Poisson Formula Research Articles

Unified Treatment of Explicit and Trace Formulas via Poisson–Newton Formula

We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson-Newton formula.

A Poisson formula for the sparse resultant

We present a Poisson formula for sparse resultants and a formula for the product of the roots of a family of Laurent polynomials, which are valid for arbitrary families of supports. To obtain these formulae, we show that the sparse resultant associated to a family of supports can be identified with the resultant of a suitable multiprojective toric cycle in the sense of Remond. This connection allows to study sparse resultants using multiprojective elimination theory and intersection theory of toric varieties.

A variant of Schwarz-Pick lemma for planar harmonic mappings

Let w(z) = P[F](z) be a harmonic mapping defined in the unit disk D with the boundary function F satisfying w(0) = 0 and w(D) C D.In this paper by using Poisson formula and directional derivation,we prove a variant of Schwarz-Pick lemma for w(z) as follows:Λ_w(z) ≤ max h(x,r),0≤x≤1where h(x,r) is a continuous function of x which is given by(3.2).Furthermore,for some boundary functions F we prove that the above estimate is sharp.

A generalization of the Poisson integral formula for the functions harmonic and biharmonic in a ball

We construct an analytic solution to the problem of extension to the unit N-dimensional ball of the potential on its values on an interior sphere. The formula generalizes the conventional Poisson formula. Bavrin’s results obtained for the two-dimensional case by methods of function theory are transferred to the N-dimensional case (N ≥ 3). We also exhibit a solution to a similar extension problem for some operator expressions depending on a potential known on an interior sphere. A connection is established between solutions to the moment problem on a segment and on a semiaxis.

Analytical functions of generalized complex variables and some appllications

The object of this work is to use functions of a generalized complex variable to solve the problems of fluid dynamics and elasticity theory. In this paper, we obtain Cauchy-Riemann conditions, generalized Laplace equation and the generalized Poisson formula for such functions.

ERROR TERM IMPROVEMENTS FOR VAN DER CORPUT TRANSFORMS

We improve the error term in the van der Corput transform for exponential sums \sum_{a \le n \le b} g(n) exp(2\pi i f(n)). For many functions g and f, we can extract the next term in the asymptotic, showing that previous results, such as those of Karatsuba and Korolev, are sharp. Of particular note, the methods of this paper avoid the use of the truncated Poisson formula, and thus can be applied to much longer intervals [a,b] with far better results. We provide a detailed analysis of the error term in the case g(x)=1 and f(x)=(x/3)^{3/2}.

Noyaux du transfert automorphe de Langlands et formules de Poisson non linéaires

The main purpose of this paper is to show that some type of explicit nonlinear Poisson formulas, which is implied by Langlands’ functoriality principle, allows to build “kernels” of automorphic transfer. So, Langlands’ functoriality principle is equivalent to these nonlinear Poisson formulas.

Dynamic Reliability of Stochastic Truss Structures under Random Load

Aim at the dynamic reliability of stochastic truss structures under stationary random excitation. In consideration of the randomness of structural physical parameters and the geometric parameters and the random excitations were first transformed into sinusoidal ones in terms of the pseudo excitation method ( PEM) , that turned the joint-random problem into a single random problem for which only structural parameters remain random, and thus the analysis is greatly simplified. The mean value and the variance of the mean square value of the structural displacement and velocity were computed by using the theory of spectral analysis. And then the expressions of stochastic truss structural dynamic reliability were educed from the Poisson formula. Finally, the influence of the randomness of structural parameters on the structural dynamic reliability was analyzed by an example; the method of this paper was proved feasible and available comparing with the Monte Carlo method.

A unified solution of Laplace's equation in an arbitrary region

An exact solution for the Laplace's equation with Dirichlet boundary conditions in an arbitrary region is presented in this work. The solution is in the form of an integral in terms of the potential on the boundary, and depends on the geometry of the region. The method is valid for bounded as well as unbounded regions. Poisson's integral formula and Schwarz's integral rep resentation for the half-plane potential problem are obtained as a verification of the approach. The method also presents a generalized mean-value theorem for harmonic functions at any interior point of a circular disk in terms of weighted values at the boundary of the region.

