■混合序列的强大数律

讨论了不同分布■混合序列的强大数律,推广了Kolmogorov强大数定律和Marcinkiewicz强大数定律.     

Approximation numbers of weighted composition operators

We study the approximation numbers of weighted composition operators $f?w?(f°\phi )$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight w can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).     

Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition

This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables, the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.     

Compact representation of the full Broyden class of quasi‐Newton updates

In this paper, we present the compact representation for matrices belonging to the Broyden class of quasi‐Newton updates, where each update may be either rank one or rank two. This work extends previous results solely for the restricted Broyden class of rank‐two updates. In this article, it is not assumed that the same Broyden update is used in each iteration; rather, different members of the Broyden class may be used in each iteration. Numerical experiments suggest that a practical implementation of the compact representation is able to accurately represent matrices belonging to the Broyden class of updates. Furthermore, we demonstrate how to compute the compact representation for the inverse of these matrices and a practical algorithm for solving linear systems with members of the Broyden class of updates. We demonstrate through numerical experiments that the proposed linear solver is able to efficiently solve linear systems with members of the Broyden class of matrices with high accuracy. As an immediate consequence of this work, it is now possible to efficiently compute the eigenvalues of any limited‐memory member of the Broyden class of matrices, allowing for the computation of condition numbers and the ability to perform sensitivity analysis.     

基于灰色线性回归组合模型的长沙、武汉房地产投资预测研究

本文通过对长沙、武汉2001-2007年房地产投资额的历史数据进行分析,分别建立了两市的灰色线性回归组合模型.模型较好地拟合了长沙、武汉的房地产投资情况,运用该模型得到了两市2008、2009年的预测值,并对其房地产投资发展趋势的异同进行比较分析,对房地产市场的发展具有一定的指导意义.     

A Polyhedral Model in Euclidean 3-Space of the Six-Pentagon Map of the Projective Plane

In a private communication, Branko Grünbaum asked: “I wonder whether you know anything about the possibility of realizing as a polyhedron in Euclidean 3-space the family of six pentagons, that is a model of the projective plane arising by identifying antipodal points of the regular dodecahedron. Naturally, any realization must have some self-intersections—but is there any realization that is not completely contained in a plane?”We show that it is possible to realize this polyhedron; in our realization five of the six faces are simple polygons. In this model there are sets of three faces, which form a realization of the Möbius strip without self-intersections. There are four variants of the model. We conjecture that in any model of this polyhedron there must be at least one self-intersecting face.     

对两道线性代数考研试题的讨论

对一道线性代数考研题在复数域上进行了解答和实现,解释了另一道线性代数考研题中正交向量选取的任意性.     

New identities in the Catalan triangle

In this paper we prove new identities in the Catalan triangle whose (n,p) entry is defined by

     

Weighted covering numbers of convex sets

In this paper we define the notions of weighted covering number and weighted separation number for convex sets, and compare them to the classical covering and separation numbers. This sheds new light on the equivalence of classical covering and separation. We also provide a formula for computing these numbers via a limit of classical covering numbers in higher dimensions.     

Wall crossings for double Hurwitz numbers

Double Hurwitz numbers count covers of P¹ by genus g curves with assigned ramification profiles over 0 and ∞, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification (Goulden et al., 2005) [10], and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0 (Shadrin et al., 2008) [15]. This paper gives a unified approach to these results and strengthens them in several ways — the most important being the extension of the results of Shadrin et al. (2008) [15] to arbitrary genus.The main tool is the authors? previous work (Cavalieri et al., 2010) [6] expressing double Hurwitz number as a sum over certain labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987) [17], and could have broader applications.