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Introdução: A perda dentária é um dos problemas de saúde bucal mais prevalentes população mundial, que acomete principalmente adultos e idosos. Objetivo: Determinar o estado de saúde oral e investigar a associação da condição de saúde bucal com variáveis sóciodemográficas de indivíduos com transtornos psiquiátricos. Método: Os pacientes eram assistidos em Centros de Atenção Psicossocial (GE) e assinaram um termo de Consentimento Livre e Esclarecido. A amostra do estudo foi de 78 pacientes psiquiátricos de idades entre 20 e 50 anos. Experiência com cárie, condições de higiene oral e periodontal foram avaliadas pelos índices CPOD, IPV (Índice de placa visível), ISG (sangramento gengival), respectivamente. Aplicou-se os testes qui-quadrado de Pearson, teste de Mann-Whitney e regressão logística na análise estatística. Como grupo controle (GC), foram recrutados pacientes que procuravam tratamento odontológico numa Unidade Básica de Saúde de Teresina-PI, os quais foram pareados segundo gênero e renda. Resultados: GC apresentou menos dentes hígidos (p=0,002). A diferença entre os grupos com relação ao CPOD foi significativa (p=0,004), especialmente no componente perdidos (p=0,004). A condição periodontal não foi significativa entre os grupos. Conclusão: Os pacientes psiquiátricos avaliados tem uma condição de saúde oral mais comprometida (número maior de dentes perdidos).

The aim of this in vitro study was evaluated the amount of color change and degree of conversion of the luting systems for ceramic veneer cementation. Forty-eight human incisors were randomly divided into six groups (n=8) according to the bonding procedure: RelyX Veneer/Single Bond 2; RelyX ARC/Single Bond 2; Filtek Z350 XT Flowable /Single Bond 2; Variolink Veneer/Tetric N-Bond; Variolink II/Tetric N-Bond; Tetric N-Flow/Tetric N-Bond. Ceramic veneers (IPS Empress Esthetic) were cemented using the different bonding procedures. Color differences of the ceramic surface after cementation and 300 h(150kJ/m2) of UV aging test, were examined with a colorimeter. Fourier transform infrared spectroscopy (FT-IR) was used to evaluate the degree of conversion for each luting system. Data were analyzed with one-way ANOVA and Tukey's test (α=0.05). All the luting systems demonstrated significant changes in color stability. The ΔE of the systems ranged from 6,11 to 9,33. Tetric N-Flow and RelyX Veneer showed the highest degree of conversion, while lowest values were found for RelyX ARC. All luting systems demonstrated clinically unacceptable (ΔE≥ 3.3) color changes. Despite to the dual cure of the RelyX ARC luting cement, it showed the lowest degree of conversion values, while Tetric N-Flow and RelyX Veneer presented the highest and similar degree of conversion.

Introdução: A perda dentária é considerada um problema de saúde pública que gera impactos negativos na qualidade de vida do indivíduo. Objetivo: Avaliar os fatores determinantes da perda dentária em usuários de um posto de saúde em Teresina - Piauí. Método: Foram entrevistados 384 usuários, de ambos os gêneros, com faixa etária de 15 a 60 anos de idade, através de um questionário. A amostra foi constituída por pessoas que haviam perdido pelo menos um elemento dentário. Resultados: Segundo o grau de escolaridade, predominaram os usuários com nível fundamental incompleto (47,92%), a faixa salarial mais prevalente foi entre 1 e 3 salários mínimos (68,23%) e aqueles que ganham menos de 1 salário mínimo (21,61%). Perguntou-se sobre a função dos dentes, a maioria (88,81%) respondeu triturar, mastigar, comer e morder alimentos. Em relação à forma de higienização bucal, 99,48% responderam que utilizavam escova e dentifrício, 45,31% responderam que utilizavam o fio dental, 69,53 % relataram uso de soluções para enxágue bucal e 60,16% responderam que utilizam o palito. Conclusão: Predomínio de usuários com nível fundamental incompleto e renda familiar de 1 a 3 salários mínimos. Percebe-se a necessidade de incentivo para propostas de promoção e autocuidado em saúde bucal.

