Some details of the description of a disordered condensed system using the theory of defect states of orientational order

Using the theory of defect states of orientational order, we describe a disordered condensed system as an elastic medium with linear topological singularities. We show that elastic stress fields produced by linear disclinations are Abelian. In the quasistationary linear approximation, we obtain expressions for linear dislocation and disclination tensor potentials. We show that using the theory of defect states of orientational order, we can describe the α and β relaxations in a supercooled liquid as relaxation processes in the respective disclination and dislocation subsystems.     

Spin-One <Emphasis Type="Italic">p</Emphasis>-Spin Glass: Exact Solution for Large <Emphasis Type="Italic">p</Emphasis>

We study the low-temperature properties of the p-spin spin glass model in the spin-one (three-state) case for large values of p. We show that the one-step replica symmetry-breaking phase is unstable at a very low temperature, and we calculate the explicit boundary of the stability interval, the Gardner temperature, analytically for large values of p. This temperature for the spin-one model has the same form of dependence on p as in the case of Ising spins (two states). In the one-step replica symmetrybreaking state, a quadrupolar orientational glass coexists with the spin glass and also with a regular quadrupole ordering.     

Orientation ordering and the transition to the orientational glass state in fullerite C<Subscript>60</Subscript>

We propose a model for describing the low-temperature transition to the orientational glass state in solid molecular C₆₀ in the framework of a theory similar to spin glass theory. We find a replica symmetric solution and also break the replica symmetry. The obtained results agree with experimental data on the partial retention of the orientational long-range order in the glass phase and on the presence of a broad maximum on the curve corresponding to the temperature dependence of the orientational part of the heat capacity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 356–368, May, 2008.     

关于辛算法稳定性的若干注记

我们讨论辛算法的线性稳定性和非线性稳定性,从动力系统和计算的角度论述了研究辛算法的这两类稳定性问题的重要性,分析总结了相关重要结果.我们给出了解析方法的明确定义,证明了稳定函数是亚纯函数的解析辛方法是绝对线性稳定的.绝对线性稳定的辛方法既有解析方法（如Runge-Kutta辛方法）,也有非解析方法（如基于常数变易公式对线性部分进行指数积分而对非线性部分使用其它数值积分的方法）.我们特别回顾并讨论了R.I.McLachlan,S.K.Gray和S.Blanes,F.Casas,A.Murua等关于分裂算法的线性稳定性结果,如通过选取适当的稳定多项式函数构造具有最优线性稳定性的任意高阶分裂辛算法和高效共轭校正辛算法,这类经优化后的方法应用于诸如高振荡系统和波动方程等线性方程或者线性主导的弱非线性方程具有良好的数值稳定性.我们通过分析辛算法在保持椭圆平衡点的稳定性,能量面的指数长时间慢扩散和KAM不变环面的保持等三个方面阐述了辛算法的非线性稳定性,总结了相关已有结果.最后在向后误差分析基础上,基于一个自由度的非线性振子和同宿轨分析法讨论了辛算法的非线性稳定性,提出了一个新的非线性稳定性概念,目的是为辛算法提供一个实际可用的非线性稳定性判别法.     

预估-校正方法的绝对稳定性讨论

预估-校正方法,即PECE方法,常被用于求解常微分方程的初值问题.而一般文献中常只讨论了单个线性多步法公式的稳定性问题,很少涉及由一个显式公式和一个隐式公式组合而成的PECE方法的稳定性.本文应用根轨迹法和对分法讨论了常用的PECE方法的稳定性,求出了一些常用PECE方法的组合公式的绝对稳定区间和绝对稳定区域,并用数值...     

NOTE ON THE STABILITY PROPERTY OF THE VANISHING EQUILIBRIUM POINT OF AN ECOLOGICAL SYSTEM CONSISTING OF A PREDATOR AND STAGE STRUCTURE PREY

We revisit the stability property of an ecological system consisting of a predator and stage structure prey which was proposed by Raid Kamel Naji and Salam Jasim Majeed. By constructing some suitable Lyapunov function and applying the differential inequality theory, we show that the conditions which ensure the local stability of the vanishing point are enough to ensure its global stability. Our result supplements and complements some known results.     

EXISTENCE AND STABILITY OF SOLUTIONS OF STOCHASTIC SEMILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study stochastic semilinear functional differential equations in a Hilbert space. First, we prove the existence and uniqueness of a mild solution under two sets of hypotheses. We then consider the exponential stability of the second moment of the solution process of such equations as well as the exponential stability and asymptotic stability in probability of its sample paths. We further consider global stability in the mean. Such results are obtained using both local Lipschitz and non-Lipschitz nonlinearities. Our method is an interplay of the method of successive approximations and a comparison principle. Two applications are included to motivate this study.     

Stability of flow of a liquid crystal layer on an inclined plane : PMM vol. 38, n 6, 1974, pp. 1031–1040

The framework of the linear mechanics of liquid crystal media [1] is used to study propagation of waves in a layer of a nematic liquid crystal (NLC) on an inclined plane, in a magnetic field, for three different cases of orientation of the anisotropy axis, namely orthogonal to the inclined plane, parallel to the inclined plane and orthogonal to the plane of flow. Such orientations of the anisotropy axis are realized in practice in the course of special machining of solid surfaces [2]. Exact solutions of the equations of motion are obtained describing the steady flow of the layer, and the behavior of small plane perturbations is studied. It is shown that two types of plane waves can propagate in a layer of the nematic mesophase, namely, the surface and the orientational waves. In the case of long surface waves the formulas for the critical Reynolds number are obtained. For the orientational waves a sufficient criterion of stability of the flow in the layer is obtained for two cases. The influence of the magnetic field and of the rheological parameters of NLC on the character of propagation of the first and second type waves is investigated.From amongst the papers dealing with wave propagation in NLC, we draw the readers' attention to [3] which deals with the longitudinal, shear and torsional waves in a liquid crystal domain and obtains the corresponding dispersion relationships.     

