Convergence of Noncommutative Triangular Arrays of Probability Measures on a Lie Group

A measure-theoretic approach to the central limit problem for noncommutative infinitesimal arrays of random variables taking values in a Lie group G is given. Starting with an array of probability measures on G and instance 0st one forms the finite convolution products . The authors establish sufficient conditions in terms of Lévy-Hunt characteristics for the sequence to converge towards a convolution hemigroup (generalized semigroup) of measures on G which turns out to be of bounded variation. In particular, conditions are stated that force the limiting hemigroup to be a diffusion hemigroup. The method applied in the proofs is based on properly chosen spaces of difierentiable functions and on the solution of weak backward evolution equations on G.     

Weierstrass Representation for Surfaces in the Three-Dimensional Heisenberg Group

The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature. Furthermore, a second order partial differential equation for the Gauss map is obtained, and it is shown that this equation is the complete integrability condition of the representation.     

Duality with expanding maps and shrinking maps, and its applications to Gauss maps

We study expanding maps and shrinking maps of subvarieties of Grassmann varieties in arbitrary characteristic. The shrinking map was studied independently by Landsberg and Piontkowski in order to characterize Gauss images. To develop their method, we introduce the expanding map, which is a dual notion of the shrinking map and is a generalization of the Gauss map. Then we give a characterization of separable Gauss maps and their images, which yields results for the following topics: (1) Linearity of general fibers of separable Gauss maps; (2) Generalization of the characterization of Gauss images; (3) Duality on one-dimensional parameter spaces of linear subvarieties lying in developable varieties.     

A Note on Gaussian BV Function and Its Heat Semigroup Characterization

In this note, we investigate the properties of Gaussian BV functions and give a heat semigroup characterization of BV functions in Gauss space. In particular, the latter is the nontrivial generalization of classical De Giorgi's heat kernel characterization of function of bounded variation on Euclidean space to the case of Gauss space.     

欧氏空间中具有共形Gauss映照的曲面

本文考虑欧氏空间中具有共形Ｇａｕｓｓ映照的曲面．从Ｇａｕｓｓ映照的观点给出了 ｖｅｒｏｎｅｓｅ曲面的一个新特征．     

抛物型积分-微分方程有限元近似的超收敛性质

1　引　　言有限元超收敛性质在有限元方法的研究中占有重要的地位 .利用超收敛性不仅可提高有限元实际计算的精度 ,而且还可得到后验误差估计 .对于椭圆问题有限元超收敛性质的研究目前已有了较丰富的结果 [1 - 3] ,而对于近年来引起广泛关注的发展型积分 -微分方程[4- 6] ,这方面的研究尚不成熟 .本文将研究一维抛物型积分 -微分方程半离散有限元近似的超收敛性质 ,证明了剖分单元上的 Lobatto点、Gauss点和拟 Lobatto点分别是函数、一阶和二阶导数逼近的超收敛点 ;并且在一定条件下证明了强超收敛二择一定理 ;在每个单元上 ,单元中点或…     

Deformation of surfaces in three-dimensional space induced by means of integrable systems

The correspondence between different versions of the Gauss–Weingarten equation is investigated. The compatibility condition for one version of the Gauss–Weingarten equation gives the Gauss–Mainardi–Codazzi system. A deformation of the surface is postulated which takes the same form as the original system but contains an evolution parameter. The compatibility condition of this new augmented system gives the deformed Gauss–Mainardi–Codazzi system. A Lax representation in terms of a spectral parameter associated with the deformed system is established. Several important examples of integrable equations based on the deformed system are then obtained. It is shown that the Gauss–Mainardi–Codazzi system can be obtained as a type of reduction of the self-dual Yang–Mills equations.     

Gauss小波变换像空间的描述

在连续小波变换像空间是再生核Hilbert空间的基础上,针对经常用于边界检测并且使用效果非常好的Gauss小波,给出了其小波变换像空间的再生核具体表达式.并且当固定尺度因子和固定平移因子时,利用再生核空间理论,对Gauss小波变换像空间做了具体描述,分别给出了Gauss小波变换像空间中的等距恒等式和反演公式,这为进一步研究一般的小波变换像空间提供了理论基础.     

