two noncompact "components"
are tangent.
see also Cross-Cap, Roman Surface, Veronese Va-
riety
References
Coffman, A. "Steiner Surfaces." http://www.ipfw.edu/
math/Coffman/st einersurface.html.
Coffman, A.; Schwartz, A.; and Stanton, C "The Alge-
bra and Geometry of Steiner and Other Quadratically
Parametrizable Surfaces." Computer Aided Geom. Design
13, 257-286, 1996.
Nordstrand, T. "Steiner Relative." http://www.uib.no/
people/nf ytn/stmtxt .htm.
Nordstrand, T. "Steiner Relative [2]." http://www.uib.no/
people/nf ytn/stm2txt . htm.
Steiner System
A Steiner system is a set X of v points, and a collection
of subsets of X of size k (called blocks), such that any
t points of X are in exactly one of the blocks. The
special case t = 2 and k — 3 corresponds to a so-called
Steiner Triple System. For a Projective Plane,
v — n 2 + n + 1, fc = n + l,i = 2, and the blocks are
simply lines.
see also Steiner Quadruple System, Steiner
Triple System.
References
Colbourn, C J. and Dinitz, J. H. (Eds.) CRC Handbook
of Combinatorial Designs. Boca Raton, FL: CRC Press,
1996.
Woolhouse, W. S. B. "Prize Question 1733." Lady's and
Gentleman's Diary. 1844.
Steiner's Theorem
Let Lines x and y join a variable point on a Conic Sec-
tion to two fixed points on the same Conic Section.
Then x and y are PROJECTIVELY related.
see also CONIC SECTION, PROJECTION
Steiner Triple System
Let X be a set of v > 3 elements together with a set B
of 3-subset (triples) of X such that every 2-SUBSET of
X occurs in exactly one triple of B. Then B is called a
Steiner triple system and is a special case of a STEINER
System with t = 2 and k = 3. A Steiner triple system
S(v) = S(v,k = 3, A = 1) of order v exists IFF v =
1,3 (mod 6) (Kirkman 1847). In addition, if Steiner
triple systems Si and S2 of orders v\ and V2 exist, then
so does a Steiner triple system S of order ^1^2 (Ryser
1963, p. 101).
Examples of Steiner triple systems S(v) of small orders
v are
S 3 = {{1,2,3}}
S 7 = {{1,2,4}, {2, 3,5}, {3, 4,6}, {4, 5, 7},
{5, 6,1}, {6, 7, 2}, {7, 1,3}}
5 9 = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {1, 4, 7},
{2,5,8},{3,6,9},{1,5,9},{2,6,7}}.
The number of nonisomorphic Steiner triple systems
S(v) of orders v = 7, 9, 13, 15, 19, ... (i.e., 6k + 1,3)
are 1, 1, 20, 80, > 1.1 x 10 9 , . . . (Colbourn and Dinitz
1996, pp. 14-15; Sloane's A030129). 5(7) is the same
as the finite Projective Plane of order 2. 5(9) is a
finite Affine Plane which can be constructed from the
array
a b c
d e f.
9 h i
One of the two 5(13)s is a finite Hyperbolic Plane.
The 80 Steiner triple systems 5(15) have been studied
by Tonchev and Weishaar (1997). There are more than
1.1 x 10 9 Steiner triple systems of order 19 (Stinson and
Ferch 1985; Colbourn and Dinitz 1996, p. 15).
see also Hadamard Matrix, Kirkman Triple Sys-
tem, Steiner Quadruple System, Steiner System
References
Colbourn, C. J. and Dinitz, J. H. (Eds.) "Steiner Triple Sys-
tems." §4.5 in CRC Handbook of Combinatorial Designs.
Boca Raton, FL: CRC Press, pp. 14-15 and 70, 1996.
Kirkman, T. P. "On a Problem in Combinatorics." Cam-
bridge Dublin Math. J. 2, 191-204, 1847.
Lindner, C. C. and Rodger, C- A. Design Theory. Boca
Raton, FL: CRC Press, 1997.
Ryser, H. J. Combinatorial Mathematics. Buffalo, NY:
Math. Assoc. Amer., pp. 99-102, 1963.
Sloane, N. J. A. Sequence A030129 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Stinson, D. R. and Ferch, H. "2000000 Steiner Triple Systems
of Order 19." Math. Comput. 44, 533-535, 1985.
Steinerian Curve
Steinmetz Solid 1733
Tonchev, V. D. and Weishaar, R. S. "Steiner Triple Systems
of Order 15 and Their Codes." J. Stat. Plan. Inference
58, 207-216, 1997.
Steinerian Curve
The LOCUS of points whose first POLARS with regard to
the curves of a linear net have a common point. It is also
the LOCUS of points of CONCURRENCE of line POLARS
of points of the JACOBIAN CURVE. It passes through
all points common to all curves of the system and is of
order 3(ra - l) 2 .
see also Cayleyian Curve, Jacobian Curve
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 150, 1959.
Steinhaus-Moser Notation
A Notation for Large Numbers defined by Steinhaus
(1983, pp. 28-29). In this notation, /£\ denotes n 71 , [n]
denotes "n in n TRIANGLES," and © denotes "n in n
Squares." A modified version due to Moser eliminates
the circle notation, continuing instead with POLYGONS
of ever increasing size, so n in a PENTAGON is n with n
Squares around it, etc.
see also Circle Notation, Large Number, Mega,
Moser
References
Steinhaus, H. Mathematical Snapshots, 3rd American ed.
New York: Oxford University Press, 1983.
Steinitz's Theorem
A Graph G is the edge graph of a Polyhedron Iff G
is a Simple, Planar Graph which is 3-connected.
see also PLANAR GRAPH, SIMPLE GRAPH
If the two right CYLINDERS are of different RADII a and
b with a > 6, then the VOLUME common to them is
V 2 (a,b) = |a[(a 2 + b 2 )E{k) - (a 2 - b 2 )K(k)] 9 (2)
where K(k) is the complete ELLIPTIC INTEGRAL OF THE
First Kind, E(k) is the complete Elliptic Integral
of the Second Kind, and k = b/a is the Modulus.
The curves of intersection of two cylinders of Radii a
and 6, shown above, are given by the parametric equa-
tions
x(t) = acosi (3)
y{t) = asini (4)
z{t) = ±\/*> 2 -a 2 sm 2 t (5)
(Gray 1993).
The Volume common to two Elliptic Cylinders
x 2 z 2 ,
2 2
y %
b 2 d 2
(6)
Steinmetz Solid
with c < c' is
The solid common to two (or three) right circular
Cylinders of equal Radii intersecting at Right An-
gles is called the Steinmetz solid. (Two CYLINDERS
intersecting at Right Angles are sometimes called a
BICYLINDER, and three intersecting CYLINDERS a Tri-
CYLINDER.)
The VOLUME common to two intersecting right CYLIN-
DERS of Radius r is
V 2 (r,r)
16 r 3
(1)
8ab r
V a (a,c;6,c') = ^[(c' 2 +c 2 )E(k)-(c' 2 -c 2 )K{k% (7)
where k = c/c' (Bowman 1961, p. 34).
For three Cylinders of Radii r intersecting at Right
Angles, the Volume of intersection is
V r 3 (r,r,r) = 8(2-v / 2)r 3 .
see also Bicylinder, Cylinder
(8)
References
Bowman, F. Introduction to Elliptic Functions, with Appli-
cations. New York: Dover, 1961.
Gardner, M. The Unexpected Hanging and Other Mathemat-
ical Diversions. Chicago, IL: Chicago University Press,
pp. 183-185, 1991.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 149-150, 1993.
Wells, D. G. #555 in The Penguin Book of Curious and
Interesting Puzzles. London: Penguin Books, 1992.
1734 Stella Octangula
Stella Octangula
A Polyhedron Compound composed of a Tetrahe-
dron and its RECIPROCAL (a second TETRAHEDRON
rotated 180° with respect to the first). The stella oct-
angula is also called a Stellated Tetrahedron. It
can be constructed using the following Net by cutting
along the solid lines, folding back along the plain lines,
and folding forward along the dotted lines.
Another construction builds a single TETRAHEDRON,
then attaches four tetrahedral caps, one to each face.
The edges of the two tetrahedra form the 12 DIAGONALS
of a CUBE. The solid common to both tetrahedra is an
Octahedron (Ball and Coxeter 1987).
see also Cube, Octahedron, Polyhedron Com-
pound, Tetrahedron
Stellated Tetrahedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 135—
137, 1987.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, p. 158, 1969.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 129, 1989.
Stella Octangula Number
A Figurate Number of the form,
StOct n = O n + STn-i = n(2n 2 - 1).
The first few are 1, 14, 51, 124, 245, ... (Sloane's
A007588). The Generating Function for the stella
octangula numbers is
X(X 2 + lOz + 1) O O A
K , ,,7 ; = x + 14z 2 + 51Z 3 + 124a; 4 + . . . .
(x - l) 4
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 51, 1996.
Sloane, N. J. A. Sequence A007588/M4932 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Stellated Polyhedron
A convex regular POLYHEDRON. Stellated polyhedra in-
clude the Kepler-Poinsot Solids, which consist of
three Dodecahedron Stellations and one of the
Icosahedron Stellations. Coxeter (1982) shows
that 59 Icosahedron Stellations exist. The Cube
and the TETRAHEDRON cannot be stellated. The OCT-
AHEDRON has only one stellation, the Stella OCTAN-
GULA which is a compound of two Tetrahedra.
There are therefore a total of 3 -f 1 + (59 - 1) + 1 = 63
stellated POLYHEDRA, although some are COMPOUND
Polyhedra and therefore not Uniform Polyhedra.
The set of all possible EDGES of the stellations can be
obtained by finding all intersections on the facial planes.
see also ARCHIMEDEAN SOLID STELLATION, DODEC-
AHEDRON Stellations, Icosahedron Stellations,
Kepler-Poinsot Solid, Polyhedron, Stella Oc-
tangula, Stellated Truncated Hexahedron,
Stellation, Uniform Polyhedron
References
Coxeter, H. S. M. The Fifty-Nine Icosahedra. New York:
Springer-Verlag, 1982.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Publications, 1989.
Wenninger, M. J. Polyhedron Models. Cambridge, England:
University Press, 1974.
Stellated Tetrahedron
see Stella Octangula
Stellated Truncated Hexahedron
Stellated Truncated Hexahedron
The Uniform Polyhedron LT 19 , also called the Quasi-
truncated Hexahedron, whose Dual Polyhedron
is the Great Triakis Octahedron. It has Schlafli
Symbol t'{4,3} and Wythoff Symbol 23 ||. Its
faces are 8{3} + 6{|}. For a = 1, its CIRCUMRADIUS
is
R= \^7-AV2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 144, 1989.
Stellation
The process of constructing POLYHEDRA by extending
the facial Planes past the Edges of a given Polyhe-
dron.
see also Archimedean Solid Stellation, Dodec-
ahedron Stellations, Faceting, Icosahedron
Stellations, Kepler-Poinsot Solid, Polyhedron,
Stella Octangula, Stellated Polyhedron, Stel-
lated Truncated Hexahedron, Stellation Trun-
cation, Uniform Polyhedron
References
Fleurent, G. M. "Symmetry and Polyhedral Stellation la and
lb. Symmetry 2: Unifying Human Understanding, Part 1."
Comput. Math. Appl. 17, 167-193, 1989.
Messer, P. W. "Les etoilements du rhombitricontaedre et
plus." Structural Topology 21, 25-46, 1995.
Messer, P. W. and Wenninger, M. J. "Symmetry and Poly-
hedral Stellation. II. Symmetry 2: Unifying Human Un-
derstanding, Part 1." Comput Math. Appl. 17, 195-201,
1989.
Stem-and-Leaf Diagram
The "stem" is a column of the data with the last digit
removed. The final digits of each column are placed
next to each other in a row next to the appropriate col-
umn. Then each row is sorted in numerical order. This
diagram was invented by John Tukey.
References
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, pp. 7-16, 1977.
Step
1.5 times the H-SPREAD.
see also FENCE, H-Spread
References
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, p. 44, 1977.
Stereogram 1 735
Step Function
A function on the REALS K is a step function if it can
be written as a finite linear combination of semi-open
intervals [a, 6) C R. Therefore, a step function / can be
written as
f(x) = a±fi(x) H h a n fn(x),
where oti £ R, fi(x) = 1 if x G [a*, &») and otherwise,
for i = 1, . . . , n.
see also HEAVISIDE STEP FUNCTION
Step Polynomial
see Hermite's Interpolating Fundamental Poly-
nomial
Steradian
The unit of Solid Angle. The Solid Angle corre-
sponding to all of space being subtended is 4n steradian.
see also Radian, Solid Angle
Stereogram
A plane image or pair of 2-D images which, when ap-
propriately viewed using both eyes, produces an image
which appears to be three-dimensional. By taking a pair
of photographs from slightly different angles and then al-
lowing one eye to view each image, a stereogram is not
difficult to produce.
Amazingly, it turns out that the 3-D effect can be pro-
duced by both eyes looking at a single image by defo-
cusing the eyes at a certain distance. Such stereograms
are called "random-dot stereograms."
References
Bar-Natan, D. "Random-Dot Stereograms." Math. J. 1, 69-
71, 1991.
Fineman, M. The Nature of Visual Illusion. New York:
Dover, pp. 89-93, 1996.
Julesz, B. Foundations of Cyclopean Perception. Chicago,
IL: University of Chicago Press, 1971.
Julesz, B. "Stereoscopic Vision." Vision Res. 26, 1601-1611,
1986.
Terrell, M. S. and Terrell, R.
Random Dot Stereogram."
715-724, 1994.
Tyler, C. "Sensory Processing of Binocular Disparity." In
Vergence Eye Movements: Basic and Clinical Aspects.
Boston, MA: Butterworth, pp. 199-295, 1983.
E. "Behind the Scenes of a
Amer. Math. Monthly 101,
1736 Stereographic Projection
Stereographic Projection
A Map Projection in which Great Circles are Cir-
cles and LOXODROMES are LOGARITHMIC SPIRALS.
x = A;cos0sin(A — Ao) (1)
y = k[cos<pi sin0 — sin^i cos0cos(A — Ao)], (2)
where
k
1 + sin 0i sin(/> + cos <f>i cos^cos(A — Ao) '
The inverse FORMULAS are given by
. _i / . , . ysinccos0i\
q> = sin cos c sin 0i H
V p J
A = Ao + tan -1
p cos 0i cos c — y sin <pi sin c
where
(3)
(4)
(5)
(6)
(7)
p = sjx 2 + y 2
c = 2tan~ 1 (|p).
see a/so GALL'S STEREOGRAPHIC PROJECTION
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. 154-163, 1987.
Stereology
The exploration of 3-D space from 2-D sections of PRO-
JECTIONS of solid bodies.
see also Axonometry, Cork Plug, Cross-Section,
Projection, Trip-Let
Stern-Brocot Tree
Stick Number
A special type of Binary Tree obtained by starting
with the fractions j and £ and iteratively inserting (m+
m')/(n + n') between each two adjacent fractions m/n
and m! /n f . The result can be arranged in tree form as
illustrated above. The Farey Sequence F n defines a
subtree of the Stern-Brocot tree obtained by pruning off
unwanted branches (Vardi 1991, Graham et al. 1994).
see also BINARY TREE, FAREY SEQUENCE, FORD CIR-
CLE
References
Brocot, A. "Calcul des rouages par approximation, nouvelle
methode." Revue Chonometrique 6, 186—194, 1860.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science, 2nd
ed. Reading, MA: Addison-Wesley, pp. 116-117, 1994.
Stern, M. A. "Uber eine zahlentheoretische Funktion." J.
reine angew. Math. 55, 193-220, 1858.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison-Wesley, p. 253, 1991.
Stevedore's Knot
The 6-crossing KNOT
der Polynomial
■ooi having CONWAY-ALEXAN-
A(t) = 2t 2 -5t + 2.
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 225, 1976.
Stewart's Theorem
where
i(p H- run) = b m + c n,
a = m + n.
References
Altshiller-Court, N. "Stewart's Theorem." §6B in College
Geometry: A Second Course in Plane Geometry for Col-
leges and Normal Schools, 2nd ed., rev. enl. New York:
Barnes and Noble, pp. 152-153, 1952.
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., p. 6, 1967.
Stick Number
Let the stick number s(K) of a Knot K be the least
number of straight sticks needed to make a KNOT K,
The smallest stick number of any Knot is s(T) = 6,
where T is the TREFOIL Knot. If J and K are Knots,
then
s(J + K) < s{J) + s(K) + l.
Stickelberger Relation
Stieltjes Constants 1737
For a nontrivial Knot K, let c(K) be the CROSSING
NUMBER (i.e., the least number of crossings in any pro-
jection of K). Then
|[5 + v/25 + S(c(K) - 2)] < s(K) < 2c(K).
The following table gives the stick number for some com-
mon knots.
Knot
trefoil knot 6
Whitehead link 8
see also CROSSING NUMBER (LINK), TRIANGLE COUNT-
ING
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 27-30, 1994.
Stickelberger Relation
Let P be a PRIME Ideal in D m not containing m. Then
(§(p)) = pE't 1 ,
where the sum is over all 1 < t < m which are RELA-
TIVELY Prime to m. Here Dm is the Ring of integers in
Q(Cm), $(P) ~ 9{P) m 'i an d other quantities are defined
by Ireland and Rosen (1990).
see also Prime Ideal
References
Ireland, K. and Rosen, M. "The Stickelberger Relation and
the Eisenstein Reciprocity Law." Ch. 14 in A Classical In-
troduction to Modern Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 203-227, 1990.
Stiefel Manifold
The Stiefel manifold of ORTHONORMAL A;-frames in R n
is the collection of vectors (i>i, . . . , Vk) where vi is in W 1
for all i, and the fc-tuple (vi } . . . , Vk) is ORTHONORMAL.
This is a submanifold of E nfc , having DIMENSION nk -
(k + l)fc/2.
Sometimes the "orthonormal" condition is dropped in
favor of the mildly weaker condition that the fc-tuple (vi,
. . . , Vk) is linearly independent. Usually, this does not
affect the applications since Stiefel manifolds are usually
considered only during HOMOTOPY THEORETIC consid-
erations. With respect to HOMOTOPY THEORY, the
two definitions are more or less equivalent since GRAM-
SCHMIDT ORTHONORMALIZATION gives rise to a smooth
deformation retraction of the second type of Stiefel man-
ifold onto the first.
see also Grassmann Manifold
Stiefel- Whitney Class
The ith Stiefel- Whitney class of a Real Vector Bun-
dle (or Tangent Bundle or a Real Manifold) is in
the ith cohomology group of the base SPACE involved.
It is an Obstruction to the existence of (n — i + 1)
REAL linearly independent VECTOR FIELDS on that
Vector Bundle, where n is the dimension of the
Fiber. Here, Obstruction means that the zth Stiefel-
Whitney class being NONZERO implies that there do not
exist (n — i + 1) everywhere linearly dependent VECTOR
Fields (although the Stiefel- Whitney classes are not al-
ways the Obstruction).
In particular, the nth Stiefel- Whitney class is the ob-
struction to the existence of an everywhere NONZERO
Vector Field, and the first Stiefel- Whitney class of a
Manifold is the obstruction to orientability.
see also Chern Class, Obstruction, Pontryagin
Class, Stiefel- Whitney Number
Stiefel- Whitney Number
The Stiefel- Whitney number is defined in terms of the
Stiefel- Whitney Class of a Manifold as follows.
For any collection of Stiefel- Whitney Classes such
that their cup product has the same DIMENSION as
the Manifold, this cup product can be evaluated on
the Manifold's Fundamental Class. The result-
ing number is called the PONTRYAGIN NUMBER for that
combination of Pontryagin classes.
The most important aspect of Stiefel- Whitney numbers
is that they are COBORDISM invariant. Together, PON-
TRYAGIN and Stiefel- Whitney numbers determine an ori-
ented Manifold's Cobordism class.
see also Chern Number, Pontryagin Number,
Stiefel- Whitney Class
Stieltjes Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Expanding the Riemann Zeta Function about z — 1
gives
^) = ^r+E^r^- 1 )"-
(i)
where
7n = lim
E
(\nk) n (mm)
n+l
n + 1
An alternative definition is given by
(2)
(3)
1738 Stieltjes Constants
Stieltjes' Theorem
The case n — gives the Euler-Mascheroni Con-
stant 7. The first few numerical values are given in the
following table.
n
In
0.5772156649
1
-0.07281584548
2
-0.009690363192
3
0.002053834420
4
0.002325370065
5
0.0007933238173
Briggs (1955-1956) proved that there infinitely many y n
of each SIGN. Berndt (1972) gave upper bounds of
l7n|<
) 2(n-l)!
V 7T n
for n even
for n odd.
(4)
Vacca (1910) proves that the
Constant may be expressed as
Euler-Mascheroni
= £
(-1)*
ugfcj
(5)
fc = l
where \_x\ is the FLOOR FUNCTION. Hardy (1912) gave
the Formula
27:
^71 = y^
In 2 ^
(-1)
-[21g*-|.lg(2*)J]Llg*J. (6)
k=i
Kluyver (1927) gave similar series for j n with n > 1.
A set of constants related to 7 n is
5 n = lim
^(lnfc) n - / (lnz) n dz- §(lnm) n
(T)
(Sitaramachandrarao 1986, Lehmer 1988).
References
Berndt, B. C. "On the Hurwitz Zet a- Function." Rocky
Mountain J. Math. 2, 151-157, 1972.
Bohman, J. and Froberg, C.-E. "The Stieltjes Function —
Definitions and Properties." Math. Comput. 51, 281-289,
1988.
Briggs, W. E. "Some Constants Associated with the Riemann
Zeta-Function." Mich. Math. J. 3, 117-121, 1955-1956.
Finch, S. "Favorite Mathematical Constants." http://wwv.
maths of t . com/asolve/constant/stlt j s/stlt j s . html.
Hardy, G. H. "Note on Dr. Vacca's Series for 7." Quart. J.
Pure Appl. Math. 43, 215-216, 1912.
Kluyver, J. C. "On Certain Series of Mr. Hardy." Quart. J.
Pure Appl. Math. 50, 185-192, 1927.
Knopfmacher, J. "Generalised Euler Constants." Proc. Ed-
inburgh Math. Soc. 21, 25-32, 1978.
Lehmer, D. H. "The Sum of Like Powers of the Zeros of the
Riemann Zeta Function." Math. Comput. 50, 265-273,
1988.
Liang, J. J. Y. and Todd, J. "The Stieltjes Constants." J.
Res. Nat. Bur. Standards— Math. Sci. 76B, 161-178,
1972.
Sitaramachandrarao, R. "Maclaurin Coefficients of the Rie-
mann Zeta Function." Abstracts Amer. Math. Soc. 7, 280,
1986.
Vacca, G. "A New Series for the Eulerian Constant." Quart.
J. Pure Appl. Math. 41, 363-368, 1910.
Stieltjes Integral
The Stieltjes integral is a generalization of the RlEMANN
Integral. Let f(x) and a(x) be real-values bounded
functions defined on a Closed Interval [a, b]. Take a
partition of the INTERVAL
a = xo < Xi < X2, . . . < x n -i < x n = b, (1)
and consider the Riemann sum
n-l
E
f(£i)[a{x i+1 ) - a(xi)]
(2)
with £i 6 [xij x»+i]. If the sum tends to a fixed number
/ as max(xi+i — xi) — »■ 0, then / is called the Stieltjes
integral, or sometimes the RlEMANN- STIELTJES INTE-
GRAL. The Stieltjes integral of P with respect to F is
denoted
/
P(x)dF(x),
where
Jp(x)dF(x)=^
f{x)dx for x continuous
f(x) for x discrete.
(3)
(4)
If P and F have a common point of discontinuity, then
the integral does not exist. However, if the Stieltjes
integral exists and F has a derivative F\ then
j P(x) dF(x) = j P(x)F'(x) dx. (5)
For enumeration of many of the integral's properties, see
Dresher (1981, p. 105).
see also RlEMANN INTEGRAL
References
Dresher, M. The Mathematics of Games of Strategy: Theory
and Applications. New York: Dover, 1981.
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 152-155, 1988.
Kestelman, H. "Riemann-Stieltjes Integration." Ch. 11 in
Modern Theories of Integration, 2nd rev. ed. New York:
Dover, pp. 247-269, 1960.
Stieltjes' Theorem
The m + 1 ELLIPSOIDAL HARMONICS when «i, k 2 » and
K3 are given can be arranged in such a way that the
rth function has r — 1 zeros between —a 2 and — b 2 and
the remaining m + r — 1 zeros between — b 2 and — c 2
(Whittaker and Watson 1990).
see also ELLIPSOIDAL HARMONIC
References
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, 4th ed. Cambridge, England: Cambridge Uni-
versity Press, pp. 560-562, 1990.
Stieltjes-Wigert Polynomial
Stieltjes-Wigert Polynomial
Orthogonal POLYNOMIALS associated with WEIGHTING
Function
w(x) = 7r~ 1 ^ 2 fcexp(— k 2 In 2 x) = n~ l ' 2 kx~ nx (1)
for x G (0, oo) and k > 0. Using
_(l-g w )(l-g n ~ 1 )---(l-g n ~ v+1 )
(1-«)(1-* 2 )--- (!-«")
(2)
Stirling's Approximation 1739
The integrand is sharply peaked with the contribution
important only near x = n. Therefore, let x = n + £
where («n, and write
\n(x n e~ x ) = nlna? — x = nln(n + £) — (n + £). (4)
Now,
ln(n + f) -In L (l+ £)1 = Inn + In (l + £)
where < f < n,
(5)
and
9 = exp[-(2fc 2 )- 1 ].
Then
p n (x) = (-l) n q n/2+1/4 l(l-q)(l-q 2 )
(3)
(4)
(l-? n )]
n-n-1/2
£
for n > and
References
po{x)
n 1 / 4
q v Wx)" (5)
(6)
Szego, G. Orthogonal Polynomials, ^th ed. Providence, RI:
Amer. Math. Soc, p. 33, 1975.
Stirling's Approximation
Stirling's approximation gives an approximate value for
the Factorial function n! or the Gamma Function
P(n) for n ^> 1. The approximation can most simply
be derived for n an INTEGER by approximating the sum
over the terms of the FACTORIAL with an INTEGRAL, so
that
In n! = In 1 + In 2 + . . . + In n = y. m & ~ / m x dx
= [x\nx — x]i = nlnn — n + 1 « nlnn — n. (1)
The equation can also be derived using the integral def-
inition of the FACTORIAL,
Jo
e x dx.
(2)
Note that the derivative of the LOGARITHM of the inte-
grand can be written
— \n(e~ x x n ) = — (n\nx — x) = 1. (3)
dx dx x
ln(x n e" x ) = nln(n + £) - (n + £)
u 2
nlnn + £ n — £ -f .
2 n
= n In n — n — - — h
2n
Taking the EXPONENTIAL of each side then gives
n —a: ^ nlnn — n — £ /2n n — n — £ /2n
x e
e e
(6)
(7)
Plugging into the integral expression for n! then gives
/oo
n
n
n -n -£* Ili\
e e ^ /
d£ ;
n —n I
' n e
J —a
B -r/2n
Evaluating the integral gives
il « n"e n v27rn,
(8)
(9)
(10)
Taking the LOGARITHM of both sides then gives
Inn! « nlnn — n+| ln(27rn) = (n+|) Inn — n+\ ln(27r).
(11)
This is Stirling's Series with only the first term re-
tained and, for large n, it reduces to Stirling's approxi-
mation
Inn! ^ n In n — n. (12)
Gosper notes that a better approximation to n! (i.e.,
one which approximates the terms in STIRLING'S SERIES
instead of truncating them) is given by
n!^ ^/(2n+|)7rn n e" n .
(13)
This also gives a much closer app roxi mation to the FAC-
TORIAL of 0, 0! = 1, yielding <Jii~/?> « 1.02333 instead
of obtained with the conventional Stirling approxima-
tion.
see also Stirling's Series
1740 Stirling Cycle Number
Stirling Cycle Number
see Stirling Number of the First Kind
Stirling's Finite Difference Formula
U = /0 + b(*l/2 + £-1/2) + \P*61
+ Sz(5 1 / 2 + ^-1/2) + S±5q + ■ ■ ■ j
for p £ [-1/2,1/2], where 5 is the Central Differ-
ence and
£271+1 —
#271+2 —
1 / p + n
2 \2n + 1
p J p + n
2n-\- 2 V 2n+ 1
with (]J) a Binomial Coefficient.
see a/so CENTRAL DIFFERENCE, STEFFENSON'S FOR-
MULA
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 433, 1987.
Stirling's Formula
see Stirling's Series
Stirling Number of the First Kind
The definition of the (signed) Stirling number of the first
kind is a number S« such that the number of permu-
tations of n elements which contain exactly m CYCLES
is
(-l) n " m Si m) . (1)
This means that s£ m) = for m > n and s£ n) = 1. The
Generating Function is
x(x - 1) • • • (x - n + 1) = 2^ S n } * m ( 2 )
This is the Stirling number of the first kind returned
by the Mathematica® (Wolfram Research, Champaign,
IL) command StirlingSl[n,m] . The triangle of signed
Stirling numbers of the first kind is
1
-1 1
2 -3 1
-6 11 6 1
24 - 50 35 - 10 1
(Sloane's A008275).
The NONNEGATIVE version simply gives the number of
Permutations of n objects having m Cycles (with
Stirling Number of the First Kind
cycles in opposite directions counted as distinct) and is
obtained by taking the Absolute VALUE of the signed
version. The nonnegative Stirling number of the first
kind is denoted Si(n,ra) = \Sn I or
Diagrams
illustrating Si (5,1) = 24, Si (5, 3) = 35, Si (5, 4) = 10,
and Si (5, 5) — 1 (Dickau) are shown below.
5,(5, 1)
1J(D>)I
5,(5, 3)
5,(5,4)
5,(5,5)
©©©©©©©©©©
The nonnegative Stirling numbers of the first kind sat-
isfy the curious identity
E
fc=0
(e x -x-l) k+1 Si(n jn -k)
(k+l)\
e~ xn = ]n(x+l)
(3)
(Gosper) and have the GENERATING FUNCTION
71
(1 + x)(l + 2x) • • • (1 + nx) = ^ 5l ( n ' m ) x * ( 4 )
k=i
and satisfy
i(n + 1,/s) = nSi(n,k) + Si(n,fc - 1). (5)
The Stirling numbers can be generalized to nonintegral
arguments (a sort of "Stirling polynomial") using the
identity
r(j + h) _ ^ S i(M-fc)
3 h T(3)
=E
fc=0
(h-l)h (h - 2)(3h - l)(h - l)h
(h-3)(h-2)(h-l) 2 h 2
48p
+ ..., (6)
which is a generalization of an ASYMPTOTIC SERIES for
a ratio of Gamma Functions T(j + 1/2) /T(j) (Gosper).
see also Cycle (Permutation), Harmonic Number,
Permutation, Stirling Number of the Second
Kind
Stirling Number of the Second Kind
Stirling Number of the Second Kind 1741
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Stirling Num-
bers of the First Kind." §24.1.3 in Handbook of Mathemat-
ical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing. New York: Dover, p. 824, 1972.
Adamchik, V. "On Stirling Numbers and Euler Sums." J.
Comput. Appl. Math. 79, 119-130, 1997. http://www.
wolfram.com/-victor/axticles/stirling.html.
Conway, J. H. and Guy, R. K. In The Booh of Numbers. New
York: Springer- Verlag, pp. 91-92, 1996.
Dickau, R. M. "Stirling Numbers of the First Kind."
http:// forum . swarthmore . edu / advanced / robertd /
stirlingl.html.
Knuth, D. E. "Two Notes on Notation." Amer. Math.
Monthly 99, 403-422, 1992.
Sloane, N. J. A. Sequence A008275 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
with (£) a Binomial Coefficient, or the Generat-
ing Functions
c n = \J s(n, m)x(x — 1) • • • (x — m + 1), (6)
and
x n 1
E'(»-*)|r = ^ c '- 1 ^
i>k
(7)
-,)(i- 2 I)-(i-M = ^ s(w ' fc)x "- (8)
Stirling Number of the Second Kind
The number of ways of partitioning a set of n ele-
ments into m nonempty SETS (i.e., m BLOCKS), also
called a Stirling Set Number. For example, the Set
{1,2,3} can be partitioned into three SUBSETS in one
way: {{1}, {2}, {3}}; into two SUBSETS in three ways:
{{1,2}, {3}}, {{1,3}, {2}}, and {{1},{2,3}}; and into
one Subset in one way: {{1, 2, 3}}.
The Stirling numbers of the second kind are denoted
Sn , £2(n,m), s(n,m), or < >, so the Stirling num-
bers of the second kind for three elements are
a(3,l) = l
s(3,2) = 3
s(3,3) = l.
(1)
(2)
(3)
Since a set of n elements can only be partitioned in a
single way into 1 or n SUBSETS,
s(n, 1) = s(n,n) = 1. (4)
The triangle of Stirling numbers of the second kind is
1
1 1
1 3 1
17 6 1
1 15 25 10 1
1 31 90 65 15 1
(Sloane's A008277).
The Stirling numbers of the second kind can be com-
puted from the sum
(5)
The following diagrams (Dickau) illustrate the definition
of the Stirling numbers of the second kind s{n,m) for
n — 3 and 4.
Sf'>=l S< 2 >=3 S?>=1
>
sl l) = 1
5? J =7
Stirling numbers of the second kind obey the RECUR-
RENCE Relations
s(n, k) = s(n - 1, k - 1) + ks{n - 1, k). (9)
An identity involving Stirling numbers of the second
kind is
oo . m
f{m,n) = ^k n (^—^ =(m+l)^fe!s(n,ifc)m fc .
fc=i fc=i
(10)
It turns out that /(l,n) can have only 0, 2, or 6 as a
last Digit (Riskin 1995).
see also Bell Number, Combination Lock, Leng-
yel's Constant, Minimal Cover, Stirling Number
of the First Kind
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Stirling Numbers
of the Second Kind." §24.1.4 in Handbook of Mathemat-
ical Functions with Formulas, Graphs, and Mathematical
Tables, 9th printing. New York: Dover, pp. 824-825, 1972.
1742 Stirling's Series
Stochastic Group
Comtet, L. Advanced Combinatorics. Boston, MA: Reidel,
1974.
Conway, J. H. and Guy, R. K. In The Book of Numbers. New
York: Springer- Verlag, pp. 91-92, 1996.
Dickau, R. M. "Stirling Numbers of the Second Kind."
http:// forum . swarthmore . edu / advanced / robertd /
stirling2.html.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science, 2nd
ed. Reading, MA: Addison- Wesley, 1994.
Knuth, D. E. "Two Notes on Notation." Amer. Math.
Monthly 99, 403-422, 1992.
Riordan, J. An Introduction to Combinatorial Analysis. New
York: Wiley, 1958.
Riordan, J. Combinatorial Identities. New York: Wiley,
1968.
Riskin, A. "Problem 10231." Amer. Math. Monthly 102,
175-176, 1995.
Sloane, N. J. A. Sequence A008277 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cam-
bridge, England: Cambridge University Press, 1997.
Stirling's Series
The Asymptotic Series for the Gamma Function is
given by
r W -.-v-'VE(, + ^ + ^
139
2SSz 2 51840z 3
571
2488320z 4
+ .
) <•>
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, p. 443, 1953.
Sloane, N. J. A. Sequences A001163/M5400 and A001164/
M4878 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Stirling Set Number
see Stirling Number of the Second Kind
Stirrup Curve
A plane curve given by the equation
(s a -l) 2 = y a (y-l)(y-2)(y + 5).
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 72, 1989.
(Sloane's A001163 and A001164). The series for z\ is
obtained by adding an additional factor of z,
...). (2)
139
571
51840z 3 2488320^ 4
The expansion of lnT(z) is what is usually called Stir-
ling's series. It is given by the simple analytic expression
inr(*) = 53
Bin
2n(2n - l)z 2 "- J
= iln(27r) + (z+i)lnz-z+-L-;^U + : *
Viz 360z 3 1260z 5
(3)
(4)
where B n is a Bernoulli Number.
see also Bernoulli Number, iiT-FuNCTiON, Stir-
ling's Approximation
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 257, 1972.
Arfken, G. "Stirling's Series." §10.3 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 555-559, 1985.
Conway, J. H. and Guy, R. K. "Stirling's Formula." In The
Book of Numbers. New York: Springer- Verlag, pp. 260-
261, 1996.
Stochastic
see Random Variable
Stochastic Calculus of Variations
see MALLIAVIN CALCULUS
Stochastic Group
The Group of all nonsingular n x n STOCHASTIC MA-
TRICES over a FIELD F. It is denoted S(n } F). Up is
PRIME and F is the GALOIS Field of ORDER q = p m ,
S(n,q) is written instead of S(n,F). Particular exam-
ples include
5(2,2) =Z 2
5(2,3) = 5 3
5(2,4) = A 4
5(3,2) = 5 4
5(2,5) = Z 4 xeZ 5 ,
where Z 2 is an Abelian GROUP, S n are SYMMETRIC
GROUPS on n elements, and Xo denotes the semidirect
product with 6 : Z 4 -> Aut(Z 5 ) (Poole 1995).
see also STOCHASTIC Matrix
References
Poole, D. G. "The Stochastic Group." Amer. Math. Monthly
102, 798-801, 1995.
Stochastic Matrix
Stokes' Theorem 1743
Stochastic Matrix
A Stochastic matrix is the transition matrix for a finite
Markov Chain, also called a Markov Matrix. El-
ements of the matrix must be REAL NUMBERS in the
Closed Interval [0, 1].
A completely independent type of stochastic matrix is
defined as a Square Matrix with entries in a Field F
such that the sum of elements in each column equals 1.
There are two nonsingular 2x2 STOCHASTIC MATRICES
over Z2 (i.e., the integers mod 2),
Stohr Sequence
Let a\ = 1 and define a n+ i to be the least Integer
greater than a n for n > k which cannot be written as
the Sum of at most h addends among the terms ai, a2,
. . . , a n .
see also Greedy Algorithm, s-Additive Sequence,
Ulam Sequence
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 233, 1994.
1
1
and
1
1
There are six nonsingular stochastic 3x3 MATRICES
"1 0"
1
1
"0 1*
1
'
"2 1"
2
)
"2 O"
2 1
)
"0 2"
1 2
'
"l 2"
2
Stokes Phenomenon
The asymptotic expansion of the AlRY FUNCTION Ai(z)
(and other similar functions) has a different form in dif-
ferent sectors of the Complex Plane.
see also AlRY FUNCTIONS
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 609-611, 1953.
In fact, the set S of all nonsingular stochastic nxn ma-
trices over a FIELD F forms a GROUP under MATRIX
Multiplication. This Group is called the Stochas-
tic Group.
see also MARKOV CHAIN, STOCHASTIC GROUP
References
Poole, D. G. "The Stochastic Group." Amer, Math. Monthly
102, 798-801, 1995.
Stochastic Process
A stochastic process is a family of Random Vari-
ables {x(t, •),£ G J} from some PROBABILITY SPACE
(5,S,P) into a STATE Space (S',§'). Here, J is the
Index Set of the process.
see also Index Set, Probability Space, Random
Variable, State Space
References
Doob, J. L. "The Development of Rigor in Mathematical
Probability (1900-1950)." Amer. Math. Monthly 103,
586-595, 1996.
Stochastic Resonance
A stochastic resonance is a phenomenon in which a non-
linear system is subjected to a periodic modulated signal
so weak as to be normally undetectable, but it becomes
detectable due to resonance between the weak determin-
istic signal and stochastic NOISE. The earliest definition
of stochastic resonance was the maximum of the out-
put signal strength as a function of NOISE (Bulsara and
Gammaitoni 1996).
see also Kramers Rate, Noise
References
Benzi, R.; Sutera, A.; and Vulpiani, A. "The Mechanism of
Stochastic Resonance." J. Phys. A 14, L453-L457, 1981.
Bulsara, A. R. and Gammaitoni, L. "Tuning in to Noise."
Phys. Today 49, 39-45, March 1996.
Stokes' Theorem
For w a DIFFERENTIAL (n — l)-FORM with compact sup-
port on an oriented n-dimensional MANIFOLD M,
Jm Jdi\
w,
(1)
where dw is the Exterior Derivative of the differ-
ential form w. This connects to the "standard" Gra-
dient, Curl, and Divergence Theorems by the fol-
lowing relations. If / is a function on R ,
grad(/) = c- 1 d/,
(2)
where c
-►
3 * (the dual space) is the duality
isomorphism between a VECTOR SPACE and its dual,
given by the Euclidean Inner Product on R 3 . If / is
a Vector Field on a R 3 ,
div(/) = *d*c(/),
(3)
where * is the HODGE Star operator. If / is a Vector
Field on R 3 ,
curl(/) -c _1 *dc(/).
(4)
With these three identities in mind, the above Stokes'
theorem in the three instances is transformed into the
Gradient, Curl, and Divergence Theorems re-
spectively as follows. If / is a function on R and 7
is a curve in R 3 , then
I grad(/) -d\= I df = /( 7 (1)) - / ( 7 (0)),
(5)
1744 Stolarsky Array
Stomachion
which is the Gradient Theorem. If / : R 3 -> R 3
is a Vector Field and M an embedded compact 3-
manifold with boundary in M. , then
■ / f.dA= [ *c/= [ d*c/ = / div(f)dV, (6)
JdM JdM J M J M
which is the DIVERGENCE THEOREM. If / is a VEC-
TOR Field and M is an oriented, embedded, compact
2-MANIFOLD with boundary in R 3 , then
f fdl= [ c/= / dc(f)= f curl(/).dA, (7)
JdM JdM Jm Jm
which is the CURL THEOREM.
Physicists generally refer to the CURL THEOREM
The number of ODD elements in the first n rows of Pas-
cal's Triangle is
/ (V x F) ■ da = / F • da
J s Jas
(8)
as Stokes' theorem.
see also CURL THEOREM, DIVERGENCE THEOREM,
Gradient Theorem
Stolarsky Array
A INTERSPERSION array given
by
1 2
3
5
8
13
21
34
55
4 6
10
16
26
42
68
110
178
7 11
18
29
47
76
123
199
322
9 15
24
39
63
102
165
267
432
12 19
31
50
81
131
212
343
555
14 23
37
60
97
157
254
411
665
17 28
45
73
118
191
309
500
809
20 32
52
84
136
220
356
576
932
22 36
58
94
152
246
398
644
1042
the first row of which is the Fibonacci Numbers.
see also INTERSPERSION, WYTHOFF ARRAY
References
Kimberling, C. "Interspersions and Dispersions." Proc.
Amer. Math. Soc. 117, 313-321, 1993.
Morrison, D. R. "A Stolarsky Array and Wythoff Pairs." In
A Collection of Manuscripts Related to the Fibonacci Se-
quence. Santa Clara, CA: Fibonacci Assoc, pp. 134-136,
1980.
Stolarsky-Harborth Constant
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let b(k) be the number of Is in the Binary expression of
k. Then the number of Odd Binomial Coefficients
(*) where < j < k is 2 6(fc) (Glaisher 1899, Fine 1947).
/(n) = ^2 1
b(k)
(1)
This function is well approximated by n 6 \ where
0=^ = 1.58496....
In 2
Stolarsky and Harborth showed that
(2)
u.oi.zooo v umini — r—
n-J-oo n y
< 0.812557 < limsup ^V = 1-
(3)
The value
SH = hmin£ /( ?
(4)
is called the Stolarsky-Harborth constant.
References
Finch, S. "Favorite Mathematical Constants," http://www.
mathsoft.com/asolve/constant/stlrsky/stlrsky.html.
Fine, N. J. "Binomial Coefficients Modulo a Prime." Amer.
Math. Monthly 54, 589-592, 1947.
Wolfram, S. "Geometry of Binomial Coefficients." Amer.
Math. Monthly 91, 566-571, 1984.
Stolarsky's Inequality
If < g(x) < 1 and g is nonincreasing on the INTERVAL
[0,1], then for all possible values of a and 6,
/ g(x 1/(a+b) )dx> f g(x 1/a )dx f g(x 1/b )dx.
Jo Jo Jo
Stomachion
m. ^ „_! iy j....,f..y^ .
A Dissection game similar to TANGRAMS described in
fragmentary manuscripts attributed to Archimedes and
was referred to as the LocULUS OF Archimedes (Arch-
imedes' box) in Latin texts. The word Stomachion has
as its root the Greek word for stomach. The game con-
sisted of 14 flat pieces of various shapes arranged in the
shape of a square. Like TANGRAMS, the object is to
rearrange the pieces to form interesting shapes.
see also DISSECTION, TANGRAM
Stone Space
Strange Attractor 1745
References
Rorres, C. "Stomachion Introduction." http:// www . mcs .
drexel . edu / - crorres / Archimedes / Stomachion /
intro.html.
Rorres, C. "Stomachion Construction." http://www . mcs .
drexel . edu / - crorres / Archimedes / Stomachion /
construct ion . html .
Stone Space
Let P(L) be the set of all PRIME IDEALS of L, and define
r(a) = {P\a £ P}. Then the Stone space of L is the
Topological Space defined on P(L) by postulating
that the sets of the form r(a) are a subbase for the open
sets.
see also PRIME IDEAL, TOPOLOGICAL SPACE
References
Gratzer, G. Lattice Theory: First Concepts and Distributive
Lattices. San Francisco, CA: W. H. Freeman, p. 119, 1971.
Stone- von Neumann Theorem
A theorem which specifies the structure of the generic
unitary representation of the Weyl relations and thus
establishes the equivalence of Heisenberg's matrix me-
chanics and Schrodinger's wave mechanics formulations
of quantum mechanics.
References
Neumann, J. von. "Die Eindeutigkeit der Schrodingerschen
Operationen." Math. Ann. 104, 570-578, 1931.
Stopper Knot
A Knot used to prevent the end of a string from slipping
through a hole.
References
Owen, P. Knots. Philadelphia, PA: Courage, p. 11, 1993.
St0rmer Number
A St0rmer number is a Positive Integer n for which
the largest Prime factor p of n 2 + 1 is at least 2n. Every
Gregory Number t x can be expressed uniquely as a
sum of t n s where the ns are St0rmer numbers. Conway
and Guy (1996) give a table of St0rmer numbers repro-
duced below (Sloane's A005529). In a paper on Inverse
Tangent relations, Todd (1949) gives a similar compi-
lation.
n p
n
P
n
P
n
P
n
P
1 2
10
101
19
181
26
617
35
613
2 5
11
61
20
401
27
73
36
1297
4 17
12
29
22
97
28
157
37
137
5 13
14
197
23
53
29
421
39
761
6 37
15
113
24
577
33
109
40
1601
9 41
16
257
25
313
34
89
42
353
see also GREGORY NUMBER, INVERSE TANGENT
References
Conway, J. H. and Guy, R. K. "St0rmer's Numbers." The
Book of Numbers. New York: Springer- Verlag, pp. 245-
248, 1996=
Sloane, N. J. A. Sequence A005529/M1505 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Todd, J. "A Problem on Arc Tangent Relations." Amer.
Math. Monthly 56, 517-528, 1949.
Straight Angle
An Angle of 180° = it Radians.
see also Digon, Right Angle
Straight Line
see Line
Straight Polyomino
□
B
The straight polyomino of order n is the n-POLYOMlNO
in which all squares are placed along a line.
see also L-POLYOMINO, SKEW POLYOMINO, SQUARE
Polyomino, T-Polyomino
Straightedge
An idealized mathematical object having a rigorously
straight edge which can be used to draw a Line Seg-
ment. Although Geometric Constructions are
sometimes said to be performed with a Ruler and Com-
pass, the term straightedge is preferable to Ruler since
markings on the straightedge (usually assumed to be
present on a Ruler) are not allowed by the classical
Greek rules.
see also COMPASS, GEOMETRIC CONSTRUCTION, Ge-
ometrography, Mascheroni Constant, Polygon,
Poncelet-Steiner Theorem, Ruler, Simplicity,
Steiner Construction
Strange Attractor
An attracting set that has zero MEASURE in the em-
bedding Phase Space and has Fractal dimension.
Trajectories within a strange attractor appear to skip
around randomly.
see also CORRELATION EXPONENT, FRACTAL
References
Benmizrachi, A.; Procaccia, L; and Grassberger, P. "Char-
acterization of Experimental (Noisy) Strange Attractors."
Phys. Rev. A 29, 975-977, 1984.
Grassberger, P. "On the Hausdorff Dimension of Fractal At-
tractors." J. Stat Phys. 26, 173-179, 1981.
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983a.
Grassberger, P. and Procaccia, I. "Characterization of
Strange Attractors." Phys. Rev. Let. 50, 346-349, 1983b.
1746 Strange Loop
Strassen Formulas
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. 137-
138, 1991.
Sprott, J. C. Strange Attractors: Creating Patterns in Chaos.
New York: Henry Holt, 1993.
Strange Loop
A phenomenon in which, whenever movement is made
upwards or downwards through the levels of some heirar-
chial system, the system unexpectedly arrives back
where it started. Hofstadter (1987) uses the strange loop
as a paradigm in which to interpret paradoxes in logic
(such as Grelling's Paradox and Russell's Para-
dox) and calls a system in which a strange loop appears
a Tangled Hierarchy.
see also Grelling's Paradox, Russell's Paradox,
Tangled Hierarchy
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid, New York: Vintage Books, p. 10, 1989.
Strangers
Two numbers which are Relatively Prime.
References
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 145, 1983.
Strassen Formulas
The usual number of scalar operations (i.e., the total
number of additions and multiplications) required to
perform n x n Matrix Multiplication is
M(n) = 2n 3 - n
(1)
(i.e., n z multiplications and n 3 — n 2 additions). How-
ever, Strassen (1969) discovered how to multiply two
Matrices in
S(n) = 7.7 lgn -6-4 ]
lgn
(2)
scalar operations, where Ig is the LOGARITHM to base 2,
which is less than M(n) for n > 654. For n a power of
two (n = 2 k )> the two parts of (2) can be written
7-7 lgn = 7-7 lg2fe =7-7* = 7>2 fclg7 = 7(2 fc ) lg7 = 7n lg7
(3)
6 - 4 lgn = 6 • 4 lg2 " = 6 - 4* lg2 = 6 • 4 fc = 6(2 fc ) 2 = 6n\
(4)
so (2) becomes
5(2 fc ) = 7n lg7
6n
(5)
Two 2x2 matrices can therefore be multiplied
C = AB (6)
Cll
Ci2
an
0,12
'bu
6l2
C21
C22
CL21
CL22
&21
&22
with only
S(2) = 7 • 2 igT - 6 ■ 2* = 49 - 24 = 25
(7)
(8)
scalar operations (as it turns out, seven of them are
multiplications and 18 are additions). Define the seven
products (involving a total of 10 additions) as
Qi = (aii-ba 22 )(&ii+&22) (9)
<32 = (a2i+a 22 )&ii (10)
Qz = an(6i2 -622) (11)
Q4 = a 2 2(-&ii+6 21 ) (12)
Qs = (an+ai 2 )fe22 (13)
Qe = (-an + a 12 )(&n + 612) (14)
Q 7 = (ai2 - a 22 )(&2i + 622). (15)
Then the matrix product is given using the remaining
eight additions as
en =Qi+Q 4 -Q 5 + Q7 (16)
C21 = Q 2 + Q 4 (17)
C12 = Qs + Q 5 (18)
C22 = Qi+Qs-Q2+Qe (19)
(Strassen 1969, Press et al. 1989).
Matrix inversion of a 2 x 2 matrix A to yield C = A"
can also be done in fewer operations than expected using
the formulas
Ri = Q>u
(20)
JX2 = Q>2lRl
(21)
Rz = RlQ>\2
(22)
R4 = CL21R3
(23)
R5 = R4 — 0,22
(24)
Rq = R5
(25)
C\2 = R^Rq
(26)
C 2 1 = RqRz
(27)
R 7 = R 3 c 2 i
(28)
cn = Ri - R7
(29)
C22 = — Rq
(30)
(Strassen 1969, Press et al. 1989). The leading exponent
for Strassen's algorithm for a POWER of 2 is lg 7 « 2.808.
The best leading exponent currently known is 2,376
(Coppersmith and Winograd 1990). It has been shown
that the exponent must be at least 2.
see also Complex Multiplication, Karatsuba Mul-
tiplication
Strassman's Theorem
String Rewriting 1747
References
Coppersmith, D. and Winograd, S. "Matrix Multiplication
via Arithmetic Programming." J. Symb. Comput. 9, 251-
280, 1990.
Pan, V. How to Multiply Matrices Faster. New York:
Springer- Verlag, 1982.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Is Matrix Inversion an N 3 Process?" §2.11
in Numerical Recipes in FORTRAN: The Art of Scien-
tific Computing, 2nd ed. Cambridge, England: Cambridge
University Press, pp. 95-98, 1989.
Strassen, V. "Gaussian Elimination is Not Optimal." Nu-
merische Mathematik 13, 354-356, 1969.
Strassman's Theorem
Let (K,\ • |) be a complete non-ARCHlMEDEAN VALU-
ated Field, with Valuation Ring R, and let f(X) be
a POWER series with COEFFICIENTS in R. Suppose at
least one of the COEFFICIENTS is NONZERO (so that / is
not identically zero) and the sequence of COEFFICIENTS
converges to with respect to | • |. Then f(X) has only
finitely many zeros in R.
see also Archimedean Valuation, Mahler-Lech
Theorem, Valuation, Valuation Ring
Strassnitzky's Formula
The Machin-Like Formula
cot * 2 + cot x 5 + cot x 8.
see also Machin's Formula, Machin-Like Formu-
las
Strategy
A set of moves which a player plans to follow while play-
ing a Game.
see also Game, Mixed Strategy
Stratified Manifold
A set that is a smooth embedded 2-D MANIFOLD except
for a subset that consists of smooth embedded curves,
except for a set of ISOLATED POINTS.
References
Morgan, F. "What is a Surface?" Amer. Math. Monthly 103,
369-376, 1996.
Strehl Identity
The sum identity
where (£) is a BINOMIAL COEFFICIENT.
see also BINOMIAL COEFFICIENT
Striction Curve
A NONCYLINDRICAL Ruled SURFACE always has a pa-
rameterization of the form
x(it, v) = c(u) + v6(u)j
(1)
where \S\ = 1, & - 8 — 0, and a is called the striction
curve of x. Furthermore, the striction curve does not
depend on the choice of the base curve. The striction
and DIRECTOR CURVES of the HELICOID
x(w,v) =
r o i
~cosu~
+ av
sinti
_bu_
a{u) =
bu
acosu
asinti
S(u) =
For the HYPERBOLIC PARABOLOID
x(ii, v) —
the striction and DIRECTOR CURVES are
<t(u) =
' ' u~
ro]
+ v
i
.0.
_u_
S(u) =
(2)
(3)
(4)
(5)
(6)
(7)
see also DIRECTOR CURVE, DISTRIBUTION PARAME-
TER, NONCYLINDRICAL RULED SURFACE, RULED SUR-
FACE,
References
Gray, A. "Noncylindrical Ruled Surfaces" and "Examples of
Striction Curves of Noncylindrical Ruled Surfaces." §17.3
and 17.4 in Modern Differential Geometry of Curves and
Surfaces. Boca Raton, FL: CRC Press, pp. 345-350, 1993.
String Rewriting
A Substitution Map in which rules are used to oper-
ate on a string consisting of letters of a certain alpha-
bet. String rewriting is a particularly useful technique
for generating successive iterations of certain types of
Fractals, such as the Box Fractal, Cantor Dust,
Cantor Square Fractal, and Sierpinski Carpet.
see also Rabbit Sequence, Substitution Map
References
Peitgen, H.-O. and Saupe, D. (Eds.). "String Rewriting Sys-
tems." §C.l in The Science of Fractal Images. New York:
Springer- Verlag, pp. 273-275, 1988.
Wagon, S. "Recursion via String Rewriting." §6.2 in Mathe-
matica in Action. New York: W. H. Freeman, pp. 190-196,
1991.
1748
Strip
Strong Pseudoprime
Strip
see Critical Strip, Mobius Strip
Strong Convergence
Strong convergence is the type of convergence usually
associated with convergence of a Sequence. More for-
mally, a Sequence {x n } of Vectors in an Inner
Product Space E is called convergent to a Vector x
in £7 if
\\x n — x\\ — > as n -> oo.
see also Convergent Sequence, Inner Product
Space, Weak Convergence
Strong Elliptic Pseudoprime
Let n be an Elliptic Pseudoprime associated with
(E,P), and let ra-hl = 2 3 k with k ODD and s > 0. Then
n is a strong elliptic pseudoprime when either kP =
(mod n) or 2 r kP = (mod n) for some r with 1 <
r < s.
see also ELLIPTIC PSEUDOPRIME
References
Ribenboim, P. The New Book of Prime Number Records, 3rd
ed. New York: Springer- Verlag, pp. 132-134, 1996.
Strong Frobenius Pseudoprime
A Pseudoprime which obeys an additional restriction
beyond that required for a FROBENIUS PSEUDOPRIME.
A number n with (n, 2a) = 1 is a strong Frobenius pseu-
doprime with respect to x — a IFF n is a STRONG PSEU-
DOPRIME with respect to f(x). Every strong Frobenius
pseudoprime with respect to x — a is an Euler Pseu-
doprime to the base a.
Every strong Frobenius pseudoprime with respect to
/(*)
bx — c such that ((& + 4c) /n) = —1 is a
Strong Lucas Pseudoprime with parameters (6, c).
Every strong Frobenius pseudoprime n with respect to
X 2 - bx + 1 is an EXTRA STRONG LUCAS PSEUDOPRIME
to the base b.
see also Frobenius Pseudoprime
References
Grantham, J. "Frobenius Pseudoprimes." 1996. http://
www.clark.net/pub/grantham/pseudo/pseudo.ps
Strong Law of Large Numbers
For a set of random variates xi from a distribution hav-
ing unit Mean,
p( l im *i + ---+*" ) =P ( lim(x) ) =1 ,
This result is due to Kolmogorov.
see also Law of Truly Large Numbers, Strong
Law of Small Numbers, Weak Law of Large
Numbers
Strong Law of Small Numbers
There aren't enough small numbers to meet the many
demands made of them.
References
Gardner, M. "Patterns in Primes are a Clue to the Strong
Law of Small Numbers." Set. Amer. 243, 18-28, Dec.
1980.
Guy, R. K. "The Strong Law of Small Numbers." Amer.
Math. Monthly 95, 697-712, 1988.
Strong Lucas Pseudoprime
Let U(P,Q) and V(P,Q) be LUCAS SEQUENCES gener-
ated by P and Q, and define
D
4Q.
Let n be an Odd Composite Number with (n, D) = 1,
and n—(D/n) = 2 s d with d Odd and s > 0, where (a/b)
is the Legendre Symbol. If
Ud = (mod n)
or
Vrd = (mod n)
for some r with < r < s, then n is called a strong
Lucas pseudoprime with parameters (P y Q).
A strong Lucas pseudoprime is a Lucas PSEUDOPRIME
to the same base. Arnault (1997) showed that any COM-
POSITE NUMBER n is a strong Lucas pseudoprime for at
most 4/15 of possible bases (unless n is the PRODUCT
of TWIN PRIMES having certain properties).
see also EXTRA STRONG LUCAS PSEUDOPRIME, LUCAS
PSEUDOPRIME
References
Arnault, F. "The Rabin-Monier Theorem for Lucas Pseudo-
primes." Math. Comput. 66, 869-881, 1997.
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, 3rd ed. New York:
Springer- Verlag, pp. 130-131, 1996.
Strong Pseudoprime
A strong pseudoprime to a base a is an ODD COMPOSITE
Number n with n - 1 = d ■ 2 s (for d Odd) for which
either
a = 1 (mod n)
a = — 1 (mod n)
(i)
(2)
for some r € [0, s).
The definition is motivated by the fact that a FERMAT
Pseudoprime n to the base b satisfies
in-l
1 = (mod n) .
(3)
But since n is ODD, it can be written n = 2m + 1, and
b 2m - 1 = (b m - l)(6 m + 1) = (mod n). (4)
Strong Pseudoprime
Strongly Connected Component 1749
If n is Prime, it must Divide at least one of the Fac-
tors, but can't Divide both because it would then Di-
vide their difference
(6 m + 1) - (6 m - 1) = 2.
Therefore,
(5)
(6)
6 m = ±1 (mod n) ,
so write n = 2 a t + 1 to obtain
ft"" 1 - 1 = (&* - 1)(6* + l)(fe 2t + 1) • • ■ (& 2a_lf + 1). (7)
If n Divides exactly one of these Factors but is Com-
posite, it is a strong pseudoprime. A COMPOSITE num-
ber is a strong pseudoprime to at most 1/4 of all bases
less than itself (Monier 1980, Rabin 1980). The strong
pseudoprimes provide the basis for MILLER'S Primal-
ity Test and Rabin-Miller Strong Pseudoprime
Test.
A strong pseudoprime to the base a is also an Euler
Pseudoprime to the base a (Pomerance et ah 1980).
The strong pseudoprimes include some EULER PSEU-
DOPRIMES, Fermat Pseudoprimes, and Carmichael
Numbers.
There are 4842 strong psp(2) less than 2.5x 10 10 , where a
psp(2) is also known as a POULET NUMBER. The strong
fc-pseudoprime test for k = 2, 3, 5 correctly identifies all
Primes below 2.5 x 10 10 with only 13 exceptions, and if
7 is added, then the only exception less than 2.5 x 10 10
is 315031751. Jaeschke (1993) showed that there are
only 101 strong pseudoprimes for the bases 2, 3, and
5 less than 10 12 , nine if 7 is added, and none if 11 is
added. Also, the bases 2, 13, 23, and 1662803 have no
exceptions up to 10 12 .
If n is Composite, then there is a base for which n is not
a strong pseudoprime. There are therefore no "strong
Carmichael Numbers." Let ipk denote the smallest
strong pseudoprime to all of the first k PRIMES taken
as bases (i.e, the smallest Odd Number for which the
Rabin-Miller Strong Pseudoprime Test on bases
less than or equal to k fails). Jaeschke (1993) computed
ipk from k = 5 to 8 and gave upper bounds for k = 9 to
11.
i/>i = 2047
V> 2 = 1373653
V> 3 = 25326001
^ 4 = 3215031751
<0 5 = 2152302898747
^ 6 = 3474749660383
<0 7 = 34155071728321
<0 8 = 34155071728321
V> 9 < 41234316135705689041
V>io < 1553360566073143205541002401
V>n < 56897193526942024370326972321
(Sloane's A014233). A seven-step test utilizing these
results (Riesel 1994) allows all numbers less than 3.4 x
10 14 to be tested.
Pomerance et at. (1980) have proposed a test based on
a combination of Strong PSEUDOPRIMES and LUCAS
Pseudoprimes. They offer a $620 reward for discovery
of a Composite Number which passes their test (Guy
1994, p. 28).
see also Carmichael Number, Miller's Primal-
ity Test, Poulet Number, Rabin-Miller Strong
Pseudoprime Test, Rotkiewicz Theorem, Strong
Elliptic Pseudoprime, Strong Lucas Pseudo-
prime
References
Baillie, R. and Wagstaff, S. "Lucas Pseudoprimes." Math.
Comput. 35, 1391-1417, 1980.
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.
Jaeschke, G. "On Strong Pseudoprimes to Several Bases."
Math. Comput. 61, 915-926, 1993.
Monier, L. "Evaluation and Comparison of Two Efficient
Probabilistic Primality Testing Algorithms." Theor. Com-
put. Sci. 12, 97-108, 1980.
Pomerance, C.; Selfridge, J. L.; and Wagstaff, S. S. Jr. "The
Pseudoprimes to 25 -10 9 ." Math. Comput 35, 1003-1026,
1980. Available electronically from ftp://sable.ox.ac.
uk/pub/math/primes/ps2 . Z.
Rabin, M. O. "Probabilistic Algorithm for Testing Primal-
ity." J. Number Th. 12, 128-138, 1980.
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Basel: Birkhauser, p. 92, 1994.
Sloane, N. J. A. Sequence A014233 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Strong Pseudoprime Test
see Rabin-Miller Strong Pseudoprime Test
Strong Subadditivity Inequality
4>{A) + <f>(B) - 4>{A UB)> 4>{A n B).
References
Doob, J. L. "The Development of Rigor in Mathematical
Probability (1900-1950)." Amer. Math. Monthly 103,
586-595, 1996.
Strong Triangle Inequality
\x + y\ p < max(|x| p ,|y|p)
for all x and y.
see also p-ADic Number, Triangle Inequality
Strongly Connected Component
A maximal subgraph of a Directed Graph such that
for every pair of vertices it, v in the SUBGRAPH, there is
a directed path from u to v and a directed path from v
to u.
see also Bl-CONNECTED COMPONENT
1750 Strongly Embedded Theorem
Struve Differential Equation
Strongly Embedded Theorem
The strongly embedded theorem identifies all SIMPLE
Groups with a strongly 2-embedded Subgroup. In
particular, it asserts that no Simple Group has a
strongly 2-embedded 2'-local SUBGROUP.
see also Simple Group, Subgroup
Strongly Independent
An infinite sequence {a;} of Positive Integers is
called strongly independent if any relation ^2 e i a iy with
€i = 0, ±1, or ±2 and e* = except finitely often, Im-
plies €i = for all i.
see also Weakly Independent
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer-Verlag, p. 136, 1994.
Strongly Triple- Free Set
see Triple-Free Set
Strophoid
Let C be a curve, let O be a fixed point (the Pole),
and let O' be a second fixed point. Let P and P f be
points on a line through O meeting C at Q such that
P'Q = QP = QO'. The LOCUS of P and P' is called
the strophoid of C with respect to the POLE O and
fixed point O l . Let C be represented parametrically by
(/(*)> P(*))» and let O — (xo,yo) and O' = (a?i,yi). Then
the equation of the strophoid is
-/ W w -T + + ir"' (i)
»-» ± Y TTrt •
where
_ 9~ .
f - Xq
(2)
(3)
The name strophoid means "belt with a twist," and was
proposed by Montucci in 1846 (MacTutor Archive). The
polar form for a general strophoid is
6sin(a~ 20)
sin(a — 9)
(4)
If a = 7r/2, the curve is a Right Strophoid. The
following table gives the strophoids of some common
curves.
Curve Pole
Fixed Point Strophoid
line not on line on line oblique strophoid
line not on line foot of _L right strophoid
origin to line
circle center on circumf. Preeth's nephroid
References
Lawrence, J, D. A Catalog of Special Plane Curves. New
York: Dover, pp. 51-53 and 205, 1972.
Lockwood, E. H. "Strophoids." Ch. 16 in A Book of
Curves. Cambridge, England: Cambridge University
Press, pp. 134-137, 1967.
MacTutor History of Mathematics Archive. "Right." http:
//www -groups . dcs , st-and . ac . uk/ -history /Curves/
Right.html.
Yates, R. C "Strophoid." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 217-220,
1952.
Structurally Stable
A Map <f> : M -> M where M is a Manifold is C r
structurally stable if any C r perturbation is TOPOLOG-
ically Conjugate to <f>. Here, C r perturbation means
a Function ip such that tp is close to <fi and the first r
derivatives of ip are close to those of <j>.
see also TOPOLOGICALLY CONJUGATE
Structure
see Lattice
Structure Constant
The structure constant is defined as icijfcj where e»jfe
is the Permutation Symbol. The structure constant
forms the starting point for the development of Lie AL-
GEBRA.
see also LIE ALGEBRA, PERMUTATION SYMBOL
Structure Factor
The structure factor £r of a discrete set V is the FOUR-
IER Transform of J-scatterers of equal strengths on all
points of r,
xer
xGT
References
Baake, M.; Grimm, U.; and Warrington, D. H. "Some Re-
marks on the Visible Points of a Lattice." J. Phys. A:
Math. General 27, 2669-2674, 1994.
Struve Differential Equation
The ordinary differential equation
y +zy +(z -v)y.
<W
VSFlV+i)'
see also RIGHT STROPHOID
where T(z) is the Gamma Function. The solution is
y = aJ u (z) + bY v (z) + Hv{z),
where J v (z) and Y v {z) are BESSEL FUNCTIONS OF THE
First and Second Kinds, and % u (z) is a Struve
Function (Abramowitz and Stegun 1972).
Struve Function
see also Bessel Function of the First Kind, Bes-
sel Function of the Second Kind, Struve Func-
tion
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 496, 1972.
Struve Function
Abramowitz and Stegun (1972, pp. 496-499) define the
Struve function as
7Mz) = (H" +1 5i
(-l) fc (^) 2
Imi , , 3C. (!)
r(* + f)r(* + !/+§)'
where T(z) is the Gamma Function. Watson (1966,
p. 338) defines the Struve function as
n " {z) s n^WW) f (1 " tY " 1/2 sHzt) dM2)
The series expansion is
1 „\2m+*+l
(H
r(m+|)r(i/ + m+|)
• (3)
m=0
For half integral orders,
-y„ +1/2 (z) + -^ r(n + i-m) (4)
m=0
%. {n+1/2) {z) = (-l)V n+1/a (z). (5)
The Struve function and its derivatives satisfy
H v -i{z)-H v +i{z) = 2H' v {z)-
^^. (6)
0Fr(v + f)
see also Anger Function, Bessel Function, Modi-
fied Struve Function, Weber Functions
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Struve Func-
tion H u (x)" §12.1 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 496-498, 1972.
Spanier, J. and Oldham, K. B. "The Struve Function."
Ch. 57 in An Atlas of Functions. Washington, DC: Hemi-
sphere, pp. 563-571, 1987.
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.
Student's t-Distribution
Student's ^-Distribution
1751
A Distribution published by William Gosset in 1908.
His employer, Guinness Breweries, required him to pub-
lish under a pseudonym, so he chose "Student." Given
n independent measurements X{, let
(i)
s/y/n }
where \l is the population Mean, x is the sample MEAN,
and s is the ESTIMATOR for population STANDARD DE-
VIATION (i.e., the Sample Variance) defined by
71
- aTTT 5>* - s)2 -
(2)
Student's t-distribution is defined as the distribution of
the random variable t which is (very loosely) the "best"
that we can do not knowing a. If a — s, t — z and
the distribution becomes the NORMAL DISTRIBUTION.
As N increases, Student's t-distribution approaches the
Normal Distribution.
Student's ^-distribution is arrived at by transforming to
Student's ^-Distribution with
X — {1
Then define
t
zy/n — 1.
(3)
(4)
The resulting probability and cumulative distribution
functions are
fr(t) =
rm
/ r \(l + r)/2
\r+t 2 )
F r (t)
-I.
* r(rji)
(5)
v^r(D(i + £) ( ' +1)/2
i
dt
^(M)(i + £)
t 2\(r + l)/2
2 + 2
^iM-'Gr^.S".*)].
where
r = n — 1
(6)
(7)
1752
Student's t-Distribution
Student's z-Distribution
is the number of Degrees OF Freedom, — oo <t<oo,
T(z) is the Gamma Function, B{a,b) is the Beta
Function, and I{z\a,b) is the Regularized Beta
Function defined by
I(z] o, b) =
B(z;a,b)
B(a,b) '
(8)
The Mean, Variance, Skewness, and Kurtosis of
Student's ^-distribution are
fi =
2 r
r-2
6
72 = T-
r — 4
(9)
(10)
(11)
(12)
Beyer (1987, p. 514) gives 60%, 70%, 90%, 95%,
97.5%, 99%, 99.5%, and 99.95% confidence intervals,
and Goulden (1956) gives 50%, 90%, 95%, 98%, 99%,
and 99.9% confidence intervals. A partial table is given
below for small r and several common confidence inter-
vals.
r
80%
90%
95%
99%
1
3.08
6.31
12.71
63.66
2
1.89
2.92
4.30
9.92
3
1.64
2.35
3.18
5.84
4
1.53
2.13
2.78
4.60
5
1.48
2.01
2.57
4.03
10
1.37
1.81
2.23
4.14
30
1.31
1.70
2.04
2.75
100
1.29
1.66
1.98
2.63
oo
1.28
1.65
1.96
2.58
The so-called A(t\n) distribution is useful for testing if
two observed distributions have the same MEAN. i4(i|n)
gives the probability that the difference in two observed
Means for a certain statistic t with n Degrees of
FREEDOM would be smaller than the observed value
purely by chance:
A(t\n)= \ , f (l+^\
2 x -(l + n)/2
dx.
(13)
Let X be a Normally Distributed random variable
with Mean and Variance <t 2 , let Y 2 /a 2 have a Chi-
Squared Distribution with n Degrees of Free-
dom, and let X and Y be independent. Then
__ X^Jn
(14)
is distributed as Student's t with n DEGREES OF FREE-
DOM.
P(x)
The noncentral Student's ^-distribution is given by
n n > 2 n\
■ 2"e* 2 / 2 r(in)
( V2Xx(n + x »)-li^"\ Fl (i + l n; |. _*%.)
X \ r[l(l + n)]
r[Hi + n)]
(15)
where T(z) is the Gamma Function, 1 F 1 {a\b\z) is a
Confluent Hypergeometric Function, and L™(x)
is an associated Laguerre Polynomial.
see also Paired £-Test, Student's ^-Distribution
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 948-949, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, p. 536, 1987.
Fisher, R. A. "Applications of 'Student's' Distribution."
Metron 5, 3-17, 1925,
Fisher, R. A. "Expansion of 'Student's' Integral in Powers of
n - 1." Metron 5, 22-32, 1925.
Fisher, R. A. Statistical Methods for Research Workers, 10th
ed. Edinburgh: Oliver and Boyd, 1948.
Goulden, C. H. Table A-3 in Methods of Statistical Analysis,
2nd ed. New York: Wiley, p. 443, 1956.
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. 116-117, 1992.
Student. "The Probable Error of a Mean." Biometrika 6,
1-25, 1908.
Student's ^-Distribution
The probability density function and cumulative distri-
bution functions for Student's z-distribution are given
by
f ^ = v^¥) {1 + z2yn/2
(i)
D(z) =
V-TO) a Fi(i(n - 1), f n; \{n + 1); -z" 2 )
2V^Fr[i(n+l)]
(2)
The MEAN is 0, so the MOMENTS are
Mi =o
(3)
1
n — 3
(4)
(is =
(5)
3
(6)
^ ' (n - 3)(n -
-5)"
Study's Theorem
The Mean, Variance, Skewness, and Kurtosis are
2 1
Letting
n — 3
7i =0
6
72 = -.
(g-M)
(7)
(8)
(9)
(10)
(11)
where x is the sample Mean and /x is the population
Mean gives Student's ^-Distribution.
see also Student's ^-Distribution
Study's Theorem
Given three curves <£i, </>2, 4>3 with the common group
of ordinary points G (which may be empty), let their
remaining groups of intersections #23 , 531, and £12 also
be ordinary points. If <j>[ is any other curve through
£23, then there exist two other curves <f/ 2l $$ such that
the three combined curves (f)^ are of the same order
and Linearly Dependent, each curve <f> k contains the
corresponding group gij, and every intersection of <j>% or
(p'i with <f>j or cf/j lies on <pk or <fr' k .
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 34, 1959.
Sturm Chain
The series of Sturm Functions arising in application
of the Sturm Theorem.
see also Sturm Function, Sturm Theorem
Sturm Function
Given a function f(x) = fo(x), write /1 = fix) and
define the Sturm functions by
f n (x) ~ - < f n -2(x) - /„_]
(x)
fn-2(x)
U-i{x)
}■
(i)
where [P(x)/Q(x)] is a polynomial quotient. Then con-
struct the following chain of Sturm functions,
/o = go/i - h
h = Qih - h
h = Q2fs - h (2)
fs-2 = q 3 -2fs-l — fsy
known as a Sturm Chain. The chain is terminated
when a constant —f s (x) is obtained.
Sturm functions provide a convenient way for finding
the number of real roots of an algebraic equation with
Sturm Function 1753
real coefficients over a given interval. Specifically, the
difference in the number of sign changes between the
Sturm functions evaluated at two points x — a and x = b
gives the number of real roots in the interval (a, 6). This
powerful result is known as the STURM THEOREM.
As a specific application of Sturm functions toward find-
ing Polynomial Roots, consider the function fo(x) =
x 5 — 3x — 1, plotted above, which has roots —1.21465,
-0.334734, 0.0802951 ± 1.32836z, and 1.38879 (three of
which are real). The Derivative is given by f(x) =
bx 4 - 3, and the Sturm Chain is then given by
/o = x 5 - 3x - 1
/i = 5x 4 - 3
f 2 = |(12x + 5)
(3)
(4)
(5)
(6)
The following table shows the signs of fi and the number
of sign changes A obtained for points separated by Ax =
2.
X
/o
h
h
h
A
-2
-1
1
-1
1
3
-1
-1
1
1
1
2
1
1
1
1
This shows that 3 — 1 = 2 real roots lie in ( — 2, 0), and
1 — = 1 real root lies in (0,2). Reducing the spacing
to Ax = 0.5 gives the following table.
X
/o
h
h
fs
A
-2.0
-1
-1
3
-1.5
-1
-1
3
-1.0
1
-1
2
-0.5
1
_i
-1
2
0.0
-1
-i
1
1
0.5
-1
-i
1
1
1.0
-1
1
1
1.5
1
1
2.0
1
1
This table isolates the three real roots and shows that
they lie in the intervals ( — 1.5,-1.0), (—0.5,0.0), and
(1.0, 1.5). If desired, the intervals in which the roots fall
could be further reduced.
The Sturm functions satisfy the following conditions:
1754 Sturm-Liouville Equation
Subanalytic
1. Two neighboring functions do not vanish simultane-
ously at any point in the interval.
2. At a null point of a Sturm function, its two neigh-
boring functions are of different signs.
3. Within a sufficiently small Area surrounding a zero
point of fo{x)i fi{%) is everywhere greater than zero
or everywhere smaller than zero.
see also Descartes' Sign Rule, Sturm Chain,
Sturm Theorem
References
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., p. 334, 1990.
Dorrie, H. "Sturm's Problem of the Number of Roots." §24
in 100 Great Problems of Elementary Mathematics: Their
History and Solutions. New York: Dover, pp. 112-116,
1965.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing 7 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 469, 1992.
Rusin, D. "Known Math." http : //www . math . niu . edu . /
-rusin/known-math/polynomials/sturm.
Sturm, C. "Memoire sur la resolution des equations
numeriques." Bull, des sciences de Ferussac 11, 1929.
Sturm-Liouville Equation
A second-order Ordinary Differential Equation
d_
dx
p(*)-£\ + [M*) - q(*)]v = o,
where A is a constant and w(x) is a known function
called either the density or WEIGHTING Function. The
solutions (with appropriate boundary conditions) of A
are called EIGENVALUES and the corresponding u\(x)
ElGENFUNCTIONS. The solutions of this equation satisfy
important mathematical properties under appropriate
boundary conditions (Arfken 1985).
see also Adjoint Operator, Self-Adjoint Opera-
tor
References
Arfken, G. "Sturm-Liouville Theory — Orthogonal Func-
tions." Ch. 9 in Mathematical Methods for Physicists, 3rd
ed. Orlando, FL: Academic Press, pp. 497-538, 1985.
Sturm-Liouville Theory
see Sturm-Liouville Equation
Sturm Theorem
The number of Real Roots of an algebraic equation
with Real Coefficients whose Real Roots are sim-
ple over an interval, the endpoints of which are not
ROOTS, is equal to the difference between the number
of sign changes of the Sturm Chains formed for the
interval ends.
see also Sturm Chain, Sturm Function
References
Dorrie, H. "Sturm's Problem of the Number of Roots." §24
in 100 Great Problems of Elementary Mathematics: Their
History and Solutions. New York: Dover, pp. 112-116,
1965.
Rusin, D. "Known Math." http: //www. math. niu. edu./
-rusin/known-math/polynomials/sturm.
Sturmian Separation Theorem
Let A r = aij be a Sequence of N Symmetric Matri-
ces of increasing order with i,j — 1, 2, . . . , r and r = 1,
2, . . . , N. Let A fc (A r ) be the kth EIGENVALUE of A r for
k ~ 1, 2, ...,r, where the ordering is given by
Ai(A r )> A 2 (A r ) > ...> A P (A P ).
Then it follows that
A fe+1 (A i+ i) < A fc (Ai) < A fc (A i+ i).
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.
Sturmian Sequence
If a Sequence has the property that the Block
Growth function B(n) = n + 1 for all n, then it is
said to have minimal block growth, and the sequence is
called a Sturmian sequence. An example of this is the
sequence arising from the SUBSTITUTION Map
0->01
l->0,
yielding -► 01 -► 010 -> 01001 -» 01001010 ->...,
which gives us the Sturmian sequence 01001010. . . .
Sturm Functions are sometimes also said to form a
Sturmian sequence.
see also Sturm Function, Sturm Theorem
Subalgebra
An Algebra S" which is part of a large Algebra S
and shares its properties.
see also Algebra
Subanalytic
X C W 1 is subanalytic if, for all x € M n , there is an
open U and Y C R n+m a bounded SEMIANALYTIC set
such that X n U is the projection of Y into U.
see also SEMIANALYTIC
References
Bierstone, E. and Milman, P. "Semialgebraic and Subanalytic
Sets." IHES Pub. Math. 67, 5-42, 1988.
Marker, D. "Model Theory and Exponentiation." Not.
Amer. Math. Soc. 43, 753-759, 1996.
Subfactorial
Subscript 1755
Subfactorial
The number of PERMUTATIONS of n objects in which no
object appear in its natural place (i.e., so-called "DE-
RANGEMENTS").
:„!£
("I)"
k =
fc!
-[7]
(i)
(2)
where k\ is the usual Factorial and [x] is the Nint
function. The first few values are !1 = 0, !2 = 1, !3 = 2,
14 - 9, !5 = 44, !6 = 265, !7 = 1854, !8 = 14833,
... (Sloane's A000166). For example, the only DE-
RANGEMENTS of {1,2,3} are {2,3,1} and {3,1,2}, so
!3 = 2. Similarly, the DERANGEMENTS of {1, 2, 3, 4} are
{2,1,4,3}, {2,3,4,1}, {2,4,1,3}, {3,1,4,2}, {3,4,1,2},
{3,4,2,1}, {4,1,2,3}, {4,3,1,2}, and {4,3,2,1}, so
!4 = 9.
The subfactorials are also called the RENCONTRES NUM-
BERS and satisfy the RECURRENCE RELATIONS
!n-n-!(n-l) + (-l) n
!(n + l) = n[!n+!(n-l)].
(3)
(4)
The subfactorial can be considered a special case of a
restricted ROOKS PROBLEM.
The only number equal to the sum of subfactorials of its
digits is
148,349 =!l+!4+!8+!3+!4+!9 (5)
(Madachy 1979).
see also Derangement, Factorial, Married Cou-
ples Problem, Rooks Problem, Superfactorial
References
Dorrie, H. §6 in 100 Great Problems of Elementary Mathe-
matics: Their History and Solutions. New York: Dover,
pp. 19-21, 1965.
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, p. 167, 1979.
Sloane, N. J. A. Sequences A000166/M1937 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.
Stanley, R. P. Enumerative Combinatorics, Vol 1. Cam-
bridge, England: Cambridge University Press, p. 67, 1997.
Subfield
If a subset 5 of the elements of a FIELD F satisfies the
Field Axioms with the same operations of F, then S
is called a subfield of F. Let F be a Finite Field of
order p n , then there exists a subfield of ORDER p m for
Prime p Iff m Divides n.
see also Field, Submanifold, Subspace
Subgraph
A Graph G' whose Vertices and Edges form subsets
of the Vertices and Edges of a given Graph G. If G'
is a subgraph of G, then G is said to be a Supergraph
of G'.
see also Graph (Graph Theory), Supergraph
Subgroup
A subset of GROUP elements which satisfies the four
GROUP requirements. The ORDER of any subgroup of a
Group Order h must be a Divisor of h.
see also CARTAN SUBGROUP, COMPOSITION SERIES,
Fitting Subgroup, Group
Sublime Number
Let r(n) and a(n) denote the number and sum of the di-
visors of n, respectively (i.e., the zeroth- and first-order
Divisor Functions). A number N is called sublime if
t(N) and a(N) are both PERFECT NUMBERS. The only
two known sublime numbers are 12 and
60865556702383789896703717342431696- ■ -
• • • 22657830773351885970528324860512791691264.
It is not known if any Odd sublime number exists.
see also DIVISOR FUNCTION, PERFECT NUMBER
Submanifold
A C°° (infinitely differ entiable) MANIFOLD is said to be
a submanifold of a C°° MANIFOLD M' if M is a SUB-
SET of M' and the Identity Map of M into M' is an
embedding.
see also MANIFOLD, SUBFIELD, SUBSPACE
Submatrix
Anp x q submatrix of an m x n MATRIX (with p < m,
n < q) is a p x q MATRIX formed by taking a block of
the entries of this size from the original matrix.
see also MATRIX
Subnormal
L is a subnormal SUBGROUP of H if there is a a "normal
series" (in the sense of Jordan-Holder) from L to H.
Subordinate Norm
see Natural Norm
Subscript
A quantity displayed below the normal line of text (and
generally in a smaller point size), as the "i" in a*, is
called a subscript. Subscripts are commonly used to
indicate indices (aij is the entry in the ith row and jth
column of a MATRIX A), partial differentiation (y x is an
abbreviation for dy/dx), and a host of other operations
and notations in mathematics.
see also SUPERSCRIPT
1756 Subsequence
Successes
Subsequence
A subsequence of a Sequence S = {xi}i =1 is a derived
sequence {yi}iLi — {%i+j} for some j > and N < n —
j. More generally, the word subsequence is sometimes
used to mean a sequence derived from a sequence S by
discarding some of its terms.
see also Lower-Trimmed Subsequence, Upper-
Trimmed Subsequence
Subset
A portion of a Set. B is a subset of A (written B C A)
Iff every member of B is a member of A. If B is a
Proper Subset of A (i.e., a subset other than the set
itself), this is written B C A.
A Set of n elements has 2 n subsets (including the set
itself and the Empty Set). For sets of n = 1, 2, . . .
elements, the numbers of subsets are therefore 2, 4, 8,
16, 32, 64, ... (Sloane's A000079). For example, the
set {1} has the two subsets and {1}. Similarly, the
set {1,2} has subsets (the Empty Set, {1}, {2}, and
{1,2}.
see also Empty Set, Implies, /s-Subset, Proper Sub-
set, Superset, Venn Diagram
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. 109, 1996.
Ruskey, F. "Information of Subsets of a Set." http: // sue .
esc. uvic.ca/~cos/inf /comb/Subset Info. html.
Sloane, N. J. A. Sequence A000079/M1129 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Subspace
Let V be a Real Vector Space (e.g., the real con-
tinuous functions C(I) on a CLOSED INTERVAL /, 2-D
EUCLIDEAN SPACE R 2 , the twice differentiate real func-
tions C^ 2) (I) on /, etc.). Then W is a real SUBSPACE
of V if W is a Subset of ¥ and, for every wi, w 2 G W
and t e R (the Reals), wi + w 2 € W and twi G W.
Let (H) be a homogeneous system of linear equations in
asi, . . . , x n . Then the Subset S of M n which consists of
all solutions of the system (if) is a subspace of R n .
More generally, let F q be a Field with q — p a , where p
is PRIME, and let F q , n denote the n-D VECTOR SPACE
over F q . The number of k-D linear subspaces of F q ^ n is
where
N(F q ,„)
G):
where this is the g-BlNOMlAL COEFFICIENT (Aigner
1979, Exton 1983). The asymptotic limit is
N(F q>n ) :
f c e q n2/4 [l + o(l)] for n even
1 c q n2/4 [l + o(l)] for n odd,
c e =
Eoo
k=-c
„-**
nr =1 d-
Eoo
fc=-oo9
-(fc+l/2) 2
(Finch). The case q = 2 gives the q- ANALOG of the
Wallis Formula.
see also g-BlNOMIAL COEFFICIENT, SUBFIELD, SUB-
MANIFOLD
References
Aigner, M. Combinatorial Theory. New York: Springer-
Verlag, 1979.
Exton, H. q-Hypergeometric Functions and Applications.
New York: Halstead Press, 1983.
Finch, S. "Favorite Mathematical Constants." http: //www.
mathsoft .com/asolve/constant/dig/dig.html.
Substitution Group
see Permutation Group
Substitution Map
A Map which uses a set of rules to transform ele-
ments of a sequence into a new sequence using a set
of rules which "translate" from the original sequence to
its transformation. For example, the substitution map
{1 -» 0, -> 11} would take 10 to Oil.
see also GOLDEN RATIO, MORSE-THUE SEQUENCE,
String Rewriting, Thue Constant
Subtend
Given a geometric object O in the PLANE and a point P t
let A be the ANGLE from one edge of O to the other with
Vertex at P. Then O is said to subtend an Angle A
from P.
see also ANGLE, VERTEX ANGLE
Subtraction
Subtraction is the operation of taking the DIFFERENCE
x — y of two numbers x and y. Here, the symbol between
the x and y is called the MINUS SlGN and x - y is read
"x Minus y."
see also Addition, Division, Minus, Minus Sign,
Multiplication
Succeeds
The relationship x succeeds (or FOLLOWS) y is written
x y y. The relation x succeeds or is equal to y is written
x>y.
see also PRECEDES
Successes
see Difference of Successes
Sufficient
Sum 1757
Sufficient
A Condition which, if true, guarantees that a result
is also true. (However, the result may also be true if
the Condition is not met.) If a Condition is both
Necessary and Sufficient, then the result is said to
be true Iff ("if and only if") the Condition holds.
For example, the condition that a decimal number n
end in the Digit 2 is a sufficient but not NECESSARY
condition that n be Even.
see also Iff, Implies, Necessary
Suitable Number
see Idoneal Number
Sum
A sum is the result of an ADDITION. For example,
adding 1, 2, 3, and 4 gives the sum 10, written
1 + 2 + 3 + 4 = 10.
(1)
The numbers being summed are called ADDENDS, or
sometimes SUMMANDS. The summation operation can
also be indicated using a capital sigma with upper and
lower limits written above and below, and the index in-
dicated below. For example, the above sum could be
written
£*
10.
(2)
t
n+1
A simple graphical proof of the sum X^/La ^ = n ( n +
l)/2 can also be given. Construct a sequence of stacks of
boxes, each 1 unit across and ft units high, where ft = 1,
2, . . . , n. Now add a rotated copy on top, as in the
above figure. Note that the resulting figure has Width
n and HEIGHT n + 1, and so has Area n(n + 1). The
desired sum is half this, so the AREA of the boxes in the
sum is n(n + l)/2. Since the boxes are of unit width,
this is also the value of the sum.
The sum can also be computed using the first EULER-
Maclaurin Integration Formula
£/(*) = J" f{x)dx+\f{l) + \f{n)
fc=i Jl
+ ±B 3 [/'(n) -/'(!)] + ... (3)
with /(ft) = ft. Then
^ft=/ xdx+\-l + \-n+ |(1-1) + ...
= \{n 2 -l)-\+h+\n= \n(n + 1). (4)
The general finite sum of integral POWERS can be given
by the expression
± kP= (B + n + l^-B^ t (5)
where the NOTATION B^ means the quantity in ques-
tion is raised to the appropriate Power ft and all terms
of the form B m are replaced with the corresponding
Bernoulli Numbers S m . It is also true that the Co-
efficients of the terms in such an expansion sum to 1,
as stated by Bernoulli without proof (Boyer 1943).
An analytic solution for a sum of POWERS of integers is
n
£V = C(-p)-C(-P.l + n), (6)
fc = l
where £(z) is the RlEMANN Zeta FUNCTION and £(z; a)
is the Hurwitz Zeta Function. For the special case
of p a Positive integer, Faulhaber's Formula gives
the Sum explicitly as
k=l fc=l V 7
> p +i- k n , (7)
where S kp is the Kronecker Delta, (£) is a Bino-
mial Coefficient, and B k is a Bernoulli Number.
Written explicitly in terms of a sum of POWERS,
£* p
Bkpl n p ~ k+1
ft!(p-ft+l)!
(8)
Computing the sums for p = 1, . . . , 10 gives
n
J]fe=i(n 2 +n) (9)
fc=l
n
^V = i(2n 3 + 3n 2 +n) (10)
k=l
n
^fc 3 = i(n 4 + 2n 3 +n 2 ) (11)
fc = l
n
J^ ft 4 = ^ (6n 5 + 15n 4 + 10n 3 - n) (12)
fc=i
n
]TV = i(2n 6 + 6n 5 + 5n 4 -n 2 ) (13)
1758 Sum
n
][]fc 6 = £(6n 7 + 21n 8 + 21n 5 -7n 3 +n) (14)
fc=i
^fc 7 = £(3n 8 + 12n 7 + 14n 6 - 7n 4 + 2n 2 ) (15)
k = l
n
Y, fc 8 = m( 10 " 9 + 45 " 8 + 60 " 7 - 42 " 5
+ 20n 3 - 3n) (16)
f> 9 = i(2n 10 + 10n 9 + 15n 8 - 14n 6
k = l
+ 10n 4 - 3n 2 ) (17)
^ A; 10 = £ (6n n + 33n 10 + 55n 9 - 66n 7
fc=i
+ 66n 5 -33n 3 + 5n). (18)
Factoring the above equations results in
^k = I n ( n +1)
fc=i
n
^Jfe 2 = ±ra(n + l)(2n+l)
fc=i
£*» = !„'(„ + I) 2
(19)
(20)
(21)
]Tfc 4 = in(n + l)(2n + l)(3n 2 +3n-l) (22)
^fc 5 = JLn 2 (n + l) 2 (2n 2 +2n-l) (23)
fc=i
n
^ k * = ^ n ( n + i)(2n + l)(3n 4 + 6n 3 - 3n + 1)
J2 k 7 = ^n 2 (n + l) 2 (3n 4 + 6n 3 - n 2 - 4n + 2)
fc=i
(24)
(25)
^V = ^n(n + l)(2n + l)(5n 6 + 15n 5 +5n 4
fc=i
-15n 3 -n 2 +9n-3) (26)
Tl
^fc 9 = ^n 2 (n + l) 2 (n 2 +n-l)
fc=i
x (2n 4 + 4n 3 - n 2 - 3n + 3) (27)
n
53 fcl ° = ^^ + 1)(2n + ^^ + n " ^
fe = l
x(3n 6 4- 9n 5 + 2n 4 - lln 3 + 3n 2 + lOn - 5). (28)
Sum
From the above, note the interesting identity
x> 8 =(£>)'■ ^
^=1 \fc=i /
Sums of the following type can also be done analytically.
(oo \ 2 oo / n \ oo
E* fe = E E 1 *- = E< n+1 >* n < 30)
k-Q / n~0 \ fc=0 / n=0
(oo \ 3 oo / n \
E* fc = E E* K
fe=0 / n=0 \ fc=0 /
oo
= ^(n + l)(n + 2K (31)
/ oo \ 4 oo
fc=0
^e(x> 2 + 3 *+ 2 V
n=Q \ k=0 /
oo
= ^Et6"( n+1 x 2n+1)
71 =
+ 3|n(n + l) + 2(n+l)]x"
oo
= — V(n + l)[n(2n + 1) + 9n + 12]s n
LA ■
71 =
oo
= — ^(n + l)(2n 2 + lOra + 12)x n
n=0
oo
= iV(n+l)(n + 2)(n + 3)x n . (32)
6 ^— '
n=0
By INDUCTION, the sum for an arbitrary POWER p is
ffvY i f- ("+p-i)' _»
(p_l)!^ „■
n=0
(33)
Other analytic sums include
(n- |n- fcj +p- 1)! fc
(p-l)!f- (n-ln-fel)! *
fc=0
(34)
Em ~ ro-i)iE
oo
^a„x n ) = ^a„V" + 2 ^ aiojx". (35)
n=l
\J XJ/ = XiJ/l + Si J/2 + . • • + X2J/1 + #2*/2 + ■ • •
= (xi 4- x 2 + - - -)2/i + fai + ^2 + . . .)V2
Sum
Super-3 Number 1759
SS iBl » =
i=l j-1
3=0
E* •
(37)
3=1
c n+2 - (n + l)x" +1 + x
(x - 1)*
£
(38)
for < r < n — 1
for r = n — 1
? II $5} (**"**) Ie; =1 ^ forr = n
" n" =1 (* + * - r)
E
r*fc
- IIr=i(*-r)
= 1
{n + l)^2m k = J2
H*
p=l \m=l >
(39)
(40)
• (41)
To minimize the sum of a set of squares of numbers {xi}
about a given number xo
S = ^(xi - x ) 2 = ^2%i 2 ~ 2xo ^2 Xi + iVx ° 2 ' ( 42 )
i i
take the Derivative.
-5 = -2 V\i + 2iVxo = 0.
Solving for xq gives
dxo
(43)
Xq
(44)
so S is maximized when xo is set to the MEAN.
see also Arithmetic Series, Bernoulli Number,
Clark's Triangle, Convergence Improvement,
Dedekind Sum, Double Sum, Euler Sum, Facto-
rial Sum, Faulhaber's Formula, Gabriel's Stair-
case, Gaussian Sum, Geometric Series, Gosper's
Method, Hurwitz Zeta Function, Infinite Prod-
uct, Kloosterman's Sum, Legendre Sum, Lerch
Transcendent, Pascal's Triangle, Product, Ra-
manujan's Sum, Riemann Zeta Function, Whitney
Sum
References
Boyer, C. B. "Pascal's Formula for the Sums of the Powers
of the Integers." Scripta Math. 9, 237-244, 1943.
Courant, R. and Robbins, H. "The Sum of the First n
Squares." §1.4 in What is Mathematics?: An Elementary
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 14-15, 1996.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A-B. Welles-
ley, MA: A. K. Peters, 1996.
Sum-Product Number
A sum-product number is a number n such that the sum
of n's digits times the product of n's digit is n itself, for
example
135 = (l + 3 + 5)(l-3-5).
The only sum-product numbers less than 10 7 are 1, 135,
and 144.
see also Amenable Number
Sum Rule
dx
[f(x)+g(x)] = f'(x) + g'(x),
where d/dx denotes a derivative and f{x) and g'(x) are
the derivatives of / and g, respectively.
see also DERIVATIVE
Summand
see Addend
Summatory Function
For an discrete function /(n), the summatory function
is defined by
F(n) = ^/(fc),
where D is the DOMAIN of the function.
see also Divisor Function, Mangoldt Function,
Mertens Function, Rudin-Shapiro Sequence, Tau
Function, Totient Function
Sup
see Supremum, Supremum Limit
Super-3 Number
An Integer n such that 3n 3 contains three consecutive
3s in its DECIMAL representation. The first few super-
3 numbers are 261, 462, 471, 481, 558, 753, 1036, ...
(Sloane's A014569). A. Anderson has conjectured that
all numbers ending in 471, 4710, or 47100 are super-3
(Pickover 1995).
For a digit d, super-3 numbers can be generalized to
super-d numbers n such that dn d contains d ds in its
DECIMAL representation. The following table gives the
first few super-d numbers for small d.
d Sloane Super-d numbers
2 032743 19, 31, 69, 81, 105, 106, 107, 119, ...
3 014569 261, 462, 471, 481, 558, 753, 1036, . . .
4 032744 1168, 4972, 7423, 7752, 8431, 10267, ...
5 032745 4602, 5517, 7539, 12955, 14555, 20137, ...
6 032746 27257, 272570, 302693, 323576, . . .
7 032747 140997, 490996, 1184321, 1259609, ...
8 032748 185423, 641519, 1551728, 1854230, . . .
9 032749 17546133, 32613656, 93568867, . . .
1760 Super Catalan Number
Superellipse
References
Pickover, C. A. Keys to Infinity. New York: Wiley, p. 7,
1995.
Sloane, N. J. A. Sequence A014569 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Super Catalan Number
While the Catalan Numbers are the number of p-
GOOD Paths from (n, n) to (0,0) which do not cross
the diagonal line, the super Catalan numbers count the
number of LATTICE PATHS with diagonal steps from
(n,n) to (0,0) which do not touch the diagonal line
x = y.
The super Catalan numbers are given by the RECUR-
RENCE Relation
Superegg
A superegg is a solid described by the equation
x 2 +y 2
1.
Supereggs will balance on either end for any a, 6, and
n.
see also EGG, SUPERELLIPSE
References
Gardner, M. "Pier Hein's Superellipse." Ch. 18 in Math-
ematical Carnival: A New Round-Up of Tantalizers and
Puzzles from Scientific American. New York: Vintage,
1977.
S(n) =
3(2n - 3)5(n - 1) - (n - 3)S(n - 2)
Superellipse
(Comtet 1974), with 5(1) = 5(2) = 1. (Note that the
expression in Vardi (1991, p. 198) contains two errors.)
A closed form expression in terms of LEGENDRE POLY-
NOMIALS P n (x) is
S(n)
3Pn-i(3)-P w -a(3)
An
(Vardi 1991, p. 199). The first few super Catalan num-
bers are 1, 1, 3, 11, 45, 197, . . . (Sloane's A001003).
see also Catalan Number
References
Comtet, L. Advanced Combinatorics. Dordrecht, Nether-
lands: Reidel, p. 56, 1974.
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Exercise
7.50 in Concrete Mathematics: A Foundation for Com-
puter Science, 2nd ed. Reading, MA: Addis on- Wesley,
1994.
Motzkin, T. "Relations Between Hypersurface Cross Ratios
and a Combinatorial Formula for Partitions of a Poly-
gon for Permanent Preponderance and for Non-Associative
Products." Bull. Amer. Math. Soc. 54, 352-360, 1948.
