The usual number of scalar operations (i.e., the total number of additions and multiplications) required to perform matrix multiplication is(1)(i.e., multiplications and additions). However, Strassen (1969) discovered how to multiply two matrices in(2)scalar operations, where is the logarithm to base 2, which is less than for . For a power of two (), the two parts of (2) can be written(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)so (◇) becomes(13)Two matrices can therefore be multiplied(14)(15)with only(16)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(17)(18)(19)(20)(21)(22)(23)Then the matrix product is given using the remaining eight additions as(24)(25)(26)(27)(Strassen 1969, Press et al. 1989).Matrix inversion of a matrix to yield can also be done in fewer operations than expected using the formulas(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(Strassen..
The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...
Given points, find the line segments with the shortest possible total length which connect the points. The segments need not necessarily be straight from one point to another.For three points, if all angles are less than , then the line segments are those connecting the three points to a central point which makes the angles , , and all . If one angle is greater that , then coincides with the offending angle.For four points, is the intersection of the two diagonals, but the required minimum segments are not necessarily these diagonals.A modified version of the problem is, given two points, to find the segments with the shortest total length connecting 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...
The game of billiards is played on a rectangular table (known as a billiard table) upon which balls are placed. One ball (the "cue ball") is then struck with the end of a "cue" stick, causing it to bounce into other balls and reflect off the sides of the table. Real billiards can involve spinning the ball so that it does not travel in a straight line, but the mathematical study of billiards generally consists of reflections in which the reflection and incidence angles are the same. However, strange table shapes such as circles and ellipses are often considered. The popular 1959 animated short film Donald in Mathmagic Land features a tutorial by Donald Duck on how to win at billiards using the diamonds normally inscribed around the edge of a real billiard table.Many interesting problems can arise in the detailed study of billiards trajectories. For example, any smooth plane convex set has at least two double normals, so there are..
An antelope graph is a graph formed by all possible moves of a hypothetical chess piece called an "antelope" which moves analogously to a knight except that it is restricted to moves that change by three squares along one axis of the board and four squares along the other. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable antelope moves are considered edges. The plots above show the graphs corresponding to antelope graph on chessboards for to 7.The antelope graph is connected for , Hamiltonian for (trivially) and 14 (but for no odd or other even values ), and traceable for and 21 (with the status for unknown and unknown).Precomputed properties of antelope graphs are implemented in the Wolfram Language as GraphData["Antelope", m, n]...
A giraffe graph is a graph formed by all possible moves of a hypothetical chess piece called a "giraffe" (a.k.a. -leaper) which moves analogously to a knight except that it is restricted to moves that change by one square along one axis of the board and four squares along the other. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable giraffe moves are considered edges.The smallest board allowing a closed tour for the giraffe (i.e., the giraffe graph is Hamiltonian) is the , first solved by A. H. Frost in 1886.
If is a class of recursively enumerable sets, then the set of Gödel numbers of functions whose domains belong to is called its index set. If the index set of is a recursive set, then either is empty or contains all recursively enumerable sets.Rice's theorem is an important result for computer science because it sets up boundaries for research in that area. It basically states that only trivial properties of programs are algorithmically decidable.
Let denote the recursive function of variables with Gödel number , where (1) is normally omitted. Then if is a partial recursive function, there exists an integer such thatwhere is Church's lambda notation. This is the variant most commonly known as Kleene's recursion theorem.Another variant generalizes the first variant by parameterization, and is the strongest form of the recursion theorem. This form states that for each , there exists a recursive function of variables such that is a injection and if is a total function, then for all , ..., , and ,Yet another and weaker variant of the recursion theorem guarantees the existence of a recursive function that is a fixed point for a recursive functional.
Determination of whether predicate is true or false for any given values of , ..., is called its decision problem. The decision problem for predicate is called recursively decidable if there is a total recursive function such that(1)Given the equivalence of computability and recursiveness, this definition may be restated with reference to computable functions instead of recursive functions.The halting problem was one of the first to be shown recursively undecidable. The formulation of recursive undecidability of the halting problem and many other recursively undecidable problems is based on Gödel numbers. For instance, the problem of deciding for any given whether the Turing machine whose Gödel number is is total is recursively undecidable. Hilbert's tenth problem is perhaps the most famous recursively undecidable problem.Most proofs of recursive undecidability use reduction. They show that recursive decidability..
Turing machines are defined by sets of rules that operate on four parameters: (state, tape cell color, operation, state). Let the states and tape cell colors be numbered and represented by quadruples of ordinal numbers. Then there exist algorithmic procedures that sequentially list all consistent sets of Turing machine rules. A set of rules is called consistent if any two quadruples differ in the first or second element out of the four. Any such procedure gives both an algorithm for going from any integer to its corresponding Turing machine and an algorithm for getting the index of any consistent set of Turing machine rules.Assume that one such procedure is selected. If Turing machine is defined by the set of quadruples whose index is , then is called the Gödel number of . The result of application of Turing machine with Godel number to is usually denoted .Given the equivalence of computability and recursiveness, it is common to use Gödel..
A set of integers is said to be recursively enumerable if it constitutes the range of a recursive function, i.e., if there exists a recursive function that can eventually generate any element in (Wolfram 2002, p. 1138). Any recursive set is also recursively enumerable.The union and intersection of two recursively enumerable sets are also recursively enumerable.Recursively undecidable problems give examples of recursively enumerable sets that are not recursive. For example, convergence of is known to be recursively undecidable, where denotes the Turing machine with Gödel number . Hence the set of all for which is convergent is not recursive. However, this set is recursively enumerable because it is the range of defined as follows:(1)A set is recursive iff both and its complement are recursively enumerable. This provides an approach to constructing additional sets that are not recursively enumerable. In particular, the..
A set of integers is said to be recursive if there is a total recursive function such that for and for . Any recursive set is also recursively enumerable.Finite sets, sets with finite complements, the odd numbers, and the prime numbers are all examples of recursive sets. The union and intersection of two recursive sets are themselves recursive, as is the complement of a recursive set.
A set of integers is productive if there exists a partial recursive function such that, for any , the following holds: If the domain of is a subset of , then is convergent, belongs to , and does not belong to the domain of , where denotes a recursive function whose Gödel number is .Productive sets are not recursively enumerable.
A recursively enumerable set is creative if its complement is productive. Creative sets are not recursive. The property of creativeness coincides with completeness. Namely, set is creative iff if it is many-one complete.Elementary arithmetic formulas are built up from 0, 1, 2, ..., , , , variables, connectives, and quantifiers. The set of all true arithmetic formulas is productive. Informally speaking, this means that no axiomatization of arithmetic can capture all true formulas and nothing else. For example, consider Peano arithmetic. Under the assumption that no false arithmetic formulas are provable in this theory, provable Peano arithmetic formulas form a creative set.
It is possible to perform multiplication of large numbers in (many) fewer operations than the usual brute-force technique of "long multiplication." As discovered by Karatsuba (Karatsuba and Ofman 1962), multiplication of two -digit numbers can be done with a bit complexity of less than using identities of the form(1)Proceeding recursively then gives bit complexity , where (Borwein et al. 1989). The best known bound is steps for (Schönhage and Strassen 1971, Knuth 1998). However, this algorithm is difficult to implement, but a procedure based on the fast Fourier transform is straightforward to implement and gives bit complexity (Brigham 1974, Borodin and Munro 1975, Borwein et al. 1989, Knuth 1998).As a concrete example, consider multiplication of two numbers each just two "digits" long in base ,(2)(3)then their product is(4)(5)(6)Instead of evaluating products of individual digits, now write(7)(8)(9)The..
A simplex, sometimes called a hypertetrahedron (Buekenhout and Parker 1998), is the generalization of a tetrahedral region of space to dimensions. The boundary of a -simplex has 0-faces (polytope vertices), 1-faces (polytope edges), and -faces, where is a binomial coefficient. An -dimensional simplex can be denoted using the Schläfli symbol . The simplex is so-named because it represents the simplest possible polytope in any given space.The content (i.e., hypervolume) of a simplex can be computedusing the Cayley-Menger determinant.In one dimension, the simplex is the line segment . In two dimensions, the simplex is the convex hull of the equilateral triangle. In three dimensions, the simplex is the convex hull of the tetrahedron. The simplex in four dimensions (the pentatope) is a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The regular simplex in dimensions with is denoted..
