Unified Field Theory and Principle of Representation Invariance

The main objectives of this article are to postulate a new principle of representation invariance (PRI), and to refine the unified field model of four interactions, derived using the principle of interaction dynamics (PID). Intuitively, PID takes the variation of the action functional under energy-momentum conservation constraint, and PRI requires that physical laws be independent of representations of the gauge groups. One important outcome of this unified field model is a natural duality between the interacting fields (g,A,W ^a ,S ^k ), corresponding to graviton, photon, intermediate vector bosons W ^± and Z and gluons, and the adjoint bosonic fields $(\varPhi_{\mu}, \phi^{0}, \phi^{a}_{w}, \phi^{k}_{s})$ . This duality predicts two Higgs particles of similar mass with one due to weak interaction and the other due to strong interaction. The unified field model can be naturally decoupled to study individual interactions, leading to (1) modified Einstein equations, giving rise to a unified theory for dark matter and dark energy (Ma and Wang in Discrete Contin. Dyn. Syst., Ser A. 34(2):335–366, 2014), (2) three levels of strong interaction potentials for quark, nucleon/hadron, and atom respectively (Ma and Wang in Duality theory of strong interaction, 2012), and (3) two weak interaction potentials (Ma and Wang in Duality theory of weak interaction, 2012). These potential/force formulas offer a clear mechanism for both quark confinement and asymptotic freedom—a longstanding problem in particle physics (Ma and Wang in Duality theory of strong interaction, 2012).     

A Unified Approach to Algorithms Generating Unrestricted and Restricted Integer Compositions and Integer Partitions

An original algorithm is presented that generates both restricted integer compositions and restricted integer partitions that can be constrained simultaneously by (a) upper and lower bounds on the number of summands (“parts”) allowed, and (b) upper and lower bounds on the values of those parts. The algorithm can implement each constraint individually, or no constraints to generate unrestricted sets of integer compositions or partitions. The algorithm is recursive, based directly on very fundamental mathematical constructs, and given its generality, reasonably fast with good time complexity. A general, closed form solution to the open problem of counting the number of integer compositions doubly restricted in this manner also is presented; its formulaic link to an analogous solution for counting doubly-restricted integer partitions is shown to mirror the algorithmic link between these two objects.     

拓扑中分离公理的统一处理

在[1]中,我们从泛拓扑概念出发,提出了拓扑中分离公理的一个一般模式,并给出了相应的非标准条件。本文完全采用经典观点对分离公理作出一个新的统一处理,建立了若干命题用以指明分离性质具有遗传性、可积性、商保持性以及拓扑空间到某个幂空间的可嵌入性的一般性条件;得到了联系分离性质与复盖性质的几个一般结果;指明了邻域分离性与函数分离性在本文的观点之下完全得到统一。     

Local Convergence Analysis of Projection-Type Algorithms: Unified Approach

In this paper, we use a unified approach to analyze the local convergence behavior of a wide class of projection-type methods for solving variational inequality problems. Under certain conditions, it is shown that, in a finite number of iterations, either the sequence of iterates terminates at a solution of the concerned problem or all iterates enter and remain in the relative interior of the optimal face and, hence, the subproblem reduces to a simpler form.     

On Invariance Factors and Invariance Vectors for Difference Equations

In this paper, the concept of invariance factors and invariance vectors to obtain invariants (or first integrals) for difference equations will be presented. It will be shown that all invariance factors and invariance vectors have to satisfy a functional equation. This concept turns out to be analogous to the concept of integrating factors and integrating vectors for ordinary differential equations.     

A Nonconventional Invariance Principle for Random Fields

In Kifer and Varadhan (Ann Probab, to appear), we obtained a nonconventional invariance principle (functional central limit theorem) for sufficiently fast mixing stochastic processes with discrete and continuous time. In this article, we derive a nonconventional invariance principle for sufficiently well-mixing random fields.     

A Unified Treatment of Half-Discrete Hilbert-Type Inequalities with a Homogeneous Kernel

In this paper we provide a unified treatment of half-discrete Hilbert-type inequalities with a general homogeneous kernel. The main results are proved for the case of non-conjugate exponents. A special emphasis is given to determining conditions under which these inequalities include the best possible constants. As an application, we consider some operator expressions closely connected to established inequalities. Finally, we also provide improvements of derived half-discrete inequalities by virtue of the Hermite-Hadamard inequality.     

Stochastic Invariance for Differential Inclusions

The aim of this paper is to combine two ways for representing uncertainty through stochastic differential inclusions: a 'stochastic uncertainty", driven by a Wiener process, and a 'contingent uncertainty", driven by a set-valued map. The paper is also devoted to the invariance of closed under stochastic differential inclusions with a Lipschitz right-hand side, characterized in terms of stochastic tangent sets to closed subsets.     

Morita Invariance for Infinitesimal Deformations

Algebras and Representation Theory - Let A and B be two Morita equivalent finite dimensional associative algebras over a field ��. It is well known that Hochschild cohomology is...     

约束Birkhoff系统的形式不变性

约束Birkhoff系统的形式不变性是约束Birkhoff方程在无限小变换下的一种不变性。给出约束Birkhoff系统形式不变性的定义与判据,并研究了这种形式不变性与Noether对称性之间的关系。     

A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

The concept of acombinatorial(W; P; U)-geometryfor a Coxeter groupW, a subsetPof its generating involutions and a subgroupUofWwithP  Uyields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even Δ-matroids. The concept of a (W, P)-matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes ofWandPone gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial (W; P; U)-geometries and suggest two equivalent definitions of a (W; P; U)-matroid with coefficients in a fuzzy ringK. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that (W; P; U)-matroids with coefficients encompass matroids with coefficients and Δ-matroids with coefficients.     

Evolutors and Invariance

This paper concerns invariance of multifunctions under evolutors and its characterization through tangency of evolutors to multifunctions. The theory is illustrated by its applications to identity evolutors and to evolutors generated by semilinear perturbed equations.     

Invariance principles for von Mises and U-statistics

Summary The almost sure approximation of von Mises-statistics and U-statistics by appropriate stochastic integrals with respect to Kiefer processes is obtained. In general these integrals are non-Gaussian processes. As applications we get almost sure versions for the estimator of the variance and for the ²-test of goodness of fit.This work was done while the last author was a visiting professor at the Institut für Mathematische Stochastik at the University of Göttingen during the Spring of 1982. He thanks the Institut and its members for their hospitality     

Invariance principles for Gaussian sequences

An almost sure invariance principle is proved for stationary Gaussian sequences whose covariances r(n) satisfy r(n) = O (n ^–1–)for some >0.     

相伴的高斯随机变量序列的一个强不变原理

本文利用鞅的Skorohod表示, 在序列是高斯的且序列的协方差系数以幂指数速度递减的条件下,证明了相伴高斯随机变量序列的一个强不变原理\bd 作为推论得到了相伴高斯随机变量序列的重对数律和钟重对数律