Michael J. Dinneen, Geoffrey Pritchard and Mark C. Wilson, Networks 39 (2002), 1–9.
Abstract: We consider the problem of constructing networks with as many nodes as possible, subject to upper bounds on the degree and broadcast time. The paper includes the results of an extensive empirical study of broadcasting in small regular graphs using a stochastic search algorithm to approximate the broadcast time. Significant improvements on known results are obtained for cubic broadcast networks.
Jeffrey Bergen and Mark C. Wilson, Journal of Algebra 220, (1999), 152-173.
Abstract: Many important quantum algebras such as quantum symplectic space, quantum euclidean space, quantum matrices, $latex q$-analogs of the Heisenberg algebra and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras $latex U(L_+)$ of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the X-inner automorphisms of such algebras.
The theorems and examples in our paper show that for algebras $latex R$ of this type, the non-identity X-inner automorphisms of $latex R$ tend to have infinite order. Thus if $latex G$ is a finite group of automorphisms of $latex R$, then the action of $latex G$ will be X-outer and this immediately gives useful information about crossed products $latex R*_t G$.
D. M. Riley and Mark C. Wilson, Communications in Algebra 27 (1999), 3545-3556.
Abstract: Let $latex K$ be a field of characteristic $latex p>0$. Denote by $latex omega(R)$ the augmentation ideal of either a group algebra $latex R=K[G]$ or a restricted enveloping algebra $latex R=u(L)$ over $latex K$. We first characterize those $latex R$ for which $latex omega(R)$ satisfies a polynomial identity not satisfied by the algebra of all $latex 2times 2$ matrices over $latex K$. Then, we examine those $latex R$ for which $latex omega(R)$ satisfies a semigroup identity (that is, a polynomial identity which can be written as the difference of two monomials).
Michael J. Dinneen, Jose A. Ventura, Mark C. Wilson and Golbon Zakeri, Parallel Processing Letters 9 (1999), 53-68.
Abstract: In broadcasting, or one-to-all communication, a message originally held in one node of the network must be transmitted to all the other nodes. A minimal broadcast network is a communication network that can transmit a message originated at any node to all other nodes of the network in minimum time. In this paper, we present a compound method to construct sparse, time-relaxed, minimal broadcast networks ($latex t$-mbn), in which broadcasting can be accomplished in slightly more than the minimum time. The proposed method generates a new network by connecting a subset of nodes from several copies of a $latex t_1$-mbn using the structure of another $latex t_2$-mbn. The objective is to construct a network as sparse as possible satisfying the desired broadcasting time constraint. Computational results illustrate the effectiveness of the proposed method.
D.M. Riley and Mark C. Wilson, Proceedings of the American Mathematical Society 127 (1999), 973-976.
Abstract: Let $latex C$ be a commutative ring, and let $latex R$ be an associative $latex C$-algebra generated by elements $latex {x_1,ldots,x_d}$. We show that if $latex R$ satisfies the Engel condition of degree $latex n$ then $latex R$ is nilpotent as a Lie algebra of class bounded by a function that depends only on $latex d$ and $latex n$. We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.
Geoffrey Pritchard and Mark C. Wilson, Archiv der Mathematik 70 (1998), 187-196.
Abstract: A primeness criterion due to Bell is shown to apply to the universal enveloping algebra of the Cartan type Lie superalgebras $latex S(V)$ and $latex widetilde{S}(V;t)$ when $latex dim V$ is even.
Mark C. Wilson, Journal of Pure and Applied Algebra 133 (1998), 241-260.
Abstract: We determine all finite-dimensional simple Lie superalgebras $latex L$ such that $latex U(L)$ satisfies a primeness criterion due to Bell. Some open problems related to primeness of enveloping algebras are listed.
Geoffrey Pritchard and Mark C. Wilson, Experimental Mathematics 6 (1997), 77-85.
Abstract: On the basis of experimental work involving matrix computations, we conjecture that a criterion due to Bell for primeness of the universal enveloping algebra of a Lie superalgebra applies to the Cartan type Lie superalgebras $latex W(n)$ for $latex n=3$ but does not apply for odd $latex ngeq 5$. The experiments lead to a rigorous proof, which we present.
Mark C. Wilson, Bulletin of the Australian Mathematical Society 56 (1997), 483-488.
Abstract: In 1990 Allen Bell presented a sufficient condition for the primeness of the universal enveloping algebra of a Lie superalgebra. Let $latex Q$ be a nonsingular bilinear form on a finite-dimensional vector space over a field of characteristic zero. In this paper we show that Bell’s criterion applies to the Hamiltonian Cartan type superalgebras determined by $latex Q$, and hence obtain some new prime noetherian rings.
Mark C. Wilson, Communications in Algebra 25 (1997), 487-496.
Abstract:
Let $latex K$ be a field of characteristic $latex p>0$, let $latex L$ be a restricted Lie algebra and let $latex R$ be an associative $latex K$-algebra. It is shown that the various constructions in the literature of crossed product of $latex R$ with $latex u(L)$ are equivalent. We calculate explicit formulae relating the parameters involved and obtain a formula which hints at a noncommutative version of the Bell polynomials.