Poisson Summation Research Articles

Theoretical solutions for degree distribution of decreasing random birth-and-death networks

In this paper, theoretical solutions for degree distribution of decreasing random birth-and-death networks [Formula: see text] are provided. First, we prove that the degree distribution has the form of Poisson summation, for which degree distribution equations under steady state and probability generating function approach are employed. Then, based on the form of Poisson summation, we further confirm the tail characteristic of degree distribution is Poisson tail. Finally, simulations are carried out to verify these results by comparing the theoretical solutions with computer simulations.

Ray Tracing and Modal Methods for Modeling Radio Propagation in Tunnels With Rough Walls.

At the ultrahigh frequencies common to portable radios, tunnels such as mine entries are often modeled by hollow dielectric waveguides. The roughness condition of the tunnel walls has an influence on radio propagation, and therefore should be taken into account when an accurate power prediction is needed. This paper investigates how wall roughness affects radio propagation in tunnels, and presents a unified ray tracing and modal method for modeling radio propagation in tunnels with rough walls. First, general analytical formulas for modeling the influence of the wall roughness are derived, based on the modal method and the ray tracing method, respectively. Second, the equivalence of the ray tracing and modal methods in the presence of wall roughnesses is mathematically proved, by showing that the ray tracing-based analytical formula can converge to the modal-based formula through the Poisson summation formula. The derivation and findings are verified by simulation results based on ray tracing and modal methods.

The distance between the general Poisson summation formula and that for bandlimited functions; applications to quadrature formulae

The general Poisson summation formula of harmonic analysis and analytic number theory can be regarded as a quadrature formula with remainder. The purpose of this investigation is to give estimates for this remainder based on the classical modulus of smoothness and on an appropriate metric for describing the distance of a function from a Bernstein space. Moreover, to be more flexible when measuring the smoothness of a function, we consider Riesz derivatives of fractional order. It will be shown that the use of the above metric in connection with fractional order derivatives leads to estimates for the remainder, which are best possible with respect to the order and the constants.

Computing dimensions of spaces of Arakelov divisors of number fields

The function [Formula: see text] for a number field is analogous to the dimension of the Riemann–Roch spaces at divisors on an algebraic curve. We provide a method to compute this function for number fields with unit group of rank at most 2, even with large discriminant. This method is based on using LLL-reduced bases, the “jump algorithm” and Poisson summation formula.

Beyond Endoscopy via the trace formula, II: Asymptotic expansions of Fourier transforms and bounds towards the Ramanujan conjecture

We continue the analysis of the elliptic part of the trace formula for $GL(2)$ initiated in the earlier paper of the author with the same title. In that paper Poisson summation was applied to the elliptic part and the dominant term was analyzed. The main aim of this paper is to study the remaining terms after Poisson summation. We analyze the Fourier transforms of (smoothed) orbital integrals and obtain exact asymptotic expansions. As an application we recover, using the Arthur-Selberg trace formula, Kuznetsov's result that the trace of the $p$th Hecke operator on cuspidal automorphic representations is bounded by $p^{\frac14}$.

Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform

This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform.

Abstract Poisson summation formulas over homogeneous spaces of compact groups

This paper presents the abstract notion of Poisson summation formulas for homogeneous spaces of compact groups. Let G be a compact group, H be a closed subgroup of G, and mu be the normalized G-invariant measure over the left coset space G / H associated to the Weil’s formula. We prove that the abstract Fourier transform over G / H satisfies a generalized version of the Poisson summation formula.

