An energy law preserving C finite element scheme for simulating the kinematic effects in liquid crystal dynamics

In this paper, we use finite element methods to simulate the hydrodynamical systems governing the motions of nematic liquid crystals in a bounded domain Ω. We reformulate the original model in the weak form which is consistent with the continuous dissipative energy law for the flow and director fields in W^1,2+σ(Ω) (σ > 0 is an arbitrarily small number). This enables us to use convenient conformal C⁰ finite elements in solving the problem. Moreover, a discrete energy law is derived for a modified midpoint time discretization scheme. A fixed iterative method is used to solve the resulted nonlinear system so that a matrix free time evolution may be achieved and velocity and director variables may be solved separately. A number of hydrodynamical liquid crystal examples are computed to demonstrate the effects of the parameters and the performance of the method.     

On a Class of Integrable Systems with a Cubic First Integral

A few years ago Selivanova gave an existence proof for some integrable models, in fact geodesic flows on two dimensional manifolds, with a cubic first integral. However the explicit form of these models hinged on the solution of a nonlinear third order ordinary differential equation which could not be obtained. We show that an appropriate choice of coordinates allows for integration and gives the explicit local form for the full family of integrable systems. The relevant metrics are described by a finite number of parameters and lead to a large class of models mainly on the manifolds \mathbb S²{{\mathbb S}^2} and \mathbb H²{{\mathbb H}^2} . Many of these systems are globally defined and contain as special cases integrable systems due to Goryachev, Chaplygin, Dullin, Matveev and Tsiganov.     

Nonlinear Instability for the Navier-Stokes Equations

It is proved, using a bootstrap argument, that linear instability implies nonlinear instability for the incompressible Navier-Stokes equations in L^p for all p ∈ (1,∞) and any finite or infinite domain in any dimension n.     

Nonlinear Transformation Group of CAR Fermion Algebra

Based on our previous work on the recursive fermion system in the Cuntz algebra, it is shown that a nonlinear transformation group of the CAR fermion algebra is induced from a U(2 ^p ) action on the Cuntz algebra ₂ ^p with an arbitrary positive integer p. In general, these nonlinear transformations are expressed in terms of finite polynomials in generators. Some Bogoliubov transformations are involved as special cases.     

Nonlinear heat transport near the λ transition in4He

The effect of a finite stationary heat current is investigated in the spatially inhomogenous situation where the heat current induces an interface between normal-liquid and superfluid⁴He. The nonlinear temperature profile in the vicinity of the interface and the local thermal conductivity are calculated forT>T within modelF up to oneloop order. The field-theoretic renormalization-group approach is employed to describe the critical behavior both in the linear and nonlinear response regime. The finite heat currentQ causes a finite temperature gradient atT and therefore suppresses the critical divergence of the thermal conductivity. Quantitative predictions are made on the nonlinearQ dependence of the temperature profile and of the thermal conductivity which should be experimentally observable.     

一种新型的多模虚偶相干态光场的等阶N次方Y压缩与等阶N次方H压缩

本文利用新近建立的多模压缩态理论,详细研究了一种新型的多模虚偶相干态光场|Ψ_i,e⁽²⁾>_q的广义非线性等阶N次方Y压缩与等阶N次方H压缩特性.结果发现:1)态|Ψ_i,e⁽²⁾>_q是一种典型的多模非经典光场,当压缩阶数N为奇数时,态|Ψ_i,e⁽²⁾>_q在一定条件下总可呈现出周期性变化的、任意阶的等阶N次方Y压缩效应;当腔模总数q与压缩阶数N这两者的乘积q·N为奇数时,则在一定条件下态|Ψ_i,e⁽²⁾>_q又可呈现出周期性变化的、任意阶的等阶N次方H压缩效应.2)态|Ψ_i,e⁽²⁾>_q的等阶N次方Y压缩与等阶N次方H压缩效应这两者的压缩程度和压缩深度分别与几率幅γ_q^(e)、压缩参数R_j、各模的初始相位φ_j(或者各模的初始相位和 φ_j)、压缩阶数N以及腔模(指纵模)总数q等呈较强的非线性关联,等阶N次方H压缩效应与上述诸参量之间的非线性关联程度要比等阶N次方Y压缩效应的更强.3)多模虚偶相干态光场|Ψ_i,e⁽²⁾>_q与多模偶相干态光场|Ψ_,e>_q及多模复共轭偶相干态光场|Ψ*_,e⁽²⁾>_q这后两者的等阶N次方Y压缩效应和等阶N次方H压缩效应的压缩条件和压缩特性正好相反,这种现象就称为相反压缩.     

