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.

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.

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 Fermat's principle for causal curves in time oriented Finsler spacetimes

In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the second variation of the time arrival functional along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4765066]

Greening the United Nations' Human Development Index?

The Human Development Index (HDI) introduced by the United Nations Development Programme (UNDP) in 1990 has helped facilitate widespread debate amongst development researchers, practitioners and policy makers. The HDI is an aggregate index, calculated on an annual basis by the UNDP and published in its Human Development Reports, comprising measures of three components deemed by them to be central to development: W income (the gross domestic product per capita), (ii) education (adult literacy rate) and (iii) health (life expectancy at birth). The results of calculating the HDI are typically presented as country/regional league tables, and provide a quick means for policy makers and others to judge performance. Perhaps partly because of the relative simplicity of the index, the HDI has managed to achieve a level of acceptance and use amongst politicians and policy makers that has yet to emerge with any indicator of sustainability. Indeed, despite its existence for 11 years, including nine years after the Rio Earth Summit, the HDI has not even been modified to take on board wider issues of sustainability. This paper will critically examine the potential for 'greening' the HDI so as to include environmental and resource-consumption dimensions. Copyright (C) 2003 John Wiley & Sons, Ltd and ERP Environment.

For better or for worse, till the human development index do us part?

This paper describes the results of research intended to explore the volatility inherent in the United Nations Development Programme's (UNDP) Human Development Index (HDI). The HDI is intended to be a simple and transparent device for comparing progress in human development, and is an aggregate of life expectancy, education and GDP per capita. Values of the HDI for each country are presented in the Human Development Reports (HDRs), the first being published in 1990. However, while the methodology is consistent for all countries in each year there are notable differences between years that make temporal comparisons of progress difficult. The paper presents the results of recalculating the HDI for a simplified sample of 114 countries using various methodologies employed by the UNDP. The results are a set of deviations of recalculated HDI ranks compared to the original ranks given in the HDRs. The volatility that can result from such recalculation is shown to be substantial (+/-10-15 ranks), yet reports in the popular press are frequently sensitive to movements of only a few ranks. Such movement can easily be accounted for by changes in the HDI methodology rather than genuine progress in human development. While the HDRs often carry warnings about the inadvisability of such year-on-year comparisons, it is argued that the existence of such a high-profile index and the overt presentation within league tables do encourage such comparison. Assuming that the HDI will be retained as a focal point within the HDRs, then it is suggested that greater focus be upon more meaningful and robust categories of human development (e.g. low, medium and high) rather than league tables where shifts of a few places, perhaps as a result of nothing more than a methodological or data artefact, may be highlighted in the press and by policy makers. (C) 2003 Elsevier Science B.V. All rights reserved.

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.

Alternativa protética para correção da angulação com componentes não indexados sobre implantes Cone Morse indexados: sistema tubo parafuso

From a recent perspective the morse-taper dental implants connections are increasingly being used as an alternative for replacement of a missing teeth. Nevertheless, there are a large variety of prosthetic components available on the market with some limitations regarding the final prothesis. This article demonstrated the difficulties and limitations of prosthesis implant-retained connections when using morse-taper implants (with a prosthetic index) case in which the surgical placement of the implant wasn’t successfully performed. The alternative to overcome this scenario was the technique using the tube screw over the top of a mini abutment component. It was possible to manufacture and to have satisfactory adaptation, achieving the satisfaction of the patient, restoring function and esthetics.

Avaliação comparativa da remodelação óssea periimplantar em implantes com junção cone-morse e hexágono externo: estudo clínico randomizado controlado, duplo cego, boca dividida

Background: Several theories, such as the biological width formation, the inflammatory reactions due to the implant-abutment microgap contamination, and the periimplant stress/strain concentration causing bone microdamage accumulation, have been suggested to explain early periimplant bone loss. However, it is yet not well understood to which extent the implant-abutment connection type may influence the remodeling process around dental implants. Aim: to evaluate clinical, bacteriological, and biomechanical parameters related to periimplant bone loss at the crestal region, comparing external hexagon (EH) and Morse-taper (MT) connections. Materials and methods: Twelve patients with totally edentulous mandibles received four custom made Ø 3.8 x 13 mm implants in the interforaminal region of the mandible, with the same design, but different prosthetic connections (two of them EH or MT, randomly placed based on a split-mouth design), and a immediate implant- supported prosthesis. Clinical parameters (periimplant probing pocket depth, modified gingival index and mucosal thickness) were evaluated at 6 sites around the implants, at a 12 month follow-up. The distance from the top of the implant to the first bone-to-implant contact – IT-FBIC was evaluated on standardized digital peri-apical radiographs acquired at 1, 3, 6 and 12 months follow-up. Samples of the subgingival microbiota were collected 1, 3 and 6 months after implant loading. DNA were extracted and used for the quantification of Tanerella forsythia, Porphyromonas gingivalis, Aggragatibacter actinomycetemcomitans, Prevotella intermedia and Fusobacterium nucleatum. Comparison among multiple periods of observation were performed using repeated-measures Analysis of Variance (ANOVA), followed by a Tukey post-hoc test, while two-period based comparisons were made using paired t- test. Further, 36 computer-tomographic based finite element (FE) models were accomplished, simulating each patient in 3 loading conditions. The results for the peak EQV strain in periimplant bone were interpreted by means of a general linear model (ANOVA). Results: The variation in periimplant bone loss assessed by means of radiographs was significantly different between the connection types (P<0.001). Mean IT-FBIC was 1.17±0.44 mm for EH, and 0.17±0.54 mm for MT, considering all evaluated time periods. All clinical parameters presented not significant differences. No significant microbiological differences could be observed between both connection types. Most of the collected samples had very few pathogens, meaning that these regions were healthy from a microbiological point of view. In FE analysis, a significantly higher peak of EQV strain (P=0.005) was found for EH (mean 3438.65 µ∑) compared to MT (mean 840.98 µ∑) connection. Conclusions: Varying implant-abutment connection type will result in diverse periimplant bone remodeling, regardless of clinical and microbiological conditions. This fact is more likely attributed to the singular loading transmission through different implant-abutment connections to the periimplant bone. The present findings suggest that Morse-taper connection is more efficient to prevent periimplant bone loss, compared to an external hexagon connection.