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List Of Things Named After Norbert Wiener
{{Short description, none In mathematics, there are a large number of topics named in honor of Norbert Wiener (1894 – 1964). * Abstract Wiener space * Classical Wiener space * Paley–Wiener integral * Paley–Wiener theorem * Wiener algebra * Wiener amalgam space * Wiener chaos expansion * Wiener criterion * Wiener deconvolution * Wiener definition * Wiener entropy * Wiener equation * Wiener filter **Generalized Wiener filter * Wiener's lemma * Wiener process ** Generalized Wiener process * Wiener sausage * Wiener series * Wiener–Hopf method * Wiener–Ikehara theorem * Wiener–Khinchin theorem * Wiener–Kolmogorov prediction * Wiener–Lévy theorem * Weiner–Rosenblueth model * Wiener–Wintner theorem * Wiener's tauberian theorem In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932. They provide a necessary and sufficient condition under which any function in or can be approximated by linear com ... [...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] |
Generalized Wiener Filter
The Wiener filter as originally proposed by Norbert Wiener is a signal processing filter which uses knowledge of the statistical properties of both the signal and the noise to reconstruct an optimal estimate of the signal from a noisy one-dimensional time-ordered data stream. The generalized Wiener filter generalizes the same idea beyond the domain of one-dimensional time-ordered signal processing, with two-dimensional image processing being the most common application. Description Consider a data vector d which is the sum of independent signal and noise vectors d = s+n with zero mean and covariances \langle ss^T\rangle=S and \langle nn^T\rangle=N. The generalized Wiener Filter is the linear operator G which minimizes the expected residual between the estimated signal and the true signal, e = \langle(Gd-s)^T(Gd-s)\rangle. The G that minimizes this is G = S(S+N)^, resulting in the Wiener estimator \hat s = S(S+N)^d. In the case of Gaussian distributed signal and noise, this esti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Wintner Theorem
In mathematics, the Wiener–Wintner theorem, named after Norbert Wiener and Aurel Wintner, is a strengthening of the ergodic theorem, proved by . Statement Suppose that ''τ'' is a measure-preserving transformation of a measure space ''S'' with finite measure. If ''f'' is a real-valued integrable function on ''S'' then the Wiener–Wintner theorem states that there is a measure 0 set ''E'' such that the average : \lim_\frac\sum_^\ell e^ f(\tau^j P) exists for all real λ and for all ''P'' not in ''E''. The special case for ''λ'' = 0 is essentially the Birkhoff ergodic theorem Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ..., from which the existence of a suitable measure 0 set ''E'' for any fixed ''λ'', or any countable set of values ''λ'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greenberg–Hastings Cellular Automaton
The Greenberg–Hastings Cellular Automaton (abbrev. GH model) is a three state two dimensional cellular automaton (abbrev CA) named after James M. Greenberg and Stuart Hastings, designed to model excitable media An excitable medium is a nonlinear dynamical system which has the capacity to propagate a wave of some description, and which cannot support the passing of another wave until a certain amount of time has passed (known as the refractory time). A fo ..., One advantage of a CA model is ease of computation. The model can be understood quite well using simple "hand" calculations, not involving a computer. Another advantage is that, at least in this case, one can prove a theorem characterizing those initial conditions which lead to repetitive behavior. Informal description As in a typical two dimensional cellular automaton, consider a rectangular grid, or checkerboard pattern, of "cells". It can be finite or infinite in extent. Each cell has a set of "neighbors". In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Lévy Theorem
Wiener–Lévy theorem is a theorem in Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after Norbert Wiener and Paul Lévy. Norbert Wiener first proved Wiener's 1/''f'' theorem, see Wiener's theorem. It states that if has absolutely convergent Fourier series and is never zero, then its inverse also has an absolutely convergent Fourier series. Wiener–Levy theorem Paul Levy generalized Wiener's result, showing that Let F(\theta ) = \sum\limits_^\infty c_k e^, \quad\theta \in ,2\pi /math> be an absolutely convergent Fourier series with : \, F\, = \sum\limits_^\infty , c_k, has an absolutely convergent Fourier series. The proof can be found in the Zygmund's classic book ''Trigonometric Series''. Example Let H(\theta )=\ln(\theta ) and F(\theta ) = \sum\limits_^\infty p_k e^,(\sum\limits_^\infty p_k = 1 ) is characteristic function of discr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kriging
