Topological Graph Polynomial and Quantum Field Theory Part II: Mehler Kernel Theories

We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.     

Itsy Bitsy Topological Field Theory

We construct an elementary, combinatorial kind of topological quantum field theory (TQFT), based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda–Kazez–Mati?. This topological field theory stores information in binary format on a surface and has “digital” creation and annihilation operators, giving a toy-model embodiment of “it from bit”.     

拓扑量子场理论与相交指标

该文构造了一新的上同调型拓扑量子场理论并证明了其配分函数是相交指标 (crossingindex)     

色多项式对图的刻画

By means of the chromatic polynomials, this paper provided a necessary and sufficient condition for the graph G being a mono-cycle graph(the Theorem 1), a first class bi-cycle graph and a second class bicycle graph(the Theorem 2), respectively.     

Group Extensions and Hall Polynomials

In this paper, we give two proofs of a formula containing the numbers of automorphisms of an Abelian group, of its subgroups, and of its quotient groups. The first proof is based on the use of the theory of Hall polynomials, while the second one uses extension theory for Abelian groups.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 180–185.Original Russian Text Copyright © 2005 by G. V. Voskresenskaya.     

Polynomials with Cyclic Galois Group

It is proved that certain polynomials of degree ?3 have a cyclic group of order n as Galois group over all rational function fields k(t) with characteristic not dividing n. Moreover, the extension fields of k(t) generated by the polynomials have k as precise field of constants, and possess an unramified rational point. For all 3≤?20 with the exceptions of 17 and 19 the polynomials are calculated explicitly     

On the Location of Roots of Graph Polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.     

Directed Graph Pattern Matching and Topological Embedding

Pattern matching in directed graphs is a natural extension of pattern matching in trees and has many applications to different areas. In this paper, we study several pattern matching problems in ordered labeled directed graphs. For the rooted directed graph pattern matching problem, we present an efficient algorithm which, given a pattern graphPand a target graphT, runs in time and spaceO(|E_P| |V_T| + |E_T|). It is faster than the best known method by a factor ofmin{|E_T|, |E_P| |V_T|}. This algorithm can also solve the directed graph pattern matching problem without increasing time or space complexity. Our solution to this problem outperforms the best existing method by Katzenelson, Pinter and Schenfeld by a factor ofmin{|V_P| |E_T|, |V_P| |E_P| |V_T|}. We also present an algorithm for the directed graph topological embedding problem which runs in timeO(|V_P| |E_T| + |E_P|) and spaceO(|V_P| |V_T| + |E_P| + |E_T|). To our knowledge, this algorithm is the first one for this problem.     

Nonlinear Representations of the Lorentz Group in Quantum Field Theory

Formulas for the nonlinear representation of the Lorentz group are generalized for the wave functions of Wigner particles whose trajectories begin at phase-space points with nonzero coordinates.     

不分明化拓扑群的一致结构

本文引入了不分明化拓扑群的左、右、双一致结构，讨论了此类一致结构在一致连续下的一些性质，给出了此类结构在其子群上的相对结构和在其乘积上的来积结构。     

Topological Constructions in the o–Graph Calculus

Benedetti and Petronio developed in [1] a so called o–Graph Calculus, where a compact oriented 3–manifold with nonempty boundary could be described by a quadrivalent graph together with some extra structure. In this paper, we will show how topological constructions such as puncturing, connected sums, attaching handles, closing boundary components and product and mapping tori constructions can be translated into the o–graph calculus.     

Chromatic Polynomials and the Symmetric Group

In this paper we give first a new combinatorial interpretation of the coefficients of chromatic polynomials of graphs in terms of subsets of permutations. Motivated by this new interpretation, we introduce next a combinatorially defined polynomial associated to a directed graph, and prove that it is related to chromatic polynomials. These polynomials are a specialization of cover polynomials of digraphs.I am grateful to the Swiss National Science Foundation for its partial financial supportFinal version received: June 25, 2003     

Fuzzy拓扑体

引入了 Fuzzy拓扑体的定义 ,研究了它的性质 ,得到了借助于θλ的重域系及开重域基刻画 Fuzz拓扑体的充要条件 ;为 Fuzzy拓扑体理论研究提供了框架和工具 .     

Separation Theorems for Group Invariant Polynomials

We study the existence of separation theorems by polynomials that are invariant under a group action. We show that if G is a finite subgroup of \(\textit{GL}(n,{\mathbb {C}})\), K is a set in \({\mathbb {C}}^{n}\) that is invariant under the action of G and z is a point in \({\mathbb {C}}^{n}\setminus K\) that can be separated from K by a polynomial Q, then z can be separated from K by a G-invariant polynomial P. Furthermore, if Q is homogeneous then P can be chosen to be homogeneous. As a particular case, if K is a symmetric polynomially convex compact set in \({\mathbb {C}}^{n}\) and \(z\notin K\) then there exists a symmetric polynomial that separates z and K.     

Symplectic Field Theory of a Disk,Quantum Integrable Systems,and Schur Polynomials

We consider commuting operators obtained by quantization of Hamiltonians of the Hopf (aka dispersionless KdV) hierarchy. Such operators naturally arise in the setting of Symplectic Field Theory (SFT). A complete set of common eigenvectors of these operators is given by Schur polynomials. We use this result for computing the SFT potential of a disk.     

图映射拓扑序列熵的可交换性

主要研究图上连续自映射拓扑序列熵的可交换性,证明了对任意无界的正整数递增序列A=(ai)∞i=1和任意的连续图映射f,g都有hA(fog):hA(g of).解决了Balibrea F等人在相关文献中提出的一个猜想.     

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck’s program of ‘Dessins d’ enfants’, aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups. We would like to thank Fabio Tonoli for helping us with the pictures.