Concise Encyclopedia Of Mathematics.pdf

Numerals

(0, 1)-Matrix

A(0 ; 1) / - INTEGER MATRIX , i.e., a matrix each of whose

elements is 0 or 1, also called a binary matrix.

The numbers of binary matrices with no adjacent 1s

(in either columns or rows) for n1, 2, ..., are given

by 2, 7, 63, 1234, ... (Sloane’s A006506). For example,

the binary matrices with no adjacent 1s are

(1, 0, 1)-Matrix

The number of distinct (1 ; 0 ; 1) / - / nn matrices

(counting row and column permutations, the trans-

pose, and multiplication by 1 as equivalent) having

2n different row and column sums for n2, 4, 6, ...

are 1, 4, 39, 2260, 1338614, ... (Kleber). For example,

the 22 matrix is given by

;

1 1

;

To get the total number from these counts (assuming

that 0 is not the missing sum, which is true for n5

10) ; multiply by (2n!) 2 : In general, if an -matrix which

has different column and row sums (collectively called

line sums), then

These numbers are closely related to the HARD

SQUARE ENTROPY CONSTANT . The numbers of binary

matrices with no three adjacent 1s for , 2, ..., are given

by 2, 16, 265, 16561, ... (Sloane’s A050974).

Wilf (1997) considers the complexity of transforming

an mn binary matrix A into a TRIANGULAR MATRIX

by permutations of the rows and columns of , and

concludes that the problem falls in difficulty between

a known easy case and a known hard case of the

general NP -COMPLETE PROBLEM .

1. n is even,

2. The number in f n ; 1n ; 2n ; ... ; n g that

does not appear as a line sum is either n or , and

3. Of the largest line sums, half are column sums

and half are row sums

See also A DJACENCY M ATRIX , F ROBENIUS- K O ¨ NIG T HE -

OREM , G ALE- R YSER T HEOREM , H ADAMARD’S M AXIMUM

D ETERMINANT P ROBLEM , H ARD S QUARE E NTROPY

C ONSTANT , I DENTITY M ATRIX , I NCIDENCE M ATRIX ,

I NTEGER M ATRIX , L AM’S P ROBLEM , S- C LUSTER , S- R UN

(Bodendiek and Burosch 1995, F. Galvin).

See also A LTERNATING S IGN M ATRIX , C -MATRIX ,

I NTEGER M ATRIX

References

Bodendiek, R. and Burosch, G. "Solution to the Antimagic

0 ; 1 ; 1 Matrix Problem." Aufgabe 5.30 in Streifzu ¨ ge

durch die Kombinatorik: Aufgaben und Lo ¨ sungen aus

dem Schatz der Mathematik-Olympiaden. Heidelberg,

Germany: Spektrum Akademischer Verlag, pp. 250 / 253,

1995.

References

Brualdi, R. A. "Discrepancy of Matrices of Zeros and Ones."

Electronic J. Combinatorics 6, No. 1, R15, 1 / 12, 1999.

http://www.combinatorics.org/Volume_6/v6i1toc.html.

Ehrlich, H. "Determinantenabsch ¨ tzungen f ¨ r bin ¨ re Ma-

trizen." Math. Z. 83, 123 / 132, 1964.

Ehrlich, H. and Zeller, K. "Bin ¨ re Matrizen." Z. angew.

Math. Mechanik 42, T20 / 21, 1962.

Koml´s, J. "On the Determinant of -Matrices." Studia Math.

Hungarica 2,7 / 21 1967.

Metropolis, N. and Stein, P. R. "On a Class of Matrices with

Vanishing Determinants." J. Combin Th. 3, 191 / 198,

1967.

Ryser, H. J. "Combinatorial Properties of Matrices of Zeros

and Ones." Canad. J. Math. 9, 371 / 377, 1957.

Sloane, N. J. A. Sequences A006506/M1816 and A050974 in

"An On-Line Version of the Encyclopedia of Integer

Sequences." http://www.research.att.com/~njas/se-

quences/eisonline.html.

Wilf, H. "On Crossing Numbers, and Some Unsolved

Problems." In Combinatorics, Geometry, and Probability:

A Tribute to Paul Erdos. Papers from the Conference in

Honor of Erdos’ 80th Birthday Held at Trinity College,

Cambridge, March 1993 (Ed. B. Bollob ´ s and A. Thoma-

son). Cambridge, England: Cambridge University Press,

pp. 557 / 562, 1997.

Williamson, J. "Determinants Whose Elements Are 0 and 1."

Amer. Math. Monthly 53, 427 / 434, 1946.

(1, 1)-Matrix

See also H ADAMARD M ATRIX , I NTEGER M ATRIX

References

Kahn, J.; Komlo´s, J.; and Szemeredi, E. "On the Probability

that a Random 91 Matrix is Singular." J. Amer. Math.

Soc. 8, 223 / 240, 1995.

