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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 

_£ 

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2 

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o_ 

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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. 



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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.