Inhomogeneous Ballistic Aggregation

The one-dimensional ballistic aggregation process is considered when the initial mass density or the initial particle velocities vanish outside of a finite or semi-infinite interval. In all cases, we compute the mass distributions in closed analytical form and study their long time asymptotics. The relevant length scales are found different (of the order t, t ^2/3, t ^1/2) if, at the initial time, particles occupy a finite (or semi-infinite) interval and if a finite (or infinite) number of them are set into motion.     

Analysis of the Karmarkar-Karp differencing algorithm

The Karmarkar-Karp differencing algorithm is the best known polynomial time heuristic for the number partitioning problem, fundamental in both theoretical computer science and statistical physics. We analyze the performance of the differencing algorithm on random instances by mapping it to a nonlinear rate equation. Our analysis reveals strong finite size effects that explain why the precise asymptotics of the differencing solution is hard to establish by simulations. The asymptotic series emerging from the rate equation satisfies all known bounds on the Karmarkar-Karp algorithm and projects a scaling n ^{−c ln n} , where c = 1/(2 ln 2) = 0.7213 .... Our calculations reveal subtle relations between the algorithm and Fibonacci-like sequences, and we establish an explicit identity to that effect.     

On the fundamentals of extended thermodynamics (ET) of a one-dimensional rarefied gas

Extended thermodynamics (ET) of degreer for a one-dimensional rarefied gas based, by definition, on a finite set A^r={a⁰, a²,..., a^r} of the firstr–1(3r) direct internal moments of the one-point distribution functionf is carefully investigated. With the aid of the second axiom of thermodynamics, the new representation forf, depending in a local and nonlinear way onA ^r , is explicitly derived. It is demonstrated that in ET of degreer an infinite sequence {b^r+1, b^r+2,...} ofhigher order Hermite coefficients, which normally drops out of Grad's proposition forf fashioned by mathematical apparatus such as the Hermite polynomials, cannot be considered negligible in the case when nonlinear constitutive functions are established. Using Ma's kinetic equation corresponding to a one-dimensional rarefied gas as well as the generalized representation forf, collision productions in the nonconservative moment equations are then calculated for a special choice of the rate of collisions between particles.     

Superparamagnetism in the 1D Ising model

The temperature dependence of the order parameter (〈M ²〉, the mean-square spin excess per spin) for a system with a finite number of spins of the 1D Ising model was investigated. It was shown that there is a nonzero temperature T _s below which 〈M ²〉 rapidly tends to unity. The analogy between the 1D ensemble of a finite number of spins and a system of superparamagnetic particles is discussed.     

A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings

Let S:[0,1]→[0,1] be a nonsingular transformation such that the corresponding Frobenius-Perron operator P _S :L ¹(0,1)→L ¹(0,1) has a stationary density f ^∗. We propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f ^∗. An advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply Newton’s method, the Jacobian matrices are positive-definite and tri-diagonal. The numerical experiments show that the new maximum entropy method is more accurate than the Markov finite approximation method, which also uses piecewise linear functions, provided that the involved moments are known. This is supported by the convergence rate analysis of the method.     

