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

Chiral Quasicrystalline Order and Dodecahedral Geometry in Exceptional Families of Viruses

On the example of exceptional families of viruses we i) show the existence of a completely new type of matter organization in nanoparticles, in which the regions with a chiral pentagonal quasicrystalline order of protein positions are arranged in a structure commensurate with the spherical topology and dodecahedral geometry, ii) generalize the classical theory of quasicrystals (QCs) to explain this organization, and iii) establish the relation between local chiral QC order and nonzero curvature of the dodecahedral capsid faces.Comment: 8 pages, 3 figure

Isotropic Conductivity of Two-Dimensional Three-Component Symmetric Composites

The effective dc-conductivity problem of isotropic, two-dimensional (2D), three-component, symmetric, regular composites is considered. A simple cubic equation with one free parameter for $\sigma_{e}(\sigma_1,\sigma_2,\sigma_3)$ is suggested whose solutions automatically have all the exactly known properties of that function. Numerical calculations on four different symmetric, isotropic, 2D, three-component, regular structures show a non-universal behavior of $\sigma_{e}(\sigma_1,\sigma_2,\sigma_3)$ with an essential dependence on micro-structural details, in contrast with the analogous two-component problem. The applicability of the cubic equation to these structures is discussed. An extension of that equation to the description of other types of 2D three-component structures is suggested, including the case of random structures. Pacs: 72.15.Eb, 72.80.Tm, 61.50.AhComment: 8 pages (two columns), 8 figures. J. Phys. A - submitte

Breakdown of universality in multi-cut matrix models

We solve the puzzle of the disagreement between orthogonal polynomials methods and mean field calculations for random NxN matrices with a disconnected eigenvalue support. We show that the difference does not stem from a Z2 symmetry breaking, but from the discreteness of the number of eigenvalues. This leads to additional terms (quasiperiodic in N) which must be added to the naive mean field expressions. Our result invalidates the existence of a smooth topological large N expansion and some postulated universality properties of correlators. We derive the large N expansion of the free energy for the general 2-cut case. From it we rederive by a direct and easy mean-field-like method the 2-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.Comment: 35 pages, Latex (1 file) + 3 figures (3 .eps files), revised to take into account a few reference

Tiling groupoids and Bratteli diagrams

Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. Let O be its tiling space with canonical transversal X. The tiling equivalence relation R_X is the set of pairs of tilings in X which are translates of each others, with a certain (etale) topology. In this paper R_X is reconstructed as a generalized "tail equivalence" on a Bratteli diagram, with its standard AF-relation as a subequivalence relation. Using a generalization of the Anderson-Putnam complex, O is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is built from this sequence, and its set of infinite paths dB is homeomorphic to X. The diagram B is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an etale equivalence relation R_B on dB which is homeomorphic to R_X, and contains the AF-relation of "tail equivalence".Comment: 34 pages, 4 figure

Glassy Random Matrix Models

This paper discusses Random Matrix Models which exhibit the unusual phenomena of having multiple solutions at the same point in phase space. These matrix models have gaps in their spectrum or density of eigenvalues. The free energy and certain correlation functions of these models show differences for the different solutions. Here I present evidence for the presence of multiple solutions both analytically and numerically. As an example I discuss the double well matrix model with potential $V(M)= -{\mu \over 2}M^2+{g \over 4}M^4$ where $M$ is a random $N\times N$ matrix (the $M^4$ matrix model) as well as the Gaussian Penner model with $V(M)={\mu\over 2}M^2-t \ln M$. First I study what these multiple solutions are in the large $N$ limit using the recurrence coefficient of the orthogonal polynomials. Second I discuss these solutions at the non-perturbative level to bring out some differences between the multiple solutions. I also present the two-point density-density correlation functions which further characterizes these models in a new university class. A motivation for this work is that variants of these models have been conjectured to be models of certain structural glasses in the high temperature phase.Comment: 25 pages, Latex, 7 Figures, to appear in PR

The Short Range RVB State of Even Spin Ladders: A Recurrent Variational Approach

Using a recursive method we construct dimer and nondimer variational ansatzs of the ground state for the two-legged ladder, and compute the number of dimer coverings, the energy density and the spin correlation functions. The number of dimer coverings are given by the Fibonacci numbers for the dimer-RVB state and their generalization for the nondimer ones. Our method relies on the recurrent relations satisfied by the overlaps of the states with different lengths, which can be solved using generating functions. The recurrent relation method is applicable to other short range systems. Based on our results we make a conjecture about the bond amplitudes of the 2-leg ladder.Comment: REVTEX file, 32 pages, 10 EPS figures inserted in text with epsf.st