On the order of starlikeness of the shifted Gauss hypergeometric function

We determine for several ranges of real parameters the order of starlikeness of the shifted Gauss hypergeometric function and we give some consequences of our results, in particular some mapping properties of the Carlson-Shaffer convolution operator.     

3维双曲空间中给定平均曲率曲面的Weierstrass表示

首先叙述双曲空间中曲面的Ｇａｕｓｓ映照的定义;导出Ｇａｕｓｓ映照所满足的Ｂｅｌｔｒａｍｉ方程;给出了给定平均曲率曲面的Ｗｅｉｅｒｓｔｒａｓｓ表示公式;并讨论了这种表示的完全可积条件．     

Legendre–Gauss collocation methods for ordinary differential equations

In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential equations. The first algorithm is the Legendre–Gauss collocation method, which is easy to be implemented and possesses the spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can be regarded as a specific implicit Legendre–Gauss Runge–Kutta method, with the global convergence and the spectral accuracy. Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.      

On the Convergence of Decoupled Optimal Power Flow Methods

This paper investigates the convergence of decoupled optimal power flow (DOPF) methods used in power systems. In order to make the analysis tractable, a rigorous mathematical reformation of DOPF is presented first to capture the essence of conventional heuristic decompositions. By using a nonlinear complementary problem (NCP) function, the Karush–Kuhn–Tucker (KKT) systems of OPF and its subproblems of DOPF are reformulated as a set of semismooth equations, respectively. The equivalent systems show that the sequence generated by DOPF methods is identical to the sequence generated by Gauss–Seidel methods with respect to nonsmooth equations. This observation motivates us to extend the classical Gauss–Seidel method to semismooth equations. Consequently, a so-called semismooth Gauss–Seidel method is presented, and its related topics such as algorithm and convergence are studied. Based on the new theory, a sufficient convergence condition for DOPF methods is derived. Numerical examples of well-known IEEE test systems are also presented to test and verify the convergence theorem.     

Gauss–Newton methods with approximate projections for solving constrained nonlinear least squares problems

This paper is concerned with algorithms for solving constrained nonlinear least squares problems. We first propose a local Gauss–Newton method with approximate projections for solving the aforementioned problems and study, by using a general majorant condition, its convergence results, including results on its rate. By combining the latter method and a nonmonotone line search strategy, we then propose a global algorithm and analyze its convergence results. Finally, some preliminary numerical experiments are reported in order to illustrate the advantages of the new schemes.     

Interior-boundary value problem with Saigo operators for the gellerstedt equation

For an equation of mixed elliptic-parabolic type, we consider an interior-boundary value problem in which the Dirichlet condition is posed on the elliptic part of the boundary and a point condition relating generalized derivatives and fractional integrals with the Gauss hypergeometric function of the values of the solution on the characteristics to the values of the solution and its derivative on the parabolic degeneration line is posed on the hyperbolic part.     

H~2×R中给定平均曲率曲面的Weierstrass表示

定义了H~2×R中曲面的法高斯映照,给出了给定平均曲率曲面的Weierstrass表示,证明了法高斯映照满足一个二阶偏微分方程,并且该方程是所得到的Weierstrass表示的完全可积条件.     

辛甫生公式的推广

给出了具有3次代数精度的条件,对辛甫生求积公式进行了推广,利用待定系数方法推导出了一般性的公式,并从该推广的方法导出了三点高斯公式.     

Rates of convergence for continued fractions of irrational numbers

The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.     

H~2×R中给定平均曲率曲面的Weierstrass表示

定义了H~2×R中曲面的法高斯映照,给出了给定平均曲率曲面的Weierstrass表示,证明了法高斯映照满足一个二阶偏微分方程,并且该方程是所得到的Weierstrass表示的完全可积条件.     

On gauss spaces

Summary Theorem 1 gives a characterization of the Gauss spaces. As a consequence of this theorem we construct some examples and counter-examples of such spaces. Finally, we define and sludy some properties of the canonical standard family of coverings, Gauss subspace, Guss product space, etc. Entrata in Redazione il 6 luglio 1968.     

An equivalent characterization of BMO with Gauss measure

Let γ be the Gauss measure on ? ⁿ : We establish a Calderón-Zygmund type decomposition and a John-Nirenberg type inequality in terms of the local sharp maximal function and the median value of function over cubes. As an application, we obtain an equivalent characterization of known BMO space with Gauss measure.