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Free Probability
Free probability is a mathematics, mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of statistical independence, independence, and it is connected with free products. This theory was initiated by Dan Voiculescu (mathematician), Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras. Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra of a locally compact group, group algebra, which is a type II1 von Neumann algebra#Factors, factor. The isomorphism problem asks whether these are isomorphic for different numbers of generators. It is not even known if any two free group factors are isomorphic. This is similar to Tarski's free group problem, which asks whether two different non-abelian finitely generated free groups have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Large Deviations
In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insurance mathematics, namely ruin theory with Cramér and Lundberg. A unified formalization of large deviation theory was developed in 1966, in a paper by Varadhan. Large deviations theory formalizes the heuristic ideas of ''concentration of measures'' and widely generalizes the notion of convergence of probability measures. Roughly speaking, large deviations theory concerns itself with the exponential decline of the probability measures of certain kinds of extreme or ''tail'' events. Introductory examples An elementary example Consider a sequence of independent tosses of a fair coin. The possible outcomes could be heads or tails. Let us denote the possible outcome of the i-th trial by where we encode head as 1 and tail as 0. Now let ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Circular Law
In probability theory, more specifically the study of random matrices, the circular law concerns the distribution of eigenvalues of an n \times n random matrix with independent and identically distributed entries in the limit n \to \infty. It asserts that for any sequence of random matrices whose entries are independent and identically distributed random variables, all with mean zero and variance equal to , the limiting spectral distribution is the uniform distribution over the unit disc. Ginibre ensembles The complex Ginibre ensemble is defined as X = \frac Z_1+\frac Z_2 for Z_1, Z_2 \in \R^ , with all their entries sampled IID from the standard normal distribution \mathcal N (0, 1) . The real Ginibre ensemble is defined as X = Z_1. Eigenvalues The eigenvalues of X are distributed according to\rho_n\left(z_1, \ldots, z_n\right)=\frac \exp \left(-\sum_^n\left, z_k\^2\right) \prod_\left, z_j-z_k\^2 Global law Let (X_n)_^\infty be a sequence sampled from the complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Wigner Semicircle Distribution
The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution defined on the domain minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle, i.e. a semi-ellipse, centered at (0, 0): :f(x)=\sqrt\, for −''R'' ≤ ''x'' ≤ ''R'', and ''f''(''x'') = 0 if '', x, '' > ''R''. The parameter R is commonly referred to as the "radius" parameter of the distribution. The distribution arises as the limiting distribution of the eigenvalues of many random symmetric matrices, that is, as the dimensions of the random matrix approach infinity. The distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise. General properties Because of symmetry, all of the odd-order moments of the Wigner distribution are zero. For positive integers , the -th moment of this distribution is :\frac\left(\right)^ \, In the typical special case that , this sequence coincides with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Random Matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the study of properties of random matrices, often as they become large. RMT provides techniques like mean-field theory, diagrammatic methods, the cavity method, or the replica method to compute quantities like traces, spectral densities, or scalar products between eigenvectors. Many physical phenomena, such as the spectrum of nuclei of heavy atoms, the thermal conductivity of a lattice, or the emergence of quantum chaos, can be modeled mathematically as problems concerning large, random matrices. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into Empty set, non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a Set (mathematics), set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., the subsets are nonempty mutually disjoint sets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union (set theory), union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Noncrossing Partition
In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory of free probability. The number of noncrossing partitions of a set of ''n'' elements is the ''n''th Catalan number. The number of noncrossing partitions of an ''n''-element set with ''k'' blocks is found in the Narayana number triangle. Definition A partition of a set ''S'' is a set of non-empty, pairwise disjoint subsets of ''S'', called "parts" or "blocks", whose union is all of ''S''. Consider a finite set that is linearly ordered, or (equivalently, for purposes of this definition) arranged in a cyclic order like the vertices of a regular ''n''-gon. No generality is lost by taking this set to be ''S'' = . A noncrossing partition of ''S'' is a partition in which no two blocks "cross" each other, i.e., if ''a'' and ''b'' belong to one block and ''x'' and ''y'' to another, they are not arranged in the order ''a x b y''. If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Roland Speicher
Roland Speicher (born 12 June 1960) is a German mathematician, known for his work on free probability theory. He is a professor at the Saarland University. After winning the 1979 German national competition Jugend forscht in the field of mathematics and computer science, Speicher studied physics and mathematics at the Universities of Saarbrücken, Freiburg and Heidelberg. He received in 1989 his doctorate from Heidelberg University Heidelberg University, officially the Ruprecht Karl University of Heidelberg (; ), is a public research university in Heidelberg, Baden-Württemberg, Germany. Founded in 1386 on instruction of Pope Urban VI, Heidelberg is Germany's oldest unive ... under the supervision of Wilhelm Freiherr von Waldenfels with thesis ''Quantenstochastische Prozesse auf der Cuntz-Algebra'' (Quantum Stochastic Processes on the Cuntz Algebra). From 2000 to 2010 Speicher was a professor at Queen's University at Kingston, Queen's University in Kingston, Ontario. Since 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cumulant
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the '' moments'' of the distribution. Any two probability distributions whose moments are identical will have identical cumulants as well, and vice versa. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment. But fourth and higher-order cumulants are not equal to central moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments. In particular, when two or more random variables are statistically independent, the th-order cumulant of their sum is equal to the sum of their th-order cumulants. As well, the third and higher-order cumulants of a normal distribution are zero, and it is the only distribution with this property. Just as for moments, where ''joint moments'' are used for collections of random variables ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Free Dimension
Free may refer to: Concept * Freedom, the ability to act or change without constraint or restriction * Emancipate, attaining civil and political rights or equality * Free (''gratis''), free of charge * Gratis versus libre, the difference between the two common meanings of the adjective "free". Computing * Free (programming), a function that releases dynamically allocated memory for reuse * Free software, software usable and distributable with few restrictions and no payment *, an emoji in the Enclosed Alphanumeric Supplement block. Mathematics * Free object ** Free abelian group ** Free algebra ** Free group ** Free module ** Free semigroup * Free variable People * Free (surname) * Free (rapper) (born 1968), or Free Marie, American rapper and media personality * Free, a pseudonym for the activist and writer Abbie Hoffman * Free (active 2003–), American musician in the band FreeSol Arts and media Film and television * ''Free'' (film), a 2001 American dramedy * '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left). Examples T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |