Sort by:

The complete products of a Boolean algebra of subsets generated by a set of cardinal number are the Boolean functions(1)where each may equal or its complement . For example, the complete products of are(2)Each Boolean function has a unique representation(up to order) as a union of complete products. For example,(3)(4)(5)(Comtet 1974, p. 186).

The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof.In Hilbert's day, formal logic sought to devise a complete, consistent formulation of mathematics such that propositions could be formally stated and proved using a small number of symbols with well-defined meanings. The difficulty of formal logic was demonstrated in the monumental Principia Mathematica (1925) of Whitehead and Russell's, in which hundreds of pages of symbols were required before the statement could be deduced.The foundations of this program were obliterated in the mid 1930s when Gödel unexpectedly proved a result now known as Gödel's second incompleteness theorem. This theorem not only showed Hilbert's goal to be impossible, but also proved to be only the first in a series of deep and counterintuitive statements about rigor and provability in mathematics.A very simple form of logic is the study of "truth..

The shear matrix is obtained from the identity matrix by inserting at , e.g.,(1)Bolt and Hobbs (1998) define a shear matrix as a matrix(2)such that(3)(4)

A magic square is a square array of numbers consisting of the distinct positive integers 1, 2, ..., arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number (Kraitchik 1942, p. 142; Andrews 1960, p. 1; Gardner 1961, p. 130; Madachy 1979, p. 84; Benson and Jacoby 1981, p. 3; Ball and Coxeter 1987, p. 193), known as the magic constantIf every number in a magic square is subtracted from , another magic square is obtained called the complementary magic square. A square consisting of consecutive numbers starting with 1 is sometimes known as a "normal" magic square.The unique normal square of order three was known to the ancient Chinese, who called it the Lo Shu. A version of the order-4 magic square with the numbers 15 and 14 in adjacent middle columns in the bottom row is called Dürer's magic square. Magic squares of order 3 through 8 are shown..

Two nonisomorphic graphs can share the same graph spectrum, i.e., have the same eigenvalues of their adjacency matrices. Such graphs are called cospectral. For example, the graph union and star graph , illustrated above, both have spectrum (Skiena 1990, p. 85). This is the smallest pair of simple graphs that are cospectral. Determining which graphs are uniquely determined by their spectra is in general a very hard problem.Only a small fraction of graphs are known to be so determined, but it is conceivablethat almost all graphs have this property (van Dam and Haemers 2002).In the Wolfram Language, graphs knownto be determined by their spectra are identified as GraphData["DeterminedBySpectrum"].The numbers of simple graphs on , 2, ... nodes that are determined by spectrum are 1, 2, 4, 11, 32, 146, 934, 10624, 223629, ... (OEIS A178925), while the corresponding numbers not determined by spectrum are 0, 0, 0, 0, 2, 10, 110, 1722,..

An integral graph is defined as a graph whose graph spectrum consists entirely of integers. The notion was first introduced by Harary and Schwenk (1974). The numbers of simple integral graphs on , 2, ... nodes are 0, 2, 3, 6, 10, 20, 33, 71, ... (OEIS A077027), illustrated above for small .The numbers of connected simple integral graphs on , 2, ... nodes are 1, 1, 1, 2, 3, 6, 7, 22, 24, 83, ... (OEIS A064731), illustrated above for small .The following table lists common graph classes and the their members which are integral.graphintegral for of the formantiprism graph3complete graph allcycle graph 2, 3, 4, 6empty graphallprism graph3, 4, 6star graph wheel graph 4The following table lists some special named graphs that are integral and gives their spectra.graphgraph spectrum16-cell24-cellClebsch graphcubical graphcuboctahedral graphDesargues graphHall-Janko graphHoffman graphHoffman-Singleton graphLevi graphM22 graphMcLaughlin..

The set of graph eigenvalues of the adjacency matrix is called the spectrum of the graph. (But note that in physics, the eigenvalues of the Laplacian matrix of a graph are sometimes known as the graph's spectrum.) The spectrum of a graph with -fold degenerate eigenvalues is commonly denoted (van Dam and Haemers 2003) or (Biggs 1993, p. 8; Buekenhout and Parker 1998).The product over the elements of the spectrum of a graph is known as the characteristic polynomial of , and is given by the characteristic polynomial of the adjacency matrix of with respect to the variable .The largest absolute value of a graph's spectrum is known as its spectralradius.The spectrum of a graph may be computed in the Wolfram Language using Eigenvalues[AdjacencyMatrix[g]]. Precomputed spectra for many named graphs can be obtained using GraphData[graph, "Spectrum"].A graph whose spectrum consists entirely of integers is known as an integralgraph.The..

An antimagic square is an array of integers from 1 to such that each row, column, and main diagonal produces a different sum such that these sums form a sequence of consecutive integers. It is therefore a special case of a heterosquare. It was defined by Lindon (1962) and appeared in Madachy's collection of puzzles (Madachy 1979, p. 103), originally published in 1966. Antimagic squares of orders 4-9 are illustrated above (Madachy 1979). For the square, the sums are 30, 31, 32, ..., 39; for the square they are 59, 60, 61, ..., 70; and so on.Let an antimagic square of order have entries 0, 1, ..., , , and letbe the magic constant. Then if an antimagic square of order exists, it is either positive with sums , or negative with sums (Madachy 1979).Antimagic squares of orders one, two, and three are impossible. In the case of the square, there is no known method of proof of this fact except by case analysis or enumeration by computer. There are 18 families of..

A heterosquare is an array of the integers from 1 to such that the rows, columns, and diagonals have different sums. (By contrast, in a magic square, they have the same sum.) There are no heterosquares of order two, but heterosquares of every odd order exist. They can be constructed by placing consecutive integers in a spiral pattern (Fults 1974, Madachy 1979).An antimagic square is a special case of a heterosquare for which the sums of rows, columns, and main diagonals form a sequence of consecutive integers.

Three types of matrices can be obtained by writing Pascal's triangle as a lower triangular matrix and truncating appropriately: a symmetric matrix with , a lower triangular matrix with , and an upper triangular matrix with , where , 1, ..., . For example, for , these would be given by(1)(2)(3)The Pascal -matrix or order is implemented in the Wolfram Language as LinearAlgebra`PascalMatrix[n].These matrices have some amazing properties. In particular, their determinants are all equal to 1(4)and(5)(Edelman and Strang).Edelman and Strang give four proofs of the identity (5), themost straightforward of which is(6)(7)(8)(9)where Einstein summation has been used.

The term "wedge" has a number of meanings in mathematics. It is sometimes used as another name for the caret symbol, as well as being the notation () for logical AND.In solid geometry, a wedge is a right triangular prism turned so that it rests on one of its lateral rectangular faces (left figure). Harris and Stocker (1998) define a more general type of oblique wedge in which the top edge is symmetrically shortened, causing the end triangles to slant obliquely (right figure).For an oblique wedge of base lengths and , top edge length , and height (right figure), the volume of the wedge is(1)(2)In the case of a right wedge , this simplifies to(3)The geometric centroid is located at a height(4)above the base, which simplifies to for a right wedge .

A semialgebraic set is a subset of which is a finite Boolean combination of sets of the form and , where and are polynomials in , ..., over the reals.By Tarski's theorem, the solution set of a quantified system of real algebraic equations and inequalities is a semialgebraic set (Strzebonski 2000).

Let be a finite partially ordered set, then an antichain in is a set of pairwise incomparable elements. Antichains are also called Sperner systems in older literature (Comtet 1974).For example, consider to be a family of subsets together with the subset relation (i.e., if is a subset of ). The following table gives the antichains on the set of subsets (i.e., the power set) of the -set for small .antichains123The number of antichains on the -set for , 1, 2, ..., are 1, 2, 5, 19, 167, ... (OEIS A014466). If the empty set is not considered a valid antichain, then these reduce to 0, 1, 4, 18, 166, ... (OEIS A007153; Comtet 1974, p. 273). The numbers obtained by adding one to OEIS A014466, 2, 3, 6, 20, 168, 7581, 7828354, ... (OEIS A000372), are also frequently encountered (Speciner 1972).The number of antichains on the -set are equal to the number of monotonic increasing Boolean functions of variables, and also the number of free distributive lattices..