Schroder, E. "Vier combinatorische Probleme." Z. Math.
Phys. 15, 361-376, 1870.
Sloane, N. J. A. Sequence A001003/M2898 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 198-199, 1991.
Super- Poulet Number
A Poulet Number whose Divisors d all satisfy d\2 d -
2.
see also Poulet Number
Superabundant Number
see Highly Composite Number
A curve of the form
where r > 2. "The" superellipse is sometimes taken as
the curve of the above form with r = 5/2. Superellipses
with a = b are also known as Lame Curves. The above
curves are for a = 1, b — 2, and r = 2.5, 3.0, and 3.5.
A degenerate superellipse is a superellipse with r < 2.
The above curves are for a = 1, 6 = 2, and r = 0.5, 1.0,
1.5, and 2.0.
see also Ellipse, Lame Curve, Superegg
References
Gardner, M. "Piet Hein's Superellipse." Ch. 18 in Math-
ematical Carnival: A New Round-Up of Tantalizers and
Puzzles from Scientific American. New York: Vintage,
1977.
Superfactorial
Superset 1761
Superfactorial
The superfactorial of n is defined by Pickover (1995) as
The first two values are 1 and 4, but subsequently grow
so rapidly that 3$ already has a huge number of digits.
Sloane and Plouffe (1995) define the superfactorial by
n
n$ = Y[i\,
i=i
which is equivalent to the integral values of the G-
FUNCTION. The first few values are 1, 1, 2, 12, 288,
34560, . . . (Sloane's A000178).
see also Factorial, G-Function, Large Number,
SUBFACTORIAL
References
Pickover, C. A. Keys to Infinity. New York: Wiley, p. 102,
1995.
Sloane, N. J. A. Sequence A000178/M2049 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Supergraph
If G f is a SUBGRAPH of G, then G is said to be a super-
graph of G f .
see also GRAPH (GRAPH THEORY), SUBGRAPH
Supernormal
Trials for which the Lexis Ratio
satisfies L > 1, where a is the Variance in a set of s
Lexis Trials and a B is the Variance assuming Ber-
noulli Trials.
see also BERNOULLI TRIAL, LEXIS TRIALS, SUBNORMAL
Superperfect Number
A number n such that
a (n) = o~(a(n)) ~ 2n,
where a(n) is the Divisor Function. Even superper-
fect numbers are just 2 P_1 , where M p = 2 P - 1 is a
MERSENNE PRIME. If any ODD superperfect numbers
exist, they are SQUARE NUMBERS and either n or a(n)
is Divisible by at least three distinct Primes.
More generally, an m-superperfect number is a number
for which <r m (n) = 2n. For m > 3, there are no Even
m-superperfect numbers.
see also Mersenne Number
References
Guy, R. K. "Superperfect Numbers." §B9 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 65-66, 1994.
Kanold, H.-J. "Uber 'Super Perfect Numbers.'" Elem. Math.
24, 61-62, 1969.
Lord, G. "Even Perfect and Superperfect Numbers." Elem.
Math. 30, 87-88, 1975.
Suryanarayana, D. "Super Perfect Numbers." Elem. Math.
20, 16-17, 1969.
Suryanarayana, D. "There is No Odd Super Perfect Number
of the Formp 2a ." Elem. Math. 24, 148-150, 1973.
Superposition Principle
For a linear homogeneous ORDINARY DIFFERENTIAL
Equation, if yi(x) and y2(x) are solutions, then so is
yi(x) + y2{x).
Superregular Graph
For a Vertex # of a Graph, let T x and A x denote the
Subgraphs of r - x induced by the Vertices adjacent
to and nonadjacent to #, respectively. The empty graph
is defined to be superregular, and T is said to be super-
regular if r is a Regular Graph and both T x and A x
are superregular for all x.
The superregular graphs are precisely C5, mK n (m, n >
1), G n (n > 1), and the complements of these graphs,
where C n is a Cyclic Graph, K n is a Complete
Graph and mKn is m disjoint copies of K n , and G n
is the Cartesian product of K n with itself (the graph
whose Vertex set consists of n 2 Vertices arranged in
an n x n square with two Vertices adjacent Iff they
are in the same row or column).
see also Complete Graph, Cyclic Graph, Regular
Graph
References
Vince, A. "The Superregular Graph." Problem 6617. Amer.
Math. Monthly 103, 600-603, 1996.
West, D. B. "The Superregular Graphs." J. Graph Th. 23,
289-295, 1996.
Superscript
A quantity displayed above the normal line of text (and
generally in a smaller point size), as the "i" in x z , is
called a superscript. Superscripts are commonly used
to indicate raising to a POWER (x 3 means x • x ■ x or x
Cubed), multiple differentiation (/ (3) (x) is an abbrevi-
ation for f"{x) = d s f/dx s ), and a host of other opera-
tions and notations in mathematics.
see also SUBSCRIPT
Superset
A Set containing all elements of a smaller SET. If B is a
Subset of A, then A is a superset of B } written AD B.
If A is a Proper Superset of B, this is written Ad B.
see also Proper Subset, Proper Superset, Subset
1762 Supplementary Angle
Surface
Supplementary Angle
Two ANGLES a and tt — a which together form a
Straight Angle are said to be supplementary.
see also Angle, Complementary Angle, Digon,
Straight Angle
Support
The Closure of the Set of arguments of a Function
/ for which / is not zero.
see also CLOSURE
Support Function
Let M be an oriented Regular Surface in M 3 with
normal N. Then the support function of M is the func-
tion h : M -> R defined by
fc(p)=p-N(p).
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, p. 293, 1993.
Supremum
The supremum of a set is the least upper bound of the
set. It is denoted
sup.
s
On the Real Line, the supremum of a set is the same
as the supremum of its CLOSURE.
see also INFIMUM, SUPREMUM LIMIT
Supremum Limit
The limit supremum is used for sequences and nets (as
opposed to sets) and is denoted
lim sup .
s
see also SUPREMUM
Surd
An archaic term for a SQUARE ROOT.
see also Quadratic Surd, Square Root
Surface
The word "surface" is an important term in mathe-
matics and is used in many ways. The most common
and straightforward use of the word is to denote a 2-D
SUBMANIFOLD of 3-D EUCLIDEAN Space. Surfaces can
range from the very complicated (e.g., FRACTALS such
as the MANDELBROT Set) to the very simple (such as
the PLANE). More generally, the word "surface" can be
used to denote an (n - 1)-D SUBMANIFOLD of an n-D
MANIFOLD, or in general, any co-dimension 1 subob-
ject in an object (like a BANACH Space or an infinite-
dimensional Manifold).
Even simple surfaces can display surprisingly counterin-
tuitive properties. For example, the SURFACE OF REVO-
LUTION of y — 1/x around the x-AxiS for x > 1 (called
Gabriel's Horn) has Finite Volume but Infinite
Surface Area.
see also Algebraic Surface, Barth Decic, Barth
Sextic, Bernstein Minimal Surface Theorem,
Bohemian Dome, Boy Surface, Catalan's Sur-
face, Cayley's Ruled Surface, Chair, Cleb-
sch Diagonal Cubic, Compact Surface, Cone,
Conical Wedge, Conocuneus of Wallis, Cork
Plug, Corkscrew Surface, Cornucopia, Costa
Minimal Surface, Cross-Cap, Crossed Trough,
Cubic Surface, Cyclide, Cylinder, Cylindroid,
Darwin-de Sitter Spheroid, Decic Surface, Del
Pezzo Surface, Dervish, Desmic Surface, De-
velopable Surface, Dini's Surface, Eight Sur-
face, Ellipsoid, Elliptic Cone, Elliptic Cylin-
der, Elliptic Helicoid, Elliptic Hyperboloid,
Elliptic Paraboloid, Elliptic Torus, Enneper's
Surfaces, Enriques Surfaces, Etruscan Venus
Surface, Flat Surface, Fresnel's Elasticity Sur-
face, Gabriel's Horn, Handkerchief Surface,
Helicoid, Henneberg's Minimal Surface, Hoff-
man's Minimal Surface, Horn Cyclide, Horn
Torus, Hunt's Surface, Hyperbolic Cylinder,
Hyperbolic Paraboloid, Hyperboloid, Ida Sur-
face, Immersed Minimal Surface, Kiss Surface,
Klein Bottle, Kuen Surface, Kummer Sur-
face, LlCHTENFELS SURFACE, MAEDER'S OWL MIN-
IMAL Surface, Manifold, Menn's Surface, Min-
imal Surface, Miter Surface, Mobius Strip,
Monge's Form, Monkey Saddle, Nonorientable
Surface, Nordstrand's Weird Surface, NURBS
Surface, Oblate Spheroid, Octic Surface, Ori-
entable surface, parabolic cylinder, parabolic
Horn Cyclide, Parabolic Ring Cyclide, Para-
bolic Spindle Cyclide, Paraboloid, Peano Sur-
face, Piriform, Plane, Plucker's Conoid, Poly-
hedron, Prism, Prismatoid, Prolate Spheroid,
Pseudocrosscap, Quadratic Surface, Quartic
Surface, Quintic Surface, Regular Surface,
Rembs' Surfaces, Riemann Surface, Ring Cy-
clide, Ring Torus, Roman Surface, Ruled Sur-
face, Scherk's Minimal Surfaces, Seifert Sur-
face, Sextic Surface, Shoe Surface, Sievert's
Surface, Smooth Surface, Solid, Sphere, Spher-
oid, Spindle Cyclide, Spindle Torus, Steinbach
Screw, Steiner Surface, Swallowtail Catastro-
phe, Symmetroid, Tanglecube, Tetrahedral Sur-
face, Togliatti Surface, Tooth Surface, Tri-
noid, Unduloid, Veronese Surface, Veronese Va-
riety, Wallis's Conical Edge, Wave Surface,
Wedge, Whitney Umbrella
References
Endrafi, S. "Home Page of S. Endraft." http://www.
mathematik.uni-mainz.de/-endrass/.
Surface Area
Surface Integral 1763
Fischer, G. (Ed). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, 1986.
Francis, G. K. A Topological Picturebook. New York:
Springer- Verlag, 1987.
Geometry Center. "The Topological Zoo." http://www.
geom.umn.edu/zoo/.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, 1993.
Hunt, B. "Algebraic Surfaces." http://www.mathematik.
uni-kl.de/-wwwagag/Galerie.html.
Morgan, F. "What is a Surface?" Amer. Math. Monthly 103,
369-376, 1996.
Nordstrand, T. "Gallery." http://www.uib.no/people/
nf ytn/mathgal .htm.
Nordstrand, T. "Surfaces." http://www.uib.no/people/
nfytn/surf aces. htm.
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, 1993.
Wagon, S. "Surfaces." Ch. 3 in Mathematica in Action. New
York: W. H. Freeman, pp. 67-91, 1991.
Yamaguchi, F. Curves and Surfaces in Computer Aided Ge-
ometric Design, New York: Springer- Verlag, 1988.
Surface Area
Surface area is the AREA of a given surface. Roughly
speaking, it is the "amount" of a surface, and has units
of distance squares. It is commonly denoted S for a
surface in 3-D, or A for a region of the plane (in which
case it is simply called "the" Area).
If the surface is PARAMETERIZED using u and v, then
Surface
1 = f \t u xt v
dudv,
(i)
7rr\/r 2 + h 2
conical frustum 7r(i2i + R 2 )y / (R 1 - R2) 2 + h?
cube 6a 2
cylinder 2-xrh
lune 2r 2
oblate spheroid 2ira 2 + ^ In (±±§ )
prolate spheroid 2?r6 2 + ^ sin -1 e
pyramid \ps
pyramidal frustum \ps
sphere 47rr
torus 47r 2 Rr
zone 2itrh
Surface T
cone 7rr(r + y/r 2 + h 2 )
conical frustum it[Ri 2 + R2 2
cylinder
+(ft + R2) v / (Ri-R2) 2 + h 2 ]
2nr(r + h)
Even simple surfaces can display surprisingly counterin-
tuitive properties. For instance, the surface of revolu-
tion of y = 1/x around the a;- Axis for x > 1 is called
Gabriel's Horn, and has Finite Volume but Infi-
nite surface AREA.
see also AREA, SURFACE INTEGRAL, SURFACE OF REV-
OLUTION, Volume
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 127-132, 1987.
where T u and T v are tangent vectors and a x b is the
Cross Product.
The surface area given by rotating the curve y = f(x)
from x = a to x — b about the cc-axis is
/a
l + [f'(x)}*dx.
(2)
If z — f(x,y) is defined over a region R, then
-IU£H%)»«-
where the integral is taken over the entire surface.
(3)
The following tables gives surface areas for some com-
mon SURFACES. In the first table, S denotes the lateral
surface, and in the second, T denotes the total surface.
In both tables, r denotes the Radius, h the height, p
the base Perimeter, and s the Slant Height (Beyer
1987).
Surface Integral
For a Scalar Function / over a surface parameterized
by u and v, the surface integral is given by
$= / /da= / f(u y v)\T u x T v \dudv,
(1)
where T u and T v are tangent vectors and a x b is the
Cross Product.
For a Vector Function over a surface, the surface
integral is given by
$= JF'dsi= f(F-h)da (2)
= / f*dydz + f y dz dx + f z dx dy, (3)
where a- b is a DOT PRODUCT and n is a unit NORMAL
VECTOR. If z = f(x, y), then da is given explicitly by
d * = ± [-^*-%* + i ) dxdy - (4)
If the surface is SURFACE PARAMETERIZED using u and
v, then
(5)
$ = / F-(T U x T v )dudv.
see also SURFACE PARAMETERIZATION
1764 Surface Parameterization
Surface Parameterization
A surface in 3-SPACE can be parameterized by two vari-
ables (or coordinates) u and v such that
x = x(u,v)
y = y(u,v)
z = z(u, v).
(1)
(2)
(3)
If a surface is parameterized as above, then the tangent
Vectors
- -**:*+ ft* + £i (4)
du du du
rp _ & x - dy - dz ~
dv dv dv
(5)
are useful in computing the Surface Area and Sur-
face Integral.
see also SMOOTH SURFACE, SURFACE AREA, SURFACE
Integral
Surface of Revolution
A surface of revolution is a SURFACE generated by rotat-
ing a 2-D Curve about an axis. The resulting surface
therefore always has azimuthal symmetry. Examples of
surfaces of revolution include the APPLE, Cone (exclud-
ing the base), Conical FRUSTUM (excluding the ends),
Cylinder (excluding the ends), Darwin-de Sitter
Spheroid, Gabriel's Horn, Hyperboloid, Lemon,
Oblate Spheroid, Paraboloid, Prolate Spheroid,
PSEUDOSPHERE, SPHERE, SPHEROID, and TORUS (and
its generalization, the Toroid).
The standard parameterization of a surface of revolution
is given by
x(uj v) = 4>{v) cos u
y(u,v) = <f>(v) smu
z(u,v) = i/j(v).
a)
(2)
(3)
For a curve so parameterized, the first FUNDAMENTAL
Form has
F =
(4)
(5)
(6)
Wherever <p and (f> t2 + ip n are nonzero, then the surface
is regular and the second FUNDAMENTAL FORM has
\4>W
yj<j> 12 + V' 2
/ = o
y/V 2 + </>' 2
(7)
(8)
(9)
Surface of Revolution
Furthermore, the unit NORMAL VECTOR is
sgn(<£)
N(u,u) =
yV 2 + V>' 2
and the PRINCIPAL CURVATURES are
<j> cosu
ip' sin u
<t>'
«i
«2
g _ sgn^^'V-^")
G
e
(0' 2 +^'2)3/2
E \<t>W4> t2 + ^' 2 '
The Gaussian and Mean Curvatures are
—ip' <f>" + (p'lp'ip"
0(0' 2 + <0' 2 ) 2
4>(<f>"1>'-<t>'1>")-1>'{<l>' 2 + il>' 2 )
H =
(Gray 1993).
2\<t>\(<t> ,2 +^' 2 ) 3 / 2
(10)
(11)
(12)
(13)
(14)
Pappus's Centroid Theorem gives the Volume of a
solid of rotation as the cross-sectional Area times the
distance traveled by the centroid as it is rotated.
Calculus of Variations can be used to find the curve
from a point (#1,2/1) to a point (#2,2/2) which, when
revolved around the a?- Axis, yields a surface of smallest
Surface Area A (i.e., the Minimal Surface). This
is equivalent to finding the MINIMAL SURFACE passing
through two circular wire frames. The AREA element is
dA = 2-Kyds = 2-Ky^Jx + y' 2 dx, (15)
so the Surface Area is
A = 2tt y^l + y ,2 dx,
and the quantity we are minimizing is
(16)
f = vVi + y' 2 -
(17)
This equation has f x = 0, so we can use the BELTRAMI
Identity
r Of
f -y x ~ — = a
to obtain
yy/i + y*
y yy
y y/i + y' 2
v(i + y' 2 )-
- yy' 2 = ay/l +
y' 2
y-
a^l + y' 2
y
= a
V^
(18)
(19)
(20)
(21)
(22)
Surface of Revolution
Surface of Revolution
1765
2--l = y"
a
dx
d V V f sjy 2 - a 2
../
dy
\/y 2 - o?
= a cosh'
-(f)
y = a cosh I J ,
(23)
(24)
(25)
(26)
which is called a Catenary, and the surface generated
by rotating it is called a CATENOID. The two constants
a and b are determined from the two implicit equations
, f x x -b \
2/i = a cosh I I
, (x 2 -b\
yi = a cosh [ I ,
which cannot be solved analytically.
(27)
(28)
The general case is somewhat more complicated than
this solution suggests. To see this, consider the MINIMAL
Surface between two rings of equal Radius yo. With-
out loss of generality, take the origin at the midpoint of
the two rings. Then the two endpoints are located at
(-xo,yo) and (x ,yo), and
yo — a cosh I I = a cosh ( J .
But cosh(— x) = cosh(x), so
-xq — b^
f-x -b\ f-x + b\
cosh I J = cosh ( J .
Inverting each side
— xq — b = — Xq + 6,
(29)
(30)
(31)
so b = (as it must by symmetry, since we have chosen
the origin between the two rings), and the equation of
the Minimal Surface reduces to
y = a cosh f — I .
At the endpoints
yo = a cosh
(?)•
(32)
(33)
but for certain values of xq and yo, this equation has
no solutions. The physical interpretation of this fact is
that the surface breaks and forms circular disks in each
ring to minimize Area. Calculus of Variations can-
not be used to find such discontinuous solutions (known
in this case as GOLDSCHMIDT SOLUTIONS). The mini-
mal surfaces for several choices of endpoints are shown
above. The first two cases are CATENOIDS, while the
third case is a Goldschmidt Solution.
To find the maximum value of xo/yo at which CATE-
NARY solutions can be obtained, let p = 1/a. Then (31)
gives
y p = cosh(p#o). (34)
Now, denote the maximum value of xo as xjjj. Then it
will be true that dxo/dp = 0. Take d/dp of (34),
yo = sinh(pzo) [ x + p
dxo
dp
Now set dxo/dp =
yo = ic sinh(px5).
Prom (34),
pyo* = cosh(pz *).
Take (37) ~ (36),
P X*o^COth(pXr]).
Defining u = pa?o*,
u = coth^.
(35)
(36)
(37)
(38)
(39)
This has solution u = 1.1996789403.... From (36),
yop = coshw. Divide this by (39) to obtain yo/xo =
sinhu, so the maximum possible value of xo/j/o is
Xo
yo
cschu = 0.6627434193...
(40)
Therefore, only Goldschmidt ring solutions exist for
xo/yo > 0.6627....
The Surface Area of the minimal CATENOID surface
is given by
A = 2(2tt) J y^l+y f2 dx, (41)
Jo
but since
y = y/l + y' 2 a
y — a cosh I — J ,
(42)
(43)
1766
Surface of Revolution
Surface of Revolution
— 2-zra
A = — / y 2 dx = 47ra / cosh 2 ( — )
a Jo Jo W
= 4ira / | [cosh f — J + ll da:
= 2-ira I cosh ( — \ dx + I dx
'a . , /2#\ 1 X0
/2cc\ ^c]*
V a / a Jo
dec
: na \ sinh I
2
(44)
Some caution is needed in solving (33) for a. If we take
xq = 1/2 and yo = 1 then (33) becomes
1 = a cosh
UJ'
(45)
which has two solutions: a\ = 0.2350... ("deep"), and
0,2 = 0.8483. . . ("flat"). However, upon plugging these
into (44) with x = 1/2, we find Ai = 6.8456... and
A2 = 5.9917 So A\ is not, in fact, a local minimum,
and A2 is the only true minimal solution.
The Surface Area of the Catenoid solution equals
that of the Goldschmidt Solution when (44) equals
the Area of two disks,
V [sinh (^) + —}= 2*Vo 2 (46)
a 2 [2 sinh (^) cosh (5> ) + ?5>] - 2y 2 = (47)
a2 [ cosh (?)v cosh2 (?)
Plugging in
1 +
Xq
yo
o.
(48)
yo
a
cosh
(?)■
(49)
?V(5)- ,+ ~""(?)-(?)- a <50>
Defining
2/o
gives
uyu 2
1 + cosh u ~ u = 0.
(51)
(52)
This has a solution u = 1.2113614259. The value of
xo/yo for which
-^catenary = -A 2 disks \p*J)
is therefore
Xo _ ^ cosh- 1 (f ) __ cosh- 1 u
yo
m.
m.
u
0.5276973967.
(54)
For xo/yo G (0.52770,0.6627), the CATENARY solution
has larger Area than the two disks, so it exists only as
a Relative Minimum.
There also exist solutions with a disk (of radius r) be-
tween the rings supported by two CATENOIDS of rev-
olution. The Area is larger than that for a simple
Catenoid, but it is a Relative Minimum. The equa-
tion of the Positive half of this curve is
At (0,r),
At (aj ,yo),
y = ci cosh ( h c 3 j .
r = ci cosh(c3).
j/o = ci cosh f h C3 j -
The Area of the two Catenoids is
^catenoids
(55)
(56)
(57)
2(2tt) / yi/l + y'tdx = — / y dx
Jo Cl Jo
47rci / cosh 2 ( 1- C3 J dx. (58)
Now let u = x/c± + C3, so du = dx/ci
pxq/xi+cs
A = 4ttci z / cosh z u du
rvo/xx+cs
4tvc±
7
'\ I [cosh(2u) + 1] du
Jc 3
2wci [~smh(2u) + u\
= 2ttci 2 { | sinh [2 (5o + Ca )] - I sinh(2c 3 ) + ^}
- TTd 2 {sinh [2 (^ + cs)] - sinh(2c 3 ) + ^} •
(59)
The Area of the central Disk is
^4disk = 7T7- 2 = 7TCi 2 COsh 2 C3 ,
(60)
Surface of Revolution
so the total Area is
A = ttci 2 jsinh [ 2 (^r +<*)]
+ [cosh 2 c 3 - sinh(2c 3 )] + ^} • ( 61 )
By Plateau's Laws, the Catenoids meet at an An-
gle of 120°, so
(62)
Surface of Revolution 1767
tan30°=[^'
lax.
= sinh ( h cz
x =o L Vci
= sinh C3 = — f=
and
ca^sinh" 1 ^].
This means that
cosh 2 C3 — sinh(2c3)
= [1 + sinh 2 c 3 ] - 2
sinh C3 v 1 + sinh 2 c 3
= (l + i)-2(^)^
4 2 2 Q
3 v^v^
(63)
(64)
(65)
Now examine xo/yo,
XQ
= w sech(u + C3), (66)
XQ
XQ
#o _ ci _
m " cosh(^+c 3 )
yo
ci
where w = xo/ci. Finding the maximum ratio of xo/yo
gives
d J xo
du \y
sech(u+C3)-^tanh(^+C3)sech(u+C3) =
(67)
utanh(n + c 3 ) = 1, (68)
with C3 = sinh~ 1 (l/v / 3) as given above. The solu-
tion is u — 1.0799632187, so the maximum value of
xo/yo for two Catenoids with a central disk is j/o =
0.4078241702.
If we are interested instead in finding the curve from a
point (xi,yi) to a point (£2,3/2) which, when revolved
around the y-AxiS (as opposed to the x-Axis), yields
a surface of smallest Surface Area A, we proceed as
above. Note that the solution is physically equivalent
to that for rotation about the x-Axis, but takes on a
different mathematical form. The Area element is
dA = 2-kx ds = 2ivxy 1 + y' 2 dx
- [xy/l + y' 2 dx,
A = 2n
and the quantity we are minimizing is
f^x^/l + y' 2 .
Taking the derivatives gives
d df _ d I xy'
dx dy' ~ dx I ^/i + y '2
(70)
(71)
(72)
(73)
so the Euler-Lagrange Differential Equation be-
comes
df
dx dy'
dx
f xy'
dy
[Vi + y' 2
xy f
— = a
a/i + y' 2
xV 2 = a 2 (l+y' 2 )
/2/ 2 2\ 2
y (x — a ) = a
dy _ a
dx y/x 2 — a?
= 0. (74)
(75)
(76)
(77)
(78)
, = a /— jL=+6 = acosh- 1 (- > )+&. (79)
J Vx 2 - a 2 W
Solving for x then gives
= a cosh ( I ,
(80)
which is the equation for a Catenary. The Surface
Area of the Catenoid product by rotation is
A = 2tt
= 2tt
/ xy/l + y* 2 dx = 2tt / xJ:
i
x 2 — a 2
dx
y/x 2 — a 2
2
\/(x 2 — a?) + a 2 dx
f x 2 dx
J y/x 2 - a 2
yjx 2 - a 2 + y In (x + \A 2 - « 2 )
x 2
a/x 2 2 — a 2 - xi yxi 2 — a 2
. 2, / £2 + \/x2 2 - a 2
+a ln ' ■ / 2 f
#1 + v^i - o
(81)
(69)
Isenberg (1992, p. 80) discusses finding the MINIMAL
Surface passing through two rings with axes offset from
each other.
1768
Surface of Section
Survivorship Curve
see also Apple, Catenoid, Cone Conical Frustum,
Cylinder, Darwin-de Sitter Spheroid, Eight
Surface, Gabriel's Horn, Hyperboloid, Lemon,
Meridian, Oblate Spheroid, Pappus's Centroid
Theorem, Paraboloid, Parallel (Surface of
Revolution), Prolate Spheroid, Pseudosphere,
Sinclair's Soap Film Problem, Solid of Revolu-
tion, Sphere, Spheroid, Toroid, Torus
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 931-937, 1985.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA:
Addison-Wesley, p. 42, 1980.
Gray, A. "Surfaces of Revolution." Ch. 18 in Modern Dif-
ferential Geometry of Curves and Surfaces. Boca Raton,
FL: CRC Press, pp. 357-375, 1993.
Isenberg, C. The Science of Soap Films and Soap Bubbles.
New York: Dover, pp. 79-80 and Appendix III, 1992.
Surface of Section
A surface (or "space") of section is a way of presenting a
trajectory in n-D PHASE SPACE in an (n - 1)-D SPACE.
By picking one phase element constant and plotting the
values of the other elements each time the selected el-
ement has the desired value, an intersection surface is
obtained. If the equations of motion can be formulated
as a MAP in which an explicit FORMULA gives the values
of the other elements at successive passages through the
selected element value, the time required to compute the
surface of section is greatly reduced.
see also Phase Space
Surgery
In the process of attaching a fc-HANDLE to a MANI-
FOLD M, the Boundary of M is modified by a process
called (k — l)-surgery. Surgery consists of the removal
of a Tubular Neighborhood of a (k - 1)-Sphere
S fe_1 from the BOUNDARIES of M and the dim(M) - 1
standard SPHERE, and the gluing together of these two
scarred-up objects along their common BOUNDARIES.
see also Boundary, Dehn Surgery, Handle, Mani-
fold, Sphere, Tubular Neighborhood
Surjection
An Onto (Surjective) Map.
see also BlJECTION, INJECTION, ONTO
Surjective
see Onto
Surprise Examination Paradox
see Unexpected Hanging Paradox
Surreal Number
The most natural collection of numbers which includes
both the REAL NUMBERS and the infinite ORDINAL
NUMBERS of Georg Cantor. They were invented by John
H. Conway in 1969. Every Real Number is surrounded
by surreals, which are closer to it than any REAL NUM-
BER. Knuth (1974) describes the surreal numbers in a
work of fiction.
The surreal numbers are written using the NOTATION
{a|6}, where {|} = 0, {0|} = 1 is the simplest number
greater than 0, {1|} = 2 is the simplest number greater
than 1, etc. Similarly, {|0} = —1 is the simplest number
less than 1, etc. However, 2 can also be represented by
{1|3}, {3/2|4}, {l|o,}, etc.
see also Omnific Integer, Ordinal Number, Real
Number
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.
Conway, J. H. On Numbers and Games. New York: Aca-
demic Press, 1976.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 283-284, 1996.
Conway, J. H. and Jackson, A. "Budding Mathematician
Wins Westinghouse Competition." Not. Amer. Math. Soc.
43, 776-779, 1996.
Gonshor, H. An Introduction to Surreal Numbers. Cam-
bridge: Cambridge University Press, 1986.
Knuth, D. Surreal Numbers: How Two Ex-Students Turned
on to Pure Mathematics and Found Total Happiness.
Reading, MA: Addison-Wesley, 1974. http://vww-cs-
faculty.stanford.edu/-knuth/sn.html.
Surrogate
Surrogate data are artificially generated data which
mimic statistical properties of real data. Isospectral
surrogates have identical POWER SPECTRA as real data
but with randomized phases. Scrambled surrogates have
the same probability distribution as real data, but with
white noise POWER SPECTRA.
see also POWER SPECTRUM
Surveying Problems
see Hansen's Problem, Snellius-Pothenot Prob-
lem
Survivorship Curve
ype II
Suslin's Theorem
Swastika 1769
Plotting l x from a LIFE EXPECTANCY table on a loga-
rithmic scale versus x gives a curve known as a survivor-
ship curve. There are three general classes of survivor-
ship curves, illustrated above.
1. Type I curves are typical of populations in which
most mortality occurs among the elderly (e.g., hu-
mans in developed countries).
2. Type II curves occur when mortality is not depen-
dent on age (e.g., many species of large birds and
fish). For an infinite type II population, eo = ei =
. . ., but this cannot hold for a finite population.
3. Type III curves occur when juvenile mortality is ex-
tremely high (e.g., plant and animal species produc-
ing many offspring of which few survive). In type
III populations, it is often true that e^+i > e* for
small i. In other words, life expectancy increases for
individuals who survive their risky juvenile period.
see also LIFE EXPECTANCY
Suslin's Theorem
A Set in a Polish Space is a Borel Set Iff it is both
Analytic and Coanalytic. For subsets of u;, a set is
5\ Iff it is "hyperarithmetic."
see also Analytic Set, Borel Set, Coanalytic Set,
Polish Space
Suspended Knot
An ordinary KNOT in 3-D suspended in 4-D to create a
knotted 2-sphere. Suspended knots are not smooth at
the poles.
see also Spun Knot, Twist-Spun Knot
Suspension
The Join of a Topological Space X and a pair of
points S°, E(A") = X * 5°.
see also Join (Spaces), Topological Space
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 6, 1976.
Suzanne Set
The nth Suzanne set S n is defined as the set of COMPOS-
ITE Numbers x for which n\S(x) and n\S p (x), where
x = a 4- ai(K) 1 ) + . . . + a d (10 d ) = p x p 2 •• -p n ,
and
S(x) = ^2 a.
Every Suzanne set has an infinite number of elements.
The Suzanne set S n is a superset of the MONICA Set
M„.
see also MONICA SET
References
Smith, M. "Cousins of Smith Numbers: Monica and Suzanne
Sets." Fib. Quart. 34, 102-104, 1996.
Suzuki Group
The Sporadic Group Suz.
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat.bham.ac.uk/atlas/Suz.html.
Swallowtail Catastrophe
A Catastrophe which can occur for three control fac-
tors and one behavior axis. The equations
x = uv + Zv
y = — 2uv — 4v 3
z = u
display such a catastrophe (von Seggern 1993, Nord-
strand). The above surface uses u E [—2,2] and v €
[-0.8,0.8].
References
Nordstrand, T. "Swallowtail." http://www.uib.no/people/
nf ytn/stltxt .htm.
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 94, 1993.
Swastika
3=0
S P (aO = X>(j>i).
An irregular ICOSAGON, also called the gammadion or
fylfot, which symbolized good luck in ancient Arabic and
Indian cultures. In more recent times, it was adopted as
the symbol of the Nazi Party in Hitler's Germany and
has thence come to symbolize anti-Semitism.
see also CROSS, DISSECTION
1770 Swastika Curve
Swastika Curve
The plane curve with Cartesian equation
and polar equation
4 4
y - x = xy
2 _ sin 9 cos 6
~ sin 4 - cos 4 6 '
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 71, 1989.
Sweep Signal
The general function
a, 6, Cjd) = c sin < ( (b — a) — + a J — a 2 >
References
von Seggern, D. CRC Standard Curves and Surfaces.
Raton, FL: CRC Press, p, 160, 1993,
Boca
Swinnerton-Dyer Conjecture
In the early 1960s, B. Birch and H. P. F. Swinnerton-
Dyer conjectured that if a given Elliptic Curve has
an infinite number of solutions, then the associated L-
function has value at a certain fixed point. In 1976,
Coates and Wiles showed that elliptic curves with COM-
PLEX multiplication having an infinite number of solu-
tions have L-functions which are zero at the relevant
fixed point (COATES-WlLES THEOREM), but they were
unable to prove the converse. V. Kolyvagin extended
this result to modular curves.
see also Coates-Wiles Theorem, Elliptic Curve
References
Sci-
Cipra, B. "Fermat Prover Points to Next Challenges."
ence 271, 1668-1669, 1996.
Ireland, K. and Rosen, M. "New Results on the Birch-
Swinnerton-Dyer Conjecture." §20.5 in A Classical Intro-
duction to Modern Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 353-357, 1990.
Mazur, B. and Stevens, G, (Eds.). p-Adic Monodromy and
the Birch and Swinnerton-Dyer Conjecture. Providence,
RI: Amer. Math. Soc, 1994.
Sylow Theorems
Swinnerton-Dyer Polynomial
The minimal POLYNOMIAL S n (x) whose ROOTS are
sums and differences of the SQUARE ROOTS of the first
n Primes,
Sn{x) = Y[(x ± V2 ± \/3 ± y/l ± . . . ± y/p^).
References
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 11 and 225-226, 1991.
Swirl
A swirl is a generic word to describe a function having
arcs which double back swirl around each other. The
plots above correspond to the function
/(r, 6) — sin(6 cos r — nO)
for n = 0, 1, . . . , 5.
see also Daisy, Whirl
Sylow p-Subgroup
If p k is the highest Power of a Prime p dividing the
Order of a finite Group G, then a Subgroup of G of
Order p k is called a Sylow p-subgroup of G.
see also ABHYANKAR'S CONJECTURE, SUBGROUP, Sy-
low Theorems
Sylow Theorems
Let p be a Prime Number, G a Group, and \G\ the
order of G.
1. If p divides |G|, then G has a SYLOW p-SUBGROUP.
2. In a Finite Group, all the Sylow ^-Subgroups
are isomorphic for some fixed p.
3. The number of SYLOW p-SuBGROUPS for a fixed p is
Congruent to 1 (mod p).
Sylvester Cyclotomic Number
Sylvester's Sequence 1771
Sylvester Cyclotomic Number
Given a LUCAS SEQUENCE with parameters P and Q,
discriminant D / 0, and roots a and /?, the Sylvester
cyclotomic numbers are
Q n = H(a-CP),
Sylvester's Inertia Law
The numbers of Eigenvalues that are Positive, Neg-
ative, or do not change under a congruence trans-
formation. Gradshteyn and Ryzhik (1979) state it as
follows: when a Quadratic Form Q in n variables is
reduced by a nonsingular linear transformation to the
form
where
/2tt\ . . /2tt
C = cos — J + i sin —
V n / \ n
2tt\
Q
2 i 2 .
yx + 2/2 + .
. . + Vp
Pp+1 - VP2 - • * • - Vr ,
is a Primitive Root of Unity and the product is
over all exponents r RELATIVELY PRIME to n such that
r G [l,n).
see also LUCAS SEQUENCE
References
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, p. 69, 1989.
Sylvester's Determinant Identity
where A u>w is the submatrix of A formed by the inter-
section of the subset w of columns and u of rows.
Sylvester's Four-Point Problem
Let q(R) be the probability that four points chosen at
random in a region R have a Convex Hull which is
a Quadrilateral. For an open, convex subset of the
Plane of finite Area,
0.667 ;
< q(R) < 1
35
12tt 2
0.704.
References
Schneinerman, E. and Wilf, H. S. "The Rectilinear Crossing
Number of a Complete Graph and Sylvester's 'Four Point'
Problem of Geometric Probability." Amer. Math. Monthly
101, 939-943, 1994.
the number p of POSITIVE SQUARES appearing in the
reduction is an invariant of the QUADRATIC FORM Q
and does not depend on the method of reduction.
see also Eigenvalue, Quadratic Form
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1105, 1979.
Sylvester's Line Problem
It is not possible to arrange a finite number of points so
that a Line through every two of them passes through
a third, unless they are all on a single Line.
see also Collinear, Sylvester's Four-Point Prob-
lem
Sylvester Matrix
For Polynomials of degree m and n, the Sylvester ma-
trix is an (m+n) x (m+n) matrix whose Determinant
is the Resultant of the two Polynomials.
see also RESULTANT
Sylvester's Sequence
The sequence defined by e = 2 and the RECURRENCE
Relation
£n = 1 + J^ I e» = e n -i — e n -i + 1.
(i)
Sylvester Graph
The Sylvester graph of a configuration is the set of OR-
DINARY Points and Ordinary Lines.
see also Ordinary Line, Ordinary Point
References
Guy, R. K. "Monthly Unsolved Problems, 1969-1987."
Amer. Math. Monthly 94, 961-970, 1987.
Guy, R. K. "Unsolved Problems Come of Age." Amer. Math.
Monthly 96, 903-909, 1989.
This sequence arises in Euclid's proof that there are an
Infinite number of Primes. The proof proceeds by
constructing a sequence of PRIMES using the RECUR-
RENCE Relation
e n +i = eoei * • * e n 4- 1 (2)
(Vardi 1991). Amazingly, there is a constant
E w 1.264084735306 (3)
such that
E z
+ 1
(4)
1772 Sylvester's Signature
Symmetric Function
(Vardi 1991, Graham et aL 1994). The first few numbers
in Sylvester's sequence are 2, 3, 7, 43, 1807, 3263443,
10650056950807, . . . (Sloane's A000058). The e n satisfy
oo
(5)
In addition, if < x < 1 is an Irrational Number,
then the nth term of an infinite sum of unit fractions
used to represent x as computed using the GREEDY AL-
GORITHM must be smaller than l/e n .
The n of the first few PRIME e n are 0, 1, 2, 3, 5,
Vardi (1991) gives a lists of factors less than 5 x 10 7 of
e n for n < 200 and shows that e n is COMPOSITE for
6 < n < 17. Furthermore, all numbers less than 2.5 x
10 15 in Sylvester's sequence are SQUAREFREE, and no
SQUAREFUL numbers in this sequence are known (Vardi
1991).
see also EUCLID'S THEOREMS, GREEDY ALGORITHM,
SQUAREFREE, SQUAREFUL
References
Graham, R. L.; Knuth, D. E.; and Patashnik, O. Research
problem 4.65 in Concrete Mathematics: A Foundation
for Computer Science, 2nd ed. Reading, MA: Addison-
Wesley, 1994.
Sloane, N. J. A. Sequence A000058/M0865 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Vardi, I. "Are All Euclid Numbers Squarefree?" and
"PowerMod to the Rescue." §5.1 and 5.2 in Computational
Recreations in Mathematica. Reading, MA: Addison-
Wesley, pp. 82-89, 1991.
Sylvester's Signature
Diagonalize a form over the RATIONALS to
diag[p°-A,p b .J5,...],
where all the entries are INTEGERS and A, £?, ... are
Relatively Prime to p. Then Sylvester's signature is
the sum of the — 1-parts of the entries.
see also p-SlGNATURE
Sylvester's Triangle Problem
The resultant of the vectors represented by the three
RADII from the center of a TRIANGLE'S ClRCUMClRCLE
to its VERTICES is the segment extending from the ClR-
cumcenter to the Orthocenter.
see also Circumcenter, Circumcircle, Orthocen-
ter, Triangle
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 142,
1965.
Symbolic Logic
The study of the meaning and relationships of state-
ments used to represent precise mathematical ideas.
Symbolic logic is also called Formal Logic.
see also Formal Logic, Logic, Metamathematics
References
Carnap, R. Introduction to Symbolic Logic and Its Applica-
tions. New York: Dover, 1958.
Symmedian Line
The lines Isogonal to the Medians of a Triangle
are called the triangle's symmedian lines. The symme-
dian lines are concurrent in a point called the LEMOINE
Point.
see also Isogonal Conjugate, Lemoine Point, Me-
dian (Triangle)
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 213-218, 1929.
Symmedian Point
see Lemoine Point
Symmetric
A quantity which remains unchanged in SIGN when in-
dices are reversed. For example, A^ = a* + a,j is sym-
metric since A*.
A<
see also ANTISYMMETRIC
Symmetric Block Design
A symmetric design is a BLOCK DESIGN (u, k 7 A, r, b)
with the same number of blocks as points, so b = v (or,
equivalently, r = k). An example of a symmetric block
design is a PROJECTIVE PLANE.
see also Block Design, Projective Plane
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.
Symmetric Design
see Symmetric Block Design
Symmetric Function
A symmetric function on n variables a?i, . . . , x n is a
function that is unchanged by any PERMUTATION of its
variables. In most contexts, the term "symmetric func-
tion" refers to a polynomial on n variables with this fea-
ture (more properly called a "symmetric polynomial").
Another type of symmetric functions is symmetric ra-
tional functions, which are the RATIONAL FUNCTIONS
that are unchanged by PERMUTATION of variables.
Symmetric Group
Symmetric Tensor 1773
The symmetric polynomials (respectively, symmetric ra-
tional functions) can be expressed as polynomials (re-
spectively, rational functions) in the Elementary Sym-
metric Functions. This is called the Fundamental
Theorem of Symmetric Functions.
A function f(x) is sometimes said to be symmetric about
the y- Axis if f(—x) = f(x). Examples of such func-
tions include |a:| (the ABSOLUTE VALUE) and x 2 (the
Parabola).
see also ELEMENTARY SYMMETRIC FUNCTION, FUNDA-
MENTAL Theorem of Symmetric Functions, Ra-
tional Function
References
Macdonald, I. G. Symmetric Functions and Hall Polynomi-
als, 2nd ed. Oxford, England: Oxford University Press,
1995.
Macdonald, I. G. Symmetric Funtions and Orthogonal Poly-
nomials. Providence, RI: Araer. Math. Soc, 1997.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "Symmet-
ric Function Identities." §1.7 in A=B. Wellesley, MA:
A. K. Peters, pp. 12-13, 1996.
Symmetric Group
The symmetric group 5„ of DEGREE n is the GROUP
of all Permutations on n symbols. S n is therefore of
ORDER n! and contains as SUBGROUPS every GROUP of
Order n. The number of Conjugacy Classes of S n
is given by the PARTITION FUNCTION P.
Netto's Conjecture states that the probability that
two elements Pi and Pi of a symmetric group generate
the entire group tends to 3/4 as n ->■ oo. This was
proven by Dixon in 1967.
see also Alternating Group, Conjugacy Class,
Finite Group, Netto's Conjecture, Partition
Function P, Simple Group
References
Lomont, J. S. "Symmetric Groups." Ch. 7 in Applications of
Finite Groups. New York: Dover, pp. 258-273, 1987.
Wilson, R. A. "ATLAS of Finite Group Representation."
http : //for . mat , bham . ac . uk/atlas#alt .
Symmetric Matrix
A symmetric matrix is a SQUARE MATRIX which sat-
isfies A T = A where A T denotes the TRANSPOSE, so
dij = aji. This also implies
A^A 1 " = I, (1)
where I is the Identity Matrix. Written explicitly,
(2)
The symmetric part of any MATRIX may be obtained
from
A 3 = |(A + A T ). (3)
an
ai2
•• a\
a2i
G22
• • <Z2
flnl
a n 2
* * d n
A MATRIX A is symmetric if it can be expressed in the
form
A = QDQ T , (4)
where Q is an ORTHOGONAL MATRIX and D is a DI-
AGONAL Matrix. This is equivalent to the Matrix
equation
AQ = QD, (5)
which is equivalent to
AQ n = AnQ.
(6)
for all n, where A n = D nn . Therefore, the diagonal ele-
ments of D are the Eigenvalues of A, and the columns
of Q are the corresponding EIGENVECTORS.
see also Antisymmetric Matrix, Skew Symmetric
Matrix
References
Nash, J. C. "Real Symmetric Matrices." Ch. 10 in Compact
Numerical Methods for Computers: Linear Algebra and
Function Minimisation, 2nd ed. Bristol, England: Adam
Hilger, pp. 119-134, 1990.
Symmetric Points
Two points z and z 6 C* are symmetric with respect to
a CIRCLE or straight LINE L if all Circles and straight
LINES passing through z and z s are orthogonal to L.
Mobius Transformations preserve symmetry. Let a
straight line be given by a point zq and a unit VECTOR
e t0 , then
Z = e (Z - Zq) + Zq.
Let a CIRCLE be given by center zq and RADIUS r, then
z = Zq +
(z-zo)*'
see also MOBIUS TRANSFORMATION
Symmetric Relation
A Relation R on a Set S is symmetric provided that
for every x and y in S we have xRy IFF yRx.
see also RELATION
Symmetric Tensor
A second-RANK symmetric TENSOR is defined as a TEN-
SOR A for which
Any TENSOR can be written as a sum of symmetric and
Antisymmetric parts
A mn = ±( A mn + j^nm^ + 1 ^mn _ ^m^
= ±(Bs mn +B A rnn ).
(2)
The symmetric part of a TENSOR is denoted by paren-
theses as follows:
T(a t b) = 2^ ab + Tba)
(3)
1 774 Symmetroid
Symmetry Operation
1 (a 1) o 2 ,...,a n ) — n , 2^ "^
o 1 a 2 -*-a n •
(4)
permutations
The product of a symmetric and an ANTISYMMETRIC
TENSOR is 0. This can be seen as follows. Let a a/3 be
Antisymmetric, so
a 11 = a 22 =
„ 21 ~ 12
a = —a .
Let b a /3 be symmetric, so
&12 = &21-
Then
(5)
(6)
(7)
a o a ,g = a on + a 012 + a 021 + a 022
= + a 12 6i 2 - a 12 6 12 +0 = 0. (8)
A symmetric second-RANK Tensor A m n has Scalar
invariants
5i = An + A 22 + A 22 (9)
s 2 - A22A33 + A33A11 + A11A22 - A23 2
-A 3 i 2 -Ai2 2 . (10)
Symmetroid
A Quartic Surface which is the locus of zeros of the
Determinant of a Symmetric 4x4 matrix of linear
forms. A general symmetroid has 10 Ordinary Dou-
ble Points (Jessop 1916, Hunt 1996).
References
Hunt, B. "Algebraic Surfaces." http://vvw.mathematik.
uni-kl . de/-wwwagag/Galerie . html.
Hunt, B. "Symmetroids and Weddle Surfaces." §B.5.3 in
The Geometry of Some Special Arithmetic Quotients. New
York: Springer- Verlag, pp. 315-319, 1996.
Jessop, C. Quartic Surfaces with Singular Points. Cam-
bridge, England: Cambridge University Press, p. 166,
1916.
Symmetry
An intrinsic property of a mathematical object which
causes it to remain invariant under certain classes of
transformations (such as ROTATION, REFLECTION, IN-
VERSION, or more abstract operations). The mathemat-
ical study of symmetry is systematized and formalized
in the extremely powerful and beautiful Area of math-
ematics called Group Theory.
Symmetry can be present in the form of coefficients of
equations as well as in the physical arrangement of ob-
jects. By classifying the symmetry of polynomial equa-
tions using the machinery of GROUP THEORY, for ex-
ample, it is possible to prove the unsolvability of the
general Quintic EQUATION.
In physics, an extremely powerful theorem of Noether
states that each symmetry of a system leads to a phys-
ically conserved quantity. Symmetry under TRANSLA-
TION corresponds to momentum conservation, symme-
try under ROTATION to angular momentum conserva-
tion, symmetry in time to energy conservation, etc.
see also Group Theory
References
Eppstein, D. "Symmetry and Group Theory." http://vww.
ics.uci.edu/-eppstein/junkyard/sym.html.
Farmer, D. Groups and Symmetry. Providence, RI: Amer.
Math. Soc, 1995.
Pappas, T. "Art & Dynamic Symmetry." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 154-155, 1989.
Rosen, J. Symmetry in Science: An Introduction to the Gen-
eral Theory. New York: Springer- Verlag, 1995.
S chat t Schneider, D. Visions of Symmetry: Notebooks, Peri-
odic Drawings, and Related Work of M. C. Escher. New
York: W. H. Freeman, 1990.
Stewart, I. and Golubitsky, M. Fearful Symmetry. New York:
Viking Penguin, 1993.
Symmetry Group
see Group
Symmetry Operation
Symmetry operations include the IMPROPER ROTATION,
Inversion Operation, Mirror Plane, and Rota-
tion. Together, these operations create 32 crystal
classes corresponding to the 32 POINT GROUPS.
The Inversion Operation takes
(x,y t z) -> (-£, -y, -z)
and is denoted i. When used in conjunction with a RO-
TATION, it becomes an IMPROPER ROTATION. An IM-
PROPER Rotation by 360° /n is denoted n (or S n ). For
periodic crystals, the Crystallography Restriction
allows only the IMPROPER ROTATIONS 1, 2, 3, 4, and 6.
The MIRROR Plane symmetry operation takes
(z,2/,z) -> (x,y,-z),{x J -y,z) -> (z,-y,z),
etc., which is equivalent to 2. Invariance under reflection
can be denoted ncr v or nah- The ROTATION symmetry
operation for 360° jn is denoted n (or C n ). For periodic
crystals, CRYSTALLOGRAPHY RESTRICTION allows only
1, 2, 3, 4, and 6.
Symmetry operations can be indicated with symbols
such as C n , S ny E, i, na v , and nah-
1. C n indicates Rotation about an n-fold symmetry
axis.
2. S n indicates IMPROPER ROTATION about an n-fold
symmetry axis.
3. E (or I) indicates invariance under TRANSLATION.
4. i indicates a center of symmetry under INVERSION.
Symmetry Principle
Synergetics 1 775
5. na v indicates invariance under n vertical Reflec-
tions.
6. nan indicates invariance under n horizontal REFLEC-
TIONS.
see also CRYSTALLOGRAPHY RESTRICTION, IMPROPER
Rotation, Inversion Operation, Mirror Plane,
Point Groups, Rotation, Symmetry
Symmetry Principle
Symmetric Points are preserved under a Mobius
Transformation.
see also Mobius Transformation, Symmetric
Points
Symplectic Diffeomorphism
A Map T : (Mi,u;i) -> (M 2 ,u> 2 ) between the Sym-
plectic Manifolds (Mi,wi) and (M 2i uj2) which is a
Diffeomorphism and T*(oj 2 ) = wi (where T* is the
PULLBACK MAP induced by T, i.e., the derivative of
the Diffeomorphism T acting on tangent vectors). A
symplectic diffeomorphism is also known as a SYMPLEC-
tomorphism or Canonical Transformation.
see also Diffeomorphism, Pullback Map, Symplec-
tic Manifold
References
Guillemin, V. and Sternberg, S. Symplectic Techniques in
Physics. New York: Cambridge University Press, p. 34,
1984.
Symplectic Form
A symplectic form on a SMOOTH MANIFOLD M is a
smooth closed 2-FORM uj on M which is nondegenerate
such that at every point m, the alternating bilinear form
u>m on the TANGENT SPACE T m M is nondegenerate.
A symplectic form on a VECTOR Space V over F q is
a function f{x,y) (defined for all x, y € V and taking
values in F q ) which satisfies
/(Aixi -\-X 2 X2,y) = Ai/(xi,y) + A 2 /(a?2,y),
f(y,x) = -f( x ,y),
and
f(x,x) = 0.
Symplectic forms can exist on M (or V) only if M (or
V) is EVEN-dimensional.
Symplectic Group
The symplectic group Sp n (q) for n Even is the GROUP
of elements of the General Linear Group GL n that
preserve a given nonsingular Symplectic Form. Any
SUCh MATRIX has DETERMINANT 1.
see also General Linear Group, Lie-Type Group,
Projective Symplectic Group, Symplectic Form
References
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R A.;
and Wilson, R. A. "The Groups Sp n (q) and PSp n (q) =
S n {q)" §2.3 in Atlas of Finite Groups: Maximal Sub-
groups and Ordinary Characters for Simple Groups. Ox-
ford, England: Clarendon Press, pp. x-xi, 1985.
Wilson, R. A. "ATLAS of Finite Group Representation."
http : //for . mat . bham . ac . uk/atlas#symp.
Symplectic Manifold
A pair (M,o?), where M is a MANIFOLD and a; is a
Symplectic Form on M. The Phase Space R 2n =
R n x W 1 is a symplectic manifold. Near every point
on a symplectic manifold, it is possible to find a set of
local "Darboux coordinates" in which the SYMPLECTIC
Form has the simple form
uj = 2_] dqk A dpk
(Sjamaar 1996), where dq k Adp k is a WEDGE PRODUCT.
see also Manifold, Symplectic Diffeomorphism,
Symplectic Form
References
Sjamaar, R. "Symplectic Reduction and Riemann-Roch For-
mulas for Multiplicities." Bull Amer. Math. Soc. 33,
327-338, 1996.
Symplectic Map
A Map which preserves the sum of Areas projected
onto the set of (pi,qi) planes. It is the generalization of
an Area-Preserving Map.
see also Area-Preserving Map, Liouville's Phase
Space Theorem
Symplectomorphism
see Symplectic Diffeomorphism
Synclastic
A surface on which the Gaussian Curvature K is ev-
erywhere Positive. When K is everywhere Negative,
a surface is called ANTICLASTIC. A point at which the
Gaussian Curvature is Positive is called an Ellip-
tic Point.
see also Anticlastic, Elliptic Point, Gaus-
sian Quadrature, Hyperbolic Point, Parabolic
Point, Planar Point
Synergetics
Synergetics deals with systems composed of many sub-
systems which may each be of a very different nature.
In particular, synergetics treats systems in which co-
operation among subsystems creates organized struc-
ture on macroscopic scales (Haken 1993). Examples
of problems treated by synergetics include BIFURCA-
TIONS, phase transitions in physics, convective instabili-
ties, coherent oscillations in lasers, nonlinear oscillations
in electrical circuits, population dynamics, etc.
1776 Synthesized Beam
Szilassi Polyhedron
see also Bifurcation, Chaos, Dynamical System
References
Haken, H. Synergetics, an Introduction: Nonequilibrium
Phase Transitions and Self- Organization in Physics,
Chemistry, and Biology, 3rd rev. enl. ed. New York:
Springer- Verlag, 1983.
Haken, H. Advanced Synergetics: Instability Hierarchies
of Self- Organizing Systems and Devices. New York:
Springer- Verlag, 1993.
Mikhailov, A. S. Foundations of Synergetics: Distributed Ac-
tive Systems, 2nd ed. New York: Springer- Verlag, 1994.
Mikhailov, A. S. and Loskutov, A. Y. Foundations of Syner-
getics II: Complex Patterns, 2nd ed., enl. rev. New York:
Springer- Verlag, 1996.
Synthesized Beam
see Dirty Beam
Syntonic Comma
see Comma of Didymus
Syracuse Algorithm
see Collatz Problem
Syracuse Problem
see Collatz Problem
System of Differential Equations
see Ordinary Differential Equation
System of Equations
Let a linear system of equations be denoted
AX = Y, (1)
where A is a Matrix and X and Y are Vectors. As
shown by Cramer's RULE, there is a unique solution if
A has a Matrix Inverse A -1 . In this case,
A _1 Y.
(2)
If Y = 0, then the solution is X = 0. If A has no MA-
TRIX Inverse, then the solution SUBSPACE is either a
Line or the Empty Set. If two equations are multiples
of each other, solutions are of the form
X = A + tB, (3)
for t a Real Number.
see also Cramer's Rule, Matrix Inverse
Syzygies Problem
The problem of finding all independent irreducible alge-
braic relations among any finite set of QUANTICS.
see also QUANTIC
Syzygy
A technical mathematical object defined in terms of a
Polynomial Ring of n variables over a Field k.
see also Fundamental System, Hilbert Basis The-
orem, Syzygies Problem
References
Hilbert, D. "Uber die Theorie der algebraischen Formen."
Math. Ann. 36, 473-534, 1890.
Iyanaga, S. and Kawada, Y. (Eds.). "Syzygy Theory." §364F
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, p. 1140, 1980.
Szilassi Polyhedron
A Polyhedron which is topologically equivalent to a
TORUS and for which every pair of faces has an Edge in
common. This polyhedron was discovered by L. Szilassi
in 1977. Its SKELETON is equivalent to the seven-color
torus map illustrated below.
Szpiro's Conjecture
Szpiro's Conjecture 1777
The Szilassi polyhedron has 14 VERTICES, seven faces,
and 21 EDGES, and is the DUAL POLYHEDRON of the
Csaszar Polyhedron.
see also CSASZAR POLYHEDRON, TOROIDAL POLYHE-
DRON
References
Eppstein, D. "Polyhedra and Polytopes." http://www.ics.
uci.edu/-eppstein/junkyard/polytope.html.
Gardner, M. Fractal Music, HyperCards, and More Mathe-
matical Recreations from Scientific American Magazine.
New York: W. H. Freeman, pp. 118-120, 1992.
Hart, G. "Toroidal Polyhedra." http : //www . li .net/
-george/virtual-polyhedra/toroidal.html.
Szpiro's Conjecture
A conjecture which relates the minimal DISCRIMINANT
of an Elliptic Curve to the Conductor. If true, it
would imply FERMAT'S LAST THEOREM for sufficiently
large exponents.
see also Conductor, Discriminant (Elliptic
Curve), Elliptic Curve
References
Cox, D. A. "Introduction to Fermat's Last Theorem." Amer.
Math. Monthly 101, 3-14, 1994.
t-Distribution
T
^-Distribution
see Student's £-Distribution
T-Polyomino
Tait Flyping Conjecture 1779
Tacnode
&
The order n T-polyomino consists of a vertical line of
n— 3 squares capped by a horizontal line of three squares
centered on the line,
see also L-Polyomino, Skew Polyomino, Square
Polyomino, Straight Polyomino
T-Puzzle
The Dissection of the four pieces shown at left into the
capital letter "T" shown at right.
see also DISSECTION
References
Pappas, T. "The T Problem." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 35 and 230,
1989.
T2-Separation Axiom
Finite SUBSETS are CLOSED.
see also Closure
Tableau
see Young Tableau
Tabu Search
A heuristic procedure which has proven efficient at solv-
ing Combinatorial optimization problems.
References
Glover, F.; Taillard, E.; and De Werra, D. "A User's Guide
to Tabu Search." Ann. Oper. Res. 41, 3-28, 1993.
Piwakowski, K. "Applying Tabu Search to Determine New
Ramsey Numbers." Electronic J. Combinatorics 3, R6,
1-4, 1996. http : //www . combinatorics . org/VolumeJ/
volume3.html#R6.
A Double Point at which two Osculating Curves
are tangent. The above plot shows the tacnode of the
curve 2x 4 -3x 2 y + y 2 -2y 3 +y 4 = 0. The LINKS CURVE
also has a tacnode at the origin.
see also Acnode, Crunode, Double Point Spinode
References
Walker, R. J. Algebraic Curves. New York: Springer-Verlag,
pp. 57-58, 1978.
Tacpoint
A tangent point of two similar curves.
Tactix
see Nim
Tail Probability
Define T as the set of all points t with probabilities
P(x) such that a > t =J> P(a < x < a + da) < P
or a < t => P(a < x < a + da < Po, where Po is
a Point Probability (often, the likelihood of an ob-
served event). Then the associated tail probability is
given by J T P{x)dx,
see also P- VALUE, POINT PROBABILITY
Tait Coloring
A 3-coloring of Graph EDGES so that no two EDGES
of the same color meet at a VERTEX (Ball and Coxeter
1987, pp. 265-266).
see also Edge (Graph), Tait Cycle, Vertex
(Graph)
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, 1987.
Tait Cycle
A set of circuits going along the Edges of a Graph,
each with an EVEN number of EDGES, such that just
one of the circuits passes through each VERTEX (Ball
and Coxeter 1987, pp. 265-266).
see also EDGE (GRAPH), EULERIAN CYCLE, HAMILTON-
ian Cycle, Tait Coloring, Vertex (Graph)
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, 1987.
Tait Flyping Conjecture
see Flyping Conjecture
1780
Tait's Hamiltonian Graph Conjecture
Talisman Hexagon
Tait's Hamiltonian Graph Conjecture
Every 3-connected cubic GRAPH (each VERTEX has VA-
LENCY 3) has a HAMILTONIAN Circuit. Proposed by
Tait in 1880 and refuted by W. T. Tutte in 1946 with a
counterexample, TUTTE's GRAPH. If it had been true,
it would have implied the Four-Color Theorem. A
simpler counterexample was later given by Kozyrev and
Grinberg.
see also Hamiltonian Circuit, Tutte's Graph,
Vertex (Graph)
References
Honsberger, R. Mathematical Gems I. Washington,
Math. Assoc. Amer., pp. 82-89, 1973.
DC:
References
Gabriel, R. P. Performance and Implementation of Lisp Sys-
tems, Cambridge, MA: MIT Press, 1985.
Kmith, D. E. Textbook Examples of Recursion. Preprint
1990.
Vardi, I. "The Running Time of TAK." Ch. 9 in Computa-
tional Recreations in Mathematica. Redwood City, CA:
Addison- Wesley, pp. 179-199, 1991.
Takagi Fractal Curve
see Blancmange Function
Take- Away Game
see Nim-Heap
Tait's Knot Conjectures
P. G. Tait undertook a study of Knots in response to
Kelvin's conjecture that the atoms were composed of
knotted vortex tubes of ether (Thomson 1869). He cat-
egorized Knots in terms of the number of crossings in a
plane projection. He also made some conjectures which
remained unproven until the discovery of JONES Poly-
nomials.
Tait's Flyping Conjecture states that the number of
crossings is the same for any diagram of an ALTERNAT-
ING Knot. This was proved true in 1986.
see also Alternating Knot, Flyping Conjecture,
Jones Polynomial, Knot
References
Tait, P. G. "On Knots I, II, III." Scientific Papers, Vol. 1.
London: Cambridge University Press, pp. 273-347, 1900.
Thomson, W. H. "On Vortex Motion." Trans. Roy. Soc.
Edinburgh 25, 217-260, 1869.
Takeuchi Function
see TAK Function
Talbot's Curve
A curve investigated by Talbot which is the Negative
Pedal Curve of an Ellipse with respect to its center.
It has four Cusps and two Nodes, provided the Ec-
centricity of the Ellipse is greater than l/\/2. Its
Cartesian Equation is
(a 2 + f sin 2 t) cost
x = -^- '
a
_ (a 2 -2/ 2 + / 2 sin 2 £)sin£
y _ ?
TAK Function
A Recursive Function devised by I. Takeuchi. For
Integers #, y, and z y and a function h, it is
TAK h (x,y,z)
{h(x,y,z) fovx<y
h(h(x - 1, y, z), h(y - 1, z, x), for x > y.
h(z- l,x,y))
The number of function calls Fo(a y b) required to com-
pute TAK (a, 6, 0) for a > b > is
b / x
^ a ' & > = 4 E^^H b-k )-
L- — n V /
a — b ( a + b — 2k
^ a + b-\
2k \ b-k
(Vardi 1991).
The TAK function is also connected with the Ballot
Problem (Vardi 1991).
see also Ackermann Function, Ballot Problem
where / is a constant.
References
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 157, 1967.
MacTutor History of Mathematics Archive. "Talbot's
Curve." http : //www-groups . dcs . st-and. ac .uk/ -history
/Curves/Talbots .html.
Talisman Hexagon
An (n, ft)-talisman hexagon is an arrangement of nested
hexagons containing the integers 1, 2, ... , H n = 3n(n —
Talisman Square
Tangent 1781
1) + 1, where H n is the nth Hex Number, such that
the difference between all adjacent hexagons is at least
as large as k. The hexagon illustrated above is a (3,
5)-talisman hexagon.
see also Hex Number, Magic Square, Talisman
Square
References
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 111-112, 1979,
Talisman Square
1
5
3
7
5
15
9
12
9
11
13
15
10
1
6
3
2
6
4
8
13
16
11
14
10
12
14
16
2
8
4
7
15
1
12
4
9
28
10
31
13
34
16
19
1
22
4
25
7
20
7
22
18
24
29
11
32
14
35
17
16
2
13
5
10
20
2
23
5
26
8
21
8
23
19
25
30
12
33
15
36
18
17
3
14
6
11
21
3
24
6
27
9
An n x n Array of the integers from 1 to n 2 such that
the difference between any one integer and its neighbor
(horizontally, vertically, or diagonally, without wrapping
around) is greater than or equal to some value k is called
a (n, fc)-talisman square. The above illustrations show
(4, 2)-, (4, 3)-, (5, 4)-, and (6, 8)-talisman squares.
see also Antimagic Square, Heterosquare, Magic
Square, Talisman Hexagon
References
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 110-113, 1979.
$& Weisstein, E. W. "Magic Squares." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/MagicSquares.m.
Tame Algebra
Let A denote an I
Space over R and
-algebra, so that A is a VECTOR
tame, but a 4-D 4-ASSOCIATIVE algebra and a 3-D 1-
ASSOCIATIVE algebra need not be tame. It is conjec-
tured that a 3-D 2-Associative algebra is tame, and
proven that a 3-D 3-ASSOCIATIVE algebra is tame if it
possesses a multiplicative IDENTITY ELEMENT.
References
Finch, S. "Zero Structures in Real Algebras." http: //www.
mathsof t . com/ asolve/zerodiv/zerodiv. html.
Tame Knot
A Knot equivalent to a Polygonal Knot.
which are not tame are called Wild Knots.
Knots
References
Rolfsen, D. Knots and Links.
Perish Press, p. 49, 1976.
Wilmington, DE: Publish or
Tangency Theorem
The external (internal) SIMILARITY POINT of two fixed
Circles is the point at which all the Circles homoge-
neously (nonhomogeneously) tangent to the fixed CIR-
CLES have the same POWER and at which all the tan-
gency secants intersect.
References
Dorrie, H. 100 Great Problems of Elementary Mathematics:
Their History and Solutions. New York: Dover, p. 157,
1965.
Tangent
A x A -> A
(x,y) *-+x-y,
where x • y is vector multiplication which is assumed to
be Bilinear. Now define
Z = {x € a : x ■ y = for some nonzero y € A},
where € Z. A is said to be tame if Z is a finite union
of SUBSPACES of A. A 2-D 0-Associative algebra is
The tangent function is defined by
tan# :
sin0
cos#'
(i)
where since is the Sine function and cosz is the Cosine
function. The word "tangent," however, also has an
important related meaning as a LINE or PLANE which
touches a given curve or solid at a single point. These
geometrical objects are then called a Tangent Line or
Tangent Plane, respectively.
1782 Tangent Bifurcation
The Maclaurin Series for the tangent function is
(-l) n - 1 2 2n (2 2n -l)B 2n
tancc
£
(2n)!
x 2n ~ 1 + ...
X + 3 X + \f, X + 315 X + 283S 3: +-■■! ( 2 )
where B n is a Bernoulli Number.
tan x is IRRATIONAL for any RATIONAL x ^ 0, which can
be proved by writing tan x as a Continued Fraction
tana; =
(3)
7-,
Lambert derived another CONTINUED FRACTION ex-
pression for the tangent,
tanx =
1
1
X
3
X
1
5 1
x 7
x
(4)
An interesting identity involving the PRODUCT of tan-
gents is
L(n-l)/2j
kn \ _ j y/n for n odd
n -(?)-{r i:
even,
(5)
where [x\ is the FLOOR FUNCTION. Another tangent
identity is
tanfntan 1 x) = -±- r^ 7- —r- (6)
(Beeler e£ al. 1972, Item 16).
see also Alternating Permutation, Cosine, Co-
tangent, Inverse Tangent, Morrie's Law, Sine,
Tangent Line, Tangent Plane
References
Abramowitz, M. and Stegun, C. A, (Eds.). "Circular Func-
tions." §4.3 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Spanier, J. and Oldham, K. B. "The Tangent tan (x) and
Cotangent cot(z) Functions." Ch. 34 in An Atlas of Func-
tions. Washington, DC: Hemisphere, pp. 319-330, 1987.
Tangent Line
Tangent Bundle
The tangent bundle TM of a SMOOTH MANIFOLD M
is the Space of Tangent Vectors to points in the
manifold, i.e., it is the set (x,v) where x € M and v is
tangent to x 6 M. For example, the tangent bundle to
the Circle is the Cylinder.
see also COTANGENT BUNDLE, TANGENT VECTOR
Tangent Developable
A Ruled Surface M is a tangent developable of a
curve y if M can be parameterized by x(zt, v) = y(u) -\-
vy'(u). A tangent developable is a Flat Surface.
see also Binormal Developable, Normal Devel-
opable
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 341-343, 1993.
Tangent Hyperbolas Method
see Halley's Method
Tangent Indicatrix
Let the Speed <t of a closed curve on the unit sphere
S 2 never vanish. Then the tangent indicatrix
T =
is another closed curve on S 2 . It is sometimes called the
Tantrix. If cr Immerses in S 2 , then so will r.
References
Solomon, B. "Tantrices of Spherical Curves." Amer. Math.
Monthly 103, 30-39, 1996.
Tangent Line
tangent line
A tangent line is a Line which meets a given curve at a
single Point.
see also Circle Tangents, Secant Line, Tangent,
Tangent Plane, Tangent Space, Tangent Vector
References
Yates, R. C. "Instantaneous Center of Rotation and the Con-
struction of Some Tangents." A Handbook on Curves and
Their Properties. Ann Arbor, Ml: J. W. Edwards, pp. 119-
122, 1952.
Tangent Bifurcation
see Fold Bifurcation
Tangent Map
Tangent Vector 1783
Tangent Map
If / : M — > iV, then the tangent map Tf associated to
/ is a Vector Bundle Homeomorphism Tf : TM ->
TN (i.e., a Map between the Tangent Bundles of M
and N respectively). The tangent map corresponds to
Differentiation by the formula
T/(u) = (/o0)'(O),
(1)
where <^>'(0) = v (i.e., <fi is a curve passing through the
base point to v in TM at time with velocity v). In
this case, if / : M -> N and g : N -* O, then the Chain
Rule is expressed as
T(fog)=TfoTg.
(2)
In other words, with this way of formalizing differenti-
ation, the Chain Rule can be remembered by saying
that "the process of taking the tangent map of a map is
functorial." To a topologist, the form
(f°9Y{a) = f'{g(a))og'(a) 1
(3)
for all a, is more intuitive than the usual form of the
Chain Rule.
see also DlFFEOMORPHISM
References
Gray, A. "Tangent Maps." §9.3 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 168-171, 1993.
Tangent Number
A number also called a Zag Number giving the number
of Even Alternating Permutations. The first few
are 1, 2, 16, 272, 7936, . . . (Sloane's A000182).
see also ALTERNATING PERMUTATION, EULER ZIGZAG
Number, Secant Number
References
Knuth, D. E. and Buckholtz, T. J. "Computation of Tangent,
Euler, and Bernoulli Numbers." Math. Comput. 21, 663-
688, 1967.
Sloane, N. J. A. Sequence A000182/M2096 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Tangent Plane
A tangent plane is a Plane which meets a given SUR-
FACE at a single Point. Let (xo,yo) be any point of a
surface function z = f{x,y). The surface has a nonver-
tical tangent plane at (#o,yo) with equation
z = /(so, 3/o ) + fx{xo,yo){x - xq) 4- f y (xo i yo)(y - y ).
see also Normal Vector, Tangent, Tangent Line,
Tangent Space, Tangent Vector
Tangent Space
Let x be a point in an n-dimensional COMPACT MANI-
FOLD M, and attach at x a copy of R n tangential to M.
The resulting structure is called the TANGENT SPACE
of M at x and is denoted T X M. If 7 is a smooth curve
passing through x, then the derivative of 7 at x is a
Vector in T X M.
see also TANGENT, TANGENT BUNDLE, TANGENT
Plane, Tangent Vector
Tangent Vector
For a curve with POSITION VECTOR r(£), the unit tan-
gent vector T(t) is defined by
*(*) = ^7 = T3T7 (!)
(2)
(3)
where f is a parameterization variable and s is the
Arc Length. For a function given parametrically
by (/(*)»<?(*))> the tangent vector relative to the point
(f{t),g(t)) is therefore given by
p'(«)
dv
dt
Ir'MI
Idr
| dt
dr
dt
ds
dt
dr
ds }
x(t) =
»(*) =
/'
Vf' 2 +9' 2
9'
y/f' 2 +9' 2 '
(4)
(5)
To actually place the vector tangent to the curve, it must
be displaced by (f(t),g(t)). It is also true that
dT
ds
acN
dT _ ds*
~dt~ K Tt
[T,T,T] = k 5 ^
G)
(6)
(7)
(8)
where N is the Normal Vector, k is the Curvature,
and r is the TORSION.
see also CURVATURE, NORMAL VECTOR, TANGENT,
Tangent Bundle, Tangent Plane, Tangent
Space, Torsion (Differential Geometry)
References
Gray, A. "Tangent and Normal Lines to Plane Curves." §5.5
in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 85-90, 1993.
1784 Tangential Angle
Tangle
Tangential Angle
For a PLANE Curve, the tangential angle <f> is defined
by
pd<p — ds, (1)
where s is the Arc Length and p is the Radius of
Curvature. The tangential angle is therefore given by
Jo
(t) dt,
(2)
where K,(t) is the CURVATURE. For a plane curve r(£),
the tangential angle <f>(t) can also be defined by
|f(*)l
cos[<j>(t)]
sin[<P(t)}
(3)
Gray (1993) calls <f> the Turning Angle instead of the
tangential angle.
see also Arc Length, Curvature, Plane Curve, Ra-
dius of Curvature, Torsion (Differential Geom-
etry)
References
Gray, A. "The Turning Angle." §1.6 in Modern Differential
Geometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 13-14, 1993.
Tangential Triangle
The Triangle A7\ T 2 T 3 formed by the lines tangent to
the ClRCUMClRCLE of a given Triangle AAiA 2 A 3 at
its Vertices. It is the Pedal Triangle of AAiA 2 A 3
with the Circumcenter as the Pedal Point. The
Trilinear Coordinates of the Vertices of the tan-
gential triangle are
A — —a : b : c
B = a : — b : c
C' = a : b : — c.
The Contact Triangle and tangential triangle are
perspective from the Gergonne Point.
see also Circumcircle, Contact Triangle, Ger-
gonne Point, Pedal Triangle, Perspective
Tangential Triangle Circumcenter
A Point with Triangle Center Function
a = a[b 2 cos(2 J B) + c cos(2C) - a 2 cos(2A)].
It lies on the Euler LINE.
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Tangents Law
see Law of Tangents
Tangle
A region in a Knot or LINK projection plane surrounded
by a Circle such that the Knot or Link crosses the
circle exactly four times. Two tangles are equivalent if a
sequence of REIDEMEISTER MOVES can be used to trans-
form one into the other while keeping the four string
endpoints fixed and not allowing strings to pass outside
the Circle.
The simplest tangles are the oo- tangle and 0- tangle,
shown above. A tangle with n left-handed twists is
called an n-tangle, and one with n right-handed twists
is called a —n-tangle. By placing tangles side by side,
more complicated tangles can be built up such as ( — 2, 3,
2), etc. The link created by connecting the ends of the
tangles is now described by the sequence of tangle sym-
bols, known as Conway's KNOT NOTATION. If tangles
are multiplied by and then added, the resulting tangle
symbols are separated by commas. Additional symbols
which are used are the period, colon, and asterisk.
Amazingly enough, two tangles described in this NOTA-
TION are equivalent Iff the Continued Fractions of
the form
1
2 +
1
3+
-2
are equal (Burde and Zieschang 1985)! An ALGEBRAIC
TANGLE is any tangle obtained by ADDITIONS and MUL-
TIPLICATIONS of rational tangles (Adams 1994). Not all
tangles are ALGEBRAIC.
see also Algebraic Link, Flype, Pretzel Knot
References
Adams, C C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman pp. 41-51, 1994.
Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter,
1985.
Tanglecube
Tanglecube
Taniyama-Shimura Conjecture
1785
A QUARTIC SURFACE given by the implicit equation
x 4 - 5rr 2 + y 4 - by 2 + z 4 - 5z 2 + 11.8 = 0.
References
Banchoff, T. "The Best Homework Ever?" http:// www .
brown . edu / Administration / Brown _ Alumni _ Monthly/
12-96/f eatures/homework.html.
Nordstrand, T. "Tangle." http : //www . uib . no/people/
nf ytn/tangltxt .htm.
Tangled Hierarchy
A system in which a Strange Loop appears.
see also STRANGE LOOP
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, p. 10, 1989.
Tangram
A combination of the above plane polygonal pieces such
that the Edges are coincident. There are 13 convex
tangrams (where a "convex tangram" is a set of tangram
pieces arranged into a Convex Polygon).
see also ORIGAMI, STOMACHION
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 19-20, 1989.
Gardner, M. "Tangrams, Parts 1 and 2." Ch. 3-4 in Time
Travel and Other Mathematical Bewilderments. New
York: W. H. Freeman, 1988.
Johnston, S. Fun with Tangrams Kit: 120 Puzzles with Two
Complete Sets of Tangram Pieces. New York: Dover, 1977.
Pappas, T. "Tangram Puzzle." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, p. 212, 1989.
Tanh
see Hyperbolic Tangent
Taniyama Conjecture
see Taniyama-Shimura Conjecture
Taniyama-Shimura Conjecture
A conjecture which arose from several problems pro-
posed by Taniyama in an international mathematics
symposium in 1955. Let E be an ELLIPTIC CURVE
whose equation has INTEGER COEFFICIENTS, let A^ be
the CONDUCTOR of E and, for each n, let a n be the num-
ber appearing in the L-function of E. Then there exists
a Modular Form of weight two and level N which is
an eigenform under the HECKE OPERATORS and has a
Fourier Series ^a n g n .
The conjecture says, in effect, that every rational ELLIP-
TIC Curve is a Modular Form in disguise. Stated for-
mally, the conjecture suggests that, for every ELLIPTIC
Curve y 2 = Ax 3 + Bx 2 + Cx + D over the Rationals,
there exist nonconstant MODULAR FUNCTIONS f{z) and
g(z) of the same level N such that
[f(z)] 2 =A[g(z)} 2 + Cg(z)+D.
Equivalently, for every ELLIPTIC Curve, there is a
Modular Form with the same Dirichlet L-Series.
In 1985, starting with a fictitious solution to Fermat's
Last Theorem, G. Prey showed that he could create
an unusual ELLIPTIC CURVE which appeared not to be
modular. If the curve were not modular, then this would
show that if Fermat's Last Theorem were false, then
the Taniyama-Shimura conjecture would also be false.
Furthermore, if the Taniyama-Shimura conjecture were
true, then so would be Fermat's Last Theorem!
However, Frey did not actually prove whether his curve
was modular. The conjecture that Prey's curve was
modular came to be called the "epsilon conjecture," and
was quickly proved by Ribet (Ribet's Theorem) in
1986, establishing a very close link between two math-
ematical structures (the Taniyama-Shimura conjecture
and Fermat's Last Theorem) which appeared previ-
ously to be completely unrelated.
As of the early 1990s, most mathematicians believed
that the Taniyama-Shimura conjecture was not accessi-
ble to proof. However, A. Wiles was not one of these. He
attempted to establish the correspondence between the
set of Elliptic Curves and the set of modular elliptic
curves by showing that the number of each was the same.
Wiles accomplished this by "counting" Galois represen-
tations and comparing them with the number of mod-
ular forms. In 1993, after a monumental seven-year ef-
fort, Wiles (almost) proved the Taniyama-Shimura con-
jecture for special classes of curves called SEMISTABLE
Elliptic Curves.
1786
Tank
Tan Conjectnre
Wiles had tried to use horizontal Iwasawa theory to cre-
ate a so-called CLASS NUMBER formula, but was initially
unsuccessful and therefore used instead an extension of
a result of Flach based on ideas from Kolyvagin. How-
ever, there was a problem with this extension which
was discovered during review of Wiles' manuscript in
September 1993. Former student Richard Taylor came
to Princeton in early 1994 to help Wiles patch up this
error. After additional effort, Wiles discovered the rea-
son that the Flach/Kolyvagin approach was failing, and
also discovered that it was precisely what had prevented
Iwasawa theory from working.
With this additional insight, he was able to success-
fully complete the erroneous portion of the proof us-
ing Iwasawa theory, proving the SEMISTABLE case of the
Taniyama-Shimura conjecture (Taylor and Wiles 1995,
Wiles 1995) and, at the same time, establishing FER-
MAT'S Last THEOREM as a true theorem.
see also Elliptic Curve, Fermat's Last Theorem,
Modular Form, Modular Function, Ribet's The-
orem
References
Lang, S. "Some History of the Shimura-Taniyama Conjec-
ture." Not. Amer. Math. Soc. 42, 1301-1307, 1995.
Taylor, R. and Wiles, A. "Ring-Theoretic Properties of Cer-
tain Hecke Algebras," Ann. Math. 141, 553-572, 1995.
Wiles, A. "Modular Elliptic-Curves and Fermat's Last The-
orem." Ann. Math. 141, 443-551, 1995.
Tank
see Cylindrical Segment
Tantrix
see Tangent Indicatrix
Tapering Function
see Apodization Function
Tarry-Escott Problem
For each POSITIVE INTEGER /, there exists a POSITIVE
Integer n and a Partition of {1, . . . , n} as a disjoint
union of two sets A and B, such that for 1 < i < I,
X>' = £»'-
aeA
b£B
The results extended to three or more sets of Integers
are called Prouhet's Problem.
see also Prouhet's Problem
References
Dickson, L. E. History of the Theory of Numbers, Vol 2:
Diophantine Analysis. New York: Chelsea, pp. 709-710,
1971.
Hahn, L. "The Tarry-Escott Problem." Problem 10284.
Amer. Math. Monthly 102, 843-844, 1995.
Tarry Point
The point at which the lines through the VERTICES of a
Triangle Perpendicular to the corresponding sides
of the first BROCARD TRIANGLE, are CONCURRENT.
The Tarry point lies on the ClRCUMClRCLE opposite the
Steiner Point. It has Triangle Center Function
be
6 4 + c 4 - a 2 b 2 - a 2 c 2
= sec(A + w),
where oj is the Brocard Angle. The Simson Line
of the Tarry point is PERPENDICULAR to the line OK,
when O is the ClRCUMCENTER and K is the LEMOINE
Point.
see also Brocard Angle, Brocard Triangles, Cir-
cumcircle, Lemoine Point, Simson Line, Steiner
Points
References
Gallatly, W. The Modern Geometry of the Triangle, 2nd ed.
London: Hodgson, p. 102, 1913.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 281-282, 1929.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Tarski's Theorem
Tarski's theorem states that the first-order theory of the
Field of Real Numbers is Decidable. However, the
best-known ALGORITHM for eliminating QUANTIFIERS
is doubly exponential in the number of QUANTIFIER
blocks (Heintz et al 1989).
References
Heintz, J.; Roy, R.-F.; and Solerno, P. "Complexite du
principe de Tarski-Seidenberg." C. R. Acad. Sci. Paris
Ser. I Math. 309, 825-830, 1989.
Marker, D. "Model Theory and Exponentiation." Not.
Amer. Math. Soc. 43, 753-759, 1996.
Tarski, A. "Sur les ensembles definissables de nombres reels."
Fund. Math. 17, 210-239, 1931.
Tarski, A. "A Decision Method for Elementary Algebra and
Geometry." RAND Corp. monograph, 1948.
Tau Conjecture
Also known as Ramanujan'S HYPOTHESIS. Ramanujan
proposed that
r(n)~0(n 11/2+e ),
where r(n) is the Tau Function, defined by
oo
]P r(n)x n = x(l - 3x + 5z 3 - 7x G + . . .) 8 .
n=l
This was proven by Deligne (1974), who was subse-
quently awarded the FIELDS MEDAL for his proof.
see also Tau Function
References
Deligne, P. "La conjecture de Weil. I." Inst. Hautes Etudes
Sci. PubL Math. 43, 273-307, 1974.
Tau-Dirichlet Series
Tau Function 1787
Deligne, P. "La conjecture de Weil. II." Inst. Hautes Etudes
Sci. Publ. Math. 52, 137-252, 1980.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
p. 169, 1959.
then
Tau-Dirichlet Series
r(n)
where r(n) is the TAU FUNCTION. Ramanujan conjec-
tured that all nontrivial zeros of f(z) lie on the line
R[s] = 6, where
f( s ) = X^ r ( n ) n ~
and r(n) is the TAU FUNCTION.
see also Tau Function
References
Spira, R. "Calculation of the Ramanujan Tau-Dirichlet Se-
ries." Math. Comput. 27, 379-385, 1973.
Yoshida, H. "On Calculations of Zeros of L-Functions Related
with Ramanujan's Discriminant Function on the Critical
Line." X Ramanujan Math. Soc. 3, 87-95, 1988.
Tau Function
A function r(n) related to the Divisor Function
<j fc (n), also sometimes called Ramanujan's Tau Func-
tion. It is given by the Generating Function
gr(n)x» = jja-x") 24 ,
(1)
and the first few values are 1, -24, 252, -1472, 4380,
. . . (Sloane's A000594). r(n) is also given by
9(-x) = ^(-l)"r(n)a
ff(* 2 ) = $>(±n)* B
71 = 1
(2)
(3)
Y^ r{n)x n = x(l - 3x + 5x 3 - 7x 6 + . . .) 8 - (4)
T(pn) = (mod p) .
(7)
Values of p for which the first equation holds are p = 2,
3, 5, 7, 23.
Ramanujan also studied
oo
f( x ) = ^2 T ( n ) n ~ 3 <
(8)
which has properties analogous to the RlEMANN Zeta
Function. It satisfies
/(*)!» = /(12-*)
(2tt)* (2tt) 12 - 3 '
(9)
and Ramanujan's Tau-Dirichlet Series conjecture al-
leges that all nontrivial zeros of f(s) lie on the line
SR[s] = 6. / can be split up into
f(6 + it)=z(t)e- i6it \
where
z(t) = r(6 + it)f(6 + i*)(2?r)" i *
(10)
sinh(7rt)
7rf(l + t 2 )(4 + t 2 )(9 + t 2 )(16 + £ 2 )(25 + t 2 )
0{t) = -\i\n
r(6 + zt)'
r(6 - it)
■tln(27r).
(11)
(12)
The Summatory tau function is given by
T(n) = J^r(n). (13)
Here, the prime indicates that when x is an INTEGER,
the last term r(x) should be replaced by \r(x).
Ramanujan's tau theta function Z(t) is a REAL function
for Real t and is analogous to the Riemann-Siegel
Function Z. The number of zeros in the critical strip
from t — to T is given by
N{t) = e(T) + <s{ln[T DS (6 + iT)]} ^
In Ore's Conjecture, the tau function appears as the
number of DIVISORS of n. Ramanujan conjectured and
Mordell proved that if {n,n'), then
r(nn) = r(n)r(n).
(5)
Ramanujan conjectured and Watson proved that r(n) is
divisible by 691 for almost all n. If
where is the RlEMANN THETA FUNCTION and r D s is
the Tau-Dirichlet Series, defined by
r(n)
(15)
Ramanujan conjectured that the nontrivial zeros of the
function are all real.
r(p) = (mod p) ,
(6)
1788
Tauberian Theorem
Ramanujan's r z function is denned by
r(6 + it)(27r)~ it
r z (t) =
tds{6 + it)
smh(7r£)
"T3 —
(16)
"nLx fca+ta
where r D s(z) is the Tau-Dirichlet Series.
see also Ore's Conjecture, Tau Conjecture, Tau-
Dirichlet Series
References
Hardy, G. H. "Ramanujan's Function r(n)." Ch. 10 in Ra-
manujan: Twelve Lectures on Subjects Suggested by His
Life and Work, 3rd ed. New York: Chelsea, 1959.
Sloane, N. J. A. Sequence A000594/M5153 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Tauberian Theorem
A Tauberian theorem is a theorem which deduces the
convergence of an Infinite Series on the basis of the
properties of the function it defines and any kind of aux-
iliary HYPOTHESIS which prevents the general term of
the series from converging to zero too slowly.
see also HARDY-LlTTLEWOOD TAUBERIAN THEOREM
Tautochrone Problem
Find the curve down which a bead placed anywhere will
fall to the bottom in the same amount of time. The solu-
tion is a CYCLOID, a fact first discovered and published
by Huygens in Horologium oscillatorium (1673). Huy-
gens also constructed the first pendulum clock with a
device to ensure that the pendulum was isochronous by
forcing the pendulum to swing in an arc of a CYCLOID.
The parametric equations of the CYCLOID are
x — a(9 — sinO)
y = a(l — cos#).
(i)
(2)
To see that the Cycloid satisfies the tautochrone prop-
erty, consider the derivatives
and
x — a{\ — cos0)
y = asin#,
1 + y ' 2 = a 2 [(1 - 2 cos 6 + cos 2 6) + sin 2 6]
(3)
(4)
Now
= 2a 2 (l-cos0).
1 2
^mv = mgy
(5)
(6)
(7)
Tautology
dt =
ds _ y/dx 1 + dy 2
nm
a^2(l- cos 0)d0 _ fE
y/2ga{l - cosB) \ 9
(8)
so the time required to travel from the top of the Cy-
cloid to the bottom is
-£-£'■
However, from an intermediate point #o>
ds
v = — = y/2g(y-yo),
(9)
(10)
yj2a 2 (l-cos6)
2ag(cos0o — cosO)
dO
cos 6
cos #o - cos
sin(\0)d6
dd
9o vW^-cos 2 ^)
Now let
du ■
cos(±0)
cos(^o)
sm(±0)d9
~ 2cos(6> ) '
(11)
(12)
(13)
T= -2
a r ■ -1 il a
sin u\o = ir * -
9~ V 9
(14)
and the amount of time is the same from any point!
see also Brachistochrone Problem, Cycloid
References
Muterspaugh, J.; Driver, T.; and Dick, J. E. "The Cycloid
and Tautochronism." http : //ezinf o . ucs . indiana.edu/
-jedick/project/intro.html.
Muterspaugh, J.; Driver, T.; and Dick, J. E. "P221 Tau-
tochrone Problem." http : //ezinf o . ucs . indiana . edu/
~jedick/project /project, html.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 54-60 and 384-385, 1991.
Tautology
A logical statement in which the conclusion is equivalent
to the premise. If p is a tautology, it is written \= p.
Taxicab Number
Taylor Circle 1789
Taxicab Number
The nth taxicab number Ta(n) is the smallest num-
ber representable in n ways as a sum of POSITIVE
Cubes. The numbers derive their name from the
Hardy-Ramanujan Number
Ta(2) = 1729
= I 3 + 12 3
= 9 3 + 10 3 ,
(1)
which is associated with the following story told about
Ramanujan by G. H. Hardy. "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 de-
clared, '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).
However, this property was also known as early as 1657
by F. de Bessy (Berndt and Bhargava 1993, Guy 1994).
Leech (1957) found
Ta(3) = 87539319
= 167 3 + 436 3
= 228 3 + 423 3
= 255 3 +414 3 .
(2)
Rosenstiel et ah (1991) recently found
Ta(4) = 6963472309248
= 2421 3 + 19083 3
= 5436 3 + 18948 3
= 10200 3 + 18072 3
= 13322 3 + 16630 3 .
D. Wilson found
Ta(5) = 48988659276962496
= 38787 3 + 3657S7 3
= 107839 3 + 362753 s
= 205292 3 + 342952 3
= 221424 3 + 336588 s
= 231518 3 + 331954 3 .
(3)
(4)
The first few taxicab numbers are therefore 2, 1729,
87539319, 6963472309248, ... (Sloane's A011541).
Hardy and Wright (Theorem 412, 1979) show that the
number of such sums can be made arbitrarily large but,
updating Guy (1994) with Wilson's result, the least ex-
ample is not known for six or more equal sums.
Sloane defines a slightly different type of taxicab num-
bers, namely numbers which are sums of two cubes in
two or more ways, the first few of which are 1729, 4104,
13832, 20683, 32832, 39312, 40033, 46683, 64232, ...
(Sloane's A001235).
see also Diophantine Equation — Cubic, Hardy-
Ramanujan Number
References
Berndt, B. C. and Bhargava, S. "Ramanujan — For Low-
brows." Am. Math. Monthly 100, 645-656, 1993.
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.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.
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.
Leech, J. "Some Solutions of Diophantine Equations." Proc.
Cambridge Phil Soc. 53, 778-780, 1957.
Plouffe, S. "Taxicab Numbers." http://www.lacim.uqam.
ca/pi/problem.html.
Rosenstiel, E.; Dardis, J. A.; and Rosenstiel, C. R. "The
Four Least Solutions in Distinct Positive Integers of the
Diophantine Equation s — x 3 + y z = z 3 + w z = u 3 + v 3 =
m 3 +n 3 ." Bull. Inst. Math. Appl. 27, 155-157, 1991.
Silverman, J. H. "Taxicabs and Sums of Two Cubes." Amer.
Math. Monthly 100, 331-340, 1993.
Sloane, N. J. A. Sequences A001235 and A011541 in "An On-
Line Version of the Encyclopedia of Integer Sequences."
Snow, C. P. Foreword to A Mathematician's Apology,
reprinted with a foreword by C. P. Snow (by G. H. Hardy).
New York: Cambridge University Press, p. 37, 1993.
Wooley, T. D. "Sums of Two Cubes." Intemat. Math. Res.
Not., 181-184, 1995.
Taylor Center
The center of the TAYLOR CIRCLE, which is the Spieker
Center of Ai7ii^ 2 #3, where Hi are the Altitudes.
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 277, 1929.
Taylor Circle
Prom the feet of each Altitude of a Triangle, draw
lines Perpendicular to the adjacent sides. Then the
feet of these perpendiculars lie on a CIRCLE called the
Taylor Circle.
see also TUCKER CIRCLES
References
Johnson, R, A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, p. 277, 1929.
1790 Taylor's Condition
Taylor's Condition
n=2
n=3
n=4
n=6
For a given POSITIVE INTEGER n, does there exist a
Weighted Tree with n Vertices whose paths have
weights 1, 2, ... , ( £ ) , where (™) is a Binomial COEF-
FICIENT? Taylor showed that no such TREE can exist
unless it is a Perfect SQUARE or a Perfect SQUARE
plus 2. No such TREES are known except n = 2, 3, 4,
and 6.
see also Golomb Ruler, Perfect Difference Set
References
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 56-60, 1985.
Leech, J. "Another Tree Labeling Problem." Amer. Math.
Monthly 82, 923-925, 1975.
Taylor, H. "Odd Path Sums in an Edge-Labeled Tree." Math.
Mag. 50, 258-259, 1977.
Taylor Expansion
see Taylor Series
Taylor-Greene-Chirikov Map
see Standard Map
Taylor Polynomial
see Taylor Series
Taylor Series
A Taylor series is a series expansion of a FUNCTION
about a point. A 1-D Taylor series is an expansion of a
Scalar Function /(x) about a point x = a. If a ~ 0,
the expansion is known as a Maclaurin Series.
a
(i)
1 \ f (n) (x)dx dx= J [f^- 1) (x)-f^ n - 1) (a)}dx
= / ( - 2) (x) - / ( - 2) (a) -(x- <*)/<"-%). (2)
Continuing,
///" f{n){x) {dxf = f(n3){a) ~ {x ~ a)/(n_2)(a)
-^(x-aff^ia) (3)
/■/>
Taylor Series
(x)(dx) n = f(x)-f(a)-(x-a)f'(a)
4(x- S ) 2 /"(«)-...-^(x-ar I / ( "" 1, (a). (4)
Therefore, we obtain the 1-D Taylor series
f{x) = f(a) + (x- a)f{a) + ±(x - off" (a) + ...
+ T ^y.(x-ar- 1 f^- 1 \a)+R n , (5)
where R n is a remainder term defined by
R --I-£'" )
(x) (dx) n .
(6)
Using the Mean- Value Theorem for a function g, it
must be true that
J a
g(x) dx — (x — a)g(x*)
(7)
for some x* 6 [a, x}> Therefore, integrating n times gives
the result
R n —
(x - a) n f(n)
n!
f M (x).
(8)
The maximum error is then the maximum value of (8)
for all possible x* 6 [a, x].
An alternative form of the 1-D Taylor series may be
obtained by letting
— a = Ax
so that
x = a + Ax = xq + Ax.
Substitute this result into (5) to give
/(so + Ax) = f( Xo ) + Ax/'(x ) + ±(Ax) 2 f"(x ) +
(9)
(10)
(11)
A Taylor series of a Function in two variables /(x,y)
is given by
/(x + Ax,y + Ay) - /(x, y) + [f x (x } y)Ax + f v (x y y)Ay]
+ ^N 2 /-(^ V) + 2AxAyf xy (x, y) + (Ay) 2 f yy (x, y)}
+ ±[(Ax) z f xxx {x,y) + 3(Ax) 2 Ayf xxy (x,y)
+3Ax(Ay) 2 f xyy (x,y) + {Ayff yyy {x, y)] + . . . . (12)
This can be further generalized for a FUNCTION in n
variables,
f(x 1 , . . . ,x n )
■eu
Z) (il -° 4) ^r
/«-■
,«D
(13)
Taylor Series
Rewriting,
f(xr +ai,...,x n + a„)
=EUE^ ^ <)
j = Q \ \k = l
Taking n = 2 in (13) gives
(14)
f(xi,X2
7 = ^
+( ^- a2) a^
/(a:i,a?2) ^
= Xl,X 2 =X2
= /(ai,a 2 ) +
1
+
2!
x 0/ / x 0/ 1
(xi " ai) ^7 + 2{ " 1 - ai)(a?2 ~ a2) ^ 1 ^ 2
+ (X 2 - 2 )
2 9V
dx2 2
+ .... (15)
Taking n = 3 in (14) gives
f(xi + cii, X? + a 2 ,x 3 + a 3 )
^fifd d d y
xf{x lt x 2 ,x 3 )
(16)
x 1 —x\,x 2 = X2,x A ~ £3
or, in Vector form
00
i(a.V r -)V(r')
(17)
r'=r
The zeroth- and first-order terms are
/(r)
(18)
and
(a-
V r 0/(r')|,'= r ,
(19)
respectively. The second-order term is
i(a-V r 0(a-V P 0/(r')|,'=r
= §a-V r ,[a-(V/(r'))] r , =r
= ia.[a.V r .(V r //(r'))]| r , =r , (20)
so the first few terms of the expansion are
/(r + a) = /(r) + (a-V r .)/(r')|,'=r
+ ia.[a-V r ,(V r -/(r'))]| r , =r . (21)
Tchebycheff 1791
Taylor series can also be defined for functions of a Com-
plex variable. By the Cauchy Integral Formula,
JK ' 2iriJ c z'-z 2TriJ c (z'
= J_ f f(z')dz'
2« Jc (*' - *o) (1 - jf^) '
f{z')dz'
In the interior of C,
\z-zp\
\z f -z \
< 1
■ z ) - (z - zo)
(22)
(23)
so, using
it follows that
m =
1 °°
1^7 =E<".
(24)
71 =
(z-zo) n f(z')dz'
{z 1 - z Q ) n + x
1 r °°
7^1 J 1^
JC n =
)dz
(25)
Using the the Cauchy Integral FORMULA for deriva-
tives,
J (n) (z )
/(*) = £(,-*)»:?_£
(26)
see also CAUCHY REMAINDER FORM, LAGRANGE EX-
PANSION, Laurent Series, Legendre Series, Mac-
laurin Series, Newton's Forward 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,
p. 880, 1972.
Arfken, G, "Taylor's Expansion." §5.6 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic
Press, pp. 303-313, 1985.
Morse, P. M. and Feshbach, H. "Derivatives of Analytic Func-
tions, Taylor and Laurent Series." §4.3 in Methods of The-
oretical Physics, Part I. New York: McGraw-Hill, pp. 374-
398, 1953.
Tchebycheff
see Chebyshev Approximation Formula, Cheby-
shev Constants, Chebyshev Deviation, Cheby-
shev Differential Equation, Chebyshev Func-
tion, Chebyshev-Gauss Quadrature, Cheby-
shev Inequality, Chebyshev Inequality, Cheby-
shev Integral, Chebyshev Phenomenon, Cheby-
shev Polynomial of the First Kind, Cheby-
shev Polynomial of the Second Kind, Cheby-
shev Quadrature, Chebyshev- Radau Quadra-
ture, Chebyshev-Sylvester Constant
1792 Teardrop Curve
Tennis Ball Theorem
Teardrop Curve
A plane curve given by the parametric equations
x = cos t
y = sinisin m (|t).
see also Pear-Shaped Curve
References
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 174, 1993.
