NORMALITY CRITERIA FOR FAMILIES OF MEROMORPHIC ALGEBROID FUNCTIONS

By using the definition of Hausdorff distance, we prove some normality criteria for families of meromorphic algebroid functions. Some examples are given to complement the theory in this article.     

代数体函数的性质

本文研究了亚纯函数的几个性质推广到多值的代数体函数.利用代数体函数的循环加,循环乘运算及特征的性质,证明了代数体函数的级与下级的几个性质,并研究了两个代数体函数的公共a值点的度量,且得到代数体函数的两个唯一性定理.     

ON THE SHARING VALUES OF ALGEBROID FUNCTIONS AND THEIR DERIVATIVES

In this paper,we define the shared value of an algebroid function and its derivative on its Riemann surface.By considering the relationship between the shared values and the branch points of algebroid functions and their derivatives,we obtain some uniqueness theorems of algebroid functions sharing values with their derivatives,which extend 3 IM shared values theorem of nonconstant meromorphic functions and their derivatives obtained by Mues-Steinmetz and Gundersen.     

On biconjugates of infimal functions

We deliver formulae for the biconjugate functions of some infimal functions that hold provided the fulfilment of weak regularity conditions of both closedness and interiority types. As special cases, we obtain biconjugates of infimal convolutions of finitely many functions, of optimal value functions of both constrained and unconstrained optimization problems as well as of marginal functions associated with multifunctions (that can be, for instance, convex processes) and some given functions. Moreover, we rediscover or extend different results on biconjugate functions from the literature.     

THE UNIQUENESS FOR ALGEBROID FUNCTIONS OF FINITE ORDER

In this article, we discuss, by Nevanlinna theory, the influence of multiple values and deficiencies on the uniqueness problem of algebroid functions. We get several uniqueness theorems of algebroid functions which include an at most 3 v-valued theorem. These results extend the existing achievements of some scholars.     

代数体函数涉及重值的Borel点

研究了单位圆内代数体函数涉及重值的Borel点的存在性,利用型函数统一证明了单位圆内无穷级,有限正级和部分零级代数体函数存在涉及重值的Borel点,并对有限正级的代数体函数得到两个必要条件.     

On the topological entropy of continuous and almost continuous functions

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.     

Measurability and countable separation

We introduce several types of countable separation and we give necessary and sufficient conditions, in terms of measurability of some sets constructed by means of multifunctions, in order that a topological space satisfies such a property. We give a characterization of measurability of multifunctions in terms of their graphs.     

ON UNIQUENESS FOR ALGEBROID FUNCTIONS OF FINITE ORDER

Guided by Lo.Yang's method,we concern the question that how algebroid functions are determined by their multiple values and deficient values and we prove an at most3v-valued theorem for algebroid functions.The results are complemented by an example for completeness.     

An algebroid function and its derivative

In this paper, we investigate the value distribution of an algebroid function and its derivative, and obtain two inequations between Nevanlinna characteristic function of an algebroid function and that of its derivative. We extend Chuang Chitai's theorem of meromorphic functions to algebroid functions.     

微分多项式的值分布

文[2-6]讨论了亚纯函数或代数体函数的微分多项式ω‘(z)-αω^n(z)及ω（z)ω^n(z)的例外值问题。本文利用代数体函数的Nevanlinna值分布理论方法，研究了代数体函数的微分多项式P（ω^(k)(z))-aΠk=1^s(ω(z)-ak)^pk,及a(w^(k)(z)^tw^n(t)的Picard值问题，推广了他们的结果。     

The second main theorem concerning small algebroid functions

In this paper, we firstly give the definition of meromorphic function element and algebroid mapping. We also construct the algebroid function family in which the arithmetic, differential operations are closed. On basis of these works, we firstly prove the Second Main Theorem concerning small algebroid functions for v-valued algebroid functions.     

Sieve M-estimator for a semi-functional linear model

We propose sieve M-estimator for a semi-functional linear model in which the scalar response is explained by a linear operator of functional predictor and smooth functions of some real-valued random variables.Spline estimators of the functional coefficient and the smooth functions are considered,and by selecting appropriate knot numbers the optimal convergence rate and the asymptotic normality can be obtained under some mild conditions.Some simulation results and a real data example are presented to illustrate the performance of our estimation method.     

Some Notes on Normality Criteria of Meromorphic Functions

In this paper, we study the normality of families of meromorohic functions related to a Hayman conjecture. We prove that the conditions in Hayman conjecture and in other criterions can be relaxed. The results in this paper improve some previous results.     

On the C. Chang type inequality of algebroid functions

In this paper,we investigate the growth relations between algebroid functions and their derivatives,and extend famous C.Chang inequality(see[1,4])of meromorphic functions to algebroid functions.     

��-Lines of Algebroid Functions

In this article several theorems of the theory of Γ-lines for meromorphic functions are extended to the more general setting of algebroid functions. We recall the definition of algebroid function of order k and how it can be considered as a function defined on a Riemann surface of k sheets. In this way, we prove the so called tangent variation principle for algebroid functions, previously proved for meromorphic functions by Barsegian, and we get several consequences of this result. We also extend a proposition on proximity properties of meromorphic functions.     

Toward A geometric theory of minimax equalities

We present new ideas and concepts in minimax equalities. Two important classes of multifunctions will be singled out, the Weak Passy-Prisman multifunctions and multifunctions possessing the finite simplex property. To each class of multifunctions corresponds a class of functions. We obtain necessary and sufficient conditions for a multifunction to have the finite intersection property, and necessary and sufficient conditions for a function to be a minimax function. All our results specialize to sharp improvements of known theorems, Sion, Tuy, Passy-Prisman, Flåm-Greco. One feature of our approach is that no topology is required on the space of the maximization variable. In a previous paper [6] we presented a “method of reconstruction of polytopes” from a given family of subsets, this in turn lead to a “principle of reconstruction of convex sets” Theorem 3, which plays a major role in this paper. Our intersection theorems bear no obvious relationship to other results of the same kind, like K.K.M. or other more elementary approaches based on connectedness. We conclude our work with a remark on the role of upper and lower semicontinuous regularization in mimmax equalities     

代数体函数的唯一性定理

本文研究了代数体函数的唯一性问题,推广了某些已有结果.     

Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values

We present some extreme continuous selector theorems, synthesizing the author's results; namely, we study existence and properties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the space of Bochner integrable functions.     

Normal criteria on the family of meromorphic functions with shared set

In this paper, we investigate the general normality criteria for a family of meromorphic functions using a geometric method. We obtain the necessary and sufficient conditions used for determining the normality of a family of meromorphic functions and some normal criteria on the family of meromorphic functions with shared set by using the fundamental theorem of a covering surface.