A change of variables formula for mappings in

A change of variables formula for mappings in is obtained, where the usual jacobian is replaced by the determinant of the approximate differential.

     

On a result of Aumann and Shapley about values of non-atomic games

We focus our attention on the argument developed by R.J. Aumann and L.S. Shapley in a proposition used to prove the existence of a value on a certain class of games. Since we have found such an argument to be inexact, we have devised a supplementary construction which should be incorporated in the original proof to make it run in a satisfactory way.     

A new theory of regular functions of a quaternionic variable

In this paper we develop the fundamental elements and results of a new theory of regular functions of one quaternionic variable. The theory we describe follows a classical idea of Cullen, but we use a more geometric formulation to show that it is possible to build a rather complete theory. Our theory allows us to extend some important results for polynomials in the quaternionic variable to the case of power series.     

Characterization of Sobolev and BV spaces

The main results of this paper are new characterizations of W1,p(Ω), 1<p<∞, and BV(Ω) for Ω⊂RN an arbitrary open set. Using these results, we answer some open questions of Brezis (2002) [10] and Ponce (2004) [25].     

On a Characterization of the Gamma Distribution: The Independence of the Sample Mean and the Sample Coefficient of Variation

Let n 3 and let X ₁,...,X _n be positive i.i.d. random variables whose common distribution function f has a continuous p.d.f. Using earlier work of the present authors and a method due to Anosov for solving certain integro-functional equations, it is shown that the independence of the sample mean and the sample coefficient of variation is equivalent to that f is a gamma function. While the proof is of methodological interest, this conclusion can also be arrived at without any assumptions by appealing to the Laplace-Stieltjes transform, as in the Concluding Romark (Section 3).     

Image Restoration and Decomposition via Bounded Total Variation and Negative Hilbert-Sobolev Spaces

We propose a new class of models for image restoration and decomposition by functional minimization. Following ideas of Y. Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, our model decomposes a given (degraded or textured) image u ₀ into a sum u+v. Here u∈BV is a function of bounded variation (a cartoon component), while the noisy (or textured) component v is modeled by tempered distributions belonging to the negative Hilbert-Sobolev space H ^−s . The proposed models can be seen as generalizations of a model proposed by S. Osher, A. Solé, L. Vese and have been also motivated by D. Mumford and B. Gidas. We present existence, uniqueness and two characterizations of minimizers using duality and the notion of convex functions of measures with linear growth, following I. Ekeland and R. Temam, F. Demengel and R. Temam. We also give a numerical algorithm for solving the minimization problem, and we present numerical results of denoising, deblurring, and decompositions of both synthetic and real images.     

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion, a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.     

Some critical minimization problems for functions of bounded variations

Using a new elementary method, we prove the existence of minimizers for various critical problems in BV(Ω) and also in W1,p(Ω), 1<p<∞.     

On the Unavoidable Uncertainty of Truth in Dynamic Geometry Proving

The aim of this note is to discuss some issues posed by the emergency of universal interfaces able to decide on the truth of geometric statements. More specifically, we consider a recent GeoGebra module allowing general users to verify standard geometric theorems. Working with this module in the context of Varignon’s theorem, we were driven—by the characteristics of the GeoGebra interface—to perform a quite detailed study of the very diverse fate of attempting to automatically prove this statement, when using two different construction procedures. We highlight the relevance—for the theorem proving output—of expression power of the dynamic geometry interface, and we show that the algorithm deciding about the truth of some—even quite simple—statements can fall into a not true and not false situation, providing a source of confusion for a standard user and an interesting benchmark for geometers interested in discovering new geometric facts.     

On the subdifferential of a submodular function

The author recently introduced a concept of a subdifferential of a submodular function defined on a distributive lattice. Each subdifferential is an unbounded polyhedron. In the present paper we determine the set of all the extreme points and rays of each subdifferential and show the relationship between subdifferentials of a submodular function and subdifferentials, in an ordinary sense of convex analysis, of Lovász's extension of the submodular function. Furthermore, for a modular function on a distributive lattice we give an algorithm for determining which subdifferential contains a given vector and finding a nonnegative linear combination of extreme vectors of the subdifferential which expresses the given vector minus the unique extreme point of the subdifferential.     

Characterizations and Examples of Hidden Regular Variation

Random vectors on the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn (1996, 1997). We provide characterizations and examples of such distribution in terms of mixture models and product models.Sidney Resnicks research was partially supported by NSF grant DMS-0303493 and NSA grant MSPF-02G-183 at Cornell University.This revised version was published online in March 2005 with corrections to the cover date.     

The landscape: Through a Different Lens A New Angle on the Price Spiral in Higher Education

Abstract

“Academic institutions now find themselves confronting hard questions about whether their spending patterns match their priorities and about how to communicate the choices they have made to the public.”

