A cellular automaton is a collection of "colored" cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells. The rules are then applied iteratively for as many time steps as desired. von Neumann was one of the first people to consider such a model, and incorporated a cellular model into his "universal constructor." Cellular automata were studied in the early 1950s as a possible model for biological systems (Wolfram 2002, p. 48). Comprehensive studies of cellular automata have been performed by S. Wolfram starting in the 1980s, and Wolfram's fundamental research in the field culminated in the publication of his book A New Kind of Science (Wolfram 2002) in which Wolfram presents a gigantic collection of results concerning automata, among which are a number of groundbreaking new discoveries.The Season 2 episode..
Triangle geometry is the study of the properties of triangles, including associated triangle centers, triangle lines, central circles, triangle cubics, and many others. These geometric objects often have remarkable properties with respect to the triangle.An amazing number of connections between geometric structures occur in triangle geometry, prompting Crelle (1821) to state, "It is indeed a wonder that so simple a figure as the triangle is so inexhaustible in its properties," and J. Wetzel to remark that triangle geometry "has more miracles per square meter than any other area of mathematics" (Kimberling 1998, p. 1).Triangle geometry lay dormant for most of the middle of the 20th century, but has recently arisen "from the dust and ashes that history has piled on it" (Davis 1995) by the use of computers to systematically study and geometric structures and their properties (Davis 1995,..
The Feigenbaum constant is a universal constant for functions approaching chaos via period doubling. It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function(1)and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter is increased for fixed . The plot above is made by iterating equation (1) with several hundred times for a series of discrete but closely spaced values of , discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining.A similar plot that more directly shows the cycle may be constructed by plotting as a function of . The plot above (Trott, pers. comm.) shows the resulting curves for , 2, and 4.Let be the point at which a period -cycle appears, and denote the converged value by . Assuming geometric convergence, the difference between this value and..
Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. It is the only integer (and, in fact, the only real number) that is neither negative nor positive. A number which is not zero is said to be nonzero. A root of a function is also sometimes known as "a zero of ."The Schoolhouse Rock segment "My Hero, Zero" extols the virtues of zero with such praises as, "My hero, zero Such a funny little hero But till you came along We counted on our fingers and toes Now you're here to stay And nobody really knows How wonderful you are Why we could never reach a star Without you, zero, my hero How wonderful you are."Zero is commonly taken to have the factorization (e.g., in the Wolfram Language's FactorInteger[n] command). On the other hand, the divisors and divisor function are generally taken to be undefined, since by convention, (i.e., divides 0) for every except zero.Because the number of..
The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. has numerical value(1)(OEIS A002162).The irrationality measure of is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990).It is not known if is normal (Bailey and Crandall 2002).The alternating series and BBP-typeformula(2)converges to the natural logarithm of 2, where is the Dirichlet eta function. This identity follows immediately from setting in the Mercator series, yielding(3)It is also a special case of the identity(4)where is the Lerch transcendent.This is the simplest in an infinite class of such identities, the first few of which are(5)(6)(E. W. Weisstein, Oct. 7, 2007).There are many other classes of BBP-type formulas for , including(7)(8)(9)(10)(11)Plouffe (2006) found the beautiful sum(12)A rapidly converging Zeilberger-type sum..
In response to a letter from Goldbach, Euler considered sums ofthe form(1)(2)with and and where is the Euler-Mascheroni constant and is the digamma function. Euler found explicit formulas in terms of the Riemann zeta function for with , and E. Au-Yeung numerically discovered(3)where is the Riemann zeta function, which was subsequently rigorously proven true (Borwein and Borwein 1995). Sums involving can be re-expressed in terms of sums the form via(4)(5)(6)and(7)where is defined below.Bailey et al. (1994) subsequently considered sums ofthe forms(8)(9)(10)(11)(12)(13)(14)(15)where and have the special forms(16)(17)(18)where is a generalized harmonic number.A number of these sums can be expressed in terms of the multivariatezeta function, e.g.,(19)(Bailey et al. 2006a, p. 39, sign corrected; Bailey et al. 2006b).Special cases include(20)(P. Simone, pers. comm., Aug. 30, 2004).Analytic single..