The pentatope is the simplest regular figure in four dimensions, representing the four-dimensional analog of the solid tetrahedron. It is also called the 5-cell, since it consists of five vertices, or pentachoron. The pentatope is the four-dimensional simplex, and can be viewed as a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The pentatope has Schläfli symbol .It is one of the six regular polychora.The skeleton of the pentatope is isomorphic to the complete graph , known as the pentatope graph.The pentatope is self-dual, has five three-dimensional facets (each the shape of a tetrahedron), 10 ridges (faces), 10 edges, and five vertices. In the above figure, the pentatope is shown projected onto one of the four mutually perpendicular three-spaces within the four-space obtained by dropping one of the four vertex components (R. Towle)...
An independent vertex set of a graph is a subset of the vertices such that no two vertices in the subset represent an edge of . The figure above shows independent sets consisting of two subsets for a number of graphs (the wheel graph , utility graph , Petersen graph, and Frucht graph).The polynomial whose coefficients give the number of independent vertex sets of each cardinality in a graph is known as its independence polynomial.A set of vertices is an independent vertex set iff its complement forms a vertex cover (Skiena 1990, p. 218). The counts of vertex covers and independent vertex sets in a graph are therefore the same.The empty set is trivially an independent vertex setsince it contains no vertices, and therefore no edge endpoints.A maximum independent vertex set is an independent vertex set of a graph containing the largest possible number of vertices for the given graph, and the cardinality of this set is called the independence number..
The characteristic polynomial is the polynomial left-hand side of the characteristicequation(1)where is a square matrix and is the identity matrix of identical dimension. Samuelson's formula allows the characteristic polynomial to be computed recursively without divisions. The characteristic polynomial of a matrix may be computed in the Wolfram Language as CharacteristicPolynomial[m, x].The characteristic polynomial of a matrix(2)can be rewritten in the particularly nice form(3)where is the matrix trace of and is its determinant.Similarly, the characteristic polynomial of a matrix is(4)where Einstein summation has been used, whichcan also be written explicitly in terms of traces as(5)In general, the characteristic polynomial has the form(6)(7)where is the matrix trace of the matrix , , and is the sum of the -rowed diagonal minors of the matrix (Jacobson 1974, p. 109).Le Verrier's algorithm for computing the characteristic..
The problem of determining how many nonattacking kings can be placed on an chessboard. For , the solution is 16, as illustrated above (Madachy 1979). In general, the solutions are(1)(Madachy 1979), giving the sequence of doubled squares 1, 1, 4, 4, 9, 9, 16, 16, ... (OEIS A008794). This sequence has generating function(2)The minimal number of kings needed to occupy or attack every square on an chessboard (i.e., domination numbers for the king graphs) are given for , 2, ... by 1, 1, 1, 4, 4, 4, 9, 9, 9, 16, ... (OEIS A075561), with the case illustrated above and noted by (Madachy 1979, p. 39). In general, for an chessboard,(3)
If the three straight lines joining the corresponding vertices of two triangles and all meet in a point (the perspector), then the three intersections of pairs of corresponding sides lie on a straight line (the perspectrix). Equivalently, if two triangles are perspective from a point, they are perspective from a line.The 10 lines and 10 3-line intersections form a configuration sometimes called Desargues' configuration.Desargues' theorem is self-dual.
Consider an (0, 1)-matrix such as(1)for . Call two elements adjacent if they lie in positions and , and , or and for some . Call the number of such arrays with no pairs of adjacent 1s. Equivalently, is the number of configurations of nonattacking kings on an chessboard with regular hexagonal cells.The first few values of for , 2, ... are 2, 6, 43, 557, 14432, ... (OEIS A066863).The hard square hexagon constant is then given by(2)(3)(OEIS A085851).Amazingly, is algebraic and is given by(4)where(5)(6)(7)(8)(9)(10)(11)(Baxter 1980, Joyce 1988ab).The variable can be expressed in terms of the tribonacci constant(12)where is a polynomial root, as(13)(14)(15)(T. Piezas III, pers. comm., Feb. 11, 2006).Explicitly, is the unique positive root(16)where denotes the th root of the polynomial in the ordering of the Wolfram Language...
A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet. A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself.A finite projective plane of order is formally defined as a set of points with the properties that: 1. Any two points determine a line,2. Any two lines determine a point,3. Every point has lines on it, and 4. Every line contains points. (Note that some of these properties are redundant.) A projective plane is therefore a symmetric (, , 1)..
A nexus number is a figurate number built up of the nexus of cells less than steps away from a given cell. The th -dimensional nexus number is given by(1)(2)where is a binomial coefficient. The symbolic representations and sequences for first few -dimensional nexus numbers are given in the table below.name01unit1odd number2hex number3rhombic dodecahedral number4nexus numberOEIS, , , ...01, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...1A0054081, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...2A0032151, 7, 19, 37, 61, 91, 127, 169, 217, ...3A0059171, 15, 65, 175, 369, 671, 1105, 1695, 2465, ...4A0225211, 31, 211, 781, 2101, 4651, 9031, 15961, ...
The -roll mill curve is given by the equationwhere is a binomial coefficient.
The th central fibonomial coefficient is defined as(1)(2)where is a fibonomial coefficient, is a Fibonacci number, is the golden ratio, and is a q-Pochhammer symbol (E. W. Weisstein, Dec. 8, 2009).For , 2, ..., the first few are 1, 6, 60, 1820, 136136, ... (OEIS A003267).
As originally stated by Gould (1972),(1)where GCD is the greatest common divisor and is a binomial coefficient. This was subsequently extended by D. Singmaster to(2)(Sato 1975), and generalized by Sato (1975) to(3)An even larger generalization was obtained by Hitotumatu and Sato (1975), who defined(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)with(16)and showed that each of the twelve binomial coefficients , , , , , , , , , , , and has equal greatest common divisor.A second star of David theorem states that if two triangles are drawn centered on a given element of Pascal's triangle as illustrated above, then the products of the three numbers in the associated points of each of the two stars are the same (Butterworth 2002). This follows from the fact that(17)(18)(19)The second star of David theorem holds true not only for the usual binomial coefficients, but also for q-binomial coefficients, where the common product is given by(20)In..
The idempotent numbers are given bywhere is a Bell polynomial and is a binomial coefficient. A table of the first few is given below.A000027A001788A036216A040075A050982A050988A050989112213361442412155809020166240540240301776722835224052542188179213608179207000100856994608612361290247875018144176410101152026244086016078750027216041160
(1)The number triangle illustrated above (OEIS A008949) composed of the partial sums of binomial coefficients,(2)(3)where is a gamma function and is a hypergeometric function.The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Bernoulli triangle.
The fibonomial coefficient (sometimes also called simply the Fibonacci coefficient) is defined by(1)where and is a Fibonacci number. This coefficient satisfies(2)for , where is a Lucas number.The triangle of fibonomial coefficients is given by(3)(OEIS A010048). may be called the central fibonomial coefficient by analogy with the central binomial coefficient.
Pascal's triangle is a number triangle with numbersarranged in staggered rows such that(1)where is a binomial coefficient. The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám. It is therefore known as the Yanghui triangle in China. Starting with , the triangle is(2)(OEIS A007318). Pascal's formula shows that each subsequent row is obtained by adding the two entries diagonally above,(3)The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Pascal's triangle.The first number after the 1 in each row divides all other numbers in that row iff it is a prime.The sums of the number of odd entries in the first rows of Pascal's triangle for , 1, ... are 0, 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45, 49, ... (OEIS A006046). It is then..
The nesting of two or more functions to form a single new function is known as composition. The composition of two functions and is denoted , where is a function whose domain includes the range of . The notation(1)is sometimes used to explicitly indicate the variable.Composition is associative, so that(2)If the functions is continuous at and is continuous at , then is also continuous at .A function which is the composition of two other functions, say and , is sometimes said to be a composite function.Faà di Bruno's formula gives an explicit formula for the th derivative of the composition .A combinatorial composition is defined as an ordered arrangement of nonnegative integers which sum to (Skiena 1990, p. 60). It is therefore a partition in which order is significant. For example, there are eight compositions of 4,(3)(4)(5)(6)(7)(8)(9)(10)A positive integer has compositions.The number of compositions of into parts (where..