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp() is defined to be , where is the unique Lie group homeomorphism from the real numbers to the Lie group such that its velocity at time 0 is .On a Riemannian manifold, exp is a map from the tangent bundle of the manifold to the manifold, and exp() is defined to be , where is the unique geodesic traveling through the base-point of such that its velocity at time 0 is .The three notions of exp (exp from complex analysis, exp from Lie groups, and exp from Riemannian geometry) are all linked together, the strongest link being between the Lie groups and Riemannian geometry definition. If is a compact Lie group, it admits a left and right invariant Riemannian metric. With respect to that metric, the two exp maps agree on their common domain. In other words, one-parameter subgroups are geodesics. In the case of the manifold..

Let represent "or", represent "and", and represent "not." Then, for two logical units and ,These laws also apply in the more general context of Boolean algebra and, in particular, in the Boolean algebra of set theory, in which case would denote union, intersection, and complementation with respect to any superset of and .

The law appearing in the definition of Boolean algebrasand lattice which states thatfor binary operators and (which most commonly are logical OR and logical AND). The two parts of the absorption law are sometimes called the "absorption identities" (Grätzer 1971, p. 5).

The directional derivative is the rate at which the function changes at a point in the direction . It is a vector form of the usual derivative, and can be defined as(1)(2)where is called "nabla" or "del" and denotes a unit vector.The directional derivative is also often written in the notation(3)(4)where denotes a unit vector in any given direction and denotes a partial derivative.Let be a unit vector in Cartesian coordinates, so(5)then(6)

In general, a cross is a figure formed by two intersecting line segments. In linear algebra, a cross is defined as a set of mutually perpendicular pairs of vectors of equal magnitude from a fixed origin in Euclidean -space.The word "cross" is also used to denote the operation of the cross product, so would be pronounced " cross ."

A vector derivative is a derivative taken with respect to a vector field. Vector derivatives are extremely important in physics, where they arise throughout fluid mechanics, electricity and magnetism, elasticity, and many other areas of theoretical and applied physics.The following table summarizes the names and notations for various vector derivatives.symbolvector derivativegradientLaplacian or vector Laplacian or directional derivativedivergencecurlconvective derivativeVector derivatives can be combined in different ways, producing sets of identities that are also very important in physics.Vector derivative identities involving the curl include(1)(2)(3)(4)(5)In Cartesian coordinates(6)(7)In spherical coordinates,(8)(9)(10)Vector derivative identities involving the divergenceinclude(11)(12)(13)(14)(15)In Cartesian coordinates,(16)(17)(18)(19)(20)(21)In spherical coordinates,(22)(23)(24)(25)(26)(27)By..

For a scalar function over a surface parameterized by and , the surface integral is given by(1)(2)where and are tangent vectors and is the cross product.For a vector function over a surface, the surfaceintegral is given by(3)(4)(5)where is a dot product and is a unit normal vector. If , then is given explicitly by(6)If the surface is surface parameterized using and , then(7)

Given a Seifert form , choose a basis , ..., for as a -module so every element is uniquely expressible as(1)with integer. Then define the Seifert matrix as the integer matrix with entries(2)For example, the right-hand trefoil knot has Seifertmatrix(3)A Seifert matrix is not a knot invariant, but it can be used to distinguish between different Seifert surfaces for a given knot.

Let be an real square matrix with such that(1)for all real numbers , , ..., and , , ..., such that . Then Grothendieck showed that there exists a constant satisfying(2)for all vectors and in a Hilbert space with norms and . The Grothendieck constant is the smallest possible value of . For example, the best values known for small are(3)(4)(5)(Krivine 1977, 1979; König 1992; Finch 2003, p. 236).Now consider the limit(6)which is related to Khinchin's constant and sometimes also denoted . Krivine (1977) showed that(7)and postulated that(8)(OEIS A088367). The conjecture was refuted in 2011 by Yury Makarychev, Mark Braverman, Konstantin Makarychev, and Assaf Naor, who showed that is strictly less than Krivine's bound (Makarychev 2011).Similarly, if the numbers and and matrix are taken as complex, then a similar set of constants may be defined. These are known to satisfy(9)(10)(11)(Krivine 1977, 1979; König 1990, 1992; Finch..

Combinatorial matrix theory is a rich branch of mathematics that combines combinatorics, graph theory, and linear algebra. It includes the theory of matrices with prescribed combinatorial properties, including permanents and Latin squares. It also comprises combinatorial proof of classical algebraic theorems such as Cayley-Hamilton theorem.As mentioned in Season 4 episodes 407 "Primacy" and 412 "Power" of the television crime drama NUMB3RS, professor Amita Ramanujan's primary teaching interest is combinatorial matrix theory.

The square matrix with entries given by(1)for , 1, 2, ..., , where i is the imaginary number , and normalized by to make it a unitary. The Fourier matrix is given by(2)and the matrix by(3)(4)In general,(5)with(6)where is the identity matrix and is the diagonal matrix with entries 1, , ..., . Note that the factorization (which is the basis of the fast Fourier transform) has two copies of in the center factor matrix.

The set of linear Möbius transformations which satisfywhere and are odd and and are even. is a subgroup of the modular group Gamma, and is also called the theta subgroup. The fundamental region of the modular lambda group is illustrated above.

The group of all Möbius transformations of the form(1)where , , , and are integers with . The group can be represented by the matrix(2)where . Every can be expressed in the form(3)where(4)(5)although the representation is not unique (Apostol 1997, pp. 28-29).

Given a knot diagram, it is possible to construct a collection of variables and equations, and given such a collection, a group naturally arises that is known as the group of the knot. While the group itself depends on the choices made in the construction, any two groups that arise in this way are isomorphic (Livingston 1993, p. 103).For example, the knot group of the trefoil knot is(1)or equivalently(2)(Rolfsen 1976, pp. 52 and 61), while that of Solomon'sseal knot is(3)(Livingston 1993, p. 127).The group of a knot is not a complete knot invariant (Rolfsen 1976, p. 62). Furthermore, it is often quite difficult to prove that two knot group presentations represent nonisomorphic groups (Rolfsen 1976, p. 63).

Symmetry operations include the improper rotation, inversion operation, mirror plane, and rotation. Together, these operations create 32 crystal classes corresponding to the 32 point groups.The inversion operation takesand is denoted . When used in conjunction with a rotation, it becomes an improper rotation. An improper rotation by is denoted (or ). For periodic crystals, the crystallography restriction allows only the improper rotations , , , , and .The mirror plane symmetry operation takesetc., which is equivalent to . Invariance under reflection can be denoted or . The rotation symmetry operation for is denoted (or ). For periodic crystals, crystallography restriction allows only 1, 2, 3, 4, and 6.Symmetry operations can be indicated with symbols such as , , , , , and . 1. indicates rotation about an -fold symmetry axis. 2. indicates improper rotation about an -fold symmetry axis. 3. (or ) indicates invariance under translation. 4...

For any sequence of integers , there is a flag manifold of type (, ..., ) which is the collection of ordered sets of vector subspaces of (, ..., ) with and a subspace of . There are also complex flag manifolds with complex subspaces of instead of real subspaces of a real -space.These flag manifolds admit the structure of manifoldsin a natural way and are used in the theory of Lie groups.

A positive definite function on a group is a function for which the matrix is always positive semidefinite Hermitian.

Wallis's constant is the real solution (OEIS A007493) to the cubic equationIt was solved by Wallis to illustrate Newton's methodfor numerical equation solving.

