Critical points in coupled Potts models and critical phases in coupled loop models

We show how to couple two critical Q-state Potts models to yield a new self-dual critical point. We also present strong evidence of a dense critical phase near this critical point when the Potts models are defined in their completely packed loop representations. In the continuum limit, the new critical point is described by an SU(2) coset conformal field theory, while in this limit of the the critical phase, the two loop models decouple. Using a combination of exact results and numerics, we also obtain the phase diagram in the presence of vacancies. We generalize these results to coupling two Potts models at different Q.Comment: 23 pages, 10 figure

Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory

Lambda<0 Quantum Gravity in 2+1 Dimensions I: Quantum States and Stringy S-Matrix

We consider the theory of pure gravity in 2+1 dimensions, with negative cosmological constant. The theory contains simple matter in the form of point particles; the later are classically described as lines of conical singularities. We propose a formalism in which quantum amplitudes for process involving black holes and point particles are obtained as Liouville field theory (LFT) correlation functions on Riemann surfaces X. Point particles are described by LFT vertex operators, black holes (asymptotic regions) are in correspondence with boundaries of X. We analyze two examples: the amplitude for emission of a particle by the BTZ black hole, and the amplitude of black hole creation by two point particles. We then define an inner product between quantum states. The value of this inner product can be interpreted as the amplitude for one set of point particles to go into another set producing black holes. The full particle S-matrix is then given by the sum of all such amplitudes. This S-matrix is that of a non-critical string theory, with the world-sheet CFT being essentially the Liouville theory. Lambda<0 quantum gravity in 2+1 dimensions is thus a string theory.Comment: v1: 22+1 pages, figures, v2: 25+1 pages, remarks on entropy and unitarity adde

Associations between genetic variations in the FURIN gene and hypertension

<p>Abstract</p> <p>Background</p> <p>Hypertension is a complex disease influenced by multiple genetic and environmental factors. The Kazakh ethnic group is characterized by a relatively high prevalence of hypertension. Previous research indicates that the FURIN gene may play a pivotal role in the renin-angiotensin system and maintaining the sodium-electrolyte balance. Because these systems influence blood pressure regulation, we considered FURIN as a candidate gene for hypertension. The purpose of this study was to systematically investigate the association between genetic variations in the FURIN gene and essential hypertension in a Xinjiang Kazakh population.</p> <p>Methods</p> <p>We sequenced all exons and the promoter regions of the FURIN gene in 94 hypertensive individuals to identify genetic variations associated with the disorder. Genotyping was performed using the TaqMan polymerase chain reaction method for four representative common single nucleotide polymorphisms (SNPs, -7315C > T, 1970C > G, 5604C > G, 6262C > T) in 934 Kazakh Chinese people. One SNP (1970C > G) was replicated in 1,219 Uygur Chinese people.</p> <p>Results</p> <p>Nine novel and seven known single nucleotide polymorphisms were identified in the FURIN gene. The results suggest that 1970C > G was associated with a hypertension phenotype in Kazakh Chinese (additive model, <it>P </it>= 0.091; dominant model, <it>P = </it>0.031, allele model, <it>P </it>= 0.030), and after adjustment with logistic regression analysis, ORs were 1.451 (95%CI 1.106-1.905, <it>P </it>= 0.008) and 1.496 (95% 1.103-2.028, <it>P </it>= 0.01) in additive and dominant models, respectively. In addition, the association between 1970C > G and hypertension was replicated in Uygur subjects (additive model, <it>P </it>= 0.042; dominant model, <it>P </it>= 0.102; allele model, <it>P </it>= 0.027) after adjustment in additive and dominant models, ORs were 1.327 (95% 1.07-1.646), <it>P </it>= 0.01 and 1.307 (95%CI 1.015-1.681, <it>P </it>= 0.038), respectively. G allele carriers exhibited significant lower urinary Na⁺excretion rate than non-carriers in the Kazakh Chinese population (152.45 ± 76.04 uM/min vs 173.33 ± 90.02 uM/min, <it>P </it>= 0.007).</p> <p>Conclusion</p> <p>Our results suggest that the FURIN gene may be a candidate gene involved in human hypertension, and that the G allele of 1970C > G may be a modest risk factor for hypertension in Xinjiang Kazakh and Uygur populations.</p

Kleinian groups and uniformization in examples and problems

Aimed at researchers, graduate students and undergraduates alike, this book presents a unified exposition of all the main areas and methods of the theory of Kleinian groups and the theory of uniformization of manifolds. The past 20 years have seen a rejuvenation of the field, due to the development of powerful new methods in topology, the theory of functions of several complex variables, and the theory of quasiconformal mappings. Thus this new book should provide a valuable resource, listing the basic facts regarding Kleinian groups and serving as a general guide to the primary literature, particularly the Russian literature in the field. In addition, the book includes a large number of examples, problems, and unsolved problems, many of them presented for the first time