For an matrix, let denote any permutation , , ..., of the set of numbers 1, 2, ..., , and let be the character of the symmetric group corresponding to the partition . Then the immanant is defined aswhere the summation is over the permutations of the symmetric group and
A set partition of a set is a collection of disjoint subsets of whose union is . The number of partitions of the set is called a Bell number.
Let denote the partition lattice of the set . The maximum element of is(1)and the minimum element is(2)Let denote the number of chains of any length in containing both and . Then satisfies the recurrence relation(3)where and is a Stirling number of the second kind. The first few values of for , 2, ... are then 1, 1, 4, 32, 436, 9012, 262760, ... (OEIS A005121).Lengyel (1984) proved that the quotient(4)is bounded between two constants as , and Flajolet and Salvy (1990) improved the result of Babai and Lengyel (1992) to show that(5)(OEIS A086053).
An isolated point of a graph is a node of degree 0 (Hartsfield and Ringel 1990, p. 8; Harary 1994, p. 15; D'Angelo and West 2000, p. 212; West 2000, p. 22). The number of -node graphs with no isolated points are 0, 1, 2, 7, 23, 122, 888, ... (OEIS A002494), the first few of which are illustrated above. The number of graphical partitions of length is equal to the number of -node graphs that have no isolated points.Connected graphs have no isolated points.An isolated point on a curve is more commonly known as an acnode.An isolated point of a discrete set is a member of (Krantz 1999, p. 63).
A path is a continuous mapping , where is the initial point, is the final point, and denotes the space of continuous functions. The notation for a path parametrized by is commonly denoted .A graph path is a sequence such that , , ..., are graph edges of the graph and the are distinct.A path in a topological space is a continuous map . Often the name path is given to the image of .
Krall and Fink (1949) defined the Bessel polynomials as the function(1)(2)where is a modified Bessel function of the second kind. They are very similar to the modified spherical bessel function of the second kind . The first few are(3)(4)(5)(6)(7)(OEIS A001497). These functions satisfy thedifferential equation(8)Carlitz (1957) subsequently considered the related polynomials(9)This polynomial forms an associated Sheffer sequencewith(10)This gives the generating function(11)The explicit formula is(12)(13)where is a double factorial and is a confluent hypergeometric function of the first kind. The first few polynomials are(14)(15)(16)(17)(OEIS A104548).The polynomials satisfy the recurrence formula(18)
The term "poweroid" has at least two meanings. Sheffer sequences are sometimes called poweroids (Steffensen 1941, Shiu 1982, Di Bucchianico and Loeb 2000). Jackway and Deriche (1996) and Jackway (2002) use the term to mean a function of the formThe case corresponds to the paraboloid and the case is sometimes called a quartoid (Jackway and Deriche 1996).
Faà di Bruno's formula gives an explicit equation for the th derivative of the composition . If and are functions for which all necessary derivatives are defined, then(1)where and the sum is over all partitions of , i.e., values of , ..., such that(2)(Roman 1980).It can also be expressed in terms of Bell polynomial as(3)(M. Alekseyev, pers. comm., Nov. 3, 2006).Faà di Bruno's formula can be cast in a framework that is a special case of a Hopf algebra (Figueroa and Gracia-Bondía 2005).The first few derivatives for symbolic and are given by(4)(5)(6)
Let denote the independence number of a graph . Then the Shannon capacity , sometimes also denoted , of is defined aswhere denoted the graph strong product (Shannon 1956, Alon and Lubetzky 2006). The Shannon capacity is an important information theoretical parameter because it represents the effective size of an alphabet in a communication model represented by a graph (Alon 1998). satisfiesThe Shannon capacity is in general very difficult to calculate (Brimkov et al. 2000). In fact, the Shannon capacity of the cycle graph was not determined as until 1979 (Lovász 1979), and the Shannon capacity of is perhaps one of the most notorious open problems in extremal combinatorics (Bohman 2003).Lovász (1979) showed that the Shannon capacity of the -Kneser graph is , that of a vertex-transitive self-complementary graph (which includes all Paley graphs) is , and that of the Petersen graph is 4.All graphs whose Shannon capacity is known..
In machine learning theory and artificial intelligence, a concept over a domain is a Boolean function . A collection of concepts is called a concept class.In context-specific applications, concepts are usually thought to assign either a "positive" or "negative" outcome (corresponding to range values of 1 or 0, respectively) to each element of the domain . In that way, concepts are the fundamental component of learning theory.
A Hadamard matrix is a type of square (-1,1)-matrix invented by Sylvester (1867) under the name of anallagmatic pavement, 26 years before Hadamard (1893) considered them. In a Hadamard matrix, placing any two columns or rows side by side gives half the adjacent cells the same sign and half the other sign. When viewed as pavements, cells with 1s are colored black and those with s are colored white. Therefore, the Hadamard matrix must have white squares (s) and black squares (1s).A Hadamard matrix of order is a solution to Hadamard's maximum determinant problem, i.e., has the maximum possible determinant (in absolute value) of any complex matrix with elements (Brenner and Cummings 1972), namely . An equivalent definition of the Hadamard matrices is given by(1)where is the identity matrix.A Hadamard matrix of order corresponds to a Hadamard design (, , ), and a Hadamard matrix gives a graph on vertices known as a Hadamard graphA complete set of Walsh..
The disdyakis dodecahedron is the dual polyhedron of the Archimedean great rhombicuboctahedron and Wenninger dual . It is also called the hexakis octahedron (Unkelbach 1940; Holden 1971, p. 55).If the original great rhombicuboctahedronhas unit side lengths, then the resulting dual has edge lengths(1)(2)(3)The inradius is(4)Scaling the disdyakis dodecahedron so that gives a solid with surface area and volume(5)(6)
The Icosian game, also called the Hamiltonian game (Ball and Coxeter 1987, p. 262), is the problem of finding a Hamiltonian cycle along the edges of an dodecahedron, i.e., a path such that every vertex is visited a single time, no edge is visited twice, and the ending point is the same as the starting point (left figure). The puzzle was distributed commercially as a pegboard with holes at the nodes of the dodecahedral graph, illustrated above (right figure). The Icosian Game was invented in 1857 by William Rowan Hamilton. Hamilton sold it to a London game dealer in 1859 for 25 pounds, and the game was subsequently marketed in Europe in a number of forms (Gardner 1957).A graph having a Hamiltonian cycle, i.e., on which the Icosian game may be played, is said to be a Hamiltonian graph. While the skeletons of all the Platonic solids and Archimedean solids (i.e., the Platonic graphs and Archimedean graphs, respectively) are Hamiltonian, the same is..
The number of colors sufficient for map coloring on a surface of genus is given by the Heawood conjecture,where is the floor function. The fact that (which is called the chromatic number) is also necessary 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 bottle.A -holed torus therefore requires colors. For , 1, ..., the first few values of are 4, 7 (illustrated above, M. Malak, pers. comm., Feb. 22, 2006), 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, ... (OEIS A000934). A set of regions requiring the maximum of seven regions is shown above for a normal torusThe above figure shows the relationship between the Heawoodgraph and the 7-color torus coloring.
Define a valid "coloring" to occur when no two faces with a common edge share the same color. Given two colors, there is a single way to color an octahedron (Ball and Coxeter 1987, pp. 238-239). Given three colors, there is one way to color a cube (Ball and Coxeter 1987, pp. 238-239) and 144 ways to color an icosahedron (Ball and Coxeter 1987, pp. 239-242). Given four colors, there are two distinct ways to color a tetrahedron (Ball and Coxeter 1987, p. 238) and four ways to color a dodecahedron, consisting of two enantiomorphous ways (Steinhaus 1999, pp. 196-198; Ball and Coxeter 1987, p. 238). Given five colors, there are four ways to color an icosahedron. Given six colors, there are 30 ways to color a cube (Steinhaus 1999, p. 167). These values are related to the chromatic polynomial of the corresponding dual skeleton graph, which however overcounts since it does not take rotational equivalence..
