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Cauchy Matrix
In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an ''m''×''n'' matrix with elements ''a''''ij'' in the form : a_=;\quad x_i-y_j\neq 0,\quad 1 \le i \le m,\quad 1 \le j \le n where x_i and y_j are elements of a field \mathcal, and (x_i) and (y_j) are injective sequences (they contain ''distinct'' elements). The Hilbert matrix is a special case of the Cauchy matrix, where :x_i-y_j = i+j-1. \; Every submatrix of a Cauchy matrix is itself a Cauchy matrix. Cauchy determinants The determinant of a Cauchy matrix is clearly a rational fraction in the parameters (x_i) and (y_j). If the sequences were not injective, the determinant would vanish, and tends to infinity if some x_i tends to y_j. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles: The determinant of a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as : \det \mathbf= (Schechter 1959, eqn 4; Cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Displacement Rank
Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path covered to reach the final position is irrelevant. **Particle displacement, a measurement of distance of the movement of a particle in a medium as it transmits a wave (represented in mathematics by the lower-case Greek letter ξ) **Displacement field (mechanics), an assignment of displacement vectors for all points in a body that is displaced from one state to another **Electric displacement field, as appears in Maxwell's equations *Wien's displacement law, a relation concerning the spectral distribution of blackbody radiation *Angular displacement, a change in orientation of a rigid body, the amount of rotation about a fixed axis. Engineering *Engine displacement, the total volume of air/fuel mixture an engine can draw in during one compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fay's Trisecant Identity
In algebraic geometry, Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by . Fay's identity holds for theta functions of Jacobians of curves, but not for theta functions of general abelian varieties. The name "trisecant identity" refers to the geometric interpretation given by , who used it to show that the Kummer variety of a genus ''g'' Riemann surface, given by the image of the map from the Jacobian to projective space of dimension 2''g'' – 1 induced by theta functions of order 2, has a 4-dimensional space of trisecants. Statement Suppose that *''C'' is a compact Riemann surface *''g'' is the genus of ''C'' *θ is the Riemann theta function of ''C'', a function from C''g'' to C *''E'' is a prime form In algebraic geometry, the Schottky–Klein prime form ''E''(''x'',''y'') of a compact Riemann surface ''X'' depends on two elements ''x'' and ''y'' of ''X'', and vanishes if and only if ''x'' = ''y''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toeplitz Matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: :\qquad\begin a & b & c & d & e \\ f & a & b & c & d \\ g & f & a & b & c \\ h & g & f & a & b \\ i & h & g & f & a \end. Any ''n'' × ''n'' matrix ''A'' of the form :A = \begin a_0 & a_ & a_ & \cdots & \cdots & a_ \\ a_1 & a_0 & a_ & \ddots & & \vdots \\ a_2 & a_1 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & a_ & a_ \\ \vdots & & \ddots & a_1 & a_0 & a_ \\ a_ & \cdots & \cdots & a_2 & a_1 & a_0 \end is a Toeplitz matrix. If the ''i'', ''j'' element of ''A'' is denoted ''A''''i'', ''j'' then we have :A_ = A_ = a_. A Toeplitz matrix is not necessarily square. Solving a Toeplitz system A matrix equation of the form :Ax = b is called a Toeplitz system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LU Factorization
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish mathematician Tadeusz Banachiewicz in 1938. Definitions Let ''A'' be a square matrix. An LU factorization refers to the factorization of ''A'', with proper row and/or column orderings or permutations, into two factors – a lower triangular matrix ''L'' and an upper triangular matrix ''U'': : A = LU. In the lower triangular matrix all elements above the diagonal are zero, in the upper triangular matrix, all the el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pivot Element
The pivot or pivot element is the element of a matrix, or an array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called pivoting. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error. It is often used for verifying row echelon form. Pivoting might be thought of as swapping or sorting rows or columns in a matrix, and thus it can be represented as multiplication by permutation matrices. However, algorithms rarely move the matrix elements because this would cost too much time; instead, they just keep track of the permutations. Overall, pivoting adds more operations to the computational cost of an algorithm. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Multipole Method
__NOTOC__ The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the ''n''-body problem. It does this by expanding the system Green's function using a multipole expansion, which allows one to group sources that lie close together and treat them as if they are a single source. The FMM has also been applied in accelerating the iterative solver in the method of moments (MOM) as applied to computational electromagnetics problems. The FMM was first introduced in this manner by Leslie Greengard and Vladimir Rokhlin Jr. and is based on the multipole expansion of the vector Helmholtz equation. By treating the interactions between far-away basis functions using the FMM, the corresponding matrix elements do not need to be explicitly stored, resulting in a significant reduction in required memory. If the FMM is then applied in a hierarchical manner, it can improve the complexity of matrix-vector products in an ite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FLOPS
In computing, floating point operations per second (FLOPS, flops or flop/s) is a measure of computer performance, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate measure than measuring instructions per second. Floating-point arithmetic Floating-point arithmetic is needed for very large or very small real numbers, or computations that require a large dynamic range. Floating-point representation is similar to scientific notation, except everything is carried out in base two, rather than base ten. The encoding scheme stores the sign, the exponent (in base two for Cray and VAX, base two or ten for IEEE floating point formats, and base 16 for IBM Floating Point Architecture) and the significand (number after the radix point). While several similar formats are in use, the most common is ANSI/IEEE Std. 754-1985. This standard defines the format for 32-bit numbers called ''single precision'', as well as 6 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Displacement Structure
Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path covered to reach the final position is irrelevant. **Particle displacement, a measurement of distance of the movement of a particle in a medium as it transmits a wave (represented in mathematics by the lower-case Greek letter ξ) **Displacement field (mechanics), an assignment of displacement vectors for all points in a body that is displaced from one state to another **Electric displacement field, as appears in Maxwell's equations *Wien's displacement law, a relation concerning the spectral distribution of blackbody radiation *Angular displacement, a change in orientation of a rigid body, the amount of rotation about a fixed axis. Engineering *Engine displacement, the total volume of air/fuel mixture an engine can draw in during one compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Polynomials
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + 1 nodes \, which must all be distinct, x_j \neq x_m for indices j \neq m, the Lagrange basis for polynomials of degre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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