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.     

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.     

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.     

Projections of normed linear spaces with closed subspaces of finite codimension as kernels

Summary It follows from [1], [4] and [7] that any closed <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>n$-codimensional subspace ($n \ge 1$ integer) of a real Banach space $X$ is the kernel of a projection $X \to X$, of norm less than $f(n) + \varepsilon$~($\varepsilon > 0$ arbitrary), where \[ f (n) = \frac{2 + (n-1) \sqrt{n+2}}{n+1}. \] We have $f(n) < \sqrt{n}$ for $n > 1$, and \[ f(n) = \sqrt{n} - \frac{1}{\sqrt{n}} + O \left(\frac{1}{n}\right). \] (The same statement, with $\sqrt{n}$ rather than $f(n)$, has been proved in [2]. A~small improvement of the statement of [2], for $n = 2$, is given in [3], pp.~61--62, Remark.) In [1] for this theorem a deeper statement is used, on approximations of finite rank projections on the dual space $X^*$ by adjoints of finite rank projections on $X$. In this paper we show that the first cited result is an immediate consequence of the principle of local reflexivity, and of the result from [7].     

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.     

A short Proof of Bezout's Theorem in P2

We will give a new short proof of Bezout's Theorem for complex algebraic curves in P ² which is local.     

On the moduli of convexity

It is known that, given a Banach space , the modulus of convexity associated to this space is a non-negative function, non-decreasing, bounded above by the modulus of convexity of any Hilbert space and satisfies the equation for every , where is a constant. We show that, given a function satisfying these properties then, there exists a Banach space in such a way its modulus of convexity is equivalent to , in Figiel's sense. Moreover this Banach space can be taken to be two-dimensional.

     

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.     

Common Fixed Point Theory for Multivalued Contractive Maps of Reich Type in Uniform Spaces

This article presents common fixed point theory for multivalued maps defined on complete gauge spaces. Our analysis is elementary and relies only on properties of the generalized Hausdorff pseudometric.     

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.     

Estimates of difference norms for functions in anisotropic Sobolev spaces

We investigate the spaces of functions on ?ⁿ for which the generalized partial derivatives Dequation/tex2gif-sup-2.gif_kf exist and belong to different Lorentz spaces Lequation/tex2gif-sup-3.gif . For the functions in these spaces, the sharp estimates of the Besov type norms are found. The methods used in the paper are based on estimates of non‐increasing rearrangements. These methods enable us to cover also the case when some of the p_k's are equal to 1. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of coercive approximations for strictly monotone quasistatic models in inelastic deformation theory

This article studies coercive approximation procedures in the infinitesimal inelastic deformation theory. For quasistatic, strictly monotone, viscoplastic models using the energy method and the Young measures approach a convergence theorem in generalized Orlicz spaces is proved. The main step in the proof is a characterization of the weak limit of non‐linear terms by the convergence in measure. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

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.     

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.