Acoustic radiation from shear deformable ring-stiffened laminated composite cylindrical shell submerged in flowing fluid

Acoustic radiation from a point driven, infinite, periodically ring-stiffened, laminated composite cylindrical shell submerged in flowing fluid is investigated theoretically. Both the effects of in-plane and out-of-plane vibrations of the ring-stiffeners and the effects of fluid convection on far field acoustic radiation behaviors are concerned. The equations of motion of the laminated composite cylindrical shell is presented on the basis of the first order shear deformation theory. Fourier transform and Poisson summation formula are used to transform the equations into a set of infinite algebraic equations expressed in the wavenumber domain. After truncation, the response of the laminated composite cylindrical shell is solved, and the stationary phase approximate is employed to find the expression for the far field sound pressure. Convergence analysis of the numerical solutions is conducted. The theoretical model and numerical method proposed in this paper are validated by comparison with those presented in available literature. Finally, numerical results are presented to demonstrate the effects of various parameters such as the size and spacing of the ring-stiffener, the thickness and the radius of the cylindrical shell, the lamination angle and the lamination scheme of the composite materials as well as the Mach number on the far field sound pressure.

Fermi’s golden rule: its derivation and breakdown by an ideal model

Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from to , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time, even when the first order perturbation approximation itself is still valid.

Hamiltonian finite-temperature quantum field theory from its vacuum on partially compactified space

The partition function of a relativistic invariant quantum field theory is expressed by its vacuum energy calculated on a spatial manifold with one dimension compactified to a 1-sphere $S^1 (\beta)$, whose circumference $\beta$ represents the inverse temperature. Explicit expressions for the usual energy density and pressure in terms of the energy density on the partially compactified spatial manifold $\mathbb{R}^2 \times S^1 (\beta)$ are derived. To make the resulting expressions mathematically well-defined a Poisson resummation of the Matsubara sums as well as an analytic continuation in the chemical potential are required. The new approach to finite-temperature quantum field theories is advantageous in a Hamilton formulation since it does not require the usual thermal averages with the density operator. Instead, the whole finite-temperature behaviour is encoded in the vacuum wave functional on the spatial manifold $\mathbb{R}^2 \times S^1 (\beta)$. We illustrate this approach by calculating the pressure of a relativistic Bose and Fermi gas and reproduce the known results obtained from the usual grand canonical ensemble. As a first non-trivial application we calculate the pressure of Yang-Mills theory as function of the temperature in a quasi-particle approximation motivated by variational calculations in Coulomb gauge.

Probabilistic trace and Poisson summation formulae on locally compact abelian groups

Abstract We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d-dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on L 2 ${L^{2}}$ -space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p-adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.

Poisson Summation Formula in Hardy Spaces $$H^p(T_\Gamma )$$ H p ( T Γ ) , $$p\in (0,1]$$ p ∈ ( 0 , 1

The Poisson summation formula for Hardy spaces $$H^p\left( T_\Gamma \right) $$ in tubes $$T_\Gamma \subset \mathbb {C}^n$$ for $$p\in \left( 0,1\right] $$ is obtained. Unlike the case of $$L^p\left( \mathbb {R}^n\right) $$ spaces, the formula holds everywhere in $$T_\Gamma $$ without any additional assumptions. To the best of our knowledge, the result is new even for the univariate case—Hardy spaces in the upper half-plane.

Rapid computation of directional wellbore drawdown in a confined aquifer via Poisson resummation

We have derived a rapidly computed analytical solution for drawdown caused by a partially or fully penetrating directional wellbore (vertical, horizontal, or slant) via Green's function method. The mathematical model assumes an anisotropic, homogeneous, confined, box-shaped aquifer. Any dimension of the box can have one of six possible boundary conditions: 1) both sides no-flux; 2) one side no-flux – one side constant-head; 3) both sides constant-head; 4) one side no-flux; 5) one side constant-head; 6) free boundary conditions. The solution has been optimized for rapid computation via Poisson Resummation, derivation of convergence rates, and numerical optimization of integration techniques. Upon application of the Poisson Resummation method, we were able to derive two sets of solutions with inverse convergence rates, namely an early-time rapidly convergent series (solution-A) and a late-time rapidly convergent series (solution-B). From this work we were able to link Green's function method (solution-B) back to image well theory (solution-A). We then derived an equation defining when the convergence rate between solution-A and solution-B is the same, which we termed the switch time. Utilizing the more rapidly convergent solution at the appropriate time, we obtained rapid convergence at all times. We have also shown that one may simplify each of the three infinite series for the three-dimensional solution to 11 terms and still maintain a maximum relative error of less than 10−14.

