Poisson Summation Research Articles

Superstatistical functions of modified shifted Morse potential for diatomic molecules

The application of the molecular potential model to the study of different thermodynamic systems has been a focus of interest in recent time due to its importance in molecular and chemical physics. Hence, the asymptotic iteration method and the Laguerre polynomials are employed to obtain the eigensolutions of the Schrödinger equation with the modified shifted Morse potential model. Vibrational energy results for different diatomic molecules have been presented numerically, and these results are compared with experimental data and other results from available literature. The variation of normalised wave function and probability density of modified shifted Morse potential for some selected diatomic molecules at the ground- and first-excited states have been considered. Results obtained show that the equally spaced sinusoidal wave curves and varying nodes depend on the level of electron concentrations in the diatomic molecules considered. Partition function and other superstatistical function expressions of modified shifted Morse potential for some diatomic molecules within the modified Dirac delta distribution are obtained, using the Poisson summation formula. The superstatistical plots show a high level of dependence on the temperature parameter, maximum vibrational quantum number and deformation parameter. The superstatistical functions for a deformation parameter of zero correspond to the conventional thermodynamic functions.

Irrational Outputs from Rational Inputs in Theta-3 Functions Using Dirichlet’s Theorem

Theta Functions are not an easy subject of mathematics because new researches about this wide topic are still being published regularly. The findings about Theta Functions lead directly to many applications in different fields of applied mathematics and help also to develop old interesting mathematical topics such as Poisson summation and Riemann Zeta Function. This work proves by using a theorem of Dirichlet and a previously demonstrated useful approximation that the rational inputs of Theta-3 Functions give only irrationals as outputs of these functions. This paper applies only simple notions of sequences and represents an opportunity for the students of mathematics to easily understand the steps of the demonstrations.

Localization and resummation of unstable instantons in 2d Yang-Mills

We compute the exact all-orders perturbative expansion for the partition function of 2d SU(2) Yang-Mills theory on closed surfaces around higher critical points of the classical action. We demonstrate that the expansion can be derived from the lattice partition function for all genera using a distributional generalization of the Poisson summation formula. We then recompute the expansion directly, using a stationary phase version of supersymmetric localization. The result of localization is a novel effective action which is itself a distribution rather than a function of the supersymmetric moduli. We comment on possible applications to A-twisted models and their analogs in higher dimensions.

An Elementary Proof of the Transformation Formula for the Dedekind Eta Function

In this work, we give an elementary proof of the transformation formula for the Dedekind eta function under the action of the modular group $\text{PSL}\,(2,\mathbb{Z})$. We start by giving a proof of the transformation formula $\eta(\tau)$ under the transformation $\tau\to -1/\tau$, using the Jacobi triple product identity and the Poisson summation formula. After we establish some identities for the Dedekind sum, the transformation formula for $\eta(\tau)$ under the transformation induced by a general element of the modular group $\text{PSL}\,(2,\mathbb{Z})$ is derived by induction.

Certain Fourier operators and their associated Poisson summation formulae on GL1

Certain Fourier operators and their associated Poisson summation formulae on GL1

Information theory and thermodynamic properties of diatomic molecules using molecular potential.

Owing to the devise applications of molecules in industries, the bound state solution of the non-relativistic wave equation with a molecular potential function has been obtained in a closed-form using the Nikiforov-Uvarov method. The solutions of the bound state are then applied to study the information-theoretic measures such as the one-dimensional Shannon and Renyi entropic densities. The expectation values for the position and momentum spaces were obtained to verify the Heisenberg's uncertainty principle. Utilizing the energy spectrum equation, the thermodynamic vibrational partition function is obtained via the Poisson summation. Other thermodynamic function variations with absolute temperature have been obtained numerically for four diatomic molecules (H2, N2, O2, and HF) using Maple 18 software. The Shannon global entropic sum inequality has also been verified. The Renyi sum for constrained index parameters satisfies the global entropic inequality. The thermodynamic properties of the four molecules are similar and conform to works reported in the existing literature. The obtained vibrational energies are in fair agreement with the ones obtained using other forms of potential energy. The result further indicates that the lowest bounds for the Shannon, Renyi, and Heisenberg inequalities are ground states phenomena.

The pointwise behavior of Riemann’s function

We present a new and simple method for the determination of the pointwise Hölder exponent of Riemann's function \sum_{n=1}^{\infty} n^{-2}\sin(\pi n^{2} x) at every point of the real line. In contrast to earlier approaches, where wavelet analysis and the theta modular group were needed for the analysis of irrational points, our method is direct and elementary, being only based on the following tools from number theory and complex analysis: the evaluation of quadratic Gauss sums, the Poisson summation formula, and Cauchy's theorem.

Mean value of real Dirichlet characters using a double Dirichlet series

AbstractWe study the double character sum $\sum \limits _{\substack {m\leq X,\\m\mathrm {\ odd}}}\sum \limits _{\substack {n\leq Y,\\n\mathrm {\ odd}}}\left (\frac {m}{n}\right )$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey, Farmer, and Soundararajan by applying the Poisson summation formula. The result is interesting because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages—it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.

