The Hyperbolic Volume of Knots from the Quantum Dilogarithm

The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number N. By the analysis of particularexamples, it is argued that, for a hyperbolic knot (link), the absolute valueof this invariant grows exponentially at large N, the hyperbolic volume of the knot (link) complement being the growth rate.     

Ising Models on Hyperbolic Graphs II

We consider Ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph (v, f), where v is the number of neighbors of each vertex and f is the number of sides of each face. Let T _c be the critical temperature and T _c =supTT _c: ^f=( ⁺+ ^–)/2, where ^f is the free boundary condition (b.c.) Gibbs state, ⁺ is the plus b.c. Gibbs state and ^– is the minus b.c. Gibbs state. We prove that if the hyperbolic graph is self-dual (i.e., v=f) or if v is sufficiently large (how large depends on f, e.g., v35 suffices for any f3 and v17 suffices for any f17) then 0<T _c <T _c, in contrast with that T _c =T _c for Ising models on the hypercubic lattice Z ^d with d2, a result due to Lebowitz.⁽²²⁾ While whenever T<T _c , ^f=( ⁺+ ^–)/2. The last result is an improvement in comparison with the analogous statement in refs. 28 and 33, in which it was only proved that ^f=( ⁺+ ^–)/2 when TT _c and it remains to show in both papers that ^f =( ⁺+ ^–)/2 whenever T<T _c . Therefore T _c and T _c divide [0, ] into three intervals: [0, T _c ), (T _c , T _c), and (T _c, ] in which ^{+ –} but ^f =( ⁺+ ^–)/2, ^{+ –} and ^f ( ⁺+ ^–)/2, and ⁺= ^–, respectively.     

A Note on Dimer Models and McKay Quivers

We give one formulation of a procedure of Hanany and Vegh (J High Energy Phys 0710(029):35, 2007) which takes a lattice polygon as an input and produces a set of isoradial dimer models. We study the case of lattice triangles in detail and discuss the relation with coamoebas following Feng et al. (Adv Theor Math Phys 12(3):489–545, 2008).     

Bootstrap Percolation and Kinetically Constrained Models on Hyperbolic Lattices

We study bootstrap percolation (BP) on hyperbolic lattices obtained by regular tilings of the hyperbolic plane. Our work is motivated by the connection between the BP transition and the dynamical transition of kinetically constrained models, which are in turn relevant for the study of glass and jamming transitions. We show that for generic tilings there exists a BP transition at a nontrivial critical density, 0<ρ _c <1. Thus, despite the presence of loops on all length scales in hyperbolic lattices, the behavior is very different from that on Euclidean lattices where the critical density is either zero or one. Furthermore, we show that the transition has a mixed character since it is discontinuous but characterized by a diverging correlation length, similarly to what happens on Bethe lattices and random graphs of constant connectivity.     

W-algebras and superalgebras from constrained WZW models: A group theoretical classification

We present a classification ofW algebras and superalgebras arising in Abelian as well as non Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with anSl(2) subalgebra (resp.OSp(1/2) superalgebra) of a simple Lie algebra (resp. superalgebra)G. However, the determination of anU(1) _Y factor, commuting withSl(2) (resp.OSp(1/2)), appears, when it exists, particularly useful to characterize the correspondingW algebra. The (super) conformal spin contents of eachW (super) algebra is performed. The class of all the superconformal algebras (i.e. with conformal spinss<=2) is easily obtained as a byproduct of our general results.     

A Note on Equivalence Among Various Scalar Field Models of Dark Energies

In this work, we have tried to find out similarities between various available models of scalar field dark energies (e.g., quintessence, k-essence, tachyon, phantom, quintom, dilatonic dark energy, etc). We have defined an equivalence relation from elementary set theory between scalar field models of dark energies and used fundamental ideas from linear algebra to set up our model. Consequently, we have obtained mutually disjoint subsets of scalar field dark energies with similar properties and discussed our observation.     

