几个改进的矩阵不等式（英文）

本文研究了矩阵不等式的问题.利用两个新的标量不等式,得到了矩阵的加权几何均值不等式和Hilbert-Schmidt范数不等式,所得的结果改进了相应的不等式.     

反向基本不等式及应用

运用导数方法,对两个正数的基本不等式中的权数作了分析研究,给出了基本不等式的三个反向不等式;运用反向不等式,通过选取适当的权数,给出了指数平均的反向不等式,同时给出了涉及A,G,L,E的几个新的不等式.     

变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题,给出了不等式组及变分不等式问题近似解的可微优化方法,得到了不等式组和变分不等式问题的解集合的示性函数.     

Orlicz等周不等式

本文利用经典的Popoviciu不等式和Orlicz-Minkowski混合体积不等式,建立了凸体的广义Orlicz等周不等式.这个新的Orlicz等周不等式在特殊情况下,分别产生了经典的等周不等式,L_p-等周不等式和Orlicz等周不等式.     

不等式高考试题分析与教学策略分析

高中数学由众多内容组成,其中不等式是高中数学的重要组成部分.学生对不等式的学习从初中就开始了,掌握了不等式的基本性质和解法,为学生在高中进一步学习不等式知识打下了基础.此外,现实生活中蕴含着丰富的不等式知识,随处可见不等式在生活中的应用.如今,不等式在数学高考中更是占据了重要的位置,同时与函数、方程、三角等高中数学知识联系紧密.分析     

从积分几何的观点看几何不等式——纪念李国平院士吴新谋教授诞辰100周年

该文先介绍一些中国数学家在几何不等式方面的工作.作者用积分几何中著名的Poincarè公式及Blaschke公式估计一随机凸域包含另一域的包含测度, 得到了经典的等周不等式和Bonnesen -型不等式.还得到了一些诸如对称混合等周不等式、Minkowski -型和Bonnesen -型对称混合等似不等式在内的一些新的几何不等式.最后还研究了Gage -型等周不等式以及Ros -型等周不等式.     

含参数的一元二次不等式的轻松破解

含参数的不等式解法,涉及到分类讨论,于是也就成了学生一遇到就头疼的问题,甚至是恐惧,在后面的利用导数求函数单调区间的问题时,也就变成了部分学生的难题.针对学生在此类问题中出现的问题,笔者做一梳理,对轻松求解含参数的不等式,乃至分类讨论问题进行了思考.一、熟练掌握两类特殊不等式的解法,形成固定套路即会解两类特殊不等式,一类是一元一次不等式,另一类是一元二次不等式.解不等式,从代数角度上看就是利用不等式的性质,找已知不等式的同解不等式的过程,这个过程的主要任务是化简,即化简到一元一次不等式;从几     

一个函数不等式的加强与延伸

从一个常见的不等式谈起,分析了多种证明方法,运用该不等式推导出了多个重要结论,对不等式进行了扩充和加强,解释了蕴含的意义,显示了该不等式的重要性和深刻性.     

Hardy平均及其不等式

本文研究了涉及Hardy平均的不等式问题.借助于优超理论,获得了2维Hardy平均不等式成立的充要条件及n维Hardy平均不等式的充分条件.并为建立解析不等式和高维结果提供了一些方法.     

完全主正矩阵的行列式不等式

给出了完全主正矩阵的凸性不等式和Minkowski型不等式,并推出了M矩阵,亚正定矩阵等类型的矩阵在一定条件下的凸性不等式和Minkowski型不等式.     

Active control of secondary resonances piezoelectric sandwich beams

Secondary resonances of piezoelectric/elastic/piezoelectric sandwich beams submitted to active control are studied in this paper. The proportional and derivative nonlinear potential feedback controls via piezoelectric sensor and actuator layers are used. The dynamics of the beam is modelled by a highly nonlinear ordinary-differential equation. The method of multiple scales is applied and approximate solutions are obtained for hard excitations. Analytical frequency and phase-amplitude relationships as well as the time response are explicitly given for various super- and subharmonic resonances. Static and dynamic stability criteria are elaborated and critical displacement and excitation amplitudes associated to the resulting unstable zones are analytically given. The feedback parameters effects on the subharmonic and superharmonic resonances and on their stability are investigated.     