Lyons and Sullivan have shown how to discretize harmonic functions on a Riemannian manifold $M$ whose Brownian motion satisfies a certain recurrence property called $\ast$-recurrence. We study analogues of this discretization for tensor fields which are harmonic in the sense of the covariant Laplacian. We show that, under certain restrictions on the holonomy of the connection, the lifted diffusion on the orthonormal frame bundle has the same $\ast$-recurrence property as the original Brownian motion. This observation permits us to reduce to the discretization of ordinary harmonic functions by a device called scalarization., Comment: 15 pages

The purpose of this paper is to compare a classical non-holonomic system---a sphere rolling against the inner surface of a vertical cylinder under gravity---and a class of discrete dynamical systems known as no-slip billiards in similar configurations. A well-known notable feature of the non-holonomic system is that the rolling sphere does not fall; its height function is bounded and oscillates harmonically up and down. The central issue of the present work is whether similar bounded behavior can be observed in the no-slip billiard counterpart. Our main results are as follows: for circular cylinders in dimension $3$, the no-slip billiard has the bounded orbits property, and very closely approximates rolling motion, for a class of initial conditions which we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to an overall downward acceleration. Considering cylinders with different cross-section shapes, we show that no-slip billiards between two parallel hyperplanes in Euclidean space of arbitrary dimension are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept that depends on a factorization of the motion into transversal and longitudinal components) has period two---a very common occurrence in planar no-slip billiards---the motion in the longitudinal direction, under no forces, is generically not bounded. This is shown using a formula for a longitudinal linear drift that we prove in arbitrary dimensions. While the systems for which we can prove the existence of bounded orbits have relatively simple transverse dynamics, we also briefly explore numerically a no-slip billiard system, namely the stadium cylinder billiard, that can exhibit chaotic transversal dynamics., Comment: 34 pages, 13 figures

Introduction: In restorative dentistry, the faceting of teeth with aesthetic compromise is considered an effective procedure in which the clinician must choose the ideal treatment and select appropriate materials to obtain satisfactory and predictable aesthetic results. In this sense, the use of the Advanced Polymerization System (APS) has advantages, such as aesthetics, durability, degree of conversion and minimal color change. Objective: Demonstrate the aesthetic-functional restoration using adhesive system and composite resin with APS polymerization technology. Case report: A female patient attended the Integrated Clinic of the Faculty of Dentistry of UFRJ with a complaint of blackening of teeth related to ceramic laminates. After clinical and radiographic examination, it was decided to use composite resin veneers. Results: The composite resin veneers associated with the adhesive system promoted an excellent aesthetic result. Conclusion: The improvement of direct restorations with acid etching techniques and the use of new adhesive systems, combined with the constant development and scientific improvement of composite resins, enabled dentists to perform restorations in the most diverse clinical indications. Introducción: En odontología restauradora, el tallado de dientes con compromiso estético se considera un procedimiento efectivo en el que el clínico debe elegir el tratamiento ideal y seleccionar los materiales adecuados para obtener resultados estéticos satisfactorios y predecibles. En este sentido, el uso del Sistema de Polimerización Avanzado (APS) tiene ventajas, como la estética, durabilidad, grado de conversión y mínimo cambio de color. Objetivo: Demostrar la restauración estético-funcional mediante sistema adhesivo y resina compuesta con tecnología de polimerización APS. Caso clínico: Paciente de sexo femenino que acudió a la Clínica Integrada de la Facultad de Odontología de la UFRJ por quejarse de ennegrecimiento de dientes relacionado con carillas cerámicas. Tras el examen clínico y radiográfico, se decidió utilizar carillas de resina compuesta. Resultados: Las facetas de resina compuesta asociadas con el sistema adhesivo promovieron un excelente resultado estético. Conclusión: La mejora de las restauraciones directas con técnicas de grabado ácido y el uso de nuevos sistemas adhesivos, combinado con el constante desarrollo y mejora científica de las resinas compuestas, permitió a los odontólogos realizar restauraciones en las más diversas indicaciones clínicas. Introdução: Na Odontologia Restauradora, o facetamento de dentes com comprometimento estético é considerado um procedimento eficaz no qual o clínico deve escolher o tratamento ideal e selecionar materiais apropriados para obter resultados estéticos satisfatórios e previsíveis. Nesse sentido, a utilização do sistema Advanced Polymerization System (APS) apresenta vantagens, como estética, perdurabilidade, grau de conversão e mínima alteração de cor. Objetivo: Demonstrar o restabelecimento estético-funcional utilizando sistema adesivo e resina composta com a tecnologia de polimerização APS. Relato de caso: Paciente do sexo feminino compareceu à Clínica Integrada da Faculdade de Odontologia da UFRJ com queixa do escurecimento dos dentes referente aos laminados cerâmicos. Após exame clínico e radiográfico, fez-se a opção por utilizar facetas em resina composta. Resultados: As facetas de resina composta associadas ao sistema adesivo promoveu excelente resultado estético. Conclusão: O aprimoramento das restaurações diretas com técnicas de condicionamento ácido e utilização de novos sistemas adesivos, aliados ao constante desenvolvimento e aperfeiçoamento científico das resinas compostas, possibilitaram ao cirurgião dentista realizar restaurações nas mais diversas indicações clínicas.