0-Free

Z EROFREE

D IVISION BY Z ERO ,F ALLACY ,N AUGHT ,Z ERO ,Z ERO

D IVISOR ,Z ERO- F ORM ,Z ERO M ATRIX ,Z ERO- S UM G AME ,

Z EROFREE

The number one (1), also called "unity" is the first

POSITIVE INTEGER .Itisan ODD NUMBER . Although the

number 1 used to be considered a PRIME NUMBER ,it

F ALLACY

;

requires special treatment in so many definitions and

applications involving primes greater than or equal to

2 that it is usually placed into a class of its own (Wells

1986, p. 31). The number 1 is sometimes also called

"unity," so the th roots of 1 are often called the th

ROOTS OF UNITY . FRACTIONS having 1 as a NUMERATOR

are called UNIT FRACTIONS . If only one root, solution,

etc., exists to a given problem, the solution is called

UNIQUE .

The GENERATING FUNCTION having all COEFFICIENTS

1 is given by

References

Daiev, V. "Problem 636: Greatest Divisors of Even Integers."

Math. Mag. 40, 164 / 165, 1967.

Guy, R. K. "Residues of Powers of Two." §F10 in Unsolved

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

Verlag, p. 250, 1994.

Montgomery, P.-L. "New solution to 2^n 3 (mod n)."

NMBRTHRY@listserv.nodak.edu posting, 24 Jun 1999.

Sloane, N. J. A. Sequences A036236 and A050259 in "An

On-Line Version of the Encyclopedia of Integer Se-

quences." http://www.research.att.com/~njas/sequences/

eisonline.html.

Wells, D. The Penguin Dictionary of Curious and Interesting

Numbers. Middlesex, England: Penguin Books, pp. 41 /

44, 1986.

1 x 1xx 2 x 3 x 4 ... :

See also F ALLACY , O NE- F ORM , O NE- M OUTH T HEOREM ,

O NE- N INTH C ONSTANT , O NE- S HEETED H YPERBOLOID ,

O NE-TO- O NE , O NE- W AY F UNCTION , 2 , 3 , C OMPLEXITY

(N UMBER ) , E XACTLY O NE , R OOT OF U NITY , U NIQUE ,

U NIT F RACTION , Z ERO

2x mod 1 Map

Let x 0 be a RATIONAL NUMBER in the CLOSED INTERVAL

[0 ; 1] ; and generate a SEQUENCE using the MAP

x n1 2x n (mod 1) :

(1)

Then the number of periodic ORBITS of period p (for

PRIME ) is given by

References

Wells, D. The Penguin Dictionary of Curious and Interesting

Numbers. Middlesex, England: Penguin Books, pp. 30 /

32, 1986.

N p 2 p 2

(2)

(i.e, the number of period- repeating bit strings,

modulo shifts). Since a typical ORBIT visits each point

with equal probability, the NATURAL INVARIANT is

given by

The number two (2) is the second POSITIVE INTEGER

and the first PRIME NUMBER .Itis EVEN , and is the

only EVEN PRIME (the PRIMES other than 2 are called

the ODD PRIMES ). The number 2 is also equal to its

FACTORIAL since 2!2 : A quantity taken to the

POWER 2 is said to be SQUARED . The number of times

k a given BINARY number b n b 2 b 1 b 0 is divisible by 2

is given by the position of the first b k 1 ; counting

from the right. For example, 121100 is divisible by

2 twice, and 131101 is divisible by 2 zero times.

The only known solutions to the CONGRUENCE

r(x)1 :

(3)

See also T ENT M AP

References

Ott, E. Chaos in Dynamical Systems. Cambridge, England:

Cambridge University Press, pp. 26 / 31, 1993.

3 is the only INTEGER which is the sum of the

preceding POSITIVE INTEGERS (123) and the only

number which is the sum of the FACTORIALS of the

preceding POSITIVE INTEGERS ( / 1!2!3) : It is also

the first ODD PRIME . A quantity taken to the POWER 3

is said to be CUBED .

The sequence 1, 31, 331, 3331, 33331, ... (Sloane’s

A033175) consisting of n0, 1, ... 3s followed by a 1.

The th tern is given by

2 n 3 (mod n)

are n4700063497 (Sloane’s A050259; Guy 1994)

and

63130707451134435989380140059866138830623361447484274774099906755

(P.-L. Montgomery 1999). In general, the least satis-

fying

2 n k (mod n)

for k2, 3, ... are n3, 4700063497, 6, 19147, 10669,

25, 9, 2228071, ... (Sloane’s A036236).

a(n) 10 n1 7

The result is prime for , 2, 3, 4, 5, 6, 7, 17, 39, ...

(Sloane’s A055520); i.e., for 3, 31, 331, 3331, 33331,

333331, 3333331, 33333331, ... (Sloane’s A051200), a

fact which Gardner (1997) calls "a remarkable pat-

tern that is entirely accidental and leads nowhere."

