Polyharmonic Functions Research Articles

Polyharmonic Functions in the Quarter Plane

Polyharmonic Functions in the Quarter Plane

Logarithmic terms in discrete heat kernel expansions in the quadrant

In the context of lattice walk enumeration in cones, we consider the number of walks in the quarter plane with fixed starting and ending points, prescribed step-set, and given length. After renormalisation, this number may be interpreted as a discrete heat kernel in the quadrant. We propose a new method to compute complete asymptotic expansions of these numbers of walks as their length tends to infinity, based on two main ingredients: explicit expressions for the underlying generating functions in terms of elliptic Jacobi theta functions along with a duality known as Jacobi transformation. This duality allows us to pass from a classical Taylor expansion of the series to an expansion at the critical point of the model. We work through two examples. First, we present our approach on the well-known Kreweras model, which is algebraic, and show how to obtain a complete asymptotic expansion in this case. We then consider a more generic (so-called infinite group) model and find the associated complete asymptotic expansion. In this second case, we prove the existence of logarithmic terms in the asymptotic expansion, and we relate the coefficients appearing in the expansion to polyharmonic functions. To our knowledge, this is the first time that logarithmic terms have been observed in the asymptotics of a class of lattice walks confined to a quadrant.

Full asymptotic expansion for orbit-summable quadrant walks and discrete polyharmonic functions

Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start- and endpoints for orbit-summable models with finite group, up to arbitrary precision. The resulting representation greatly resembles one conjectured for walks starting from the origin in 2020 by Chapon, Fusy and Raschel, differing only in terms appearing due to the periodicity of the model. We will see that the dependency on start- and endpoint is given by discrete polyharmonic functions, which are solutions of △nv=0 for a discretization △ of a Laplace–Beltrami operator. They can be decomposed into a sum of products of lower order polyharmonic functions of either the start- or the endpoint only, which leads to a partial extension of a theorem by Denisov and Wachtel.

Mean value characterizations of the Dunkl polyharmonic functions

Mean value characterizations of the Dunkl polyharmonic functions

Lattice sums for double periodic polyanalytic functions

In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, employing the Eisenstein summation, that the lattice sum S_2 is equal to pi for the square array. Further, it became clear that such equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions, including the conditionally convergent Eisenstein series. In 1935, Natanzon used a polyharmonic function to solve the plane elasticity problem. This paper is devoted to the extension of the classic lattice sums to the lattice sums for double periodic (pseudoperiodic) polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums.

The Single Degree of Freedom Simulation Model of Underwater Explosion Impact

The hulls of naval ships are exposed to forces and moments coming from internal and external sources. Usually, these are interactions that can be described mathematically by harmonic and polyharmonic functions. The shock of UNDEX type (underwater explosion) works completely differently and its time waveform is difficult to describe with mathematical functions as pressure vs. time. The paper presents a simplification of physical and mathematical models of 1-D kickoff pressure whose aim is performance the simulation of the external force of the detonation wave. The proposed models were verified and tuned on naval, sea trials. The main goals of the proposed models are to perform simulation calculations of the detonation pressure for different explosion charge weights from different distances of the UNDEX epicentre for the design process of machine foundation. The effects of pressure are transformed as impulses exposed on shock absorber mounted at light shock machine.

The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the S-spectrum. The function spaces and the associated functional calculi define the so-called fine structure of the spectral theories on the S-spectrum. Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

The universal approximation theorem for complex-valued neural networks

We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function σ:C→C in which each neuron performs the operation CN→C,z↦σ(b+wTz) with weights w∈CN and a bias b∈C, and with σ applied componentwise. We completely characterize those activation functions σ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of Cd arbitrarily well. Unlike the classical case of real networks, the set of “good activation functions”—which give rise to networks with the universal approximation property—differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as σ is neither a polynomial, a holomorphic function, or an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of σ is not a polyharmonic function.

Harnack inequality for polyharmonic equations

Some new types of mean value formulas for the polyharmonic functions were established. Based on the formulas, the Harnack inequality for the nonnegative solutions to the polyharmonic equations was proved.

