This paper contains material presented on 31 January 2002 to staff and students of the Department of Mathematics of the University of Utrecht. The talk was one in a series of lectures titled Basic Notions in Mathematics. The paper follows closely the talk, that consisted of six parts; basic definitions, a short survey of important results, some remarks in connection with the Maximum Clique Prob...

The varieties W equivalent to a given variety V are characterized in a purely categorical way. In fact they are described as the models of those Lawvere theories which are Morita equivalent to the Lawvere theory of V which therefore are characterized first. Along this way the conceptual meanings of the n–th matrix power construction of a variety and McKenzie’s σ– modification of classes of alge...

let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.

Our goal is to give a list of rewriting properties, and then automatically find a term rewriting system (TRS) satisfying these properties. In earlier work we did this for finite abstract reduction systems; in this paper we extend the approach to ground term rewriting systems over constants and one unary symbol. In particular, we fully automatically find a TRS that is locally confluent but not c...

How useful are probabilistic forecasts of the outcomes of particular situations? Potentially, they contain more information than unequivocal forecasts and, as they allow a more realistic representation of the relative likelihood of different outcomes, they might be more accurate and therefore more useful to decision makers. To test this proposition, I first compared a SquaredError Skill Score (...

In many cavitating liquid flows, when the number and concentration of the bubbles exceeds some critical level, the flow becomes unsteady and large clouds of cavitating bubbles are periodically formed and then collapse when convected into regions of higher pressure. This phenomenon is known as cloud cavitation and when it occurs it is almost always associated with a substantial increase in the c...