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

Quasi-exact-solution of the Generalized Exe Jahn-Teller Hamiltonian

We consider the solution of a generalized Exe Jahn-Teller Hamiltonian in the context of quasi-exactly solvable spectral problems. This Hamiltonian is expressed in terms of the generators of the osp(2,2) Lie algebra. Analytical expressions are obtained for eigenstates and eigenvalues. The solutions lead to a number of earlier results discussed in the literature. However, our approach renders a new understanding of ``exact isolated'' solutions

Quaternionic Root Systems and Subgroups of the Aut(F4)Aut(F_{4})

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4),SP(2)= SO(5),SO(8),SO(9),F_{4} and E_{8} in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E_{8} which form a closed octonion algebra. The automorphism group Aut(F_{4}) of the Dynkin diagram of F_{4} of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F_{4}.The Weyl groups of many Lie algebras, such as, G_{2},SO(7),SO(8),SO(9),SU(3)XSU(3) and SP(3)X SU(2) have been constructed as the subgroups of Aut(F_{4}). We have also classified the other non-parabolic subgroups of Aut(F_{4}) which are not Weyl groups. Two subgroups of orders192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group $G_{2}$ of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e_{1},e_{2},e_{3} is used to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and F_{4} respectively

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

Effect of lactic acid bacteria and the potential probiotic Hafnia alvei on growth and survival rates of narrow clawed crayfish (Astacus leptodactylus Esch., 1823) stage II juveniles

The aim of this study was to screen potential probiotic bacteria against Aeromonas hydrophila and determine the effects of antagonistic bacteria and a commercial product containing lactic acid bacteria on the survival and growth of stage II Astacus leptodactylus juveniles. For this purpose, a total of 110 bacterial strains were isolated from adult, stage II crayfish juveniles and rearing water screened for antagonistic activities against A. hydrophila with well diffusion agar assay. Hafnia alvei strain from stage II crayfish juveniles displayed the inhibition zone (10mm) against A. hydrophila. The experiment was conducted in a completely randomized design with four treatments for 60 days: (I) crayfish fed with live food without probiotics (control group); (II) crayfish fed with live food enriched with lactic acid bacteria (0.015 gL^-1); (III) crayfish fed with live food enriched with Hafnia alvei (10^6 CFU mL^−1); (IV) crayfish fed with control diet and H. alvei added to rearing water (10^6 CFU mL^−1). As a result of this study, lactic acid bacteria and Hafnia alvei applications did not positively affect growth and survival of stage II A. leptodactylus juveniles. In the future, studies on screening potential probiotic bacteria should be used in vitro and in vivo tests. In addition, it will be useful to investigate the lactic acid bacteria and Bacillus spp. from indigenous microflora of crayfish

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

Cilostazol enhances atorvastatin-induced vasodilation of female rat aorta during aging

Statins have cholesterol-independent effects including an increased vascular nitric oxide activity and are commonly used by patients with cardiovascular disease. Such patients frequently have cardiovascular diseases, which may be treated with cilostazol, a platelet aggregation inhibitor. This study was designed to investigate whether combined use of cilostazol would increase the inhibitory effect of statin on vascular smooth muscle and how maturation would affect these responses. Female Wistar rats, aged 3–4 months (young) and 14–15 months (adult), were sacrificed by cervical dislocation and the thoracic aorta was dissected and cut into 3- to 4-mm-long rings. The rings were mounted under a resting tension of 1 g in a 20-ml organ bath filled with Krebs–Henseleit solution. Rings were precontracted with phenylephrine (10−6 M), and the presence of endothelium was confirmed with acetylcholine (10−6 M). Then, the concentration–response curves were obtained for atorvastatin alone (10−10 to 3 × 10−4 M; control) and in the presence of cilostazol (10−6 M) in young and adult rat aortas. This experimental protocol was also carried out in aorta rings, which had been pretreated with NG-nitro-l-arginine methyl ester (l-NAME, 10−4 M). Atorvastatin induced concentration-dependent relaxations in young and adult rat thoracic aorta rings precontracted with phenylephrine. The pIC50 value of atorvastatin was significantly decreased in adult rat aortas. In addition, pretreatment of aortas with cilostazol enhanced the potency of atorvastatin in both young and adult aortas. Incubation with l-NAME did not completely eliminate the relaxations to atorvastatin in the presence of cilostazol. These results suggest that combined application of cilostazol with atorvastatin was significantly more potent than atorvastatin alone. Combined drug therapy may be efficacious in delaying the occurrence of cardiovascular events

Spatiotemporal graph indicators for air traffic complexity analysis

There has been extensive research in formalising air traffic complexity, but existing works focus mainly on a metric to tie down the peak air traffic controllers workload rather than a dynamic approach to complexity that could guide both strategical, pre-tactical and tactical actions for a smooth flow of aircraft. In this paper, aircraft interdependencies are formalized using graph theory and four complexity indicators are described, which combine spatiotemporal topological information with the severity of the interdependencies. These indicators can be used to predict the dynamic evolution of complexity, by not giving one single score, but measuring complexity in a time window. Results show that these indicators can capture complex spatiotemporal areas in a sector and give a detailed and nuanced view of sector complexity

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

Dynamics of Ku and bacterial non-homologous end-joining characterized using single DNA molecule analysis

We use single-molecule techniques to characterize the dynamics of prokaryotic DNA repair by non-homologous end-joining (NHEJ), a system comprised only of the dimeric Ku and Ligase D (LigD). The Ku homodimer alone forms a ∼2 s synapsis between blunt DNA ends that is increased to ∼18 s upon addition of LigD, in a manner dependent on the C-terminal arms of Ku. The synapsis lifetime increases drastically for 4 nt complementary DNA overhangs, independently of the C-terminal arms of Ku. These observations are in contrast to human Ku, which is unable to bridge either of the two DNA substrates. We also demonstrate that bacterial Ku binds the DNA ends in a cooperative manner for synapsis initiation and remains stably bound at DNA junctions for several hours after ligation is completed, indicating that a system for removal of the proteins is active in vivo. Together these experiments shed light on the dynamics of bacterial NHEJ in DNA end recognition and processing. We speculate on the evolutionary similarities between bacterial and eukaryotic NHEJ and discuss how an increased understanding of bacterial NHEJ can open the door for future antibiotic therapies targeting this mechanism