Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

The curve selection lemma and the Morse-Sard theorem

We use an inequality due to Bochnak and Lojasiewicz, which follows from the Curve Selection Lemma of real algebraic geometry in order to prove that, given a C(r) function f : U subset of R(m) -> R, we have lim(y -> xy is an element of crit(f)) vertical bar f(y) - f(x)vertical bar/vertical bar y - x vertical bar(r) = 0, for all x is an element of crit(f)` boolean AND U, where crit( f) = {x is an element of U vertical bar df ( x) = 0}. This shows that the so-called Morse decomposition of the critical set, used in the classical proof of the Morse-Sard theorem, is not necessary: the conclusion of the Morse decomposition lemma holds for the whole critical set. We use this result to give a simple proof of the classical Morse-Sard theorem ( with sharp differentiability assumptions).

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

GENERICITY OF NONDEGENERATE GEODESICS WITH GENERAL BOUNDARY CONDITIONS

Let M be a possibly noncompact manifold. We prove, generically in the C(k)-topology (2 <= k <= infinity), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of L. Biliotti, M. A. Javaloyes and P. Piccione [6] for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P subset of M x M that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal Delta subset of M x M. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are C(k)-generic are given.

Stability of gradient semigroups under perturbations

In this paper we prove that gradient-like semigroups (in the sense of Carvalho and Langa (2009 J. Diff. Eqns 246 2646-68)) are gradient semigroups (possess a Lyapunov function). This is primarily done to provide conditions under which gradient semigroups, in a general metric space, are stable under perturbation exploiting the known fact (see Carvalho and Langa (2009 J. Diff. Eqns 246 2646-68)) that gradient-like semigroups are stable under perturbation. The results presented here were motivated by the work carried out in Conley (1978 Isolated Invariant Sets and the Morse Index (CBMS Regional Conference Series in Mathematics vol 38) (RI: American Mathematical Society Providence)) for groups in compact metric spaces (see also Rybakowski (1987 The Homotopy Index and Partial Differential Equations (Universitext) (Berlin: Springer)) for the Morse decomposition of an invariant set for a semigroup on a compact metric space).

Spectral flow and iteration of closed semi-Riemannian geodesics

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.

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.

On the variations of the Betti numbers of regular levels of Morse flows

We generalize results in Cruz and de Rezende (1999) [7] by completely describing how the Beth numbers of the boundary of an orientable manifold vary after attaching a handle, when the homology coefficients are in Z, Q, R or Z/pZ with p prime. First we apply this result to the Conley index theory of Lyapunov graphs. Next we consider the Ogasa invariant associated with handle decompositions of manifolds. We make use of the above results in order to obtain upper bounds for the Ogasa invariant of product manifolds. (C) 2011 Elsevier B.V. All rights reserved.

A further exploration of a nucleophilicity index based on the gas-phase ionization potentials

An empirical nucleophilicity index based on the gas-phase ionization potentials has been recently shown to be useful categorizing and settling the nucleophilicity power of a series of captodative ethylenes reacting in cycloaddition reactions (L.R. Domingo, E. Chamorro, P. Perez, Journal of Organic Chemistry 73 (2008) 4615-4624). In the present work, the applicability of such model is tested within a broader series of substituted alkenes, substituted aromatic compounds and simple nucleophilic molecules. This index obtained within a Koopman`s theorem framework has been evaluated here in both gas and solution phases for several well-known nucleophiles. These results are found to be linearly correlated. Finally, the feasibility of the predictive character of this index has been discussed in comparison to the available experimental nucleophilicities of some amines in water. These results further support and validate the usefulness of such approximation in the modeling of the global nucleophilicity. (C) 2008 Elsevier B.V. All rights reserved.

Genericity of Nondegenerate Critical Points and Morse Geodesic Functionals

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard-Smale theorem, along the lines of an analogous result of B. White [29]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.

On the semi-Riemannian bumpy metric theorem

We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold M, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the C(k)-topology, k=2, ..., infinity, in the set of metrics of a given index on M. A higher-order genericity Riemannian result of Klingenberg and Takens is extended to semi-Riemannian geometry.

