On C(r)-closing for flows on orientable and non-orientable 2-manifolds

We provide an affirmative answer to the C(r)-Closing Lemma, r >= 2, for a large class of flows defined on every closed surface.

3-manifolds in Euclidean space from a contact viewpoint

We study the geometry of 3-manifolds generically embedded in R(n) by means of the analysis of the singularities of the distance-squared and height functions on them. We describe the local structure of the discriminant (associated to the distribution of asymptotic directions), the ridges and the flat ridges.

Aerosol inorganic composition at a tropical site: Discrepancies between filter-based sampling and a semi-continuous method

The concentrations of the water-soluble inorganic aerosol species, ammonium (NH4+), nitrate (NO3-), chloride (Cl-), and sulfate (SO42-), were measured from September to November 2002 at a pasture site in the Amazon Basin (Rondnia, Brazil) (LBA-SMOCC). Measurements were conducted using a semi-continuous technique (Wet-annular denuder/Steam-Jet Aerosol Collector: WAD/SJAC) and three integrating filter-based methods, namely (1) a denuder-filter pack (DFP: Teflon and impregnated Whatman filters), (2) a stacked-filter unit (SFU: polycarbonate filters), and (3) a High Volume dichotomous sampler (HiVol: quartz fiber filters). Measurements covered the late dry season (biomass burning), a transition period, and the onset of the wet season (clean conditions). Analyses of the particles collected on filters were performed using ion chromatography (IC) and Particle-Induced X-ray Emission spectrometry (PIXE). Season-dependent discrepancies were observed between the WAD/SJAC system and the filter-based samplers. During the dry season, when PM2.5 (D-p <= 2.5 mu m) concentrations were similar to 100 mu g m(-3), aerosol NH4+ and SO42- measured by the filter-based samplers were on average two times higher than those determined by the WAD/SJAC. Concentrations of aerosol NO3- and Cl- measured with the HiVol during daytime, and with the DFP during day- and nighttime also exceeded those of the WAD/SJAC by a factor of two. In contrast, aerosol NO3- and Cl- measured with the SFU during the dry season were nearly two times lower than those measured by the WAD/SJAC. These differences declined markedly during the transition period and towards the cleaner conditions during the onset of the wet season (PM2.5 similar to 5 mu g m(-3)); when filter-based samplers measured on average 40-90% less than the WAD/SJAC. The differences were not due to consistent systematic biases of the analytical techniques, but were apparently a result of prevailing environmental conditions and different sampling procedures. For the transition period and wet season, the significance of our results is reduced by a low number of data points. We argue that the observed differences are mainly attributable to (a) positive and negative filter sampling artifacts, (b) presence of organic compounds and organosulfates on filter substrates, and (c) a SJAC sampling efficiency of less than 100%.

A semi-Markov multistate model for estimation of the mean quality-adjusted survival for non-progressive processes

We discuss the estimation of the expected value of the quality-adjusted survival, based on multistate models. We generalize an earlier work, considering the sojourn times in health states are not identically distributed, for a given vector of covariates. Approaches based on semiparametric and parametric (exponential and Weibull distributions) methodologies are considered. A simulation study is conducted to evaluate the performance of the proposed estimator and the jackknife resampling method is used to estimate the variance of such estimator. An application to a real data set is also included.

SUPPORT OF MAXIMIZING MEASURES FOR TYPICAL C(0) DYNAMICS ON COMPACT MANIFOLDS

Given a compact manifold X, a continuous function g : X -> IR, and a map T : X -> X, we study properties of the T-invariant Borel probability measures that maximize the integral of g. We show that if X is a n-dimensional connected Riemaniann manifold, with n >= 2, then the set of homeomorphisms for which there is a maximizing measure supported on a periodic orbit is meager. We also show that, if X is the circle, then the ""topological size"" of the set of endomorphisms for which there are g maximizing measures with support on a periodic orbit depends on properties of the function g. In particular, if g is C(1), it has interior points.

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged. leads to a linear system of equations for the inter-face configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin`s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system`s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve. (C) 2009 Elsevier Inc. All rights reserved.

EQUIVARIANT PATH FIELDS ON TOPOLOGICAL MANIFOLDS

A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf`s result to topological manifolds, replacing vector fields with path fields. In this note, we give an equivariant analog of Brown`s theorem for locally smooth G-manifolds where G is a finite group.

ON CLOSED GEODESICS IN THE LEAF SPACE OF SINGULAR RIEMANNIAN FOLIATIONS

In this paper we prove the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.fs. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171-206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic gamma, we prove the existence of a locally constant integer valued map Lambda(gamma) on the unit circle with the property that the Morse index of the iterated gamma(N) is equal, up to a correction term epsilon(gamma) is an element of {0,1}, to the sum of the values of Lambda(gamma) at the N-th roots of unity. The discontinuities of Lambda(gamma) occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincare map of gamma. We discuss some applications of the theory.

On a Gromoll-Meyer type theorem in globally hyperbolic stationary spacetimes

Following the lines of the celebrated Riemannian result of Gromoll and Meyer, we use infinite dimensional equivariant Morse theory to establish the existence of infinitely many geometrically distinct closed geodesics in a class of globally hyperbolic stationary Lorentzian manifolds.

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface (Sigma, g), its tangent bundle T Sigma enjoys a natural pseudo-Kahler structure, that is the combination of a complex structure 2, a pseudo-metric G with neutral signature and a symplectic structure Omega. We give a local classification of those surfaces of T Sigma which are both Lagrangian with respect to Omega and minimal with respect to G. We first show that if g is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in R(3) or R(1)(3) induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in TS(2) or TH(2) respectively. We relate the area of the congruence to a second-order functional F = f root H(2) - K dA on the original surface. (C) 2010 Elsevier B.V. All rights reserved.

Desingularization of singular Riemannian foliation

Let F be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F we construct a regular Riemannian foliation (F) over cap on a compact Riemannian manifold (M) over cap and a desingularization map (rho) over cap : (M) over cap -> M that projects leaves of (F) over cap into leaves of F. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F are compact, then, for each small epsilon > 0, we can find (M) over cap and (F) over cap so that the desingularization map induces an epsilon-isometry between M/F and (M) over cap/(F) over cap. This implies in particular that the space of leaves M/F is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {((M) over cap (n)/(F) over cap (n))}.

On Polar Foliations and the Fundamental Group

In this work we investigate the relation between the fundamental group of a complete Riemannian manifold M and the quotient between the Weyl group and reflection group of a polar action on M, as well as the relation between the fundamental group of M and the quotient between the lifted Weyl group and lifted reflection group. As applications we give alternative proofs of two results. The first one, due to the author and Toben, implies that a polar action does not admit exceptional orbits, if M is simply connected. The second result, due to Lytchak, implies that the orbits are closed and embedded if M is simply connected. All results are proved in the more general case of polar foliations.

Birkhoffian Systems in Infinite Dimensional Manifolds

We generalize the theory of Kobayashi and Oliva (On the Birkhoff Approach to Classical Mechanics. Resenhas do Instituto de Matematica e Estatistica da Universidade de Sao Paulo, 2003) to infinite dimensional Banach manifolds with a view towards applications in partial differential equations.