The Diagram Approach in Knot Theory and Applications to Graph Theory

The paper is a survey of a series of publications of the authors awarded by I. I Shuvalov First Prize of Lomonosov Moscow State University for scientific activity, 2017.     

图论在网络出版传播中的应用

利用图论理论研究了网络连通度与广播时间的关系问题 ,获得了网络单信息广播时间的一个界和最少时间广播网的几个充分条件 ,并建立了一个有效的广播方案 .     

Some Recent Progress and Applications in Graph Minor Theory

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the ``rough' structure of graphs excluding a fixed minor. This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed. Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, Grant number 16740044, by Sumitomo Foundation, by C & C Foundation and by Inoue Research Award for Young Scientists Supported in part by the Research Grant P1–0297 and by the CRC program On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia     

On Optimal Defuzzification and Learning Algorithms: Theory and Applications

In this paper, the issue of optimal defuzzification which is advocated in the Optimality Principle of Defuzzification (Song and Leland (1996)) is addressed. It was shown that defuzzification can be treated as a mapping from a high dimensional space to the real line. When system performance indices are considered, the defuzzification mapping which optimizes the performance indices for the given fuzzy sets is known as the optimal defuzzification mapping. Thus, finding this optimal defuzzification mapping becomes the essence of defuzzification. The problem with this idea, however, is that the space formed by all possible continuous defuzzification mappings is so large to search that the only recourse is an approximation to the optimal defuzzification mapping. With this, learning algorithms can be devised to find the optimal parameters of defuzzifiers with fixed structures. The proposed method is rigorously examined and compared with some well-known defuzzification methods. To overcome the resultant enormous computational load problem with this algorithm, the concept of defuzzification filter is additionally proposed. An application of the method to the power system stabilization problem is presented.     

Parallel Algorithms for Hierarchical Clustering and Applications to Split Decomposition and Parity Graph Recognition

We present efficient (parallel) algorithms for two hierarchical clustering heuristics. We point out that these heuristics can also be applied to solving some algorithmic problems in graphs, including split decomposition. We show that efficient parallel split decomposition induces an efficient parallel parity graph recognition algorithm. This is a consequence of the result of S. Cicerone and D. Di Stefano [[7]] that parity graphs are exactly those graphs that can be split decomposed into cliques and bipartite graphs.     

Hamilton-Jacobi Equations on Graph and Applications

This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand’s transport inequality.     

Applications of Finite Fields to Combinatorics and Finite Geometries

We survey some applications of finite fields to finite geometries in part A and to combinatorics and error-correcting codes in parts B and C.     

Modular Data: The Algebraic Combinatorics of Conformal Field Theory

This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It tries to refine, modernise, and bridge the gap between papers [6] and [55]. Our paper is essentially self-contained, apart from some of the background motivation (Section 1) and examples (Section 3) which are included to give the reader a sense of the context. Detailed proofs will appear elsewhere. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems.     

基于图论的MPPT控制方法及仿真研究

太阳能发电系统中需要对输出最大功率点进行跟踪.介绍了几种常见的最大功率跟踪方法,分析了太阳能电池的系统原理,讨论了无阴影和在局部阴影下太阳能电池的电气特性曲线.在此基础上提出了基于图论的最大功率跟踪的新方法,方法可以有效地跟踪无阴影和局部阴影下太阳能电池输出的最大功率点.最后,根据太阳能电池的直流物理模型,在Simulink仿真环境下建立了太阳能电池的仿真模型并进行了仿真.     

The Combinatorics of Laguerre,Charlier, and Hermite Polynomials

This paper describes several combinatorial models for Laguerre, Charlier, and Hermite polynomials, and uses them to prove combinatorially some classical formulas. The so-called “Italian limit formula” (from Laguerre to Hermite), the Appel identity for Hermite polynomials, and the two Sheffer identities for Laguerre and Charlier polynomials are proved. We also give bijective proofs of the three-term recurrences. These three families form the bottom triangle in R. Askey's chart classifying hypergeometric orthogonal polynomials.     

图论中矩阵的可实现性

对图的关联矩阵,邻接矩阵,基本割集矩阵,基本圈矩阵的可实现性分别进行了论证,并将邻接矩阵的可实现性推广到一般形式.得到了同一个基本割集矩阵的奥凯达图形是不唯一的;以及这些奥凯达图形所对应的图是互相同构的结果;并且指出了基本圈矩阵的可实现性可以依靠基本割集矩阵的可实现性来解决.