A problem of determining a special spatial part of 3D memory kernel in an integro‐differential hyperbolic equation

The inverse problem of determining 2D spatial part of integral member kernel in integro‐differential wave equation is considered. It is supposed that the unknown function is a trigonometric polynomial with respect to the spatial variable y with coefficients continuous with respect to the variable x. Herein, the direct problem is represented by the initial‐boundary value problem for the half‐space x>0 with the zero initial Cauchy data and Neumann boundary condition as Dirac delta function concentrated on the boundary of the domain . Local existence and uniqueness theorem for the solution to the inverse problem is obtained.     

Some properties of the Owen T-function

Some new relations for the Owen function are obtained including differentiation and integration formulas, integral representations and series. Applications of the Owen function in probability theory and in communication theory are shown.     

Simple Equations Method (SEsM): Algorithm,Connection with Hirota Method,Inverse Scattering Transform Method,and Several Other Methods

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.     

Weight distributions of a class of cyclic codes with arbitrary number of nonzeros in quadratic case

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three nonzeros. Recently, the authors of [37] constructed a class of cyclic codes with arbitrary number of nonzeros, and computed the weight distribution for several cases. In this paper, we determine the weight distribution for a new family of such codes. This is achieved by introducing certain new methods, such as the theory of Jacobi sums over finite fields and subtle treatment of some complicated combinatorial identities.     

A novel two‐dimensional CdII coordination polymer: poly[aqua[μ4‐2‐(4‐carboxylatobenzoyl)benzoato]cadmium(II)]

The title Cd^II coordination framework, [Cd(C₁₅H₈O₅)(H₂O)]_n or [Cd(bpdc)(H₂O)]_n [H₂bpdc is 2‐(4‐carboxybenzoyl)benzoic acid], has been prepared and characterized using IR spectroscopy, elemental analysis, thermal analysis and single‐crystal X‐ray diffraction. Each Cd^II centre is six‐coordinated by two O atoms from one 2‐(4‐carboxylatobenzoyl)benzoate (bpdc²⁻) ligand in chelating mode, three O‐donor atoms from three other bpdc²⁻ anions and one O atom from a coordinated water molecule in an octahedral coordination environment. Two crystallographically equivalent Cd^II cations are bridged by one O atom of the 2‐carboxylate group of one bpdc²⁻ ligand and by both O atoms of the 4‐carboxylate group of a second bpdc²⁻ ligand to form a binuclear [(Cd)₂(O)(OCO)] secondary building unit. Adjacent secondary building units are interlinked to form a one‐dimensional [Cd(OCO)₂]_n chain. The bpdc²⁻ ligands link these rod‐shaped chains to give rise to a complex two‐dimensional [Cd(bpdc)]_n framework with a 4,4‐connected binodal net topology of point symbol {4³.6².8}. The compound exhibits a strong fluorescence emission and typical ferroelectric behaviour in the solid state at room temperature.     

Soliton and other solutions to the (1+2)-dimensional chiral nonlinear Schrödinger equation

The (1+2)-dimensional chiral nonlinear Schrödinger equation (2D-CNLSE) as a nonlinear evolution equation is considered and studied in a detailed manner. To this end, a complex transform is firstly adopted to arrive at the real and imaginary parts of the model, and then, the modified Jacobi elliptic expansion method is formally utilized to derive soliton and other solutions of the 2D-CNLSE. The exact solutions presented in this paper can be classified as topological and nontopological solitons as well as Jacobi elliptic function solutions.     

JACOBI PSEUDOSPECTRAL METHOD FOR FOURTH ORDER PROBLEMS

In this paper, we investigate Jacobi pseudospectral method for fourth order problems. We establish some basic results on the Jacobi-Gauss-type interpolations in non-uniformly weighted Sobolev spaces, which serve as important tools in analysis of numerical quadratures, and numerical methods of differential and integral equations. Then we propose Jacobi pseudospectral schemes for several singular problems and multiple-dimensional problems of fourth order. Numerical results demonstrate the spectral accuracy of these schemes, and coincide well with theoretical analysis.     

椭圆曲线y~2=x~3-21x+90的正整数点

利用初等方法证明了椭圆曲线y~2=x~3-21x+90无正整数点.