Acoustic Radiation From Stiffened Double Concentric Large Cylindrical Shells: Part I Circumferential Harmonic Waves

Abstract Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically investigated via circumferential harmonic waves driven by a point force. The vibration of double isotropic circular cylindrical shells is described by the first-order shear deformation shell theory. One set of uniformly spaced annular bulkheads connects the inner and outer cylindrical shells. In-plane motion equations of the annular bulkheads are expressed by two displacement potential functions. Sound pressure loadings of periodic cavities exerting on the inner and outer cylindrical shells are derived according to the Fourier transform and Poisson summation formula. Far-field sound pressure of the stiffened double cylindrical shells is obtained using the stationary phase method and acoustic radiation features of stiffened double concentric large cylindrical shells with periodic cavities are analyzed in terms of sound pressure power spectra and sound pressure level. Acoustic propagation features of stiffened double cylindrical shells with or without acoustic cavities are shown.

Motivic Euler Products and Motivic Height Zeta Functions

A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of Hrushovski and Kazhdan’s motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with exponentials constructed using Denef and Loeser’s motivic vanishing cycles.

Strain and magnetic field effect on thermodynamic properties of a two dimensional GaAs quantum dot

The strain and magnetic field effect on thermodynamic properties of a two dimensional GaAs quantum dot (QD) is investigated within the effective mass approximation. To this end, first we have obtained the wave functions and energy levels of the system in the presence of Bychkov-Rashba and Dressalhaus terms. Then using the obtained energy levels, we have computed the partition function by the Poisson summation formalism. Afterward, we have deduced some thermodynamic properties such as mean energy, entropy, specific heat and free energy using the canonical ensemble approach. Finally, the thermodynamic properties for a two dimensional GaAs QD have been discussed in detail.

Double square moments and bounds for resonance sums of cusp forms

Let f and g be holomorphic cusp forms for the modular group SL2(Z) of weight k1 and k2 with Fourier coefficients λf(n) and λg(n), respectively. For real α≠0 and 0<β≤1, consider a smooth resonance sum SX(f,g;α,β) of λf(n)λg(n) against e(αnβ) over X≤n≤2X. Double square moments of SX(f,g;α,β) over both f and g are nontrivially bounded when their weights k1 and k2 tend to infinity together. By allowing both f and g to move, these double moments are indeed square moments associated with automorphic forms for GL(4). By taking out a small exceptional set of f and g, bounds for individual SX(f,g;α,β) shall then be proved. These individual bounds break the resonance barrier of X5/8 for 1/6<β<1 and achieve a square-root cancellation for 1/3<β<1 for almost all f and g as an evidence for Hypothesis S for cusp forms over integers. The methods used in this study include Petersson's formula, Poisson's summation formula, and stationary phase integrals.

Relative Reshetikhin–Turaev Invariants, Hyperbolic Cone Metrics and Discrete Fourier Transforms I

We propose the Volume Conjecture for the relative Reshetikhin–Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern–Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient 3-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston’s result that all compact oriented 3-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen–Yang’s Volume Conjecture for the Reshetikhin–Turaev invariants of closed oriented hyperbolic 3-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented 3-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin–Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

Comparing Compound Poisson Distributions by Deficiency: Continuous-Time Case

In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the number of summands guaranteeing the prescribed value of the (1−α)-quantile of the normalized Poisson sum for a given α∈(0,1). This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic (1−α)-quantile of the sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution.

Half-waves and spectral Riesz means on the 3-torus

For a full rank lattice Lambda subset mathbb {R}^d and textbf{A}in mathbb {R}^d, consider N_{d,0;Lambda ,textbf{A}}(Sigma ) = # ([Lambda +textbf{A}] cap Sigma mathbb {B}^d) = # {textbf{k}in Lambda : |textbf{k}+textbf{A}| le Sigma }. Consider the iterated integrals Nd,k+1;Λ,A(Σ)=∫0ΣNd,k;Λ,A(σ)dσ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} N_{d,k+1;\\Lambda ,\ extbf{A}}(\\Sigma ) = \\int _0^\\Sigma N_{d,k;\\Lambda ,\ extbf{A}}(\\sigma ) \\,\ extrm{d}\\sigma , \\end{aligned}$$\\end{document}for kin mathbb {N}. After an elementary derivation via the Poisson summation formula of the sharp large-Sigma asymptotics of N_{3,k;Lambda ,textbf{A}}(Sigma ) for kge 2 (these having an O(Sigma ) error term), we discuss how they are encoded in the structure of the Fourier transform mathcal {F}N_{3,k;Lambda ,textbf{A}}(tau ). The analysis is related to Hörmander’s analysis of spectral Riesz means, as the iterated integrals above are weighted spectral Riesz means for the simplest magnetic Schrödinger operator on the flat d-torus. That the N_{3,k;Lambda ,textbf{A}}(Sigma ) obey an asymptotic expansion to O(Sigma ^2) is a special case of a general result holding for all magnetic Schrödinger operators on all manifolds, and the subleading polynomial corrections can be identified in terms of the Laurent series of the half-wave trace at tau =0. The improvement to O(Sigma ) for kge 2 follows from a bound on the growth rate of the half-wave trace at late times.

