Bialgebra cohomology, pointed Hopf algebras, and deformations

We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all bialgebra two-cocycles of certain Radford biproducts (bosonizations). These two-cocycles are precisely those associated to the finite dimensional pointed Hopf algebras in the recent classification of Andruskiewitsch and Schneider, in an interpretation of these Hopf algebras as graded bialgebra deformations of Radford biproducts.     

完备格的关系表示理论及其应用

主要讨论完备格的关系表示问题,分别建立了完全分配格的正则表示定理、超连续格的有限正则表示定理、λ-超连续格的λ-正则关系表示定理、区间拓扑T2完备格的广义有阴正则表示定理,给出了正则关系、完全分配格、超连续格、λ-超连续格、区间拓扑T2完备格的内蕴式刻划;给出了本文所建立的完备格的关系表示理论在Domain理论、格论和拓扑学中的若干应用.     

Representations of Reductive p-Adic Groups: Localization of Hecke Algebras and Applications

Let F be a non-Archimedean local field and G be the group ofF-points of a connected reductive group defined over F. LetM be an F-Levi subgroup of G and P = MN be a parabolic subgroupwith Levi decomposition P = MN. Jacquet, or truncated, restrictiongives a functor from the category of smooth representationsof G to that of M. The main result describes this functor interms of homomorphisms and localizations of Hecke algebras attachedto certain compact open subgroups of G and M. This leads tonew and straightforward proofs of some fundamental results.The first computes the smooth dual of a Jacquet module of asmooth representation of G, generalizing the corresponding resultfor admissible representations due to Harish-Chandra and Casselman.The second identifies the co-adjoint of the Jacquet functorrelative to P as the induction functor relative to the M-oppositeof P, an unpublished result of J.-N. Bernstein.     

模糊粗糙集的表示及应用

一个模糊粗糙集是一对模糊集,它可以用一簇经典粗糙集表示出来.本文研究了模糊粗糙集的表示问题,利用模糊集的分解定理证明了一个模糊粗糙集可以用一簇粗糙模糊集表示出来,利用这个结果可以证明模糊粗糙集的一些重要性质.     

Stirling数的概率表示和应用

本文证明了第二类Ｓｔｉｒｌｉｎｇ数Ｓ（ｎ，ｋ）和第一类Ｓｔｉｒｌｉｎｇ数ｓ（ｎ，ｋ）都是常见的随机变量和的矩，从而使概率论的方法和技巧，在组合和式计算、恒等式的发现与证明以及渐近估计等方面得到许多新的结果．     

Some Equivalent Representations of Nonsquare Constants and Its Applications

Let(X,‖·‖ ) be a normed space.Let S(X) ={ x∈X:‖x‖ =1 } and B(X) ={ x∈X:‖ x‖≤ 1 } be the unit sphere and unit ball of X,respectively.In 1 990 ,Gao andLau[1 ] introduced the following definition.　 Definition　 The Parameter CJ(X) of a normed space X,which will be called a non-square constant in the sense of James in this paper,is defined byCJ(X) =sup{‖ x + y‖∧‖ x -y‖ :x,y∈ S(X) } .The nonsquare constant CS(X) of a normed space X in the sense of Schaffer is definedbyC…     

有限域的子空间表示及其应用

本文研究了有限域在其子域上向量子空间的表示问题,推广了文献[3]中的一个结果,并给出了关于Singer差集的一个构造性证明；最后,利用有限域的所有超平面作为集合,构作了一类结合方案,并计算了它们的参数.     

Representations of the Absolute Value Function and Applications in Gaussian Estimates

We study the expectation of unsigned Gaussian quadratic forms and negative absolute moments of Gaussian products. The main tool we use is the integral representation of the absolute value function.     

Explicit Martingale Representations for Brownian Functionals and Applications to Option Hedging

Abstract

Using Clark-Ocone formula, explicit martingale representations for path-dependent Brownian functionals are computed. As direct consequences, explicit martingale representations of the extrema of geometric Brownian motion and explicit hedging portfolios of path-dependent options are obtained.     

Pointed Wiedersehen Metrics on Exotic Spheres and Diffeomorphisms of S6

Using a Kaluza–Klein-type procedure, an explicit metric h on an exotic sphere ⁷ is constructed, satisfying the Wiedersehen condition at a set of points diffeomorphic to S ¹. The formulas for the geodesics allows the writing down of formulas for an explicit degree 1 diffeomorphism : S ⁶ S ⁶ that is not isotopic to the identity.     