Technique
A specific method of performing an operation. The
terms Algorithm, Method, and Procedure are also
used interchangeably.
see also ALGORITHM, METHOD, PROCEDURE
Teichmiiller Space
Teichmuller's Theorem asserts the Existence and
Uniqueness of the extremal quasiconformal map be-
tween two compact Riemann Surfaces of the same
Genus modulo an Equivalence Relation. The
equivalence classes form the Teichmiiller space T p of
compact Riemann Surfaces of Genus p.
see also Riemann's Moduli Problem
Teichmuller's Theorem
Asserts the Existence and Uniqueness of the ex-
tremal quasiconformal map between two compact RIE-
MANN Surfaces of the same Genus modulo an Equiv-
alence Relation.
see also Teichmuller Space
Telescoping Sum
A sum in which subsequent terms cancel each other,
leaving only initial and final terms. For example,
Temperature
The "temperature" of a curve V is defined as
n — 1 / \
*-~< V ai di+x }
= ( 1 --) + (---) +
\a\ a2 / Vct2 as/
, 0>n-2 Q>n-1
a\ a n
1 \ / 1
+
&n-l Q>n
T =
1
^ (sfe) '
where / is the length of T and h is the length of the
Perimeter of the Convex Hull. The temperature
of a curve is only if the curve is a straight line, and
increases as the curve becomes more "wiggly."
see also CURLICUE FRACTAL
References
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
pp. 164-165, 1995.
Templar Magic Square
s
A
T
R
A
R
E
P
T
E
N
E
T
P
E
R
A
R
T
A
S
is a telescoping sum.
see also Zeilberger's Algorithm
A MAGIC SQUARE-type arrangement of the words in the
Latin sentence "Sator Arepo tenet opera rotas" ("the
farmer Arepo keeps the world rolling"). This square has
been found in excavations of ancient Pompeii.
see also MAGIC SQUARE
References
Bouisson, S. M. La Magie: Ses Grands Rites, Son Histoire.
Paris, pp. 147-148, 1958.
Grosser, F. "Ein neuer Vorschlag zur Deutung der Sator-
Formel." Archiv. f. Relig. 29, 165-169, 1926.
Heietala, H. "The Templar Magic Square." http://www.
trantex.fi/staff/heikkih/knights/pubsator.htm.
Hocke, G. R. Manierismus in der Literatur: Sprach-Alchimie
und esoterische Kombinationskunst. Hamburg, Germany:
Rowohlt, p. 24, 1967.
Tennis Ball Theorem
A closed simple smooth spherical curve dividing the
SPHERE into two parts of equal areas has at least four
inflection points.
see also Ball, Baseball Cover
References
Arnold, V. I. Topological Invariants of Plane Curves and
Caustics. Providence, RI: Amer. Math. Soc, 1994.
Martinez- Maure, Y. "A Note on the Tennis Ball Theorem."
Amer. Math. Monthly 103, 338-340, 1996.
Tensor
Tensor Spherical Harmonic 1793
Tensor
An nth-RANK tensor of order m is a mathematical ob-
ject in m- dimensional space which has n indices and
m n components and obeys certain transformation rules.
Each index of a tensor ranges over the number of dimen-
sions of Space. If the components of any tensor of any
RANK vanish in one particular coordinate system, they
vanish in all coordinate systems.
Zeroth-RANK tensors are called SCALARS, and first-
RANK tensors are called VECTORS. In tensor notation,
a vector v would be written Vi, where i = 1, . . . , m.
Tensor notation can provide a very concise way of writ-
ing vector and more general identities. For example,
in tensor notation, the DOT PRODUCT u • v is simply
written
U- V = UiVi, (1)
where repeated indices are summed over (ElN STEIN
Summation) so that uiVi stands for u\v\ + . . . + UmV m -
Similarly, the CROSS PRODUCT can be concisely written
as
u x v = djku v , (2)
where e ijk is the LEVl-ClVITA TENSOR.
Second-RANK tensors resemble square MATRICES. CON-
TRAVARIANT second-RANK tensors are objects which
transform as
A ,ij = dx'j dx> J A ki^
dxk dxi
COVARIANT second-RANK tensors are objects which
transform as
(3)
Mixed second-RANK tensors are objects which trans-
form as
b'{
dx'j dxi k
dx k dx'j
(5)
If two tensors A and B have the same RANK and the
same COVARIANT and Contravariant indices, then
References
Abraham, R.; Marsden, J. E.; and Ratiu, T. S. Manifolds,
Tensor Analysis, and Applications. New York: Springer-
Verlag, 1991.
Akivis, M. A. and Goldberg, V. V. An Introduction to Linear
Algebra and Tensors. New York: Dover, 1972.
Arfken, G. "Tensor Analysis." Ch. 3 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 118-167, 1985.
Aris, R. Vectors, Tensors, and the Basic Equations of Fluid
Mechanics. New York: Dover, 1989.
Bishop, R. and Goldberg, S. Tensor Analysis on Manifolds.
New York: Dover, 1980.
Jeffreys, H. Cartesian Tensors. Cambridge, England: Cam-
bridge University Press, 1931.
Joshi, A. W. Matrices and Tensors in Physics, 3rd ed. New
York: Wiley, 1995.
Lass, H. Vector and Tensor Analysis. New York: McGraw-
Hill, 1950.
Lawden, D. F. An Introduction to Tensor Calculus, Relativ-
ity, and Cosmology, 3rd ed. Chichester, England: Wiley,
1982.
McConnell, A. J. Applications of Tensor Analysis. New
York: Dover, 1947.
Morse, P. M. and Feshbach, H. "Vector and Tensor Formal-
ism." §1.5 in Methods of Theoretical Physics, Part I. New
York: McGraw-Hill, pp. 44-54, 1953.
Simmonds, J. G. A Brief on Tensor Analysis, 2nd ed. New
York: Springer- Verlag, 1994.
Sokolnikoff, I. S. Tensor Analysis — Theory and Applications,
2nd ed. New York: Wiley, 1964.
Synge, J. L. and S child, A. Tensor Calculus. New York:
Dover, 1978.
Wrede, R. C. Introduction to Vector and Tensor Analysis.
New York: Wiley, 1963.
Tensor Calculus
The set of rules for manipulating and calculating with
Tensors.
Tensor Density
A quantity which transforms like a Tensor except for
a scalar factor of a Jacobian.
Tensor Dual
see Dual Tensor
A ij + B u = c n
A)+B}=C}.
(6)
(7)
(8)
A transformation of the variables of a tensor changes the
tensor into another whose components are linear HOMO-
GENEOUS Functions of the components of the original
tensor.
see also Antisymmetric Tensor, Curl, Diver-
gence, Gradient, Irreducible Tensor, Isotropic
Tensor, Jacobi Tensor, Ricci Tensor, Riemann
Tensor, Scalar, Symmetric Tensor, Torsion
Tensor, Vector, Weyl Tensor
Tensor Product
see Direct Product (Tensor)
Tensor Space
Let E be a linear space over a FIELD K. Then the
Direct Product <8)J=i & is cauec ^ a tensor space of
degree k.
References
Yokonuma, T. Tensor Spaces and Exterior Algebra. Provi-
dence, RJ: Amer. Math. Soc, 1992.
Tensor Spherical Harmonic
see Double Contraction Relation
1794 Tensor Transpose
Tessellation
Tensor Transpose
see Transpose
Tent Map
A piece wise linear, 1-D Map on the interval [0, 1] ex-
hibiting Chaotic dynamics and given by
Xn+l = fi(l - 2\x n - \ I).
The case fi = 1 is equivalent to the LOGISTIC EQUATION
with r = 4, so the Natural Invariant in this case is
p(x) =
1
7ry x(l — x)
see also 2x MOD 1 MAP, LOGISTIC EQUATION, LOGISTIC
Equation with r = 4
Terminal
see Sink (Directed Graph)
Ternary
The BASE 3 method of counting in which only the digits
0, 1, and 2 are used. These digits have the following
multiplication table.
X
1
2
1
1
2
2
2
11
Erdos and Graham (1980) conjectured that no POWER
of 2, 2 n , is a SUM of distinct powers of 3 for n > 8.
This is equivalent to the requirement that the ternary
expansion of 2 n always contains a 2. This has been
verified by Vardi (1991) up to n = 2-3 20 . N. J. A. Sloane
has conjectured that any POWER of 2 has a in its
ternary expansion (Vardi 1991, p. 28).
see also Base (Number), Binary, Decimal, Hexa-
decimal, Octal, Quaternary
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, 1980.
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, pp. lu-
ll, 1991.
Vardi, I. "The Digits of 2 n in Base Three." Computational
Recreations in Mathematica. Reading, MA: Addison-
Wesley, pp. 20-25, 1991.
^ Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/-ewv6n/math/notebooks/Bases.m.
Tessellation
A regular TILING of POLYGONS (in 2-D), POLYHEDRA
(3-D), or POLYTOPES (n-D) is called a tessellation. Tes-
sellations can be specified using a SCHLAFLI SYMBOL.
Consider a 2-D tessellation with q regular p-gons at each
Vertex. In the Plane,
1-
P 9
2n
(1)
(2)
(p-2)(</-2) = 4 (3)
(Ball and Coxeter 1987), and the only factorizations are
4 = 4 ■ 1 = (6 - 2)(3 - 2) => {6,3}
(4)
= 2 ■ 2 = (4 - 2)(4 - 2) =>• {4,4}
(5)
= 1 • 4 = (3 - 2)(6 - 2) =* {3, 6}.
(6)
Therefore, there are only three regular tessellations
(composed of the Hexagon, Square, and Triangle),
illustrated as follows.
{6,3}
{4,4}
{3,6}
There do not exist any regular STAR POLYGON tes-
sellations in the PLANE. Regular tessellations of the
Sphere by Spherical Triangles are called Trian-
gular Symmetry Groups.
Regular tilings of the plane by two or more convex reg-
ular Polygons such that the same Polygons in the
same order surround each VERTEX are called semireg-
ular tilings. In the plane, there are eight such tessella-
tions, illustrated below.
Tessellation
Tetrachoric Function
1795
AAAAAA
\AAAAAA
vwvw
In 3-D, a POLYHEDRON which is capable of tessellating
space is called a'SPACE-FlLLING POLYHEDRON. Exam-
ples include the Cube, Rhombic Dodecahedron, and
Truncated Octahedron. There is also a 16-sided
space-filler and a convex Polyhedron known as the
SCHMITT-CONWAY BlPRlSM which fills space only ape-
riodically.
A tessellation of n-D polytopes is called a Honeycomb.
see also Archimedean Solid, Cell, Honey-
comb, Schlafli Symbol, Semiregular Polyhe-
dron, Space-Filling Polyhedron, Tiling, Trian-
gular Symmetry Group
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 105-
107, 1987.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 60-63, 1989.
Gardner, M. Martin Gardner's New Mathematical Diver-
sions from Scientific American. New York: Simon and
Schuster, pp. 201-203, 1966.
Gardner, M. "Tilings with Convex Polygons." Ch. 13
in Time Travel and Other Mathematical Bewilderments.
New York: W. H. Freeman, pp. 162-176, 1988.
Kraitchik, M. "Mosaics." §8.2 in Mathematical Recreations.
New York: W. W. Norton, pp. 199-207, 1942.
Lines, L. Solid Geometry. New York: Dover, pp. 199 and
204-207 1965.
Pappas, T. "Tessellations." The Joy of Mathematics. San
Carlos, CA: Wide World Publ./Tetra, pp. 120-122, 1989.
Peterson, I. The Mathematical Tourist: Snapshots of Modern
Mathematics. New York: W. H. Freeman, p. 75, 1988.
Rawles, B. Sacred Geometry Design Sourcebook: Universal
Dimensional Patterns. Nevada City, CA: Elysian Pub.,
1997.
Walsh, T. R. S. "Characterizing the Vertex Neighbourhoods
of Semi-Regular Polyhedra." Geometriae Dedicatal, 117—
123, 1972.
Tesseract
^
«^
/ \
The Hypercube in R is called a tesseract. It
has the SCHLAFLI Symbol {4,3,3}, and VERTICES
(±l,±l,±l,=bl). The above figures show two visual-
izations 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 TESSER-
ACT symmetrically projected into the PLANE (Coxeter
1973). A Tesseract has 16 Vertices, 32 Edges, 4
Squares, and 8 Cubes.
see also Hypercube, Polytope
References
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, p. 123, 1973.
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.gGom.umn.edu/docs/outreach/4-cube/.
Tesseral Harmonic
A Spherical Harmonic which is expressible as prod-
ucts of factors linear inn 2 , y 2 , and z 2 multiplied by one
of 1, cc, y, z, yz, zx, xy, and xyz.
see also ZONAL HARMONIC
Tethered Bull Problem
Let a bull be tethered to a silo whose horizontal CROSS-
Section is a Circle of Radius J? by a leash of length
L. Then the Area which the bull can graze if L < Rir
is
irL 2 L 3
A =
2 + 3R'
References
Hoffman, M. E. "The Bull and the Silo: An Application of
Curvature." Amer. Math. Monthly 105, 55-58, 1998.
Tetrabolo
A 4-Polyabolo.
Tetrachoric Function
The function denned by
T n , t^ Z
(n-l)
(*),
1796 Tetracontagon
where
see also NORMAL DISTRIBUTION
References
Kenney, J. F. and Keeping, E. S. "Tetrachoric Correlation."
§8.5 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton,
NJ: Van Nostrand, pp. 205-207, 1951.
Tetracontagon
A 40-sided Polygon.
Tetracuspid
see Hypocycloid — 4-Cusped
Tetrad
A Set of four, also called a Quartet.
see also Hexad, Monad, Pair, Quartet, Quintet,
Triad, Triple, Twins
Tetradecagon
A 14-sided POLYGON, sometimes called a Tetrakai-
decagon.
Tetradecahedron
A 14-sided POLYHEDRON, sometimes called a Tetra-
kaidecahedron.
see also Cuboctahedron, Truncated Octahedron
References
Ghyka, M. The Geometry of Art and Life. New York: Dover,
p. 54, 1977.
Tetradic
Tetradics transform DYADICS in much the same way
that DYADICS transform VECTORS. They are repre-
sented using Hebrew characters and have 81 compo-
nents (Morse and Feshbach 1953, pp. 72-73). The use
of tetradics is archaic, since TENSORS perform the same
function but are notationally simpler.
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Vol. 1. New York: McGraw-Hill, 1953,
Tetradyakis Hexahedron
The Dual Polyhedron of the Cubitruncated Cub-
octahedron.
Tetrafiexagon
Tetraflexagon
A Flexagon made with Square faces. Gardner (1961)
shows how to construct a tri-tetraflexagon,
l
l
2]X
2_(7
2 2
2 2
tetra-tetraflexagon,
1
3
1
1
2
3
1
3
4
2
4
4
3
3
4
2
2
1
3
4
1
2
4
3
1
1
ir
^
2
u
i
\2
mi
and hexa-tetraflexagon.
6 5
2 1
1 2
1
5
51
3 \<
T
I
vAA
5 || 2
2T3
t
see also Flexagon, Flexatube, Hexaflexagon
References
Chapman, P. B. "Square Flexagons." Math. Gaz. 45, 192-
194, 1961.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 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 Book
of Mathematical Puzzles & Diversions: A New Selection.
New York: Simon and Schuster, 1961.
Pappas, T. "Making a Tri-Tetra Flexagon." The Joy of
Mathematics, San Carlos, CA: Wide World Publ./Tetra,
p. 107, 1989.
Tetragon
Tetrahedral Surface
1797
Tetragon
see Quadrilateral
Tetrahedral Coordinates
Coordinates useful for plotting projective 3-D curves of
the form f(x ,xi,X2,x 3 ) = which are denned by
xq = 1 — z — V2x
xi = 1 — z + \/2x
x 2 = l + z + V2y
x s = l + z- V2y.
see also Cayley Cubic, Kummer Surface
Tetrahedral Graph
A000292). The GENERATING FUNCTION of the tetrahe-
dral numbers is
A Polyhedral Graph which is also the Complete
Graph K±.
see also CUBICAL GRAPH, DODECAHEDRAL GRAPH,
Icosahedral Graph, Octahedral Graph, Tetra-
hedron
Tetrahedral Group
The Point Group of symmetries of the Tetrahe-
dron, denoted Tj. The tetrahedral group has symmetry
operations E, 8C 3 , 3C 2 , 6S 4 , and 6a d (Cotton 1990).
see also Icosahedral Group, Octahedral Group,
Point Groups, Tetrahedron
References
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 47, 1990.
Lomont, J. S. "Icosahedral Group." §3.10.C in Applications
of Finite Groups. New York: Dover, p. 81, 1987.
(x - 1)
X = x + Ax 2 + lOz 3 + 20z 4 + . . . .
(2)
Tetrahedral numbers are EVEN, except for every fourth
tetrahedral number, which is Odd (Conway and Guy
1996).
The only numbers which are simultaneously SQUARE
and Tetrahedral are Te\ = 1, Te 2 = 4, and Te 4 s —
19600 (giving £1 = 1, S 2 = 4, and S 14 o = 19600), as
proved by Meyl (1878; cited in Dickson 1952, p. 25).
Numbers which are simultaneously TRIANGULAR and
tetrahedral satisfy the BINOMIAL COEFFICIENT equa-
tion
(3)
«•
the only solutions of which are (m,n) = (10,16), (22,
56), and (36, 120) (Guy 1994, p. 147). Beukers (1988)
has studied the problem of finding numbers which are
simultaneously tetrahedral and PYRAMIDAL via INTE-
GER points on an ELLIPTIC CURVE, and finds that the
only solution is the trivial Te\ = P\ = 1.
see also Pyramidal Number, Truncated Tetrahe-
dral Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 59, 1987.
Beukers, F. "On Oranges and Integral Points on Certain
Plane Cubic Curves." Nieuw Arch. Wish. 6, 203-210,
1988.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 44-46, 1996.
Dickson, L. E. History of the Theory of Numbers, Vol. 2:
Diophantine Analysis. New York: Chelsea, 1952.
Guy, R. K. "Figurate Numbers." §D3 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 147-150, 1994.
Meyl, A.-J.-J. "Solution de Question 1194." Nouv. Ann.
Math. 17, 464-467, 1878.
Sloane, N. J. A. Sequence A000292/M3382 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Tetrahedral Number
A Figurate Number Te n of the form
n
n + 2
(1)
where T n is the nth TRIANGULAR Number and (^) is a
Binomial Coefficient. These numbers correspond to
placing discrete points in the configuration of a TETRA-
HEDRON (triangular base pyramid). Tetrahedral num-
bers are PYRAMIDAL NUMBERS with r = 3, and are
the sum of consecutive TRIANGULAR Numbers. The
first few are 1, 4, 10, 20, 35, 56, 84, 120, ... (Sloane's
Tetrahedral Surface
A SURFACE given by the parametric equations
x = A(u-a) m (v-a) n
y = B{u-b) m {v~b) n
z = C(u-c) 7n (v~c) n .
References
Eisenhart, L. P. A Treatise on. the Differential Geometry of
Curves and Surfaces. New York: Dover, p. 267, 1960.
1798
Tetrahedroid
Tetrahedroid
A special case of a quartic KUMMER SURFACE.
References
Fischer, G. (Ed.). Mathematical Models from the Collections
of Universities and Museums. Braunschweig, Germany:
Vieweg, pp. 17-19, 1986.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 183, 1994.
Tetrahedron
The regular tetrahedron, often simply called "the" tetra-
hedron, is the Platonic Solid Pi with four Vertices,
six EDGES, and four equivalent EQUILATERAL TRIAN-
GULAR faces (4{3». It is also Uniform Polyhedron
Ui. It is described by the SCHLAFLl Symbol {3,3} and
the Wythoff Symbol is 3 | 2 3. It is the prototype of
the Tetrahedral Group T d ,
The tetrahedron is its own Dual POLYHEDRON. It
is the only simple POLYHEDRON with no DIAGONALS,
and cannot be Stellated. The Vertices of a
tetrahedron are given by (0,0, \/3), (0, |V6, -fv^),
(-V2,-|a/6,-|V3), and (y/2, -fv^, -|V3), or by
(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0). In the latter case,
the face planes are
x+y+z=2
x — y — z =
-x + y — z =
x + y — z = 0.
(i)
(2)
(3)
(4)
Let a tetrahedron be length a on a side. The Vertices
are located at (x, 0, 0), (-d, ±a/2, 0), and (0, 0, h).
From the figure,
Perspective View Bottom View Side View
- 19
—7Tx= £-7=0=^0.
cos(f ) 2 VZ 3
(5)
d is then
d=y/x>- (|a)* = aj\^\ = ay j*~*
a
12
Tetrahedron
This gives the Area of the base as
A = \a{R + x) = \a[—a+—a
= I, 2 (4 + ^
6 J
! 2 3y3 1/^2
2 a ~q~ = 4 V3a ■
(7)
The height is
h = \/a 2 - x 2 — aJ\ — | = \\/§a.
(8)
The ClRCUMRADlus R is found from
x 2 + (fe - fl) 2 = fl 2
(9)
x 2 + h 2 - 2hR 4- R 2 = R 2 .
(10)
Solving gives
r. x +h
2h *Vl
The Inradius r is
^ + l-iJ|=iN/6a« 0.61237a.
(11)
= h-R= \\a-^a= iV6a « 0.20412a, (12)
y o o
which is also
r=±fc=|*.
(13)
The MlDRADlUS is
P = V 7 ^ 2 + <* 2 = a\/S + i = \[l* = \ y/ia
« 0.35355a. (14)
Plugging in for the VERTICES gives
(a\/3,0,0),(-|\/3a,±|a,0), and (0,0, § a/6 a). (15)
Since a tetrahedron is a PYRAMID with a triangular base,
V = ~Abh, and
The Dihedral Angle is
= tan -1 (2\/2) = 2sin _1 (iv / 6) = cos _1 (i). (17)
= 1^3a.
(6)
Tetrahedron
Tetrahedron 5-Compound 1799
By slicing a tetrahedron as shown above, a SQUARE can
be obtained. This cut divides the tetrahedron into two
congruent solids rotated by 90°.
Now consider a general (not necessarily regular) tetra-
hedron, defined as a convex POLYHEDRON consisting of
four (not necessarily identical) TRIANGULAR faces. Let
the tetrahedron be specified by its VERTICES at (xi,yi)
where i = 1, . . . , 4. Then the VOLUME is given by
V =
3!
xi y\ z x 1
x 2 y-i z 2 1
X3 2/3 Z 3 1
X4 V4 Z4 1
(18)
Specifying the tetrahedron by the three Edge vectors
a, b, and c from a given Vertex, the Volume is
V=i|a-(bxc)|.
(19)
If the faces are congruent and the sides have lengths a,
6, and c, then
v = J {<* + 6 2 - c 2 )(" 2 + c 2 - & 2 )(6 2 + c* - a 2 ) (2Q)
(Klee and Wagon 1991, p. 205). Let a, 6, c, and d be
the areas of the four faces, and define
B ~ led
C= Lbd
D = Lbc,
(21)
(22)
(23)
where Ljk means here the ANGLE between the PLANES
formed by the Faces j and &, with Vertex along their
intersecting Edge. Then
a 2 = b 2 +c 2 +d 2 -2cdcos B -2bdcosC -2bccos D. (24)
The analog of GAUSS'S Circle PROBLEM can be asked
for tetrahedra: how many LATTICE POINTS lie within a
tetrahedron centered at the ORIGIN with a given INRA-
DIUS (Lehmer 1940, Granville 1991, Xu and Yau 1992,
Guy 1994).
see also AUGMENTED TRUNCATED TETRAHEDRON,
Bang's Theorem, Ehrhart Polynomial, Heronian
Tetrahedron, Hilbert's 3rd Problem, Isosceles
Tetrahedron, Sierpinski Tetrahedron, Stella
Octangula, Tetrahedron 5-Compound, Tetrahe-
dron 10-Compound, Truncated Tetrahedron
References
Davie, T. "The Tetrahedron." http://www.dcs, st-and.ac.
uk/-ad/mathrecs/polyhedra/tetrahedron.html.
Granville, A. "The Lattice Points of an n- Dimensional Tet-
rahedron." Aequationes Math. 41, 234-241, 1991.
Guy, R. K. "Gaufi's Lattice Point Problem." §F1 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 240-241, 1994.
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.
Lehmer, D. H. "The Lattice Points of an n- Dimensional Tet-
rahedron." Duke Math. J. 7, 341-353, 1940.
Xu, Y. and Yau, S. "A Sharp Estimate of the Number of
Integral Points in a Tetrahedron." J. reine angew. Math.
423, 199-219, 1992.
Tetrahedron 5-Compound
A Polyhedron Compound composed of 5 Tetra-
hedra. Two tetrahedron 5-compounds of opposite
CHIRALITY combine to make a TETRAHEDRON 10-
COMPOUND. The following diagram shows pieces which
can be assembled to form a tetrahedron 5-compound
(Cundy and Rollett 1989).
AA »»s&
see also Polyhedron Compound, Tetrahedron 10-
COMPOUND
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 135,
1987.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 139-141, 1989.
Wang, P. "Renderings." http : //www.ugcs . caltech.edu/
-pet erw/portf olio/renderings/.
Wenninger, M. J. Polyhedron Models. New York: Cambridge
University Press, p. 44, 1989.
1800 Tetrahedron 10-Compound
Tetrahedron 10-Compound
Tetrakaidecahedron
Two Tetrahedron 5-Compounds of opposite Chi-
RALITY combined.
see also Polyhedron Compound, Tetrahedron 5-
Compound
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 135,
1987.
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 141-142, 1989,
Wenninger, M. J. Polyhedron Models. New York: Cambridge
University Press, p. 45, 1989.
Tetrahedron Inscribing
Pick four points at random on the surface of a unit
Sphere. Find the distribution of possible volumes of
(nonregular) Tetrahedra. Without loss of generality,
the first point can be chosen as (1, 0, 0). Designate the
other points a, b, and c. Then the distances from the
first Vertex are
cos 0i — 1
sin 0i
" cos 02 sin 02 — 1
b = sin 62 sin 02
COS 02
cos 03 sin 03 — 1
sin 03 sin 03
COS 03
The average volume is then
p2ir /*2tt /»2tt /•tt/2 /*t/2
9 = h / / / / - ia(b
^ JO JO JO J-n/2 J-n/2
(1)
(2)
(3)
dfodfodOsdOiddi, (4)
where
pZTT 1*21* PZ-K pTT/4
C= / / / dfodfc d9 3 d0 2 dB± = 8?r 5
JO JO Jo J-ir/2
(5)
and
a • (b x c) = — cos 02 sin 0i + cos 03 sin 0i
— cos 03 cos 02 sin 02 sin 0i 4- cos 02 cos 03 sin 03 sin 0i
— cos 03 sin 02 sin 02 + cos 03 cos 0i sin 2 sin 2
-t- cos 02 sin 03 sin 03 — cos 2 cos 0i sin 03 sin 03 . (6)
The integrals are difficult to compute analytically, but
10 7 computer Trials give
{V) « 0.1080
(V 2 ) « 0.02128
av 2 = (V 2 ) - {V) 2 « 0.009937.
(7)
(8)
(9)
see also POINT-POINT DISTANCE — 1-D, TRIANGLE IN-
SCRIBING in a Circle, Triangle Inscribing in an
Ellipse
References
Buchta, C "A Note on the Volume of a Random Poly tope in
a Tetrahedron." Ill J. Math. 30, 653-659, 1986.
Tetrahemihexacron
The Dual Polyhedron of the Tetrahemihexahe-
dron.
Tetrahemihexahedron
The Uniform Polyhedron U 4 whose Dual Polyhe-
dron is the Tetrahemihexacron. It has Schlafli
Symbol r'-j^} and Wythoff Symbol § 3 | 2. Its faces
are 4{3} + 3{4}. It is a faceted form of the OCTAHE-
DRON. Its ClRCUMRADIUS is
R
§V2.
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 101-102, 1971.
Tetrakaidecagon
see Tetradecagon
Tetrakaidecahedron
see Tetradecahedron
Tetrakis Hexahedron
Tetrakis Hexahedron
Theorem
1801
The Dual Polyhedron of the Truncated Octahe-
dron.
Tetranacci Number
The tetranacci numbers are a generalization of the FI-
BONACCI Numbers defined by T = 0, T± = 1, T 2 = 1,
T 3 = 2, and the Recurrence Relation
T n = T n _i + T n -2 4" ^-3 + T n _4
for n > 4. They represent the n = 4 case of the FI-
BONACCI u-Step Numbers. The first few terms are 1,
1, 2, 4, 8, 15, 29, 56, 108, 208, ... (Sloane's A000078).
The ratio of adjacent terms tends to 1.92756, which is
the Real Root of x 5 - 2x 4 + 1 = 0.
see also Fibonacci ti-Step Number, Fibonacci Num-
ber, Tribonacci Number
References
Sloane, N. J. A. Sequence A000078/M1108 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Tetrix
The 3-D analog of the Sierpinski Sieve illustrated
above, also called the Sierpinski Sponge or Sierpinski
Tetrahedron. Let N n be the number of tetrahedra,
L n the length of a side, and A n the fractional VOLUME
of tetrahedra after the nth iteration. Then
N n = A n
(1)
L n = (I)» = 2-
(2)
A n = L n 3 N n = (|) n .
(3)
The Capacity Dimension is therefore
,. lniV n ,. ln(4")
d c ._ = — hm - — = — = lim , .„ .
CaP n-*oo lnL„ n^ooln(2-")
ln4 _ 21n2
_ ln2 ~ ln2 - '
(4)
SO the tetrix has an INTEGRAL CAPACITY DIMENSION
(albeit one less than the DIMENSION of the 3-D TETRA-
HEDRA from which it is built), despite the fact that it is
a Fractal.
The following illustration demonstrates how this coun-
terintuitive fact can be true by showing three stages of
the rotation of a tetrix, viewed along one of its edges.
In the last frame, the tetrix "looks" like the 2-D PLANE.
see also Menger Sponge, Sierpinski Sieve
References
Dickau, R. M. "Sierpinski Tetrahedron." http:// forum .
swarthmore.edu/advanced/robertd/tetrahedron.html.
Eppstein, D. "Sierpinski Tetrahedra and Other Fractal
Sponges." http : //www . ics . uci . edu/~eppstein/ junkyard
/sierpinski. html.
Tetromino
The five 4-POLYOMINOES, known as STRAIGHT, L-, T-,
Square, and Skew.
References
Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer-
ican Book of Mathematical Puzzles & Diversions. New
York: Simon and Schuster, pp. 124-140, 1959.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 80-81, 1975.
Thales' Theorem
An ANGLE inscribed in a SEMICIRCLE is a RIGHT AN-
GLE.
see also Right Angle, Semicircle
Theorem
A statement which can be demonstrated to be true by
accepted mathematical operations and arguments. In
general, a theorem is an embodiment of some general
principle that makes it part of a larger theory.
According to the Nobel Prize-winning physicist Richard
Feynman (1985), any theorem, no matter how diffi-
cult to prove in the first place, is viewed as "TRIVIAL"
1802 Theorema Egregium
by mathematicians once it has been proven. There-
fore, there are exactly two types of mathematical ob-
jects: Trivial ones, and those which have not yet been
proven.
see also Axiom, Axiomatic System, Corollary,
Deep Theorem, Porism, Lemma, Postulate, Prin-
ciple, Proposition
References
Feynman, R. P. and Leighton, R. Surely You're Joking, Mr.
Feynman! New York: Bantam Books, 1985.
Theorema Egregium
see Gauss's Theorema Egregium
Theta Function
The theta functions are the elliptic analogs of the Ex-
ponential FUNCTION, and may be used to express the
Jacobi Elliptic Functions. Let t be a constant Com-
plex Number with 9f[t] > 0. Define the Nome
where
q = e i7Tt = e^'CO/*^
t,-i K 'W
K{k)
(1)
(2)
and K(k) is a complete ELLIPTIC INTEGRAL OF THE
First Kind, k is the MODULUS, and k' is the comple-
mentary MODULUS. Then the theta functions are, in
the NOTATION of Whittaker and Watson,
CO
^(z, 9 ) = 2j](-l)V n+1/2)2 sin[(2n+l) 2 ]
n=0
oo
= zq 1/4 ^(-l) n g n(n+1) sin[(2n + l)z] (3)
n—
oo
tf a (z, q) = 2 Y, <? (n+1/2)2 cos[(2n + l)z]
oo
= 2</ 1/4 Y, <7 n(n+1) cos K 2 ™ + 1)*] (4)
oo
#3 (z 7 q) = 1 + 2 ^ q n2 cos(2nz) (5)
71=1
OO
M*,q)= £ (-!)V 2 e 2 " iz
n= — co
OO
-1 + 2 ^(-1) V 2 cos(2n^). (6)
n=l
Written in terms of £,
Mt,Q)= £ q ^V\*W (7)
n= — oo
oo
Theta Function
These functions are sometimes denoted ©i or 0i, and a
number of indexing conventions have been used. For a
summary of these notations, see Whittaker and Watson
(1990). The theta functions are quasidoubly periodic,
as illustrated in the following table.
#i
#i(* + 7r)/tfi(z)
0i(z + t7r)/tfi(;s)
til
-1
-TV
$2
-1
N
$S
1
N
T?4
1
-N
Here,
N = g - x e- 2 ".
(9)
The quasiperiodicity can be established as follows for
the specific case of $4,
oo
o / . \ \ ^ / -f\n n 2niz 2niir
$4{z + ir,q) = 2_ (- 1 ) Q e e
n= — oo
co
= J2 (-lVq n2 e 2niz =Mz,<l) (10)
n= — co
oo
o / , , \ \ A / i \n n 2ni7vt 2niz
$4(^-f 7T£,<?) - 2_^ ( _1 ) ^ e e
7l= — OO
OO
E/ i\n n 2n 2niz
(-1) 3 q e
= -,-V"* £ (-1)
-1 -2iz V^ 1 f -i\n n 2 2
= -q- 1 e- 2i '#4{*>Q)-
n+1 (n+1) 2 2(n+l)z*
(ii)
The theta functions can be written in terms of each
other:
M**Q) = -*e" +,rit/4 i?4(z+ ±7rt,qr) (12)
3 (*,«)=04(* +§*,$). (14)
Any theta function of given arguments can be expressed
in terms of any other two theta functions with the same
arguments.
0.4
0.2
-1 -0.5
-0.2
10
0.5 1 4
2
0.20.40.60.8 1
10
-1 -0.5 0.5 1 -1 -0.5 0.5 1
Theta Function
Define
■&i = i?i(z = 0), (15)
which are plotted above. Then we have the identities
0i 2 (*)0 4 2 = -&3 2 (z)* 2 - i?2 2 (z)^3 2 (16)
tf 2 2 (z)l?4 2 = 1?4 2 Wl?2 2 - 1?1 2 (Z)1?3 2 (17)
tf 3 2 (z)tf4 2 = 1?4 2 Wt? 3 2 " 1?l 2 (z)tf 2 2 (18)
1 ?4 2 (^)^4 2 = tf 3 2 (z)l?3 2 " t>2 2 (z)t?2 2 . (19)
Taking z = in the last gives the special case
tf 4 4 = tf 3 4 - tf 2 4 . (20)
In addition,
tf 3 (a:) = ^ x n2 =3 l + 2x + 2x 4 + 2x 9 + ... (21)
7l= — CO
/a; x s x 5 x 7 \
1 9 3 2 (x) = l + 4(- - + - --- +...)
\ 1 - x l-i 3 1 - a: 5 l-i 7 /
(22)
., , ( x 2x 2 Zx 3 4i 4 \
(23)
The theta functions obey addition rules such as
3 (* + y)M* ~ y)#3 2 = $3 2 {y)#s 2 (z) + &i 2 (y)<di 2 (z).
Letting y = z gives a duplication FORMULA
$z(2z)$ % 3 =tf 3 4 (2) + tfi 4 (z).
(24)
(25)
For more addition FORMULAS, see Whittaker and Wat-
son (1990, pp. 487-488). Ratios of theta function deriva-
tives to the functions themselves have the simple forms
tfi o
tf 4 (z
04 (*
w 2n
■ cot z + 4 > — sin(2n^)
^^ 1 — q 2n
n—X
:-tan,z + 4^(-l) n _ 2n sin(2n2)
71—1
(26)
(27)
(28)
= £r
q 2n - l sin(2z)
2q 2n ~ 1 cos(2z) + q 4
•E^
4g n sin(2nz)
(29)
Theta Function 1803
The theta functions can be expressed as products in-
stead of sums by
oo
0i {z) = 2Gq 1/4 sin z JJ [1 - 2q 2n cos(2^) + g 4n ] (30)
71=1
OO
2 (*) = 2Gq 1/4 cos z J| [1 + 2q 2n cos(2z) + q 4n ] (31)
n=l
oo
*)(«) = G JJ[1 + 29 2 "- 1 cos(2z) + q 4n ~ 2 ] (32)
71 = 1
OO
4 (*) - G JJ[1 - Sg 271 " 1 cos(2*) + g 4n " 2 ], (33)
n=l
where
G= JJU-^")
(34)
(Whittaker and Watson 1990, pp. 469-470).
The theta functions satisfy the PARTIAL DIFFERENTIAL
Equation
h& +§?=»• <»>
where y = , dj(z\t). Ratios of the theta functions with 04
in the DENOMINATOR also satisfy differential equations
d
r*i(*)i
dz
M*)_
d
'M*)]
dz
m M*).
d
*»wl
dz
M*)_
04 2 ( Z )
3 #4 2 (Z)
2 tf 4 2 (z) '
(36)
(37)
(38)
Some additional remarkable identities are
0i =0 2 0304 (39)
3 M) = -(^V 2 /^ (f ,-±) , (40)
which were discovered by Poisson in 1827 and are equiv-
alent to
y^ e ~t( a; +n) 2 _ fi[ y^ 2
r > 27rifcx-(7T 2 fe 2 /t)
(41)
71= — OO
Another amazing identity is
20i{§(-&+c+d+e)]0a[i(6-c+d+e)]0s[i(6+c-d+e)]
x^ 4 [i(6 + c + <2 - c)] = M*>)Mc)Md)Me)
+t?a(6)t»i(c)tf 4 (i)i»3(c) - tfi(6)tfa(c)tfs(d)*4(e)
+i? 4 (6)^3(c)7? 2 (d)i?i(e) (42)
(Whittaker and Watson 1990, p. 469).
1804 Theta Function
Third Curvature
The complete Elliptic Integrals OF the First and
Second KINDS can be expressed using theta functions.
Let
_ * (*)
*-
M*y
(43)
and plug into (36)
(J) 2 = (tf 2 2 -^3 2 )(tf3 2 -^2 2 ). (44)
Now write
and
V2
Z&3 = U.
(45)
(46)
Then
(J) =(l- 2/ 2 )(l-fcV), (47)
where the MODULUS is defined by
k = k(q)
tV(5)
Define also the complementary MODULUS
*4 2 (-9)
V(<?)
Now, since
we have shown
k' = h f (q)~ Q _ 2
k 2 + k' 2 = 1.
(48)
(49)
(50)
(51)
sn(u,fc), (52)
The solution to the equation is
= ^3 M^3' 2 \t) =
which is a JACOBI ELLIPTIC FUNCTION with periods
4K(k) = 27r$3 2 {q) (53)
and
2iK'(k) = Trt&z(q).
(54)
Here, K is the complete Elliptic Integral of the
First Kind,
K{k) = \irt* 2 (q). (55)
see also Blecksmith-Brillhart-Gerst Theorem,
Elliptic Function, Eta Function, Euler's Pen-
tagonal Number Theorem, Jacobi Elliptic Func-
tions, Jacobi Triple Product, Landen's For-
mula, Mock Theta Function, Modular Equation,
Modular Transformation, Mordell Integral,
Neville Theta Function, Nome, Poincare-Fuchs-
Klein Automorphic Function, Prime Theta
Function, Quintuple Product Identity, Ramanu-
jan Theta Functions, Schroter's Formula, We-
ber Functions
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 577, 1972.
Bellman, R. E. A Brief Introduction to Theta Functions,
New York: Holt, Rinehart and Winston, 1961.
Berndt, B. C. "Theta-Functions and Modular Equations."
Ch. 25 in Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 138-244, 1994.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 430-432, 1953.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, fth ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Theta Operator
In the Notation of Watson (1966),
= *-^-.
dz
References
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.
Theta Subgroup
see Lambda Group
Thiele's Interpolation Formula
Let p be a Reciprocal Difference. Then Thiele's
interpolation formula is the CONTINUED FRACTION
f(x) = f(x 1 ) +
x — Xi
X — X2
p(xi,X 2 )+ p2(Xl,X2 } Xs) ~ f{Xi) +
X — X3
Ps(xi,X2jX3,X4) — p(xi,X2) + • • •
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.
Milne-Thomson, L. M. The Calculus of Finite Differences.
London: Macmillan, 1951.
Thiessen Polytope
see VORONOI POLYGON
Third Curvature
Also known as the TOTAL CURVATURE. The linear ele-
ment of the INDICATRIX
dsp = ydsx 2 + dsB 2 -
see also Lancret Equation
Thirteenth
Thomson Problem
1805
Thirteenth
see Friday the Thirteenth
Thorn's Eggs
EGG-shaped curves constructed using multiple CIRCLES
which Thorn (1967) used to model Megalithic stone rings
in Britain.
see also EGG, OVAL
References
Dixon, R. Mathographics. New York: Dover, p. 6, 1991.
Thorn, A. "Mathematical Background." Ch. 4 in Megalithic
Sites in Britain. Oxford, England: Oxford University
Press, pp. 27-33, 1967.
Thomae's Theorem
r(x + y + 3 + l) f ~ a ,-b,x + y + s + l \
r(x + 5 + l)r(y + 5 + l) 3 2 \ x + 3 + l,y + 5 + l ' J
= r(a + fr+s + 1) / —a;, — t/, a + 6 + s + 1 \
r(a + 5+ l)r(6+3 + l) 3 2 ^ a+a+l,6 + s + l ' ^ »
where T(z) is the GAMMA FUNCTION and the function
3F2 (a, 6, c; d, e; z) is a GENERALIZED Hypergeometric
Function.
see also Generalized Hypergeometric Function
References
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
pp. 104-105, 1959.
Thomassen Graph
The Graph illustrated above.
see also Thomsen Graph
Thompson's Functions
see BEI, BER, KELVIN FUNCTIONS
Thompson Group
The Sporadic Group Th.
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http://for.mat.bham.ac.uk/atlas/Th.htnil.
Thomsen's Figure
Take any TRIANGLE with VERTICES A, B, and C. Pick
a point A\ on the side opposite A, and draw a line Par-
allel to AB. Upon reaching the side AC at i?i, draw
the line PARALLEL to BC . Continue (left figure). Then
A 3 = Ai for any Triangle. If A x is the MIDPOINT of
BC, then A 2 = A\ (right figure).
see also MIDPOINT, TRIANGLE
References
Madachy, J. S. Madachy's Mathematical Recreations.
York: Dover, pp. 234, 1979.
New
Thomsen Graph
The Complete Bipartite Graph if 3 ,3, which is
equivalent to the UTILITY GRAPH. It has a CROSSING
Number 1.
see also Complete Bipartite Graph, Crossing
Number (Graph), Thomassen Graph, Utility
Graph
Thomson Lamp Paradox
A lamp is turned on for 1/2 minute, off for 1/4 minute,
on for 1/8 minute, etc. At the end of one minute, the
lamp switch will have been moved Ho times, where Ho is
Aleph-0. Will the lamp be on or off? This PARADOX
is actually nonsensical, since it is equivalent to asking if
the "last" Integer is Even or Odd.
New York: Wiley, pp. 19-
References
Pickover, C. A. Keys to Infinity.
23, 1995.
Thomson's Principle
see DlRICHLET'S PRINCIPLE
Thomson Problem
Determine the stable equilibrium positions of N clas-
sical electrons constrained to move on the surface of a
Sphere and repelling each other by an inverse square
law. Exact solutions for N = 2 to 8 are known, but
N = 9 and 11 are still unknown.
1806
Thousand
Thue-Morse Constant
In reality, Earnshaw's theorem guarantees that no sys-
tem of discrete electric charges can be held in stable
equilibrium under the influence of their electrical inter-
action alone (Aspden 1987).
see also FEJES TOTH'S PROBLEM
References
Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Dowla, F.;
and Wooten, F. "Method of Constrained Global Optimiza-
tion." Phys. Rev. Let. 72, 2671-2674, 1994.
Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Dowla,
F.; and Wooten, F. "Method of Constrained Global
Optimization— Reply." Phys. Rev. Let 74, 1483, 1995.
Ashby, N. and Brittin, W. E. "Thomson's Problem." Amer.
J. Phys. 54, 776-777, 1986.
Aspden, H. "Earnshaw's Theorem." Amer. J. Phys. 55,
199-200, 1987.
Berezin, A. A. "Spontaneous Symmetry Breaking in Classical
Systems." Amer. J. Phys. 53, 1037, 1985.
Calkin, M. G.; Kiang, D.; and Tindall, D. A. "Minimum
Energy Configurations." Nature 319, 454, 1986.
Erber, T. and Hockney, G. M. "Comment on 'Method of
Constrained Global Optimization.'" Phys. Rev. Let. 74,
1482-1483, 1995.
Marx, E. "Five Charges on a Sphere." J. Franklin Inst. 290,
71-74, Jul. 1970.
Melnyk, T. W.; Knop, O.; and Smith, W. R. "Extremal Ar-
rangements of Points and Unit Charges on a Sphere: Equi-
librium Configurations Revisited." Canad. J. Chem. 55,
1745-1761, 1977.
Whyte, L. L. "Unique Arrangement of Points on a Sphere."
Amer. Math. Monthly 59, 606-611, 1952.
Thousand
1,000 = 10 3 . The word "thousand" appears in com-
mon expressions in a number of languages, for example,
"a thousand pardons" in English and "tusen takk" ("a
thousand thanks") in Norwegian.
see also HUNDRED, LARGE NUMBER, MILLION
Three
see 3
Three-Colorable
see Colorable
Three- Valued Logic
A logical structure which does not assume the EX-
CLUDED Middle Law. Three possible truth values are
possible: true, false, or undecided. There are 3072 such
logics.
see also EXCLUDED MIDDLE LAW, FUZZY LOGIC, LOGIC
Threefoil Knot
see Trefoil Knot
Thue Constant
The base-2 Transcendental Number
0.11011011111011011111. .. 2 ,
where the nth bit is 1 if n is not divisible by 3 and is
the complement of the (n/3)th bit if n is divisible by 3.
It is also given by the SUBSTITUTION MAP
111
110.
In decimal, the Thue constant equals 0.8590997969
see also RABBIT CONSTANT, THUE-MORSE CONSTANT
References
Thue-Morse Constant
The constant also called the Parity Constant and
defined by
oo
P=±^2 p ( n ) 2 ~ n = 0.4124540336401075977 ... (1)
Tl =
(Sloane's A014571), where P(n) is the Parity of n.
Dekking (1977) proved that the Thue-Morse constant
is Transcendental, and Allouche and Shallit give a
complete proof correcting a minor error of Dekking.
The Thue-Morse constant can be written in base 2 by
stages by taking the previous iteration a n , taking the
complement a^", and appending, producing
Three-In-A-Row
see Tic-Tac-Toe
Three Jug Problem
Given three jugs with x pints in the first, y in the second,
and z in the third, obtain a desired amount in one of the
vessels by completely filling up and/or emptying vessels
into others. This problem can be solved with the aid of
Trilinear Coordinates.
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 89-93, 1967.
a = 0.0 2
a x = O.OI2
a 2 = O.OIIO2
a s = O.OIIOIOOI2
a 4 = 0.0110100110010110 2 .
This can be written symbolically as
_ n -2 n
Q>n + 1 — CL n *+- a n ■ &
(2)
(3)
with ao = 0. Here, the complement is the number a n
such that a„+a^" = 0.11...2, which can be found from
a n + a n = y^(|) fe
G) 2 " .
1 -^
(4)
Thue-Morse Sequence
Therefore,
a n = 1 — a n
and
— 2 n \ — 2 n
a n +i = a n + (1 — 2 — a n )2
(5)
(6)
The regular CONTINUED FRACTION for the Thue-Morse
constant is [0 2221435214215 44 141241
115 14 1 50 15 511142141 43 141213 16 1
2121 50 12 424 1252111552 22 5111 1274
352111411 15 154 721221211 50 141
2 867374 111551161272 1650 23 3 1 1 1 2 5
3 84 1 1 1 1284 , . .] (Sloane's A014572), and seems to
continue with sporadic large terms in suspicious-looking
patterns. A nonregular CONTINUED FRACTION is
1
(7)
2-
4-
16
15
256-
255
65536 - . .
A related infinite product is
1-3- 15 -255 -65535 •
4P:
2 • 4 ■ 16 • 256 ■ 65536 •
(8)
The Sequence aoo = 0110100110010110100101100...
(Sloane's A010060) is known as the Thue-Morse SE-
QUENCE.
see also Rabbit Constant, Thue Constant
References
Allouche, J. P.; Arnold, A.; Berstel, J.; Brlek, S.; Jockusch,
W.; Plouffe, S.; and Sagan, B. "A Relative of the Thue-
Morse Sequence." Discr. Math. 139, 455-461, 1995.
Allouche, J. P. and Shallit, J. In preparation.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Item 122, Feb. 1972.
Dekking, F. M. "Transcendence du nombre de Thue-Morse."
Comptes Rendus de I'Academie des Sciences de Paris 285,
157-160, 1977.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/cntf rc/cntf re .html.
Sloane, N. J. A. Sequences A010060, A014571, and A014572
in "An On-Line Version of the Encyclopedia of Integer Se-
quences."
Thue-Morse Sequence
The Integer Sequence (also called the Morse-Thue
Sequence)
01101001100101101001011001101001.
(1)
(Sloane's A010060) which arises in the Thue-Morse
Constant. It can be generated from the Substitution
Map
0->01
1 -+ 10
(2)
(3)
Thue-Morse Sequence 1807
starting with as follows:
-» 01 -> 0110 -> 01101001 ->.... (4)
Writing the sequence as a POWER SERIES over the GA-
LOIS Field GF(2),
F(x) = + lx + lx 2 + Ox 3 + lx A + . . . , (5)
then F satisfies the quadratic equation
(1 + x)F 2 + F - — ^ (mod 2). (6)
I + x*
This equation has two solutions, F and F', where F* is
the complement of F, i.e.,
F + F' = l + x + x 2 +x z + ..
1 + x'
(7)
which is consistent with the formula for the sum of the
roots of a quadratic. The equality (6) can be demon-
strated as follows. Let (abedef. . . ) be a shorthand for
the Power series
a + bx + ex 2 + dx + . . . ,
(8)
so F{x) is (0110100110010110...). To get F 2 , simply
use the rule for squaring Power Series over GF(2)
(A + B) 2 =A 2 + B 2 (mod 2),
(9)
which extends to the simple rule for squaring a POWER
Series
(ao + aix-f a2# 2 + . . .) = ao + a±x +a,2X +... (mod 2),
(10)
i.e., space the series out by a factor of 2, (0 110100
1 . . . ), and insert zeros in the Odd places to get
F 2 = (0010100010000010 . . .).
(11)
Then multiply by x (which just adds a zero at the front)
to get
xF 2 = (00010100010000010 . . .). (12)
Adding to F 2 gives
(1 + x)F 2 = (0011110011000011 . . .). (13)
This is the first term of the quadratic equation, which
is the Thue-Morse sequence with each term doubled up.
The next term is F, so we have
(1 + x)F 2 = (0011110011000011 . . .) (14)
F = (0110100110010110 . . .). (15)
1808 Thue Sequence
Thurston's Geometrization Conjecture
The sum is the above two sequences XORed together
(there are no CARRIES because we're working over
GF(2)), giving
(1 + x)F 2 + F = (0101010101010101 . . .). (16)
We therefore have
(l + :z)F 2 +F= x
l + x 2
= x + x 3 + x 5 + x 7 -f x 9 + x 11 + . . . (mod 2). (17)
The Thue-Morse sequence is an example of a cube-
free sequence on two symbols (Morse and Hedlund
1944), i.e., it contains no substrings of the form VFV^VF,
where W is any WORD. For example, it does not con-
tain the Words 000, 010101 or 010010010. In fact,
the following stronger statement is true: the Thue-
Morse sequence does not contain any substrings of the
form W^Wa, where a is the first symbol of W. We
can obtain a SQUAREFREE sequence on three sym-
bols by doing the following: take the Thue-Morse se-
quence 0110100110010110... and look at the sequence
of WORDS of length 2 that appear: 01 11 10 01 10 00
01 11 10 ... . Replace 01 by 0, 10 by 1, 00 by 2 and
11 by 2 to get the following: 021012021. . . . Then this
SEQUENCE is Squarefree (Morse and Hedlund 1944).
The Thue-Morse sequence has important connections
with the Gray Code. Kindermann generates fractal
music using the Self- Similarity of the Thue-Morse
sequence.
see also Gray Code, Parity Constant, Rabbit Se-
quence, Thue Sequence
References
Kindermann, L. "MusiNum — The Music in the Numbers."
http:// www . forwiss . uni - erlangen . de/ - kinderma /
musinum/.
Morse, M. and Hedlund, G. A. "Unending Chess, Symbolic
Dynamics, and a Problem in Semigroups." Duke Math. J.
11, 1-7, 1944.
Schroeder, M. R. Fractals, Chaos, and Power Laws: Minutes
from an Infinite Paradise. New York: W. H. Freeman,
1991.
Sloane, N. J. A. Sequence A010060 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Thue Sequence
The Sequence of Binary Digits of the Thue Con-
stant, 0.110110111110110111110110110... 2 (Sloane's
A014578).
see also Rabbit Constant, Thue Constant
References
Guy, R. K. "Thue Sequences." §E21 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 223-224, 1994.
Sloane, N. J. A. Sequence A014578 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Thue-Siegel-Roth Theorem
If a is a Transcendental Number, it can be approx-
imated by infinitely many RATIONAL NUMBERS m/n to
within n~ r , where r is any POSITIVE number.
see also LlOUVILLE'S RATIONAL APPROXIMATION THE-
OREM, LlOUVILLE-ROTH CONSTANT, ROTH'S THEO-
REM
Thue-Siegel-Schneider-Roth Theorem
see Thue-Siegel-Roth Theorem
Thue's Theorem
If n > 1, (a, n) = 1 (i.e., a and n are RELATIVELY
Prime), and m is the least integer > y/n, then there
exist an x and y such that
ay = ±x (mod n)
where < x < m and < y < m.
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 161, 1993.
Thurston's Geometrization Conjecture
Thurston's conjecture has to do with geometric struc-
tures on 3-D Manifolds. Before stating Thurston's
conjecture, some background information is useful. 3-
dimensional MANIFOLDS possess what is known as a
standard 2-level DECOMPOSITION. First, there is the
Connected Sum Decomposition, which says that ev-
ery Compact 3-Manifold is the Connected Sum of
a unique collection of PRIME 3-Manifolds.
The second DECOMPOSITION is the JACO-SHALEN-
Johannson Torus Decomposition, which states that
irreducible orientable COMPACT 3-MANIFOLDS have a
canonical (up to ISOTOPY) minimal collection of dis-
joint ly Embedded incompressible Tori such that each
component of the 3-MANIFOLD removed by the TORI is
either "atoroidal" or "Seifert-fibered."
Thurston's conjecture is that, after you split a 3-
MANIFOLD into its CONNECTED Sum and then JACO-
Shalen-Johannson Torus Decomposition, the re-
maining components each admit exactly one of the fol-
lowing geometries:
1. Euclidean Geometry,
2. Hyperbolic Geometry,
3. Spherical Geometry,
4. the Geometry of S 2 x M,
5. the Geometry of H 2 x R,
6. the Geometry of SL 2 R,
7. Nil Geometry, or
8. Sol Geometry.
Thwaites Conjecture
Tiling 1809
Here, § 2 is the 2-SPHERE and H 2 is the HYPERBOLIC
PLANE. If Thurston's conjecture is true, the truth of
the Poincare CONJECTURE immediately follows.
see also Connected Sum Decomposition, Euclid-
ean Geometry, Hyperbolic Geometry, Jaco-
Shalen-Johannson Torus Decomposition, Nil Ge-
ometry, Poincare Conjecture, Sol Geometry,
Spherical Geometry
Thwaites Conjecture
see Collatz Problem
Tic-Tac-Toe
The usual game of tic-tac-toe (also called TlCKTACK-
TOE) is 3-in-a-row on a 3 x 3 board. However, a gen-
eralized n-lN-A-Row on an n x m board can also be
considered. For n = 1 and 2 the first player can always
win. If the board is at least 3x4, the first player can
win for n = 3.
However, for TlC-TAC-TOE which uses a 3 X 3 board,
a draw can always be obtained. If the board is at least
4 x 30, the first player can win for n = 4. For n = 5, a
draw can always be obtained on a 5 x 5 board, but the
first player can win if the board is at least 15 X 15. The
cases n = 6 and 7 have not yet been fully analyzed for
an n x n board, although draws can always be forced
for n = 8 and 9. On an oo x co board, the first player
can win for n = 1, 2, 3, and 4, but a tie can always be
forced for n > 8. For 3x3x3 and 4x4x4, the first
player can always win (Gardner 1979).
see also PONG HAU K'l
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 103-
104, 1987.
de Fouquieres, B. Ch. 18 in Les Jeux des Anciens } 2nd ed.,
Paris, 1873.
Gardner, M. "Mathematical Games: The Diverse Pleasures
of Circles that Are Tangent to One Another." Sci. Amer.
240, 18-28, Jan. 1979a.
Gardner, M. "Ticktacktoe Games." Ch. 9 in Wheels, Life,
and Other Mathematical Amusements. New York: W. H,
Freeman, 1983.
Stewart, I. "A Shepherd Takes A Sheep Shot." Sci. Amer.
269, 154-156, 1993.
Ticktacktoe
see Tic-Tac-Toe
Tight Closure
The application of characteristic p methods in COMMU-
TATIVE ALGEBRA, which is a synthesis of some areas of
Commutative Algebra and Algebraic Geometry.
see also Algebraic Geometry, Commutative Alge-
bra
References
Bruns, W. "Tight Closure." Bull. Amer. Math. Soc. 33,
447-457, 1996.
Huneke, C. "An Algebraist Commuting in Berkeley." Math.
Intell. 11, 40-52, 1989.
Tightly Embedded
Q is said to be tightly embedded if \QC\Q g \ is Odd for
all g € G - Ng{Q), where N G {Q) is the NORMALIZER
of Q in G.
Tiling
A plane-filling arrangement of plane figures or its gener-
alization to higher dimensions. Formally, a tiling is a col-
lection of disjoint open sets, the closures of which cover
the plane. Given a single tile, the so-called first Corona
is the set of all tiles that have a common boundary point
with the tile (including the original tile itself).
Wang's Conjecture (1961) stated that if a set of tiles
tiled the plane, then they could always be arranged to
do so periodically. A periodic tiling of the PLANE by
Polygons or Space by Polyhedra is called a Tes-
sellation. The conjecture was refuted in 1966 when
R. Berger showed that an aperiodic set of 20,426 tiles
exists. By 1971, R. Robinson had reduced the num-
ber to six and, in 1974, R. Penrose discovered an aperi-
odic set (when color- matching rules are included) of two
tiles: the so-called PENROSE TILES. (Penrose also sued
the Kimberly Clark Corporation over their quilted toi-
let paper, which allegedly resembles a Penrose aperiodic
tiling; Mirsky 1997.)
It is not known if there is a single aperiodic tile.
n-gon
tilings
3
4
5
6
any
any
14
3
The number of tilings possible for convex irregular
Polygons are given in the above table. Any TRIAN-
GLE or convex QUADRILATERAL tiles the plane. There
are at least 14 classes of convex PENTAGONAL tilings.
There are at least three aperiodic tilings of HEXAGONS,
given by the following types:
A -r B + C = 360°
A + B + D = 360°
A = C = E
a — d
a = d,c = e
a — b>c = d,e = f
(i)
(Gardner 1988). Note that the periodic hexagonal TES-
SELLATION is a degenerate case of all three tilings with
A = B = C = D = E = F a = b = c^=d = e = f.
(2)
d
1810 Tiling Theorem
Toeplitz Matrix
There are no tilings for convex n-gons for n>7.
see also Anisohedral Tiling, Corona (Tiling),
Gosper Island, Heesch's Problem, Isohedral
Tiling, Koch Snowflake, Monohedral Tiling,
Penrose Tiles, Polyomino Tiling, Space-Filling
Polyhedron, Tiling Theorem, Triangle Tiling
References
Eppstein, D. "Tiling." http://www.ics.uci.edu/-eppstein
/junkyard/tiling. html.
Gardner, M. "Tilings with Convex Polygons." Ch. 13
in Time Travel and Other Mathematical Bewilderments.
New York: W. H. Freeman, pp. 162-176, 1988,
Gardner, M. Chs. 1-2 in Penrose Tiles to Trapdoor
Ciphers. . . and the Return of Dr. Matrix, reissue ed.
Washington, DC: Math. Assoc. Amer.
Grunbaum, B. and Shepard, G. C. "Some Problems on Plane
Tilings." In The Mathematical Gardner (Ed. D. Klarner).
Boston, MA: Prindle, Weber, and Schmidt, pp. 167-196,
1981.
Grunbaum, B. and Sheppard, G. C. Tilings and Patterns.
New York: W. H. Freeman, 1986.
Lee, X. "Visual Symmetry." http://www.best.com/-xah/
MathGraphicsGalleryjiir/Tiling_dir/t iling.html.
Mirsky, S. "The Emperor's New Toilet Paper." Sci. Amer.
277, 24, July 1997.
Pappas, T. "Mathematics & Moslem Art." The Joy of Math-
ematics. San Carlos, CA: Wide World Publ./Tetra, p. 178,
1989.
Peterson, I. The Mathematical Tourist: Snapshots of Modern
Mathematics. New York: W. H. Freeman, pp. 82-85, 1988.
Rawles, B. Sacred Geometry Design Sourcebook: Uni-
versal Dimensional Patterns. Nevada City, CA:
Elysian Pub., 1997. http://www.oro.net/-elysian/
bruce_rawles_books .html.
S chat t Schneider, D. "In Praise of Amateurs." In The Math-
ematical Gardner (Ed. D. Klarner). Boston, MA: Prindle,
Weber, and Schmidt, pp. 140-166, 1981.
Seyd, J. A. and Salman, A. S. Symmetries of Islamic Geo-
metrical Patterns. River Edge, NJ: World Scientific, 1995.
Stein, S. and Szabo, S. Algebra and Tiling. Washington, DC:
Math. Assoc. Amer., 1994.
Tiling Theorem
Due to Lebesgue and Brouwer. If an n-D figure is cov-
ered in any way by sufficiently small subregions, then
there will exist points which belong to at least n + 1 of
these subareas. Moreover, it is always possible to find a
covering by arbitrarily small regions for which no point
will belong to more than n + 1 regions.
see also TESSELLATION, TILING
Times
The operation of MULTIPLICATION, i.e., a times b. Vari-
ous notations are a x 6, a* 6, a&, and (a)(6). The "multi-
plication sign" x is based on Saint Andrew's C „oss
(Bergamini 1969). Floating point MULTIPLICATION is
sometimes denoted <g>.
see also Cross Product, Dot Product, Minus,
Multiplication, Plus, Product
References
Bergamini, D. Mathematics. New York: Time-Life Books,
p. 11, 1969.
Tit-for-Tat
A strategy for the iterated PRISONER'S DILEMMA in
which a prisoner cooperates on the first move, and there-
after copies the previous move of the other prisoner. Any
better strategy has more complicated rules.
see also Prisoner's Dilemma
References
Goetz, P. "Phil's Good Enough Complexity Dictionary."
http : //www . cs .buf f alo . edu/~goetz/dict .html.
Titanic Prime
A Prime with > 1000 Digits. As of 1990, there were
more than 1400 known (Ribenboim 1990). The table
below gives the number of known titanic primes as a
function of year end.
Year Titanic Primes
1992
1993
1994
1995
2254
9166
9779
12391
References
Caldwell, C. "The Ten Largest Known Primes." http: //www.
utm.edu/research/primes/largest .html#largest.
Morain, F. "Elliptic Curves, Primality Proving and Some
Titanic Primes." Asterique 198-200, 245-251, 1992.
Ribenboim, P. The Little Book of Big Primes. Berlin:
Springer- Verlag, p. 97, 1990.
Yates, S. "Titanic Primes." J. Recr. Math. 16, 250-262,
1983-84.
Yates, S. "Sinkers of the Titanics." J. Recr. Math. 17, 268-
274, 1984-85.
Titchmarsh Theorem
If f(uj) is Square Integrable over the Real o> axis,
then any one of the following implies the other two:
1. The Fourier Transform of /(a;) is for t < 0.
2. Replacing cj by z, the function f(z) is analytic in
the Complex Plane z for y > and approaches
f(x) almost everywhere as y — y 0. Furthermore,
ST \f( x +iy)\ 2 dx < k for some number k and y >
(i.e., the integral is bounded).
3. The Real and Imaginary Parts of f(z) are
Hilbert Transforms of each other.
Tits Group
A finite Simple Group which is a Subgroup of the
Twisted Chevalley Group 2 F 4 (2).
Toeplitz Matrix
Given 2N — 1 numbers r^ where k = —N + 1, . . . , — 1,
0, 1, . . . , N - 1, a Matrix of the form
7*1
r-i
r
r-2
r-i
r-n+i
T-n+2
ro
Togliatti Surface
Topological Entropy 1811
is called a Toeplitz matrix. MATRIX equations of the
form
N
/ ^ r i-j x j = Vi
3 = 1
can he solved with G(N 2 ) operations.
see also Vandermonde Matrix
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Vandermonde Matrices and Toeplitz Matri-
ces." §2.8 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 82-89, 1992.
Togliatti Surface
Togliatti (1940, 1949) showed that QuiNTIC SURFACES
having 31 ORDINARY DOUBLE POINTS exist, although
he did not explicitly derive equations for such sur-
faces. Beauville (1978) subsequently proved that 31 dou-
ble points are the maximum possible, and quintic sur-
faces having 31 ORDINARY DOUBLE Points are there-
fore sometimes called Togliatti surfaces, van Straten
(1993) subsequently constructed a 3-D family of solu-
tions and in 1994, Barth derived the example known as
the Dervish.
see also Dervish, Ordinary Double Point, Quintic
Surface
References
Beauville, A. "Surfaces algebriques complexes." Asterisque
54, 1-172, 1978.
Endrafl, S. "Togliatti Surfaces." http://www . matheraatik .
uni - mainz . de / Algebraische Geometrie / docs /
Etogliatti . shtml.
Hunt, B. "Algebraic Surfaces." http://www.mathematik.
uni-kl . de/~wwwagag/Galerie .html.
Togliatti, E. G. "Una notevole superficie de 5° ordine con
soli punti doppi isolati." Vierteljschr. Naturforsch. Ges.
Zurich 85, 127-132, 1940.
Togliatti, E. "Sulle superficie monoidi col massimo numero di
punti doppi." Ann. Mat Pura AppL 30, 201-209, 1949.
van Straten, D. "A Quintic Hypersurface in P with 130
Nodes." Topology 32, 857-864, 1993.
Tomography
Tomography is the study of the reconstruction of 2- and
3-dimensional objects from 1-dimensional slices. The
RADON TRANSFORM is an important tool in tomogra-
phy.
Rather surprisingly, there exist certain sets of four direc-
tions in Euclidean n-space such that X-rays of a convex
body in these directions distinguish it from all other
convex bodies.
see also Aleksandrov's Uniqueness Theorem,
Brunn-Minkowski Inequality, Busemann-Petty
Problem, Dvoretzky's Theorem, Radon Trans-
form, Stereology
References
Gardner, R. J. "Geometric Tomography." Not. Amer. Math,
Soc. 42, 422-429, 1995.
Gardner, R. J. Geometric Tomography. New York: Cam-
bridge University Press, 1995.
Tooth Surface
The Quartic Surface given by the equation
-« ■ ~« ■ z 4 -(x 2 +y 2 + 2?) = 0.
References
Nordstrand, T. "Surfaces."
nfytn/ surf aces. htm.
http : //www . uib . no/people/
Topological Basis
A topological basis is a SUBSET B of a Set T in which
all other Open Sets can be written as UNIONS or finite
Intersections of B. For the Real Numbers, the Set
of all Open Intervals is a basis.
Topological Completion
The topological completion C of a FIELD F with respect
to the Absolute Value | • | is the smallest Field con-
taining F for which all Cauchy Sequences or rationals
converge.
References
Burger, E. B. and Struppeck, T. "Does ]T)~ =0 ^ Reall y Con "
verge? Infinite Series and p-adic Analysis." Amer. Math.
Monthly 103, 565-577, 1996.
Topologically Conjugate
Two Maps <fi,ip : M —* M are said to be topologically
conjugate if there Exists a Homeomorphism h : M ->
M such that <j> o h = h o -0, i.e., h maps ^-orbits onto
</>-orbits. Two maps which are topologically conjugate
cannot be distinguished topologically.
see also ANOSOV DlFFEOMORPHISM, STRUCTURALLY
Stable
Topological Dimension
see LEBESGUE COVERING DIMENSION
Topological Entropy
The topological entropy of a MAP M is defined as
kr(M) = sup fe(M,{Wi}),
{Wi}
where {Wi} is a partition of a bounded region W con-
taining a probability measure which is invariant under
M, and sup is the Supremum.
References
Ott, E. Chaos in Dynamical Systems. New York: Cambridge
University Press, pp. 143-144, 1993,
1812
Topological Groupoid
Topology
Topological Groupoid
A topological groupoid over B is a GROUPOID G such
that B and G are TOPOLOGICAL SPACES and a, j3, and
multiplication are continuous maps. Here, a and f3 are
maps from G onto R with a : (z,7,y) \-> x and (3 :
(a;, 7,2/) *-► y-
References
Weinstein, A. "Groupoids: Unifying Internal and External
Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996.
Topological Manifold
A TOPOLOGICAL Space M satisfying some separability
(i.e., it is a Hausdorff Space) and countability (i.e., it
is a PARACOMPACT Space) conditions such that every
point p £ M has a NEIGHBORHOOD homeomorphic to
an Open Set in R n for some n > 0. Every Smooth
Manifold is a topological manifold, but not necessarily
vice versa. The first nonsmooth topological manifold
occurs in 4-D.
Nonparacompact manifolds are of little use in math-
ematics, but non-Hausdorff manifolds do occasionally
arise in research (Hawking and Ellis 1975). For man-
ifolds, Hausdorff and second countable are equivalent
to Hausdorff and paracompact, and both are equiva-
lent to the manifold being embeddable in some large-
dimensional Euclidean space.
see also HAUSDORFF SPACE, MANIFOLD, PARACOM-
PACT Space, Smooth Manifold, Topological
Space
References
Hawking, S. W. and Ellis, G. F. R. The Large Scale Structure
of Space-Time. New York: Cambridge University Press,
1975.
Topological Space
A Set X for which a Topology T has been specified
is called a topological space (Munkres 1975, p. 76).
see also Kuratowski's Closure-Component Prob-
lem, Open Set, Topological Vector Space
References
Berge, C. Topological Spaces Including a Treatment of Multi-
Valued Functions, Vector Spaces and Convexity. New
York: Dover, 1997.
Munkres, J. R. Topology: A First Course. Englewood Cliffs,
NJ: Prentice-Hall, 1975.
Topological Vector Space
A Topological Space such that the two algebraic op-
erations of Vector Space are continuous in the topol-
ogy-
References
Kothe, G. Topological Vector Spaces. New York: Springer-
Verlag, 1979.
Topologically Transitive
A Function / is topologically transitive if, given any
two intervals U and V, there is some Positive Integer
k such that f k {U) n V = 0. Vaguely, this means that
neighborhoods of points eventually get flung out to "big"
sets so that they don't necessarily stick together in one
localized clump.
see also Chaos
Topology
Topology is the mathematical study of properties of ob-
jects which are preserved through deformations, twist-
ings, and stretchings. (Tearing, however, is not allowed.)
A Circle is topologically equivalent to an Ellipse (into
which it can be deformed by stretching) and a SPHERE
is equivalent to an ELLIPSOID. Continuing along these
lines, the Space of all positions of the minute hand on
a clock is topologically equivalent to a CIRCLE (where
Space of all positions means "the collection of all po-
sitions"). Similarly, the Space of all positions of the
minute and hour hands is equivalent to a TORUS. The
Space of all positions of the hour, minute and second
hands form a 4-D object that cannot be visualized quite
as simply as the former objects since it cannot be placed
in our 3-D world, although it can be visualized by other
means.
There is more to topology, though. Topology began with
the study of curves, surfaces, and other objects in the
plane and 3-space. One of the central ideas in topology
is that spatial objects like CIRCLES and SPHERES can
be treated as objects in their own right, and knowledge
of objects is independent of how they are "represented"
or "embedded" in space. For example, the statement
"if you remove a point from a CIRCLE, you get a line
segment" applies just as well to the CIRCLE as to an
Ellipse, and even to tangled or knotted CIRCLES, since
the statement involves only topological properties.
Topology has to do with the study of spatial objects
such as curves, surfaces, the space we call our universe,
the space-time of general relativity, fractals, knots, man-
ifolds (objects with some of the same basic spatial prop-
erties as our universe), phase spaces that are encoun-
tered in physics (such as the space of hand-positions of
a clock), symmetry groups like the collection of ways of
rotating a top, etc.
The "objects" of topology are often formally defined as
Topological Spaces. If two objects have the same
topological properties, they are said to be HOMEOMOR-
PHIC (although, strictly speaking, properties that are
not destroyed by stretching and distorting an object are
really properties preserved by ISOTOPY, not HOMEO-
MORPHISM; ISOTOPY has to do with distorting embed-
ded objects, while HOMEOMORPHISM is intrinsic).
Topology is divided into Algebraic TOPOLOGY (also
called Combinatorial Topology), Differential
Topology, and Low-Dimensional Topology.
Topology
Torispherical Dome 1813
There is also a formal definition for a topology defined in
terms of set operations. A Set X along with a collection
T of Subsets of it is said to be a topology if the Subsets
in T obey the following properties:
1. The (trivial) subsets X and the EMPTY Set are
inT.
2. Whenever sets A and B are in T, then so is An B.
3. Whenever two or more sets are in T, then so is their
Union
(Bishop and Goldberg 1980).
A Set X for which a topology T has been specified
is called a TOPOLOGICAL Space (Munkres 1975, p. 76).
For example, the SetX — {0, 1, 2, 3} together with the
Subsets T = {0}, {1, 2, 3}, 0, {0, 1, 2, 3}} comprises
a topology, and X is a Topological Space.
Topologies can be built up from TOPOLOGICAL BASES.
For the REAL NUMBERS, the topology is the UNION of
Open Intervals.
see also ALGEBRAIC TOPOLOGY, DIFFERENTIAL TO-
POLOGY, Genus, Klein Bottle, Kuratowski Re-
duction Theorem, Lefshetz Trace Formula,
Low-Dimensional Topology, Point-Set Topol-
ogy, Zariski Topology
References
Adamson, I. A General Topology Workbook. Boston, MA:
Birkhauser, 1996.
Armstrong, M. A. Basic Topology, rev. New York: Springer-
Verlag, 1997.
Barr, S. Experiments in Topology. New York: Dover, 1964.
Berge, C. Topological Spaces Including a Treatment of Multi-
Valued Functions, Vector Spaces and Convexity. New
York: Dover, 1997.
Bishop, R. and Goldberg, S. Tensor Analysis on Manifolds.
New York: Dover, 1980.
Blackett, D. W. Elementary Topology: A Combinatorial and
Algebraic Approach. New York: Academic Press, 1967.
Bloch, E. A First Course in Geometric Topology and Differ-
ential Geometry. Boston, MA: Birkhauser, 1996.
Chinn, W. G. and Steenrod, N. E. First Concepts of To-
pology: The Geometry of Mappings of Segments, Curves,
Circles, and Disks. Washington, DC: Math. Assoc. Amer.,
1966.
Eppstein, D. "Geometric Topology." http://www.ics.uci.
edu/*eppstein/junkyard/topo.html.
Francis, G. K. A Topological Picturebooh. New York:
Springer- Verlag, 1987.
Gemignani, M. C. Elementary Topology. New York: Dover,
1990.
Greever, J. Theory and Examples of Point- Set Topology. Bel-
mont, CA: Brooks/Cole, 1967.
Hirsch, M. W. Differential Topology. New York: Springer-
Verlag, 1988.
Hocking, J. G. and Young, G. S. Topology. New York: Dover,
1988.
Kahn, D. W. Topology: An Introduction to the Point-Set and
Algebraic Areas, New York: Dover, 1995.
Kelley, J. L. General Topology. New York: Springer- Verlag,
1975.
Kinsey, L. C. Topology of Surfaces. New York: Springer-
Verlag, 1993.
Lipschutz, S. Theory and Problems of General Topology.
New York: Schaum, 1965.
Mendelson, B. Introduction to Topology. New York: Dover,
1990.
Munkres, J. R. Elementary Differential Topology. Princeton,
NJ: Princeton University Press, 1963.
Munkres, J. R. Topology: A First Course. Englewood Cliffs,
NJ: Prentice-Hall, 1975.
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.
Shakhmatv, D. and Watson, S. "Topology Atlas." http://
www.unipissing. ca/topology/.
Steen, L. A. and Seebach, J. A. Jr. Counterexamples in To-
pology, New York: Dover, 1996.
Thurston, W. P. Three- Dimensional Geometry and Topology,
Vol. 1. Princeton, NJ: Princeton University Press, 1997.
van Mill, J. and Reed, G. M. (Eds.). Open Problems in To-
pology. New York: Elsevier, 1990.
Veblen, O. Analysis Situs, 2nd ed. New York: Amer. Math.
Soc, 1946.
Top os
A Category modeled after the properties of the Cat-
egory of sets.
see also CATEGORY, LOGOS
References
Freyd, P. J. and Scedrov, A. Categories, Allegories. Amster-
dam, Netherlands: North-Holland, 1990.
McLarty, C. Elementary Categories, Elementary Toposes.
New York: Oxford University Press, 1992.
Toric Variety
Let mi, m 2 , . . . , m n be distinct primitive elements of
a 2-D Lattice M such that det(mi,rai+i) > for i ~
1, . . . , n. Each collection V = {mi,m2, . . . , m n } then
forms a set of rays of a unique complete fan in M, and
therefore determines a 2-D toric variety Xr-
Russ.
References
Danilov, V. I. "The Geometry of Toric Varieties."
Math. Surv, 33, 97-154, 1978.
Fulton, W. Introduction to Toric Varieties. Princeton, NJ:
Princeton University Press, 1993.
Morelli, R. "Pick's Theorem and the Todd Class of a Toric
Variety." Adv. Math. 100, 183-231, 1993.
Oda, T. Convex Bodies and Algebraic Geometry. New York:
Springer- Verlag, 1987.
Pommersheim, J. E. "Toric Varieties, Lattice Points, and
Dedekind Sums." Math. Ann. 295, 1-24, 1993.
Torispherical Dome
A torispherical dome is the surface obtained from the
intersection of a SPHERICAL Cap with a tangent TORUS,
as illustrated above. The radius of the sphere R is called
1814
Torn Square Fractal
Toroidal Field
the "crown radius," and the radius of the torus is called
the "knuckle radius." Torispherical domes are used to
construct pressure vessels.
see also Dome, Spherical Cap
Torn Square Fractal
see CESARO FRACTAL
Toroid
A Surface of Revolution obtained by rotating a
closed Plane Curve about an axis parallel to the plane
which does not intersect the curve. The simplest toroid
is the Torus.
see also PAPPUS'S CENTROID THEOREM, SURFACE OF
Revolution, Torus
Toroidal Coordinates
A system of CURVILINEAR COORDINATES for which sev-
eral different notations are commonly used. In this work
(u, v y <j>) is used, whereas Arfken (1970) uses (£, 77, ip).
The toroidal coordinates are defined by
y =
a sinh v cos <fi
coshv — cosu
a sinh v sin <f>
cosh v — cos u
asinu
cosh v — cos u '
(i)
(2)
(3)
where sinh 2 is the HYPERBOLIC SINE and coshz is the
Hyperbolic Cosine. The Scale Factors are
h u
h v
h<j> ■
cosh v — cos u
a
cosh v — cos u
a sinh v
cosh v — cos u
(4)
(5)
(6)
The Laplacian is
(coshv — cosuf d
V 2 /
d/>
du
+
.cosht; -
sinhv
(coshv — cosw) 3 d ,
a 2 sinh v dv \ cosh v -
(cosh?; — cosu) 2 d 2 f
cos u du j
iu dv J
a 2 sinh v
d<j> 2
(7)
+
—3 cos coth v + cosh v coth v
cosh v — cos u
+3 cos 2 u coth v csch v — cos 3 u csch 2
cosh v — cos u
v\eP_
) W
+(cos u — cosh v) sin u — — h (cosh v — cos u) 2 — — -
du du 2
+ (cosh v — cos u) (cosh v coth v — sinh v
— cos u coth v
5 _L. f V, 2 \* ^
+ (COSh V — COSU) -^-r-.
dt?
<% 2 '
(8)
The Helmholtz Differential Equation is not sepa-
rable in toroidal coordinates, but LAPLACE'S EQUATION
is,
see also Bispherical Coordinates, Laplace's
Equation — Toroidal Coordinates
References
Arfken, G. "Toroidal Coordinates (£, r?, <£)." §2.13 in Math-
ematical Methods for Physicists, 2nd ed. Orlando, FL:
Academic Press, pp. 112-115, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, p. 666, 1953.
Toroidal Field
A Vector Field resembling a TORUS which is purely
circular about the z-Axis of a SPHERE (i.e., follows lines
of Latitude). A toroidal field takes the form
1 or
sin 6 d<j>
dT
de
see also Divergenceless Field, Poloidal Field
References
Stacey, F. D. Physics of the Earth, 2nd ed. New York: Wiley,
p. 239, 1977.
Toroidal Function
Torsion (Differential Geometry) 1815
Toroidal Function
A class of functions also called Ring Functions which
appear in systems having toroidal symmetry. Toroidal
functions can be expressed in terms of the Legendre
Functions and Second Kinds (Abramowitz and Ste-
gun 1972, p. 336):
PZ_ 1/2 (coshr,) = [T(l -/Or^l - e-^r^e
x 2*1(5 - & \ + v ~ w 1 - 2 M ; 1 - e~ 2r) )
-( 1 /+l/2)r ?
Pn- 1/2 (cOSh I?)
T(n + m+ |)(sinh77) n
F
Jo
T(n - m + \)2™y/^T{m + \)
sin 2
" 7n <fid(fi
/0 (cosh rj + cos0sinh7?) n+m+1 / 2
Q^_ 1/2 (cosh^) = [T(l + ^p^e^r^ + Z/ + /x)
(-l)"T(n+|)
<2n-l/2(cOsh77)
JO
r(n-m+i)
cosh(mt) d£
/Q (cosh 77 + coshisinhr7) n+1 / 2
for n > m. Byerly (1959) identifies
— P n (cothz) - csch n x dnPm(c ° thx)
. n/2 P m (coth^-csch x d{cQthx)n
as a Toroidal Harmonic.
see also Conical Function
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Toroidal Func-
tions (or Ring Functions)." §8.11 in Handbook of Mathe-
matical Functions with Formulas, Graphs, and Mathemat-
ical Tables, 9th printing. New York: Dover, p, 336, 1972.
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. 266, 1959.
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1468,
1980.
Toroidal Harmonic
see Toroidal Function
Toroidal Polyhedron
A toroidal polyhedron is a POLYHEDRON with Genus
g > 1 (i.e., having one or more Holes). Examples of
toroidal polyhedra include the CSASZAR POLYHEDRON
and Szilassi Polyhedron, both of which have Genus
1 (i.e., the TOPOLOGY of a TORUS).
The only known Toroidal Polyhedron with no Di-
agonals is the CSASZAR POLYHEDRON. If another ex-
ists, it must have 12 or more Vertices and Genus
g > 6. The smallest known single-hole toroidal POLY-
HEDRON made up of only Equilateral Triangles is
composed of 48 of them.
see also Csaszar Polyhedron, Szilassi Polyhedron
References
Gardner, M. Time Travel and Other Mathematical Bewilder-
ments. New York: W. H. Freeman, p. 141, 1988.
Hart, G. "Toroidal Polyhedra." http://www.li.net/
-george/virtual-polyhedr a/toroidal. html.
Stewart, B. M. Adventures Among the Toroids, 2nd rev. ed.
Okemos, Ml: B. M. Stewart, 1984.
Toronto Function
TV , _ r(|m +|) . 2 ^
T{m,n,r) = * ^ fl - ~-
fjiiM£;t + n;r 2 ),
where 1 F 1 (a;b-z) is a CONFLUENT HYPERGEOMET-
ric Function and F(z) is the Gamma Function
(Abramowitz and Stegun 1972).
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 509, 1972.
Torricelli Point
see Fermat Point
Torsion (Differential Geometry)
The rate of change of the OSCULATING PLANE of a
Space Curve. The torsion r is Positive for a right-
handed curve, and Negative for a left-handed curve.
A curve with Curvature k # is planar Iff r = 0.
The torsion can be defined by
tee-N-B',
where N is the unit Normal VECTOR and B is the
unit Binormal Vector. Written explicitly in terms of
a parameterized VECTOR FUNCTION x,
XXX a.. « »..
T = .. .. =p |XXX|,
where |abc| denotes a SCALAR TRIPLE PRODUCT and
p is the Radius of Curvature. The quantity 1/r is
called the RADIUS OF TORSION and is denoted a or <j>.
see also Curvature, Radius of Curvature, Radius
of Torsion
References
Gray, A. "Drawing Space Curves with Assigned Curvature."
§7.8 in Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 145-147, 1993.
Kreyszig, E. "Torsion," §14 in Differential Geometry. New
York: Dover, pp. 37-40, 1991.
1816 Torsion (Group Theory)
Torus
Torsion (Group Theory)
If G is a GROUP, then the torsion elements Tor(G) of G
(also called the torsion of G) are denned to be the set
of elements g in G such that g n = e for some NATURAL
NUMBER n, where e is the IDENTITY ELEMENT of the
Group G.
In the case that G is Abelian, Tor(G) is a Subgroup
and is called the torsion subgroup of G. If Tor(G) con-
sists only of the Identity Element, the GROUP G is
called torsion-free.
see also Abelian Group, Group, Identity Element
Torsion Number
One of a set of numbers denned in terms of an invariant
generated by the finite cyclic covering spaces of a Knot
complement. The torsion numbers for KNOTS up to 9
crossings were cataloged by Reidemeister (1948).
References
Reidemeister, K. Knotentheorie. New York: Chelsea, 1948.
Rolfsen, D. "Torsion Numbers." §6 A in Knots and Links.
Wilmington, DE: Publish or Perish Press, pp. 145-146,
1976.
Torsion Tensor
The Tensor defined by
t jk = — (r jk - r fcj),
where T l jk are CONNECTION COEFFICIENTS.
see also CONNECTION COEFFICIENT
Torus
radius of the tube be a. Then the equation in Carte-
sian Coordinates is
The parametric equations of a torus are
x — (c + a cos v) cos u
y = (c + a cos v) sin u
z = a sin v
(i)
(2)
(3)
(4)
for ti, v € [0, 27r). Three types of torus, known as the
Standard Tori, are possible, depending on the relative
sizes of a and c. c > a corresponds to the RING Torus
(shown above), c = a corresponds to a HORN TORUS
which is tangent to itself at the point (0, 0, 0), and
c < a corresponds to a self-intersecting Spindle Torus
(Pinkall 1986).
If no specification is made, "torus" is taken to mean
Ring Torus. The three Standard Tori are illustrated
below, where the first image shows the full torus, the
second a cut-away of the bottom half, and the third a
Cross-Section of a plane passing through the z- Axis.
full view cutaway cross-section
e+o
ring
torus
horn
torus
spindle
torus
The STANDARD TORI and their inversions are CY-
CLIDES. If the coefficient of sint> in the formula for z
is changed to b ^ a, an Elliptic Torus results.
A torus is a surface having GENUS 1, and therefore pos-
sessing a single "Hole." The usual torus in 3-D space is
shaped like a donut, but the concept of the torus is ex-
tremely useful in higher dimensional space as well. One
of the more common uses of n-D tori is in DYNAMICAL
SYSTEMS. A fundamental result states that the PHASE
Space trajectories of a Hamiltonian System with n
Degrees of Freedom and possessing n Integrals of
Motion lie on an n-D Manifold which is topologically
equivalent to an n-torus (Tabor 1989).
The usual 3-D "ring" torus is known in older literature
as an "ANCHOR Ring." Let the radius from the center
of the hole to the center of the torus tube be c, and the
■a-frj— |Lx
To compute the metric properties of the ring torus, de-
fine the inner and outer radii by
r — c — a
R = c+a.
(5)
(6)
Torus
Torus
1817
Solving for a and c gives
a= \(R-r)
c=\{R + r).
Then the Surface Area of this torus is
S = (27ro)(27rc) = 4w 2 ac
= n 2 (R + r)(R-r),
(7)
(8)
0)
(10)
and the VOLUME can be computed from PAPPUS'S Cen-
troid Theorem
V = (7ra 2 ) 2 7rc = 27rVc
= \<x 2 {R + r)(R-r) 2 .
(11)
(12)
The coefficients of the first and second FUNDAMENTAL
FORMS of the torus are given by
e = — (c + a cos v) cos v
(13)
/ = o
(14)
9 = -a
(15)
E = (c + acosi;) 2
(16)
F =
(17)
G = a 2 ,
(18)
giving Riemannian Metric
ds 2 = (c + acosv) 2 du + a dv 2 , (19)
Area Element
dA = a(c + a cos v) du A dv (20)
(where du A dv is a WEDGE PRODUCT), and GAUSSIAN
and Mean Curvatures as
K :
H = -
a(c + a cos v )
c + 2a cos v
(21)
(22)
2a(c + acosv)
(Gray 1993, pp. 289-291).
A torus with a HOLE in its surface can be turned inside
out to yield an identical torus. A torus can be knotted
externally or internally, but not both. These two cases
are Ambient Isotopies, but not Regular Isotopies.
There are therefore three possible ways of embedding a
torus with zero or one Knot.
An arbitrary point P on a torus (not lying in the xy-
plane) can have four CIRCLES drawn through it. The
first circle is in the plane of the torus and the second
is Perpendicular to it. The third and fourth CIR-
CLES are called Villarceau Circles (Villarceau 1848,
Schmidt 1950, Coxeter 1969, Melnick 1983).
To see that two additional Circles exist, consider a
coordinate system with origin at the center of torus, with
z pointing up. Specify the position of P by its ANGLE (j)
measured around the tube of the torus. Define <j> — for
the circle of points farthest away from the center of the
torus (i.e., the points with x 2 + y 2 = R 2 ), and draw the
x-AxiS as the intersection of a plane through the z-axis
and passing through P with the ccy-plane. Rotate about
the y- AXIS by an ANGLE 0, where
= sin" 1 (-\ . (23)
In terms of the old coordinates, the new coordinates are
x = xi cos — z\ sin 6
z = x\ sin 6 + z\ cos 0.
(24)
(25)
So in (xi,j/i,2i) coordinates, equation (1) of the torus
becomes
[y(x\ cos — z\ sin 0) 2 -f yi 2 — c] 2
+ (xi sin 6 + z\ cos 0) 2
Squaring both sides gives
(26)
(x\ cos# — z\ sin#) 2 + y\ 2 + c 2
— 2cy (x± cosO — z\ sin#) 2 + y\ 2
+(xi sin + z x cos 0) 2 = a 2 . (27)
But
(xi cos 6 — z\ sin#) 2 + (x\ sin# + z\ cos#) 2 = x\ 2 -f z\ 2 ,
(28)
Xi 2 -\-yi 2 +zi 2 +c 2 -2cy/(xi cos0 - z\ sin#) 2 + y± 2 = a 2 .
(29)
In the z\ = plane, plugging in (23) and factoring gives
[xi 2 + (yi - a) 2 - c 2 ][x! 2 + (yi + a) - c 2 } = 0. (30)
This gives the CIRCLES
and
2 i / \2 2
Xi + (yi - a) = c
Xi 2 + (2/i+a) 2 =c 2
(31)
(32)
1818
Torus
Torus Coloring
in the z\ plane. Written in Matrix form with parameter
t 6 [0, 27r), these are
C COS t
c sin t-\- a
c cos t
c sin t — a
In the original {x,y, z) coordinates,
(33)
(34)
cos0 -
- sin
*1
c cos t
Ci =
10
c sin £ + a
m — sin 9 cos 9
c cos cost
=
c sin £ + a
— c sin cost _
cos 9 sin#~
c cos £
c 2 =
1
c sin t — a
_ — sin 9 cos 9 _
c cos cost
c sin t — a
_ — csin9cost_
The point P
must satisfy
z = asin^> = c sin 9 cost,
so
cost —
a sin
(35)
csin#
(36)
(37)
(38)
Plugging this in for x± and y± gives the ANGLE ip by
which the CIRCLE must be rotated about the 2-Axis in
order to make it pass through P,
?/> = tan'
■©
c sin t -\- a c\/l — cos 2 t + a
c cos 9 cos £
c cos 9 cos £
The four CIRCLES passing through P are therefore
(39)
Ci
c 2 =
Cz
C 4
cos^
— sinV>
sin^
cos?/*
0"
1.
c cos 9 cost
c sin £ + a
_ —c sin 9 cost
cosV*
— sin^
sin^
costp
0"
1.
c cos 9 cost
csint — a
_ — csin#cos£
(c + acos(f>) cos
(c + a cos (f>) sin
asin0
t~
t
c + a cos t '
asint
(40)
(41)
(42)
(43)
Ring Torus, Spindle Torus, Spiric Section, Stan-
dard Tori, Toroid, Torus Coloring, Torus Cut-
ting
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 131-132, 1987.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, pp. 132-133, 1969.
Geometry Center. "The Torus." http://www.geom.umn.edu/
zoo/toptype/torus/.
Gray, A. "Tori." §11.4 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 218-220 and 289-290, 1993.
Melzak, Z. A. Invitation to Geometry. New York: Wiley,
pp. 63-72, 1983.
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.
Schmidt, H. Die Inversion und ihre Anwendungen. Munich:
Oldenbourg, p. 82, 1950.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics:
An Introduction. New York: Wiley, pp. 71-74, 1989.
Villarceau, M. "Theoreme sur le tore." Nouv. Ann. Math. 7,
345-347, 1848.
Torus Coloring
The number of colors SUFFICIENT for Map COLORING
on a surface of GENUS g is given by the HEAWOOD CON-
JECTURE,
X(9)= [1(7+^485 + 1)],
where [x\ is the FLOOR FUNCTION. The fact that x(g)
(which is called the Chromatic Number) is also Nec-
essary was proved by Ringel and Youngs (1968) with
two exceptions: the Sphere (which requires the same
number of colors as the Plane) and the Klein Bot-
tle. A g-holed TORUS therefore requires x(g) colors.
For # = 0, 1, . . . , the first few values of x(g) are 4, 7,
8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, . . . (Sloane's
A000934).
see also CHROMATIC NUMBER, FOUR-COLOR THEO-
REM, Heawood Conjecture, Klein Bottle, Map
Coloring
References
Gardner, M. "Mathematical Games: The Celebrated Four-
Color Map Problem of Topology." Sci. Amer. 203, 218-
222, Sep. 1960.
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.
see also Apple, Cyclide, Elliptic Torus, Genus
(Surface), Horn Torus, Klein Quartic, Lemon,
Torus Cutting
Total Space 1819
Torus Cutting
With n cuts of a TORUS of Genus 1, the maximum
number of pieces which can be obtained is
N(n) = |(n 3 +3n 2 + 8n).
The first few terms are 2, 6, 13, 24, 40, 62, 91, 128, 174,
230, . . . (Sloane's A003600).
see also Cake Cutting, Circle Cutting, Cylinder
Cutting, Pancake Cutting, Plane Cutting, Pie
Cutting, Square Cutting
References
Gardner, M, Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, pp. 149-150, 1978.
Sloane, N. J. A. Sequence A003600/M1594 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Torus Knot
A (p, g)-torus Knot is obtained by looping a string
through the Hole of a TORUS p times with q revolutions
before joining its ends, where p and q are RELATIVELY
PRIME. A (p, g)-torus knot is equivalent to a (qr,p)-torus
knot. The CROSSING NUMBER of a (p, <j)-torus knot is
c = mm{p(q~ l),<?(p- 1)}
(1)
(Murasugi 1991). The Unknotting Number of a
(p, g)-torus knot is
u=±(p-l)(q-l)
(2)
(Adams 1991).
Torus knots with fewer than 11 crossings are the TRE-
FOIL Knot 03 oi (3, 2), Solomon's Seal Knot 05 00 i
(5, 2), 07ooi (7, 2), O8019 (4, 3), 09 oi (9, 2), and 10i 24
(5, 3) (Adams et al. 1991). The only Knots which are
not Hyperbolic Knots are torus knots and SATEL-
LITE Knots (including Composite Knots). The (2, q),
(3, 4), and (3, 5)-torus knots are Almost Alternating
Knots.
The Jones Polynomial of an (m, n)-ToRUS Knot is
,(m-l)(n-l)/2/-t _ im+1 _ f^ + 1 . .m + n
1-t 2
The Bracket Polynomial for the torus knot K n
(2,n) is given by the Recurrence Relation
n-l A -Zn+2
(3)
where
(K n )=A{K n - 1 ) + (-l) n - l A
<*i> = -A 3
(4)
(5)
see also ALMOST ALTERNATING KNOT, HYPERBOLIC
Knot, Knot, Satellite Knot, Solomon's Seal
Knot, Trefoil Knot
References
Adams, C; Hildebrand, M.; and Weeks, J. "Hyperbolic In-
variants of Knots and Links." Trans. Amer. Math. Soc.
326, 1-56, 1991.
Gray, A. "Torus Knots." §8.2 in Modern Differential Geom-
etry of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 155-161, 1993.
Murasugi, K. "On the Braid Index of Alternating Links."
Trans. Amer. Math. Soc. 326, 237-260, 1991.
Total Angular Defect
see Descartes Total Angular Defect
Total Curvature
The total curvature of a curve is the quantity vV 2 -f k 2 ,
where r is the TORSION and k is the CURVATURE. The
total curvature is also called the Third Curvature.
see also Curvature, Torsion (Differential Geom-
etry)
Total Differential
see Exact Differential
Total Function
A Function defined for all possible input values.
Total Intersection Theorem
If one part of the total intersection group of a curve
of order n with a curve of order m + ri2 constitutes
the total intersection with a curve of order m, then the
other part will constitute the total intersection with a
curve of order n^.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 32, 1959.
Total Order
A total order satisfies the conditions for a Partial Or-
der plus the comparability condition. A Relation <
is a partial order on a Set S if
1. Reflexivity: a < a for all a 6 S
2. Antisymmetry: a < b and b < a implies a = b
3. Transitivity: a < b and b < c implies a < c,
and is a total order if, in addition,
4. Comparability: For any a, b 6 S, either a < b or
b< a.
see also PARTIAL ORDER, RELATION
Total Space
The Space E of a Fiber Bundle given by the Map
/ : E -> B, where B is the Base Space of the FIBER
Bundle.
see also BASE SPACE, FIBER BUNDLE, SPACE
1820
Totative
Totient Function
Totative
A Positive Integer less than or equal to a number
n which is also Relatively Prime to n, where 1 is
counted as being Relatively Prime to all numbers.
The number of totatives of n is the value of the TOTIENT
Function <j>(n).
see also Relatively Prime, Totient Function
Totient Function
and
20 40 60 80 100
The totient function 0(n), also called Euler's totient
function, is defined as the number of POSITIVE INTE-
GERS < n which are Relatively Prime to (i.e., do
not contain any factor in common with) n, where 1 is
counted as being Relatively Prime to all numbers.
Since a number less than or equal to and RELATIVELY
Prime to a given number is called a Totative, the to-
tient function <j>{n) can be simply defined as the number
of Totatives of n. For example, there are eight Tota-
tives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so 0(24) = 8.
By convention, 0(0) = 1. The first few values of (j>(n)
for n = 1, 2, . . . are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ...
(Sloane's A000010). <f>{n) is plotted above for small n.
For a Prime p,
<t>(p) =P- 1,
(1)
since all numbers less than p are Relatively Prime to
p. If m = p a is a POWER of a PRIME, then the numbers
which have a common factor with m are the multiples of
p: p, 2p, . . . , (p a-1 )p. There arep a_1 of these multiples,
so the number of factors RELATIVELY PRIME to p a is
Hp a )=p a
■p -p
1 (p-i)=p q
l
• (2)
Now take a general m divisible by p. Let 4> p (m) be the
number of POSITIVE INTEGERS < m not DIVISIBLE by
p. As before, p, 2p, . . . , (m/p)p have common factors,
so
<t) P {m)
m
m
P
(3)
Now let q be some other Prime dividing m. The Inte-
gers divisible by q are q, 2<?, . . . , (m/q)q. But these du-
plicate pcjf, 2pqr, . . . , (m/pq)pq. So the number of terms
which must be subtracted from 4> p to obtain 4> pq is
A(j) q (m)
m
pq
™ (1 .
