Generalized S-KKM Theorems and Minimax Inequalities in FC-Spaces

The concept of FC-closed subset in FC-space without any linear structure was introduced. Then, the generalized KKM theorem is proved for FC-closed value mappings under some conditions. The FC-space in the theorem is the generalization of L-convex space and the condition of the mapping with finitely FC-closed value is weaker than that with finitely L-closed value.

GENERALIZED BOCHNER'S THEOREM FOR RADIAL FUNCTION

A radial function can be expressed by its generator through The positive definite of the function plays an important rote in the radial basis interpolation. We can naturally use Bochner's Theorem to check if is positive definite. This requires however a n-dhnensiotial Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-di-mensional space. The completely monotone function:, which is discussed in [4] is positive definite for arbitrary space dimensions. With this technique tve can very easily characterize the positive definite, of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, whcih are very useful in application. Can we find some conditions (as the completely monotone function) only for the \-dimen simial Fourier transform of the generator epto characterize a radial function 9, which is positivedefinite in n-dimensional (fixed n) spacel In this paper we defined a kind of incompletelymonotone function of order a, for a= 0,,1/2 ,1,3/2,(we denote the function class by ICM) ,in this sence a normal positvie function is in ICM a positive monotone decreasing function is inICM and a positive monotone decreasing and convex function is in ICM2- Based on this definition we get a generalized Bochner's Theorem for radial function-. If dimensional Fouriertransform of the generator of a radial function can be written as , then corre-spending radial function (x) is positive definite as a n-variate function iff F is an incomplete-ly monotone function of order a= (n- 1 )/2 (or simply In this way we have characterized the positive definite of the radial function as a n-vari-ate function through its generator in the sense of the Bocher's Theorem.

HYPERCYCLICITY, SUPERCYCLICITY AND GENERALIZED RAKOEVIC'S PROPERTY

A Banach space operator satisfies generalized RakoSevi5＇s property （gw） if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the spectrum of T. In this note, we characterize hypecyclic and supercyclic operators satisfying the property （gw）.

A Generalized Form of Whitney's Theorem

Whitney＇s theorem is a famous theorem in the local singularity theory. In this paper, as an application of Malgrange preparation theorem, a generalized form of Whitney＇s theorem will be derived.

Some Remarks for the Relationships between the Generalized Bernoulli and Euler Polynomials

In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.

Uncertainty Principles for the Generalized Fourier Transform Associated with Spherical Mean Operator

The aim of this paper is to establish an extension of qualitative and quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator.

Existence of Solutions to a Generalized System

In this paper, we introduce a generalized system (for short, GS) in real Banach spaces. Using Brouwer’s fixed point theorem, we establish some existence theorems for the generalized system without monotonicity. Further, we extend the concept of C-strong pseudomonotonicity and extend Minty’s lemma for the generalized system. And using the Minty lemma and KKM-Fan lemma, we establish an existence theorem for the generalized system with monotonicity in real reflexive Banach spaces. As the continuation of existing studies, our paper present a series of extended results based on existing corresponding results.

Biased Random Number Generator Based on Bell's Theorem

We propose a biased random number generation protocol whose randomness is based on the violation of the Clauser Home inequality. Non-maximally entangled state is used to maximize the Bell violation. Due to the rotational asymmetry of the quantum state, the ratio of Os to ls varies with the measurement bases. The experimental partners can then use their measurement outcomes to generate the biased random bit string. The bias of their bit string can be adjusted by altering their choices of measurement bases. When this protocol is implemented in a device-independent way, we show that the bias of the bit string can still be ensured under the collective attack.

Investigation of Probability Generating Function in an Interdependent <i>M/M/</i>1:(∞;GD) Queueing Model with Controllable Arrival Rates Using Rouche’s Theorem

Present paper deals a M/M/1:(∞;GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration;which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.

Deriving some quantum optical identities using the general ordering theorem

Using the newly introduced general ordering theorem by Sh/ihandeh and Bazrafkan, we derive and generalize some quantum optical identities and give their applications.

SOME PROPERTIES OF SOLUTIONS OF ONE-DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH GENERAL COEFFICIENTS

In this paper, the one-dimensional time-homogenuous lto’s stochastic differential equations, which have degenerate and discontinuous diffusion coefficients, are considered. The non-confluent property of solutions is showed under some local integrability condition on the diffusion and drift coefficients. The strong comparison theorem for solutions is also established.