We provide stochastic foundations for the analysis of a class of reaction–diffusion systems using as an example the known Temporal Analysis of Products (TAP) experiments, showing how to effectively obtain explicit solutions to the associated equations by approximating the 3-dimensional domain of diffusion U (the reactor) by 1-dimensional network models. In a typical TAP experiment a pulse of reactant gas of species A is injected into U , which is filled with chemically inert material, permeable to gas diffusion. Particles of catalyst are placed amid this inert medium, forming active sites where the reaction A → B may occur. On part of the boundary of U designated as the exit, a mixture of A and B can escape. We study the problem of determining the (molar) fraction of product gas in the mixture after U is fully evacuated. This fraction is identified with the reaction probability –that is, the probability of a single diffusing molecule reacting before leaving U . Specifically, we are interested in how this probability depends on the reaction rate constant k . After giving a stochastic formulation of the problem and the boundary value problem whose solution is this probability, we study a class of domains, called fat graphs , comprising a network of thin tubes with active sites at junctures. The main result of the paper is that in the limit, as the thin tubes approach curves in 3-dimensional space, reaction probability converges to functions of the point of gas injection that can be computed explicitly in terms of a rescaled parameter k . By this 3D to 1D reduction, the simpler processes on metric graphs can be used as model systems for more realistic 3-dimensional configurations. This is illustrated with analytic and numerical examples. One example, in particular, illustrates an important application of our method: finding the catalyst configuration that maximizes reaction probability at a given reaction rate constant.

Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive (collision) forces. When it is desired or necessary to account for linear/angular momentum exchange in collisions involving a spherical body, a type of billiard system often referred to as no-slip has been used. In recent work, it has become apparent that no-slip billiards resemble non-holonomic mechanical systems in a number of ways. Based on an idea by Borisov, Kilin and Mamaev, we show that no-slip billiards very generally arise as limits of non-holonomic (rolling) systems, in a way that is akin to how ordinary billiards arise as limits of geodesic flows through a flattening of the Riemannian manifold., Comment: 27 pages, 11 figures

Made available in DSpace on 2022-04-29T08:45:29Z (GMT). No. of bitstreams: 0 Previous issue date: 2020-01-01 To evaluate the effect of accelerated artificial aging (AAA) on chemical and physical properties of four different resin cements and two flowable resins. Forty-eight human incisors were prepared and the IPS Empress Esthetic laminate veneers were fabricated. The samples were randomly divided into the groups: dual-polymerizing cements (RelyX ARC and Variolink II), light-polymerizing cements (RelyX Veneer and Variolink Veneer) or flowable composite resins (Filtek Z350 Flow and Tetric N-Flow). The degree of conversion was determined in a Fourier transform infrared spectroscopy. The first cylinder was measuring the initial bond strength and, the remaining cylinder was submitted to AAA before bond strength test. The fracture pattern was analyzed by stereomicroscopy. The results were analyzed by one-way and two-way split-spot ANOVA/ Tukey's test (α =0.05). There were clinically perceptible color changes for all materials. The degree of conversion, the following decreasing sequence was found: RelyX ARC> Variolink II> RelyX Veneer > Filtek Z350 Flow > Variolink Veneer = Tetric N-Flow. For the microshear bond strength, RelyX ARC = Variolink II > Variolink Veneer = RelyX Veneer > flowable composite resins. AAA induced color change and decreased microshear bond strength for all luting systems evaluated. Department of Restorative Dentistry Piracicaba Dental School State University of Campinas Department of Dentistry Faculty of Sciences of Health Federal University of Jequitinhonha and Mucuri Valley Department of Biological Sciences Bauru School of Dentistry University of São Paulo Department of Dental Materials and Prosthesis Paulista State University Alberto Luiz Coimbra Institute for Graduate Studies and Engineering Research Federal University of Rio de Janeiro Department of Dental Clinic Federal University of Piauí Department of Pediatric Dentistry and Orthodontics Federal University of Rio de Janeiro Department of Dental Clinic Federal University of Rio de Janeiro Department of Dental Materials and Prosthesis Paulista State University

