Identities for Catalan's Constant Arising from Integrals Depending on a Parameter

In this paper we provide some relationships between Catalan's constant and the ₃F₂ and ₄F₃ hypergeometric functions, deriving them from some parametric integrals. In particular, using the complete elliptic integral of the first kind, we found an alternative proof of a result of Ramanujan for ₃F₂, a second identity related to ₄F₃ and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.     

具有特征参数多项式边界条件的Sturm-Liouville 方程的逆结点问题

逆结点问题是通过特征函数的零点重构算子. 本文主要讨论具有特征参数多项式边界条件的 Sturm-Liouville 方程的逆结点问题. 20世纪50年代以后,人们发现在许多工程领域, Sturm-Liouville 问题的谱参数不仅出现在方程中, 而且也出现在边界条件中,因此带参数边界条件的逆结点问题对数学物理方面的研究有重要意义. 本文讨论区间 $[0,1]$ 上边界条件为参数多项式的 Sturm-Liouville 方程的逆结点问题, 并证明在 $[0,b]$ \big($ b\in \big(\frac{1}{2},1\big]$\big) 上结点的稠密子集可唯一确定 $[0,1]$ 上的势函数和边界条件中多项式的未知系数.     

A Sturm-Liouville Inverse Spectral Problem with Boundary Conditions Depending on the Spectral Parameter

We consider a boundary value problem generated by the Sturm-Liouville equation on a finite interval. Both the equation and the boundary conditions depend quadratically on the spectral parameter. This boundary value problem occurs in the theory of small vibrations of a damped string. The inverse problem, i.e., the problem of recovering the equation and the boundary conditions from the given spectrum, is solved.     

Direct and Inverse Problems of Electromagnetic Control

A mathematic model is suggested of the direct and universe problems of flaw detection. The direct problem consists in determining the complex amplitude of EMF in the through-type-induction- transducer–air gap–ferromagnetic cylinder system. In the direct problem, the distribution laws of magnetic permeability and electrical conduction are assumed to be known. A formula is obtained for calculating the complex amplitude of EMF in the class of piecewise-constant dependences of electromagnetic parameters. The inverse problem of flaw detection for a ferromagnetic cylinder consists in determining the magnetic permeability based on the measured values of the moduli of the EMF amplitudes in the system on a fixed frequency grid. An approximate solution of the inverse problem is searched in the class of piecewise-constant functions. Tikhonov’s method of regularization is used to solve the inverse problem. The results of numerical and physical modeling are presented.     

Direct and Inverse Problems for a Phase-Field Model with Memory

We consider a semilinear integrodifferential system in non-normal form. Such a system is a generalization of the one that arises in the phase-field theory with memory. We prove an abstract existence and uniqueness theorem and a continuous dependence result for the direct problem. Reformulating the direct problem in a suitable way we prove that the identification problem admits a unique solution.     

Analysis of Direct and Inverse Cavity Scattering Problems

Consider the scattering of a time-harmonic electromagnetic plane wave by an arbitrarily shaped and filled cavity embedded in a perfect electrically conducting infinite ground plane.A method of symmetric coupling of finite element and boundary integral equations is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases.Given the incident field,the direct problem is to determine the field distribution from the known shape of the cavity; while the inverse problem is to determine the shape of the cavity from the measurement of the field on an artificial boundary enclosing the cavity.In this paper,both the direct and inverse scattering problems are discussed based on a symmetric coupling method.Variational formulations for the direct scattering problem are presented,existence and uniqueness of weak solutions are studied,and the domain derivatives of the field with respect to the cavity shape are derived.Uniqueness and local stability results are established in terms of the inverse problem.     

Leray-Schauder Degree Method in Functional Boundary Value Problems Depending on the Parameter

Using the Leray-Schauder degree method sufficient conditions for the one-parameter boundary value problem x″ = f(t, x, x′, λ), α(x) = A, x(0) ? x(1) = B, x′(0) ? x′(1) = C, are stated. The application is given for a class of functional boundary value problems for nonlinear third-order functional differential equations depending on the parameter.     

On the Eigenvalue Accumulation of Sturm-Liouville Problems Depending Nonlinearly on the Spectral Parameter

A nonlinear spectral problem for a Sturm-Liouville equation-(p(x, λ)y'(x, λ))' + q(x, λ) y(x, λ) = 0 on a finite interval [a, b] with λ-dependent boundary conditions is considered. The spectral parameter λ is varying in an interval ∧ and p(x, λ), q(x, A) are real, continuous functions on [a, b] × ∧ Some criteria to the eigenvalue accumulation at the endpoints of A will be established. The results are applied to concrete problems arising in magnetohydrodynamics.     