A graphoid consists of a set of elements together with two collections and of nonempty subsets of , called circuits and cocircuits respectively, such that 1. For any and , , 2. No circuit properly contains another circuit and no cocircuit properly contains another cocircuit, 3. For any painting of with colors exactly one element green and the rest either red or blue, there exists either (a) a circuit containing the green element and no red elements, or (b) a cocircuit containing the green element and no blue elements.
An edge-magic graph is a labeled graph with graph edges labeled with distinct elements so that the sum of the graph edge labels at each graph vertex is the same.A vertex-magic graph labeled graph vertices which give the same sum along every straight line segment. No magic pentagrams can be formed with the number 1, 2, ..., 10 (Trigg 1960; Langman 1962, pp. 80-83; Dongre 1971; Richards 1975; Buckley and Rubin 1977-1978; Trigg 1998), but 168 almost magic pentagrams (in which the sums are the same for four of the five lines) can. The figure above show a magic pentagram with sums 24 built using the labels 1, 2, 3, 4, 5, 6, 8, 9, 10, and 12 (Madachy 1979).
A combinatorial conjecture formulated by Kneser (1955). It states that whenever the -subsets of a -set are divided into classes, then two disjoint subsets end up in the same class.Lovász (1978) gave a proof based on graph theory. In particular, he showed that the Kneser graph, whose vertices represent the -subsets, and where each edge connects two disjoint subsets, is not -colorable. More precisely, his results says that the chromatic number is equal to , and this implies that Kneser's conjecture is always false if the number of classes is increased to .An alternate proof was given by Bárány (1978).
A rook polynomial is a polynomial(1)whose number of ways nonattacking rooks can be arranged on an chessboard. The rook polynomials are given by(2)where is an associated Laguerre polynomial.The first few rook polynomials on square boards are(3)(4)(5)(6)(OEIS A021010).As an illustration, note that the case has two ways to place two rooks (i.e., the rook number ), four ways to place one rook (), and one way to place no rooks (), hence .
The numerators and denominators obtained by taking the ratios of adjacent terms in the triangular array of the number of "bordered" alternating sign matrices with a 1 at the top of column are, respectively, the numbers in the (2, 1)- and (1, 2)-Pascal triangles which are different from 1. This conjecture was proven by Zeilberger (1996).
The rascal triangle is a number triangle withnumbers arranged in staggered rows such that(1)The published study of this triangle seems to have originated relatively recently, having been added to Sloane's Online Encyclopedia of Integer Sequences (OEIS) as recently as 2002--where it was cataloged as [t]riangle with diagonal congruent to mod --and having been the subject of scholarly mathematical publication as recently as 2010 (Anggoro et al. 2010). The triangle is sometimes written without capitalization as the rascal triangle.One common point of exposition among literature regarding the rascal triangle is its similarity to Pascal's triangle. Indeed, the rascal triangle is topically similar to Pascal's triangle in that the configuration starting with begins(2)and that the rows afterwards have their first and last entries equal to(3)respectively.The similarities don't end there, however. One well-known fact about Pascal's..
A polygonal diagonal is a line segment connecting two nonadjacent polygon vertices of a polygon. The number of ways a fixed convex -gon can be divided into triangles by nonintersecting diagonals is (with diagonals), where is a Catalan number. This is Euler's polygon division problem. Counting the number of regions determined by drawing the diagonals of a regular -gon is a more difficult problem, as is determining the number of -tuples of concurrent diagonals (Kok 1972).The number of regions which the diagonals of a convexpolygon divide its center if no three are concurrent in its interior is(1)(2)The first few values are 0, 0, 1, 4, 11, 25, 50, 91, 154, 246, ... (OEIS A006522).
An alternating sign matrix is a matrix of 0s, 1s, and s in which the entries in each row or column sum to 1 and the nonzero entries in each row and column alternate in sign. The first few for , 2, ... are shown below:(1)(2)(3)(4)Such matrices satisfy the additional property that s in a row or column must have a "outside" it (i.e., all s are "bordered" by s). The numbers of alternating sign matrices for , 2, ... are given by 1, 2, 7, 42, 429, 7436, 218348, ... (OEIS A005130).The conjecture that the number of is explicitly given by the formula(5)now proven to be true, was known as the alternating sign matrix conjecture. can be expressed in closed form as a complicated function of Barnes G-functions, but additional simplification is likely possible.A recurrence relation for is given by(6)where is the gamma function.Let be the number of alternating sign matrices with one in the top row occurring in the th position. Then(7)The result(8)for..
In complex analysis, a branch (also called a sheet) is a portion of the range of a multivalued function over which the function is single-valued. Combining all the sheets gives the full structure of the function. It is often convenient to choose a particular branch of a function to work with, and this choice is often designated the "principal branch" (or "principal sheet").In graph theory, a branch at a point in a tree is a maximal subtree containing as an endpoint (Harary 1994, p. 35).
The Euler numbers, also called the secant numbers or zig numbers, are defined for by(1)(2)where is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating permutations and are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.A different sort of Euler number, the Euler number of a finite complex , is defined by(3)This Euler number is a topological invariant.To confuse matters further, the Euler characteristic is sometimes also called the "Euler number" and numbers produced by the prime-generating polynomial are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47). In this work, primes generated by that polynomial are termed Euler primes.Some values of the (secant) Euler numbers are(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(OEIS A000364).The slightly different convention defined by(16)(17)is..
Let there be integers with . The values represent the denominations of different coins, where these denominations have greatest common divisor of 1. The sums of money that can be represented using the given coins are then given by(1)where the are nonnegative integers giving the numbers of each coin used. If , it is obviously possibly to represent any quantity of money . However, in the general case, only some quantities can be produced. For example, if the allowed coins are , it is impossible to represent and 3, although all other quantities can be represented.Determining the function giving the greatest for which there is no solution is called the coin problem, or sometimes the money-changing problem. The largest such for a given problem is called the Frobenius number .The result(2)(3)(Nijenhuis and Wilf 1972) is mathematical folklore. The total number of such nonrepresentable amounts is given by(4)The largest nonrepresentable amounts for..
The Frobenius number is the largest value for which the Frobenius equation(1)has no solution, where the are positive integers, is an integer, and the solutions are nonnegative integer. As an example, if the values are 4 and 9, then 23 is the largest unsolvable number. Similarly, the largest number that is not a McNugget number (a number obtainable by adding multiples of 6, 9, and 20) is 43.Finding the Frobenius number of a given problem is known as the coinproblem.Computation of the Frobenius number is implemented in the Wolfram Language as FrobeniusNumber[a1, ..., an].Sylvester (1884) showed(2)(3)
For every positive integer , there exists a sphere which has exactly lattice points on its surface. The sphere is given by the equation(1)where and are the coordinates of the center of the so-called Schinzel circle(2)and is its radius.
For every positive integer , there exists a circle in the plane having exactly lattice points on its circumference. The theorem is based on the number of integral solutions to the equation(1)given by(2)where is the number of divisors of of the form and is the number of divisors of the form . It explicitly identifies such circles (the Schinzel circles) as(3)Note, however, that these solutions do not necessarily have the smallest possible radius.
A circle having a given number of lattice points on its circumference. The Schinzel circle having lattice points is given by the equation(1)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 has nine lattice points on its circumference, so does the circle centered at (1/3, 0) with radius 65/3.A table of minimal circles to is given by Pegg (2008).
Let be the area of a simply closed lattice polygon. Let denote the number of lattice points on the polygon edges and the number of points in the interior of the polygon. ThenThe formula has been generalized to three- and higherdimensions using Ehrhart polynomials.
Two sets and are said to be independent if their intersection , where is the empty set. For example, and are independent, but and are not. Independent sets are also called disjoint or mutually exclusive.An independent vertex set of a graph is a subset of the vertices such that no two vertices in the subset represent an edge of . The figure above shows independent vertex sets consisting of two subsets for a number of graphs (the wheel graph , utility graph , Petersen graph, and Frucht graph).An independent edge set can be defined similarly(Skiena 1990, p. 219).
