Presentations of a numerical semigroup

In this paper, we mainly study the minimal presentations of numerical semigroups. Moreover, we examine the concept of gluing, complete intersection, catenary degree, elasticity of some numerical semigroups.

New algorithm method for solving the variational inequality problem in Hilbert space

Theipurpose of,thisipaper,is toiintroduce,aiconcept of generalizedinon_spreading,and define a new algorithm,for infinite,families of generalizedinon_spreading,and finite families of resolvent,mappings. Also, We study,the existence,solution of variational inequality,to a commonifixedipoint in Hilbertispaces. The main,results in this paper extendiand generalized,of many knowniresults initheiliterature.

Mathematical study of the small oscillations of a finite cylindrical column liquid-gas under zero gravity

This paper deals with the mathematical study of the small motions of a system formed by a cylindrical liquid column bounded by two parallel circular rings and an internal cylindrical column constituted by a barotropic gas under zero gravity. From the equations of motion, the authors deduce a variational equation. Then, the study of the small oscillations depends on the coerciveness of a hermitian form that appears in this equation. It is proved that this last problem is reduced to an auxiliary eigenvalues problem. The discussion shows that, under a simple geometric condition, the problem is a classical vibration problem.

A new Alzer type Inequality Related to Binomial Function

In this paper, we establish a new Alzer type inequality related to binomial function by using Sitnik methods.

Simple forms for coefficients in two families of ordinary differential equations

In the paper, by virtue of the Faá di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion formula for the Stirling numbers of the first and second kinds, the author finds simple, meaningful, and significant forms for coefficients in two families of ordinary differential equations.

Inverse obstacle scattering with non-over-determined data

t is proved that the scattering amplitude \(A(\beta, \alpha_0, k_0)\), known for all \(\beta\in S^2\), where \(S^2\) is the unit sphere in \(\mathbb{R}^3\), and fixed \(\alpha_0\in S^2\) and \(k_0>0\), determines uniquely the surface \(S\) of the obstacle \(D\) and the boundary condition on \(S\). The boundary condition on \(S\) is assumed to be the Dirichlet, or Neumann, or the impedance one. The uniqueness theorem for the solution of multidimensional inverse scattering problems with non-over-determined data was not known for many decades.A detailed proof of such a theorem is given in this paper for inverse scattering by obstacles for the first time. It follows from our results that the scattering solution vanishing on the boundary \(S\) of the obstacle cannot have closed surfaces of zeros in the exterior of the obstacle different from \(S\). To have a uniqueness theorem for inverse scattering problems with non-over-determined data is of principal interest because these are the minimal scattering data that allow one to uniquely recover the scatterer.

Existence of the solutions to convolution equations with distributional kernels

It is proved that a class of convolution integral equations of the Volterra type has a globalsolution, that is, solutions defined for all \(t\ge 0\). Smoothness of the solution is studied.

Alternative proofs for summation formulas of some trigonometric series

In the paper, the authors supply alternative proofs for some summation formulas of rigonometric series.

Completeness of the set \(\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}\)

Let \(S^2\) be the unit sphere in \(\mathbb{R}^3\), \(k>0\) be a fixed constant, \(s\in S\), and \(S\) is a smooth, closed, connected surface, the boundary of a bounded domain \(D\) in \(\mathbb{R}^3\). It is proved that the set \(\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}\) is total in \(L^2(S)\) if and only if \(k^2\) is not a Dirichlet eigenvalue of the Laplacian in \(D\).

On Properties of meromorphic solutions of difference Painlevé I and II equation

In this paper, we investigate some properties of finite order transcendental meromorphic solutions of difference Painlev \(\)\acute{e}\) I and II equations, and obtain precise estimations of exponents of convergence of poles of difference \(\)\Delta w(z)=w(z+1)-w(z)\) and divided difference \(\)\frac{\Delta w(z)}{w(z)}\), and of fixed points of \(\)w(z+\eta)$ ($\eta\in \mathbb{C}\setminus\{0\}\)).