Ángel Giménez

Geometrical particle models on 3D null curves

Abstract The simplest (2+1)-dimensional mechanical systems associated with light-like curves, already studied by Nersessian and Ramos, are reconsidered. The action is linear in the curvature of the particle path and the moduli spaces of solutions are completely exhibited in 3-dimensional Minkowski background, even when the action is not proportional to the pseudo-arc length of the trajectory.

Null generalized helices in Lorentz–Minkowski spaces

We obtain a Lancret-type theorem for null generalized helices in Lorentz– Minkowski spaces L n .I nL 3 we find that the only null generalized helices are the ordinary null helices. However, in L 5 we have to consider two types of null generalized helices according to whether the axis is non-null or null. In both cases we obtain the solutions to the natural equations problem.

New designs of ventricular catheters for hydrocephalus by 3-D computational fluid dynamics

IntroductionBased on a landmark study by Lin et al. of the two-dimensional flow in ventricular catheters (VCs) via computational fluid dynamics (CFD), we studied in a previous paper the three-dimensional flow patterns of five commercially available VC. We found that the drainage of the cerebrospinal fluid (CSF) mostly occurs through the catheter’s most proximal holes. In this paper, we design five VC prototypes with equalized flow characteristics.MethodsWe study five prototypes of VC by means of CFD in three-dimensional (3-D) automated models and compare the fluid-mechanical results with our previous study of currently in use VC. The general procedure for the development of a CFD model calls for transforming the physical dimensions of the system to be studied into a virtual wire-frame model, which provides the coordinates for the virtual space of a CFD mesh. The incompressible Navier–Stokes equations, a system of strongly coupled, nonlinear, partial differential equations governing the motion of the flow field, are then solved numerically.ResultsBy varying the number of drainage holes and the ratio hole/segment, we improved flow characteristics in five prototypes of VC. Models 1, 2, and 3 have a distal to proximal decreasing flow. Model 4 has an inverse flow to the previous ones, that is, a distal to proximal increasing flow, while model 5 has a constant flow over the segments.ConclusionsNew catheter designs with variable hole diameter, number of holes, and ratio hole/segment along the catheter allow the fluid to enter the catheter more uniformly along its length, thus reducing the chance that the catheter becomes occluded.

s-Degenerate curves in Lorentzian space forms

In this paper we introduce s-degenerate curves in Lorentzian space forms as those ones whose derivative of order s is a null vector provided that s>1 and all derivatives of order less than s are space-like (see the exact definition in Section 2). In this sense classical null curves are 1-degenerate curves. We obtain a reference along an s-degenerate curve in an n-dimensional Lorentzian space with the minimum number of curvatures. That reference generalizes the reference of Bonnor for null curves in Minkowski space–time and it will be called the Cartan frame of the curve. The associated curvature functions are called the Cartan curvatures of the curve. We characterize the s-degenerate helices (i.e. s-degenerate curves with constant Cartan curvatures) in n-dimensional Lorentzian space forms and we obtain a complete classification of them in dimension four.

ATTRACTORS FOR A LATTICE DYNAMICAL SYSTEM GENERATED BY NON-NEWTONIAN FLUIDS MODELING SUSPENSIONS

In this paper we consider a lattice dynamical system generated by a parabolic equation modeling suspension flows. We prove the existence of a global compact connected attractor for this system and the upper semicontinuity of this attractor with respect to finite-dimensional approximations. Also, we obtain a sequence of approximating discrete dynamical systems by the implementation of the implicit Euler method, proving the existence and the upper semicontinuous convergence of their global attractors.

Computing the Topological Entropy of Multimodal Maps via Min-Max Sequences

We derive an algorithm to recursively determine the lap number (minimal number of monotonicity segments) of the iterates of twice differentiable l-modal map, enabling to numerically calculate the topological entropy of these maps. The algorithm is obtained by the min-max sequences—symbolic sequences that encode qualitative information about all the local extrema of iterated maps.

Basic cerebrospinal fluid flow patterns in ventricular catheters prototypes.

ObjectA previous study by computational fluid dynamics (CFD) of the three-dimensional (3-D) flow in ventricular catheters (VC) disclosed that most of the total fluid mass flows through the catheter’s most proximal holes in commercially available VC. The aim of the present study is to investigate basic flow patterns in VC prototypes.MethodsThe general procedure for the development of a CFD model calls for transforming the physical dimensions of the system to be studied into a virtual wire-frame model which provides the coordinates for the virtual space of a CFD mesh, in this case, a VC. The incompressible Navier–Stokes equations, a system of strongly coupled, nonlinear, partial differential conservation equations governing the motion of the flow field, are then solved numerically. New designs of VC, e.g., with novel hole configurations, can then be readily modeled, and the corresponding flow pattern computed in an automated way. Specially modified VCs were used for benchmark experimental testing.ResultsThree distinct types of flow pattern in prototype models of VC were obtained by varying specific parameters of the catheter design, like the number of holes in the drainage segments and the distance between them. Specifically, we show how to equalize and reverse the flow pattern through the different VC drainage segments by choosing appropriate parameters.ConclusionsThe flow pattern in prototype catheters is determined by the number of holes, the hole diameter, the ratio hole/segment, and the distance between hole segments. The application of basic design principles of VC may help to develop new catheters with better flow circulation, thus reducing the possibility of becoming occluded.

A Simplified Algorithm for the Topological Entropy of Multimodal Maps

A numerical algorithm to compute the topological entropy of multimodal maps is proposed. This algorithm results from a closed formula containing the so-called min-max symbols, which are closely related to the kneading symbols. Furthermore, it simplifies a previous algorithm, also based on min-max symbols, which was originally proposed for twice differentiable multimodal maps. The new algorithm has been benchmarked against the old one with a number of multimodal maps, the results being reported in the paper. The comparison is favorable to the new algorithm, except in the unimodal case.

Switching systems and entropy

Switching systems are non-autonomous dynamical systems obtained by switching between two or more autonomous dynamical systems as time goes on. They can be mainly found in control theory, physics, economy, biomathematics, chaotic cryptography and of course in the theory of dynamical systems, in both discrete and continuous time. Much of the recent interest in these systems is related to the emergence of new properties by the mechanism of switching, a phenomenon known in the literature as Parrondos paradox. In this paper we consider a discrete-time switching system composed of two affine transformations and show that the switched dynamics has the same topological entropy as the switching sequence. The complexity of the switching sequence, as measured by the topological entropy, is fully transferred, for example, to the switched dynamics in this particular case.

Parametric study of ventricular catheters for hydrocephalus

BackgroundTo drain the excess of cerebrospinal fluid in a hydrocephalus patient, a catheter is inserted into one of the brain ventricles and then connected to a valve. This so-called ventricular catheter is a standard-size, flexible tubing with a number of holes placed symmetrically around several transversal sections or “drainage segments”. Three-dimensional computational dynamics shows that most of the fluid volume flows through the drainage segment closest to the valve. This fact raises the likelihood that those holes and then the lumen get clogged by the cells and macromolecules present in the cerebrospinal fluid, provoking malfunction of the whole system. In order to better understand the flow pattern, we have carried out a parametric study via numerical models of ventricular catheters.MethodsThe parameters chosen are the number of drainage segments, the distances between them, the number and diameter of the holes on each segment, as well as their relative angular position.ResultsThese parameters were found to have a direct consequence on the flow distribution and shear stress of the catheter. As a consequence, we formulate general principles for ventricular catheter design.ConclusionsThese principles can help develop new catheters with homogeneous flow patterns, thus possibly extending their lifetime.