拟亚纯映射的奇异方向

该文定义了平面上拟亚纯映射的Ｎｅｖａｎｌｉｎｎａ方向,证明了无穷级拟亚纯映射ｆ（ｚ）至少有一条Ｎｅｖａｎｌｉｎｎａ方向并且它还是关于型函数Ｕ（ｒ）的Ｂｏｒｅｌ方向．     

关于代数体函数的最大型Borel方向

本文得到下列结果：（１）有限正级代数体函数存在最大型Ｂｏｒｅｌ方向；（２）有限正级代数体函数最大型Ｂｏｒｅｌ方向上存在充满圆序列．     

关于拟共形映射的基本不等式及其应用

本文建立了平面上拟共形映射的一个基本不等式，构造了拟共形映射的充满圆序列，证明了其Ｂｏｒｅｌ方向的存在性。     

拟亚纯映射的Borel方向

对于平面上的Ｋ－拟亚纯映射时,建立了一个角域的基本不等式,由此证明了Ｋ－拟亚纯映射的Ｂｏｒｅｌ方向的存在性及其相应性质。     

代数体亚纯函数的最大型Borel方向

本文参考李国平的半纯函数聚值线的统一理论，证明了当ｒ→∞时，Ｔ（ｒ，Ｗ）／ｌｏｇ＾２ｒ的上极限等于∞的有限级（包含零级）的代数亚纯函数的最大型Ｂｏｒｅｌ方向的存在性，所得结果推广了本国平在亚纯函数中的结果。     

亚纯函数亏值总数与Borel方向总数的一般关系

对于级较高的有穷级亚纯函数，本文给出了亏值总数与Ｂｏｒｅｌ方向总数的一般关系。     

拟共形映射的重值

本文建立了一个关于拟共形射重值的基本不等式,应用它研究了涉及其重值的亏量关系,充满圆及Ｂｏｒｅｌ方向。     

覆盖曲面定理与代数体函数的重值

本文改进了Ｔｓｕｊｉ著名的球面有限连通覆盖定理,应用它建立了关于涉及代数体函数重值的基本不等式,并由此导出了有限正级代数体函数涉及其重值的充满圆序列及Ｂｏｒｅｌ方向的精确结果．     

关于单位圆内零级亚纯函数的Borel点

本文证明，对于单位圆内的零级亚纯函数，在其圆周上至少存在一个关于其型函数的Ｂｏｒｅｌ点．     

乘积σ—代数的一种生成方法

本文得到了有限及可数无限维乘积σ—代数的一种新的生成方法，并证明了可数无限维Ｂｏｒｅｌ域具有连续统的势．     

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.     

模型论在代数上的应用

给出了模型论在代数上的两个应用,得到了下列定理：定理A：如环R的任何有限生成子环均是局部环,则R是局部环.定理B：存在自然数的真扩张R使其具有下列特征：（1）虽然R有无限多零因子,但R中有无限多零因子,但R中的首1多项式的根的个数可以得到很好的控制.（2）R不仅将自然数的素数特征保留下来,而且还可在其上定义指数函数。     

Foundations for an iteration theory of entire quasiregular maps

The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have also been obtained for non-uniformly quasiregular maps. The purpose of this paper is to extend the iteration theory of transcendental entire functions to the quasiregular setting. As no examples of uniformly quasiregular maps of transcendental type are known, we work without the assumption of uniform quasiregularity. Here the Julia set is defined as the set of all points such that the complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type, the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.     

Cercles De Remplissage for Entire Functions

It is shown that every transcendental entire function f growstranscendentally in a sequence of cercles de remplissage. Anexample shows that if then there may be no sequence of cercles de remplissage theunion of which contains infinitely many zeros of f. It is alsoshown that every transcendental entire function f has a Haymandirection, that is, a direction such that, in every open sectorcontaining , either f assumes all complex values infinitelyoften, or else every derivative of f assumes all complex values,except possibly zero, infinitely often. 1991 Mathematics SubjectClassification 30D20.     

Some transcendental functions over function fields with positive characteristicCertaines fonctions transcendantes sur des corps de fonctions de caractéristique positive

In this work we shall define two families of functions over function fields with positive characteristic and show that such a function is transcendental if and only if its generating sequence is not ultimately zero. As a result, the Carlitz exponential and the Carlitz logarithm are transcendental functions. Our proof is elementary in the sense that we only use a theorem due to H. Sharif and C. Woodcock, and to T. Harase which generalizes the theorem of Christol about automatic sequences. To cite this article: J.-Y. Yao, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 939–943.     

