The Projective Cover of the Trivial Module Over a Group Algebra of a Finite Group

We determine all finite groups G such that the Loewy length (socle length) of the projective cover P(k _G ) of the trivial kG-module k _G is four, where k is a field of characteristic p > 0 and kG is the group algebra of G over k, by using previous results and also the classification of finite simple groups. As a by-product we prove also that if p = 2 then all finite groups G such that the Loewy lengths of the principal block algebras of kG are four, are determined.     

Chip-firing via open covers

We investigate chip-firing with respect to open covers of discrete graphs and metric graphs. For the case of metric graphs we show that given an open cover and a sink q, stabilization of a divisor D is unique and that there is a distinguished configuration equivalent to D, which we call the critical configuration. Also, we show that given a double cover of the metric graph by stars, which is the continuous analogue of the sandpile model, the critical configurations are in bijection with reduced divisors. Passing to the discrete case, we interpret open covers of a graph as simplicial complexes on the vertex and observe that chip-firing with respect to a simplicial complex is equivalent to the model introduced by Paoletti [G. Paoletti. July 11 2007: Master in Physics at University of Milan, defending thesis “Abelian sandpile models and sampling of trees and forests”; supervisor: Prof. S. Caracciolo. http://pcteserver.mi.infn.it/caraccio/index.html]. We generalize this setup for directed graphs using weighted simplicial complexes on the vertex set and show that the fundamental results extend. In the undirected case we present a generalization of the Cori-Le Borgne algorithm for chip-firing models via open covers, giving an explicit bijection between the critical configurations and the spanning trees of a graph.(http://www.elsevier.com/locate/endm)     

循环多胞形$C^{3}(6)$的对偶和单形乘积上的小覆盖

计算了循环多胞形$C^{3}(6)$的对偶和单形乘积上(可定向)小覆盖的等变同胚类的个数     

On Some Conditions for Complete Convergence for Arrays of Rowwise Negatively Dependent Random Variables

Abstract

This note contains some sufficient conditions for the complete convergence in the strong law of large numbers for arrays of rowwise negatively dependent random variables. Moreover, the rowwise sums are randomly indexed.     

Weak Global Dimension of Coherent Rings

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ? ? the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ? ?-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n ? 2)th ? ?-syzygy in a minimal ? ?-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.     

Semiperfect Modules with Respect to a Preradical

In an earlier paper [8] the authors introduced strongly and properly semiprime modules. Here properly semiprime modules M are investigated under the condition that every cyclic submodule is M-projective (self-pp-modules). We study the idempotent closure of M using the techniques of Pierce stalks related to the central idempotents of the self-injective hull of M. As an application of our theory we obtain several results on (not necessarily associative) biregular, properly semiprime, reduced and Firings. An example is given of an associative semiprime PSP ring with polynomial identity which coincides with its central closure and is not biregular (see 3.6). Another example shows that a semiprime left and right FP-injective Pl-ring need not be regular (see 4.8). Some of the results were already announced in [7].     

空间微小碎片对低轨道航天器太阳电池表面撞击损伤研究

低轨道空间微小碎片密度相对较高,对太阳电池等大面积暴露材料的累积撞击损伤效应是航天器设计中应考虑的主要问题之一.以典型的太阳同步轨道为例分析和计算了太阳电池表面所遭遇的微小碎片通量,进行了微小碎片的撞击损伤模拟实验,并建立了撞击损伤方程.根据碎片通量分布及撞击损伤方程计算了微小碎片的超高速撞击所产生的太阳电池表面的面积损伤率,通过光学透射率的实验测试并结合理论模型,对碎片导致的太阳电池光学透射率衰减进行了计算和分析.结果表明,10a累积撞击导致的太阳电池表面的面积损伤率平均约为0.61%,严重时达到2.3 关键词： 微小碎片 超高速撞击 太阳电池 玻璃盖片     

水流损失和热补偿共同作用对增强型地热系统(EGS)产能影响的研究

基于青海共和盆地-3705m地热田实测数据,结合流固耦合传热理论并运用Comsol软件,建立了离散型裂隙岩体流体传热模型。考虑水流损失和热补偿共同作用,模拟得到了开采过程中上、下岩层(盖层和垫层)为绝热不渗透、传热不渗透、渗透传热时,储层(上、下岩层和压裂层)温度场的变化特征,分析了产出流量、水流损失、产出温度、产热速率的变化规律。研究结果表明:采热过程中产出流量始终小于注入流量;产出流量增幅速率先增大后减小,最后趋于稳定,前3a产出流量增幅超过总增幅量的3/4;忽略水流损失,将高估产热速率,采热初期甚至达到考虑水流损失时产热速率的3倍以上;考虑水流损失,产热速率呈先快速上升再趋于稳定后逐渐下降的趋势,最优开采时间为3a^11a;研究上、下岩层对产出温度的影响,仅考虑传热,采热寿命延长5.43%,同时考虑渗流传热时,采热寿命延长2.71%;采热前9a,水流损失占主导作用,即流入上、下岩层水流损失对产热速率的影响高于热补偿效应,开采10a后,热补偿效应占主导作用;同时考虑水流损失和热补偿效应得到的产热速率变化规律与实际工程更为符合,建议选择低渗透能力的上、下岩层延长增强型地热系统(EGS)运行时间。     

Finite groups with a given number of relative centralizers

We will study all finite groups and their subgroups with at most four relative centralizers.     

Decision and approximation complexity for identifying codes and locating-dominating sets in restricted graph classes

An identifying code is a subset of vertices of a graph with the property that each vertex is uniquely determined (identified) by its nonempty neighbourhood within the identifying code. When only vertices out of the code are asked to be identified, we get the related concept of a locating-dominating set. These notions are closely related to a number of similar and well-studied concepts such as the one of a test cover. In this paper, we study the decision problems Identifying Code and Locating-Dominating Set (which consist in deciding whether a given graph admits an identifying code or a locating-dominating set, respectively, with a given size) and their minimization variants Minimum Identifying Code and Minimum Locating-Dominating Set. These problems are known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Moreover, it is known that they are approximable within a logarithmic factor, but hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite: both problems remain hard in these cases. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), the two problems are hard to approximate within some constant factor, a question which was open. We summarize all known results in the area, and we compare them to the ones for the related problem Dominating Set. In particular, our work exhibits important graph classes for which Dominating Set is efficiently solvable, but Identifying Code and Locating-Dominating Set are hard (whereas in all previous works, their complexity was the same). We also introduce graph classes for which the converse holds, and for which the complexities of Identifying Code and Locating-Dominating Set differ.