Santiago López de Medrano

A new family of complex, compact, non-symplectic manifolds

In this paper we study a family of complex, compact, non-symplectic manifolds arising from linear complex dynamical systems. For every integern>3, and an ordered partition ofn into an odd numberk of positive integers we construct such a manifold together with an (n-2)-dimensional space of complex structures. We show that, under mild additional hypotheses, these deformation spaces are universal. Some of these manifolds are holomorphically equivalent to some known examples and we stablish the identification with them. But we also obtain new manifolds admitting a complex structure, and we describe the differentiable type of some of them.

A qualitative model of a motor circadian rhythm

In Nature it is possible to observe diverse rhythms. Because of their adaptive characteristics, the circadian rhythms are of major importance and have been the subject of numerous experimental and theoretical studies. In this article, we give a presentation of the main results we have obtained about the motor circadian rhythm along some years of collaboration between biologists and mathematicians. We present a mathematical model simulating changes in frequency, synchronization and amplitude of the circadian oscillation during two developmental stages of the crayfish, namely, the juvenile and the adult stages. We report also some work in progress on the simulation of the phase response curve and on a simplified model of the rhythm.

Topology of the intersections of quadrics II

We study the topology of the affine real variety given by the intersection with the unit sphere of the zero set in

Perturbation of a circadian rhythm by single and periodic signals and its mathematical simulation

{\mathbb R}^n

Mutants of compactified representations revisited

Rn of a pair of quadratic forms. We give a complete topological description of this variety in the generic case: when non-empty, it is always diffeomorphic to either the unit tangent bundle of a sphere, the product of two or three spheres, or the connected sum of an odd number of manifolds, each of them a product of two spheres. With this we complete the partial description given in (López de Medrano, 1989). Starting with the cases described in that article and other elementary ones, the proofs are based on the study of three geometric operations that give new of these varieties from simpler ones. For each operation, the topology of the new variety can be described, under appropriate conditions, from that of the old one. The global structure of the proof consists in an elaborate partial ordering of all the cases, in such a way that at each step those conditions are satisfied.

Some open book and contact structures on moment-angle manifolds

Based on experimental work on the ontogeny of the electroretinogram circadian rhythm in crayfish, we present a mathematical model simulating changes in both frequency and amplitude of the electroretinogram oscillation during several developmental stages until shortly before the adult age. Simultaneously, we propose a hypothetical oscillation in the hormonal release whose frequency is imposed on the electroretinogram oscillation. The model consists of two coupled nonlinear oscillators in which a dynamical response is obtained mainly through an Andronov-Hopf bifurcation. Through the construction of the model, a biological hypothesis about the essential elements underlying the ERG circadian rhythm and their interrelations is formulated and discussed.

Generalizing the May-Leonard System to Any Number of Species

Adaptive characteristics of circadian rhythm are based on their capacity to be synchronized by external signals, particularly light signals. The effect of both single and periodic light signals on the electroretinogram (ERG) circadian rhythm in crayfish is studied. In a previous work (Lara-Aparicioet al., Bull math. Biol. 55, 97–110, 1993) we developed a mathematical model simulating the emergence of the ERG circadian rhythm during the ontogeny of the crayfish. In the present work we have tested the familiar wave-shift behaviour of an oscillator with a single limit cycle. Two new facts, not present in a simpler model, now appear, which simulate adequately the experimental results, i.e. the presence of a transient stage and the shape of the perturbed wave which changes according to the characteristics of the external light signals.

Modeling of biological rhythms - to René Thom, in memoriam

We consider the affine real variety