Features of the hydrodynamic pressure field of an electric explosion in an equilibrium gas-liquid medium

The hydrodynamic features of an electric explosion in a bubble gas-liquid mixture are studied in the equilibrium approximation of the medium for the case of fine gas bubbles when the initial size of the latter ranges from units to tens of micrometers, as is observed when actual liquids contain natural gas. In mathematically modeling the electric-explosion processes, the characteristics of the hydrodynamic field were calculated, taking into account the finite size of the plasma piston for which the quasi-wave equation with the nonlinear barotropic equation of state of the mixture was numerically integrated, using an explicit finite-difference scheme in an ellipsoidal coordinate system. It is established that the presence of gaseous inclusions manifests itself when the gas concentration is ɛ ₀≳10⁻⁴, whereas appreciable nonlinear effects appear when ɛ ₀≳5×10⁻³. Zh. Tekh. Fiz. 68, 7–12 (July 1998)     

多模复共轭虚偶相干态光场的等阶N次方Y压缩与等阶N次方H压缩

本文利用新近建立的多模压缩态理论,详细研究了一种新型的多模复共轭虚偶相干态光场|Ψ_i,e^*(2)>_q的广义非线性等阶N次方Y压缩与等阶N次方H压缩特性.结果发现:1)当压缩阶数N为奇数时,态Ψ_i,e^*(2)>_q在一定条件下总可呈现出周期性变化的、任意阶的等阶N次方Y压缩效应;当腔模总数q与压缩阶数N两者之乘积q·N为奇数时,则在一定条件下态|Ψ_i,e^*(2)>_q又可呈现出周期性变化的、任意阶的等阶N次方H压缩效应.2)态|Ψ_i,e^*(2)>_q与多模偶相干态光场|Ψ_i,e^*(2)>_q与多模虚偶相干态光场|Ψ_i,e⁽²⁾>_q的压缩幅度、压缩条件和压缩特征等完全相同.这一结果再次从理论上证实了多模压缩光中的确存在着“相反压缩”和“压缩简并”这两类新的物理现象.     

Étude numérique de l'influence du transfert radiatif sur la convection naturelle laminaire,bidimensionnelle, permanente,dans un espace annulaire d'axe horizontal,délimité par deux cylindres circulaires isothermes

Combined radiation and natural convection in a participating medium between concentric or vertically eccentric horizontal cylinders is investigated numerically. The annular medium is considered as a gray, emitting, absorbing, and isotropically scattering gas. The equations of steady, laminar, two-dimensional, thermal, natural convection are written by using a two-cylindrical coordinate system, the stream function, and the vorticity. The finite volumes method is used to discretize the coupled equations of momentum, energy, and radiative transfer. To solve the global nonlinear algebraic equations the successive-over-relaxation iterating scheme is applied. Numerical solutions are obtained for a Rayleigh number in the range 10³–10⁵ and radiation-conduction parameter ranging from 0 to ∞. The influences of radiation-conduction parameter, Rayleigh number and other parameters on flow and temperature distributions and heat transfer are discussed.     

The dynamics of coupled nonlinear model Boltzmann equations

A multispecies gas described by coupled nonlinear Boltzmann equations is studied as a dynamical system. Properties are determined of theN coupled nonlinear ODEs for the number densities obtained from the Boltzmann equations for the spatially uniform system ofN species undergoing binary scattering, removal, and regeneration in the presence of an external force field and a reservoir of background gas. The physically realizable setQ, the nonnegative cone in theN-dimensional phase space of species number densities, is established as invariant under the flow. The fixed-point equations for the ODEs are shown to be equivalent to 2 ^N linear systems, and conditions for the stability and instability of the fixed points are then established. Stable fixed points are demonstrated to exist inQ by showing that they enter via a sequence of transcritical bifurcations as physical parameters are varied. For the two-species case the typical global structure of the solutions is established. Various particular cases are described including one which possesses an infinite family of periodic solutions and one that depends delicately upon initial conditions due to a separatrix that separatesQ into two invariant sets.     