(4)
(j> pq (m) = 0,(m) - A(j> q (m)
~ m \ ~p)~l\p)
-»R)H)- <*>
By induction, the general case is then
,(„) = n[l-l 1-i .- (l-l). (6)
An interesting identity relates <j)(n 2 ) to </>(n),
4>(n 2 ) = n4>{n). (7)
Another identity relates the DIVISORS d of n to n via
£>(<*) = "■ (8)
d
The Divisor Function satisfies the Congruence
na{n) = 2 (mod 0(n)) (9)
for all Primes and no Composite with the exceptions of
4, 6, and 22 (Subbarao 1974), where a(n) is the DIVISOR
Function. No Composite solution is currently known
to
n- 1 = (mod <f>(n)) (10)
(Honsberger 1976, p. 35).
Walfisz (1963), building on the work of others, showed
that
N
Y<t>(n) = ^L + 0[JV(lniV) 2/3 (lnlniV) 4/3 ] 1 (11)
n=l
and Landau (1900, quoted in Dickson 1952) showed that
EsR!-^ w+B+0 (Tr)- (12 >
where
^E
K&)j 2 _ C(2)C(3) __ 315
k<t>(k) C(6) 2tt 4
C(3)
1.9435964368...
315^, ^> [/x(fc)] 2 lnfc
fc=l
= -0.0595536246 . .
k<f>{k)
(13)
(14)
Totient Function
Totient Valence Function
1821
ti(k) is the Mobius Function, ((z) is the Riemann
Zeta Function, and 7 is the Euler-Mascheroni
Constant (Dickson). A can also be written
n^
i-Pk"
(i- Pfc -^)(i-p fc - 3 )
fc=l k~l
n
1 +
1
p k (p k -i)_
(15)
Note that this constant is similar to ARTIN'S CONSTANT.
If the GOLDBACH CONJECTURE is true, then for every
number m, there are PRIMES p and q such that
cj>{p)+<j>{q) = 2m (16)
(Guy 1994, p. 105).
Curious equalities of consecutive values include
0(5186) = 0(5187) = 0(5188) = 2 5 3 4 (17)
0(25930) = 0(25935) = 0(25940) = 0(25942) = 2 7 3 4
(18)
0(404471) = 0(404473) = 0(404477) = 2 8 3 2 5 2 7 (19)
(Guy 1994, p. 91).
20 40 60 80 100
The SUMMATORY totient function, plotted above, is de-
fined by
$(n) = X>(fc)
(20)
References
Abramowitz, M. and Stegun, C. A, (Eds.). "The Euler
Totient Function." §24.3.2 in Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Ta-
bles, 9th printing. New York: Dover, p. 826, 1972.
Beiler, A. H. Ch. 12 in Recreations in the Theory of Numbers:
The Queen of Mathematics Entertains. New York: Dover,
1966.
Conway, J. H. and Guy, R. K. "Euler's Totient Num-
bers." The Book of Numbers. New York: Springer- Verlag,
pp. 154-156, 1996.
Courant, R. and Robbins, H. "Euler's tp Function. Fermat's
Theorem Again." §2.4.3 in Supplement to Ch. 1 in What
is Mathematics?: An Elementary Approach to Ideas and
Methods, 2nd ed. Oxford, England: Oxford University
Press, pp. 48-49, 1996.
DeKoninck, J.-M. and Ivic, A. Topics in Arithmetical Func-
tions: Asymptotic Formulae for Sums of Reciprocals of
Arithmetical Functions and Related Fields. Amsterdam,
Netherlands: North-Holland, 1980.
Dickson, L. E. History of the Theory of Numbers, Vol. 1:
Divisibility and Primality. New York: Chelsea, pp. 113-
158, 1952.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/asolve/constant/totient/totient .html.
Guy, R. K. "Euler's Totient Function," "Does <j>(n) Properly
Divide n — 1," "Solutions of 4>(m) = cr(n)," "Carmichael's
Conjecture," "Gaps Between Totatives," "Iterations of <fc
and <r," "Behavior of 4>(a(n)) and <r(<f>(n)). n §B36-B42 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 90-99, 1994.
Halberstam, H. and Richert, H.-E. Sieve Methods. New York:
Academic Press, 1974.
Honsberger, R. Mathematical Gems II. Washington, DC:
Math. Assoc. Amer., p. 35, 1976.
Perrot, J. 1811. Quoted in Dickson, L. E. History of the
Theory of Numbers, Vol. 1: Divisibility and Primality.
New York: Chelsea, p. 126, 1952.
Shanks, D. "Euler's <f> Function." §2.27 in Solved and Un-
solved Problems in Number Theory, ^th ed. New York:
Chelsea, pp. 68-71, 1993.
Sloane, N. J. A. Sequences A000010/M0299 and A002088/
M1008 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Subbarao, M. V. "On Two Congruences for Primality." Pa-
cific J. Math. 52, 261-268, 1974.
Totient Function Constants
see Silverman Constant, Totient Function
and has the asymptotic series
1
#(a?).
2C(2)
x +(xlnx)
Q
~ — z-z 2 + Q(xlnx),
(21)
(22)
Totient Valence Function
N^irn) is the number of INTEGERS n for which <f>(n) =
m, also called the Multiplicity of m (Guy 1994). The
table below lists values for <f>(N) < 50.
where £(z) is the Riemann Zeta Function (Perrot
1881). The first values of $(n) are 1, 2, 4, 6, 10, 12, 18,
22, 28, . . . (Sloane's A002088).
see also Dedekind Function, Euler's Totient
Rule, Fermat's Little Theorem, Lehmer's Prob-
lem, Leudesdorf Theorem, Noncototient, Non-
totient, Silverman Constant, Totative, Totient
Valence Function
1822
Totient Valence Function
Tournament Matrix
<t>(N)
m
N
1
2
1, 2
2
3
3, 4, 6
4
4
5, 8, 10, 12
6
4
7, 9, 14, 18
8
5
15, 16, 20, 24, 30
10
2
11, 22
12
6
13, 21, 26, 28, 36, 42
16
6
17, 32, 34, 40, 48, 60
18
4
19, 27, 38, 54
20
5
25, 33, 44, 50, 66
22
2
23, 46
24
10
35, 39, 45, 52, 56, 70, 72, 78, 84, 90
28
2
29, 58
30
2
31, 62
32
7
51, 64, 68, 80, 96, 102, 120
36
8
37, 57, 63, 74, 76, 108, 114, 126
40
9
41, 55, 75, 82, 88, 100, 110, 132, 150
42
4
43, 49, 86, 98
44
3
69, 92, 138
46
2
47, 94
48
11
65, 104, 105, 112, 130, 140, 144,
156, 168, 180, 210
A table listing the first value of <j>(N) with multiplicities
up to 100 follows (Sloane's A014573).
M
<t>
M
4>
M
<t>
M
d>
3
26
2560
51
4992
76
21840
2
1
27
384
52
17640
77
9072
3
2
28
288
53
2016
78
38640
4
4
29
1320
54
1152
79
9360
5
8
30
3696
55
6000
80
81216
6
12
31
240
56
12288
81
4032
7
32
32
768
57
4752
82
5280
8
36
33
9000
58
2688
83
4800
9
40
34
432
59
3024
84
4608
10
24
35
7128
60
13680
85
16896
11
48
36
4200
61
9984
86
3456
12
160
37
480
62
1728
87
3840
13
396
38
576
63
1920
88
10800
14
2268
39
1296
64
2400
89
9504
15
704
40
1200
65
7560
90
18000
16
312
41
15936
66
2304
91
23520
17
72
42
3312
67
22848
92
39936
18
336
43
3072
68
8400
93
5040
19
216
44
3240
69
29160
94
26208
20
936
45
864
70
5376
95
27360
21
144
46
3120
71
3360
96
6480
22
624
47
7344
72
1440
97
9216
23
1056
48
3888
73
13248
98
2880
24
1760
49
720
74
11040
99
26496
25
360
50
1680
75
27720
100
34272
It is thought that N^(m) > 2 (i.e., the totient valence
function never takes on the value 1), but this has not
been proven. This assertion is called CARMlCHAEL's
Totient Function Conjecture and is equivalent to
the statement that for all n, there exists m ^ n such
that <f>(n) = 4>{m) (Ribenboim 1996, pp. 39-40). Any
counterexample must have more than 10,000,000 Digits
(Schlafly and Wagon 1994, Conway and Guy 1996).
see also Carmichael's Totient Function Conjec-
ture, Totient Function
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New-
York: Springer- Verlag, p. 155, 1996.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 94, 1994.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, 1996.
Schlafly, A. and Wagon, S. "Carmichaers Conjecture on the
Euler Function is Valid Below 10 10 > 000 > 000 .» Math. Corn-
put 63, 415-419, 1994.
Sloane, N. J. A. Sequence A014573 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Touchard's Congruence
B p+k = B k + Bk+i (mod p) ,
when p is Prime and B n is a BELL NUMBER.
see also Bell Number
Tour
A sequence of moves on a chessboard by a CHESS piece
in which each square of a CHESSBOARD is visited exactly
once.
see also Chess, Knight's Tour, Magic Tour, Trav-
eling Salesman Constants
Tournament
A Complete Directed Graph. A so-called Score
SEQUENCE can be associated with every tournament.
Every tournament contains a HAMILTONIAN Path.
see also Complete Graph, Directed Graph, Ham-
iltonian Path, Score Sequence
References
Chartrand, G. "Tournaments." §27.2 in Introductory Graph
Theory. New York: Dover, pp. 155-161, 1985.
Moon, J. W. Topics on Tournaments. New York: Holt, Rine-
hart, and Winston, 1968.
Ruskey, F. "Information on Score Sequences." http://sue.
esc .uvic . ca/~cos/inf /nump/ScoreSequence .html.
Tournament Matrix
A matrix for a round-robin tournament involving n play-
ers competing in n(n — l)/2 matches (no ties allowed)
having entries
{1 if player i defeats player j
— 1 if player i loses to player j
if i = j-
The Matrix satisfies
A + A T + I = J,
where I is the Identity Matrix, J is an n x n Matrix
of all Is, and A T is the Matrix Transpose of A.
Tower of Power
Trace (Matrix) 1823
The tournament matrix for n players has zero Deter-
minant Iff n is Odd (McCarthy and Benjamin 1996).
The dimension of the NULLSPACE of an n-player tour-
nament matrix is
dim [nullspace] = I
for n even
for n odd
(McCarthy 1996).
References
McCarthy, C. A. and Benjamin, A. T. "Determinants of the
Tournaments." Math. Mag. 69, 133-135, 1996.
Michael, T, S. "The Ranks of Tournament Matrices," Amer.
Math. Monthly 102, 637-639, 1995.
Tower of Power
see Power Tower
Towers of Hanoi
\
A PUZZLE invented by E. Lucas in 1883. Given a stack of
n disks arranged from largest on the bottom to smallest
on top placed on a rod, together with two empty rods,
the towers of Hanoi puzzle asks for the minimum number
of moves required to reverse the order of the stack (where
moves are allowed only if they place smaller disks on top
of larger disks). The problem is ISOMORPHIC to finding
a Hamiltonian Path on an u-Hypercube.
For n disks, the number of moves h n required is given
by the RECURRENCE RELATION
Solving gives
h n — 2h n ~i 4- 1-
h n = 2 U - 1.
The number of disks moved after the kth step is the
same as the element which needs to be added or deleted
in the kth Addend of the RYSER Formula (Gardner
1988, Vardi 1991).
A Hanoi Graph can be constructed whose Vertices
correspond to legal configurations of n towers of Hanoi,
where the Vertices are adjacent if the corresponding
configurations can be obtained by a legal move. It can
be solved using a binary GRAY CODE.
Poole (1994) gives Mathematical (Wolfram Research,
Champaign, IL) routines for solving an arbitrary disk
configuration in the fewest possible moves. The proof
of minimality is achieved using the LUCAS CORRESPON-
DENCE which relates PASCAL'S TRIANGLE to the HANOI
Graph. Algorithms are known for transferring disks
for four pegs, but none has been proved minimal. For
additional references, see Poole (1994).
see also Gray Code, Ryser Formula
References
Bogomolny, A. "Towers of Hanoi." http://vww.cut-the-
knot . com/recurrence/hanoi .html.
Chartrand, G. "The Tower of Hanoi Puzzle." §6.3 in Intro-
ductory Graph Theory. New York: Dover, pp. 135-139,
1985.
Dubrovsky, V. "Nesting Puzzles, Part I: Moving Oriental
Towers." Quantum 6, 53-57 (Jan.) and 49-51 (Feb.),
1996.
Gardner, M. "The Icosian Game and the Tower of Hanoi."
Ch. 6 in The Scientific American Book of Mathematical
Puzzles & Diversions. New York: Simon and Schuster,
1959.
Kasner, E. and Newman, J. R. Mathematics and the Imagi-
nation. Redmond, WA: Tempus Books, pp. 169-171, 1989.
Kolar, M. "Towers of Hanoi." http://www.pangea.ca/
kolar/ javascript /Hanoi /Hanoi. html.
Poole, D. G. "The Towers and Triangles of Professor Glaus
(or, Pascal Knows Hanoi)." Math. Mag. 67, 323-344,
1994.
^ Poole, D. G. "Towers of Hanoi." http: //www. astro.
virginia.edu/-eww6n/math/notebooks/Hanoi.in.
Ruskey, F. "Towers of Hanoi." http://sue.csc.uvic.ca/-
cos/inf /comb/Subset Info. html#Hanoi.
Schoutte, P. H. "De Ringen van Brahma." Eigen Haard 22,
274-276, 1884.
Kraitchik, M. "The Tower of Hanoi." §3.12.4 in Mathematical
Recreations. New York: W. W. Norton, pp. 91-93, 1942.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 111-112, 1991.
Trace (Complex)
The image of the path 7 in C under the Function / is
called the trace. This term is unrelated to that applied
to Matrices and Tensors.
Trace (Group)
see Character (Group)
Trace (Map)
Let a Patch be given by the map x : U -> R", where U
is an open subset of R 2 , or more generally by x : A -»■
R n , where A is any Subset of R 2 . Then x(*7) (or more
generally, x(A)) is called the trace of x.
see also PATCH
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 183-184, 1993.
Trace (Matrix)
The trace of an n x n SQUARE MATRIX A is defined by
Tr(A) = an,
(1)
where EINSTEIN SUMMATION is used (i.e., the an is
summed over i = 1, ..., n). For SQUARE MATRICES
A and B, it is true that
Tr(A) = Tr(A T )
Tr(A+B)-Tr(A) + Tr(B)
Tr(aA) = aTr(A)
(2)
(3)
(4)
1824 Trace (Matrix)
Tractrix
(Lange 1987, p. 40). The trace is invariant under a Sim-
ilarity Transformation
A' = BAB" 1
(Lange 1987, p. 64). Since
(bab~ )ij = buaikb^j ,
(5)
(6)
Trace (Tensor)
The trace of a second-RANK TENSOR T is a SCALAR
given by the CONTRACTED mixed TENSOR equal to Tj.
The trace satisfies
Tr
d
M " 1(x fe MW
= ^Mdet(z)],
and
Tlr(BAB- 1 ) = 6«a, fc 6- 1 w
= (b~ b)kiaik — Skidik
= a kk =Tr(A),
(7)
where Sij is the Kronecker Delta.
The trace of a product of square matrices is independent
of the order of the multiplication since
Tr(AB) = (ab)a = aijbji — bjidij
= (6a) w = TV(BA). (8)
Therefore, the trace of the Commutator of A and B is
given by
Tr([A, B]) = Tr(AB) - Tr(BA) = 0.
(9)
The product of a SYMMETRIC and an ANTISYMMETRIC
Matrix has zero trace,
Tr{A s B A ) = 0.
(10)
The value of the trace can be found using the fact that
the matrix can always be transformed to a coordinate
system where the z- Axis lies along the axis of rotation.
In the new coordinate system, the MATRIX is
A'
cos <p sin (f)
— sin <f> cos 4>
1
so the trace is
Tr(A') = Tr(A) = an = 1 + 2 cos <j>.
(11)
(12)
References
Lang, S. Linear Algebra, 3rd ed. New York: Springer- Verlag,
pp. 40 and 64, 1987.
<Hn[detM] = ln[det(M + 5M)) - ln(detM)
f det(Af + 3Af) "
~ n [ detM
= ln[detM~ 1 (M + < 5M)]
= ln[det(l + M~ 1 5M)]
K\n[l + Tr(M~ l 5M)]
^Tr(M _1 5M).
see also Contraction (Tensor)
Tractory
see Tractrix
Tractrix
The tractrix is the CATENARY INVOLUTE described by a
point initially on the vertex. It has a constant NEGATIVE
Curvature and is sometimes called the Tractory or
Equitangential Curve. The tractrix was first studied
by Huygens in 1692, who gave it the name "tractrix."
Later, Leibniz, Johann Bernoulli, and others studied the
curve.
The tractrix arises from the following problem posed to
Leibniz: What is the path of an object starting off with
a vertical offset when it is dragged along by a string of
constant length being pulled along a straight horizontal
line? By associating the object with a dog, the string
with a leash, and the pull along a horizontal line with
the dog's master, the curve has the descriptive name
HUNDKURVE (hound curve) in German. Leibniz found
the curve using the fact that the axis is an asymptote
to the tractrix (MacTutor Archive).
In Cartesian Coordinates the tractrix has equation
= a sech f - j — sj a 2 - y 2 .
(i)
Tractrix
One parametric form is
x(t) = a(t — tanhi)
y(t) = asech£.
(2)
(3)
The Arc Length, Curvature, and Tangential An-
gle are
s(t) = ln(cosht)
n(i) = csch t
4>{t) = 2tarr 1 [tanh(|t)].
(4)
(5)
(6)
A second parametric form in terms of the ANGLE <j> of
the straight line tangent to the tractrix is
x = a{ln[tan(^0)] + cos0}
(7)
(8)
(Gray 1993). This parameterization has CURVATURE
K(<£) = |tan0|. (9)
A parameterization which traverses the tractrix with
constant speed a is given by
x{t ) = h e ~ v/a ^we[o^)
K } \ae v/a for^G (-oo,0]
(10)
y(*) - {
' a[t3inh~ 1 (Vl-e- 2v / a ) - Vl - e-Wa]
for v € [0, oo)
a[- tanh _1 (>/l-e 2w / a ) + Vl - e 2v / a ]
for v € ( — oo,0].
(ii)
When a tractrix is rotated around its asymptote, a
Pseudosphere results. This is a surface of constant
Negative Curvature. For a tractrix, the length of
a Tangent from its point of contact to an asymptote
is constant. The Area between the tractrix and its
asymptote is finite.
see also CURVATURE, DlNl'S SURFACE, MICE PROBLEM,
Pseudosphere, Pursuit Curve, Tractroid
References
Geometry Center. "The Tractrix." http://www.geora.umn.
edu/zoo/dif fgeom/pseudosphere/tractrix.html.
Gray, A. "The Tractrix" and "The Evolute of a Tractrix is a
Catenary." §3.5 and 5.3 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 46-50 and 80-81, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 199-200, 1972.
Lee, X. "Tractrix." http://www.best.com/-xah/Special
PlaneCurves_dir/Tractrix_dir/tractrix.html.
Transcendental Equation 1825
Lockwood, E. H. "The Tractrix and Catenary." Ch, 13 in A
Book of Curves. Cambridge, England: Cambridge Univer-
sity Press, pp. 118-124, 1967.
MacTutor History of Mathematics Archive. "Tractrix."
http : //www-groups . dcs . st-and . ac . uk/ -history /Curves
/Tractrix . html.
Yates, R. C. "Tractrix." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J, W. Edwards, pp. 221-224,
1952.
Tractrix Evolute
The Evolute of the Tractrix is the Catenary.
Tractrix Radial Curve
The Radial Curve of the Tractrix is the Kappa
Curve.
Tractroid
The Surface of Revolution produced by revolving
the Tractrix
x ■
z ■
■ sech u
■ u — tanh u
about the 2-AxiS is a tractroid given by
x ■
y-
z ■
sech u cos v
sech u sin v
u — tanhw.
(i)
(2)
(3)
(4)
(5)
see also PSEUDOSPHERE, SURFACE OF REVOLUTION,
Tractrix
Transcendental Curve
A curve which intersects some straight line in an infin-
ity of points (but for which not every point lies on this
curve).
References
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.
Transcendental Equation
An equation or formula involving TRANSCENDENTAL
FUNCTIONS.
1826
Transcendental Function
Transcendental Number
Transcendental Function
A function which "transcends," i.e., cannot be expressed
in terms of, the usual Elementary Functions. Define
The number e was proven to be transcendental by Her-
mite in 1873, and Pi (n) by Lindemann in 1882. e 7 " is
transcendental by GELFOND'S THEOREM since
; l{z) =
e(z)
ln(z)
Z
-. e
Cl /(z) = ?/(z)
/
f(z)dz,
and let fa = Z(/(^)), etc. These are called the "elemen-
tary" transcendental functions (Watson 1966, p. 111).
see also Algebraic Function, Elementary Func-
tion
References
Watson, G. N. A Treatise on the Theory of Bessel Functions,
2nd ed. Cambridge, England: Cambridge University Press,
1966.
Transcendental Number
A number which is not the ROOT of any POLYNOMIAL
equation with Integer Coefficients, meaning that it
not an ALGEBRAIC NUMBER of any degree, is said to be
transcendental. This definition guarantees that every
transcendental number must also be IRRATIONAL, since
a Rational Number is, by definition, an Algebraic
Number of degree one.
Transcendental numbers are important in the history
of mathematics because their investigation provided the
first proof that CIRCLE SQUARING, one of the GEOMET-
RIC Problems of Antiquity which had baffled math-
ematicians for more than 2000 years was, in fact, insolu-
ble. Specifically, in order for a number to be produced by
a Geometric Construction using the ancient Greek
rules, it must be either RATIONAL or a very special kind
of Algebraic Number known as a Euclidean Num-
ber. Because the number tv is transcendental, the con-
struction cannot be done according to the Greek rules.
Georg Cantor was the first to prove the Existence of
transcendental numbers. Liouville subsequently showed
how to construct special cases (such as LlOUVlLLE's
Constant) using Liouville's Rational Approxima-
tion Theorem. In particular, he showed that any num-
ber which has a rapidly converging sequence of ratio-
nal approximations must be transcendental. For many
years, it was only known how to determine if special
classes of numbers were transcendental. The determi-
nation of the status of more general numbers was con-
sidered an important enough unsolved problem that it
was one of Hilbert's Problems.
Great progress was subsequently made by GELFOND'S
THEOREM, which gives a general rule for determining if
special cases of numbers of the form or are transcen-
dental. Baker produced a further revolution by proving
the transcendence of sums of numbers of the form a In/?
for Algebraic Numbers a and 0.
(-l)- i = (e i7r )- i =:e 7r .
The Gelfond-Schneider Constant 2^ is also trans-
cendental. Other known transcendentals are sin 1 where
sinz is the Sine function, Jo(l) where Jo(x) is a BES-
sel Function of the First Kind (Hardy and Wright
1985), In 2, In 3/ In 2, the first zero x = 2.4048255... of
the Bessel Function Jo(x ) (Le Lionnais 1983, p. 46),
7T + In 2 + y/2 In 3 (Borwein et al. 1989), the Thue-
Morse Constant P = 0.4124540336... (Dekking
1977, Allouche and Shallit), the Champernowne Con-
stant 0.1234567891011. . . , the Thue Constant
0.110110111110110111110110110...,
r(|) (Le Lionnais 1983, p. 46), F(\)tv- 1/4 (Davis 1959),
and T(~) (Chudnovsky, Waldschmidt), where F(x) is the
Gamma Function. At least one of 7re and iz + e (and
probably both) are transcendental, but transcendence
has not been proven for either number on its own.
It is not known if e e , 7T 71 " , 7r e , 7 (the Euler-Mascheroni
Constant), J (2), or h{2) (where I n (x) is a Modified
Bessel Function of the First Kind) are transcen-
dental.
The "degree" of transcendence of a number can be char-
acterized by a so-called Liouville-Roth Constant.
There are still many fundamental and outstanding prob-
lems in transcendental number theory, including the
Constant Problem and Schanuel's Conjecture.
see also Algebraic Number, Constant Prob-
lem, Gelfond's Theorem, Irrational Num-
ber, Lindemann- WeierstraB Theorem, Liouville-
Roth Constant, Roth's Theorem, Schanuel's
Conjecture, Thue-Siegel-Roth Theorem
References
Allouche, J. P. and Shallit, J. In preparation.
Baker, A. "Approximations to the Logarithm of Certain Ra-
tional Numbers." Acta Arith. 10, 315-323, 1964.
Baker, A. "Linear Forms in the Logarithms of Algebraic
Numbers I." Mathematika 13, 204-216, 1966,
Baker, A. "Linear Forms in the Logarithms of Algebraic
Numbers II." Mathematika 14, 102-107, 1966.
Baker, A. "Linear Forms in the Logarithms of Algebraic
Numbers III." Mathematika 14, 220-228, 1966.
Baker, A. "Linear Forms in the Logarithms of Algebraic
Numbers IV." Mathematika 15, 204-216, 1966.
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.
Chudnovsky, G. V, Contributions to the Theory of Trans-
cendental Numbers. Providence, Rl: Amer. Math. Soc,
1984.
Courant, R. and Robbins, H. "Algebraic and Transcendental
Numbers." §2.6 in What is Mathematics?: An Elementary
Transcritical Bifurcation
Approach to Ideas and Methods, 2nd ed. Oxford, England:
Oxford University Press, pp. 103-107, 1996.
Davis, P. J. "Leonhard Euler's Integral: A Historical Profile
of the Gamma Function." Amer. Math. Monthly 66, 849-
869, 1959.
Dekking, F. M. "Transcendence du nombre de Thue-Morse."
Comptes Rendus de VAcademie des Sciences de Paris 285,
157-160, 1977.
Gray, R. "Georg Cantor and Transcendental Numbers."
Amer. Math. Monthly 101, 819-832, 1994.
Hardy, G. H. and Wright, E. M. An Introduction to the The-
ory of Numbers, 5th ed. Oxford, England: Oxford Univer-
sity Press, 1985.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 46, 1983.
Siegel, C. L. Transcendental Numbers. New York: Chelsea,
1965.
Transcritical Bifurcation
Let / : R x R -
maps satisfying
= 1
/(0,/0
L&cJ M -o,«=o
idxl ^^ LcteJ M= o,z
be a one-parameter family of C
(i)
(2)
>0.
(3)
(4)
(5)
M =0,x =
Then there are two branches, one stable and one unsta-
ble. This Bifurcation is called a transcritical bifurca-
tion. An example of an equation displaying a transcrit-
ical bifurcation is
x = fix — x 2 . (6)
(Guckenheimer and Holmes 1997, p. 145).
see also BIFURCATION
References
Guckenheimer, J. and Holmes, P. Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields, 3rd
ed. New York: Springer- Verlag, pp. 145 and 149-150, 1997.
Rasband, S. N. Chaotic Dynamics of Nonlinear Systems.
New York: Wiley, pp. 27-28, 1990.
Transfer Function
The engineering terminology for one use of FOURIER
Transforms. By breaking up a wave pulse into its
frequency spectrum
U = F(v)e a
(1)
the entire signal can be written as a sum of contributions
from each frequency,
/(*)
f u du = / F{v)e 2 * iut du.
-oo </-oo
Transform 1827
If the signal is modified in some way, it will become
g v (t) = 4>iy)Mt) = ct>{v)F{ V )e 2 - ivt (3)
/oo poo
g„{t)dt= I <My)F{v)e*' ivt dv,
-oo J —oo
(4)
where <p(u) is known as the "transfer function." FOUR-
IER Transforming <j> and F,
/oo
*(0e -a,ri, "<tt
-OO
(5)
(6)
f(t)e- 2 * ivt dt.
•oo
From the CONVOLUTION THEOREM,
/oo
/(*)$(* -t) dr. (7)
-OO
see also CONVOLUTION THEOREM, FOURIER TRANS-
FORM
Transfinite Diameter
Let
<fi(z) — cz -h co + c±z~ l + c 2 z~ 2 + ...
be an ANALYTIC FUNCTION, REGULAR and UNIVALENT
for \z\ > 1, which maps \z\ > 1 CONFORMALLY onto the
region T preserving the POINT AT INFINITY and its di-
rection. Then the function <f>(z) is uniquely determined
and c is called the transfinite diameter, sometimes also
known as Robin's Constant or the Capacity of <j>(z).
see also ANALYTIC FUNCTION, REGULAR FUNCTION,
Univalent Function
Transfinite Number
One of Cantor's Ordinal Numbers u;, w + l, a; + 2, . . . ,
a; + aj, a; + w + 1, ... which is "larger" than any WHOLE
Number.
see also N , Ni, Cardinal Number, Continuum, Or-
dinal Number, Whole Number
References
Pappas, T. "Transfinite Numbers." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 156-158,
1989.
Transform
A shortened term for INTEGRAL TRANSFORM.
Geometrically, if S and T are two transformations, then
the Similarity Transformation TST' 1 is some-
times called the transform (Woods 1961).
(2)
1828
Transformation
Transposition Group
see also Abel Transform, Boustrophedon Trans-
form, Discrete Fourier Transform, Fast Four-
ier Transform, Fourier Transform, Frac-
tional Fourier Transform, Hankel Trans-
form, Hartley Transform, Hilbert Transform,
Laplace-Stieltjes Transform, Laplace Trans-
form, Mellin Transform, Number Theoretic
Transform, Poncelet Transform, Radon Trans-
form, Wavelet Transform, z-Transform, Z-
Transform
References
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, p. 5,
1961.
Transformation
see Function, Map
Transitive
A RELATION R on a Set S is transitive provided that
for all x } y and z in 5 such that xRy and yRz, we also
have xRz.
see also Associative, Commutative, Relation
Transitive Closure
The transitive closure of a binary Relation R on a
SET X is the minimal TRANSITIVE relation R' on X
that contains R. Thus aR'b for any elements a and b of
X, provided either that aRb or that there exists some
element c of X such that aRc and cRb.
see also Reflexive Closure, Transitive Reduc-
tion
Transitive Reduction
The transitive reduction of a binary RELATION R on
a SET X is the minimum relation R' on X with the
same TRANSITIVE CLOSURE as R. Thus aR'b for any
elements a and b of X, provided that aRb and there
exists no element c of X such that aRc and cRb,
see also REFLEXIVE REDUCTION, TRANSITIVE CLO-
SURE
Transitivity Class
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).
see also Monohedral Tiling
References
Berglund, J. "Is There a fc-Anisohedral Tile for k > 5?"
Amer. Math. Monthly 100, 585-588, 1993.
Translation
A transformation consisting of a constant offset with no
Rotation or distortion. In n-D Euclidean Space, a
translation may be specified simply as a VECTOR giving
the offset in each of the n coordinates.
see also Affine Group, Dilation, Euclidean
Group, Expansion, Glide, Improper Rotation, In-
version Operation, Mirror Image, Reflection,
Rotation
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 211, 1987.
Translation Relation
A mathematical relationship transforming a function
f(x) to the form f{x 4- a).
see also Argument Addition Relation, Argument
Multiplication Relation, Recurrence Relation,
Reflection Relation
Transpose
The object obtained by replacing all elements a,ij with
a,ji. For a second- RANK Tensor aij, the tensor trans-
pose is simply aji. The matrix transpose, written A ,
is the MATRIX obtained by exchanging A's rows and
columns, and satisfies the identity
(A-)" 1 = (A" 1 )-.
The product of two transposes satisfies
(B A )ij = (6 )ik(a )kj — bkiajk = a^bki = (AB)jj
= (AB)5.
Therefore,
(AB) T = B T A T .
Transpose Map
see Pullback Map
Transposition
An exchange of two elements of a Set with all others
staying the same. A transposition is therefore a PER-
MUTATION of two elements. For example, the swapping
of 2 and 5 to take the list 123456 to 153426 is a trans-
position.
see also Permutation, Transposition Order
Transposition Group
A Permutation Group in which the Permutations
are limited to TRANSPOSITIONS.
see also PERMUTATION GROUP
Transposition Order
Trapezoidal Hexecontahedron 1829
Transposition Order
An ordering of PERMUTATIONS in which each two adja-
cent permutations differ by the TRANSPOSITION of two
elements. For the permutations of {1,2,3} there are
two listings which are in transposition order. One is
123, 132, 312, 321, 231, 213, and the other is 123, 321,
312, 213, 231, 132.
see also Lexicographic Order, Permutation
References
Ruskey, F. "Information on Combinations of a Set."
http://sue .esc .uvic . ca/ -cos/ inf / comb /Combinations
Info.html.
Transversal Array
A set of n cells in an n x n SQUARE such that no two
come from the same row and no two come from the same
column. The number of transversals of an n x n SQUARE
is n! (n FACTORIAL).
Transversal Design
A transversal design TD\(k,n) of order n, block size &,
and index A is a triple (V, G, B) such that
1. V is a set of kn elements,
2. G is a partition of V into k classes, each of size n
(the "groups"),
3. B is a collection of fc-subsets of V (the "blocks"),
and
4. Every unordered pair of elements from V is contained
in either exactly one group or in exactly A blocks, but
not both.
References
Colbourn, C. J. and Dinitz, J. H. (Eds.) CRC Handbook
of Combinatorial Designs. Boca Raton, FL: CRC Press,
p. 112, 1996.
Transversal Line
A transversal line is a Line which intersects each of a
given set of other lines. It is also called a Semisecant.
see also LINE
Transylvania Lottery
A lottery in which three numbers are picked at random
from the INTEGERS 1-14.
see also Fano Plane
Trapdoor Function
An easily computed function whose inverse is extremely
difficult to compute. An example is the multiplication
of two large PRIMES. Finding and verifying two large
PRIMES is easy, as is their multiplication. But factoriza-
tion of the resultant product is very difficult.
see also RSA Encryption
References
Gardner, M. Chs. 13-14 in Penrose Tiles and Trapdoor
Ciphers. . . and the Return of Dr. Matrix, reissue ed. New
York: W. H. Freeman, pp. 299-300, 1989.
Trapezium
There are two common definitions of the trapezium. The
American definition is a QUADRILATERAL with no PAR-
ALLEL sides. The British definition for a trapezium is
a Quadrilateral with two sides Parallel. Such a
trapezium is equivalent to a TRAPEZOID and therefore
has Area
A= \{a + b)h.
see also DIAMOND, LOZENGE, PARALLELOGRAM,
Quadrilateral, Rhomboid, Rhombus, Skew Quad-
rilateral, Trapezoid
Trapezohedron
The trapezohedra are the DUAL Polyhedra of the Ar-
chimedean ANTIPRISMS. However, their faces are not
Trapezoids.
see also Antiprism, Dipyramid, Hexagonal Scalen-
ohedron, Prism, Trapezoid
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 117, 1989.
Trapezoid
A Quadrilateral with two sides Parallel. The
trapezoid depicted above satisfies
m — | (a + b)
and has Area
A — \{a + b)h = mh.
The trapezoid is equivalent to the British definition of
Trapezium.
see also PYRAMIDAL FRUSTUM, TRAPEZIUM
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 123, 1987.
Trapezoidal Hexecontahedron
see Deltoidal Hexecontahedron
1830 Trapezoidal Icositetrahedron
Trapezoidal Icositetrahedron
see Deltoidal Icositetrahedron
Traveling Salesman Constants
0.34207 < 74 < 0(4) < 12 1/8 6" 1/2 < 0.55696
< 0.59460 < 2~ 3/4 < a(4) < 0.8364 (5)
Trapezoidal Rule
fix)
t
The 2-point Newton-Cotes Formula
r x 2
J Xl
f(x)dx=±h(f 1 + f 2 )- 1 ih 3 f"(t),
where fi = f(xi)> h is the separation between the points,
and £ is a point satisfying x\ < £ < xi. Picking £ to
maximize /"(£) gives an upper bound for the error in
the trapezoidal approximation to the INTEGRAL.
see also Bode'S Rule, Hardy's Rule, Newton-
Cotes Formulas, Simpson's 3/8 Rule, Simpson's
Rule, Weddle's Rule
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 885, 1972.
Traveling Salesman Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let L(n>d) be the smallest TOUR length for n points in a
d-T> HYPERCUBE. Then there exists a smallest constant
a(d) such that for all optimal TOURS in the HYPER-
CUBE,
L(n,d)
lim sup
n {d-i)/d^/2
< a(d),
(1)
and a constant j3(d) such that for almost all optimal
tours in the HYPERCUBE,
lim L(n ' d V
■.p{d).
(2)
These constants satisfy the inequalities
0.44194 < 72 = jqV2<0(2)
< 5 < 0.6508 < 0.75983 < 3~ 1/4 < a(2)
< <f> < 0.98398 (3)
0.37313 < 73 < 0(3) < 12 i/b 6 -1 ^ < 0.61772 < 0.64805
< 2 1/6 3" 1/2 < a(3) < 0.90422 (4)
(Fejes Toth 1940, Verblunsky 1951, Few 1955, Beard-
wood et al. 1959), where
Id =
r {3+ l 2 )[T(±d+l)}^ d
2 y /^(d 1 / 2 + d-Va) '
(6)
T(z) is the Gamma Function, S is an expression involv-
ing Struve Functions and Neumann Functions,
280(3-^)
840 - 280 V3 + 4^ - \/l0
(Karloff 1989), and
$= |3 _2/3 (4 + ln3) 2/3
(Goddyn 1990). In the LIMIT d -» oo,
1
(?)
(8)
0.24197 < lim y d =
d— >oo
< lim inf j3(d)
27re d-»-oo
<limsup/3(d) < lim i2 xK2d h' 1/2
d-+oo d ->°°
= 4= < 0.40825 (9)
v6
and
0.24197 <
where
-== < lim a(d)
/27re rf-+oo
< 2(3-v5)g <0-4052>
/27re
| <6>= lim [6{d)] 1/d < 0.6602,
(11)
and 9(d) is the best SPHERE PACKING density in d-D
space (Goddyn 1990, Moran 1984, Kabatyanskii and
Levenshtein 1978). Steele and Snyder (1989) proved
that the limit a(d) exists.
Now consider the constant
L(n,2) _
k ~ lim
Tl— J-OO
y/n
0(2)V2,
| = 7 2 V2 < k < 5V2 < 0.9204.
The best current estimate is k « 0.7124.
(12)
(13)
A certain self-avoiding SPACE-FILLING CURVE is an op-
timal TOUR through a set of n points, where n can be
arbitrarily large. It has length
A = lim
4(l + 2^/2)y / 5l
m^-oo yJTlm
153
0.7147827...,
(14)
Traveling Salesman Problem
Tree
1831
where L m is the length of the curve at the mth iteration
and n m is the point-set size (Moscato and Norman).
References
Beardwood, J.; Halton, J. H.; and Hammersley, J. M. "The
Shortest Path Through Many Points." Proc. Cambridge
Phil. Soc. 55, 299-327, 1959.
Chartrand, G. "The Salesman's Problem: An Introduction to
Hamiltonian Graphs." §3.2 in Introductory Graph Theory.
New York: Dover, pp. 67-76, 1985.
Fejes Toth, L. "Uber einen geometrischen Satz." Math. Zeit
46, 83-85, 1940.
Few, L. "The Shortest Path and the Shortest Road Through
n Points." Mathematika 2, 141-144, 1955.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ asolve/constant/sales/sales .html.
Flood, M. "The Travelling Salesman Problem." Operations
Res. 4, 61-75, 1956.
Goddyn, L. A. "Quantizers and the Worst Case Euclidean
Traveling Salesman Problem." J. Combin. Th. Ser. B 50,
65-81, 1990.
Kabatyanskii, G. A. and Levenshtein, V. I. "Bounds for Pack-
ing on a Sphere and in Space." Problems Inform. Transm.
14, 1-17, 1978.
KarlofT, H. J. "How Long Can a Euclidean Traveling Sales-
man Tour Be?" SIAM J. Disc. Math. 2, 91-99, 1989.
Moran, S. "On the Length of Optimal TSP Circuits in Sets of
Bounded Diameter." J. Combin. Th. Ser. B 37, 113-141,
1984.
Moscato, P. "Fractal Instances of the Traveling Sales-
man Constant." http : //www . ing . unlp . edu . ar/cetad/
mos/FRACTAL_TSP Jiome . html
Steele, J. M. and Snyder, T. L. "Worst-Case Growth Rates of
Some Classical Problems of Combinatorial Optimization."
SIAM J. Comput. 18, 278-287, 1989.
Verblunsky, S. "On the Shortest Path Through a Number of
Points." Proc. Amer. Math. Soc. 2, 904-913, 1951.
Traveling Salesman Problem
A problem in GRAPH THEORY requiring the most effi-
cient (i.e., least total distance) Tour (i.e., closed path)
a salesman can take through each of n cities. No gen-
eral method of solution is known, and the problem is
NP-Hard.
see also TRAVELING SALESMAN CONSTANTS
References
Platzman, L. K. and Bartholdi, J. J. "Spacefilling Curves
and the Planar Travelling Salesman Problem." J. Assoc.
Comput. Mach. 46, 719-737, 1989.
Trawler Problem
A fast boat is overtaking a slower one when fog suddenly
sets in. At this point, the boat being pursued changes
course, but not speed. How should the pursuing vessel
proceed in order to be sure of catching the other boat?
The amazing answer is that the pursuing boat should
continue to the point where the slow boat would be if it
had set its course directly for the pursuing boat when the
fog set in. If the boat is not there, it should proceed in
a Spiral whose origin is the point where the slow boat
was when the fog set in. The Spiral can be constructed
in such a way that the two boats will intersect before a
complete turn is made.
References
Ogilvy, C. S. Excursions in Mathematics. New York: Dover,
pp. 84 and 148, 1994.
Trebly Magic Square
see Trimagic Square
Tredecillion
In the American system, 10
see also Large Number
Tree
• • <»— #
A tree is a mathematical structure which can be viewed
as either a Graph or as a Data Structure. The two
views are equivalent, since a tree Data Structure con-
tains not only a set of elements, but also connections
between elements, giving a tree graph.
A tree graph is a set of straight line segments connected
at their ends containing no closed loops (cycles). A tree
with n nodes has n — 1 Edges. The points of connection
are known as FORKS and the segments as BRANCHES.
Final segments and the nodes at their ends are called
Leaves. A tree with two Branches at each Fork and
with one or two LEAVES at the end of each branch is
called a BINARY TREE.
When a special node is designated to turn a tree into
a Rooted Tree, it is called the Root (or sometimes
"Eve." ) In such a tree, each of the nodes which is one
Edge further away from a given Edge is called a Child,
and nodes connected to the same node are then called
Siblings.
Note that two BRANCHES placed end-to-end are equiva-
lent to a single BRANCH which means, for example, that
there is only one tree of order 3. The number t(n) of
nonisomorphic trees of order n = 1, 2, . . . (where trees
1832
TVee
Tree Searching
of orders 1, 2, . . . , 6 are illustrated above), are 1, 1, 1,
2, 3, 6, 11, 23, 47, 106, 235, . . . (Sloane's A000055).
Otter showed that
*»**¥-=*
n— ><x> OL
(1)
(Otter 1948, Harary and Palmer 1973, Knuth 1969),
where the constants a and /3 are sometimes called Ot-
ter's Tree Enumeration Constants. Write the
Generating Function for Rooted Trees as
f(z) = J2f i z\
(2)
where the Coefficients are
7$Z 5Z# d ft-** 1 '
3 = 1
d\j
with /o = and /i = 1. Then
a = 2.955765,.
is the unique POSITIVE ROOT of
and
(3)
(4)
(5)
\fa
^— ' \a k J at
k = 2
OL h
3/2
0.5349485...
(6)
see also jB-TREE, BINARY TREE, CATERPILLAR GRAPH,
Cayley Tree, Child, Dijkstra Tree, Eve, Forest,
Kruskal's Algorithm, Kruskal's Tree Theorem,
Leaf (Tree), Orchard-Planting Problem, Or-
dered Tree, Path Graph, Planted Planar Tree,
Polya Enumeration Theorem, Quadtree, Red-
Black Tree, Root (Tree), Rooted Tree, Sibling,
Star Graph, Stern-Brocot Tree, Weakly Binary
Tree, Weighted Tree
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/otter/otter.html.
Chauvin, B.; Cohen, S.; and Rouault, A. (Eds.). Trees:
Workshop in Versailles, June 1^-16, 1995. Basel, Switzer-
land: Birkhauser, 1996.
Gardner, M. "Trees." Ch. 17 in Mathematical Magic Show:
More Puzzles, Games, Diversions, Illusions and Other
Mathematical Sleight-of-Mind from Scientific American.
New York: Vintage, pp. 240-250, 1978.
Harary, F. Graph Theory. Reading, MA: Addis on- Wesley,
1994.
Harary, F. and Manvel, B. "Trees." Scripta Math. 28, 327-
333, 1970.
Harary, F. and Palmer, E. M. Graphical Enumeration. New
York: Academic Press, 1973.
Knuth, D. E. The Art of Computer Programming, Vol. 1:
Fundamental Algorithms, 2nd ed. Reading, MA: Addison-
Wesley, 1973.
Otter, R. "The Number of Trees." Ann. Math. 49, 583-599,
1948.
Sloane, N. J. A. Sequences A000055/M0791 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.
Tree-Planting Problem
see Orchard-Planting Problem
Tree Searching
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
In database structures, two quantities are generally of
interest: the average number of comparisons required to
1. Find an existing random record, and
2. Insert a new random record into a data structure.
Some constants which arise in the theory of digital tree
searching are
oo
fc=i
^=E
(2» - l) 2
1.6066951524... (1)
1.1373387363.... (2)
Erdos (1948) proved that a is IRRATIONAL. The ex-
pected number of comparisons for a successful search
is
In 2 In 2
a+§ + <5(n) + C?(n- 1/2 ) (3)
(4)
~lgn- 0.716644... + 6(n),
and for an unsuccessful search is
a+!+<5(n)-fO(n- 1/2 ) (5)
Inn j^
In 2 In 2
• lgn- 0.273948... +J(ra).
(6)
Here S(n) y e(s), and p(n) are small- amplitude periodic
functions, and Lg is the base 2 Logarithm. The Vari-
ance for searching is
V ~ —
1 7T 2 +6
12 6(ln2) 2
and for inserting is
-a-/M-e(a) -2.844383... +e(s) (7)
12 6(ln2) 2
■a - + e(s) ~ 0.763014 . . . + e(s).
(8)
Tree Searching
Trefoil Knot
1833
The expected number of pairs of twin vacancies in a
digital search tree is
(Ar,
+ 1 -h{ih +a2 - a ) +p{n)
+ o(VH),
(9)
where
oo
Q = Y[( 1 ^^k) = 0-2887880950 . . . (10)
+ ...(11)
1 J_
3 ~ 3~7 ' 3-5-15 3 • 5 ■ 15 • 21
+
1
exp
oo
^ n(2 n -
(2- - 1)
(12)
In 2 _ 7T 2
In 2 eXP I 24 61n2
n
1 — exp
47r 2 n
" In 2
(13)
and
^ 1 ■ 3 ■ 7 • 16 ■ • • (2 fc - 1) 2-, 2J : - 1
fc=i
7.7431319855....
(14)
(Flajolet and Sedgewick 1986). The linear COEFFICIENT
of (A n ) fluctuates around
c = + 1 _ 1 (-L + a 2 - a) = 0.3720486812 . . . ,
Q \ln2 /
(15)
which can also be written
1 Z 100 x
ln2j Q 1 + ;
dz
-. (16)
(l + x)(l + §x)(l+H(l+f:c)<
(Flajolet and Richmond 1992).
References
Finch, S. "Favorite Mathematical Constants." http://wwv.
mathsof t . c om/ as olve/ const ant /bin/bin. html.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ as olve /const ant /dig/dig. html.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . com/ asolve/constant/qdt/qdt .html.
Flajolet, P. and Richmond, B. "Generalized Digital Trees and
their Difference- Differential Equations." Random Struc-
tures and Algorithms 3, 305-320, 1992.
Flajolet, P. and Sedgewick, R. "Digital Search Trees Revis-
ited." SUM Review 15, 748-767, 1986.
Knuth, D. E. The Art of Computer Programming, Vol. 3:
Sorting and Searching, 2nd ed. Reading, MA: Addison-
Wesley, pp. 21, 134, 156, 493-499, and 580, 1973.
Trefoil Curve
The plane curve given by the equation
4,22,4 / 2 2\
x + x y +y = x \ x ~ V )•
Trefoil Knot
The knot 03 oi, also called the Threefoil Knot, which
is the unique Prime Knot of three crossings. It has
Braid Word <t x 3 . The trefoil and its Mirror Image
are not equivalent. The trefoil has ALEXANDER POLY-
NOMIAL -x 2 + x - 1 and is a (3, 2)-TORUS KNOT. The
Bracket Polynomial can be computed as follows.
(L)
A'd 2 - 1
+ A 2 Bd 1 - 1 + A 2 Bd 1 - 1 +AB 2 d 2
+ A 2 Bd 1 ~ 1 + AB 2 d 2 ' 1 + AB'd
2j2 - 1 +B 3 d 3 - 1
A 3 d 1 + SA 2 Bd° + 3ABV + B 3 d 2 .
Plugging in
gives
B^A- 1
d=-A 2
(L) = A- 7
1-3
A 5 .
The normalized one-variable KAUFFMAN POLYNOMIAL
X is then given by
X L = {-A 3 )- w(L) (L) = (-A 3 )- 3 (A- 7 - A
3 - A 5 )
A^ + A- 13 -^- 16 ,
where the WRITHE w(L) = 3. The JONES POLYNOMIAL
is therefore
V{t) = L(A = i
-1/4
) = t + t 3 -t 4 = t(l + t 2 -* 3 ).
Since V(t 1 ) ^ V"(t), we have shown that the mirror
images are not equivalent.
References
Claremont High School. "Trefoil_Knot Movie." Binary
encoded QuickTime movie, ftp: //chs.cusd. claremont .
edu/pub/knot/tref oil . cptbin.
Crandall, R. E. Mathematica for the Sciences. Redwood
City, CA: Addison-Wesley, 1993.
Kauffman, L. H. Knots and Physics. Singapore: World Sci-
entific, pp. 29-35, 1991.
Nordstrand, T. "Threefoil Knot." http : //www . uib . no/
people/nf ytn/tknottxt .htm.
Pappas, T, "The Trefoil Knot." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, p. 96, 1989.
1834 Trench Diggers' Constant
Triangle
Trench Diggers' Constant
see Beam Detector
Triabolo
A 3-POLYABOLO.
Triacontagon
A 30-sided POLYGON.
Trial
In statistics, a trial is a single measurable random event,
such as the flipping of a COIN, the generation of a Ran-
dom Number, the dropping of a ball down the apex of
a triangular lattice and having it fall into a single bin at
the bottom, etc.
see also Bernoulli Trial, Lexis Trials, Poisson
Trials
Triacontahedron
A 30-sided Polyhedron such as the Rhombic Tria-
contahedron.
Triad
A SET with three elements.
see also HEXAD, MONAD, QUARTET, QUINTET,
Tetrad
Triakis Icosahedron
The Dual Polyhedron of the Truncated Dodeca-
hedron Archimedean Solid. The triakis icosahedron
is also Icosahedron Stellation #2.
References
Wenninger, M. J. Polyhedron Models. New York: Cambridge
University Press, p. 46, 1989.
Triakis Octahedron
see Great Triakis Octahedron, Small Triakis
Octahedron
Triakis Tetrahedron
Trial Division
A brute-force method of finding a DIVISOR of an INTE-
GER n by simply plugging in one or a set of INTEGERS
and seeing if they DIVIDE n. Repeated application of
trial division to obtain the complete Prime Factor-
ization of a number is called Direct Search Factor-
ization. An individual integer being tested is called a
Trial Divisor.
see also DIRECT SEARCH FACTORIZATION, DIVISION,
Prime Factorization
Trial Divisor
An INTEGER n which is tested to see if it divides a given
number.
see also TRIAL DIVISION
Triamond
The unique 3-POLYIAMOND, illustrated above.
see also POLYIAMOND, TRAPEZOID
Triangle
Acute
Scalene Triangle
Equilateral
Triangle
Obtuse
Scalene Triangle
Right
Triangle
A triangle is a 3-sided POLYGON sometimes (but not
very commonly) called the TRIGON. All triangles are
convex. An Acute Triangle is a triangle whose three
angles are all ACUTE. A triangle with all sides equal is
called Equilateral. A triangle with two sides equal
is called ISOSCELES. A triangle having an OBTUSE AN-
GLE is called an OBTUSE TRIANGLE. A triangle with a
Right Angle is called Right. A triangle with all sides
a different length is called SCALENE.
D A E
The Dual Polyhedron of the Truncated Tetrahe-
dron Archimedean Solid.
B C
The sum of ANGLES in a triangle is 180°. This can be es-
tablished as follows. Let DAE\ \BC (DAE be Parallel
to BC) in the above diagram, then the angles a and j3
Triangle
Triangle 1835
satisfy a = LDAB = LABC and f3 = LEAC = LBCE,
as indicated. Adding 7, it follows that
If the coordinates of the triangle Vertices are given in
3-D by (xi,yi,Zi) where i = 1, 2, 3, then
a + /3 + 7= 180°
(1)
since the sum of angles for the line segment must equal
two Right Angles. Therefore, the sum of angles in the
triangle is also 180°.
Let S stand for a triangle side and A for an angle, and
let a set of 5s and As be concatenated such that adja-
cent letters correspond to adjacent sides and angles in a
triangle. Triangles are uniquely determined by specify-
ing three sides (SSS Theorem), two angles and a side
(A AS THEOREM), or two sides with an adjacent angle
(SAS Theorem). In each of these cases, the unknown
three quantities (there are three sides and three angles
total) can be uniquely determined. Other combinations
of sides and angles do not uniquely determine a trian-
gle: three angles specify a triangle only modulo a scale
size (AAA Theorem), and one angle and two sides not
containing it may specify one, two, or no triangles (ASS
Theorem).
Equilateral Triangle
The Ruler and Compass construction of the triangle
can be accomplished as follows. In the above figure, take
OPq as a Radius and draw OB ± OP . Then bisect OB
and construct P 2 Pi\\OPo. Extending BO to locate P 3
then gives the EQUILATERAL TRIANGLE AP1P2P3.
In Proposition IV. 4 of the Elements, Euclid showed how
to inscribe a CIRCLE (the INCIRCLE) in a given triangle
by locating the Center as the point of intersection of
Angle Bisectors. In Proposition IV. 5, he showed how
to circumscribe a CIRCLE (the ClRCUMClRCLE) about a
given triangle by locating the CENTER as the point of
intersection of the perpendicular bisectors.
If the coordinates of the triangle VERTICES are given by
{xi.yi) where i = 1, 2, 3, then the Area A is given by
the Determinant
A =
2!
X\
yi
1
X 2
yi
1
X3
ys
1
(2)
\
yi
Z\
1
2
2/2
Z2
1
+
2/3
zz
1
Z\
Xl
1
2
Z2
X 2
1
+
Z3
X3
1
Xl
2/i
1
X 2
yi
1
X3
2/3
1
(3)
(1,0)
In the above figure, let the ClRCUMClRCLE passing
through a triangle's VERTICES have Radius r, and de-
note the Central Angles from the first point to the
second B\ % and to the third point by 2 - Then the AREA
of the triangle is given by
A = 2r 2 |sin(^i)sin(^ 2 )sin[i(^i - 2 )]\ . (4)
If a triangle has sides a, 6, c, call the angles opposite
these sides A, B, and C, respectively. Also define the
Semiperimeter s as Half the Perimeter:
s=|p=^(a + 6 + c).
(5)
The Area of a triangle is then given by Heron's FOR-
MULA
A = y/s(s-a)(s-b){8-c), (6)
as well by the FORMULAS
A = I y/(a + b + c)(b + c - a)(c + a-b)(a + b-c)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14).
= \ y/2(a 2 b 2 + a 2 c 2 + b 2 c 2 ) - (a 4 + 6 4 + c 4 )
= W[{a + bY - c>][c> - {a-b) 2 ]
= jVp(p-2a)(p-26)(p-2c) f
= 2R 2 sin A sin B sin C
abc
= tt: = rs
4R
^bc sin A.
1836 Triangle
Triangle
In the above formulas, hi is the Altitude on side i, R
is the ClRCUMRADIUS, and r is the INRADIUS (Johnson
1929, p. 11). Expressing the side lengths a, 6, and c in
terms of the radii a', b' , and c' of the mutually tangent
circles centered on the TRIANGLE vertices (which define
the Soddy Circles),
+ c
= a + c
c = a + o ,
gives the particularly pretty form
A = A/a'&'c'(a' + fo'-hc').
(15)
(16)
(17)
(18)
For additional FORMULAS, see Beyer (1987) and Baker
(1884), who gives 110 FORMULAS for the Area of a
triangle.
The Angles of a triangle satisfy
cot^l
b 2 +c 2 -a 2
4A
(19)
where A is the Area (Johnson 1929, p. 11, with missing
squared symbol added). This gives the pretty identity
cot A + cot B + cot C ■
a 2 + b 2 + <?
4A
(20)
Let a triangle have Angles A, B, and C. Then
sin A sin B sin C < kABC, (21)
where
-m
(22)
(Abi-Khuzam 1974, Le Lionnais 1983). This can be used
to prove that
8u; d < ABC,
where w is the Brocard Angle.
(23)
Trigonometric Functions of half angles can be ex-
pressed in terms of the triangle sides:
cos(|A) =
sin(fA) =
.,/<•
— a)
be
(24)
■1/ fe
-b)(s-
bc
c)
(25)
tan(iA) =
-^
-b)(s-
s(s — a)
c)
(26)
where s
is the SEMIPERIMETER.
The number of different triangles which have Integral
sides and PERIMETER n is
T(n) = P 3 (n) - Yl P *W
1<3"<L«/2J
(27)
for n even
for n odd,
where P2 and P3 are PARTITION FUNCTIONS P, [x] is
the NlNT function, and |_;cj is the FLOOR FUNCTION
(Jordan et al. 1979, Andrews 1979, Honsberger 1985).
The values of T(n) for n = 1, 2, . . . are 0, 0, 1, 0, 1, 1, 2,
1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, . . .
(Sloane's A005044), which is also Alcuin's Sequence
padded with two initial Os. T(n) also satisfies
T(2n) =T(2n-3) = ft(n).
(28)
It is not known if a triangle with INTEGER sides, ME-
DIANS, and Area exists (although there are incorrect
PROOFS of the impossibility in the literature). How-
ever, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz
have shown that there are infinitely many triangles with
Rational sides (Heronian Triangles) with two Ra-
tional Medians (Guy 1994).
In the following paragraph, assume the specified sides
and angles are adjacent to each other. Specifying three
ANGLES does not uniquely define a triangle, but any two
triangles with the same ANGLES are similar (the AAA
Theorem). Specifying two Angles A and B and a side
a uniquely determines a triangle with AREA
a 2 sin B sin C _ a 2 sin B sin(7r — A — B)
2 sin A
2 sin A
(29)
(the A AS Theorem). Specifying an Angle A, a side
c, and an Angle B uniquely specifies a triangle with
Area
(30)
2(cot>l + cot5)
(the ASA THEOREM). Given a triangle with two sides,
a the smaller and c the larger, and one known Angle
A, ACUTE and opposite a, if sin A < a/c, there are two
possible triangles. If sin A — a/c, there is one possible
triangle. If sin A > a/c, there are no possible triangles.
This is the ASS Theorem. Let a be the base length
and h be the height. Then
A = \ah = I ac sin B
(31)
Triangle
Triangle 1837
(the SAS Theorem). Finally, if all three sides are spec-
ified, a unique triangle is determined with AREA given
by Heron's Formula or by
abc
45'
(32)
where R is the CIRCUMRADIUS. This is the SSS THEO-
REM.
There are four CIRCLES which are tangent to the sides
of a triangle, one internal and the rest external. Their
centers are the points of intersection of the ANGLE BI-
SECTORS of the triangle.
Any triangle can be positioned such that its shadow un-
der an orthogonal projection is EQUILATERAL.
see also AAA Theorem, AAS Theorem, Acute Tri-
angle, Alcuin's Sequence, Altitude, Angle Bi-
sector, Anticevian Triangle, Anticomplemen-
tary Triangle, Antipedal Triangle, ASS The-
orem, Bell Triangle, Brianchon Point, Bro-
card Angle, Brocard Circle, Brocard Mid-
point, Brocard Points, Butterfly Theorem,
Centroid (Triangle), Ceva's Theorem, Cevian,
Cevian Triangle, Chasles's Theorem, Circum-
center, Circumcircle, Circumradius, Contact
Triangle, Crossed Ladders Problem, Crucial
Point, D-Triangle, de Longchamps Point, Desar-
gues' Theorem, Dissection, Elkies Point, Equal
Detour Point, Equilateral Triangle, Euler
Line, Euler's Triangle, Euler Triangle For-
mula, EXCENTER, EXCENTRAL TRIANGLE, EXCIR-
cle, exeter point, exmedian, exmedian point,
Exradius, Exterior Angle Theorem, Fagnano's
Problem, Far-Out Point, Fermat Point, Fer-
mat's Problem, Feuerbach Point, Feuerbach's
Theorem, Fuhrmann Triangle, Gergonne Point,
Grebe Point, Griffiths Points, Griffiths' The-
orem, Harmonic Conjugate Points, Heilbronn
Triangle Problem, Heron's Formula, Hero-
nian Triangle, Hofstadter Triangle, Homoth-
etic Triangles, Incenter, Incircle, Inradius,
Isodynamic Points, Isogonal Conjugate, Iso-
gonic Centers, Isoperimetric Point, Isosceles
Triangle, Kabon Triangles, Kanizsa Triangle,
Kiepert's Hyperbola, Kiepert's Parabola, Law
of Cosines, Law of Sines, Law of Tangents, Leib-
niz Harmonic Triangle, Lemoine Circle, Lemoine
Point, Line at Infinity, Malfatti Points, Medial
Triangle, Median (Triangle), Median Triangle,
Menelaus' Theorem, Mid-Arc Points, Mitten-
punkt, mollweide's formulas, morley centers,
morley's theorem, nagel point, napoleon's
Theorem, Napoleon Triangles, Newton's For-
mulas, Nine-Point Circle, Number Triangle,
Obtuse Triangle, Orthic Triangle, Orthocen-
ter, Orthologic, Paralogic Triangles, Pas-
cal's Triangle, Pasch's Axiom, Pedal Trian-
gle, Perpendicular Bisector, Perspective Tri-
angles, Petersen-Shoute Theorem, Pivot Theo-
rem, Power Point, Power (Triangle), Prime Tri-
angle, Purser's Theorem, Quadrilateral, Ratio-
nal Triangle, Routh's Theorem, SAS Theorem,
Scalene Triangle, Schiffler Point, Schwarz
Triangle, Schwarz's Triangle Problem, Seidel-
Entringer-Arnold Triangle, Seydewitz's The-
orem, Simson Line, Spieker Center, SSS Theo-
rem, Steiner-Lehmus Theorem, Steiner Points,
Stewart's Theorem, Symmedian Point, Tangen-
tial Triangle, Tangential Triangle Circumcen-
ter, Tarry Point, Thomsen's Figure, Torricelli
Point, Triangle Tiling, Triangle Transforma-
tion Principle, Yff Points, Yff Triangles
References
Abi-Khuzam, F. "Proof of Yff 's Conjecture on the Brocard
Angle of a Triangle." Elem. Math. 29, 141-142, 1974.
Andrews, G. "A Note on Partitions and Triangles with Inte-
ger Sides." Amer. Math. Monthly 86, 477, 1979.
Baker, M. "A Collection of Formulae for the Area of a Plane
Triangle." Ann. Math. 1, 134-138, 1884.
Berkhan, G. and Meyer, W. F. "Neuere Dreiecksgeometrie."
In Encyklopaedie der Mathematischen Wissenschaften,
Vol. 3AB 10 (Ed. F. Klein). Leipzig: Teubner, pp. 1173-
1276, 1914.
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, pp. 123-124, 1987.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, 1969.
Davis, P. "The Rise, Fall, and Possible Transfiguration of Tri-
angle Geometry: A Mini-History." Amer. Math. Monthly
102, 204-214, 1995.
Eppstein, D. "Triangles and Simplices." http://www . ics .
uci . edu/~eppstein/junkyard/triangulation.html.
Feuerbach, K. W. Eigenschaften einiger merkwurdingen
Punkte des geradlinigen Dreiecks, und mehrerer durch
die bestimmten Linien und Figuren. Nurnberg, Germany,
1822.
Guy, R. K. "Triangles with Integer Sides, Medians, and
Area." §D21 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 188-190, 1994.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 39-47, 1985.
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, 1929.
Jordan, J. H.; Walch, R.; and Wisner, R. J. "Triangles with
Integer Sides." Amer. Math. Monthly 86, 686-689, 1979.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Kimberling, C. "Triangle Centers and Central Triangles."
Congr. Numer. 129, 1-295, 1998.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 28, 1983.
Schroeder. Das Dreieck und seine Beruhungskreise.
Sloane, N. J. A. Sequence A005044/M0146 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
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.
$ Weisstein, E. W. "Plane Geometry." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/PlaneGeometry.m.
1838 Triangle Arcs
Triangle Arcs
B P \ \Q C
In the above figure, the curves are arcs of a CIRCLE and
a = BC
b^CA^CP
c = BA = BQ.
Then
PQ 2 = 2BP ■ QC.
The figure also yields the algebraic identity
(i)
(2)
(3)
(4)
(b + c- sjK 2 + c 2 f = 2(vWc 2 - b)(y/b 2 + c 2 - c).
(5)
see also Arc
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York:
Springer- Verlag, pp. 8-9, 1994.
Dharmarajan, T. and Srinivasan, P. K. An Introduction to
Creativity of Ramanujan, Part III. Madras: Assoc. Math.
Teachers, pp. 11-13, 1987.
Triangle Center
A triangle center is a point whose Triune AR Coordi-
nates are defined in terms of the side lengths and an-
gles of a Triangle. The function giving the coordinates
a : f3 : 7 is called the TRIANGLE CENTER FUNCTION.
The four ancient centers are the CENTROID, Incenter,
ClRCUMCENTER, and ORTHOCENTER. For a listing of
these and other triangle centers, see Kimberling (1994).
A triangle center is said to be REGULAR Iff there is a
Triangle Center Function which is a Polynomial
in A, a, b, and c (where A is the Area of the Triangle)
such that the Trilinear Coordinates of the center
are
f(a,b,c) : f(b,c,a) : /(c,a,6).
A triangle center is said to be a Major Triangle Cen-
ter if the Triangle Center Function a is a function
of Angle A alone, and therefore f3 and 7 of B and C
alone, respectively.
see also MAJOR TRIANGLE CENTER, REGULAR TRIAN-
GLE Center, Triangle, Triangle Center Func-
tion, Trilinear Coordinates, Trilinear Polar
References
Davis, P. J. "The Rise, Fall, and Possible Transfiguration
of Triangle Geometry: A Mini- History." Amer. Math.
Monthly 102, 204-214, 1995.
Dixon, R. "The Eight Centres of a Triangle." §1.5 in Math-
ographics. New York: Dover, pp. 55-61, 1991.
Triangle Condition
Gale, D. "From Euclid to Descartes to Mathematica to Obliv-
ion?" Math. Intell. 14, 68-69, 1992.
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-167, 1994.
Kimberling, C. "Triangle Centers and Central Triangles."
Congr. Numer. 129, 1-295, 1998.
Triangle Center Function
A Homogeneous Function /(a, 6,c), i.e., a function
/ such that
/(ta, tb, tc) = t n /(a, 6, c),
which gives the Trilinear Coordinates of a Trian-
gle Center as
a : /? : 7 = /(a, 6, c) : /(&, c, a) : /(c, a, b).
The variables may correspond to angles (A, 5, C) or
side lengths (a, 6, c), since these can be interconverted
using the Law OF COSINES,
see also Major Triangle Center, Regular Trian-
gle Center, Triangle Center, Trilinear Coor-
dinates
References
Kimberling, C "Triangle Centers as Functions." Rocky Mtn.
J. Math. 23, 1269-1286, 1993.
Kimberling, C. "Triangle Centers." http://www.
evansville . edu/~ck6/tcenters/.
Kimberling, C. "Triangle Centers and Central Triangles."
Congr. Numer. 129, 1-295, 1998.
Triangle Coefficient
A function of three variables written A(abc) = A (a, 6, c)
and denned by
A(abc)
(a + b- c)l(a -b + c)\(-a + 6 + c)!
(a + 6 + c+l)!
References
Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra.
New York: Wiley, p. 273, 1968.
Triangle Condition
The condition that j takes on the values
j = h + J2J1 + h - l, . • . > \h - J2I,
denoted &(jiJ2J)-
References
Sobelman, I. I. Atomic Spectra and Radiative Transitions,
2nd ed. Berlin: Springer- Verlag, p. 60, 1992.
Triangle Counting
Triangle Inscribing in a Circle 1839
Triangle Counting
Given rods of length 1, 2, ..., n, how many distinct
triangles T(n) can be made? Lengths for which
ti = lj i tfc
obviously do not give triangles, but all other combina-
tions of three rods do. The answer is
T< n \-l £n(n-2)(2n-5) for n
I M n ~ !)( n " 3 )( 2n " !) for n
even
odd.
The values for n = 1, 2, . . . are 0, 0, 0, 1, 3, 7, 13, 22, 34,
50, . . . (Sloane's A002623). Somewhat surprisingly, this
sequence is also given by the Generating FUNCTION
/(*) =
= x 4 + Sx 5 + 7x 6 + 13x 7 + . . . .
(l-x) 3 (l-z 2 )
References
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 278-282, 1991.
Sloane, N. J. A. Sequence A002623/M2640 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Triangle of Figurate Numbers
see Figurate Number Triangle
Triangle Function
y^o.8
/ 0.6
/ 0.4
/ 0.2
0.5
{ } - \ 1 - \x\ \x\<l
(1)
= n(x)*n(x) (2)
= U(x) * H (x + |) - U(x) * H{x - \ ), (3)
where II is the RECTANGLE FUNCTION and H is the
HEAVISIDE Step FUNCTION. An obvious generalization
used as an APODIZATION FUNCTION goes by the name
of the Bartlett Function.
There is also a three- argument function known as the
triangle function:
A(x, y, z) = x 2 + y 2 + z 2 - 2xy - 2xz - 2yz. (4)
It follows that
A(a 2 ,6 2 ,c 2 ) = (a+6+c)(a+6-c)(a-6+c)(a-6-c). (5)
see also Absolute Value, Bartlett Function,
Heaviside Step Function, Ramp Function, Sgn,
Triangle Coefficient
Triangle Inequality
Let x and y be vectors
|x|-|y|<|x + y|<|x| + |y|. (1)
Equivalently, for Complex Numbers z\ and z 2 ,
l*i|-M < \zi + z*\ < \z x \ + \z 2 \. (2)
A generalization is
£
dk
<-T,
ait .
(3)
see also p-ADic Number, Strong Triangle Inequal-
ity
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.
Triangle Inscribing in a Circle
(1,0)
Select three points at random on a unit Circle. Find
the distribution of possible areas. The first point can
be assigned coordinates (1, 0) without loss of generality.
Call the central angles from the first point to the second
and third 9\ and #2- The range of 0\ can be restricted
to [0, tt] because of symmetry, but 62 can range from
[0,2?r). Then
^i^2) = 2|sin(^ 1 )sin(^ 2 )sin[|(^-^)]|, (1)
._ m^A(e u e 2 )d0 2 de 1
A ~ c ;
where
C =
pre p2
Jo Jo
de 2 d0i =2tt 2 .
(2)
(3)
1840 Triangle Inscribing in a Circle
Therefore,
A=^ J J |sin(^ 1 )sin(^ 2 )sin[i(0 1 -0 2 )}\d0 2 d9 1
= ^ J sind*!) J sin(^ 2 ) |sin[|(^ 2 -^i)]| d0 2
i r f 27r
7T 2 e 2 -$ l >o
i r r 2 "
+ Jo Jo s in(^e 1 )sm{^e 2 )sm[He 1 -e 2 )}de 2 de 1
7r 2 e 2 -^i<o
= ^ J sin(§0!) A sin(^ 2 )sin[§(^-^)]d»2 <
+ ^ / sin(^0 / sin(±0 2 )sin[§(0 2 - 001^2
d0i.
(4)
But
/■
(ie 2 )s\n[^{e 2 -e l )]de 2
= / sm(i/9 2 ) [8in(ifl 2 )cos(ie a )-sm(i« 1 )cos(^ 2 )] d^ a
= cos(^ 1 ) sin 2 ( 1 1 9 2 )d9 2 -sin(^e 1 ) / sin^) cos(±0 2 ) <W a
= icos(^i) / (1 -cosd 2 )dB 2 - ±sin(j0 2 ) lsiud 2 dd 2
= \ cos(§0i)(02 - sin<9 2 ) + \ sin(f 0i) cos(<9 2 ).
(5)
Write (4) as
then
(6)
(7)
(8)
A = i / sin(ifli)/idfli + / sindflO/jdfli
/*2tt
/!= / sin(§0 2 )sin[±(0 2 -0i)]d0 2 ,
and
I 2 = sin(f0 2 )sin[f(0i-0 2 )]d0 2 .
Prom (6),
J x = | cos(£0 2 )[0 2 - sin0 3 ]£ + | sin(i0i)[cos0 2 ]£
= icos(^i)(27r-^i+sin(9i)
+ §sin(f<9i)(l-cos<9i)
= 7rcos(|<9i)- |6>icos(|6>i) + i[cos(|^i)sin^i
-cos6»isin(^i)] + §sin(§0i)
= ttcos(^i)- §0icos(§0i) + | + §sin(0i - ±0i)
+ isin(^i)
= ttcos(^i)- f6»iCOs(^x)+sin(|(9i), (9)
Triangle Inscribing in an Ellipse
so
/"
./o
7isin(§0i)d0i = §7r.
(10)
Also,
h - f cos(!<9i)[sini9 2 -^lo 1 - | sin(^i)[cos(9 2 ]^
= | cos(f 2 )(sin6>i - (9i) - ± sin(f 0i)(cos0i - 1)
--|(9 lC os(|(9i)
+ |[sin<9iCOs(|^i) -cos<9isin(|<9 2 )]
+ |sin(^0
= -|(9icos(|(9i) + sin(^(9i),
/"
Jo
I 2 sin(±O 1 )d0 1 = \Tr.
(11)
(12)
Combining (10) and (12) gives
T 1 /57T 7T\ 3 .„„
The Variance is
= 2^ / / [ 2 | sin (^i)sin(§0 2 )sin[i(^-^)]|
*/ »/o
3 l 2
d6 2 d6\
2ttJ
= i {4sin 2 (i9 1 )sin 2 (ie 2 )sin 2 [i(e 2 -e 1 )]
-^|sin(ie 1 )sm(^ 2 )sin[|(^ 1 -0 9 )]| + _L j d6» 2 d0 :
6 /5tt tt\ 9 , ,1
-;(T + 4) + 4^ (2 ->]
~ 2~^ \~4 9+ 2/ _ 2^\4 2/
(13)
3(tt 2 - 6)
8tt 2
; 0.1470.
(14)
see also POINT-POINT DISTANCE — 1-D, TETRAHEDRON
Inscribing
Triangle Inscribing in an Ellipse
(«,0)
(-*,y)
(*,J0
Triangle Postulate
Triangular Cupola 1841
To inscribe an EQUILATERAL TRIANGLE in an ELLIPSE,
place the top VERTEX at (0,6), then solve to find the
(x,y) coordinate of the other two VERTICES.
y/x 2 + (b-y) 2 = 2x
x + (b — y) —Ax
3x 2 = (b-y)\
Now plugging in the equation of the ELLIPSE
2 2
X v
a 2 + 6 2 '
gives
3a 2 ( 1 - V
b 2
b 2 -2by + y 2
y 2 (l+3^) - 2by + (b 2 - 3a 2 ) =
(1)
(2)
(3)
(4)
(5)
(6)
26
^46' - 4(6* _ 3a2) (x + 3 £)
2(l + 3f|)
1 + 3
6,
6*
and
a; = ±a<i/l- -^ .
(7)
(8)
Triangle Postulate
The sum of the Angles of a Triangle is two Right
Angles. This Postulate is equivalent to the Paral-
lel 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.
Triangle Squaring
A D B
Let CD be the Altitude of a Triangle AABC and
let E be its MIDPOINT. Then
area(A,4£C) = \AB <CD = AB- DE,
and UDABFG can be squared by RECTANGLE SQUAR-
ING. The general POLYGON can be treated by draw-
ing diagonals, squaring the constituent triangles, and
then combining the squares together using the PYTHAG-
OREAN Theorem.
see also Pythagorean Theorem, Rectangle
Squaring
References
Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1
in Journey Through Genius: The Great Theorems of
Mathematics. New York: Wiley, pp. 14-15, 1990.
Triangle Tiling
A
n=l n~2 n=3
The total number of triangle (including inverted ones)
in the above figures is given by
N(n)
\|[n(n
+ 2)(2n+ 1) for n even
+ 2)(2n+l) - 1] for n odd.
The first few values are 1, 5, 13, 27, 48, 78, 118, 170,
235, 315, 411, 525, 658, 812, 988, 1188, 1413, 1665, ...
(Sloane's A002717).
References
Conway, J. H. and Guy, R. K. "How Many Triangles." In The
Book of Numbers. New York: Springer- Verlag, pp. 83-84,
1996.
Sloane, N. J. A. Sequence A002717/M3827 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Triangle Transformation Principle
The triangle transformation principle gives rules for
transforming equations involving an INCIRCLE to equa-
tions about ExciRCLES.
see also EXCIRCLE, INCIRCLE
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 191-192, 1929.
Triangular Cupola
Johnson Solid J 3 . The bottom six Vertices are
(iiA±5,0),(0,±l,0),
1842 Triangular Dipyramid
and the top three VERTICES are
2V3 7± 2 J V 3 J '
see also JOHNSON SOLID
Triangular Dipyramid
The triangular (or TRIGONAL) dipyramid is one of the
convex Deltahedra, and JOHNSON SOLID J i2 .
see also Deltahedron, Dipyramid, Johnson Solid,
Pentagonal Dipyramid
Triangular Graph
The triangular graph with n nodes on a side is denoted
T(n). Tutte (1970) showed that the Chromatic Poly-
nomials of planar triangular graphs possess a ROOT
close to <j> 2 = 2.618033..., where <j> is the Golden
Mean. More precisely, if n is the number of VERTICES
of (3, then
(Le Lionnais 1983, p. 46). Every planar triangular graph
possesses a Vertex of degree 3, 4, or 5 (Le Lionnais
1983, pp. 49 and 53).
see also LATTICE GRAPH
References
Le Lionnais, F. Les nombres remarquables . Paris: Hermann,
1983.
Tutte, W. T. "On Chromatic Polynomials and the Golden
Ratio." J. Combin. Theory 9, 289-296, 1970.
Triangular Hebesphenorotunda
see Johnson Solid
Triangular Matrix
An upper triangular Matrix U is defined by
Ua = {
aij for i < j
for i > j.
a)
Written explicitly,
an a\2
0,22
Q>2n
(2)
Triangular Number
A lower triangular MATRIX L is defined by
L . _ / a H for i ^ 3
\ for i < j.
j %3
(3)
Written explicitly,
an
•
•
OL21
Q>22
•
.
dnl
CLn2
a n n
(4)
see also Hessenberg Matrix, Hilbert Matrix, Ma-
trix, Vandermonde Matrix
Triangular Number
A Figurate Number of the form T n = n(n + l)/2 ob-
tained by building up regular triangles out of dots. The
first few triangle numbers are 1, 3, 6, 10, 15, 21, ...
(Sloane's A000217). T 4 = 10 gives the number and ar-
rangement of Bowling pins, while T& = 15 gives the
number and arrangement of balls in BILLIARDS. Trian-
gular numbers satisfy the RECURRENCE RELATION
T„+i a -T„ 2 = (n+l) s ,
(1)
as well as
and
3T n + T n _i =T 2n (2)
3T n + T n+ i = T2n+i (3)
1 + 3 + 5 + . . . + (2n - 1) = T n + T n -x (4)
(2n + l) 2 = 8T + 1 = T„_i + 6T n + T n+1 (5)
(Conway and Guy 1996). They have the simple Gen-
erating Function
m
(1-x) 3
^x+3x 2 -\-6x 3 + 10x 4 + 15x 5 + .... (6)
Every triangular number is also a HEXAGONAL NUM-
BER, since
|r(r + l)
(r±l) [ 2 (r±i)-i] for r odd
("§) [2 ("§)-!] for r even.
(7)
Triangular Number
Also, every PENTAGONAL NUMBER is 1/3 of a triangular
number. The sum of consecutive triangular numbers is
a Square Number, since
T r +T,-i = |r(r + l) + i(r-l)r
-ir[(r + l) + (r-l)] = r 2 . (8)
Interesting identities involving triangular numbers and
Square Numbers are
52(-l) k+l T k =;
(9)
Tj = Y,^ = \n 2 {n+\f (10)
k = l
J2 k3 = T » ( u >
for g Odd and
fc = l,3 ( ...,q
n=\{q 2 + 2q-\).
(12)
All Even Perfect Numbers are triangular T v with
Prime p. Furthermore, every Even Perfect Number
P > 6 is of the form
P = 1 + 9T n
t3n+lj
(13)
where T n is a triangular number with n — Sj 4- 2 (Eaton
1995, 1996). Therefore, the nested expression
9(9 •• • (9(9(9(9T n + 1) + 1) + 1) + 1) . . . + 1) + 1 (14)
generates triangular numbers for any T n . An Integer k
is a triangular number IFF Sk + 1 is a SQUARE NUMBER
> 1.
The numbers 1, 36, 1225, 41616, 1413721, 48024900,
. . . (Sloane's A001110) are SQUARE TRIANGULAR NUM-
BERS, i.e., numbers which are simultaneously triangular
and SQUARE (Pietenpol 1962). Numbers which are si-
multaneously triangular and TETRAHEDRAL satisfy the
Binomial Coefficient equation
(:)•
(15)
the only solutions of which are (m,n) = (10,16), (22,
56), and (36, 120) (Guy 1994, p. 147).
The smallest of two INTEGERS for which n 3 - 13 is four
times a triangular number is 5 (Cesaro 1886; Le Lionnais
1983, p. 56). The only FIBONACCI NUMBERS which are
triangular are 1, 3, 21, and 55 (Ming 1989), and the only
Triangular Number 1843
PELL NUMBER which is triangular is 1 (McDaniel 1996).
The Beast Number 666 is triangular, since
T 6 . 6 = T 36 = 666.
(16)
In fact, it is the largest REPDIGIT triangular number
(Bellew and Weger 1975-76).
Fermat's Polygonal Number Theorem states that
every Positive Integer is a sum of most three Tri-
angular Numbers, four Square Numbers, five Pen-
tagonal Numbers, and n ti-Polygonal Numbers.
Gauss proved the triangular case, and noted the event
in his diary on July 10, 1796, with the notation
* * ETRHKA num = A + A + A.
(17)
This case is equivalent to the statement that every num-
ber of the form 8m + 3 is a sum of three ODD SQUARES
(Duke 1997). Dirichlet derived the number of ways in
which an Integer m can be expressed as the sum of
three triangular numbers (Duke 1997). The result is
particularly simple for a PRIME of the form 8m + 3, in
which case it is the number of squares mod 8m +3 minus
the number of nonsquares mod 8m + 3 in the INTERVAL
4m + 1 (Deligne 1973).
The only triangular numbers which are the PRODUCT of
three consecutive INTEGERS are 6, 120, 210, 990, 185136,
258474216 (Guy 1994, p. 148).
see also Figurate Number, Pronic Number,
Square Triangular Number
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 59, 1987.
Bellew, D. W. and Weger, R. C. "Repdigit Triangular Num-
bers." J. Recr. Math. 8, 96-97, 1975-76.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 33-38, 1996.
Deligne, P. "La Conjecture de Weil." Inst Hautes Etudes
Sci. Pub. Math. 43, 273-308, 1973.
Dudeney, H. E. Amusements in Mathematics. New York:
Dover, pp. 67 and 167, 1970.
Duke, W. "Some Old Problems and New Results about Quad-
ratic Forms." Not. Amer. Math. Soc. 44, 190-196, 1997.
Eaton, C. F. "Problem 1482." Math. Mag. 68, 307, 1995.
Eaton, C. F. "Perfect Number in Terms of Triangular Num-
bers." Solution to Problem 1482. Math. Mag. 69, 308-
309, 1996.
Guy, R. K. "Sums of Squares" and "Figurate Numbers."
§C20 and §D3 in Unsolved Problems in Number Theory,
2nd ed. New York: Springer- Verlag, pp. 136-138 and 147-
150, 1994.
Hindin, H. "Stars, Hexes, Triangular Numbers and Pythag-
orean Triples." J. Recr. Math. 16, 191-193, 1983-1984.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
p. 56, 1983.
McDaniel, W. L. "Triangular Numbers in the Pell Sequence."
Fib. Quart. 34, 105-107, 1996.
Ming, L. "On Triangular Fibonacci Numbers." Fib. Quart.
27, 98-108, 1989.
Pappas, T. "Triangular, Square & Pentagonal Numbers."
The Joy of Mathematics. San Carlos, CA: Wide World
Publ./Tetra, p. 214, 1989.
1844 Triangular Orthobicupola
Tribar
Pietenpol, J. L "Square Triangular Numbers." Amer. Math.
Monthly 169, 168-169, 1962.
Satyanarayana, U. V. "On the Representation of Numbers as
the Sum of Triangular Numbers." Math. Gaz. 45, 40-43,
1961.
Sloane, N. J. A. Sequences A000217/M2535 and A001110/
M5259 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Triangular Orthobicupola
see Johnson Solid
Triangular Pyramid
see Tetrahedron
Triangular Square Number
see Square Triangular Number
Triangular Symmetry Group
(2, 3, 3) (2, 3, 4) (2, 3, 5)
Given a TRIANGLE with angles (7r/p, 7r/g, 7r/r), the
resulting symmetry GROUP is called a (p, q, r) triangle
group (also known as a Spherical Tessellation). In
3-D, such GROUPS must satisfy
111,
- + - + -> 1,
p q r
and so the only solutions are (2,2,n), (2,3,3), (2,3,4),
and (2, 3, 5) (Ball and Coxeter 1987). The group (2, 3, 6)
gives rise to the semiregular planar TESSELLATIONS of
types 1, 2, 5, and 7. The group (2, 3, 7) gives hyperbolic
tessellations.
see also GEODESIC DOME
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 155-
161, 1987.
Coxeter, H. S. M. "The Partition of a Sphere According to
the Icosahedral Group." Scripta Math 4, 156-157, 1936.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, 1973.
Kraitchik, M. "A Mosaic on the Sphere." §7.3 in Mathemat-
ical Recreations. New York: W. W. Norton, pp. 208-209,
1942.
Triangulation
Triangulation is the division of a surface into a set of
Triangles, usually with the restriction that each TRI-
ANGLE side is entirely shared by two adjacent TRIAN-
GLES. It was proved in 1930 that every surface has a
triangulation, but it might require an infinite number
of TRIANGLES. A surface with a finite number of trian-
gles in its triangulation is called COMPACT. B. Chazelle
showed that an arbitrary Simple Polygon can be tri-
angulated in linear time.
see also Compact Surface, Delaunay Triangula-
tion, Japanese Triangulation Theorem, Simple
Polygon
Triaugmented Dodecahedron
see Johnson Solid
Triaugmented Hexagonal Prism
see Johnson Solid
Triaugmented Triangular Prism
One of the convex Deltahedra and Johnson Solid
J 5 i. The Vertices are (±1,±1,0), (0,0,^2),
(0,±l,-^/3), (±(1 + x/6)/2,0,-(V2 + a/3)/2), where
the x and z coordinates of the last are found by solving
z 2 + l 2 + (z + ^) 2 = 2 2
(x- l) 2 + l 2 + 2 2 = 2 2 .
see also Deltahedron, Johnson Solid
Triaugmented Truncated Dodecahedron
see Johnson Solid
Triaxial Ellipsoid
see Ellipsoid
Tribar
D
\L
D
D
An Impossible Figure published by R. Penrose (1958).
It also exists as a Tribox.
References
Draper, S. W. "The Penrose Triangle and a Family of Related
Figures." Perception 7, 283-296, 1978.
Fineman, M. The Nature of Visual Illusion. New York:
Dover, p. 119, 1996.
Tribox
Trident
1845
Jablan, S. "Set of Modular Elements 'Space Tiles'." http://
members . tripod . com/ -modularity/space . htm.
Pappas, T. "The Impossible Tribar." The Joy of Mathemat-
ics. San Carlos, CA: Wide World PubL/Tetra, p. 13, 1989.
Penrose, R. "Impossible Objects: A Special Type of Visual
Illusion." Brit. J. Psychology 49, 31-33, 1958.
Tribox
/
/
/
/
An Impossible Figure.
see also Impossible Figure, Tribar
References
Jablan, S. "Are Impossible Figures Possible?"
members . tripod . com/ -modular it y/kulpa . htm.
http://
Tribonacci Number
The tribonacci numbers are a generalization of the FI-
BONACCI Numbers defined by T\ = 1, T 2 = 1, T 3 = 2,
and the RECURRENCE RELATION
T n — T n -l + Tn-2 + T n
(1)
for n > 4. The represent the n = 3 case of the FI-
BONACCI n-STEP Numbers. The first few terms are 1,
1, 2, 4, 7, 13, 24, 44, 81, 149, ... (Sloane's A000073).
The ratio of adjacent terms tends to 1.83929, which is
the REAL ROOT of x 4 - 2x 3 + 1 = 0. The Tribonacci
numbers can also be computed using the GENERATING
Function
= 1 + z + 2z 2 + 4z 3 + 7z 4
: - Z 2 _
1 - z — z* — z°
+13/ + 24/ + 44/ + 81/ + 149/ + . . . . (2)
An explicit FORMULA for T n is also given by
{|(19 + 3^/33) 1/3 + ^(19-3V33) 1/3 + |} rt (586 + 102y / 33) 1/3
(586 + 102^/33) 2 / 3 + 4 - 2(586 + 102 v / 33) 1/3
('3)
where [x] denotes the NlNT function (Plouffe). The first
part of a Numerator is related to the Real root of
x 3 — x 2 — x — 1, but determination of the DENOMINATOR
requires an application of the LLL Algorithm. The
numbers increase asymptotically to
T n
where
(1| + 1^33)1/3 + | ( li + lV33)- 1 / 3 +l
^27
= 1.83928675521.
(4)
(5)
see also Fibonacci n-STEP Number, Fibonacci Num-
ber, Tetranacci Number
References
Plouffe, S. "Tribonacci Constant." http://lacim.uqam:ca/
piDATA/tribo.txt.
Sloane, N. J. A. Sequence A000073/M1074 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Trichotomy Law
Every Real Number is Negative, 0, or Positive.
Tricolorable
A projection of a Link is tricolorable if each of the
strands in the projection can be colored in one of three
different colors such that, at each crossing, all three col-
ors come together or only one does and at least two dif-
ferent colors are used. The TREFOIL KNOT and trivial
2-link are tricolorable, but the UNKNOT, WHITEHEAD
Link, and Figure-of-Eight Knot are not.
If the projection of a knot is tricolorable, then REIDE-
MEISTER Moves on the knot preserve tricolorability, so
either every projection of a knot is tricolorable or none
is.
Tricomi Function
see Confluent Hypergeometric Function of the
Second Kind, Gordon Function
Tricuspoid
see Deltoid
Tricylinder
see Steinmetz Solid
Tridecagon
A 13-sided POLYGON, sometimes also called the
Triskaidecagon.
Trident
The plane curve given by the equation
3 3
xy = x — a .
see also Trident of Descartes, Trident of New-
ton
(Plouffe).
1846 Trident of Descartes
Trident of Descartes
Trigonal Dodecahedron
The plane curve given by the equation
(a + x)(a — x)(2a — x) = x 3 — 2ax 2 — a 2 x + 2o? = axy
(a + x)(a — x)(2a — x)
The above plot has a = 2.
Trident of Newton
The Cubic Curve defined by
ax 3 + 6x 2 + cz + d — xy
with a ^ 0. The curve cuts the axis in either one or
three points. It was the 66th curve in Newton's classi-
fication of CUBICS. Newton stated that the curve has
four infinite legs and that the y-axis is an ASYMPTOTE
to two tending toward contrary parts.
References
New
Lawrence, J* D, A Catalog of Special Plane Curves.
York: Dover, pp. 109-110, 1972.
MacTutor History of Mathematics Archive. "Trident of New-
ton." http: //www-groups . dcs . st-and.ac .uk/ -history/
Curves/Trident .html.
Tridiagonal Matrix
A Matrix with Nonzero elements only on the diagonal
and slots horizontally or vertically adjacent the diagonal.
A general 4x4 tridiagonal Matrix has the form
an ai2
0>21 &22 G>23
az2 «33 &34
(243 &44
Inversion of such a matrix requires only n (as opposed
to n 3 ) arithmetic operations (Acton 1990).
see also Diagonal Matrix, Jacobi Algorithm
References
Acton, F. S. Numerical Methods That Work, 2nd printing.
Washington, DC: Math. Assoc. Amer., p. 103, 1990.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Tridiagonal and Band Diagonal Systems of
Equations." §2.4 in Numerical Recipes in FORTRAN: The
Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 42-47, 1992.
Tridiminished Icosahedron
see Johnson Solid
Tridiminished Rhombicosidodecahedron
see Johnson Solid
Tridyakis Icosahedron
The Dual Polyhedron of the Icositruncated Do-
DECADODECAHEDRON.
Trifolium
Lawrence (1972) defines a trifolium as a FOLIUM with
b € (0,4a). However, the term "the" trifolium is some-
times applied to the FOLIUM with b = a, which is then
the 3-petalled ROSE with Cartesian equation
(a; +y )[y + x(x + a)} — 4axy
and polar equation
r = a cos 0(4 sin 2 - 1) = -a cos(30),
The trifolium with b — a is the Radial Curve of the
Deltoid.
see also BlFOLIUM, FOLIUM, QUADRIFOLIUM
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York; Dover, pp. 152-153, 1972.
MacTutor History of Mathematics Archive. "Trifolium."
http: //www-groups .dcs . st-and. ac ,uk/~history7 Curves
/Trifolium, html.
Trigon
see Triangle
Trigonal Dipyramid
see Triangular Dipyramid
Trigonal Dodecahedron
An irregular DODECAHEDRON.
Trigonometric Functions
Trigonometry 1847
see also DODECAHEDRON, PYRITOHEDRON, RHOMBIC
Dodecahedron
References
Cotton, F. A. Chemical Applications of Group Theory, 3rd
ed. New York: Wiley, p. 62, 1990.
Trigonometric Functions
see Trigonometry
Trigonometric Series
Trigonometric Substitution
Integrals of the form
/
f (cos 9, sin 9) d9
(1)
can be solved by making the substitution z = e % so that
dz = ie 1 dO and expressing
e + e
-%9
z + z' 1
(2)
A sin(20) + B sin(4<£) + C sin(6<£) + D sin(8<£)
= sin(20)(A' + cos(2<j>)(B' + cos(20)(C' + D' cos(20)))),
where
A' = A-C
B f = 2B- 4D
C f = 4C
D' = SD.
A sin + B sin(3<£) + C sin(5<£) + D sm(7<p)
= sin0(A' + sin 2 4>(B' + sin 2 0(C" + £>' sin 2 <£))),
where
4' = A + SB + 5C + ID
B' = -45 - 20C - 56L>
C' = 16C+112D
D' = -64D.
A -h B cos(20) + C cos(40) + D cos(60) + E cos(8</>)
= A' + cos(20)(B' + cos(20)(C' + cos(20)
x(Z>' + :E'cos(20)))),
where
A' =A-C + E
B f =B-3D
C' = 2C - 8£
£>' = 4L>
£7' = 8E.
References
Snyder, J. P. Map Projections— A Working Manual. U, S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, p. 19, 1987.
= . (3)
2i 2% w
The integral can then be solved by CONTOUR INTEGRA-
TION.
Alternatively, making the substitution t = tan(#/2)
transforms (1) into
/ { ( 2t 1 ~ t2 \ 2dt (A)
J T \ 1 + t 2 ' 1 + t 2 ) 1 + t 2 * [ }
The following table gives trigonometric substitutions
which can be used to transform integrals involving
square roots.
Form
Substitution
x — a sin
x = a tan#
x = a sec 8
y/a 2 — x 2
Va 2 + x 2
yjx 2 — a 2
see also HYPERBOLIC SUBSTITUTION
Trigonometry
The study of Angles and of the angular relationships
of planar and 3-D figures is known as trigonometry.
The trigonometric functions (also called the CIRCULAR
FUNCTIONS) comprising trigonometry are the COSE-
CANT esc z, Cosine cosx, Cotangent cotx, Secant
secx, Sine sin a;, and Tangent tanx. The inverses of
these functions are denoted esc" 1 a;, cos" 1 x, cot -1 x,
sec -1 x, sin -1 x, and tan -1 x. Note that the / _1 Nota-
tion here means INVERSE FUNCTION, not f to the -1
Power.
sin
The trigonometric functions are most simply defined us-
ing the Unit Circle. Let be an Angle measured
counterclockwise from the z-AxiS along an Arc of the
CIRCLE. Then cos is the horizontal coordinate of the
Arc endpoint, and sin# is the vertical component. The
Ratio sin 6/ cos is defined as tan#. As a result of this
1848 Trigonometry
definition, the trigonometric functions are periodic with
period 27r, so
func(27m + 0) — func(<9),
(1)
where n is an INTEGER and tunc is a trigonometric func-
tion.
Prom the Pythagorean Theorem,
sin + cos 0=1.
Therefore, it is also true that
tan 2 6 + 1 = sec 2
1 + cot 2 = esc 2 0.
(2)
(3)
(4)
The trigonometric functions can be defined algebraically
in terms of COMPLEX EXPONENTIALS (i.e., using the
Euler Formula) as
2%
cscz =
2%
smz e lz — e~
COSZ =
e + e
secz =
tanz =
cosz
sinz
e xz + e -i
e iz - e~
cosz i(e iz + e~ iz )
cotz =
tanz
i(e iz +e~ iz ) _ i{l + e~ 2iz )
e iz _ e ~iz ~ I _ e -2iz
(5)
(6)
(7)
(8)
(9)
• (10)
Osborne's Rule gives a prescription for converting
trigonometric identities to analogous identities for HY-
PERBOLIC Functions.
The ANGLES nit/m (with m, n integers) for which the
trigonometric function may be expressed in terms of fi-
nite root extraction of real numbers are limited to val-
ues of m which are precisely those which produce con-
structible POLYGONS. Gauss showed these to be of the
form
m = 2 pip2 • * -Pa,
(ii)
where k is an INTEGER > and the p% are distinct FER-
mat Primes. The first few values are m = 1, 2, 3, 4,
5, 6, 8, 10, 12, 15, 16, 17, 20, ... (Sloane's A003401).
Although formulas for trigonometric functions may be
found analytically for other m as well, the expressions
involve ROOTS of COMPLEX NUMBERS obtained by solv-
ing a Cubic, Quartic, or higher order equation. The
cases m — 1 and m = 9 involve the Cubic Equation
and Quartic Equation, respectively. A partial table
of the analytic values of Sine, Cosine, and Tangent
for arguments iv/m is given below. Derivations of these
formulas appear in the following entries.
Trigonometry
° rad
tana;
22.5
30.0
10'
0.0 1
15.0 ^7T KV6-V2) \(y/E+y/2) 2 - V$
18.0 ^tt \{Vh-l) |v / 10 + 2v / 5 |V25- lOy^
V2 - 1
V / 5^2\/5
1
„ \ \^
36.0 \ix |\/l0-2v / 5 |(1 + V5)
45.0 W
lv/2
60.0 \-k
1V3
90.0 \-k
1
80.0 7T
1
2
CO
The Inverse Trigonometric Functions are gener-
ally defined on the following domains.
Function
Domain
sin -1 x
-!*■ < y < \*
cos -1 X
< y < 7v
tan -1 x
-\<k <y<\n
esc -1 X
Q<y<\ir or 7T <y <^
sec -1 x
< y < 7T
cot -1 X
< y < |?r or — 7r <y< — \tt
Inverse-forward identities are
tan -1 (cot x) = \ix — x
sin - (cos a:) = \-r — x
sec - (esc a:) = ^n — x,
and forward-inverse identities are
cos(sin - x) =
cos(tan - x) =
\/l-x 2
1
Vl + x 2
sin(cos - x) =
sin(tan - x) =
\/l-x 2
X
Vl-r-x 2
tan(cos - x) —
tan(sin~ 1 x) =
Vl-x 2
X
X
Inverse sum identities include
sin l x + cos l x = \ir
tan l x + cot x x = ^7r
sec x + esc a? =
>*"»
where (20) follows from
x = sin(sin -1 x) = cos(|-7r — sin" 1 x).
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
Trigonometry
Complex inverse identities in terms of LOGARITHMS in-
clude
sin 1 (z) = — i\n(iz ± yl — z 1 )
cos -1 (jz) = — i\n(z ± iy 1 — z 2 )
tan 1 (z) = — iln
1 + iz
x/T+z~i
2 \l + iz)
For Imaginary arguments,
sin(iz) = isinhz
cos(iz) = cosh 2.
