Saso Strle

The cardiovascular system as coupled oscillators

Based on physiological knowledge, and on an analysis of signals related to its dynamics, we propose a model of the cardiovascular system. It consists of coupled oscillators. Each of them describes one of the subsystems involved in the regulation of one passage of blood through the circulatory system. The flow of blood through the system of closed tubes-the blood vessels-is described by wave equations.

Knot concordance and Heegaard Floer homology invariants in branched covers

By studying the Heegaard Floer homology of the preimage of a knot K in S^3 inside its double branched cover, we develop simple obstructions to K having finite order in the classical smooth concordance group. As an application, we prove that all 2-bridge knots of crossing number at most 12 for which the smooth concordance order was previously unknown have infinite smooth concordance order.

ON THE OVERESTIMATION OF THE CORRELATION DIMENSION

The effect of non-linear mapping from the time domain to the phase space may result in an overestimation of the correlation dimension. We analyse the origin of the overestimation and suggest a criterion for the number of points necessary to approach the true scaling region in the correlation integral.

Nonorientable surfaces in homology cobordisms

We investigate constraints on embeddings of a nonorientable surface in a 4–manifold with the homology of M I , where M is a rational homology 3–sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsvath–Szabo d –invariants or Atiyah–Singer – invariants of M . One consequence is that the minimal genus of a smoothly embedded surface in L.2k; q/ I is the same as the minimal genus of a surface in L.2k; q/ . We also consider embeddings of nonorientable surfaces in closed 4–manifolds.

Revealing oscillators calculation of the dimension of attractors

The monitoring of various natural phenomena yields data sets sustaining the dynamics of usually very complex systems. The question then is: how to reveal the characteristics and the nature of a system? In approaching this question, the calculation of the dimension of its attractor may give an indication of its complexity. The present review is an account of the analyses of the calculation of attractor dimension. Various quasi-periodic systems with different degrees of freedom are simulated and, by adding noise and nonstationarity, experimental conditions are mimicked. By increasing the complexity of the system an overestimation of attractor dimension is profound, even on noise-free signals. The presented analysis points out that for experimental data sets, resulting from systems already having more than one degree of freedom, the calculated value of attractor dimension cannot be used to determine whether the system dynamics is chaotic. It may be firmly determined whether it is deterministic or not, since in the case of a deterministic system its surrogate has a higher dimension than the original signal.