Polar foliations and isoparametric maps

A singular Riemannian foliation F on a complete Riemannian manifold M is called a polar foliation if, for each regular point p, there is an immersed submanifold Sigma, called section, that passes through p and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of M into the orbits of a polar action, i.e., an isometric action with sections. In this article we prove that the leaves of H : M -> Sigma, coincide with the level sets of a smooth map H: M -> Sigma, if M is simply connected. In particular, the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.

The analytic torsion of a disc

In this article, we study the Reidemeister torsion and the analytic torsion of the m dimensional disc, with the Ray and Singer homology basis (Adv Math 7:145-210, 1971). We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-Muller theorem. We use a formula proved by Bruning and Ma (GAFA 16:767-873, 2006) that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary (Luck, J Diff Geom 37:263-322, 1993). Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang (Asian J Math 4:695-714, 2000), and we compare the results. The results of these work were announced in the study of Hartmann et al. (BUMI 2:529-533, 2009).

Characterization of Stability Region for General Autonomous Nonlinear Dynamical Systems

The existing characterization of stability regions was developed under the assumption that limit sets on the stability boundary are exclusively composed of hyperbolic equilibrium points and closed orbits. The characterizations derived in this technical note are a generalization of existing results in the theory of stability regions. A characterization of the stability boundary of general autonomous nonlinear dynamical systems is developed under the assumption that limit sets on the stability boundary are composed of a countable number of disjoint and indecomposable components, which can be equilibrium points, closed orbits, quasi-periodic solutions and even chaotic invariant sets.

A continuously differentiable upwinding scheme for the simulation of fluid flow problems

This paper deals with the numerical solution of complex fluid dynamics problems using a new bounded high resolution upwind scheme (called SDPUS-C1 henceforth), for convection term discretization. The scheme is based on TVD and CBC stability criteria and is implemented in the context of the finite volume/difference methodologies, either into the CLAWPACK software package for compressible flows or in the Freeflow simulation system for incompressible viscous flows. The performance of the proposed upwind non-oscillatory scheme is demonstrated by solving two-dimensional compressible flow problems, such as shock wave propagation and two-dimensional/axisymmetric incompressible moving free surface flows. The numerical results demonstrate that this new cell-interface reconstruction technique works very well in several practical applications. (C) 2012 Elsevier Inc. All rights reserved.

Modeling and simulation of severe slugging with mass transfer effects

A mathematical model and numerical simulations are presented to investigate the dynamics of gas, oil and water flow in a pipeline-riser system. The pipeline is modeled as a lumped parameter system and considers two switchable states: one in which the gas is able to penetrate into the riser and another in which there is a liquid accumulation front, preventing the gas from penetrating the riser. The riser model considers a distributed parameter system, in which movable nodes are used to evaluate local conditions along the subsystem. Mass transfer effects are modeled by using a black oil approximation. The model predicts the liquid penetration length in the pipeline and the liquid level in the riser, so it is possible to determine which type of severe slugging occurs in the system. The method of characteristics is used to simplify the differentiation of the resulting hyperbolic system of equations. The equations are discretized and integrated using an implicit method with a predictor-corrector scheme for the treatment of the nonlinearities. Simulations corresponding to severe slugging conditions are presented and compared to results obtained with OLGA computer code, showing a very good agreement. A description of the types of severe slugging for the three-phase flow of gas, oil and water in a pipeline-riser system with mass transfer effects are presented, as well as a stability map. (C) 2011 Elsevier Ltd. All rights reserved.

Effects of melatonin on ovarian follicles

Objective: To evaluate the histomorphometry and expression of Ki-67 and c-kit in ovarian follicles of pinealectomized or melatonin-treated pinealectomized rats. Study design: Forty adult rats were randomly divided into four groups of 10 animals: Group I – control; Group II – sham-pinealectomized; Group III – pinealectomized (Px), and Group IV – Px treated with melatonin (10 mg/night, per animal). After two months’ treatment, on the night of proestrous, the animals were placed in metabolic cages for night urine collection and subsequent measurement of 6-sulfatoxymelatonin (6-SMT). The rats were anesthetized, blood samples were taken for estrogen and progesterone determinations, and they were then euthanized. The ovaries were dissected out for further histological and immunohistochemical analyses. Data were first submitted to analysis of variance (ANOVA) complemented with the Tukey–Kramer test for multiple comparisons (P < 0.05). Results: The urinary levels of 6-SMT and serum progesterone were lower in the Px group (GIII). Exogenous melatonin treatment restored both blood melatonin and 6-SMT urinary levels. The histomorphometric data in Group III revealed a significant increase of degenerating antral and nonantral follicles with regard to the other groups. In addition no corpora lutea were observed in this group. No significant differences were noticed regarding the number of corpora lutea among the other groups (I, II and IV), but the number of cells and the thickness of the theca interna of Px animals (Group III) were higher than in the other groups. Conversely, the density of progesterone receptors (fmol/g) in the ovaries of Group III was significantly lower than in the other groups. Conclusion: Our data indicate that melatonin exerts a role on the maintenance of a proper follicular function, and is thus important for ovulation and progesterone production.