Theory of Computation of Multidimensional Entropy with an Application to the Monomer-Dimer Problem

We outline the most recent theory for the computation of the exponential growth rate of the number of configurations on a multi-dimensional grid. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 8 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3-dimensional grid with an error smaller than 1.35%.Comment: 35 pages, one pstricks and two eps figures, submitte

Cones of closed alternating walks and trails

Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the \emph{alternating cone}. The integral (respectively, $\{0,1\}$) vectors in the alternating cone are sums of characteristic vectors of closed alternating walks (respectively, trails). We study the basic properties of the alternating cone, determine its dimension and extreme rays, and relate its dimension to the majorization order on degree sequences. We consider whether the alternating cone has integral vectors in a given box, and use residual graph techniques to reduce this problem to searching for a closed alternating trail through a given edge. The latter problem, called alternating reachability, is solved in a companion paper along with related results.Comment: Minor rephrasing, new pictures, 14 page

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Motivated by L\'{e}vy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be ``continuous'' in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator $f\mapsto{1/2}f''$. We then consider the special case where the state space is the self-similar set $\{\pm q^k:k\in \mathbb{Z}\}\cup\{0\}$ for some $q>1$. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan's ``lost'' notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, $q$-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding It\^{o} excursion measure and the Laplace exponent of the inverse local time -- both again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve $q$-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.Comment: Published in at http://dx.doi.org/10.1214/193940307000000383 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The mechanism of radiation action in leukaemogenesis. Isolation of a leukaemogenic filtrable agent from tissues of irradiated and normal C57BL mice.

IRRADIATION of C57B1 mice induced a high incidence of lymphatic leukaemia, while being refractory to the spontaneous development of the disease. Several investigators have isolated a leukaemogenic agent from these radiation-induced tumours, which produces lymphoid leukaemia when injected into isologous newborn or young adult non-irradiated mice (Lieberman and Kaplan, 1959; Latarjet and Duplan, 1962; Laznicka and Smetanova, 1963; Ilbery and Winn, 1964). It has been assumed that the leukaemogenic agent is present during post-natal life in non-irradiated C57B1 mice, and that ionizing irradiation causes the release of a leukaemogenic agent, in addition to thymus and bone marrow injury, which are essential factors in radiation leukaemogenesis (Kaplan, 1964). Experimental support for this hypothesis was provided by demonstrating the presence of a leukaemogenic agent, for a limited period after completion of the irradiation treatment, in centrifugates prepared from pooled, irradiated, non-]eukaemic thymus and bone marrow (Haran-Ghera, 1966). The aim of the present studies was to isolate a leukaemogenic filtrate fro

The polytope of dual degree partitions

AbstractWe determine the extreme points and facets of the convex hull of all dual degree partitions of simple graphs on n vertices. (This problem was raised in the Laplace Energy group of the Workshop Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns held at the American Institute of Mathematics Research Conference Center on October 23–27, 2006 [R. Brualdi, Leslie Hogben, Brian Shader, AIM Workshop – Spectra of Families of Matrices Described by Graphs, Digraphs, and Sign Patterns, Final Report: Mathematical Results, November 17, 2006].

Approximating Nearest Neighbor Distances

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we consider the most natural metric with this property, which we call the nearest neighbor metric. Given a point set P and a path $\gamma$, our metric charges each point of $\gamma$ with its distance to P. The total charge along $\gamma$ determines its nearest neighbor length, which is formally defined as the integral of the distance to the input points along the curve. We describe a $(3+\varepsilon)$-approximation algorithm and a $(1+\varepsilon)$-approximation algorithm to compute the nearest neighbor metric. Both approximation algorithms work in near-linear time. The former uses shortest paths on a sparse graph using only the input points. The latter uses a sparse sample of the ambient space, to find good approximate geodesic paths.Comment: corrected author nam