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Vandermonde Polynomial
In algebra, the Vandermonde polynomial of an ordered set of ''n'' variables X_1,\dots, X_n, named after Alexandre-Théophile Vandermonde, is the polynomial: :V_n = \prod_ (X_j-X_i). (Some sources use the opposite order (X_i-X_j), which changes the sign \binom times: thus in some dimensions the two formulas agree in sign, while in others they have opposite signs.) It is also called the Vandermonde determinant, as it is the determinant of the Vandermonde matrix. The value depends on the order of the terms: it is an alternating polynomial, not a symmetric polynomial. Alternating The defining property of the Vandermonde polynomial is that it is ''alternating'' in the entries, meaning that permuting the X_i by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the polynomial – in fact, it is the basic alternating polynomial, as will be made precise below. It thus depends on the order, and is zero if two entries are equal â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Polynomials
In algebra, an alternating polynomial is a polynomial f(x_1,\dots,x_n) such that if one switches any two of the variables, the polynomial changes sign: :f(x_1,\dots,x_j,\dots,x_i,\dots,x_n) = -f(x_1,\dots,x_i,\dots,x_j,\dots,x_n). Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation: :f\left(x_,\dots,x_\right)= \mathrm(\sigma) f(x_1,\dots,x_n). More generally, a polynomial f(x_1,\dots,x_n,y_1,\dots,y_t) is said to be ''alternating in'' x_1,\dots,x_n if it changes sign if one switches any two of the x_i, leaving the y_j fixed. Relation to symmetric polynomials Products of symmetric and alternating polynomials (in the same variables x_1,\dots,x_n) behave thus: * the product of two symmetric polynomials is symmetric, * the product of a symmetric polynomial and an alternating polynomial is alternating, and * the product of two alternating polynomials is symmetric. This is exactly the addition table for parity, with "symmetric" co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Capelli Polynomial
Capelli is an Italian surname meaning hair (plural). Notable people with the surname include: *Adler Capelli (born 1973), Italian former track cyclist *Alfredo Capelli (1855–1910), Italian mathematician *Andy Capelli, fictional character on ''General Hospital'' * Ather Capelli (1902–1944), Italian journalist * Camillo Capelli, also called Camillo Mantovano, (active 16th century), Italian painter of the Renaissance period * Claudio Capelli (born 1986), Swiss artistic gymnast * Daniele Capelli (born 1986), Italian footballer *Ermanno Capelli (born 1985), Italian professional road racing cyclist *Francis Alphonse Capelli the real name of Frank A. Capell (1907–1980), American author and essayist *Francesco Capelli (active c. 1568), Italian painter * Ivan Capelli (born 1963), Italian former Formula One driver *Javier Capelli (born 1985), Argentine footballer *Joseph Capelli, fictional character in Resistance, and main protagonist in ''Resistance 3 '' *Monia Capelli (born 1969), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Unitary Group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on \mathbb^n. It is itself a subgroup of the general linear group, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics. The groups are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with n qubits and thus 2^n basis states. (Alternatively, the more general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Representation
In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of ''V'' to the zero vector. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations. Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