Solving Inverse Problems of Stationary Convection–Diffusion Equation Using the Radial Basis Function Method with Polyharmonic Polynomials

In this article, the radial basis function method with polyharmonic polynomials for solving inverse problems of the stationary convection–diffusion equation is presented. We investigated the inverse problems in groundwater pollution problems for the multiply-connected domains containing a finite number of cavities. Using the given data on the part of the boundary with noises, we aim to recover the missing boundary observations, such as concentration on the remaining boundary or those of the cavities. Numerical solutions are approximated using polyharmonic polynomials instead of using the certain order of the polyharmonic radial basis function in the conventional polyharmonic spline at each source point. Additionally, highly accurate solutions can be obtained with the increase in the terms of the polyharmonic polynomials. Since the polyharmonic polynomials include only the radial functions. The proposed polyharmonic polynomials have the advantages of a simple mathematical expression, high precision, and easy implementation. The results depict that the proposed method could recover highly accurate solutions for inverse problems with cavities even with 5% noisy data. Moreover, the proposed method is meshless and collocation only such that we can solve the inverse problems with cavities with ease and efficiency.

Mean value theorems for polyharmonic functions

The problem of characterization of polyharmonic functions by mean value expressions is analyzed. A sufficient condition for the biharmonic function is formulated and proved in the paper. Mean value theorems and their operators have various applications in function theory (approximation of functions, description of functional spaces) and in the qualitative theory of linear differential equations with partial derivatives (boundary properties, elimination of features, differential properties of solutions, etc.). The main properties of polyharmonic functions, in particular, properties with mean value are investigated in the work. The results of L. Salcman and V. Volchkov, which generalize the classical mean value theorems, are analyzed. These results also reveal deep connections between complex analysis, the theory of linear differential equations with partial derivatives, harmonic analysis, integral geometry and the theory of special functions. The new directions of research in the theory of polyharmonic functions, differential equations, as well as in computational mathematics are established with the mean value theorems (algorithms "walk on spheres"\ , application of the Monte Carlo method). Particular attention is paid to the case of biharmonic functions. It is found that when performing the equality with the mean value on the circle from the set on the complex plane, the function of class $C^2$ on the specified set satisfies the Laplace equation of the second order, i.e. the function is polyharmonic of the second order. By using a smooth function with a compact support, the required function is obtained as a weak solution of the corresponding Laplace equation. In the future, this result can be used to establish a sufficient condition of biharmonicity of a twice continuously differentiated function on a plane that satisfies the above-mentioned relation only for a pair of positive values of $r_1$, $r_2$. Thus, it is possible to formulate a new theorem on two radii.

Weighted integrability of polyharmonic functions in the higher-dimensional case

This paper is concerned with the $L^p$ integrability of $N$-harmonic functions with respect to the standard weights $(1-|x|^2)^{\alpha}$ on the unit ball $\mathbb{B}$ of $\mathbb{R}^n$, $n\geq 2$. More precisely, our goal is to determine the real (negative) parameters $\alpha$, for which $(1-|x|^2)^{\alpha/p} u(x) \in L^p(\mathbb{B})$ implies that $u\equiv 0$, whenever $u$ is a solution of the $N$-Laplace equation on $\mathbb{B}$. This question is motivated by the uniqueness considerations of the Dirichlet problem for the $N$-Laplacian $\Delta^N$. Our study is inspired by a recent work of Borichev and Hedenmalm [Adv. Math., 264(2014), pp. 464-505], where a complete answer to the above question in the case $n=2$ is given for the full scale $0<p<\infty$. When $n\geq 3$, we obtain an analogous characterization for $\frac{n-2}{n-1}\leq p<\infty$, and remark that the remaining case can be genuinely more difficult. Also, we extend the remarkable cellular decomposition theorem of Borichev and Hedenmalm to all dimensions.

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.

О ЗАДАЧЕ ДИРИХЛЕ ДЛЯ НЕЛОКАЛЬНОГО ПОЛИГАРМОНИЧЕСКОГО УРАВНЕНИЯ

The paper studies the solvability conditions for one class of boundary value problems for a nonlocal polyharmonic equation in the unit ball with Dirichlet conditions on the boundary generated by a certain orthogonal matrix. The existence and uniqueness of the solution to the posed Dirichlet problem are investigated and the Green's function is constructed. First, some auxiliary statements are established: the inversability of the Vandermonde matrix of the m-th roots of unity is investigated, then the eigenvectors and eigenvalues of the auxiliary matrix generated by the coefficients of the nonlocal operator of the problem are found, and then the inverse matrix to it is obtained. To prove the uniqueness of the solution to the problem, the commutativity of the boundary operators and the nonlocal operator of the problem is established, and it is shown that if a solution to the problem exists, then this solution is a polyharmonic function. Then the conditions for the uniqueness of the solution to the problem under consideration are obtained. Further, on the basis of the auxiliary statements obtained above, conditions for the existence of a solution to the nonlocal problem are found. The solution to this problem is written out through the solution of auxiliary Dirichlet problems for the polyharmonic equation in the unit ball. Finally, using the well-known Green's function of the Dirichlet problem for the polyharmonic equation in the unit ball, the Green's function of the original nonlocal problem is constructed.

