On Cauchy differences of all orders

Summary This paper deals with the problem of characterizing higher order Cauchy differences of mappings on groups and semigroups. Symmetric, first order Cauchy differencesf(x + y)–f(x)–f(y) for mapsf between groups were characterized by Jessen, Karpf, and Thorup [8] through the use of first partial Cauchy differences. Our results are similar and extend their result to higher order differences. Our results also extend those of Heuvers [6] for mappings between vector spaces over the rationals.     

Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in Frith and Schauerte (Quaest Math 33:507–530, 2010) a completion (called a quasi-completion) was constructed for such structures and in Frith and Schauerte (Quaest Math, 2012) completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this paper questions of functoriality for this quasi-completion are considered and one sees that having enough regular Cauchy bifilters plays an important rôle. The quasi-complete strong quasi-nearness biframes with enough regular Cauchy bifilters are seen to form a coreflective subcategory of the strong quasi-nearness biframes with enough regular Cauchy bifilters. Here a significant difference between the symmetric and asymmetric cases emerges: a strong (even quasi-uniform) quasi-nearness biframe need not have enough regular Cauchy bifilters. The Cauchy filter quotient leads to further characterizations of those quasi-nearness biframes having enough regular Cauchy bifilters. The fact that the Cauchy filter quotient of a totally bounded quasi-nearness biframe is compact shows that any totally bounded quasi-nearness biframe with enough regular Cauchy bifilters is in fact quasi-uniform. The paper concludes with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases.     

On the quadrature methods for the numerical solution of singular integral equations

A Cauchy type singular integral equation can be numerically solved by the use of an appropriate numerical integration rule and the reduction of this equation to a system of linear algebraic equations, either directly or after the reduction of the Cauchy type singular integral equation to an equivalent Fredholm integral equation of the second kind. In this paper two fundamental theorems on the equivalence (under appropriate conditions) of the aforementioned methods of numerical solution of Cauchy type singular integral equations are proved in sufficiently general cases of Cauchy type singular integral equations of the second kind.     

Incomplete cauchy numbers

By using the restricted Stirling numbers and associated Stirling numbers, we introduce two kinds of incomplete Cauchy numbers, which are generalizations that of the classical Cauchy numbers. We also study several arithmetical and combinatorial properties.     

The generalized Cauchy problem with data on two surfaces for a quasilinear analytic system

We consider the generalized Cauchy problem with data on two surfaces for a second-order quasilinear analytic system. The distinction of the generalized Cauchy problem from the traditional statement of the Cauchy problem is that the initial conditions for different unknown functions are given on different surfaces: for each unknown function we pose its own initial condition on its own coordinate axis. Earlier, the generalized Cauchy problem was considered in the works of C. Riquier, N. M. Gyunter, S. L. Sobolev, N. A. Lednev, V. M. Teshukov, and S. P. Bautin. In this article we construct a solution to the generalized Cauchy problem in the case when the system of partial differential equations additionally contains the values of the derivatives of the unknown functions (in particular outer derivatives) given on the coordinate axes. The last circumstance is a principal distinction of the problem in the present article from the generalized Cauchy problems studied earlier.     

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.     

Lipschitz Continuous Solutions to the Cauchy Problem for Quasi-linear Hyperbolic Systems

Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasilinear hyperbolic systems are considered. Based on the methods of approximation and integral equations,the author gives two...     

Difference of generalized integration operators on the space of Cauchy transforms

In this paper, the boundedness and compactness of the differences of generalized integration operators from the space of Cauchy integral transforms to the Bloch-type spaces and the weighted Dirichlet spaces are investigated.     

BREAKDOWN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C1 solution to Cauchy problem.     

拟线性双曲组的经典解的奇性形成

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C1 solution to Cauchy problem.     

On the Cauchy Problem for D t 2-D xa(t,x)D x in the Gevrey Class of Order s > 2

We study the Cauchy problem for second order hyperbolic equations with non negative characteristic form of two independent variables. We show that for such equations in divergence-free form, the Cauchy problem is well posed in the Gevrey class of order less than 5/2.     

A Cauchy kernel for the Hermitian submonogenic system

Hermitian monogenic functions are the null solutions of two complex Dirac type operators. The system of these complex Dirac operators is overdetermined and may be reduced to constraints for the Cauchy datum together with what we called the Hermitian submonogenic system (see [8], [9]). This last system is no longer overdetermined and it has properties that are similar to those of the standard Dirac operator in Euclidean space, such as a Cauchy–Kowalevski extension theorem and Vekua type solutions. In this paper, we investigate plane wave solutions of the Hermitian submonogenic system, leading to the construction of a Cauchy kernel. We also establish a Stokes type formula that, when applied to the Cauchy kernel provides an integral representation formula for Hermitian submonogenic functions.     

q-Difference equation and the Cauchy operator identities

In this paper, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function for Rogers-Szegö polynomials. Applying the technique of parameter augmentation to two multiple generalizations of q-Chu-Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins-Miller transformation.     

Formation of Singularities of Solutions for Cauchy Problem of Quasilinear Hyperbolic Systems with Dissipative Terms

ln this paper, for a class of 2 × 2 quasilinear hyperbolic systems, we get existence theorems of the global smooth solutions of its Cauchy problem, under a certain hypotheses. In addition, Tor two concrete quasilinear hyperbolic systems, we study the formation of the singularities of the C¹-solution to its Cauchy problem.     

Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients

We study Gevrey properties and summability of power series in two variables that are formal solutions of a Cauchy problem for general linear partial differential equations with constant coefficients. In doing so, we extend earlier results in two articles of Balser and Lutz, Miyake, and Schäfke for the complex heat equation, as well as in a paper of Balser and Miyake, who have investigated the same questions for a certain class of linear PDE with constant coefficients subject to some restrictive assumptions. Moreover, we also present an example of a PDE where the formal solution of the Cauchy problem is not k-summable for whatever value of k, but instead is multisummable with two levels under corresponding conditions upon the Cauchy data. That this can occur has not been observed up to now.     

On the restricted evolution problem of the Einstein relativistic mechanics

This work is focused on the interior Cauchy problem for the Einstein’s field equations. Precisely, in the relativistic study of the evolution of a continuum reversible system, the Cauchy problem is broken into two separate problems: the initial data problem and the restricted problem of evolution.     

关于线性偏微分方程的Cauchy问题的唯一性

<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零.     

Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series

In this paper we obtain two new characterizations of completeness of a normed space through the behaviour of its weakly unconditionally Cauchy series. We also prove that barrelledness of a normed space X can be characterized through the behaviour of its weakly-* unconditionally Cauchy series in X*.     

泛Clifford分析中无界域上的Cauchy积分公式和Cauchy-Pompeiu公式

本文研究了泛Clifford分析中的Cauchy积分公式和Cauchy-Pompeiu公式.通过引入修正的Cauchy核,得出了取值在泛Clifford代数上的两公式在无界域上的表达式.此两公式是有界域上的相应结果的推广,并为研究无界域上的边值问题打下了基础.     

Divided differences and a unified approach to evaluating determinants involving generalized factorials

By employing divided differences, a unified approach to the evaluation of some determinant involving generalized factorials is proposed. Previous generalizations of the Vandermonde determinant and the Cauchy determinant due to Chu-Claudio, Chu-Wang-Zhang and Johnson are included as special cases of our unified treatment.