Stochastic Constructions of Mixing Systems of Positive Local Rank

Mathematical Notes -     

A Theorem of the Alternative and Its Application to the Optimization of Set-Valued Maps

In this paper, we establish a theorem of the alternative in ordered linear topological spaces. Then, optimality conditions for the optimization of set-valued maps are obtained.     

对博弈论中Shaked ＆ Sutton结论的一个证明

本文运用博弈论基本原理 ,运用Rubinstern(1982 )讨价还价模型 ,针对贴现系数的不同情况 ,运用数学归纳法对Shaked&Sutton(1984 )的结论进行了证明     

微积分学中中值定理的统一证明

由一个定理的结论,给出Lagrange中值定理,Cauchy中值定理,积分中值定理和Taylor中值定理的统一证明及一个计算待定型极限的方法.     

A Sharp Form of the Cramér–Wold Theorem

The Cramér–Wold theorem states that a Borel probability measure P on ℝ ^d is uniquely determined by its one-dimensional projections. We prove a sharp form of this result, addressing the problem of how large a subset of these projections is really needed to determine P. We also consider extensions of our results to measures on a separable Hilbert space. First author partially supported by the Spanish Ministerio de Ciencia y Tecnología, grant BFM2002-04430-C02-02. Second author partially supported by Instituto de Cooperación Iberoamericana, Programa de Cooperación Interuniversitaria AL-E 2003. Third author partially supported by grants from NSERC and the Canada research chairs program.     

关于叶果洛夫定理证明的一个注记

给出了一种函数列收敛的等价刻画,并利用此等价刻画来给出实变函数中叶果洛夫定理的另一种证明方法.     

Characterization of the minimal and maximal operator ideals associated to the tensor norm defined by a sequence space

We characterize the minimal and maximal operator ideals associated, in the sense of Defant and Floret, to a wide class of tensor norms derived from a Banach sequence space. Our results are extensions of classical ones about tensor norms of Saphar [Studia Math. 38 (1972) 71-100] and show the key role played by the structure of finite-dimensional subspaces in this kind of problems.     

Decay of geometry for Fibonacci critical covering maps of the circle

We study the growth of Dfⁿ(f(c)) when f is a Fibonacci critical covering map of the circle with negative Schwarzian derivative, degree d2 and critical point c of order ℓ>1. As an application we prove that f exhibits exponential decay of geometry if and only if ℓ2, and in this case it has an absolutely continuous invariant probability measure, although not satisfying the so-called Collet–Eckmann condition.     

An Analog of the Poincarée Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem

We establish an analog of the Cauchy–Poincarée separation theorem for normal matrices in terms of majorization. A solution to the inverse spectral problem (Borg type result) is also presented. Using this result, we generalize and extend the Gauss–Lucas theorem about the location of roots of a complex polynomial and of its derivative. The generalization is applied to prove old conjectures due to de Bruijn–Springer and Schoenberg.     

n重积分换元公式的证明

在建立微分线性空间基础上、引入向量外积和有向体积,找到了微分线性变换与有向体积之间的关系,由此给出了n重积分换元公式的一个简单证法.     

On a Relation between Intrinsic and Extrinsic Dirichlet Forms for Interacting Particle Systems

In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic pre–Dirichlet form ℰ^Γ of the Poisson measure π_σ in terms of the extrinsic one ℰ^P. More precisely, replacing π_σ by a Gibbs measure μ on the configuration space Γ_X we derive a relation between the intrinsic prend–Dirichlet form ℰ^Γ_μ of the measure μ and the extrinsic one ℰ^P. As a consequence we prove the closability of ℰ^Γ_μ on L²(Γ_X, μ) under very general assumptions on the interaction potential of the Gibbs measures μ.     

Rolle's Theorem for Polynomials of Degree Four in a Hilbert Space

In an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem.     

单调有界数列必有极限定理的一个直接证明及其作用

利用实数十进制无限小数表示直接构造性地给出"单调有界数列必有极限"定理的一种简洁的新证明,并且从新视角揭示数学分析中的实数完备性和高等数学中的数列极限存在准则.     

On the Central Limit Theorem and Its Weak Invariance Principle for Strongly Mixing Sequences with Values in a Hilbert Space via Martingale Approximation

In this paper we not only prove an extension to Hilbert spaces of a sharp central limit theorem for strongly real-valued mixing sequences, but also slightly improve it. The proof is mainly based on the Bernstein blocking technique and approximations by martingale differences. Moreover, we derive also the corresponding functional central limit theorem.     

Banach空间一阶非线性微分方程终值问题解的存在性

在Fréchet空间中利用推广的Tychonov不动点定理研究了Banach空间中一阶非线性微分方程终值问题解的存在性.     

锥映象的K.Fan逼近定理及其应用

本文在Banach空间中分别就锥内的闭球、环形区域及球面，对半紧的1－集压缩映象证明了著名的K.Fan逼近定理，本质上改进和推广了文[2]的所有结果。并且应用所得结果给出了关于Banach空间中两点边值问题正固有元和正解的几个存在定理，这也是文[3]主要结果的推广和补充。     

An Asymptotic Independence Theorem for the Number of Matchings in Graphs

Let z(G) be the number of matchings (independent edge subsets) of a graph G. For a set M of edges and/or vertices, the ratio represents the probability that a randomly picked matching of G does not contain an edge or cover a vertex that is an element of M. We provide estimates for the quotient , depending on the sizes of the disjoint sets A and B, their distance and the maximum degree of the underlying graph G. It turns out that this ratio approaches 1 as the distance of A and B tends to ∞, provided that the size of A and B and the maximum degree are bounded, showing asymptotic independence. We also provide an application of this theorem to an asymptotic enumeration problem related to the dimer-monomer model from statistical physics. This material is based upon work supported by the German Research Foundation DFG under grant number 445 SUA-113/25/0-1 and the South African National Research Foundation under grant number 65972.     

On the Rate of Almost Everywhere Convergence of Certain Classical Integral Means, II

In this sequel to previous work of A. Stokolos and W. Trebels (1999, J. Approx. Theory98, 203–222) we indicate at the example of the Gauss–Weierstrass and the Abel–Poisson means the sharpness of some results obtained there. This is achieved by modifying methods of K. I. Oskolkov (1977, Math. USSR-Sb.32, 489–514) and A. A. Soljanik (1986, Ph.D. Thesis, Odessa) developed for the periodic case.     

“复数域上微分中值定理新证”注记

所谓“实数域上的微分中值定理可以推广到复数域上”的论断值得商榷．通过给出实例的方法,具体分析所谓的“解析函数的微分中值定理”之错误所在．