ON EXISTENCE OF SOLUTIONS OF DIFFERENCE RICCATI EQUATION

Consider the difference Riccati equation f(z+1) =(A(z)f(z)+B(z))/(C(z)f(z)+D(z)),where A,B, C,D are meromorphic functions, we give its solution family with one-parameter H(f(z))={f_0(z),f(z)=((f_1(z)-f_0(z))(f_2(z)-f_0(z)))/(Q(z)(f_2(z)-f_1(z))+(f_2(z)-f_0(z)))}, where Q(z) is any constant in C or any periodic meromorphic function with period 1, and f_0(z),f_1(z),f_2(z) are its three distinct meromorphic solutions.     

On Algebraic Limits and Topological Properties of Sets of Möbius groups in $ \bar {R}^n $

Let G be the algebraic limit of a sequence $ \{ G_m\} $ of r generator subgroups of $ M(\bar {R}^n) $ . We prove that (i) if $ G_m $ is elementary, then G is elementary; (ii) if $ G_m $ is not non-elementary and non-discrete and G satisfies the Condition A, then G is not non-elementary and non-discrete. Let $ E_l = \{ (f_1, f_2,\ldots ,f_l): f_1,f_2,\ldots ,f_l\in M(\bar {R}^2), \langle \,f_1,f_2,\ldots , \,f_l \rangle\ elementary\} $ and $ D_l = \{ (f_1,f_2,\ldots ,f_l): f_1,f_2,\ldots ,f_l\in M(\bar {R}^2), \langle \,f_1,f_2,\ldots ,f_l \rangle\ discrete\} $ . we obtain that the set $ E_l , D_l\cup E_l $ and $ D_l\cap E_l^c $ are closed in $ \underbrace {M(\bar {R}^2)\times \cdots \times M(\bar {R}^2)}\limits _{l} $ .     

Beukers-like supercongruences for generalized Apéry numbers

For positive integers \(f_1,f_2,m,l\), the author and Chetry defined a generalization of Apéry numbers \(A(f_1,f_2,m,l,\lambda )\) given by

$$\begin{aligned} A(f_1,f_2,m,l,\lambda ):=\sum _{j=0}^{f_2}{f_1+j\atopwithdelims ()j}^m{f_2\atopwithdelims ()j}^l\lambda ^j. \end{aligned}$$

In this article, we prove certain Beukers-like supercongruences for these generalized Apéry numbers.

     

单位圆上的有界单叶函数

<正> 1.引言设函数在单位圆|z|<1上是正则的,单叶的.它映照|z|<于|w|<1中.这种f_k(z)的全体形成一函数族 B_k,乃是 k 称的有界单叶函数族.对于 B_1中的函数 f_1(z),劳宝生讨论了|a|,|z_0|<1,|f_1(z_0)|和|f′(z_0)|四者之间的关系.利用关系式(?),他的许多结果可以直接推广到函数族B_k中来.但是关于f_k(z),还有些应该直接研讨的问题.例如当|a|,|z|取定值或|a|,     

Compact products of Hankel operators

In this paper we completely characterize compact products of three Hankel operators on the Hardy space. We obtain a necessary and sufficient condition for that , and are simultaneously compact.This work was partly supported by NSF grants. The second author was also partly supported by the Research Council of Vanderbilt University.     

复振荡理论中关于超级的角域分布

设f_1和f_2是微分方程f″+A(z)f=0的两个线性无关的解,其中A(z)是无穷级整函数且超级σ_2(A)=0．令E=f_1f_2．本文研究了微分方程f″+A(z)f=0的解在角域中的零点分布,得出E的超级为+∞的Borel方向与零点聚值线的关系．     

一类二阶变系数线性微分方程的积分因子解法

通过寻求积分因子f1(x),f2(x),求解一类二阶线性微分方程,包括二阶常系数线性微分方程和二阶Euler方程.     

Proximal and syndetical properties in nonautonomous discrete systems

This paper is mainly concerned with a class of nonautonomous discrete systems $(X, f_{1,\infty})$. New definitions of proximity relations and sensitivity in nonautonomous discrete systems are given. Some relations among $P(f_{1,\infty})$, $L(f_{1,\infty})$, $R(f_{1,\infty})$, $S(f_{1,\infty})$ and $P(f_{1,\infty})(x)$ are derived. And some chaotic properties of $f_{1,\infty}$ are proved.     

RANDOM ITERAtION OF HOLOMORPHIC SELF-MAPS OVER BOUNDED DOMAINS IN C~N

ＲＡＮＤＯＭＩＴＥＲＡＴＩＯＮＯＦＨＯＬＯＭＯＲＰＨＩＣＳＥＬＦ－ＭＡＰＳＯＶＥＲＢＯＵＮＤＥＤＤＯＭＡＩＮＳＩＮＣ~Ｎ￥ＺＨＡＮＧＷＥＮＪＵＮ；ＲＥＮＦＵＹＡＯ（ＤｅｐａｒａｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｎａｌＵｎｉｖｅｒｓｉｔｙｔＫａ?..     

Hyers–Ulam–Rassias Stability of Derivations in Proper JCQ*–triples

In this paper, we investigate stability of derivations in proper JCQ*–triples associated to the following Pexiderized functional equation $$f(x + y + z) = f_{0}(x) + f_{1}(y) + f_{2}(z)$$ .     

