Les actions de groupes en géométrie symplectique et l'application moment

Ce mémoire porte sur quelques notions appropriées d'actions de groupe sur les variétés symplectiques, à savoir en ordre décroissant de généralité : les actions symplectiques, les actions faiblement hamiltoniennes et les actions hamiltoniennes. Une connaissance des actions de groupes et de la géométrie symplectique étant prérequise, deux chapitres sont consacrés à des présentations élémentaires de ces sujets. Le cas des actions hamiltoniennes est étudié en détail au quatrième chapitre : l'importante application moment y est définie et plusieurs résultats concernant les orbites de la représentation coadjointe, tels que les théorèmes de Kirillov et de Kostant-Souriau, y sont démontrés. Le dernier chapitre se concentre sur les actions hamiltoniennes des tores, l'objectif étant de démontrer le théorème de convexité d'Atiyha-Guillemin-Sternberg. Une discussion d'un théorème de classification de Delzant-Laudenbach est aussi donnée. La présentation se voulant une introduction assez exhaustive à la théorie des actions hamiltoniennes, presque tous les résultats énoncés sont accompagnés de preuves complètes. Divers exemples sont étudiés afin d'aider à bien comprendre les aspects plus subtils qui sont considérés. Plusieurs sujets connexes sont abordés, dont la préquantification géométrique et la réduction de Marsden-Weinstein.

REDUCTION OF TODA LATTICE HIERARCHY TO GENERALIZED KDV HIERARCHIES AND THE 2-MATRIX MODEL

Toda lattice hierarchy and the associated matrix formulation of the 2M-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which Abelianize the second KP Hamiltonian structure, we are able to obtain a unified formalism for the reduced SL(M + 1, M - k) KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded SL(M + 1, M - k) matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free field representations of the associated W(M, M - k) Poisson bracket algebras generalising the familiar nonlinear W-M+1 algebra. Discrete Backlund transformations for SL(M + 1, M - k) KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the SL(M + 1, 1) KdV hierarchy.

On wave action and phase in the non-canonical Hamiltonian formulation

The long time–evolution of disturbances to slowly–varying solutions of partial differential equations is subject to the adiabatic invariance of the wave action. Generally, this approximate conservation law is obtained under the assumption that the partial differential equations are derived from a variational principle or have a canonical Hamiltonian structure. Here, the wave action conservation is examined for equations that possess a non–canonical (Poisson) Hamiltonian structure. The linear evolution of disturbances in the form of slowly varying wavetrains is studied using a WKB expansion. The properties of the original Hamiltonian system strongly constrain the linear equations that are derived, and this is shown to lead to the adiabatic invariance of a wave action. The connection between this (approximate) invariance and the (exact) conservation laws of pseudo–energy and pseudomomentum that exist when the basic solution is exactly time and space independent is discussed. An evolution equation for the slowly varying phase of the wavetrain is also derived and related to Berry's phase.

A rescaling of the phase space for Hamiltonian map: Applications on the Kepler map and mappings with diverging angles in the limit of vanishing action

A rescale of the phase space for a family of two-dimensional, nonlinear Hamiltonian mappings was made by using the location of the first invariant Kolmogorov-Arnold-Moser (KAM) curve. Average properties of the phase space are shown to be scaling invariant and with different scaling times. Specific values of the control parameters are used to recover the Kepler map and the mapping that describes a particle in a wave packet for the relativistic motion. The phase space observed shows a large chaotic sea surrounding periodic islands and limited by a set of invariant KAM curves whose position of the first of them depends on the control parameters. The transition from local to global chaos is used to estimate the position of the first invariant KAM curve, leading us to confirm that the chaotic sea is scaling invariant. The different scaling times are shown to be dependent on the initial conditions. The universality classes for the Kepler map and mappings with diverging angles in the limit of vanishing action are defined. © 2013 Published by Elsevier Inc. All rights reserved.

Gauge-invariant actions from constraint Hamiltonian dynamics

Dirac's constraint Hamiltonian formalism is used to construct a gauge-invariant action for the massive spin-one and -two fields.

