Asymptotically Hyperbolic Metrics on Unit Ball Admitting Multiple Horizons

In this paper, we construct an asymptotically hyperbolic metric with scalar curvature -6 on unit ball $\mathbf{D}^3$, which contains multiple horizons.Comment: 11page

Dynamic Left-turn Phase Optimization Using Fuzzy Logic Control

The left-turn movement at an intersection has long been a concern of traffic engineers as it is a major capacity reduction factor. Different left-turn signal phasings have been shown to result in significant differences in delay, intersection capacity, and even safety level. First, past studies about leading and lagging signal phases and signal control application are overviewed. Then this research gives a theoretical analysis of signal left-turn phase operations at both isolated and coordinated signalized intersections, compares the difference in delay based on leading and lagging left-turn signal phase designs, analyzes the influences of traffic control delay components for leading and lagging left-turn, identifies the main control factors, and gives a new model to guide the choosing between the leading and lagging left-turn phases. In the third part of this research, some basic mathematical definitions and rules of fuzzy logic control are described. A four-level fuzzy logic control model is designed. To implement this control model, observed approaching traffic flows are used to estimate relative traffic intensities in the competing approaches. These traffic intensities are then used to determine whether a leading or lagging signal phase should be selected or terminated. Finally, this research designs a dynamic traffic signal left-turn phase control system, and implements the four-level fuzzy logic control model to optimize signalized intersection operation. The performance of this dynamic traffic signal left-turn phase fuzzy logic control system compared favorably in all categories to fixed time control, actuated control, and traditional fuzzy control based on simulation using field data. The results suggest that the proposed dynamic traffic signal left-turn phase fuzzy logic control system is a superior and efficient tool for reducing intersection traffic delay. The study also demonstrated that the successful implementation of the proposed model does not rely on the installation of expensive or complicated equipment

Statistical and Deterministic Dynamics of Maps with Memory

We consider a dynamical system to have memory if it remembers the current state as well as the state before that. The dynamics is defined as follows: $x_{n+1}=T_{\alpha}(x_{n-1},x_{n})=\tau (\alpha \cdot x_{n}+(1-\alpha)\cdot x_{n-1}),$ where $\tau$ is a one-dimensional map on $I=[0,1]$ and $0<\alpha <1$ determines how much memory is being used. $T_{\alpha}$ does not define a dynamical system since it maps $U=I\times I$ into $I$. In this note we let $\tau$ to be the symmetric tent map. We shall prove that for $0<\alpha <0.46,$ the orbits of $\{x_{n}\}$ are described statistically by an absolutely continuous invariant measure (acim) in two dimensions. As $\alpha$ approaches $0.5$ from below, that is, as we approach a balance between the memory state and the present state, the support of the acims become thinner until at $\alpha =0.5$, all points have period 3 or eventually possess period 3. For $0.5<\alpha <0.75$, we have a global attractor: for all starting points in $U$ except $(0,0)$, the orbits are attracted to the fixed point $(2/3,2/3).$ At $\alpha=0.75,$ we have slightly more complicated periodic behavior.Comment: 37 page

Instability of Isolated Spectrum for W-shaped Maps

In this note we consider $W$-shaped map $W_0=W_{s_1,s_2}$ with $\frac {1}{s_1}+\frac {1}{s_2}=1$ and show that eigenvalue 1 is not stable. We do this in a constructive way. For each perturbing map $W_a$ we show the existence of the "second" eigenvalue $\lambda_a$, such that $\lambda_a\to 1$, as $a\to 0$, which proves instability of isolated spectrum of $W_0$. At the same time, the existence of second eigenvalues close to 1 causes the maps $W_a$ behave in a metastable way. They have two almost invariant sets and the system spends long periods of consecutive iterations in each of them with infrequent jumps from one to the other.Comment: 12 pages, 2 figure

Online Action Detection

In online action detection, the goal is to detect the start of an action in a video stream as soon as it happens. For instance, if a child is chasing a ball, an autonomous car should recognize what is going on and respond immediately. This is a very challenging problem for four reasons. First, only partial actions are observed. Second, there is a large variability in negative data. Third, the start of the action is unknown, so it is unclear over what time window the information should be integrated. Finally, in real world data, large within-class variability exists. This problem has been addressed before, but only to some extent. Our contributions to online action detection are threefold. First, we introduce a realistic dataset composed of 27 episodes from 6 popular TV series. The dataset spans over 16 hours of footage annotated with 30 action classes, totaling 6,231 action instances. Second, we analyze and compare various baseline methods, showing this is a challenging problem for which none of the methods provides a good solution. Third, we analyze the change in performance when there is a variation in viewpoint, occlusion, truncation, etc. We introduce an evaluation protocol for fair comparison. The dataset, the baselines and the models will all be made publicly available to encourage (much needed) further research on online action detection on realistic data.Comment: Project page: http://homes.esat.kuleuven.be/~rdegeest/OnlineActionDetection.htm

Slope Conditions for Stability of ACIMs of Dynamical Systems

The family of $W-$shaped maps was introduced in 1982 by G. Keller. Based on the investigation of the properties of the maps, it was conjectured that instability of the absolutely continuous invariant measure (acim) can result only from the existence of small invariant neighbourhoods of the fixed critical point of the limiting map. We show that the conjecture is not true by constructing a family of $W-$shaped maps with acims supported on the entire interval, whose limiting dynamical behavior is described by a singular measure. We then generalize the above result by constructing families of $W-$shaped maps $\{W_a\}$ with a turning fixed point having slope $s_1$ on one side and $-s_2$ on the other. Each such $W_a$ map has an acim $\mu_a$. Depending on whether $\frac{1}{s_1}+\frac{1}{s_2}$ is larger, equal, or smaller than 1, we show that the limit of $\mu_a$ is a singular measure, a combination of singular and absolutely continuous measure or an acim, respectively. We also consider $W$-shaped maps satisfying $\frac 1{s_1}+\frac 1{s_2}=1$ and show that the eigenvalue $1$ of the associated Perron-Frobenius operator is not stable, which in turn implies the instability of the isolated spectrum. Motivated by the above results, we introduce the harmonic average of slopes condition, with which we obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Moreover, we prove stability results using Rychlik's Theorem together with the harmonic average of slopes condition for piecewise expanding maps