A lattice reduction algorithm, named after discoverers Lenstra, Lenstra, and Lovasz (1982), that produces a lattice basis of "short" vectors. It was noticed by Lenstra et al. (1982) that the algorithm could be used to obtain factors of univariate polynomials, which amounts to the determination of integer relations. However, this application of the algorithm, which later came to be one of its primary applications, was not stressed in the original paper.For a complexity analysis of the LLL algorithm, see Storjohann (1996).The Wolfram Language command LatticeReduce[matrix] implements the LLL algorithm to perform lattice reduction. The Wolfram Language's implementation requires the input to consist of rational numbers, so Rationalize may need to be called first.More recently, other algorithms such as PSLQ, which can be significant faster than LLL, have been developed for finding integer relations. PSLQ achieves its performance..
The dilogarithm is a special case of the polylogarithm for . Note that the notation is unfortunately similar to that for the logarithmic integral . There are also two different commonly encountered normalizations for the function, both denoted , and one of which is known as the Rogers L-function.The dilogarithm is implemented in the Wolfram Language as PolyLog[2, z].The dilogarithm can be defined by the sum(1)or the integral(2)Plots of in the complex plane are illustrated above.The major functional equations for the dilogarithm are given by (3)(4)(5)(6)(7)A complete list of which can be evaluated in closed form is given by(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)where is the golden ratio (Lewin 1981, Bailey et al. 1997; Borwein et al. 2001).There are several remarkable identities involving the dilogarithm function. Ramanujan gave the identities (20)(21)(22)(23)(24)(25)(26)(Berndt 1994, Gordon and McIntosh 1997) in..
A definite integral is an integral(1)with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral (which is the usual definition encountered in elementary textbooks). However, a general definite integral is taken in the complex plane, resulting in the contour integral(2)with , , and in general being complex numbers and the path of integration from to known as a contour.The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if is the indefinite integral for a continuous function , then(3)This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Definite integrals may be evaluated in the Wolfram Language using Integrate[f, x, a, b].The question of which definite..
An algorithm which can be used to find integer relations between real numbers , ..., such thatwith not all . Although the algorithm operates by manipulating a lattice, it does not reduce it to a short vector basis, and is therefore not a lattice reduction algorithm. PSLQ is based on a partial sum of squares scheme (like the PSOS algorithm) implemented using QR decomposition. It was developed by Ferguson and Bailey (1992). A much simplified version of the algorithm was subsequently developed by Ferguson et al. (1999), which also extends the algorithm to complex numbers and quaternions. Ferguson et al. (1999) also demonstrated that PSLQ is distinct from the HJLS algorithm.The PSLQ algorithm terminates after a number of iterations bounded by a polynomial in and uses a numerically stable matrix reduction procedure (Ferguson and Bailey 1992). PSLQ tends to be faster than the Ferguson-Forcade algorithm and LLL algorithm because of clever techniques..
The inverse tangent is the multivalued function (Zwillinger 1995, p. 465), also denoted (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 311; Jeffrey 2000, p. 124) or (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127), that is the inverse function of the tangent. The variants (e.g., Bronshtein and Semendyayev, 1997, p. 70) and are sometimes used to refer to explicit principal values of the inverse cotangent, although this distinction is not always made (e.g,. Zwillinger 1995, p. 466).The inverse tangent function is plotted above along the real axis.Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation (commonly used in North America and in pocket calculators worldwide), denotes the tangent and the..
