Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions I

In two series of papers we construct quasi regular polyhedra and their duals which are similar to the Catalan solids. The group elements as well as the vertices of the polyhedra are represented in terms of quaternions. In the present paper we discuss the quasi regular polygons (isogonal and isotoxal polygons) using 2D Coxeter diagrams. In particular, we discuss the isogonal hexagons, octagons and decagons derived from 2D Coxeter diagrams and obtain aperiodic tilings of the plane with the isogonal polygons along with the regular polygons. We point out that one type of aperiodic tiling of the plane with regular and isogonal hexagons may represent a state of graphene where one carbon atom is bound to three neighboring carbons with two single bonds and one double bond. We also show how the plane can be tiled with two tiles; one of them is the isotoxal polygon, dual of the isogonal polygon. A general method is employed for the constructions of the quasi regular prisms and their duals in 3D dimensions with the use of 3D Coxeter diagrams.Comment: 22 pages, 16 figure

A hierarchical solution approach for a multicommodity distribution problem under a special cost structure

Cataloged from PDF version of article.Motivated by the spare parts distribution system of a major automotive manufacturer in Turkey, we consider a multicommodity distribution problem from a central depot to a number of geographically dispersed demand points. The distribution of the items is carried out by a set of identical vehicles. The demand of each demand point can be satisfied by several vehicles and a single vehicle is allowed to serve multiple demand points. For a given vehicle, the cost structure is dictated by the farthest demand point from the depot among all demand points served by that vehicle. The objective is to satisfy the demand of each demand point with the minimum total distribution cost. We present a novel integer linear programming formulation of the problem as a variant of the network design problem. The resulting optimization problem becomes computationally infeasible for real-life problems due to the large number of integer variables. In an attempt to circumvent this disadvantage of using the direct formulation especially for larger problems, we propose a Hierarchical Approach that is aimed at solving the problem in two stages using partial demand aggregation followed by a disaggregation scheme. We study the properties of the solution returned by the Hierarchical Approach. We perform computational studies on a data set adapted from a major automotive manufacturer in Turkey. Our results reveal that the Hierarchical Approach significantly outperforms the direct formulation approach in terms of both the running time and the quality of the resulting solution especially on large instances. © 2012 Elsevier Ltd. All rights reserved

The Prevalence of Social Science in Gay Rights Cases: The Synergistic Influences of Historical Context, Justificatory Citation, and Dissemination Efforts

Disjunctive legal change is often accompanied by a period of frantic activity as the competing forces of stasis and evolution vie for domination. Nowhere is the battle for legal change likely to be more sharply joined than when the findings of modern science, in their varied and multifarious forms, are pitted directly against prevailing moral or societal precepts. One of the latest incarnations of this trend is the battle over the legal recognition of gay rights. In recent history, the courts have been inundated by gay litigants seeking the rights and protections already afforded other discrete groups within society. In the resulting legal skirmishes, gay individuals are resorting with increasing regularity to the sciences in an effort to overcome the moral opprobrium surrounding homosexuality. The judicial opinions which have resulted from the onslaught of gay litigants have not remained untouched by the scientific information adduced. Rather, as this Article will demonstrate, a disproportionally large number of gay rights opinions contain citations and references to social science information. These judicial opinions have become artifacts of the battle between modern science and existing moral conceptions of homosexuality and provide a discrete microcosm within which to examine science\u27s contribution to legal change. The lessons derived from gay rights cases may help to elucidate other contexts in which science and morality meet head-on

The effect of diclofenac sodium on neural tube development in the early stage of chick embryos

Background: Neural tube defects are congenital malformations of the central nervous system. Genetic predisposition and some environmental factors play an important role in the development of neural tube defects. This study aimed to investigate the effects of diclofenac sodium on the neural tube development in a chick embryo model that corresponds to the first month of vertebral deve- lopment in mammals.  Materials and methods: Seventy-five fertile, specific pathogen-free eggs were incubated for 28 h and were divided into five groups of 15 eggs each. Diclofenac sodium was administered via the sub-blastodermic route at this stage. Incubation was continued till the end of the 48th h. All eggs were then opened and embryos were dissected from embryonic membranes and evaluated morphologically and histopathologically.  Results: It was determined that the use of increasing doses of diclofenac sodium led to defects of midline closure in early chicken embryos. There were statistically significant differences in neural tube positions (open or close) among the groups. In addition; crown–rump length, somite number were significantly decreased in high dose experimental groups compared with control group.  Conclusions: This study showed that development of neurons is affected in chi- cken embryos after administration of diclofenac sodium. The exact teratogenic mechanism of diclofenac sodium is not clear; therefore it should be investigated.

Non-crystallographic reduction of generalized Calogero-Moser models

We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero–Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic types, we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio. We explicitly show for the simplest of these non-crystallographic models, how the corresponding classical equations of motion can be derived from a Lie algebraic Lax pair based on the larger, crystallographic Coxeter group

Spectrum of the Relativistic Particles in Various Potentials

We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials becomes exactly and quasi-exactly solvable potentials of non-relativistic quantum mechanics when they are transformed into a Schr\"{o}dinger-like equation. For the exactly solvable potentials, eigenvalues are calculated and eigenfunctions are given by confluent hypergeometric functions. It is shown that, our formulation also leads to the study of those potentials in the framework of the supersymmetric quantum mechanics

Family Unification in Five and Six Dimensions

In family unification models, all three families of quarks and leptons are grouped together into an irreducible representation of a simple gauge group, thus unifying the Standard Model gauge symmetries and a gauged family symmetry. Large orthogonal groups, and the exceptional groups $E_7$ and $E_8$ have been much studied for family unification. The main theoretical difficulty of family unification is the existence of mirror families at the weak scale. It is shown here that family unification without mirror families can be realized in simple five-dimensional and six-dimensional orbifold models similar to those recently proposed for SU(5) and SO(10) grand unification. It is noted that a family unification group that survived to near the weak scale and whose coupling extrapolated to high scales unified with those of the Standard model would be evidence accessible in principle at low energy of the existence of small (Planckian or GUT-scale) extra dimensions.Comment: 13 pages, 2 figures, minor corrections, references adde

Saliency for animated meshes with material properties

We propose a technique to calculate the saliency of animated meshes with material properties. The saliency computation considers multiple features of 3D meshes including their geometry, material and motion. Each feature contributes to the final saliency map which is view independent; and therefore, can be used for view dependent and view independent applications. To verify our saliency calculations, we performed an experiment in which we use an eye tracker to compare the saliencies of the regions that the viewers look with the other regions of the models. The results confirm that our saliency computation gives promising results. We also present several applications in which the saliency information is used. © 2010 ACM