Asymptotic Expansions for lambda_d of the Dimer and Monomer-Dimer Problems

In the past few years we have derived asymptotic expansions for lambda_d of the dimer problem and lambda_d(p) of the monomer-dimer problem. The many expansions so far computed are collected herein. We shine a light on results in two dimensions inspired by the work of M. E. Fisher. Much of the work reported here was joint with Shmuel Friedland.Comment: 4 page

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We compute the first three terms in the formal asymptotic expansion of (lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching conjecture is satisfied for (lambda_d)(p).Comment: 15 pages, much more about d=1,2,

Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics

Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis often falls short of explaining this performance. Because of this, "beyond worst-case analysis" of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained by Bringmann et al. (Algorithmica, 2013), who have used random shortest path metrics on complete graphs to analyze heuristics. The goal of this paper is to generalize these findings to non-complete graphs, especially Erd\H{o}s-R\'enyi random graphs. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances, we prove that the greedy heuristic for the minimum distance maximum matching problem, the nearest neighbor and insertion heuristics for the traveling salesman problem, and a trivial heuristic for the $k$-median problem all achieve a constant expected approximation ratio. Additionally, we show a polynomial upper bound for the expected number of iterations of the 2-opt heuristic for the traveling salesman problem.Comment: An extended abstract appeared in the proceedings of WALCOM 201

Critical Dimension for Stable Self-gravitating Stars in AdS

We study the self-gravitating stars with a linear equation of state, $P=a \rho$, in AdS space, where $a$ is a constant parameter. There exists a critical dimension, beyond which the stars are always stable with any central energy density; below which there exists a maximal mass configuration for a certain central energy density and when the central energy density continues to increase, the configuration becomes unstable. We find that the critical dimension depends on the parameter $a$, it runs from $d=11.1429$ to 10.1291 as $a$ varies from $a=0$ to 1. The lowest integer dimension for a dynamically stable self-gravitating configuration should be $d=12$ for any $a \in [0,1]$ rather than $d=11$, the latter is the case of self-gravitating radiation configurations in AdS space.Comment: Revtex, 11 pages with 7 eps figure

New Lower Bounds on the Self-Avoiding-Walk Connective Constant

We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice $Z^d$. The method is based on loop erasure and restoration, and does not require exact enumeration data. Our bounds are best for high $d$, and in fact agree with the first four terms of the $1/d$ expansion for the connective constant. The bounds are the best to date for dimensions $d \geq 3$, but do not produce good results in two dimensions. For $d=3,4,5,6$, respectively, our lower bound is within 2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.Comment: 35 pages, 388480 bytes Postscript, NYU-TH-93/02/0

Producing the docile body: analysing Local Area Under-performance Inspection (LAUI)

Sir Michael Wilshaw, the head of the Office for Standards in Education (OfSTED), declared a 'new wave' of Local Area Under-performance Inspections (LAUI) of schools 'denying children the standard of education they deserve'. This paper examines how the threat of LAUI played out over three mathematics lessons taught by a teacher in her first year in the profession. A Foucauldian approach is mobilised with regard to disciplinary power and 'docile bodies'. The paper argues that, in the case in point, LAUI was a tool mediating performative conditions and, ultimately, the docile body. The paper will be of concern to policy sociologists, teachers, school leaders, and those interested in school inspection

Weighted distances in scale-free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear in Random Structures and Algorithm

Probabilistic Analysis of Facility Location on Random Shortest Path Metrics

The facility location problem is an NP-hard optimization problem. Therefore, approximation algorithms are often used to solve large instances. Such algorithms often perform much better than worst-case analysis suggests. Therefore, probabilistic analysis is a widely used tool to analyze such algorithms. Most research on probabilistic analysis of NP-hard optimization problems involving metric spaces, such as the facility location problem, has been focused on Euclidean instances, and also instances with independent (random) edge lengths, which are non-metric, have been researched. We would like to extend this knowledge to other, more general, metrics. We investigate the facility location problem using random shortest path metrics. We analyze some probabilistic properties for a simple greedy heuristic which gives a solution to the facility location problem: opening the $\kappa$ cheapest facilities (with $\kappa$ only depending on the facility opening costs). If the facility opening costs are such that $\kappa$ is not too large, then we show that this heuristic is asymptotically optimal. On the other hand, for large values of $\kappa$, the analysis becomes more difficult, and we provide a closed-form expression as upper bound for the expected approximation ratio. In the special case where all facility opening costs are equal this closed-form expression reduces to $O(\sqrt[4]{\ln(n)})$ or $O(1)$ or even $1+o(1)$ if the opening costs are sufficiently small.Comment: A preliminary version accepted to CiE 201

A major star formation region in the receding tip of the stellar Galactic bar

We present an analysis of the optical spectroscopy of 58 stars in the Galactic plane at $l=27$\arcdeg, where a prominent excess in the flux distribution and star counts have been observed in several spectral regions, in particular in the Two Micron Galactic Survey (TMGS) catalog. The sources were selected from the TMGS, to have a $K$ magnitude brighter than +5 mag and be within 2 degrees of the Galactic plane. More than 60% of the spectra correspond to stars of luminosity class I, and a significant proportion of the remainder are very late giants which would also be fast evolving. This very high concentration of young sources points to the existence of a major star formation region in the Galactic plane, located just inside the assumed origin of the Scutum spiral arm. Such regions can form due to the concentrations of shocked gas where a galactic bar meets a spiral arm, as is observed at the ends of the bars of face-on external galaxies. Thus, the presence of a massive star formation region is very strong supporting evidence for the presence of a bar in our Galaxy.Comment: 13 pages (latex) + 4 figures (eps), accepted in ApJ Let

CASTNet: Community-Attentive Spatio-Temporal Networks for Opioid Overdose Forecasting

Opioid overdose is a growing public health crisis in the United States. This crisis, recognized as "opioid epidemic," has widespread societal consequences including the degradation of health, and the increase in crime rates and family problems. To improve the overdose surveillance and to identify the areas in need of prevention effort, in this work, we focus on forecasting opioid overdose using real-time crime dynamics. Previous work identified various types of links between opioid use and criminal activities, such as financial motives and common causes. Motivated by these observations, we propose a novel spatio-temporal predictive model for opioid overdose forecasting by leveraging the spatio-temporal patterns of crime incidents. Our proposed model incorporates multi-head attentional networks to learn different representation subspaces of features. Such deep learning architecture, called "community-attentive" networks, allows the prediction of a given location to be optimized by a mixture of groups (i.e., communities) of regions. In addition, our proposed model allows for interpreting what features, from what communities, have more contributions to predicting local incidents as well as how these communities are captured through forecasting. Our results on two real-world overdose datasets indicate that our model achieves superior forecasting performance and provides meaningful interpretations in terms of spatio-temporal relationships between the dynamics of crime and that of opioid overdose.Comment: Accepted as conference paper at ECML-PKDD 201