The polylogarithm , also known as the Jonquière's function, is the function(1)defined in the complex plane over the open unit disk. Its definition on the whole complex plane then follows uniquely via analytic continuation.Note that the similar notation is used for the logarithmic integral.The polylogarithm is also denoted and equal to(2)where is the Lerch transcendent (Erdélyi et al. 1981, p. 30). The polylogarithm arises in Feynman diagram integrals (and, in particular, in the computation of quantum electrodynamics corrections to the electrons gyromagnetic ratio), and the special cases and are called the dilogarithm and trilogarithm, respectively. The polylogarithm is implemented in the Wolfram Language as PolyLog[n, z].The polylogarithm also arises in the closed form of the integrals of the Fermi-Diracdistribution(3)where is the gamma function, and the Bose-Einstein distribution(4)The special case..
A set of real numbers , ..., is said to possess an integer relation if there exist integers such thatwith not all . For historical reasons, integer relation algorithms are sometimes called generalized Euclidean algorithms or multidimensional continued fraction algorithms.An interesting example of such a relation is the 17-vector (1, , , ..., ) with , which has an integer relation (1, 0, 0, 0, , 0, 0, 0, , 0, 0, 0, , 0, 0, 0, 1), i.e.,This is a special case of finding the polynomial of degree satisfied by .Integer relations can be found in the Wolfram Language using FindIntegerNullVector[x1, x2, ...].Integer relation algorithms can be used to solve subset sum problems, as well as to determine if a given numerical constant is equal to a root of a univariate polynomial of degree or less (Bailey and Ferguson 1989, Ferguson and Bailey 1992).One of the simplest cases of an integer relation between two numbers is the one inherent in the definition of the greatest..
There is a series of BBP-type formulas for in powers of ,(1)(2)(3)(4)(5)(6),(7)(8)(9)(10)some of which are noted by Bailey et al. (1997), and ,(11)(12)Another identity is(13)where is the polylogarithm. (13) is equivalent to(14)(Bailey et al. 1997).
Catalan's constant is a constant that commonly appears in estimates of combinatorial functions and in certain classes of sums and definite integrals. It is usually denoted (this work), (e.g., Borwein et al. 2004, p. 49), or (Wolfram Language).Catalan's constant may be defined by(1)(Glaisher 1877, who however did not explicitly identify the constant in this paper). It is not known if is irrational.Catalan's constant is implemented in the WolframLanguage as Catalan.The constant is named in honor of E. C. Catalan (1814-1894), who first gave an equivalent series and expressions in terms of integrals. Numerically,(2)(OEIS A006752). can be given analytically by the following expressions(3)(4)(5)where is the Dirichlet beta function, is Legendre's chi-function, is the Glaisher-Kinkelin constant, and is the partial derivative of the Hurwitz zeta function with respect to the first argument.Glaisher (1913) gave(6)(Vardi..
There are many formulas of of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. is intimately related to the properties of circles and spheres. For a circle of radius , the circumference and area are given by(1)(2)Similarly, for a sphere of radius , the surface area and volume enclosed are(3)(4)An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula(5)There are three other Machin-like formulas,as well as thousands of other similar formulas having more terms.Gregory and Leibniz found(6)(7)(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The error after the th term of this series in the Gregory series is larger than so this sum converges so slowly that 300 terms are not sufficient to calculate correctly to two decimal places! However, it can be transformed..
The Gregory series is a pi formula found by Gregory and Leibniz and obtained by plugging into the Leibniz series,(1)(Wells 1986, p. 50). The formula converges very slowly, but its convergence can be accelerated using certain transformations, in particular(2)where is the Riemann zeta function (Vardi 1991).Taking the partial series gives the analytic result(3)Rather amazingly, expanding about infinity gives the series(4)(Borwein and Bailey 2003, p. 50), where is an Euler number. This means that truncating the Gregory series at half a large power of 10 can give a decimal expansion for whose decimal digits are largely correct, but where wrong digits occur with precise regularity. For example, taking gives where the sequence of differences is precisely twice the Euler (secant) numbers. In fact, just this pattern of digits was observed by J. R. North in 1988 before the closed form of the truncated series was known..
