A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.Comment: Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue 1-2, pp 405-42

A toral diffeomorphism with a non-polygonal rotation set

We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon

Evaluating weaknesses of "perceptual-cognitive training" and "brain training" methods in sport: An ecological dynamics critique

The recent upsurge in "brain training and perceptual-cognitive training," proposing to improve isolated processes, such as brain function, visual perception, and decision-making, has created significant interest in elite sports practitioners, seeking to create an "edge" for athletes. The claims of these related "performance-enhancing industries" can be considered together as part of a process training approach proposing enhanced cognitive and perceptual skills and brain capacity to support performance in everyday life activities, including sport. For example, the "process training industry" promotes the idea that playing games not only makes you a better player but also makes you smarter, more alert, and a faster learner. In this position paper, we critically evaluate the effectiveness of both types of process training programmes in generalizing transfer to sport performance. These issues are addressed in three stages. First, we evaluate empirical evidence in support of perceptual-cognitive process training and its application to enhancing sport performance. Second, we critically review putative modularized mechanisms underpinning this kind of training, addressing limitations and subsequent problems. Specifically, we consider merits of this highly specific form of training, which focuses on training of isolated processes such as cognitive processes (attention, memory, thinking) and visual perception processes, separately from performance behaviors and actions. We conclude that these approaches may, at best, provide some "general transfer" of underlying processes to specific sport environments, but lack "specificity of transfer" to contextualize actual performance behaviors. A major weakness of process training methods is their focus on enhancing the performance in body "modules" (e.g., eye, brain, memory, anticipatory sub-systems). What is lacking is evidence on how these isolated components are modified and subsequently interact with other process "modules," which are considered to underlie sport performance. Finally, we propose how an ecological dynamics approach, aligned with an embodied framework of cognition undermines the rationale that modularized processes can enhance performance in competitive sport. An ecological dynamics perspective proposes that the body is a complex adaptive system, interacting with performance environments in a functionally integrated manner, emphasizing that the inter-relation between motor processes, cognitive and perceptual functions, and the constraints of a sport task is best understood at the performer-environment scale of analysis

Motor preparation of spatially and temporally defined movements: Evidence from startle

This article is available open access through the publisher’s website at the link below. Copyright © 2011 the American Physiological Society.Previous research has shown that the preparation of a spatially targeted movement performed at maximal speed is different from that of a temporally constrained movement (Gottlieb et al. 1989b). In the current study, we directly examined preparation differences in temporally vs. spatially defined movements through the use of a startling stimulus and manipulation of the task goals. Participants performed arm extension movements to one of three spatial targets (20°, 40°, 60°) and an arm extension movement of 20° at three movement speeds (slow, moderate, fast). All movements were performed in a blocked, simple reaction time paradigm, with trials involving a startling stimulus (124 dB) interspersed randomly with control trials. As predicted, spatial movements were modulated by agonist duration and timed movements were modulated by agonist rise time. The startling stimulus triggered all movements at short latencies with a compression of the kinematic and electromyogram (EMG) profile such that they were performed faster than control trials. However, temporally constrained movements showed a differential effect of movement compression on startle trials such that the slowest movement showed the greatest temporal compression. The startling stimulus also decreased the relative timing between EMG bursts more for the 20° movement when it was defined by a temporal rather than spatial goal, which we attributed to the disruption of an internal timekeeper for the timed movements. These results confirm that temporally defined movements were prepared in a different manner from spatially defined movements and provide new information pertaining to these preparation differences

Four layer bandage compared with short stretch bandage for venous leg ulcers: systematic review and meta-analysis of randomised controlled trials with data from individual patients

<p><b>Objective:</b> To compare the effectiveness of two types of compression treatment (four layer bandage and short stretch bandage) in people with venous leg ulceration.</p> <p><b>Design:</b> Systematic review and meta-analysis of patient level data.</p> <p><b>Data:</b> sources Electronic databases (the Cochrane Central Register of Controlled Trials, the Cochrane Wounds Group Specialised Register, Medline, Embase, CINAHL, and National Research Register) and reference lists of retrieved articles searched to identify relevant trials and primary investigators. Primary investigators of eligible trials were invited to contribute raw data for re-analysis.</p> <p><b>Review:</b> methods Randomised controlled trials of four layer bandage compared with short stretch bandage in people with venous leg ulceration were eligible for inclusion. The primary outcome for the meta-analysis was time to healing. Cox proportional hazards models were run to compare the methods in terms of time to healing with adjustment for independent predictors of healing. Secondary outcomes included incidence and number of adverse events per patient.</p> <p><b>Results:</b> Seven eligible trials were identified (887 patients), and patient level data were retrieved for five (797 patients, 90% of known randomised patients). The four layer bandage was associated with significantly shorter time to healing: hazard ratio (95% confidence interval) from multifactorial model based on five trials was 1.31 (1.09 to 1.58), P=0.005. Larger ulcer area at baseline, more chronic ulceration, and previous ulceration were all independent predictors of delayed healing. Data from two trials showed no evidence of a difference in adverse event profiles between the two bandage types.</p> <p><b>Conclusions:</b> Venous leg ulcers in patients treated with four layer bandages heal faster, on average, than those of people treated with the short stretch bandage. Benefits were consistent across patients with differing prognostic profiles.</p&gt