Hamiltonian formalism for path-dependent Lagrangians

A presymplectic structure for path-dependent Lagrangian systems is set up such that, when applied to ordinary Lagrangians, it yields the familiar Legendre transformation. It is then applied to derive a Hamiltonian formalism and the conserved quantities for those predictive invariant systems whose solutions also satisfy a Fokker-type action principle.

On Hamiltonian balanced dynamics and the slowest invariant manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.

Critical exponents for a transition from integrability to non-integrability via localization of invariant tori in the Hamiltonian system

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Scaling invariance of the diffusion coefficient in a family of two-dimensional Hamiltonian mappings

We consider a family of two-dimensional nonlinear area-preserving mappings that generalize the Chirikov standard map and model a variety of periodically forced systems. The action variable diffuses in increments whose phase is controlled by a negative power of the action and hence effectively uncorrelated for small actions, leading to a chaotic sea in phase space. For larger values of the action the phase space is mixed and contains a family of elliptic islands centered on periodic orbits and invariant Kolmogorov-Arnold-Moser (KAM) curves. The transport of particles along the phase space is considered by starting an ensemble of particles with a very low action and letting them evolve in the phase until they reach a certain height h. For chaotic orbits below the periodic islands, the survival probability for the particles to reach h is characterized by an exponential function, well modeled by the solution of the diffusion equation. On the other hand, when h reaches the position of periodic islands, the diffusion slows markedly. We show that the diffusion coefficient is scaling invariant with respect to the control parameter of the mapping when h reaches the position of the lowest KAM island. © 2013 American Physical Society.

Time-dependent properties in two-dimensional and Hamiltonian mappings

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Direct Visualization Of The Action Of Triton X-100 On Giant Vesicles Of Erythrocyte Membrane Lipids.

The raft hypothesis proposes that microdomains enriched in sphingolipids, cholesterol, and specific proteins are transiently formed to accomplish important cellular tasks. Equivocally, detergent-resistant membranes were initially assumed to be identical to membrane rafts, because of similarities between their compositions. In fact, the impact of detergents in membrane organization is still controversial. Here, we use phase contrast and fluorescence microscopy to observe giant unilamellar vesicles (GUVs) made of erythrocyte membrane lipids (erythro-GUVs) when exposed to the detergent Triton X-100 (TX-100). We clearly show that TX-100 has a restructuring action on biomembranes. Contact with TX-100 readily induces domain formation on the previously homogeneous membrane of erythro-GUVs at physiological and room temperatures. The shape and dynamics of the formed domains point to liquid-ordered/liquid-disordered (Lo/Ld) phase separation, typically found in raft-like ternary lipid mixtures. The Ld domains are then separated from the original vesicle and completely solubilized by TX-100. The insoluble vesicle left, in the Lo phase, represents around 2/3 of the original vesicle surface at room temperature and decreases to almost 1/2 at physiological temperature. This chain of events could be entirely reproduced with biomimetic GUVs of a simple ternary lipid mixture, 2:1:2 POPC/SM/chol (phosphatidylcholine/sphyngomyelin/cholesterol), showing that this behavior will arise because of fundamental physicochemical properties of simple lipid mixtures. This work provides direct visualization of TX-100-induced domain formation followed by selective (Ld phase) solubilization in a model system with a complex biological lipid composition.

Presynaptic Neuromuscular Action Of A Methanolic Extract From The Venom Of Rhinella Schneideri Toad.

Rhinella schneideri, previously known as Bufo paracnemis, is a common toad in many regions of Brazil. Its venom exerts important cardiovascular effects on humans and other animals. Although this toad venom has been the subject of intense investigations, little is known about its neuromuscular activity. The neurotoxicity of a methanolic extract of R. schneideri venom was tested on mouse phrenic nerve-diaphragm (PND) preparations mounted for conventional twitch tension recording - in response to indirect stimulation - and for electrophysiological measurements. Venom extract (50 μg/mL) increased the muscle twitch tension in PND preparations but did not significantly alter the resting membrane potential values. Electrophysiological evaluations showed that the extract (50 μg/mL) significantly augmented the frequency of miniature end-plate potential (from 38 ± 3.5 to 88 ± 15 after 60 minutes; n = 5; p < 0.05) and quantal content (from 128 ± 13 to 272 ± 34 after five minutes; n = 5; p < 0.05). Pretreatment with ouabain (1 μg/mL) for five minutes prevented the increase in quantal content (117 ± 18 and 154 ± 33 after five and 60 minutes, respectively). These results indicate that the methanolic extract of R. schneideri venom acts primarily presynaptically to enhance neurotransmitter release in mouse phrenic-diaphragm preparations.