Metal-insulator transition in the two-orbital Hubbard model at fractional band fillings: Self-energy functional approach

We investigate the infinite-dimensional two-orbital Hubbard model at arbitrary band fillings. By means of the self-energy functional approach, we discuss the stability of the metallic state in the systems with same and different bandwidths. It is found that the Mott insulating phases are realized at commensurate band fillings. Furthermore, it is clarified that the orbital selective Mott phase with one orbital localized and the other itinerant is stabilized even at fractional band fillings in the system with different bandwidths.Comment: 7 pages, 10 figure

Evolution of Linear Absorption and Nonlinear Optical Properties in V-Shaped Ruthenium(II)-Based Chromophores

In this article, we describe a series of complexes with electron-rich cis-{Ru^(II)(NH_3)_4}^(2+) centers coordinated to two pyridyl ligands bearing N-methyl/arylpyridinium electron-acceptor groups. These V-shaped dipolar species are new, extended members of a class of chromophores first reported by us (Coe, B. J. et al. J. Am. Chem. Soc. 2005, 127, 4845−4859). They have been isolated as their PF_6− salts and characterized by using various techniques including ^1H NMR and electronic absorption spectroscopies and cyclic voltammetry. Reversible Ru^(III/II) waves show that the new complexes are potentially redox-switchable chromophores. Single crystal X-ray structures have been obtained for four complex salts; three of these crystallize noncentrosymmetrically, but with the individual molecular dipoles aligned largely antiparallel. Very large molecular first hyperpolarizabilities β have been determined by using hyper-Rayleigh scattering (HRS) with an 800 nm laser and also via Stark (electroabsorption) spectroscopic studies on the intense, visible d → π^* metal-to-ligand charge-transfer (MLCT) and π → π^* intraligand charge-transfer (ILCT) bands. The latter measurements afford total nonresonant β_0 responses as high as ca. 600 × 10^(−30) esu. These pseudo-C_(2v) chromophores show two substantial components of the β tensor, β_(zzz) and β_(zyy), although the relative significance of these varies with the physical method applied. According to HRS, β_(zzz) dominates in all cases, whereas the Stark analyses indicate that β_(zyy) is dominant in the shorter chromophores, but β_(zzz) and β_(zyy) are similar for the extended species. In contrast, finite field calculations predict that β_(zyy) is always the major component. Time-dependent density functional theory calculations predict increasing ILCT character for the nominally MLCT transitions and accompanying blue-shifts of the visible absorptions, as the ligand π-systems are extended. Such unusual behavior has also been observed with related 1D complexes (Coe, B. J. et al. J. Am. Chem. Soc. 2004, 126, 3880−3891)

Blood Pressure Lowering With Nilvadipine in Patients With Mild-to-Moderate Alzheimer Disease Does Not Increase the Prevalence of Orthostatic Hypotension

BACKGROUND: Hypertension is common among patients with Alzheimer disease. Because this group has been excluded from hypertension trials, evidence regarding safety of treatment is lacking. This secondary analysis of a randomized controlled trial assessed whether antihypertensive treatment increases the prevalence of orthostatic hypotension (OH) in patients with Alzheimer disease. METHODS AND RESULTS: Four hundred seventy‐seven patients with mild‐to‐moderate Alzheimer disease were randomized to the calcium‐channel blocker nilvadipine 8 mg/day or placebo for 78 weeks. Presence of OH (blood pressure drop ≥20/≥10 mm Hg after 1 minute of standing) and OH‐related adverse events (dizziness, syncope, falls, and fractures) was determined at 7 follow‐up visits. Mean age of the study population was 72.2±8.2 years and mean Mini‐Mental State Examination score was 20.4±3.8. Baseline blood pressure was 137.8±14.0/77.0±8.6 mm Hg. Grade I hypertension was present in 53.4% (n=255). After 13 weeks, blood pressure had fallen by −7.8/−3.9 mm Hg for nilvadipine and by −0.4/−0.8 mm Hg for placebo (P<0.001). Across the 78‐week intervention period, there was no difference between groups in the proportion of patients with OH at a study visit (odds ratio [95% CI]=1.1 [0.8–1.5], P=0.62), nor in the proportion of visits where a patient met criteria for OH, corrected for number of visits (7.7±13.8% versus 7.3±11.6%). OH‐related adverse events were not more often reported in the intervention group compared with placebo. Results were similar for those with baseline hypertension. CONCLUSIONS: This study suggests that initiation of a low dose of antihypertensive treatment does not significantly increase the risk of OH in patients with mild‐to‐moderate Alzheimer disease. CLINICAL TRIAL REGISTRATION: URL: https://www.clinicaltrials.gov. Unique identifier: NCT02017340