Mixed Convection Heat Transfer in a Lid-Driven Cavity under the Effect of a Partial Magnetic Field

Mixed convection flow in a lid-driven cavity in the presence of a uniform partial magnetic field is investigated utilizing the pseudo spectral method based on polyharmonic spline radial basis functions. The two-dimensional problem is concerned in a unit square cavity designed by adiabatic horizontal walls and vertical walls with constant temperatures. The top wall moves either in the positive x or in the negative x directions. The partial magnetic field is taken into account in the positive y direction. Working fluid is chosen as air. Efficiency of dimensionless Hartmann number and Richardson number on fluid flow and heat transfer is presented as well as the length and the central location of partial magnetic field. Flow is affected in the direction of vertical partial magnetic field, and is propelled to the other parts of the cavity regarding the central location of magnetic field. In case of middle centered partial magnetic field. The increase in the length of the partial magnetic field and the middle centered partial magnetic field as well reduce the convective heat transfer. The negative x directed lid-driven cavity has less influence on convection than the positive x directed lid-driven cavity.

Boundary behaviour of lambda -polyharmonic functions on regular trees

This paper studies the boundary behaviour of lambda -polyharmonic functions for the simple random walk operator on a regular tree, where lambda is complex and |lambda |> rho , the ell ^2-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential Fatou theorem is proved.

Almansi-type theorems for slice-regular functions on Clifford algebras

We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with polyharmonic functions, the elements of the kernel of an iterated Laplacian. Here, we consider polynomials of the form , with Clifford coefficients , and get an analogous decomposition related to zonal polyharmonics. We show the relation between such decomposition and the Dirac (or Cauchy–Riemann) operator and extend the results to slice-regular functions.

GROWING POLYHARMONIC FUNCTIONS AND THE CAUCHY PROBLEM

In this paper, the orderly polyharmonic function and its normal derivatives are studied by continuing it to the internal points of this field on the given values in a part of the boundary of the field, i.e. the solution of the Cauchy problem is given. This is one of the most inaccurate issues.

Boundary value problems for systems of inhomogeneous polyharmonic equations with applications in the theory of thin shells and plates

A number of problems in the theory of elasticity, the theory of heterogeneous media, the theory of thin shells and plates is reduced to solving boundary value problems for systems of inhomogeneous polyharmonic equations. The paper proposes a numerical algorithm for solving systems of polyharmonic equations of the form in single-connected and multi-connected areas with a piecewise smooth contour with specified boundary conditions. Two cases are considered when the function is a known polyharmonic function and when the function is also the desired polyharmonic function. The boundary conditions can have the form similar to Dirichlet conditions, Neiman conditions, and can have a mixed form when on one part of the boundary conditions of the Dirichlet type are given, and on the other - hand, the conditions of the Neiman type. On the basis of multiple applications of the Laplace operator and the boundary element method, which is based on the green integral identity, the given system is reduced to a system of integral identities. After approximating the boundary by an inscribed n-gon and discretizing the system of integral identities, the latter is reduced to a system of linear algebraic equations, which is conveniently represented as a system of matrix equations. The existence and uniqueness of the solution follows from the existence of a unique solution of a system of linear algebraic equations. Special attention is paid to the application of the algorithm to the solution of problems on the bending of thin plates, and the bending load can be a known function, and can be an unknown polyharmonic function of an arbitrary order with given boundary conditions. The problem of bending a thin plate of elliptic shape with a known load on the surface is solved, as well as the problem of bending a thin square plate with an unknown load, which is the solution of a harmonic equation with given boundary conditions. The level lines are constructed and the forms of curved plates are given.

Interpolation of data functions on parallel hyperplanes

We provide necessary and sufficient conditions for functions f1,…,f2N defined on the Euclidean space Rd such that the following interpolation problem can be solved: for ε>0 and real numbers t1<...<t2N there exists a polyharmonic function u of order N defined on t1−ε,t2N+ε×Rd→ℂ satisfying the interpolation condition utj,y=fjy for all y∈Rd. An important ingredient of the proof is the study of the interpolation problem for a special class of exponential polynomials which is interesting in its own right.