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This paper presents the uncoupled anisotropic thermoelasticity formulation for Boundary Element using the Dual Reciprocity Method. The proposed formulation demands two interpolation functions, one for the particular solution of the elastic problem and another to interpolate the derivatives of the thermal field. For the last one, the shape parameter of multiquadric Radial Basis Functions (RBF) was demonstrated to be a problem to be solved. Some polyharmonic RBFs were tested and a modified function was proposed. Singularity at derivatives of some functions made them impossible to be used and some of them demonstrated great sensibility to saturation. The use of internal points was mandatory under temperature fields irregularly distributed. Few polyharmonic functions were suitable to be used for thermoelasticity and low order functions presented reliable solutions, with the proposed modified RBF presenting slightly better performance over the previously used function. [ABSTRACT FROM AUTHOR]

In the present paper, the relatively new method of Radial Basis Function-Generated Finite Difference (RBF-FD) is used to solve a class of Partial Differential Equations (PDEs) with Dirichlet and Robin boundary conditions. For this approximation, Polyharmonic Splines (PHS) are used alongside Polynomials. This combination has many benefits. On the other hand, Polyharmonic Splines have no shape parameter and therefore relieve us of the hassle of calculating the optimal shape parameter. As the first problem, a two-dimensional Poisson equation with the Dirichlet boundary condition is investigated in various domains. Then, an elliptic PDE with the Robin boundary condition is solved by the proposed method. The results of numerical studies indicate the excellent efficiency, accuracy and high speed of the method, while for these studies, very fluctuating and special test functions have been used. [ABSTRACT FROM AUTHOR]

We give characterizations of the Dunkl polyharmonic functions, i.e., solutions to the iteration of the Dunkl-Laplace operator Δ κ which is a differential-reflection operator associated with a Coxeter–Weil group W generated by a finite set of reflections and an invariant multiplicity function κ , in terms of integral means over Euclidean balls and spheres. [ABSTRACT FROM AUTHOR]

Vibratory conveying of a material point by harmonic longitudinal oscillations and normal oscillations with a piecewise constant acceleration of an inclined conveying surface is considered. The dependence of dimensionless conveying velocity (a ratio of velocity to the product of frequency and amplitude of longitudinal oscillations) on several dimensionless parameters is investigated in the moving modes without hopping. Maximal conveying velocity is achieved at a certain value of phase difference angle between the longitudinal and normal oscillations, which is called optimal. The equations for determining the optimal phase difference angle are obtained. The values of dimensionless conveying velocity and optimal phase difference angles depend on dimensionless parameters: the inclination angle parameter, the intense vibration parameter, and the index of asymmetry of normal oscillations -- the ratio of the maximal acceleration to the gravitational acceleration. Comparison of vibratory conveying by normal oscillations with piecewise constant acceleration to conveying by normal polyharmonic oscillations shows an increase in conveying velocity with the index of asymmetry, equal to the number of harmonics, especially at large inclination angles of a conveying track. For further research, it is proposed to verify the obtained theoretical results through experimental studies. [ABSTRACT FROM AUTHOR]

In the paper, an elementary solution of polyharmonic equation is determined and, with the help of it, an integral representation of solutions of polyharmonic equation in a bounded domain is presented. [ABSTRACT FROM AUTHOR]

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 C N → C , z ↦ σ (b + w T z) with weights w ∈ C N and a bias b ∈ C. 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 C d 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, nor 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. [ABSTRACT FROM AUTHOR]

The authors consider the existence, multiplicity, and uniqueness for polyharmonic problem with Navier boundary conditions. One of the interesting features in our proof is that we give a new attempt to consider the uniqueness of nontrivial solution for the above polyharmonic problem by using the theory of monotone mappings. This is probably the first time this theory is used to solve polyharmonic problems. Then we apply the fixed point theorems on cones to analyze the existence and multiplicity of positive solutions for the above polyharmonic problem. This is very difficult for partial differential equations, especially for polyharmonic equations. The main reason is that the Green's function for the above polyharmonic problem is unbounded. We overcome the difficulties by using some new techniques. The uniqueness of nontrivial solution and the existence of positive solutions for polyharmonic equations with Dirichlet boundary conditions are also considered. [ABSTRACT FROM AUTHOR]

