Human upright posture exhibits an everlasting oscillatory behavior of complex nature, called as human postural sway. Variations in the position of the Center-of-Pressure (CoP) were used to describe the human postural sway. In this study / CoP data, which has experimentally been collected from 28 different subjects (14 males and 14 females with their ages ranging from 6 to 84), who were divided into 4 groups according to their ages has been analyzed. The data collection from each of the subjects was performed in 5 successive trials, each of which has lasted for 180-seconds long. Linear analysis methods such as the variance/standard deviation, Fast Fouri&eacute / r Transformation, and Power Spectral Density estimates were applied to the detrended CoP signal of human postural sway. Also the Run test and Ensemble averages methods were used to search for stationarity and ergodicity of the CoP signal respectively. Furthermore, in order to reveal the nonlinear characteristics of the human postural sway, its dynamics were reconstructed in m-dimensional state space from the CoPx signals. Then, the correlation dimension (D2) estimates from the embedded dynamics were calculated. Additionally, the statistical and dynamical measures computed were checked against any significant changes, which may occur during aging. The results of the study suggested that human postural sway is a stationary process when 180-second long biped quiet stance data is considered. In addition, it exhibits variable dynamical structure complex in nature (112 deterministic chaos versus 28 stochastic time series of human postural sway) for five successive trials of 28 different subjects. Moreover, we found that groups were significantly different in the correlation dimension (D2) measure (p&amp / #8804 / 0.0003). Finally, the behavior of the experimental CoPx signals was checked against two types of linear processes by using surrogate data method. The shuffled CoPx signals (Surrogate I) suggested that temporal order of CoPx is important / however, phase-randomization (Surrogate II) did not change the behavioral characteristics of the CoPx signal.

<p>We consider stochastic hybrid systems that stem from evolution equations with right-hand sides that stochastically switch between a given set of right-hand sides. To begin our study, we consider a linear ordinary differential equation whose right-hand side stochastically switches between a collection of different matrices. Despite its apparent simplicity, we prove that this system can exhibit surprising behavior.</p><p>Next, we construct mathematical machinery for analyzing general stochastic hybrid systems. This machinery combines techniques from various fields of mathematics to prove convergence to a steady state distribution and to analyze its structure.</p><p>Finally, we apply the tools from our general framework to partial differential equations with randomly switching boundary conditions. There, we see that these tools yield explicit formulae for statistics of the process and make seemingly intractable problems amenable to analysis.</p> / Dissertation

In this thesis, we study two problems. The first one is the renorming problem in Banach lattices. We state the problem and give some known results related to it. Then we pass to construct a positive doubly power bounded operator with a nonpositive inverse on an infinite dimensional AL-space which generalizes the result of [10]. The second problem is related to the mean ergodicity of positive operators on KBspaces. We prove that any positive power bounded operator T in a KB-space E which satisfies lim n!1 dist1 n n&amp / #8722 / 1 Xk=0 Tkx, [&amp / #8722 / g, g] + BE= 0 (8x 2 E, kxk 1), () where BE is the unit ball of E, g 2 E+, and 0 &lt / 1, is mean ergodic and its fixed space Fix(T) is finite dimensional. This generalizes the main result of [12]. Moreover, under the assumption that E is a -Dedekind complete Banach lattice, we prove that if, for any positive power bounded operator T, the condition () implies that T is mean ergodic then E is a KB-space.

<div>Markov jump processes are continuous-time stochastic processes widely used in a variety of applied disciplines. Inference typically proceeds via Markov chain Monte Carlo (MCMC), the state-of-the-art being a uniformization-based auxiliary variable Gibbs sampler. This was designed for situations where the process parameters are known, and Bayesian inference over unknown parameters is typically carried out by incorporating it into a larger Gibbs sampler. This strategy of sampling parameters given path, and path given parameters can result in poor Markov chain mixing.</div><div><br></div><div>In this thesis, we focus on the problem of path and parameter inference for Markov jump processes.</div><div><br></div><div>In the first part of the thesis, a simple and efficient MCMC algorithm is proposed to address the problem of path and parameter inference for Markov jump processes. Our scheme brings Metropolis-Hastings approaches for discrete-time hidden Markov models to the continuous-time setting, resulting in a complete and clean recipe for parameter and path inference in Markov jump processes. In our experiments, we demonstrate superior performance over Gibbs sampling, a more naive Metropolis-Hastings algorithm we propose, as well as another popular approach, particle Markov chain Monte Carlo. We also show our sampler inherits geometric mixing from an ‘ideal’ sampler that is computationally much more expensive.</div><div><br></div><div>In the second part of the thesis, a novel collapsed variational inference algorithm is proposed. Our variational inference algorithm leverages ideas from discrete-time Markov chains, and exploits a connection between Markov jump processes and discrete-time Markov chains through uniformization. Our algorithm proceeds by marginalizing out the parameters of the Markov jump process, and then approximating the distribution over the trajectory with a factored distribution over segments of a piecewise-constant function. Unlike MCMC schemes that marginalize out transition times of a piecewise-constant process, our scheme optimizes the discretization of time, resulting in significant computational savings. We apply our ideas to synthetic data as well as a dataset of check-in recordings, where we demonstrate superior performance over state-of-the-art MCMC methods.</div><div><br></div>

