Time scale separation and dynamic heterogeneity in the low temperature East model

We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbor is 1. We focus on the glassy effects caused by the kinetic constraint as $q\downarrow 0$, where $q$ is the equilibrium density of the 0's. In the physical literature this limit is equivalent to the zero temperature limit. We first prove that, for any given $L=O(1/q)$, the divergence as $q\downarrow 0$ of three basic characteristic time scales of the East process of length $L$ is the same. Then we examine the problem of dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale $L=O(q^{-\gamma})$, $\gamma<1$, we show that the characteristic time scale of two East processes of length $L$ and $\lambda L$ respectively are indeed separated by a factor $q^{-a}$, $a=a(\gamma)>0$, provided that $\lambda \geq 2$ is large enough (independent of $q$, $\lambda=2$ for $\gamma<1/2$). In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form $111..10$, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. A key result for this part is a very precise computation of the relaxation time of the chain as a function of $(q,L)$, well beyond the current knowledge, which uses induction on length scales on one hand and a novel algorithmic lower bound on the other. Finally we show that no form of time scale separation occurs for $\gamma=1$, i.e. at the equilibrium scale $L=1/q$, contrary to what was assumed in the physical literature based on numerical simulations.Comment: 40 pages, 4 figures; minor typographical corrections and improvement

Support Network Responses to Acquired Brain Injury

Acquired brain injury (ABI) affects social relationships; however, the ways social and support networks change and evolve as a result of brain injury is not well understood. This study explored ways in which survivors of ABI and members of their support networks perceive relationship changes as recovery extends into the long-term stage. Two survivors of ABI and members of their respective support networks participated in this case study integrating information from interviews, field notes, and artifacts. Inductive data analysis revealed themes of adjustment to impairments and compensations, connection changes with other people, feelings of protectiveness toward the survivor, emotional intensity, and the influence of personality traits on the recovery process. Application of these themes to intervention suggests health care professionals might benefit from shifting their focus from the survivor alone to the survivor functioning within a social support network

Instability of condensation in the zero-range process with random interaction

The zero-range process is a stochastic interacting particle system that is known to exhibit a condensation transition. We present a detailed analysis of this transition in the presence of quenched disorder in the particle interactions. Using rigorous probabilistic arguments, we show that disorder changes the critical exponent in the interaction strength below which a condensation transition may occur. The local critical densities may exhibit large fluctuations, and their distribution shows an interesting crossover from exponential to algebraic behavior

Cutoff for the East process

The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-window. The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $\nu$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $\nu$, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen ('82) implies a CLT for the location of the front, yielding the cutoff result. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an $O(1)$-window.Comment: 33 pages, 2 figure

Spanning Tree Coverage Algorithm on Large Spaces for Multi-UAV Systems

V této práci jsou navrženy algoritmy Improved Artificially Weighted Spanning Tree Coverage (IAWSTC) a Cycle Growing With Event Partitioning (CGWEP). Tyto nové přístupy jsou vhodné pro prohledávání prostorů se známými překážkami, kde se mohou objevit také nečekané překážky. Pro vyřešení tohoto problému je také navržen algoritmus pro vyhlazování trajektorií a algoritmus přeplánování při detekci neočekávaných překážek. Byly provedeny simulační experimenty spolu s experimenty na opravdových dronech. Algoritmy jsou porovnány a zhodnoceny.In this work, the Improved Artificially Weighted Spanning Tree Coverage (IAWSTC) and the Cycle Growing With Event Partitioning (CGWEP) algorithms are proposed. These novel approaches are suitable for the exploration of environments with cluttered regions, where unexpected obstacles can appear. In order to solve these problems, a smoothing algorithm together with an online replanner to avoid unexpected detected obstacles is proposed. Experiments are carried out in simulation and on real drones. The algorithms are then compared and conclusions are deduced

On fuzzy input data and the worst scenario method

summary:In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set $\mathcal U_{\mathrm ad}$ of admissible input data instead of a fixed and unique input. The worst scenario method takes into account all states generated by $\mathcal U_{\mathrm ad}$ and maximizes a functional criterion reflecting a particular feature of the state solution, as local stress, displacement, or temperature, for instance. An increase in the criterion value indicates a deterioration in the featured quantity. The method takes all the elements of $\mathcal U_{\mathrm ad}$ as equally important though this can be unrealistic and can lead to too pessimistic conclusions. Often, however, additional information expressed through a membership function of $\mathcal U_{\mathrm ad}$ is available, i.e., $\mathcal U_{\mathrm ad}$ becomes a fuzzy set. In the article, infinite-dimensional $\mathcal U_{\mathrm ad}$ are considered, two ways of introducing fuzziness into $\mathcal U_{\mathrm ad}$ are suggested, and the worst scenario method operating on fuzzy admissible sets is proposed to obtain a fuzzy set of outputs

The Sublime & the Picturesque in Art

Is it possible to create classical\u27\u27 landscape pictures of today? The genre of drawing and painting landscapes is the subject for this thesis. For comparing for relevancy, the binary topic is two aesthetic terms, the Sublime and the Picturesque. I will show what the art critics and artists say about those terms with several historical and contemporary artists a\u3c; examples. I will explain what I did with some of my drawings and paintings in relation to the terms within the classical composition format. The challenge for my work is to compose the contemporary landscape using the classical composing principles used by Old Masters. A contemporary German artist has succeeded very well in composing the \u27·classical·\u27 landscapes of today with modern cultural attributes, but they are fictional digital composites. Her approach to classical composing today\u27s landscape is innovative. It is a rehash concept- that is, re-use of an old idea yet new. It is like re-contextualizing. By applying my mental template of one of the classical compositions to drawing and painting, I compose an actual landscape with minor modifications because I want to see how different the picture of today is from the ones of the classical period from the seventeenth century to the early nineteenth century which did not have modern trappings, such as telephone poles, cars, highways. I seek for either the Sublime or the Picturesque or both-more or less--in the actual landscape. I will continue composing the landscapes classically. Mastering a repertoire of pictorial techniques is my constant objective