The asymptotic properties of Poisson-type integrals on the classes of differentiable functions are analyzed using modern methods of the optimal solution theory and approximation theory. Exact values of the upper bound of the deviation of functions of the Sobolev classes from Poisson-type integrals in the uniform metric are found. The research method used in the article makes it possible to estimate the error of the deviation of the classes of differentiable functions from their polyharmonic Poisson integrals with a predetermined accuracy. The results obtained in the study will contribute to the construction of higher-quality mathematical models of natural and social phenomena and, therefore, to more efficient solution of many problems of applied mathematics. [ABSTRACT FROM AUTHOR]

The asymptotic properties of integrals of the Poisson type on the classes of differentiable functions are analyzed with the use of modern methods of the theory of optimal solutions and the theory of approximation of functions. Namely, the exact values of the upper bound of the deviation of the functions of the Sobolev classes from integrals of the Poisson type in the uniform metric are found. The research method used in the study makes it possible to estimate the deviation error of the classes of differentiable functions from their polyharmonic Poisson integrals with predetermined accuracy. The results obtained in the study will further contribute to the construction of higher-quality mathematical models of natural and social phenomena and therefore to more efficient solution of many problems of applied mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a conformally flat and complete manifold (M , g) = (R n , e 2 u | d x | 2) with finite total Q-curvature. We introduce a new volume entropy, incorporating the background Euclidean metric, and demonstrate that the metric g is normal if and only if the volume entropy is finite. Furthermore, we establish an identity for the volume entropy utilizing the integrated Q-curvature. Additionally, under normal metric assumption, we get a result concerning the behavior of the geometric distance at infinity compared with Euclidean distance. With help of this result, we prove that each polynomial growth polyharmonic function on such manifolds is of finite dimension. Meanwhile, we prove several rigidity results by imposing restrictions on the sign of the Q-curvature. Specifically, we establish that on such manifolds, the Cohn-Vossen inequality achieves equality if and only if each polynomial growth polyharmonic function is a constant. [ABSTRACT FROM AUTHOR]

Iterated Dirichlet problem, also called as Riquier or Navier problem, and polyharmonic Dirichlet problem are studied for n-Poisson equation in Clifford analysis using iterated polyharmonic Green function and polyharmonic Green-Almansi type function appropriate for the boundary conditions of the problems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 △ n v = 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. [ABSTRACT FROM AUTHOR]

We consider the open unit disk D equipped with the hyperbolic metric and the associated hyperbolic Laplacian L. For λ ∈ C and n ∈ N , a λ -polyharmonic function of order n is a function f : D → C such that (L − λ I) n f = 0. If n = 1 , one gets λ -harmonic functions. Based on a Theorem of Helgason on the latter functions, we prove a boundary integral representation theorem for λ -polyharmonic functions. For this purpose, we first determine n th -order λ -Poisson kernels. Subsequently, we introduce the λ -polyspherical functions and determine their asymptotics at the boundary ∂ D , i.e., the unit circle. In particular, this proves that, for eigenvalues not in the interior of the L 2 -spectrum, the zeroes of these functions do not accumulate at the boundary circle. Hence the polyspherical functions can be used to normalise the n th -order Poisson kernels. By this tool, we extend to this setting several classical results of potential theory: namely, we study the boundary behaviour of λ -polyharmonic functions, starting with Dirichlet and Riquier type problems and then proceeding to Fatou type admissible boundary limits. [ABSTRACT FROM AUTHOR]

