Starting with a quadrilateral $q_0=(A_1,A_2,A_3,A_4)$ of $\m R^2$, one constructs a sequence of quadrilaterals $q_n=(A_{4n+1},\ldots ,A_{4n+4})$ by iteration of foldings~: $q_n= \ph_4\circ\ph_3\circ\ph_2\circ\ph_1(q_{n-1})$ where the folding $\ph_j$ replaces the vertex number $j$ by its symmetric with respect to the opposite diagonal. We have studied [1] the dynamical behavior of this sequence. In particular, we have seen that the drift ${\D v(q_0):= \lim_{n\ra\infty}}\frac1n q_n$ exists and, for Lebesgue almost all $q_0$, the sequence $( q_n -nv(q_0))_{n\geq 1}$ is dense on a bounded analytic curve. Here, we prove that, for Baire generic $q_0$, the closure of the same sequence $( q_n -nv(q_0))_{n\geq 1}$ contains all the translates of $q_0$.

The paradigm of computer science is nowadays shifting from computation to communication. In this paper we aim to give an impression of the algebra of actions and communications as it has been developed during the past quarter of a century. We present specifically the system ACP, Algebra of Communicating Processes. Here $a, b, c, \ldots$ denote atomic events (or actions, or steps). For example, $a{\cdot}a{\cdot}(b + c)$ is the process able to perform two $a$-steps followed by a $b$-step or a $c$-step. This differs from the process $a{\cdot}a{\cdot}b + a{\cdot}a{\cdot}c$, due to the different timing of the choice between a $b$- and a $c$-step. Process $x$ can operate in parallel with process $y$, notation $x \| y$, where some actions in $x$ may happen simultaneously with some actions in $y$; the resulting actions are called communication actions. Such actions are `half actions', needing their counterpart for the execution as a full action. In daily life a handshake is an example of a pair of half actions. We even have ternary communication actions on the keyboard of a PC: control-alt-delete. In music we find also quaternary actions, etc. The results of such internal communications we want to abstract away. This is formalised with the invisible action $\tau$ (`silent move'), facilitating the composition of modular hierarchies of process systems. The classical triple specification-implementation-verification then takes a straightforward form. The specification of the desired external behaviour is a simple process SPEC. The implementation is a complicated process IMP. The verification that the implementation is indeed correct, can be a purely algebraic, equational computation that a suitable abstraction of IMP yields SPEC: ABS(IMP) = SPEC. Next to an overview of the process algebraic framework, we present a soundness and completeness proof for the axiom system underlying this framework, with respect to a fundamental semantics called rooted branching bisimulation. Thus a derivation of ABS(IMP) = SPEC implies that the implementation and its specification satisfy the same behavioural properties (soundness), while on the other hand the absence of such a derivation implies that the implementation and its specification are not rooted branching bisimulation equivalent (completeness). Computing with actions and communications, process algebra, is logically and mathematically interesting. On top of that, process algebra contributes to security and comfort: current process algebra tools are used to prove correctness of railway emplacements, helicopter software, and communication protocols in the remote control of a television set. This paper has grown out of an invited lecture of the second author at the Joint Mathematical BeNeLuxFra Conference in Ghent, May 2005. Slides of that lecture can be found at www.cs.vu.nl/~jwk.

Let $X$ be a compact manifold with boundary $\partial X$, and suppose that $\partial X$ is the total space of a fibration \[ Z\rightarrow \partial X \rightarrow Y\, . \] Let $D_\Phi$ be a generalized Dirac operator associated to a $\Phi$-metric $g_\Phi$ on $X$. Under the assumption that $D_\Phi$ is fully elliptic we prove an index formula for $D_\Phi$. The proof is in two steps: first, using results of Melrose and Rochon, we show that the index is unchanged if we pass to a certain $b$-metric $g_b (\epsilon)$. Next we write the $b-$ (i.e. the APS) index formula for $g_b(\ep)$; the $\Phi$-index formula follows by analyzing the limiting behaviour as $\epsilon\searrow 0$ of the two terms in the formula. The interior term is studied directly whereas the adiabatic limit formula for the eta invariant follows from work of Bismut and Cheeger.

We present a simple and explicit multivariate procedure for testing homogeneity of two independent samples of size $n$. The test statistic $T_n$ is the $L_1$ distance between the two empirical distributions restricted to a finite partition. We first discuss Chernoff-type large deviation properties of $T_n$. This results in a distribution-free strongly consistent test of homogeneity, which rejects the null if $T_n$ becomes large. Then the asymptotic null distribution of the test statistic is obtained, leading to a new consistent test procedure.