A Summation Formula for Macdonald Polynomials

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases \({t = 1}\) and \({q = 0}\), we recover known expressions for the monomial symmetric and Hall–Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q–Whittaker polynomials.     

Derivative Moments for Characteristic Polynomials from the CUE

We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity. The latter asymptotic calculation allows us to prove a long-standing conjecture from random matrix theory.     

A Conjecture about Raising Operators for Macdonald Polynomials

We present a conjecture giving Macdonald polynomials as an hypergeometric-type series in terms of raising operators. It is proved that this conjecture agrees with Jing and Józefiak’s expression in the two-row case, and with Lassalle and Schlosser’s formula in the three-row case.Mathematics Subject Classifications (2000). 33D52, 33D67, 81R50     

A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.     

A Volume Comparison Theorem for Asymptotically Hyperbolic Manifolds

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least ?6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter–Schwarzschild metrics.     

A Note on the Guerra and Talagrand Theorems for Mean Field Spin Glasses: The Simple Case of Spherical Models

The aim of this paper is to discuss the main ideas of the Talagrand proof of the Parisi Ansatz for the free-energy of Mean Field Spin Glasses with a physicist's approach. We consider the case of the spherical p-spin model, which has the following advantages: (1) the Parisi Ansatz takes the simple “one step replica symmetry breaking form,” (2) the replica free-energy as a function of the order parameters is simple enough to allow for numerical maximization with arbitrary precision. We present the essential ideas of the proof, we stress its connections with the theory of effective potentials for glassy systems, and we reduce the technically more difficult part of the Talagrand's analysis to an explicit evaluation of the solution of a variational problem.     

广州西村窑彩绘瓷无损分析检测

利用微聚焦X射线荧光探针与拉曼光谱仪2种方法,对西村窑彩绘瓷进行了系统的分析测试,结果表明西村窑彩绘瓷胎料选择了一种高质量的高岭土,瓷釉中氧化磷的含量远高于相应的瓷胎,暗示瓷釉中添加了草木灰作为助熔剂;瓷器彩绘的显色物相为赤铁矿(α-Fe2O3),彩绘的不同颜色主要与其中氧化铁含量有关,彩绘表面条状物经鉴定为钙长石(CaAl2Si2O8)。     

A Note on Palindromicity

Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin–Shapiro sequence. The second provides a constructive method to build palindromic minimal sequences based upon regular, generic model sets with centro-symmetric window. These give rise to diagonal tight-binding models in one dimension with purely singular continuous spectrum.     

Torus Knots and Links from Eikonal Equations and Knot Invariants for Classification of Atoms

The history of knot theory and physics has a deep roots. It started by Lord Kelvin, in 1867, when he conjectured that atoms were knotted vortex tubes of ether. In 1997, Faddeev and Niemi suggested that knots might exist as stable soliton solution in a simple three dimensional classical field theory. That opening up a wide range of possible applications in physics. In this work we consider the Eikonal equation, which is a partial differential equation describing the traveltime propagation, which is an important part of seismic imaging algorithms. We will follow the work of Wereszczynski of solving the Eikonal equation in cylindrical coordinates. We show that only torus knots and links do occur, so figure eight knot does not occur. We show that these solutions are not unique, which means the possible occurrence of the same knot type for different configurations. Using the idea of framed knots, it is shown that two Eikonal knots are equivalent if and only if they are ambient isotopic as a framed knots, i.e. if and only if they are of the same knot type and of the same twisting number.     

双曲型守恒律的一种五阶半离散中心迎风格式

给出一种求解双曲型守恒律的五阶半离散中心迎风格式.对一维问题,该格式以五阶中心WENO重构为基础;对二维问题,用逐维计算的方法将五阶中心WENO重构进行推广.时间方向的离散采用Runge-Kutta方法.格式保持了中心差分格式简单的优点,即不用求解Riemann问题,避免进行特征分解.用该格式对一维和二维Euler方程进行数值试验,结果表明该格式是高精度、高分辨率的.