On Match Lengths, Zero Entropy and Large Deviations - with Application to Sliding Window Lempel-Ziv Algorithm

The Sliding Window Lempel-Ziv (SWLZ) algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature. In this paper, we undertake a finer study of these quantities under two different scenarios, i) \emph{zero entropy} sources that are characterized by strong long-term memory, and ii) the processes with weak memory as described through various mixing conditions. For zero entropy sources, a general statement on match length is obtained. It is used in the proof of almost sure optimality of Fixed Shift Variant of Lempel-Ziv (FSLZ) and SWLZ algorithms given in literature. Through an example of stationary and ergodic processes generated by an irrational rotation we establish that for a window of size $n_w$, a compression ratio given by $O(\frac{\log n_w}{{n_w}^a})$ where $a$ depends on $n_w$ and approaches 1 as $n_w \rightarrow \infty$, is obtained under the application of FSLZ and SWLZ algorithms. Also, we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy. Next, we extend the study of Ornstein and Weiss on the asymptotic behavior of the \emph{normalized} version of recurrence times and establish the \emph{large deviation property} (LDP) for a class of mixing processes. Also, an estimator of entropy based on recurrence times is proposed for which large deviation principle is proved for sources satisfying similar mixing conditions.Comment: accepted to appear in IEEE Transactions on Information Theor

Fragility of the Commons under Prospect-Theoretic Risk Attitudes

We study a common-pool resource game where the resource experiences failure with a probability that grows with the aggregate investment in the resource. To capture decision making under such uncertainty, we model each player's risk preference according to the value function from prospect theory. We show the existence and uniqueness of a pure Nash equilibrium when the players have heterogeneous risk preferences and under certain assumptions on the rate of return and failure probability of the resource. Greater competition, vis-a-vis the number of players, increases the failure probability at the Nash equilibrium; we quantify this effect by obtaining bounds on the ratio of the failure probability at the Nash equilibrium to the failure probability under investment by a single user. We further show that heterogeneity in attitudes towards loss aversion leads to higher failure probability of the resource at the equilibrium.Comment: Accepted for publication in Games and Economic Behavior, 201

Learning in the Repeated Secretary Problem

In the classical secretary problem, one attempts to find the maximum of an unknown and unlearnable distribution through sequential search. In many real-world searches, however, distributions are not entirely unknown and can be learned through experience. To investigate learning in such a repeated secretary problem we conduct a large-scale behavioral experiment in which people search repeatedly from fixed distributions. In contrast to prior investigations that find no evidence for learning in the classical scenario, in the repeated setting we observe substantial learning resulting in near-optimal stopping behavior. We conduct a Bayesian comparison of multiple behavioral models which shows that participants' behavior is best described by a class of threshold-based models that contains the theoretically optimal strategy. Fitting such a threshold-based model to data reveals players' estimated thresholds to be surprisingly close to the optimal thresholds after only a small number of games

Stationary mass distribution and nonlocality in models of coalescence and shattering

We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and small mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses), or non-local (collision between particles of widely different masses give the main contribution to the mass flux). We show that the non-local regime is further divided into two sub-regimes corresponding to weak and strong non-locality. We also observe that at the boundaries between the local and non-local regimes, the mass distribution acquires logarithmic corrections to scaling and calculate these corrections. Exact solutions for special kernels and numerical simulations are used to validate some non-rigorous steps used in the analysis. Our results show that for local kernels, the scaling solutions carry a constant flux of mass due to aggregation, whereas for the non-local case there is a correction to the constant flux exponent. Our results suggest that for general scale-invariant kernels, the universality classes of mass distributions are labeled by two parameters: the homogeneity degree of the kernel and one further number measuring the degree of the non-locality of the kernel.Comment: Published versio

Trunk Robot for Extended Environments

We describe the design and physical realization of a novel type of large-scale continuum robot. The design, based on a hybrid concentric-tube/tendon actuated structure, is realized at a significantly larger scale than previous concentric tube continuum robots, with an extended length well over one meter. While operation at this scale opens up new types of potential applications, realization at this scale presents interesting challenges. We detail and discuss the associated issues via the prototyping and testing of the physical system with the help of experiments

A Search for Dark Matter Annihilation in Galaxy Groups

We use 413 weeks of publicly-available $\textit{Fermi}$ Pass 8 gamma-ray data, combined with recently-developed galaxy group catalogs, to search for evidence of dark matter annihilation in extragalactic halos. In our study, we use luminosity-based mass estimates and mass-to-concentration relations to infer the $J$-factors and associated uncertainties for hundreds of galaxy groups within a redshift range $z \lesssim 0.03$. We employ a conservative substructure boost-factor model, which only enhances the sensitivity by an $\mathcal{O}(1)$ factor. No significant evidence for dark matter annihilation is found and we exclude thermal relic cross sections for dark matter masses below $\sim$30 GeV to 95% confidence in the $b\bar{b}$ annihilation channel. These bounds are comparable to those from Milky Way dwarf spheroidal satellite galaxies. The results of our analysis increase the tension, but do not rule out, the dark matter interpretation of the Galactic Center excess. We provide a catalog of the galaxy groups used in this study and their inferred properties, which can be broadly applied to searches for extragalactic dark matter.Comment: 5+18 pages, 1+14 figures, catalog available at: https://github.com/bsafdi/DMCat; v2 updated to journal version with several updates, results and conclusions unchange