预拟不变凸函数判别准则的注记

Yang et al gave some criteria of prequasi-invex functions,semistrictly prequasi-invex functions and strictly prequasi-invex functions in 2001 ,under a certain set of conditions. In this note,some of these conditions can be weakened to get the same results ,and another simplified proof for a criterion of prequasi-invex functions established under the condition of lower semicontinuity is given.     

WEAK CONVERGENCE OF HENSTOCK INTEGRABLE SEQUENCES

Some relationships between pointwise and weak convergence of a sequence of Henstock integrable functions are studied, In particular it is provided an example of a sequence of Henstock integrable functions whose pointwise limit is different from the weak one. By introducing an asymptotic version of the Henstock equiintegrability notion it is given a necessary and sufficient condition in order that a pointwisely convergent sequence of Henstock integrable functions is weakly convergent to its pointwise limit.     

On the Rational Spline Functions

Since Schoenberg's fundamental paper was published in 1946, the theory of polynomial spline functions developed rapldly and was rich in contents. Spline functions have wide range of applications to numerical approximation.But to certain functions, for example, to the function with singular points, the interpolation by polynomial spline is not always efficient and reasonable. Since 1973 some authors have investigated the rational spline functions, and abtained a lot of results. But     

A Partial Differential Inequality in Geological Models

Sedimentation and erosion processes in sedimentary basins can be modeled by a parabolic equation with a limiter on the fluxes and a constraint on the time variation.This limiter happens to satisfy a stationary scalar hyperbolic inequality,within a constraint,for which the authors prove the existence and the uniqueness of the solution.Actually,this solution is shown to be the maximal element of a convenient convex set of functions.The existence proof is obtained thanks to the use of a numerical scheme.     

PLANE ELASTIC PROBLEMS OF DIFFERENT MEDIA WITH CRACKS ON THE INTERFACE

The equllibrium problem for the infinite elastic plane consisting of two different media with many cracks on the interface is discussed.It is transferred to a boundary value problem for analytic functions and then further reduced to a singular integral equation,the unique solvability and an effective method of solution for which are eatablished.A practical example in applications is illustrated,the solution of which is obtained in closed form.     

Teaching Power Series

<正> Power series is an important topic in calculus. Many well-known elementary functions can be expanded into power series. Furthermore, the derivative of a power series is equal to the sum of its termwise derivatives. The proof involves the use of uniform convergence and is often omitted in a calculus course. In this note, we shall demonstrate how the subject can be     

The Stable Farkas Lemma for composite convex functions in infinite dimensional spaces

In this paper, we consider a general composite convex optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. Some Fenchel conjugate transforms for the composite convex functions are derived to obtain the equivalent condition of the Stable Farkas Lemma, which is formulated by using the epigraph of the conjugates for the convex functions involved and turns out to be weaker than the classic Slater condition. Moreover, we get some necessary and sufficient conditions for stable duality results of the composite convex functions and present an example to illustrate that the monotonic increasing property of the outer convex function in the objective function is essential. Our main results in this paper develop some recently results.     

Harmonic L~p-functions on Complete Riemannian Manifolds

This survey paper concerns some existence theorems of harmonic functions belonging to LP (M), M being a complete Riemannian manifold. It is well known that a function which is analytic and bounded on the whole complex plane must reduce to a constant.This classical result, known as Liouville's theorem, is also true on a higher-dimensional Euclidean spaces. The generalization of this theorem to other Riemannian manifolds is very interesting. Besides its beauty, the proof usally requires sharp estimates which provide deeper understanding of the Laplacian and hence give broad applications to problems in global analysis.The basic problem in this paper is to study how the geometric conditions of a complete Riemannian manifold affect the validity of the Liouville theorem. The paper consists of two parts. Part I describes the results systematically and Part I will be more technical and will contain the detailed proofs of the results given in the first part.     

EQUIVALENCE OF SEMI-LAGRANGIAN AND LAGRANGE-GALERKIN SCHEMES UNDER CONSTANT ADVECTION SPEED

We compare in this paper two major implementations of large time-step schemes for advection equations, i.e., Semi-Lagrangian and Lagrange-Galerkin techniques. We show that SL schemes are equivalent to exact LG schemes via a suitable definition of the basis functions. In this paper, this equivalence will be proved assuming some simplifying hypoteses, mainly constant advection speed, uniform space grid, symmetry and translation invariance of the cardinal basis functions for interpolation. As a byproduct of this equivalence, we obtain a simpler proof of stability for SL schemes in the constant-coefficient case.     

An integral representation for the weighted geometric mean and its applications

By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.