The Power of Group Generators and Relations: An Examination of the Concept and Its Applications

This paper investigates the approach of presenting groups by generators and relations from an original angle. It starts by interpreting this familiar concept with the novel notion of “formal words” created by juxtaposing letters in a set. Taking that as basis, several fundamental results related to free groups, such as Dyck’s Theorem, are proven. Then, the paper highlights three creative applications of the concept in classifying finite groups of a fixed order, representing all dihedral groups geometrically, and analyzing knots topologically. All three applications are of considerable significance in their respective topic areas and serve to illustrate the advantages and certain limitations of the approach flexibly and comprehensively.

一类广义Douglas-Weyl度量的特征

研究了一类重要的由黎曼度量α和1-形式β定义的Finsler度量——(α,β)-度量——成为广义Douglas-Weyl度量的条件。在度量具有迷向S-曲率的条件下,给出了非Randers型的正则(α,β)-度量是广义Douglas-Weyl度量的条件。

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.

Generalized KKM-Type Theorems for Weakly Generalized KKM Mapping with Some Applications

In this paper, we establish some new generalized KKM-type theorems based on weakly generalized KKM mapping without any convexity structure in topological spaces. As applications, some minimax inequalities and an existence theorem of equilibrium points for an abstract generalized vector equilibrium problem are proved in topological spaces. The results presented in this paper unify and generalize some known results in recent literature.

S-粗集的副集α-生成与α-生成定理

给出了S 粗集的副集α 生成 ,副集的集 -生成概念 ;给出了副集α 生成特性 ,副集的集 -生成特性 ;提出了副集的α 生成定理 ,副集的集 -生成定理 ;给出了S 粗集的属性非空原理 ,S 粗集的属性基数原理 .

On Generalized Douglas–Weyl(α, β)-Metrics

In this paper, we study generalized Douglas-Weyl （α,β）-metrics. Suppose that a regular （α,β）-metric F is not of Randers type. We prove that F is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas-Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.

S-粗集的副集α-生成粒度

利用S-粗集理论,给出单向副集As(Xσ)的α-生成粒度,双向副集As(X*)的α-生成粒度等概念;提出单向副集As(Xσ)的α-生成粒度特性定理,双向副集As(X*)的α-生成粒度特性定理;给出副集α-生成粒度的存在背景及在风险投资决策估计中的应用.

函数S-粗集,函数粗集与信息系统规律拆分-合成

给出函数单向S-粗集(function one direction singular rough sets)、函数单向S-粗集对偶(dual of function one direction singular rough sets)、函数双向S-粗集(function two direction singular rough sets)与函数粗集(function roughsets)。它们都是把函数概念引入到S-粗集中,改进S-粗集得到的。函数粗集是把函数概念引入到Z.Pawlak粗集中,改进Z.Pawlak粗集得到的。函数单向S-粗集、函数单向S-粗集对偶、函数双向S-粗集是函数S-粗集的三类形式。给出函数S-粗集与S-粗集的关系;给出函数粗集与Z.Pawlak粗集的关系;给出函数S-粗集与函数粗集的关系。利用这些结果,给出函数的区间离散与有限元素集的生成、函数离散-元素集合生成原理;给出函数S-粗集生成的信息规律、函数等价类动态特性-属性补充与删除原理;给出数据拆分-合成原理、信息规律动态拆分-合成的属性特征;给出信息规律动态拆分-合成不变性原理;利用这些概念与结果,给出信息规律拆分-合成与信息图像嵌入-分离的应用,给出嵌入信息图像的分离-辨识。函数S-粗集、函数粗集是粗集理论与应用研究中的一个新的研究方向。

基于S—R分解定理的极矩弹性力学增量型能量原理和广义变分原理

采用拖带坐标系的应力张量增量和变形梯度S—R分解定理意义下的应变张量增量,本文证明如在大位移、大变形途径中,存在有唯一的势能与余能函数,则可建立大变形极矩弹性理论的最小势能原理和最小余能原理。在此基础上用泛函的扩宗量变换与Lagrange乘子法结合建立了具有 _j^i、 _j^i、 _i、 ~i、五类独立变分宗量的广义变分原理,它是目前极矩弹性力学中最一般的广义变分原理。