tempered%20distributions
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1: 1.16 Distributions
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is called a distribution, or generalized function, if it is a continuous linear functional on , that is, it is a linear functional and for every in ,
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§1.16(v) Tempered Distributions
… ► … ►§1.16(vii) Fourier Transforms of Tempered Distributions
… ►The Fourier transform of a tempered distribution is again a tempered distribution, and …2: 2.6 Distributional Methods
§2.6 Distributional Methods
… ►To each function in this equation, we shall assign a tempered distribution (i. …, a continuous linear functional) on the space of rapidly decreasing functions on . …Motivated by the definition of distributional derivatives, we can assign them the distributions defined by … ►§2.6(iv) Regularization
…3: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution.
Justus von Liebig Verlag, Darmstadt (German, English).
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)].
J. Chem. Phys. 121 (2), pp. 1167.
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4: 8.23 Statistical Applications
§8.23 Statistical Applications
►The functions and are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). The function and its normalization play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …5: Foreword
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►That 1046-page tome proved to be an invaluable reference for the many scientists and engineers who use the special functions of applied mathematics in their day-to-day work, so much so that it became the most widely distributed and most highly cited NIST publication in the first 100 years of the institution’s existence.
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►November 20, 2009
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6: 20 Theta Functions
Chapter 20 Theta Functions
…7: 27.2 Functions
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27.2.1
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
Tables of primes (§27.21) reveal great irregularity in their distribution.
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27.2.3
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8: 24.18 Physical Applications
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►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)).
►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).
9: 12.11 Zeros
10: 32.14 Combinatorics
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32.14.1
►where the distribution function
is defined here by
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32.14.2
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►The distribution function given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of Hermitian matrices; see Tracy and Widom (1994).
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