(24)
(25)
(26)
(27)
(28)
(29)
For Complex arguments,
sin(x + iy) = sin x cosh y + i cos a: sinh y (30)
cos(a; + it/) = cos x cosh t/ — i sin a: sinh y. (31)
For the ABSOLUTE Square of COMPLEX arguments z =
| sin(# + iy) | = sin 2 x + sinh 2 y
(32)
| cos(x + iy)\ 2 = cos 2 x + sinh 2 y. (33)
The MODULUS also satisfies the curious identity
| sin(x + iy)\ = | sin x + sin(iy) \ . (34)
The only functions satisfying identities of this form,
\f(x + iy)\ = \f(z) + f(iy)\ (35)
are f(z) — Az, f(z) = Asm(bz), and f(z) — ^4sinh(6z)
(Robinson 1957).
Trigonometric product formulas can be derived using
the following figure (Kung 1996).
y
fr sin 0)
In the figure,
0=\{a-(3)
7= §(« + /?),
(36)
(37)
Trigonometry 1849
s = |(sina + sin/3) = cos[£(a - /3)]sin[±(a + /3)]
(38)
£ = |(cosa + cos/3) = cos[|(a — /3)]cos[|(a + /?)].
(39)
A sin /?)
-1 1 1
With 6 and 7 as previously defined, the above figure
(Kung 1996) gives
u = cos /3 — cos a = 2 sin[|(a — /?)] sin[| (a + j3)]
(40)
= sin a -sin/3 = 2sin[|(a - /3)] cos[i(a + /?)].
(41)
Angle addition FORMULAS express trigonometric func-
tions of sums of angles a ± j3 in terms of functions of
a and /3. They can be simply derived used COMPLEX
exponentials and the Euler FORMULA,
sin(a+/3) =
,<(«+<9> _ e -<<«+<8) e ia e^ -
e e — e e
2i 2i
(cos a -\- i sin a) (cos (3 -\- i sin /3)
_
(cos a — i sin a) (cos /3 — isin/3)
—
cos a cos (3 + i sin /3 cos a + i sin a cos (3 — sin a sin (3
— cos a cos /3 + i cos a sin (3 -\- i sin a cos /3 4- sin a sin /3
2i
= sin a: cos (3 -f- sin /3 cos a
(42)
cos(a+/3) =
s *(«+*> + e -H«+P) e - e ^ +(
2 2
(cos a + isina)(cos/3 + isin/3)
_
(cos a — i sin a) (cos /3 — i sin /3)
2
cos a cos /3 + i cos a sin f3 -\- i sin a cos /3 — sin a sin /3
cos a cos /3 — i cos a sin /? — i sin a cos (3 — sin a sin (3
= cos a cos (3 — sin a sin j3.
(43)
1850
Trigonometry
Trigonometry
Taking the ratio gives the tangent angle addition For-
mula
_ sin(o: 4- 0) sin a cos + sin cos a
tan(a-fp) = — „^„^__^^^
cos(a: + 0) cos a cos — sin a sin /?
~+~ „„ a fl tana + tanp
cos/3
1 _ sinaain/3 ! _ tanatan/3*
cos Q! cos ap
(44)
The angle addition FORMULAS can also be derived
purely algebraically without the use of COMPLEX NUM-
BERS. Consider the following figure.
tain /J
Lco&fi
From the large RIGHT TRIANGLE,
sin(o; + 0) =
cos(ct + 0) =
L sin + a
L cos a + b
Lcos/3
L cos a + b
But, from the small triangle (inset at upper right),
Lsina
a = ; -rr-
cos(a + 0)
b = L sin a tan(a + 0).
(45)
(46)
(47)
(48)
Plugging a and 6 from (47) and (48) into (45) and (46)
gives
sin(a + 0) =
^ cos(a+pj
L, L sina sin(o!+j3)
C0Sa + co»(a +J 9)
sin cos(a + 0) + sin a
cos a cos(a -f- /3) + sin a sin(a + /3) '
(49)
and
cos(a + 0)
Lcos0
L cos a +
L sin a sin(cx+p")
cos(a+/9)
cos/3
rnq r* -I- si " " sirl ("+/ 3 ) '
COba-h cos (a-h/3)
Now solve (50) for cos(a + 0),
cos(a -h 0) cos a + sin a sin(a + 0) = cos /?
to obtain
/ rt . cos /3 — sin a sin(a + 0)
cos(oj + p) = -.
(50)
(51)
(52)
Plugging (52) into (49) gives
sin(a + (3) =
Sin/3 r cos/3-sin QS in (a + ^) -| .
^ |_ cos ot J
r cos ff — sin
: L
a sin(or + /3) "1
+ sin ct sin(a -f 0)
sin cos /3 — sin a sin /3 sin(a + 0) + sin a cos a
cos a cos — sin a cos a sin(a + 0) + sin a cos a sin(a + 0)
sin cos — sin a sin sin(a + /3) + sin a cos a
cos a: cos
sin a cos a + sin /3 cos sin a sin /3
cos a cos /3
cos a cos /3
sin(a+/?), (53)
sin(a + 0) 1 +
t a sin /3 \
s a cos /3 y
sin a cos a + sin cos /3
cos a cos /3
(54)
sin (a + 0) (cos a cos + sin a sin 0)
= sin a cos a 4- sin /3 cos /3, (55)
and
sin(a + 0)
sin a cos a -f sin cos /3
sin a sin /3 + cos a cos /3
__ sin a cos a + sin cos /3 sin oc cos /3 + sin cos a
sin a sin + cos a cos sin a cos -f sin /3 cos a
(56)
Multiplying out the DENOMINATOR gives
(cos a cos + sin a sin /3) (sin a cos /3 + sin cos a:)
= sin a cos a cos + cos a sin cos /3
+ sin a sin /3 cos 4- sin a cos a sin /3
= sin a cos a + sin cos /3,
so
sin(a + 0) = sin a cos/3 + sin /3 cos a.
Multiplying out (50),
(57)
(58)
cos(a + 0) cos a + sin a sin(a + 0) = cos /3 (59)
cos(a+/3) =
cos /3 — sin a sin(a + /3)
cos a
_ cos — sin a(sin a cos + sin /3 cos a)
cos a
_ cos 0(1 — sin 2 a) + sin a cos a sin /3
cos a
_ cos 2 a cos /3 + sin a cos a sin
cos a
= cos a cos + sin a sin /?.
(60)
Trigonometry
Summarizing,
sin(a + 0) = sin a cos -f sin cos a
sin(a — 0) = sin a cos — sin /? cos a
cos(a + 0) — cos a cos — sin a sin /?
cos(a — 0) — cos a cos /? + sin a sin
tana + tan/3
tan(o; + j3)
tan(a — (3)
1 — tan a tan f3
tana — tan/?
1 + tana tan f3
(61)
(62)
(63)
(64)
(65)
(66)
The sine and cosine angle addition identities can be sum-
marized by the MATRIX EQUATION
cos x sin x
— sin x cos x
cosy
— sin y
sin y
cosy
cos(ic + y) sin(x + y)
— sin(a: + y) cos(as- -f y)
The double angle formulas are
sin(2a) = 2 sin a cos a
cos(2a) = cos 2 a — sin 2 a
= 2 cos a — 1
= 1-2 sin 2 a
tan(2a)
2 tan a
1 — tan 2 a
General multiple angle formulas are
sin(na) = 2sin[(n - l)a]cosa — sin[(n - 2)a]
sin(na;) = ncos n_1 a; since
n(n- l)(n- 2) n _ 3 . 3
i l± l cos jsin x + .
1-2-3
cos(na) = 2 cos[(n — l)a] cos a - cos[(n - 2)a]
, x n n(n- 1) n -2 . 2
cos(nx) = cos x — ■— cos xsm x
n(n-l)(n-2)(n-3) ^-4 .4
-( — r^-4 — ^-r cos ccsin X
tan(na)
1-2- 3-4
tan[(n — l)a] + tana
1 — tan[(n — l)a] tana'
(67)
(68)
(69)
(70)
(71)
(72)
(73)
(74)
(75)
(76)
(77)
Therefore, any trigonometric function of a sum can be
broken up into a sum of trigonometric functions with
sin a cos a cross terms. Particular cases for multiple an-
gle formulas up to n = 4 are given below.
sin(3a) = 3 sin a - 4 sin 3 a (78)
cos(3a) = 4 cos 3 a - 3 cos a (79)
tn N 3 tan a -tan 3 a /onX
tan ( 3 «) = l-3fn»a (80)
sin(4a) = 4 sin a cos a — 8 sin 3 a cos a (81)
cos(4a) = 8 cos 4 a - 8 cos 2 a + 1 (82)
. . 4 tan a — 4 tan 3 a /ooX
tan(4a) = — — 5 — 7—. (83)
v ; 1-6 tan 2 a + tan 4 a v '
Trigonometry 1851
Beyer (1987, p. 139) gives formulas up to n — 6.
Sum identities include
tan(a - (3) _ sin(a - f3) cos(a + /?)
tan(a + f3) cos(a - f3) sin(a + /3)
(sin a cos j3 — sin (3 cos a) (cos a cos (3 — sin a sin f3)
(cos a cos + sin a sin f3) (sin a cos /? + sin f3 cos a)
sin a cos a — sin cos /?
sin a cos a 4- sin cos /3
• (84)
Infinite sum identities include
~ kx sm(ky) 1
£
fc-1. 3, 5,..,
= - tan
2
_! / sin y
(S)- (85 >
Trigonometric half-angle formulas include
sin
/ a \ / 1 — cos a
cos
tan
/ a N / 1 -f cos a
UJ = v 2
/ a \ sin a
\2/ 1 + cosa
1 — cos a
sin a
1 ± \Zl + tan 2 a
tana
tan a sin a
tan a + sin a
(86)
(87)
(88)
(89)
(90)
(91)
The Prosthaphaeresis Formulas are
sin a + sin£ = 2sin[|(a + 0)] cos[f (a - 0)] ( 92 )
sina - sin/3 = 2 cos[|(a + 0)] sin[|(a - /?)] (93)
sin a + cos/3 = 2 cos[f (a + 0)] cos [|( a ~ 0)1 ( 94 )
cosa - cos/? = -2 sin[f (a + /?)] sin [§(a " /?)]• ( 95 )
Related formulas are
sin a cos /3 = § [sin(a - /?) + sin(a + /?)] (96)
cos a cos —\ [cos(a — 0) + cos(a + /3)] (97)
cos a sin = § [sin(a + /?) - sin(a - /?)] (98)
sinasin/?= |[cos(a-/3) -cos(a + /3)]. (99)
Multiplying both sides by 2 gives the equations some-
times known as the WERNER FORMULAS.
Trigonometric product/sum formulas are
sin(a + 0) sin (a - 0) = sin 2 a - sin 2 — cos 2 - cos a
(100)
1852 Trigonometry
Trigonometry Values — n/2
cos(a + P) cos(a - p) = cos 2 a - sin 2 P — cos 2 P - sin a.
Power formulas include
sin x = | [1 ■
s(2x)]
sin 3 x = \ [3 sin x - sin(3x)]
sin 4 x = | [3 — 4 cos(2cc) + cos(4cc)]
(101)
(102)
(103)
(104)
and
cos 2 x = |[1 + cos(2x)]
cos 3 x — | [3 cos x -f cos(3a?)]
(105)
(106)
cos 4 x = I [3 + 4 cos(2x) + cos(4z)] (107)
(Beyer 1987, p. 140). Formulas of these types can also
be given analytically as
sin 2n x ■
2n
2 2n V n
+
cos[2(n - k)x] (108)
sin 2 " +1 = ^ £>l) fc ( 2n + *) sin[(2n + 1 - 2*jx]
(109)
cos
n — 1 / \
1 y> /2n\
+ 2 2n-l l^\k )
fc=0 v 7
! " +1 i = ^E ( 2n fc + x ) cos K 2n + J - 2fc)x] (111)
cos[2(n- fc)x] (110)
(Kogan), where (™) is a BINOMIAL COEFFICIENT.
see also Cosecant, Cosine, Cotangent, Euclidean
Number, Inverse Cosecant, Inverse Cosine, In-
verse Cotangent, Inverse Secant, Inverse Sine,
Inverse Tangent, Inverse Trigonometric Func-
tions, Osborne's Rule, Polygon, Secant, Sine,
Tangent, Trigonometry Values: 7r, 7r/2, tt/3, tt/4,
tt/5, tt/6, tt/7, tt/8, tt/9, tt/IO, tt/11, tt/12, tt/15, tt/16,
tt/17, tt/18, tt/20, 0, Werner Formulas
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Circular Func-
tions." §4.3 in Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972.
Bahm, L. B. The New Trigonometry on Your Own. Patter-
son, NJ: Littlefield, Adams & Co., 1964.
Beyer, W. H. "Trigonometry." CRC Standard Mathematical
Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 134-152,
1987.
Dixon, R. "The Story of Sine and Cosine." § 4.4 in Matho-
graphics. New York: Dover, pp. 102-106, 1991.
Hobson, E. W. A Treatise on Plane Trigonometry. London:
Cambridge University Press, 1925.
Kells, L. M.; Kern, W. F.; and Bland, J. R. Plane and Spher-
ical Trigonometry. New York: McGraw-Hill, 1940.
Kogan, S. "A Note on Definite Integrals Involving Trigono-
metric Functions." http : //www .mathsof t . com/asolve/
constant /pi/s in/ sin . html.
Kung, S. H. "Proof Without Words: The Difference-Product
Identities" and "Proof Without Words: The Sum-Product
Identities." Math. Mag. 69, 269, 1996.
Maor, E. Trigonometric Delights. Princeton, NJ: Princeton
University Press, 1998.
Morrill, W. K. Plane Trigonometry, rev. ed. Dubuque, IA:
Wm. C. Brown, 1964.
Robinson, R. M. "A Curious Mathematical Identity." Amer.
Math. Monthly 64, 83-85, 1957.
Sloane, N. J. A. Sequence A003401/M0505 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Thompson, J. E. Trigonometry for the Practical Man.
Princeton, NJ: Van Nostrand.
$ Weisstein, E. W. "Exact Values of Trigonometric Func-
tions." http : //www . astro . Virginia . edu/ ~eww6n/math/
notebooks/TrigExact .m.
Yates, R. C. "Trigonometric Functions." A Handbook on
Curves and Their Properties. Ann Arbor, MI: J. W. Ed-
wards, pp. 225-232, 1952.
Zill, D. G. and Dewar, J. M. Trigonometry. New York:
McGraw-Hill 1990.
Trigonometry Values — n
By the definition of the trigonometric functions,
sin 7r =
COS7T = — 1
tan 7r —
CSC 7T = oo
sec7r = — 1
COt7T = OO.
(1)
(2)
(3)
(4)
(5)
(6)
Trigonometry Values — tt/2
By the definition of the trigonometric functions,
(!)-
(§)-
(0 =
G)-
(!) =
G)-
cos
tan
cot
oo
0.
(1)
(2)
(3)
(4)
(5)
(6)
see also DlGON
Trigonometry Values — n/3
Trigonometry Values — 7r/3
Prom Trigonometry Values: 7r/6
Trigonometry Values — rr/5 1853
-(!)-»
(i)
-(1) = *^
(2)
together with the trigonometric identity
sin(2ai) = 2 sin a cos a,
(3)
the identity
sin(|)=2sin(^)cos(^)=2(i)(|V3)
= *vs
is obtained. Using the identity
(4)
cos(2a) = 1 — 2 sin 2 a,
(5)
then gives
cos(0=l-2sin 2 (0=l-2(!) 2 =
§• ( 6 )
Summarizing,
*(!) = »<«
(7)
~(JH
(8)
tan (^\ = Vs.
(9)
see also Equilateral Triangle
Trigonometry Values — 7r/4
For a Right Isosceles Triangle, symmetry requires
that the angle at each VERTEX be given by
~7T + 2a = 7T,
so a = 7r/4. The sides are equal, so
sin a + cos 2 a = 2 sin 2 a = 1.
(i)
(2)
Solving,
sin (i)
= iv/2
(3)
cos (i)
= ^
(4)
tan(^)
= 1.
(5)
see also SQUARE
Trigonometry Values — 7r/5
Use the identity
sin(5a) = 5 sin a — 20 sin 3 a + 16 sin 5 a.
Now, let a = 7r/5 and x = sin a. Then
sinTr = = 5z - 20x 3 + 16x 5
16a; 4 - 20z 2 + 5 = 0.
Solving the Quadratic Equation for x 2 gives
j _ 20 ± A /(-20) 2 - 4 • 16 - 5
(1)
(2)
(3)
*•(!)
2*16
20 ± V80
= H5±V5).
32 8 '
Now, sin(7r/5) must be less than
-m(J) = *V5,
so taking the MINUS SIGN and simplifying gives
sm
(f)-^-l>/5^
cos(7r/5) can be computed from
Summarizing,
(D-i/»-*'(f)-i('+^-
/2tt
/37T
47T
cos
tan
tan
tan
tan
(
(i
(2-k
1 (t
/3tt
'(t
'(t
(I.
It
/3tt
V 5
/47T
= |V / 10-2%/5
= i\/l0 + 2v / 5
= i\/l0-2%/5
= K 1 + v / 5)
= |(-l + V / 5)
= 1(1-^5)
= %/5-2\/5
= \/5 + 2v/5
= -V5 + 2\/5
= -^5-2^5.
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
see also Dodecahedron, Icosahedron, Pentagon,
Pentagram
1854 Trigonometry Values — rr/6
Trigonometry Values — 7r/6
Given a Right Triangle with angles defined to be a
and 2a, it must be true that
a -\- 2a + \tv — 7r,
(i)
so a = 7r/6. Define the hypotenuse to have length 1
and the side opposite a to have length a?, then the side
opposite 2a has length \/l — z 2 . This gives sin a = z
and
' (2)
sin
(2a) = yjl-x 2 .
But
sin(2a) — 2 sin a cos a — 2xy 1 — £ 2 , (3)
so we have
sfl-x 2 = 2x^1 -a; 2 - (4)
This gives 2x = 1, or
sin(j)=|. (5)
cos(7r/6) is then computed from
cos (0 = ^/1-^ (J) = ^-(^ = 173. (6)
Summarizing,
-(5) = I (7)
cos(j) = |v / 3 (8)
tan(j)=iV3. (9)
see also HEXAGON, HEXAGRAM
Trigonometry Values — tt/7
Trigonometric functions of nir/7 for n an integer cannot
be expressed in terms of sums, products, and finite root
extractions on real rational numbers because 7 is not a
Fermat Prime. This also means that the Heptagon
is not a CONSTRUCTIBLE POLYGON.
However, exact expressions involving roots of complex
numbers can still be derived using the trigonometric
identity
sin(na) = 2 sin[(n — l)a] cos a — sin[(n — 2)a]. (1)
The case n — 7 gives
sin(7a) = 2 sin(6a) cos a — sin(5a)
= 2(32 cos a sin a — 32 cos a sin a+6 cos a sin a) cos a
- (5 sin a - 20 sin 3 a -f 16 sin 5 a)
= 64 cos a sin a — 64 cos a sin a + 12 cos a sin a
—5 sin a + 20(1 — cos a) sin a
— 16(1 — 2 cos a + cos a) sin a
= sin a(64 cos 6 a - 80 cos 4 a + 24 cos 2 a - 1) . (2)
Trigonometry Values — ty/7
Rewrite this using the identity cos 2 a — 1 — sin 2 a,
sin I — J = sin a(7 — 56 sin a + 112 sin a — 64 sin a)
= -64sina(sin 6 a- ^ sin 4 a + || sin 2 a - ^). (3)
Now, let a = tt/7 and x = sin 2 a, then
• / \ n 3 72,7 7
Sin(7T) = = 2 - Z X +8 X_ 64'
(4)
which is a CUBIC EQUATION in x. The ROOTS are
numerically found to be x « 0.188255, 0.611260...,
0.950484 — But sin a = y^, so these ROOTS corre-
spond to sin a « 0.4338, sin(2a) « 0.7817, sin(3a) ss
0.9749. By Newton's Relation
we have
or
J_J_ r * = " a 0'
£i£2#3 = 64,
(5)
(6)
'2tt\ .
- sin
(t)
\^-
sin(0sin( 7;
Similarly,
/tt\ /2?r\ /3tt\ 1
cos^jcos( y jcos( y j = -.
The constants of the CUBIC EQUATION are given by
(7)
(8)
Q=i(3a 1 -a 2 2 ) = i[3-|-(-|) 2 ] = -
R = i(9a 2 ai - 2a\ - 27a )
= ^[9(-D(78)-2(-|) 3 -27(-i)]
_ __7_
~ 3456'
(9)
(10)
The Discriminant is then
D = Q 3 + R 2
343
49
2,985,984 ' 11,943,936
<0,
(11)
so there are three distinct Real Roots. Finding the
first one,
Writing
■= Vr + Vd+Vr-Vd-\
^ = 3" 3/2 tU
a 2 .
(12)
(13)
plugging in from above, and anticipating that the solu-
tion we have picked corresponds to sin(37r/7),
Trigonometry Values — rr/8
sin 1 — =
V 7 J
V£ =
VviSi
;+3"
7
3/2 j
128
V 3456
- 3-
7 1 / 7\
3/2 i_ - (-- )
128 3\ 4/
-H-
7
3456
+ 3" 3 / 2
— i + V^
128 V :
7
J456
7 7
_ 3-3/2 i+ _
128 12
= V V 3^ ( "
-1 + 3 3 '
' V 3456
(1 + 3-/-0 + 1
12
yi(_l + 3»/»0 - Y2CI + 3»/»i) + 7
(14)
see a/50 Heptagon
Trigonometry Values — tt/S
sin(^)=sin(i.j) = yi(l-cos|)
= ) /l(l-i^) = iV'2->/2. (1)
Now, checking to see if the SQUARE ROOT can be sim-
plified gives
a 2 - 6 2 c = 2 2 - l 2 ■ 2 = 4 - 2 = 2,
(2)
which is not a PERFECT Square, so the above expres-
sion cannot be simplified. Similarly,
■(§).-- (H)-yiO^fJ
(3)
tan —
(i)
I 2-V2 _ J(2~V2)
2 + v^ "
4-2
4+2-4^
But
= /^p=v^^.
a 2 - b 2 c = 3 2 - 2 2 2 = 9 - 8 = 1
(4)
(5)
is a Perfect Square, so we can find
d= §(3±1) = 1,2.
Rewrite the above as
V2 + 1
tan
V8/ V2-1 2" 1
+ 1.
(6)
(7)
Ifr'gonomefcry Values — tt/9 1855
Summarizing,
sin (f)
1^2-^2
iV2+71
'(f) -*^^
^ = 1^
cos U;
V2
tan
see ateo Octagon
(8)
(9)
(10)
(11)
(12)
(13)
Trigonometry Values — 7r/9
Trigonometric functions of mr/9 radians for n an in-
teger not divisible by 3 (e.g., 40° and 80°) cannot be
expressed in terms of sums, products, and finite root
extractions on real rational numbers because 9 is not a
product of distinct Fermat Primes. This also means
that the NONAGON is not a CONSTRUCTIBLE POLYGON,
However, exact expressions involving roots of complex
numbers can still be derived using the trigonometric
identity
sin(3a) = 3 sin a — 4 sin a.
(i)
Let a = 7r/9 and x = sin a. Then the above identity
gives the Cubic EQUATION
4x 3 -3x + 1^3 =
s 3 -!* = -iV3.
This cubic is of the form
where
x +px = q,
(2)
(3)
(4)
(5)
(6)
The Discriminant is then
k 3 / „\ 2
-H)
256
+
<0.
16
+
16-4 16-16
-4 + 3
256
(7)
1856 Trigonometry Values — 7r/10
Trigonometry Values — 7r/ll
There are therefore three Real distinct roots, which are
approximately -0.9848, 0.3240, and 0.6428. We want
the one in the first QUADRANT, which is 0.3240.
Summarizing,
^3
(i) = V ~if + v ~i + V "if " V "a!
256
V3 , 1 .
16+I6 1 -
V3 , 1 .
I6 + I6 J
:2- 4/3 (t/iW5-\/i+7l)
: 0.3240....
(8)
Similarly,
« 0.7660....
(9)
Because of the Newton's Relations, we have the iden-
tities
* (|) *(?)*(*)., (1 „,
~(i)~(T)-(f)-»^ <">
t m (l)ta„(f)t»(f) = V3. (12)
see a/so NONAGON, STAR OF GOLIATH
Trigonometry Values — k/10
sin(^)=sin(i.|) = v /i[l-cos(|)]
= ^/l[l- 1(1 + ^5)] = 1(^-1). (1)
So we have
Go-) = cos (H) = vT
= ^/l[l + 1(1 + ^5)]
= i\/l0 + 2v / 5,
1 + cos
(I)]
(2)
and
tan
(s)-^-*^ 7 ^- < 3 >
cos(^) -|\/l0 + 2v / 5
tan (~ ) = |a/25-10V5
sin
(4)
(5)
(6)
(7)
(8)
(9)
cos (fj) =1(10-2^5)
tan (^) = |\/25 + W5.
An interesting near-identity is given by
\ [cos(^) + cosh(^) + 2cos(^^)cosh(^x/2)] « 1.
(10)
In fact, the left-hand side is approximately equal to 1 +
2.480 x 10 -13 .
see also DECAGON, DECAGRAM
Trigonometry Values — 7r/ll
Trigonometric functions of nir/ll for n an integer cannot
be expressed in terms of sums, products, and finite root
extractions on real rational numbers because 11 is not a
Fermat Prime. This also means that the Undecagon
is not a CONSTRUCTIBLE POLYGON.
However, exact expressions involving roots of complex
numbers can still be derived using the trigonometric
identity
sin(llo:) = sin(12o: — a) cos a — cos(12a) sin a
= 2 sin(6a) cos(6a) cos a — [1 — 2 sin (6a)] sin a. (1)
Using the identities from Beyer (1987, p. 139),
sin(6a) = sin a cos a[32 cos 4 a — 32 cos 2 a + 6] (2)
cos(6a) = 32 cos 6 a - 48 cos 4 a + 18 cos 2 a - 1 (3)
gives
sin(lla) = 2 cos asina(32cos a — 32 cos a + 6)
x (32 cos 6 a - 48 cos 4 a + 18 cos 2 a - 1)
- sin a[l - 2 sin 2 a cos 2 a(32 cos 4 a - 32 cos 2 a + 6) 2 ]
= sin a(ll - 220 sin 2 a + 1232 sin 4 aa
-2816 sin 6 a + 2816 sin 8 -1024 sin 10 a). (4)
Now, let a = 7r/ll and x = sin 2 a, then
sin7T =
= 11 - 220z + 1232x 2 - 2816x 3 + 2816x 4 - 1024z 5 . (5)
Trigonometry Values — tt/12
This equation is an irreducible QuiNTlC EQUATION, so
an analytic solution involving FINITE ROOT extractions
does not exist. The numerical ROOTS are x = 0.07937,
0.29229, 0.57115, 0.82743, 0.97974. So sin a = 0.2817,
sin(2a:) = 0.5406, sin(3a) = 0.7557, sin(4a) = 0.9096,
sin(5a) = 0.9898. Prom one of Newton's Identities,
sin (n) sin (it) sin (it) sin (it) Kit)
11 ^ (6)
1024 32
(n) cos ( fi) cos (it) cos (it) cos (it)
32
(7)
tan (£) tan (£) tan (£) tan (£) tan (£)
= y/U. (8)
The trigonometric functions of 7r/ll also obey the iden-
tity
tan(^)+4 S in(^)=x/n. (9)
see also Undecagon
References
Beyer, W. H. "Trigonometry." CRC Standard Mathematical
Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.
Trigonometry Values — tt/12
sin (^)= sin (| -J)
= -sin(0cos(|)+sin(0cos(j)
= -i>/2(i) + f>/3(i>/2)
= i(V6-V2). (1)
Similarly,
cos (S) =cos (f"i)
= cos (I) cos (f) -sin (f) sin (j)
= 1(^/6 + V2). (2)
Summarizing,
sin (~) - |(V6 - a/2) w 0.25881 (3)
cos(^) - 1(^+^)^0.96592 (4)
tan (^A = 2 - V3 « 0.26794 (5)
esc (^) = v^ 4- v 7 ^ « 3.86370 (6)
sec f~) = \/6 - >/2 « 1.03527 (7)
cot (—\ =2 + V3^ 3.73205. (8)
Trigonometry Values — 7r/16 1857
Trigonometry Values — 7r/15
7T 7T \
/ 7T \ . (TT 7T
ll5J =Sln U-10
= sin © cos (lu)- sin (lr]) cos ©
^(2^3 - 2\/l5+ V4oToV^) (1)
1 (1
2
and
(S) =cos (J-S)
= cos(^)cos(^)+sin(|)sin(^)
= ^1(5 + ^ + 51(^-1)
2 V 8
= i(\/30 + 6v^+V / 5-l).
Summarizing,
sin (J) = ^(2\/3-2v / 15 + V / 4oT8v^)
: 0.20791
Sin (lf) = 5(^+^- Vl0-2>/5)
: 0.40673
(2)
(3)
(4)
cos (^) = 1(^30 + 6^5 + V5 - 1) « 0.97814 (5)
cos f^) = I(>/30-6>/5 + 1) « 0.91354 (6)
tan (l5") = K 3 ^ ~ Vl5 - \/50 - 22\/5 )
: 0.21255.
(7)
Trigonometry Values — 7r/16
sin (io) =sin (H)
tan
= ^/|-iv / 2T^ = i v 2 - ^ 2 +^ (1)
(ie-) =cos (H)
-^( 1 + COS ?) = ^( 1 + l ^ T ^)
= \A + j\/2 + v^=iV2+\/2 + V2 (2)
/jr\ _ / 2 - -s/2 + a/2
Vie/Va + V^TTl
= \/4 + 2\/2-\/2-l.
(3)
1858 Trigonometry Values — 7r/17
Summarizing,
sinfy^) = |V2- Vu^w 0.19509
Sin (lf) = 2V 2 ~ \/2 - \/2 ?y 0.55557
COS (l^) = 2V 2 +^ /2 + v/ 2^ - 98079
COS (lf) = lv 2 + ^2-^2^0.83147
tan (^-) == ^4 + 2^ - v^ - 1 « 0.19891. (8)
(4)
(5)
(6)
(7)
Trigonometry Values — 7r/17
Rather surprisingly, trigonometric functions of rnr/17
for n an integer can be expressed in terms of sums, prod-
ucts, and finite root extractions because 17 is a Fer-
mat Prime. This makes the Heptadecagon a Con-
STRUCTIBLE, as first proved by Gauss. Although Gauss
did not actually explicitly provide a construction, he did
derive the trigonometric formulas below using a series of
intermediate variables from which the final expressions
were then built up.
Let
c= V / 17 + v / 17
e* .= \/l7- VT7
a = y v / 34 + 6v / 17+(v / 34- V2)e* -8\/2e
j3 = 2\/l7 + 3\/l7- 2\/2e - V2e* .
Then
'{tt) = IVh- 2 ^- 2 ^**
-2^68 + 12v / 17 + 2v / 2(v / 17- l)e* - 16\/2e] 1
: 0.18375
iL J = j[30 + 2v / 17 + 2v / 2e*
+2^68 + 12\/l7+2\/2(VT7 - l)e* - 16V2e]
0.98297
2tt
~ j = ^[136-8VT7 + 4x/2(l - \/l7)e' + 16\/2e
+2(\/2- v / 34-2e*) V /34 + 6VT7+(V34- V2)e* - 8V2e] 1/2
« 0.36124
(~)-*[- 1 + ^ / 17 + V / 2^
+ \/ 68 + 12\ / 17 - 2\/2(l - \/l7)e* - 16\/2e]
s 0.0.93247
sin
(~) = i i (-v / 2 + V / 34 + 26* + 2a)
X V / 68-4yi7-2(v / 34- V2)e* + 8^2 c + a(V2 ~ ^34 - 2e*)
w 0.0.67370
Trigonometry Values — 7r/18
V~J = i[136-8vTT + 8>/2c-2(\/34-3v / 2)e*
-2/3(1 - vTY- v^e*)] 1 ' 2
w 0.99573
-2\/l7 + 3\/l7- \/2e'- 2^2 e),
sb 0.09227
There are some interesting analytic formulas involving
the trigonometric functions of nir/17. Define
P(x) = (x- l)(x-2)(x 2 + l)
ffiW =
54 (a;)
2+^P^)
1-s
2-y^fr)
1-x
/i(x) = 1^(^-1]
a = | tan" 1 4,
where i = 1 or 4. Then
/i(tana) = cos ^j
f 4 (tan a) = cos ( — J .
see also CONSTRUCTIBLE POLYGON, FERMAT PRIME,
Heptadecagon
References
Casey, J. Plane Trigonometry. Dublin: Hodges, Figgis, &
Co., p. 220, 1888.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 192-194 and 229-230, 1996.
Dorrie, H. "The Regular Heptadecagon." §37 in 100 Great
Problems of Elementary Mathematics: Their History and
Solutions. New York: Dover, pp. 177-184, 1965.
Ore, 0. Number Theory and Its History. New York: Dover,
1988.
Smith, D. E, A Source Book in Mathematics. New York:
Dover, p. 348, 1994.
Trigonometry Values — tt/18
The exact values of cos(7r/18) and sin(7r/18) are given
by infinite NESTED RADICALS.
sin (^) = iy 2 - ^+^2 + V2^.
« 0.17365,
where the sequence of signs +, +, - repeats with period
3, and
cos ( j) - I V3 j y 8 - ^/s-x/s + v/S^+l
« 0.98481,
where the sequence of signs — , — , + repeats with period
3.
Trigonometry Values — iv/20
Trigonometry Values — 7r/20
Trilinear Coordinates
1859
sin (^) =sin GS) = v^( 1 -
= iy8-2\/l0 + 2\/5
= JV8-2\/lO +
~, 0.15643
7T
cos io
i)
cos(j)=cos(^)=yi(l
(1)
+ cos^
i)
s 0.98768 . . .
'l0 + 2a/5
tan (^-) = 1 + v/5 - ^5 + 2^
w 0.15838.
(2)
(3)
« 0.15838
An interesting near-identity is given by
\ [cos(i)+cosh(i) + 2cos(i^)cosh(^A/2)] «1.
(4)
In fact j the left-hand side is approximately equal to 1 +
2.480 x 10" 13 .
Trigonometry Values —
By the definition of the trigonometric functions,
sinO =
cos = 1
tan =
esc = oo
sec = 1
cot = oo.
Trigyrate Rhombicosidodecahedron
see Johnson Solid
Trihedron
The Triple of unit Orthogonal Vectors T, N, and
B (Tangent Vector, Normal Vector, and Binor-
mal Vector).
see also BlNORMAL VECTOR, NORMAL VECTOR, TAN-
GENT Vector
Trilinear Coordinates
Given a Triangle AABC, the trilinear coordinates of
a point P with respect to AABC are an ordered TRIPLE
of numbers, each of which is PROPORTIONAL to the di-
rected distance from P to one of the side lines. Trilinear
coordinates are denoted a : f3 : 7 or (a, £,7) and also
are known as Barycentric Coordinates, Homoge-
neous Coordinates, or "trilinears."
In trilinear coordinates, the three VERTICES A, B, and
C are given by 1 : : 0, : 1 : 0, and : : 1. Let the
point P in the above diagram have trilinear coordinates
a : (3 : 7 and lie at distances a', b' , and c' from the
sides BC, AC, and AB, respectively. Then the distances
a — ka> b' = k/3, and c f = ky can be found by writing
A a for the Area of ABPC, and similarly for A 6 and
A c . We then have
A = A a + A 6 + A c = \aa + \bb* + ~cc
= \{akoc + bk(3 + ckj) = \k(aoc + b(3 + erf). (1)
2A
aa + b(3 + cj '
(2)
where A is the Area of AABC and a, b, and c are the
lengths of its sides. When the values of the coordinates
are taken as the actual lengths (i.e., the trilinears are
chosen so that k = 1), the coordinates are known as
Exact Trilinear Coordinates.
Trilinear coordinates are unchanged when each is mul-
tiplied by any constant jit, so
h :t2 '. ts = \it\ : \iti : ^3*
When normalized so that
t\ + 1 2 + t 3 = 1,
(3)
(4)
trilinear coordinates are called AREAL COORDINATES.
The trilinear coordinates of the line
ux + vy + wz —
u : v : w = ad a '• bds ' ede,
(5)
(6)
where di is the Point-Line Distance from Vertex A
to the Line.
Trilinear coordinates for some common POINTS are sum-
marized in the following table, where A, B, and C are
the angles at the corresponding vertices and a, 6, and c
are the opposite side lengths.
1860
Trilinear Coordinates
Point
Triangle Center Function
centroid M
circumcenter O
de Longchamps point
equal detour point
Feuerbach point F
incenter I
isoperimetric point
Lemoine point
nine-point center N
orthocenter H
vertex A
vertex B
vertex C
esc A, 1/a
cos A
cos A — cos B cos C
sec(^A) cos{\B) cos(§C) + 1
1 - cos(B - C)
1
sec{\A) cos(§B) cos(f C) - 1
a
cos(£ - C)
cos B cos C
1:0:0
0:1:0
0:0:1
To convert trilinear coordinates to a vector position for
a given triangle specified by the x- and y-coordinates of
its axes, pick two UNIT VECTORS along the sides. For
instance, pick
(7)
(8)
where these are the Unit Vectors BC and AB. As-
sume the Triangle has been labeled such that A = xi
is the lower rightmost VERTEX and C = X2. Then the
VECTORS obtained by traveling l a and l c along the sides
and then inward PERPENDICULAR to them must meet
- yl .
+ /c
C 2
— &7
c 2
-ci
=
X
V
2
2
+
la
ax
a 2
— ka
OL2
-a x
• (9)
Solving the two equations
Xi + l c ci — kjC2 ~ X2l a o>\ ™- kaa2 (10)
yx + JcC 2 + k-ycx = y 2 laCi2 + kaax, (11)
gives
_ fca(q 1 c 1 + CL2C2) - 7fc(ci 2 + c 2 2 ) + ^2(^1 - x 3 ) + ^(3/3 - Vi)
CL1C2 — d 2 C\
(12)
ka{a x 2 + a 2 2 ) - 7/2(0x0! + a 2 c 2 ) + a 2 (x! - x 2 ) + ai(y 2 - J/i )
aiC 2 ~- 02^1
(13)
But a and c are Unit Vectors, so
lc
f« =
ka(axcx + a 2 c 2 ) - 7k + c 2 (xi - x 2 ) + ci(y 2 - 2/i)
aiC2 — &2Cl
(14)
ka -jk(axcx + a 2 c 2 ) + a 2 (xx - x 2 ) + ai(y 2 - yi)
axc 2 — a 2 cx
Trimagic Square
And the VECTOR coordinates of the point a : f3 : 7 are
then
xi + lc
— &7
C2
-ci
(16)
(15)
see also AREAL COORDINATES, EXACT TRILINEAR CO-
ORDINATES, Orthocentric Coordinates, Power
Curve, Quadriplanar Coordinates, Triangle,
Trilinear Polar
References
Boyer, C. B. History of Analytic Geometry. New York:
Yeshiva University, 1956.
Casey, J. "The General Equation — Trilinear Co-Ordinates."
Ch. 10 in A Treatise on the Analytical Geometry of the
Point, Line, Circle, and Conic Sections, Containing an
Account of Its Most Recent Extensions, with Numerous
Examples, 2nd ed,, rev. enl. Dublin: Hodges, Figgis, &
Co., pp. 333-348, 1893.
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, pp. 67-71, 1959.
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New
York: Wiley, 1969.
Coxeter, H. S. M. "Some Applications of Trilinear Coordi-
nates." Linear Algebra Appl. 226-228, 375-388, 1995.
Kimberling, C "Triangle Centers and Central Triangles."
Congr. Numer. 129, 1-295, 1998.
Trilinear Line
A Line is given in Trilinear Coordinates by
la + mj3 4- iry = 0.
see also Line, Trilinear Coordinates
Trilinear Polar
Given a TRIANGLE CENTER X — I : m : n, the line
la + m/3 + nj =
is called the trilinear polar of X' 1 and is denoted L.
see also Chasles's Polars Theorem
Trillion
The word trillion denotes different numbers in Amer-
ican and British usage. In the American system, one
trillion equals 10 12 . In the British, French, and German
systems, one trillion equals 10 18 .
see also Billion, Large Number, Million
Trimagic Square
If replacing each number by its square or cube in a
Magic Square produces another Magic Square, the
square is said to be a trimagic square. Trimagic squares
of order 32, 64, 81, and 128 are known. Tarry gave a
method for constructing a trimagic square of order 128,
Cazalas a method for trimagic squares of orders 64 and
81, and R. V. Heath a method for constructing an or-
der 64 trimagic square which is different from Cazalas's
(Kraitchik 1942).
Trimean
Trinomial Triangle 1861
Trimagic squares are also called Trebly Magic
Squares, and are 3-Multimagic Squares.
see also Bimagic Square, Magic Square, Mul-
timagic Square
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 212-
213, 1987.
Kraitchik, M. "Multimagic Squares." §7.10 in Mathematical
Recreations. New York: W. W. Norton, pp. 176-178, 1942.
Trimean
The trimean is defined to be
TM= i(ff 1 +2M + fr a ),
where Hi are the Hinges and M is the Median. Press
et al. (1992) call this TUKEY's TRIMEAN. It is an L-
ESTIMATE.
see also HlNGE, L-ESTIMATE, MEAN, MEDIAN (STATIS-
TICS)
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. 694, 1992.
Tukey, J. W. Explanatory Data Analysis. Reading, MA:
Addison- Wesley, pp. 46-47, 1977.
Trimorphic Number
A number n such that the last digits of n 3 are the same
as n. 49 is trimorphic since 49 3 = 117649 (Wells 1986,
p. 124). The first few are 1, 4, 5, 6, 9, 24, 25, 49, 51, 75,
76, 99, 125, 249, 251, 375, 376, 499, ....
see also Automorphic Number, Narcissistic Num-
ber, Super-3 Number
References
Wells, D. The Penguin Dictionary of Curious and Interesting
Numbers. Middlesex, England: Penguin Books, 1986.
A Minimal Surface discovered by L. P. M. Jorge and
W. Meeks III in 1983 with Enneper-WeierstraB Pa-
rameterization
f (C 3 - i)
2
(i)
9 = C 2
(2)
(Dickson 1990). Explicitly, it is given by
x = R
re ie 41n(re**-l)
S(l + re ie +r 2 e 2ie ) 9
2\n(l + re ie +r 2 e 2i9 )~
+ 9
(3)
» = -!»
~-3re i6 (l-\-re ie )
r 3 e sie _ i
4
+ -
V3(r s e Sie ljtan" 1 ^-^")]
(4)
r 3 e 3i9 _ 1
z = $l
-i
I 2
(5)
J 3{r*e 3ie - 1)
?
for € [0,2tt) and r € [0,4].
see also Minimal Surface
References
Dickson, S. "Minimal Surfaces." Mathematica J. 1, 38-40,
1990.
Wolfram Research "Mathematica Version 2.0 Graphics
Gallery." http : // www . mathsource . com / cgi - bin / Math
Source/Applications/Graphics/3D/0207-155.
Trinomial
A Polynomial with three terms.
see also BINOMIAL, MONOMIAL, POLYNOMIAL
Trinomial Identity
Trinoid
(x 2 + axy + by 2 )(t 2 + atu + bu 2 ) = r 2 + ars + bs 2 , (1)
where
r = xt - byu
s = yt-\- xu + ayu.
(2)
(3)
Trinomial Triangle
The Number Triangle obtained by starting with a
row containing a single "1" and the next row containing
three Is and then letting subsequent row elements be
1862
Triomino
Triple Scalar Product
computed by summing the elements above to the left,
directly above, and above to the right:
1
111
12 3 2 1
13 6 7 6 3 1
1 4 10 16 19 16 10 4 1
(Sloane's A027907). The nth row can also be obtained
by expanding (1 + x + x 2 ) n and taking coefficients:
(l + x + x 2 )° = l
(1 + x + x 2 ) 1 = l + x + x 2
(1 + x + x 2 ) 2
1 + 2x + Sx 2 + 2x 3 + x 4
and so on.
see also Pascal's Triangle
References
Sloane, N. J. A. Sequence A027907 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
Triomino
The two 3-POLYOMINOES are called triominoes, and are
also known as the TROMINOES. The left triomino above
is "Straight," while the right triomino is called "right"
or L-.
see also L-Polyomino, Polyomino, Straight Poly-
omino
References
Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer-
ican Book of Mathematical Puzzles & Diversions. New
York: Simon and Schuster, pp. 124-140, 1959.
Hunter, J. A. H. and Madachy, J. S. Mathematical Diver-
sions. New York: Dover, pp. 80-81, 1975.
Lei, A. "Tromino." http://www.cs.ust.hk/-philipl/
omino/tromino . html
Trip-Let
A 3-dimensional solid which is shaped in such a way that
its projections along three mutually perpendicular axes
are three different letters of the alphabet. Hofstadter
(1989) has constructed such a solid for the letters G, E,
and B.
see also CORK PLUG
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, cover and pp. xiv, 1,
and 273, 1989.
Triple
A group of three elements, also called a TRIAD.
see also Amicable Triple, Monad, Pair, Pythag-
orean Triple, Quadruplet, Quintuplet, Tetrad,
Triad, Twins
Triple-Free Set
A Set of Positive integers is called weakly triple-free
if, for any integer x, the SET {x, 2x, 3x} £ S. It is called
strongly triple- free if x £ 5 IMPLIES 2x £ S and 3x 5.
Define
p(n) = max{|S| : S C {1, 2, . . . , n}
is weakly triple-free}
q(n) ==max{|S| : S C {1,2,. .. ,n}
is strongly triple-free},
where \S\ denotes the CARDINALITY of 5, then
lim E^ > |
and
lim ^^ =0.6134752692.
n~+oo n
(Finch).
see also Double-Free Set
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/triple/triple.html.
Triple Jacobi Product
see Jacobi Triple Product
Triple Point
A point where a curve intersects itself along three arcs.
The above plot shows the triple point at the ORIGIN of
the Trifolium (x 2 + y 2 ) 2 + ?>x 2 y - y 3 = 0.
see also Double Point, Quadruple Point
References
Walker, R. J. Algebraic Curves. New York: Springer- Verlag,
pp. 57-58, 1978.
Triple Scalar Product
see Scalar Triple Product
Triple Vector Product
Tritangent 1863
Triple Vector Product
see Vector Triple Product
Triplet
see Triple
Triplicate- Ratio Circle
see Lemoine Circle
Trisected Perimeter Point
A triangle center which has a Triangle Center Func-
tion
a = bc(v - c + a) (v - a -f b) ,
where v is the unique Real Root of
2x 3 - 3(a -h b + c)x 2 -f (a 2 + b 2 + c 2 + 86c + Sea + 8afe)x
-(6 2 c + ca + a 2 6 -f 56c 2 + 5ca 2 + 5a6 2 + 9abc) = 0.
References
Kimberling, C. "Central Points and Central Lines in the
Plane of a Triangle." Math. Mag. 67, 163-187, 1994.
Trisection
Angle trisection is the division of an arbitrary ANGLE
into three equal Angles. It was one of the three Geo-
metric Problems of Antiquity for which solutions
using only COMPASS and STRAIGHTEDGE were sought.
The problem was algebraically proved impossible by
Wantzel (1836).
Although trisection is not possible for a general Angle
using a Greek construction, there are some specific an-
gles, such as 7r/2 and it radians (90° and 180°, respec-
tively), which can be trisected. Furthermore, some AN-
GLES are geometrically trisect able, but cannot be con-
structed in the first place, such as 37r/7 (Honsberger
1991). In addition, trisection of an arbitrary angle can
be accomplished using a marked Ruler (a Neusis Con-
struction) as illustrated below (Courant and Robbins
1996).
An ANGLE can also be divided into three (or any WHOLE
Number) of equal parts using the Quadratrix of Hip-
PIAS or TRISECTRIX.
see also ANGLE BISECTOR, MACLAURIN TRISECTRIX,
Quadratrix of Hippias, Trisectrix
References
Bogomolny, A. "Angle Trisection." http://vvv.cut-the-
knot . com/pythagoras/archi .html.
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 190-191, 1996.
Courant, R. and Robbins, H. "Trisecting the Angle." §3.3.3
in What is Mathematics?: An Elementary Approach to
Ideas and Methods, 2nd ed. Oxford, England: Oxford Uni-
versity Press, pp. 137-138, 1996.
Coxeter, H. S.M. "Angle Trisection." §2,2 in Introduction to
Geometry, 2nd ed. New York: Wiley, p. 28, 1969.
Dixon, R. Mathographics. New York: Dover, pp. 50-51, 1991.
Dorrie, H. "Trisection of an Angle." §36 in 100 Great Prob-
lems of Elementary Mathematics: Their History and So-
lutions. New York: Dover, pp. 172-177, 1965.
Dudley, U. The Trisectors. Washington, DC: Math. Assoc.
Amer., 1994.
Geometry Center. "Angle Trisection." http://www.geom.
umn.edu : 80/docs/f orum/angtri/.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., pp. 25-26, 1991.
Ogilvy, C. S. "Angle Trisection." Excursions in Geometry.
New York: Dover, pp. 135-141, 1990.
Wantzel, P. L. "Recherches sur les moyens de reconnaitre si
un Probleme de Geometrie peut se resoudre avec la regie
et le compas." J. Math, pures appliq. 1, 366—372, 1836.
Trisectrix
see Catalan's Trisectrix, Maclaurin Trisectrix
Trisectrix of Catalan
see Catalan's Trisectrix
Trisectrix of Maclaurin
see Maclaurin Trisectrix
Triskaidecagon
see Tridecagon
Triskaidekaphobia
The number 13 is traditionally associated with bad luck.
This superstition leads some people to fear or avoid
anything involving this number, a condition known as
triskaidekaphobia. Triskaidekaphobia leads to interest-
ing practices such as the numbering of floors as 1, 2,
. . . , 11, 12, 14, 15, . . . , omitting the number 13, in many
high-rise hotels.
see also 13, Baker's Dozen, Friday the Thir-
teenth, Triskaidekaphobia
Tritangent
The tritangent of a CUBIC SURFACE is a PLANE which
intersects the surface in three mutually intersecting
lines. Each intersection of two lines is then a tangent
point of the surface.
see also CUBIC SURFACE
References
Hunt, B. "Algebraic Surfaces." http://www.mathematik.
uni-kl . de/-wwwagag/Galerie .html.
1864
Tritangent Triangle
Truncated Dodecadodecahedron
Tritangent Triangle
see Excentral Triangle
Trivial
According to the Nobel Prize- winning physicist Richard
Feynman (1985), mathematicians designate any THE-
OREM as "trivial" once a proof has been obtained — no
matter how difficult the theorem was to prove in the
first place. There are therefore exactly two types of
true mathematical propositions: trivial ones, and those
which have not yet been proven.
see also Proof, Theorem
References
Feynman, R. P. and Leighton, R. Surely You're Joking, Mr.
Feynman! New York: Bantam Books, 1985.
Trivialization
In the definition of a Fiber Bundle / : E ->> B, the
homeomorphisms gu : f~ x (U) — ¥ U x F that commute
with projection are called local trivializations for the
Fiber Bundle /.
see also Fiber Bundle
Trochoid
The curve described by a point at a distance b from the
center of a rolling CIRCLE of RADIUS a.
x — a<j) — b sin <j>
y = a — b cos 0.
If b < a, the curve is a Curtate Cycloid. If b = a, the
curve is a CYCLOID. If 6 > a, the curve is a PROLATE
Cycloid.
see also Curtate Cycloid, Cycloid, Prolate Cy-
cloid
References
Lee, X. "Trochoid." http://www.best.com/-xah/Special
PlaneCurves_dir/Trochoid_dir/trochoid.html.
Wagon, S. Mathematica in Action. New York: W. H. Free-
man, pp. 46-50, 1991.
Yates, R. C. "Trochoids." A Handbook on Curves and Their
Properties. Ann Arbor, MI: J. W. Edwards, pp. 233-236,
1952.
Tromino
see Triomino
True
A statement which is rigorously known to be correct. A
statement which is not true is called FALSE, although
certain statements can be proved to be rigorously UN-
DECIDABLE within the confines of a given set of assump-
tions and definitions. Regular two-valued LOGIC allows
statements to be only true or FALSE, but FUZZY LOGIC
treats "truth" as a continuum which can have any value
between and 1.
see also Alethic, False, Fuzzy Logic, Logic, Truth
Table, Undecidable
Truncate
To truncate a Real Number is to remove its nonin-
tegral part. Truncation of a number x therefore corre-
sponds to taking the Floor Function [x\.
see also Ceiling Function, Floor Function, Nint,
Round
Truncated Cone
see Conical Frustum
Truncated Cube
An Archimedean Solid whose Dual Polyhedron is
the Triakis Octahedron. It has Schlafli Symbol
t{4,3}. It is also Uniform Polyhedron Ug and has
Wythoff Symbol 2 3 | 4. Its faces are 8{3}+6{8}. The
Inradius, Midradius, and Circumradius for a = 1
r = ^5 + 2v / 2)V / 7 + 4\/2^ 1.63828
p= f(2 + >/2)« 1.70711
R=\ \/7 + 4\/2 « 1.77882.
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 138,
1987.
Truncated Cuboctahedron
see Great Rhombicuboctahedron (Archimedean)
Truncated Dodecadodecahedron
The Uniform Polyhedron U 59 , also called the Qua-
sitruncated Dodecahedron, whose Dual Polyhe-
dron is the Medial Disdyakis Triacontahedron.
It has Schlafli Symbol W | i and Wythoff Sym-
bol 2 | | 5. Its faces are 12{10} + 30{4} + 12{^}. Its
Circumradius for a = 1 is
r= \VrL.
Truncated Dodecahedron
Truncated Octahedral Number 1865
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, pp. 152-153, 1989.
Truncated Dodecahedron
An Archimedean Solid whose Dual Polyhedron is
the Triakis Icosahedron. It has Schlafli Symbol
t{5,3}. It is also Uniform Polyhedron C/ 2 e and has
Wythoff Symbol 2 3 | 5. Its faces are 20{3} + 12{10}.
The Inradius, Midradius, and Circumradius for a =
1 are
r= Jg (17\/2 + 3VT0)\/37 + 15\/5 « 2.88526
p= |(5 + 3y/E)n 2.92705
R=\ V74 + 30v/5 « 2.96945.
Truncated Great Dodecahedron
The Uniform Polyhedron U37 whose Dual Polyhe-
dron is the Small Stellapentakis Dodecahedron.
It has Schlafli Symbol t{5, §}. It has Wythoff
Symbol 2 § 5. Its faces are 12{§} + 12{10}. Its Cir-
cumradius for a = 1 is
R= ±\/34+l(h/5.
see also Great Icosahedron
References
Wenninger, M. J. Polyhedron Models. Cambridge, England:
Cambridge University Press, p. 115, 1971.
Truncated Great Icosahedron
see Great Truncated Icosahedron
Truncated Hexahedron
see Truncated Cube
Truncated Icosahedron
An Archimedean Solid used in the construction of
Soccer Balls. Its Dual Polyhedron is the Pen-
takis Dodecahedron. It has Schlafli Symbol
t{3,5}. It is also Uniform Polyhedron U 2 b and has
Wythoff Symbol 2 5 1 3. Its faces are 20{6} + 12{5}.
The Inradius, Midradius, and Circumradius for
a = 1 are
r= gfj (21 + VE)Vw + W5 « 2.37713
p= |(1 + \/5) as 2.42705
R=\ VSS + ISVS « 2.47802.
Truncated Icosidodecahedron
see Great Rhombicosidodecahedron (Archimed-
ean)
Truncated Octahedral Number
A Figurate Number which is constructed as an Oct-
ahedral Number with a Square Pyramid removed
from each of the six VERTICES,
TO n = 3ti _ 2 - 6F„_i = |(3n - 2)[2(3n - 2) 2 + 1],
where O n is an OCTAHEDRAL Number and P n is a
Pyramidal Number. The first few are 1, 38, 201, 586,
... (Sloane's A005910). The Generating Function
for the truncated octahedral numbers is
x(6x 3 + 55x 2 + 34s + 1)
(x - iy
= x + 38x 2 + 201s 3 + . . . .
1866
Truncated Octahedron
Truncated Tetrahedral Number
see also Octahedral Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 52, 1996.
Sloane, N. J. A. Sequence A005910/M5266 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Truncated Octahedron
An Archimedean Solid, also known as the Mecon,
whose Dual Polyhedron is the Tetrakis Hexa-
hedron. It is also Uniform Polyhedron Us and
has Schlafli Symbol t{3,4} and Wythoff Sym-
bol 2 4 | 3. The faces of the truncated octahedron are
8{6}+6{4}. The truncated octahedron has the Oh OCT-
AHEDRAL GROUP of symmetries.
: V2 a/2
The solid can be formed from an OCTAHEDRON via
Truncation by removing six Square Pyramids, each
with edge slant height a = s/3 and height h, where s is
the side length of the original OCTAHEDRON. From the
above diagram, the height and base area of the SQUARE
Pyramid are
A h — a .
(i)
(2)
The VOLUME of the truncated octahedron is then given
by the VOLUME of the OCTAHEDRON
V <
h^/2s z
9V2a 3
^octahedron = fV^S =yVZa (3)
minus six times the volume of the Square Pyramid,
V = ^octahedron " 6(|^/l) = (9^2 - ^2)^ = 8^2 d\
(4)
The truncated octahedron is a SPACE-FILLING POLYHE-
DRON. The INRADIUS, MlDRADIUS, and ClRCUMRADIUS
for a — 1 are
see also Octahedron, Square Pyramid, Trunca-
tion
References
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York:
Dover, pp. 29-30 and 257, 1973.
Truncated Polyhedron
A polyhedron with truncated faces, given by the
Schlafli Symbol t{ p }.
see also RHOMBIC POLYHEDRON, SNUB POLYHEDRON
Truncated Pyramid
see Pyramidal Frustum
Truncated Square Pyramid
The truncated square pyramid is a special case of a
Pyramidal Frustum for a Square Pyramid. Let
the base and top side lengths of the truncated pyramid
be a and 6, and let the height be h. Then the Volume
of the solid is
V ^ \{a +ab + b 2 )h.
This FORMULA was known to the Egyptians ca. 1850
BC. The Egyptians cannot have proved it without calcu-
lus, however, since Dehn showed in 1900 that no proof of
this equation exists which does not rely on the concept of
continuity (and therefore some form of INTEGRATION).
see also Frustum, Pyramid, Pyramidal Frustum,
Square Pyramid
Truncated Tetrahedral Number
A Figurate Number constructed by taking the (3n -
2)th Tetrahedral Number and removing the (n -
l)th Tetrahedral Number from each of the four cor-
ners,
Ttet„
Tesn-3 - 4Te n _i = |n(23n 2 - 27n + 10).
The first few are 1, 16, 68, 180, 375, ... (Sloane's
A005906). The Generating Function for the trun-
cated tetrahedral numbers is
a?(10x 2 + 12x+ 1) n 2 _ 3 rtrt 4
-^ — , HSA } - x + 16x 2 + 89z 3 + 180z 4 + . . . .
(x - l) 4
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 46-47, 1996.
Sloane, N. J. A. Sequence A005906/M5002 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
r= ^%/lOw 1.42302
(5)
p=§ = 1.5
(6)
R= ±Vl0« 1.58114.
(7)
Truncated Tetrahedron
Tschirnhausen Cubic Caustic 1867
Truncated Tetrahedron
An Archimedean Solid whose dual is the Triakis
Tetrahedron. It has Schlafli Symbol t{3,3}. It
is also Uniform Polyhedron £/ 2 and has Wythoff
Symbol 23 | 3. Its faces are 4{3} + 4{6}. The Inra-
DIUS, MlDRADIUS, and ClRCUMRADIUS for a truncated
tetrahedron with a — 1 are
r = ^\/22^ 0.95940
p= |V2 « 1.06066
R= J\/22« 1-17260.
Truncation
The removal of portions of SOLIDS falling outside a set
of symmetrically placed planes. The five PLATONIC
SOLIDS belong to one of the following three truncation
series (which, in the first two cases, carry the solid to its
Dual Polyhedron).
% «
Cube Truncated Cuboctahedron Truncated Octahedron
Cube Octahedron
Icosahedron
© e>
Icosidodec- Truncated Dodecahedron
ahedron Dodecahedron
Tetrahedron
Truncated
Tetrahedron
Octahedron
see also Stellation, Truncated Cube, Truncated
Dodecahedron, Truncated Icosahedron, Trun-
cated Octahedron, Truncated Tetrahedron,
Vertex Figure
Truth Table
A truth table is a 2-D array with n + 1 columns. The
first n columns correspond to the possible values of n
inputs, and the last column to the operation being per-
formed. The rows list all possible combinations of inputs
together with the corresponding outputs. For example,
the following truth table shows the result of the binary
AND operator acting on two inputs A and B, each of
which may be true or false.
A
B
AAB
F
F
F
F
T
F
T
F
F
T
T
T
see also And, Multiplication Table, Or, XOR
Tschebyshev
An alternative spelling of the name "Chebyshev."
see also CHEBYSHEV APPROXIMATION FORMULA,
Chebyshev Constants, Chebyshev Deviation,
Chebyshev Differential Equation, Chebyshev
Function, Chebyshev-Gauss Quadrature, Cheby-
shev Inequality, Chebyshev Integral, Cheby-
shev Phenomenon, Chebyshev Polynomial of the
First Kind, Chebyshev Polynomial of the Sec-
ond Kind, Chebyshev Quadrature, Chebyshev-
Radau Quadrature, Chebyshev-Sylvester Con-
stant
Tschirnhausen Cubic
The Tschirnhausen cubic is a plane curve given by
r cos
(¥)
and is also known as Catalan's Trisectrix and
L'HOSPITAL'S Cubic. The name Tschirnhaus's cubic
is given in R. C. Archibald's 1900 paper attempting to
classify curves (MacTutor Archive). Tschirnhaus's cu-
bic is the Negative Pedal Curve of a Parabola with
respect to the FOCUS.
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 87-90, 1972.
MacTutor History of Mathematics Archive. "Tschirnhaus's
Cubic." http : //www-groups . dcs . st-and . ac.uk/-history
/Curves/Tschirnhaus .html.
Tschirnhausen Cubic Caustic
The Caustic of the Tschirnhausen Cubic taking the
Radiant Point as the pole is Neile's Parabola.
1868
Tschirnhausen Cubic Pedal Curve
Tukey's Biweight
Tschirnhausen Cubic Pedal Curve
The Pedal Curve to the Tschirnhausen Cubic for
Pedal Point at the origin is the Parabola
x = 1 - t 2
y = 2t.
see also Parabola, Pedal Curve, Pedal Point,
Tschirnhausen Cubic
Tschirnhausen Transformation
A transformation of a POLYNOMIAL equation f(x) =
which is of the form y = g(x)/h(x) where g and h are
POLYNOMIALS and h{x) does not vanish at a root of
f(x) = 0. The Cubic Equation is a special case of such
a transformation. Tschirnhaus (1683) showed that a
POLYNOMIAL of degree n > 2 can be reduced to a form in
which the x n ~ x and x n ~ 2 terms have COEFFICIENTS.
In 1786, E. S. Bring showed that a general QuiNTlC
Equation can be reduced to the form
x h -j-px + q = 0.
In 1834, G. B. Jerrard showed that a Tschirnhaus trans-
formation can be used to eliminate the # n_1 , # n_2 , and
x n ~ 3 terms for a general Polynomial equation of de-
gree n > 3.
see also Bring Quintic Form, Cubic Equation
References
Boyer, C. B. A History of Mathematics. New York: Wiley,
pp. 472-473, 1968.
Tschirnhaus. Acta Eruditorum. 1683.
Tubular Neighborhood
The tubular embedding of a Submanifold M m c N n
of another MANIFOLD N n is an EMBEDDING t : M x
B n_m —¥ N such that t(x, 0) = x whenever x 6 M,
where B n_m is the unit BALL in K n_m centered at 0.
The tubular neighborhood is also called the PRODUCT
Neighborhood.
see also BALL, EMBEDDING, PRODUCT NEIGHBORHOOD
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 34-35, 1976.
Tucker Circles
Let three equal lines P\Q\, P2Q2, and P3Q3 be drawn
Antiparallel to the sides of a triangle so that two (say
P2Q2 and P3Q3) are on the same side of the third line as
A2P2Q3A3. Then P2Q3P3Q2 is an isosceles TRAPEZOID,
i.e., P3Q2, P1Q3, and P2Q1 are parallel to the respective
sides. The Midpoints d, C 2 , and C 3 of the antiparal-
lels are on the respective symmedians and divide them
proportionally.
If T divides KO in the same ratio, TCi, TC 2 , TC 3 are
parallel to the radii OA\, OA2, and OA3 and equal.
Since the antiparallels are perpendicular to the symme-
dians, they are equal chords of a circle with center T
which passes through the six given points. This circle is
called the Tucker circle.
If
KC\ KC2 KC3
KAi KA 2 KA3
then the radius of the Tucker circle is
KT
KO'
i2V / c 2 + (l-c) 2 tano;,
where uj is the Brocard Angle.
The Cosine Circle, Lemoine Circle, and Taylor
Circle are Tucker circles.
see also Antiparallel, Brocard Angle, Cosine
Circle, Lemoine Circle, Taylor Circle
References
Johnson, R. A. Modern Geometry: An Elementary Treatise
on the Geometry of the Triangle and the Circle. Boston,
MA: Houghton Mifflin, pp. 271-277 and 300-301, 1929.
Tukey's Biweight
The function
*(*)
={: M)i
for \z\ < c
for \z\ > c
sometimes used in ROBUST ESTIMATION. It has a min-
imum at z = — c/a/3 and a maximum at z = c/\/3,
where
_ 3x^_
n 2
1>'(z)
0,
Tukey's Trimean
and an inflection point at z = 0, where
^"(z) = -|=0.
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. 697, 1992.
Tukey's Trimean
see Trimean
Tunnel Number
Let a KNOT K be n-EMBEDDABLE. Then its tunnel
number is a Knot invariant which is related to n.
see also Embeddable Knot
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, p. 114, 1994.
Turan Graph
The (n,fc)-Turan graph is the EXTREMAL GRAPH on n
Vertices which contains no /c-Clique. In other words,
the Turan graph has the maximum possible number of
EDGES of any n-vertex graph not containing a COM-
PLETE Graph K k . Turan's Theorem gives the maxi-
mum number of edges i(n, k) for the (n, fc)-Turan graph.
For k = 3,
t(n,3) = |n 4 ,
so the Turan graph is given by the Complete Bipar-
tite Graphs
f K n/2 , n /2 n even
I JK"(„_i)/2,(n+i)/2 n odd-
see also Clique, Complete Bipartite Graph,
Turan's Theorem
References
Aigner, M. "Turan's Graph Theorem."
Monthly 102, 808-816, 1995.
Amer. Math.
Turan's Inequalities
For a set of POSITIVE 7*, k = 0, 1, 2..., Turan's in-
equalities are given by
7fc
■7fc-i7fc+i > °
for k = 1, 2, ... .
see also JENSEN POLYNOMIAL
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.
Szego, G. Orthogonal Polynomials, ^th ed. Providence, RI:
Amer. Math. Soc, p. 388, 1975.
Turning Angle 1869
Turan's Theorem
Let G(V,E) be a Graph with VERTICES V and Edges
E on n VERTICES without a fc-CLIQUE. Then
t(n,k) <
(k - 2)n 2
2(fc - 1)
where t(n,k) = \E\ is the EDGE NUMBER. More pre-
cisely, the if -Graph nrn 1 ,...,n h _i with l n * ~ n o\ < 1 for
i ^ j is the unique Graph without a fc-CLlQUE with the
maximal number of EDGES t(n,k).
see also CLIQUE, K-GRAPH, TURAN GRAPH
References
Aigner, M. "Turan's Graph
Monthly 102, 808-816, 1995.
Theorem." Amer. Math.
Turbine
A Vector Field on a Circle in which the directions of
the Vectors are all at the same Angle to the Circle.
see also CIRCLE, VECTOR FIELD
Turing Machine
A theoretical computing machine which consists of an
infinitely long magnetic tape on which instructions can
be written and erased, a single-bit register of memory,
and a processor capable of carrying out the following
instructions: move the tape right, move the tape left,
change the state of the register based on its current value
and a value on the tape, and write or erase a value on the
tape. The machine keeps processing instructions until
it reaches a particular state, causing it to halt. Deter-
mining whether a Turing machine will halt for a given
input and set of rules is called the HALTING PROBLEM.
see also BUSY BEAVER, CELLULAR AUTOMATON,
Chaitin's Omega, Church-Turing Thesis, Com-
putable Number, Halting Problem, Universal
Turing Machine
References
Penrose, R. "Algorithms and Turning Machines." Ch. 2
in The Emperor's New Mind: Concerning Computers,
Minds, and the Laws of Physics. Oxford, England: Oxford
University Press, pp. 30-73, 1989,
Turing, A. M. "On Computable Numbers, with an Applica-
tion to the Entscheidungsproblem." Proc. London Math.
Soc. Ser. 2 42, 230-265, 1937.
Turing, A. M. "Correction to: On Computable Numbers,
with an Application to the Entscheidungsproblem." Proc.
London Math. Soc. Ser. 2 43, 544-546, 1938.
Turning Angle
see Tangential Angle,
1870 Tutte's Graph
Tutte's Graph
A counterexample to Tait's Hamiltonian Graph
Conjecture given by Tutte (1946). A simpler coun-
terexample was later given by Kozyrev and Grinberg.
see also Hamiltonian Circuit, Tait's Hamiltonian
Graph Conjecture
References
Honsberger, R. Mathematical Gems I. Washington, DC:
Math. Assoc. Amer., pp. 82-89, 1973.
Saaty, T. L. and Kainen, P. C. The Four-Color Problem:
Assaults and Conquest. New York: Dover, p. 112, 1986.
Tutte, W. T. "On Hamiltonian Circuits." J. London Math.
Soc. 21, 98-101, 1946.
Tutte Polynomial
Let G be a Graph, and let ea(T) denote the cardinality
of the set of externally active edges of a spanning tree
T of G and ia(T) denote the cardinality of the set of
internally active edges of T. Then
tc(x,y) = ^ x a(T) y
ea(T)
TCG
References
Gessel, I. M. and Sagan, B. E. "The Tutte Polynomial
of a Graph, Depth-First Search, and Simplicial Complex
Partitions." Electronic J. Combinatorics 3, No. 2, R9,
1-36, 1996. http : //www . combinatorics . org/Volume_3/
volume3_2 . html#R9.
Tutte, W. T. "A Contribution to the Theory of Chromatic
Polynomials." Canad. J. Math. 6, 80-91, 1953.
Tutte's Theorem
Let G be a Graph and S a Subgraph of G. Let the
number of Odd components in G — 5 be denoted S',
and \S\ the number of VERTICES of S. The condition
\S\ > S' for every SUBSET of VERTICES is NECESSARY
and Sufficient for G to have a I-Factor.
see also Factor (Graph)
References
Honsberger, R. "Lovasz' Proof of a Theorem of Tutte."
Ch. 14 in Mathematical Gems II. Washington, DC: Math.
Assoc. Amer., pp. 147-157, 1976.
Tutte, W. T, "The Factorization of Linear Graphs." J. Lon-
don Math. Soc. 22, 107-111, 1947.
Twin Peaks
Twin Peaks
For an INTEGER n > 2, let lpf(z) denote the LEAST
Prime Factor of n. A Pair of Integers (x,y) is called
a twin peak if
1. x < y,
2. lpf( a ;) = lpf(y),
3. For all z, x < z < y IMPLIES lpf(z) < lpf(x).
A broken-line graph of the least prime factor function
resembles a jagged terrain of mountains. In terms of
this terrain, a twin peak consists of two mountains of
equal height with no mountain of equal or greater height
between them. Denote the height of twin peak (x, y) by
p = lpf (x) = lpf (y). By definition of the LEAST PRIME
Factor function, p must be Prime.
Call the distance between two twin peaks (x,y)
s = y — x.
Then 5 must be an Even multiple of p; that is, s — kp
where k is EVEN. A twin peak with s = kp is called a
fcp-twin peak. Thus we can speak of 2p-twin peaks, 4p-
twin peaks, etc. A A;p-twin peak is fully specified by &,
p, and x, from which we can easily compute y = x -f- kp.
The set of ftp- twin peaks is periodic with period q = p#,
where p# is the PRIMORIAL of p. That is, if (x,y) is a
fcp-twin peak, then so is (x -f q, y + q). A fundamental
kp-twin peak is a twin peak having x in the fundamental
period [0, q). The set of fundamental &p-twin peaks is
symmetric with respect to the fundamental period; that
is, if (x, y) is a twin peak on [0, <?), then so is (q — y, q — x).
The question of the EXISTENCE of twin peaks was first
raised by David Wilson in the math-fun mailing list on
Feb. 10, 1997. Wilson already had privately showed the
Existence of twin peaks of height p < 13 to be unlikely,
but was unable to rule them out altogether. Later that
same day, John H. Conway, Johan de Jong, Derek Smith,
and Manjul Bhargava collaborated to discover the first
twin peak. Two hours at the blackboard revealed that
p = 113 admits the 2p-twin peak
x = 126972592296404970720882679404584182254788131,
which settled the EXISTENCE question. Immediately
thereafter, Fred Helenius found the smaller 2p-twin peak
with p = 89 and
x = 9503844926749390990454854843625839.
The effort now shifted to finding the least PRIME p ad-
mitting a 2p-twin peak. On Feb. 12, 1997, Fred Helenius
found p — 71, which admits 240 fundamental 2p-twin
peaks, the least being
x = 7310131732015251470110369.
Twin Prime Conjecture
Twin Primes
1871
Helenius's results were confirmed by Dan Hoey, who also
computed the least 2p-twin peak L(2p) and number of
fundamental 2p-twin peaks N(2p) for p = 73, 79, and
83. His results are summarized in the following table.
L{2p)
N(2p)
71 7310131732015251470110369 240
73 2061519317176132799110061 40296
79 3756800873017263196139951 164440
83 6316254452384500173544921 6625240
The 2p-twin peak of height p — 73 is the smallest known
twin peak. Wilson found the smallest known 4j?-twin
peak with p= 1327, as well as another very large 4p-twin
peak with p = 3203. Richard Schroeppel noted that the
latter twin peak is at the high end of its fundamental
period and that its reflection within the fundamental
period [0,p#) is smaller.
Many open questions remain concerning twin peaks,
e.g.,
1. What is the smallest twin peak (smallest n)?
2. What is the least Prime p admitting a 4p-twin peak?
3. Do 6p-twin peaks exist?
4. Is there, as Conway has argued, an upper bound on
the span of twin peaks?
5. Let p < q < r be Prime. If p and r each admit kp-
twin peaks, does q then necessarily admit a fcp-twin
peak?
see also Andrica's Conjecture, Divisor Function,
Least Common Multiple, Least Prime Factor
Twin Prime Conjecture
Adding a correction proportional to 1/lnp to a compu-
tation of Brun's Constant ending with ... + 1/p +
l/(p + 2) will give an estimate with error less than
c{yjp lnp) -1 . An extended form of the conjecture states
that
where n 2 is the Twin Primes Constant. The twin
prime conjecture is a special case of the more general
Prime Patterns Conjecture corresponding to the
set 5 = {0,2}.
see also Brun's Constant, Prime Arithmetic Pro-
gression, Prime Constellation, Prime Patterns
Conjecture, Twin Primes
Twin Primes
Twin primes are PRIMES (p, q) such that p — q — 2. The
first few twin primes are n ± 1 for n = 4, 6, 12, 18, 30,
42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240,
270, 282, ... (Sloane's A014574). Explicitly, these are
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ...
(Sloane's A001359 and A006512).
Let 7T2 (n) be the number of twin primes p and p-\- 2 such
that p < n. It is not known if there are an infinite num-
ber of such Primes (Shanks 1993), but all twin primes
except (3, 5) are of the form 6n±l. J. R. Chen has shown
there exists an Infinite number of Primes p such that
p+ 2 has at most two factors (Le Lionnais 1983, p. 49).
Bruns proved that there exists a computable INTEGER
Xq such that if x > Xo, then
7T2{x) <
lOOx
(lnx) 2
(Ribenboim 1989, p. 201). It has been shown that
K2(x) < C JJ
P>2
(P-1) 2
(lnx) 2
h°(^
(i)
)]•
(2)
where c has been reduced to 68/9 ft* 7.5556 (Fouvry and
Iwaniec 1983), 128/17 ^ 7.5294 (Fouvry 1984), 7 (Bom-
bieri et al. 1986), 6.9075 (Fouvry and Grupp 1986), and
6.8354 (Wu 1990). The bound on c is further reduced
to 6.8324107886 in a forthcoming thesis by Haugland
(1998). This calculation involved evaluation of 7-fold in-
tegrals and fitting of three different parameters. Hardy
and Littlewood conjectured that c = 2 (Ribenboim 1989,
p. 202).
Define
E = Umin£ Pn + 1 - pn .
(3)
If there are an infinite number of twin primes, then
E = 0. The best upper limit to date is E < \ + 7r/16 =
0.44634... (Huxley 1973, 1977). The best previous
values were 15/16 (Ricci), (2 + \/3)/8 = 0.46650...
(Bombieri and Davenport 1966), and (2^2 - l)/4 =
0.45706... (Pil'Tai 1972), as quoted in Le Lionnais
(1983, p. 26).
Some large twin primes are 10, 006, 428 ± 1, 1, 706, 595 x
2 11235 ± 1, and 571, 305 x 2 7701 ± 1. An up-to-date table
of known twin primes with 2000 or more digits follows.
An extensive list is maintained by Caldwell.
(p,p + l)
dig.
Reference
260,497,545 X 2 6625 ± 1
2003
Atkin & Rickert 1984
43,690,485,351,513 X 10 1995 ± 1
2009
Dubner, Atkin 1985
2,846!!!! ±1
2151
Dubner 1992
10,757,0463 X 10 2250 ± 1
2259
Dubner, Atkin 1985
663,777 x 2 7650 ± 1
2309
Brown et al 1989
75,188,117,004 X 10 2298 ± 1
2309
Dubner 1989
571305 x 2 7701 ± 1
2324
Brown et al. 1989
1,171,452,282 x 10 2490 ± 1
2500
Dubner 1991
459 - 2 8529 ± 1
2571
Dubner 1993
1,706,595 ■ 2 11235 ± 1
3389
Noll et al. 1989
4,655,478,828- 10 3429 ±1
3439
Dubner 1993
1,692,923,232- 10 4020 ± 1
4030
Dubner 1993
6,797,727 • 2 15328 ± 1
4622
Forbes 1995
697,053,8132 16352 ± 1
4932
Indlekofer & Ja'rai 1994
570,918,348 • 10 5120 ± 1
5129
Dubner 1995
242,206,083 ■ 2 3888t} ± 1
11713
Indlekofer & Ja'rai 1995
1872
Twin Primes
Twin Primes Constant
The last of these is the largest known twin prime
pair. In 1995, Nicely discovered a flaw in the Intel®
Pentium™ microprocessor by computing the recip-
rocals of 824,633,702,441 and 824,633,702,443, which
should have been accurate to 19 decimal places but were
incorrect from the tenth decimal place on (Cipra 1995,
1996; Nicely 1996).
If n > 2, the INTEGERS n and n + 2 form a pair of twin
primes Iff
4[(n - 1)! + 1] + n = (mod n{n + 2)) . (4)
n — pp' where (p,£>') is a pair of twin primes IFF
<f){n)(T{n) = (n - 3)(n + 1) (5)
(Ribenboim 1989). The values of 7r 2 (n) were found by
Brent (1976) up to n = 10 11 . T. Nicely calculated them
up to 10 14 in his calculation of BRUN'S CONSTANT. The
following table gives the number less than increasing
powers of 10 (Sloane's A007508).
7r 2 (n)
10 3 35
10 4 205
10 5 1224
10 6 8,169
10 7 58,980
10 8 440,312
10 9 3,424,506
10 10 27,412,679
10 11 224,376,048
10 12 1,870,585,220
10 13 15,834,664,872
10 14 135,780,321,665
see also BRUN'S CONSTANT, DE POLIGNAC'S CONJEC-
TURE Prime Constellation, Sexy Primes, Twin
Prime Conjecture, Twin Primes Constant
References
Bombieri, E. and Davenport, H. "Small Differences Between
Prime Numbers." Proc. Roy. Soc. Ser. A 293, 1-8, 1966.
Bombieri, E.; Friedlander, J. B.; and Iwaniec, H. "Primes
in Arithmetic Progression to Large Moduli." Acta Math.
156, 203-251, 1986.
Bradley, C. J. "The Location of Twin Primes." Math. Gaz.
67, 292-294, 1983.
Brent, R. P. "Irregularities in the Distribution of Primes and
Twin Primes." Math. Comput. 29, 43-56, 1975.
Brent, R. P. "UMT 4." Math. Comput 29, 221, 1975.
Brent, R. P. "Tables Concerning Irregularities in the Distri-
bution of Primes and Twin Primes to 10 11 ." Math. Corn-
put 30, 379, 1976.
Caldwell, C. http://www.utm.edu/cgi-bin/caldwell/
primes . cgi/twin.
Cipra, B. "How Number Theory Got the Best of the Pentium
Chip." Science 267, 175, 1995.
Cipra, B. "Divide and Conquer." What's Happening in the
Mathematical Sciences, 1995-1996, Vol. 3. Providence,
RI: Amer. Math. Soc, pp. 38-47, 1996.
Fouvry, E. "Autour du theoreme de Bombieri- Vinogradov."
Acta. Math. 152, 219-244, 1984.
Fouvry, E. and Grupp, F. "On the Switching Principle in
Sieve Theory." J. Reine Angew. Math. 370, 101-126,
1986.
Fouvey, E. and Iwaniec, H. "Primes in Arithmetic Progres-
sion." Acta Arith. 42, 197-218, 1983.
Guy, R. K. "Gaps between Primes. Twin Primes." §A8 in
Unsolved Problems in Number Theory, 2nd ed. New York:
Springer- Verlag, pp. 19-23, 1994.
Haugland, J. K. Topics in Analytic Number Theory. Ph.D.
thesis. Oxford, England: Oxford University, Oct. 1998.
Huxley, M. N. "Small Differences between Consecutive
Primes." Mathematica 20, 229-232, 1973.
Huxley, M. N. "Small Differences between Consecutive
Primes. II." Mathematica 24, 142-152, 1977.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
1983.
Nicely, T. R. "The Pentium Bug.' http://www . lynchburg .
edu / public / academic / math / nicely / pent bug /
pentbug.htm.
Nicely, T. "Enumeration to 10 14 of the Twin Primes
and Brun's Constant." Virginia J. Sci. 46, 195-
204, 1996. http://www.lynchburg.edu/public/academic/
math/nicely/twins/twins . htm.
Parady, B. K.; Smith, J. F.; and Zarantonello, S. E. "Largest
Known Twin Primes." Math. Comput 55, 381-382, 1990.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 199-204, 1989.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 30, 1993.
Sloane, N. J. A. Sequences A014574, A001359/M2476,
A006512/M3763, and A007508/M1855 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Weintraub, S. "A Prime Gap of 864." J. Recr. Math. 25,
42-43, 1993.
Wu, J. "Sur la suite des nombres premiers jumeaux." Acta.
Arith. 55, 365-394, 1990.
Twin Primes Constant
The twin primes constant II2 is defined by
n, s n
p prirr
In(in a )= £ In
p prirr
- E
1 -
p>3
p prime
(p-1) 2
(P-1) 2
2
(1)
In 1
21n| 1- -
P
= -£
2 J -2
J=2
£;
P>3
p prime
(2)
where the ps in sums and products are taken over
PRIMES only. Flajolet and Vardi (1996) give series with
accelerated convergence
00
n 2 = [][c( n )(i-2-T 7 " (3)
71 = 2
= I lift fltCWCi - 2-)(i - 3-)(i - 5-)
n=2
x(i-r n )]- f ", (4)
Twins
Two-Ears Theorem
1873
with
!5>(d>2«",
(5)
where \i{x) is the MOBIUS FUNCTION. (4) has conver-
gence like - (ll/2)" n .
The most accurately known value of Eh is
n 2 = 0.6601618158...,
(6)
Le Lionnais (1983, p. 30) calls C 2 the Shah-Wilson
Constant, and 2C 2 the twin prime constant (Le Lion-
nais 1983, p. 37).
see also Brun's Constant, Goldbach Conjecture,
Mertens Constant
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/hrdyltl/hrdyltl.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.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
1983.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, p. 202, 1989.
Ribenboim, P. The Little Book of Big Primes. New York:
Springer- Verlag, p. 147, 1991.
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, pp. 61-66,
1994.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 30, 1993.
Wrench, J. W. "Evaluation of Artin's Constant and the Twin
Prime Constant." Math. Comput. 15, 396-398, 1961.
Twins
see Brothers, Pair
Twirl
A Rotation combined with an Expansion or Dila-
tion.
see also Screw, Shift
Twist
The twist of a ribbon measures how much it twists
around its axis and is defined as the integral of the in-
cremental twist around the ribbon. Letting Lk be the
linking number of the two components of a ribbon, Tw
be the twist, and Wr be the Writhe, then
Lk(fl) = Tw(i2) + Wr(rt)
(Adams 1994, p. 187).
see also Screw, Writhe
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, 1994.
Twist Map
A class of Area-Preserving Maps of the form
6 i+ i =0i + 2iTa{ri)
which maps CIRCLES into CIRCLES but with a twist re-
sulting from the a = a(r») term.
Twist Move
twist
untwist
The Reidemeister Move of type II.
see also Reidemeister Moves
Twist Number
see Writhe
Twist-Spun Knot
A generalization of SPUN KNOTS due to Zeeman. This
method produces 4-D KNOT types that cannot be pro-
duced by ordinary spinning.
see also SPUN KNOT
Twisted Chevalley Groups
Finite Simple Groups of Lie-Type of Orders 14,
52, 78, 133, and 248. They are denoted *D 4 (q), E 6 (q),
E 7 (q), £ 8 (<z), *i(«), 2 Ft(2 n Y, G 2 (<z), 2 G 2 (3"), 2 B(2 n ).
see also Chevalley Groups, Finite Group, Simple
Group, Tits Group
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http : //for . mat . bham . ac . uk/atlas#twi.
Twisted Conic
see Skew Conic
Twisted Sphere
see Corkscrew Surface
Two
see 2
Two-Ears Theorem
Except for Triangles, every Simple Polygon has at
least two nonoverlapping Ears.
see also Ear, One-Mouth Theorem, Principal Ver-
tex
References
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.
1874 Two-Point Distance Type II Error
Two-Point Distance
see Point-Point Distance — 1-D, Point-Point
Distance — 2-D, Point-Point Distance — 3-D
Two Triangle Theorem
see Desargues' Theorem
Tychonof Compactness Theorem
The topological product of any number of COMPACT
Spaces is Compact.
Type
Whitehead and Russell (1927) devised a hierarchy of
"types" in order to eliminate self-referential statements
from Principia Mathematical which purported to derive
all of mathematics from logic. A set of the lowest type
contained only objects (not sets), a set of the next higher
type could contain only objects or sets of the lower type,
and so on. Unfortunately, GODEL'S INCOMPLETENESS
THEOREM showed that both Principia Mathematica and
all consistent formal systems must be incomplete.
see also GODEL'S INCOMPLETENESS THEOREM
References
Hofstadter, D. R. Godel, Escher, Bach: An Eternal Golden
Braid. New York: Vintage Books, pp. 21-22, 1989.
Whitehead, A. N. and Russell, B. Principia Mathematica.
New York: Cambridge University Press, 1927.
Type I Error
An error in a STATISTICAL Test which occurs when a
true hypothesis is rejected (a false negative in terms of
the Null Hypothesis).
see also NULL HYPOTHESIS, SENSITIVITY, SPECIFICITY,
Statistical Test, Type II Error
Type II Error
An error in a STATISTICAL TEST which occurs when a
false hypothesis is accepted (a false positive in terms of
the Null Hypothesis).
see also Null Hypothesis, Sensitivity, Specificity,
Statistical Test, Type I Error
U-Number
u
U-Number
see ULAM SEQUENCE
Ulam Map
Ultrametric
1875
f(x) = 1 - 2a; 2
for x G [—1,1]- Fixed points occur at x = —1, 1/2, and
order 2 fixed points at x = (1 ± \/5 )/4. The INVARIANT
Density of the map is
p(y) =
tta/1 - 2/ 2
References
Beck, C. and Schlogl, F. Thermodynamics of Chaotic Sys-
tems: An Introduction. Cambridge, England: Cambridge
University Press, p. 194, 1995.
Ulam Number
see Ulam Sequence
Ulam's Problem
see Collatz Problem
Ulam Sequence
The Ulam sequence {a;} = (u, v) is defined by a\ = u,
a 2 = f, with the general term a n for n > 2 given by
the least INTEGER expressible uniquely as the Sum of
two distinct earlier terms. The numbers so produced
are sometimes called U-Numbers or Ulam Numbers.
The first few numbers in the (1, 2) Ulam sequence are
1, 2, 3, 4, 6, 8, 11, 13, 16, . . . (Sloane's A002858). Here,
the first term after the initial 1, 2 is obviously 3 since
3 = 1 + 2. The next term is 4 = 1 + 3. (We don't
have to worry about 4 = 2 + 2 since it is a sum of a
single term instead of unique terms.) 5 is not a member
of the sequence since it is representable in two ways,
5 = 1 + 4 = 2 + 3, but 6 = 2 + 4 is a member.
Proceeding in the manner, we can generate Ulam se-
quences for any (u, v) 1 examples of which are given be-
low.
(1.2) = {1,2,3,4,6,8,11,13,16,18,...}
(1.3) = {1,3,4,5,6,8,10,12,17,21,...}
(1.4) = {1,4,5,6,7,8,10,16,18,19,...}
(1.5) = {1,5, 6, 7,8, 9, 10, 12, 20,22,...}
(2. 3) = {2, 3, 5, 7, 8, 9, 13, 14, 18, 19, . . .}
(2.4) = {2,4,6,8,12,16,22,26,32,36,...}
(2. 5) = {2, 5, 7, 9, 11, 12, 13, 15, 19, 23, . . .}.
Schmerl and Spiegel (1994) proved that Ulam sequences
(2,v) for ODD v > 5 have exactly two EVEN terms.
Ulam sequences with only finitely many Even terms
eventually must have periodic successive differences
(Finch 1991, 1992abc). Cassaigne and Finch (1995)
proved that the Ulam sequences (4, v) for 5 < v = 1
(mod 4) have exactly three Even terms.
The Ulam sequence can be generalized by the s-
Additive Sequence.
see also Greedy Algorithm, s- Additive Sequence,
Stohr Sequence
References
Cassaigne, J. and Finch, S. "A Class of 1-Additive Sequences
and Quadratic Recurrences." Exper. Math 4, 49-60, 1995.
Finch, S. "Conjectures About 1-Additive Sequences." Fib.
Quart. 29, 209-214, 1991.
Finch, S. "Are 0- Additive Sequences Always Regular?"
Amer. Math. Monthly 99, 671-673, 1992a.
Finch, S. "On the Regularity of Certain 1-Additive Se-
quences." J. Combin. Th. Ser. A 60, 123-130, 1992b.
Finch, S. "Patterns in 1-Additive Sequences." Exper. Math.
1, 57-63, 1992c.
Finch, S. "Ulam s-Additive Sequences." http://www.
maths of t . com/asolve/s add/ sadd.html.
Guy, R. K. "A Quarter Century of Monthly Unsolved Prob-
lems, 1969-1993." Amer. Math. Monthly 100, 945-949,
1993.
Guy, R. K. "Ulam Numbers." §C4 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 109-110, 1994.
Guy, R. K. and Nowakowski, R. J. ^Monthly Unsolved Prob-
lems, 1969-1995." Amer. Math. Monthly 102, 921-926,
1995.
Recaman, B. "Questions on a Sequence of Ulam." Amer.
Math. Monthly 80, 919-920, 1973.
Schmerl, J. and Spiegel, E. "The Regularity of Some 1-
Additive Sequences." J. Combin. Theory Ser. A 66, 172-
175, 1994.
Sloane, N. J. A. Sequence A002858/M0557 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Ultrametric
An ultrametric is a Metric which satisfies the following
strengthened version of the TRIANGLE INEQUALITY,
d(x t z) < max(d(x,y),d(y,z))
1876
Ultraradical
Ultraspherical Polynomial
for all x, y, z. At least two of d(x, y), d(y, z), and d(x, z)
are the same.
Let X be a Set, and let X^ (where N is the Set of
Natural Numbers) denote the collection of sequences
of elements of X (i.e., all the possible sequences xi, £2,
X3, . . . ). For sequences a = (ai, a2, . . .), b = (61,62, ■ • •),
let n be the number of initial places where the sequences
agree, i.e., a\ = 61, a 2 = 6 2 , . . . , a n = 6 n , but a n +i ^
6 n +i- Take n = if a\ ^ b\. Then defining d(a, b) = 2 _n
gives an ultrametric.
The p-ADlC Number metric is another example of an
ultrametric.
see also Metric, p-ADic Number
Ultraradical
A symbol which can be used to express solutions not
obtainable by finite ROOT extraction. The solution to
the irreducible QuiNTic EQUATION
x 5 -h x ■
Ultraspherical Function
A function defined by a POWER SERIES whose coeffi-
cients satisfy the RECURRENCE RELATION
CLj + 2 — 0>i
(k + j)(k + j + 2a) -n(n + 2a)
(fc + j + l)(fc + j + 2) '
For x y£ — 1, the function converges for a < 1/2 and
diverges for a > 1/2.
Ultraspherical Polynomial
The ultraspherical polynomials are solutions Pn *(x) to
the Ultraspherical Differential Equation for In-
teger n and a < 1/2. They are generalizations of LEG-
endre Polynomials to (n + 2)-D space and are pro-
portional to (or, depending on the normalization, equal
to) the Gegenbauer Polynomials C^ x) (x), denoted
in Mathematical (Wolfram Research, Champaign, IL)
Gegenbauer C [n , lambda , x] . The ultraspherical polyno-
mials are also Jacobi Polynomials with a — f3. They
are given by the Generating Function
is written ja .
see also Radical
Ultraspherical Differential Equation
(1 - x 2 )y" - (2a -f l)xy + n[n 4- 2a)y = 0. (1)
Alternate forms are
(l-x 2 )y // + (2A-3)xr / H-(n+l)(n+2A-l)y = 0, (2)
where
Y
= (l-x 2 ) x -^P^(x),
(3)
d 2 u
dx 2 +
"(n + A) 2 \ + \-\ 2 + \x 2 ~
1-x 2 ' (1-x 2 ) 2
u = 0,
(4)
where
u = (l-x 2 ) A/2+1/ M A) (x),
(5)
and
d
a
2 u
02 +
x , 2 A(l-A)"
[ (n + A) + sinVJ
u ■■
= 0,
(6)
where
u = sin OP^ (cos 0).
(?)
The solutions are the ULTRASPHERICAL FUNCTIONS
Pi (x). For integral n with a < 1/2, the function con-
verges to the Ultraspherical Polynomials cl?\x).
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 547-549, 1953.
(l-2xt + t 2 ) x ^
n=0
y>i A) (*)*",
(i)
and can be given explicitly by
p(»( x ) - r ( A +5) T(n + 2A) ( A-i/2,A-i/2) ( -,
^ [X> - T(2A) r(n + A+i) n () '
(2)
where p^- 1 / 2 ^- 1 / 2 ) . g & j AC0BI polynomial (Szego
1975, p. 80). The first few ultraspherical polynomials
are
P CA) (x) = l
P 1 (A) (z) = 2Ax
P 2 (A) (x) = -A + 2A(l + A)a; 2
(3)
(4)
(5)
P£ A) (x) = -2A(1 + X)x + |A(1 + A)(2 + A);r 3 . (6)
p(A)
In terms of the HYPERGEOMETRIC FUNCTIONS,
P!f ) (x) =
n -f 2A - 1
n
x 2 F 1 (-n,n + 2A; A + \ ; |(1 - x)) (7)
x 2 Fi (-n, -n - A + \ ; -In - 2A + 1; YZ~)
(8)
=C +2 . A+i )(^r
Fx(-n,-n-A+i;A+i;|^i). (9)
X 2
Ultraspherical Polynomial
They are normalized by
L 11 -
,2^-V5[pWl2
prr dx
2 1 - 2 V
r(n + 2A)
(n + A)r 2 (A)r(n + l)'
Derivative identities include
(l-z 2 )£[P< A) ] = [2(n + A)]- 1 [(n + 2A-l)
(10)
(11)
(n + 2\)P?J 1 {x) - n(n + lJP&^x)] (12)
(A)
-nxP( A) ( a; ) + (n + 2A- lJPj^a:)
(n + 2A)xP^ A) (x) - (n + lJP&^s)
3(A)
«/>**>(*) = x^[P^(x)} - £[PrJi(x)]
(A)
dx
dx
(13)
(14)
(15)
(n + 2A)P< A) (z) = ^[P^^)] - xj^[Pi"\x)] (16)
^ttft'iOO " Pn-iW = 2(" + A)Pl A) P,i A) (x) (17)
da?
= 2A[P^ +1 )( ;C )-P 7 ( l A „t 1) (^)]
(18)
(Szego 1975, pp. 80-83).
A Recurrence Relation is
nP^\x) = 2(n + \-l)xP^ l (x)-(n + 2\-2)P^ 2 (x)
(19)
for n = 2, 3, ....
Special double-i/ FORMULAS also exist
Umbral Calculus 1877
see also BIRTHDAY PROBLEM, CHEBYSHEV POLYNOM-
IAL of the Second Kind, Elliptic Function, Hy-
PERGEOMETRIC FUNCTION, JACOBI POLYNOMIAL
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Orthogonal
Polynomials." Ch. 22 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 771-802, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 643, 1985.
Iyanaga, S. and Kawada, Y. (Eds.). "Gegenbauer Polyno-
mials (Gegenbauer Functions)." Appendix A, Table 20.1
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, pp. 1477-1478, 1980.
Koschmieder, L. "Uber besondere Jacobische Polynome."
Math. Zeztschrift 8, 123-137, 1920.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 547-549 and 600-
604, 1953.
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, 1975.
Umbilic Point
A point on a surface at which the CURVATURE is the
same in any direction.
Umbral Calculus
The study of certain properties of FINITE DIFFERENCES.
The term was coined by Sylvester from the word "um-
bra" (meaning "shadow" in Latin), and reflects the fact
that for many types of identities involving sequences of
polynomials with POWERS a 71 , "shadow" identities are
obtained when the polynomials are changed to discrete
values and the exponent in a n is changed to the POCH-
HAMMER Symbol (a) n = a(a - 1) • * • (a -n + 1).
For example, NEWTON'S FORWARD DIFFERENCE FOR-
MULA written in the form
^'(*>=l 2 „ |2
(A)^_/2 V + 2A-l v _ Fi( _ i/ji/ + A . A+ i. 1 _ a . 3)
^t+r 1 )
(20)
2 F 1 (-u,u + X;i;x i )
(21)
,(A) t „,_f^ + 2X\ 2jFi( _ 1/)I/ + a + 1;A+ i ;1 _ x2)
PZUW
2v + \
(22)
= (-1)"2A
^W^-^ + A+ljf;* 2 ).
(23)
Special values are given in the following table.
A Special Polynomial
Legendre
Chebyshev polynomial of the second kind
Koschmieder (1920) gives representations in terms of
Elliptic Functions for a = -3/4 and a = -2/3.
WnA B /W
(1)
n=0
with f(x-\-a) = fx+a looks suspiciously like a finite
analog of the Taylor Series expansion
/(x + a) = ^ -j .
(2)
where D is the DIFFERENTIAL OPERATOR. Similarly,
the Chu-Vandermonde Identity
i.—n \ /
(3)
with (£) a Binomial Coefficient, looks suspiciously
like an analog of the Binomial Theorem
(«+-)" = Eft)-'
n\ k n-k
" X
(4)
1878
Umbrella
Undulating Number
(Di Bucchianico and Loeb).
see also BINOMIAL Theorem, Chu-Vandermonde
Identity, Finite Difference
References
Roman, S. and Rota, G.-C. "The Umbral Calculus." Adv.
Math. 27, 95-188, 1978.
Roman, S. The Umbral Calculus. New York: Academic
Press, 1984.
Umbrella
see Whitney Umbrella
Unambiguous
see Well-Defined
Unbiased
A quantity which does not exhibit BIAS. An ESTIMATOR
is an Unbiased Estimator of 9 if
Undecagon
<*>
see also Bias (Estimator), Estimator
The unconstructible 11-sided POLYGON with SCHLAFLI
Symbol {11}.
see also Decagon, Dodecagon, Trigonometry
Values — 7r/ll
Undecidable
Not Decidable as a result of being neither formally
provable nor unprovable.
see also GODEL'S INCOMPLETENESS THEOREM,
Richardson's Theorem
Undecillion
In the American system, 10 36 .
see also Large Number
Uncia
1 uncia = ^ .
The word uncia was Latin for a unit equal to 1/12 of
another unit called the as. The words "inch" (1/12 of a
foot) and "ounce" (originally 1/12 of a pound and still
1/12 of a "Troy pound," now used primarily to weigh
precious metals) are derived from the word uncia.
see also CALCUS, HALF, QUARTER, SCRUPLE, UNIT
Fraction
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 4, 1996.
Uncorrelated
Variables Xi and Xj are said to be uncorrelated if their
Covariance is zero:
COv(Xi,Xj) — 0.
