Tilings,packings, coverings,and the approximation of functions

A packing (resp. covering) ? of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of ?. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls. Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real‐valued functions that rests on so‐called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non‐compact spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discontinuous Solutions of Linearized, Steady-State, Viscous, Compressible Flows

In thIs paper we prove the existence and uniqueness of the solution to a linearized, two dimensional, steady state, viscous, compressible Navier-Stokes equations in a strip. We obtain a regularity result and a new a priori estimate of the solution. We establish the discontinuity of the solution when the boundary data of pressure have a jump discontinuity. We also derive a formula for the discontinuous part and show that the remainder is smooth in the strip.     

需求不确定的强势零售商的闭环供应链决策

建立了需求不确定下两个竞争的生产商和一个强势零售商组成的具有产品再制造的闭环供应链模型.利用逆向归纳法对不同废旧品回收渠道下的模型进行分析,然后通过算例分析了不同模型的回收成本和新增回收率对零售价格、回收模式决策、供应链系统及成员期望利润的影响.结果表明:回收成本的增加导致生产商和系统的期望利润降低,零售商领头下只有在零售商负责回收的模式中,零售商的期望利润才是增加的;新增回收率的增加,可降低产品的零售价,增加供应链成员的期望利润,继而提高整个供应链系统的利润.     

Magnitude,Diversity, Capacities,and Dimensions of Metric Spaces

Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact spaces, in an equivalent but more natural and direct manner than in previous works by Leinster, Willerton, and the author. The new definition uncovers a previously unknown relationship between magnitude and capacities of sets. Exploiting this relationship, it is shown that for a compact subset of Euclidean space, the magnitude dimension considered by Leinster and Willerton is equal to the Minkowski dimension.     

Development of elementary teachers' science,technology, engineering,arts, and mathematics planning practices

The purpose of this instrumental case study was to understand elementary teachers' planning of Science, Technology, Engineering, Arts, and Mathematics (STEAM) curriculum during a two-year professional development experience. This research was guided by the following question: What planning practices developed or changed during a professional development experience? Analysis of 25 teachers' planning documents indicated growth in the following areas: (a) tighter alignment to fewer standards, (b) more meaningful integration and broader definitions of arts and technology, and (c) increased use of formative assessment but persistent difficulties in summatively assessing the multiple disciplines of STEAM. The growth in teachers' STEAM instruction from the beginning to the end of this two-year professional development was important in building our understanding of what constitutes sound design of STEAM inquiries.     

Sums,differences, products,and ratios of hypergeometric beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, differences, products and ratios of independent beta random variables. In this Note we extend Professor Pham-Gia's results when $X_1$ and $X_2$ are independent random variables distributed according to the confluent and Gauss hypergeometric distributions (which are generalizations of the beta distribution). For each of these distributions, we derive exact expressions for the densities of $S=X_1+X_2$, $D=X_1?X_2$, $P=X_1{X}_2$, and $R=X_2/X_1$. The expressions turn out to involve the hypergeometric functions of one and two variables. To cite this article: S. Nadarajah, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Trees, parking functions, syzygies, and deformations of monomial ideals

For a graph , we construct two algebras whose dimensions are both equal to the number of spanning trees of . One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to -parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

     

Center,Centroid, Extended Centroid,and Quotients of Jordan Systems

In this article we prove that the extended centroid of a nondegenerate Jordan system is isomorphic to the centroid (and to the center in the case of Jordan algebras) of its maximal Martindale-like system of quotients with respect to the filter of all essential ideals.