A tensor category is strict if the maps , , and are always identities.A related notion is that of a tensor R-category.

In the above figure, let be a right triangle, arcs and be segments of circles centered at and respectively, and define(1)(2)(3)Then(4)This can be seen by letting , , and and then solving the equations(5)(6)(7)to obtain(8)(9)(10)Plugging in the above gives(11)by the Pythagorean theorem, so plugging in , the figure yields the algebraic identity(12)The area of intersection formed (inside the triangle) by the circular sectors determined by arcs is given by(13)

If moves along a known curve, then describes a pursuit curve if is always directed toward and and move with uniform velocities. Pursuit curves were considered in general by the French scientist Pierre Bouguer in 1732, and subsequently by the English mathematician Boole.Under the name "path minimization," pursuit curves are alluded to by math genius Charlie Eppes in the Season 2 episode "Dark Matter" of the television crime drama NUMB3RS when considering the actions of the mysterious third shooter.The equations of pursuit are given by(1)which specifies that the tangent vector at point is always parallel to the line connecting and , combined with(2)which specifies that the point moves with constant speed (without loss of generality, taken as unity above). Plugging (2) into (1) therefore gives(3)The case restricting to a straight line was studied by Arthur Bernhart (MacTutor Archive). Taking the parametric equation..

Given a ship with a known constant direction and speed , what course should be taken by a chase ship in pursuit (traveling at speed ) in order to intercept the other ship in as short a time as possible? The problem can be solved by finding all points which can be simultaneously reached by both ships, which is an Apollonius circle with . If the circle cuts the path of the pursued ship, the intersection is the point towards which the pursuit ship should steer. If the circle does not cut the path, then it cannot be caught.

A matrix for a round-robin tournament involving players competing in matches (no ties allowed) having entries(1)This scoring system differs from that used to compute a score sequence of a tournament, in which a win gives one point and a loss zero points. The matrix satisfies(2)where is the transpose of (McCarthy and Benjamin 1996).The tournament matrix for players has zero determinant iff is odd (McCarthy and Benjamin 1996). Furthermore, the dimension of the null space of an -player tournament matrix is(3)(McCarthy and Benjamin 1996).

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.

In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors and of the real plane or the real space are orthogonal iff their dot product . This condition has been exploited to define orthogonality in the more abstract context of the -dimensional real space .More generally, two elements and of an inner product space are called orthogonal if the inner product of and is 0. Two subspaces and of are called orthogonal if every element of is orthogonal to every element of . The same definitions can be applied to any symmetric or differential k-form and to any Hermitian form.

A projective space is a space that is invariant under the group of all general linear homogeneous transformation in the space concerned, but not under all the transformations of any group containing as a subgroup.A projective space is the space of one-dimensional vector subspaces of a given vector space. For real vector spaces, the notation or denotes the real projective space of dimension (i.e., the space of one-dimensional vector subspaces of ) and denotes the complex projective space of complex dimension (i.e., the space of one-dimensional complex vector subspaces of ). can also be viewed as the set consisting of together with its points at infinity.

When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, mirror images are equivalent for free objects.The word "free" is also used in technical senses to refer to a free group, free semigroup, free tree, free variable, etc.In algebraic topology, a free abstract mathematical object is generated by elements in a "free manner" ("freely"), i.e., such that the elements satisfy no nontrivial relations among themselves. To make this more formal, an algebraic gadget is freely generated by a subset if, for any function where is any other algebraic gadget, there exists a unique homomorphism (which has different meanings depending on what kind of gadgets you're dealing with) such that restricted to is .If the algebraic gadgets are vector spaces, then freely generates iff is a basis for . If the algebraic gadgets are..

Let and be nonincreasing sequences of real numbers. Then majorizes if, for each , 2, ..., ,with equality if . Note that some caution is needed when consulting the literature, since the direction of the inequality is not consistent from reference to reference. An order-free characterization along the lines of Horn's theorem is also readily available. majorizes iff there exists a doubly stochastic matrix such that . Intuitively, if majorizes , then is more "mixed" than . Horn's theorem relates the eigenvalues of a Hermitian matrix to its diagonal entries using majorization. Given two vectors , then majorizes iff there exists a Hermitian matrix with eigenvalues and diagonal entries .

Let be a group, and let be a set of group elements such that the identity element . The Cayley graph associated with is then defined as the directed graph having one vertex associated with each group element and directed edges whenever . The Cayley graph may depend on the choice of a generating set, and is connected iff generates (i.e., the set are group generators of ).Care is needed since the term "Cayley graph" is also used when is implicitly understood to be a set of generators for the group, in which case the graph is always connected (but in general, still dependent on the choice of generators). This sort of Cayley graph of a group may be computed in the Wolfram Language using CayleyGraph[G], where the generators used are those returned by GroupGenerators[G].To complicate matters further, undirected versions of proper directed Cayley graphs are also known as Cayley graphs.An undirected Cayley graph of a particular generating set of..

The functions , , ..., are linearly dependent if, for some , , ..., not all zero,(1)for all in some interval . If the functions are not linearly dependent, they are said to be linearly independent. Now, if the functions and in (the space of functions with continuous derivatives), we can differentiate (1) up to times. Therefore, linear dependence also requires(2)(3)(4)where the sums are over , ..., . These equations have a nontrivial solution iff the determinant(5)where the determinant is conventionally called the Wronskian and is denoted .If the Wronskian for any value in the interval , then the only solution possible for (2) is (, ..., ), and the functions are linearly independent. If, on the other hand, over some range, then the functions are linearly dependent somewhere in the range. This is equivalent to stating that if the vectors , ..., defined by(6)are linearly independent for at least one , then the functions are linearly independent in ...

Schmidt (1993) proposed the problem of determining if for any integer , the sequence of numbers defined by the binomial sums(1)are all integers.The following table gives the first few values of for small .OEISvalues1A0018501, 3, 13, 63, 321, 1683, 8989, 48639, ...2A0052591, 5, 73, 1445, 33001, 819005, ...3A0928131, 9, 433, 36729, 3824001, 450954009, ...4A0928141, 17, 2593, 990737, 473940001, ...5A0928151, 33, 15553, 27748833, 61371200001, ...This was proved by Strehl (1993, 1994) and Schmidt (1995) for the case , corresponding to the Franel numbers. Strehl (1994) also found an explicit expression for the case . The resulting identities for are therefore known as the Strehl identities. The problem was restated in Graham et al. (1994, pp. 256 and 549), who indicated that H. S. Wilf had shown to be an integer for any for (Zudilin 2004).The problem was answered in the affirmative by Zudilin (2004), who found explicit expressions..

where is an Eulerian number and is a binomial coefficient (Worpitzky 1883; Comtet 1974, p. 242).

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..

Closed forms are known for the sums of reciprocals of even-indexed Lucasnumbers(1)(2)(3)(4)(5)(OEIS A153415), where is the golden ratio, is a q-polygamma function, and is a Jacobi theta function, and odd-indexed Lucas numbers(6)(7)(8)(9)(10)(11)(OEIS A153416), where is a Lambert series (Borwein and Borwein 1987, pp. 91-92). This gives the reciprocal Lucas constant as(12)(13)(14)(15)(16)(OEIS A093540), where is the golden ratio and is a Fibonacci number.Borwein and Borwein (1987, pp. 94-101) give a number of related beautiful formulas.

Closed forms are known for the sums of reciprocals of even-indexed Fibonaccinumbers(1)(2)(3)(4)(5)(6)(7)(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is the golden ratio, is a q-polygamma function, and is a Lambert series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci numbers(8)(9)(10)(11)(12)(13)(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi elliptic function. Together, these give a closed form for the reciprocal Fibonacci constant of(14)(15)(16)(17)(18)(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of was formally raised by Paul Erdős and this sum was proved to be irrational by André-Jeannin (1989).Borwein..