A specialized mesh-free radial basis function-based finite difference (RBF-FD) discretization is used to solve the large eigenvalue problems arising in hydrodynamic stability analyses of flows in complex domains. Polyharmonic spline functions with polynomial augmentation (PHS+poly) are used to construct the discrete linearized incompressible and compressible Navier-Stokes operators on scattered nodes. Rigorous global and local eigenvalue stability studies of these global operators and their constituent RBF stencils provide a set of parameters that guarantee stability without the need for hyperviscosity or other ad hoc regularizations, while balancing accuracy and computational efficiency. Specialized elliptical stencils to compute boundary-normal derivatives are introduced, and the treatment of reflectional and rotational symmetries is discussed. In particular, treating the pole singularity in cylindrical coordinates permits dimensionality reduction from 3D to 2D in rotational flows like jets. The numerical framework is demonstrated and validated on several hydrodynamic stability methods ranging from classical linear theory of laminar flows to state-of-the-art non-modal approaches that are applicable to turbulent mean flows. The examples include linear stability, resolvent, and wavemaker analyses of cylinder flow at Reynolds numbers ranging from 47 to 180, and resolvent and wavemaker analyses of the self-similar flat-plate boundary layer at a Reynolds number as well as the turbulent mean of a high-Reynolds-number transonic jet at Mach number 0.9. All previously-known results are found in close agreement with the literature. Finally, the resolvent-based wavemaker analyses of the Blasius boundary layer and turbulent jet flows offer new physical insight into the modal and non-modal growth in these flows. • Radial basis function framework for hydrodynamic stability analysis. • Accurate, stable, and efficient discretizations of large eigenvalue problems. • Treatment of boundary conditions and pole singularity for scattered nodes. • Validated for various stability theoretical methods and three benchmark problems. • Resolvent-based wavemaker analysis for Blasius boundary layers and turbulent jets. [ABSTRACT FROM AUTHOR]

The calculation of potential energy surfaces for quantum dynamics can be a time consuming task--especially when a high level of theory for the electronic structure calculation is required. We propose an adaptive interpolation algorithm based on polyharmonic splines combined with a partition of unity approach. The adaptive node refinement allows to greatly reduce the number of sample points by employing a local error estimate. The algorithm and its scaling behavior are evaluated for a model function in 2, 3, and 4 dimensions. The developed algorithm allows for a more rapid and reliable interpolation of a potential energy surface within a given accuracy compared to the non-adaptive version. [ABSTRACT FROM AUTHOR]

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 P (x) = ∑ k = 0 d x k a k , with Clifford coefficients a k ∈ R n , 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. [ABSTRACT FROM AUTHOR]

We consider a weighted version of Bergman type spaces with respect to iterated parabolic operators of fractional order on the upper half space. We discuss reproducing properties and the orthogonal projection. [ABSTRACT FROM AUTHOR]

We explore a class of quantum control operations based on a wide family of harmonic magnetic fields that vary softly in time. Depending on the magnetic field amplitudes taking part, these control operations can produce either squeezing or loop (orbit) effects, and even parametric resonances, on the canonical variables. For these purposes we focus our attention on the evolution of observables whose dynamical picture is ascribed to a quadratic Hamiltonian that depends explicitly on time. In the first part of this work we survey such operations in terms of biharmonic magnetic fields. The dynamical analysis is simplified using a stability diagram in the amplitude space, where the points of each region will characterise a specific control operation. We discuss how the evolution loop effects are formed by fuzzy (non-commutative) trajectories that can be closed or open, in the latter case, even hiding some features that can be used to manipulate the operational time. In the second part, we generalise the case of biharmonic fields and translate the discussion to the case of polyharmonic fields. Using elementary properties of the Toeplitz matrices, we can derive exact solutions of the problem in a symmetric evolution interval, leading to the temporal profile of those magnetic fields suitable to achieve specific control operations. Some of the resulting fuzzy orbits can be destroyed by the influence of external forces, while others simply remain stable. [ABSTRACT FROM AUTHOR]

We introduce intrinsic interpolatory bases for data structured on graphs and derive properties of those bases. Polyharmonic Lagrange functions are shown to satisfy exponential decay away from their centers. The decay depends on the density of the zeros of the Lagrange function, showing that they scale with the density of the data. These results indicate that Lagrange-type bases are ideal building blocks for analyzing data on graphs, and we illustrate their use in kernel-based machine learning applications. [ABSTRACT FROM AUTHOR]