Singularités et transfert

A transfer similar to that for endoscopy is introduced in the context of stably invariant harmonic analysis on reductive groups. For the group \(SL(2),\) the existence of the transfer is verified and some aspects of the passage from the trace formula to the Poisson formula are examined.

An Arithmetic Poisson Formula for the multi-variate resultant

In these pages we compute the expectation of several functions of multi-variate complex polynomials. The common thread of all our outcomes is the basic technique used in their proofs. The used techniques combine essentially the unitary invariance of Bombieri–Weyl’s Hermitian product and some elementary Integral Geometry. Using different combinations of these techniques we compute the expectation of the logarithm of the absolute value of an affine polynomial and we compute the expected value of Akatsuka Zeta Mahler’s measure. As main consequences of these results and techniques, we show a probabilistic answer to question (d) in Armentano and Shub (2012) [2], concerning the complexity of one point homotopy, and an Arithmetic Poisson Formula for the multi-variate resultant. These two last statements and bounds are related to the complexity of algorithms for polynomial equation solving.

Notes on the Poisson formula

These notes are a part of my lectures on representations of adelic groups attached to two-dimensional schemes. They contain a study of the one-dimensional case as a preliminary step to the case of dimension two. We consider the following issues: the Tate--Iwasawa method for algebraic curves; a discrete version and holomorphic duality; the Poisson formula and residues; explicit formulas; relation with the Artin representation; analogues for the number fields. With appendix on the Dedekind zeta-functions by Irina Rezvjakova.

Stochastic Boundary Methods of Fundamental Solutions for solving PDEs

Abstract A mesh-free Stochastic Boundary Method (SBM) based on randomized versions of the Method of Fundamental Solutions (MFS) is suggested. The randomization is used in the following steps of MFS: (1) the singular source positions are randomly distributed outside the domain, (2) the large system of linear equations for the weights in the expansion over the fundamental solutions is resolved by a randomized SVD method we introduced in [56] , or the randomized projection method we developed in [54] . We construct also a new method of stochastic boundary method based on the inversion of the Poisson formula representing the solution in a disc (a sphere, in R 3 ). We present a series of applications of the suggested SBM: we combine SBM with the Random Walk on Spheres and Random Walk on Boundary algorithms which results in methods giving the solution in any set of arbitrary points, without introducing any mesh in the domain. The Laplace, biharmonic, and the system of elasticity equations are involved in our analysis. We present some numerical results and give a brief discussion of the performance of the suggested methods. The numerical experiments carried out for the Laplace and Lame equations confirm our conclusion that the best results are obtained with the overdetermined systems generated by MFS where the number of source points is considerably smaller than the number of collocation points.

Two basic boundary-value problems for the inhomogeneous Cauchy–Riemann equation in an infinite sector

The object of the present article is to investigate the Schwarz and Dirichlet boundary value problems for the inhomogeneous Cauchy–Riemann equation in an infinite sector. Firstly, we obtain the Schwarz–Poisson formula in a sector with angle (). Secondly, boundary behaviors of some linear integrals will be studied, especially at the corner point. Finally, the solutions and the conditions of solvability are explicitly obtained.

Stochastic boundary collocation and spectral methods for solving PDEs

We develop a stochastic boundary method (SBM) which can be considered as a randomized version of the method of fundamental solutions (MFS). We suggest solving the large system of linear equations for the weights in the expansion over the fundamental solutions by a randomized SVD method introduced by Sabelfeld and Mozartova (2011). In addition, we also deal with solving inhomogeneous problems where we use the integral representation through the Green integral formula. The relevant volume integrals are calculated by a Monte Carlo integration technique which uses the symmetry of the Green function. We construct also a stochastic boundary method based on the spectral inversion of the Poisson formula representing the solution in a disc. This is done for the Laplace equation, and the system of elasticity equations. We stress that the stochastic boundary method proposed is of high generality; it can be applied to any bounded and unbounded domain with any boundary condition provided the existence and uniqueness of the solution are proven. We present a series of numerical results which illustrate the performance of the suggested methods.