Bone Remodeling Adjacent to Morse Cone-Connection Implants with Platform Switch: A Fluorescence Study in the Dog Mandible

Purpose: The aim of this study was to evaluate, through fluorescence analysis, the effect that different interimplant distances, after prosthetic restoration, will have on bone remodeling in submerged and nonsubmerged implants restored with a ""platform switch."" Materials and Methods: Fifty-six Ankylos implants were placed 1.5 mm subcrestally in seven dogs. The implants were placed so that two fixed prostheses, with three interimplant contacts separated by 1-mm, 2-mm, and 3-mm distances, could be fabricated for each side of the mandible. The sides and the positions of the groups were selected randomly. To better evaluate bone remodeling, calcein green was injected 3 days before placement of the prostheses at 12 weeks postimplantation. At 3 days before sacrifice (8 weeks postloading), alizarin red was injected. The amounts of remodeled bone within the different interimplant areas were compared statistically before and after loading in submerged and nonsubmerged implants. Results: Statistically significant differences existed in the percentage of remodeled bone seen in the different regions. Mean percentages of remodeled bone in the submerged and nonsubmerged groups, respectively, were as follows: for the 1-mm distance, 23.0% +/- 0.05% and 23.1% +/- 0.03% preloading and 27.0% +/- 0.03% and 25.2% +/- 0.04% postloading, for the 2-mm distance, 18.2% +/- 0.05% and 18.1% +/- 0.04% preloading and 21.3% +/- 0.07% and 19.9% +/- 0.03% postloading, for the 3-mm distance, 18.3% +/- 0.03% and 18.3% +/- 0.03% preloading and 18.8% +/- 0.04% and 19.8% +/- 0.04% postloading, for distal-extension regions, 16.6% +/- 0.02% and 17.4% +/- 0.04% preloading and 17.0% +/- 0.04% and 18.4% +/- 0.04% postloading. Conclusions: Based upon this animal study, loading increases bone formation for submerged or nonsubmerged implants, and the interimplant distance of 1 mm appears to result in more pronounced bone remodeling than the 2-mm or 3-mm distances in implants with a ""platform switch."" INT J ORAL MAXILLOFAC IMPLANTS 2009;24:257-266

Dietary Quality Index and Associated Factors among Adolescents of the State of Sao Paulo, Brazil

Objective To assess dietary quality and associated factors in adolescents. Study design We conducted a population-based cross-sectional study in a sample of 1584 adolescents living in areas of the state of Sao Paulo, Brazil. Dietary intake was measured with the 24-hour recall method, and dietary quality was assessed by means of the Health Eating Index (HEI), adapted to fit to the local requirements. Linear regression analyses were performed to assess the association between the HEI and demographic, socioeconomic, and lifestyle variables. Results A total of 97.1% of the adolescents studied had an inadequate diet or a diet that needed improvement. The mean overall HEI score was 59.7. Lower mean HEI scores were found for fruits, dairy products, and vegetables. Male adolescents who were physically active and lived in a house or apartment had higher HEI scores. The multiple regression analyses showed that the quality of the diet improved as age decreased. Adolescents who lived in houses or apartments had higher HEI scores than adolescents living in shacks or slums, regardless of age and energy intake. Conclusions Dietary quality is associated with income and age. A better understanding of the factors associated can provide input to the formulation of policies and development of nutritional actions. (J Pediatr 2010; 156:456-60).

Effects of processing methods on amaranth starch digestibility and predicted glycemic index

Amaranth has attracted a great deal of interest in recent decades due to its valuable nutritional, functional, and agricultural characteristics. Amaranth seeds can be cooked, popped, roasted, flaked, or extruded for consumption. This study compared the in vitro starch digestibility of processed amaranth seeds to that of white bread. Raw seeds yielded rapidly digestible starch content (RDS) of 30.7% db and predicted glycemic index (pGI) of 87.2, the lowest among the studied products. Cooked, extruded, and popped amaranth seeds had starch digestibility similar to that of white bread (92.4, 91.2, and 101.3, respectively), while flaked and roasted seeds generated a slightly increased glycemic response (106.0 and 105.8, respectively). Cooking and extrusion did not alter the RDS contents of the seeds. No significant differences were observed among popped, flaked, and roasted RDS contents (38.0%,46.3%, and 42.9%, respectively), which were all lower than RDS content of bread (51.1%). Amaranth seed is a high glycemic food most likely because of its small starch granule size, low resistant starch content (< 1%), and tendency to completely lose its crystalline and granular starch structure during those heat treatments.