组合拓扑方法在组合学和图论中的应用

本文介绍组合拓扑方法在图论和组合学中的应用，探索一些新的离散问题和连续问题的关系，介绍目前有关这方面的新结果及发展动向。本文主要介绍同调理论在图论中的应用，与图有关的复形及性质，不动点定理在离散问题中的应用等。文中提出了一些新结果及可供研究的新问题。     

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.     

Free Component Analysis: Theory,Algorithms and Applications

Foundations of Computational Mathematics - We describe a method for unmixing mixtures of freely independent random variables in a manner analogous to the independent component analysis (ICA)-based...     

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.     

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.     

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.     

Graph Theory and Economic Models: from Small to Large Size Applications

This empirical study explores the structure of macroeconomic models using major concepts and algorithms of the graph theory. Different sizes of applications with dynamic effects are considered. We will firstly examine the matching problem when assigning the equations to the variables. We'll also propose a simple method for improving the regular circular embedding of graphs on the basis of one of the longest circuit and adequate permutations. The determination of the maximal list of edge-disjoint circuits also produces an useful insight into the structure. A typology of the interdependent variables is proposed using the all- pairs shortest paths matrix. This classification is based on both the emissions of nodes towards the rest of the directed graph and the perturbations that the rest of the graph exerts on these nodes. The computations have been done using the softwares MATHEMATICA^® 5.1, LINDO 6.1 and our own programs in Fortran 77L and C++.