In statistics, originally in geostatistics, kriging or Kriging, also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging gives the best linear unbiased prediction (BLUP) at unsampled locations. Interpolating methods based on other criteria such as smoothness (e.g., smoothing spline) may not yield the BLUP. The method is widely used in the domain of spatial analysis and computer experiments. The technique is also known as Wiener–Kolmogorov prediction, after Norbert Wiener and Andrey Kolmogorov. The theoretical basis for the method was developed by the French mathematician Georges Matheron in 1960, based on the master's thesis of Danie G. Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex in South Africa. Krige sought to estimate the most likely distribution of gold based on samples from a few boreholes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Khinchin Theorem
In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process. History Norbert Wiener proved this theorem for the case of a deterministic function in 1930; Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published that probabilistic analogue in 1934. Albert Einstein explained, without proofs, the idea in a brief two-page memo in 1914. The case of a continuous-time process For continuous time, the Wiener–Khinchin theorem says that if x is a wide-sense stochastic process whose autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value, r_(\tau) = \mathbb\big (t)^*x(t - \tau)\big/math> (the asterisk denotes complex co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Ikehara Theorem
The Wiener–Ikehara theorem is a Tauberian theorem introduced by . It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (Chandrasekharan, 1969). Statement Let ''A''(''x'') be a non-negative, monotonic nondecreasing function of ''x'', defined for 0 ≤ ''x'' 1 to the function ''ƒ''(''s'') and that, for some non-negative number ''c'', :f(s) - \frac has an extension as a continuous function for ℜ(''s'') ≥ 1. Then the limit as ''x'' goes to infinity of ''e''−''x'' ''A''(''x'') is equal to c. One Particular Application An important number-theoretic application of the theorem is to Dirichlet series of the form :\sum_^\infty a(n) n^ where ''a''(''n'') is non-negative. If the series converges to an analytic function in :\Re(s) \ge b with a simple pole of residue ''c'' at ''s'' = ''b'', then :\sum_a(n) \sim \frac X^b. Applying this to the logarithmic derivative of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Hopf Method
The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed boundary conditions on the same boundary. In general, the method works by exploiting the complex-analytical properties of transformed functions. Typically, the standard Fourier transform is used, but examples exist using other transforms, such as the Mellin transform. In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively analytic in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the complex plane, typically, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Series
In mathematics, the Wiener series, or Wiener G-functional expansion, originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Volterra series and having the same relation to it as an orthogonal Hermite polynomial expansion has to a power series. For this reason it is also known as the Wiener–Hermite expansion. The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise. This property allows the terms to be identified in applications by the ''Lee–Schetzen method''. The Wiener series is important in nonlinear system identification. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience. The name Wiener series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Sausage
In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time ''t'', given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" is German for "Viennese". The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by , and it was used by to explain results of a Bose–Einstein condensate, with proofs published by . Definitions The Wiener sausage ''W''δ(''t'') of radius δ and length ''t'' is the set-valued random variable on Brownian paths b (in some Euclidean space) defined by :W_\delta(t)() is the set of points within a distance δ of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Wiener Process
In statistics, a generalized Wiener process (named after Norbert Wiener) is a continuous time random walk with drift and random jumps at every point in time. Formally: :a(x,t) dt + b(x,t) \eta \sqrt where a and b are deterministic functions, t is a continuous index for time, x is a set of exogenous variables that may change with time, dt is a differential in time, and η is a random draw from a standard normal distribution at each instant. See also *Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ... Wiener process {{statistics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