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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. [ABSTRACT FROM AUTHOR]

In the paper we consider the polyharmonic Bergman space for the union of the rotated unit Euclidean balls. Using so called zonal polyharmonics we derive the formulas for the kernel of this space. Moreover, we study the weighted polyharmonic Bergman space. By the same argument we get the Bergman kernel for this space. [ABSTRACT FROM AUTHOR]

Let H denote the class of functions which are harmonic and univa- lent in the open unit disc D = {z : |z| < 1g. In this paper, we define and investigate a family of complex-valued harmonic functions that are sense-preserving and univalent in D: We obtain growth result, extreme points, convolution, convex combinations and the closure property under certain integral operator for this family of functions. [ABSTRACT FROM AUTHOR]

On a countable tree T, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator P. We provide a boundary integral representation for general eigenfunctions of P with eigenvalue λ ∈ C. This is possible whenever λ is in the resolvent set of P as a self-adjoint operator on a suitable ℓ2-space and the diagonal elements of the resolvent ("Green function") do not vanish at λ. We show that when P is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition holds for all λ≠ 0 in the resolvent set. These results extend and complete previous results by Cartier, by Figà-Talamanca and Steger, and by Woess. For those eigenvalues, we also provide an integral representation of λ-polyharmonic functions of any order n, that is, functions f : T → ℂ for which (λ ⋅ I − P)nf = 0. This is a far-reaching extension of work of Cohen et al., who provided such a representation for the simple random walk on a homogeneous tree and eigenvalue λ = 1. Finally, we explain the (much simpler) analogous results for "forward only" transition operators, sometimes also called martingales on trees. [ABSTRACT FROM AUTHOR]

We provide necessary and sufficient conditions for functions f 1 , ... , f 2 N defined on the Euclidean space R d such that the following interpolation problem can be solved: for ε > 0 and real numbers t 1 <... < t 2 N there exists a polyharmonic function u of order N defined on t 1 − ε , t 2 N + ε × R d → ℂ satisfying the interpolation condition u t j , y = f j y for all y ∈ R d. 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. [ABSTRACT FROM AUTHOR]

We introduce and develop the notion of spherical polyharmonics, which are a natural generalisation of spherical harmonics. In particular we study the theory of zonal polyharmonics, which allows us, analogously to zonal harmonics, to construct Poisson kernels for polyharmonic functions on the union of rotated balls. We find the representation of Poisson kernels and zonal polyharmonics in terms of the Gegenbauer polynomials. We show the connection between the classical Poisson kernel for harmonic functions on the ball, Poisson kernels for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua kernel for holomorphic functions on the Lie ball. [ABSTRACT FROM AUTHOR]

Highlights • Large PHS+poly based RBF-FD stencils can lead to high orders of accuracy without numerical ill-conditioning. • It can also combine high orders of accuracy near boundaries with an absence of Runge-phenomenon-type boundary errors. • Numerical explanations to this behavior are provided based on a closed-form expression for the RBF+poly cardinal functions. • It explains the role of polynomials and RBFs in RBF+poly approximations. Abstract Radial basis function generated finite difference (RBF-FD) approximations generalize grid-based regular finite differences to scattered node sets. These become particularly effective when they are based on polyharmonic splines (PHS) augmented with multi-variate polynomials (PHS+poly). One key feature is that high orders of accuracy can be achieved without having to choose an optimal shape parameter and without having to deal with issues related to numerical ill-conditioning. The strengths of this approach were previously shown to be especially striking for approximations near domain boundaries, where the stencils become highly one-sided. Due to the polynomial Runge phenomenon, regular FD approximations of high accuracy will in such cases have very large weights well into the domain. The inclusion of PHS-type RBFs in the process of generating weights makes it possible to avoid this adverse effect. With that as motivation, this study aims at gaining a better understanding of the behavior of PHS+poly generated RBF-FD approximations near boundaries, illustrating it in 1-D, 2-D and 3-D. [ABSTRACT FROM AUTHOR]