The first Strehl identity is the binomial sum identity(Strehl 1993, 1994; Koepf 1998, p. 55), which are the so-called Franel numbers. For , 2, ..., the first few terms are 1, 2, 10, 56, 346, 2252, 15184, 104960, ... (OEIS A000172).The second Strehl identity is the binomial sum identity(Strehl 1993, 1994; Koepf 1998, p. 55) that is the case of Schmidt's problem. For , 1, 2, ..., these give the Apéry numbers 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).

The important binomial theorem states that(1)Consider sums of powers of binomial coefficients(2)(3)where is a generalized hypergeometric function. When they exist, the recurrence equations that give solutions to these equations can be generated quickly using Zeilberger's algorithm.For , the closed-form solution is given by(4)i.e., the powers of two. obeys the recurrence relation(5)For , the closed-form solution is given by(6)i.e., the central binomial coefficients. obeys the recurrence relation(7)Franel (1894, 1895) was the first to obtain recurrences for ,(8)(Riordan 1980, p. 193; Barrucand 1975; Cusick 1989; Jin and Dickinson 2000), so are sometimes called Franel numbers. The sequence for cannot be expressed as a fixed number of hypergeometric terms (Petkovšek et al. 1996, p. 160), and therefore has no closed-form hypergeometric expression.Franel (1894, 1895) was also the first to obtain the recurrence..

A conservative vector field (for which the curl ) may be assigned a scalar potentialwhere is a line integral.

The doublestruck capital letter Z, , denotes the ring of integers ..., , , 0, 1, 2, .... The symbol derives from the German word Zahl, meaning "number" (Dummit and Foote 1998, p. 1), and first appeared in Bourbaki's Algèbre (reprinted as Bourbaki 1998, p. 671). The ring of integers is sometimes also denoted using the double-struck capital I, .

An integer is a fundamental discriminant if it is not equal to 1, not divisible by any square of any odd prime, and satisfies or . The function FundamentalDiscriminantQ[d] in the Wolfram Language version 5.2 add-on package NumberTheory`NumberTheoryFunctions` tests if an integer is a fundamental discriminant.It can be implemented as: FundamentalDiscriminantQ[n_Integer] := n != 1&& (Mod[n, 4] == 1 \[Or] ! Unequal[Mod[n, 16], 8, 12])&& SquareFreeQ[n/2^IntegerExponent[n, 2]]The first few positive fundamental discriminants are 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, ... (OEIS A003658). Similarly, the first few negative fundamental discriminants are , , , , , , , , , , , ... (OEIS A003657).

The ring of integers is the set of integers ..., , , 0, 1, 2, ..., which form a ring. This ring is commonly denoted (doublestruck Z), or sometimes (doublestruck I).More generally, let be a number field. Then the ring of integers of , denoted , is the set of algebraic integers in , which is a ring of dimension over , where is the extension degree of over . is also sometimes called the maximal order of .The Gaussian integers is the ring of integers of , and the Eisenstein integers is the ring of integers of , where is a primitive cube root of unity.

A class number formula is a finite series giving exactly the class number of a ring. For a ring of quadratic integers, the class number is denoted , where is the discriminant. A class number formula is known for the full ring of cyclotomic integers, as well as for any subring of the cyclotomic integers. This formula includes the quadratic case as well as many cubic and higher-order rings.

Given a factor of a number , the cofactor of is .A different type of cofactor, sometimes called a cofactor matrix, is a signed version of a minor defined byand used in the computation of the determinant of a matrix according toThe cofactor can be computed in the WolframLanguage using Cofactor[m_List?MatrixQ, {i_Integer, j_Integer}] := (-1)^(i+j) Det[Drop[Transpose[ Drop[Transpose[m], {j}]], {i} ]]which is the equivalent of the th component of the CofactorMatrix defined below. MinorMatrix[m_List?MatrixQ] := Map[Reverse, Minors[m], {0, 1}] CofactorMatrix[m_List?MatrixQ] := MapIndexed[#1 (-1)^(Plus @@ #2)&, MinorMatrix[m],{2}]Cofactors can be computed using Cofactor[m, i, j] in the Wolfram Language package Combinatorica` .

The logical axiomwhere denotes NOT and denotes OR, that, when taken together with associativity and commutativity, is equivalent to the axioms of Boolean algebra.The Robbins operator can be defined in the WolframLanguage by Robbins := Function[{x, y}, ! (! (! y \[Or] x) \[Or] ! (x \[Or] y))]That the Robbins axiom is a true statement in Booleanalgebra can be verified by examining its truth table.TTTTFTFTFFFF

The classification theorem of finite simple groups, also known as the "enormous theorem," which states that the finite simple groups can be classified completely into 1. Cyclic groups of prime group order, 2. Alternating groups of degree at least five, 3. Lie-type Chevalley groups given by , , , and , 4. Lie-type (twisted Chevalley groups or the Tits group) , , , , , , , , , 5. Sporadic groups , , , , , , Suz, HS, McL, , , , He, , , , HN, Th, , , , O'N, , Ly, Ru, . The "proof" of this theorem is spread throughout the mathematical literature and is estimated to be approximately pages in length.

The conjugate transpose of an matrix is the matrix defined by(1)where denotes the transpose of the matrix and denotes the conjugate matrix. In all common spaces (i.e., separable Hilbert spaces), the conjugate and transpose operations commute, so(2)The symbol (where the "H" stands for "Hermitian") gives official recognition to the fact that for complex matrices, it is almost always the case that the combined operation of taking the transpose and complex conjugate arises in physical or computation contexts and virtually never the transpose in isolation (Strang 1988, pp. 220-221).The conjugate transpose of a matrix is implemented in the Wolfram Language as ConjugateTranspose[A].The conjugate transpose is also known as the adjoint matrix, adjugate matrix, Hermitian adjoint, or Hermitian transpose (Strang 1988, p. 221). Unfortunately, several different notations are in use as summarized in the..

An matrix which gives the possible outcome of a two-person zero-sum game when player A has possible moves and player B moves. The analysis of the matrix in order to determine optimal strategies is the aim of game theory. The so-called "augmented" payoff matrix is defined as follows:

Long multiplication is the method of multiplication that is commonly taught to elementary school students throughout the world. It can be used on two numbers of arbitrarily large size or number of decimal digits. The numbers to be multiplied are placed vertically over one another with their least significant digits aligned. The top number is named the multiplicand and the lower number is the multiplier. The result of the multiplication is the product.For example, we can multiply . The number with more digits is usually selected as the multiplicand: The long multiplication algorithm starts with multiplying the multiplicand by the least significant digit of the multiplier to produce a partial product, then continuing this process for all higher order digits in the multiplier. Each partial product is right-aligned with the corresponding digit in the multiplier. The partial products are then summed: Implicit in using this method is the following..

There are at least three theorems known as Jensen's theorem.The first states that, for a fixed vector , the functionis a decreasing function of (Cheney 1999).The second states that if is a real polynomial not identically constant, then all nonreal zeros of lie inside the Jensen disks determined by all pairs of conjugate nonreal zeros of (Walsh 1955, 1961; Householder 1970; Trott 2004, p. 22). This theorem is a sharpening of Lucas's root theorem.The third theorem considers a function defined and analytic throughout a disk and supposes that has no zeros on the bounding circle , that inside the disk it has zeros , , ..., (where a zero of order is included times in the list, and that . Then(Edwards 2001, p. 40).

Let be a real entire function of the form(1)where the s are positive and satisfy Turán's inequalities(2)for , 2, .... The Jensen polynomial associated with is then given by(3)where is a binomial coefficient.