Cuntz-Pimsner C*-algebras associated with subshifts

By using C*-correspondences and Cuntz-Pimsner algebras, we associate to every subshift (also called a shift space) $X$ a C*-algebra $O_X$, which is a generalization of the Cuntz-Krieger algebras. We show that $O_X$ is the universal C*-algebra generated by partial isometries satisfying relations given by $X$. We also show that $O_X$ is a one-sided conjugacy invariant of $X$.Comment: 28 pages. This is a slightly updated version of a preprint from 2004. Submitted for publication. In version 2 the Introduction has been changed, two remarks (Remark 7.6 and 7.7) have been added and the list of references has been update

Computations on Sofic S-gap Shifts

Let $S=\{s_{n}\}$ be an increasing finite or infinite subset of $\mathbb N \bigcup \{0\}$ and $X(S)$ the $S$-gap shift associated to $S$. Let $f_{S}(x)=1-\sum\frac{1}{x^{s_{n}+1}}$ be the entropy function which will be vanished at $2^{h(X(S))}$ where $h(X(S))$ is the entropy of the system. Suppose $X(S)$ is sofic with adjacency matrix $A$ and the characteristic polynomial $\chi_{A}$. Then for some rational function $Q_{S}$, $\chi_{A}(x)=Q_{S}(x)f_{S}(x)$. This $Q_{S}$ will be explicitly determined. We will show that $\zeta(t)=\frac{1}{f_{S}(t^{-1})}$ or $\zeta(t)=\frac{1}{(1-t)f_{S}(t^{-1})}$ when $|S|<\infty$ or $|S|=\infty$ respectively. Here $\zeta$ is the zeta function of $X(S)$. We will also compute the Bowen-Franks groups of a sofic $S$-gap shift.Comment: This paper has been withdrawn due to extending results about SFT shifts to sofic shifts (Theorem 2.3). This forces to apply some minor changes in the organization of the paper. This paper has been withdrawn due to a flaw in the description of the adjacency matrix (2.3

Topological Entropy of Braids on the Torus

A fast method is presented for computing the topological entropy of braids on the torus. This work is motivated by the need to analyze large braids when studying two-dimensional flows via the braiding of a large number of particle trajectories. Our approach is a generalization of Moussafir's technique for braids on the sphere. Previous methods for computing topological entropies include the Bestvina--Handel train-track algorithm and matrix representations of the braid group. However, the Bestvina--Handel algorithm quickly becomes computationally intractable for large braid words, and matrix methods give only lower bounds, which are often poor for large braids. Our method is computationally fast and appears to give exponential convergence towards the exact entropy. As an illustration we apply our approach to the braiding of both periodic and aperiodic trajectories in the sine flow. The efficiency of the method allows us to explore how much extra information about flow entropy is encoded in the braid as the number of trajectories becomes large.Comment: 19 pages, 44 figures. SIAM journal styl

Strictly Toral Dynamics

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus $\mathbb{T}^2$ which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points $ine(f)$ is shown to be a disjoint union of periodic topological disks ("elliptic islands"), while the set of essential points $ess(f)$ is an essential continuum, with typically rich dynamics (the "chaotic region"). This generalizes and improves a similar description by J\"ager. The key result is boundedness of these "elliptic islands", which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in $ess(f)$ is as rich as in $\mathbb{T}^2$ from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.Comment: Incorporates suggestions and corrections by the referees. To appear in Inv. Mat

Estimating urban flood risk - uncertainty in design criteria

The design of urban stormwater infrastructure is generally performed assuming that climate is static. For engineering practitioners, stormwater infrastructure is designed using a peak flow method, such as the Rational Method as outlined in the Australian Rainfall and Runoff (AR&R) guidelines and estimates of design rainfall intensities. Changes to Australian rainfall intensity design criteria have been made through updated releases of the AR&R77, AR&R87 and the recent 2013 AR&R Intensity Frequency Distributions (IFDs). The primary focus of this study is to compare the three IFD sets from 51 locations Australia wide. Since the release of the AR&R77 IFDs, the duration and number of locations for rainfall data has increased and techniques for data analysis have changed. Updated terminology coinciding with the 2013 IFD release has also resulted in a practical change to the design rainfall. For example, infrastructure that is designed for a 1: 5 year ARI correlates with an 18.13 % AEP, however for practical purposes, hydraulic guidelines have been updated with the more intuitive 20 % AEP. The evaluation of design rainfall variation across Australia has indicated that the changes are dependent upon location, recurrence interval and rainfall duration. The changes to design rainfall IFDs are due to the application of differing data analysis techniques, the length and number of data sets and the change in terminology from ARI to AEP. Such changes mean that developed infrastructure has been designed to a range of different design criteria indicating the likely inadequacy of earlier developments to the current estimates of flood risk. In many cases, the under-design of infrastructure is greater than the expected impact of increased rainfall intensity under climate change scenarios