逐段单调函数的高度与拓扑共轭

迭代运算下,函数值可以交叉于不同的子区间,使得逐段单调函数的高度异常复杂.本文考虑一个非单调点的连续函数类.首先给出高度的充分必要条件,以此获得此类函数的一种划分.其次针对函数类的一个非空子集,给出判定拓扑共轭的充分必要条件和构造拓扑共轭的新方法.进一步地,我们阐明这样的事实:两个逐段单调函数拓扑共轭是其高度相等的充分不必要条件,最后举例说明.     

A linear perturbed Palais-Smale condition for lower semicontinuous functions on Banach spaces

This paper introduces a notion of linear perturbed Palais-Smale condition for real-valued functions on Banach spaces. In terms of strongly exposed points, it presents a characterization which guarantees linear perturbed Palais-Smale condition holds for lower semicontinuous functions with bounded effective domains defined on a Banach space with the Radon-Nikody＇m property; and gives an example showing that linear perturbed P-S condition is strictly weaker than the P-S condition.     

A new geometric condition for Fenchel's duality in infinite dimensional spaces

In 1951, Fenchel discovered a special duality, which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions, using conjugates. Fenchel's duality is central to the study of constrained optimization. It requires an existence of an interior point of a convex set which often has empty interior in optimization applications. The well known relaxations of this requirement in the literature are again weaker forms of the interior point condition. Avoiding an interior point condition in duality has so far been a difficult problem. However, a non-interior point type condition is essential for the application of Fenchel's duality to optimization. In this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions. We also establish a necessary and sufficient condition for the ε-subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions. Our results offer further insight into Fenchel's duality. Dedicated to Terry Rockafellar on his 70th birthday     

Lipschitz B-Vex Functions and Nonsmooth Programming

In this paper, the equivalence between the class of B-vex functions and that of quasiconvex functions is proved. Necessary and sufficient conditions, under which a locally Lipschitz function is B-vex, are established in terms of the Clarke subdifferential. Regularity of locally Lipschitz B-vex functions is discussed. Furthermore, under appropriate conditions, a necessary optimality condition of the Slater type and a sufficient optimality condition are obtained for a nonsmooth programming problem involving B-vex functions.     

A weak condition of globally asymptotic stability for neural networks

In this work we consider a general class of continuous activation functions which may be neither bounded nor differentiable; however, many sigmoidal functions are included as special cases. With this class of activation functions we give a result on asymptotic stability for neural networks under a weak condition of nonnegative definiteness. Then we show that differentiability is a condition for its exponential stability.     

一类自相似集重分形分支的量纲

本文首先定义具有量纲函数的重分形测度,然后证明当Euclid空间中的两个重分形测度具有等价的量纲函数时,它们也等价.进一步,对于直线上满足强分离条件(SSC)的自相似集,在某些加倍条件下,本文给出判断其重分形分支的量纲函数的充要条件.     

An approximate method of calculating the oscillations of a liquid in inclined elastic cavities and channels

Small oscillations of an ideal incompressible liquid, which partially fills an inclined elastic container (a mobile cavity or channel) of arbitrary form and having a longitudinal plane of symmetry, are considered. The integral condition of continuity of the liquid is obtained by integrating the differential condition of incompressibility and exact satisfaction of the kinematic boundary conditions on the wetted side walls. Using this equation, systems of coordinate functions are constructed which represent the kinematically possible displacements of the liquid, for calculating the oscillations by the Ritz method and the finite-element method. The basic unknown functions, which describe the displacements of the liquid in cross-sections, are approximated by power functions and Legendre functions. Transverse layers of liquid, within the limits of the thickness of which a linear approximation for the unknown functions can be used, are considered as finite elements.     

The extension problem for functions with zero weighted spherical means

We study functions on a sphere with a pricked point having zero integrals with a given weight over all admissible “hemispheres”. We find a condition under which the point is a removable set for such a class of functions. We show that this condition cannot be dropped or substantially weakened.     