Let be a pair consisting of finite, connected CW-complexes where is a subcomplex of . Define the associated chain complex group-wise for each by setting(1)where denotes singular homology with integer coefficients and where denotes the union of all cells of of dimension less than or equal to . Note that is free Abelian with one generator for each -cell of .Next, consider the universal covering complexes of and , respectively. The fundamental group of can be identified with the group of deck transformations of so that each determines a map(2)which then induces a chain map(3)The chain map turns each chain group into a module over the group ring which is -free with one generator for each -cell of and which is finitely generated over due to the finiteness of .Hence, there is a free chain complex(4)over , the homology groups of which are zero due to the fact that deformation retracts onto . A simple argument shows the existence of a so-called preferred basis..

The wedge product is the product in an exterior algebra. If and are differential k-forms of degrees and , respectively, then(1)It is not (in general) commutative, but it is associative,(2)and bilinear(3)(4)(Spivak 1999, p. 203), where and are constants. The exterior algebra is generated by elements of degree one, and so the wedge product can be defined using a basis for :(5)when the indices are distinct, and the product is zero otherwise.While the formula holds when has degree one, it does not hold in general. For example, consider :(6)(7)(8)If have degree one, then they are linearly independent iff .The wedge product is the "correct" type of product to use in computinga volume element(9)The wedge product can therefore be used to calculate determinants and volumes of parallelepipeds. For example, write where are the columns of . Then(10)and is the volume of the parallelepiped spanned by ...

A natural transformation is called unital if the leftmost diagram above commutes. Similarly, a natural transformation is called unital if the diagram on the right-hand side above commutes.Note that in these definitions, , , and are all objects in a tensor category , is the neutral (or identity) object in , and the juxtaposition is shorthand for the tensor product in . What's more, the subscripts attached to the transformations and denote the components of the functors (indexed with respect to the objects in ) in question.

A set in which can be reduced to one of its points, say , by a continuous deformation, is said to be contractible. The transformation is such that each point of the set is driven to through a path with the properties that 1. Each path runs entirely inside the set. 2. Nearby points move on "neighboring" paths. Condition (1) implies that a disconnected set,i.e., a set consisting of separate parts, cannot be contractible.Condition (2) implies that the circumference of a circle is not contractible. The latter follows by considering two near points and lying on different sides of a point . The paths connecting and with are either opposite each other or have different lengths. A similar argument shows that, in general, for all , the -sphere (i.e., the boundary of the -dimensional ball) is not contractible.A gap or a hole in a set can be an obstruction to contractibility. There are, however, examples of contractible sets with holes, for example,..

The homomorphism which, according to the snake lemma, permits construction of an exact sequence(1)from the above commutative diagram with exact rows. The homomorphism is defined by(2)for all , denotes the image, and is obtained through the following construction, based on diagram chasing.1. Exploit the surjectivity of to find such that . 2. Since because of the commutativity of the right square, belongs to , which is equal to due to the exactness of the lower row at . This allows us to find such that . While the elements and are not uniquely determined, the coset is, as can be proven by using more diagram chasing. In particular, if and are other elements fulfilling the requirements of steps (1) and (2), then and , and(3)hence because of the exactness of the upper row at . Let be such that(4)Then(5)because the left square is commutative. Since is injective, it follows that(6)and so(7)..

The interior product is a dual notion of the wedge product in an exterior algebra , where is a vector space. Given an orthonormal basis of , the forms(1)are an orthonormal basis for . They define a metric on the exterior algebra, . The interior product with a form is the adjoint of the wedge product with . That is,(2)for all . For example,(3)and(4)where the are orthonormal, are two interior products.An inner product on gives an isomorphism with the dual vector space . The interior product is the composition of this isomorphism with tensor contraction.

A commutative diagram is a collection of maps in which all map compositions starting from the same set and ending with the same set give the same result. In symbols this means that, whenever one can form two sequences(1)and(2)the following equality holds:(3)Commutative diagrams are usually composed by commutative triangles and commutative squares.Commutative triangles and squares can also be combined to form plane figures or space arrangements.A commutative diagram can also contain multiple arrows that indicate different maps between the same two sets.A looped arrow indicates a map from a set to itself.The above commutative diagram expresses the fact that is the inverse map to , since it is a pictorial translation of the map equalities and .This can also be represented using two separate diagrams.Many other mathematical concepts and properties, especially in algebraic topology, homological algebra, and category theory, can be formulated..

In category theory, a tensor category consists of a category , an object of , a functor , and a natural isomorphism(1)(2)(3)where the data are subject to the following axioms: 1. Given four objects , , , and of , the top diagram above commutes. 2. Given two objects and of , the bottom diagram above commutes. In the above, is called the tensor product, is called the associator, is called the right unit, and is called the left unit of the tensor category. The object is referred to as the neutral element or the identity of the tensor product.If the maps , , and are always identities, the tensor category in question is said to be strict.A related notion is that of a tensor R-category.

In algebraic topology, the Reidemeister torsion is a notion originally introduced as a topological invariant of 3-manifolds which has now been widely adapted to a variety of contexts. At the time of its discovery, the Reidemeister torsion was the first 3-manifold invariant able to distinguish between manifolds which are homotopy equivalent but not homeomorphic. Since then, the notion has been adapted to higher-dimensional manifolds, knots and links, dynamical systems, Witten's equations, and so on. In particular, it has a number of different definitions for various contexts.For a commutative ring , let be a finite acyclic chain complex of based finitely generated free R-modules of the form(1)The Reidemeister torsion of is the value defined by(2)where is the set of units of , is a chain contraction, is the boundary map, and(3)is a map from to . In this context, Reidemeister torsion is sometimes referred to as the torsion of the complex (Nicolaescu..

Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. Informally, the Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces (Gardner 1984, pp. 9-10). Formally, the th Betti number is the rank of the th homology group of a topological space. The following table gives the Betti number of some common surfaces.surfaceBetti numbercross-cap1cylinder1klein bottle2Möbius strip1plane lamina0projective plane1sphere0torus2Let be the group rank of the homology group of a topological space . For a closed, orientable surface of genus , the Betti numbers are , , and . For a nonorientable surface with cross-caps, the Betti numbers are , , and .The Betti number of a finitely generated Abelian group is the (uniquely determined) number such thatwhere , ..., are finite cyclic..

Let be a compact -dimensional oriented Riemannian manifold without boundary, let be a group representation of by orthogonal matrices, and let be the associated vector bundle. Suppose further that the Laplacian is strictly negative on where is the linear space of differential k-forms on with values in . In this context, the analytic torsion is defined as the positive real root ofwhere the -function is defined byfor the collection of eigenvalues of , the restriction of to the collection of bundle sections of the sheaf .Intrinsic to the above computation is that is a real manifold. However, there is a collection of literature on analytic torsion for complex manifolds, the construction of which is nearly identical to the construction given above. Analytic torsion on complex manifolds is sometimes called del bar torsion...

The th exterior power of an element in an exterior algebra is given by the wedge product of with itself times. Note that if has odd degree, then any higher power of must be zero. The situation for even degree forms is different. For example, if(1)then(2)(3)(4)

Let be functors between categories and . A natural transformation from to consists of a family of morphisms in which are indexed by the objects of so that, for each morphism between objects in , the equalityholds. The elements are called the components of the natural transformation.If all the components are isomorphisms in , then is called a natural isomorphism between and . In this case, one writes .

Let four lines in a plane represent four roads in general position, and let one traveler be walking along each road at a constant (but not necessarily equal to any other traveler's) speed. Say that two travelers and have "met" if they were simultaneously at the intersection of their two roads. Then if has met all other three travelers (, , and ) and , in addition to meeting , has met and , then and have also met!

Conjugation is the process of taking a complex conjugate of a complex number, complex matrix, etc., or of performing a conjugation move on a knot.Conjugation also has a meaning in group theory. Let be a group and let . Then, defines a homomorphism given byThis is a homomorphism becauseThe operation on given by is called conjugation by .Conjugation is an important construction in group theory. Conjugation defines a group action of a group on itself and this often yields useful information about the group. For example, this technique is how the Sylow Theorems are proven. More importantly, a normal subgroup of a group is a subgroup which is invariant under conjugation by any element. Normal groups are extremely important because they are the kernels of homomorphisms and it is possible to take the quotient of a group and one of its normal subgroups...

