The Wave Equation with Computable Initial Data Whose Unique Solution Is Nowhere Computable

We give a rough statement of the main result. Let D be a compact subset of ?³× ?. The propagation u(x, y, z, t) of a wave can be noncomputable in any neighborhood of any point of D even though the initial conditions which determine the wave propagation uniquely are computable. A precise statement of the result appears below.     

Iterative Characterizations of Computable Unary Functions: A General Method

Iterative characterizations of computable unary functions are useful patterns for the definition of programming languages based on iterative constructs. The features of such a characterization depend on the pairing producing it: this paper offers an infinite class of pairings involving very nice features.     

Interpolation Formula for Functions of Several Variables

Mathematical Notes -     

加权(C)eby(s)ev-Ostrowski型不等式

关于著名的Cebysev不等式，已有众多的研究成果．通过建立积分不等式，来建立全新的加权Cebysev型积分不等式．给予了独立的证明，并给出了此类不等式的新评价．     

Computable Real-Valued Functions on Recursive Open and Closed Subsets of Euclidean Space

In this paper we study intrinsic notions of “computability” for open and closed subsets of Euclidean space. Here we combine together the two concepts, computability on abstract metric spaces and computability for continuous functions, and delineate the basic properties of computable open and closed sets. The paper concludes with a comprehensive examination of the Effective Riemann Mapping Theorem and related questions.     

Computable choice functions for computable linear orderings

A choice set for a computable linear ordering is a set which contains one element from each maximal block of the ordering. We obtain a partial characterization of the computable linear order‐types for which each computable model has a computable choice set, and a full characterization in the relativized case; Every model of the linear order‐type α of degree ≤ d has a choice set of degree ≤ d iff α can written as a finite sum of order‐types, each of which either has finitely many blocks, or has order‐type n · η for some integer n.     

On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space of differentiable functions on a bounded domain to the Lebesgue space . The values of operators on a function are locally determined by the values of both the function itself and all of its partial derivatives up to order inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.     

Lp-Computability

In this paper we investigate conditions for L^p-computability which are in accordance with the classical Grzegorczyk notion of computability for a continuous function. For a given computable real number p ≥ 1 and a compact computable rectangle I ? ?^q, we show that an L^p function f ∈ L^p(I) is L^P-computable if and only if (i) f is sequentially computable as a linear functional and (ii) the L^p-modulus function of f is effectively continuous at the origin of ?^q.     

Approaches to Effective Semi-Continuity of Real Functions

For semi-continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi-computable of type one, if its open hypograph hypo(f) is recursively enumerably open in dom(f) × ?; we call f lower semi-computable of type two, if its closed epigraph Epi(f) is recursively enumerably closed in dom(f) × ?; we call f lower semi-computable of type three, if Epi(f) is recursively closed in dom(f) × ?. We show that type one and type two semi-computability are independent and that type three semi-computability plus effectively uniform continuity implies computability, which is false for type one and type two instead of type three. We show also that the integral of a type three semi-computable real function on a computable interval is not necessarily computable.     

亚纯函数及其导数在角域内的唯一性

In this paper, applying the Nevanlinna theory of meromorphic function in one angular domain, we deal with a problem of uniqueness for meromorphic functions and their derivatives sharing three finite value ignoring multiplicities in an angular domain instead of the whole complex plane. Obtained results improve a recent result of Lin Weichuan and Seiki Mori.     

某些新广义函数类(英文)

给出由非标准离散函数及其差商所定义的新广义函数的某些类,它们密切联系于通常的直至某阶为连续可微的函数．周的一个深刻的定理被用来建立这些类与非标准Ｓｏｂｏｌｅｖ空间之间的关系．     

连续函数压缩映射问题的讨论

就连续函数应用问题的教学内容进行探讨.介绍数列和连续函数的压缩映射原理及其在方程求解,数学建模等方面的应用,并给出压缩映射的一般形式.续而通过压缩映射原理在图像处理和导航系统中的应用,阐明创新思想的重要性,以求开拓学生的思路和想象空间.     

Nonuniqueness Sequences for Weighted Algebras of Holomorphic Functions in the Unit Circle

Suppose that is a system of continuous subharmonic functions in the unit disk and is the class of holomorphic functions f in such that log|f(z)| ≤ B _f p _f (z) + C _f , z ∈ , where B _f and C _f are constants and p _f ∈ . We obtain sufficient conditions for a given number sequence Λ = { λ_n} ⊂ to be a subsequence of zeros of some nonzero holomorphic function from , i.e., Λ is a nonuniqueness sequence for .__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 775–787.Original Russian Text Copyright ©2005 by L. Yu. Cherednikova.     

Functions operating from a complex Banach space to its real part

We consider functions operating from a complex Banach function space to its real part. We show among other things, that if for all in an ultraseparating Banach function space , then Re.

     

导数具有公共小函数的亚纯函数

进一步讨论亚纯函数的k阶导数具有公共小函数的唯一性问题,得到两个亚纯函数唯一性问题的结果,改进了李平的有关结果.     

下半连续函数的逼近性质

讨 论了下半连续的 广义实值函数 通过 Ｌｉｐ ｓｃｈ ｉｔｚ 函数逼近 的基本性 质,并由 此导出了 实值函数的广义连 续性定理     

Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ?-ideals contained in the maximal ?-ideal M _x  = {f ? C(X):f(x) = 0}. The space X has finite rank if there is an n ? N such that every point x ? X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ? X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.     

伪概周期函数二十年

伪概周期函数是1992年在我的博士论文中定义的,大多数人是在1994年我的两篇文章得知此函数的．从那以来,伪概周期函数引起国内外许多数学工作者的兴趣,成为一个活跃的研究领域．文中将介绍伪概周期函数是如何定义的,并且综述近20年它的发展．     

A Unified Approach to Several Inequalities Involving Functions and Derivatives

There are many inequalities measuring the deviation of the average of a function over an interval from a linear combination of values of the function and some of its derivatives. A general setting is given from which the desired inequalities are obtained using Holder's inequality. Moreover, sharpness of the constants is usually easy to prove by studying the equality cases of Holder's inequality. Comparison of averages, extension to weighted integrals and n-dimensional results are also given.