Independent Statistics are always uncorrelated, but
the converse is not necessarily true.
see also Covariance, Independent Statistics
Uncountable Set
see Uncountably Infinite Set
Uncountably Infinite Set
An Infinite Set which is not a Countably Infinite
Set.
see also Aleph-O, Aleph-1, Countable Set, Count-
ably Infinite Set, Finite, Infinite
Undetermined Coefficients Method
Given a nonhomogeneous Ordinary Differential
Equation, select a differential operator which will an-
nihilate the right side, and apply it to both sides. Find
the solution to the homogeneous equation, plug it into
the left side of the original equation, and solve for con-
stants by setting it equal to the right side. The solution
is then obtained by plugging the determined constants
into the homogeneous equation.
see also ORDINARY DIFFERENTIAL EQUATION
Undulating Number
A number of the form aba • • •, abab • • •, etc. The first few
nontrivial undulants (with the stipulation that a / b)
are 101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212,
. . . (Sloane's A046075). Including the trivial 1- and 2-
digit undulants and dropping the requirement that a/fe
gives Sloane's A033619.
The first few undulating SQUARES are 121, 484, 676,
69696, . . . (Sloane's A016073), with no larger such num-
bers of fewer than a million digits (Pickover 1995). Sev-
eral tricks can be used to speed the search for square un-
dulating numbers, especially by examining the possible
patterns of ending digits. For example, the only possible
sets of four trailing digits for undulating SQUARES are
0404, 1616, 2121, 2929, 3636, 6161, 6464, 6969, 8484,
and 9696.
The only undulating POWER n p = aba • • • for 3 < p < 31
and up to 100 digits is 7 3 = 343 (Pickover 1995). A
large undulating prime is given by 7 + 720(100 49 - 1)/99
(Pickover 1995).
Unduloid
Uniform Apodization Function 1879
A binary undulant is a POWER of 2 whose base-10 rep-
resentation contains one or both of the sequences 010 • • •
and 101 ■ ■ ■. The first few are 2 n for n = 103, 107, 138,
159, 179, 187, 192, 199, 205, ... (Sloane's A046076).
The smallest n for which an undulating sequence of ex-
actly d-digit occurs for d = 3, 4, ... are n = 103, 138,
875, 949, 6617, 1802, 14545, ... (Sloane's A046077).
An undulating binary sequence of length 10 occurs for
n = 1,748,219 (Pickover 1995).
References
Pickover, C. A. "Is There a Double Smoothly Undulating
Integer?" In Computers, Pattern, Chaos and Beauty. New-
York: St. Martin's Press, 1990.
Pickover, C. A. "The Undulation of the Monks." Ch. 20 in
Keys to Infinity. New York: W. H. Freeman, pp. 159-161
1995.
Sloane, N. J. A. Sequences A016073, A033619, A046075,
A046076, and A046077 in "An On-Line Version of the En-
cyclopedia of Integer Sequences."
Unduloid
A Surface of Revolution with constant Nonzero
Mean Curvature also called an Onduloid. It is a
Roulette obtained from the path described by the
Foci of a Conic Section when rolled on a Line. This
curve then generates an unduloid when revolved about
the Line. These curves are special cases of the shapes
assumed by soap film spanning the gap between pre-
scribed boundaries. The unduloid of a PARABOLA gives
a Catenoid.
see also Calculus of Variations, Catenoid,
Roulette
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., p. 48, 1989.
Delaunay, C. "Sur la surface de revolution dont la courbure
moyenne est constante." J. math, pures appl. 6, 309-320,
1841.
do Carmo, M. P. "The Onduloid." §3.5G in Mathematical
Models from the Collections of Universities and Muse-
ums (Ed. G. Fischer). Braunschweig, Germany: Vieweg,
pp. 47-48, 1986.
Fischer, G. (Ed.). Plate 97 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, p. 93, 1986.
Thompson, D'A. W. On Growth and Form, 2nd ed., compl.
rev. ed. New York: Cambridge University Press, 1992.
Yates, R. C. A Handbook on Curves and Their Properties.
Ann Arbor, MI: J. W. Edwards, p. 184, 1952.
Unexpected Hanging Paradox
A PARADOX also known as the Surprise Examination
Paradox or Prediction Paradox.
A prisoner is told that he will be hanged on some day
between Monday and Friday, but that he will not know
on which day the hanging will occur before it happens.
He cannot be hanged on Friday, because if he were still
alive on Thursday, he would know that the hanging will
occur on Friday, but he has been told he will not know
the day of his hanging in advance. He cannot be hanged
Thursday for the same reason, and the same argument
shows that he cannot be hanged on any other day. Nev-
ertheless, the executioner unexpectedly arrives on some
day other than Friday, surprising the prisoner.
This PARADOX is similar to that in Robert Louis Steven-
son's "The Imp in the Bottle," in which you are offered
the opportunity to buy, for whatever price you wish, a
bottle containing a genie who will fulfill your every de-
sire. The only catch is that the bottle must thereafter
be resold for a price smaller than what you paid for it, or
you will be condemned to live out the rest of your days
in excrutiating torment. Obviously, no one would buy
the bottle for 1$ since he would have to give the bottle
away, but no one would accept the bottle knowing he
would be unable to get rid of it. Similarly, no one would
buy it for 2^, and so on. However, for some reasonably
large amount, it will always be possible to find a next
buyer, so the bottle will be bought (Paulos 1995).
see also SORITES PARADOX
References
Chow, T. Y. "The Surprise Examination or Unexpected
Hanging Paradox." Amer. Math. Monthly 105, 41-51,
1998.
Clark, D. "How Expected is the Unexpected Hanging?"
Math. Mag. 67, 55-58, 1994.
Gardner, M. "The Paradox of the Unexpected Hanging."
Ch. 1 in The Unexpected Hanging and Other Mathematical
Diversions. Chicago, IL: Chicago University Press, 1991.
Margalit, A. and Bar-Hillel, M. "Expecting the Unexpected."
Philosophia 13, 263-288, 1983.
Pappas, T. "The Paradox of the Unexpected Exam." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, p. 147, 1989.
Paulos, J. A. A Mathematician Reads the Newspaper. New
York: BasicBooks, p. 97, 1995.
Quine, W. V. O. "On a So-Called Paradox." Mind 62, 65-67,
1953.
Unfinished Game
see Sharing Problem
Unhappy Number
A number which is not Happy is said to be unhappy.
see also HAPPY Number
Unicursal Circuit
A Circuit in which an entire Graph is traversed in
one route. An example of a curve which can be traced
unicursally is the MOHAMMED SIGN.
Uniform Apodization Function
2
k 0.6
1.5
l.i
\ 0. 1
A *
0.5
0./5
\ A ^ * 1 -
yvuv
-V. 5
** -0II4
-1
-0.5
0.5
1
V
An Apodization Function
/(*) = 1,
(l)
1880
Uniform Boundedness Principle
Uniform Distribution
having Instrument Function
1. The series sum
I(x)= P e~ 27rikx dx
J —a
1 / — 27rifca 27rifcx\
(e -e )
27rik
sin(27r/ca)
7r/c
= 2asinc(27rfca).
(2)
The peak (in units of a) is 2. The extrema are given by
letting j3 = 2nka and solving
d ( , sin/3-/3cos/3 ( .
d0 i0Sm0) = & = ° (3)
sin/3- /3cos/3 = (4)
tan/3 = 0. (5)
Solving this numerically gives 0o = 0, j3± — 4.49341,
fo = 7.72525, ... for the first few solutions. The second
of these is the peak Positive sidelobe, and the third is
the peak NEGATIVE sidelobe. As a fraction of the peak,
they are 0.128375 and -0.217234. The Full Width AT
Half Maximum is found by setting I(x) = 1
sinc(x) = §, (6)
and solving for X1/2, yielding
x 1/2 = 27rk 1/2 a = 1.89549. (7)
Therefore, with L = 2a,
n „ TTTW nf 0.603353 1.20671 , ,
FWHM = 2k 1/2 = = = ■ (8)
see also APODIZATION FUNCTION
Uniform Boundedness Principle
If a "pointwise-bounded" family of continuous linear
Operators from a Banach Space to a Normed
Space is "uniformly bounded." Symbolically, if
sup ||Ti(:c)|| is FINITE for each x in the unit Ball, then
sup||T;|| is Finite. The theorem is also called the
Banach-Steinhaus Theorem.
References
Zeidler, E. Applied Functional Analysis: Applications to
Mathematical Physics. New York: Springer- Verlag, 1995.
Uniform Convergence
A Series Y^=i Un ( x ) is uniformly convergent to S(x)
for a set E of values of x if, for each e > 0, an Integer
N can be found such that
\S n (x)-S{x)\ <e
(1)
for n > N and all x 6 E. To test for uniform conver-
gence, use Abel's Uniform Convergence Test or
the WEIERSTRAft M-TEST. If individual terms u n (x) of
a uniformly converging series are continuous, then
f(x) = Yl Un ^
(2)
is continuous,
2. The series may be integrated term by term
r b °o pb
f{x) dx —y I u n (x) dx, (3)
/t> _ _ pt>
f(x) dx = 2J / Un
and
3. The series may be differentiated term by term
n=l
(4)
see also Abel's Theorem, Abel's Uniform Conver-
gence Test, WeierstraB M-Test
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 299-301, 1985.
Uniform Distribution
A distribution which has constant probability is called a
uniform distribution, sometimes also called a RECTAN-
GULAR Distribution. The probability density function
and cumulative distribution function for a continuous
uniform distribution are
P(x)
D{x)
•g— for a < x < b
for x < a, x > b
for x < a
for a < x < b
x~~a
6 — a
1 for x > b.
(1)
(2)
(3)
D(x) = I[l - (1 - xf sgn(l - x) + x sgn(x)]. (4)
The Characteristic Function is
2
With a = and b = 1, these can be written
P(x) = § sgn(x) - sgn(x - 1)
4>{t)
ht
s\n{\ht)e x
where
b — m + ^h.
The Moment-Generating Function is
ob
M(t)
/*> xt xt
(5)
(6)
(7)
(8)
Uniform Distribution
Uniform Polyhedron 1881
and
M'(t)
M(t)
b — a
( e *b„ e ta
< t(b-a)
for t /
for t = 0,
(9)
— [be — ae )
--(e bt
at\
e )
e bt (ta-l)-e at (g*-l)
(b - a)t 2
(10)
The function is not differentiate at zero, so the Mo-
ments cannot be found using the standard technique.
They can, however, be found by direct integration. The
Moments about are
(11)
(12)
(13)
(14)
Mi = |(a + 6)
^ = I(a 2 + a6 + fe 2 )
M3 = i(a + 6)(a 2 + & 2 )
& = l(a 4 + a 3 6 + a 2 6 2 + a& 3 + 6 4 ).
The Moments about the Mean are
yn -0
M2 = ~(b-a) 2
Ms -0
^ = 8o( b ~ a ) 4 ^
so the Mean, Variance, Skewness, and Kurtosis are
(15)
(16)
(17)
(18)
M=f(a + 6)
2 1 /i \2
cr = \i2 = j2 ( b - a )
72 = -f.
(19)
(20)
(21)
(22)
The probability distribution function and cumulative
distributions function for a discrete uniform distribution
are
P(n) =
1_
N
*(») = #
(23)
(24)
for n — 1,
tion is
N
N. The Moment-Generating Func-
__ _ i e * _ C *(^+D
' TV 6 ~ ]
i
ra = l
iV
1 -e*
iV(l-e*) '
The Moments about are
Mn
1 V^ rn
(25)
(26)
/xi = §(JV + l)
(27)
(28)
4 -v" ■ ^ (29)
M4 = ^(A r +l)(2AT+l)(3JV 2 + 3AT-l), (30)
ti' 2 = \{N+l){2N + l)
li' 3 = \N{N + \f
and the Moments about the Mean are
/x a = &(JV-l)(JV + l) (31)
Ms = (32)
^ = ^ (iV - 1)(AT + 1)(3JV 2 - 7). (33)
The Mean, Variance, Skewness, and Kurtosis are
H=\{N+l)
M3 „
72
6(N 2 + 1)
5(JV- l)(iV + l)'
(34)
(35)
(36)
(37)
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 531 and 533, 1987.
Uniform Polyhedron
The uniform polyhedra are POLYHEDRA with identical
Vertices. Coxeter et al. (1954) conjectured that there
are 75 such polyhedra in which only two faces are al-
lowed to meet at an EDGE, and this was subsequently
proven. (However, when any Even number of faces may
meet, there are 76 polyhedra.) If the five pentagonal
PRISMS are included, the number rises to 80.
The VERTICES of a uniform polyhedron all lie on a
Sphere whose center is their Centroid. The Ver-
tices joined to another VERTEX lie on a CIRCLE.
Source code and binary programs for generat-
ing and viewing the uniform polyhedra are also
available at http : //www . math . technion . ac . il/~ rl/
kaleido/. The following depictions of the polyhedra
were produced by R. Maeder's UniformPolyhedra.m
package for Mathematical (Wolfram Research, Cham-
paign, IL). Due to a limitation in Mathematical s Ten-
derer, uniform polyhedra 69, 72, 74, and 75 cannot be
displayed using this package.
1882
Uniform Polyhedron
Uniform Polyhedron
n
Name/Dual
1
tetrahedron
tetrahedron
2
truncated tetrahedron
triakis tetrahedron
3
octahemioctahedron
octahemioctacron
4
tetrahemihexahedron
tetrahemihexacron
5
octahedron
cube
6
cube
octahedron
7
cuboctahedron
rhombic dodecahedron
8
truncated octahedron
tetrakis hexahedron
9
truncated cube
triakis octahedron
10
small rhombicuboctahedron
deltoidal icositetrahedron
11
truncated cuboctahedron
disdyakis dodecahedron
12
snub cube
pentagonal icositetrahedron
13
small cubicuboctahedron
small hexacronic icositetrahedron
14
great cubicuboctahedron
great hexacronic icositetrahedron
15
cubohemioctahedron
hexahemioctahedron
16
cubitruncated cuboctahedron
tetradyakis hexahedron
17
great rhombicuboctahedron
great deltoidal icositetrahedron
18
small rhombihexahedron
small rhombihexacron
19
stellated truncated hexahedron
great triakis octahedron
20
great truncated cuboctahedron
great disdyakis dodecahedron
21
great rhombihexahedron
great rhombihexacron
22
icosahedron
dodecahedron
23
dodecahedron
xcosahedron
24
icosidodecahedron
rhombic triacontahedron
25
truncated icosahedron
pentakis dodecahedron
26
27
46
47
48
49
50
Name/Dual
truncated dodecahedron
triakis icosahedron
small rhombicosidodecahedron
deltoidal hexecontahedron
truncated icosidodecahedron
disdyakis triacontahedron
snub dodecahedron
pentagonal hexecontahedron
small ditrigonal icosidodecahedron
small triambic icosahedron
small icosicosidodecahedron
small icosacronic hexecontahedron
small snub icosicosidodecahedron
small hexagonal hexecontahedron
small dodecicosidodecahedron
small dodecacronic hexecontahedron
small stellated dodecahedron
great dodecahedron
great dodecahedron
small stellated dodecahedron
dodecadodecahedron
medial rhombic triacontahedron
truncated great dodecahedron
small stellapentakis dodecahedron
rhombidodecadodecahedron
medial deltoidal hexecontahedron
small rhombidodecahedron
small rhombidodecacron
snub dodecadodecahedron
medial pentagonal hexecontahedron
ditrigonal dodecadodecahedron
medial triambic icosahedron
great ditrigonal dodecicosidodecahedron
great ditrigonal dodecacronic hexecontahedron
small ditrigonal dodecicosidodecahedron
small ditrigonal dodecacronic hexecontahedron
icosidodecadodecahedron
medial icosacronic hexecontahedron
icositruncated dodecadodecahedron
tridyakis icosahedron
snub icosidodecadodecahedron
medial hexagonal hexecontahedron
great ditrigonal icosidodecahedron
great triambic icosahedron
great icosicosidodecahedron
great icosacronic hexecontahedron
small icosihemidodecahedron
small icosihemidodecacron
small dodecicosahedron
small dodecicosacron
Uniform Polyhedron
Uniform Polyhedron
1883
n
Name/Dual
51
small dodecahemidodecahedron
small dodecahemidodecacron
52
great stellated dodecahedron
great icosahedron
53
great icosahedron
great stellated dodecahedron
54
great icosidodecahedron
great rhombic triacontahedron
55
great truncated icosahedron
great stellapentakis dodecahedron
56
rhombicosahedron
rhombicosacron
57
great snub icosidodecahedron
great pentagonal hexecontahedron
58
small stellated truncated dodecahedron
great pentakis dodecahedron
59
truncated dodecadodecahedron
medial disdyakis triacontahedron
60
inverted snub dodecadodecahedron
medial inverted pentagonal hexecontahedron
61
great dodecicosidodecahedron
great dodecacronic hexecontahedron
62
small dodecahemicosahedron
small dodecahemicosacron
63
great dodecicosahedron
great dodecicosacron
64
great snub dodecicosidodecahedron
great hexagonal hexecontahedron
65
great dodecahemicosahedron
great dodecahemicosacron
66
great stellated truncated dodecahedron
great triakis icosahedron
67
great rhombicosidodecahedron
great deltoidal hexecontahedron
68
great truncated icosidodecahedron
great disdyakis triacontahedron
69
great inverted snub icosidodecahedron
great inverted pentagonal hexecontahedron
70
great dodecahemidodecahedron
great dodecahemidodecacron
71
great icosihemidodecahedron
great icosihemidodecacron
72
small retrosnub icosicosidodecahedron
small hexagrammic hexecontahedron
73
great rhombidodecahedron
great rhombidodecacron
74
great retrosnub icosidodecahedron
great pentagrammic hexecontahedron
75
great dirhombicosidodecahedron
great dirhombicosidodecacron
n
Name/Dual
76
pentagonal prism
pentagonal dipyramid
77
pentagonal antiprism
pentagonal deltahedron
78
pentagrammic prism
pentagrammic dipyramid
79
pentagrammic antiprism
pentagrammic deltahedron
80
pentagrammic crossed antiprism
pentagrammic concave deltahedron
see also ARCHIMEDEAN SOLID, AUGMENTED POLYHE-
DRON, Johnson Solid, Kepler-Poinsot Solid, Pla-
tonic Solid, Polyhedron, Vertex Figure, Wyth-
off Symbol
1884
Uniform Variate
Unimodular Matrix
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, p. 136,
1987,
Bulatov, V.v "Compounds of Uniform Polyhedra." http://
www . physics . orst . edu/ -bulatov /polyhedra /uniform,
compounds/.
Bulatov, V. "Dual Uniform Polyhedra." http: //www.
physics . orst . edu/ -bulatov /polyhedra/ dual/.
Bulatov, V. "Uniform Polyhedra." http: //www. physics,
orst . edu/-bulatov/polyhedra/unif orm/.
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.
Har'El, Z. "Uniform Solution for Uniform Polyhedra." Ge-
ometriae Dedicata 47, 57-110, 1993.
Har'El, Z. "Kaleido." http://www.math.technion.ac.il/
-rl/kaleido/.
Har'El, Z. "Eighty Dual Polyhedra Generated by Kaleido."
http:// www ♦ math . technion .ac.il/~rl/ kaleido /
dual.html.
Har'El, Z. "Eighty Uniform Polyhedra Generated by Ka-
leido." http: //www. math. technion. ac . il/ -rl/kaleido/
poly.html.
Hume, A. "Exact Descriptions of Regular and Semi-
Regular Polyhedra and Their Duals." Computing Science
Tech. Rept. No. 130. Murray Hill, NJ: AT&T Bell Lab.,
1986.
Hume, A. Information files on polyhedra. http://netlib.
bell-labs . com/netlib/polyhedra/.
Johnson, N. W. "Convex Polyhedra with Regular Faces."
Canad. J. Math. 18, 169-200, 1966,
Maeder, R. E. "Uniform Polyhedra." Mathematica J.
3, 1993. ftp : //ftp . inf . ethz . ch/doc/papers/ti/scs/
unipoly.ps. gz.
Maeder, R. E. Polyhedra. m and PolyhedraExamples
Mathematica® notebooks. http : //www . inf . ethz . ch/
department/TI/rm/programs .html.
Maeder, R. E, "The Uniform Polyhedra." http://www.inf.
ethz . ch/department/TI/rm/unipoly/.
Skilling, J. "The Complete Set of Uniform Polyhedron." Phil
Trans. Roy. Soc. London, Ser. A 278, 111-136, 1975.
Virtual Image. "The Uniform Polyhedra CD-ROM." http : //
ourworld . CompuServe . com/homepages/vir_image/html/
unif ormpolyhedr a . html .
Wenninger, M. J. Polyhedron Models. New York; Cambridge
University Press, pp. 1-10 and 98, 1989.
Zalgaller, V. Convex Polyhedra with Regular Faces. New-
York: Consultants Bureau, 1969.
Ziegler, G. M. Lectures on Polytopes. Berlin: Springer-
Verlag, 1995.
Uniform Variate
A Random Number which lies within a specified range
(which can, without loss of generality, be taken as [0,
1]), with a Uniform Distribution.
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Uniform Deviates." §7.1 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 267-277, 1992.
Unimodal Distribution
A Distribution such as the Gaussian Distribution
which has a single "peak."
see also Bimodal Distribution
Unimodal Sequence
A finite SEQUENCE which first increases and then de-
creases. A Sequence {si, S2, ■••, s n } is unimodal if
there exists a t such that
and
Si < S2 < . . . < s t
St > St+i > . . . > 5 n .
Unimodular Group
A group whose left HAAR MEASURE equals its right
Haar Measure.
see also HAAR MEASURE
References
Knapp, A. W. "Group Representations and Harmonic Anal-
ysis, Part II." Not. Amer. Math. Soc. 43, 537-549, 1996.
Unimodular Matrix
A Matrix A with Integer elements and Determi-
nant det(A) = ± 1, also called a UNIT MATRIX.
The inverse of a unimodular matrix is another uni-
modular matrix. A POSITIVE unimodular matrix has
det(A) = +1. The nth Power of a Positive Unimod-
ular Matrix
M =
7T121 TU22
(1)
mnU n -i(a) - U n -2(a) mi2U n -i(a>)
m2iU n -i(a) m 2 2Un-i(a) — U n -2(a)
where
a = i(mn +77122)
(2)
(3)
and the U n are CHEBYSHEV POLYNOMIALS OF THE SEC-
OND Kind,
U m (x) —
sin[(m + 1) cos 1 x]
(4)
see also CHEBYSHEV POLYNOMIAL OF THE SECOND
Kind
References
Born, M. and Wolf, E. Principles of Optics: Electromagnetic
Theory of Propagation, Interference, and Diffraction of
Light, 6th ed. New York: Pergamon Press, p. 67, 1980.
Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA:
Addison- Wesley, p. 149, 1980.
Unimodular Transformation
Unit Fraction
1885
Unimodular Transformation
A transformation x' = Ax is unimodular if the DETER-
MINANT of the Matrix A satisfies
det(A) = ±1.
A Necessary and Sufficient condition that a linear
transformation transform a lattice to itself is that the
transformation be unimodular.
Union
The union of two sets A and B is the set obtained by
combining the members of each. This is written AUB,
and is pronounced "A union B" or "A cup £." The
union of sets A± through A n is written IJILi ^*-
Let A, B, C, . . . be sets, and let P(S) denote the prob-
ability of S. Then
P(A UB) = P(A) + P(B) - P(A n B). (1)
Similarly,
P(A ITS U C) = P[A U(BU C)}
= P(A) + P(B U C) - P[A n(BU C)]
= P(A) + [P(B) + P(C) - P(B n C)]
-ppnB)u(inC)]
= P(A) + P(S) + P(C) - p(b n c)
-{P(A n B) + P(A n c) - P[(A n 5) n (A n c)]}
= P(A) + p[b) + P(C) - P(A n s)
-p(A n C) - P(JB n C) + P(A n P n C). (2)
If A and 5 are DISJOINT, by definition P(A n P) = 0,
so
P(AUP) = P(A) + P(P). (3)
Continuing, for a set of n disjoint elements Ei } £"2, * * • >
Pn
p U* =E P ^)-
(4)
which is the Countable Additivity Probability
Axiom. Now let
Ei = ADBi, (5)
then
p( {JehbA =J2 p ( EnBi ^ ( 6 )
see also Intersection, Or
Uniplanar Double Point
see Isolated Singularity
Unipotent
Ap-Element x of a Group Q is unipotent if F*(Cg(x))
is ap-GROUP, where F* is the generalized Fitting Sub-
group.
see also FITTING SUBGROUP, p-ELEMENT, p-GROUP
Unique
The property of being the only possible solution (per-
haps modulo a constant, class of transformation, etc.).
see also Aleksandrov's Uniqueness Theorem, Ex-
istence, May-Thomason Uniqueness Theorem
Unique Factorization Theorem
see Fundamental Theorem of Arithmetic
Unit
A unit is an element in a Ring that has a multiplicative
inverse. If n is an ALGEBRAIC INTEGER which divides
every ALGEBRAIC INTEGER in the FIELD, n is called a
unit in that Field. A given Field may contain an in-
finity of units. The units of Z n are the elements Rela-
tively Prime to n. The units in Z n which are Squares
are called Quadratic Residues.
see also Eisenstein Unit, Fundamental Unit,
Prime Unit, Quadratic Residue
Unit Circle
A Circle of Radius 1, such as the one used to defined
the functions of TRIGONOMETRY.
see also Unit Disk, Unit Square
Unit Disk
A Disk with Radius 1.
see also Five Disks Problem, Unit Circle, Unit
Square
Unit Fraction
A unit fraction is a FRACTION with NUMERATOR 1, also
known as an EGYPTIAN FRACTION. Any RATIONAL
Number has infinitely many unit fraction representa-
tions, although only finitely many have a given fixed
number of terms. Each FRACTION x/y with y Odd has
a unit fraction representation in which each DENOMINA-
TOR is Odd (Breusch 1954; Guy 1994, p. 160). Ever y
x/y has a t-term representation where t = 0(y/\ogy )
(Vose 1985).
1886
Unit Matrix
Unit Vector
There are a number of ALGORITHMS (including the
Binary Remainder Method, Continued Frac-
tion Unit Fraction Algorithm, Generalized Re-
mainder Method, Greedy Algorithm, Reverse
Greedy Algorithm, Small Multiple Method, and
Splitting Algorithm) for decomposing an arbitrary
Fraction into unit fractions.
see also Calcus, Half, Quarter, Scruple, Uncia
References
Beck, A.; Bleicher, M. N.; and Crowe, D. W. Excursions into
Mathematics. New York: Worth Publishers, 1970.
Beeckmans, L. "The Splitting Algorithm for Egyptian Frac-
tions." J. Number Th. 43, 173-185, 1993.
Bleicher, M. N. "A New Algorithm for the Expansion of Con-
tinued Fractions." J. Number Th. 4, 342-382, 1972.
Breusch, R. "A Special Case of Egyptian Fractions." Solution
to advanced problem 4512. Amer. Math. Monthly 61, 200-
201, 1954.
Brown, K. S. "Egyptian Unit Fractions." http://wvv.
seanet . com/~ksbrown.
Eppstein, D. "Ten Algorithms for Egyptian Fractions."
Math. Edu. Res. 4, 5-15, 1995.
Eppstein, D. "Egyptian Fractions." http://www. ics . uci .
edu/~eppstein/numth/egypt/.
^Eppstein, D. Egypt. ma Mathematica notebook, http://
www.ics.uci.edu/-eppstein/numth/egypt/egypt.ma.
Graham, R. "On Finite Sums of Unit Fractions." Proc. Lon-
don Math. Soc. 14, 193-207, 1964.
Guy, R. K. "Egyptian Fractions." §D11 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 87-93 and 158-166, 1994.
Klee, V. and Wagon, S. Old and New Unsolved Problems in
Plane Geometry and Number Theory. Washington, DC:
Math. Assoc. Amer., pp. 175-177 and 206-208, 1991,
Niven, I. and Zuckerman, H. S. An Introduction to the Theory
of Numbers, 5th ed. New York: Wiley, p. 200, 1991.
Stewart, I. "The Riddle of the Vanishing Camel." Sci. Amer.,
122-124, June 1992.
Tenenbaum, G. and Yokota, H. "Length and Denominators
of Egyptian Fractions." J. Number Th. 35, 150-156, 1990.
Vose, M. "Egyptian Fractions." Bull. London Math. Soc.
17, 21, 1985.
Wagon, S. "Egyptian Fractions." §8.6 in Mathematica in
Action. New York: W. H. Freeman, pp. 271-277, 1991.
Unit Matrix
see Unimodular Matrix
Unit Point
The point in the PLANE with Cartesian coordinates (1,
!)•
References
Woods, F. S. Higher Geometry: An Introduction to Advanced
Methods in Analytic Geometry. New York: Dover, p. 9,
1961.
Unit Ring
A unit ring is a set together with two BINARY OPERA-
TORS 5(+, *) satisfying the following conditions:
1. Additive associativity: For all a,b,c 6 5, (a+6) + c =
a-h(b-fc),
2. Additive commutativity: For all a, b £ 5, a + b =
b + a,
3. Additive identity: There exists an element G S
such that for all a G S : + a = a + = a,
4. Additive inverse: For every a G S, there exists a
—a £ S such that a + (—a) = (—a) + a = 0,
5. Multiplicative associativity: For all a,b,c 6 S, (a*
b) * c = a * (b * c),
6. Multiplicative identity: There exists an element 1 G
5 such that for all a £ 5, 1 * a — a * 1 = a,
7. Left and right distributivity: For all a, 6, c G 5, a *
(b + c) = (a*b) + (a*c) and (b + c)*a = (b*a) + (c*a).
Thus, a unit ring is a Ring with a multiplicative identity.
see also Binary Operator, Ring
References
Rosenfeld, A. An Introduction to Algebraic Structures. New
York: Holden-Day, 1968.
Unit Sphere
A Sphere of Radius 1.
see also Sphere, Unit Circle
Unit Square
A Square with side lengths 1. The unit square usually
means the one with coordinates (0, 0), (1, 0), (1, 1), (0,
1) in the real plane, or 0, 1, 1 + z, and i in the COMPLEX
Plane.
see also Heilbronn Triangle Problem, Unit Cir-
cle, Unit Disk
Unit Step
see HEAVISIDE STEP FUNCTION
Unit Vector
A VECTOR of unit length. The unit vector v having the
same direction as a given (nonzero) vector v is defined
by
V
v= 1 — r ,
|v|
where |v| denotes the NORM of v, is the unit vector in
the same direction as the (finite) VECTOR v. A unit
VECTOR in the x n direction is given by
I dr
I dx n
where r is the RADIUS VECTOR.
see also NORM, RADIUS VECTOR, VECTOR
Unital
Unitary Multiplicative Character 1887
Unital
A Block Design of the form (q 3 + 1, q -f 1, 1).
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.
Unitary Aliquot Sequence
An Aliquot Sequence computed using the analog of
the Restricted Divisor Function s*(n) in which
only Unitary Divisors are included.
see also Aliquot Sequence, Unitary Sociable
Numbers
References
Guy, R. K. "Unitary Aliquot Sequences." §B8 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 63-65, 1994.
Unitary Amicable Pair
A Pair of numbers m and n such that
a* (m) = a* (n) = m + n,
where cr*(n) is the sum of UNITARY DIVISORS. Hagis
(1971) and Garcia (1987) give 82 such pairs. The first
few are (114, 126), (1140, 1260), (18018, 22302), (32130,
40446), . . . (Sloane's A002952 and A002953).
References
Garcia, M. "New Unitary Amicable Couples." J. Recr, Math.
19, 12-14, 1987.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 57, 1994.
Hagis, P. "Relatively Prime Amicable Numbers of Opposite
Parity." Math. Comput. 25, 915-918, 1971.
Sloane, N. J. A. Sequences A002952/M5372 and A002953/
M5389 in "An On-Line Version of the Encyclopedia of In-
teger Sequences."
Unitary Divisor
A Divisor d of c for which
GCD(d,c/d) = 1,
where GCD is the GREATEST COMMON DIVISOR.
see also Divisor, Greatest Common Divisor, Uni-
tary Perfect Number
References
Guy, R. K. "Unitary Perfect Numbers." §B3 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 53-59, 1994.
Unitary Group
The unitary group U n (q) is the set ofnxn Unitary
Matrices.
see also Lie-Type Group, Unitary Matrix
References
Wilson, R. A. "ATLAS of Finite Group Representation."
http : fit or . mat . bham . ac . uk/atlas#unit .
Unitary Matrix
A unitary matrix is a MATRIX U for which
(i)
where f denotes the Adjoint Operator. This guaran-
tees that
U f U = 1. (2)
Unitary matrices leave the length of a Complex vector
unchanged. The product of two unitary matrices is itself
unitary. If U is unitary, then so is U _1 . A SIMILARITY
Transformation of a Hermitian Matrix with a uni-
tary matrix gives
(nan' 1 )* = [(uaXiT 1 )]* - (iT^M* = (uV(aV)
= uav/ = uavT . (3)
For Real Matrices, Hermitian is the same as Or-
thogonal. Unitary matrices are NORMAL MATRICES.
If M is a unitary matrix, then the PERMANENT
|perm(M)| < 1 (4)
(Mine 1978, p. 25, Vardi 1991).
see also Adjoint Operator, Hermitian Matrix,
Normal Matrix, Orthogonal Matrix, Perma-
nent
References
Arfken, G. "Hermitian Matrices, Unitary Matrices." §4.5 in
Mathematical Methods for Physicists, 3rd ed. Orlando,
FL: Academic Press, pp. 209-217, 1985.
Mine, H. Permanents. Reading, MA: Addison- Wesley, 1978.
Vardi, I. "Permanents." §6.1 in Computational Recreations
in Mathematica. Reading, MA: Addison- Wesley, pp. 108
and 110-112, 1991.
Unitary Multiperfect Number
A number n which is an INTEGER multiple k of the Sum
of its Unitary Divisors <r*(n) is called a unitary k-
multiperfect number. There are no Odd unitary multi-
perfect numbers.
References
Guy, R. K. "Unitary Perfect Numbers." §B3 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 53-59, 1994.
Unitary Multiplicative Character
A Multiplicative Character is called unitary if it
has Absolute Value 1 everywhere.
see also CHARACTER (MULTIPLICATIVE)
1888 Unitary Perfect Number
Unknotting Number
Unitary Perfect Number
A number n which is the sum of its Unitary Divisors
with the exception of n itself. There are no Odd unitary
perfect numbers, and it has been conjectured that there
are only a FINITE number of EVEN ones. The first few
are 6, 60, 90, 87360, 146361946186458562560000, . . .
(Sloane's A002827).
References
Guy, R. K. "Unitary Perfect Numbers." §B3 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 53-59, 1994.
Sloane, N. J. A. Sequence A002827/M4268 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Wall, C. R. "On the Largest Odd Component of a Unitary
Perfect Number." Fib. Quart 25, 312-316, 1987.
Unitary Sociable Numbers
Sociable Numbers computed using the analog of the
Restricted Divisor Function s*(n) in which only
Unitary Divisors are included.
see also SOCIABLE NUMBERS
References
Guy, R. K. "Unitary Aliquot Sequences." §B8 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 63-65, 1994.
Unitary Transformation
A transformation of the form
A' = UAU f ,
where f denotes the ADJOINT OPERATOR.
see also Adjoint Operator, Transformation
Unitary Unimodular Group
see Special Unitary Group
Unity
The number 1. There are n nth ROOTS OF UNITY,
known as the de Moivre Numbers.
see also 1, Primitive Root of Unity
Univalent Function
A function or transformation / in which f(z) does not
overlap z.
Univariate Function
A Function of a single variable (e.g., /(#), g(z), #(£),
etc.).
see also MULTIVARIATE FUNCTION
Univariate Polynomial
A POLYNOMIAL in a single variable. In common usage,
univariate POLYNOMIALS are sometimes simply called
"POLYNOMIALS."
see also POLYNOMIAL
Universal Graph
see Complete Graph
Universal Statement
A universal statement S is a FORMULA whose FREE vari-
ables are all in the scope of universal quantifiers.
Universal Turing Machine
A TURING MACHINE which, by appropriate program-
ming using a finite length of input tape, can act as any
Turing Machine whatsoever.
see Chaitin's Constant, Halting Problem, Turing
Machine
References
Penrose, R. The Emperor's New Mind: Concerning Com-
puters, Minds, and the Laws of Physics. Oxford: Oxford
University Press, pp. 51-57, 1989.
Unknot
A closed loop which is not KNOTTED. In the 1930s,
by making use of Reidemeister Moves, Reidemeister
first proved that KNOTS exist which are distinct from
the unknot. He proved this by COLORING each part of
a knot diagram with one of three colors.
The KNOT Sum of two unknots is another unknot.
The Jones Polynomial of the unknot is defined to give
the normalization
V(t) = 1.
Haken (1961) devised an Algorithm to tell if a knot
projection is the unknot. The ALGORITHM is so com-
plicated, however, that it has never been implemented.
Although it is not immediately obvious, the unknot is a
Prime Knot.
see also COLORABLE, KNOT, KNOT THEORY, LINK,
Reidemeister Moves, Unknotting Number
References
Haken, W. "Theorie der Normalnachen.'
245-375, 1961.
Acta Math. 105,
Unknotting Number
The smallest number of times a KNOT must be passed
through itself to untie it. Lower bounds can be com-
puted using relatively straightforward techniques, but it
is in general difficult to determine exact values. Many
unknotting numbers can be determined from a knot's
Signature. A Knot with unknotting number 1 is a
Prime Knot (Scharlemann 1985). It is not always true
that the unknotting number is achieved in a projection
with the minimal number of crossings.
The following table is from Kirby (1997, pp. 88-89), with
the values for IO139 and IO152 taken from Kawamura.
The unknotting numbers for IO154 and lOiei can be
found using Menasco's Theorem (Stoimenow 1998).
Unless
Unstable Spiral Point 1889
3i
1
89
1
9io
2 or 3
9 3 2
1 or 2
4i
1
810
1 or \
2 9n
2
9 3 3
1
5i
2
811
1
9l2
1
9 34
1
5 2
1
812
2
9l3
2 or 3
9 35
2 or 3
61
1
813
1
9l4
1
9 3 6
2
62
1
814
1
9l5
2
9 3 7
2
63
1
815
2
9l6
3
938
2 or 3
7i
3
816
2
9l7
2
9 3 9
1
7 2
1
817
1
9i 8
2
9 4 o
2
7 3
2
818
2
9l9
1
941
2
7 4
2
819
3
9 2 o
2
9 4 2
1
7 5
2
820
1
9 2 1
1
9 4 3
2
7 6
1
821
1
922
1
9 4 4
1
7 7
1
9i
4
9 2 3
2
9 4 5
1
81
1
9 2
1
9 2 4
1
9 46
2
8 2
2
9 3
3
9 2 5
2
9 4 7
2
83
2
9 4
2
9 2 6
1
9 48
2
84
2
9 5
2
9 2 7
1
9 49
2 or 3
8 5
2
9 6
3
9 2 8
1
IO139
4
8 6
2
9 7
2
9 2 9
1
IO152
4
87
1
9 8
2
9 3 o
1
IO154
3
8 8
2
9 9
3
9 31
2
lOiei
3
see also Bennequin's Conjecture, Menasco's The-
orem, Milnor's Conjecture, Signature (Knot)
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 57-64, 1994.
Cipra, B. "From Knot to Unknot." What's Happening in
the Mathematical Sciences, Vol. 2. Providence, RI: Amer.
Math. Soc, pp. 8-13, 1994.
Kawamura, T. "The Unknotting Numbers of IO139 and 10i 52
are 4." To appear in Osaka J. Math. http://ms421sun.
ms.u-tokyo.ac.jp/-kawaraura/worke.html.
Kirby, R. (Ed.) "Problems in Low-Dimensional Topol-
ogy." AMS/IP Stud. Adv. Math., 2.2, Geometric Topology
(Athens, GA, 1993). Providence, RI: Amer. Math. Soc,
pp. 35-473, 1997.
Scharlemann, M. "Unknotting Number One Knots are
Prime." Invent. Math. 82, 37-55, 1985.
Stoimenow, A. "Positive Knots, Closed Braids and the Jones
Polynomial." Rev. May, 1997. http://www.inf ormatik.
hu-berlin.de/ ~st oimeno/pos.ps.gz.
$$ Weisstein, E. W. "Knots and Links." http: //www. astro.
Virginia. edu/-eww6n/math/notebooks/Knots.m,
Unless
If A is true unless B, then not-B implies A, but B does
not necessarily imply not- A.
see also PRECISELY UNLESS
Unlesss
see Precisely Unless
Unmixed
A homogeneous IDEAL defining a projective ALGEBRAIC
Variety is unmixed if it has no embedded Prime divi-
sors.
Unpoke Move
see Poke Move
Unsafe
A position in a Game is unsafe if the person who plays
next can win. Every unsafe position can be made SAFE
by at least one move.
see also Game, Safe
Unsolved Problem
see Problem
Unstable Improper Node
A Fixed Point for which the Stability Matrix has
equal POSITIVE EIGENVALUES.
see also Elliptic Fixed Point (Differential
Equations), Fixed Point, Hyperbolic Fixed
Point (Differential Equations), Stable Im-
proper Node, Stable Node, Stable Spiral Point,
Unstable Node, Unstable Spiral Point, Unsta-
ble 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.
Unstable Node
A Fixed Point for which the Stability Matrix has
both Eigenvalues Positive, so Ai > A 2 > 0.
see also Elliptic Fixed Point (Differential
Equations), Fixed Point, Hyperbolic Fixed
Point (Differential Equations), Stable Im-
proper Node, Stable Node, Stable Spiral Point,
Stable Star, Unstable Improper 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.
Unstable Spiral Point
A Fixed Point for which the Stability Matrix has
Eigenvalues of the form A± = a ± i/3 (with a,/3 > 0).
see also Elliptic Fixed Point (Differential
Equations), Fixed Point, Hyperbolic Fixed
Point (Differential Equations), Stable Im-
proper Node, Stable Node, Stable Spiral Point,
Stable Star, Unstable Improper Node, Unstable
Node, 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.
1890
Unstable Star
Urchin
Unstable Star
A Fixed Point for which the Stability Matrix has
one zero Eigenvector with Positive Eigenvalue
A> 0.
see also Elliptic Fixed Point (Differential
Equations), Fixed Point, Hyperbolic Fixed
Point (Differential Equations), Stable Im-
proper Node, Stable Node, Stable Spiral Point,
Stable Star, Unstable Improper Node, Unstable
Node, Unstable Spiral Point
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.
Untouchable Number
An untouchable number is an INTEGER which is not the
sum of the PROPER DIVISORS of any other number. The
first few are 2, 5, 52, 88, 96, 120, 124, 146, . . . (Sloane's
A005114). Erdos has proven that there are infinitely
many. It is thought that 5 is the only Odd untouchable
number.
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 840, 1972.
Guy, R. K. "Untouchable Numbers." §B10 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, pp. 66-67, 1994.
Sloane, N. J. A. Sequence A005114/M1552 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Upper Bound
see Least Upper Bound
Upper Integral
The limit of an Upper Sum, when it exists, as the Mesh
Size approaches 0.
see also LOWER INTEGRAL, RlEMANN INTEGRAL, UP-
PER Sum
Upper Limit
Let the greatest term if of a SEQUENCE be a term which
is greater than all but a finite number of the terms which
are equal to H. Then H is called the upper limit of the
Sequence.
An upper limit of a SERIES
upper lim S n = lim S n = k
n— >oo n— ► oo
is said to exist if, for every e > 0, \S n — k\ < e for
infinitely many values of n and if no number larger than
k has this property.
see also Limit, LOWER Limit
References
Bromwich, T. J. I'a and MacRobert, T. M. "Upper and Lower
Limits of a Sequence." §5.1 in An Introduction to the The-
ory of Infinite Series, 3rd ed. New York: Chelsea, p. 40,
1991.
Upper Sum
For a given function f(x) over a partition of a given in-
terval, the upper sum is the sum of box areas f(x^)Axk
using the greatest value of the function /(#£) in each
subinterval Axk.
see also Lower Sum, Riemann Integral, Upper In-
tegral
Upper- Trimmed Subsequence
The upper-trimmed subsequence of x = {x n } is the se-
quence X(x) obtained by dropping the first occurrence
of n for each n. If z is a FRACTAL SEQUENCE, then
X(x) = x.
see also Lower-Trimmed Subsequence
References
Kimberling, C. "Fractal Sequences and Interspersions." Ars
Combin. 45, 157-168, 1997.
Upward Drawing
see HASSE DIAGRAM
Urchin
Kepler's original name for the SMALL STELLATED DO-
DECAHEDRON.
Utility Graph
Utility Graph
Utility Problem 1891
The utility problem asks, "Can a PLANAR GRAPH be
constructed from each of three nodes ('house owners') to
each of three other nodes ('wells')?" The answer is no,
and the proof can be effected using the Jordan Curve
THEOREM, while a more general result encompassing
this one is the Kuratowski Reduction Theorem.
The utility graph UG is the graph showing the rela-
tionships described above. It is identical to the THOM-
SEN Graph and, in the more formal parlance of Graph
Theory, is known as the Complete Bipartite Graph
#3,3-
see also Complete Bipartite Graph, Kuratowski
Reduction Theorem, Planar Graph, Thomsen
Graph
References
Chartrand, G. "The Three Houses and Three Utilities Prob-
lem: An Introduction to Planar Graphs." §9.1 in Intro-
ductory Graph Theory. New York: Dover, pp. 191-202,
1985.
Ore, 0. Graphs and Their Uses. New York: Random House,
pp. 14-17, 1963,
Pappas, T. "Wood, Water, Grain Problem." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
pp. 175 and 233, 1989.
Utility Problem
see Utility Graph
Valence
V
Valence
see Valency
Valency
The number of EDGES at a GRAPH VERTEX.
Valuation
A generalization of the p-ADic Numbers first proposed
by Kiirschak in 1913. A valuation | • | on a FIELD K is a
Function from K to the Real Numbers R such that
the following properties hold for all x, y e K:
1. |x| > 0,
Id = IFF x = 0,
Valuation
1893
\xy\ = \x\\y\,
\x\ < 1 Implies |l+a;| < C for some constant C > 1
(independent of x).
If (4) is satisfied for C = 2, then | • | satisfies the TRI-
ANGLE Inequality,
4a. \x + y\ < \x\ + \y\ for all x,y £ K.
If (4) is satisfied for C = 1 then | ■ | satisfies the stronger
Triangle Inequality
4b. \x + y\ < max(|rr|, |y|).
The simplest valuation is the ABSOLUTE VALUE for
Real Numbers. A valuation satisfying (4b) is called
non-ARCHlMEDEAN Valuation; otherwise, it is called
Archimedean.
If | ■ |i is a valuation on K and A > 1, then we can define
a new valuation | ■ I2 by
Z2
\x\l
(1)
This does indeed give a valuation, but possibly with a
different constant C in Axiom 4. If two valuations are
related in this way, they are said to be equivalent, and
this gives an equivalence relation on the collection of
all valuations on K. Any valuation is equivalent to one
which satisfies the triangle inequality (4a). In view of
this, we need only to study valuations satisfying (4a),
and we often view axioms (4) and (4a) as interchange-
able (although this is not strictly true).
If two valuations are equivalent, then they are both non-
Arciiimedean or both Archimedean. Q, M, and C
with the usual Euclidean norms are Archimedean val-
uated fields. For any Prime p, the p-ADic Numbers
Q with the p-adic valuation | ■ \ p is a non- Archimedean
valuated field.
If K is any FIELD, we can define the trivial valuation
on K by |x| = 1 for all x ^ and |0| = 0, which is
a non- Archimedean valuation. If K is a Finite Field,
then the only possible valuation over K is the trivial one.
It can be shown that any valuation on Q is equivalent
to one of the following: the trivial valuation, Euclidean
absolute norm | - |, or p-adic valuation | • | p .
The equivalence of any nontrivial valuation of Q to ei-
ther the usual ABSOLUTE VALUE or to a p-ADIC NUM-
BER absolute value was proved by Ostrowski (1935).
Equivalent valuations give rise to the same topology.
Conversely, if two valuations have the same topology,
then they are equivalent. A stronger result is the fol-
lowing: Let I ■ |i, I • I2, • ■ - , I • |fc be valuations over K
which are pairwise inequivalent and let a\ , a^ , • • . , cik
be elements of K. Then there exists an infinite sequence
(xi, X2, • • • ) of elements of K such that
lim
+ 00 w.r.t. |-|i
CL\
lim x n — a,2,
+ 00 w.r.t. |-J2
(2)
(3)
etc. This says that inequivalent valuations are, in some
sense, completely independent of each other. For exam-
ple, consider the rationals Q with the 3-adic and 5-adic
valuations | • [3 and | • |s, and consider the sequence of
numbers given by
43 • 5 71 + 92 • 3"
3 n -h 5 n
(4)
Then x n — > 43 as n —> 00 with respect to | * | 3 , but
x n -¥ 92 as n -* 00 with respect to | - | 5 , illustrating
that a sequence of numbers can tend to two different
limits under two different valuations.
A discrete valuation is a valuation for which the VALUA-
TION Group is a discrete subset of the Real Numbers
UL Equivalently, a valuation (on a FIELD K) is discrete
if there exists a Real Number e > such that
\x\ e (l-e,l + e)
= 1 for all x e K.
(5)
The p-adic valuation on Q is discrete, but the ordinary
absolute valuation is not.
if 1
a valuation on K. then it induces a metric
d(x,y)
|ar — s/|
(6)
on f£T, which in turn induces a TOPOLOGY on K, If
I ■ I satisfies (4b) then the metric is an Ultrametric.
We say that (K % \ ■ | ) is a complete valuated field if the
Metric Space is complete.
see also Absolute Value, Local Field, Metric
Space, p-adic Number, Strassman's Theorem, Ul-
trametric, Valuation Group
References
Cassels, J. W. S. Local Fields. Cambridge, England: Cam-
bridge University Press, 1986.
Ostrowski, A. "Untersuchungen zur aritmetischen Theorie
der Korper." Math. Zeit. 39, 269-404, 1935.
1894
Valuation Group
Vampire Number
Valuation Group
Let (K, | * |) be a valuated field. The valuation group G
is defined to be the set
G = {\x\ :xeK,x^0},
with the group operation being multiplication. It is
a Subgroup of the Positive Real Numbers, under
multiplication.
Valuation Ring
Let (K, | * |) be a non- Archimedean valuated field. Its
valuation ring R is defined to be
R= {x eK : \x\ < 1}.
The valuation ring has maximal IDEAL
Vampire Number
A number v = xy with an Even number n of DIG-
ITS formed by multiplying a pair of n/2-DlGlT numbers
(where the Digits are taken from the original number
in any order) x and y together. Pairs of trailing zeros
are not allowed. If v is a vampire number, then x and
y are called its "fangs." Examples of vampire numbers
include
1260 = 21 X 60
1395 = 15 x 93
1435 = 35 x 41
1530 = 30 x 51
1827 = 21 x 87
2187 = 27 x 81
6880 = 80 x 86
M = {x€K: \x\ < 1},
and the field R/M is called the residue field, class field,
or field of digits. For example, if if = Q (p-adic num-
bers), then R = Z p (p-adic integers), M = pZ p (p-adic
integers congruent to mod p), and R/M = GF(p), the
Finite Field of order p.
Valuation Theory
The study of VALUATIONS which simplifies class field
theory and the theory of algebraic function fields.
see also VALUATION
References
lyanaga, S. and Kawada, Y. (Eds.). "Valuations." §425
in Encyclopedic Dictionary of Mathematics. Cambridge,
MA: MIT Press, pp. 1350-1353, 1980.
Value
The quantity which a Function / takes upon applica-
tion to a given quantity.
see also Value (Game)
Value (Game)
The solution to a Game in Game THEORY. When a
Saddle Point is present
minmin aij = mm max a^ = v,
i<m j<n j<n i<m
and v is the value for pure strategies.
see also ABSOLUTE VALUE, GAME THEORY, MlNIMAX
Theorem, Valuation
(Sloane's A0 14575). There are seven 4-digit vampires,
155 6-digit vampires, and 3382 8-digit vampires. Gen-
eral formulas can be constructed for special classes of
vampires, such as the fangs
x = 25 • 10 fc + 1
y = 100(10 fc+1 +52)/25,
giving the vampire
v = X y = (10* +1 + 52)10 fc+2 + 100(10 fc+1 + 52)/25
= x*.10 k+2 +t
= 8(26 + 5- 10 fc )(l + 25-10 fc ),
where x* denotes x with the DIGITS reversed (Roushe
and Rogers).
Pickover (1995) also defines pseudovampire numbers, in
which the multiplicands have different number of digits.
References
Pickover, C. A. "Vampire Numbers." Ch. 30 in Keys to In-
finity. New York: W. H. Freeman, pp. 227-231, 1995.
Pickover, C. A. "Vampire Numbers." Theta 9, 11-13, Spring
1995.
Pickover, C. A. "Interview with a Number." Discover 16,
136, June 1995.
Roushe, F. W. and Rogers, D. G. "Tame Vampires." Un-
dated manuscript.
Sloane, N. J. A. Sequence A014575 in "An On-Line Version
of the Encyclopedia of Integer Sequences."
van der Grinten Projection
van der Grinten Projection
van Kampen's Theorem 1895
References
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 239-242, 1987.
van der Pol Equation
An Ordinary Differential Equation which can be
derived from the Rayleigh DIFFERENTIAL EQUATION
by differentiating and setting y = y '. It is an equation
describing self-sustaining oscillations in which energy is
fed into small oscillations and removed from large os-
cillations. This equation arises in the study of circuits
containing vacuum tubes and is given by
y
M (l-z/V+2/ = 0.
A Map Projection given by the transformation
x — sgn(A — A )
tv\A{G - P 2 ) - y/A*{G - F 2 ) 2 - (P 2 + A 2 )(G 2 - P 2 )\
P 2 + A 2
y = sgn(c£)
where
tt\PQ - AyJ{A 2 + \){P 2 + A 2 ) -Q 2
P 2 + A 2
A:
G
A-Ao
A-Ao
COS0
7T
sin + cos — 1
V sin 6 J
i
= sin
2<j>\
7T |
Q = A 2 + G.
The inverse FORMULAS are
<f> - sgn(y)7r -mi cos(0 x + |tt) - ~^-
0C3 J
A =
n\X 2 + Y 2 - 1 + v / i + 2(x 2 -r 2 ) + (x 2 + y 2 ) 2
2X
where
x=*
7C
Cl = -|y|(i + x 2 + y 2 )
c 2 = ci - 2F 2 + X 2
c 3 = -2ci + 1 + 2Y 2 + (X 2 + y 2 ) 2
C3
1
+
(2c 2 z _ 9cic 2 \
V C3 3 c 3 2 y
ai = — I ci
c 3 V 3c 3
27 \ c 3 3
c 2 2
mi = 2
fi
m
1 -1
3 COS
/ 3d
\aimi
)•
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
+ A ,
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
see also Rayleigh Differential Equation
References
Kreyszig, E. Advanced Engineering Mathematics, 6th ed.
New York: Wiley, pp. 165-166, 1988.
van der Waerden Number
The threshold numbers proven to exist by VAN DER
Waerden's Theorem. The first few are 1, 3, 9, 35,
178, ... (Sloane's A005346).
References
Goodman, J. E. and O'Rourke, J. (Eds.). Handbook of Dis-
crete & Computational Geometry. Boca Raton, FL: CRC
Press, p. 159, 1997.
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., p. 29, 1991.
Sloane, N. J. A. Sequence A005346/M2819 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
van der Waerden's Theorem
For any given POSITIVE INTEGERS k and r, there exists
a threshold number n{k,r) (known as a VAN DER WAER-
DEN Number) such that no matter how the numbers 1,
2, . . . , n are partitioned into k classes, at least one of
the classes contains an Arithmetic Progression of
length at least r. However, no Formula for n(k,r) is
known.
see also ARITHMETIC PROGRESSION
References
Honsberger, R. More Mathematical Morsels. Washington,
DC: Math. Assoc. Amer., p. 29, 1991.
Khinchin, A. Y. "Van der Waerden's Theorem on Arithmetic
Progressions." Ch. 1 in Three Pearls of Number Theory.
New York: Dover, pp. 11-17, 1998.
van der Waerden, B. L. "Beweis einer Baudetschen Vermu-
tung." Nieuw Arch. Wiskunde 15, 212-216, 1927.
van Kampen's Theorem
In the usual diagram of inclusion homeomorphisms, if
the upper two maps are injective, then so are the other
two.
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, pp. 74-75 and 369-373, 1976.
1896 van Wijngaarden-Deker-Brent Method
Vanishing Point
van Wijngaarden-Deker-Brent Method
see Brent's Method
Vandermonde Determinant
#1,.
• j X n ) —
1
1
Xi
X2
X! 2 -
X 2 2 •
1
x n
_ 2
Xn
_ n-1
=n
[Xi -
Xj)
(Sharpe 1987). For INTEGERS a x , . . . , a n , A(ai, .
is divisible by 117=1 (* ~ X ) ! (Chapman 1996).
see a/so Vandermonde Matrix
■ ,a n )
References
Chapman, R. "A Polynomial Taking Integer Values." Math.
Mag. 69, 121, 1996.
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed, San Diego, CA: Academic
Press, p. 1111, 1979.
Sharpe, D. §2.9 in Rings and Factorization. Cambridge, Eng-
land: Cambridge University Press, 1987.
Vandermonde Identity
see Chu- Vandermonde Identity
Vandermonde Matrix
A type of matrix which arises in the LEAST SQUARES
Fitting of Polynomials and the reconstruction of a
Distribution from the distribution's Moments. The
solution of an n x n Vandermonde matrix equation re-
quires 0(n 2 ) operations. A Vandermonde matrix of or-
der n is of the form
Xi
x 2
Xi
x, 2
Xl
X2
n-1
1
see also Toeplitz Matrix, Tridiagonal Matrix,
Vandermonde Determinant
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Vandermonde Matrices and Toeplitz Matri-
ces." §2.8 in Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 82-89, 1992.
Vandermonde's Sum
see Chu- Vandermonde Identity
Vandermonde Theorem
A special case of GAUSS'S THEOREM with a a NEGATIVE
Integer — n:
2 F 1 (-n i b]c;l)
(c - b) n
(c)„
where 2 Fi(a, b\c\z) is a HYPERGEOMETRIC FUNCTION
and (a) n is a POCHHAMMER SYMBOL (Bailey 1935, p. 3).
see also Gauss's Theorem
References
Bailey, W. N. Generalised Hypergeometric Series. Cam-
bridge, England: Cambridge University Press, 1935.
Vandiver's Criteria
Let p be a Irregular Prime, and let P = rp + 1 be a
PRIME with P < p 2 —p. Also let t be an INTEGER such
that t 3 ^ 1 (mod P). For an IRREGULAR PAIR (p,2&),
form the product
where
Q«=r"" 2 IJ(t rt -i) 6P -
6=1
m=i(pl-l)
m
If Qik ^ 1 (mod P) for all such IRREGULAR PAIRS,
then Fermat's Last Theorem holds for exponent p.
see also Fermat's Last Theorem, Irregular Pair,
Irregular Prime
References
Johnson, W. "Irregular Primes and Cyclotomic Invariants."
Math. Comput. 29, 113-120, 1975.
Vanishing Point
vanishing points
71 / ^
one-point
perspective
The point or points to which the extensions of PARALLEL
lines appear to converge in a PERSPECTIVE drawing.
see also PERSPECTIVE, PROJECTIVE GEOMETRY
References
Dixon, R. "Perspective Drawings." Ch. 3 in Mathographics.
New York: Dover, pp. 79-88, 1991.
Varga's Constant
Varga's Constant
V= - =9.2890254919...,
where A is the ONE-NINTH CONSTANT.
see also One-Ninth Constant
Variance
For N samples of a variate having a distribution with
known MEAN //, the "population variance" (usually
called "variance" for short, although the word "popu-
lation" should be added when needed to distinguish it
from the SAMPLE Variance) is defined by
var (z) = jj 5^0» - M) 2 = (x 2 ~ ^x + /x 2 )
= (^ 2 }-{2 M x) + (m 2 )
= {x 2 )-2ti{x)+n\ (1)
where
(2)
But since (x) is an UNBIASED ESTIMATOR for the Mean
Ai = (x) , (3)
it follows that the variance
a = var(x) = ( x ) — \i . (4)
The population STANDARD DEVIATION is then defined
as
a = y / var(x) = \/(x 2 ) - y? . (5)
A useful identity involving the variance is
var(/(x) + g(x)) = var (/(a)) + van(g(x)). (6)
Therefore,
var (ax + b) = ([(ax + b) — {ax + 6)] )
= ((ax + b- a(x) - b) 2 )
= ((ax — afi) 2 ^ = {^a 2 (x — fi) 2 ^
= a 2 ((x - /i) 2 ) = a 2 var(z) (7)
var(fe) = 0. (8)
If the population MEAN is not known, using the sample
mean x instead of the population mean \i to compute
(9)
gives a Biased Estimator of the population variance.
In such cases, it is appropriate to use a Student's t~
Distribution instead of a Gaussian Distribution.
Variance 1897
However, it turns out (as discussed below) that an Un-
biased Estimator for the population variance is given
by
N
s 2 = <r£ = -jy— j- 5Z(a* - xf. (10)
The Mean and Variance of the sample standard de-
viation for a distribution with population mean \i and
Variance are
N
(11)
a '»>* = n^rK* " ^ " ( N ~ 3) ^ 2 1- (12)
The quantity Ns N 2 /a 2 has a Chi-Squared Distribu-
tion.
For multiple variables, the variance is given using the
definition of COVARIANCE,
n m
var
x<\
\i=l / \ i=l 3 = 1
n m
> xt ] = cov I > Xi, > ;
\ i=l j = l
n m
y]y^cov(x i ,a; J )
i=l j = l
n m
— /. y, co v(xi,Xj) + 2_. / jCOv(xj,%j)
1=1 3=1 i=l 3 = 1
j = i j^i
n n m
= >^COv(Xi,Xj) -f 2. / ^COv(Xj,Xj)
i=l i-1 j=l
3*i
n n m
= \ var(cci) + 2 \ \ cov(xi, Xj).
i=l j = l
n m n ro
i=l j=i+l
(13)
A linear sum has a similar form:
n rn
var [ > a^i I — cov I > ai^i, N flj^j I
V i=l / \i=l j = l /
n m
= 2. /. a i a j cov(z;, Xj)
n n m
= 2, a i var(xi) + 2 N^ N^ a^a-,- cov(£i,£j). (14)
i=l 1 = 1 j— i+1
These equations can be expressed using the Co VARI-
ANCE Matrix.
To estimate the population VARIANCE from a sample
of iV elements with a priori unknown MEAN (i.e., the
Mean is estimated from the sample itself), we need an
1898 Variance
Variance
Unbiased Estimator for a. This is given by the k-
STATISTIC k 2 , where
i. N
(15)
and m 2 = s 2 is the Sample Variance
t=i
Note that some authors prefer the definition
i=l
since this makes the sample variance an Unbiased Es-
timator for the population variance.
When computing numerically, the MEAN must be com-
puted before s 2 can be determined. This requires stor-
ing the set of sample values. It is possible to calculate
s' 2 using a recursion relationship involving only the last
sample as follows. Here, use fij to denote \i calculated
from the first j samples (not the jth Moment)
fij
£U*«
(18)
and s 2 denotes the value for the sample variance s' 2
calculated from the first j samples. The first few values
calculated for the Mean are
Mi = xi
1 ■ Ml + #2
M2
M3
2/X2 + xz
Therefore, for j = 2, 3 it is true that
_ (j - l)fij-l+Xj
to~ J •
Therefore, by induction,
[U + 1) - 1]M(j+D-i + x j+i
(19)
(20)
(21)
(22)
Mi+i =
J + l
__ 3to + gj+i
J + l
(jL j+1 (j + 1) = 0" + i)m; + (iEj+i - to) ( 24 )
#i+i ~ Mi
Mi+i = Mi +
and
i-i
(23)
(24)
(25)
(26)
for j > 2, so
x2 i + i
J*.
»=i V x * - Mi+i) v^, \2
i=l
i + 1
i+i i+i
= 5^(«* - Mi) 2 + X^' ~ Mj+i ^ 2
t=i *=i
i+i
+ 2 ^(a* - to)(to ~ Mi+0- ( 27 )
Working on the first term,
3 + 1
y^( x j ~ to) 2 = X^ x ' " to) 2 + (sj+i - to) 2
0'-l)^ 2 + (^+i-Mi) 2 - (28)
Use (24) to write
hi - to = (i + !)(Mi+i " to)> ( 29 )
i+i
$]( x * " W) 2 = 0" - !) 5 i 2 + W + ^(w+i ~ to) 2 - (30)
Now work on the second term in (27),
j+i
y^ito - to+i) 2 = U + x )(to ~ Mi+i) 2 - (31)
i=l
Considering the third term in (27),
i+i i+i
^2(xi - to)(to ~ Mi+i) = (to ~ to+i) z2( x i ~ Mi)
= (to -Mi+0
y^pE* -Mi) + (^i+i -Mi)
= (Mi - Mi+0 x J+i ~ to ~ 3to + $^ x » J ' ^ 32 ^
But
so
i+1
^2 x i =Jto,
i~i
(33)
y^(Mi - Mi+o(^i+i - to)
i+1
~ X^ Mj ~ ^+1)0 + !)(Mi+i - Mi)
2 = 1
= -(j + l)(Mi-Mi+i) a - (34)
Variance
Plugging (30), (31), and (34) into (27),
J s i+i 2 = [U ~ !)*/ + U + !) 2 (w+i - Mi) 2 ]
+ IU + 1)(W " Mi+i) + 2[-0' + l)0*i " Mi+i)]
= (j - i)sj 2 + 0' + !) 2 (mj+i - to) 2
-U + 1 )(Pi~ to+if
= (i - l)«i a + 0' + 1)[0' + 1) - l](Mi+i - W) 2
0' - 1)*/ + J'O' + 1)(Mj+i - Mi)
(35)
s j+ S =[l--)s/ + U + l)(/ij+i - Mi) • (36)
To find the variance of s 2 itself, remember that
2\2
and
var(s 2 ) = ^s 4 ) — (s 2 )
/ 2\ jV ~ 1
(37)
(38)
Now find (s 4 ).
( S 4 ) = (( S 2 ) 2 ) = ((( 2; 2 )-( a; ) 2 ) 2 )
^((E«.) a H(E-'(E-)
+ ^((E*) 4 )- < 39 >
Working on the first term of (39),
((!> 2 ) 2 ) = (2> 4+ 5> v )
= (E* 4 ) + (E*v)
= AT (xi 4 ) + iV(7V - 1) ( Xi 2 ) (xj 2 )
N^ 4 + N(N - 1) M 2 .
(40)
The second term of (39) is known from /c-STATlSTICS,
(E xi2 (E^) 2 ) = N ^ + N( > N - 1 ^ 2 ' ( 41 )
as is the third term,
^^^^iV^^+SiV^-l)^^/)
AT M ; + 3 iV(7V-l)M2 •
(42)
Variance 1899
Combining (39)-(42) gives
(s 4 ) = ^[N^ + N(N-l)»' 2 2 }
- A [iVA ii + JV(Ar-i)/ii 2 ]
+ ^[^4 + 3iV(7V~l)^ 2 ]
" Vat iv^ + iW 4
^ JV-1 _ 2(JV-1) 3(JV-1)
jV AT 2 + iV 3
/*2
AT 2 - 2N + 1
TV 3
M4
(N - 1){N 2 -2N + 3) l2
+ Jp ^ 2
(TV _ i)[(jy _ i)^ + (at 2 - 2 JV + 3)/4 2 ]
TV 3
(43)
so plugging in (38) and (43) gives
2\3
var(s 2 ) = (s 4 ) — (s 2 ^
(N - 1)[(N - l)/xi + (AT 2 -2Ar + 3) M2 2 ]
TV 3
(N-1) 2 N f2
N 3
V<2
N-l
N 3
{(A^-l)^ + [(iV 2 -2iV + 3)
-7V(iV-l)] M2 2 }
( i V_l )[ ( A r-i)^-(Ar-3)// 2 2 ]
N 3
(44)
Student calculated the Skewness and KURTOSIS of the
distribution of s 2 as
7i
72
8
N-l
12
JV-1
(45)
(46)
and conjectured that the true distribution is PEARSON
Type III Distribution
f(s 2 ) = C(s 2 ) (N - 3)/2 e- Ns2/2a2
where
2
Ns 2
N-l
N x(JV-l)/2
(&Y
r(^) •
This was proven by R. A. Fisher.
(47)
(48)
(49)
1900 Variate
The distribution of s itself is given by
f{s) = 2 ^/., ,, e-"" 3 *'*"- 2
{*)■
where
Tr(f)
6(iV) =
jvr(^i)'
The Moments are given by
_/2W'E(z£)
^"UJ r(V) '
(50)
(51)
(52)
(53)
and the variance is
var(s) = v% — v\
N -1
* _ ^ " V - [6(iVH 2
1_
N
N
2r 2 (f )
(54)
An Unbiased Estimator of a is s/b(N). Romanovsky
showed that
b(N) = 1 -
139
47V 32AT2 518497V 3
+ ..
(55)
see a/so Correlation (Statistical), Covariance,
Covariance Matrix, ^-Statistic, Mean, Sample
Variance
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Moments of a Distribution: Mean, Vari-
ance, Skewness, and So Forth." §14.1 in Numerical Recipes
in FORTRAN: The Art of Scientific Computing, 2nd
ed. Cambridge, England: Cambridge University Press,
pp. 604-609, 1992.
Variate
A Random Variable in statistics.
Variation
The A-variation is a variation in which the varied path
over which an integral is evaluated may end at different
times than the correct path, and there may be variation
in the coordinates at the endpoints.
The 5-variation is a variation in which the varied path
in configuration space terminates at the endpoints rep-
resenting the system configuration at the same time t\
and £2 as the correct path; i.e., the varied path always
returns to the same endpoints in configuration space, so
<M*i) = <M*2) = o-
see also Calculus of Variations, Variation of Ar-
gument, Variation of Parameters
Variation Coefficient
Variation of Argument
Let [argf(z)] denote the change in argument of a func-
tion f(z) around a closed loop 7. Also let N denote the
number of ROOTS of f(z) in 7 and P denote the number
of Poles of f(z) in 7. Then
[axg f(z)] = —(N-P).
(1)
To find [arg f(z)] in a given region i?, break R into paths
and find [arg/(z)] for each path. On a circular ARC
z = Re ie ,
(2)
let f(z) be a Polynomial P(z) of degree n. Then
arg
('^)1
[argP(z)] =
= [arg z n ] +
Plugging in z = Re %e gives
[argP(z)] = [argPe ien ] +
P(Re ie )
arg
mi
arg
P{Re ie )
lim n .„
h-voo Re t0n
P{Re ie )
Re i9n
Re i6n
[constant] ,
= 0,
and
[aigP(z)] = [arge^] = n(9 2 - 0i).
For a Real segment z = x,
/(*)
[arg f(x)] = tan
For an IMAGINARY segment z — iy,
[arg /(it/)]
(3)
(4)
(5)
(6)
(7)
(8)
J. -i2£M\' 2
l tan wmj ex -
(9)
Note that the ARGUMENT must change continuously, so
"jumps" occur across inverse tangent asymptotes.
Variation Coefficient
If s x is the Standard Deviation of a set of samples xi
and x its MEAN, then
V= — -
x
Variation of Parameters
Variation of Parameters
For a second-order ORDINARY DIFFERENTIAL EQUA-
TION,
y" + p(x)y + q(x)y = g(x). (1)
Assume that linearly independent solutions yi(x) and
2/2(2) are known. Find v\ and v 2 such that
y*(x) = vi(x)yi(x) + V2(x)y 2 (x) (2)
y*'(x) = (ui + ^2) + (uiyl + V23/2)- (3)
Now, impose the additional condition that
v[yi +v' 2 y2 = (4)
so that
y*'(x) = (uiyi +V22/2) (5)
y [ x ) = U12/1 + ^22/2 + uiyi + ^y 2 - (6)
Plug y*, y*', and y*" back into the original equation to
obtain
vi(yi+pyi+qyi)+v 2 (y2 -\-py 2 +qy2)+v , iy[-\-v 2 y2 = g(x)
(7)
^iyi + V22/2 = 0(s). (8)
Therefore,
v[yi + v' 2 y 2 =
viyi +v 2 y 2 =g(x).
(9)
(10)
Generalizing to an nth degree ODE, let t/i , ... , y n be
the solutions to the homogeneous ODE and let v[(x) y
. . . , v' n (x) be chosen such that
' yivl + 2/2^2 + ■>- + ynv' n =
yivi + yW2 + ... + y'nVn = o
(11)
(n — 1) / , (n — 1) / . . , ,
2/1 ^1+1/2 «2+---+2/»
( "- 1) «;= 5 (x).
Then the particular solution is
y*(x) = vi(x)yi(x) + . . . + u n (a;)y„(a;). (12)
Variety
see Algebraic Variety
Vassiliev Polynomial 1901
Varignon Parallelogram
The figure formed when the BlMEDIANS (MIDPOINTS
of the sides) of a convex QUADRILATERAL are joined.
Varignon's Theorem demonstrated that this figure is
a Parallelogram. The center of the Varignon paral-
lelogram is the Centroid if four point masses are placed
on the Vertices of the Quadrilateral.
see also Midpoint, Parallelogram, Quadrilat-
eral, Varignon's Theorem
Varignon's Theorem
The figure formed when the BlMEDIANS (MIDPOINTS of
the sides) of a convex QUADRILATERAL are joined in
order is a PARALLELOGRAM. Equivalently, the BlME-
DIANS bisect each other. The Area of this Varignon
Parallelogram is half that of the Quadrilateral.
The Perimeter is equal to the sum of the diagonals of
the original QUADRILATERAL.
see also BlMEDIAN, MIDPOINT, QUADRILATERAL,
Varignon Parallelogram
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited.
Washington, DC: Math. Assoc. Amer., pp. 51-56, 1967.
Vassiliev Polynomial
Vassiliev (1990) introduced a radically new way of look-
ing at KNOTS by considering a multidimensional space
in which each point represents a possible 3-D knot con-
figuration. If two Knots are equivalent, a path then
exists in this space from one to the other. The paths
can be associated with polynomial invariants.
Birman and Lin (1993) subsequently found a way to
translate this scheme into a set of rules and list of po-
tential starting points, which makes analysis of Vassiliev
polynomials much simpler. Bar-Natan (1995) and Bir-
man and Lin (1993) proved that JONES POLYNOMIALS
and several related expressions are directly connected
(Peterson 1992). In fact, substituting the POWER se-
ries for e x as the variable in the JONES POLYNOMIAL
yields a Power Series whose Coefficients are Vas-
siliev polynomials (Birman and Lin 1993). Bar-Natan
(1995) also discovered a link with Feynman diagrams
(Peterson 1992).
References
Bar-Natan, D. "On the Vassiliev Knot Invariants." Topology
34, 423-472, 1995.
Birman, J, S. "New Points of View in Knot Theory." Bull.
Amer. Math. Soc. 28, 253-287, 1993.
1902
Vault
Birman, J. S. and Lin, X.-S. "Knot Polynomials and Vas-
siliev's Invariants." Invent. Math. Ill, 225-270, 1993.
Peterson, I. "Knotty Views: Tying Together Different Ways
of Looking at Knots." Sci. News 141, 186-187, 1992.
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.
Stoimenow, A. "Degree-3 Vassiliev Invariants." http://www.
informatik.hu-berlin.de/-stoimeno/vas3.html.
Vassiliev, V. A. "Cohomology of Knot Spaces." In Theory
of Singularities and Its Applications (Ed. V. I. Arnold).
Providence, RI: Amer. Math. Soc, pp. 23-69, 1990.
Vassiliev, V. A. Complements of Discriminants of Smooth
Maps: Topology and Applications. Providence, RI: Amer.
Math. Soc, 1992.
Vault
Let a vault consist of two equal half- CYLINDERS of
length and diameter 2a which intersect at RIGHT
ANGLES so that the lines of their intersections (the
"groins") terminate in the Vertices of a Square.
Then the SURFACE AREA of the vault is given by
A = 4(7r-2)a 2 .
see also DOME
References
Lines, L. Solid Geometry. New York: Dover, pp. 112-113,
1965.
Vector
A vector is a set of numbers ^4o, • • • , A n that transform
as
(i)
J±i — a% j J±j .
This makes a vector a TENSOR of Rank 1. Vectors
are invariant under TRANSLATION, and they reverse sign
upon inversion.
A vector is uniquely specified by giving its DIVERGENCE
and CURL within a region and its normal component
over the boundary, a result known as HELMHOLTZ'S
Theorem (Arfken 1985, p. 79). A vector from a point
A to a point B is denoted A§, and a vector v may be
denoted v, or more commonly, v.
A vector with unit length is called a Unit Vector and
is denoted with a Hat. An arbitrary vector may be
converted to a Unit Vector by dividing by its Norm,
i.e.,
♦-H- < 2 >
Let n be the Unit Vector defined by
cos 9 sin (f)'
sin sin <f>
coscb
(3)
Vector Bundle
Then the vectors n, a, b, c, d satisfy the identities
'"-> = / /
t/0 Jo
= [sinflr /
(cos sin <f>) sin <j> dO d(j)
sm<pd(j) = (4)
(m) = (5)
{mrij) = \6ij (6)
{nin k n k ) = (7)
{niTlkninm) = j^(SikSlm + SuSkm + SimSkl) (8)
((a-n) 2 ) = |a 2 (9)
<(a.n)(b.fi)) = ia.b (10)
<(a-n)n> = |o (11)
((a x n) 2 ) = fa 2 (12)
((axn)-(b xn)) = fab, (13)
and
((a-n)(b-n)(c-n)(d-n)>
= i[(6/a-b)(6/cd) + (6/a-c)(6/6-d) + (6/a-d)(6/6-c)].
(14)
where <5»j is the Kronecker Delta, a - b is a Dot
Product, and Einstein Summation has been used.
see also Four- Vector, Helmholtz's Theorem,
Norm, Pseudovector, Scalar, Tensor, Unit Vec-
tor, Vector Field
References
Arfken, G. "Vector Analysis." Ch. 1 in Mathematical Meth-
ods for Physicists, 3rd ed. Orlando, FL: Academic Press,
pp. 1-84, 1985.
Aris, R. Vectors, Tensors, and the Basic Equations of Fluid
Mechanics. New York: Dover, 1989.
Crowe, M. J. A History of Vector Analysis: The Evolution
of the Idea of a Vectorial System. New York: Dover, 1985.
Gibbs, J. W. and Wilson, E. B. Vector Analysis: A Text-
Book for the Use of Students of Mathematics and Physics,
Founded Upon the Lectures of J. Willard Gibbs. New York:
Dover, 1960.
Marsden, J. E. and Tromba, A. J. Vector Calculus, J^th ed.
New York: W. H. Freeman, 1996.
Morse, P. M. and Feshbach, H. "Vector and Tensor Formal-
ism." §1.5 in Methods of Theoretical Physics, Part I. New
York: McGraw-Hill, pp. 44-54, 1953.
Schey, H. M. Div, Grad, Curl, and All That: An Informal
Text on Vector Calculus. New York: Norton, 1973.
Schwartz, M.; Green, S.; and Rutledge, W. A. Vector Analy-
sis with Applications to Geometry and Physics. New York:
Harper Brothers, 1960.
Spiegel, M. R. Theory and Problems of Vector Analysis. New
York: Schaum, 1959.
Vector Bundle
A special class of Fiber Bundle in which the Fiber
is a Vector Space. Technically, a little more is re-
quired; namely, if / : E -> B is a BUNDLE with FIBER
M n , to be a vector bundle, all of the Fibers f~ 1 (x) for
Vector Derivative
Vector Derivative 1903
x e B need to have a coherent Vector Space struc-
ture. One way to say this is that the "trivializations"
h : / _1 (C7) 4[/xl n , are FlBER-for-FlBER VECTOR
Space Isomorphisms.
see also Bundle, Fiber, Fiber Bundle, Lie Alge-
broid, Stable Equivalence, Tangent Map, Vec-
tor Space, Whitney Sum
Vector Derivative
The basic types of derivatives operating on a VECTOR
Field are the Curl Vx, Divergence V-, and Gradi-
ent V.
Vector derivative identities involving the CURL include
V x (fcA) = lfcVxA (1)
V x (/A) = /(V x A) + (V/) x A (2)
V x (A x B) = (B ■ V)A - (A • V)B
+A(V-B)-B(V- A) (3)
'A^ = /(VxA)Ux(V/)
V x
Vx(A + B) = VxA + VxB.
In Spherical Coordinates,
V x r =
Vxf =
V x [rf(r)\ = /(r)(V x r) + [V/(r)] x r
(4)
(5)
= /W(0) + |rxr:
+ = 0.
(6)
(?)
(8)
Vector derivative identities involving the DIVERGENCE
include
V * (fcA) = A:V ■ A
V-(/A) = /(V-A) + (V/).A
V • (A x B) ^ B ■ (V x A) - A ■ (V x B)
/(V.A)-(V/)-A
'•(*)
V-(A + B) = V-A + V-B
V(uv) = uV v 4- (Vu) • v. -
In Spherical Coordinates,
V-r:
V'f :
3
2
r
V • [r/(r)] = ^[x/(r)] + ^[vf(r)] + ^W(r)]
|wwi-*g + '"!fs+'
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
9r 8 ( 2 , 2 , 2sl/2 / 2 . 2 . 2\-l/2 X
_ = _ (x +y +Z )/ =x{x + y +z) =-
(19)
£[*/mi = t! + '- (20)
By symmetry,
V-[r/(r)] = 3/(r) + J(x 2 +y 2 + 2 2 )|=3/(r) + r|
V • (f/(r)) = ?/(r) + f
n-l
(21)
(22)
V ■ (fr n ) = 3r n - x + (n - l)r n_1 = (n + 2)r n '\ (23)
Vector derivative identities involving the GRADIENT in-
clude
V(*/) = kVf (24)
V(fg) = fVg + gVf (25)
V(A • B) = A x (V x B) + B x (V x A)
+(A- V)B + (B- V)A (26)
V(A • V/) = A x (V x V/) + V/ x (V x A)
+A-V(V/) + V/-VA
= V/ x (V x A) + A ■ V(V/) + V/ ■ VA (27)
7\ = gv/-/vg (28)
V(/ + y)-V/ + V 5 (29)
V(A • A) = 2A x (V x A) + 2(A • V)A (30)
(A • V)A = V(|A 2 ) - A x (V x A). (31)
Vector second derivative identities include
V 2 t = V ■ (Vt)
V 2 A = V(V - A) - V x (V x A).
d 2 t d 2 t d 2 t
dtf + dy 2 + dz 2
(32)
(33)
This very important second derivative is known as the
Laplacian.
V x (Vt) = (34)
V(V ■ A) = V 2 A + V x (V x A) (35)
V - (V x A) = (36)
V x (V x A) = V(V • A) - V 2 A
V x (V 2 A) = V x [V(V • A)] - V x [V x (V x A)]
= -V x [V x (V x A)]
= -{V[V • (V x A)] - V 2 (V x A)]}
- V 2 (V x A) (37)
V 2 (V-A) = V*[V(V- A)]
= V ■ [V 2 A + V x (V x A)] = V - (V 2 A) (38)
V 2 [V x (V x A)] = V 2 [V(V • A) - V 2 A]
= V 2 [V(V-A)]-V 4 A (39)
V x [V 2 (V x A)] = V 2 [V(V • A)] - V 4 A (40)
V 4 A = -V 2 [V x (V x A)] + V 2 [V(V • A)]
= V x [V 2 (V x A)] - V 2 [V x (V x A)]. (41)
1904
Vector Direct Product
Vector Norm
Combination identities include
A x (VA) = ±V(A • A) - (A ■ V)A (42)
V x (<£V0) = 0V x (V0) + (V0) x (V<£) = (43)
( A .V)r = A - f(A -^ (44)
r
V/-A = V-(/A)-/(V-A) (45)
/(V • A) = V • (/A) - AV/, (46)
where (45) and (46) follow from divergence rule (2).
see also Curl, Divergence, Gradient, Laplacian,
Vector Integral, Vector Quadruple Product,
Vector Triple Product
References
Gradshteyn, I. S. and Ryzhik, I. M. "Vector Field Theorem."
Ch. 10 in Tables of Integrals, Series, and Products, 5th ed.
San Diego, CA: Academic Press, pp. 1081-1092, 1980.
Morse, P. M. and Feshbach, H. "Table of Useful Vector and
Dyadic Equations." Methods of Theoretical Physics, Part
I. New York: McGraw-Hill, pp. 50-54 and 114-115, 1953.
Vector Direct Product
Given VECTORS u and v, the vector direct product is
Vector Function
A function of one or more variables whose Range is
3-dimensional, as compared to a SCALAR FUNCTION,
whose Range is 1-dimensional.
see also Complex Function, Real Function,
Scalar Function
Vector Harmonic
see Vector Spherical Harmonic
Vector Integral
The following vector integrals are related to the CURL
Theorem. If
F = cxP0r,t/,2), (1)
then
If
then
/ ds x P = / (da. x V) x P.
Jc J s
F = cF,
//*-/.
da x VF.
(2)
(3)
(4)
r T n
T
U2V
=
T
- U 3V
where <g> is the Matrix Direct Product and v T is
the matrix TRANSPOSE. For 3x3 vectors
UiVi U±V2 UiVz
U2V1 U2V2 U2V3
U3V1 U3V2 U3V3
Note that if u = x*, then Uj = 5ij, where 8%j is the
Kronecker Delta.
see also MATRIX DIRECT PRODUCT, SHERMAN-
Morrison Formula, Woodbury Formula
Vector Division
There is no unique solution A to the MATRIX equation
y = Ax unless x is PARALLEL to y, in which case A is a
SCALAR. Therefore, vector division is not denned.
see also Matrix, SCALAR
The following are related to the Divergence Theo-
rem. If
F = c xP(z,y,z),
(5)
then
/ V xFdV = / daxF.
Jv J s
(6)
Finally, if
F = cF,
(7)
then
/> /*
/ VFdV = / Fdsi.
Jv J s
(8)
see also CURL THEOREM, DIVERGENCE THEOREM,
Gradient Theorem, Green's First Identity,
Green's Second Identity, Line Integral, Surface
Integral, Vector Derivative, Volume Integral
Vector Field
A MAP f : W 1 i-» W 1 which assigns each x a VECTOR
Function f(x). Flows are generated by vector fields
and vice versa. A vector field is a Section of its Tan-
gent Bundle.
see also Flow, Scalar Field, Seifert Conjecture,
Tangent Bundle, Vector, Wilson Plug
References
Gray, A. "Vector Fields IR 71 " and "Derivatives of Vector
Fields IR n ." §9.4-9.5 in Modern Differential Geometry
of Curves and Surfaces. Boca Raton, FL: CRC Press,
pp. 171-174 and 175-178, 1993.
Morse, P. M. and Feshbach, H. "Vector Fields." §1,2 in Meth-
ods of Theoretical Physics, Part I. New York: McGraw-
Hill, pp. 8-21, 1953.
Vector Norm
Given an n-D Vector
Xi
X 2
a vector norm ||x|| (sometimes written simply |x|) is a
NONNEGATIVE number satisfying
1. ||x|| > when x ^ and ||x|| = IFF x = 0,
2. ||fcx|| = |fc| ||x|| for any SCALAR fc,
3. ||x + y||<||x|| + ||y||.
Vector Ordering
see also Compatible, Matrix Norm, Natural
Norm, Norm
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1114, 1980,
Vector Ordering
If the first NONZERO component of the vector difference
A - B is > 0, then A >- B. If the first Nonzero
component of A — B is < 0, then A -< B.
see also PRECEDES, SUCCEEDS
Vector Potential
A function A such that
B = V x A.
The most common use of a vector potential is the rep-
resentation of a magnetic field. If a Vector Field has
zero DIVERGENCE, it may be represented by a vector
potential.
see also Divergence, Helmholtz'S Theorem, Po-
tential Function, Solenoidal Field, Vector
Field
Vector Quadruple Product
(A x B) • (C x D) - (A ■ C)(B • D) - (A • D)(B - C)
(1)
(A x B) 2 = (A x B) ■ (A x B)
= (A.A)(B.B)-(A-B)(B.A)
= A 2 B 2 -(A-B) 2 (2)
Ax(Bx(CxD)) = B(A • (C x D)) - (A . B)(C x D)
(3)
(AxB) x (C xD) = [A,B,D]C-[A,B,C]D
= (C x D) x (B x A) = [C, D, A]D - [C, D, B]A, (4)
where [A,B,D] denotes the Vector Triple Prod-
uct. Equation (1) is known as Lagrange's Identity.
see also Lagrange's Identity, Vector Triple
Product
Vector Space
A vector space over R n is a set of VECTORS for which
any VECTORS X, Y, and Z G W 1 . and any SCALARS r,
s G M have the following properties:
1. COMMUTATIVITY:
X + Y = Y + X.
2. Associativity of vector addition:
(X + Y) + Z = X + (Y + Z).
Vector Spherical Harmonic 1905
3. Additive identity: For all X,
o + x = x + o = x.
4. Existence of additive inverse: For any X, there exists
a — X such that
X+(-X) = 0.
5. Associativity of scalar multiplication:
r(sX) - (rs)X.
6. Distributivity of scalar sums:
(r + s)X - rX + sX.
7. Distributivity of vector sums:
r(X + Y) =rX + rY.
8. Scalar multiplication identity:
IX = X.
An n-D vector space of characteristic two has
S(k,n) = (2 n - 2°)(2 n - 2 1 ) • • ■ (2 n - 2 k ~ 1 )
distinct SUBSPACES of DIMENSION k.
A Module is abstractly similar to a vector space, but
it uses a RING to define COEFFICIENTS instead of the
Field used for vector spaces. MODULES have Coeffi-
cients in much more general algebraic objects.
see also Banach Space, Field, Function Space,
Hilbert Space, Inner Product Space, Module,
Ring, Topological Vector Space
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 530-534, 1985.
Vector Spherical Harmonic
The Spherical Harmonics can be generalized to vec-
tor spherical harmonics by looking for a SCALAR FUNC-
TION ip and a constant VECTOR c such that
MeVx (e0) = ip{V x c) + (Vip) x c
= (Vip) x c = -c x V,i> (1)
V ■ M = 0.
Now use the vector identities
V 2 M = V 2 (V x M) = V x (V 2 M)
= V x (V 2 c0) = Vx (cVV)
k 2 M = k 2 V x (c0) = Vx (cVV),
(2)
(3)
(4)
1906 Vector Spherical Harmonic
Vector Transformation Law
so
V 2 M + k 2 M = V x [c(V 2 V> + A; 2 ^)],
(5)
and M satisfies the vector HELMHOLTZ DIFFERENTIAL
Equation if ip satisfies the scalar Helmholtz Differ-
ential Equation
V 2 V + A;V = 0.
Construct another vector function
V xM
N:
(6)
(7)
which also satisfies the vector Helmholtz Differen-
tial Equation since
V 2 N = r V 2 (V x M) = -V x (V 2 M)
k k
= ivx(-fc 2 M) = 4VxM=:-fc 2 N, (8)
k
which gives
We have the additional identity
V 2 N + fc 2 N = 0.
(9)
V x N = I V x (V x M) = iv(V • M)
k k
= I V 2 M - yV 2 M = ^^ = jfeM. (10)
k k k
In this formalism, ip is called the generating function and
c is called the Pilot Vector. The choice of generating
function is determined by the symmetry of the scalar
equation, i.e., it is chosen to solve the desired scalar
differential equation. If M is taken as
A number of conventions are in use. Hill (1954) defines
V, m = -
J + l
Y t m r +
dYj"
e
2/ + 1 V / (/ + l)(2/ + l) d$
+ iMy/(l + l)(2l + 1) sin 0Y, m (15)
wr
Xm
I
-Y l m T +
dY"
2Z + 1 vWTi) oe
iM
y/l(2l + l)sm$
M
Yr<t>
(16)
y/l(l + I)sin9
vre- Jl^ ^I^
\/J(J + l)
d9
(17)
Morse and Feshbach (1953) define vector harmonics
called B, C, and P using rather complicated expres-
References
Arfken, G. "Vector Spherical Harmonics." §12.11 in Mathe-
matical Methods for Physicists, 3rd ed. Orlando, FL: Aca-
demic Press, pp. 707-711, 1985.
Blatt, J. M. and Weisskopf, V. "Vector Spherical Harmonics."
Appendix B, §1 in Theoretical Nuclear Physics. New York:
Wiley, pp. 796-799, 1952.
Bohren, C F. and Huffman, D. R. Absorption and Scattering
of Light by Small Particles. New York: Wiley, 1983.
Hill, E. H. "The Theory of Vector Spherical Harmonics."
Amer. J. Phys. 22, 211-214, 1954.
Jackson, J. D. Classical Electrodynamics, 2nd ed. New York:
Wiley, pp. 744-755, 1975.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part II. New York: McGraw-Hill, pp. 1898-1901, 1953.
Vector Transformation Law
The set of n quantities Vj are components of an n-D
Vector v Iff, under Rotation,
M^Vx (n/>),
(11)
where r is the radius vector, then M is a solution to
the vector wave equation in spherical coordinates. If we
want vector solutions which are tangential to the radius
vector,
M ■ r = r ■ (Vi/> x c) = (V^)(c x r) = 0, (12)
c x r =
and we may take
(13)
(14)
(Arfken 1985, pp. 707-711; Bohren and Huffman 1983,
p. 88).
v i = a ij v j
for i = 1, 2, . . . , n. The DIRECTION COSINES between
x'i and Xj are
_ dx'i _ dxj
13 ~ dxj dx\ '
They satisfy the orthogonality condition
dxj dx'i dxj s
aijaik = d^d^ = d^ = Sjk '
where 5jk is the Kronecker Delta.
see also TENSOR, VECTOR
Vector Triple Product
Veronese Surface
1907
Vector Triple Product
The triple product can be written in terms of the LEVI-
Civita Symbol e ijk as
Venn Diagram
A-(B x C) = e ijk A i B j C k .
The BAC-CAB Rule can be written in the form
(1)
A x (B x C) = B(A • C) - C(A ■ B) (2)
(A x B) x C = -C x (A x B)
= -A(B-C) + B(A-C). (3)
Addition identities are
A • (B x C) = B • (C x A) = C ■ (A x B)
[A, B, C]D = [D, B, C] A + [A, D, C]B + [A, B, D]C
[q,q,q ][r,r ,r ]
q ■ r q • r
q*r
q • r q • r q • r
q • r q • r q * r
(4)
(5)
(6)
see also BAC-CAB Rule, Cross Product, Dot
Product, Levi-Civita Symbol, Scalar Triple
Product, Vector Quadruple Product
References
Arfken, G. "Triple Scalar Product, Triple Vector Product."
§1.5 in Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 26-33, 1985.
Vee
The symbol V variously means "disjunction" (in Logic)
or "join" (for a Lattice).
see also Wedge
Velocity
dr
dV
where r is the POSITION VECTOR and d/dt is the de-
rivative with respect to time. Expressed in terms of the
Arc Length,
v=-f
dt '
where T is the unit TANGENT VECTOR, so the SPEED
(which is the magnitude of the velocity) is
ds
dt
W(t)\.
see also Angular Velocity, Position Vector,
Speed
The simplest Venn diagram consists of three symmetri-
cally placed mutually intersecting CIRCLES. It is used
in LOGIC theory to represent collections of sets. The
region of intersection of the three CIRCLES A D B n C,
in the special case of the center of each being located at
the intersection of the other two, is called a REULEAUX
Triangle.
In general, an order n Venn diagram is a collection of n
simple closed curves in the Plane such that
1. The curves partition the Plane into 2 n connected
regions, and
2. Each Subset S of {1, 2, . .., n} corresponds to a
unique region formed by the intersection of the inte-
riors of the curves in S (Ruskey).
see also Circle, Flower of Life, Lens, Magic Cir-
cles, Reuleaux Triangle, Seed of Life
References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed.
Stradbroke, England: Tarquin Pub., pp. 255-256, 1989.
Ruskey, F. "A Survey of Venn Diagrams." Elec. J. Corn-
bin. 4, DS#5, 1997. http://www.combinatorics.org/
Surveys/ds5/VennEJC . html.
Ruskey, F. "Venn Diagrams." http : // sue . esc . uvic . ca/ -
cos/inf /comb/Subset Inf o. html # Venn.
Verging Construction
see NEUSIS CONSTRUCTION
Verhulst Model
see Logistic Map
Veronese Surface
A smooth 2-D surface given by embedding the PROJEC-
TIVE Plane into projective 5-space by the homogeneous
parametric equations
v(x,y,z) = (x 2 ,y 2 ,z 2 ,xy y xz,yz).
The surface can be projected smoothly into 4-space, but
all 3-D projections have singularities (CofTman). The
projections of these surfaces in 3-D are called Steiner
Surfaces. The Volume of the Veronese surface is 2tt 2 .
see also STEINER SURFACE
References
CofFman, A. "Steiner Surfaces." http://www.ipfw.edu/
math/Cof f man/steinersurf ace .html.
1908 Veronese Variety
Vertex (Parabola)
Veronese Variety
see Veronese Surface
Versed Sine
see Versine
Versiera
see Witch of Agnesi
Versine
vers(z) = 1 — cosz,
where cosz is the COSINE. Using a trigonometric iden-
tity, the versine is equal to
vers(z) = 2 sin {\z).
see also Cosine, COVERSINE, Exsecant, Haversine
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.
Vertex Angle
vertex
The point about which an Angle is measured is called
the angle's vertex, and the angle associated with a given
vertex is called the vertex angle.
see also ANGLE
Vertex Coloring
Brelaz's Heuristic Algorithm can be used to find
a good, but not necessarily minimal, Vertex coloring
of a Graph.
see also BRELAZ'S HEURISTIC ALGORITHM, COLORING
Vertex Connectivity
The minimum number of VERTICES whose deletion from
a Graph disconnects it.
see also EDGE CONNECTIVITY
Vertex Cover
see Hitting Set
Vertex Degree
The degree of a VERTEX of a GRAPH is the number of
Edges which touch the Vertex, also called the Local
Degree. The Vertex degree of a point A in a Graph,
denoted p(A), satisfies
5>(A) =
■ 2E,
where E is the total number of EDGES. Directed
GRAPHS have two types of degrees, known as the In-
degree and the Outdegree.
see also Directed Graph, Indegree, Local De-
gree, Outdegree
Vertex Enumeration
A Convex Polyhedron is defined as the set of solu-
tions to a system of linear inequalities
mx < b,
where m is a REAL s x d MATRIX and b is a REAL s-
VECTOR. Given m and b, vertex enumeration is the
determination of the polyhedron's VERTICES.
see also Convex Polyhedron, Polyhedron
References
Avis, D. and Fukuda, K. "A Pivoting Algorithm for Con-
vex Hulls and Vertex Enumeration of Arrangements and
Polyhedra." In Proceedings of the 1th ACM Symposium
on Computational Geometry, North Conway, NH, 1991,
pp. 98-104, 1991.
Fukada, K. and Mizukosh, I. "Vertex Enumeration
Package for Convex Polytopes and Arrangements, Version
0.41 Beta." http://www.mathsource.com/cgi-bin/Math
Source/ Applications/Mathematics/0202-633.
Vertex Figure
The line joining the MIDPOINTS of adjacent sides in a
POLYGON is called the polygon's vertex figure. For a
regular n-gon with side length s,
v = s cos
(;)■
For a Polyhedron, the faces that join at a Vertex
form a solid angle whose section by the plane is the
vertex figure.
see also Truncation
Vertex (Graph)
A point of a GRAPH, also called a NODE.
see also Edge (Graph), Null Graph, Tait Color-
ing, Tait Cycle, Tait's Hamiltonian Graph Con-
jecture, Vertex (Polygon)
Vertex (Parabola)
For a Parabola oriented vertically and opening up-
wards, the vertex is the point where the curve reaches a
minimum.
Vertex (Polygon)
Vertex (Polygon)
edge
A point at which two EDGES of a POLYGON meet.
see also Principal Vertex, Vertex (Graph), Ver-
tex (Polyhedron)
Vertex (Polyhedron)
face
A point at which three of more EDGES of a POLYHE-
DRON meet. The concept can also be generalized to a
POLYTOPE.
see also Vertex (Graph), Vertex (Polygon)
Vertex (Polytope)
The vertex of a POLYTOPE is a point where edges of the
Polytope meet.
Vertical
Oriented in an up-down position.
see also Horizontal
Vertical-Horizontal Illusion
The Horizontal line segment in the above figure ap-
pears to be shorter than the VERTICAL line segment,
despite the fact that it has the same length.
see also Illusion, Muller-Lyer Illusion, Poggen-
dorff Illusion, Ponzo's Illusion
References
Fineman, M. The Nature of Visual Illusion. New York:
Dover, p. 153, 1996.
Vibration Problem
Vertical Perspective Projection
1909
A Map Projection given by the transformation equa-
tions
x = fc'cos<£sin(A - Ao) (1)
y = k'[cos4>i sm(f> — sin 0i cos0cos(A — Ao)], (2)
where P is the distance of the point of perspective in
units of Sphere Radii and
*' =
P-l
(3)
P — cos c
cose = sin 0i sin0 + cos^i cos0cos(A — Ao). (4)
References
Snyder, J. P. Map Projections — A Working Manual. U. S.
Geological Survey Professional Paper 1395. Washington,
DC: U. S. Government Printing Office, pp. 173-178, 1987.