The happy end problem, also called the "happy ending problem," is the problem of determining for the smallest number of points in general position in the plane (i.e., no three of which are collinear), such that every possible arrangement of points will always contain at least one set of points that are the vertices of a convex polygon of sides. The problem was so-named by Erdős when two investigators who first worked on the problem, Ester Klein and George Szekeres, became engaged and subsequently married (Hoffman 1998, p. 76).Since three noncollinear points always determine a triangle, .Random arrangements of points are illustrated above. Note that no convex quadrilaterals are possible for the arrangements shown in the fifth and eighth figures above, so must be greater than 4. E. Klein proved that by showing that any arrangement of five points must fall into one of the three cases (left top figure; Hoffman 1998,..
A magic hexagon of order is an arrangement of close-packed hexagons containing the numbers 1, 2, ..., , where is the th hex number such that the numbers along each straight line add up to the same sum. (Here, the hex numbers are i.e., 1, 7, 19, 37, 61, 91, 127, ...; OEIS A003215). In the above magic hexagon of order , each line (those of lengths 3, 4, and 5) adds up to 38.It was discovered independently by Ernst von Haselberg in 1887 (Bauch 1990, Hemme 1990), W. Radcliffe in 1895 (Tapson 1987, Hemme 1990, Heinz), H. Lulli (Hendricks, Heinz), Martin Kühl in 1940 (Gardner 1963, 1984; Honsberger 1973), Clifford W. Adams, who worked on the problem from 1910 to 1957 (Gardner 1963, 1984; Honsberger 1973), and Vickers (1958; Trigg 1964).This problem and the solution have a long history. Adams came across the problem in 1910. He worked on the problem by trial and error and after many years arrived at the solution which he transmitted to M. Gardner,..
A one-person game played on a rectangular lattice of lamps which can be turned on and off. A move consists of flipping a "switch" inside one of the squares, thereby toggling the on/off state of this and all four vertically and horizontally adjacent squares. Starting from a randomly chosen light pattern, the aim is to turn all the lamps off. The problem of determining if it is possible to start from set of all lights being on to all lights being off is known as the "all-ones problem." As shown by Sutner (1989), this is always possible for a square lattice (Rangel-Mondragon).This can be translated into the following algebraic problem. 1. Each lamp configuration can be viewed as a matrix with entries in (i.e., a (0,1)-matrix, where each 1 represents a burning light and 0 represents a light turned off. For example, for the case,(1)2. The action of the switch placed at can be interpreted as the matrix addition , where is the matrix in which..
Given a (0,1)-matrix, fill 11 spaces in each row in such a way that all columns also have 11 spaces filled. Furthermore, each pair of rows must have exactly one filled space in the same column. This problem is equivalent to finding a projective plane of order 10. Using a computer program, Lam et al. (1989) showed that no such arrangement exists.Lam's problem is equivalent to finding nine orthogonal Latinsquares of order 10.
The problem of determining how many nonattacking knights can be placed on an chessboard. For , the solution is 32 (illustrated above). In general, the solutions are(1)giving the sequence 1, 4, 5, 8, 13, 18, 25, ... (OEIS A030978,Dudeney 1970, p. 96; Madachy 1979).The minimal number of knights needed to occupy or attack every square on an chessboard (i.e., domination numbers for the knight graphs) are given for , 2, ... by 1, 4, 4, 4, 5, 8, 10, 12, 14, ... (OEIS A006075), with corresponding numbers of such solutions given by 1, 1, 2, 3, 8, 22, 3, ... (OEIS A006076).
In continuum theory, a dendrite is a locally connected continuum that contains no simple closed curve. A semicircle is therefore a dendrite, while a triangle is not.The term dendrite is used by Steinhaus (1999, pp. 120-125) to refer to a system of line segments connecting a given set of points, where the total length of paths is as short as possible (therefore implying that no closed cycles are permitted) and the paths are not allowed to cross. This definition differs from the one in continuum theory since a semicircle is a dendritic continuum but is not a line segment.
In algebraic topology, a -skeleton is a simplicial subcomplex of that is the collection of all simplices of of dimension at most , denoted .The graph obtained by replacing the faces of a polyhedron with its edges and vertices is therefore the skeleton of the polyhedron. The polyhedral graphs corresponding to the skeletons of Platonic solids are illustrated above. The number of topologically distinct skeletons with graph vertices for , 5, 6, ... are 1, 2, 7, 18, 52, ... (OEIS A006869).
The "15 puzzle" is a sliding square puzzle commonly (but incorrectly) attributed to Sam Loyd. However, research by Slocum and Sonneveld (2006) has revealed that Sam Loyd did not invent the 15 puzzle and had nothing to do with promoting or popularizing it. The puzzle craze that was created by the 15 puzzle began in January 1880 in the United States and in April in Europe and ended by July 1880. Loyd first claimed in 1891 that he invented the puzzle, and he continued until his death a 20 year campaign to falsely take credit for the puzzle. The actual inventor was Noyes Chapman, the Postmaster of Canastota, New York, and he applied for a patent in March 1880.The 15 puzzle consists of 15 squares numbered from 1 to 15 that are placed in a box leaving one position out of the 16 empty. The goal is to reposition the squares from a given arbitrary starting arrangement by sliding them one at a time into the configuration shown above. For some initial arrangements,..
If is an integer, then for every residue class (mod ), there are infinitely many nonnegative integers for which , where is the partition function P.
Given a sum and a set of weights, find the weights which were used to generate the sum. The values of the weights are then encrypted in the sum. This system relies on the existence of a class of knapsack problems which can be solved trivially (those in which the weights are separated such that they can be "peeled off" one at a time using a greedy-like algorithm), and transformations which convert the trivial problem to a difficult one and vice versa. Modular multiplication is used as the trapdoor one-way function. The simple knapsack system was broken by Shamir in 1982, the Graham-Shamir system by Adleman, and the iterated knapsack by Ernie Brickell in 1984.
Consider a set of positive integer-denomination postage stamps sorted such that . Suppose they are to be used on an envelope with room for no more than stamps. The postage stamp problem then consists of determining the smallest integer which cannot be represented by a linear combination with and .Without the latter restriction, this problem is known as the Frobenius problem or Frobenius postage stamp problem.The number of consecutive possible postage amounts is given by(1)where is called an -range.Exact solutions exist for arbitrary for and 3. The solution is(2)for . It is also known that(3)(Stöhr 1955, Guy 1994), where is the floor function, the first few values of which are 2, 4, 7, 10, 14, 18, 23, 28, 34, 40, ... (OEIS A014616; Guy 1994, p. 123).Hofmeister (1968, 1983) showed that for ,(4)where and are functions of (mod 9), and Mossige (1981, 1987) showed that(5)(Guy 1994, p. 123).Shallit (2002) proved that the (local) postage..
The word "rigid" has two different meaning when applied to a graph. Firstly, a rigid graph may refer to a graph having a graph automorphism group containing a single element.A framework (or graph) is rigid iff continuous motion of the points of the configuration maintaining the bar constraints comes from a family of motions of all Euclidean space which are distance-preserving. A graph that is not rigid is said to be flexible (Maehara 1992).For example, the cycle graph is rigid, while is flexible. An embedding of the bipartite graph in the plane is rigid unless its six vertices lie on a conic (Bolker and Roth 1980, Maehara 1992).A graph is (generically) -rigid if, for almost all (i.e., an open dense set of) configurations of , the framework is rigid in .Cauchy (1813) proved the rigidity theorem, one of the first results in rigidity theory. Although rigidity problems were of immense interest to engineers, intensive mathematical study of..
Let a graph have exactly graph edges, where is the number of graph vertices in . Then is "generically" rigid in iff for every subgraph of having graph vertices and graph edges.
Any bounded planar region with positive area placed in any position of the unit square lattice can be translated so that the number of lattice points inside the region will be at least (Blichfeldt 1914, Steinhaus 1999). The theorem can be generalized to dimensions.