Nonseparable,Compactly Supported Interpolating Refinable Functions with Arbitrary Smoothness

In this paper we construct families of compactly supported nonseparable interpolating refinable functions with arbitrary smoothness (or regularity). The symbols for the newly constructed scaling functions are given by a simple formula related to the Bernstein polynomials. The emphasis of the paper is to show that under an easy-to-verify geometric condition these families satisfy Cohenrs condition, and they have arbitrarily high regularity. Furthermore, the constructed scaling functions satisfy, under the same geometrical condition, the Strang–Fix conditions of arbitrarily high order, which implies that corresponding interpolating schemes have arbitrarily high accuracy.     

Functions of small growth with no unbounded Fatou components

We prove a form of the cos πρ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand, we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen’s condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.     

Congruence preserving functions in the residue class rings of polynomials over finite fields

In this paper, as an analogue of the integer case, we define congruence preserving functions over the residue class rings of polynomials over finite fields. We establish a counting formula for such congruence preserving functions, determine a necessary and sufficient condition under which all congruence preserving functions are also polynomial functions, and characterize such functions.     

Toward a Unified Theory of Generalized Functions: Convergence

Generalized functions are usually treated as bounded linear functionals on spaces of test functions. Here they are considered as objects which ?convolute”? with test functions and satisfy a certain associativity condition. In this setting simple definitions are given for convergence of sequences of test functions and generalized functions, definitions which do not depend on particular properties of the space of test functions. It is shown that for the most commonly used spaces, these general definitions reduce to the standard ones.     

Theory of submodular programs: A fenchel-type min-max theorem and subgradients of submodular functions

We consider submodular programs which are problems of minimizing submodular functions on distributive lattices with or without constraints. We define a convex (or concave) conjugate function of a submodular (or supermodular) function and show a Fenchel-type min-max theorem for submodular and supermodular functions. We also define a subgradient of a submodular function and derive a necessary and sufficient condition for a feasible solution of a submodular program to be optimal, which is a counterpart of the Karush-Kuhn-Tucker condition for convex programs. This work is supported by the Alexander von Humboldt fellowship (1982/83), West Germany.     

Measure free martingales

We give a necessary and sufficient condition on a sequence of functions on a set Ω under which there is a measure on Ω which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities.     

The low of the iterated logarithm for empirical processes under absolute regularity

The compact law of the iterated logarithm for empirical processes whose underlying sequence satisfies a -mixing condition is considered. In particular, we show a compact law of the iterated logarithm for VC subgraph classes of functions, for classes of functions which satisfy the bracketing condition in Doukhanet al. ⁽⁶⁾ and for some classes of smooth functions.Research partially supported by NSF Grant DMS-93-02583.     

An integral type characterization of constant functions on metric-measure spaces

The main purpose of this article is to generalize a characterization of constant functions to the context of metric-measure spaces. In fact, we approximate a measurable function, in terms of a certain integrability condition, by Lipschitz functions. Then, similar to Brezis (2002) [2], we establish a necessary and sufficient condition in order that any measurable function which satisfies an integrability condition to be constant a.e. Also, we provide a different proof for the main result of Pietruska-Pa?uba (2004) [7] in the setting of Dirichlet forms.     

Invex-convexlike functions and duality

We define a class of invex-convexlike functions, which contains all convex, pseudoconvex, invex, and convexlike functions, and prove that the Kuhn-Tucker sufficient optimality condition and the Wolfe duality hold for problems involving such functions. Applications in control theory are given.The author is grateful to Professor W. Stadler and the referees for many valuable remarks and suggestions, which have enabled him to improve considerably the paper.     

一类亚解析函数的复合边值问题

本文研究了一类亚解析函数的复合边值问题.利用经典的消去法和对称扩张,得到了边值问题的可解条件和解的表达式.推广了解析函数的相应理论.     

An iterative substructuring method for the p-version of the boundary element method for hypersingular integral operators in three dimensions

Summary. We study preconditioners for the -version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner. In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the mesh size. For all these methods we prove that the resulting condition numbers are bounded by . Here, is the polynomial degree of the ansatz functions and is a constant which is independent of and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported. Received July 8, 1996 / Revised version received January 8, 1997