Vertical Tangent
A function f(x) has a vertical tangent line at xq if / is
continuous at xo and
lim f(x) = ±oo.
Vesica Piscis
see Lens
Vibration Problem
Solution of a system of second-order homogeneous ordi-
nary differential equations with constant COEFFICIENTS
of the form
where B is a POSITIVE DEFINITE MATRIX. To solve the
vibration problem,
1. Solve the CHARACTERISTIC Equation of B to get
Eigenvalues Ai, . . . , X n . Define &i = ^f\i-
2. Compute the corresponding EIGENVECTORS ei, . . . ,
3. The normal modes of oscillation are given by xi =
Ai sm{u)it + ai)ei, . . . , x n = A n sin{uj n t + a n )e n ,
where A± , . . . , A n and a± , . . . , a n are arbitrary con-
stants.
4. The general solution is x = XI 7= l Xi *
1910 Vickery Auction
Visible Point
Vickery Auction
An AUCTION in which the highest bidder wins but pays
only the second-highest bid. This variation over the nor-
mal bidding procedure is supposed to encourage bidders
to bid the largest amount they are willing to pay.
see also AUCTION
Viergruppe
The mathematical group Z4 (g> Z4 , also denoted D2 . Its
multiplication table is
V
I
Vi
v 2
v 3
I
Vl
v 2
v 3
v A
Vi
Vt
I
v 3
v 2
v 2
v 2
v 3
I
Vi
v 3
v 3
V2
Vi
I
see also Dihedral Group, Finite Group— Z4
Vieta's Substitution
The substitution of
Zw
into the standard form CUBIC EQUATION
x 3 + px — q,
which reduces the cubic to a QUADRATIC EQUATION in
( W 8 ) a -ipV)-g = 0.
see also Cubic Equation
Vigesimal
The base-20 notational system for representing Real
NUMBERS. The digits used to represent numbers using
vigesimal Notation are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B,
C, D, E, F, G, H, I, and J. A base-20 number system was
used by the Aztecs and Mayans. The Mayans compiled
extensive observations of planetary positions in base-20
notation.
see also Base (Number), Binary, Decimal, Hexa-
decimal, Octal, Quaternary, Ternary
References
A Weisstein, E. W. "Bases." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks /Bases, m.
Vigintillion
In the American system, 10 63 .
see also Large Number
Villarceau Circles
Given an arbitrary point on a TORUS, four CIRCLES can
be drawn through it. The first is in the plane of the torus
and the second is PERPENDICULAR to it. The third and
fourth CIRCLES are called Villarceau circles.
see also Torus
References
Melzak, Z. A. Invitation to Geometry, New York: Wiley,
pp. 63-72, 1983.
Villarceau, M. "Theoreme sur le tore." Nouv. Ann. Math. 7,
345-347, 1848.
Vinculum
A horizontal line placed above multiple quantities to
indicate that they form a unit. It is most commonly
used to denote Roots (\/l2345) and repeating decimals
(O.TTT).
Vinogradov's Theorem
Every sufficiently large Odd number is a sum of three
Primes. Proved in 1937.
see also Goldbach Conjecture
Virtual Group
see GROUPOID
Visibility
see Visible Point
Visible Point
Two LATTICE Points (x,y) and (x\y r ) are mutually
visible if the line segment joining them contains no fur-
ther Lattice Points. This corresponds to the require-
ment that (x' — x, y' - y) — 1, where (m y n) denotes the
Greatest Common Divisor. The plots above show
the first few points visible from the ORIGIN.
If a Lattice Point is selected at random in 2-D, the
probability that it is visible from the origin is 6/-7T 2 . This
is also the probability that two Integers picked at ran-
dom are Relatively Prime. If a Lattice Point is
picked at random in n-D, the probability that it is visible
Visible Point Vector Identity
VivianVs Theorem 1911
from the ORIGIN is 1/C(™)> where C(n) is the RlEMANN
Zeta Function.
An invisible figure is a POLYGON all of whose corners are
invisible. There are invisible sets of every finite shape.
The lower left-hand corner of the invisible squares with
smallest x coordinate of AREAS 2 and 3 are (14, 20) and
(104, 6200).
see also LATTICE POINT, ORCHARD VISIBILITY PROB-
LEM, Riemann Zeta Function
References
Apostol, T. §3.8 in Introduction to Analytic Number Theory.
New York: Springer- Verlag, 1976.
Baake, M.; Grimm, U.; and Warrington, D. H. "Some Re-
marks on the Visible Points of a Lattice." J. Phys. A:
Math. General 27, 2669-2674, 1994.
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Feb. 1972.
Herzog, F. and Stewart, B. M. "Patterns of Visible and Non-
visible Lattice Points." Amer. Math. Monthly 78, 487-496,
1971.
Mosseri, R. "Visible Points in a Lattice." J. Phys. A: Math.
Gen. 25, L25-L29, 1992.
Schroeder, M. R. "A Simple Function and Its Fourier Trans-
form." Math. Intell. 4, 158-161, 1982.
Schroeder, M. R. Number Theory in Science and Communi-
cation, 2nd ed. New York: Springer- Verlag, 1990
Visible Point Vector Identity
A set of identities involving n-D visible lattice points
was discovered by Campbell (1994). Examples include
Viviani's Curve
J] (l-yVr^^l-z)- 1 ^
y)
a>0,6<l
for \yz\, \z\ < 1 and
n/-, a b c\-l/c /-, \-l/[(l-a0(l-
(1 -x y Z ) ' = (1 -Z) /LV A
(a,6,c) = l
a,6>0,c<l
y)\
for \xyz\,\xz\,\yz\,\z\ < 1.
References
Campbell, G. B. "Infinite Products Over Visible Lattice
Points." Internal. J. Math. Math. Sci. 17, 637-654, 1994.
Campbell, G. B. "Visible Point Vector Identities." http://
www . geocities . com/ Cape Canaveral /Launchpad/ 9416 /
vpv.html.
Vitali's Convergence Theorem
Let f n (z) be a sequence of functions, each regular in a
region D, let |/n(^)| < M for every n and z in D, and let
fn(z) tend to a limit as n — > co at a set of points having
a Limit Point inside D. Then f n {z) tends uniformly
to a limit in any region bounded by a contour interior
to £>, the limit therefore being an analytic function of
z.
see also MONTEL'S THEOREM
References
Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford,
England: Oxford University Press, p. 168, 1960.
The Space Curve giving the intersection of the Cyl-
inder
(i)
(2)
(x
-«)'
+ y 2
2
= a
and the SPHERE
2
X
+ y 2
+ z 2 --
= 4 2 .
It is given by the parametric equations
x = a(l + cos<)
(3)
y = a sin t
(4)
z = 2asin(|t).
(5)
The Curvature and Torsion are given by
,. V13 + 3COS*
K(, ~a(3 + cost)3/2
(6)
T(t) „ ecos(ii)
w o(13 + 3cos<)'
(7)
see also Cylinder, Sphere, Steinmetz Solid
References
Gray, A. "Viviani's Curve." §7.6 in Modern Differential Ge-
ometry of Curves and Surfaces. Boca Raton, FL: CRC
Press, pp. 140-142, 1993.
von Seggern, D. CRC Standard Curves and Surfaces. Boca
Raton, FL: CRC Press, p. 270, 1993.
Viviani's Theorem
For a point P inside an EQUILATERAL TRIANGLE
AABC, the sum of the perpendiculars pi from P to
the sides of the Triangle is equal to the Altitude h.
This result is simply proved as follows,
AABC = APBC + APCA + APAB. (1)
With s the side length,
\sh = \sp a + \sp b + \spa (2)
so
h = p a +pb +Pc- (3)
see also ALTITUDE, EQUILATERAL TRIANGLE
1912 Vojta's Conjecture
Volume Element
Vojta's Conjecture
A conjecture which treats the heights of points relative
to a canonical class of a curve defined over the INTE-
GERS.
References
Cox, D. A. "Introduction to Fermat's Last Theorem." Amer.
Math. Monthly 101, 3-14, 1994.
Volterra Integral Equation of the First Kind
An Integral Equation of the form
f{x) = / k(x,t)<f>(t)dt.
-I
see also Fredholm Integral Equation of the
First Kind, Fredholm Integral Equation of the
Second Kind, Integral Equation, 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.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Volterra Equations." §18.2 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 786-788, 1992.
Volterra Integral Equation of the Second
Kind
An Integral Equation of the form
px
J a
<p(x) = f(x)+ / k(x,t)4>(t)dt
see also FREDHOLM INTEGRAL EQUATION OF THE
First Kind, Fredholm Integral Equation of the
Second Kind, Integral Equation, Volterra In-
tegral Equation of the First Kind
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, p. 865, 1985.
Press, VV\ H.; Flannery, B. P.; Teukolsky, S. A.; and Vet-
terling, W. T. "Volterra Equations." §18.2 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 786-788, 1992.
Volume
The volume of a solid body is the amount of "space" it
occupies. Volume has units of Length cubed (i.e., cm 3 ,
m 3 , in 3 , etc.) For example, the volume of a box (RECT-
ANGULAR Parallelepiped) of Length L, Width W,
and Height H is given by
V = L x W x H .
The volume can also be computed for irregularly-shaped
and curved solids such as the CYLINDER and CUBE. The
volume of a Surface of Revolution is particularly
simple to compute due to its symmetry.
The following table gives volumes for some common
Surfaces. Here r denotes the Radius, h the height, A
the base Area, and s the Slant Height (Beyer 1987).
Surface
V
cone
\nr 2 h
conical frustum
\nhiRx 2 + R 2 2 + R1R2)
cube
a 3
cylinder
7rr 2 h
ellipsoid
^Ttabc
oblate spheroid
|iro 2 6
prolate spheroid
|7ra6 2
pyramid
\Ah
pyramidal frustum
f /i(4i + A 2 + VA1A2 )
sphere
W
spherical sector
2 2 l
|7rr h
spherical segment
lnh 2 r(3r-h)
torus
2ir 2 Rr 2
Even simple SURFACES can display surprisingly coun-
terintuitive properties. For instance, the SURFACE OF
Revolution of y = 1/x around the cc-axis for x > 1
is called GABRIEL'S HORN, and has finite volume, but
infinite SURFACE AREA.
The generalization of volume to n DIMENSIONS for n > 4
is known as CONTENT.
see also Arc Length, Area, Content, Height,
Length (Size), Surface Area, Surface of Revo-
lution, Volume Element, Width (Size)
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed.
Boca Raton, FL: CRC Press, pp. 127-132, 1987.
Volume Element
A volume element is the differential element dV whose
VOLUME INTEGRAL over some range in a given coordi-
nate system gives the VOLUME of a solid,
V
III
dxdydz.
(i)
In R n , the volume of the infinitesimal n-HYPERCUBE
bounded by dxi, ..., dx n has volume given by the
Wedge Product
dV = dx\ A ... A dx n
(2)
(Gray 1993).
The use of the antisymmetric WEDGE PRODUCT instead
of the symmetric product dx\ . . . dx n is a technical re-
finement often omitted in informal usage. Dropping the
Volume Integral
von Staudt-Clausen Theorem 1913
wedges, the volume element for CURVILINEAR COORDI-
NATES in R 3 is given by
dV — |(/iiui dui) ■ (/12U2 du2) x (/13U3 duz)\
= /11/12/13 du\ du2 duz
dr dr dr
dx
dux
dy
du\
dz
du2
dx
du 2
dy
du2
dz
du2
duz
dx
du 3
dy
duz
dz
duz
du\ dui duz
du± du2 dus
d(x,y,z)
d(m, 112,113)
du\ du2 dus ,
(3)
(4)
(5)
(6)
(7)
where the latter is the Jacobian and the hi are Scale
Factors.
see also Area Element, Jacobian, Line Element,
Riemannian Metric, Scale Factor, Surface In-
tegral, Volume Integral
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.
Volume Integral
A triple integral over three coordinates giving the VOL-
UME within some region R,
III
dx dy dz.
see also Integral, Line Integral, Multiple Inte-
gral, Surface Integral, Volume, Volume Ele-
ment
von Aubel's Theorem
Given an arbitrary QUADRILATERAL, place a SQUARE
outwardly on each side, and connect the centers of op-
posite SQUARES. Then the two lines are of equal length
and cross at a Right Angle.
see also Quadrilateral, Right Angle, Square
References
Kitchen, E. "Dorrie Tiles and Related Miniatures." Math.
Mag. 67, 128-130, 1994.
von Dyck's Theorem
Let a GROUP G have a presentation
G — (x 1 ,...,x n \rj(x 1 ,...,x n )J e J)
so that G = F/Rj where F is the FREE GROUP with ba-
sis {xi,.. .,x n } and R is the NORMAL SUBGROUP gen-
erated by the rj. If if is a GROUP with H = (yi, . . . ,y n )
and if rj(j/i, . . . , y„) = 1 for all j, then there is a surjec-
tive homomorphism G —> H with Xi h* yi for all i.
see also Free Group, Normal Subgroup
References
Rotman, J. J. An Introduction to the Theory of Groups, J^th
ed. New York: Springer- Verlag, p. 346, 1995.
von Mangoldt Function
see Mangoldt Function
von Neumann Algebra
A Group "with bells and whistles." It was while study-
ing von Neumann algebras that Jones discovered the
amazing and highly unexpected connections with KNOT
THEORY which led to the formulation of the JONES
Polynomial.
References
Iyanaga, S. and Kawada, Y. (Eds.). "Von Neumann Alge-
bras." §430 in Encyclopedic Dictionary of Mathematics.
Cambridge, MA: MIT Press, pp. 1358-1363, 1980.
von Staudt-Clausen Theorem
B2n
? J*'
Pk
Pfc-l|2n
where B 2n is a BERNOULLI NUMBER, A n is an INTEGER,
and the pkS are the PRIMES satisfying pk - l\2k. For
example, for k = 1, the primes included in the sum are
2 and 3, since (2-l)|2 and (3-l)|2. Similarly, for k = 6,
the included primes are (2, 3, 5, 7, 13), since (1, 2, 3,
6, 12) divide 12 = 2-6. The first few values of A n for
n = 1, 2, ... are 1, 1, 1, 1, 1, 1, 2, -6, 56, -528, ...
(Sloane's A000164).
The theorem was rediscovered by Ramanujan (Hardy
1959, p. 11) and can be proved using p-ADIC NUMBERS.
see also Bernoulli Number, p-ADic Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, p. 109, 1996.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Sug-
gested by His Life and Work, 3rd ed. New York: Chelsea,
1959.
Hardy, G. H. and Wright, E. M. "The Theorem of von
Staudt" and "Proof of von Staudt's Theorem." §7.9-7.10
in An Introduction to the Theory of Numbers, 5th ed. Ox-
ford, England: Clarendon Press, pp. 90-93, 1979.
Sloane, N. J. A. Sequence A000146/M1717 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Staudt. "Beweis eines Lehrsatzes, die Bernoullischen Zahlen
betreffend." J. reine angew. Math. 21, 372-374, 1840.
1914
von Staudt Theorem
Vulgar Series
von Staudt Theorem
see von Staudt-Clausen Theorem
Voronoi Cell
The generalization of a VORONOI POLYGON to n-D, for
n > 2.
Voronoi Diagram
VR Number
A "visual representation" number which is a sum of
some simple function of its digits. For example,
/• y\
/ * T^ / *
1 •[• / \ \ •
\ ^^^C • / / n. * / ^v
\ • \ ^/^
\ \r *j
/\ * ^\.x"^C^ * /
X / • 1
The partitioning of a plane with n points into n con-
vex Polygons such that each Polygon contains ex-
actly one point and every point in a given POLYGON is
closer to its central point than to any other. A Voronoi
diagram is sometimes also known as a DlRICHLET TES-
SELLATION. The cells are called DlRICHLET REGIONS,
Thiessen Polytopes, or Voronoi Polygons.
see also Delaunay Triangulation, Medial Axis,
Voronoi Polygon
References
Eppstein, D. "Nearest Neighbors and Voronoi Diagrams."
http://www.ics.uci.edu/-eppstein/junkyard/nn.html.
Voronoi Polygon
A POLYGON whose interior consists of all points in the
plane which are closer to a particular Lattice Point
than to any other. The generalization to n-D is called a
Dirichlet Region, Thiessen Polytope, or Voronoi
Cell.
1233 = 12 2 + 33 2
2661653 = 1653 2 - 266 2
221859 = 22 3 + 18 3 + 59 3
40585 + 41 + 0! + 5! + 8! + 5!
148349 =!l+!4+!8+!3+!4+!9
4913= (4 + 9 + 1 + 3) 3
are all VR numbers given by Madachy (1979).
References
Madachy, J. S. Madachy's Mathematical Recreations. New
York: Dover, pp. 165-171, 1979.
Vulgar Series
see Farey Series
Voting
It is possible to conduct a secret ballot even if the
votes are sent in to a central polling station (Lipton
and Widgerson, Honsberger 1985).
see also ARROW'S PARADOX, BALLOT PROBLEM,
May's Theorem, Quota System, Social Choice
Theory
References
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 157-162, 1985.
Lipton, R. G.; and Widgerson, A. "Multi-Party Crypto-
graphic Protocols."
W2- Constant
W
W2-Constant
W 2 = 1.529954037....
References
Plouffe, S. "W2 Constant." http://lacim.uqam.ca/piDATA/
v2.txt.
W-Function
see Lambert's V7-Function
Wada Basin
A Basin of Attraction in which every point on the
common boundary of that basin and another basin is
also a boundary of a third basin. In other words, no
matter how closely a boundary point is zoomed into, all
three basins appear in the picture.
see also BASIN OF ATTRACTION
References
Nusse, H. E. and Yorke, J. A. "Basins of Attraction.'
271, 1376-1380, 1996.
Science
Walk
A sequence of Vertices and Edges such that the Ver-
tices and Edges are adjacent. A walk is therefore
equivalent to a graph Cycle, but with the Vertices
along the walk enumerated as well as the EDGES.
see also CIRCUIT, CYCLE (GRAPH), PATH, RANDOM
Walk
Wallace-Bolyai-Gerwein Theorem
Two Polygons are congruent by Dissection Iff they
have the same Area. In particular, any POLYGON is
congruent by DISSECTION to a SQUARE of the same
AREA. Laczkovich (1988) also proved that a CIRCLE
is congruent by DISSECTION to a SQUARE (furthermore,
the DISSECTION can be accomplished using TRANSLA-
TIONS only).
see also DISSECTION
References
Klee, V. and Wagon, S. Old and New Unsolved Problems in
Plane Geometry and Number Theory. Washington, DC:
Math. Assoc. Amer., pp. 50-51, 1991.
Laczkovich, M. "Von Neumann's Paradox with Translation."
Fund. Math. 131, 1-12, 1988.
Wallace-Simson Line
see Simson Line
Wallis Formula 1915
Wallis's Conical Edge
The Right Conoid surface given by the parametric
equations
x(u, v) = vcosu
y(u,v) = vsinu
z(u, v) = cya 2 — b 2 cos 2 w.
see also RIGHT CONOID
References
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 354-355, 1993.
Wallis Cosine Formula
tt l-3-5-(n-l) f _ 2 4
prr/2 f 7T 1-3-5 — (n-
/ cos n xdx=\ L.s 2 :^
Jo K l-3-5---n
^ for n = 3, 5,
see also WALLIS FORMULA, WALLIS SINE FORMULA
Wallis Formula
The Wallis formula follows from the INFINITE PRODUCT
representation of the Sine
sinx = x
ft('-i)
n = l X 7
(1)
Taking x = tt/2 gives
n
(2n) 2
n
(2n) 2
(2n) 2
f=n
(2n) 2
(2n-l)(2n + l)
2-2 4-4 6-6
1-3 3-5 5-7
(2)
(3)
1916 Wallis's Problem
Walsh Function
A derivation due to Y. L. Yung uses the RlEMANN ZETA
Function. Define
(-i)"
F(,) = -Li.(-l) = £i-
= (i-2 1 - 3 KW
(4)
(5)
Wallis Sieve
A compact set W&
with Area
M(^oc) =
8 24 48
9 25 49
7T
4
created by punching a square hole of length 1/3 in the
center of a square. In each of the eight squares remain-
ing, punch out another hole of length 1/(3 * 5), and so
Wallis Sine Formula
F'(0) = ^(-l) n lnn= -Inl + ln2~ln3 + ...
MtttJ-
(6)
Taking the derivative of the zeta function expression
gives
^(1 - 2 l -)CW = 2 1 -(ln2)C( S ) + (1 - 2 1 -)C'(*) (7)
[^(l-2 1 - s )C( S )] s=o = -ln2-C'(0)
(8)
Equating and squaring then gives the Wallis formula,
which can also be expressed
7T
2
4 C(o) e -C'(o)
(9)
The q- ANALOG of the Wallis formula for q = 2 is
CO
JJ(1 - q~ k )~ l = 3.4627466194. . . (10)
fc=i
(Finch).
see also WALLIS COSINE FORMULA, WALLIS SlNE FOR-
MULA
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 258, 1972.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/dig/dig.html.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 63-64,
1951.
Wallis's Problem
Find solutions to o~(x 2 ) — cr(y 2 ) other than (x y y) =
(4,5), where a is the Divisor Function.
see also Fermat's Sigma Problem
nn/2
x dx ■
( it 1-3-5 — (n-l)
1 2-4-6--(n-l)
L l-3-5---n
for n = 2, 4, ...
for n = 3, 5,
see also WALLIS COSINE FORMULA, WALLIS FORMULA
Wallpaper Groups
The 17 Plane Symmetry Groups. Their symbols are
pi, p2, pm, pg, cm, pmm, pmg, pgg, cmm, p4, p4m,
p4g, p3, p31m, p3ml, p6, and p6m. For a description
of the symmetry elements present in each space group,
see Coxeter (1969, p. 413).
References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New-
York: Wiley, 1969.
Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagina-
tion. New York: Chelsea, 1952.
Joyce, D. E. "Wallpaper Groups (Plane Symmetry Groups)."
http://alephO.clarku.edu/-djoyce/wallpaper/.
Lee, X. "The Discontinuous Groups of Rotation and
Translation in the Plane." http://www.best .com/ ~xah/
Wallpaper _dir/cO_WallPaper. html.
Schattschneider, D. "The Plane Symmetry Groups: Their
Recognition and Notation." Amer. Math. Monthly 85,
439-450, 1978.
Weyl, H. Symmetry. Princeton, NJ: Princeton University
Press, 1952.
Walsh Function
Functions consisting of a number of fixed-amplitude
square pulses interposed with zeros. Following Harmuth
(1969), designate those with Even symmetry Cal(fc,£)
and those with Odd symmetry Sal(fc,t). Define the Se-
QUENCY k as half the number of zero crossings in the
time base. Walsh functions with nonidentical SEQUEN-
CIES are ORTHOGONAL, as are the functions Cal(fc,£)
and Sal(k,t). The product of two Walsh functions is
also a Walsh function. The Walsh functions
Wal(M)
fCal(fc/2,t)
\Sal((fc + l)/2,t)
for k = 0, 2, 4, . . .
for k = 1, 3, 5,
The Walsh functions Cal(fc, t) for k = 0, 1, ... , n/2 - 1
and Sal(fc, t) for k = 1, 2, ... , n/2 are given by the rows
of the Hadamard Matrix H n .
see also Hadamard Matrix, Sequency
Walsh Index
Waring's Problem 1917
References
Beauchamp, K. G. Walsh Functions and Their Applications.
London: Academic Press, 1975.
Harmuth, H. F. "Applications of Walsh Functions in Com-
munications." IEEE Spectrum 6, 82-91, 1969.
Thompson, A. R.; Moran, J. M.; and Swenson, G. W. Jr.
Interferometry and Synthesis in Radio Astronomy. New
York: Wiley, p. 204, 1986.
Tzafestas, S. G. Walsh Functions in Signal and Systems
Analysis and Design. New York: Van Nostrand Reinhold,
1985.
Walsh, J. L. "A Closed Set of Normal Orthogonal Functions."
Amer. J. Math. 45, 5-24, 1923.
2. N — n 2 - 1, with n - 1 and n + 1 PRIME.
see also LUCAS SEQUENCE, SYLVESTER CYCLOTOMIC
Number
References
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, pp. 69-70, 1989.
Waring's Conjecture
see Waring's Prime Conjecture, Waring's Sum
Conjecture
Walsh Index
The statistical Index
Pw =
Waring Formula
[n/2j
where p n is the price per unit in period n and q n is the
quantity produced in period n.
see also INDEX
References
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics,
PL 1, 3rd ed. Princeton, NJ: Van Nostrand, p. 66, 1962.
Wang's Conjecture
Wang's conjecture states that if a set of tiles can tile
the plane, then they can always be arranged to do so
periodically (Wang 1961). The CONJECTURE was re-
futed when Berger (1966) showed that an aperiodic set
of tiles existed. Berger used 20,426 tiles, but the number
has subsequently been greatly reduced.
see also TILING
References
Adler, A. and Holroyd, F. C. "Some Results on One-
Dimensional Tilings." Geom. Dedicata 10, 49-58, 1981.
Berger, R. "The Undecidability of the Domino Problem."
Mem. Amer. Math. Soc. No. 66, 1-72, 1966.
Griinbaum, B. and Sheppard, G. C. Tilings and Patterns.
New York: W. H. Freeman, 1986.
Hanf, W. "Nonrecursive Tilings of the Plane. I." J. Symbolic
Logic 39, 283-285, 1974.
Mozes, S. "Tilings, Substitution Systems, and Dynamical
Systems Generated by Them." J. Analyse Math. 53, 139-
186, 1989.
Myers, D. "Nonrecursive Tilings of the Plane. II." J. Sym-
bolic Logic 39, 286-294, 1974.
Robinson, R. M. "Undecidability and Nonperiodicity for
Tilings of the Plane." Invent. Math. 12, 177-209, 1971.
Wang, H. Bell Systems Tech. J. 40, 1-41, 1961.
Ward's Primality Test
Let TV be an ODD INTEGER, and assume there exists
a Lucas Sequence {U n } with associated Sylvester
Cyclotomic Numbers {Q n } such that there is an n >
\fN (with n and TV Relatively Prime) for which TV
Divides Q n . Then TV is a Prime unless it has one of
the following two forms:
1. TV = (n - l) 2 , with n-1 PRIME and n > 4, or
A n +B n = Y(-l) j ^-i n ~ J )(AB) j (A + B) n - 2j ,
where \x\ is the Floor Function and (£) is a Bino-
mial Coefficient.
see also Fermat's Last Theorem
Waring's Prime Conjecture
Every Odd Integer is a Prime or the sum of three
Primes.
Waring's Problem
Waring proposed a generalization of LAGRANGE'S
Four-Square Theorem, stating that every Ratio-
nal Integer is the sum of a fixed number g(n) of nth
Powers of Integers, where n is any given Positive
Integer and g(n) depends only on n. Waring origi-
nally speculated that g(2) = 4, g(3) = 9, and 5(4) = 19.
In 1909, Hilbert proved the general conjecture using an
identity in 25-fold multiple integrals (Rademacher and
Toeplitz 1957, pp. 52-61).
In Lagrange's Four-Square Theorem, Lagrange
proved that g{2) = 4, where 4 may be reduced to 3
except for numbers of the form 4 n (8fc + 7) (as proved
by Legendre). In the early twentieth century, Dickson,
Pillai, and Niven proved that p(3) = 9. Hilbert, Hardy,
and Vinogradov proved g(4) < 21, and this was sub-
sequently reduced to g(4) = 19 by Balasubramanian
et al. (1986). Liouville proved (using Lagrange's
Four-Square Theorem and Liouville Polynomial
Identity) that g(5) < 53, and this was improved to
47, 45, 41, 39, 38, and finally g(b) < 37 by Wieferich.
See Rademacher and Toeplitz (1957, p. 56) for a simple
proof. J.-J. Chen (1964) proved that g(b) = 37.
Dickson, Pillai, and Niven also conjectured an explicit
formula for g(s) for s > 6 (Bell 1945), based on the
relationship
(i)"-L(i)"J- l -(!)"{L(l)' + 'J}- <•>
1918 Waring 7 s Problem
Waring 7 s Problem
If the DlOPHANTINE (i.e., n is restricted to being an
Integer) inequality
is true, then
'(»>= 2 "+[(|)"J- 2 -
(2)
(3)
This was given as a lower bound by Euler, and has been
verified to be correct for 6 < n < 200,000. Since 1957,
it has been known that at most a Finite number of k
exceed Euler's lower bound.
There is also a related problem of finding the least In-
teger n such that every POSITIVE Integer beyond a
certain point (i.e., all but a Finite number) is the Sum
of G(n) nth POWERS. From 1920-1928, Hardy and Lit-
tlewood showed that
G(n) < (n-2)2 n_1 -h5
and conjectured that
n(1 \ ( 2k + 1 for k not a power of 2
G(/eJ < |
The best currently known bound is
G(k) <ck\nk
. Ak for A; a power of 2.
(4)
(5)
(6)
for some constant c. Heilbronn (1936) improved Vino-
gradov's results to obtain
G(n) < 6nlnn +
4 + 3
-K)]
n + 3.
(7)
It has long been known that G(2) = 4. Dickson and
Landau proved that the only INTEGERS requiring nine
CUBES are 23 and 239, thus establishing G(3) < 8.
Wieferich proved that only 15 INTEGERS require eight
CUBES: 15, 22, 50, 114, 167, 175, 186, 212, 213, 238,
303, 364, 420, 428, and 454, establishing G(3) < 7. The
largest number known requiring seven CUBES is 8042.
In 1933, Hardy and Littlewood showed that G(4) < 19,
but this was improved in 1936 to 16 or 17, and shown to
be exactly 16 by Davenport (1939b). Vaughan (1986)
greatly improved on the method of Hardy and Little-
wood, obtaining improved results for n > 5. These
results were then further improved by Briidern (1990),
who gave G(5) < 18, and Wooley (1992), who gave G(n)
for n = 6 to 20. Vaughan and Wooley (1993) showed
G(S) < 42.
Let G + (n) denote the smallest number such that almost
all sufficiently large INTEGERS are the sum of G^(n)
nth POWERS. Then <3 + (3) = 4 (Davenport 1939a),
G+(4) = 15 (Hardy and Littlewood 1925), G + (8) = 32
(Vaughan 1986), and G + (16) = 64 (Wooley 1992). If
the negatives of POWERS are permitted in addition to
the powers themselves, the largest number of nth POW-
ERS needed to represent an aribtrary integer are denoted
eg(n) and EG(n) (Wright 1934, Hunter 1941, Gardner
1986). In general, these values are much harder to cal-
culate than are g(n) and G(n),
The following table gives g(n), G?(n), (3 + (n), e#(n), and
EG(n) for n < 20. The sequence of g(n) is Sloane's
A002804.
n
g( n )
G(n)
G+(n)
eg(n)
EG(n)
2
4
4
3
3
3
9
< 7
<4
[4,5]
4
19
16
< 15
[9, 10]
5
37
< 18
6
73
< 27
7
143
< 36
8
279
<42
< 32
9
548
< 55
10
1079
<63
11
2132
< 70
12
4223
<79
13
8384
<87
14
16673
< 95
15
33203
< 103
16
66190
< 112
< 64
17
132055
< 120
18
263619
< 129
19
526502
< 138
20
1051899
< 146
see also EULER'S CONJECTURE, SCHNIRELMANN'S THE-
OREM, Vinogradov's Theorem
References
Balasubramanian, R.; Deshouillers, J.-M.; and Dress, F.
"Probleme de Waring por les bicarres 1, 2." C. R. Acad.
Sci. Paris Sir. I Math. 303, 85-88 and 161-163, 1986.
Bell, E. T. The Development of Mathematics, 2nd ed. New
York: McGraw-Hill, p. 318, 1945.
Briidern, J. "On Waring's Problem for Fifth Powers and
Some Related Topics." Proc. London Math. Soc. 61, 457—
479, 1990.
Davenport, H. "On Waring's Problem for Cubes." Acta
Math. 71, 123-143, 1939a.
Davenport, H. "On Waring's Problem for Fourth Powers,"
Ann. Math. 40, 731-747, 1939b.
Dickson, L. E. "Waring's Problem and Related Results."
Ch. 25 in History of the Theory of Numbers, Vol. 2: Dio-
phantine Analysis. New York: Chelsea, pp. 717-729, 1952.
Gardner, M. "Waring's Problems." Ch. 18 in Knotted Dough-
nuts and Other Mathematical Entertainments. New York:
W. H. Freeman, 1986.
Guy, R. K. "Sums of Squares." §C20 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
pp. 136-138, 1994.
Hardy, G. H. and Littlewood, J. E. "Some Problems of Parti-
tio Numerorum (VI): Further Researches in Waring's Prob-
lem." Math. Z. 23, 1-37, 1925.
Hunter, W. "The Representation of Numbers by Sums of
Fourth Powers." J. London Math. Soc. 16, 177-179, 1941.
Khinchin, A. Y. "An Elementary Solution of Waring's Prob-
lem." Ch. 3 in Three Pearls of Number Theory. New York:
Dover, pp. 37-64, 1998.
Waring's Sum Conjecture
Watt's Curve 1919
Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, 1957.
Stewart, I. "The Waring Experience." Nature 323, 674, 1986.
Vaughan, R. C. "On Waring's Problem for Smaller Expo-
nents." Proc. London Math. Soc. 52, 445-463, 1986.
Vaughan, R. C. and Wooley, T. D. "On Waring's Problem:
Some Refinements." Proc. London Math. Soc. 63, 35-68,
1991.
Vaughan, R. C. and Wooley, T. D. "Further Improvements
in Waring's Problem." Phil. Trans. Roy. Soc. London A
345, 363-376, 1993a.
Vaughan, R. C. and Wooley, T. D. "Further Improvements in
Waring's Problem III. Eighth Powers." Phil Trans. Roy.
Soc. London A 345, 385-396, 1993b.
Wooley, T. D. "Large Improvements in Waring's Problem."
Ann. Math. 135, 131-164, 1992.
Wright, E. M. "An Easier Waring's Problem." J. London
Math. Soc. 9, 267-272, 1934.
Waring's Sum Conjecture
see Waring's Problem
Waring's Theorem
If each of two curves meets the Line at Infinity in
distinct, nonsingular points, and if all their intersections
are finite, then if to each common point there is attached
a weight equal to the number of intersections absorbed
therein, the CENTER OF MASS of these points is the
center of gravity of the intersections of the asymptotes.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 166, 1959.
References
Gradshteyn, 1. S. and Ryzhik, I. U. Eqns. 6.617.1 and 6.617.2
in Tables of Integrals, Series, and Products, 5th ed. San
Diego, CA: Academic Press, p. 710, 1979.
Ito, K. (Ed.), Encyclopedic Dictionary of Mathematics, 2nd
ed. Cambridge, MA: MIT Press, p. 1806, 1987.
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1476,
1980.
Watson-Nicholson Formula
Let hI l) be a Hankel Function of the First or
Second Kind, let x,v > 0, and define
w
-m 7
Then
V
- tan" 1 w)]}H[%(±vw) + 0\v'
Ili L \x) = 3- 1/2 wexp{(-l) L+1 i[7r/6 + v{w
^1/3(3
References
Iyanaga, S. and Kawada, Y. (Eds.), Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1475,
1980.
Watson Quintuple Product Identity
see Quintuple Product Identity
Watson's Theorem
Watchman Theorem
see Art Gallery Theorem
Watson's Formula
Let J v (z) be a Bessel Function of the First Kind,
Y v (z) a Bessel Function of the Second Kind, and
K u (z) a Modified Bessel Function of the First
Kind. Also let dt[z] > and require 5R[jz — 1/] < 1. Then
Mz)Y v {z) - J„(z)Y»{z
_ 4sin[(^ -
n Jo
t)e
-(t* + v)t
dt.
The fourth edition of Gradshteyn and Ryzhik (1979),
Iyanaga and Kawada (1980), and Ito (1987) erroneously
give the exponential with a Plus Sign. A related inte-
gral is given by
4 f°
* Jo
K (2z sinh t)e~ 2vt dt
for R[z] > 0.
see also DlXON-FERRAR FORMULA, NICHOLSON'S FOR-
MULA
3^2
a,b,c
|(a + 6+l),c
r(i)r(§ + c )r[|(i + fl + fe)]r(|-|a-|& + c )
r[i(l + a)]T[\{l + b)]T{\ - \a + c)T{\ - \b + c) '
where 3 F 2 (a,[>,c; d, e; z) is a GENERALIZED HYPERGEO-
metric Function and T(z) is the Gamma Function.
Watt's Curve
A curve named after James Watt (1736-1819), the Scot-
tish engineer who developed the steam engine (MacTu-
tor Archive). The curve is produced by a Linkage of
1920
Watt's Parallelogram
rods connecting two wheels of equal diameter. Let the
two wheels have RADIUS b and let their centers be lo-
cated a distance 2a apart. Further suppose that a rod
of length 2c is fixed at each end to the CIRCUMFERENCE
of the two wheels. Let P be the Midpoint of the rod.
Then Watt's curve C is the LOCUS of P.
The POLAR equation of Watt's curve is
r 2 = b 2 -(asmO±^c 2 -a 2 cos 2 0) 2 .
If a = c, then C is a CIRCLE of Radius b with a figure
of eight inside it.
References
Lockwood, E. H. A Book of Curves. Cambridge, England:
Cambridge University Press, p. 162, 1967.
MacTutor History of Mathematics Archive. "Watt's Curve."
http : //www-groups . dcs . st-and . ac . uk/ -history /Curves
/Watts. html.
Watt's Parallelogram
A LINKAGE used in the original steam engine to turn
back- and- forth motion into approximately straight-line
motion.
see also LINKAGE
References
Rademacher, H. and Toeplitz, O. The Enjoyment of Math-
ematics: Selections from Mathematics for the Amateur.
Princeton, NJ: Princeton University Press, pp. 119-121,
1957.
Wave
A 4-POLYHEX.
References
Gardner, M. Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, p. 147, 1978.
Wave Equation
The wave equation is
(i)
where V 2 is the LAPLACIAN.
The 1-D wave equation is
d 2 ip _ 1 d 2 j>
dx 2 v 2 dt 2
(2)
In order to specify a wave, the equation is subject to
boundary conditions
Wave Equation
^(i,*) = o, (4)
and initial conditions
i,(x,0) = f(x)
^{x,0)=g(x).
(5)
(6)
The wave equation can be solved using the so-called
d'Alembert's solution, a FOURIER TRANSFORM method,
or Separation of Variables.
d'Alembert devised his solution in 1746, and Euler sub-
sequently expanded the method in 1748. Let
f = x — at
7] = x + at.
By the CHAIN RULE,
d 2 ip _ d 2 tp d 2 i> d2tf>
(7)
(8)
(9)
v 2 at 2 de dtdr? dr) 2 ' { }
The wave equation then becomes
d 2 vb
= 0.
(11)
Any solution of this equation is of the form
V>(£> n) = f{n) + g{0 = f(x + vt) + g (x - vt), (12)
where / and g are any functions. They represent two
waveforms traveling in opposite directions, / in the
Negative x direction and g in the Positive x direc-
tion.
The 1-D wave equation can also be solved by applying
a Fourier Transform to each side,
/
00 d 2 <ip(x,t) [ e - 27ri kx dx
dx 2
v 2 /
#*&<Le-™-dx t (13)
iKo,t) = o
(3)
which is given, with the help of the FOURIER TRANS-
FORM Derivative identity, by
(«) 3 *(M)4», (14)
where
/oo
i>{x,t)e~ 2 * ikx dx. (15)
■ 00
Wave Equation
This has solution
¥(M) = A(k)e 2 * ikvt + B(k)e~ 27Tikvt .
Taking the inverse FOURIER TRANSFORM gives
(16)
il>{x,t)
i:
i:
*(M)e dx
[A(k)e 2
+ B(k)e- 2 " ikvt }e~ 2 " ikx dk
/OO
A(k)e- 2 " ik <- x - vi) dk
-OO
/OO
B(k)e-™ kl ' +vt) dk
-oo
= fi{x-vt) + b(k)f 2 (x + vt),
where
h{u)=T{A{k)]= J
J — oo
/OO
B{k)e- 2ltiku
■oo
A(k)e- 27riku dk
dk.
(17)
(18)
(19)
This solution is still subject to all other initial and
boundary conditions.
The 1-D wave equation can be solved by SEPARATION
of Variables using a trial solution
This gives
${x,t) = X(x)T{t).
d 2 X = 1 d 2 T
dx 2 v 2 dt 2
1 d 2 X 1 1 d 2 T , 2
— = — k .
X dx 2 v 2 T dt 2
So the solution for X is
X(x) = C cos{kx) + Dsln(kx),
Rewriting (22) gives
1 d 2 T
so the solution for T is
T(t) - Ecos(uit) + Fsin(wi),
(20)
(21)
(22)
(23)
(24)
(25)
where v = w/k. Applying the boundary conditions
it(Q,t) = VCM) = to (23) gives
Wave Equation 1921
where m is an INTEGER. Plugging (23), (25) and (26)
back in for ip in (21) gives, for a particular value of m,
(TfVKX \
~1~~ )
= [Am cos(umt) + B m sin(u; m £)] sin ( —— J ■
(27)
The initial condition ^(x,0) = then gives B m = 0, so
(27) becomes
ipm(x,t) = Am cos(aj m t) sin (— —J . (28)
The general solution is a sum over all possible values of
m, so
oo
VK^i *) = X^ Am cos ( w ***) sin ( ~rr ) ' ^
Using Orthogonality of sines again,
J 8m(^)sm(^) < fa=IW Jm> (30)
where Si m is the Kronecker Delta defined by
1 m = n
-{;
m ^ n '
(31)
gives
// 777,71" 33 \
i/)(xj 0) sin I — - — J dx
oo
4 sin (^—J an (-£-)<**
OO
= ^A l \L8 l m = \LAm, (32)
i=i
so we have
The computation of i4 m s for specific initial distortions
is derived in the Fourier Sine Series section. We
already have found that B m = 0, so the equation of
motion for the string (29), with
OJ m = Vkm
VTUTT
(34)
C = kL = 7717T,
(26)
oo
^(x,t) = ^2 Am cos ( VTn ^ \ S i n f V^E\ t (35)
1922 Wave Equation
Wave Equation
where the Am COEFFICIENTS are given by (33).
A damped 1-D wave
dx" 2 ~ ^~d1? + ~dt'
given boundary conditions
1>(o,t) = o
rKL,t)=0,
initial conditions
dt
(x,0) = g(x),
and the additional constraint
2tt
0<b< r v '
(36)
(37)
(38)
(39)
(40)
(41)
can also be solved as a FOURIER SERIES.
ip{x,t)
oo
— J^sin f — — j e~ v bt/2 [a n s'm( fi n t) + b n cos(/j n £)],
n=l
(42)
where
V4v 2 n 2 7v 2 - b 2 L 2 v 4 vV4n 2 7r 2 - b 2 L 2 v 2 /An .
Mn = ^t = ^7 ( 43 )
2L
2L
b n - j I sin ( —— ) f(x) dx
2 f f L . (mvx\
/ x v 2 b .. N
fa) + -yf( x )
(44)
dx.
(45)
To find the motion of a rectangular membrane with sides
of length L x and L y (in the absence of gravity), use the
2-D wave equation
d 2 z d 2 z 1 d 2 z
+
dx 2
dy 2
dt 2
(46)
where z{x,y,t) is the vertical displacement of a point
on the membrane at position (x, y) and time t. Use
Separation of Variables to look for solutions of the
form
z(x,y,t) = X(x)Y{y)T(t). (47)
Plugging (47) into (46) gives
,d 2 X
J 2 Y _1_
v 2
YT H^ + XT W=-> XY ^T' W
. d 2 T
dt 2
where the partial derivatives have now become complete
derivatives. Multiplying (48) by v /XYT gives
v 2 d 2 X v 2 d 2 Y
X dx 2 Y dy 2
1 d 2 T
T dt 2 '
(49)
The left and right sides must both be equal to a con-
stant, so we can separate the equation by writing the
right side as
T dt 2 " '
This has solution
T(t) = Cu cos(wt) + £>„ sin(w*).
Plugging (50) back into (49),
v 2 d 2 X v 2 d 2 Y _ _ 2
X dx 2 + Y dy 2 W '
which we can rewrite as
i d 2 x _ _i^y; _ ^ ___, 2
X dx 2 Y dy 2 v 2 ~ x
(50)
(51)
(52)
(53)
since the left and right sides again must both be equal to
a constant. We can now separate out the Y(y) equation
Y dy 2 ~ x v 2 ~ y ' ( '
where we have defined a new constant k y satisfying
,2
rCx ~T~ fcy
'v 2
(55)
Equations (53) and (54) have solutions
X(x) = Ecos(k x x) + Fsin(k x x)
Y(y) = Gcos(k y y) + Hsin(k y y).
(56)
(57)
We now apply the boundary conditions to (56) and (57).
The conditions 2(0, y, t) ~ and z(x, 0, t) = mean that
£ =
G = 0.
(58)
Similarly, the conditions z(L Xi y y t) = and z(x,L y ,t) —
give sin(k x L x ) = and sm(k y L y ) — 0, so L x k x = p-K
and L y ky — qn, where p and q are INTEGERS. Solving
for the allowed values of k x and k y then gives
k EJL
k -£L
L, y
(59)
Plugging (52), (56), (57), (58), and (59) back into (22)
gives the solution for particular values of p and g,
z pq{ x iy,t) = [C u cos(cji) + Duj sin(ujt)]
>,-(^)]k*
qiry
(60)
Wave Equation
Wave Equation 1923
Lumping the constants together by writing A pq ^
Cu>F p H q (we can do this since a; is a function of p and
g, so Cu can be written as C pq ) and B pq = DuF p H q , we
obtain
z pq (x,y,t) = [A pq cos(u) pq t) + B pq sin(tj pq t)]
, (piTX\ .
x sin I — — sin.
(61)
Plots of the spatial part for modes (1, 1), (1, 2), (2, 1),
and (2, 2) follow.
The general solution is a sum over all possible values of
p and q, so the final solution is
z(x, y, t) = 2_, 7_.[-^p9 cos(uj pq t) -f B pq sin(u; pq t)]
P=l q = l
where to is defined by combining (55) and (59) to yield
Ll) vq = 7TV\
'(£)'
+
(63)
Given the initial conditions z(x, y, 0) and ^| (x, y> 0), we
can compute the A pq s and B pq s explicitly. To accom-
plish this, we make use of the orthogonality of the SINE
function in the form
T f . (m-nx\ . (rnvx\ !
I = / sin I ) sin I — — - J dx = ^Ld mn ,
(64)
where J mn is the KRONECKER Delta. This can be
demonstrated by direct INTEGRATION. Let u = ttx/L
so du = (n/L) dx in (64), then
^ Jo
sin(mu) sin(nu) du.
(65)
Now use the trigonometric identity
sin a sin j3 = \ [cos(a — j3) ~ cos(a + f3)] (66)
to write
I = — / cos[(m — n)u] du+l cos[(m -f n)u] du.
2?r Jo Jo
(67)
Note that for an INTEGER / ^ 0, the following INTEGRAL
vanishes
/ cos(Zu) dn = y [sin(/u)]o = y[sin(j7r) — sinO]
Jo l l
sin(/7r) = 0,
(68)
since sin(/7r) = when I is an INTEGER. Therefore,
1 = when / = m — n ^ 0. However, I does not vanish
when / = 0, since
/ cos(0 * u) du = /
Jo Jo
COS(0 • U)du — I du = 7T.
(69)
We therefore have that I — L5 mn /2, so we have derived
(64). Now we multiply z(x y y, 0) by two sine terms and
integrate between and L x and between and L yy
I I z{x,y^)sm[~-j dx
x sin | ^ | dy. (70)
Now plug in z(x,y,t), set t = 0, and prime the indices
to distinguish them from the p and q in (70),
9 '-r° Lp'=i (/0
sin
dy.
) sin ( E S L]dx
(71)
Making use of (64) in (71)
L..
E/ e
^pvy'W
-(£)-(£)* (72 >
so the sums over p' and g' collapse to a single term
1 = -y 2J ^ P9 ' if**'*' =
LixLi-u
(73)
9=1
Equating (72) and (73) and solving for A pq then gives
L/ x L/i
/ / z(aj,y,0)sin f ^~j dx
. ( qny\
dy. (74)
1924 Wave Operator
An analogous derivation gives the B pq s as
4 r L v f* r L * q z /p7rx\
x sin | ^ ) dy. (75)
The equation of motion for a membrane shaped as a
Right Isosceles Triangle of length c on a side and
with the sides oriented along the POSITIVE x and y axes
is given by
ip(x, y, t) = [C pq cos(uj pq t) + D pq sm(u; pq t)]
sin
(76)
where
7VV
C
y/pTi
(77)
and p, q INTEGERS with p > q. This solution can be
obtained by subtracting two wave solutions for a square
membrane with the indices reversed. Since points on
the diagonal which are equidistant from the center must
have the same wave equation solution (by symmetry),
this procedure gives a wavefunction which will vanish
along the diagonal as long as p and q are both Even or
Odd. We must further restrict the modes since those
with p < q give wavefunctions which are just the NEG-
ATIVE of (q,p) and (p,p) give an identically zero wave-
function. The following plots show (3, 1), (4, 2), (5, 1),
and (5,3).
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Wave Equa-
tion in Prolate and Oblate Spheroidal Coordinates." §21.5
in Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables, 9th printing. New
York: Dover, pp. 752-753, 1972.
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part I. New York: McGraw-Hill, pp. 124-125, 1953.
Wave Operator
An Operator relating the asymptotic state of a Dy-
namical System governed by the Schrodinger equation
*f t m = WW
to its original asymptotic state.
see also SCATTERING OPERATOR
Wavelet
Wave Surface
A Surface represented parametrically by Elliptic
Functions.
Wavelet
Wavelets are a class of a functions used to localize a
given function in both space and scaling. A family of
wavelets can be constructed from a function ^(a:), some-
times known as a "mother wavelet," which is confined in
a finite interval. "Daughter wavelets" ip a,b (x) are then
formed by translation (b) and contraction (a). Wavelets
are especially useful for compressing image data, since a
Wavelet Transform has properties which are in some
ways superior to a conventional FOURIER TRANSFORM.
An individual wavelet can be defined by
r*(x) = \*\- 1/a i>(^)-
(i)
Then
W*(/)(a, b ) = ^J_ /(*)* ( L ^) dt, (2)
and Calderon's Formula gives
/oo poo
I {f,^ h )^ b {x)a- 2 dadb. (3)
■oo J — oo
A common type of wavelet is defined using Haar FUNC-
TIONS.
see also Fourier Transform, Haar Function,
Lemarie's Wavelet, Wavelet Transform
References
Benedetto, J. J. and Frazier, M. (Eds.). Wavelets: Math-
ematics and Applications. Boca Raton, FL: CRC Press,
1994.
Chui, C. K. An Introduction to Wavelets. San Diego, CA:
Academic Press, 1992.
Chui, C. K. (Ed.). Wavelets: A Tutorial in Theory and
Applications. San Diego, CA: Academic Press, 1992.
Chui, C. K.; Montefusco, L.; and Puccio, L. (Eds.). Wavelets:
Theory, Algorithms, and Applications. San Diego, CA:
Academic Press, 1994.
Daubechies, I. Ten Lectures on Wavelets. Philadelphia, PA:
Society for Industrial and Applied Mathematics, 1992.
Erlebacher, G. H.; Hussaini, M. Y.; and Jameson, L. M.
(Eds.). Wavelets: Theory and Applications. New York:
Oxford University Press, 1996.
Foufoula-Georgiou, E. and Kumar, P. (Eds.). Wavelets in
Geophysics. San Diego, CA: Academic Press, 1994.
Hernandez, E. and Weiss, G. A First Course on Wavelets.
Boca Raton, FL: CRC Press, 1996.
Hubbard, B. B. The World According to Wavelets: The Story
of a Mathematical Technique in the Making. New York:
A. K. Peters, 1995.
Jawerth, B. and Sweldens, W. "An Overview of Wavelet
Based Multiresolution Analysis." SIAM Rev. 36, 377-
412, 1994.
Kaiser, G. A Friendly Guide to Wavelets. Cambridge, MA:
Birkhauser, 1994.
Massopust, P. R. Fractal Functions, Fractal Surfaces, and
Wavelets. San Diego, CA: Academic Press, 1994.
Wavelet Matrix
Weak Law of Large Numbers 1925
Meyer, Y. Wavelets: Algorithms and Applications. Philadel-
phia, PA: SIAM Press, 1993.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Wavelet Transforms." §13.10 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed, Cambridge, England: Cambridge University Press,
pp. 584-599, 1992.
Schumaker, L. L. and Webb, G. (Eds.). Recent Advances in
Wavelet Analysis. San Diego, CA: Academic Press, 1993.
Stollnitz, E. J.; DeRose, T. D.; and Salesin, D. H. "Wavelets
for Computer Graphics: A Primer, Part 1." IEEE Com-
puter Graphics and Appl 15, No. 3, 76-84, 1995.
Stollnitz, E. J.; DeRose, T. D.; and Salesin, D. H. "Wavelets
for Computer Graphics: A Primer, Part 2." IEEE Com-
puter Graphics and Appl 15, No. 4, 75-85, 1995.
Strang, G. "Wavelets and Dilation Equations: A Brief Intro-
duction." SIAM Rev. 31, 614-627, 1989.
Strang, G. "Wavelets." Amer. Sci. 82, 250-255, 1994.
Taswell, C. Handbook of Wavelet Transform Algorithms.
Boston, MA: Birkhauser, 1996.
Teolis, A. Computational Signal Processing with Wavelets.
Boston, MA: Birkhauser, 1997.
Walter, G. G. Wavelets and Other Orthogonal Systems with
Applications, Boca Raton, FL: CRC Press, 1994.
"Wavelet Digest." http://www.math. sc.edu/ -wavelet/.
Wickerhauser, M. V. Adapted Wavelet Analysis from Theory
to Software. Wellesley, MA: Peters, 1994.
Wavelet Matrix
A Matrix composed of Haar Functions which is used
in the WAVELET TRANSFORM. The fourth-order wavelet
matrix is given by
Wa
1
1
i
1
1
-l
1 -
-1
1
1 -
-1
-]
ri
1
=
i
-1
1
1
1
A wavelet matrix can be computed in 0(n) steps, com-
pared to 0(nlg2) for the FOURIER MATRIX.
see also Fourier Matrix, Wavelet, Wavelet
Transform
Wavelet Transform
A transform which localizes a function both in space
and scaling and has some desirable properties compared
to the Fourier Transform. The transform is based
on a Wavelet Matrix, which can be computed more
quickly than the analogous FOURIER MATRIX.
see also Daubechies Wavelet Filter, Lemarie's
Wavelet
References
Blair, D. and MathSoft, Inc. "Wavelet Resources." http://
www.mathsof t . com/ wavelets .html.
Daubechies, I. Ten Lectures on Wavelets. Philadelphia, PA:
SIAM, 1992.
DeVore, R.; Jawerth, B.; and Lucier, B. "Images Compres-
sion through Wavelet Transform Coding." IEEE Trans.
Information Th. 38, 719-746, 1992.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Wavelet Transforms." §13.10 in Numerical
Recipes in FORTRAN: The Art of Scientific Computing,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 584-599, 1992.
Strang, G. "Wavelet Transforms Versus Fourier Transforms."
Bull. Amer. Math. Soc. 28, 288-305, 1993.
Weak Convergence
Weak convergence is usually either denoted x n ^x or
x n — ^ x. A Sequence {x n } of Vectors in an In-
ner Product Space E is called weakly convergent to
a Vector in E if
(x n ,y) -* (x, y) as n -> oo, for all y e E.
Every STRONGLY CONVERGENT sequence is also weakly
convergent (but the opposite does not usually hold).
This can be seen as follows. Consider the sequence
{x n } that converges strongly to x y i.e., \\x n — x\\ — > as
n -» oo. Schwarz's Inequality now gives
| (x n -x,y)\< \\x n - x\\ \\y\\ as n ^ oo.
The definition of weak convergence is therefore satisfied.
see also INNER PRODUCT SPACE, SCHWARZ'S INEQUAL-
ITY, Strong Convergence
Weak Law of Large Numbers
Also known as Bernoulli's THEOREM. Let xi, . . . , x n
be a sequence of independent and identically distributed
random variables, each having a MEAN {xi) = \i and
Standard Deviation a. Define a new variable
xi + . . . + x n
(i)
Then, as n — > oo, the sample mean (x) equals the pop-
ulation Mean ^i of each variable.
. . /xi + ... + a n \ 1// Vl ,/ u n ^
(X) = ( ) = -((»!> + . • .+ <*„» = - = /*
(2)
/Xl + ... + x 2 \
var(x) = var I — I
■G)+"-+'«(t)
var I
n 2
a
n 2
+ ^r = — •
a
n
(3)
Therefore, by the Chebyshev Inequality, for all e >
0,
var(x) _ a
P(l* -/*!>«)<
«2 -
(4)
1926 Weakly Binary Tree
As n — > oo, it then follows that
lim P(\x - n\ > e) =
(5)
for e arbitrarily small; i.e., as n — > oo, the sample MEAN
is the same as the population MEAN.
Stated another way, if an event occurs x times in s
TRIALS and if p is the probability of success in a sin-
gle Trial, then the probability that the relative fre-
quency of successes is x/s differs from p by less than
any arbitrary POSITIVE quantity e which approaches 1
as s — >> oo.
see also LAW OF TRULY LARGE NUMBERS, STRONG
Law of Large Numbers
Weakly Binary Tree
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
A Rooted Tree for which the Root is adjacent to
at most two Vertices, and all nonroot Vertices are
adjacent to at most three VERTICES. Let b(n) be the
number of weakly binary trees of order n, then 6(5) = 6.
Let
9( z ) = £^ 9iZ%
oo
i=0
where
90 =
gi = 92 — 93 — l
(1)
(2)
(3)
g2i+i = 7 , fl2t+i- jgj (4)
3 = 1
i-1
92i = \gi{9i + l) + 5^ff2*-j-yj. (5)
j=i
Otter (Otter 1948, Harary and Palmer 1973, Knuth
1969) showed that
b{n)\
3/2
lim
n-*oo f"
where
£ = 2.48325.
is the unique Positive Root of
(6)
(7)
a)
1,
and
77 = 0.7916032....
£ is also given by
£ = lim (c n ) 2 ,
(8)
(9)
where c n is given by
Co = 2
C n = (C n -l) + 2,
Web Graph
(10)
(ii)
giving
' = W"\/ 3 + — + — + — — + ■
2 V *" V Cl c i C2 C1C2C3
(12)
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/otter/otter.html.
Harary, F. Graph Theory. Reading, MA: Addis on- Wesley,
1969.
Harary, F. and Palmer, E. M. Graphical Enumeration. New
York: Academic Press, 1973.
Knuth, D. E. The Art of Computer Programming, Vol. 1:
Fundamental Algorithms, 2nd ed. Reading, MA: Addison-
Wesley, 1973.
Otter, R. "The Number of Trees." Ann. Math. 49, 583-599,
1948.
Weakly Complete Sequence
A Sequence of numbers V = {^n} is said to be weakly
complete if every POSITIVE INTEGER n beyond a cer-
tain point N is the sum of some SUBSEQUENCE of V
(Honsberger 1985). Dropping two terms from the FI-
BONACCI Numbers produces a Sequence which is not
even weakly complete. However, the SEQUENCE
F'
(-1)"
is weakly complete, even with any finite subsequence
deleted (Graham 1964).
see also COMPLETE SEQUENCE
References
Graham, R. "A Property of Fibonacci Numbers." Fib.
Quart 2, 1-10, 1964.
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., p. 128, 1985.
Weakly Independent
An infinite sequence {a*} of Positive Integers is
called weakly independent if any relation ^Ciai with
€i = or ±1 and u = 0, except finitely often, IMPLIES
€i — for all i.
see also STRONGLY INDEPENDENT
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p. 136, 1994.
Weakly Triple-Free Set
see Triple-Free Set
Web Graph
A graph formed by connecting several concentric
Wheel Graphs along spokes.
see also Wheel Graph
Weber Differential Equations
Weber Functions 1927
Weber Differential Equations
Consider the differential equation satisfied by
<1 -v 2 \
-1/2
w — z
w fc ,_ 1/4 (§A
where W is a WHITTAKER FUNCTION.
zdz
d(wz 1/2 )
zdz
d 2 w
R
+ (2fc-
2fc
^2
3 \ 1/2
4z 4 /
(1)
(2)
dz 2
This is usually rewritten
d 2 D n (z)
h 2 )w = 0.
(3)
+ {n+\-\z 2 )D n {z) = Q.
dz 2 ■ v- ■ 2 4
The solutions are Parabolic Cylinder Functions.
The equations
d 2 U
~ckt?
d 2 V
(4)
(c+k 2 u)U =
(5)
dv 3
+ (c-fcV)V = 0,
(6)
which arise by separating variables in LAP LACE'S EQUA-
TION in Parabolic Cylindrical Coordinates, are
also known as the Weber differential equations. As
above, the solutions are known as PARABOLIC CYLIN-
DER Functions.
Weber's Discontinuous Integrals
r™ (
/ Jq(clx) cos(cie) dx = <
Jo(ax) sin(cx) dx = < v <
I
y/al-cl
1
a < c
a > c
a < c
a > c,
where J (z) is a zeroth order Bessel Function of the
First Kind.
References
Bowman, F. Introduction to Bessel Functions. New York:
Dover, pp. 59-60, 1958.
Weber's Formula
1 (a 2 + 6 2 )/(4p 2 ) r
■■(*)
where 5R[i/] > -1, |argp| < 7r/4, and a, 6 > 0, J w (^) is
a Bessel Function of the First Kind, and I u {z) is
a Modified Bessel Function of the First Kind.
see also BESSEL FUNCTION OF THE FIRST KIND, MOD-
IFIED Bessel Function of the First Kind
References
lyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1476,
1980.
Weber Functions
Although Bessel Functions of the Second Kind
are sometimes called Weber functions, Abramowitz and
Stegun (1972) define a separate Weber function as
i r
S v (z) = - / s\n(v9 - z sin 6) d9.
* Jo
(1)
Letting Cn = e 2wi/m be a Root OF Unity, another set
of Weber functions is defined as
/(*) =
/!(*) =
/a(z) = \/2
*?(£(* +1))
r){\z)
r)(z)
V(2z)
ri(z)
72
73
[f M (z) + 8][h*(z)-f,*(*)]
(2)
(3)
(4)
(5)
(6)
(Weber 1902, Atkin and Morain 1993), where 77(2) is
the Dedekind Eta Function. The Weber functions
satisfy the identities
/(* + D = 'i w
C48
(7)
/i(* + D = ^
C48
(8)
/ 2 (z + l) = < 24 / 2 (z)
(9)
f(-l)-m
(10)
a(~) = /»w
(11)
h{-\)=Mz)
(12)
(Weber 1902, Atkin and Morain 1993).
see also ANGER FUNCTION, BESSEL FUNCTION OF
the Second Kind, Dedekind Eta Function, j-
FUNCTION, JACOBI IDENTITIES, JACOBI TRIPLE PROD-
UCT, Modified Struve Function, Q-Function,
Struve Function
e~» * Mat)Mbt)tdt, References
Abramowitz, M. and Stegun, C. A. (Eds.). "Anger and We-
ber Functions." §12.3 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 498-499, 1972.
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, pp. 68-69, 1987.
Weber, H. Lehrbuch der Algebra, Vols. I-II. New York:
Chelsea, pp. 113-114, 1902.
1928
Weber-Sonine Formula
Weekday
Weber-Sonine Formula
For 3t[pt, + nu] > 0, | argp| < 7r/4, and a > 0,
f
Jo
J u {at)e~ p * t
-p*t*.ti-i
dt
a\T[i(«/ + M )]
(*)
2pl 2p^r(i/ + 1)
iFi (l^ + ^iz+l;-
2p 2
where J^z) is a Bessel Function of the First
Kind, F(z) is the Gamma Function, and iFi(a;6;z)
is a Confluent Hypergeometric Function.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1474,
1980.
Weber's Theorem
If two curves of the same Genus (Curve) > 1 are in
rational correspondence, then that correspondence is Bl-
RATIONAL.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New-
York: Dover, p. 135, 1959.
Wedderburn's Theorem
A Finite Division Ring is a Field.
Weddle's Rule
The wedge product is ASSOCIATIVE
(s At) Au ~ s A (t Au), (6)
and Bilinear
(aisi + a 2 s 2 ) At = ai(si At) + a 2 (s 2 At) (7)
s A (aih + a 2 t 2 ) = ai(s A ti) +a 2 (sAi 2 ), (8)
but not (in general) COMMUTATIVE
sAt=(-l) pq {tAs), (9)
where s is a p-form and t is a g-form. For a 0-form s
and 1-form t,
(sAt)^ = st^. (10)
For a 1-form s and 1-form t,
yS A tjfj,i/ — 2 \S yXv Siyt^ij.
(ii)
The wedge product is the "correct" type of product to
use in computing a Volume Element
dV — dx\ A ... A dx n -
(12)
f{x)dx= ±h(fi +5/ 2 + h
see also Differential Form, Exterior Derivative,
Inner Product, Volume Element
Weekday
The day of the week W for a given day of the month D,
month M, and year 100C + Y can be determined from
the simple equation
+6/4 + 5/ 6 + fe + . . . + 5/ 6 „_i + hn) W = D+ [2.6M - 0.2J + [\y\ + [\C\ - 2C (mod 7) ,
see also Bode's Rule, Hardy's Rule, Newton-
Cotes Formulas, Simpson's 3/8 Rule, Simpson's
Rule, Trapezoidal Rule, Weddle's Rule
Wedge
A right triangular Prism turned so that it rests on one
of its lateral faces.
see also CONICAL WEDGE, CYLINDRICAL WEDGE,
Prism
Wedge Product
An antisymmetric operation on DIFFERENTIAL FORMS
(also called the Exterior Derivative)
where months are numbered beginning with March and
W = for Sunday, W = 1 for Monday, etc. (Uspensky
and Heaslet 1939, Vardi 1991).
A more complicated form is given by
W ^D + M + C + Y (mod 7) ,
where W = 1 for Sunday, W — 2 for Monday, etc. and
the numbers assigned to months, centuries, and years
are given in the tables below (Kraitchik 1942, pp. 110-
111).
Month
M
dxi A dxj = —dxj A dxi,
(i)
which Implies
dxi Adxi = (2)
hi A dxj = dxj Abi = bi dxj (3)
dxi A (bi dxj) = bi dxi A dxj (4)
#i A #2 = (61 dx\ + b 2 dx 2 ) A (a dxi + c 2 dx 2 )
= (&1C2 — b 2 ci) dx\ A dx 2
= -0 2 A6 1 . (5)
January 1
February 4
March 3
April 6
May 1
June 4
July 6
August 2
September 5
October
November 3
December 5
Weibull Distribution
Gregorian
Century C
15,
19, 23 1
16,
20, 24
17,
21, 25 5
18,
22, 26 3
Julian
Century C
00,
07, 14 5
01,
08, 15 4
02,
09, 16 3
03,
10, 17 2
04,
11, 18 1
05,
12, 19
06,
13, 20 6
Year
Y
00
06
17
23 28 34
45
01
07
12
18
29 35
40
46
1
02
13
19
24 30
41
47
2
03
08
14
25 31 36
42
3
09
15
20
26 37
43
48
4
04
10
21
27 32 38
49
5
05
11
16
22
33 39
44
50
6
51
56
62
73 79 84
90
57
63
68
74 85
91
96
1
52
58
69
75 80 86
97
2
53
59
64
70
81 87
92
98
3
54
65
71
76 82
93
99
4
55
60
66
77 83 88
94
5
61
67
72
78 89
95
6
see also FRIDAY THE THIRTEENTH
References
Kraitchik, M. "The Calendar." Ch. 5 in Mathematical Recre-
ations. New York: W. W. Norton, pp. 109-116, 1942.
Uspensky, J. V. and Heaslet, M. A. Elementary Number The-
ory. New York: McGraw-Hill, pp. 206-211, 1939.
Vardi, L Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, pp. 237-238, 1991.
Weibull Distribution
The Weibull distribution is given by
P(x) = af3 a x a 'e
D(x) = 1 - e- (l/ " )a
(1)
(2)
for x 6 [0,oo) (Mathematica® Statistics' Continuous
Distributions f WeibullDistribution[a,b] , Wolfram
Research, Champaign, IL). The Mean, Variance,
SKEWNESS, and KURTOSIS of this distribution are
M = /3r(l + a- 1 ) (3)
a 2 = /3 2 [T(1 + 2a" 1 ) - T 2 (l + a" 1 )] (4)
2r 3 (l + a" 1 ) - 3T(1 + a _1 )r(l + 2a" 1 )
Weierstrati-Casorati Theorem 1929
r(l + 3a" 1 )
+ [T(l + 2a- 1 ) - r 2 (i + a- 1 )] 3 / 2
, = m
72 [r(i + 2a- 1 )-r 2 (i-ha- 1 )] 2 '
where Y(z) is the Gamma Function and
(5)
(6)
/(a) = -6r 4 (l + a -1 ) + 12r 2 (l + a _1 )r(l + 2a" 1 )
-3r 2 (l + 2a -1 ) - 4r(l + a _1 )r(l + 3a"" 1 )
+r(l+4a~ 1 ). (7)
A slightly different form of the distribution is
P{x) = V-'e-"'^
D(x) = 1 - e-* a/0
(8)
(9)
(Mendenhall and Sincich 1995). The Mean and Vari-
ance for this form are
^/^rxi + O (io)
a 1 = f5 2/<x [V{l + 2a" 1 ) - T 2 (l + a" 1 )]. (11)
The Weibull distribution gives the distribution of life-
times of objects. It was originally proposed to quantify
fatigue data, but it is also used in analysis of systems
involving a "weakest link."
see also FlSHER-TlPPETT DISTRIBUTION
References
Mendenhall, W. and Sincich, T, Statistics for Engineering
and the Sciences, J^th ed. Englewood Cliffs, NJ: Prentice
Hall, 1995.
Spiegel, M. R. Theory and Problems of Probability and
Statistics. New York: McGraw-Hill, p. 119, 1992.
Weierstrafi Approximation Theorem
If / is continuous on [a, 6], then there exists a POLY-
NOMIAL p on [a, b] such that
\f(x)-P( X )\<€
for all x e [a, b] and e > 0. In words, any continuous
function on a closed and bounded interval can be uni-
formly approximated on that interval by POLYNOMIALS
to any degree of accuracy.
see also Muntz's Theorem
WeierstraB-Casorati Theorem
An Analytic Function approaches any given value
arbitrarily closely in any e-NEIGHBORHOOD of an Es-
sential Singularity.
7i
[r(i + 2a- 1 )-r 2 (i + a- 1 )p/2
1930 Weierstrafi Constant
Weierstrafi Constant
<\)=\ n
i
(0,0)
2 5 / 4 v^e-/ 8
2(m + ni)
l/[2(m + nz)]+l/[8(m + ni) 2 ]
= 0.4749493799....
r 2 (|)
References
Le Lionnais, F. Les nombres remarquobles. Paris: Hermann,
p. 62, 1983.
Plouffe, S. "Weierstrass Constant." http://lacim.uqam.ca/
piDATA/weier.txt.
Waldschmidt, M. "Fonctions entieres et nombres transcen-
dants." Cong. Nat. Soc. Sav. Nancy 5, 1978.
Waldschmidt, M. "Nombres transcendants et fonctions sigma
de Weierstrass." C. R. Math. Rep. Acad. Sci. Canada 1,
111-114, 1978/79.
Weierstrafi Elliptic Function
[wpp z\
The Weierstrafi elliptic functions are elliptic functions
which, unlike the JACOBI ELLIPTIC FUNCTIONS, have a
second-order POLE at z = 0. The above plots show the
Weierstrafi elliptic function p(z) and its derivative p'(z)
for invariants (defined below) of gi =0 and £3 = 0.
Weierstrafi elliptic functions are denoted p(z) and can
be defined by
m,TL= — 00
(z — 2muji — 2nuj2) 2
1
{2mw\ + 2nuj2) 2
• (i)
Write Qmn = 2rau>i + 2nu}2- Then this can be written
P (z) = z~ 2 + J^[(z - a™)" 2 - n~ 2 n ]. (2)
Weierstrafi Elliptic Function
An equivalent definition which converges more rapidly
is
of OO
00
E' 2 (nu 2 \
csc K^V
(3)
p(z) is an Even Function since p{—z) gives the
same terms in a different order. To specify p com-
pletely, its periods or invariants, written p(z\cji , ^2) and
p{ z \ 92,93)1 respectively, must also be specified.
The differential equation from which Weierstrafi elliptic
functions arise can be found by expanding about the
origin the function f(z) = p(z) — z~ 2 .
p(z) - z- 2 = /(0) + f(0)z + ±f"(0)z 2
+ ii/'"(0)/ + |/ (4) (0) Z 4 + .... (4)
But /(0) — and the function is even, so /'(0) =
/'"(0) = 0and
f(z) = p(z) - z- 2 = ±f"(0)z 2 + |/ (4) (0)z 4 + .
Taking the derivatives
/ / = ~2s / [(z-a mn )- 3 ]
/" = 6x f (z-n rnn )- 4
/'" = _24E'(z-n m „)- B
/ (4) = 120E'(z-Q mn )- 6 .
So
/"(0) = 6E'n,
/ (4) (0) = 120£'fl
mn
6
mn '
(5)
(6)
(7)
(8)
(9)
(10)
(11)
Plugging in,
p(z) - z~ 2 = 3E'n- 4 n2 2 + 5E'Q- 6 n z 4 + 0(z*). (12)
Define the Invariants
52 = 60E'fi-i
g 3 = 140E'n- 6 „,
then
(13)
(14)
(15)
(16)
p(z) = z- 2 + ±g 2 z 2 + ±g 3 z 4 + 0(z 6 )
p'(z) = -2z~ 3 + ±g 2 z + \g 3 z z + 0{z 5 ).
Now cube (15) and square (16)
p 3 (z) = z- 6 + ±g 2 z- 2 + ±g 3 + 0(z 2 ) (17)
WeierstraQ Elliptic Function
p'\z) = 4z~ 6 - \g 2 z- 2 - lg z + 0(z 2 ). (18)
Taking (18) - 4 x (17) cancels out the z~ 6 term, giving
p' 2 {z) -4p 3 (z)
= H-i)^- 2 + H-!)s3 + o(z 2 )
= -92Z-' 2 - 93 + 0(z 2 ) (19)
p'\z) - Ap\z) + g 2 z~ 2 + g 3 = 0{z 2 ). (20)
But, from (5)
p(z) = z~ 2 + £/'W + i/ t4) (0)z 4 + . . . , (21)
so p(z) = 2™ 2 + G(z 2 ) and (20) can be written
p t2 (z) - 4p 3 (z)+g 2 p(z)+g 3 - G(z 2 ). (22)
The WeierstraB elliptic function is analytic at the ori-
gin and therefore at all points congruent to the origin.
There are no other places where a singularity can oc-
cur, so this function is an ELLIPTIC FUNCTION with no
Singularities. By Liouville's Elliptic Function
Theorem, it is therefore a constant. But as z — > 0,
G(z 2 ) -+ 0, so
p 2 (z) = 4p 3 (z) - g 2 p(z) - g 3 . (23)
The solution to the differential equation
' 2 A 3
y =4y -g2y~gs
(24)
is therefore given by y = p(z -f a), providing that num-
bers a>i and U2 exist which satisfy the equations defin-
ing the Invariants. Writing the differential equation
in terms of its roots ei, e 2) and e 3;
V 2 = 4 y 3 - 92y ~ g 3 = 4(y - ei)(y - e 2 )(y - e 3 ) (25)
3
2 ln(y') = In 4 + ]T ln(y - c P ) (26)
^=y , E(»-er)- 1 (27)
2y"
r=l
3
S-Ec-*)- 1
y
(28)
. 'V-.W.-) .., £„_„,-. (29)
-^--j^- = -2>-
r— 1
(30)
WeierstraB Elliptic Function 1931
Now take (30)/4+ [(30)/4] 2 ,
J/ y__
2*/' 3 y /4
+
. " 2
4y>*
16
^(tf-Cr)- 1
(31)
3y^_j/l
V 4 2y> 3
i^( y - Cr )- 2 -fyJJ(y-c r )- 1 . (32)
The term on the right is half the Schwarzian Deriv-
ative.
The Derivative of the Weierstrafi elliptic function is
given by
= -2z" 3 -2^'(2-n m „)-
(33)
This is an Odd FUNCTION which is itself an elliptic func-
tion with pole of order 3 at z = 0. The Integral is
given by
poo
z= (4t 3 - g 2 t - g 3 )~ 1/2 dt. (34)
A duplication formula is obtained as follows.
p(2z) = lim p(y + z) - - lim
- p(*) - lim p(y)
p'(*) ~ p'
P(*) - p(v)
= ilhnf-
-Hb
P(z) ~ p'(z-\-h)
p(z) - p(z + h)
p'{z) - p'{z + h)
■ 2p(z)
r Um h iy
[h^o p(z) - p(« + /i)J J
■ 2p(z)
= 1 \ p"{z) '
4 Lp'W-
■2p(z).
(35)
A general addition theorem is obtained as follows. Given
p'(z)=Ap(z)+B
p'(y) = Ap(y) + B
(36)
(37)
with zero y and z where z ^ ±y (mod 2u>i, 2u> 2 ), find the
third zero £. Consider p'(() - Ap(Q - B. This has a
pole of order three at £ = 0, but the sum of zeros (= 0)
equals the sum of poles for an ELLIPTIC FUNCTION, so
z + y + C = and C = —z - y.
o(-z-y) = Ap(-z-y) + B
(38)
1932 Weierstraft Elliptic Function
-P(z + V) =Ap(z + y) + B.
Combining (36), (37), and (39) gives
p(z) p'(z) 1
p{y) p'(y) l
p{z + y) -p{z + y) i.
r a i
ro]
-l
=
. B .
_o_
so
p{z) p'{z) 1
p(y) p'(y) i
p{z + y) -p{z + y) i
= o.
(39)
(40)
(41)
Denning u + v + w = where n = z and u = y gives the
symmetric form
p(u) p'(u) 1
p(v) p'(v) 1
p(w) p(w) 1
= 0.
To get the expression explicitly, start again with
(42)
(43)
p'(C) - Ap(C) -B = 0,
where £ = z, y, — z — y.
p'\O-[Ap(<:) + B] 2 =0. (44)
But p 2 (C) = 4p 4 (C) - <?2P(C) - A3, so
4p 3 (0-A 2 p 2 (0-(2AB+g 2 )p(0-(B 2 +g 3 ) = 0. (45)
The solutions p(C) = ^ are given by
4/ - AV - (2AB + g 2 )z - (B 2 + g 3 ) = 0. (46)
But the sum of roots equals the Coefficient of the
squared term, so
i a*
pi*) + p(y) + p(* + y) = £a
p'W - p'(y) = ^b(^) - p(y)]
^_ p'(z)-p'(a)
p(*) - p(y)
(47)
(48)
(49)
(* + y) =
pQ) - p'(y)
p(*0 - p(s/)
-pW-p(y). (50)
Half-period identities include
— /i \ / i. . \ , ( e i - e 2)(ei - e 3 )
p(-2 w 0-ei
:ei +
(ei - e 2 )(ei - e 3 )
x — ei
Multiplying through,
x - eix = eix - ei 2 + (ei - e 2 )(ei - e 3 )
(51)
(52)
WeierstraR Elliptic Function
which gives
P(|^i) = § { 2e i ± V 4e i 2 " 4 I e i 2 " ( e i " e 2)(ei - e 3 )]}
= ei ± \/( e i -e 2 )(ei - e 3 ). (54)
From Whittaker and Watson (1990, p. 445),
p'(|^i) = ~2^(e! - e 2 )(ei - e 3 )
x(Vei - e 2 + \/ei - el). (55)
The function is HOMOGENEOUS,
p(A;z|Au;i, Au/ 2 ) = \~ 2 p(z\w l7 U2) (56)
p(A^; A~ 4 # 2 , A~ 6 p 3 ) = A" 2 p(z;5 2 ,5 3 ). (57)
To invert the function, find 2u>i and 2u> 2 of p(z|u;i,a; 2 )
when given p(z;g2,gz). Let ei, e 2 , and e 3 be the roots
such that (ei — e 2 )/(ei — e 3 ) is not a Real Number > 1
or < 0. Determine the Parameter t from
Now pick
ei~e 2 = ^4 4 (0|r)
ei-e 3 tf 3 4 (0|r)*
_ Vei - e 2
" ^4 2 (0|r)'
As long as # 2 3 7^ 27p 3 , the periods are then
2u;i = 7T^4
2u; 2
(58)
(59)
(60)
(61)
WeierstraB elliptic functions can be expressed in terms
of Jacobi Elliptic Functions by
p(u;g2,gz) = e 3 + (ei - e 3 )
xns I u^/ei — e 3 ,
where
e 2 - e 3
ei - e 3
p(^i) = ei
p(^ 2 ) = e 2
p(v 3 ) — -p(~oj± - (jj 2 ) = e 3 ,
and the INVARIANTS are
<? 2 = 60E'n- 4 n
<? 3 = 140S'Q- 6 n .
(62)
(63)
(64)
(65)
(66)
(67)
2e x + [ ei 2 - (ei - e 2 )(ei - e 3 )] = 0, (53)
Weierstraft Elliptic Function
An addition formula for the Weierstrafi elliptic function
can be derived as follows.
p(z + wi) + p(z) + p(wi)
p'(z) -p'(u>i)
p(z) - p{u>x)
1 P%)
4 [p(z) - ei ]2
• (68)
Use
r=l
(69)
(z + wi) = -p(^) - ei +
= -p(z) - ei +
4 [p(*) - ei ]2
[p(^)-e2][p(g)-e 3 ] ^
(70)
Use X)Li ^ = °>
p(z + u)i) =ei +
-2ei-p(z)][p(z)-ei]
p(*0 - ei
|2 (^) ~ P( z )( e 2 + e 3 ) + e 2 e 3
ei +
-p(z)(ei + e2 + e 3 ) + e 2 e 3 +2ei 2
(71)
p(*) - e i
But X^ 3 =i e T- = an d
2ei 2 + e 2 e 3 = ei -ei(e 2 +e3) + e2e 3 = (ei -e 2 )(ei -63),
(72)
SO
/ , \ 1 ( ei ~ e 2)(ei -e3) f „ Q ,
P(* + "0 = ei + p(z)- ei • (73)
The periods of the Weierstrafi elliptic function are given
as follows. When <? 2 and g$ are Real and # 2 3 — 27# 3 2 >
0, then ei, e 2) and e 3 are Real and defined such that
ei > e 2 > e 3 -
wi
*/ ei
(4i J -g 2 t-g z )- l,i dt
(74)
«3
ua = -i {93 + 92t - 4t 3 y 1/2 dt (75)
J — oo
a; 2 = — uj\ — W3*
(76)
The roots of the Weierstrafi elliptic function satisfy
ei = p(wi) (77)
e 2 - p(w a ) (78)
Weierstrafi Function 1933
where o; 3 = -u;i-u;2. The e^s are ROOTS of 4t 3 -Q2t-Qz
and are unequal so that ei ^ e 2 ^ e 3 . They can be
found from the relationships
ei + e 2 + e 3 = -a 2 =
(80)
e 2 e 3 + e 3 ei + eie 2 = ai = - \g2 (81)
eie 2 e3 = — a = ^^3.
(82)
see also Equianharmonic Case, Lemniscate Case,
PSEUDOLEMNISCATE CASE
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Weierstrass
Elliptic and Related Functions." Ch. 18 in Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 627-671, 1972.
Fischer, G. (Ed.)- Plates 129-131 in Mathematische Mod-
elle/ Mathematical Models, Bildband/ Photograph Volume.
Braunschweig, Germany: Vieweg, pp. 126-128, 1986.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, ^th ed, Cambridge, England: Cambridge Uni-
versity Press, 1990.
Weierstrafi- Erdman Corner Condition
In the Calculus of Variations, the condition
fy'(x,y,y(B-)) = /y'(a»3M/'(3+))
must hold at a corner (s,y) of a minimizing arc £?i2.
Weierstrafi Extreme Value Theorem
see Extreme Value Theorem
Weierstrafi Form
A general form into which an ELLIPTIC CURVE over any
Field K can be transformed is called the Weierstrafi
form, and is given by
y 2 + ay = x 3 + bx 2 + cxy + dx + e,
where a, 6, c, d, and e are elements of K.
Weierstrafi Function
e 3 = p(u; 3 ),
(79)
1934
Weierstra&s Gap Theorem
WeierstraB Sigma Function
A Continuous Function which is nowhere Differ-
ENTIABLE. It is given by
such that
\u n (x)\ < M n
f(x) =y b n cos(a n 7rx)
where n is an Odd Integer, b G (0, 1), and ab > 1 +
3tt/2. The above plot is for a = 10 and 6 = 1/2.
see also Blancmange Function, Continuous Func-
tion, DlFFERENTIABLE
References
Darboux, G. "Memoir sur les fonctions discontinues." Ann.
lEcole Normale, Ser. 2 4, 57-112, 1875.
Darboux, G. "Memoir sur les fonctions discontinues." Ann.
lEcole Normale, Ser. 2 8, 195-202, 1879.
du Bois-Reymond, P. "Versuch einer Klassification der
willkiirlichen Functionen reeller Argumente nach ihren
Anderungen in den kleinsten Intervallen." J. fur Math.
79, 21-37, 1875.
Faber, G. "Einfaches Beispiel einer stetigen nirgends differen-
tiierbaren Funktion." Jahresber. Deutschen Math. Verein.
16 538-540, 1907,
Hardy, G. H. "Weierstrass's Non-Differentiable Function,"
Trans. Amer. Math. Soc. 17, 301-325, 1916.
Landsberg, G. "Uber Differentzierbarkeit stetiger Funktio-
nen." Jahresber. Deutschen Math. Verein. 17, 46—51,
1908.
Lerch, M. "Uber die Nichtdifferentiirbarkeit gewisser Func-
tionen." J. reine angew. Math. 13, 126-138, 1888.
Pickover, C. A. Keys to Infinity. New York: W. H. Freeman,
p. 190, 1995.
WeierstraB, K. Abhandlungen aus der Functionenlehre. Ber-
lin: J. Springer, p. 97, 1886.
WeierstraB's Gap Theorem
Given a succession of nonsingular points which are on a
nonhyperelliptic curve of Genus p, but are not a group
of the canonical series, the number of groups of the first
k which cannot constitute the group of simple POLES
of a Rational Function is p. If points next to each
other are taken, then the theorem becomes: Given a
nonsingular point of a nonhyperelliptic curve of GENUS
p, then the orders which it cannot possess as the single
pole of a Rational Function are p in number.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 290, 1959.
WeierstraB Intermediate Value Theorem
If a continuous function defined on an interval is some-
times Positive and sometimes Negative, it must be
at some point.
for all x £ E, then the series exhibits ABSOLUTE CON-
VERGENCE for each x € E as well as UNIFORM CON-
VERGENCE in E.
see also Absolute Convergence, Uniform Conver-
gence
References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Or-
lando, FL: Academic Press, pp. 301-303, 1985.
WeierstraB Point
A Pole of multiplicity less than p + 1.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, pp. 290-291, 1959.
WeierstraB's Polynomial Theorem
A function, continuous in a finite close interval, can be
approximated with a preassigned accuracy by POLYNO-
MIALS. A function of a Real variable which is continu-
ous and has period 2ix can be approximated by trigono-
metric Polynomials.
References
Szego, G. Orthogonal Polynomials, J^th ed. Providence, RI:
Amer. Math. Soc, p. 5, 1975.
WeierstraB Product Inequality
If < a, 6, c, d < 1, then
(l-a)(l-&)(l-c)(l-d) + a + 6 + c + d> 1.
References
Honsberger, R. Mathematical Gems III. Washington, DC:
Math. Assoc. Amer., pp. 244-245, 1985.
WeierstraB Sigma Function
The QUASIPERIODIC FUNCTION defined by
dz
In«r(z) = C(*),
(1)
where £(z) is the WeierstraB Zeta Function and
lim
(2)
Then
WeierstraB M-Test
Let Xlfcli Un ( x ) b e a SERIES of functions all defined for
a set E of values of x. If there is a CONVERGENT series
of constants
£
M n
w-'nfc-sb)"*
a(z + 2wi)
a{z + 2w 2 )
2tji(z+wi)
Z Z
-I —
a{z)
«x(z)
(3)
(4)
(5)
Weierstrass Theorem
Weighings 1935
r(z) =
e~ VrZ a(z + a; r )
for r = 1, 2, 3.
<t(z|u/i,o;2) = ^T ex P
where v = 7r;z/(2u;i), and
»7i = -
6#i
7T 2 <
12wit?i
V2 = -
7r 2 u) 2 -i?'i" _ _™_
(6)
(7)
(8)
(9)
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Weierstrass
Elliptic and Related Functions." Ch. 18 in Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 627-671, 1972.
Weierstrafi's Theorem
The only hypercomplex number systems with commu-
tative multiplication and addition are the algebra with
one unit such that e = e 2 and the GAUSSIAN INTEGERS.
see also Gaussian Integer, Peirce's Theorem
Weierstrafi Zeta Function
The Quasiperiodic Function defined by
dC(z)
dz
-P(z)
with
lim[({z) - z' 1 } = 0.
(1)
(2)
Then
cw
- z l = - / [p(z) - z 2 ]dz
Jo
= -s' / [(* - n mn y 2 - n~ 2 n ] dz (3)
JO
oo
C(z) = z- 1 + J^ [(z - Q^y 1 + U^ n + zQ- 2 n ] (4)
m,n=^~- oo
so £(z) is an Odd Function. Integrating p(z-\-2u;i) =
p{z) gives
C(* + 2wi) = C(*) + 2i7i. (5)
Letting z — -u>i gives ((— u>i) + 2??i = ~C(^i) + 2??i, so
771 = C{oj\). Similarly, 772 = C(^2). From Whittaker and
Watson (1990),
1
p(s)
P 2 {x)
1
p(y)
P 2 (y)
1
P(z)
P 2 (z)
1
p(a0
P(x)
1
p(y)
p'(v)
1
pM
p'W
If a: + y + z = 0, then
K(*) + C(v) + C(*)] 2 + C'(x) + C'(y)C'(z) = 0. (7)
Also,
= t( x + y + z)-t(x)-{(y)-C(z)
(8)
(Whittaker and Watson 1990, p. 446).
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Weierstrass
Elliptic and Related Functions." Ch. 18 in Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
pp. 627-671, 1972.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, J^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Weighings
n weighings are Sufficient to find a bad Coin among
(3 n - l)/2 Coins, vos Savant (1993) gives an algorithm
for finding a bad ball among 12 balls in three weighings
(which, in addition, determines if the bad ball is heavier
or lighter than the other 11).
Bachet's weights problem asks for the minimum number
of weights (which can be placed in either pan of a two-
arm balance) required to weigh any integral number of
pounds from 1 to 40. The solution is 1, 3, 9, and 27: 1,
2 = -1 + 3, 3, 4 = 1 + 3, 5 = -1 - 3 + 9, 6 = -3 + 9,
7= 1-3 + 9, 8= -1 + 9, 9, 10= 1 + 9, 11 = -1 + 3 + 9,
12 = 3 + 9, 13 = 1 + 3 + 9, 14 = -1-3-9 + 27,
15 = -3 - 9 + 27, 16 = 1 - 3 - 9 + 27, 17 = -1 - 9 + 27,
and so on.
see also Golomb Ruler, Perfect Difference Set,
Three Jug Problem
References
Bachet, C. G. Problem 5, Appendix in Problemes plaisans et
delectables, 2nd ed. p. 215, 1624.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 50-52,
1987.
Kraitchik, M. Mathematical Recreations. New York:
W. W. Norton, pp. 52-55, 1942.
Pappas, T. "Counterfeit Coin Puzzle." The Joy of Mathe-
matics. San Carlos, CA: Wide World Publ./Tetra, p. 181,
1989.
Tartaglia. Book 1, Ch. 16, §32 in Trattato de' numeri e
misure, Vol. 2. Venice, 1556.
vos Savant, M. The World's Most Famous Math Problem.
New York: St. Martin's Press, pp. 39-42, 1993.
77l (jJ2 — 772<^i
(6)
1936 Weight
Welch Apodization Function
Weight
The word weight has many uses in mathematics. It
can refer to a function w(x) (also called a WEIGHTING
Function or Weight Function) used to normalize
ORTHONORMAL Functions. It can also be used to in-
dicate one of a set of a multiplicative constants placed in
front of terms in a Moving Average, Newton-Cotes
Formulas, edge or vertex of a Graph or Tree, etc.
see also WEIGHTED TREE, WEIGHTING FUNCTION
Weight Function
see Weighting Function
Weighted Tree
A Tree in which each branch is given a numerical
Weight (i.e., a labelled Tree).
see also Labelled Graph, Taylor's Condition,
Tree
Weighting Function
A function w(x) used to normalize ORTHONORMAL
Functions
/
[f n (x)] 2 w(x) dx = N n .
see also Weight
Weingarten Equations
The Weingarten equations express the derivatives of the
NORMAL using derivatives of the position vector. Let
x : U -> E 3 a Regular Patch, then the Shape Op-
erator S of x is given in terms of the basis {x u ,x v }
by
m n tvt fF-eG eF-fE
EG-F 2
-S(x„) = N v = ^ x u +
EG-F 2 '
fF-gE^
EG-F 2 '
(1)
»> (2)
where N is the NORMAL VECTOR, E, F, and G the
coefficients of the first Fundamental Form
ds 2 = E du + IF dudv + G dv 2 ,
(3)
and e, /, and g the coefficients of the second FUNDA-
MENTAL FORM given by
e = -N u ■ x u = N • x uu (4)
J —- IN v ' X-n ^ IN * Xyu
g — -N v ■ x v = N • x. vv . (6)
see also Fundamental Forms, Shape Operator
References
Gray, A. "Calculation of the Shape Operator." §14.3 in Mod-
ern Differential Geometry of Curves and Surfaces. Boca
Raton, FL: CRC Press, pp. 274-277, 1993.
Weingarten Map
see Shape Operator
Weird Number
A number which is ABUNDANT without being Semiper-
fect. (A Semiperfect Number is the sum of any
set of its own DIVISORS.) The first few weird numbers
are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, . . .
(Sloane's A006037). No Odd weird numbers are known,
but an infinite number of weird numbers are known to
exist. The SEQUENCE of weird numbers has POSITIVE
Schnirelmann Density.
see also ABUNDANT NUMBER, SCHNIRELMANN DEN-
SITY, Semiperfect Number
References
Benkoski, S. "Are All Weird Numbers Even?" Amer. Math.
Monthly 79, 774, 1972.
Benkoski, S. J. and Erdos, P. "On Weird and Pseudoperfect
Numbers." Math. Comput. 28, 617-623, 1974.
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. Sequence A006037/M5339 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Welch Apodization Function
I(k) = a2V27r
The Apodization Function
x 2
A{x) = l-^.
a 2
Its Full Width at Half Maximum is a/2 a. Its In-
strument Function is
J 3 / 2 (27r/ca)
(27rfca) 3 / 2
sin(27rfca) — 27rafccos(27ra/c)
= a 2a 3 fc 3 7r 3 '
where J u (z) is a Bessel Function of the First
Kind. It has a width of 1.59044, a maximum of |, maxi-
mum Negative sidelobe of -0.0861713 times the peak,
and maximum POSITIVE sidelobe of 0.356044 times the
peak.
see also Apodization Function, Instrument Func-
tion
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. Numerical Recipes in FORTRAN: The Art of
Scientific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, p. 547, 1992.
Well-Defined
Wheel 1937
Well-Defined
An expression is called well-defined (or UNAMBIGUOUS)
if its definition assigns it a unique interpretation or
value. Otherwise, the expression is said to not be well
defined or to be AMBIGUOUS.
For example, the expression abc (the PRODUCT) is well-
defined if a, 5, and c are integers. Because integers are
ASSOCIATIVE, abc has the same value whether it is in-
terpreted to mean (ab)c or a(bc). However, if a, b, and
c are Matrices or Cayley Numbers, then the expres-
sion abc is not well-defined, since Matrices and Cay-
ley Number are not, in general, Associative, so that
the two interpretations (ab)c and a(bc) can be different.
Sometimes, ambiguities are implicitly resolved by no-
tat ional convention. For example, the conventional in-
terpretation of a A b A c = a b is a^ b \ never (a 6 ) c , so
that the expression a A b A c is well-defined even though
exponentiation is nonassociative.
Well-Ordered Set
A SET having the property that every nonempty SUBSET
has a least member.
see also Axiom of Choice, Hubert's Problems,
Subset, Well-Ordering Principle
Well-Ordering Principle
Every nonempty set of POSITIVE integers contains a
smallest member.
see also Well-Ordered Set
References
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, p. 149, 1993.
Werner Formulas
2 sin a cos = sin(a — 0) + sin(a + 0) (1)
2 cos a cos = cos(a — 0) + cos(a + 0) (2)
2 cos a sin = sin(a + 0) — sin(a — 0) (3)
2 sin a sin = cos(a — 0) — cos(a + 0). (4)
see also TRIGONOMETRY
Weyl Tensor
The TENSOR
C ij ki = R'hi ~ 2* li [ fejr ] I] + §<* [ VV>
where R { j kl is the Riemann Tensor and R is the Cur-
vature Scalar. The Weyl tensor is denned so that
every CONTRACTION between indices gives 0. In partic-
ular, C X p\ K = 0. The number of independent compo-
nents for a Weyl tensor in N-D is given by
C N = ±N(N + 1)(N + 2)(N - 3).
see also Curvature Scalar, Riemann Tensor
References
Weinberg, S. Gravitation and Cosmology: Principles and
Applications of the General Theory of Relativity, New
York: Wiley, p. 146, 1972.
Weyrich's Formula
Under appropriate constraints,
2 J-ao Vr 2 +x 2
where H^\z) is a Hankel Function of the First
Kind.
References
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary
of Mathematics. Cambridge, MA: MIT Press, p. 1474,
1980.
Wheat and Chessboard Problem
Let one grain of wheat be placed on the first square of a
Chessboard, two on the second, three on the third, etc.
How many grains total are placed on an 8 x 8 Chess-
board? Since this is a GEOMETRIC SERIES, the answer
for n squares is
^2 2 i = T - 1.
Weyl's Criterion
A Sequence {xi, x 2 , . . .} is Equidistributed Iff
li m 1 y c a«™» =
n<N
for each m — 1, 2, . . . .
see also EQUIDISTRIBUTED SEQUENCE, RAMANUJAN'S
Sum
References
Polya, G. and Szego, G. Problems and Theorems in Analysis
I. New York: Springer- Verlag, 1972.
Vardi, I. Computational Recreations in Mathematica. Red-
wood City, CA: Addison- Wesley, pp. 155-156 and 254,
1991.
Plugging in n = 8 x 8 = 64 then gives 2 b - 1 =
18446744073709551615.
References
Pappas, T. "The Wheat and & Chessboard." The Joy of
Mathematics. San Carlos, CA: Wide World Publ./Tetra,
p. 17, 1989.
Wheel
see Aristotle's Wheel Paradox, Benham's
Wheel, Wheel Graph
1938 Wheel Graph
Wheel Graph
Whitehead Link
w.
w c
w.
T 4 "5 "6
A Graph W n of order n which contains a Cycle of
order n — 1, and for which every NODE in the cycle is
connected to one other NODE (known as the Hub). In
a wheel graph, the Hub has DEGREE n — 1, and other
nodes have degree 3. Wa = K±, where K<± is the Com-
plete Graph of order four.
see also Complete Graph, Gear Graph, Hub, Web
Graph
Wheel Paradox
see Aristotle's Wheel Paradox
Whewell Equation
An Intrinsic Equation which expresses a curve in
terms of its Arc Length s and Tangential Angle
0.
see also ARC LENGTH, CESARO EQUATION, INTRINSIC
Equation, Natural Equation, Tangential Angle
References
Yates, R. C. "Intrinsic Equations." A Handbook on Curves
and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 123-126, 1952.
Whipple's Transformation
iFq
a, 1 + 2 Q") b-> c, <i, e, — m
|a, 1-j-a — 6, 1 + a — c,
1 + a — d, 1 + a — e,l + a + m_
- (1 + a)m(l + a - d- e)m
(1 + a - d) m (l + a - e)m
1 + a — b — c, d, e, — m
1 + a — 6, 1 + a — c, d + e — a — m
X4F3
where 7 F 6 and 4^3 are Generalized Hypergeomet-
ric Functions and T(z) is the Gamma Function.
see also Generalized Hypergeometric Function
Whirl
Whirls are figures constructed by nesting a sequence of
polygons (each having the same number of sides), each
slightly smaller and rotated relative to the previous one.
see also Daisy, Swirl
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Fig-
ures. Princeton, NJ: Princeton University Press, p. 66,
1991.
Pappas, T. "Spider & Spirals." The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, p. 228, 1989.
^ Weisstein, E. W. "Fractals." http: //www. astro. Virginia.
edu/-eww6n/math/notebooks/Fractal.m.
Whisker Plot
see Box-and- Whisker Plot
Whitehead Double
The Satellite Knot of an Unknot twisted inside a
Torus.
see also Satellite Knot, Torus, Unknot
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, pp. 115-116, 1994.
Whitehead Link
The Link 5q?, illustrated above, with Braid Word
a 1 2 a 2 2 cr 1 ~ 1 a2~ 2 and JONES POLYNOMIAL
V(t) = r 3/2 (-i + t - 2t 2 + t 3 - 2t 4 + t 5 ).