Action Of Essential Oils From Brazilian Native And Exotic Medicinal Species On Oral Biofilms.

Essential oils (EO) obtained from twenty medicinal and aromatic plants were evaluated for their antimicrobial activity against the oral pathogens Candida albicans, Fusobacterium nucleatum, Porphyromonas gingivalis, Streptococcus sanguis and Streptococcus mitis. The antimicrobial activity of the EO was evaluates by microdilution method determining Minimal Inhibitory Concentration. Chemical analysis of the oils compounds was performed by Gas chromatography-mass spectrometry (CG-MS). The most active EO were also investigated as to their actions on the biolfilm formation. The most of the essential oils (EO) presented moderate to strong antimicrobial activity against the oral pathogens (MIC--Minimal Inhibitory Concentrations values between 0.007 and 1.00 mg/mL). The essential oil from Coriandrum sativum inhibited all oral species with MIC values from 0.007 to 0.250 mg/mL, and MBC/MFC (Minimal Bactericidal/Fungicidal Concentrations) from 0.015 to 0.500 mg/mL. On the other hand the essential oil of C. articulatus inhibited 63.96% of S. sanguis biofilm formation. Through Scanning Eletronic Microscopy (SEM) images no changes were observed in cell morphology, despite a decrease in biofilm formation and changes on biofilm structure. Chemical analysis by Gas Chromatography-Mass Spectrometry (GC-MS) of the C. sativum essential oil revealed major compounds derivatives from alcohols and aldehydes, while Cyperus articulatus and Aloysia gratissima (EOs) presented mono and sesquiterpenes. In conclusion, the crude oil from C. articulatus exhibited the best results of antimicrobial activity e ability to control biofilm formation. The chemical analysis showed the presence of terpenes and monoterpenes such as a-pinene, a-bulnesene and copaene. The reduction of biofilms formation was confirmed from SEM images. The results of this research shows a great potential from the plants studied as new antimicrobial sources.

[strategies For Tackling Absenteeism In Dental Appointments In The Family Health Units Of A Large Municipality: Action Research].

The aim of this study was to analyze the reasons for missed appointments in dental Family Health Units (FHU) and implement strategies to reduce same through action research. This is a study conducted in 12 FHUs in Piracicaba in the State of São Paulo from January, 1 to December, 31 2010. The sample was composed of 385 users of these health units who were interviewed over the phone and asked about the reasons for missing dental appointments, as well as 12 dentists and 12 nurses. Two workshops were staged with professionals: the first to assess the data collected in interviews and develop strategy, and the second for evaluation after 4 months. The primary cause for missed appointments was the opening hours of the units coinciding with the work schedule of the users. Among the strategies suggested were lectures on oral health, ongoing education in team meetings, training of Community Health Agents, participation in therapeutic groups and partnerships between Oral Health Teams and the social infrastructure of the community. The adoption of the single medical record was the strategy proposed by professionals. The strategies implemented led to a 66.6% reduction in missed appointments by the units and the motivating nature of the workshops elicited critical reflection to redirect health practices.

SOLITON SOLUTIONS TO SYSTEMS OF COUPLED SCHRODINGER EQUATIONS OF HAMILTONIAN TYPE

We study the existence of positive solutions of Hamiltonian-type systems of second-order elliptic PDE in the whole space. The systems depend on a small parameter and involve a potential having a global well structure. We use dual variational methods, a mountain-pass type approach and Fourier analysis to prove positive solutions exist for sufficiently small values of the parameter.