Consider strings, each oriented vertically from a lower to an upper "bar." If this is the least number of strings needed to make a closed braid representation of a link, is called the braid index. A general -braid is constructed by iteratively applying the () operator, which switches the lower endpoints of the th and th strings--keeping the upper endpoints fixed--with the th string brought above the th string. If the th string passes below the th string, it is denoted .The operations and on strings define a group known as the braid group or Artin braid group, denoted .Topological equivalence for different representations of a braid word and is guaranteed by the conditions(1)as first proved by E. Artin.Any -braid can be expressed as a braid word, e.g., is a braid word in the braid group . When the opposite ends of the braids are connected by nonintersecting lines, knots (or links) may formed that can be labeled by their corresponding..

Let be an entire function of finite order and the zeros of , listed with multiplicity, then the rank of is defined as the least positive integer such that(1)Then the canonical Weierstrass product is given by(2)and has degree . The genus of is then defined as , and the Hadamard factorization theory states that an entire function of finite order is also of finite genus , and(3)

The twistor equation states thatwhere the parentheses denote symmetrization, in a Lorentz transformation, primed spinors transform under the conjugate of the transformation for unprimed ones, Einstein summation is used throughout, and denotes the spinor connection, which is equivalent to the Levi-Civita connection on Minkowski space. The zero rest mass equation can be solved by twistor functions. The solution uses ideas from complex variable theory and cohomology.

The vector space generated by the columns of a matrix viewed as vectors. The column space of an matrix with real entries is a subspace generated by elements of , hence its dimension is at most . It is equal to the dimension of the row space of and is called the rank of .The matrix is associated with a linear transformation , defined byfor all vectors of , which we suppose written as column vectors. Note that is the product of an and an matrix, hence it is an matrix according to the rules of matrix multiplication. In this framework, the column vectors of are the vectors , where are the elements of the standard basis of . This shows that the column space of is the range of , and explains why the dimension of the latter is equal to the rank of .

A homogeneous space is a space with a transitive group action by a Lie group. Because a transitive group action implies that there is only one group orbit, is isomorphic to the quotient space where is the isotropy group . The choice of does not affect the isomorphism type of because all of the isotropy groups are conjugate.Many common spaces are homogeneous spaces, such as the hypersphere,(1)and the complex projective space(2)The real Grassmannian of -dimensional subspaces in is(3)The projection makes a principal bundle on with fiber . For example, is a bundle, i.e., a circle bundle, on the sphere. The subgroup(4)acts on the right, and does not affect the first column so is well-defined.

The vector space generated by the rows of a matrix viewed as vectors. The row space of a matrix with real entries is a subspace generated by elements of , hence its dimension is at most equal to . It is equal to the dimension of the column space of (as will be shown below), and is called the rank of .The row vectors of are the coefficients of the unknowns in the linear equation system(1)where(2)and is the zero vector in . Hence, the solutions span the orthogonal complement to the row space in , and(3)On the other hand, the space of solutions also coincides with the kernel (or null space) of the linear transformation , defined by(4)for all vectors of . And it also true that(5)where denotes the kernel and the image, since the nullity and the rank always add up to the dimension of the domain. It follows that the dimension of the row space is(6)which is equal to the dimension of the column space...

Given a principal bundle , with fiber a Lie group and base manifold , and a group representation of , say , then the associated vector bundle is(1)In particular, it is the quotient space where .This construction has many uses. For instance, any group representation of the orthogonal group gives rise to a bundle of tensors on a Riemannian manifold as the vector bundle associated to the frame bundle.For example, is the frame bundle on , where(2)writing the special orthogonal matrix with rows . It is a bundle with the action defined by(3)which preserves the map .The tangent bundle is the associated vector bundle with the standard group representation of on , given by pairs , with and . Two pairs and represent the same tangent vector iff there is a such that and .

A principal bundle is a special case of a fiber bundle where the fiber is a group . More specifically, is usually a Lie group. A principal bundle is a total space along with a surjective map to a base manifold . Any fiber is a space isomorphic to . More specifically, acts freely without fixed point on the fibers, and this makes a fiber into a homogeneous space. For example, in the case of a circle bundle (i.e., when ), the fibers are circles, which can be rotated, although no point in particular corresponds to the identity. Near every point, the fibers can be given the group structure of in the fibers over a neighborhood by choosing an element in each fiber to be the identity element. However, the fibers cannot be given a group structure globally, except in the case of a trivial bundle.An important principal bundle is the frame bundle on a Riemannian manifold. This bundle reflects the different ways to give an orthonormal basis for tangent vectors.Consider all..

Let be a locally compact Abelian group. Let be the group of all continuous homeomorphisms , in the compact open topology. Then is also a locally compact Abelian group, where the asterisk defines a contravariant equivalence of the category of locally compact Abelian groups with itself. The natural mapping , sending to , where , is an isomorphism and a homeomorphism. Under this equivalence, compact groups are sent to discrete groups and vice versa.

The fundamental group of an arcwise-connected set is the group formed by the sets of equivalence classes of the set of all loops, i.e., paths with initial and final points at a given basepoint , under the equivalence relation of homotopy. The identity element of this group is the set of all paths homotopic to the degenerate path consisting of the point . The fundamental groups of homeomorphic spaces are isomorphic. In fact, the fundamental group only depends on the homotopy type of . The fundamental group of a topological space was introduced by Poincaré (Munkres 1993, p. 1).The following is a table of the fundamental group for some common spaces , where denotes the fundamental group, is the first integral homology group, denotes the group direct product, denotes the free product, denotes the ring of integers, and is the cyclic group of order .space ()symbolcirclecomplex projective space00figure eightKlein bottle-torusreal projective..

Given a matrix equationthe normal equation is that which minimizes the sum of the square differences between the left and right sides:It is called a normal equation because is normal to the range of .Here, is a normal matrix.

The Casoratian of sequences , , ..., is defined by the determinantThe Casoratian is implemented in the Wolfram Language as Casoratian[y1, y2, ..., n].The solutions , , ..., of the linear difference equationfor , 1, ..., are linearly independent sequences iff their Casoratian is nonzero for (Zwillinger 1995).

A product of ANDs, denotedThe conjunctions of a Boolean algebra of subsets of cardinality are the functionswhere . For example, the 8 conjunctions of are , , , , , , , and (Comtet 1974, p. 186).A literal is considered a (degenerate) conjunction (Mendelson1997, p. 30).The Wolfram Language command Conjunction[expr, a1, a2, ...] gives the conjunction of expr over all choices of the Boolean variables .

The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and iff vertex is incident upon edge (Skiena 1990, p. 135). However, some authors define the incidence matrix to be the transpose of this, with a column for each vertex and a row for each edge. The physicist Kirchhoff (1847) was the first to define the incidence matrix.The incidence matrix of a graph (using the first definition) can be computed in the Wolfram Language using IncidenceMatrix[g]. Precomputed incidence matrices for a many named graphs are given in the Wolfram Language by GraphData[graph, "IncidenceMatrix"].The incidence matrix of a graph and adjacency matrix of its line graph are related by(1)where is the identity matrix (Skiena 1990, p. 136).For a -D polytope , the incidence matrix is defined by(2)The th row shows which s surround , and the th column shows which s bound . Incidence matrices are also..

The adjacency matrix, sometimes also called the connection matrix, of a simple labeled graph is a matrix with rows and columns labeled by graph vertices, with a 1 or 0 in position according to whether and are adjacent or not. For a simple graph with no self-loops, the adjacency matrix must have 0s on the diagonal. For an undirected graph, the adjacency matrix is symmetric.The illustration above shows adjacency matrices for particular labelings of the claw graph, cycle graph , and complete graph .Since the labels of a graph may be permuted without changing the underlying graph being represented, there are in general multiple possible adjacency matrices for a given graph. In particular, the number of distinct adjacency matrices for a graph with vertex count and automorphism group order is given bywhere is the number or permutations of vertex labels. The illustration above shows the possible adjacency matrices of the cycle graph .The adjacency..

