Transverse Laplacians for Substitution Tilings

Pearson and Bellissard recently built a spectral triple – the data of Riemannian noncommutative geometry – for ultrametric Cantor sets. They derived a family of Laplace–Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversals of tiling spaces, which are particular self-similar Cantor sets. We use Bratteli diagrams to encode the self-similarity, and Cuntz–Krieger algebras to implement it. We show that the abscissa of convergence of the ζ-function of the spectral triple gives indications on the exponent of complexity of the tiling. We determine completely the spectrum of the Laplace–Beltrami operators, give an explicit method of calculation for their eigenvalues, compute their Weyl asymptotics, and a Seeley equivalent for their heat kernels.     

平面全息光栅刻线密度的倍频式调整方法

刻线密度准确与否直接影响光栅色散及给定波长的衍射方向,进而影响光谱仪器结构设计.为了提高全息光栅刻线密度的制作精度,提出了平面全息光栅刻线密度的倍频式调整方法.将给定刻线密度的基准光栅放在干涉场曝光区域内,调节光束干涉角,干涉场曝光光束经基准光栅衍射后,根据调整光栅刻线密度的不同选择不同的衍射级次相互叠加形成莫尔条纹,...     

Interlaced Particle Systems and Tilings of the Aztec Diamond

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N, and particles on successive lines are subject to an interlacing constraint. It is shown that this particle system is exactly solvable, to the extent that not only can the partition function be computed exactly, but so too can the marginal distributions. These results in turn are used to give new derivations within the particle picture of a number of known fundamental properties of the tiling problem, for example that the number of distinct configurations is 2 ^N(N+1)/2, and that there is a limit to the GUE minor process, which we show at the level of the joint PDFs. It is shown too that the study of tilings of the half Aztec diamond—not known from earlier literature—also leads to an interlaced particle system, now with successive lines 2n−1 and 2n (n=1,…,N/2−1) having n particles. Its exact solution allows for an analysis of the half Aztec diamond tilings analogous to that given for the Aztec diamond tilings.     

Convergence to Equilibrium for Spin Glasses

We derive estimates on the thermalisation times for dynamical spin glasses at high temperature. Received: 4 November 1999 / Accepted: 15 May 2000     

Convergence Time to Equilibrium of the Metropolis Dynamics for the GREM

Journal of Statistical Physics - We study the convergence time to equilibrium of the Metropolis dynamics for the generalized random energy model with an arbitrary number of hierarchical levels, a...     

Convergence to Equilibrium for Intermittent Symplectic Maps

We investigate a class of area preserving non-uniformly hyperbolic maps of the two torus. First we establish some results on the regularity of the invariant foliations, then we use this knowledge to estimate the rate of mixing.     

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.     

Exponential Convergence to the Maxwell Distribution for Some Class of Boltzmann Equations

We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space where is the one-particle phase space and is the Liouville measure on Γ⁽¹⁾. Special solutions of these equations, called “Maxwellians,” are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted L ¹-space converge to a Maxwellian in , exponentially fast in time.     

Convergence to Equilibrium for the Discrete Coagulation-Fragmentation Equations with Detailed Balance

Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker–Döring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.     

全固化非平面单频Nd：YAG环形激光器

利用Ｎｄ：ＹＡＧ激光增益介质的法拉第效应，以及光束在非平面环形谐振腔中传播产生的偏振面旋转，使激光二极管泵浦的环形Ｎｄ：ＹＡＧ激光器单向运转，获得单频激光输出，最大单纵模激光（１．０６４μｍ）输出为３６５ｍＷ，功率波动小于２％，短期频率漂移小于４５ＭＨｚ，并利用内腔倍频技术得到单频绿光输出，最大单纵模绿光（５３２ｎｍ）输出为７５ｍＷ，功率波动小于４％。     

Transverse Properties of Light Distribution in the Focal Plane for Rectangular Annular Pupil Plane Filters

Rectangular annular pupils are designed. The transverse intensity distribution and the superresolving characteristics of the pupils in the focal plane are analyzed. The influences of total annular zone numbers and the central core magnitude on the light efficiency (the Strehl ratio S), the superresolving power (the normalized spot size G) and the intensity of sidelobe are studied in detail. The properties are different for two kinds of pupils: one with total odd annular zone numbers and another with total even annular zone numbers. Circular annular pupils are also discussed and the comparison between them is given.     