The symmetric group of degree is the group of all permutations on symbols. is therefore a permutation group of order and contains as subgroups every group of order .The th symmetric group is represented in the Wolfram Language as SymmetricGroup[n]. Its cycle index can be generated in the Wolfram Language using CycleIndexPolynomial[SymmetricGroup[n], x1, ..., xn]. The number of conjugacy classes of is given , where is the partition function P of . The symmetric group is a transitive group (Holton and Sheehan 1993, p. 27).For any finite group , Cayley's group theorem proves is isomorphic to a subgroup of a symmetric group.The multiplication table for is illustrated above.Let be the usual permutation cycle notation for a given permutation. Then the following table gives the multiplication table for , which has elements.(1)(2)(3)(1)(23)(3)(12)(123)(132)(2)(13)(1)(2)(3)(1)(2)(3)(1)(23)(3)(12)(123)(132)(2)(13)(1)(23)(1)(23)(1)(2)(3)(132)(2)(13)(3)(12)(123)(3)(12)(3)(12)(123)(1)(2)(3)(1)(23)(2)(13)(132)(123)(123)(3)(12)(2)(13)(132)(1)(2)(3)(1)(23)(132)(132)(2)(13)(1)(23)(1)(2)(3)(123)(3)(12)(2)(13)(2)(13)(132)(123)(3)(12)(1)(23)(1)(2)(3)This..
A permutation group is a finite group whose elements are permutations of a given set and whose group operation is composition of permutations in . Permutation groups have orders dividing .Two permutations form a group only if one is the identity element and the other is a permutation involution, i.e., a permutation which is its own inverse (Skiena 1990, p. 20). Every permutation group with more than two elements can be written as a product of transpositions.Permutation groups are represented in the Wolfram Language as a set of permutation cycles with PermutationGroup. A set of permutations may be tested to see if it forms a permutation group using PermutationGroupQ[l] in the Wolfram Language package Combinatorica` .Conjugacy classes of elements which are interchangedin a permutation group are called permutation cycles.Examples of permutation groups include the symmetric group (of order ), the alternating group (of order for ),..
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 maximal number of regions into which lines divide a plane arewhich, for , 2, ... gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124), the same maximal number of regions into which a circle, square, etc. can be divided by lines.
The average number of regions into which lines divide a square is(Santaló 1976; Finch 2003, p. 481).The maximum number of sequences is presumably the same as for circledivision by lines, namelyFor , 2, ..., this gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124), which is the same number into which the plane, a circle, etc. can be divided.
A problem sometimes known as Moser's circle problem asks to determine the number of pieces into which a circle is divided if points on its circumference are joined by chords with no three internally concurrent. The answer is(1)(2)(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), where is a binomial coefficient. The first few values are 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (OEIS A000127). This sequence demonstrates the danger in making assumptions based on limited trials. While the series starts off like , it begins differing from this geometric series at .
The number of ways of finding a subrectangle with an rectangle can be computed by counting the number of ways in which the upper right-hand corner can be selected for a given lower left-hand corner. For a lower left-hand corner with coordinates , there are possible upper right-hand corners, so(1)(2)Equivalently, is the number of ways of picking two lines out of sets of and lines, giving(3)(4)as before. Particular tilings are shown above for and rectangles.
Kobon Fujimura asked for the largest number of nonoverlapping triangles that can be constructed using lines (Gardner 1983, p. 170). A Kobon triangle is therefore defined as one of the triangles constructed in such a way. The first few terms are 1, 2, 5, 7, 11, 15, 21, ... (OEIS A006066).It appears to be very difficult to find an analytic expression for the th term, although Saburo Tamura has proved an upper bound on of , where is the floor function (Eppstein). For , 3, ..., the first few upper limits are therefore 2, 5, 8, 11, 16, 21, 26, 33, ... (OEIS A032765).A. Wajnberg (pers. comm., Nov. 18, 2005) found a configuration for containing 25 triangles (left figure). A different 10-line, 25-triangle construction was found by Grünbaum (2003, p. 400), and a third configuration is referenced by Honma. The upper bound on means that the maximum must be either 25 or 26 (but it is not known which). Two other distinct solutions were..
Given a set of nonnegative integers, the number partitioning problem requires the division of into two subsets such that the sums of number in each subset are as close as possible. This problem is known to be NP-complete, but is perhaps "the easiest hard problem" (Hayes 2002; Borwein and Bailey 2003, pp. 38-39).
The power polynomials are an associated Sheffer sequence with(1)giving generating function(2)and exponential generating function(3)and binomial identity(4)
A polynomial given by(1)where are the roots of unity in given by(2)and runs over integers relatively prime to . The prime may be dropped if the product is instead taken over primitive roots of unity, so that(3)The notation is also frequently encountered. Dickson et al. (1923) and Apostol (1975) give extensive bibliographies for cyclotomic polynomials.The cyclotomic polynomial for can also be defined as(4)where is the Möbius function and the product is taken over the divisors of (Vardi 1991, p. 225). is an integer polynomial and an irreducible polynomial with polynomial degree , where is the totient function. Cyclotomic polynomials are returned by the Wolfram Language command Cyclotomic[n, x]. The roots of cyclotomic polynomials lie on the unit circle in the complex plane, as illustrated above for the first few cyclotomic polynomials.The first few cyclotomic polynomials are(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)The cyclotomic..
In his Meditationes algebraicae, Waring (1770, 1782) proposed a generalization of Lagrange's four-square theorem, stating that every rational integer is the sum of a fixed number of th powers of positive integers, where is any given positive integer and depends only on . Waring originally speculated that , , and . 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 , where 4 may be reduced to 3 except for numbers of the form (as proved by Legendre; Hardy 1999, p. 12). In 1909, Wieferich proved that . In 1859, Liouville proved (using Lagrange's four-square theorem and Liouville polynomial identity) that . Hardy, and Little established , and this was subsequently reduced to by Balasubramanian et al. (1986). For the case , in 1896, Maillet began with a proof that , in 1909 Wieferich proved , and..
Consider the number of sequences that can be formed from permutations of a set of elements such that each partial sum is nonnegative. The number of sequences with nonnegative partial sums which can be formed from the permutations of 1s and s (Bailey 1996, Brualdi 1997) is given by the Catalan numbers . For example, the permutations of having nonnegative partial sums are , , , , and (1, , 1, , 1, ).Similarly, the number of nonnegative partial sums of 1s and s (Bailey 1996) is given bywhere these coefficients form Catalan's triangle(OEIS A009766) and
Let a chess piece make a tour on an chessboard whose squares are numbered from 1 to along the path of the chess piece. Then the tour is called a magic tour if the resulting arrangement of numbers is a magic square, and a semimagic tour if the resulting arrangement of numbers is a semimagic square. If the first and last squares traversed are connected by a move, the tour is said to be closed (or "re-entrant"); otherwise it is open. (Note some care with terminology is necessary. For example, Jelliss terms a semimagic tour a "magic tour" and a magic tour a "diagonally magic tour.")Magic knight graph tours are not possible on boards for odd. However, as had long been known, they are possible for all boards of size for . However, the () remained open even since it was first investigated by authors such as Beverley (1848). It was not resolved until an exhaustive computer enumeration of all possibilities was completed on August 5,..
A zebra graph is a graph formed by all possible moves of a hypothetical chess piece called a "zebra" which moves analogously to a knight except that it is restricted to moves that change by two squares along one axis of the board and three squares along the other. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable zebra moves are considered edges. The graphs above gives the positions on a square chess boards that are reachable by zebra moves.Square () zebra graphs are connected for .The smallest square board where a tour exists (i.e., for which the underlying zebra graph is Hamiltonian) is the board, first solved in 1886 by Frost (Jelliss). In fact, there are a total of inequivalent (directed) zebra's tours (Hamiltonian cycles) on this board...
A fiveleaper graph is a graph formed by all possible moves of a hypothetical chess piece called a "fiveleaper" which moves analogously to a knight except that it is restricted to moves that change by three squares along one axis of the board and four squares along the other or by five squares along one axis. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable fiveleaper moves are considered edges. The fiveleaper gets its name from the fact that all its move have a length of 5 squares.The fiveleaper is similar to the hypothetical chess piece called an "antelope," but it can make an antelope's move or a rook's move of exactly 5 squares.The plots above show the graphs corresponding to antelope graphs on chessboards for to 7.The fiveleaper graph is connected for (trivially) and , Hamiltonian for (trivially) and 8, 10, 12, 14, ...(and all other even but for no odd up to at least ), and traceable..
The rook graph (confusingly called the grid by Brouwer et al. 1989, p. 440) and also sometimes known as a lattice graph (e.g., Bouwer) is the graph Cartesian product of complete graphs, which is equivalent to the line graph of the complete bipartite graph . This is the definition adopted for example by Brualdi and Ryser (1991, p. 153), although restricted to the case . This definition corresponds to the connectivity graph of a rook chess piece (which can move any number of spaces in a straight line-either horizontally or vertically, but not diagonally) on an chessboard.The graph has vertices and edges. It is regular of degree , has diameter 3, girth 3 (for ), and chromatic number . It is also perfect (since it is the line graph of a bipartite graph) and vertex-transitive.The rook graph is also isomorphic to the Latin square graph. The vertices of such a graph are defined as the elements of a Latin square of order , with two vertices being adjacent..
There are several closely related results that are variously known as the binomial theorem depending on the source. Even more confusingly a number of these (and other) related results are variously known as the binomial formula, binomial expansion, and binomial identity, and the identity itself is sometimes simply called the "binomial series" rather than "binomial theorem."The most general case of the binomial theorem is the binomialseries identity(1)where is a binomial coefficient and is a real number. This series converges for an integer, or . This general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with the binomial theorem.When is a positive integer , the series terminates at and can be written in the form(2)This form of the identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).The differing terminologies are..
There are several related series that are known as the binomial series.The most general is(1)where is a binomial coefficient and is a real number. This series converges for an integer, or (Graham et al. 1994, p. 162). When is a positive integer , the series terminates at and can be written in the form(2)The theorem that any one of these (or several other related forms) holds is knownas the binomial theorem.Special cases give the Taylor series(3)(4)where is a Pochhammer symbol and . Similarly,(5)(6)which is the so-called negative binomial series.In particular, the case gives(7)(8)(9)(OEIS A001790 and A046161), where is a double factorial and is a binomial coefficient.The binomial series has the continued fractionrepresentation(10)(Wall 1948, p. 343).
Consider a network of resistors so that may be connected in series or parallel with , may be connected in series or parallel with the network consisting of and , and so on. The resistance of two resistors in series is given by(1)and of two resistors in parallel by(2)The possible values for two resistors with resistances and are therefore(3)for three resistances , , and are(4)and so on. These are obviously all rational numbers, and the numbers of distinct arrangements for , 2, ..., are 1, 2, 8, 46, 332, 2874, ... (OEIS A005840), which also arises in a completely different context (Stanley 1991).If the values are restricted to , then there are possible resistances for 1- resistors, ranging from a minimum of to a maximum of . Amazingly, the largest denominators for , 2, ... are 1, 2, 3, 5, 8, 13, 21, ..., which are immediately recognizable as the Fibonacci numbers (OEIS A000045). The following table gives the values possible for small .possible resistances11234If..
Let be the resistance distance matrix of a connected graph on nodes. Then Foster's theorems state thatwhere is the edge set of , andwhere the latter sum runs over all pairs of adjacent edges and is the vertex degree of the vertex common to those edges (Palacios 2001).
The fraction of odd values of the partition function P(n) is roughly 50%, independent of , whereas odd values of occur with ever decreasing frequency as becomes large. Kolberg (1959) proved that there are infinitely many even and odd values of .Leibniz noted that is prime for , 3, 4, 5, 6, but not 7. In fact, values of for which is prime are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (OEIS A046063), corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575). Numbers which cannot be written as a product of are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (OEIS A046064), corresponding to numbers of nonisomorphic Abelian groups which are not possible for any group order.Ramanujan conjectured a number of amazing and unexpected congruences involving . In particular, he proved(1)using Ramanujan's identity (Darling 1919; Hardy and Wright 1979; Drost 1997; Hardy 1999, pp. 87-88; Hirschhorn 1999). Ramanujan (1919) also showed that(2)and..
The factorial is defined for a positive integer as(1)So, for example, . An older notation for the factorial was written (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996).The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).The factorial is implemented in the Wolfram Language as Factorial[n] or n!.The triangular number can be regarded as the additive analog of the factorial . Another relationship between factorials and triangular numbers is given by the identity(2)(K. MacMillan, pers. comm., Jan. 21, 2008).The factorial gives the number of ways in which objects can be permuted. For example, , since the six possible permutations of are , , , , , . The first few factorials for , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (OEIS A000142).The..
Find the array of single digits which contains the maximum possible number of primes, where allowable primes may lie along any horizontal, vertical, or diagonal line.For the array, 11 primes are maximal and are contained in the two distinct arrays(1)giving the primes (3, 7, 13, 17, 31, 37, 41, 43,47, 71, 73) and (3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97), respectively.The best array is(2)which contains 30 primes: 3, 5, 7, 11, 13, 17, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 97, 113, 157, 179, ... (OEIS A032529). This array was found by Rivera and Ayala and shown by Weisstein in May 1999 to be maximal and unique (modulo reflection and rotation).The best arrays known are(3)all of which contain 63 primes. The first was found by C. Rivera and J. Ayala in 1998, and the other three by James Bonfield on April 13, 1999. Mike Oakes proved by computation that the 63 primes is optimal for the array.The best prime arrays known are(4)each of which contains 116 primes...
An algebra is called a lattice if is a nonempty set, and are binary operations on , both and are idempotent, commutative, and associative, and they satisfy the absorption law. The study of lattices is called lattice theory.Note that this type of lattice is distinct from the regular array of points known as a point lattice (or informally as a mesh or grid). While every point lattice is a lattice under the ordering inherited from the plane, many lattices are not point lattices.Lattices offer a natural way to formalize and study the ordering of objects using a general concept known as the partially ordered set. A lattice as an algebra is equivalent to a lattice as a partially ordered set (Grätzer 1971, p. 6) since 1. Let the partially ordered set be a lattice. Set and . Then the algebra is a lattice. 2. Let the algebra be a lattice. Set iff . Then is a partially ordered set, and the partially ordered set is a lattice. 3. Let the partially ordered set be..
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,..
There are two problems commonly known as the subset sum problem.The first ("given sum problem") is the problem of finding what subset of a list of integers has a given sum, which is an integer relation problem where the relation coefficients are 0 or 1.The ("same sum problem") is the problem of finding a set of distinct positive real numbers with as large a collection as possible of subsets with the same sum (Proctor 1982).The same sum problem was solved by Stanley (1980) using the tools of algebraic geometry, with the answer given for numbers by the first positive integers: . Proctor (1982) gave the first elementary proof of this result. The maximal numbers of subsets of having the same sum for , 2, ... are 1, 1, 2, 2, 3, 5, 8, 14, 23, ... (OEIS A025591). Similarly, the numbers of different subset sums for , 2, ... are 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, ... (OEIS A000124). For example, for , the subsets of are(1)(2)(3)(4)(5)(6)(7)(8)so..
Newton's method for finding roots of a complex polynomial entails iterating the function , which can be viewed as applying the Euler backward method with step size unity to the so-called Newtonian vector field . The rescaled and desingularized vector field then has sinks at roots of and has saddle points at roots of that are not also roots of . The union of the closures of the unstable manifolds of the saddles of defines a directed graph whose vertices are the roots of and of , and whose edges are the unstable curves oriented by the flow direction. This graph, along with the labelling of each vertex with the multiplicity of as a root of , is defined to be the Newtonian graph of (Smale 1985, Shub et al. 1988, Kozen and Stefánsson 1997).
The vertex set of a graph is simply a set of all vertices of the graph. The cardinality of the vertex set for a given graph is known as the vertex count of .The vertex set for a particular graph embedding of a graph is given in the Wolfram Language using PropertyValue[g, VertexCoordinates] or GraphEmbedding[g]. Vertex sets for many named graphs are available via GraphData[graph, "VertexCoordinates"] (for the primary embedding) and GraphData[graph, "Embeddings"] (for all available embeddings).The vertex set of an abstract simplicial complex is the union of one-point elements of (Munkres 1993, p. 15).
Stern's diatomic series is the sequence(1)... (OEIS A002487) which arises in the Calkin-Wilftree. It is sometimes also known as the fusc function (Dijkstra 1982).The th term can be given by the recurrence equation(2)with and . A sum formula is given by(3)A generating function is given by(4)(5)
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, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have showed that there are infinitely many triangles with rational sides (Heronian triangles) with two rational triangle medians (Guy 1994, p. 188).