We introduce a class of random mechanical systems called random billiards to study the problem of quantifying the irreversibility of nonequilibrium macroscopic systems. In a random billiard model, a point particle evolves by free motion through the interior of a spatial domain, and reflects according to a reflection operator, specified in the model by a Markov transition kernel, upon collision with the boundary of the domain. We derive a formula for entropy production rate that applies to a general class of random billiard systems. This formula establishes a relation between the purely mathematical concept of entropy production rate and textbook thermodynamic entropy, recovering in particular Clausius' formulation of the second law of thermodynamics. We also study an explicit class of examples whose reflection operator, referred to as the Maxwell-Smolukowski thermostat, models systems with boundary thermostats kept at possibly different temperatures. We prove that, under certain mild regularity conditions, the class of models are uniformly ergodic Markov chains and derive formulas for the stationary distribution and entropy production rate in terms of geometric and thermodynamic parameters., Comment: 30 pages, 9 figures

We consider random flights of point particles inside $n$-dimensional channels of the form $\mathbb{R}^{k} \times \mathbb{B}^{n-k}$, where $\mathbb{B}^{n-k}$ is a ball of radius $r$ in dimension $n-k$. The particle velocities immediately after each collision with the boundary of the channel comprise a Markov chain with a transition probabilities operator $P$ that is determined by a choice of (billiard-like) random mechanical model of the particle-surface interaction at the "microscopic" scale. Our central concern is the relationship between the scattering properties encoded in $P$ and the constant of diffusivity of a Brownian motion obtained by an appropriate limit of the random flight in the channel. Markov operators obtained in this way are {\em natural} (definition below), which means, in particular, that (1) the (at the surface) Maxwell-Boltzmann velocity distribution with a given surface temperature, when the surface model contains moving parts, or (2) the so-called Knudsen cosine law, when this model is purely geometric, is the stationary distribution of $P$. We show by a suitable generalization of a central limit theorem of Kipnis and Varadhan how the diffusivity is expressed in terms of the spectrum of $P$ and compute, in the case of 2-dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2)., 39 pages, 11 images

We explore some beginning steps in stochastic thermodynamics of billiard-like mechanical systems by introducing extremely simple and explicit random mechanical processes capable of exhibiting steady-state irreversible thermodynamical behavior. In particular, we describe a Markov chain model of a minimalistic heat engine and numerically study its operation and efficiency.

Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads to the study of no-slip billiards--planar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change at each collision with the boundary of the billiard domain. In this paper we greatly extend previous results on boundedness of orbits (Broomhead and Gutkin) and linear stability of periodic orbits for a Sinai-type billiard (Wojtkowski) for no-slip billiards. We show among other facts that: (i) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than $\pi$, no-slip billiard dynamics will always contain elliptic period-$2$ orbits; (ii) polygonal no-slip billiards always admit small invariant open sets and thus cannot be ergodic with respect to the canonical invariant billiard measure; (iii) the no-slip version of a Sinai billiard must contain linearly stable periodic orbits of period $2$ and, more generally, we provide a curvature threshold at which a commonly occurring period-$2$ orbit shifts from being hyperbolic to being elliptic; (iv) finally, we make a number of observations concerning periodic orbits in a class of polygonal billiards., Comment: 32 pages, 20 figures

A random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post-collision distribution of velocities for any given pre-collision velocity. We consider such collision operators for certain random billiards that we call billiards with microstructure. Collisions modeled by these operators can still be thought of as elastic and time reversible. The operators are canonically determined by a second (deterministic) billiard system that models “microscopic roughness” on the billiard table boundary. Our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards. Among the main results, we give geometric conditions for these operators to be Hilbert-Schmidt and relate their spectrum and speed of convergence to stationary Markov chains with geometric features of the microscopic billiard structure. The relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples.

