Constructing Minimal Projective Resolutions

Let Λ be a monomial algebra. For any Λ-module M, we describe graphically its minimal projective resolution as a weighted graph Δ(M). If M is finitely generated we give two algorithms for computing Δ(M). We also give a short computation-like argument to reprove a famous Syzygy Theorem.     

Analysis of a game of graph solitaire

Let G be a simple undirected graph. To each vertex assign one of two colours say red or blue. A solitaire game is played on G as follows. A move consists of selecting a blue vertex, inverting the colours of its neighbours and then deleting the chosen vertex and all edges incident upon it. The goal is to delete all the vertices. The question of finding a winning strategy in the case when G is a simple cycle was proposed in the problems section of the American Mathematical Monthly. In this article some general results on winning colour configurations are derived and the game is analysed for certain other graph classes.     

正则树的双变量色多项式研究

最近Klaus Dohmen等人提出新的双变量色多项式概念,对此,本文提出—个—般性的减边公式.通过反复运用该公式,可以方便求得任何简单图的双变量色多项式.由此减边公式,研究了一些特殊图和多分支图的双变量色多项式公式.本文还研究了由互不相连的多个子图都与某个顶点相连而成的图的双变量色多项式计算的删点公式以及简单图的双变量色多项式系数和问题.进而,本文提出—个新概念—正则树.利用这个减边公式,研究了正则树的双变量色多项式计算公式和—些性质,以及正则树整子图的双变量色多项式公式及其有关性质.     

Optimization of fat tree encoder for ultra high speed analog to digital converter using 45 nanometer technology

The design of high speed, compact and low power fat tree encoder circuits using static CMOS gates is presented. In this paper, we propose a modified 3 bit fat tree encoder (FTE) that can operate in high frequency without a sophisticated circuit structure. In addition, the technique of hardware sharing is adopted in this design to reduce the number of transistors. The study uses complementary metal oxide semiconductor (CMOS) 45 nm-technology. The proposed static design has improved delay and power compared to a conventional ROM encoder circuit implementation. The simulation result indicates that it functions successfully and works at 200-MHz speed. The average power consumption of the circuit under room temperature is 20.7 nW. The total core area is 0.011 mm². As expected, the proposed design can be easily integrated in various kind of digital application.     

Analysis of cross-correlations between financial markets after the 2008 crisis

We analyze the cross-correlation matrix C of the index returns of the main financial markets after the 2008 crisis using methods of random matrix theory. We test the eigenvalues of C for universal properties of random matrices and find that the majority of the cross-correlation coefficients arise from randomness. We show that the eigenvector of the largest deviating eigenvalue of C represents a global market itself. We reveal that high volatility of financial markets is observed at the same times with high correlations between them which lowers the risk diversification potential even if one constructs a widely internationally diversified portfolio of stocks. We identify and compare the connection and cluster structure of markets before and after the crisis using minimal spanning and ultrametric hierarchical trees. We find that after the crisis, the co-movement degree of the markets increases. We also highlight the key financial markets of pre and post crisis using main centrality measures and analyze the changes. We repeat the study using rank correlation and compare the differences. Further implications are discussed.     

介质窗口材料对高功率微波传输特性的影响

运用光学方法,分析平面波通过无限大介质平板的传输特性,进而分析单层介质窗口的引入对微波传输性能的影响。在S波段微波源上对聚四氟乙烯材料进行了真空环境下的微波放电击穿实验,通过长时间的表面放电,在聚四氟乙烯材料表面出现电痕和电树枝。在100 ns脉冲宽度下,未发生大面积表面放电前,通过光学照相,在介质表面观测到局部放电现象,认为局部放电仍能导致材料表面破坏。作为对比,进行了聚苯乙烯材料的电老化实验,实验结果表明聚苯乙烯材料具有良好的耐电痕、耐电树枝老化特性。     

The Number of Subtrees of Trees with Given Degree Sequence

This article investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalize the recent results of Kirk and Wang (SIAM J Discrete Math 22 (2008), 985–995). These trees coincide with those which were proven by Wang and independently Zhang et al. (2008) to minimize the Wiener index. We also provide a partial ordering of the extremal trees with different degree sequences, some extremal results follow as corollaries.