On a Generalization of the Stone–Weierstrass Theorem

A categorical version of the famous theorem of Stone and Weierstrass is formulated and studied in detail. Several applications and examples are given.     

A preparation theorem for Weierstrass systems

It is shown that Lion and Rolin's preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a Weierstrass system.

     

多复变量亚纯函数的第二基本定理和唯一性定理

本文给出了多复变量亚纯函数的第二基本定理的一个改进形式,并进一步得到了一个关于重值的唯一性定理(详见文[1-16])。     

Weierstrass逼近定理的应用

讨论 Weierstrass逼近定理的应用 .运用 Weierstrass逼近定理 ,我们对于连续偶函数和连续奇函数的性质进行了进一步的刻划 .     

“复数域上微分中值定理新证”注记

所谓“实数域上的微分中值定理可以推广到复数域上”的论断值得商榷．通过给出实例的方法,具体分析所谓的“解析函数的微分中值定理”之错误所在．     

Topological X-rays and MRIs

We use the Implicit Function Theorem to establish a result of non-existence of limit to a certain class of functions of several variables. We consider functions given by quotients such that both the numerator and denominator functions are null at the limit point. We show that the non-existence of the limit of such function is related with the gradient vectors of the numerator and denominator functions. We prove the limit does not exist if the dimension of the vector subspace spanned by the gradient vectors is ≥1.     

Some connections of monotonicity with Weierstrass extreme value theorem

The main scope of this article is to prove that if a continuous function attains its minimum before maximum on every compact interval, than it is nondecreasing. Finally, other similar results are established.     

复变函数中刘维尔定理的新证明

利用截断函数的性质给出了复变函数中刘维尔定理的一种新的证明方法.     

Special Pairs in the Generating Subset of the Weierstrass Semigroup at a Pair

We discuss the structure of the Weierstrass semigroup at a pair of points on an algebraic curve. It is known that the Weierstrass semigroup at a pair (P, Q) contains the unique generating subset (P, Q). We find some characterizations of the elements of (P, Q) and prove that, for any point P on a curve, (P, Q) consists of only maximal elements for all except for finitely many points QP on the given curve. Also we obtain more results concerning special and nonspecial pairs.     

关于weierstrass逼近定理的几点注记

Weierstrass逼近定理是函数逼近论中的重要定理之一,定理阐述了闭区间上的连续函数可以用一多项式去逼近.将该定理进行推广:即使一个函数是几乎处处连续的,也不一定具有与连续函数相类似的逼近性质,但是一个处处不连续的函数却有可能具有这样的性质.证明了定义在闭区间上且与连续函数几乎处处相等的函数具有类似的逼近性质,并给出了weierstrass逼近定理的一个推广应用.     

关于多元幂级数

引入了多元函数项级数的概念,给出了其收敛域及和函数的定义;通过详实的例子讨论了多元幂级数的收敛域、和函数及多元函数展开为多元幂级数的计算方法.     

Hausdorff Dimension of a Class of Weierstrass Functions

It was proved by Shen that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log \lambda/\log b$, for every integer $b\geq 2$ and every $\lambda\in (1/b,1)$ [Hausdorff dimension of the graph of the classical Weierstrass functions, Math. Z., 289 (2018), 223–266]. In this paper, we prove that the dimension formula holds for every integer $b\geq 3$ and every $\lambda\in (1/b,1)$ if we replace the function $\cos$ by $\sin$ in the definition of Weierstrass function. A class of more general functions are also discussed.     

多元函数中值定理中介值的渐近性

讨论了n元函数中值定理中介值的渐近性质.     

The density of elliptic curves having a global minimal Weierstrass equation

We show that a positive density of elliptic curves over a number field counted using their short Weierstrass equations belong to a given Weierstrass class and in particular, a positive density of elliptic curves have a global minimal Weierstrass equation. The density is given by a ratio of partial zeta functions of the number field K evaluated at 10 with some extra factors for the bad primes.     

关于有限圆盘内解析函数的Hadamard分解定理的结果

In this paper, we investigate the Hadamard factorization theorem of analytic functions in the finite disc DR = {z ∈ C : |z| = r R ∞}, where C is the whole complex plane. Examples are given to show that our results are sharp.     

Multidimensional generalization of the Korous inequality

This paper deals with some qualitative properties of orthogonal polynomials in several variables. The boundedness and relations between two sets of orthonormal polynomials associated with an arbitrary weight function and its extension are investigated. It presents an analogy to Korous' result for general orthogonal polynomials in one variable.     

The Weierstrass semigroup of a pair of GaloisWeierstrass points with prime degree on a curve

We describe the Weierstrass semigroup of a Galois Weierstrass point with prime degree and the Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree, where a Galois Weierstrass point with degree n means a total ramification point of a cyclic covering of the projective line of degree n.*Supported by Korea Research Foundation Grant (KRF-2003-041-C20010).**Partially supported by Grant-in-Aid for Scientific Research (15540051), JSPS.     

关于反函数定理

在函数解析的假设下，Ｓｍａｌｅ利用其α判据对反函数存在邻域的半径给出了一个估计．本文不仅将此估计精确化，而且把假设条件放宽为二次连续可微．