Direct and Inverse Eigenvalue Problems for Diagonal-Plus-Semiseparable Matrices

In this paper we study both direct and inverse eigenvalue problems for diagonal-plus-semiseparable (dpss) matrices. In particular, we show that the computation of the eigenvalues of a symmetric dpss matrix can be reduced by a congruence transformation to solving a generalized symmetric definite tridiagonal eigenproblem. Using this reduction, we devise a set of recurrence relations for evaluating the characteristic polynomial of a dpss matrix in a stable way at a linear time. This in turn allows us to apply divide-and-conquer eigenvalue solvers based on functional iterations directly to dpss matrices without performing any preliminary reduction into a tridiagonal form. In the second part of the paper, we exploit the structural properties of dpss matrices to solve the inverse eigenvalue problem of reconstructing a symmetric dpss matrix from its spectrum and some other informations. Finally, applications of our results to the computation of a QR factorization of a Cauchy matrix with real nodes are provided.     

General Parameter Mapping Technique--A Procedure for Solution of Non-linear Boundary Value Problems Depending on an Actual Parameter

A new general method is developed to obtain the dependence ofthe solution of a non-linear boundary value problem on an arbitraryactual parameter of the problem under consideration. This techniqueis essentially an application of the implicit function theorem(Davidenko's approach) to the solution of non-linear boundaryvalue problems. The procedure suggested requires the integrationof sets of ordinary differential equations (initial value problems)with the right-hand sides obtained from a solution of a certainset of auxiliary differential equations.     

含参量积分的一条定理及其应用

给出了含参量积分的一条定理,借助其完善了积分交换顺序定理的证明,并且通过一个具体的例子刻画了这条定理的应用.     

Parabolic Equations with Large Parameter. Inverse Problems

Mathematical Notes - For an abstract parabolic equation with initial condition and multidimensional parabolic initial-boundary value problem with absolute terms rapidly oscillating in time, inverse...     

一类含参积分的连续性定理

本文首先研究同等连续函数列所具有的性质 ,并利用所得到同等连续函数列的性质证明了一类含参积分的连续性定理 .     

Reachable Sets and Integral Funnels of Differential Inclusions Depending on a Parameter

Doklady Mathematics - A parameter-dependent control system is considered in Euclidean space $${{\mathbb{R}}^{n}}$$ . The dependence, on the parameter, of the reachable sets and integral funnels of...     

Study of Volterra Integro-Differential Equations with Kernels Depending on a Parameter

We carry out spectral analysis of operator functions that are the symbols of integro-differential equations with unbounded operator coefficients in a separable Hilbert space. The structure and localization of the spectrum of operator functions which are symbols of these equations play an important role in studies of the asymptotic behavior of their solutions.     

Conditional Expectation Depending on a Parameter And Strong Holomorphic Conditional Probability

In this paper,we have given the existence theorem of the conditional expectation depending pa-rameter and the existence theorem of the strong holomorphic conditional probability,and another a few ralated theorems.These results are to approach one of addition for correlated components in the probabil-ity theory and the measure theory.     

在混合扰动下从闭轨族分支的极限环

本文讨论了对确定微分方程组的向量场和分析其轨线穿过方向的参考闭曲线族同时进行扰动分支极限环的方法，并给出了一个平面二次微分系统在混合扰动下分支出三个极限环的例子     

Inverse Problems of Finding the Absorption Parameter in the Diffusion Equation

Mathematical Notes - The paper is devoted to the study of inverse problems of finding, together with a solution u(x, t) of the diffusion equation $${u_t} - \Delta u + [c(x,t) + a{q_0}(x,t)]u =...     

The product of a Baire space with a hereditarily Baire metric space is Baire

In this paper we prove that the product of a Baire space with a metrizable hereditarily Baire space is again a Baire space. This answers a recent question of J. Chaber and R. Pol.

     

A Complete Topological Classification of the Space of Baire Functions on Ordinals

Considering the spaces B_p[1, α] of all Baire functions x: [1, α] → ? on the ordinal segments [1, α] that are endowed with the topology of pointwise convergence, we give a complete topological classification of these spaces.