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Schwartz–Zippel Lemma
In mathematics, the Schwartz–Zippel lemma (also called the DeMillo–Lipton–Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing. Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial that ignores all its variables and always returns zero. The lemma states that evaluating a nonzero polynomial on inputs chosen randomly from a large-enough set is likely to find an input that produces a nonzero output. it was discovered independently by Jack Schwartz, Richard Zippel, and Richard DeMillo and Richard J. Lipton, although DeMillo and Lipton's version was shown a year prior to Schwartz and Zippel's result. The finite field version of this bound was proved by Øystein Ore in 1922.Ö. Ore, Über höhere Kongruenzen. Norsk Mat. Forenings Skrifter Ser. I (1922), no. 7, 15 pages. Statement and proof of the lemma Theorem 1 (Schwartz, Zippel). ''Let'' : P\in R _1,x_2,\ldots,x_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Identity Testing
In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative int ... are identical. More formally, a PIT algorithm is given an arithmetic circuit that computes a polynomial ''p'' in a Field (mathematics), field, and decides whether ''p'' is the zero polynomial. Determining the computational complexity theory, computational complexity required for polynomial identity testing, in particular Randomized algorithm#Derandomization, finding deterministic algorithms for PIT, is one of the most important open problems in algebraic complexity theory. Description The question "Does (x+y)(x-y) equal x^2 - y^2 \, ?" is a question about whether two polynomials are identical. As with any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manindra Agrawal
Manindra Agrawal (born 20 May 1966) is an Indian computer scientist and director of Indian Institute of Technology, Kanpur. He is also a professor at the Department of Computer Science and Engineering at the Indian Institute of Technology, Kanpur. He was the recipient of the first Infosys Prize for Mathematics, the Godel Prize in 2006; and the Shanti Swarup Bhatnagar Award in Mathematical Sciences in 2003. He has been honoured with Padma Shri, India's 4th highest civilian award, in 2013. Career He created the AKS primality test with Neeraj Kayal and Nitin Saxena, for which he and his co-authors won the 2006 Fulkerson Prize, and the 2006 Gödel Prize. He was also awarded 2002 Clay Research Award for this work. The test is the first unconditional deterministic algorithm to test an ''n''-digit number for primality in a time that has been proven to be polynomial in ''n''. In September 2008, Agrawal was chosen for the first Infosys Mathematics Prize for outstanding contrib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmonds Matrix
In graph theory, the Edmonds matrix A of a balanced bipartite graph G = (U, V, E) with sets of vertices U = \ and V = \ is defined by :A_ = \left\{ \begin{array}{ll} x_{ij} & (u_i, v_j) \in E \\ 0 & (u_i, v_j) \notin E \end{array}\right. where the ''xij'' are indeterminate (variable), indeterminates. One application of the Edmonds matrix of a bipartite graph is that the graph admits a perfect matching if and only if the polynomial det(''A'') in the ''xij'' is not identically zero. Furthermore, the number of perfect matchings is equal to the number of monomials in the polynomial det(''A''), and is also equal to the permanent (mathematics), permanent of A. In addition, the rank (linear algebra), rank of A is equal to the maximum matching size of G. The Edmonds matrix is named after Jack Edmonds. The Tutte matrix is a generalisation to non-bipartite graphs. References * * Algebraic graph theory {{graph-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. For example, the 3x4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2x3 block, the top-right 2x1 block, the bottom-left 1x3 block, and the bottom-right 1x1 block. : \left \begin a_ & a_ & a_ & b_ \\ a_ & a_ & a_ & b_ \\ \hline c_ & c_ & c_ & d \end \right Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned. This notion can be made more precise for an n by m matrix M by partitioning n into a collection \text, and then partitioning m into a collection \text. The original m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pfaffian
In mathematics, the determinant of an ''m''-by-''m'' skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depends on ''m''. When ''m'' is odd, the polynomial is zero, and when ''m'' is even, it is a nonzero polynomial of degree ''m''/2, and is unique up to multiplication by ±1. The convention on skew-symmetric tridiagonal matrices, given below in the examples, then determines one specific polynomial, called the Pfaffian polynomial. The value of this polynomial, when applied to the entries of a skew-symmetric matrix, is called the Pfaffian of that matrix. The term Pfaffian was introduced by , who indirectly named them after Johann Friedrich Pfaff. Explicitly, for a skew-symmetric matrix A, : \operatorname(A)^2=\det(A), which was first proved by , who cites Jacobi for introducing these polynomials in work on Pfaffian systems of differential equations. Cayley obtains this relation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew-symmetric Matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ denotes the entry in the i-th row and j-th column, then the skew-symmetric condition is equivalent to Example The matrix A = \begin 0 & 2 & -45 \\ -2 & 0 & -4 \\ 45 & 4 & 0 \end is skew-symmetric because A^\textsf = \begin 0 & -2 & 45 \\ 2 & 0 & 4 \\ -45 & -4 & 0 \end = -A . Properties Throughout, we assume that all matrix entries belong to a field \mathbb whose characteristic is not equal to 2. That is, we assume that , where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. * The sum of two skew-symmetric matrices is skew-symmetric. * A scalar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte Matrix
In graph theory, the Tutte matrix ''A'' of a graph ''G'' = (''V'', ''E'') is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is V = \ then the Tutte matrix is an ''n''-by-''n'' matrix ''A'' with entries : A_ = \begin x_\;\;\mbox\;(i,j) \in E \mbox ij\\ 0\;\;\;\;\mbox \end where the ''x''''ij'' are indeterminates. The determinant of this skew-symmetric matrix is then a polynomial (in the variables ''xij'', ''i < j'' ): this coincides with the square of the pfaffian of the matrix ''A'' and is non-zero (as a polynomial) a perfect matching exists. (This polynomial is not the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Matching
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph with edges and vertices , a perfect matching in is a subset of , such that every vertex in is adjacent to exactly one edge in . The adjacency matrix of a perfect matching is a symmetric permutation matrix. A perfect matching is also called a 1-factor; see Graph factorization for an explanation of this term. In some literature, the term complete matching is used. Every perfect matching is a maximum-cardinality matching, but the opposite is not true. For example, consider the following graphs:Alan Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985, Chapter 5. : In graph (b) there is a perfect matching (of size 3) since all 6 vertices are matched; in graphs (a) and (c) there is a maximum-cardinality matching (of size 2) which is not perfect, since some vertices are unmatched. A perfect matching is also a minimum-size edge cov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
