A Geometric characterization of the perfect Suzuki-Tits ovoids

We characterize in a geometrical way those Suzuki-Tits ovoids which are defined over a perfect fieidK (or equivalently living inside a self polar symplectic quadrangle). We simplify our axioms in the particular cases that (1) the associated Suzuki group has exactly two orbits in the set of lines ofPG(3,K), and (2) the ovoid is finite.Senior Research Associate at the Belgian National Fund for Scientific Research     

Particle motion in a spatially periodic flow

The motion of small spherical particles in two-dimensional Taylor–Green flow is investigated assuming one-way coupling. The particle dynamics is modeled by a modified Maxey–Riley equation. Trajectories are computed numerically using a Cash–Karp Runge–Kutta scheme which provides an adaptive time-step control. Depending on the Stokes number and the fluid-to-particle density ratio the particle trajectories can either be periodic or chaotic. To clarify the nature of this dynamical system further investigations are carried out by computing the Feigenbaum scenario. It is expected that the particle dynamics in this idealized flow will provide useful informations for an explanation of particle accumulation phenomena in more realistic periodic three-dimensional viscous flows. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Detecting perfect powers by factoring into coprimes

This paper presents an algorithm that, given an integer , finds the largest integer such that is a th power. A previous algorithm by the first author took time where ; more precisely, time ; conjecturally, time . The new algorithm takes time . It relies on relatively complicated subroutines--specifically, on the first author's fast algorithm to factor integers into coprimes--but it allows a proof of the bound without much background; the previous proof of relied on transcendental number theory.

The computation of is the first step, and occasionally the bottleneck, in many number-theoretic algorithms: the Agrawal-Kayal-Saxena primality test, for example, and the number-field sieve for integer factorization.

     

A new construction of the Clifford-Fourier kernel

In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels.     

The Gibbs' phenomenon

This paper analyses the answers of a group of first-year university Mathematics students to a questionnaire, with the aim of determining the difficulties they have when carrying out non-routine tasks related to improper integrals. The questionnaire consisted of nine questions including not only calculus tasks and determining the convergence of given improper integrals, but also intuitive questions and some paradoxical results too (for example, a figure with an infinite longitude which closes the same area as the unit circumference, or an infinite figure with a finite volume). In particular, the students were asked to interpret most of the results they had obtained. Answers given by the students to each of the questions were categorized, which allowed some initial conclusions to be reached regarding this research. From analyses carried out, one can conclude that there are students who have difficulties in articulating the different systems of representation, and have problems in connecting and relating this knowledge as a generalization of previous concepts, such as definite integrals, series and sequences.     

An Eberlein–Šmulian type result for the weak* topology

We present a result on relative weak* compactness in the dual of a Banach space X that allows a short proof of both the Eberlein–Šmulian theorem and Šmulian’s characterisation of weak compactness of closed convex subsets of X.     

Maximal pivots on graphs with an application to gene assembly

We consider principal pivot transform (pivot) on graphs. We define a natural variant of this operation, called dual pivot, and show that both the kernel and the set of maximally applicable pivots of a graph are invariant under this operation. The result is motivated by and applicable to the theory of gene assembly in ciliates.