Cycle Lengths In Aᵏb

Let A be a nonnegative, n n matrix, and let b be a nonnegative, nxn vector. Let S be the sequence {Akb }, k = 0, l, 2, .... Define m(A, b) to be the length of the cycle of zero-nonzero patterns into which S eventually falls. Define m(A) to be the maximum, over all nonnegative b of m(A, b). Finally, define m(n) to be the maximum, over all nonnegative, nxn matrices A of m(A). This paper shows given A and b, that m(A, b) is a divisor of a certain number, which is determined by the structure of A and b. It is also shown that log m(n) ~ (n log n) /2

On the Ramsey numbers R(3, 8) and R(3, 9)

AbstractUsing methods developed by Graver and Yackel, and various computer algorithms, we show that 28 ≤ R(3, 8) ≤ 29, and R(3, 9) = 36, where R(k, l) is the classical Ramsey number for 2-coloring the edges of a complete graph

Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records

We review how the renormalized force correlator Delta(u), the function computed in the functional RG field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a 1-dimensional random-force landscape. The limit of small velocity allows to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Frechet. For each class we obtain analytically the universal function Delta(u), the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w. We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the ABBM model, avalanche distributions and Delta(u) can be computed for any velocity. For 2-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(u), and to extract the anisotropic scaling exponents zeta_x > zeta_y. We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered e.g. in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w.Comment: 41 pages, 35 figure

Effects of an Alpha-4 Integrin Inhibitor on Restenosis in a New Porcine Model Combining Endothelial Denudation and Stent Placement

Restenosis remains the main complication of balloon angioplasty and/or stent implantation. Preclinical testing of new pharmacologic agents preventing restenosis largely rely on porcine models, where restenosis is assessed after endothelial abrasion of the arterial wall or stent implantation. We combined endothelial cell denudation and implantation of stents to develop a new clinically relevant porcine model of restenosis, and used this model to determine the effects of an α4 integrin inhibitor, ELN 457946, on restenosis. Balloon-angioplasty endothelial cell denudation and subsequent implantation of bare metal stents in the left anterior descending coronary, iliac, and left common carotid arteries was performed in domestic pigs, treated with vehicle or ELN 457946, once weekly via subcutaneous injections, for four weeks. After 1 month, histopathology and morphometric analyses of the arteries showed complete healing and robust, consistent restenotic response in stented arteries. Treatment with ELN 457946 resulted in a reduction in the neointimal response, with decreases in area percent stenosis between 12% in coronary arteries and 30% in peripheral vessels. This is the first description of a successful pig model combining endothelial cell denudation and bare metal stent implantation. This new double injury model may prove particularly useful to assess pharmacological effects of drug candidates on restenosis, in coronary and/or peripheral arteries. Furthermore, the ELN 457946 α4 integrin inhibitor, administered subcutaneously, reduced inflammation and restenosis in stented coronary and peripheral arteries in pigs, therefore representing a promising systemic therapeutic approach in reducing restenosis in patients undergoing angioplasty and/or stent implantation

On Circular Critical Graphs

A graph on n vertices is called circular if its automorphism group contains an n-cycle. Let ω(G) and α(G) be, respectively, the clique number and the independence number of the graph G. A graph G with n vertices is called an (α, ω)-graph if 1. (1) n=α(G)ω(G)+1 2. (2) every vertex is in exactly α(G) maximum independent sets and α(G) maximum cliques, and 3. (3) each maximum clique intersects all but one maximum independent set, and vice versa. A graph is called critical if it is imperfect and all of its proper induced subgraphs are perfect. Lovasz and Padberg showed that all critical graphs are (α, ω)-graphs. Only one method is known for constructing circular (α, ω)-graphs. We show that the only critical graphs which arise from this construction are the odd, chordless cycles of length at least 5, and their complements

On Medians Of Lattice Distributions And A Game With Two Dice

Let D₁ and D₂ be two dice with k and l integer faces, respectively, where k and l are two positive integers. The game G_n consists of tossing each die n times and summing the resulting faces. The die with the higher total wins the game. We examine the question of which die wins game G_n more often, for large values of n. We also give an example of a set of three dice which is non-transitive in game G_n for infinitely many values of n

The Perfect Graph Conjecture For Toroidal Graphs

This chapter discusses the perfect graph conjecture for toroidal graphs. Graphs are assumed to be finite without loops or multiple edges. The ω (G) is defined as the size of the largest complete subgraph of G, while γ (G) is defined as the vertex coloring number of G. A graph G is perfect only if G has property P. Each maximal clique of G intersects all but one maximal independent set of G, and vice versa. If G is toroidal and has property P, then G is perfect. In a critical toroidal graph G, either ω (G) \u3c 4 or G is regular of degree six and triangulates the torus