The Zariski topology is a topology that is well-suited for the study of polynomial equations in algebraic geometry, since a Zariski topology has many fewer open sets than in the usual metric topology. In fact, the only closed sets are the algebraic sets, which are the zeros of polynomials.For example, in , the only nontrivial closed sets are finite collections of points. In , there are also the zeros of polynomials such as lines and cusps .The Zariski topology is not a T2-space. In fact, any two open sets must intersect, and cannot be disjoint. Also, the open sets are dense, in the Zariski topology as well as in the usual metric topology.Because there are fewer open sets than in the usual topology, it is more difficult for a function to be continuous in Zariski topology. For example, a continuous function must be a constant function. Conversely, when the range has the Zariski topology, it is easier for a function to be continuous. In particular, the polynomials..

A singular point of an algebraic curve is a point where the curve has "nasty" behavior such as a cusp or a point of self-intersection (when the underlying field is taken as the reals). More formally, a point on a curve is singular if the and partial derivatives of are both zero at the point . (If the field is not the reals or complex numbers, then the partial derivative is computed formally using the usual rules of calculus.)The following table gives some representative named curves that have various types of singular points at their origin.singularitycurveequationacnodecuspcusp curvecrunodecardioidquadruple pointquadrifoliumramphoid cuspkeratoid cusptacnodecapricornoidtriple pointtrifoliumConsider the following two examples. For the curvethe cusp at (0, 0) is a singular point. For the curve is a nonsingular point and this curve is nonsingular.Singular points are sometimes known as singularities,and vice versa...

Let be any functions of two variables . Then the expression(1)is called a Lagrange bracket (Lagrange 1808; Whittaker 1944, p. 298).The Lagrange brackets are anticommutative,(2)(Plummer 1960, p. 136).If are any functions of variables , then(3)where the summation on the right-hand side is taken over all pairs of variables in the set .But if the transformation from to is a contact transformation, then(4)giving(5)(6)(7)(8)Furthermore, these may be regarded as partial differential equations which must be satisfied by , considered as function of in order that the transformation from one set of variables to the other may be a contact transformation.Let be independent functions of the variables . Then the Poisson bracket is connected with the Lagrange bracket by(9)where is the Kronecker delta. But this is precisely the condition that the determinants formed from them are reciprocal (Whittaker 1944, p. 300; Plummer 1960,..

Let any finite or infinite set of points having no finite limit point be prescribed, and associate with each of its points a definite positive integer as its order. Then there exists an entire function which has zeros to the prescribed orders at precisely the prescribed points, and is otherwise different from zero. Moreover, this function can be represented as a product from which one can read off again the positions and orders of the zeros. Furthermore, if is one such function, thenis the most general function satisfying the conditions of the problem, where denotes an arbitrary entire function.This theorem is also sometimes simply known as Weierstrass's theorem. A spectacularexample is given by the Hadamard product.

Primorial primes are primes of the form , where is the primorial of . A coordinated search for such primes is being conducted on PrimeGrid. is prime for , 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, ... (OEIS A057704; Guy 1994, pp. 7-8; Caldwell 1995). These correspond to with , 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, ... (OEIS A006794). The largest known primorial primes as of Nov. 2015 are summarized in the following table (Caldwell).digitsdiscoverer6845Dec. 1992365851PrimeGrid (Dec. 20, 2010)476311PrimeGrid (Mar. 5, 2012) (also known as a Euclid number) is prime for , 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, ... (OEIS A014545; Guy 1994, Caldwell 1995, Mudge 1997). These correspond to with , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547,..

Let be the th prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by(1)The values of for , 2, ..., are 2, 6, 30, 210, 2310, 30030, 510510, ... (OEIS A002110).It is sometimes convenient to define the primorial for values other than just the primes, in which case it is taken to be given by the product of all primes less than or equal to , i.e.,(2)where is the prime counting function. For , 2, ..., the first few values of are 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, ... (OEIS A034386).The logarithm of is closely related to the Chebyshev function , and a trivial rearrangement of the limit(3)gives(4)(Ruiz 1997; Finch 2003, p. 14; Pruitt), where eis the usual base of the natural logarithm.

The product of primes(1)with the th prime, is called the primorial function, by analogy with the factorial function. Its logarithm is closely related to the Chebyshev function .The zeta-regularized product over allprimes is given by(2)(3)(Muñoz Garcia and Pérez-Marco 2003, 2008), answering the question posed by Soulé et al. (1992, p. 101). A derivation proceeds by algebraic manipulation of the prime zeta function and gives the more general results(4)and(5)(Muñoz Garcia and Pérez-Marco 2003).Mertens theorem states that(6)where is the Euler-Mascheroni constant, and a closely related result is given by(7)There are amazing infinite product formulas forprimes given by(8)(Ramanujan 1913-1914; Le Lionnais 1983, p. 46) and(9)(OEIS A082020; Ramanujan 1913-1914).More general formulas are given by(10)where is the Riemann zeta function and by the Euler product(11)Named prime..

At the age of 17, Bernard Mares proposed the definite integral (Borwein and Bailey2003, p. 26; Bailey et al. 2006)(1)(2)(OEIS A091473). Although this is within of ,(3)(OEIS A091494), it is not equal to it. Apparently, no closed-form solution is known for .Interestingly, the integral(4)(5)(Borwein et al. 2004, pp. 101-102) has a value fairly close to , but no other similar relationships seem to hold for other multipliers of the form or .The identity(6)can be expanded to yield(7)In fact,(8)where is a Borwein integral.

The Hadamard product is a representation for the Riemann zeta function as a product over its nontrivial zeros ,(1)where is the Euler-Mascheroni constant and is the Gamma function (Titchmarsh 1987, Voros 1987). The constant in the exponent is given by(2)(3)(OEIS A077142). Hadamard used the Weierstrass product theorem to derive this result. The plot above shows the convergence of the formula along the real axis using the first 100 (red), 500 (yellow), 1000 (green), and 2000 (blue) Riemann zeta function zeros.The product can also be stated in the alternate form(4)where is the xi-function and(5)(Havil 2003, p. 204).

The first few values of (known as a superfactorial) for , 2, ... are given by 1, 2, 12, 288, 34560, 24883200, ... (OEIS A000178).The first few positive integers that can be written as a product of factorials are1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, ... (OEIS A001013).The number of ways that is a product of smaller factorials, each greater than 1, for , 2, ... is given by 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, ... (OEIS A034876), and the numbers of products of factorials not exceeding are 1, 2, 4, 8, 15, 28, 49, 83, ... (OEIS A101976).The only known factorials which are products of factorials in an arithmeticprogression of three or more terms are(1)(2)(3)(Madachy 1979).The only solutions to(4)are(5)(6)(7)(Cucurezeanu and Enkers 1987).There are no nontrivial identities of the form(8)for with for for except(9)(10)(11)(12)(Madachy 1979; Guy 1994, p. 80). Here, "nontrivial" means that identities with , or equivalently are excluded, since..

Any discrete finite wavelet transform can be represented as a matrix, and such a wavelet matrix can be computed in steps, compared to for the Fourier matrix, where is the base-2 logarithm. A single wavelet matrix can be built using Haar functions.

