一道硕士研究生入学试题的六种证法

费马定理,洛尔定理,拉格朗日中值定理,达布定理,泰勒公式,凸函数,这些内容几乎覆盖了一元函数微分学,应用这些知识,对吉林大学2009年硕士研究生入学考试试题中的一道试题提供六种证法.     

关于AQSI序列的几乎处处收敛性及强大数定理

建立了关于AQSI序列的Kolmogorov型不等式和AQSI序列的三级数定理,讨论了AQSI序列的几乎处处收敛性并且得到了关于AQSI序列的chung型强大数定理.     

关于weierstrass逼近定理的几点注记

Weierstrass逼近定理是函数逼近论中的重要定理之一,定理阐述了闭区间上的连续函数可以用一多项式去逼近.将该定理进行推广:即使一个函数是几乎处处连续的,也不一定具有与连续函数相类似的逼近性质,但是一个处处不连续的函数却有可能具有这样的性质.证明了定义在闭区间上且与连续函数几乎处处相等的函数具有类似的逼近性质,并给出了weierstrass逼近定理的一个推广应用.     

相依序列加权和的几乎处处中心极限定理

该文讨论了非平稳负(正)相依序列加权和的几乎处处中心极限定理,改进并推广了相依序列几乎处处中心极限定理的相关结果.     

关于树上连续状态马氏泛函的一类强偏差定理

强偏差定理一直是国际概率论界研究的中心课题之一.通过构造适当的非负鞅,将Doob鞅收敛定理应用于几乎处处收敛的研究,给出了一类特殊非齐次树上连续状态马氏泛函的若干强偏差定理.     

非齐次树上马氏信源的一类Shannon-McMillan定理

通过构造适当的非负鞅,将Doob鞅收敛定理应用于几乎处处收敛的研究,给出了非齐次树上m重非齐次马氏信源的一类Shannon-McMillan定理.     

对《罗尔定理的一个证明》一文的质疑

杨翰深同志在《数学通报》1990年第4期介绍《罗尔定理的一个证明》.此文在对罗尔定理的证明中,回避了平行弦定理和区间套定理的运用,试图另辟新径.这种尝试其愿望是良好的,遗憾的是文中出现了一些疏漏及错误.这里仅就下述问题提出质疑,以便与作者和读者商榷.     

关于随机序列几何平均的若干强偏差定理

利用似然比构造几乎处处收敛的上鞅,讨论了正值随机变量序列几何平均的一类强偏差定理,得到了与经典算术平均的强极限定理的类似结果.     

有限级亚纯函数的不变原理

研究有限级亚纯函数的统计性质,证明了有限级亚纯函数的几乎必然不变原理,如中心极限定理、重对数定律和泛函中心极限定理等统计定律是几乎必然不变原理的推论.     

树指标马氏链关于乘积二项分布的强偏差定理

通过构造适当的非负上鞅,将Doob鞅收敛定理应用于几乎处处收敛的研究,研究给出了树指标马氏链关于乘积二项分布的一类强偏差定理.     

Linear birth and death models and associated Laguerre and Meixner polynomials

We study birth and death processes with linear rates λ_n = n + α + c + 1, μ_{n + 1} = n + c, n 0 and μ₀ is either zero or c. The spectral measures of both processes are found using generating functions and the integral transforms of Laplace and Stieltjes. The corresponding orthogonal polynomials generalize Laguerre polynomials and the choice μ₀ = c generates the associated Laguerre polynomials of Askey and Wimp. We investigate the orthogonal polynomials in both cases and give alternate proofs of some of the results of Askey and Wimp on the associated Laguerre polynomials. We also identify the spectra of the associated Charlier and Meixner polynomials as zeros of certain transcendental equations.     

Affine-inscribed and affine-circumscribed polygons and polytopes

Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ³, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in ℝ³ (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder with directrix P. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298. Translated by B. M. Bekker.     

Partitions and pairwise sums and products

A seven cell partition of N is constructed with the property that no infinite set has all of its pairwise sums and products in any one cell. A related Ramsey Theory question is shown to have different answers for two and three cell partitions.     

Distributive and Multiplication Modules and Rings

We study rings in which every ideal is a finitely generated multiplication right ideal.     

Beliefs and beyond: affect and the teaching and learning of mathematics

The importance of beliefs for the teaching and learning of mathematics is widely recognized among mathematics educators. In this special issue, we explicitly address what we call “beliefs and beyond” to indicate the larger field surrounding beliefs in mathematics education. This is done to broaden the discussion to related concepts (which may not originate in mathematics education) and to consider the interconnectedness of concepts. In particular, we present some new developments at the conceptual level, address different approaches to investigate beliefs, highlight the role of student beliefs in problem-solving activities, and discuss teacher beliefs and their significance for professional development. One specific intention is to consider expertise from colleagues in the fields of educational research and psychology, side by side with perspectives provided by researchers from mathematics education.