We present some explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary. [ABSTRACT FROM AUTHOR]

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions. [ABSTRACT FROM AUTHOR]

We study the unique (non-unique) solvability of the Neumann problem for the polyharmonic equation in unbounded domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight . Depending on the value of the parameter a, we prove uniqueness (or non-uniqueness) theorems or present exact formulas for the dimension of the solution space of the Neumann problem in the exterior of a compact set and in a half space. [ABSTRACT FROM AUTHOR]

Abstract Duke, Imamoḡlu, and Tóth constructed a polyharmonic Maass form of level 4 whose Fourier coefficients encode real quadratic class numbers. A more general construction of such forms was subsequently given by Bruinier, Funke, and Imamoḡlu. Here we give a direct construction of such a form for the full modular group and study the properties of its coefficients. We give interpretations of the coefficients of the holomorphic parts of each of these polyharmonic Maass forms as inner products of certain weakly holomorphic modular forms and harmonic Maass forms. The coefficients of square index are particularly intractable; in order to address these, we develop various extensions of the usual normalized Peterson inner product using a strategy of Bringmann, Ehlen and Diamantis. [ABSTRACT FROM AUTHOR]

Abstract We present a new method for the solution of PDEs on manifolds M ⊂ R d of co-dimension one using stable scale-free radial basis function (RBF) interpolation. Our method involves augmenting polyharmonic spline (PHS) RBFs with polynomials to generate RBF-finite difference (RBF-FD) formulas. These polynomial basis elements are obtained using the recently-developed least orthogonal interpolation technique (LOI) on each RBF-FD stencil to obtain local restrictions of polynomials in R 3 to stencils on M. The resulting RBF-LOI method uses Cartesian coordinates, does not require any intrinsic coordinate systems or projections of points onto tangent planes, and our tests illustrate robustness to stagnation errors. We show that our method produces high orders of convergence for PDEs on the sphere and torus, and present some applications to reaction–diffusion PDEs motivated by biology. Highlights • We present an RBF-FD method for 2-dimensional manifolds. • The method combines RBFs with carefully chosen polynomials. • Polynomials are obtained from least orthogonal interpolation. • Our method is free of stagnation errors. • Our method shows high orders of convergence. [ABSTRACT FROM AUTHOR]

We consider the weighted polyharmonic Bergman space , where is the space of all real-valued polyharmonic functions of degree m on and . has the reproducing kernel which is called by the weighted polyharmonic Bergman kernel. [ABSTRACT FROM AUTHOR]

We study the existence and positivity of solutions to problems involving higher-order fractional Laplacians (-Δ)s for any s > 1. In particular, using a suitable variational framework and the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for s ∈ (n, n + 1) with n ∈ N odd, and we mention some particular domains where positivity preserving properties do hold. [ABSTRACT FROM AUTHOR]

Let α ∈ ℕ, α ≥ 1 and (-Δ)α = -Δ((-Δ)α-1) be the polyharmonic operator. We prove existence and non-existence results for the following Hamiltonian systems of polyharmonic equations under Dirichlet boundary conditions ... where Ω is a sufficiently smooth bounded domain, N > 2α, v is the outward pointing normal to ∂Ω and the Hamiltonian H ∈ C¹(ℝ²; ℝ) satisfies suitable growth assumptions. [ABSTRACT FROM AUTHOR]

A spline-interpolation solution of static and dynamic elasticity theory problems is suggested. The method allows to solve the problems for solids with plane sections parallel to the plane XOY. We reduce space and dynamic problems to the series of plane boundary-value problems. The recursive formulas are obtained to determine the spline coefficients. The convergence of the constructed approximate solutions to the exact solutions is proved.

This paper deals with the function u which satisfies △ku = 0, where k ≥ 2 is an integer. Such a function u is called a polyharmonic function. The author gives an upper bound of the measure of the nodal set of u, and shows some growth property of u. [ABSTRACT FROM AUTHOR]