The College Cost Crisis: A Congressional Analysis of College Costs and Implications for America's Higher Education System     

具有边梁加固的板弯曲问题的变分-差分方法研究

具有边梁加固的板的弯曲问题,其平衡方程模型为四阶椭圆型偏微分方程的边值问题,其中的自然边界条件涉及到了沿板边的切线和法线方向的高阶导数,对于非均匀、变厚度的板,该问题还具有"变系数"的特点.由问题的变分模型入手,应用变分-差分方法构造了该边值问题的一个差分格式.由于该方法能够结合平衡方程模型中的边界条件以消除沿板边的高阶导数项,因而,所得差分算子仅仅依赖于板面网格结点,并且保持了差分算子的对称、正定性质.同时,将已得算法在计算机上进行了数值模拟,并与现有文献进行了对比计算.结果显示本文所给出的算法具有较高的精确度,该算法将可用于定量地揭示板与边梁之间相互作用的规律,为工程设计提供参考依据.     

On the Rate of Convergence of Fourier Series of Functions of Bounded Variation

Mathematical Notes -     

基于差异系数的多元过程能力指数

过程能力指数表示过程能力满足产品质量标准的程度的评价指标，最普遍使用的过程能力指数有Cp、Cpk、Cpm、Cpkm等，它们已在各大工业部门得到了广泛的应用，但是多元质量特性的过程能力指数尚未得到很好的解决。许多学试图以不同的方式、方法计算多元质量特性的过程能力指数，但存在局限性。本以多元正态分布为基础，比较分析了每个质量特征的差异系数，给出了多元过程能力指数的计算公式。最后的一个算例表明，该方法是有效且符合实际的。     

Foundations of a Nonlinear Distributional Geometry

Colombeau's construction of generalized functions (in its special variant) is extended to a theory of generalized sections of vector bundles. As particular cases, generalized tensor analysis and exterior algebra are studied. A point value characterization for generalized functions on manifolds is derived, several algebraic characterizations of spaces of generalized sections are established and consistency properties with respect to linear distributional geometry are derived. An application to nonsmooth mechanics indicates the additional flexibility offered by this approach compared to the purely distributional picture.     

Rediscovering the contributions of Rodrigues on the representation of special functions

Following the pressure of Hermite in 1865 and Heine in 1868, the name of Rodrigues was definitively associated to his famous representation given in 1815. In this paper, we precise the contribution of Rodrigues, proving that his formula not only generates the Legendre polynomials, but also special Gegenbauer polynomials and even the corresponding second kind functions, which are never mentioned. The links with Hildebrandt polynomials and with the Bell polynomials are also briefly examined.     

Dual-Role Based Pricing in a Dynamic and Competitive Environment

The dual role of price, as a product attribute signaling quality and as a measure of sacrifice, serving as a benchmark for comparing the utility gains from superior product quality, is now well established in the marketing and economic literature. However, knowledge about their long-run impact and the influence of competition on these effects still remains very sparse. In the current paper, with reference to a dynamic and competitive framework, an analytical model is proposed to help determining optimal decision rules for price incorporating both roles. The main results are as follows: (i) The optimal pricing policy is determined as a Nash equilibrium strategy. (ii) The resulting equilibrium price is higher than an equilibrium that disregards the carryover price effects. (iii) For a symmetric competition, we provide normative rules on how firms should set prices as a function of the perceived quality; particularly, how the price should be set initially, when there is little product familiarity and the perceived quality is low, and how this price should vary as the perceived quality increases. (iv) At steady state, we find that the level of equilibrium margin, in percentage terms, decreases with the elasticity of demand with respect to the brand price, but this decrease is moderated by the elasticity of demand with respect to the brand perceived quality, the cross elasticity of demand with respect to the competitor’s perceived quality and the effect of the competitor’s current price on the firm’s perceived quality deterioration. The author thanks Konstantin Kogan for helpful discussions and comments.     

环境污染中具有脉冲效应的捕食系统动力学分析

针对环境问题日益严峻,考虑到种群密度制约项的影响,建立了环境污染中脉冲输入污染物,且具有HollingⅡ型的两种群捕食与被捕食系统的动力学模型.利用脉冲微分方程的积分均值法和比较定理,得到系统弱平均持续生存和种群灭绝的充分条件.理论可用来保护物种,尤其是濒临灭绝的物种,防止灭绝.     

一类脉冲微分系统的有界变差解

在比文[6]更弱的条件下讨论了固定时刻脉冲微分系统与Kurzweil广义常微分方程的关系,并建立了这类脉冲微分系统有界变差解的局部存在性和唯一性定理．