scholarly journals Convergence results for compound Poisson distributions and applications to the standard Luria–Delbrück distribution

2005 ◽  
Vol 42 (3) ◽  
pp. 620-631 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Probability Measure ◽  
Integral Representation ◽  
Convergence Result ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Convergence Results ◽  
The Fourier Transform ◽  
Related Paper

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

scholarly journals Convergence results for compound Poisson distributions and applications to the standard Luria–Delbrück distribution

2005 ◽  
Vol 42 (03) ◽  
pp. 620-631
Author(s):  
M. Möhle
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Probability Measure ◽  
Integral Representation ◽  
Convergence Result ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Convergence Results ◽  
The Fourier Transform ◽  
Related Paper

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

NOTE ON THE VLASOV EQUATION FOR PLASMAS

1963 ◽  
Vol 41 (12) ◽  
pp. 1960-1966 ◽  
Author(s):  
Ta-You Wu ◽  
M. K. Sundaresan
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Vlasov Equation ◽  
Hermite Polynomials ◽  
Infinite Matrix ◽  
Mean Square ◽  
Initial Value ◽  
Positive Ions ◽  
The Mean ◽  
The Fourier Transform

The linearized Vlasov equation is solved as an initial value problem by expanding (the Fourier components of) the distribution function in a series of Hermite polynomials in the momentum, with coefficients which are functions of time. The spectrum of frequencies is given by the eigenvalues of an infinite matrix. All the frequencies ω are real, extending from small values of order ω2 = k2(u22), where (u22) is the mean square velocity of the positive ions (of mass M), to [Formula: see text], where ω1, (u12) are the plasma frequency and mean square velocity of the electrons (of mass m). The classic work of Landau solves the Vlasov equation for (the Fourier transform of) the potential for which he obtains the "damping", whereas Van Kampen and the present writers solve the equation for (the Fourier transform of) the distribution function itself. While the present work gives results equivalent to those of Van Kampen, the method is simpler and in fact elementary.

scholarly journals Multiscale Asymptotic Analysis and Parallel Algorithm of Parabolic Equation in Composite Materials

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Xin Wang ◽  
Xi-liang Duan ◽  
Yang Gao
Keyword(s):  
Fourier Transform ◽  
Parabolic Equation ◽  
Asymptotic Analysis ◽  
Frequency Domain ◽  
Expansion Method ◽  
Convergence Result ◽  
Oscillating Coefficients ◽  
Multiscale Fem ◽  
Periodic Configuration ◽  
The Fourier Transform

An efficient parallel multiscale numerical algorithm is proposed for a parabolic equation with rapidly oscillating coefficients representing heat conduction in composite material with periodic configuration. Instead of following the classical multiscale asymptotic expansion method, the Fourier transform in time is first applied to obtain a set of complex-valued elliptic problems in frequency domain. The multiscale asymptotic analysis is presented and multiscale asymptotic solutions are obtained in frequency domain which can be solved in parallel essentially without data communications. The inverse Fourier transform will then recover the approximate solution in time domain. Convergence result is established. Finally, a novel parallel multiscale FEM algorithm is proposed and some numerical examples are reported.

AUTOMATIC THREE‐DIMENSIONAL MODELING FOR THE INTERPRETATION OF GRAVITY OR MAGNETIC ANOMALIES

Geophysics ◽  
1975 ◽  
Vol 40 (6) ◽  
pp. 1014-1034 ◽  
Author(s):  
A. Gerard ◽  
N. Debeglia
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Iterative Method ◽  
Three Dimensional ◽  
Magnetic Anomalies ◽  
Average Depth ◽  
Three Dimensional Modeling ◽  
Dimensional Modeling ◽  
The Fourier Transform ◽  
Homogeneous Media

Transformation of gravity or magnetic anomaly maps into isodepth maps of a surface separating two homogeneous media may be accomplished by (1) systematically estimating an average depth and density or magnetization contrast for the surface and (2) using an iterative method to adjust local depths compared to the average depth of the surface. Average depth, density or magnetization contrast, and iterative adjustment of local depths are determined using the Fourier transform of the field to be interpreted and that of the field generated by an equivalent surface. This leads us to propose a method of estimating the average depth of the sources and a distribution function for the depths and then a complete and very economical algorithm for the calculation of the corresponding equivalent surface.

scholarly journals Spectral Functions for the Vector-Valued Fourier Transform

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Spectral Function ◽  
Vector Function ◽  
Finite Interval ◽  
Spectral Functions ◽  
The Matrix ◽  
The Fourier Transform ◽  
Vector Valued

A scalar distribution function σ(s) is called a spectral function for the Fourier transform φ^(s)=∫Reitsφ(t)dt (with respect to an interval I⊂R) if for each function φ∈L2(R) with support in I the Parseval identity ∫Rφ^s2dσ(s)=∫Rφt2dt holds. We show that in the case I=R there exists a unique spectral function σ(s)=(1/2π)s, in which case the above Parseval identity turns into the classical one. On the contrary, in the case of a finite interval I=(0,b), there exist infinitely many spectral functions (with respect to I). We introduce also the concept of the matrix-valued spectral function σ(s) (with respect to a system of intervals {I1,I2,…,In}) for the vector-valued Fourier transform of a vector-function φ(t)={φ1(t),φ2(t),…,φn(t)}∈L2(I,Cn), such that support of φj lies in Ij. The main result is a parametrization of all matrix (in particular scalar) spectral functions σ(s) for various systems of intervals {I1,I2,…,In}.

scholarly journals On the Fourier transform of the products of M-Wright functions

2014 ◽  
Vol 33 (1) ◽  
pp. 245
Author(s):  
Alireza Ansari
Keyword(s):  
Fourier Transform ◽  
Integral Representation ◽  
Mellin Transform ◽  
Integral Representations ◽  
Wright Function ◽  
The Fourier Transform

In this  note, by applying the Bromwich's integral for the inverse Mellin transform we find a new integral representation for the   M-Wright function  $$ M_\alpha(x)=\sum _{k=0}^{\infty }\frac{(-x)^{k} }{k!\Gamma (-\alpha k+1-\alpha )},\quad  \alpha=\frac{1}{2n+1}, n\in \mathbb{N},$$ and state the Fourier transform of this function. Also, using the new integral representations for the products of the M-Wright functions, we get the Fourier transform of it.

scholarly journals Another Proof of the Harer-Zagier Formula

10.37236/5420 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Boris Pittel
Keyword(s):  
Fourier Transform ◽  
Probability Measure ◽  
Symmetric Group ◽  
Generating Function ◽  
Recurrence Equation ◽  
Young Diagrams ◽  
Group Characters ◽  
Bivariate Generating Function ◽  
The Fourier Transform

For a regular $2n$-gon there are $(2n-1)!!$ ways to match and glue the $2n$ sides. The Harer-Zagier bivariate generating function enumerates the gluings by $n$ and the genus $g$ of the attendant surface and leads to a recurrence equation for the counts of gluings with parameters $n$ and $g$. This formula was originally obtained using multidimensional Gaussian integrals. Soon after, Jackson and later Zagier found alternative proofs using symmetric group characters. In this note we give a different, characters-based, proof. Its core is computing and marginally inverting the Fourier transform of the underlying probability measure on $S_{2n}$. A key ingredient is the Murnaghan-Nakayama rule for the characters associated with one-hook Young diagrams.

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