U.S. and Chinese Teachers' Constructing,Knowing, and Evaluating Representations to Teach Mathematics

This study examined U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach mathematics. All Chinese lesson plans are very similar, because they are all based on the Chinese national unified curriculum in mathematics. However, the U.S. lesson plans are extremely varied, even for those teachers from the same school. The Chinese teachers' lessons are very detailed; the U.S. teachers' lesson plans have exclusively adopted the "outline and worksheet" format. In the Chinese lesson plans, concrete representations are used exclusively to mediate students' understanding of the concept of average. In U.S. lessons, concrete representations are not only used to model the averaging processes to foster students' understanding of the concept, but they are also used to generate data. The U.S. teachers are much more likely than the Chinese teachers to predict drawing and guess-and-check strategies. For some problems, the Chinese teachers are much more likely than are the U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score, even though the strategy is appropriate for the correct answer, because it is less generalizable. This study contributed to our understanding of the cross-national differences between U.S. and Chinese students' mathematical thinking. It also contributed to our understanding about teachers' beliefs from a cross-cultural perspective.     

U.S. and Chinese Teachers' Constructing, Knowing, and Evaluating Representations to Teach Mathematics

This study examined U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach mathematics. All Chinese lesson plans are very similar, because they are all based on the Chinese national unified curriculum in mathematics. However, the U.S. lesson plans are extremely varied, even for those teachers from the same school. The Chinese teachers' lessons are very detailed; the U.S. teachers' lesson plans have exclusively adopted the “outline and worksheet” format. In the Chinese lesson plans, concrete representations are used exclusively to mediate students' understanding of the concept of average. In U.S. lessons, concrete representations are not only used to model the averaging processes to foster students' understanding of the concept, but they are also used to generate data. The U.S. teachers are much more likely than the Chinese teachers to predict drawing and guess-and-check strategies. For some problems, the Chinese teachers are much more likely than are the U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score, even though the strategy is appropriate for the correct answer, because it is less generalizable. This study contributed to our understanding of the cross-national differences between U.S. and Chinese students' mathematical thinking. It also contributed to our understanding about teachers' beliefs from a cross-cultural perspective.     

On Mutual Representations of Symmetrically Weighted Stirling-Type Pairs with Applications

This paper is concerned with the representation problem of symmetrically weighted Stirling-type pairs.Quasi-Schlornilch formulas are proved for the mutual representations between two weighted Stirling-type pairs, which generalize previous results independently by Carlitz, Howard et al.Moreover, some arithmetic properties of weighted Stirling-type paris are discussed.     

Centroids,Representations, and Submodular Flows

The notion of centroid of a tree is generalized to apply to an arbitrary intersecting family of sets. Centroids are used to construct a compact representation for any intersecting family of sets, as well as any crossing family. The size of the representation for a family on n elements is O(n²), compared to size O(n³) for previous representations. Efficient algorithms to construct the representation are given. For example on a network of n vertices and m edges, the representation of all minimum cuts uses O(m log(n²/m)) space; it is constructed in O(nm log(n²/m)) time (this is the best-known time for finding one minimum cut). The representation is used to improve several submodular flow algorithms. For example a minimum-cost dijoin is found in time O(n²m); as a result a minimum-cost planar feedback are set is found in time O(n³). The previous best-known time bounds for these two problems are both a factor n larger.     

Exotic Quantifiers, Complexity Classes, and Complete Problems

We define new complexity classes in the Blum–Shub–Smale theory of computation over the reals, in the spirit of the polynomial hierarchy, with the help of infinitesimal and generic quantifiers. Basic topological properties of semialgebraic sets like boundedness, closedness, compactness, as well as the continuity of semialgebraic functions are shown to be complete in these new classes. All attempts to classify the complexity of these problems in terms of the previously studied complexity classes have failed. We also obtain completeness results in the Turing model for the corresponding discrete problems. In this setting, it turns out that infinitesimal and generic quantifiers can be eliminated, so that the relevant complexity classes can be described in terms of the usual quantifiers only.      

Differential-Operator Representations of S n and Singular Vectors in Verma Modules

Given a weight of sl(n, \mathbb C{\mathbb C}), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group S _n on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by {σ(1)|σ ∈ S _n }. Moreover, the singular vectors of sl(n, \mathbb C{\mathbb C}) in the Verma module are given by those σ(1) that are polynomials. The well-known results of Verma, Bernstein–Gel’fand–Gel’fand and Jantzen for the case of sl(n, \mathbb C{\mathbb C}) are naturally included in our almost elementary approach of partial differential equations.     

Semigroups,Potential Spaces and Applications to (S)PDE

The paper studies perturbed semilinear parabolic partial (pseudo-) differential equations on σ-finite measure spaces under low smoothness assumptions. We obtain results on existence, uniqueness and regularity. The hypotheses are formulated in terms of the semigroup, regularity is measured by means of abstract potential spaces. Being a priori analytic, our results allow to investigate related stochastic partial differential equations in the almost sure pathwise sense. For example we can study (fractional) semilinear heat equations driven by fractional Brownian noises on metric measure spaces.