See also 1 , B INARY , 3 , R ULER F UNCTION , S QUARED ,

T WO- E ARS T HEOREM , T WO- F ORM , T WO- G RAPH , T WO -

S CALE E XPANSION , T WO- S HEETED H YPERBOLOID ,

Z ERO

See also 1 , 2 , 3X1 M APPING , C UBED , P ERIOD T HREE

T HEOREM , T ERNARY , T HREE- C HOICE P OLYGON ,

T HREE- C HOICE W ALK , T HREE- C OLORABLE , T HREE

C ONICS T HEOREM , T HREE J UG P ROBLEM , T HREE-

V ALUED L OGIC , T REFOIL K NOT , W IGNER 3J- S YMBOL ,

Z ERO

EQUILATERAL

TRIANGLE

TETRAHEDRON PENTATOPE

SIMPLEX

POLYGON

POLYHEDRON POLYCHORON POLYTOPE

LINE SEG-

MENT

PLANE

HYPERPLANE HYPERPLANE

References

Gardner, M. The Last Recreations: Hydras, Eggs, and Other

Mathematical Mystiﬁcations. New York: Springer-Verlag,

p. 194, 1997.

Sloane, N. J. A. Sequences A033175, A051200, and A055520

in "An On-Line Version of the Encyclopedia of Integer

Sequences." http://www.research.att.com/~njas/se-

quences/eisonline.html.

Smarandache, F. Properties of Numbers. University of

Craiova, 1973.

Wells, D. The Penguin Dictionary of Curious and Interesting

Numbers. Middlesex, England: Penguin Books, pp. 46 /

48, 1986.

SQUARE

OCTAHEDRON 16-CELL

CROSS POLY-

TOPE

EDGE

FACE

FACET

AREA

VOLUME

CONTENT

The SURFACE AREA of a HYPERSPHERE in -D is given by

S n 2p n = 2

G 2 n ;

and the VOLUME by

3x1 Mapping

C OLLATZ P ROBLEM

V n

p n = 2 R n

G 1 2 n

;

where G(n) is the GAMMA FUNCTION .

See also D IMENSION , H YPERCUBE , H YPERSPHERE

See also F OUR C OINS P ROBLEM , F OUR- C OLOR T HEO -

REM , F OUR C ONICS T HEOREM , F OUR E XPONENTIALS

C ONJECTURE , F OUR T RAVELERS P ROBLEM , F OUR- V EC -

TOR , F OUR- V ERTEX T HEOREM , L AGRANGE’S F OUR-

S QUARE T HEOREM

References

Hinton, C. H. The Fourth Dimension. Pomeroy, WA: Health

Research, 1993.

Manning, H. The Fourth Dimension Simply Explained.

Magnolia, MA: Peter Smith, 1990.

Manning, H. Geometry of Four Dimensions. New York:

Dover, 1956.

Neville, E. H. The Fourth Dimension. Cambridge, England:

Cambridge University Press, 1921.

Rucker, R. von Bitter. The Fourth Dimension: A Guided

Tour of the Higher Universes. Boston, MA: Houghton

Mifﬂin, 1984.

Sommerville, D. M. Y. An Introduction to the Geometry of

Dimensions. New York: Dover, 1958.

References

Wells, D. The Penguin Dictionary of Curious and Interesting

Numbers. Middlesex, England: Penguin Books, pp. 55 /

58, 1986.

4-D Geometry

4- D IMENSIONAL G EOMETRY

4-Dimensional Geometry

4-dimensional geometry is Euclidean geometry ex-

tended into one additional DIMENSION . The prefix

"hyper-" is usually used to refer to the 4- (and higher-)

dimensional analogs of 3-dimensional objects, e.g.

H YPERCUBE , HYPERPLANE , HYPERSPHERE . -dimen-

sional POLYHEDRA are called POLYTOPES . the 4-dimen-

sional cases of general -dimensional objects are often

given special names, such as those summarized in the

following table.

See also F IVE D ISKS P ROBLEM , M IQUEL F IVE C IRCLES

T HEOREM , P ENTAGON , P ENTAGRAM , P ENTAHEDRON ,

T ETRAHEDRON 5- C OMPOUND

References

Wells, D. The Penguin Dictionary of Curious and Interesting

Numbers. Middlesex, England: Penguin Books, pp. 58 /

67, 1986.

5-Cell

P ENTATOPE

2-D

3-D

4-D

General

CIRCLE

SPHERE

GLOME

HYPERSPHERE

SQUARE

CUBE

TESSERACT

HYPERCUBE

See also 6- S PHERE C OORDINATES , H EXAGON , H EXAHE -

DRON , S IX C IRCLES T HEOREM , S IX- C OLOR T HEOREM ,

S IX E XPONENTIALS T HEOREM , W IGNER 6J- S YMBOL

which gives spheres tangent to the xy-plane at the

origin for w constant.

The metric coefficients are

References

Wells, D. The Penguin Dictionary of Curious and Interesting

Numbers. Middlesex, England: Penguin Books, pp. 67 /

69, 1986.

g uu g vv g ww

u 2 v 2 w 2

Þ 2 :

(7)

See also C ARTESIAN C OORDINATES , I NVERSION

6-Sphere Coordinates

References

Moon, P. and Spencer, D. E. "6-Sphere Coordinates

(u ; v ; w) : / " Fig. 4.07 in Field Theory Handbook, Including

Coordinate Systems, Differential Equations, and Their

Solutions, 2nd ed. New York: Springer-Verlag, pp. 122 /

123, 1988.

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