Harmonic analysis and the Riemann-Roch theorem

This paper is a continuation of papers: arXiv:0707.1766 [math.AG] and arXiv:0912.1577 [math.AG]. Using the two-dimensional Poisson formulas from these papers and two-dimensional adelic theory we obtain the Riemann-Roch formula on a projective smooth algebraic surface over a finite field.

Quaternionic Analysis and the Schrödinger Model for the Minimal Representation of O(3,3)

In the series of papers [FL1, FL2] we approach quaternionic analysis from the point of view of representation theory of the conformal group SL(2,HC) ≃ SL(4,C) and its real forms. This approach has proven very fruitful and pushed further the parallel with complex analysis and develop a rich theory. In [FL2] we study the counterparts of Cauchy-Fueter and Poisson formulas on the spaces of split quaternions HR and Minkowski space M and show that they solve the problem of separation of the discrete and continuous series on SL(2,R) and the imaginary Lobachevski space SL(2,C)/SL(2,R). In particular, we introduce an operator PlR, compute its effect on the discrete and continuous series components of the space of functions H(H + ) and obtain a surprising formula for the Plancherel measure of SL(2,R). The proof is based on a transition to the Minkowski space M and some pretty lengthy computations. In this paper we introduce an operator d dR PlR on H(H + R ) and show that its effect on the discrete and continuous series components can be easily computed

Steady-state oscillations in the continuously inhomogeneous medium described by the Ovsyannikov equation

Using the group analysis methods, for the Ovsyannikov equation with maximal symmetry which describes steady-state oscillations in a continuous inhomogeneous medium, we obtain exact solutions to boundary-value problems for some regions (generalized Poisson formulas), which in particular can serve as test solutions for simulating steady-state oscillations in continuous inhomogeneous media. We find operators acting on the set of solutions in a one-parameter family of these equations.

Split quaternionic analysis and separation of the series for [formula omitted] and [formula omitted

We extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions H R . The special role of the unit sphere in the classical quaternions H – identified with the group SU ( 2 ) – is now played by the group SL ( 2 , R ) realized by the unit quaternions in H R . As in the previous work, we use an analogue of the Cayley transform to relate the analysis on SL ( 2 , R ) to the analysis on the imaginary Lobachevski space SL ( 2 , C ) / SL ( 2 , R ) identified with the one-sheeted hyperboloid in the Minkowski space M . We study the counterparts of Cauchy–Fueter and Poisson formulas on H R and M and show that they solve the problem of separation of the discrete and continuous series. The continuous series component on H R gives rise to the minimal representation of the conformal group SL ( 4 , R ) , while the discrete series on M provides its K-types realized in a natural polynomial basis. We also obtain a surprising formula for the Plancherel measure of SL ( 2 , R ) in terms of the Poisson-type integral on the split quaternions H R . Finally, we show that the massless singular functions of four-dimensional quantum field theory are nothing but the kernels of projectors onto the discrete and continuous series on the imaginary Lobachevski space SL ( 2 , C ) / SL ( 2 , R ) . Our results once again reveal the central role of the Minkowski space in quaternionic and split quaternionic analysis as well as a deep connection between split quaternionic analysis and the four-dimensional quantum field theory.

The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals – their interconnections

It is shown that the Whittaker–Kotel'nikov–Shannon sampling theorem of signal analysis, which plays the central role in this article, as well as (a particular case) of Poisson's summation formula, the general Parseval formula and the reproducing kernel formula, are all equivalent to one another in the case of bandlimited functions. Here equivalent is meant in the sense that each is a corollary of the other. Further, the sampling theorem is equivalent to the Valiron–Tschakaloff sampling formula as well as to the Paley–Wiener theorem of Fourier analysis. An independent proof of the Valiron formula is provided. Many of the equivalences mentioned are new results. Although the above theorems are equivalent amongst themselves, it turns out that not only the sampling theorem but also Poisson's formula are in a certain sense the ‘strongest’ assertions of the six well-known, basic theorems under discussion.