Discrete Tori and Trigonometric Sums

We prove a discrete analogue of the Poisson summation formula.

Systematic derivation of angular-averaged Ewald potential

In this work we provide a step by step derivation of an angular-averaged Ewald potential suitable for numerical simulations of disordered Coulomb systems. The potential was first introduced by E Yakub and C Ronchi without a clear derivation. Two methods are used to find the coefficients of the series expansion of the potential: based on the Euler–Maclaurin and Poisson summation formulas. The expressions for each coefficient is represented as a finite series containing derivatives of Jacobi theta functions. We also demonstrate the formal equivalence of the Poisson and Euler–Maclaurin summation formulas in the three-dimensional case. The effectiveness of the angular-averaged Ewald potential is shown by the example of calculating the Madelung constant for a number of crystal lattices.

Non-Uniform Gratings That Radiate Preselected Plane Waves

A generalized Poisson summation formula produces exact expressions for the locations and strengths of line currents that radiate preselected plane waves. The line currents can be in free space or placed above a perfectly conducting ground plane. Closed-form expressions are obtained for the line current locations and amplitudes when the plane-wave parameters have periodic amplitudes and propagation vectors. A corresponding wire-impedance solution is derived that employs thin-wire approximations and takes into account all multiple interactions between wires. Remarkably, this solution holds even when the required impedance values are nonperiodic. The plane-wave parameters within one period determine the period of the impedance values and, thus, determine how many different types of loaded wires are required. For general nonperiodic plane-wave parameters, the line-source locations and amplitudes can be determined from an FFT approach involving a complex-contour integral, and the corresponding wire impedances can be computed numerically. With these impedance solutions, the number of degrees of freedom for the planar grating has been increased significantly to allow for the creation of a wide range of plane-wave fields.

Association of prenatal phthalate exposure with pubertal development in Spanish boys and girls

BackgroundPhthalates are widespread, anti-androgenic chemicals known to alter early development, with possible impact on puberty timing. AimTo investigate the association of prenatal phthalate exposure with pubertal development in boys and girls. MethodsUrinary metabolites of six different phthalate diesters (DEP, DiBP, DnBP, BBzP, DEHP, and DiNP) and non-phthalate plasticizer DINCH® were quantified in two urine samples collected during pregnancy from mothers participating in the INMA Spanish cohort study. Pubertal assessment of their children at age 7–10 years (409 boys, 379 girls) was conducted using the parent-reported Pubertal Development Scale. Modified Poisson and Weighted Quantile Sum (WQS) regression was employed to examine associations between prenatal phthalates and risk of puberty onset, adrenarche, and gonadarche. Effect modification by child weight status was explored by stratified analysis. ResultsPrenatal exposure to DEHP was associated with higher risk of puberty onset (relative risk [RR] = 1.32, 95% CI = 1.09–1.59 per each log-unit increase in concentrations) and gonadarche (RR = 1.23, 95% CI = 1.00–1.50) in boys and higher risk of adrenarche (RR = 1.25, 95% CI = 1.03–1.51) in girls at age 7–10 years. In boys, prenatal exposure to DEP, DnBP, and DEHP was also associated with higher risk of adrenarche or gonadarche (RRs = 1.49–1.80) in those with normal weight, and BBzP and DINCH® exposure with lower risk of adrenarche (RR = 0.49, 95% CI = 0.27–0.89 and RR = 0.47, 95% CI = 0.24–0.90, respectively) in those with overweight/obesity. In girls, DiBP, DnBP, and DINCH® were associated with slightly higher risk of gonadarche (RRs = 1.14–1.19) in those with overweight/obesity. In the WQS model, the phthalate mixture was not associated with puberty in boys or girls. ConclusionPrenatal exposure to certain phthalates was associated with pubertal development at age 7–10 years, especially earlier puberty in boys with normal weight and girls with overweight/obesity. However, there was no evidence of effect of the phthalate mixture on advancing or delaying puberty in boys or girls.

Solutions of Schrodinger equation and thermodynamic properties of Iodine and Scandium Fluoride molecules based on Formula method

The study presents the thermodynamic properties of the Iodine and Scandium Fluoride molecules with molecular Deng-Fan potential. The bound state energy solution of the radial Schrodinger equation is obtained via the formula method. The partition function and other thermodynamic properties are evaluated via the Poisson summation approach. The numerical values of energy of the I2 and ScF molecules are found to be in agreement with results obtained from other methods in the literature. The results further show that the partition function decreases, and then converges to a constant value as temperature increases.