On the isomorphisms and automorphism groups of circulants

Denote byC _n(S) the circulant graph (or digraph). LetM be a minimal generating element subset ofZ _n, the cyclic group of integers modulon, and In this paper, we discuss the problems about the automorphism group and isomorphisms ofC _n(S). When M S , we determine the automorphism group ofC _n(S) and prove that for any T if and only ifT = S, where is an integer relatively prime ton. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.     

Geometry of quasiminimal phase transitions

We consider the quasiminima of the energy functional

Design isomorphisms and group isomorphisms

In this paper conditions are given which imply that a design isomorphism between designs with group actions is in fact a group isomorphism. The conditions are geometric. Automorphism groups are then calculated for a family of designs using the geometric conditions.The research in the paper was conducted at Loyola University of Chicago.     

Cubic spline interpolation with quasiminimal B-spline coefficients

Summary The end conditions for cubic spline interpolation with equidistant knots will be defined so as to make the (slightly modified) B-spline coefficients minimal. This produces good approximation results as compared e.g. with the not-a-knot spline.     

Non-canonical isomorphisms

Many categorical axioms assert that a particular canonically defined natural transformation between certain functors is invertible. We give two examples of such axioms where the existence of any natural isomorphism between the functors implies the invertibility of the canonical natural transformation. The first example is distributive categories, the second (semi-)additive ones. We show that each follows from a general result about monoidal functors.     

An Abstract Cogalois Theory for profinite groups

The aim of this paper is to develop an abstract group theoretic framework for the Cogalois Theory of field extensions.     

Abstract simplicity of complete Kac-Moody groups over finite fields

Let G be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix A, as constructed by Tits. It is known that G admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group which is defined to be the closure of G in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of was previously known to hold when A is of affine type. We extend this result to many indefinite cases, including all hyperbolic generalized Cartan matrices A of rank at least four. Our proof uses Tits’ simplicity theorem for groups with a BN-pair and methods from the theory of pro-p groups.     

Pro-homology and isomorphisms of pro-groups

We extend the scope of some useful theorems of E. Dror and J. Stallings of which the following result of Stallings is easy to state: A homomorphism θ: A→B of nilpotent groups is an isomorphism if and only if θ induces an isomorphism and an epimorphism on the first and the second homology groups with integer coefficients, respectively. We extend and enlarge the algebraic framework to the category of pro-groups in a natural way so that an anaogue of these results of Dror and Stallings can be appropriately stated and proved. This is done in a natural manner so that our extended results cań be useful in pro-homotopy theory or shape theory which is analogous to the usefulness of the results of Dror and Stallings in the context of homotopy theory. As an application, we obtain a version of a Whitehead Theorem in pro-homotopy and shape theory which only involves homology pro-groups; a complete proof of this result will be given elsewhere.