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Analysis Of Boolean Functions
In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on \^n or \^n (such functions are sometimes known as pseudo-Boolean functions) from a spectral perspective. The functions studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics, social choice theory, random graphs, and theoretical computer science, especially in hardness of approximation, property testing, and PAC learning. Basic concepts We will mostly consider functions defined on the domain \^n. Sometimes it is more convenient to work with the domain \^n instead. If f is defined on \^n, then the corresponding function defined on \^n is :f_(x_1,\ldots,x_n) = f((-1)^,\ldots,(-1)^). Similarly, for us a Boolean function is a \-valued function, though often it is more convenient to consider \-valued functions instead. Fourier expansion Every real-valued function f\colon \^n \to \mathbb ... [...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] |
Continuous-time Markov Chain
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to the least value of a set of exponential random variables, one for each possible state it can move to, with the parameters determined by the current state. An example of a CTMC with three states \ is as follows: the process makes a transition after the amount of time specified by the holding time—an exponential random variable E_i, where ''i'' is its current state. Each random variable is independent and such that E_0\sim \text(6), E_1\sim \text(12) and E_2\sim \text(18). When a transition is to be made, the process moves according to the jump chain, a discrete-time Markov chain with stochastic matrix: :\begin 0 & \frac & \frac \\ \frac & 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorica
''Combinatorica'' is an international journal of mathematics, publishing papers in the fields of combinatorics and computer science. It started in 1981, with László Babai and László Lovász as the editors-in-chief with Paul Erdős as honorary editor-in-chief. The current editors-in-chief are Imre Bárány and József Solymosi. The advisory board consists of Ronald Graham, Gyula O. H. Katona, Miklós Simonovits, Vera Sós, and Endre Szemerédi. It is published by the János Bolyai Mathematical Society and Springer Verlag. The following members of the '' Hungarian School of Combinatorics'' have strongly contributed to the journal as authors, or have served as editors: Miklós Ajtai, László Babai, József Beck, András Frank, Péter Frankl, Zoltán Füredi, András Hajnal, Gyula Katona, László Lovász, László Pyber, Alexander Schrijver, Miklós Simonovits, Vera Sós, Endre Szemerédi, Tamás Szőnyi, Éva Tardos, Gábor Tardos.{{cite web, url=https://www.springer.com/ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Applied Mathematics
''Advances in Applied Mathematics'' is a peer-reviewed mathematics journal publishing research on applied mathematics. Its founding editor was Gian-Carlo Rota (Massachusetts Institute of Technology); from 1980 to 1999, Joseph P. S. Kung (University of North Texas) served as managing editor. It is currently published by Elsevier with eight issues per year and edited by Hal Schenck (Auburn University) and Catherine Yan (Texas A&M University). Abstracting and indexing The journal is abstracted and indexed by: * ACM Guide to Computing Literature * CompuMath Citation Index * Current Contents/Physics, Chemical, & Earth Sciences * ''Mathematical Reviews'' * Science Citation Index * Scopus According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.848. See also * List of periodicals published by Elsevier This is a list of scientific, technical and general interest periodicals published by Elsevier or one of its imprints or subsidiary companies. Both pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mehler Kernel
The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. Mehler's formula defined a function and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials (.) based on weight function exp(−²) as :E(x,y) = \sum_^\infty \frac ~ \mathit_n(x)\mathit_n(y) ~. This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis. Physics version In physics, the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel. It provides the fundamental solution---the most general solution to :\frac = \frac-x^2\varphi \equiv D_x \varphi ~. The orthonormal eigenfunctions of the operator are the Hermite functions, :\psi_n = \frac, with corresponding eigenvalues (2+1), furnishing particular solutions : \varphi_n(x, t)= e^ ~H_n(x) \exp(-x^2/2) ~. The general solution is then a linear comb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ornstein–Uhlenbeck Process
In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck. The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous. In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables. Over time, the process tends to drift towards its mean function: such a process is called mean-reverting. The process can be considered to be a modification of the random walk in continuous time, or Wiener process, in which the properties of the process have been changed so that there is a tendency of the walk to move back towa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the Germany, German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable ''X'' is obtained by summing a large number ''N'' of independent random variables of order 1, then ''X'' is of order \sqrt and its law is approximately Gaussian. Definitions Let ''n'' ∈ N and let ''B''0(R''n'') denote the complete measure, completion of the Borel sigma algebra, Borel ''σ''-algebra on R''n''. Let ''λ''''n'' : ''B''0(R''n'') → [0, +∞] denote the usual ''n''-dimensional Lebesgue measure. Then the standard Gaussian measure ''γ''''n'' : ''B''0(R''n'') → [0, 1] is defined by :\gamma^ (A) = \frac \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariance Principle
In cognitive linguistics, the invariance principle is a simple attempt to explain similarities and differences between how an idea is understood in "ordinary" usage, and how it is understood when used as a conceptual metaphor. Kövecses (2002: 102) provides the following examples based on the semantics of the English verb ''to give'': :She gave him a book. (source language) Based on the metaphor CAUSATION IS TRANSFER we get: :(a) She gave him a kiss. :(b) She gave him a headache. However, the metaphor does not work in exactly the same way in each case, as seen in: :(a') She gave him a kiss, ''and he still has it''. :(b') She gave him a headache, ''and he still has it''. The invariance principle offers the hypothesis that metaphor only maps components of meaning from the source language that remain coherent in the target context. The components of meaning that remain coherent in the target context retain their "basic structure" in some sense, so this is a form of invariance. Geor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric And Functional Analysis
''Geometric and Functional Analysis'' (''GAFA'') is a mathematical journal published by Birkhäuser, an independent division of Springer-Verlag. The journal is published approximately bi-monthly. The journal publishes papers on broad range of topics in geometry and analysis including geometric analysis, riemannian geometry, symplectic geometry, geometric group theory, non-commutative geometry, automorphic forms and analytic number theory, and others. ''GAFA'' is both an acronym and a part of the official full name of the journal. History ''GAFA'' was founded in 1991 by Mikhail Gromov and Vitali Milman. The idea for the journal was inspired by the long-running Israeli seminar series "Geometric Aspects of Functional Analysis" of which Vitali Milman had been one of the main organizers in the previous years. The journal retained the same acronym as the series to stress the connection between the two. Journal information The journal is reviewed cover-to-cover in Mathematical Revie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
