Generating Functions for Spherical Harmonics and Spherical Monogenics

In this paper, we study generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in \({\mathbb{R}^{m}}\) . Here spherical monogenics are polynomial solutions of the Dirac equation in \({\mathbb{R}^{m}}\) . In particular, we obtain the recurrence formula which expresses the generating function in dimension m in terms of that in dimension m–1. Hence we can find closed formulæ of generating functions in \({\mathbb{R}^{m}}\) by induction on the dimension m.     

复变量广义Airy函数

本文详尽研究了在波传播与流动稳定性问题中有广泛应用的广义Airy函数,给出了复变量广义Airy函数的一系列图表.计算结果是令人满意的.     

广义复数上的函数

本文用复变函数方法统一讨论既含椭圆型也含双曲型的一类线性偏微分方程。可以看到它们的许多个性寓于其共性之中。     

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])。     

Space-Time Foam Differential Algebras of Generalized Functions and a Global Cauchy-Kovalevskaia Theorem

The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called space-time foam structures in General Relativity, where the set of singularities can be dense. A variety of applications of these algebras have been presented elsewhere, including in de Rham cohomology, Abstract Differential Geometry, Quantum Gravity, etc. Here a global Cauchy-Kovalevskaia theorem is presented for arbitrary analytic nonlinear systems of PDEs. The respective global generalized solutions are analytic on the whole of the domain of the equations considered, except for singularity sets which are closed and nowhere dense, and which upon convenience can be chosen to have zero Lebesgue measure. In view of the severe limitations due to the polynomial type growth conditions in the definition of Colombeau algebras, the class of singularities such algebras can deal with is considerably limited. Consequently, in such algebras one cannot even formulate, let alone obtain, the global version of the Cauchy-Kovalevskaia theorem presented in this paper.     

广义算子值函数可微性的刻画

该文讨论了广义算子值函数的微分, 用算子象征刻画了其可微性, 并给出了一些例子及应用.     

Hartogs Extension Theorem for Functions with Values in Complex Clifford Algebras

A regular extension phenomenon of functions defined on Euclidean space with values in a Clifford algebra was studied by Le Hung Son in the 90’s using methods of Clifford analysis, a function theory which, is centred around the notion of a monogenic function, i.e. a null solution of the firstorder, vector-valued Dirac operator in . The isotonic Clifford analysis is a refinement of the latter, which arises for even dimension. As such it also may be regarded as an elegant generalization to complex Clifford algebra-valued functions of both holomorphic functions of several complex variables and two-sided biregular function theories. The aim of this article is to present a Hartogs theorem on isotonic extendability of functions on a suitable domain of . As an application, the extension problem for holomorphic functions and so for the two-sided biregular ones is discussed.      

On Positive Values of Spherical Harmonics and Trigonometric Polynomials

Mathematical Notes -     

?B-值广义泛函值函数可微性的刻画

In this paper, we discuss fuzzy simplex and fuzzy convex hull, and give several representation theorems for fuzzy simplex and fuzzy convex hull. In addition, by giving a new characterization theorem of fuzzy convex hull, we improve some known results about fuzzy convex hull.     

广义路余代数和广义对偶Gabriel定理

通过将箭图的每个顶点放置一个k-余代数,首先引进了广义路余代数的概念,其次给出了广义路余代数的一些基本性质,还讨论了同构问题.证明了两个正规广义路余代数是同构的当且仅当他们的箭图及对应顶点上的单余代数是同构的.对于满足Codim C_0■1余代数C,证明了对偶Wedderburn-Malcev定理成立.作为广义路余代数的一个应用,推广了点余代数的对偶Gabriel定理.     

Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions

In this paper, a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the efficiency of this approach. We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation, which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.AMS subject classifications: 65M70, 41A30, 81Q05     

Generalized Affine Functions and Generalized Differentials

We study some classes of generalized affine functions, using a generalized differential. We study some properties and characterizations of these classes and we devise some characterizations of solution sets of optimization problems involving such functions or functions of related classes.     

广义Cauchy中值定理及其逆定理

将C auchy中值定理的条件进行适当减弱,得到了广义C auchy中值定理,从而推广了C auchy中值定理,并在凸函数的条件下,证明了其逆定理亦成立.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

广义框架和框架算子

本文研究了可分的Hilbert空间H中的广义框架,应用算子论方法给出了广义框架是H中紧广义框架,对偶广义框架,独立广义框架的充要条件:证明了有关广义框架算子的一些结果。     

A-Browder＇s Theorem and Generalized a-Weyl＇s Theorem

Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for a-isoloid operators. We also study the generalized a-Weyl＇s theorem for Class A operators.     

Complex Spherical Designs and Codes

Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper, we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these association schemes are nonsymmetric.