Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture

We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the diameters of polyhedral graphs. One particular variant has a diameter which satisfies the best known upper bound on the diameters of polyhedra. Another variant has superlinear asymptotic diameter, and together with some combinatorial operations, gives a concrete approach for disproving the Linear Hirsch Conjecture.Comment: 16 pages, 4 figure

Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method

This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of non-negative real numbers satisfying certain summation conditions. Transportation problems are, in many ways, the simplest kind of linear programs and thus have a rich combinatorial structure. First, we give new results on the diameters of certain classes of transportation polytopes and their relation to the Hirsch Conjecture, which asserts that the diameter of every $d$-dimensional convex polytope with $n$ facets is bounded above by $n-d$. In particular, we prove a new quadratic upper bound on the diameter of $3$-way axial transportation polytopes defined by $1$-marginals. We also show that the Hirsch Conjecture holds for $p \times 2$ classical transportation polytopes, but that there are infinitely-many Hirsch-sharp classical transportation polytopes. Second, we present new results on subpolytopes of transportation polytopes. We investigate, for example, a non-regular triangulation of a subpolytope of the fourth Birkhoff polytope $B_4$. This implies the existence of non-regular triangulations of all Birkhoff polytopes $B_n$ for $n \geq 4$. We also study certain classes of network flow polytopes and prove new linear upper bounds for their diameters.Comment: PhD thesis submitted June 2010 to the University of California, Davis. 183 pages, 49 figure

Graphs of Transportation Polytopes

This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$.Comment: 29 pages, 7 figures. Final version. Improvements to the exposition of several lemmas and the upper bound in Theorem 1.1 is improved by a factor of tw

Entanglement entropy of the ν=1/2\nu=1/2 composite fermion non-Fermi liquid state

The so-called ``non-Fermi liquid'' behavior is very common in strongly correlated systems. However, its operational definition in terms of ``what it is not'' is a major obstacle against theoretical understanding of this fascinating correlated state. Recently there has been much interest in entanglement entropy as a theoretical tool to study non-Fermi liquids. So far explicit calculations have been limited to models without direct experimental realizations. Here we focus on a two dimensional electron fluid under magnetic field and filling fraction $\nu=1/2$, which is believed to be a non-Fermi liquid state. Using the composite fermion (CF) wave-function which captures the $\nu=1/2$ state very accurately, we compute the second R\'enyi entropy using variational Monte-Carlo technique and an efficient parallel algorithm. We find the entanglement entropy scales as $L\log L$ with the length of the boundary $L$ as it does for free fermions, albeit with a pre-factor twice that of the free fermion. We contrast the results against theoretical conjectures and discuss the implications of the results.Comment: 4+ page

Comparison of Porcelain Surface and Flexural Strength Obtained by Microwave and Conventional Oven Glazing

Statement of problem. Although the superior qualities of microwave technology are common knowledge in the industry, effects of microwave glazing of dental ceramics have not been investigated. Purpose. The purpose of this study was to investigate the surface roughness and flexural strength achieved by glazing porcelain specimens in a conventional and microwave oven. Material and methods. Thirty specimens of each type of porcelain (Omega 900 and IPS d.Sign) were fabricated and sintered in a conventional oven. The specimens were further divided into 3 groups (n=10): hand polished (using diamond rotary ceramic polishers), microwave glazed, and conventional oven glazed. Each specimen was evaluated for surface roughness using a profilometer. The flexural strength of each specimen was measured using a universal testing machine. A 2-way ANOVA and Tukey HSD post hoc analysis were used to determine significant intergroup differences in surface roughness (α=.05). Flexural strength results were also analyzed using 2-way ANOVA, and the Weibull modulus was determined for each of the 6 groups. The surfaces of the specimens were subjectively evaluated for cracks and porosities using a scanning electron microscope (SEM). Results. A significant difference in surface roughness was found among the surface treatments (P=.02). Follow-up tests showed a significant difference in surface roughness between oven-glazed and microwave-glazed treatments (P=.02). There was a significant difference in flexural strength between the 2 porcelains (P Conclusions. The surface character of microwave-glazed porcelain was superior to oven-glazed porcelain. Omega 900 had an overall higher flexural strength than IPS d.Sign. Weibull distributions of flexural strengths for Omega 900 ovenglazed and microwave-glazed specimens were similar. SEM analysis demonstrated a greater number of surface voids and imperfections in IPS d.Sign as compared to Omega 900

Penile Rehabilitation Therapy with PDE-V Inhibitors Following Radical Prostatectomy: Proceed with Caution

Penile rehabilitation therapy following radical prostatectomy is a much debated topic. Erectile dysfunction is still a significant contributor to postoperative morbidity following radical prostatectomy, despite meticulous nerve-sparing technique. Secondary smooth muscle changes in the penis have been identified as the underlying causes of penile atrophy, veno-occlusive dysfunction, and fibrosis. Initial observations that intracavernous injection therapies used on a regular basis postoperatively resulted in improvements in the return of spontaneous erectile function led to the development of penile rehabilitation protocols. Chronic dosing of PDE-V inhibitors is now commonly used by urologists after radical prostatectomy. Despite the current enthusiasm of penile rehabilitation therapy, current scientific evidence with clinical trials is still limited