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of \(L_1\) signals. This \(L_1\) assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that \(d/2+\varepsilon \) derivatives of the signal are bounded by some polynomial in the \(L_2\) sense.

Nonlinear Model for Gas Flow into a Horizontal Well from a Shale Reservoir with a Branched Fracture Network

We propose a mathematical model describing gas flow from a fractured shale reservoir into a horizontal well. Based on Langmuir adsorption theory and Darcy’s law, we have obtained a basic solution using a combination of a point source function, an integral of a Bessel’s function, and the Poisson summation formula. We plotted typical curves for the dimensionless pressure and its derivative on a log-log scale. We analyze the stages of gas flow from the shale formation to the horizontal well and the factors influencing it.

A New Transform Related to Distance From a Boolean Function

We introduce a new transform on Boolean functions generalizing the Walsh–Hadamard transform. For Boolean functions $q$ and $f$ , the $q$ -transform of $f$ measures the proximity of $f$ to the set of functions obtained from $q$ by change of basis. This has implications for security against certain algebraic attacks. In this paper, we derive the expected value and second moment (Parseval’s equation) of the $q$ -transform, leading to a notion of $q$ -bentness. We also develop a Poisson summation formula, which leads to a proof that the $q$ -transform is invertible.

Wave optics in black hole spacetimes: the Schwarzschild case

We investigate the wave optics in the Schwarzschild spacetime. Applying the standard formalism of wave-scattering problems, the Green function represented by the sum over the partial waves is evaluated using the Poisson sum formula. The effect of orbiting scattering due to the unstable circular orbit for null rays is taken into account as the contribution of the Regge poles of the scattering matrix and the asymptotic form of the scattering wave is obtained in the eikonal limit. Using this wave function, images of the black hole illuminated by a point source are reconstructed. We also discuss the wave effect in the frequency domain caused by the interference between the direct and the winding rays.

An analytical solution for quantum size effects on Seebeck coefficient

There are numerous experimental and numerical studies about quantum size effects on Seebeck coefficient. In contrast, in this study, we obtain analytical expressions for Seebeck coefficient under quantum size effects. Seebeck coefficient of a Fermi gas confined in a rectangular domain is considered. Analytical expressions, which represent the size dependency of Seebeck coefficient explicitly, are derived in terms of confinement parameters. A fundamental form of Seebeck coefficient based on infinite summations is used under relaxation time approximation. To obtain analytical results, summations are calculated using the first two terms of Poisson summation formula. It is shown that they are in good agreement with the exact results based on direct calculation of summations as long as confinement parameters are less than unity. The analytical results are also in good agreement with experimental and numerical ones in literature. Maximum relative errors of analytical expressions are less than 3% and 4% for 2D and 1D cases, respectively. Dimensional transitions of Seebeck coefficient are also examined. Furthermore, a detailed physical explanation for the oscillations in Seebeck coefficient is proposed by considering the relative standard deviation of total variance of particle number in Fermi shell.

On the Implementation of Exponential B-Splines by Poisson Summation Formula

Polynomial splines have played an important role in image processing, medical imaging and wavelet theory. Exponential splines which are of more general concept have been recently investigated.We focus on cardinal exponential splines and develop a method to implement the exponential B-splines which form a Riesz basis of the space of cardinal exponential splines with finite energy.

From ``Dirac Combs'' to Fourier-positivity

Motivated by various problems in physics and applied mathematics, we look for constraints and properties of real Fourier-positive functions, i.e. with positive Fourier transforms. Properties of the "Dirac comb" distribution and of its tensor products in higher dimensions lead to Poisson resummation, allowing for a useful approximation formula of a Fourier transform in terms of a limited number of terms. A connection with the Bochner theorem on positive definiteness of Fourier-positive functions is discussed. As a practical application, we find simple and rapid analytic algorithms for checking Fourier-positivity in 1- and (radial) 2-dimensions among a large variety of real positive functions. This may provide a step towards a classification of positive positive-definite functions.