Affine Wa(A4), Quaternions, and Decagonal Quasicrystals

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I2 (h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh of order 2h as a maximal subgroup. As a simple application we demonstrate projections of the root and weight lattices of A4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine Wa(A4) can be decomposed into two orthogonal spaces whose point groups is the dihedral group D5 which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group Wa(A3), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group Wa(An), n>=4 . We also note that the projection of the Voronoi cell of the root lattice of Wa(A4) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.Comment: 26 pages, 17 figure

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

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

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

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

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

A4 Flavor Models in Split Seesaw Mechanism

A seesaw mechanism in an extra-dimension, known as the split seesaw mechanism, provides a natural way to realize a splitting mass spectrum of right-handed neutrinos. It leads to one keV sterile neutrino as a dark matter candidate and two heavy right-handed neutrinos being responsible for leptogenesis to explain the observed baryon asymmetry of the Universe. We study models based on $A_4$ flavor symmetry in the context of the split seesaw mechanism. It is pointed out that most of known $A_4$ flavor models with three right-handed neutrinos being $A_4$ triplet suffer from a degeneracy problem for the bulk mass terms, which disturbs the split mechanism for right-handed neutrino mass spectrum. Then we construct a new $A_4$ flavor model to work in the split seesaw mechanism. In the model, the experimentally observed neutrino masses and mixing angles can be realized from both type I+II seesaw contributions. The model predicts the $\mu-\tau$ symmetry in the neutrino mass matrix at the leading order, resulting in the vanishing $\theta_{13}$ and maximal $\theta_{23}$. The flavor symmetry $A_4$ is broken via the flavon vacuum alignment which can be obtained from the orbifold compactification. The model can be consistent with all data of neutrino oscillation experiments, cosmological discussions of dark matter abundance, leptogenesis, and recent astrophysical data.Comment: 21 pages, 1 figure, version to appear in JHE

Numerical study of effect of elastomeric stress absorbers on stress reduction in bone-dental implant interface

Objective This paper focused on optimal stress distribution in the mandibular bone surrounding a dental implant and is devoted to the development of a modified Osteoplant® implant type in order to minimize stress concentration in the bone-implant interface. Material and Methods This study investigated 0.4 mm thick layers of two elastomeric stress barriers incorporated into the dental implant using 3-D finite element analysis. Results Overall, this proposed implant provoked lower load transfer in bone-implant interface due to the effect of the elastomers as stress absorbers. The stress level in the bone was reduced between 28% and 42% for three load cases: 75 N, 60 N and 27 N in corono-apical, linguo-buccal and disto-mesial direction, respectively. Conclusion The proposed model provided an acceptable solution for load transfer reduction to the mandible. This investigation also permitted to choose how to incorporate two elastomers into the Osteoplant® implant system