Gelfand–Mazur Theorems in normed algebras: A survey

The Gelfand–Mazur Theorem, a very basic theorem in the theory of Banach algebras states that: (Real version) Every real normed division algebra is isomorphic to the algebra of all real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$ or the quaternions $\mathbb{H}$; (Complex version) Every complex normed division algebra is isometrically isomorphic to $\mathbb{C}$. This theorem has undergone a large number of generalizations. We present a survey of these generalizations and also discuss some closely related unsettled issues.     

Statistics of isomorphism types in free products

Let Γ be a free product of finitely many finite- and infinite-cyclic groups. For a subgroup Δ of finite index given by its coset representation we compute its isomorphism type, i.e., its decomposition as a free product of finite- and infinite-cyclic groups. We determine the set of isomorphism types realized by finite-index subgroups, the asymptotics of the subgroup numbers with prescribed isomorphism types, and the distribution of the isomorphism types among subgroups of fixed index. Apart from group-theoretic arguments, the proofs of the present paper make use of asymptotic, combinatorial, and probabilistic ideas and techniques.     

一种求循环码对偶码的新方法

本文应用离散傅里叶变换来分析有限域上时域多项式环和它的频域多项式环之间的同构,从另一角度提出了一种由循环码单位元生成它的对偶码的新方法。本文还建立了循环码单位元与它的频域单位元之间的联系,由此可以很方便地生成它的对偶码,理论和实例表明这种方法是行之有效的。     

Exploiting orbits in symmetric ILP

This paper describes components of a branch-and-cut algorithm for solving integer linear programs having a large symmetry group. It describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group. Pruning and orbit computations are performed by backtracking procedures using a Schreier-Sims table for representing the symmetry group. Applications to hard set covering problems, generation of covering designs and error correcting codes are given.Mathematics Subject Classification (2000):90C10, 90C27, 90C57     

L-S-系中的正合列

经典模论中,正合列是研究函子问题的基本工具。在半群S-系理论中,正合列对函子的研究同样起着重要作用。用类似L-模中的方法,给出L-S-系及其正合列的定义,得到已知序列正合的一个必要条。另外,还定义L-S-系中两个短正合列之间的弱同构与同构,并得到在一般S-系范畴中短正合列μ1→μ1Цμ2→μ2弱同构于短正合列μ1→η→μ2的充分条件。     

Isomorphism in electron-pair densities of atoms

The spherically averaged electron-pair intracule (relative motion) h(u) and extracule (center-of-mass motion) d(R) densities are a couple of densities which characterize the motion of electron pairs in atomic systems. We study a generalized electron-pair density (q; a, b) that represents the probability density function for the magnitude of two-electron vector a r _j +b r _k of any pair of electrons j and k to be q, where a and b are nonzero real numbers. In particular, h(u)=g(u;1, −1) and d(R) = . It is shown that the scaling property of the Dirac delta function and the inversion symmetry of orbitals in atoms due to the central force field generate several isomorphic relations in the electron-pair density (q; a, b) with respect to the two parameters a and b. The approximate isomorphism d(R)≅8h(2R) known in the literature between the intracule and extracule densities is a special case of the present results. Received: 24 May 2000 / Accepted: 18 July 2000 / Published online: 27 September 2000     

Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition (A,B,M), where A∼M and B⁄∼M, such that G[A]∈P, G[B]∈Q and G[M] is not (P,Q)-split. Then the graph G is reconstructible.     

Largest Non-Unique Subgraphs

The reduction number r(G) of a graph G is the maximum integer m≤|E(G)| such that the graphs G−E, E⊆E(G),|E|≤m, are mutually non-isomorphic, i.e., each graph is unique as a subgraph of G. We prove that and show by probabilistic methods that r(G) can come close to this bound for large orders. By direct construction, we exhibit graphs with large reduction number, although somewhat smaller than the upper bound. We also discuss similarities to a parameter introduced by Erdős and Rényi capturing the degree of asymmetry of a graph, and we consider graphs with few circuits in some detail. Supported by a grant from the Danish Natural Science Research Council.     

直觉模糊群与它的同态像

在Ｋ,Ａｔａｎａｓｓｏｖ引进直觉模糊集的基础上,给出它的扩展原理,进而定义了直觉模糊群,并讨论了它的简单性质,最后在两个经典群同态与同构意义下,研究了这咱直觉模糊群的像、前像与逆映射等问题。