Poisson Formula Research Articles

The Algebraic Multiplicity of the Spectral Radius of a Uniform Hypertree

It is well-known that the spectral radius of a connected uniform hypergraph is an eigenvalue of the hypergraph. However, its algebraic multiplicity remains unknown. In this paper, we use the Poisson Formula and the matching polynomials to give the algebraic multiplicity of the spectral radius of a uniform hypertree.

Generalized Poisson Formula for Second Order Elliptic Systems

Generalized Poisson Formula for Second Order Elliptic Systems

Almansi-Type Decomposition for Slice Regular Functions of Several Quaternionic Variables

AbstractIn this paper we propose an Almansi-type decomposition for slice regular functions of several quaternionic variables. Our method yields $$2^n$$ 2 n distinct and unique decompositions for any slice function with domain in $$\mathbb {H}^n$$ H n . Depending on the choice of the decomposition, every component is given explicitly, uniquely determined and exhibits desirable properties, such as harmonicity and circularity in the selected variables. As consequences of these decompositions, we give another proof of Fueter’s Theorem in $$\mathbb {H}^n$$ H n , establish the biharmonicity of slice regular functions in every variable and derive mean value and Poisson formulas for them.

A progressive approach to solving a generalized CEV-type model by applying symmetry-invariant surface conditions

<abstract><p>In this paper, we examine a type of constant elasticity of variance model that is subject to its terminal condition. We prove that certain transformations may be applied to obtain a simpler equation that has known solution processes. Four cases are obtained that play a role in specifying the many unknown parameters of the model. The corresponding terminal condition is transformed into an initial condition, and we then demonstrate how to solve this Cauchy problem by using Lie symmetries and Poisson's formula. Finally, we examine the behaviour of the obtained solutions.</p></abstract>

The characteristic polynomials of uniform double hyperstars and uniform hypertriangles

A k-power hypergraph is obtained from a graph by adding k−2 new vertices to each edge of the graph. A k-uniform double hyperstar is the k-power hypergraph of a double star. The k-uniform hypertriangle is the k-power hypergraph of a triangle. We present the characteristic polynomials of uniform double hyperstars by using the Poisson formula. Moreover, we find expressions for the traces of uniform hypertriangles and use these to obtain the characteristic polynomials of uniform hypertriangles.

Subresultants and the Shape Lemma

In nice cases, a zero-dimensional complete intersection ideal over a field has a Shape Lemma. There are also cases where the ideal is generated by the resultant and first subresultant polynomials of the generators. This paper explores the relation between these representations and studies when the resultant generates the elimination ideal. We also prove a Poisson formula for resultants arising from the hidden variable method.

Asymmetric distributions of desired component mass fraction in point samples

Introduction. Desired component mass fraction distributions in point samples at preparation plants are asymmetric, and it is due to the natural heterogeneity of ores. The theory of mass fractions distribution asymmetry in samples. The theory is based on the lump sampling of point samples. Mass fraction distributions here consist of two differentsized fractions: valuable mineral and rock. These distributions are always asymmetric. Lump dispersion for asymmetric distributions depends on the mass fraction of the valuable mineral for both released grains and aggregates. Mass fraction distribution in point samples with poor product mass fraction is described by the Poisson formula. Materials and methods of research. Mass fraction distribution correspondence to the Poisson formula was experimentally confirmed based on an artificial rock mass represented by quartz grit with colored markers. Point samples of different mass were collected from the rock mass. Experimental estimates. The distributions obtained experimentally correspond to the distributions calculated by the Poisson formula. The paper introduces a method for experimental determination of valuable mineral average grain size and ore texture index. These values were determined for asbestos ore to exemplify the method proposed. Results discussion. Sampling standards do not consider the asymmetry of mass fraction distributions in point samples. At preparation plants the asymmetry manifests itself in hurricane samples, positive product imbalance, and divergence of double weights analysis results, that exceed the acceptable limits. The asymmetry of components mass fraction distributions in concentrates should be taken into account in sampling standards and methods, as well as at preparation plants in general.

The Schwartz Inequality and the Schwartz Formula for A-Analytic Functions

In this paper, we study A-analytic functions. We provide main fundamental theorems of the theory of A-analytic functions and prove analogs of the Schwartz inequality, the Schwartz formula, and the Poisson formula for A-analytic functions.

Concentration phenomena for the fractional Q‐curvature equation in dimension 3 and fractional Poisson formulas

We study the compactness properties of metrics of prescribed fractional $Q$-curvature of order $3$ in $\R^3$. We will use an approach inspired from conformal geometry, seeing a metric on a subset of $\R^3$ as the restriction of a metric on $\R^4_+$ with vanishing fourth-order $Q$-curvature. We will show that a sequence of such metrics with uniformly bounded fractional $Q$-curvature can blow up on a large set (roughly, the zero set of the trace of a nonpositive biharmonic function $\Phi$ in $\R^4_+$), in analogy with a $4$-dimensional result of Adimurthi-Robert-Struwe, and construct examples of such behaviour. In doing so, we produce general Poisson-type representation formulas (also for higher dimension), which are of independent interest.

The Harmonicity of Slice Regular Functions

In this article, we investigate harmonicity, Laplacians, mean value theorems, and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well-known Representation Formula for slice regular functions over {mathbb {H}}. Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over {mathbb {H}} (analogous to an holomorphic function over {mathbb {C}}) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over {mathbb {H}}? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over {mathbb {H}} and a Jensen’s Formula for semi-regular ones.

