二元函数连续性条件之探析

对二元函数连续性判定条件给出了详细分析，强调有关问题的关键点，纠正了常见的模糊认识，得到一系列连续性充分条件及其推广形式．     

Neumann series and Lommel functions of two variables

We obtain several new results for Neumann series of Bessel functions as well as for its various special cases. The generalization of some well-known results for these kind of series, such as the Graf's addition theorem, are also established.     

多元函数的可微性

给出了多元函数可微性的等价条件 ;利用这个条件可以简便有效地判别多元函数的可微性     

On some discrete inequalities in two independent variables

In this paper we establish some new nonlinear difference inequalities. We also present an application of one inequality to certain nonlinear sum-difference equation.     

On derivatives of Bernstein type rational functions of two variables

In this paper the author defines Bernstein type rational functions of two variables and prove the approximation theorems for the derivatives of them.     

Complex versus real orthogonal polynomials of two variables

Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex Hermite orthogonal polynomials and the disk polynomials are used as illustrating examples.     

On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals

A sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the Banach space of continuous functions, under the supremum norm, from a compact Hausdorff topological space into a Banach space equipped with its weak topology . Strongly extreme points of the unit ball of are characterized as the norm-one functions that are uniformly strongly extreme point valued on a dense subset of . It is shown that a variety of stronger types of extreme points (e.g. denting points) never exist in the unit ball of . Lastly, some naturally arising and previously known extreme points of the unit ball of are shown to actually be strongly exposed points.

     

Correlations of functions of normal variables

Let (X₁, X₂,…, X_k, Y₁, Y₂,…, Y_k) be multivariate normal and define a matrix C by C_ij = cov(X_i, Y_j). If (i) (X₁,…, X_k) = (Y₁,…, Y_k) and (ii) C is symmetric positive definite, then 0 < varf(X₁,…, X_k) < ∞ corr(f(X₁,…, X_k),f(Y₁,…, Y_k)) > 0. Condition (i) is necessary for the conclusion. The sufficiency of (i) and (ii) follows from an infinite-dimensional version, which can also be applied to a pair of jointly normal Brownian motions.     

A result on common zeros location of orthogonal polynomials in two variables

The aim of this paper is to investigate some general properties of common zeros of orthogonal polynomials in two variables for any given region D⊂R² from a view point of invariant factor. An important result is shown that if X₀ is a common zero of all the orthogonal polynomials of degree k then the intersection of any line passing through X₀ and D is not empty. This result can be used to settle the problem of location of common zeros of orthogonal polynomials in two variables. The main result of the paper can be considered as an extension of the univariate case.     

Classical orthogonal polynomials in two variables: a matrix approach

Classical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated to a two-variable moment functional satisfying a matrix analogue of the Pearson differential equation. Furthermore, we characterize classical orthogonal polynomials in two variables as the polynomial solutions of a matrix second order partial differential equation. AMS subject classification 42C05, 33C50Partially supported by Ministerio de Ciencia y Tecnología (MCYT) of Spain and by the European Regional Development Fund (ERDF) through the grant BFM2001-3878-C02-02, Junta de Andalucía, G.I. FQM 0229 and INTAS Project 2000-272.     

On isocline lines for functions and convex stratifications of two variables

Let the isoclines of a function u be the level lines of the function θ = arg(Du). Formulas for the curvature and the length of isocline lines in terms of the curvatures k, j of the level curves and of the steepest descent lines of u are given. The special case when all isoclines are straight lines is studied: in this case the steepest descent lines bend proportionally to the level lines; the support function of the level lines is linear function on the isoclines parameterized by the level values, possibly changing them. This characterization gives a new proof of a property of the developable surfaces found in [A. Fialkow, Geometric characterization of invariant partial differential equations, Amer. J. Math. 59(4) (1937), pp. 833–844]. When u is in the class of quasi convex functions, the L _p norm of the length function I _θ of the isoclines has minimizers with isoclines straight lines; the same occurs for other functionals on u depending on k, j. For a strictly regular quasi convex function isoclines may have arbitrarily large length and arbitrarily large L ₁ norm of I _θ.     

Hermite-Spline-Interpolation Für Funktionen Von Zwei Variablen

We construct a smooth surface through n points in the Euclidean space with prescribed tangent planes in these points, which is also optimal in a certain sense.     

Sobolev orthogonal polynomials in two variables and second order partial differential equations

We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form

(∗)     

Optimal-order quadratic interpolation in vertices of unstructured triangulations

We study the problem of Lagrange interpolation of functions of two variables by quadratic polynomials under the condition that nodes of interpolation are vertices of a triangulation. For an extensive class of triangulations we prove that every inner vertex belongs to a local six-tuple of vertices which, used as nodes of interpolation, have the following property: For every smooth function there exists a unique quadratic Lagrange interpolation polynomial and the related local interpolation error is of optimal order. The existence of such six-tuples of vertices is a precondition for a successful application of certain post-processing procedures to the finite-element approximations of the solutions of differential problems. This work was supported by the grant GA ČR 103/05/0292.     

Hyperplane Singularities of Analytic Functions of Several Complex Variables

A new class of non-isolated singularities called hyperplane singularities is introduced. Special deformations with simplest critical points are constructed and an algebraic expression for the number of Morse points is given. The topology of the Milnor fibre is completely studied.     

Generic exponential sums associated with polynomials of degree 3 in two variables

The generic Newton polygon of L-functions associated with the exponential sums of polynomials of degree 3 in two variables is studied by Dwork’s analytic methods. Wan’s conjecture is shown to be true for this case.