Dimension of graphoids of rational vector-functions

Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar R=R\cup\{\infty\}$ and defined on a cofinite subset $dom(f)$ of the torus $\bar R^2$. Then the family \F determines a continuous vector-function $F:dom(F)\to\bar R^F$ defined on the dense $G_\delta$-set $dom(F)=\bigcap_{f\in F}dom(F)$ of $\bar R^2$. The closure $\bar\Gamma(F)$ of its graph $\Gamma(F)=\{(x,f(x)):x\in dom(F)\}$ in $\bar R^2\times\bar R^F$ is called the {\em graphoid} of the family $F$. We prove the graphoid $\bar\Gamma(F)$ has topological dimension $dim(\bar\Gamma(F))=2$. If the family $F$ contains all linear fractional transformations $f(x,y)=\frac{x-a}{y-b}$ for $(a,b)\in Q^2$, then the graphoid $\bar\Gamma(F)$ has cohomological dimension $dim_G(\bar\Gamma(F))=1$ for any non-trivial 2-divisible abelian group $G$. Hence the space $\bar\Gamma(F)$ is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.Comment: 20 page

Z2SAL: a translation-based model checker for Z

Despite being widely known and accepted in industry, the Z formal specification language has not so far been well supported by automated verification tools, mostly because of the challenges in handling the abstraction of the language. In this paper we discuss a novel approach to building a model-checker for Z, which involves implementing a translation from Z into SAL, the input language for the Symbolic Analysis Laboratory, a toolset which includes a number of model-checkers and a simulator. The Z2SAL translation deals with a number of important issues, including: mapping unbounded, abstract specifications into bounded, finite models amenable to a BDD-based symbolic checker; converting a non-constructive and piecemeal style of functional specification into a deterministic, automaton-based style of specification; and supporting the rich set-based vocabulary of the Z mathematical toolkit. This paper discusses progress made towards implementing as complete and faithful a translation as possible, while highlighting certain assumptions, respecting certain limitations and making use of available optimisations. The translation is illustrated throughout with examples; and a complete working example is presented, together with performance data

A bijection to count (1-23-4)-avoiding permutations

A permutation is (1-23-4)-avoiding if it contains no four entries, increasing left to right, with the middle two adjacent in the permutation. Here we give a 2-variable recurrence for the number of such permutations, improving on the previously known 4-variable recurrence. At the heart of the proof is a bijection from (1-23-4)-avoiding permutations to increasing ordered trees whose leaves, taken in preorder, are also increasing.Comment: latex, 16 page

Combinatorial interpretations of the Jacobi-Stirling numbers

The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.Comment: 15 page