Transverse Properties of Light Distribution in the Focal Plane for Rectangular Annular Pupil Plane Filters

1　Ｉｎｔｒｏｄｕｃｔｉｏｎ　　Ｐｕｐｉｌ ｐｌａｎｅｆｉｌｔｅｒｓｃａｎｂｅｕｓｅｄｔｏｃｈａｎｇｅｔｈｅｌｉｇｈｔｉｎｔｅｎｓｉｔｙｄｉｓｔｒｉｂｕｔｉｏｎｉｎｔｈｅｆｏｃａｌｐｌａｎｅ ,ｓｏｔｈｅｙｈａｖｅｐｏｔｅｎｔｉａｌａｐｐｌｉｃａｔｉｏｎｓｉｎｔｈｅａｒｅａｓｏｆｏｐｔｉｃａｌｄａｔａｓｔｏｒａｇｅ[1～ 3 ] ,ｌｉｔｈｏｇｒａｐｈｙ[4] ,ａｎｄｍｉｃｒｏｓｃｏｐｙ[5～ 7] .Ｍａｎｙｐｕｐｉｌｆｉｌｔｅｒｓｈａｖｅｂｅｅｎｓｔｕｄｉｅｄ ,ｓｕｃｈａｓｆｉｌｔｅｒｓｗｉｔｈｖａｒｉｅｄａｍｐ…     

Convergence of the valence-bond calculation for methane

A minimal Slater basis set valence-bond calculation for the ground state of methane, including all possible structures not involving carbon 1s² excitations is reported. Simplifications for the computation of the matrix elements between Slater determinants are outlined. Comparisons are made with truncated expansions that include only certain valence-bond structures chosen according to physical criteria and with SCF-CI calculations. The result suggests that suitably selected valence-bond functions can be of utility for potential energy surface calculations.     

Convergence to equilibrium of the stochastic Heisenberg model

The stochastic Heisenberg model is a probabilistic model of the time evolution of a classical Heisenberg ferromagnet. It is proved that the stochastic process converges to equilibrium at sufficiently high temperatures, and that the equilibrium state is a Gibbs state of the Hamiltonian possessing the global Markov property. The principal technique employed is the expansion of observables on the state space into Laplace series.     

Convergence of solutions to the rough-surface scattering problem

Abstract

A new formulation of the rough-surface scattering problem is obtained that is closely linked to the Kirchhoff approximation. The governing equation is cast into a form amenable to solution by the method of successive approximations. The domain of convergence of this solution is established and is shown to apply also to the odd-ordered operator expansion, small-slope approximation and perturbation theory provided that the slope of the scattering surface is everywhere less than unity. The analysis is performed for scattering from one-dimensional pressure-release surfaces. Numerical examples are presented for sinusoidal and echelette gratings.     

Convergence of solutions to the rough-surface scattering problem

A new formulation of the rough-surface scattering problem is obtained that is closely linked to the Kirchhoff approximation. The governing equation is cast into a form amenable to solution by the method of successive approximations. The domain of convergence of this solution is established and is shown to apply also to the odd-ordered operator expansion, small-slope approximation and perturbation theory provided that the slope of the scattering surface is everywhere less than unity. The analysis is performed for scattering from one-dimensional pressure-release surfaces. Numerical examples are presented for sinusoidal and echelette gratings.     

Convergence of the Stochastic Six-Vertex Model to the ASEP

In this note we establish the convergence of the stochastic six-vertex model to the one-dimensional asymmetric simple exclusion process, under a certain limit regime recently predicted by Borodin-Corwin-Gorin. This convergence holds for arbitrary initial data.     

Convergence to diffusion waves of solutions for viscous conservation laws

We study the large-time behavior of solutions of viscous conservation laws. It is shown that solutions tend to diffusion waves, which are constructed based on the heat equation and Burgers equation. The convergence is in theL _p , 1p sense and is obtained as a consequence of theL ₂ decay of the difference between the solution and its asymptotic state of diffusion waves.Supported by the National Sciences Council of the Republic of China under the contract NSC-76-0208-M-001-09Supported by NSF under Grant No. DMS-84-01355