The Whitehead link has LINKING NUMBER 0.
Whitehead Manifold
Whitney-Mikhlin Extension Constants 1939
Whitehead Manifold
An open 3-MANIFOLD which is simply connected but is
topologically distinct from Euclidean 3-space.
References
Rolfsen, D. Knots and Links. Wilmington, DE: Publish or
Perish Press, p. 82, 1976.
Whitehead's Theorem
Maps between CW-Complexes that induce Isomor-
phisms on all HOMOTOPY GROUPS are actually HOMO-
TOPY equivalences.
see also CW-COMPLEX, HOMOTOPY GROUP, ISOMOR-
PHISM
Whitney-Graustein Theorem
A 1937 theorem which classified planar regular closed
curves up to regular HOMOTOPY by their WINDING
Numbers. In his thesis, S. Smale generalized this re-
sult to regular closed curves on an n-MANIFOLD.
Whitney-Mikhlin Extension Constants
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let B n (r) be the n-D closed BALL of RADIUS r > 1
centered at the ORIGIN. A function which is defined
on B(r) is called an extension to B(r) of a function /
defined on JB(1) if
F{x) = / (x)V x € B(l).
(1)
Given 2 Banach Spaces of functions defined on B(l)
and B(r), find the extension operator from one to the
other of minimal norm. Mikhlin (1986) found the best
constants x such that this condition, corresponding to
the Sobolev W(l,2) integral norm, is satisfied,
\
B(l)
im? + J2
i=i
dl
dxj
dx
<x
\
I
JB(r
X (l,r) = l. Let
then for n > 2,
X{n,r) = \ll +
i/=±(n-2),
dx . (2)
(3)
/„(!) I„(r)K v+1 {l) + K„(r)I„ +1 {l)
7„+i(l) I„(r)K v (l) - K„(t)I v (1)
(4)
where I v (z) is a Modified Bessel Function of the
First Kind and K„(z) is a Modified Bessel Func-
tion of the Second Kind. For n = 2,
x(2,r) = max
1 +
7,(1) J„(r)ir„ + i(l) + K v {r)I v+i (\)
I„+i(l) /„(r)tf„(l) - jr„(r)I„(l)
/i(l)
/i(l) + / a (l)
1 +
Ji(r)Jg- (l) + Jri(r)/o(l)
7 1 (r-)K 1 (l)-7G(r)I 1 (l)
For r -> oo,
X(n, oo) = */l +
^(1) K„{1)
i v+1 {\)K v {iy
which is bounded by
n - 1 < x(", oo) < y/(n- l) 2 +4.
For Odd n, the Recurrence Relations
afc+i = an-i — (2k — l)a,k
bk+i = b k -i + (2k - l)b k
with
e + e
e — e
-l
1 _i_ ak kfc-M
ao
a x
bo — e
bi = e' 1
where e is the constant 2.71828. . . , give
x(2fc + l,oo)
The first few are
x(3,oo) = e
x(5,oo) =
x(7,oo) =
x(9,oo) =
e 2 -7
7 V 37 -5e 2
18e 2 - 133
X(11,00) " v / 133V2431-329e 2
x(13,oo) =
2431 V 3655e 2 - 27007
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
Similar formulas can be given for even n in terms of
7o(l),Ji(l),ffo(l),tfi(l).
References
Finch, S. "Favorite Mathematical Constants." http://www.
mathsoft.com/asolve/constant/mkhln/mkhln.html.
Mikhlin, S. G. Constants in Some Inequalities of Analysis.
New York: Wiley, 1986.
1940 Whitney Singularity
Whittaker Differential Equation
Whitney Singularity
see Pinch Point
Whitney Sum
An operation that takes two VECTOR BUNDLES over a
fixed SPACE and produces a new VECTOR BUNDLE over
the same SPACE. If E x and E 2 are VECTOR BUNDLES
over B, then the Whitney sum E\ © E2 is the VECTOR
Bundle over B such that each Fiber over B is naturally
the direct sum of the E\ and E 2 FIBERS over B.
The Whitney sum is therefore the FIBER for FIBER di-
rect sum of the two BUNDLES E\ and E 2 . An easy for-
mal definition of the Whitney sum is that E\ © E 2 is
the pull-back BUNDLE of the diagonal map from B to
B x B, where the Bundle over 5xBisEixE 2 .
see also Bundle, Fiber, Vector Bundle
Whitney Umbrella
giving Gaussian Curvature and Mean Curvature
A surface which can be interpreted as a self-intersecting
Rectangle in 3-D. It is given by the parametric equa-
tions
x = uv (1)
y = u (2)
z = v 2 (3)
for u,v e [—1,1]. The center of the "plus" shape which
is the end of the line of self-intersection is a Pinch
POINT. The coefficients of the first FUNDAMENTAL
Form are
E =
F
2v
Vu 2 + 4v 2 + 4v 4
2u
Vu 2 + 4v 2 + 4v 4 '
(4)
(5)
(6)
and the coefficients of the second FUNDAMENTAL FORM
are
K = -
H = -
4v 2
(u 2 + 4v 2 +4v A ) 2
u(l + 3v 2 )
(u 2 + 4v 2 +4v 4 ) 3 / 2 '
(10)
(11)
References
Francis, G. K. A Topological Picturebook. New York:
Springer- Verlag, pp. 8-9, 1987.
Geometry Center. "Whitney's Umbrella." http://vww.
geom.uinn.edu/zoo/features/whitney/.
Gray, A. Modern Differential Geometry of Curves and Sur-
faces. Boca Raton, FL: CRC Press, pp. 225 and 309-310,
1993.
(1)
Whittaker Differential Equation
d 2 u du (k \ -m 2 \
dz 2 dz \ z z 2 J
Let u = e~ z/2 Wk ,77i (z), where Wk, m (z) denotes a Whit-
taker FUNCTION. Then (1) becomes
iL(_ \ e -* f2 W + e- z/2 W) + (-\e~ z/2 W + e~ z/2 W')
+ (* + l^).-"*-a
(2)
Rearranging,
(l e ~ z/2 W - \e~ z/2 W' - \e- z/ 'W + e- z/2 W")p
+{-\e~ z/2 W + e- z/2 W) +(* + i^) e~ z/2 W
= (3)
■\e~ z/2 W + e
-l* W » +f^ + i-^-) e~ z ' 2 W = 0,
SO
1 . k . z-'
W"+[-- + - +
4 z
(4)
W = 0, (5)
where W r = dW/dz. The solutions are known as WHIT-
TAKER Functions.
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook
of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, 9th printing. New York: Dover,
p. 505, 1972.
e = 1 + v
f = uv
9
2 , A 2
u + 4v ,
(7)
(8)
(9)
Whittaker Function
Wieferich Prime 1941
Whittaker Function
Solutions to the WHITTAKER DIFFERENTIAL EQUA-
TION. The linearly independent solutions are
M fc , m (z)EE^ 2+ ™e-*/ 2
x I ! + -2+™- k . (|+™-fc)(f+™- fc ) . a ,
1 l!(2m + 1) 2!(2m + l)(2m + 2)
■)•
(i)
and M k ,-m(z), where M fc , m (z) is a Confluent Hyper-
geometric Function. In terms of Confluent Hy-
pergeometric Functions, the Whittaker functions
are
Af fc , m (z) = e-* /2 z m+1/ Vi(| +m-fc,l + 2m;2) (2)
W k , m (z) = e- z/2 z m+1/2 U(± +m-fc,l + 2m;z) (3)
(see Whittaker and Watson 1990, pp. 339-351). How-
ever, the Confluent Hypergeometric Function
disappears when 2m is an INTEGER, so Whittaker func-
tions are often defined instead. The Whittaker functions
are related to the PARABOLIC CYLINDER FUNCTIONS.
When |arg z\ < 37r/2 and 2m is not an INTEGER,
Wk. m (z) = wi r( ~ 2m) .. M h , m {z)
T { 2 -m-k)
r(i+m-fc)
When | arg(-z)| < 37r/2 and 2m is not an INTEGER,
+ r/i 2m l^ M -"-^- x ^ (5)
T(i + m + fc)
Whittaker functions satisfy the RECURRENCE RELA-
TIONS
W k , m (z) = z 1/2 W k - 1/2 , m - 1/2 (z)+(±-k+m)W k -i, m {z)
(6)
W h , m {z) = z 1/2 W k ^ /2trn+1/2 (z) + (h-k-^) W k-iM^)
(7)
zW' k , m (z) = (*-^)Wfc, m (z)-(m a -(*-i) a ]Wfc-i, ra (z).
(8)
see also CONFLUENT HYPERGEOMETRIC FUNCTION,
Rummer's Formulas, Pearson-Cunningham Func-
tion, Schlomilch's Function, Sonine Polynomial
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Confluent Hy-
pergeometric Functions." Ch. 13 in Handbook of Mathe-
matical Functions with Formulas, Graphs, and Mathemat-
ical Tables, 9th printing. New York: Dover, pp. 503-515,
1972.
Iyanaga, S. and Kawada, Y. (Eds.). "Whittaker Functions."
Appendix A, Table 19.11 in Encyclopedic Dictionary of
Mathematics. Cambridge, MA: MIT Press, pp. 1469-1471,
1980.
Whittaker, E. T. and Watson, G. N. A Course in Modern
Analysis, l^th ed. Cambridge, England: Cambridge Uni-
versity Press, 1990.
Whole Number
One of the numbers 1, 2, 3, . . . (Sloane's A000027), also
called the Counting Numbers or Natural Numbers.
is sometimes included in the list of "whole" numbers
(Bourbaki 1968, Halmos 1974), but there seems to be no
general agreement. Some authors also interpret "whole
number" to mean "a number having FRACTIONAL Part
of zero," making the whole numbers equivalent to the
integers.
Due to lack of standard terminology, the following terms
are recommended in preference to "COUNTING NUM-
BER," "Natural Number," and "whole number."
Set
Name
Symbol
...,-2, -1,0, 1,2,...
1,2,3,4,...
0,1,2,3,4...
-1,-2, -3,-4,...
integers
positive integers
nonnegative integers
negative integers
see also Counting Number, Fractional Part, In-
teger, M, Natural Number, Z, Z + , Z + , Z*
References
Bourbaki, N. Elements of Mathematics: Theory of Sets.
Paris, France: Hermann, 1968.
Halmos, P. R. Naive Set Theory. New York: Springer- Verlag,
1974.
Sloane, N. J. A. Sequence A000027/M0472 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Width (Partial Order)
For a Partial Order, the size of the longest An-
tichain is called the width.
see also Antichain, Length (Partial Order), Par-
tial Order
Width (Size)
The width of a box is the horizontal distance from side
to side (usually defined to be greater than the DEPTH,
the horizontal distance from front to back).
see also Depth (Size), Height
References
Eppstein, D. "Width, Diameter, and Geometric Inequali-
ties." http://www . ics . uci . edu/ -eppstein/ junkyard/
diam.html.
Wiedersehen Manifold
The only Wiedersehen manifolds are the standard round
spheres, as was established by proof of the BLASCHKE
Conjecture.
see also BLASCHKE CONJECTURE
Wieferich Prime
A Wieferich prime is a Prime p which is a solution to
the Congruence equation
r>P-l
1 (mod p )
1942
Wieferich Prime
Wiener-Khintchine Theorem
Note the similarity of this expression to the special case
of Fermat's Little Theorem
■yP-l
1 (mod p) ,
which holds for all Odd PRIMES. However, the only
Wieferich primes less than 4 x 10 12 are p = 1093 and
3511 (Lehmer 1981, Crandall 1986, Crandall et al. 1997).
Interestingly, one less than these numbers have sugges-
tive periodic BINARY representations
1092 = IOOOIOOOIOO2
3510 = IIOHOHOIIO2.
A Prime factor p of a Mersenne Number M q — 2 q - 1
is a Wieferich prime Iff p 2 \2 q -l. Therefore, Mersenne
PRIMES are not Wieferich primes.
If the first case of Fermat's Last Theorem is false for
exponent p, then p must be a Wieferich prime (Wieferich
1909). If p\2 n ±l with p and n RELATIVELY PRIME, then
p is a Wieferich prime Iff p 2 also divides 2 n ± 1. The
Conjecture that there are no three Powerful Num-
bers implies that there are infinitely many Wieferich
primes (Granville 1986, Vardi 1991). In addition, the
ABC CONJECTURE implies that there are at least Clnx
Wieferich primes < x for some constant C (Silverman
1988, Vardi 1991).
see also abc Conjecture, Fermat's Last Theo-
rem, Fermat Quotient, Mersenne Number, Miri-
manoff's Congruence, Powerful Number
References
Brillhart, J.; Tonascia, J.; and Winberger, P. "On the Fer-
mat Quotient." In Computers and Number Theory (Ed.
A. O. L. Atkin and B. J. Birch). New York: Academic
Press, pp. 213-222, 1971.
Crandall, R. Projects in Scientific Computation. New York:
Springer- Verlag, 1986.
Crandall, R.; Dilcher, K; and Pomerance, C. "A search for
Wieferich and Wilson Primes." Math. Comput 66, 433-
449, 1997.
Granville, A, "Powerful Numbers and Fermat's Last Theo-
rem." C. R. Math. Rep. Acad. Sci. Canada 8, 215-218,
1986.
Lehmer, D. H. "On Fermat's Quotient, Base Two." Math.
Comput. 36, 289-290, 1981.
Ribenboim, P. "Wieferich Primes." §5.3 in The New Book
of Prime Number Records. New York: Springer- Verlag,
pp. 333-346, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 116 and 157, 1993.
Silverman, J. "Wieferich's Criterion and the abc Conjecture."
J. Number Th. 30, 226-237, 1988.
Vardi, I. "Wieferich." §5.4 in Computational Recreations in
Mathematica. Reading, MA: Addison- Wesley, pp. 59-62
and 96-103, 1991.
Wieferich, A. "Zum letzten Fermat 'schen Theorem." J. reine
angew. Math. 136, 293-302, 1909.
Wielandt's Theorem
Let the n x n MATRIX A satisfy the conditions of the
Perron-Frobenius Theorem and the n x n Matrix
C = Cij satisfy
\Cij\ < CLij
for ij - 1, 2, . . . , n. Then any EIGENVALUE A of C
satisfies the inequality |Ao| < R with the equality sign
holding only when there exists annxn MATRIX D = Sij
(where 5ij is the Kronecker Delta) and
C=^DAD-\
R
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.
Wiener Filter
An optimal FILTER used for the removal of noise from
a signal which is corrupted by the measuring process
itself.
see also Filter
References
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetter-
ling, W. T. "Optimal (Wiener) Filtering with the FFT."
§13.3 in Numerical Recipes in FORTRAN: The Art of Sci-
entific Computing, 2nd ed. Cambridge, England: Cam-
bridge University Press, pp. 539-542, 1992,
Wiener Function
see Brown Function
Wiener-Khintchine Theorem
Recall the definition of the AUTOCORRELATION function
C(t) of a function E(t),
C{t)= / E*{r)E{t + T)dT.
(1)
Also recall that the Fourier Transform of E(t) is
defined by
&v.
/oo
E v e~ 2l ' i ' JT
-OO
giving a COMPLEX CONJUGATE of
/oo
Ete 27rivT dv.
■00
(2)
(3)
Wiener Measure
Plugging E*(t) and E(t + r) into the AUTOCORRELA-
TION function therefore gives
-2^'(t + r) dl/ i
/oo r />oo
/ EU
-oo L 1 ' — oo
* [/>
/oo /»oo /»oo
/ / s;B I/ »e _2,rtT(, ''- , ' ) e- a,rt, ''*dT«ii/d«/'
■oo «/ — oo J —oo
/oo /»oo
/ e:e„,5{v' - v)
OOtZ-OO
■£
ElE v e~
' dv dv
I rn |2 — 2irivt »
1^1 e ajv
^[IK
(4)
so, amazingly, the AUTOCORRELATION is simply given
by the FOURIER TRANSFORM of the ABSOLUTE SQUARE
of E(y),
C{t) =?[&{?)?].
(5)
The Wiener-Khintchine theorem is a special case of the
Cross-Correlation Theorem with f — g.
see also AUTOCORRELATION, CROSS-CORRELATION
Theorem, Fourier Transform
Wiener Measure
The distribution which arises whenever a central limit
scaling procedure is carried out on path-space valued
random variables.
Wiener Space
see MALLIAVIN CALCULUS
Wigner 3 j- Symbol
The Wigner 3j symbols are written
3i n 3
ra\ mi m
(i)
and are sometimes expressed using the related
Clebsch-Gordon Coefficients
C 3 m irn2 = (jiJ2m 1 m 2 \jiJ2Jm) (2)
(Condon and Shortley 1951, pp. 74-75; Wigner 1959,
p. 206), or Racah ^-Coefficients
V(jiJ2J]m 1 m 2 m)
Connections among the three are
(jiJ2mim 2 \jiJ2m)
(3)
v I mi mi —ml
(4)
Wigner 3j-Symbol 1943
(jiJ2mim,2 \j1j2jm)
= (-l) i+m y/2j + lV{ji323\ mim 2 - m) (5)
V( jlJ2 y, mm ) = (-l)-^^ 2 * *).
(6)
The Wigner 3j-symbols have the symmetries
Ji 32 3
mi mi m
32 3 3i
mi m m\
3i J2 _
m ra\ mi
= ("I)
31+32+3 [32 3i 3
mi mi m
(_iyi+32+3 ( h 3 h \
\ mi m mi J
( 3 h h \
\ m mi m\ J
31+32+3 ( 31 32 3
mi — mi —m
_ f_]\3l+32+3
The symbols obey the orthogonality relations
^— ' \ mi mi ml V m x m 2 ml
(7)
Qm\m\ ^m2fTi'o
(8)
2^ I mi mo m/lmi m, m' I ~ ° n ° m i"V
7ni mi m I V mi mi m
where Sij is the Kronecker Delta.
(9)
General formulas are very complicated, but some spe-
cific cases are
( h 32 jl +32 \ _ /_ 1 \ii-J2+mi+m 2
I mi mi —mi — m,2 J
(2ji)!(2j a )l
(2ji + 2J2 + l)!(ji+mi)!
(ji +32 +mi + m 2 )!(ji + ji - mi - m 2 )\
(ji - mi)\(j 2 + m 2 )\(J2 ~m 2 )\
1/2
(10)
3i 32 3
ji —ji — m
= ( — ]\—3\+32+™>
(2ji)!(-Ji+J2+j)!
{jl+J2+j + l)KJl-J2+J) 1
(ji + J2 +m)\(j - m)\
(ji +J2 ~ j) } -(-Ji +J2 -m)\(j + m)\
nl/2
(11)
1944 Wigner 3j-Symbol
Wigner 6j-Symbol
Ji 32 3
)(2g-2j 2 )!(2g-2j)!
_SL
(9-JlV-(9~J2)K9-JY-
(12)
if J = 2#
if J = 2p + 1,
for J = ji + j 2 + j.
For Spherical Harmonics Y lm (9,</>),
y Zimi (^0)y /2m2 (^,0)
/(2/i + l)(2/ 2 + l)(2/+l) / /! l 2 I
Z,ro
47T
T7li m,2 m
xC(^)(5 o 2 [)■ (13)
For values of l 3 obeying the Triangle Condition
A(ZiW 3 ),
/
Y hrni (0, <f>)Y hm2 (<9, (f>)Y hrn3 (<9, 4>) sin d6 d</>
(2fi + l)(2t a + l)(2h + l)
47T
X| 2 ) ( mi m a m 3 ' (14)
and
*/*
(cos 0)P h (cos 0)P/ 3 (cos 6) sin d0
(15)
see also Clebsch-Gordon Coefficient, Racah V-
coefficient, racah t^-coefficient, wlgner 6j-
Symbol, Wigner 9j-Symbol
References
Abramowitz, M. and Stegun, C. A. (Eds.). "Vector-Addition
Coefficients." §27.9 in Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables,
9th printing. New York: Dover, pp. 1006-1010, 1972.
Condon, E. U. and Shortley, G. The Theory of Atomic Spec-
tra. Cambridge, England: Cambridge University Press,
1951.
de Shalit, A. and Talmi, I. Nuclear Shell Theory. New York:
Academic Press, 1963.
Gordy, W. and Cook, R. L. Microwave Molecular Spectra,
3rd ed. New York: Wiley, pp. 804-811, 1984.
Messiah, A. "Clebsch-Gordon (C.-G.) Coefficients and '3j'
Symbols," Appendix CI in Quantum Mechanics, Vol. 2.
Amsterdam, Netherlands: North-Holland, pp. 1054-1060,
1962.
Rotenberg, M.; Bivens, R.; Metropolis, N.; and Wooten, J. K.
The 3j and Qj Symbols. Cambridge, MA: MIT Press, 1959.
Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra.
New York: Wiley, pp. 275-276, 1968.
Sobel'man, I. I. "Angular Momenta." Ch. 4 in Atomic Spec-
tra and Radiative Transitions, 2nd ed. Berlin: Springer-
Verlag, 1992.
Wigner, E. P. Group Theory and Its Application to the Quan-
tum Mechanics of Atomic Spectra, expanded and improved
ed. New York: Academic Press, 1959.
Wigner 6j-Symbol
A generalization of CLEBSCH-GORDON COEFFICIENTS
and Wigner 3j-Symbol which arises in the coupling of
three angular momenta. Let tensor operators T^ and
U^ act, respectively, on subsystems 1 and 2 of a system,
with subsystem 1 characterized by angular momentum
ji and subsystem 2 by the angular momentum j 2 . Then
the matrix elements of the scalar product of these two
tensor operators in the coupled basis J = ji + j2 are
given by
(nj[r^J'M'\T w ■ U (k) \T 1 j 1 T 2 j 2 JM)
-XX ( 1V1+J2+ J J^ & J i I
-Sjj.8 MM .(-l) | & h j2 |
Ard'xWT^Wr^ir^WU^Wr^), (1)
(*)|
where
\ k ji h J
is the Wigner 6j-symbol and 7*1
and T2 represent additional pertinent quantum numbers
characterizing subsystems 1 and 2 (Gordy and Cook
1984).
Edmonds (1968) gives analytic forms of the 6j-symbol
for simple cases, and Shore and Menzel (1968) and
Gordy and Cook (1984) give
(-1)'
fa b c 1 _
1° c b )~ V(26 + l)(2c+l)
U b c\_ 2{-iy +1 x
\l c 6/
(2)
^26(26 + 1)(26 + 2)2c(2c + l)(2c + 2)
(3)
a b c 1 2(-l) s [3X(jy-l)-4b(6+l)c(c+l)]
2 c b
(a b c 1
I 2 C b )
^/(26- 1)26(26+1) {2b -f- 2)(26 + 3)
1
^{2c - l)2c(2c + l)(2c + 2)(2c + 3) '
(4)
where
s = a + 6 + c (5)
X = 6(6 + 1) + c(c + 1) - a(a + 1). (6)
see also Clebsch-Gordon Coefficient, Racah V-
Coefficient, Racah V^-Coefficient, Wigner 3j-
Symbol, Wigner 9j-Symbol
References
Carter, J. S.; Flath, D. E.; and Saito, M. The Classical and
Quantum §j-Symbols. Princeton, NJ: Princeton University
Press, 1995.
Wigner 9j-Symbol
Edmonds, A. R. Angular Momentum in Quantum Mechan-
ics, 2nd ed., rev. printing. Princeton, NJ: Princeton Uni-
versity Press, 1968.
Gordy, W. and Cook, R. L. Microwave Molecular Spectra,
3rd ed. New York: Wiley, pp. 807-809, 1984.
Messiah, A. "Racah Coefficients and '6j' Symbols." Ap-
pendix C.II in Quantum Mechanics, Vol. 2. Amsterdam,
Netherlands: North-Holland, pp. 567-569 and 1061-1066,
1962.
Rotenberg, M.; Bivens, R.; Metropolis, N.; and Wooten, J. K.
The 3j and 6j Symbols. Cambridge, MA: MIT Press, 1959.
Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra.
New York: Wiley, pp. 279-284, 1968.
Wigner 9j-Symbol
A generalization of Clebsch-Gordon Coefficients
and WIGNER 3j- and 6j-SYMBOLS which arises in the
coupling of four angular momenta and can be written in
terms of the WlGNER 3j- and 6J-SYMBOLS. Let tensor
operators T^ kl ^ and U^ 2 > act, respectively, on subsys-
tems 1 and 2. Then the reduced matrix element of the
product T^ k ^ x JJ^ 2 ^ of these two irreducible operators
in the coupled representation is given in terms of the
reduced matrix elements of the individual operators in
the uncoupled representation by
= ^{2J + l){2J> + l){2k + l)y] { j 2 32 k 2 }
T » I J' J k )
(rViji||T (fcl) ||r''ni 1 )(r"riji||^ fc2 >||rT i j 2 ). (1)
( 3x h *1
where < j 2 j 2 k 2
I J' J k
and Cook 1984).
is a Wigner 97-symbol (Gordy
Shore and Menzel (1968) give the explicit formulas
= YJ(-l) 2 *(2z + l)
}{i 1 1}{
(-1)
b+c+J+K
y/{2J+l){2K+l)
j\ ( J L S\
H J)
a d \
( a b J 1
\d c Kj
(3)
S L J
L S
+
r 2 l l\
\l 1 1}
{
5+L+J+l
(-1)
15(2L + 1)
S J L\
J S 1 J
J2 L L\'
\L 1 1/
(4)
Wigner-Eckart Theorem 1945
see also CLEBSCH-GORDON COEFFICIENT, RACAH V-
COEFFICIENT, RACAH ^-COEFFICIENT, WlGNER 3j-
Symbol, Wigner 6j-Symbol
References
Gordy, W. and Cook, R. L. Microwave Molecular Spectra,
3rd ed. New York: Wiley, pp. 807-809, 1984.
Messiah, A. ( "9j' Symbols." Appendix C.III in Quantum Me-
chanics, Vol. 2. Amsterdam, Netherlands: North- Holland,
pp. 567-569 and 1066-1068, 1962.
Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra.
New York: Wiley, pp. 279-284, 1968.
Wigner-Eckart Theorem
A theorem of fundamental importance in spectroscopy
and angular momentum theory which provides both (1)
an explicit form for the dependence of all matrix ele-
ments of irreducible tensors on the projection quantum
numbers and (2) a formal expression of the conservation
laws of angular momentum (Rose 1995).
The theorem states that the dependence of the ma-
trix element (j'm'|Ti,M|J7n) on the projection quan-
tum numbers is entirely contained in the WlGNER 3j-
Symbol (or, equivalent^, the Clebsch-Gordon Co-
efficient), given by
(j'm'\T LM \jm) = CULj'imMm'XfWTLWj),
where C(jLj';mMm f ) is a Clebsch-Gordon Coeffi-
cient and Tlm is a set of tensor operators (Rose 1995,
p. 85).
see also Clebsch-Gordon Coefficient, Wigner 3j-
Symbol
References
Cohen- Tannoudji, C; Diu, B.; and Laloe, F. "Vector Opera-
tors: The Wigner-Eckart Theorem." Complement D x in
Quantum Mechanics, Vol. 2. New York: Wiley, pp. 1048-
1058, 1977.
Edmonds, A. R. Angular Momentum in Quantum Mechan-
ics, 2nd ed., rev. printing. Princeton, NJ: Princeton Uni-
versity Press, 1968.
Gordy, W. and Cook, R. L. Microwave Molecular Spectra,
3rd ed. New York: Wiley, p. 807, 1984.
Messiah, A. "Representation of Irreducible Tensor Operators:
Wigner-Eckart Theorem." §32 in Quantum Mechanics,
Vol. 2. Amsterdam, Netherlands: North- Holland, pp. 573-
575, 1962.
Rose, M. E. "The Wigner-Eckart Theorem." §19 in Elemen-
tary Theory of Angular Momentum. New York: Dover,
pp. 85-94, 1995.
Shore, B. W. and Menzel, D. H. "Tensor Operators and the
Wigner-Eckart Theorem." §6.4 in Principles of Atomic
Spectra. New York: Wiley, pp. 285-294, 1968.
Wigner, E. P. Group Theory and Its Application to the Quan-
tum Mechanics of Atomic Spectra, expanded and improved
ed. New York: Academic Press, 1959.
Wybourne, B. G. Symmetry Principles and Atomic Spec-
troscopy. New York: Wiley, pp. 89 and 93-96, 1970.
1946
Wilbraham-Gibbs Constant
Wilf-Zeilberger Pair
Wilbraham-Gibbs Constant
N.B, A detailed on-line essay by S. Finch was the start-
ing point for this entry.
Let a piece wise smooth function / with only finitely
many discontinuities (which are all jumps) be defined
on [-7r,7r] with FOURIER SERIES
ak
b k
i r
- I f(t)cos(kt)dt
77 J -IT
i r
- / f(t)sin(kt)dt,
7T /
J — 7T
(1)
(2)
Sn(fjx) = |a + < 2_Jt ak cos (kx) + bksin(kx)] > . (3)
Let a discontinuity be at x — c, with
lim f(x) > lim /(x),
Define
D =
lim
0(c)
_ 1
~ 2
lim f(x)} - \ lim f(x)} > 0.
lim f(x) + lim f(x)
x—tc~ X— VC +
(4)
(5)
(6)
and let x = x n < c be the first local minimum and
x = in > c the first local maximum of Sn{f, x) on either
side of x n . Then
D
lim S n (f,x n ) = 0(c) H G'
TV
D
where
lim S n (f,£ n ) = <l>(c)--G',
G f = / sine 9 d6 = 1.851937052.
Jo
(7)
(8)
(9)
Here, sincx = sinx/x is the SlNC FUNCTION. The
Fourier Series of y = x therefore does not converge
to — 7T and 7r at the ends, but to —2G r and 2G f . This
phenomenon was observed by Wilbraham (1848) and
Gibbs (1899). Although Wilbraham was the first to note
the phenomenon, the constant G f is frequently (and un-
fairly) credited to Gibbs and known as the GlBBS CON-
STANT. A related constant sometimes also called the
Gibbs Constant is
v = 2 r
n Jo
sine a; da; = 1.17897974447216727.
(10)
(Le Lionnais 1983).
References
Carslaw, H. S. Introduction to the Theory of Fourier's Series
and Integrals, 3rd ed. New York: Dover, 1930.
Finch, S. "Favorite Mathematical Constants." http://www.
maths of t . c ora/ as olve/ const ant /gibbs /gibbs .html.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann,
pp. 36 and 43, 1983.
Zygmund, A. G. Trigonometric Series 1, 2nd ed. Cambridge,
England: Cambridge University Press, 1959.
Wilcoxon Rank Sum Test
A nonparametric alternative to the two-sample t-test.
see also PAIRED £-TEST, PARAMETRIC TEST
Wilcoxon Signed Rank Test
A nonparametric alternative to the PAIRED 2-TEST
which is similar to the FlSHER Sign Test. This test as-
sumes that there is information in the magnitudes of the
differences between paired observations, as well as the
signs. Take the paired observations, calculate the differ-
ences, and rank them from smallest to largest by ABSO-
LUTE Value. Add all the ranks associated with Posi-
tive differences, giving the T+ statistic. Finally, the P-
Value associated with this statistic is found from an ap-
propriate table. The Wilcoxon test is an .R-ESTIMATE.
see also FlSHER SIGN TEST, HYPOTHESIS TESTING,
Paired £-Test, Parametric Test
Wild Knot
A Knot which is not a Tame Knot.
see also TAME KNOT
References
Milnor, J. "Most Knots are Wild." Fund. Math. 54, 335-
338, 1964.
Wild Point
For any point P on the boundary of an ordinary BALL,
find a NEIGHBORHOOD of P in which the intersection
with the Ball's boundary cuts the NEIGHBORHOOD
into two parts, each HOMEOMORPHIC to a Ball. A
wild point is a point on the boundary that has no such
Neighborhood.
see also Ball, Homeomorphic, Neighborhood
Wilf-Zeilberger Pair
A pair of CLOSED Form functions (F, G) is said to be
a Wilf-Zeilberger pair if
F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k). (1)
The Wilf-Zeilberger formalism provides succinct proofs
of known identities and allows new identities to be dis-
covered whenever it succeeds in finding a proof cer-
tificate for a known identity. However, if the starting
point is an unknown hypergeometric sum, then the Wilf-
Zeilberger method cannot discover a closed form solu-
tion, while Zeilberger's Algorithm can.
Wilf-Zeilberger pairs are very useful in proving Hyper-
geometric Identities of the form
>£(ra, k) = rhs(n)
(2)
for which the SUMMAND £(n, k) vanishes for all k outside
some finite interval. Now divide by the right-hand side
to obtain
£>(n,*) = l, (3)
Wilf-Zeilberger Pair
where
F(n 9 k) =
i(n, k)
rhs(n)
(4)
Now use a Rational Function R(n,k) provided by
Zeilberger's Algorithm, define
G(n,k) = R(n,k)F(n,k).
(5)
The identity (1) then results. Summing the relation over
all integers then telescopes the right side to 0, giving
£>(n + l,fc) = 53F(n,fc).
(6)
Therefore, J^ k ^( n ' ^ is ^dependent of ra, and so must
be a constant- If F is properly normalized, then it will
be true that £ fc F(0, k) = 1.
For example, consider the BINOMIAL COEFFICIENT
identity
e-£ (;)-»■■
k fc=0 v 7
(7)
R(n, k)
the function R(n,k) returned by Zeilberger's Algo-
rithm is T
(8)
(9)
2(fc - n - 1) '
Therefore,
(".*>=(:)
F(n.Jfe\ = I '_" 12
and
G(n,fc) = R(n,k)F{n,k)
k ( n \ 2 ~ n
2(n-h
2(Jfe-ra-l) \fe,
fcn!2" n _ / n \ 2 ~n-i
1 - k)k\(n - A:)! \fc - l)
(10)
Taking
F(n + 1, fc) - F(n, fc) = G(n, A; + 1) - G(n, k) (11)
then gives the alleged identity
"VV-'-Cr
;)'-'+ (»:,)'-■'
(12)
Expanding and evaluating shows that the identity does
actually hold, and it can also be verified that
w> -(:)-{;
for fc =
otherwise,
(13)
Wilf-Zeilberger Pair 1947
For any Wilf-Zeilberger pair (F,G),
oo oo.
j^G(n.O) = ]^[F(n,n - 1) + G(n - l,n - 1)] (14)
n=0 n=l
whenever either side converges (Zeilberger 1993). In ad-
dition,
X>(«>°>=£
F(s(n+ 1), n) 4- J^ G(sn + i, n)
X)F(0,k) = ^G(n 1 0),
fc=0
(15)
(16)
and
E G ^°) = E
n=0
y]F(s(n-F-l),/:n + j)
jm>
where
-j- \^ G(sn + i, in.)
t-i
F, )t (n, fc) = ^2 F ( sn ' * fc + &
s-l
G., t (n, fc) = ]P G(sn + i, tk)
(17)
(18)
(19)
(Amdeberhan and Zeilberger 1997). The latter identity
has been used to compute Apery's CONSTANT to a large
number of decimal places (PloufTe) .
see also APERY'S CONSTANT, CONVERGENCE IMPROVE-
MENT, Zeilberger's Algorithm
References
Amdeberhan, T. and Zeilberger, D. "Hypergeometric Se-
ries Acceleration via the WZ Method." Electronic J.
Combinatorics 4, No. 2, R3, 1-3, 1997. http://www.
combinatorics.org/Volnme_4/wilf toe. html#R03. Also
available at http://www.math.temple.edu/-zeilberg/
mamarim/mamarimhtml/accel.html.
Cipra, B. A. "How the Grinch Stole Mathematics." Science
245, 595, 1989.
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. "The WZ Phe-
nomenon." Ch. 7 in A=B. Wellesley, MA: A. K. Peters,
pp. 121-140, 1996.
Plouffe, S. "32,000,279 Digits of Zeta(3)." http://lacim.
uqam. ca/piDATA/Zeta3 . txt.
Wilf, H. S. and Zeilberger, D. "Rational Functions Certify
Combinatorial Identities." J. Amer. Math. Soc. 3, 147-
158, 1990.
Zeilberger, D. "The Method of Creative Telescoping." J.
Symb. Comput. 11, 195-204, 1991.
Zeilberger, D. "Closed Form (Pun Intended!)." Contempo-
rary Math. 143, 579-607, 1993.
so £fc F (M) = 1 (Petkovsek et al. 1996, pp. 25-27).
1948
Wilkie's Theorem
Winding Number (Contour)
Wilkie's Theorem
Let <f>(xi, . . . ,x n ) be an C eKp formula, where £ e x P =
C U {e x } and £ is the language of ordered rings C =
{+, — , •, <, 0, 1}. Then there are n > m and /i, . . . , f s €
Z[a;i,... ,x n ,e X1 ,. . . ,e Xn ] such that 0(xi, . . . , x n ) is
equivalent to
dx m -f i • ■ ■ ^jXnji \xi , . . . , x n , e , . . . , e J = . . .
= /s (Xi j . . . , Xn j € , . . ,
Wilson Quotient
-) = o
(Wilkie 1996). In other words, every formula is equiva-
lent to an existential formula and every definable set is
the projection of an exponential variety (Marker 1996).
References
Marker, D. "Model Theory and Exponentiation." Not.
Amer. Math. Soc. 43, 753-759, 1996.
Wilkie, A. J. "Model Completeness Results for Expansions of
the Ordered Field of Real Numbers by Restricted Pfaffian
Functions and the Exponential Function." J. Amer. Math.
Soc. 9, 1051-1094, 1996.
Williams p+1 Factorization Method
A variant of the POLLARD p — 1 METHOD which uses
LUCAS SEQUENCES to achieve rapid factorization if some
factor p of N has a decomposition ofp+1 in small Prime
factors.
see also Lucas Sequence, Pollard p - 1 Method,
Prime Factorization Algorithms
References
Riesel, H. Prime Numbers and Computer Methods for Fac-
torization, 2nd ed. Boston, MA: Birkhauser, p. 177, 1994.
Williams, H. C. "Ap+1 Method of Factoring." Math. Corn-
put 39, 225-234, 1982.
Wilson Plug
A 3-D surface with constant VECTOR FIELD on its
boundary which traps at least one trajectory which en-
ters it.
see also Vector Field
Wilson's Primality Test
see Wilson's Theorem
Wilson Prime
A Prime satisfying
W(p) = (mod p) ,
where W(p) is the WILSON QUOTIENT, or equivalently,
(p- 1)! = -1 (modp 2 ) .
5, 13, and 563 are the only Wilson primes less than
5 x 10 8 (Crandall et al. 1997).
References
Crandall, R.; Dilcher, K; and Pomerance, C. "A search for
Wieferich and Wilson Primes." Math. Comput. 66, 433-
449, 1997.
Ribenboim, P. "Wilson Primes." §5.4 in The New Book
of Prime Number Records. New York: Springer- Verlag,
pp. 346-350, 1996.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison-Wesley, p. 73, 1991.
W(p) =
(P-1)
V
Wilson's Theorem
Iff p is a PRIME, then (p — 1)! + 1 is a multiple of p,
that is
(p — 1)! = — 1 (mod p) .
This theorem was proposed by John Wilson in 1770 and
proved by Lagrange in 1773. Unlike Fermat'S LITTLE
THEOREM, Wilson's theorem is both Necessary and
Sufficient for primality. For a Composite Number,
(n — 1)! = (mod n) except when n = 4.
see also Fermat's Little Theorem, Wilson's Theo-
rem Corollary, Wilson's Theorem (Gauss's Gen-
eralization)
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. 142-143 and 168-169, 1996.
Ore, 0. Number Theory and Its History. New York: Dover,
pp. 259-261, 1988.
Shanks, D. Solved and Unsolved Problems in Number Theory,
4th ed. New York: Chelsea, pp. 37-38, 1993.
Wilson's Theorem Corollary
Iff a Prime p is of the form Ax + 1, then
[{2x)\} 2 = -1 (modp).
Wilson's Theorem (Gauss's Generalization)
Let P be the product of INTEGERS less than or equal to
n and Relatively Prime to n. Then
n_ J — 1 (mod n
I 1 (mod n)
) for n = 4,p OL ,2p Q
otherwise.
k\n
When m = 2, this reduces to P ~ 1 (mod 2) which is
equivalent to P = — 1 (mod 2).
see also WILSON'S THEOREM, WILSON'S THEOREM
Corollary
Winding Number (Contour)
Denoted 71(7,20) and denned as the number of times a
path 7 curve passes around a point.
71(7, a)
- J_ / dz
2-ni I z — a '
The contour winding number was part of the inspiration
for the idea of the DEGREE of a Map between two Com-
pact, oriented MANIFOLDS of the same DIMENSION. In
the language of the DEGREE of a Map, if 7 : [0, 1] -¥ C
Winding Number (Map)
Witch ofAgnesi 1949
is a closed curve (i,e,, 7(0) = 7(1)), then it can be con-
sidered as a Function from S 1 to C. In that context,
the winding number of 7 around a point p in C is given
by the degree of the Map
7"
17 ~P\
from the CIRCLE to the CIRCLE.
Winding Number (Map)
The winding number of a map is defined by
W= lim
f n (9) - 9
which represents the average increase in the angle per
unit time (average frequency). A system with a RA-
TIONAL winding number W = p/q is MODE- LOCKED,
whereas a system with an IRRATIONAL winding number
is QUASIPERIODIC. Note that since the Rationals are
a set of zero MEASURE on any finite interval, almost all
winding numbers will be irrational, so almost all maps
will be QUASIPERIODIC.
Windmill
One name for the figure used by Euclid to prove the
PYTHAGOREAN THEOREM.
see Bride's Chair, Peacock's Tail
Window Function
see Rectangle Function
Winkler Conditions
Conditions arising in the study of the ROBBINS EQUA-
TION and its connection with BOOLEAN ALGEBRA. Win-
kler studied Boolean conditions (such as idempotence or
existence of a zero) which would make a ROBBINS AL-
GEBRA become a BOOLEAN ALGEBRA. Winkler showed
that each of the conditions
3C,3D,C + D = C
3C,3D,n(C + D) = n(C),
known as the first and second Winkler conditions, SUF-
FICES. A computer proof demonstrated that every Rob-
BINS Algebra satisfies the second Winkler condition,
from which it follows immediately that all ROBBINS AL-
GEBRAS are BOOLEAN.
References
McCune, W. "Robbins Algebras are Boolean." http://www.
mcs . anl . gov/home/mccune/ar/robbins/.
Winkler, S. "Robbins Algebra: Conditions that Make a Near-
Boolean Algebra Boolean." J. Automated Reasoning 6,
465-489, 1990.
Winkler, S. "Absorption and Idempotency Criteria for a
Problem in Near-Boolean Algebra." J. Algebra 153, 414-
423, 1992.
Winograd Transform
A discrete FAST FOURIER TRANSFORM ALGORITHM
which can be implemented for N — 2, 3, 4, 5, 7, 8,
11, 13, and 16 points.
see also FAST FOURIER TRANSFORM
Wirtinger's Inequality
If y has period 27T, y' is L 2 , and
Jo
y dx — 0,
then
unless
/•2"7T /»27T
/ y 2 dx< / j/' 2
Jo Jo
y = A cos x + B sin x.
dx
References
Hardy, G. H.; Littlewood, J. E.; and Polya, G. Inequalities,
2nd ed. Cambridge, England: Cambridge University Press,
pp. 184-187, 1988.
Wirtinger-Sobolev Isoperimetric Constants
Constants 7 such that
[/
f\ 9 dx
1/9
<7
dx
i/p
where / is a real-valued smooth function on a region ft
satisfying some BOUNDARY CONDITIONS.
References
Finch, S. "Favorite Mathematical Constants." http://vww.
mathsof t . com/ asolve/constant/ws/ws .html.
Witch of Agnesi
A curve studied and named "versiera" (Italian for "she-
devil" or "witch") by Maria Agnesi in 1748 in her book
Istituzioni Analitiche (MacTutor Archive). It is also
known as Cubique d'Agnesi or Agnesienne. Some
suggest that Agnesi confused an old Italian word mean-
ing "free to move" with another meaning "witch." The
curve had been studied earlier by Fermat and Guido
Grandi in 1703.
It is the curve obtained by drawing a line from the origin
through the Circle of radius 2a {OB), then picking the
point with the y coordinate of the intersection with the
circle and the x coordinate of the intersection of the
extension of line OB with the line y = 2a. The curve
1950
Witness
Wolstenholme's Theorem
has Inflection Points at y = 3a/2. The line y = is
an Asymptote to the curve.
In parametric form,
or
x = 2a cot 6
y^ o[l -cos(20)],
x = 2a£
2a
2/
1 + t 2 '
(1)
(2)
(3)
(4)
In rectangular coordinates,
see also Lichnerowicz Formula, Lichnerowicz-
Weitzenbock Formula, Seiberg-Witten Equa-
tions
References
Cipra, B. "A Tale of Two Theories." What's Happening
in the Mathematical Sciences, 1995-1996, Vol. 3. Provi-
dence, RI: Amer. Math. Soc, pp. 14-25, 1996.
Donaldson, S. K. "The Seiberg-Witten Equations and 4-
Manifold Topology." Bull. Amer. Math. Soc. 33, 45-70,
1996.
Kotschick, D. "Gauge Theory is Dead! — Long Live Gauge
Theory!" Not. Amer. Math. Soc. 42, 335-338, 1995.
Seiberg, N. and Witten, E. "Monopoles, Duality, and Chi-
ral Symmetry Breaking in N = 2 Supersymmetric QCD."
Nucl. Phys. B 431, 581-640, 1994.
Witten, E. "Monopoles and 4-Manifolds." Math. Res. Let.
1, 769-796, 1994.
d>a?
x 2 + 4a 2 '
(5)
Wittenbauer's Parallelogram
see also Lame Curve
References
Lawrence, J. D. A Catalog of Special Plane Curves. New
York: Dover, pp. 90-93, 1972.
Lee, X. "Witch of Agnesi." http://www . best . com/ - xah/
Special Plane Curves _dir / Witch Of Agnesi _dir / witch Of
Agnesi.html.
MacTutor History of Mathematics Archive. "Witch of Ag-
nesi." http : //www-groups . dcs . st-and . ac . uk/ -history/
Curves/Witch. html.
Yates, R. C. "Witch of Agnesi." A Handbook on Curves
and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 237-238, 1952.
Witness
A witness is a number which, as a result of its number
theoretic properties, guarantees either the composite-
ness or primality of a number n. Witnesses are most
commonly used in connection with Fermat'S Little
Theorem Converse. A Pratt Certificate uses
witnesses to prove primality, and Miller's Primality
Test uses witnesses to prove compositeness.
see also Adleman-Pomerance-Rumely Primality
Test, Fermat's Little Theorem Converse, Mil-
ler's Primality Test, Pratt Certificate, Primal-
ity Certificate
Witten's Equations
Also called the Seiberg-Witten Invariants. For a
connection A and a Positive Spinor <f> £ r(V+),
D A <f> =
The solutions are called monopoles and are the minima
of the functional
L
Divide the sides of a QUADRILATERAL into three equal
parts. The figure formed by connecting and extending
adjacent points on either side of a Vertex is a Paral-
lelogram known as Wittenbauer's parallelogram.
see also QUADRILATERAL, WITTENBAUER'S THEOREM
Wittenbauer's Theorem
The Centroid of a Quadrilateral Lamina is the
center of its WITTENBAUER'S PARALLELOGRAM.
see also Centroid (Geometric), Lamina, Quadri-
lateral, Wittenbauer's Parallelogram
Wolstenholme's Theorem
If p is a Prime > 3, then the Numerator of
1 ^ 2 ^ 3 ^ ■ ' ' ^ p-1
is divisible by p 2 and the NUMERATOR of
JL — i
+ 2 2 + 3 2 + " ' + {p - l) 2
is divisible by p. These imply that if p > 5 is PRIME,
then
2p- 1
p-1
= 1 (mod p 3 ).
(\Ft-i<r(<f>,4>)\ 2 + \D A <t>\ 2 ).
References
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed.
New York: Springer- Verlag, p, 85, 1994.
Ribenboim, P. The Book of Prime Number Records, 2nd ed.
New York: Springer- Verlag, p. 21, 1989.
Woodall Number
Worm
1951
Woodall Number
Numbers of the form
W n = 2 n n - 1.
The first few are 1, 7, 23, 63, 159, 383, . . . (Sloane's
A003261). The only Woodall numbers W n for n <
100,000 which are PRIME are for n = 5312, 7755, 9531,
12379, 15822, 18885, 22971, 23005, 98726, ... (Sloane's
A014617; Ballinger).
see also Cullen Number, Cunningham Number,
Fermat Number, Mersenne Number, Sierpinski
Number of the First Kind
References
Ballinger, R. "Cullen Primes: Definition and Status,"
http://ballingerr.xray.ufl.edu/proths/cullen.html.
Guy, R. K. "Cullen Numbers." §B20 in Unsolved Problems
in Number Theory, 2nd ed. New York: Springer- Verlag,
p. 77, 1994.
Leyland, P. ftp : //sable . ox. ac .uk/pub/math/f actors/
woodall.
Ribenboim, P. The New Book of Prime Number Records.
New York: Springer- Verlag, pp. 360-361, 1996.
Sloane, N. J. A. Sequences A014617 and A003261/M4379 in
"An On-Line Version of the Encyclopedia of Integer Se-
quences."
Woodbury Formula
(A + UVV 1 = A" 1 - [A-'UCl + N^A-'Ur^A' 1 ].
Word
N.B. A detailed on-line essay by S. Finch was the start-
ing point for this entry.
A finite sequence of n letters from some Alphabet is
said to be an n-ary word. A "square" word consists of
two identical subwords (for example, acbacb). A square-
free word contains no square words as subwords (for ex-
ample, abcacbabcb). The only squarefree binary words
are a, &, a&, 6a, aba, and bob. However, there are ar-
bitrarily long ternary squarefree words. The number of
ternary squarefree words of length n is bounded by
6 • 1.032 n < s{n) < 6 * 1.379 n (1)
(Brandenburg 1983). In addition,
See lim [s(n)] 1/n = 1.302 .. . (2)
n— >-oo
(Brinkhuis 1983, Noonan and Zeilberger). Binary cube-
free words satisfy
2 ■ 1.080 n < c(n) < 2 ■ 1.522"
(3)
A word is said to be overlapfree if it has no subwords of
the form xyxyx. A squarefree word is overlapfree, and an
overlapfree word is cubefree. The number t(n) of binary
overlapfree words of length n satisfies
„ 1-155 ^ ,/ x ~ 1-587
(4)
for some constants p and q (Restivo and Selemi 1985,
Kobayashi 1988). In addition, while
In t(n)
lim
n-+oo Inn
(5)
does not exist,
1.155 <T L < 1.276 < 1.332 < T v < 1.587, (6)
where
T L = lim inf
n— J- co
Tu ee lim sup
lnt(n)
Inn
lnt(n)
Inn
(7)
(8)
(Cassaigne 1993).
see also ALPHABET
References
Brandenburg, F.-J. "Uniformly Growing kth Power- Free Ho-
momorphisms." Theor. Comput Sci. 23, 69-82, 1983.
Brinkhuis, J. "Non- Repetitive Sequences on Three Symbols."
Quart J. Math. Oxford Ser. 2 34, 145-149, 1983.
Cassaigne, J. "Counting Overlap-Free Binary Words."
STAGS } 93: Tenth Annual Symposium on Theoretical As-
pects of Computer Science, Wurzburg, Germany, Febru-
ary 25-27, 1993 Proceedings (Ed. G. Goos, J. Hartma-
nis, A. Finkel, P. Enj albert, K. W. Wagner). New York:
Springer- Verlag, pp. 216-225, 1993.
Finch, S. "Favorite Mathematical Constants." http://www.
mathsof t . c om/ as olve/ const ant /words /words .html.
Kobayashi, Y. "Enumeration of Irreducible Binary Words."
Discrete Appl. Math. 20, 221-232, 1988.
Noonan, J. and Zeilberger, D. "The Goulden-Jackson Cluster
Method: Extensions, Applications, and Implementations."
Submitted.
World Line
The path of an object through PHASE SPACE.
Worm
A 4-POLYHEX.
References
Gardner, M, Mathematical Magic Show: More Puzzles,
Games, Diversions, Illusions and Other Mathematical
Sleight- of- Mind from Scientific American. New York:
Vintage, p. 147, 1978.
1952 Worpitzky's Identity
Worpitzky's Identity
where ( £ ) is an Eulerian Number and (£) is a Bi-
nomial Coefficient.
Writhe
Also called the Twist Number.
p of a Link L,
The sum of crossings
w(L)= Y, C (P)>
pec(L)
where e(p) defined to be ±1 if the overpass slants from
top left to bottom right or bottom left to top right and
C(L) is the set of crossings of an oriented Link. If a
Knot K is Amphichiral, then w(K) = (Thistle-
thwaite). Letting Lk be the LINKING NUMBER of the
two components of a ribbon, Tw be the TWIST, and Wr
be the writhe, then
Lk(K) = Tw(K) + Wr(jK").
(Adams 1994, p. 187).
see also Screw, Twist
References
Adams, C. C. The Knot Book: An Elementary Introduction
to the Mathematical Theory of Knots. New York: W. H.
Freeman, 1994.
Wronskian
W(0i,...,0n);
01
(n-1)
4>2
<f>2
fa^
<f>'n
If the Wronskian is NONZERO in some region, the func-
tions (f>i are LINEARLY INDEPENDENT. If W = over
some range, the functions are linearly dependent some-
where in the range.
see also Abel's Identity, Gram Determinant, Lin-
early Dependent Functions
References
Morse, P. M. and Feshbach, H. Methods of Theoretical Phys-
ics, Part L New York: McGraw-Hill, pp. 524-525, 1953.
Wulff Shape
An equilibrium Minimal SURFACE for a crystal which
has the least anisotropic surface energy for a given vol-
ume. It is the anisotropic analog of a SPHERE.
see also SPHERE
WythoiFs Game
Wynn's Epsilon Method
A method for numerical evaluation of SUMS and PROD-
UCTS which samples a number of additional terms in the
series and then tries to fit them to a POLYNOMIAL mul-
tiplied by a decaying exponential.
see also Euler-Maclaurin Integration Formulas
WythofF Array
A Interspersion array given by
1
2
3
5
8
13
21
34
55
4
7
11
18
29
47
76
123
199
6
10
16
26
42
68
110
178
288
9
15
24
39
63
102
165
267
432
12
20
32
52
84
136
220
356
576
14
23
37
60
97
157
254
411
665
17 28 45 73 118 191 309 500 809
19 31 50 81 131 212 343 555 898
22 36 58 94 152 246 398 644 1042
the first row of which is the FIBONACCI NUMBERS.
see also FIBONACCI NUMBER, INTERSPERSION, StO-
larsky Array
References
Kimberling, C. "Fractal Sequences and Interspersions." Ars
Combin. 45, 157-168, 1997.
WythofF Construction
A method of constructing UNIFORM POLYHEDRA.
see also UNIFORM POLYHEDRON
References
Har'El, Z. "Uniform Solution for Uniform Polyhedra." Ge-
ometriae Dedicata4:7, 57—110, 1993.
Wythoff 's Game
A game played with two heaps of counters in which a
player may take any number from either heap or the
same number from both. The player taking the last
counter wins. The rth SAFE combination is (x,x + r),
where x = [<pr\, with (f> the GOLDEN RATIO and [x\ the
Floor Function. It is also true that x + r= |_0 2r J-
The first few Safe combinations are (1, 2), (3, 5), (4,
7), (6, 10), ....
see also Nim, Safe
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 39-40,
1987.
Coxeter, H. S. M. "The Golden Section, Phyllotaxis, and
WythofTs Game." Scripta Math. 19, 135-143, 1953.
O'Beirne, T. H. Puzzles and Paradoxes. Oxford, England:
Oxford University Press, pp. 109 and 134-138, 1965.
Wythoff Symbol
Wythoff Symbol 1953
Wythoff Symbol
A symbol used to describe Uniform Polyhedra. For
example, the Wythoff symbol for the TETRAHEDRON
is 3 | 2 3. There are three types of Wythoff symbols
p\qr, pq\r and pqr\ y and one exceptional symbol
| | | 3 | used for the Great Dirhombicosidodecahe-
dron. Some special cases in terms of Schlafli Sym-
bols are
p\q2 = p\2q = {q,p}
2\pc
■-{:}
pq | 2 = r «
2q\p^t{p,q}
2pq\=%<
\2pq = s
For the symbol pqr\, permuting the letters gives the
same POLYHEDRON.
see also UNIFORM POLYHEDRON
References
Har'El, Z. "Uniform Solution for Uniform Polyhedra." Ge-
ometriae Dedicata 47, 57-110, 1993.
x-Axis
X
x-Axis
XOR
1955
z-axis
A
y-axis
The horizontal axis of a 2-D plot in CARTESIAN COOR-
DINATES, also called the ABSCISSA.
see also Abscissa, Ordinate, j/- Axis, z-Axis
x-Intercept
y-axis
y-intercept
^-intercept
The point at which a curve or function crosses the x-
Axis (i.e., when y = in 2-D).
see also Line, y-lNTERCEPT
Xi Function
0.8
0.75
0.7
0.65
0.6
0.55
-4
-2
2
4
The zeros of £(z) and of its DERIVATIVES are all located
on the Critical Strip z = a + it, where < a < 1.
Therefore, the nontrivial zeros of the RlEMANN Zeta
Function exactly correspond to those of £(2). The
function f (z) is related to what Gradshteyn and Ryzhik
(1980, p. 1074) call S(t) by
S(t)=«*),
(3)
where z = \ + it. This function can also be defined as
E(it) EE |(t 2 - I) W -*/2-l/4 r( l t+ 1 )C(4+ I )f (4)
giving
s(t) = -!(t a + ih <t/a - 1/4 r(£
Itt)C(i-tt). (5)
The de Bruun-Newman Constant is defined in terms
of the S(t) function.
see also de Bruun-Newman Constant
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Se-
ries, and Products, corr. enl. ^th ed. San Diego, CA: Aca-
demic Press, 1980.
XOR
An operation in LOGIC known as EXCLUSIVE Or. It
yields true if exactly one (but not both) of two condi-
tions is true. The Binary XOR operator has the fol-
lowing Truth Table.
A
B
A XOR B
F
F
F
F
T
T
T
F
T
T
T
F
The Binomial Coefficient (™) mod 2 can be com-
puted using the XOR operation n XOR m, making PAS-
CAL'S Triangle mod 2 very easy to construct.
see also AND, BINARY OPERATOR, BOOLEAN ALGEBRA,
Logic, Not, Or, Pascal's Triangle, Truth Table
t(z) = \z{z - l)^|#C(z) - V ~ ^ 2
(*-i)r(i* + i)C(*)
T z/2
(1)
where C,{z) is the Riemann Zeta Function and T(z) is
the Gamma Function (Gradshteyn and Ryzhik 1980,
p. 1076). The £ function satisfies the identity
t{l-z)=t(z).
(2)
y-Axis
Y
y-Axis
Yff Points
1957
z-axis
4
y-axis
The vertical axis of a 2-D plot in Cartesian Coordi-
nates, also called the Ordinate.
see also Abscissa, Ordinate, x-Axis, z-Axis
y-Intercept
y-axis
^-intercept
x-intercept
The point at which a curve or function crosses the y-
AxiS (i.e., when x — in 2-D).
see also Line, x-Intercept
Yacht
A 6-Polyiamond.
References
Golomb, S. W. Polyominoes: Puzzles, Patterns, Problems,
and Packings, 2nd ed. Princeton, NJ: Princeton University
Press, p. 92, 1994.
Yanghui Triangle
see Pascal's Triangle
Yff Center of Congruence
Let three ISOSCELIZERS, one for each side, be con-
structed on a Triangle such that the four interior
triangles they determine are congruent. Now parallel-
displace these ISOSCELIZERS until they concur in a single
point. This point is called the Yff center of congruence
and has TRIANGLE CENTER FUNCTION
a = sec(^A).
see also CONGRUENT ISOSCELIZERS POINT, ISOSCE-
LIZER
References
Kimberling, C. "Yff Center of Congruence." http://www.
evansville.edu/-ck6/tcenters/recent/yffcc.html.
Yff Points
A u C B
Let points A' , B' , and C f be marked off some fixed dis-
tance x along each of the sides £C, CA, and AB. Then
the lines AA' , BB\ and CC concur in a point U known
as the first Yff point if
(a — x)(b — x)(c — x).
(i)
This equation has a single real root u } which can by
obtained by solving the CUBIC EQUATION
where
f(x) = 2x — px + qx — r = 0,
p = a + 6 + c
q = ab + ac-\- be
r — abc.
(2)
(3)
(4)
(5)
The Isotomic Conjugate Point U' is called the sec-
ond Yff point. The Triangle Center Functions of
the first and second points are given by
and
a \b — uj
a \c — uj
1/3
1/3
(6)
(7)
respectively. Analogous to the inequality w < tt/6 for
the Brocard Angle uj, u < p/6 holds for the Yff
points, with equality in the case of an EQUILATERAL
Triangle. Analogous to
LV < Ct{ < TV — 3u>
for i = 1, 2, 3, the Yff points satisfy
u < ai < p — 3ti.
(8)
(9)
Yff (1963) gives a number of other interesting properties.
The line UU' is PERPENDICULAR to the line containing
the INCENTER J and ClRCUMCENTER O, and its length
is given by
mr=*^, (io)
u s + abc
where A is the Area of the TRIANGLE.
see also Brocard Points, Yff Triangles
References
Yff, P. "An Analog of the Brocard Points." Amer. Math.
Monthly 70, 495-501, 1963.
1958 Yff Triangles
Yff Triangles
A u C B
The TRIANGLE AA'B'C formed by connecting the
points used to construct the Yff Points is called the
first Yff triangle. The Area of the triangle is
A =
2R 1
where R is the ClRCUMRADlUS of the original TRIANGLE
A ABC. The second Yff triangle is formed by connecting
the Isotomic Conjugate Points of A 1 , B\ and C'.
see also Yff Points
References
Yff, P. "An Analog of the Brocard Points."
Monthly 70, 495-501, 1963.
Yin- Yang
Math.
A figure used in many Asian cultures to symbolize the
unity of the two "opposite" male and female elements,
the "yin" and "yang." The solid and hollow parts com-
posing the symbol are similar and combine to make a
Circle. Each part consists of two equal oppositely ori-
ented Semicircles of radius 1/2 joined at their edges,
plus a Semicircle of radius 1 joining the other edges.
see also Baseball Cover, Circle, Piecewise Cir-
cular Curve, Semicircle
References
Dixon, R. Mathographics. New York: Dover, p. 11, 1991.
Gardner, M, "Mathematical Gaines: A New Collection of
'Brain-Teasers.'" Scl Amer. 203, 172-180, Oct. 1960,
Gardner, M. "Mathematical Games: More About the Shapes
that Can Be Made with Complex Dominoes." Sci. Amer.
203, 186-198, Nov. 1960.
Young Diagram
Young's Integral
A Young diagram, also called a FERRERS DIAGRAM, rep-
resents Partitions as patterns of dots, with the nth row
having the same number of dots as the nth term in the
Partition. A Young diagram of the Partition
n — a + fe+... + c,
for a list a, 6, . . . , c of k Positive Integers with a >
b > ... > c is therefore the arrangement of n dots or
square boxes in k rows, such that the dots or boxes are
left-justified, the first row is of length a, the second row
is of length 6, and so on, with the kth row of length c.
The above diagram corresponds to one of the possible
partitions of 100.
see also DURFEE SQUARE, HOOK LENGTH FORMULA,
Partition, Partition Function P, Young Tableau
References
Messiah, A. Appendix D in Quantum Mechanics, 2 vols. Am-
sterdam, Netherlands: North- Holland, p. 1113, 1961-62.
Young Girl-Old Woman Illusion
A perceptual Illusion in which the brain switches be-
tween seeing a young girl and an old woman.
see also Rabbit-Duck Illusion
References
Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, p. 173, 1989.
Young Inequality
For < p < 1,
ab<
+ 1
-l)
b i/(i-x/p)^
Young's Integral
Let f(x) be a Real continuous monotonic strictly in-
creasing function on the interval [0, a] with /(0) =
and b < /(a), then
ab
< / f(x)dx+ / r\y)dy,
Jo Jo
where/ 1 (y) is the INVERSE Function. Equality holds
IFF b = f(a).
References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals , Se-
ries, and Products, 5th ed. San Diego, CA: Academic
Press, p. 1099, 1979.
Young Tableau
Young Tableau 1959
Young Tableau
The Young Tableau of a Young Diagram is ob-
tained by placing the numbers 1, . . . , n in the n
boxes of the diagram. A "standard" Young tableau
is a Young tableau in which the numbers form a non-
decreasing sequence along each line and along each
column. The standard Young tableaux of size three
are given by {{1,2,3}}, {{1,3}, {2}}, {{1,2}, {3}},
and {{1}, {2}, {3}}. The number of standard Young
tableaux of size 1, 2, 3, . . . are 1, 2, 4, 10, 26, 76, 232,
764, 2620, 9496, . . . (Sloane's A000085). These numbers
can be generated by the RECURRENCE RELATION
a(n) = a(n — 1) + (n — l)a(n — 2)
with a(l) = 1 and a(2) = 2.
There is a correspondence between a Permutation
and a pair of Young tableaux, known as the Schen-
sted Correspondence. The number of all standard
Young tableaux with a given shape (corresponding to a
given Young Diagram) is calculated with the Hook
Length Formula. The Bumping Algorithm is used
to construct a standard Young tableau from a permuta-
tion of {1, . . . , n}.
see also Bumping Algorithm, Hook Length For-
mula, Involution (Set), Schensted Correspon-
dence, Young Diagram
References
Fulton, W. Young Tableaux with Applications to Represen-
tation Theory and Geometry. New York: Cambridge Uni-
versity Press, 1996.
Ruskey, F. "Information on Permutations." http://sue.csc
.uvic.ca/-cos/inf /perm/Permlnf o.html#Tableau.
Skiena, S. S. The Algorithm Design Manual. New York:
Springer- Verlag, pp. 254-255, 1997.
Sloane, N. J. A. Sequence A000085/M1221 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Z -Transform
1961
The Ring of Integers . . . , -2, -1, 0,1,2,..., also
denoted I.
see also C, C*, COUNTING NUMBER, 11, N, NATURAL
Number, Q, R, Whole Number, Z~, IT
ir
The Negative Integers . . . , -3, -2, -1.
see also COUNTING NUMBER, NATURAL NUMBER, NEG-
ATIVE, Whole Number, Z, Z + , Z*
The Positive Integers 1, 2, 3, ... , equivalent to N.
see also Counting Number, N, Natural Number,
Positive, Whole Number, Z, Z~, Z*
The Nonnegative Integers 0, 1, 2,
see also Counting Number, Natural Number, Non-
negative, Whole Number Z, Z~, Z +
z-Axis
z-axis
v-axis
The axis in 3-D CARTESIAN COORDINATES which is usu-
ally oriented vertically. CYLINDRICAL COORDINATES
are defined such that the z-axis is the axis about which
the azimuthal coordinate 9 is measured.
see also Axis, z-Axis, y-AxiS
^-Distribution
see Fisher's z-Distribution, Student's z-Distribu-
TION
Z-Number
A Z-number is a REAL NUMBER f such that
< frac
(§)'<
for all k = 1, 2, . . . , where frac(cc) is the fractional part
of #. Mahler (1968) showed that there is at most one Z-
number in each interval [ra, n+ 1) for integral n. Mahler
(1968) therefore concluded that it is unlikely that any
Z-numbers exist. The .Z-numbers arise in the analysis
of the COLLATZ PROBLEM,
see also COLLATZ PROBLEM
References
Flatto, L. "Z-Numbers and j3- Transformations." Symbolic
Dynamics and its Applications, Contemporary Math. 135,
181-201, 1992.
Guy, R. K. "Mahler's Z-Numbers." §E18 in Unsolved Prob-
lems in Number Theory, 2nd ed. New York: Springer-
Verlag, p. 220, 1994.
Lagarias, J, C. "The 3x-\-l Problem and its Generalizations."
Amer. Math. Monthly 92, 3-23, 1985. http://www.cecm.
sf u . ca/organics/papers/lagarias/.
Mahler, K. "An Unsolved Problem on the Powers of 3/2."
Austral Math. Soc. 8, 313-321, 1968.
Tijdman, R. "Note on Mahler's |-Problem." Kongel. Norske
Vidensk Selsk. Skr. 16, 1-4, 1972,
z- Score
The z-score associated with the zth observation of a ran-
dom variable x is given by
where x is the MEAN and cr the STANDARD DEVIATION
of all observations asi, . . . , x n .
z- Transform
The discrete z-transforcn is defined as
N-l
Z[a] — y a n z n .
(i)
The Discrete Fourier Transform is a special case
of the z- transform with
A z-transform with
-2iri/N
-27ria/JV
(2)
(3)
for a ^ ±1 is called a FRACTIONAL FOURIER TRANS-
FORM.
see also DISCRETE FOURIER TRANSFORM, FRACTIONAL
Fourier Transform
References
Arndt, J. "The z-Transform (ZT)." Ch. 3 in "Remarks on
FFT Algorithms." http://www.jjj.de/fxt/.
z-Transform (Population)
see Population Comparison
Z- Transform
The Z-transform of F(t) is defined by
Z[F(t)] = £[F*(t)] t
(1)
where
F'(t) = F(t)d T (t) = J2F(nT)S(t - nT), (2)
1962
Z -Transform
Zaslavskii Map
S(t) is the Delta Function, T is the sampling period,
and C is the LAPLACE TRANSFORM. An alternative def-
inition is
W)l= £ (rr^r)/W.
where
The inverse ^-transform is
Z~ l [f{z)] = F'(t) = ±-, j f {z)z ^ d z.
(3)
(4)
(5)
It satisfies
Z[oF(t) + bG(t)} = aZ[F(t)] + bZ[F(t)}
Z[F(t + T)] = zZ[F(t)] - zF(0)
(6)
(7)
Z[F(t + 2T)] = z 2 Z[F{t)] - z 2 F{0) - zF(t) (8)
m — 1
Z[F(t + mT)} = z^ZlFit)] - ^ z™'* F{rt) (9)
Z[F(t-mT)] = z' rn Z{F(t)]
Z[e at F(t)] = Z[e~ aT z]
Z[e- at F(t)] = Z[e aT z]
tF(t) = -Tz±Z[F(t)]
■ >rm =-?£
/(*)
dz.
(10)
(11)
(12)
(13)
(14)
Transforms of special functions (Beyer 1987, pp. 426-
427) include
Z[5(t)\ = 1
Z[S(t - mT)] = z
Z[H(t)]
Z[H(t - mT)} =
Z[t] =
Z[t 2 ] =
Z[t 3 ] =
Z[a"*] =
Z]cos(wt)] =
Z[sin(wt)]
z-\
z
z m (z - 1)
Tz
(z - iy
T 2 z{z + 1)
(z-ir
T 3 z{z 2 + 4z + 1)
(z - 1)4
z-a» T
z sin(o;T)
z 2 -2zcos(u>T) + l
z[z - cos(o;r)]
z 2 - 2z cos(o;T) + 1 '
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
where H(t) is the Heaviside Step Function. In gen-
eral,
n \z k -*
(z - 1)"+ 1
(26)
where the ( ) are EULERIAN NUMBERS. Amazingly,
the Z-transforms of t n are therefore generators for Eu-
ler's Triangle.
see also EULER'S TRIANGLE, EULERIAN NUMBER
References
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, pp. 424-428, 1987.
Bracewell, R. The Fourier Transform and Its Applications.
New York: McGraw-Hill, pp. 257-262, 1965.
Zag Number
An Even Alternating Permutation number, more
commonly called a Tangent Number.
see also Alternating Permutation, Tangent Num-
ber, Zig Number
Zarankiewicz's Conjecture
The Crossing Number for a Complete Bigraph is
where [x\ is the FLOOR FUNCTION. This has been
shown to be true for all m,n < 7. Zarankiewicz has
shown that, in general, the FORMULA provides an up-
per bound to the actual number.
see also COMPLETE BlGRAPH, CROSSING NUMBER
(Graph)
Zariski Topology
A Topology of an infinite set whose OPEN Sets have
finite complements.
Zaslavskii Map
The 2-D map
x-n+i = [x n + v(l + fiy n ) + ei/ficos(27rx n )} (mod 1)
2/n+i = e~ r [y n +ecos(27r:r n )],
where
(Zaslavskii 1978). It has CORRELATION EXPONENT v «
1.5 (Grassberger and Procaccia 1983) and CAPACITY
Dimension 1,39 (Russell et al. 1980).
References
Grassberger, P. and Procaccia, I. "Measuring the Strangeness
of Strange Attractors." Physica D 9, 189-208, 1983.
Russell, D. A.; Hanson, J, D.; and Ott, E. "Dimension of
Strange Attractors." Phys. Rev. Let 45, 1175-1178, 1980.
Zaslavskii, G. M. "The Simplest Case of a Strange Attrac-
tor." Phys. Let. 69A, 145-147, 1978.
Zassenhaus-Berlekamp Algorithm
Zeilberger's Algorithm 1963
Zassenhaus-Berlekamp Algorithm
A method for factoring POLYNOMIALS.
Zeckendorf Representation
A number written as a sum of nonconsecutive FI-
BONACCI Numbers,
n = Y2 6kFky
k=0
where ejt are or 1 and
CfcCk+l
0.
Every POSITIVE INTEGER can be written uniquely in
such a form.
see also Zeckendorf's Theorem
References
Grabner, P. J.; Tichy, R. F.; Nemes, L; and Petho, A. "On
the Least Significant Digit of Zeckendorf Expansions." Fib.
Quart. 34, 147-151, 1996.
Vardi, I. Computational Recreations in Mathematica. Read-
ing, MA: Addison- Wesley, p. 40, 1991.
Zeckendorf, E. "Representation des nombres naturels par une
sorame des nombres de Fibonacci ou de nombres de Lucas."
Bull. Soc. Roy. Sci. Liege 41, 179-182, 1972.
Zeckendorf 's Theorem
The Sequence {F n - 1} is Complete even if restricted
to subsequences which contain no two consecutive terms,
where F n is a FIBONACCI NUMBER.
see also FIBONACCI DUAL THEOREM, ZECKENDORF
Representation
References
Brown, J. L. Jr. "Zeckendorf's Theorem and Some Applica-
tions." Fib. Quart 2, 163-168, 1964.
Keller, T. J. "Generalizations of Zeckendorf's Theorem."
Fib. Quart. 10, 95-112, 1972.
Lekkerkerker, C. G. "Voorstelling van natuurlyke getallen
door een som van Fibonacci." Simon Stevin 29, 190-195,
1951-52.
Zeeman's Paradox
There is only one point in front of a PERSPECTIVE draw-
ing where its three mutually PERPENDICULAR VANISH-
ING Points appear in mutually Perpendicular direc-
tions, but such a drawing nonetheless appears realistic
from a variety of distances and angles.
see also LEONARDO'S PARADOX, PERSPECTIVE, VAN-
ISHING Point
References
Dixon, R. Mathographics. New York: Dover, p. 82, 1991.
Zeilberger's Algorithm
An Algorithm which finds a Polynomial recurrence
for a terminating HYPERGEOMETRIC IDENTITIES of the
form
y> (n\ Ilti( a * n + a 'i k + °") !
Zk
= C
n£i(Sin + aI)!
YlL&n + Vi)
where (£) is a BINOMIAL COEFFICIENT, a», a-, a it &*,
b'i, bi are constant integers and a", a^, b", 6^, C, x, and
z are complex numbers (Zeilberger 1990). The method
was called Creative Telescoping by van der Poorten
(1979), and led to the development of the amazing ma-
chinery of Wilf-Zeilberger Pairs.
see also Binomial Series, Gosper's Algorithm, Hy-
pergeometric Identity, Sister Celine's Method,
Wilf-Zeilberger Pair
References
Krattenthaler, C. "HYP and HYPQ: The Mathematica
Package HYP." http : //radon . mat . univie . ac . at/People/
kr att /hyp_hypq/hyp . html .
Paule, P. "The Paule-Schorn Implementation of Gosper's and
Zeilberger's Algorithms." http://www.risc.uni-linz.ac.
at/research/combinat/risc/software/PauleSchorn/.
Paule, P. and Riese, A. "A Mathematica q- Analogue of Zeil-
berger's Algorithm Based on an Algebraically Motivated
Approach to qr-Hypergeometric Telescoping." In Special
Functions, q-Series and Related Topics, Fields Institute
Communications 14, 179-210, 1997.
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. "Zeilberger's
Algorithm." Ch. 6 in A=B. Wellesley, MA: A. K. Peters,
pp. 101-119, 1996.
Riese, A. "A Generalization of Gosper's Algorithm to Biba-
sic Hypergeometric Summation." Electronic J. Combina-
torics 1, R19, 1-16, 1996. http: //www. combinatorics,
org/ Volume.l /volume 1 .html#R19.
van der Poorten, A. "A Proof that Euler Missed. . . Apery's
Proof of the Irrationality of £(3)." Math. Intel. 1,196-203,
1979.
Wegschaider, K. Computer Generated Proofs of Binomial
Multi-Sum Identities. Diploma Thesis, RISC Linz, Aus-
tria: J. Kepler University, May 1997. http: //www . rise .
uni - linz . ac . at / research / combinat / rise / software /
Mult i Sum/.
Zeilberger, D. "Doron Zeilberger's Maple Packages and
Programs: EKHAD." http://www.math.temple.edu/
-zeilberg/programs . html.
Zeilberger, D. "A Fast Algorithm for Proving Terminating
Hypergeometric Series Identities." Discrete Math. 80,
207-211, 1990.
Zeilberger, D. "A Holonomic Systems Approach to Special
Function Identities." J. Comput. Appl. Math. 32, 321-
368, 1990.
Zeilberger, D. "The Method of Creative Telescoping." J.
Symb. Comput. 11, 195-204, 1991.
1964 Zeisel Number
Zermelo-Fraenkel Axioms
Zeisel Number
A number N = P1P2 - * * Pk (where the pis are distinct
Primes) such that
p n — Apn-i +B,
with A and B constants and po = 1. For example,
1885 = 1 • 5 • 13 ■ 29 and 114985 = 1 ■ 5 ■ 13 • 29 - 61
are Zeisel numbers with (A y B) = (2,3).
References
Brown, K. S. "Zeisel Numbers." http://www.seanet.com/
-ksbrown/kmath015 .htm.
Zeno's Paradoxes
A set of four Paradoxes dealing with counterintuitive
aspects of continuous space and time.
1. Dichotomy paradox: Before an object can travel a
given distance d, it must travel a distance d/2. In
order to travel d/2, it must travel d/4, etc. Since this
sequence goes on forever, it therefore appears that
the distance d cannot be traveled. The resolution of
the paradox awaited CALCULUS and the proof that
infinite GEOMETRIC SERIES such as XXi^ 1 / 2 ) 1 = *
can converge, so that the infinite number of "half-
steps" needed is balanced by the increasingly short
amount of time needed to traverse the distances.
2. Achilles and the tortoise paradox: A fleet-of-foot
Achilles is unable to catch a plodding tortoise which
has been given a head start, since during the time
it takes Achilles to catch up to a given position, the
tortoise has moved forward some distance. But this
is obviously fallacious since Achilles will clearly pass
the tortoise! The resolution is similar to that of the
dichotomy paradox.
3. Arrow paradox: An arrow in flight has an instanta-
neous position at a given instant of time. At that
instant, however, it is indistinguishable from a mo-
tionless arrow in the same position, so how is the
motion of the arrow perceived?
4. Stade paradox: A paradox arising from the assump-
tion that space and time can be divided only by a
definite amount.
References
Pappas, T. "Zeno's Paradox — Achilles & the Tortoise." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./
Tetra, pp. 116-117, 1989.
Russell, B. Our Knowledge and the External World as a Field
for Scientific Method in Philosophy. New York: Rout-
ledge, 1993.
Salmon, W. (Ed.). Zeno's Paradoxes. New York: Bobs-
Merrill, 1970.
Stewart, I. "Objections from Elea." In From Here to Infin-
ity: A Guide to Today's Mathematics. Oxford, England:
Oxford University Press, p. 72, 1996.
vos Savant, M. The World's Most Famous Math Problem.
New York: St. Martin's Press, pp. 50-55, 1993.
Zermelo's Axiom of Choice
see Axiom of Choice
Zermelo-Fraenkel Axioms
The Zermelo-Fraenkel axioms are the basis for
Zermelo-Fraenkel Set Theory. In the following,
3 stands for EXISTS, G for "is an element of," V for FOR
All, => for Implies, ^ for Not (Negation), a for And,
V for Or, ^ for "is Equivalent to," and S denotes the
union y of all the sets that are the elements of x,
1. Existence of the empty set: 3tiiu^{u G x).
2. Extensionality axiom: VzVy(Vtt(w G x ^ u G y) — V
x = y).
3. Unordered pair axiom: \/x\/y3z\fu(u G z ^ u = xV
4. Union (or "sum-set") axiom: V#3yVu(u G y ^
3v(u G v A v G x)).
5. Subset axiom: *ix3yiu(u £|/^ Wv(v G u — » v G
x)).
6. Replacement axiom: For any set-theoretic formula
A(u,v),
WuVv\/w(A(u, v) A A(u y w) — » v = w)
— > \/x3y"iv{v G y ^ 3u(u G x A A(u t v))).
7. Regularity axiom: For any set-theoretic formula
A(u), 3xA(x) -» 3x(A(x) A -<3y(A(y) Ay € x)).
8. Axiom of Choice:
Vx\Vu(u G x — > 3v(v G u))
AVttW((it exAvGxA~^u = v)
— > ~>3w(w 6uAw6v))-> 3y{y C S(x)
AWu(u G x -» 3z(z G u A z G y
AVw(w £ u Aw € y — v w = z)))}]
9. Infinity axiom: 3x(3u(u G x) A \fu(u 6 x -> 3v(v G
xAuCvA^v = u))).
If Axiom 6 is replaced by
6'. Axiom of subsets: for any set-theoretic formula A (u),
Vx3yyu(u G y ^ u G x A A(u)),
which can be deduced from Axiom 6, then the set theory
is called Zermelo Set Theory instead of Zermelo-
Fraenkel Set Theory.
Abian (1969) proved Consistency and independence
of four of the Zermelo-Fraenkel axioms.
see also Zermelo-Fraenkel Set Theory
References
Abian, A, "On the Independence of Set Theoretical Axioms."
Amer. Math. Monthly 76, 787-790, 1969.
lyanaga, S. and Kawada, Y. (Eds.). "Zermelo-Fraenkel Set
Theory." §35B in Encyclopedic Dictionary of Mathemat-
ics, Vol. 1. Cambridge, MA: MIT Press, pp. 134-135,
1980.
Zermelo-Fraenkel Set Theory
Zermelo-Fraenkel Set Theory
A version of Set Theory which is a formal system
expressed in first-order predicate LOGIC. Zermelo-
Fraenkel set theory is based on the ZERMELO-FRAENKEL
Axioms.
see also Logic, Set Theory, Zermelo-Fraenkel
Axioms, Zermelo Set Theory
Zermelo Set Theory
The version of set theory obtained if Axiom 6 of
Zermelo-Fraenkel Set Theory is replaced by
6\ Axiom of subsets: for any set-theoretic formula A(u),
\/x3yVu(u e y ^ u € x A A(u)),
which can be deduced from Axiom 6.
see also Zermelo-Fraenkel Set Theory
References
lyanaga, S. and Kawada, Y. (Eds.). "Zermelo-Fraenkel Set
Theory." §35B in Encyclopedic Dictionary of Mathemat-
ics. Cambridge, MA: MIT Press, p. 135, 1980.
Zernike Polynomial
Orthogonal Polynomials which arise in the expan-
sion of a wavefront function for optical systems with cir-
cular pupils. The Odd and Even Zernike polynomials
are given by
e T rrn
U^(p,(j>) - w »W C oB lmw
(1)
with radial function
(n-m)/2
W = £ wn^
(-l)'(n-O
1=0
[i(„ + TO ) -/]![!(„ - m )-/]! P
(2)
for n and m integers with n > rn > and n — m Even.
Otherwise,
iC(p) = 0. (3)
Here, (f> is the azimuthal angle with < <f> < 2ir and p
is the radial distance with < p < 1 (Prata and Rusch
1989). The radial functions satisfy the orthogonality
relation
J o Rn(p)R-(p)pd P =^ T y
(4)
where Sij is the Kronecker Delta, and are related to
the Bessel Function of the First Kind by
/'
Jo
x?(p)j m (vp) P dp = ( -i)(»-™)/»^M
(5)
(Born and Wolf 1989, p. 466). The radial Zernike poly-
nomials have the Generating Function
{2zp)™ x /l-2z(l-2p 2 ) + 2 2
(6)
Zernike Polynomial
and are normalized so that
i£ m (i) = 1
1965
(7)
(Born and Wolf 1989, p. 465). The first few NONZERO
radial polynomials are
j*8(p) = i
R\(p) = P
R° 2 (p) = 2p 2 -l
Rl(p)=p 2
R\(p) = 3p 3 - 2p
Rl(p) = P 3
Rl(p) = 6p 4 - 6p 2 + 1
Rl(p) = 4p 4 - 3p 2
rUp) = p 4
(Born and Wolf 1989, p. 465).
The Zernike polynomial is a special case of the Jacobi
Polynomial with
TV \ ' ^ ' nOt
(8)
and
x = 1 - 2p
(9)
=
(10)
a = m
(11)
n = |(n — m).
(12)
The Zernike polynomials also satisfy the RECURRENCE
Relations
pRn(p) =
iC +2 (p)
2(n+l)
n + 2
[(n + m+2)i^(p)
+(n-m)K^(p)] (13)
x "+2vr/ ( n + 2 )2-m 2
(n-m + 2) 2
4(n + l)p 2
{n + m) 2
n + 2
iC(p) + iC +a (p) =
-Rn-2(P) (H)
K{P) -
,n +a , , _ 1 d[R™£( P )-K^(p)]
n+ 1
dp
(15)
(Prata and Rusch 1989). The coefficients A™ and B™
in the expansion of an arbitrary radial function F(p,<f>)
in terms of Zernike polynomials
OO CO
F( ft « = ^K o !/;te«+cr(^)]
(16)
771=0 n = 77l
1966
Zernike Polynomial
Zero
are given by
Am
■Sin
where
(17)
em „ = ( e =72 form = 0,n^O (lg)
I 1 otherwise
Let a "primary" aberration be given by
* = aL
r2J + n
(0,<f>)p n cos m 6
(19)
with 21 + m + n = 4 and where y* is the COMPLEX
Conjugate of Y, and define
4™ = ai™^ 21 *™ (M),
giving
*
r ^4 Zm7l iC(p) cos(m#).
(20)
(21)
Then the types of primary aberrations are given in the
following table (Born and Wolf 1989, p. 470).
Aberration I m n A A'
spherical
aberration
coma
astigmatism
field curvature
distortion
4
I040A*
eA 040 R°(p)
A' Q31 p z cosO A 031 Rl(p)cosO
A' 022 p 2 cos 2 A 022 Rl{p) cos(2(9)
A' 120 p 2 eA 120 Rl(p)
A' xll p cos AiiiHj;(p)cos0
see also JACOBI POLYNOMIAL
References
Bezdidko, S. N. "The Use of Zernike Polynomials in Optics."
Sov. J. Opt Techn. 41, 425, 1974.
Bhatia, A. B. and Wolf, E. "On the Circle Polynomials of
Zernike and Related Orthogonal Sets." Proc. Cambridge
Phil. Soc. 50, 40, 1954.
Born, M. and Wolf, E. "The Diffraction Theory of Aber-
rations." Ch. 9 in Principles of Optics: Electromagnetic
Theory of Propagation, Interference, and Diffraction of
Light, 6th ed. New York: Pergamon Press, pp. 459-490,
1989.
Mahajan, V. N. "Zernike Circle Polynomials and Optical
Aberrations of Systems with Circular Pupils." In Engi-
neering and Lab. Notes 17 (Ed. R. R. Shannon), p. S-21,
Aug. 1994.
Prata, A. and Rusch, W. V. T. "Algorithm for Computa-
tion of Zernike Polynomials Expansion Coefficients." Appl.
Opt. 28, 749-754, 1989.
Wang, J. Y. and Silva, D. E. "Wave-Front Interpretation with
Zernike Polynomials." Appl. Opt. 19, 1510-1518, 1980.
Zernike, F. "Beugungstheorie des Schneidenverfahrens und
seiner verbesserten Form, der Phasenkontrastmethode."
Physica 1, 689-704, 1934.
Zhang, S. and Shannon, R. R. "Catalog of Spot Diagrams."
Ch. 4 in Applied Optics and Optical Engineering, Vol. 11.
New York: Academic Press, p. 201, 1992.
Zero
The Integer denoted which, when used as a counting
number, means that no objects are present. It is the only
Integer (and, in fact, the only Real Number) which
is neither NEGATIVE nor POSITIVE. A number which is
not zero is said to be NONZERO.
Because the number of PERMUTATIONS of elements is
1, 0! (zero FACTORIAL) is often defined as 1. This def-
inition is useful in expressing many mathematical iden-
tities in simple form. A number other than taken to
the POWER is defined to be 1. 0° is undefined, but
defining 0° = 1 allows concise statement of the beauti-
ful analytical formula for the integral of the generalized
Sinc Function
/°
Jo
sin'x, w i-«=(_i)L(«-»)/aJ
r — ax =
x° 2°- c (6- 1)!
|a/2j-c
x £ (- 1 )
fc=0
k(( ^)(a-2k) b - l [ln(a~2k)] c
given by Kogan, where a > b > c, c = a — b (mod 2),
and [x\ is the FLOOR FUNCTION.
The following table gives the first few numbers n such
that n k contains no zeros, for small k. The largest known
n for which 2 n contain no zeros is 86 (Madachy 1979),
with no other n < 4.6 x 10 7 (M. Cook), improving the
3.0739 x 10 7 limit obtained by Beeler et al. (1972). The
values a(n) such that the positions of the right-most
zero in 2 a(n) increases are 10, 20, 30, 40, 46, 68, 93, 95,
129, 176, 229, 700, 1757, 1958, 7931, 57356, 269518, . . .
(Sloane's A031140). The positions in which the right-
most zeros occur are 2, 5, 8, 11, 12, 13, 14, 23, 36, 38,
54, 57, 59, 93, 115, 119, 120, 121, 136, 138, 164, ...
(Sloane's A031141). The right-most zero f 2 781 ' 717 ' 865
occurs at the 217th decimal place, the farthest over for
powers up to 2.5 x 10 9 .
k Sloane n such that n contains no 0s
4, 5, 6, 7, 8, 9, 13, 14, 15, 16, ..
4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ..
4, 7, 8, 9, 12, 14, 16, 17, 18, . . .
4, 5, 6, 7, 9, 10, 11, 17, 18, 30, .
4, 5, 6, 7, 8, 12, 17, 24, 29, 44, .
6, 7, 10, 11, 19, 35, ...
5, 6, 8, 9, 11, 12, 13, 17, 24, 27,
4, 6, 7, 12, 13, 14, 17, 34, . . .
4, 6, 7, 8, 9, 12, 13, 14, 15, 16, .
2
007377
1,
2,3,
3
030700
1,
2,3,
4
030701
1,
2,3,
5
008839
1,
2,3,
6
030702
1,
2,3,
7
030703
1,
2,3,
8
030704
1,
2,3,
9
030705
1,
2,3,
11
030706
1,
2,3,
While it has not been proven that the numbers listed
above are the only ones without zeros for a given base,
the probability that any additional ones exist is van-
ishingly small. Under this assumption, the sequence of
largest n such that k n contains no zeros for k = 2, 3,
... is then given by 86, 68, 43, 58, 44, 35, 27, 34, 0, 41,
... (Sloane's A020665).
Zero Divisor
Zigzag Permutation 1967
see also 10, Naught, Negative, Nonnegative, Non-
zero, One, Positive, Two
References
Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM.
Cambridge, MA: MIT Artificial Intelligence Laboratory,
Memo AIM-239, Item 57, Feb. 1972.
Kogan, S. "A Note on Definite Integrals Involving Trigono-
metric Functions." http://www.mathsoft.com/asolve/
constant /pi/sin/sin .html.
Madachy, J. S. Madachy's Mathematical Recreations, New-
York: Dover, pp. 127-128, 1979.
Pappas, T. "Zero-Where & When." The Joy of Mathemat-
ics. San Carlos, CA: Wide World Publ./Tetra, p. 162,
1989.
Sloane, N. J. A. Sequence A007377/M0485 in "An On-Line
Version of the Encyclopedia of Integer Sequences."
Zero Divisor
A Nonzero element # of a Ring for which x • y = 0,
where y is some other NONZERO element and the vec-
tor multiplication x • y is assumed to be BILINEAR. A
Ring with no zero divisors is known as an INTEGRAL
DOMAIN. Let A denote an R-algebra, so that A is a
Vector Space over R and
A x A ^ A
(x,y) \-> x-y.
Now define
Zeta Function
A function satisfying certain properties which is com-
puted as an INFINITE SUM of NEGATIVE POWERS. The
most commonly encountered zeta function is the RiE-
mann Zeta Function,
Z = {x e A : x • y = for some NONZERO y G A},
fc=i
see also Dedekind Function, Dirichlet Beta
Function, Dirichlet Eta Function, Dirichlet
L-Series, Dirichlet Lambda Function, Epstein
Zeta Function, Jacobi Zeta Function, Nint Zeta
Function, Prime Zeta Function, Riemann Zeta
Function
References
Ireland, K. and Rosen, M. "The Zeta Function." Ch. 11 in
A Classical Introduction to Modern Number Theory, 2nd
ed. New York: Springer- Verlag, pp. 151-171, 1990.
Zeuthen's Rule
On an Algebraic Curve, the sum of the number of
coincidences at a noncuspidal point C is the sum of the
orders of the infinitesimal distances from a nearby point
P to the corresponding points when the distance PC is
taken as the principal infinitesimal.
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 131, 1959.
where 6 Z. A is said to be m-AssociATIVE if there
exists an m-dimensional Subspace 5 of A such that
(y • x) ■ z = y • (x • z) for all y,z € A and x G S. A is said
to be Tame if Z is a finite union of Subspaces of A.
References
Finch, S. "Zero Structures in Real Algebras." http://www.
mathsoft.com/asolve/zerodiv/zerodiv.html.
Zero (Root)
see Root
Zero-Sum Game
A Game in which players make payments only to each
other. One player's loss is the other player's gain, so the
total amount of "money" available remains constant.
see also Finite Game, Game
References
Dresner, M. The Mathematics of Games of Strategy: Theory
and Applications. New York: Dover, p. 2, 1981.
Zeta Fuchsian
A class of functions discovered by Poincare which are
related to the AUTOMORPHIC FUNCTIONS.
see also AUTOMORPHIC FUNCTION
Zeuthen's Theorem
If there is a (v, u') correspondence between two curves
of Genus p and p' and the number of Branch Points
properly counted are and 0\ then
+ 2i/(p-l)=/?' + 2i/(p' -1).
see also Chasles-Cayley-Brill Formula
References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New
York: Dover, p. 246, 1959.
Zig Number
An Odd Alternating Permutation number, more
commonly called an EULER NUMBER or SECANT NUM-
BER.
see also Alternating Permutation, Euler Num-
ber, Zag Number
Zig-Zag Triangle
see also Seidel-Entringer-Arnold Triangle
Zigzag Permutation
see Alternating Permutation
1968
Zillion
Zonohedron
Zillion
A generic word for a very LARGE NUMBER. The term
has no well-defined mathematical meaning. Conway and
Guy (1996) define the nth zillion as 10 3n+3 in the Ameri-
can system (million = 10 6 , billion = 10 9 , trillion = 10 12 ,
...) and 10 6n in the British system (million = 10 6 ,
billion = 10 12 , trillion = 10 18 , ...)■ Conway and Guy
(1996) also define the words n-PLEX and n-MINEX for
10 n and 10 _7 \ respectively.
see also Large Number
References
Conway, J. H. and Guy, R. K. The Book of Numbers. New
York: Springer- Verlag, pp. 13—16, 1996.
Zipf ' s Law
In the English language, the probability of encountering
the rth most common word is given roughly by P(r) =
0.1/r for r up to 1000 or so. The law breaks down for less
frequent words, since the HARMONIC SERIES diverges.
Pierce's (1980, p. 87) statement that J2 p ( r ) > 1 for
r = 8727 is incorrect. Goetz states the law as follows:
The frequency of a word is inversely proportional to its
Rank r such that
P(r)
Zone
rln(1.78 J R)'
where R is the number of different words.
see also Harmonic Series, Rank (Statistics)
References
Goetz, P. "Phil's Good Enough Complexity Dictionary."
http://www.cs.buffalo.edu/-goetz/dict.html.
Pierce, J. R. Introduction to Information Theory: Symbols,
Signals, and Noise, 2nd rev. ed. New York: Dover, pp. 86-
87 and 238-239, 1980.
Zollner's Illusion
N A
In this Illusion, the Vertical lines in the above figure
are PARALLEL, but appear to be tilted at an angle.
see also ILLUSION
References
Jablan, S. "Some Visual Illusions Occurring in Interrupted
Systems." http: //members .tripod, com/ -modularity/
interr.htm.
Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide
World Publ./Tetra, p. 172, 1989.
Zonal Harmonic
A Spherical Harmonic which is a product of factors
linear in x 2 , y 2 , and z 2 , with the product multiplied by
z when n is ODD.
see also Tesseral Harmonic
The Surface Area of a Spherical Segment. Call the
Radius of the Sphere R, the upper and lower Radii
6 and a, respectively, and the height of the SPHERICAL
Segment h. The zone is a Surface of Revolution
about the z-AxiS, so the SURFACE AREA is given by
5 = 2tt
/ xy/l + x n
l dz.
(1)
Int
of a
he xz-plane
Circle,
, the eqi
x =
lation of the
zone is
simply that
y/R 2 -z*,
(2)
so
x = -
-z(R 2 -z 2 )-
1/2
(3)
J 2
X = —
z 2
(4)
and
.V* 3 -* 2
S = 2tt I ' ^fR?
J \,
1 +
R 2 -z<
dz
= 2ttR I dz = 2nR(^R 2 -b 2 - \/R 2 - a 2 )
■i
yjB?^a?
= 2nRh.
(5)
This result is somewhat surprising since it depends only
on the height of the zone, not its vertical position with
respect to the Sphere.
see also Sphere, Spherical Cap, Spherical Seg-
ment, Zonohedron
References
Beyer, W. H. (Ed.). CRC Standard Mathematical Tables,
28th ed. Boca Raton, FL: CRC Press, p. 130, 1987.
Zonohedron
A convex POLYHEDRON whose faces are PARALLEL-sided
2m-gons. There exist n(n — 1) Parallelograms in a
nonsingular zonohedron, where n is the number of differ-
ent directions in which EDGES occur (Ball and Coxeter
Zonotype
Zsigmondy Theorem 1969
1987, pp. 141-144). Zonohedra include the Cube, En-
NEACONTAHEDRON, GREAT RHOMBIC Triacontahe-
dron, Medial Rhombic Triacontahedron, Rhom-
bic Dodecahedron, Rhombic Icosahedron, Rhom-
bic Triacontahedron, Rhombohedron, and Trun-
cated Cuboctahedron, as well as the entire class of
Parallelepipeds.
Regular zonohedra have bands of PARALLELOGRAMS
which form equators and are called "ZONES." Ev-
ery convex polyhedron bounded solely by PARALLELO-
GRAMS is a zonohedron (Coxeter 1973, p. 27). Plate
II (following p. 32 of Coxeter 1973) illustrates some
equilateral zonohedra. Equilateral zonohedra can be
regarded as 3-dimensional projections of n~D HYPER-
CUBES (Ball and Coxeter 1987).
see also HYPERCUBE
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recre-
ations and Essays, 13th ed. New York: Dover, pp. 141-
144, 1987.
Coxeter, H. S. M. "Zonohedra." §2.8 in Regular Polytopes,
3rd ed. New York: Dover, pp. 27-30, 1973.
Coxeter, H. S. M. Ch. 4 in Twelve Geometric Essays. Car-
bondale, IL: Southern Illinois University Press, 1968.
Eppstein, D. "Ukrainian Easter Egg." http://www.ics.uci
.edu/~eppstein/junkyard/ukraine.
Fedorov, E. S. Zeitschr. Krystallographie und Mineralogie
21, 689, 1893.
Fedorov, E.W. Nachala Ucheniya o Figurakh. Leningrad,
1953.
Hart, G. W. "Zonohedra." http://www.li.net/-george/
virtual -polyhedra/zonohedra-inf o.html.
Zonotype
The Minkowski Sum of line segments.
Zorn's Lemma
If 5 is any nonempty PARTIALLY ORDERED Set in
which every Chain has an upper bound, then S has
a maximal element. This statement is equivalent to the
Axiom of Choice.
see also Axiom OF CHOICE
Zsigmondy Theorem
If 1 < b < a and (a, b) = 1 (i.e., a and b are RELATIVELY
Prime), then a n - b n has a Primitive Prime Factor
with the following two possible exceptions:
1. 2
2
n ■
2 and a + b is a POWER of 2.
Similarly, if a > b > 1, then a n + b n has a PRIMITIVE
Prime Factor with the exception 2 3 + l 3 = 9.
References
Ribenboim, P. The Little Book of Big Primes. New York:
Springer- Verlag, p. 27, 1991.