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107
The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy ..."
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Cited by 149 (51 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
Scaling limit for the spacetime covariance of the stationary totally asymmetric simple exclusion process
 Comm. Math. Phys
"... The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0,t]. For j = ..."
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Cited by 38 (10 self)
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The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0,t]. For j = (1 − 2ρ)t + 2w(ρ(1 − ρ)) 1/3 t 2/3 we prove that the fluctuations of Nt(j) for large t are of order t 1/3 and we determine the limiting distribution function Fw(s), which is a generalization of the GUE TracyWidom distribution. The family Fw(s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a RiemannHilbert problem. In our work we arrive at Fw(s) through the asymptotics of a Fredholm determinant. Fw(s) is simply related to the scaling function for the spacetime covariance of the stationary TASEP, equivalently to the asymptotic transition
Representations of Orthogonal Polynomials
, 1998
"... Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computes recurrence and differential equations for hyperexponential integrals. Further versions of this algorithm all ..."
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Cited by 26 (10 self)
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Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computes recurrence and differential equations for hyperexponential integrals. Further versions of this algorithm allow the computation of recurrence and differential equations from Rodrigues type formulas and from generating functions. In particular, these algorithms can be used to compute the differential/difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations, and from their Rodrigues representations and generating functions. In recent work, we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials, in terms of the coefficients of their differential/difference equations, to give an algorithm to identify the polynomial system from a given recurrence equation....
Nonlinear fractional dynamics of lattice with longrange interaction. Eprint: nlin.SI/0512010
"... A unified approach has been developed to study nonlinear dynamics of a 1D lattice of particles with longrange powerlaw interaction. Classical case is treated in the framework of the wellknown FrenkelKontorova chain model. Qunatum dynamics is considered follow to Davydov’s approach to molecular e ..."
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Cited by 17 (4 self)
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A unified approach has been developed to study nonlinear dynamics of a 1D lattice of particles with longrange powerlaw interaction. Classical case is treated in the framework of the wellknown FrenkelKontorova chain model. Qunatum dynamics is considered follow to Davydov’s approach to molecular excitons. In the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the longrange interaction. Fractional generalizations of the sineGordon, nonlinear Schrödinger, and HilbertSchrödinger equations have been found. There exists a critical value of the powers of the longrange potential. Below the critical value (s < 3, s ̸ = 1, 2) we obtain equations with fractional derivatives while for s> 3 we have the wellknown nonlinear dynamical equations with space derivatives of integer order. Longrange interaction impact on quantum lattice propagator has been studied. We have shown that the quantum exciton propagator exhibits transition from the wellknown Gaussianlike behavior to powerlaw decay due to longrange interaction. Link between 1D quantum lattice dynamics in the imanaginary time domain and random walk model has been discussed.
An efficient spectral method for ordinary differential equations with rational function coefficients
 Math. Comp
, 1996
"... Abstract. We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These familie ..."
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Cited by 15 (3 self)
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Abstract. We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple 3term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e., matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers. 1.
Eigenvalues of the Laplacian for Bianchi groups
 EMERGING APPLICATIONS OF NUMBER THEORY, IMA SERIES NO. 109
, 1997
"... We present results concerning the computation of Laplace eigenvalue spectra for certain arithmetic, discrete subgroups of PSL(2; C) acting on the hyperbolic upper half space. These subgroups are PSL(2; O), where O is the ring of integers of an imaginary quadratic number field Q( p \GammaD ). Speci ..."
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Cited by 9 (0 self)
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We present results concerning the computation of Laplace eigenvalue spectra for certain arithmetic, discrete subgroups of PSL(2; C) acting on the hyperbolic upper half space. These subgroups are PSL(2; O), where O is the ring of integers of an imaginary quadratic number field Q( p \GammaD ). Special attention is devoted to the cases D = 1; 2; 3; 7; 11 having fundamental domains with one cusp. It is proved that the spectra are not simple. We sketch a method showing how such eigenvalues and associated eigenfunctions can be computed. Some of the eigenvalues are recognized as being lifts from the modular group PSL(2; Z). As an application to quantum chaos, we demonstrate that the spectra exhibit random fluctuations close to Poissonian. Samples of eigenvalues are listed.
Congruences involving Bernoulli and Euler numbers
 J. Number Theory
"... Abstract. Let [x] be the integral part of x. Let p> 5 be a prime. In the paper 1 we mainly determineP[p/4] x=1 xk (mod p2 p−1), (mod p [p/4] ..."
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Cited by 8 (6 self)
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Abstract. Let [x] be the integral part of x. Let p> 5 be a prime. In the paper 1 we mainly determineP[p/4] x=1 xk (mod p2 p−1), (mod p [p/4]
Positivity properties and stability of solitarywave solutions of model equations for long waves
 Comm
, 1992
"... Abstract. Sufficient conditions are given for stability of solitarywave solutions of model equations for onedimensional long nonlinear waves. These conditions differ from others which have appeared previously in that they are phrased in terms of positivity properties of the Fourier transforms of t ..."
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Cited by 8 (1 self)
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Abstract. Sufficient conditions are given for stability of solitarywave solutions of model equations for onedimensional long nonlinear waves. These conditions differ from others which have appeared previously in that they are phrased in terms of positivity properties of the Fourier transforms of the solitary waves. Their use leads to simplified proofs of existing stability results for the Kortewegde Vries, BenjaminOno, and Intermediate Long Wave equations; and to new stability results for certain solitarywave solutions of partial differential equations of Kortewegde Vries type. 1. Introduction. This article is a contribution to the theory of stability of travellingwave solutions of model equations for onedimensional long nonlinear waves. Considered herein are equations of the form (1.1) ut + u p ux − (Mu)x = 0,