关于直接调节系统的绝对稳定性

<正> 非线性自动调节系统的绝对稳定性问题最初是由讨论飞机自动驾驶仪引起的.它的一般化是讨论带有一个非线性执行机构的系统     

Stiff微分方程初值问题的一类数值方法

1.引言 实践表明,数值积分常微分方程初值问题 dx/dt=f(t,x), (1.1) x(t_0)=x_0时,若(1.1)是Stiff的,积分过程的稳定性是一个突出的问题.用传统的数值方法,比如Euler法,Adams法或Runge-Kutta法,为了保证计算稳定,积分步长受到相当地限制.即使运算速度为 100万次/秒的计算机,计算时间也将成为重大的负担.     

STABILITY OF DISCRETE-TIME COHEN-GROSSBERG BAM NEURAL NETWORKS WITH DELAYS

In this paper, we study the existence and stability of an equilibrium of discrete-time Cohen-Grossberg BAM Neural Networks with delays. We obtain several sufficient conditions ensuring the existence and stability of an equilibrium of such systems, using discrete Halanay-type inequality and vector Lyapunov methods. In addition, we show that the proposed sufficient condition is independent of the delay parameter. An example is given to demonstrate the effectiveness of the results obtained.     

THE STABILITY OF A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

1 IntroductionConsider the following functional differential equationX'(t) = A(t)x(t) [' C(t,s)x(s)ds, (1)icwhere x E m; A(t) = (ail(t))... is a n x n function matrix, which continuesin [0, co); C(t,s) = (qj(t,8))... is a n x n function matrix, which conti-nues when 0 5 s 5 t < co, and L oo IIC(u,t)lldu continues in [0, co).The problem on the stability for the zero solution of (1) has been studied bymany papers. But in the known results, the boundedness of j: IIC(t, s)lldsor L " IIC…     

带有分段常数变元的时滞微分方程解的稳定性和振动性

本文讨论带有分段常数变元的时滞微分方程解的渐性性质。我们利用Ｒａｚｕｍｉｋｈｉｍ方法证明了零解的稳定性，并且改进了文［１］得到的两上振动定理。     

Orientational phase transition in solid C60: Bifurcation approach

A simple model for the angle-dependent interaction betweenC ₆₀ molecules in the face-centered cubic lattice is proposed. The bifurcations of the solutions of the nonlinear integral equations for orientational distribution functions in the mean-field approximation are analyzed, and the orientational phase transition in solidC ₆₀ is described. The quantitative results for the orientational phase transition agree with the experimental data. Translated from Teoreticheskaya i matematicheskaya Fizika, Vol. 121, No. 3, pp. 479–491, December, 1999.     

树的稳定子集和稳定指标

一个实矩阵的符号稳定性问题在经济学、生态学等诸多领域中都有应用背景．本文利用[1]中给出的不可约矩阵的符号稳定性的有关结论，将一个实矩阵的符号稳定性判定问题转化为一个等价的图论问题，即判定无向树中一个点子集的稳定性问题．本文引入了树的稳定子集的概念并给出了稳定子集的递归判别方法．本文还提出井研究了树的稳定指标，即树中所有稳定子集的最小基数，证明了关于稳定指标的一个min—max型定理，井给出了n阶树的稳定指标的最好上界及达到上界的极树的完全刻划。     

On an integral equation in the theory of phase transitions for a system of magnetic rods

We study a nonlinear integral equation for the orientational distribution function (ODF) describing anisotropic nematic ordering in a system of magnetic rods. For highly elongated rods, we give a classification of bifurcation points and find their asymptotic behavior in the limits of small and large magnetic moments of the rods. We develop an algorithm to construct a nearly-isotropic ODF in the vicinity of the bifurcation point. We show that for both small and large magnetic moments, the ODF obtained has a left direction of bifurcation. However, for intermediate values of the magnetic moments, solutions with a right direction of bifurcation exist along with those with the left direction.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 427–440, December, 1996.     

THE STABILITY OF STATIONARY SOLUTION FOR OUTFLOW PROBLEM ON THE NAVIER-STOKES-POISSON SYSTEM

In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.     

CONSENSUS ANALYSIS AND DESIGN OF LINEAR INTERCONNECTED MULTI-AGENT SYSTEMS

We deal with the state consensus problem of a general Linear Interconnected Multi-Agent System(LIMAS) under a time-invariant and directed communication topology.Firstly,we propose a linear consensus protocol in a general form,which consists of state feedback of the agent itself and feedback form of the relative states between the agent and its neighbors.Secondly,a state-linear-transformation is applied to equivalently transform the state consensus problem into a partial stability problem.Based on the partial stability theory,we derive a sufficient and necessary criterion of consensus convergence,which is expressed via the Hurwitz stability of a real matrix constructed from the parameters of the agent models and the protocols,and present an analytical formula of the consensus function.Lastly,we propose a design procedure of the gain matrices in the protocol by solving a bilinear matrix inequality.     

单时滞类Lorenz系统的Hopf分岔分析

本文研究单时滞类Lorenz系统的稳定性问题.利用规范型理论,分析该系统在其零平衡点处的稳定性和发生Hopf分岔的条件,并通过数值仿真验证所得结论的正确性.本文的结论可以看做是对文献[5]研究成果的推广.