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Bunching (mathematics)
In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. Preliminary definitions ''a''-mean For any real vector :a=(a_1,\dots,a_n) define the "''a''-mean" 'a''of positive real numbers ''x''1, ..., ''x''''n'' by : \frac\sum_\sigma x_^\cdots x_^, where the sum extends over all permutations σ of . When the elements of ''a'' are nonnegative integers, the ''a''-mean can be equivalently defined via the monomial symmetric polynomial m_a(x_1,\dots,x_n) as : = \frac m_a(x_1,\dots,x_n), where ℓ is the number of distinct elements in ''a'', and ''k''1, ..., ''k''ℓ are their multiplicities. Notice that the ''a''-mean as defined above only has the usual properties of a mean (e.g., if the mean of equal numbers is equal to them) if a_1+\cdots+a_n=1. In the general case, one can consider instead , which is called a Muirhead mean.Bullen, P. S. Handbook of means a ... [...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] |
Doubly Stochastic Matrix
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_=1, Thus, a doubly stochastic matrix is both left stochastic and right stochastic. Indeed, any matrix that is both left and right stochastic must be square: if every row sums to one then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal. Birkhoff polytope The class of n\times n doubly stochastic matrices is a convex polytope known as the Birkhoff polytope B_n. Using the matrix entries as Cartesian coordinates, it lies in an (n-1)^2-dimensional affine subspace of n^2-dimensional Euclidean space defined by 2n-1 independent linear constraints specifying that the row and column sums all equal one. (There are 2n-1 constraints rather than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gian-Carlo Rota
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, probability theory, and phenomenology. Early life and education Rota was born in Vigevano, Italy. His father, Giovanni, an architect and prominent antifascist, was the brother of the mathematician Rosetta, who was the wife of the writer Ennio Flaiano. Gian-Carlo's family left Italy when he was 13 years old, initially going to Switzerland. Rota attended the Colegio Americano de Quito in Ecuador, and graduated with an A.B. in mathematics from Princeton University in 1953 after completing a senior thesis, titled "On the solubility of linear equations in topological vector spaces", under the supervision of William Feller. He then pursued graduate studies at Yale University, where he received a Ph.D. in mathematics in 1956 after completing a do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial Symmetric Polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has . Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every ''symmetric'' polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial. Symmetric polynomials also form an interesting structure by themse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubly Stochastic Matrix
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_=1, Thus, a doubly stochastic matrix is both left stochastic and right stochastic. Indeed, any matrix that is both left and right stochastic must be square: if every row sums to one then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal. Birkhoff polytope The class of n\times n doubly stochastic matrices is a convex polytope known as the Birkhoff polytope B_n. Using the matrix entries as Cartesian coordinates, it lies in an (n-1)^2-dimensional affine subspace of n^2-dimensional Euclidean space defined by 2n-1 independent linear constraints specifying that the row and column sums all equal one. (There are 2n-1 constraints rather than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality Of Arithmetic And Geometric Means
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number). The simplest non-trivial case – i.e., with more than one variable – for two non-negative numbers and , is the statement that :\frac2 \ge \sqrt with equality if and only if . This case can be seen from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the elementary case of the binomial formula: :\begin 0 & \le (x-y)^2 \\ & = x^2-2xy+y^2 \\ & = x^2+2xy+y^2 - 4xy \\ & = (x+y)^2 - 4xy. \end Hence , with equality precisely when , i.e. . The AM–GM inequality then follows from taking the positive square root of both sides and then dividing both ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomials
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a monomial, being equal to the empty product and to x^0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power x^n of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form x^a y^b z^c with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1). # A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a specia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Majorization
In mathematics, majorization is a preorder on vectors of real numbers. Let ^_,\ i=1,\,\ldots,\,n denote the i-th largest element of the vector \mathbf\in\mathbb^n. Given \mathbf,\ \mathbf \in \mathbb^n, we say that \mathbf weakly majorizes (or dominates) \mathbf from below (or equivalently, we say that \mathbf is weakly majorized (or dominated) by \mathbf from below) denoted as \mathbf \succ_w \mathbf if \sum_^k x_^ \geq \sum_^k y_^ for all k=1,\,\dots,\,d. If in addition \sum_^d x_i^ = \sum_^d y_i^, we say that \mathbf majorizes (or dominates) \mathbf , written as \mathbf \succ \mathbf , or equivalently, we say that \mathbf is majorized (or dominated) by \mathbf. The order of the entries of the vectors \mathbf or \mathbf does not affect the majorization, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0). As a consequence, majorization is not a partial order, since \mathbf \succ \mathbf and \mathbf \succ \mathbf do not imply \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff-von Neumann Theorem
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_=1, Thus, a doubly stochastic matrix is both left stochastic and right stochastic. Indeed, any matrix that is both left and right stochastic must be square: if every row sums to one then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal. Birkhoff polytope The class of n\times n doubly stochastic matrices is a convex polytope known as the Birkhoff polytope B_n. Using the matrix entries as Cartesian coordinates, it lies in an (n-1)^2-dimensional affine subspace of n^2-dimensional Euclidean space defined by 2n-1 independent linear constraints specifying that the row and column sums all equal one. (There are 2n-1 constraints rather ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Matrix
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when used to multiply another matrix, say , results in permuting the rows (when pre-multiplying, to form ) or columns (when post-multiplying, to form ) of the matrix . Definition Given a permutation of ''m'' elements, :\pi : \lbrace 1, \ldots, m \rbrace \to \lbrace 1, \ldots, m \rbrace represented in two-line form by :\begin 1 & 2 & \cdots & m \\ \pi(1) & \pi(2) & \cdots & \pi(m) \end, there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the ''m'' × ''m'' identity matrix, , either permute the columns or permute the rows, according to . Both methods of defining permutation matrices appear in the literature and the properties expressed in one representation can be easily converted to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Matrix
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and types of stochastic matrices: :A right stochastic matrix is a real square matrix, with each row summing to 1. :A left stochastic matrix is a real square matrix, with each column summing to 1. :A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. In the same vein, one may define a stochastic vector (also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations. As a simple example, the harmonic mean of 1, 4, and 4 is : \left(\frac\right)^ = \frac = \frac = 2\,. Definition The harmonic mean ''H'' of the positive real numbers x_1, x_2, \ldots, x_n is defined to be :H = \frac = \frac = \left(\frac\right)^. The third formula in the above equation expresses the harmonic mean as the reciprocal of the arithmetic mean of the reciprocals. From the following formula: :H = \frac. it is more apparent that the harmonic mean is related to the arithmetic and geometric means. It is the reciprocal dual of the arithmetic mean for positive inputs: :1/H(1/x_1 \ldots 1/x_n) = A(x_1 \ldots x_n) The harmonic mean is a Schur-con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