Random billiards are billiard dynamical systems for which the reflection law giving the post-collision direction of a billiard particle as a function of the pre-collision direction is specified by a Markov (scattering) operator P. Billiards with microstructure are random billiards whose Markov operator is derived from a “microscopic surface structure” on the boundary of the billiard table. The microstructure in turn is defined in terms of what we call a billiard cellQ, the shape of which completely determines the operator P. This operator, defined on an appropriate Hilbert space, is bounded self-adjoint and, for the examples considered here, a Hilbert-Schmidt operator. A central problem in the statistical theory of such random billiards is to relate the geometric characteristics of Q and the spectrum of P. We show, for a particular family of billiard cell shapes parametrized by a scale invariant curvature K (Fig. 2), that the billiard Laplacian P−I is closely related to the ordinary spherical Laplacian, and indicate, by partly analytical and partly numerical means, how this provides asymptotic information about the spectrum of P for small values of K. It is shown, in particular, that the second moment of scattering about the incidence angle closely approximates the spectral gap of P.

We obtain a general formula for reactant conversion in diffusion–reaction TAP systems over single non-porous catalyst particles as a product of two terms: α = P H × K . The first term, P H , is a purely geometric factor dependent only on the shape of the reactor and the shape and position of the catalyst particle, interpreted as the probability that an individual molecule of reactant will hit the catalyst before leaving the reactor. The second term, K = k τ H / ( 1 + k τ H ) , is a geometrical/chemical term involving the kinetic constant k and a transport characteristic (residence time in the catalyst zone, τ H ). Both P H and τ H can be effectively calculated for any given reactor-particle configuration. Our formula greatly extends the validity of a formula given in Shekhtman et al. [1999. Thin-zone TAP-reactor—theory and application. Chem. Eng. Sci. 54, 4371–4378] for thin-zone TAP-systems. It is derived by a probabilistic analysis of the residence time of individual molecular trajectories in chemically active regions. Our results are based on the theory previously developed in our paper [Feres, R., Yablonsky, G.S., Mueller, A., Baernstein, A., Zheng, X., Gleaves, J., 2009. Probabilistic analysis of transport-reaction processes over catalytic particles: theory and experimental testing. Chem. Eng. Sci. 64, 568–581].

Let (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

We developed a probabilistic theory of transport–reaction processes over various types of configurations of catalytic particles, particularly for a single particle. The theory describes a single pulse–response experiment which is similar to the single pulse temporal analysis of products (TAP) experiment. The analysis is based on the general theory of Brownian motion with “killing” and the Feynman–Kac formula. Different diffusion–reaction problems, particularly the problem “diffusion–very fast reaction” (infinite rate reaction), have been analyzed. In the latter case, the probability for a reactant to be converted equals a purely geometric characteristic, namely, the probability for a reactant molecule to hit at least one catalyst particle in a configuration of particles. Solving a boundary value problem, the probability of conversion can be found as a function of the apparent kinetic parameter. Based on experimental data on exit flow and conversion this parameter can be extracted. Experimental data in studies of CO oxidation over a single particle of Pt catalyst show a qualitative agreement with data from computational experiments based on the developed probabilistic theory. For two-particle catalyst configurations, it was found computationally a non-trivial dependence of the reactant conversion on some geometric characteristics, especially the distance between particles. A distance was found for which conversion is highest.

A computational procedure is developed for determining the conversion probability for reaction-diffusion systems in which a first-order catalytic reaction is performed over active particles. We apply this general method to systems on metric graphs, which may be viewed as 1-dimensional approximations of 3-dimensional systems, and obtain explicit formulas for conversion. We then study numerically a class of 3-dimensional systems and test how accurately they are described by model formulas obtained for metric graphs. The optimal arrangement of active particles in a 1-dimensional multiparticle system is found, which is shown to depend on the level of catalytic activity: conversion is maximized for low catalytic activity when all particles are bunched together close to the point of gas injection, and for high catalytic activity when the particles are evenly spaced.