On the Variation of a Functional when the Boundary of the Domain is not Fixed

The formula for the variation of a functional is obtained where the integrand depends on several independent variables x ^j , several functions of these variables u ^k and a finite number of partial derivatives of any order of the functions u ^k with respect to x ^j . Besides the functions u ^k , the boundary of the domain can also move. The formula generalizes the well-known case when only first-order partial derivatives are admitted.      

Determinantal Process Starting from an Orthogonal Symmetry is a Pfaffian Process

When the number of particles N is finite, the noncolliding Brownian motion (BM) and the noncolliding squared Bessel process with index ν>−1 (BESQ^(ν)) are determinantal processes for arbitrary fixed initial configurations. In the present paper we prove that, if initial configurations are distributed with orthogonal symmetry, they are Pfaffian processes in the sense that any multitime correlation functions are expressed by Pfaffians. The 2×2 skew-symmetric matrix-valued correlation kernels of the Pfaffians processes are explicitly obtained by the equivalence between the noncolliding BM and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding BM with finite duration in which all particles start from the origin, Nδ ₀, and by the equivalence between the noncolliding BESQ^(ν) and that of the noncolliding squared generalized meander starting from Nδ ₀.     

Long Time Stability of Some Small Amplitude Solutions in Nonlinear Schrödinger Equations

We consider small perturbations of the Zakharov–Shabat nonlinear Schr?dinger equation on [0,π] with vanishing or periodic boundary conditions; we prove a Nekhoroshev type result for solutions starting in the neighbourhood (in the H ¹ topology) of the majority of small amplitude finite dimensional invariant tori of the linearized system. More precisely we will prove that along the considered solutions all the actions of the linearized system are approximatively constant up to times growing exponentially with the inverse of a suitable small parameter. Received: 10 December 1996 / Accepted: 17 March 1997     

The energy spectrum of fully developed turbulence generated by random stirring

For fully developed turbulence in an incompressible fluid described by the Navier-Stokes equations with Gaussian random forces the relation between the energy spectrum and the stirring mechanism is investigated within a variational approach. Therein, the effect of nonlinear mode coupling is approximated by a wave number dependent eddy viscosity determined via a nonlinear integral equation for the energy spectrum. For various stirring spectra analytic approximations are compared with the solution obtained numerically with a cutoff in the integral kernel which ensures in eddy relaxing processes that the stirring forces exert strain only on scales larger than the eddy size. The results are compared with renormalization group calculations and closure approximations. Random forces injecting energy at a ratek ^–1 into the wave number banddk aroundk lead to a Kolmogorov distribution of energy. The spectrum of small-scale velocity fluctuations is shown to be universal in the sense that it remains unchanged under variations of the long wavelength stirring spectra.     

新型矩形点阵光子晶体光纤的高双折射负色散效应

设计了一种新型矩形点阵光子晶体光纤,该光纤纤芯缺失一根空气柱,包层沿光纤长度方向在普通矩形点阵光子晶体光纤中每两列之间隔一行插入一列空气孔而形成正方形网孔结构.采用全矢量有限元法并结合各向异性完美匹配边界条件,对该光纤的色散、双折射和约束损耗进行了数值模拟.结果发现,该光纤具有高双折射负色散效应和较强的模约束能力,约束损耗小于10-2dB·m-1,通过改变光纤结构参数(即空气孔间隔Λ和相对孔间隔d/Λ),可以调节该光纤高双折射负色散工作波长.若调整光纤结构参数Λ=2.0μm,d/Λ=0.4,该光纤在C波段(1.53—1.565μm)呈现负色散并具有负色散斜率,双折射高达10-2,非线性系数接近55km-1W-1.该光纤将在保偏光通信、色散补偿以及基于四波混频的波长转换器设计等方面具有重要的应用.