We use the Poisson cluster process structure of a Hawkes process to derive non asymptotic estimates of the tail of the extinction time, of the coupling time or of the number of points per interval. This allows us to define a family of independent Hawkes processes; each of them approximating the initial process on a particular interval. Then we can easily derive exponential inequalities for Hawkes processes which can precise the ergodic theorem.

We combine results of Vaserstein and Shalom to prove Kazhdan's Property (T) for various topological groups of the form $\textnormal{SL}(n,\mathcal{C}(X,\mathbb R))$ or $\textnormal{SL}(n,\mathcal{C}(X,\mathbb C))$, for $n\ge 3$ and $X$ a topological subspace of a Euclidean space.

This paper addresses the verification of protocol conformance between two types of state machines in model-driven software design: protocol state machines that enable specifying allowed sequences of signals and behavioral state machines that are intended for implementation specification. The contribution of the present paper, is to provide a methodology, which is based on refinement mappings, to automatically verify on protocol conformance. It helps to turn UML into a powerful and interesting tool for the development of business critical software systems.

Partial functions and "undefinedness" have been around in mathematics for a long time, without causing any trouble. It was only when mathematics and computer science met in projects on ``automatization'' of formal reasoning that some problems came up [Hoogewijs 1987]. Where humans are able to avoid the application of a partial function on an argument outside ``the domain'' of the function, formalizing the rules for this activity seems to be less trivial. In [Farmer 1996] Farmer states that there does not exist a consensus on how partial functions should be mechanized and the developer of a mechanized mathematics system must choose among many different possible ways of representing and reasoning about partial functions. We want to add one more possibility by introducing ``partially defined iota terms'' of the form $\iota x_\psi(\varphi)$ which represents the unique $x$ satisfying $\varphi$ whenever the condition $\psi$ is fulfilled. We present an extension of a two-valued first order sequent calculus for predicate logic with identity [Hermes 1973], where we are able to reason correctly about partially defined iota terms.

In recent years, there has been an enormous effort put in the definition and analysis of fractional or fractal operators. Fractional calculus is for example a flourishing field of active research. In this paper we restrict ourselves to the fractional Fourier operator and friends that are traditionally used in optics, mechanical engineering and signal processing. The book by H.M. Ozaktas, Z. Zalevsky, and M.A. Kutay, {\em The fractional Fourier transform}, John Wiley, 2001 gives a state of the art of 2001. Because this field is still in full expansion, we want to summarize in this survey paper some of the recent developments that appeared in the literature since then, revealing some unexplored aspects.

The observation and the discussion of the physical reality of phenomena, leads to bring out concepts which have to be described in a non ambiguous mathematical language. Concerning Dirac's calculus we shall introduce, besides the usual definitions for the concepts of point, number, function etc ... , additional concepts for the physical point, the physical equalities, physical infinities and infinitesimals ... etc ... In particular we introduce a new equality $=^{D}$, called Dirac-equality, which differs as well from the classical equality as from the weak equalities introduced in various theories of generalized functions. All these definitions are based on a definition in the language of Relative Set Theory of the metaconcepts of improperness used by P.A.M. Dirac in [Di], when he claimed " \textit{Strictly of course, $\delta(x)$ is not a proper function of $x$, ... , ... $\delta'(x), \delta''(x)$.... are even more discontinuous and less proper than $\delta(x)$ itself}". We defined this way a concept of observed derivative which extends the usual one to a large class of \textit{discontinuous} possibly non-standard functions. All the multiplications of improper or very improper functions, including the delta-functions and their observed derivatives, are obviously allowed. Now the problem of the multiplication is replaced by another one: under which conditions is the Dirac-equality of two functions preserved by a multiplication term by term?

Starting from the Zariski topology, a natural notion of nonstandard generic point is introduced in complex algebraic geometry. The existence of this kind of point is a strong form of the Nullstellensatz. This notion is connected with the classical concept of generic point in the spectrum $\text{Spec}({\cal A}_{n,\mathbb C})$ of the corresponding algebra ${\cal A}_{n,\mathbb C}$. The nonstandard affine space ${^*\mathbb C}^n$ appears as an affine unfolding of the geometric space $\text{Spec}({\cal A}_{n,\mathbb C})$. This affine space is the disjoint union of the sets whose elements are the nonstandard generic points of prime and proper ideals of ${\cal A}_{n,\mathbb C}$: this structure leads to the definition of algebraic points in ${^*\mathbb C}^n$. A natural extension to analytic points in ${^*\mathbb C}^n$ is given by Robinson's concept of generic point in local complex analytic geometry. The end of this paper is devoted to a generalization of this point of view to the real analytic case.