Financial Risk Modelling and Econometric Inference

In finance, the explicit modelling of uncertainty takes on a particularly important role. The values of financial derivatives increase in the return volatility of the underlying security. This notion requires a concept of volatility and hence uncertainty. In addition, the choice between modelling in discrete and continuous time is not arbitrary, since it corresponds to a distinction between incomplete and complete markets, respectively, and this distinction matters for asset pricing, financial risk modelling, and inference. Risk and volatility are closely connected, and implied volatility, volatility forecasting, volatility in term structure models, stochastic volatility, and portfolio analysis are considered and related to a more general interplay between cross-sectional and dynamic aspects in finance. Stocks, bonds, and options are considered and placed in the context of efficiency and separation in inference.     

Remarks on classifications and adjunctions

Jon Barwise and Jerry Seligman introduced the category of classifications and infomorphisms to model information theoretical concepts and information flow processes. This is the dual category of the category of Chu spaces introduced by Michael Barr and Peter Chu and studied extensively by Vaughan Pratt, Gordon Plotkin and others at Stanford. It is also very closely related to the theory of institutions introduced in a different context by Joseph Goguen and Rod Burstall. Here some aspects of the theory are reviewed and some adjunctions between the category of classifications and other related categories are studied from a more abstract point of view.     

Latroids and their representation by codes over modules

It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code is an evaluation of the Tutte polynomial of the matroid, and a standard identity relating the Tutte polynomials of dual matroids gives rise to a MacWilliams identity relating the weight enumerators of dual codes. More recently, codes over rings and modules have been considered, and MacWilliams type identities have been found in certain cases.

In this paper we consider codes over rings and modules with code duality based on a Morita duality of categories of modules. To these we associate latroids, defined here. We generalize notions of deletion, contraction, minors and duality, on both codes and latroids, and examine all natural relations among these.

We define generating functions associated with codes and latroids, and prove identities relating them, generalizing above-mentioned generating functions and identities.

     

RECENT PROGRESS ON SPHERICAL HARMONIC APPROXIMATION MADE BY BNU RESEARCH GROUP ——In memory of Professor Sun Yongsheng

As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

Range的《微积分是什么?从简单代数到深入分析》

在于向中国读者介绍Michael Range的新书.他对导数及积分都进行了深入而又简练的分析.     

Nonlinear and Geometric Programming – Current Status

Great strides have been made in nonlinear programming (NLP) in the last 5 years. In smooth NLP, there are now several reliable and efficient codes capable of solving large problems. Most of these implement GRG or SQP methods, and new software using interior point algorithms is under development. NLP software is now much easier to use, as it is interfaced with many modeling systems, including MSC/NASTRAN, and ANSYS for structural problems, GAMS and AMPL for general optimization, Matlab and Mathcad for general mathematical problems, and the widely used Microsoft Excel spreadsheet. For mixed integer problems, branch and bound and outer approximation codes are now available and are coupled to some of the above modeling systems, while search methods like Tabu Search and Genetic algorithms permit combinatorial, nonsmooth, and nonconvex problems to be attacked.     

Rectangularity and paramonotonicity of maximally monotone operators

Maximally monotone operators play a key role in modern optimization and variational analysis. Two useful subclasses are rectangular (also known as star monotone) and paramonotone operators, which were introduced by Brezis and Haraux, and by Censor, Iusem and Zenios, respectively. The former class has a useful range of properties while the latter class is of importance for interior point methods and duality theory. Both notions are automatic for subdifferential operators and known to coincide for certain matrices; however, more precise relationships between rectangularity and paramonotonicity were not known. Our aim is to provide new results and examples concerning these notions. It is shown that rectangularity and paramonotonicity are actually independent. Moreover, for linear relations, rectangularity implies paramonotonicity but the converse implication requires additional assumptions. We also consider continuous linear monotone operators, and we point out that in the Hilbert space both notions are automatic for certain displacement mappings.