Weak Convergence of Measures in Conservative Systems

Families of probability measures on the phase space of a dynamical system are considered. These measures are obtained as shifts of a given measure by the phase flow. Sufficient conditions for the existence of the weak convergence of the measures as the rate of the shift tends to infinity are suggested. The existence of such a limit leads to a new interpretation of the second law of thermodynamics. Bibliography: 5 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 194–205     

Optimization of Bilinear Systems Using a Higher-Order Variational Method

This paper derives some optimization results for bilinear systems using a higher-order method by characterizing them over matrix Lie groups. In the derivation of the results, first a bilinear system is transformed to a left-invariant system on matrix Lie groups. Then, the product of exponential representation is used to express this system in canonical form. Next, the conditions for optimality are obtained by the principles of variational calculus. It is demonstrated that closed-form analytical solutions exist for classes of bilinear systems whose Lie algebra are nilpotent.     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).     

Symmetric Bilinear Forms over Valuation Rings

本文讨论赋值环上的对称线性型、二次型和对称矩阵的合同标准形。     

Construction of Symmetric and Symplectic Bilinear Forms

Let V be a finite-dimensional vector-space. A linear mapping on V is called simple if V( - 1) is 1-dimensional. Let S be a set of simple bijections on V. We discuss conditions entraining that each element of S is orthogonal (respectively symplectic) under an appropriate symmetric (respectively symplectic) bilinear form on V.     

Weak Convergence of Solutions of the Liouville Equation for Nonlinear Hamiltonian Systems

We suggest sufficient conditions for the existence of weak limits of solutions of the Liouville equation as time increases indefinitely. The presence of the weak limit of the probability distribution density leads to a new interpretation of the second law of thermodynamics for entropy increase.     

Approximation of Weak Convergence

The relations between the k-uniform convexity of Banach spaces and both the real and the complex method of interpolation are studied. Estimates on the behaviour of the moduli of k-rotundity are given.     

辛代数的不变双线性型

考察了辛代数和与它相联系的李三系的双线性型之间的关系,并证明了辛代数的反对称不变双线性型可以唯一扩张到与它相联系的李三系中.作为这种关系的一个应用,得到了二次辛代数是单辛代数的一个充要条件,并证明二次辛代数的唯一分解定理.     

Exact Estimates for Moments of Random Bilinear Forms

The present paper concentrates on the analogues of Rosenthal's inequalities for ordinary and decoupled bilinear forms in symmetric random variables. More specifically, we prove the exact moment inequalities for these objects in terms of moments of their individual components. As a corollary of these results we obtain the explicit expressions for the best constant in the analogues of Rosenthal's inequality for ordinary and decoupled bilinear forms in identically distributed symmetric random variables in the case of the fixed number of random variables.     

On Weak Convergence of Hypermeasures

Measures on the hyperspace of the closed sets with the FLACHSMEYER-FELL topology are completely defined by their capacities. A necessary and sufficient condition is given for the weak convergence of a sequence of positive bounded σ-additive measures on the hyperspace in terms of their capacities.     

Weak Convergence of Topological Measures

Journal of Theoretical Probability - Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and...     

Weak Convergence in Circulant Matrices

Necessary and sufficient conditions for convergence in distribution of products of i.i.d. d× d random circulant matrices are established here. The important role played by matrices in SO(d) is pointed out, and the validity of this result is shown to also hold for a class of Toeplitz matrices.     

对一些弱连续性的讨论

引入了弱半连续及弱准连续性的概念,讨论了他们的一些性质,并对某些弱连续性之间的关系进行了探讨.     

强收敛与弱收敛的微小区别

对于独立同分布随机变量序列{Xn,n≥1},证明了其满足弱大数律的一个必要条件是:对于任何r∈(0,1),E|X|r<∞都成立.另外我们还举例说明了这样的条件不是充分的.     

可积鞅测度的弱收敛(英语)

本文引入了可积鞅测度弱收敛的概念,并给出了可积鞅测弱收敛的一系列条件     

Bilinear Forms and Fierz Identities for Real Spin Representations

Given a real representation of the Clifford algebra corresponding to ${\mathbb{R}^{p+q}}$ with metric of signature (p, q), we demonstrate the existence of two natural bilinear forms on the space of spinors. With the Clifford action of k-forms on spinors, the bilinear forms allow us to relate two spinors with elements of the exterior algebra. From manipulations of a rank four spinorial tensor introduced in [1], we are able to find a general class of identities which, upon specializing from four spinors to two spinors and one spinor in signatures (1,3) and (10,1), yield some well-known Fierz identities. We will see, surprisingly, that the identities we construct are partly encoded in certain involutory real matrices that resemble the Krawtchouk matrices [2][3].     

Weak Reflections and Weak Factorization Systems

We describe a one-to-one correspondence between saturated weak factorization systems and weak reflections in categories C\mathcal{C} with finite products. This actually extends to an adjunction between the category of natural weak factorization systems on C\mathcal{C} (in the sense of Grandis and Tholen, Arch Math 42:397–408, 2006, and Garner, arXiv preprint, 2007) and the category of monads on C\mathcal{C}. Explicit comparisons are made with the parallel result of Cassidy et al. (J Aust Math Soc 38:287–329, 1985), linking factorization systems and reflective subcategories.