The Poisson Formula for Solutions to Initialboundary Value Problems for B-Hyperbolic Equations with Bessel Operators with Negative Index

For the Euler–Poisson–Darboux type equations with the Bessel operators with negative index we consider the Cauchy problem with the boundary conditions. We obtain a representation of the solution in the form of the Poisson formula.

Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair h_1, h_2 of zonal harmonic functions such that f=h_1-bar{x} h_2. We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

Integral formulas of the Hilbert, Poincaré-Bertrand, Schwarz and Poisson type for the β–analytic function theory

In the present work we obtain some analogues of the Hilbert formulas on the unit circle and on the upper half-plane for the theory of solutions of a special case of the Beltrami equation in C to be referred as β-analytic functions. Furthermore, a Poincaré–Bertrand formula related to the β-Cauchy singular integral over a closed Jordan curve is derived and it is used to derive the corresponding Schwarz and Poisson formulas.

Mesh-free error integration in arbitrary dimensions: A numerical study of discrepancy functions

We are interested in mesh-free formulas based on the Monte-Carlo methodology for the approximation of multi-dimensional integrals, and we investigate their accuracy when the functions belong to a reproducing-kernel space. A kernel typically captures regularity and qualitative properties of functions “beyond” the standard Sobolev regularity class. We are interested in the issue whether quantitative error bounds can be a priori guaranteed in applications (including mathematical finance, scientific computing, and machine learning). Our main contribution is a numerical study of the error discrepancy function based on a comparison between several numerical strategies, when one varies the choice of the kernel, the number of approximation points, and the dimension of the problem. We consider two strategies in order to localize to a bounded set the standard kernels defined in the whole Euclidian space (exponential, multiquadric, Gaussian, truncated), namely, on one hand the class of periodic kernels defined via a discrete Fourier transform on a lattice and, on the other hand, a class of transport-based kernels. Relying on the Poisson formula on a lattice together with heuristic arguments, we study the derivation of theoretical bounds for the discrepancy function of periodic kernels. Then, for four kernels of particular interest we perform extensive numerical experiments and generate the optimal distributions of points and the discrepancy error functions. Our numerical results validate our theoretical observations and, importantly, provide us with quantitative estimates for the error made with a kernel-based strategy as opposed to a purely random strategy.

Boundary Value Problems for the Euler–Poisson–Darboux Equation

We consider the Cauchy problem for the Euler–Poisson–Darboux equation with boundary conditions. The Bessel operators in the equation can have different parameters. We establish a representation of the solution in the form of the Poisson formula with a special shift generated by the product of cylindrical functions of the first kind and different orders.

Explicit stencil computation schemes generated by Poisson's formula for the 2D wave equation

AbstractA new approach to building explicit time‐marching stencil computation schemes for the transient two‐dimensional acoustic wave equation is implemented. It is based on using Poisson's formula and its three time level modification combined with polynomial stencil interpolation of the solution at each time‐step and exact integration. The time‐stepping algorithm consists of two explicit stencil computation procedures: a first time‐step procedure incorporating the initial conditions and a two‐step scheme for the second and next time‐steps. Three particular explicit stencil schemes (with 5, 9, and 13 space points) are constructed using this approach. Their stability regions are presented. All of the obtained first time‐step computation expressions are different from those used in conventional finite‐difference methods. Accuracy advantages of the new schemes in comparison with conventional finite‐difference schemes are demonstrated by simulation using an exact benchmark solution.

Summation and the Poisson formula

By giving the definition of the sum of a series indexed by a set on which a group operates, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function and a few other series indexed by $${\mathbb {Z}}^k$$ has an intrinsic meaning as a complex number, independent of the requirements of analytic continuation. The definition of the sum requires nothing more than algebra and the concept of absolute convergence. The analytical significance of the algebraically defined sum is then explained by an argument that relies on the Poisson formula for tempered distributions.

Multi-scale Apparent Magnetization Inversion Imaging Method Based on the Theoretical Pseudo-gravity Anomalies

Abstract The complex superposition of magnetic anomalies on the ground caused by multi-source magnetic geologic bodies makes it difficult to invert their geometries and positions. According to the theoretical pseudo-gravity anomalies of the Poisson formula of the gravity and magnetic field, this paper converts the magnetic anomaly dipole source into the centroid anomaly of unipolar source of the gravity, adopts the multi-scale interpolation cutting method to separate the anomaly and utilizes the iterative method to downward continue the multi-scale anomaly to the close-to-source depth of the top interface of each magnetic source so as to acquire the high-resolution magnetic anomaly near the top interface of each source. This magnetization inversion of the anomaly can greatly improve the inversion accuracy and speed. However, the continuation space is always required to meet the first type of boundary conditions of the Laplace equation, which leads to the fact that the close-to-source level may not really reflect the real depth of the anomaly which contains the equivalent effect of depth. Therefore, the method is essentially a fast apparent magnetization imaging method which describes the multi-source and multi-scale geological bodies. The increased vertical resolution may help to better define the magnetic interface morphology, and it can also obtain the physical property distribution of the multi-scale magnetic source geological body in three-dimensional space. The method is demonstrated on the synthetic models and on an aeromagnetic data set from the Xisha Trough, South China Sea.

A combinatorial method for computing characteristic polynomials of starlike hypergraphs

By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike hypergraphs including hyperpaths and hyperstars,which are given recursively and explicitly.

Poisson Formulas for Boundary Value Problems for the Euler–Poisson–Darboux Equation

We consider the Cauchy problem for the Euler–Poisson–Darboux equation with boundary conditions. The Bessel operators in the equation can have different parameters. We establish a representation of the solution in the form of the Poisson formula with a special shift generated by the product of cylindrical functions of the first kind and different orders.