A new non‐polynomial univariate interpolation formula of Hermite type

A new C ^∞ interpolant is presented for the univariate Hermite interpolation problem. It differs from the classical solution in that the interpolant is of non‐polynomial nature. Its basis functions are a set of simple, compact support, transcendental functions. The interpolant can be regarded as a truncated Multipoint Taylor series. It has essential singularities at the sample points, but is well behaved over the real axis and satisfies the given functional data. The interpolant converges to the underlying real‐analytic function when (i) the number of derivatives at each point tends to infinity and the number of sample points remains finite, and when (ii) the spacing between sample points tends to zero and the number of specified derivatives at each sample point remains finite. A comparison is made between the numerical results achieved with the new method and those obtained with polynomial Hermite interpolation. In contrast with the classical polynomial solution, the new interpolant does not suffer from any ill conditioning, so it is always numerically stable. In addition, it is a much more computationally efficient method than the polynomial approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

关于求解极限的若干思考

无穷小等价替换是计算极限一种常用且好用的方法,但要求替换项在极限点的某个邻域内没有零点,文章证明了函数在极限点的任意邻域都出现零点时,满足一定条件,等价替换仍可实施.另外利用极坐标计算二元函数极限时有诸多的限制,文章针对一类特殊函数给出了应用极坐标便于计算应用的若干结论.     

TRANSCENDENTAL AND ALGEBRAIC EXTENSIONS OF COMMUTATIVE RINGS

Abstract

Transcendental and algebraic elements over commutative rings are defined. Rings with zero nil radical are considered. For a transcendental over R, necessary and sufficient conditions are derived for elements of R[α] to be algebraic or transcendental over R. For R a ring with identity and a finite number of minimal prime ideals, necessary and sufficient conditions are given for any element in a unitary overring of R to be algebraic or transcendental over R. It is proved that if α is algebraic Over R, so is every element of R[α]. It is show that if R is Noetherian, β is algebraic over R[α] and α is algebraic over R, then, under certain conditions, β is algebraic over R. If R has a finite number of minimal prime ideals, P₁,…,P_k, which are pairwise comaximal, then if t is transcendental over R, R[t] can be obtained by adjoining k algebraic elements a_i over R to R whose defining polynomials are in P_i [x], and conversely, if such elements are adjoined to R, they generate an element transcendental over R.     

EXAMPLES ON EXCEPTIONAL VALUES OF MEROMORPHIC FUNCTIONS

In this paper, by means of the definition of Borel exceptional value method, another exceptional value of meromorphic function which is a T exceptional value is defined by linking the concept of T direction. And we construct a meromorphic function with zero as Borel exceptional value, but not as T exceptional value; and another meromorphic function with zero as T exceptional value, but not as Borel exceptional value.     

Transcendental Kantianism,Naturalized Kantianism,and the Bounds of Psychology

Are there sensory states (“perceptions”) that, unlike mere sensory registrations, require an explanatory framework (“psychology”) that goes beyond biology? Based on a reconstruction of Kant’s a priori, transcendental psychology, contemporary Kantians answer this question in the positive but dramatically limit the scope of psychology. In contrast, naturalistically oriented deflationists answer it in the negative, thereby not giving psychology any explanatory role whatsoever. In his recent monumental book Origins of Objectivity, Burge argues against both of these approaches and seeks to develop an intermediate approach between them. This he does by embedding Kantian transcendental psychology in contemporary science of perception, thereby naturalizing the former and considerably broadening the scope of psychology. In this paper I critically examine Burge’s naturalized Kantianism, thereby defending transcendental Kantianism. To this end, I first outline Kantian transcendental psychology of perception, highlighting the features that distinguish it from biology. I then show how Burge naturalizes this psychology by embedding its most fundamental notions in contemporary science of perception. Based on all this, I conclude the paper by arguing for two closely related claims. First, that transformed into empirical psychology, Kantian transcendental psychology loses the features that distinguish it from biology. Second, that genuine perception starts at the high cognitive level for which transcendental psychology accounts and not at the rather low or elementary level on which Burge focuses.