Continuing the study begun in [Feres, R., Yablonsky, G., 2004. Knudsen's cosine law and random billiards. Chemical Engineering Science 59, 1541–1556], we consider transport in the Knudsen regime of inert gases through straight channels and investigate how small-scale surface geometry of a macroscopically flat channel affects the diffusion characteristics of the gas. We show that the diffusivity constant contains information about the surface micro-geometry. Our investigation is carried out partly by analytical means and also through numerical simulation of what we call time-of-escape experiment . This is a type of multi-scattering experiment that measures the mean residence time in channels of varying lengths, allowing for an analysis of the transport process at various stages of development. We focus attention on the smoothness constant ξ = D / D 0 , defined as the ratio of diffusivity for a given micro-geometry, D , and diffusivity D 0 under “ideal roughness,” i.e., under the assumption that the Knudsen cosine law holds at each collision. The dependence of ξ on small-scale surface morphology is investigated using a variety of parametrized families of micro-geometries. We show that ξ can tell the presence of certain geometric features, such as curvature or sharp angles at the microscopic level. For example, for a simple two-dimensional model of atomically smooth surface made of a linear packing of spheres of radius R probed by gas molecules of radius A (Fig. 6), we obtain the approximate relation ξ ∝ 1 + A / R , with a proportionality constant of roughly 1.3. (See Section 2 for the range of parameters considered.) This relates to rapid diffusion phenomena recently observed by [Holt, J.K., Park, H.G., Wang, Y., Stadermann, M., Artyukhin, A.B., Grigoropoulos, C.P., Noy, A., Bakajin, O., 2006. Fast mass transport through sub-2-nanometer carbon nanotubes. Science 312, (1034)]. In the classical collision model of Maxwell–Smoluchowski, with no detailed regard of surface geometry, one always has ξ ⩾ 1 , but we observe that ξ can sometimes be less than 1 (Section 2.5). We obtain exact values for the mean number and duration of collisions as function of the micro-geometry (Section 3). This is needed to determine the relative importance of these quantities on diffusivity. We also extend the analysis of the scattering operator of random billiards of [Feres, R., Yablonsky, G., 2004. Knudsen's cosine law and random billiards. Chemical Engineering Science 59, 1541–1556] by giving a general criterion for the validity of Knudsen's cosine law, and describe some of the spectral properties of this operator.

This article lays the groundwork for understanding the details of gas transport in the so-called Knudsen regime, particularly during the early, nonstationary stages of the transport process. A simple mathematical model of gas–surface interaction is introduced, which is both fairly general and sufficiently explicit to permit a detailed analysis. The model, which we call a random billiard, is an example of a random dynamical system based on the idea of a billiard system with geometric microstructure. Among the main results, we show that Knudsen's cosine law is a stationary probability distribution of post-collision velocities, although not always the unique one. Through numerical experiments with a variety of microgeometries we also provide some insight concerning the approach to stationarity.

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold without boundary is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

We show how some basic dynamical ideas can be brought to bear on the study of Cartan geometries. In our main results, we give conditions for certain types of Cartan geometries to have constant curvature. We also consider ergodic actions of `higher-rank' semisimple groups on bundles supporting (not necessarily invariant) Cartan connections and show that these are `standard locally homogeneous' actions provided that some noncompact 1-parameter subgroup `preserves' a Cartan geometry.

The main result proved here is a topological version of Zimmer's cocycle superrigidity theorem. Several applications are also given which concern lattice actions, rigid geometric structures, and the cohomology of actions semisimple groups of higher rank with coefficients in vector-bundles.

Let / be a C°° Anosov diffeomorphism of a compact manifold M, preserving a smooth measure. If / satisfies the |pinching assumption defined below, it must preserve a continuous affine connection for which the leaves of the Anosov foliations are totally geodesic, geodesically complete, and flat (its tangential curvature is defined along individual leaves). If this connection, which is the unique /-invariant affine connection on M, is Cr-differentiable, r > 2, then / is conjugate via a Cr+2-affine diffeomorphism to a hyperbolic automorphism of a geodesically complete flat manifold. If / preserves a smooth symplectic form, has C3 Anosov foliations, and satisfies the 2 : 1-nonresonance condition (an assumption that is weaker than pinching), then / is C°° conjugate to a hyperbolic automorphism of a complete flat manifold. (In the symplectic case, the invariant connection is the one previously defined by Kanai in the context of geodesic flows.) If the foliations are C2 and the holonomy pseudo-groups satisfy a certain growth conditipn, the same conclusion holds.

