Haarlet analysis of Lipschitz regularity in metric measure spaces

In this note we shall give a characterization of Lipschitz spaces on spaces of homogeneous type via Haar coefficients.     

Dirichlet series with periodic coefficients

In this paper we shall unify the results obtained so far in various scattered literature, for Dirichlet characters and the associated Dirichlet L-functions, under the paradigm of periodic arithmetic functions and the associated Dirichlet series. Notably we shall determine the Laurent coefficients of the series in question to cover Funakura’s result and proceed on to prove the Ayoub-Berndt-Carlitz-Chowla-Müller-Redmond theorem.     

Computing eigenvalues of Sturm-Liouville problems via optimal control theory

In this paper, we shall address three problems arising in the computation of eigenvalues of Sturm-Liouville boundary value problems. We first consider a well-posed Sturm-Liouville problem with discrete and distinct spectrum. For this problem, we shall show that the eigenvalues can be computed by solving for the zeros of the boundary condition at the terminal point as a function of the eigenvalue. In the second problem, we shall consider the case where some coefficients and parameters in the differential equation are continuously adjustable. For this, the eigenvalues can be optimized with respect to these adjustable coefficients and parameters by reformulating the problem as a combined optimal control and optimal parameter selection problem. Subsequently, these optimized eigenvalues can be computed by using an existing optimal control software, MISER. The last problem extends the first to nonstandard boundary conditions such as periodic or interrelated boundary conditions. To illustrate the efficiency and the versatility of the proposed methods, several non-trivial numerical examples are included.     

负系数差分方程的振动性质

研究了一类负系数差分方程的振动性质,得到了这类差分方程振动的判据.     

负系数中立型差分方程的振动性质

研究了一类负系数中立型差分方程的振动性质,得到了这类差分方程振动的判据.     

Note on the Monotonicity of Coefficients for Some q-Series Arising from Ramanujan's Lost Notebook

In this note, we shall give a partition-theoretic interpretation which explains the non-negativity of the coefficients of some q-series arising from Ramanujan's Lost Notebook and prove the almost-increasingness of the coefficients via a vector partition generating function.     

Zum Franklin—Schneiderschen Satz

In this note we shall carry on further the simultaneous approximation of a, b and exp(bloga). In a recent paper BUNDSCHUH [2] proved a theorem which appears to be a sharpening of a theorem of SCHNEIDER [10]. But there is an error in the proof. We shall show, that under a supplementary condition, the theorem of BUNDSCHUH remains valid and as well, get an improvement to this theorem. We further give some results on linear forms in logarithms of two U-numbers with algebraic coefficients.     

高次Galerkin有限元法的超收敛性

通过局部误差估计,对具有光滑边界的二阶常系数椭圆型方程,给出了高次Ｇａｌｅｒｋｉｎ 有限元法的超收敛性． 运用对称技巧和积分恒等式技巧,在局部对称矩形网格或三角形网格上,我们得到了改进的超收敛性（提高１－ ３ 阶）．     

具有可测系数的二阶线性抛物型方程在高维区域的初—斜微商边值问题

Alkhutov,Manedov在[1]中讨论了具有可测系数的线性一致抛物型方程的Dirichlet问题,其中系数满足:这里k0,k1,p(＞n 2)是非负常数,本文讨论带有可测系数的一般线性一致抛物型方程的初-斜微商边值问题.     

A note on certain divisibility properties of the Fourier coefficients of normalized Eisenstein series

In this note, we shall prove certain divisibility properties of the Fourier coefficients of a class of normalized Eisenstein series modulo certain prime powers.     

Laguerre polynomials and the inverse Laplace transform using discrete data

We consider the problem of finding a function defined on (0,∞) from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution. Error estimate is given. Numerical results are produced.     

On the Cauchy problem for second order strictly hyperbolic equations with non–regular coefficients

In this paper we shall consider some necessary and sufficient conditions for well–posedness of second order hyperbolic equations with non–regular coefficients with respect to time. We will derive some optimal regularities for well–posedness from the intensity of singularity to the coefficients by WKB representation of the solution and some counter examples which are constructed by using ideas of Floquet theory.     

On continuous solutions of a generalized Cauchy-Riemann system with more than one singularity

We study the solvability of the Riemann-Hilbert problem for a generalized Cauchy-Riemann system with several singularities and reveal several new phenomenon. For the number of continuous solutions we shall show that it depends not only on the index but also on the location and type of the singularities; moreover, it does not depend continuously on the coefficients of the equation.     

General Inhomogeneous Discrete Linear Partial Differential Equations with Constant Coefficients on the Whole Spaces

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.     

二阶椭圆方程的下解及Holder连续的一个新证明(英文)

借助于检试函数和迭代技术,在系数弱光滑性假设条件下,本文研究了二阶椭圆方程的下解和正则性.进一步借助于局部估计和Poincare不等式,我们给出了Holder连续的一个新证明.     

Finite element error estimates for non-linear elliptic equations of monotone type

Summary In this paper we shall consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient, and the derivation of error estimates for the finite element approximations. Such problems arise in many practical situations — for example, in shock-free airfoil design, seepage through coarse grained porous media, and in some glaciological problems. By making use of certain properties of the nonlinear coefficients, we shall demonstrate that the variational formulations associated with these boundary value problems are well-posed. We shall also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities. With the aid of these inequalities and some standard results from approximation theory, we show how one may derive error estimates for the finite element approximations in the energy norm.     

Asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions

We study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s, and are given by a differential recursion equation. For positive integer s, series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients, while for s=0 they satisfy certain perturbed linear ordinary differential equations. For arbitrary s?0, these series solve a differential-integral equation. Such power series, in general, are not multisummable. However, we shall prove existence of solutions of the same differential-integral equation that in sectors of, in general, maximal opening have the formal series as their asymptotic expansion. Furthermore, we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner, thus exhibiting a Stokes phenomenon.     

Fuzzy linear programming problems as bi-criteria optimization problems

In this paper, we consider fuzzy linear programming (FLP) problems which involve fuzzy numbers only in coefficients of objective function. First, we shall give concepts of optimal solutions to (FLP) problems and investigate their properties. Next, in order to find all optimal solutions, we define three types of bi-criteria optimization problems.     

Continuity properties for flat families of polynomials (I) Continuous parametrizations

Inspired by classical results in algebraic geometry, we study the continuity with respect to the coefficients, of the zero set of a system of complex homogeneous polynomials with a given pattern and when the Hilbert polynomial of the generated ideal is fixed. In this work we prove topological properties of some classifying spaces, e.g. the space of systems with given pattern, fixed Hilbert polynomial is locally compact, and we establish continuous parametrizations of Nullstellensatz formulae. In the general case we get local rational results but in the complex case we get global results using rational polynomials in the real and imaginary parts of the coefficients. In a second companion paper, we shall treat the continuity of zero sets for the Hausdorff distance, i.e., from a metric point of view.     

The Cauchy problem for hyperbolic systems with Hölder continuous coefficients with respect to time

We discuss the local existance and uniqueness of solutions of certain nonstrictly hyperbolic systems, with Hölder continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones, then we shall prove them by use of Tanabe-Sobolevski’s method.