For two polynomials and of degrees and , respectively, the Sylvester matrix is an matrix formed by filling the matrix beginning with the upper left corner with the coefficients of , then shifting down one row and one column to the right and filling in the coefficients starting there until they hit the right side. The process is then repeated for the coefficients of .The Sylvester matrix can be implemented in the WolframLanguage as: SylvesterMatrix1[poly1_, poly2_, var_] := Function[{coeffs1, coeffs2}, With[ {l1 = Length[coeffs1], l2 = Length[coeffs2]}, Join[ NestList[RotateRight, PadRight[coeffs1, l1 + l2 - 2], l2 - 2], NestList[RotateRight, PadRight[coeffs2, l1 + l2 - 2], l1 - 2] ] ] ][ Reverse[CoefficientList[poly1, var]], Reverse[CoefficientList[poly2, var]] ]For example, the Sylvester matrix for and isThe determinant of the Sylvester matrix of two polynomialsis the resultant of the polynomials.SylvesterMatrix is an (undocumented)..

The determination of a set of factors (divisors) of a given integer ("prime factorization"), polynomial ("polynomial factorization"), etc., which, when multiplied together, give the original number, polynomial, etc In many cases of interest (particularly prime factorization, factorization is unique, and so gives the "simplest" representation of a given quantity in terms of smaller parts.The terms "factorization" and "factoring" are used synonymously.The term "factorization" is occasionally misused, including by no less "authority" than The New York Times, where Fox (2006) wrote, "He was 88, which can be factored as 1, 2, 4, 8, 11, 22, 44, and 88." This usage is incorrect since the given numbers are indeed factors, but the collection of factors does not comprise a factorization...

A method of determining the maximum number of positive and negative real roots of a polynomial.For positive roots, start with the sign of the coefficient of the lowest (or highest) power. Count the number of sign changes as you proceed from the lowest to the highest power (ignoring powers which do not appear). Then is the maximum number of positive roots. Furthermore, the number of allowable roots is , , , .... For example, consider the polynomial(1)Since there are three sign changes, there are a maximumof three possible positive roots.For negative roots, starting with a polynomial , write a new polynomial with the signs of all odd powers reversed, while leaving the signs of the even powers unchanged. Then proceed as before to count the number of sign changes . Then is the maximum number of negative roots. For example, consider the polynomial(2)and compute the new polynomial(3)In this example, there are four sign changes, so there area maximum of..

Müntz's theorem is a generalization of the Weierstrass approximation theorem, which states that any continuous function on a closed and bounded interval can be uniformly approximated by polynomials involving constants and any infinite sequence of powers whose reciprocals diverge.In technical language, Müntz's theorem states that the Müntz space is dense in iff

For any ideal in a Dedekind ring, there is an ideal such that(1)where is a principal ideal, (i.e., an ideal of rank 1). Moreover, for a Dedekind ring with a finite ideal class group, there is a finite list of ideals such that this equation may be satisfied for some . The size of this list is known as the class number.Class numbers are usually studied in the context of the orders of number fields. If this order is maximal, then it is the ring of integers of the number field, in which case the class number is equal to the order of the class group of the number field; otherwise it is equal to the order of the Picard group of the nonmaximal order in question.When the class number of a ring of integers in a number field is 1, the ring corresponding to a given ideal has unique factorization and, in a sense, the class number is a measure of the failure of unique factorization in that ring.A finite series giving exactly the class number of a ring is known as a class number formula...

A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated structures and interconnections (Knopp 1996, pp. 98-99). Riemann surfaces are one way of representing multiple-valued functions; another is branch cuts. The above plot shows Riemann surfaces for solutions of the equationwith , 3, 4, and 5, where is the Lambert W-function (M. Trott).The Riemann surface of the function field is the set of nontrivial discrete valuations on . Here, the set corresponds to the ideals of the ring of integers of over . ( consists of the elements of that are roots of monic polynomials over .) Riemann surfaces provide a geometric visualization of functions elements and their analytic continuations.Schwarz proved at the end of nineteenth century that the automorphism group of a compact Riemann surface of genus is finite,..

A natural transformation between functors of categories and is said to be a natural isomorphism if each of the components is an isomorphism in .

An equalizer of a pair of maps in a category is a map such that 1. , where denotes composition. 2. For any other map with the same property, there is exactly one map such that i.e., one has the above commutative diagram. It can be shown that the equalizer is a monomorphism.Moreover, it is unique up to isomorphism.In the category of sets, the equalizer is given by thesetand by the inclusion map of the subset in .The same construction is valid in the categories of additive groups, rings, modules, and vector spaces. For these, the kernel of a morphism can be viewed, in a more abstract categorical setting, as the equalizer of and the zero map.The dual notion is the coequalizer.

Let be the set of all prime ideals of , and define . Then the Stone space of is the topological space defined on by postulating that the sets of the form are a subbase for the open sets.

The values of for which imaginary quadratic fields are uniquely factorable into factors of the form . Here, and are half-integers, except for and 2, in which case they are integers. The Heegner numbers therefore correspond to binary quadratic form discriminants which have class number equal to 1, except for Heegner numbers and , which correspond to and , respectively.The determination of these numbers is called Gauss's class number problem, and it is now known that there are only nine Heegner numbers: , , , , , , , , and (OEIS A003173), corresponding to discriminants , , , , , , , , and , respectively. This was proved by Heegner (1952)--although his proof was not accepted as complete at the time (Meyer 1970)--and subsequently established by Stark (1967).Heilbronn and Linfoot (1934) showed that if a larger existed, it must be . Heegner (1952) published a proof that only nine such numbers exist, but his proof was not accepted as complete at the time. Subsequent..

Let be a field, and a -algebra. Elements , ..., are algebraically independent over if the natural surjection is an isomorphism. In other words, there are no polynomial relations with coefficients in .

If is a root of a nonzero polynomial equation(1)where the s are integers (or equivalently, rational numbers) and satisfies no similar equation of degree , then is said to be an algebraic number of degree .A number that is not algebraic is said to be transcendental. If is an algebraic number and , then it is called an algebraic integer.In general, algebraic numbers are complex, but they may also be real. An example of a complex algebraic number is , and an example of a real algebraic number is , both of which are of degree 2.The set of algebraic numbers is denoted (Wolfram Language), or sometimes (Nesterenko 1999), and is implemented in the Wolfram Language as Algebraics.A number can then be tested to see if it is algebraic in the Wolfram Language using the command Element[x, Algebraics]. Algebraic numbers are represented in the Wolfram Language as indexed polynomial roots by the symbol Root[f, n], where is a number from 1 to the degree of the polynomial..

If is a root of the polynomial equationwhere the s are integers and satisfies no similar equation of degree , then is called an algebraic integer of degree . An algebraic integer is a special case of an algebraic number (for which the leading coefficient need not equal 1). Radical integers are a subring of the algebraic integers.A sum or product of algebraic integers is again an algebraic integer. However, Abel's impossibility theorem shows that there are algebraic integers of degree which are not expressible in terms of addition, subtraction, multiplication, division, and root extraction (the elementary operations) on rational numbers. In fact, if elementary operations are allowed on real numbers only, then there are real numbers which are algebraic integers of degree 3 that cannot be so expressed.The Gaussian integers are algebraic integers of , since are roots of..

Given a general quadratic curve(1)the quantity is known as the discriminant, where(2)and is invariant under rotation. Using the coefficients from quadratic equations for a rotation by an angle ,(3)(4)(5)(6)(7)(8)Now let(9)(10)(11)(12)and use(13)(14)to rewrite the primed variables(15)(16)(17)(18)From (16) and (18), it follows that(19)Combining with (17) yields, for an arbitrary (20)(21)(22)(23)which is therefore invariant under rotation. This invariant therefore provides a useful shortcut to determining the shape represented by a quadratic curve. Choosing to make (see quadratic equation), the curve takes on the form(24)Completing the square and defining new variablesgives(25)Without loss of generality, take the sign of to be positive. The discriminant is(26)Now, if , then and both have the same sign, and the equation has the general form of an ellipse (if and are positive). If , then and have opposite signs, and the equation..

Check the price

for your project

for your project

we accept

Money back

guarantee

guarantee

Price calculator

We've got the best prices, check out yourself!