By a random billiard we mean a billiard system in which the standard rule of specular reflection is replaced with a Markov transition probabilities operator P that gives, at each collision of the billiard particle with the boundary of the billiard domain, the probability distribution of the post-collision velocity for a given pre-collision velocity. A random billiard with microstructure, or RBM for short, is a random billiard for which P is derived from a choice of geometric/mechanical structure on the boundary of the billiard domain, as explained in the text. Such systems provide simple and explicit mechanical models of particle–surface interaction that can incorporate thermal effects and permit a detailed study of thermostatic action from the perspective of the standard theory of Markov chains on general state spaces. The main focus of this paper is on the operator P itself and how it relates to the mechanical and geometric features of the microstructure, such as mass ratios, curvatures, and potentials. The main results are as follows: (1) we give a characterization of the stationary probabilities (equilibrium states) of P and show how standard equilibrium distributions studied in classical statistical mechanics such as the Maxwell–Boltzmann distribution and the Knudsen cosine law arise naturally as generalized invariant billiard measures; (2) we obtain some of the more basic functional theoretic properties of P, in particular that P is under very general conditions a self-adjoint operator of norm 1 on a Hilbert space to be defined below, and show in a simple but somewhat typical example that P is a compact (Hilbert–Schmidt) operator. This leads to the issue of relating the spectrum of eigenvalues of P to the geometric/mechanical features of the billiard microstructure; (3) we explore the latter issue, both analytically and numerically in a few representative examples. Additionally, (4) a general algorithm for simulating the Markov chains is given based on a geometric description of the invariant volumes of classical statistical mechanics. Our description of these volumes may have independent interest.

This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators P_h, depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that (P_h -I)/h converges for small h to a second order elliptic differential operator L on compactly supported functions and that the Markov chain process associated to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L are selfadjoint (densely) defined on the space L2(H,{\eta}) of square-integrable functions over the (lower) half-space H in R^m, where {\eta} is a stationary measure. This measure's density is either (post-collision) Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes., Comment: 34 pages, 13 figures

We consider actions of lattices in certain higher rank simple Lie groups by affine (i.e. connection-preserving) transformations of a compact Riemannian manifold. When the dimension of the manifold is not too large, such actions are partially described here in terms of affine actions on the flat torus and isometric actions. The main tools are Marguils' and Zimmer's rigidity theorems.

Given an affine (i.e. connection-preserving) diffeomorphismf of a Riemannian manifoldM, we consider itscenter foliation, N, comprised by the directions that neither expand nor contract exponentially under the action generated byf. The main remarks made here (Corollary 3 and Theorem 7) are: There exists a metric compatible with the Levi-Civita connection for which the universal cover ofM decomposes isometrically as the Riemannian product of the universal cover of a leaf ofN (these covers are all isometric) and the Euclidean space; and ifN is one-dimensional,M is flat and the foliation is (up to finite cover) the fiber foliation of a Riemannian submersion onto a flat torus.

We consider connection-preserving actions of lattice subgroups ofSL nℝ on compact Riemannian manifolds, forn≥3. The main result gives a description of such actions, when dimM=n+1, as follows: Either the action preserves a smooth invariant Riemannian metric onM or it can be described in terms of linear actions on then-torus.

Phase reduction is a commonly used techinque for analyzing stable oscillators, particularly in studies concerning synchronization and phase lock of a network of oscillators. In a widely used numerical approach for obtaining phase reduction of a single oscillator, one needs to obtain the gradient of the phase function, which essentially provides a linear approximation of isochrons. In this paper, we extend the method for obtaining partial derivatives of the phase function to arbitrary order, providing higher order approximations of isochrons. In particular, our method in order 2 can be applied to the study of dynamics of a stable oscillator subjected to stochastic perturbations, a topic that will be discussed in a future paper. We use the Stuart-Landau oscillator to illustrate the method in order 2.

We improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.