Recent missions to solar system comets, such as ESA's Rosetta mission, raise interest for models and descriptions of their plasma environment. The interaction with various space phenomena such as stellar wind make the construction of an analytical description difficult. Instead, a simplified view of the comet environment is considered where the effects of magnetism and departures from radial symmetry are neglected. This is done in an effort to construct an approximation of the comet plasma behaviour later to be compared against observational accounts to find which plasma features are dependent on more complex phenomena and which plasma features arise as a result of the simpler comet view. Several attempts are made to construct an analytical description of comet plasma as based on the description within another branch of plasma physics: fusion. Previous work regarding the vacuum expansion of plasma after a stationary target is rapidly ablated via high-intensity lasers appears promising for adaptation to the comet environment. Before the comet environment can be considered the different natures of the two problems have to be considered. For example, the comet case is a stationary expansion problem as opposed to fast-ignition fusion where the expansion is treated as an initial value problem. Having accounted for the problems' inherent differences, a few methods are proposed to convert solutions of lab fusion distribution functions to the comet case. Additionally, a numerical approach to calculate the distribution function of comet electrons is presented employing ergodic invariance. Lastly, a toy-model simulation of the timescale for variations in the potential show that the error in the ergodic invariance may in practice have a faster convergent timescale dependence than theoretical bounds suggest. Optimistically, this suggest the possibility of future use in numerical attempts at modelling comet plasma.

We consider the incompressible 2D Navier-Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and extremely degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a uniquely ergodic and exponentially mixing quasi-periodic invariant measure. The result is true for any value of the viscosity ν > 0. By utilizing this quasi-periodic invariant measure, we show the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes. Estimates of the corresponding rate of convergence are also obtained, which is the same as in the time homogeneous case for the strong law of large numbers, while the convergence rate in the central limit theorem depends on the Diophantine approximation property on the quasi-periodic frequency and the mixing rate of the quasi-periodic invariant measure. We also prove the existence of a stable quasi-periodic solution in the laminar case (when the viscosity is large). The scheme of analyzing the statistical behavior of the time inhomogeneous solution process by the quasi-periodic invariant measure could be extended to other inhomogeneous Markov processes.

No description available.

[pt] Em meados do século XIX, G. Boole mostrou que a transformação x -> x − 1/x, definida em R − {0}, preserva a medida de Lebesgue (Ble). Mais de um século depois, R. Adler e B.Weiss mostraram que essa aplicação, chamada de transformação de Boole, é, de fato, ergódica com respeito à medida de Lebesgue (Adl). Nesse trabalho, apresentaremos o conceito de sistemas alternantes, definido recentemente por S. Muñoz (Mun), que consiste numa grande classe de aplicações na reta que generaliza a transformação de Boole e que torna possível uma análise abrangente de propriedades como transitividade robusta e ergodicidade. Para mostrar que, sob certas condições, sistemas alternantes são ergódicos com relação à medida de Lebesgue, mostraremos, usando o Teorema do Folclore, que a transformação induzida do sistema alternante é ergódica. / [en] In the middle of the 19th century, G. Boole proved that the transformation x -> x − 1/x, defined on R − {0}, is a Lebesgue measure preserving transformation (Ble). Over one hundred years later, R. Adler and B.Weiss proved that this map, called Boole`s map, is, in fact, ergodic with respect to the Lebesgue measure (Adl). In this work, we present the notion of alternating systems, recently introduced by S. Mu`noz (Mun), which is a large class of functions on the real line that generalizes the Boole`s map and allows us to make a wide analysis on certain properties such as robust transitivity and ergodicity. In order to show that, under certain conditions, alternating systems are ergodic with respect to the Lebesgue measure, we show, using the Folklore Theorem, that the induced transformation of an alternating system is ergodic.

No description available.

The concept of LR-nets was introduced and investigated firstly by H.P. Lotz in [27] and by F. Raebiger in [30]. Therefore we call such nets Lotz-Raebiger nets, shortly LR-nets. In this thesis we treat two problems on asymptotic behavior of these operator nets. First problem is to generalize well known theorems for Ces`aro averages of a single operator to LR-nets, namely to generalize the Eberlein and Sine theorems. The second problem is related to the strong convergence of Markov LR-nets on L1-spaces. We prove that the existence of a lower-bound functions is necessary and sufficient for asymptotic stability of LR-nets of Markov operators.