has decimal expansion given by(1)(OEIS A000796). The following table summarizes some record computations of the digits of .1999Kanada, Ushio and KurodaDec. 2002Kanada, Ushio and Kuroda (Peterson 2002, Kanada 2003)Aug. 2012A. J. Yee (Yee)Aug. 2012S. Kondo and A. J. Yee (Yee)Dec. 2013A. J. Yee and S. Kondo (Yee)The calculation of the digits of has occupied mathematicians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his life calculating to 35 places. Although he did not live to publish his result, it was inscribed on his gravestone. Wells (1986, p. 48) discusses a number of other calculations. The calculation of also figures in the Season 2 Star Trek episode "Wolf in the Fold" (1967), in which Captain Kirk and Mr. Spock force an evil entity (composed of pure energy and which feeds on fear) out of the starship..
The BBP (named after Bailey-Borwein-Plouffe) is a formula for calculating pidiscovered by Simon Plouffe in 1995,Amazingly, this formula is a digit-extraction algorithm for in base 16.Following the discovery of this and related formulas, similar formulas in other bases were investigated. This class of formulas are now known as BBP-type formulas.
The constant pi, denoted , is a real number defined as the ratio of a circle's circumference to its diameter ,(1)(2) has decimal expansion given by(3)(OEIS A000796). Pi's digits have many interesting properties, although not very much is known about their analytic properties. However, spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140-141) and digit-extraction algorithms (the BBP formula) are known for .A brief history of notation for pi is given by Castellanos (1988ab). is sometimes known as Archimedes' constant or Ludolph's constant after Ludolph van Ceulen (1539-1610), a Dutch calculator. The symbol was first used by Welsh mathematician William Jones in 1706, and subsequently adopted by Euler. In Measurement of a Circle, Archimedes (ca. 225 BC) obtained the first rigorous approximation by inscribing and circumscribing -gons on a circle using the Archimedes algorithm. Using (a 96-gon),..
The figure eight knot, also known as the Flemish knot and savoy knot, is the unique prime knot of four crossings 04-001. It has braid word .The figure eight knot is implemented in the WolframLanguage as KnotData["FigureEight"].It is a 2-embeddable knot, and is amphichiral as well as invertible. It has Arf invariant 1. It is not a slice knot (Rolfsen 1976, p. 224).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the figure eight knot are(1)(2)(3)(4)(5)(6)There are no other knots on 10 or fewer crossings sharing the same Alexander polynomial, BLM/Ho polynomial, bracket polynomial, HOMFLY polynomial, Jones polynomial, or Kauffman polynomial F.The figure eight knot has knot group(7)(Rolfsen 1976, p. 58).Helaman Ferguson's sculpture "Figure-Eight Complement II" illustrates the knot complement of the figure eight..
In most computer programs and computing environments, the precision of any calculation (even including addition) is limited by the word size of the computer, that is, by largest number that can be stored in one of the processor's registers. As of mid-2002, the most common processor word size is 32 bits, corresponding to the integer . General integer arithmetic on a 32-bit machine therefore allows addition of two 32-bit numbers to get 33 bits (one word plus an overflow bit), multiplication of two 32-bit numbers to get 64 bits (although the most prevalent programming language, C, cannot access the higher word directly and depends on the programmer to either create a machine language function or write a much slower function in C at a final overhead of about nine multiplies more), and division of a 64-bit number by a 32-bit number creating a 32-bit quotient and a 32-bit remainder/modulus.Arbitrary-precision arithmetic consists of a set of algorithms,..
A Gröbner basis for a system of polynomials is an equivalence system that possesses useful properties, for example, that another polynomial is a combination of those in iff the remainder of with respect to is 0. (Here, the division algorithm requires an order of a certain type on the monomials.) Furthermore, the set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials. For linear functions in any number of variables, a Gröbner basis is equivalent to Gaussian elimination.The algorithm for computing Gröbner bases is known as Buchberger's algorithm. Calculating a Gröbner basis is typically a very time-consuming process for large polynomial systems (Trott 2006, p. 37).Gröbner bases are pervasive in the construction of symbolic algebra algorithms, and Gröbner bases with respect to lexicographic order are very useful for solving equations and for elimination..