ALGEBROIDAL FUNCTION AND ITS DERIVED FUNCTION IN UNIT CIRCULAR DISC

In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in unit circular disc.

BOREL RADIUS AND T-RADIUS OF THE ALGEBROIDAL FUNCTION IN THE UNIT DISC

Using Ahlfors’ theory of covering surface and a type-function,we confirm the existence theorem of a Borel radius and a T-radius for the algebroidal function dealing with multiple values in the unit disc,which briefly extend some results for the algebroidal functions in the complex plane

THE BOREL DIRECTIONS OF ALGEBROIDAL FUNCTION AND ITS COEFFICIENT FUNCTIONS

In this article, the relationship between the Borel direction of algebroidal function and its coefficient functions is studied for the first time. To begin with, several theorems of algebroidal functions in unit disk are proved. By these theorems, some interesting conclusions are obtained.

Uniqueness Theorem of Algebroidal Functions in the Unit Disc

In this paper, the uniqueness of algebroidal functions in the unit disc is investigated. Suppose that W（z） and M（z） are v-valued and k-valued algebroidal functions in the unit disc, respectively. Let e^iθ be a b-cluster point of order co or order ρ（x） of the algebroidal function W（z） or M（z）. It is shown that if -↑E（aj, W（z）） = -↑E（aj,M（z）） holds in the domain {｜z｜ 〈 1}∩Ω（θ-δ,θ＋δ）, where b, aj （j = 1,…, 2v ＋ 2k ＋ 1） are complex constants, then W（z） = M（z）. The same results are obtained for the case that e^iθ is a Borel point of order co or order ρ（x） of the algebroidal function W（z） or M（z）.

THE INTEGRATION OF ALGEBROIDAL FUNCTIONS

In this paper, we introduce the integration of algebroidal functions on Riemann surfaces for the first time. Some properties of integration are obtained. By giving the defi- nition of residues and integral function element, we obtain the condition that the integral is independent of path. At last, we prove that the integral of an irreducible algebroidal function is also an irreducible algebroidal function if all the residues at critical points are zeros.

Uniqueness Theorem of Algebroidal Function in the Unit Disc

In this paper, discussed are the problems about uniqueness of algebroidal functions in the unit disc with share-values in a sector domain instead of the whole disk. Results are obtained extending some uniqueness theorems of meromorphic functions.

Normal family of algebroidal functions

The multivalue algebroidal functions are studied with geometric method. The difficulties of multivalue and branch points are overcome, and some theorems on normality are obtained.

Uniqueness Theorem of Algebroidal Functions in an Angular Domain

Let W（z） and M（z） be v-valued and k-valued algebroidal functions respectively △（θ） be a b-cluster line of order ∞ （or ρ（r）） of W（z） （or M（z））. It is shown that W（z） ≡ M（z） provided E（aj, W（z）） = E（aj,M（z）） （j = 1,..., 2v ＋ 2k ＋ 1） holds in the angular domain Ω（θ-δ,θ-δ）, where b, aj （j = 1,..., 2v -b 2k -k 1） are complex constants. The same results axe obtained for the case that△（θ） is a Borel direction of order ∞ （or ρ（τ）） of W（z） （or M（z））.

Nonlinear Conformal Gravitation

In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.

ON THE OPERATIONS OF ALGEBROID FUNCTIONS

In this article, we first investigate the operational properties of algebroid functions. Then we prove two uniqueness theorems for algebroid functions.

ON THE GROWTH OF SOLUTIONS OF HIGHER-ORDER ALGEBRAIC DIFFERENTIAL EQUATIONS

Using Nevanlinna theory of the value distribution of meromorphic functions, the author investigates the problem of the growth of solutions of two types of algebraic differential equation and obtains some results.

THE NORMALITY OF ALGEBROID MULTIFUNCTIONS AND THEIR COEFFICIENT FUNCTIONS

In this paper, we investigate the normality relationship between algebroid multifunctions and their coefficient functions. We prove that the normality of a k-valued entire algebroid multifunctions family is equivalent to their coefficient functions in some conditions. Furthermore, we obtain some new normality criteria for algebroid multifunctions families based on these results. We also provide some examples to expound that some restricted conditions of our main results are necessary.

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.

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.

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.

Generalized Higher-Order Algebraic Differential Equations with Admissible Algebroid Solutions

Using the Nevanlinna theory of the value distribution of meromorphic functions, we investigate the existence problem of admissible algebroid solutions of generalized higher order algebraic differential equations.

ON UNIQUENESS FOR ALGEBROID FUNCTIONS OF FINITE ORDER

Guided by Lo.Yang＇s method, we concern the question that how algebroid func- tions are determined by their multiple values and deficient values and we prove an at most 3v-valued theorem for algebroid functions. The results are complemented by an example for completeness.

ALGEBRAIC SOLUTIONS OF ALGEBRAIC DIFFERENTIAL EQUATIONS

Using value distribution theory and techniques,the problem of the algebroid solutions of second order algebraic differential equation is investigated.Examples show that the results are sharp.

Minkowski, Schwarzschild and Kerr Metrics Revisited

In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones. Unhappily, these physicists are neither familiar with the formal theory of systems of partial differential equations and differential modules, nor with the formal theory of Lie pseudogroups. Hence, even if they discovered a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. The purpose of this difficult computational paper is to provide differential and homological methods in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools, which are now available as computer algebra packages, question the mathematical foundations of GR and the origin of gravitational waves.

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 3v-valued theorem.These results extend the existing achievements of some scholars.