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Majorization
In mathematics, majorization is a preorder on vector space, vectors of real numbers. For two such vectors, \mathbf,\ \mathbf \in \mathbb^n, we say that \mathbf weakly majorizes (or dominates) \mathbf from below, commonly denoted \mathbf \succ_w \mathbf, when : \sum_^k x_i^ \geq \sum_^k y_i^ for all k=1,\,\dots,\,n, where x_i^ denotes ith largest entry of x. If \mathbf, \mathbf further satisfy \sum_^n x_i = \sum_^n y_i, we say that \mathbf majorizes (or dominates) \mathbf , commonly denoted \mathbf \succ \mathbf. Both weak majorization and majorization are partially ordered set, partial orders for vectors whose entries are non-decreasing, but only a preorder for general vectors, since majorization is agnostic to the ordering of the entries in vectors, e.g., the statement (1,2)\prec (0,3) is simply equivalent to (2,1)\prec (3,0). Specifically, \mathbf \succ \mathbf \wedge \mathbf \succ \mathbf if and only if \mathbf, \mathbf are permutations of each other. Similarly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Karamata's Inequality
In mathematics, Karamata's inequality, named after Jovan Karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality, and generalizes in turn to the concept of Schur-convex functions. Statement of the inequality Let be an interval of the real line and let denote a real-valued, convex function defined on . If and are numbers in such that majorizes , then Here majorization means that and satisfies and we have the inequalities and the equality If is a strictly convex function, then the inequality () holds with equality if and only if we have for all . Weak majorization case x \preceq_w y if and only if \sum g\left(x_i\right) \leq \sum g\left(y_i\right) for any continuous increasing convex function g: \R \to \R. Remarks If the convex function is non-decreasing, then the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Schur-convex Function
In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f: \mathbb^d\rightarrow \mathbb that for all x,y\in \mathbb^d such that x is majorized by y, one has that f(x)\le f(y). Named after Issai Schur, Schur-convex functions are used in the study of majorization. A function ''f'' is 'Schur-concave' if its negative, −''f'', is Schur-convex. Properties Every function that is convex and symmetric (under permutations of the arguments) is also Schur-convex. Every Schur-convex function is symmetric, but not necessarily convex. If f is (strictly) Schur-convex and g is (strictly) monotonically increasing, then g\circ f is (strictly) Schur-convex. If g is a convex function defined on a real interval, then \sum_^n g(x_i) is Schur-convex. Schur–Ostrowski criterion If ''f'' is symmetric and all first partial derivatives exist, then ''f'' is Schur-convex if and only if : (x_i - x_j)\left(\frac - \frac\righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ingram Olkin
Ingram Olkin (July 23, 1924 – April 28, 2016) was a professor emeritus and chair of statistics and education at Stanford University and the Stanford Graduate School of Education. He is known for developing statistical analysis for evaluating policies, particularly in education, and for his contributions to meta-analysis, statistics education, multivariate analysis, and majorization theory. Biography Olkin was born in 1924 in Waterbury, Connecticut. He received a B.S. in mathematics at the City College of New York, an M.A. from Columbia University, and his Ph.D. from the University of North Carolina. Olkin also studied with Harold Hotelling. Olkin's advisor was S. N. Roy and his Ph.D. thesis was "On distribution problems in multivariate analysis" submitted in 1951. Olkin died from complications of colorectal cancer at his home in Palo Alto, California on April 28, 2016, aged 91. Honors and awards Olkin was awarded the fourth biennial Elizabeth Scott Award in 1998 from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dominance Order
In discrete mathematics, dominance order (synonyms: dominance ordering, majorization order, natural ordering) is a partial order on the set of partition (number theory), partitions of a positive integer ''n'' that plays an important role in algebraic combinatorics and representation theory, especially in the context of symmetric functions and representation theory of the symmetric group. Definition If ''p'' = (''p''1,''p''2,...) and ''q'' = (''q''1,''q''2,...) are partitions of ''n'', with the parts arranged in the weakly decreasing order, then ''p'' precedes ''q'' in the dominance order if for any ''k'' ≥ 1, the sum of the ''k'' largest parts of ''p'' is less than or equal to the sum of the ''k'' largest parts of ''q'': : p\trianglelefteq q \text p_1+\cdots+p_k \leq q_1+\cdots+q_k \text k\geq 1. In this definition, partitions are extended by appending zero parts at the end as necessary. Properties of the dominance ordering * Among the partitions of ''n'', (1,...,1) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution Support (measure theory), supported on the real numbers, discrete or "mixed" as well as Continuous variable, continuous, is uniquely identified by a right-continuous Monotonic function, monotone increasing function (a càdlàg function) F \colon \mathbb R \rightarrow [0,1] satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Leximin Order
In mathematics, leximin order is a total preorder on finite-dimensional vectors. A more accurate but less common term is leximin preorder. The leximin order is particularly important in social choice theory and fair division. Definition A vector x = (''x''1, ..., ''x''''n'') is ''leximin-larger'' than a vector y = (''y''1, ..., ''y''''n'') if one of the following holds: * The smallest element of x is larger than the smallest element of y; * The smallest elements of both vectors are equal, and the second-smallest element of x is larger than the second-smallest element of y; * ... * The ''k'' smallest elements of both vectors are equal, and the (''k''+1)-smallest element of x is larger than the (''k''+1)-smallest element of y. Examples The vector (3,5,3) is leximin-larger than (4,2,4), since the smallest element in the former is 3 and in the latter is 2. The vector (4,2,4) is leximin-larger than (5,3,2), since the smallest elements in both are 2, but the second-smallest elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Integer Number
An integer is the number zero ( 0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number ( −1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Muirhead's Inequality
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 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hermitian Operator
In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for all x, y ∊ ''V''. If ''V'' is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ''A'' is a Hermitian matrix, i.e., equal to its conjugate transpose ''A''. By the finite-dimensional spectral theorem, ''V'' has an orthonormal basis such that the matrix of ''A'' relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Spectrum Of A Matrix
In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if T\colon V \to V is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars \lambda such that T-\lambda I is not invertible. The determinant of the matrix equals the product of its eigenvalues. Similarly, the trace of the matrix equals the sum of its eigenvalues. From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this quantity). In many applications, such as PageRank, one is interested in the dominant eigenvalue, i.e. that which is largest in absolute value. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix. Definition Let ''V'' be a finite-dimensional vector space over some field ''K'' and suppose ''T'' : ''V'' → ''V'' is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quantum Information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term. It is an interdisciplinary field that involves quantum mechanics, computer science, information theory, philosophy and cryptography among other fields. Its study is also relevant to disciplines such as cognitive science, psychology and neuroscience. Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the scientific method. In quantum mechanics, due to the uncertainty principle, non-commuting observables cannot be precisely measured simultaneously, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |