A Sampling Theorem for Rotation Numbers of Linear Processes in R2{\R}^{2}

We prove an ergodic theorem for the rotation number of the composition of a sequence os stationary random homeomorphisms in $S^{1}$. In particular, the concept of rotation number of a matrix $g\in Gl^{+}(2,{\R})$ can be generalized to a product of a sequence of stationary random matrices in $Gl^{+}(2,{\R})$. In this particular case this result provides a counter-part of the Osseledec's multiplicative ergodic theorem which guarantees the existence of Lyapunov exponents. A random sampling theorem is then proved to show that the concept we propose is consistent by discretization in time with the rotation number of continuous linear processes on ${\R}^{2}.

Lyapunov exponents for stochastic differential equations on semi-simple Lie groups

summary:With an intrinsic approach on semi-simple Lie groups we find a Furstenberg–Khasminskii type formula for the limit of the diagonal component in the Iwasawa decomposition. It is an integral formula with respect to the invariant measure in the maximal flag manifold of the group (i.e. the Furstenberg boundary $B=G/MAN$). Its integrand involves the Borel type Riemannian metric in the flag manifolds. When applied to linear stochastic systems which generate a semi-simple group the formula provides a diagonal matrix whose entries are the Lyapunov spectrum. Some Brownian motions on homogeneous spaces are discussed

Wiener integral in the space of sequences of real numbers

summary:Let $i:H\rightarrow W$ be the canonical Wiener space where $W$={$\sigma :[0,T]\rightarrow {R}$ continuous with $\sigma \left( 0\right) =0\rbrace$, $H$ is the Cameron-Martin space and $i$ is the inclusion. We lift a isometry $H\rightarrow l_{2}$ to a linear isomorphism $\Phi :W\rightarrow {\cal V}\subset {R}^{\infty }$ which pushes forward the Wiener structure into the abstract Wiener space (AWS) $i:l_{2}\rightarrow {\cal V}$. Properties of the Wiener integration in this AWS are studied

Asymptotic angular stability in non-linear systems: rotation numbers and winding numbers

Abstract. The asymptotic angular stability of a dynamical system may be quantified by its rotation number or its winding number. These two quantities are shown to result from different assumptions, made about the flow generating the Poincare ´ map which results from the sequence of homeomorphisms in S l. An ergodic theorem of existence a.s. of the rotation number for non-linear systems is given. The advantages and disadvantages of both the rotation and winding numbers are discussed. Numerical calculations of the distribution of rotation number and winding number arising from different initial conditions are presented for three different chaotic maps

Qualidade de vida, ponto de partida ou resultado final? Quality of life, starting point or final result?

O que é qualidade de vida e o quanto podemos medir dela? Pensa-se em qualidade de vida como resultado das políticas públicas e desenvolvimento de uma sociedade, onde os determinantes socioambientais se manifestam como atributo de seus atores. Ao mesmo tempo, pode-se entender esta idéia no outro extremo da análise, a partir da percepção de uma população protagonista de sua realidade, do que vem a ser qualidade de vida segundo ela mesma. Partindo-se dos aspectos conceituais de qualidade de vida, passou-se a adotar os conceitos de diferenciais intra-urbanos como a melhor maneira de caracterizar os desajustes e as desigualdades urbanas, para assim entender os componentes da iniqüidade desse meio. A primeira iniciativa marcou a utilização do método genebrino ou distancial. Hoje, já na segunda versão desse método, incorporou-se a esse contexto outras metodologias que possibilitam maior consistência de análise para ampliar a validade dessas medições. Soma-se a esse contexto, a análise de cluster e o Sistema de Informações Geográficas, tanto no cenário intra-urbano, quanto intermunicipal.<br>What is quality of life and how much of it can be measured? Quality of life is thought of as the result of public policies and the development of a given society, where social-environmental determinants present themselves as attributes of its stakeholders. At the same time, this idea can be understood from the other end of the analyses, from the standpoint of a given population acting out its own reality, what it considers as being quality of life. Starting from the conceptual aspects of the scenarios that express quality of life, concepts about intra-urban differential aspects were adopted as the best way to characterise urban inadequacies and inequalities for a better understanding of the components of the inequities existing in this environment. Initially, the genebrino or distance method was adopted. At present, in the second version of this method, the context has been enriched by other methodologies that provide more consistency to the analysis, thus increase the validity of the measurements. Cluster Analysis and Geographic Information System were also added to this process, both in the intra-urban and multi-centric contexts