超越整函数稳定域的某些性质

讨论了有限个超越整函数fi(i=1,2,…,m)迭代生成的函数g=fm°fm-1°…°f1(z)游荡域的性质,在某些条件下,对Brannan和Hayman提出的一个问题给出了肯定的回答.此外,给出了g(z)稳定域有界的一个条件.     

Lefschetz coincidence class for several maps

The aim of this paper is to define a Lefschetz coincidence class for several maps. More specifically, for maps \({f_{1}, \ldots , f_{k} : X \rightarrow N}\) from a topological space X into a connected closed n-manifold (even nonorientable) N, a cohomological class

$$L(f_{1}, \ldots , f_{k}) \in H^{n(k-1)}(X; (f_{1}, \ldots , f_{k}) ^{\ast} (R \times \Gamma^{\ast}_{N} \times \ldots \times \Gamma^{\ast} _{N}))$$

is defined in such a way that \({L(f_{1}, \ldots , f_{k}) \neq 0}\) implies that the set of coincidences

$${\rm Coin}(f_{1}, \ldots , f_{k}) = \{x \in X\,|\,f_{1}(x) = \ldots = f_{k}(x)\}$$

is nonempty.

     

Characterization of the projective space Pr3 by a contact form

In this paper we present a theorem, which is a specific case of a very general problem, that could be described as follows: Given a differentiable manifoldV, “What are the elements of the exterior algebra ofV which describe the topology and differentiable structure ofV?” Classic examples of this situation are: (A) De Rham’s theorems, which describe the cohomology of a manifold by means of closed forms. (B) Reeb’s theorem, which describes the topology of a sphere by means of a function having only two critical points. In the present paper, we prove the following: Theorem.The projective space Pr₃ and the sphere S ³ are the only three-dimensional, compact, manifold which has a contact form with a global expression: \(\omega = f_1 df_2 - f_2 df_1 + f_3 df_4 - f_4 df_3 + f_5 df_6 - f_6 df_5 \) where the f _i are global functions satisfying certain integrability conditions.     

Random holomorphic iterations and degenerate subdomains of the unit disk

Given a random sequence of holomorphic maps of the unit disk to a subdomain , we consider the compositions

The sequence is called the iterated function system coming from the sequence We prove that a sufficient condition on the domain for all limit functions of any to be constant is also necessary. We prove that the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.

     

酉羣上的富理埃分析 Ⅰ.富理埃級数的Abel求和及Dirichlet核

<正> §1.1.引言 一个变数的Fourier分析,現在已經有了丰富的成果,不少問題得到了圓滿而完整的解决,在很多人的研究中达到了很深刻的地步.当然,一个变数的Fourier分析在数学的不少其他領域起着重要的作用.可是多个变数的Fourier分析情况就不完全是如此,在有些內容上是有些不完整之处的.至于更一般的在任意紧致羣上的Fourier分析,那末,众所周知的,重要的結果只是Peter-Weyl定理,它告訴我們說:紧致羣上的連續函数可以用     

A Frank–Wolfe type theorem for nondegenerate polynomial programs

In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let \(f_0\) and \(f_1, \ldots , f_p :{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) be convenient polynomial functions, and let \(S := \{x \in {\mathbb {R}}^n \ : \ f_i(x) \le 0, i = 1, \ldots , p\} \ne \emptyset .\) Under the assumption that the map \((f_0, f_{1}, \ldots , f_{p}) :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^{p + 1}\) is non-degenerate at infinity, we show that if \(f_0\) is bounded from below on \(S,\) then \(f_0\) attains its infimum on \(S.\)      

Janowski星形和凸函数族相关的非线性积分算子性质

In this paper, we consider a general nonlinear integral operator H_(αi),β_i(f_1,..., f_n;g_1,..., g_n)(z). Some results including coefficient problems, univalency condition and radius of convexity for this integral operator are given. Furthermore, we discuss the mapping properties between H_(αi),β_i(f_1,..., f_n; g_1,..., g_n)(z) and subclasses of analytic functions with bounded boundary rotation. The same subjects for some corresponding classes are shown upon specializing the parameters in our main results.     

星像的典型实照函数

<正> 称这处函数 f(z)为单位圆上的一个典型实照函数,记其全体为 T_r.设 f(z)是 T_r中的一个函数,f(z)映照|z|<的像是区域 D,如果对于 D 中任意一点都可以用完全落在 D 中的直线段与ω=0相联结的话,称 f(z)是一個星像的典型实照     

Metrics of L p -type and distributional equivalence principle

Given a frequency table ${F=\{f_{jk},(j,k)\in\,J\times K\}}$ crossing two categorical variables J and K, we consider a family of metrics of L _p -type on J defined by ${d_J^p (j,j^{\prime}) = \Sigma_k g(f_{.k})|f_{jk}/f_{j.} - f_{j^{\prime}k}/f_{j^{\prime}.}|^p}$ , where g is a positive function, and a symmetrical one on K. We investigate under which conditions on g, the famous principle of distributional equivalence is fulfilled by these metrics for every rational or every real F.     

非线性系统李雅普诺夫函数构造分析

本文借助常系数线性系统的E.A巴尔巴辛公式,运用“类比法”构造了特殊类型的非线性系统的李雅普诺夫函数,从而推出非线性系统全局稳定性的一些充分条件,特别详细讨论非线性驻定系统(?)+f_1(?)+f_2(?)+f_3=0,其中f_2是二元函数的情形。