Strokes for Representing Univariate Vector Field Maps

Particle systems make an excellent tool for creating tracks (which we call strokes) in vector fields. The question addressed in this paper is how such tracks should be made to vary in size and colour in order to reveal properties such as local direction and strength of the field. We find that for strokes that vary from large to small, direction is indicated by the large end. We also find that for strokes that vary in colour, the colour of the background is the most important determinant of perceived direction

On a generalization of Abelian equivalence and complexity of infinite words

In this paper we introduce and study a family of complexity functions of infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+ \cup {+\infty} and $A$ be a finite non-empty set. Two finite words $u$ and $v$ in $A^*$ are said to be $k$-Abelian equivalent if for all $x\in A^*$ of length less than or equal to $k,$ the number of occurrences of $x$ in $u$ is equal to the number of occurrences of $x$ in $v.$ This defines a family of equivalence relations $\thicksim_k$ on $A^*,$ bridging the gap between the usual notion of Abelian equivalence (when $k=1$) and equality (when $k=+\infty).$ We show that the number of $k$-Abelian equivalence classes of words of length $n$ grows polynomially, although the degree is exponential in $k.$ Given an infinite word \omega \in A^\nats, we consider the associated complexity function \mathcal {P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of $k$-Abelian equivalence classes of factors of $\omega$ of length $n.$ We show that the complexity function $\mathcal {P}^{(k)}$ is intimately linked with periodicity. More precisely we define an auxiliary function q^k: \nats \rightarrow \nats and show that if $\mathcal {P}^{(k)}_{\omega}(n)<q^k(n)$ for some k \in \ints ^+ \cup {+\infty} and $n\geq 0,$ the $\omega$ is ultimately periodic. Moreover if $\omega$ is aperiodic, then $\mathcal {P}^{(k)}_{\omega}(n)=q^k(n)$ if and only if $\omega$ is Sturmian. We also study $k$-Abelian complexity in connection with repetitions in words. Using Szemer\'edi's theorem, we show that if $\omega$ has bounded $k$-Abelian complexity, then for every D\subset \nats with positive upper density and for every positive integer $N,$ there exists a $k$-Abelian $N$ power occurring in $\omega$ at some position $j\in D.