Category Archives: before 2000

Some old talks

Here are slides for recent talks, in reverse chronological order of delivery. If the date is incomplete, then the slides are not yet available, probably because the talk has not yet been given. Slides for several earlier talks seem no longer to be available. I will be putting up any newer talks as individual blog posts.

Asymptotics of coefficients of multivariate generating functions Workshop in Asymptotic Enumeration, ANU, Canberra, 2007-09-{10-14}.

Polytope computations in social choice theory Auckland Discrete Mathematics and Social Sciences Seminar 2007-04-23.

The diameter of random Cayley graphs 12th AofA workshop Alden Biesen 2006-07-06

Asymptotics of generalized Riordan arrays 11th AofA workshop Barcelona 2005-06-08

Multivariate generating functions II CDMTCS seminar Auckland 2005-04-26

Multivariate generating functions I CDMTCS seminar Auckland 2005-04-12

Sattolo’s algorithm INRIA Rocquencourt 2004-06-28

Towards a theory of multivariate generating functions INRIA Rocquencourt 2004-06-28

Asymptotics of multivariate generating functions Melbourne FPSAC conference 2002-07-

Superalgebras and their uses Missoula 1999-03-09

Primitive ideals in Hopf algebra extensions San Antonio 1999-01-14

Algebras of my acquaintance Auckland 1998-04-29

Associative algebras satisfying a semigroup identity

D. M. Riley and Mark C. Wilson, Glasgow Mathematical Journal 41 (1999), 453-462.

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Abstract: Denote by $latex (R,cdot)$ the multiplicative semigroup of an associative algebra $latex R$ over an infinite field, and let $latex (R,circ)$ represent $latex R$ when viewed as a semigroup via the circle operation $latex xcirc y=x+y+xy$. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of $latex R$. Namely, we prove that the following conditions on $latex R$ are equivalent: the semigroup $latex (R,circ)$ satisfies an identity; the semigroup $latex (R,cdot)$ satisfies a reduced identity; and, the associated Lie algebra of $latex R$ satisfies the Engel condition. When $latex R$ is finitely generated these conditions are each equivalent to $latex R$ being upper Lie nilpotent.

Degree- and time-constrained broadcast networks

Michael J. Dinneen, Geoffrey Pritchard and Mark C. Wilson, Networks 39 (2002), 1–9.

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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.

X-inner automorphisms of semicommutative quantum algebras

Jeffrey Bergen and Mark C. Wilson, Journal of Algebra 220, (1999), 152-173.

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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 $latex R*_t G$.

Group algebras and enveloping algebras with nonmatrix and semigroup identities

D. M. Riley and Mark C. Wilson, Communications in Algebra 27 (1999), 3545-3556.

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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).

Construction of time-relaxed minimal broadcast networks

Michael J. Dinneen, Jose A. Ventura, Mark C. Wilson and Golbon Zakeri, Parallel Processing Letters 9 (1999), 53-68.

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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.

Associative rings satisfying the Engel condition

D.M. Riley and Mark C. Wilson, Proceedings of the American Mathematical Society 127 (1999), 973-976.

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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.

Primeness of the enveloping algebra of the special Lie superalgebras

Geoffrey Pritchard and Mark C. Wilson, Archiv der Mathematik 70 (1998), 187-196.

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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.

Bell’s primeness criterion for W(2n+1)

Geoffrey Pritchard and Mark C. Wilson, Experimental Mathematics 6 (1997), 77-85.

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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.