M. Shirvani

On ergodic one-dimensional cellular automata

We show that all onto cellular automata defined on the binary sequence space are invariant with respect to the Haar measure, and that an extensive class of such maps (including many nonlinear ones) are strongly mixing with respect to the Haar measure.

On free group algebras in division rings with uncountable center

Let D be a division algebra with uncountable center k. If D contains a noncommutative free k-algebra, then D also contains the k-group algebra of the free group of rank 2.

A Survey on Free Objects in Division Rings and in Division Rings with an Involution

Let D be a division ring with center k, and let D † be its multiplicative group. We investigate the existence of free groups in D †, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D †.

Solutions of Linear Differential Algebraic Equations

We show how to solve inhomogeneous linear differential algebraic systems with constant coefficients.

Some results on the coagulation equation

Eq. (1) represents the evolution of particles c( ; t) of size ≥ 0 at time t≥ 0 undergoing a change in size governed by reaction kernel K . A physical interpretation of the terms in (1) can be found in Melzak [5] or the review article by Drake [2]. Global existence and uniqueness of solutions to (1) has been proven by Aizenman and Bak [1] for constant kernels with suitable conditions imposed on the initial particle distribution c0. For the product kernel K( ; )= , McLeod [4] has shown, again for a suitably restricted initial particle distribution, that a mass conserving solution exists for some nite time. That solutions in this case need not conserve mass for all time was demonstrated by Stewart [6]. The phenomenon whereby conservation of mass breaks down in nite time is known as gelation and is physically interpreted as being caused by the appearance of an in nite “gel” or “superparticle” [3].

The mass-conserving solutions of Smoluchowski's coagulation equation: the general bilinear kernel

SummaryIt is proved that for the general bilinear kernel with arbitrary initial conditions, the solutions to the discrete coagulation equation can exhibit one of the following types of behaviour: conservation of mass for all time, conservation of mass for a finite time only, or instantaneous gelation.

On residually finite graph products

Abstract We provide a simple set of sufficient conditions for the residual finiteness of a graph product of groups, which is a generalization of G. Baumslags residual finiteness criterion for an amalgamated free product of two groups.

Ergodic endomorphisms of compact abelian groups

We show that for a surjective endomorphism of a compact abelian group ergodicity is equivalent to a condition which impliesr-mixing for allr≧1, and we characterize such maps algebraically. This is then used in proving the ergodicity of an extensive class of endomorphisms of the binary sequence space. As a simple corollary it is found that one-dimensional linear cellular automata and the accumulator automata arer-mixing for allr≧1.

The finite inner automorphism groups of division rings

Let G be a finite group of automorphisms of an associative ring R . Then the inner automorphisms ( x ↦ u −1 xu = x u , for some unit u of R ) contained in G form a normal subgroup G 0 of G . In general, the Galois theory associated with the outer automorphism group G / G 0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G / G 0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G 0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D ), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

On conjugacy separability of fundamental groups of graphs of groups

A complete determination of when the elements of a fundamental group of a (countable) graph of profinite groups are conjugacy distinguished is given. By embedding an arbitrary fundamental group G into one with profinite vertex groups and making use of the above result, questions on conjugacy separability of G can be reduced to the solution of equations in the vertex groups of G