A family of nonempty subsets of whose union contains the given set (and which contains no duplicated subsets) is called a cover (or covering) of . For example, there is only a single cover of , namely . However, there are five covers of , namely , , , , and .A minimal cover is a cover for which removal of one member destroys the covering property. For example, of the five covers of , only and are minimal covers. There are various other types of specialized covers, including proper covers, antichain covers, -covers, and -covers (Macula 1994).The number of possible covers for a set of elements arethe first few of which are 1, 5, 109, 32297, 2147321017, 9223372023970362989, ...(OEIS A003465).
"Neighborhood" is a word with many different levels of meaning in mathematics.One of the most general concepts of a neighborhood of a point (also called an epsilon-neighborhood or infinitesimal open set) is the set of points inside an -ball with center and radius . A set containing an open neighborhood is also called a neighborhood.The graph neighborhood of a vertex in a graph is the set of all the vertices adjacent to generally including itself. More generally, the th neighborhood of is the set of all vertices that lie at the distance from . The subgraph induced by the neighborhood of a graph from vertex (again, most commonly including itself) is called the neighborhood graph (or sometimes "ego graph" in more recent literature).
Define the sequence , , and(1)for . The first few values are(2)(3)(4)(5)Janssen and Tjaden (1987) showed that this sequence converges for exactly one value , where (OEIS A085835), confirming Grossman's conjecture. However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. The plot above shows the first few iterations of for to 30, with odd shown in red and even shown in blue, for ranging from 0 to 1. As can be seen, the solutions alternate by parity. For each fixed , the red values go to 0, while the blue values go to some positive number.Nyerges (2000) has generalized the recurrence to the functional equation(6)
A Lucas polynomial sequence is a pair of generalized polynomials which generalize the Lucas sequence to polynomials is given by(1)(2)where(3)(4)Solving for and and taking the solution for with the sign gives(5)(Horadam 1996). Setting gives(6)(7)giving(8)(9)The sequences most commonly considered have , giving(10)The polynomials satisfy the recurrence relation(11)Special cases of the and polynomials are given in the following table.1Fibonacci polynomial Lucas polynomial 1Pell polynomial Pell-Lucas polynomial 1Jacobsthal polynomial Jacobsthal-Lucas polynomial Fermat polynomial Fermat-Lucas polynomial Chebyshev polynomial of the second kind Chebyshev polynomial of the first kind
The Lucas numbers are the sequence of integers defined by the linear recurrence equation(1)with and . The th Lucas number is implemented in the Wolfram Language as LucasL[n].The values of for , 2, ... are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (OEIS A000204).The Lucas numbers are also a Lucas sequence and are the companions to the Fibonacci numbers and satisfy the same recurrence.The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers (where 1 and are now consecutive) is (Honsberger 1985, p. 122).The only square numbers in the Lucas sequence are 1 and 4 (Alfred 1964, Cohn 1964). The only triangular Lucas numbers are 1, 3, and 5778 (Ming 1991). The only cubic Lucas number is 1.Rather amazingly, if is prime, . The converse does not necessarily hold true, however, and composite numbers such that are known as Lucas pseudoprimes.For , 2, ..., the numbers of decimal digits in are..
The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation(1)with . As a result of the definition (1), it is conventional to define .The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... (OEIS A000045).Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials with .Fibonacci numbers are implemented in the WolframLanguage as Fibonacci[n].The Fibonacci numbers are also a Lucas sequence , and are companions to the Lucas numbers (which satisfy the same recurrence equation).The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). (The right panel instead applies the Perrin sequence).A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 43,..
For a simple continued fraction with convergents , the fundamental recurrence relation is given by
An indicial equation, also called a characteristic equation, is a recurrence equation obtained during application of the Frobenius method of solving a second-order ordinary differential equation. The indicial equation is obtained by noting that, by definition, the lowest order term (that corresponding to ) must have a coefficient of zero. 1. If the two roots are equal, only one solution can beobtained. 2. If the two roots differ by a noninteger, two solutionscan be obtained. 3. If the two roots differ by an integer,the larger will yield a solution. The smaller may or may not. For an example of the construction of an indicial equation, see Besselfunction of the first kind.The following table gives the indicial equations for some common differential equations.differential equationindicial equationBessel differential equationChebyshev differential equationHermite differential equationJacobi differential equationLaguerre differential..
A generalization of the Fibonacci numbers defined by the four constants and the definitions and together with the linear recurrence equationfor . With , , , and , the Horadam sequence reduces to the Fibonacci numbers.
The Epstein zeta function for a matrix of a positive definite real quadratic form and a complex variable with (where denotes the real part) is defined by(1)where the sum is over all column vectors with integer coordinates and the prime means the summation excludes the origin (Terras 1973). Epstein (1903) derived the analytic continuation, functional equation, and so-called Kronecker limit formula for this function.Epstein (1903) defined this function in the course of an effort to find the most general possible function satisfying a functional equation similar to that satisfied by the Riemann zeta function (Glasser and Zucker 1980, p. 68).A slightly different notation is used in theoretical chemistry, where the Epstein zeta function arises in connection with lattice sums. Let be a positive definite quadratic form(2)where with , ... is a symmetric matrix. Then the Epstein zeta function can be defined as(3)where and are arbitrary..
Given a convex plane region with area and perimeter , thenwhere is the number of enclosed lattice points.
Let be the smallest tour length for points in a -D hypercube. Then there exists a smallest constant such that for all optimal tours in the hypercube,(1)and a constant such that for almost all optimal tours in the hypercube,(2)These constants satisfy the inequalities(3)(4)(5)(6)(7)(8)(9)(Fejes Tóth 1940, Verblunsky 1951, Few 1955, Beardwood et al. 1959),where(10) is the gamma function, is an expression involving Struve functions and Bessel functions of the second kind,(11)(OEIS A086306; Karloff 1989), and(12)(OEIS A086307; Goddyn 1990).In the limit ,(13)(14)(15)and(16)where(17)and is the best sphere packing density in -D space (Goddyn 1990, Moran 1984, Kabatyanskii and Levenshtein 1978). Steele and Snyder (1989) proved that the limit exists.Now consider the constant(18)so(19)Nonrigorous numerical estimates give (Johnson et al. 1996) and (Percus and Martin 1996).A certain self-avoiding space-filling function..
The ant colony algorithm is an algorithm for finding optimal paths that is based on the behavior of ants searching for food.At first, the ants wander randomly. When an ant finds a source of food, it walks back to the colony leaving "markers" (pheromones) that show the path has food. When other ants come across the markers, they are likely to follow the path with a certain probability. If they do, they then populate the path with their own markers as they bring the food back. As more ants find the path, it gets stronger until there are a couple streams of ants traveling to various food sources near the colony.Because the ants drop pheromones every time they bring food, shorter paths are more likely to be stronger, hence optimizing the "solution." In the meantime, some ants are still randomly scouting for closer food sources. A similar approach can be used find near-optimal solution to the traveling salesman problem.Once the food..
Let be a set and a collection of subsets of . A subset is shattered by if each subset of can be expressed as the intersection of with a subset in . Symbolically, then, is shattered by if for all , there exists some for which .If is shattered by , one says that shatters .There are a number of equivalent ways to define shattering. One can easily verify that the above definition is equivalent to saying that shatters ifwhere denotes the power set of . Yet another way to express this concept is to say that a set of cardinality is shattered by a set if where here,In the field of machine learning theory, one usually considers the set to be a sample of outcomes drawn according to a distribution with the set representing a collection of "known" concepts or laws. In this context, saying that is shattered by intuitively means that all of the constituent outcomes in can be known by knowing only the laws in ...
A recurrence plot is defined as a plot of the quantitywhere is the Heaviside step function and denotes a norm. A recurrence plot is therefore a binary plot. The figure above shows a recurrence plot for the Lorenz attractor with , , , , , , and .Recurrence plots were initially used to graphically display nonstationarity in time series (Eckmann et al. 1987, Gao and Cai 2000), but are also useful for visualizing functions.A so-called global recurrence plot or unthresholded recurrence plot of a function is a plot of (or ) in the - plane. Recurrence plots for a number of common functions are illustrated above.
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..