Dynamics of a Local Perturbation in the XY Model. I—Approach to Equilibrium

The magnetisation and spin correlation functions of the X–Y model are analysed exactly when the system is subjected to a local perturbation of the magnetic field in the z-direction. When such a perturbation is removed, these quantities approach their equilibrium values asymptotically as (time)^?1.     

Two-pattern strings II—frequency of occurrence and substring complexity

The previous paper in this series introduced a class of infinite binary strings, called two-pattern strings, that constitute a significant generalization of, and include, the much-studied Sturmian strings. The class of two-pattern strings is a union of a sequence of increasing (with respect to inclusion) subclasses T_λ of two-pattern strings of scope λ, λ=1,2,…. Prefixes of two-pattern strings are interesting from the algorithmic point of view (their recognition, generation, and computation of repetitions and near-repetitions) and since they include prefixes of the Fibonacci and the Sturmian strings, they merit investigation of how many finite two-pattern strings of a given size there are among all binary strings of the same length. In this paper we first consider the frequency f_λ(n) of occurrence of two-pattern strings of length n and scope λ among all strings of length n on {a,b}: we show that lim_n→∞f_λ(n)=0, but that for strings of lengths n2λ, two-pattern strings of scope λ constitute more than one-quarter of all strings. Since the class of Sturmian strings is a subset of two-pattern strings of scope 1, it was natural to focus the study of the substring complexity of two-pattern strings to those of scope 1. Though preserving the aperiodicity of the Sturmian strings, the generalization to two-pattern strings greatly relaxes the constrained substring complexity (the number of distinct substrings of the same length) of the Sturmian strings. We derive upper and lower bounds on C₁(k) (the number of distinct substring of length k) of two-pattern strings of scope 1, and we show that it can be considerably greater than that of a Sturmian string. In fact, we describe circumstances in which lim_k→∞(C₁(k)−k)=∞.     

On the X-Y convex hull of a set of X-Y polygons

We study the class of rectilinear polygons, calledX – Y polygons, with horizontal and vertical edges, which are frequently used as building blocks for very large-scale integrated (VLSI) circuit layout and wiring. In the paper we introduce the notion of convexity within the class ofX – Y polygons and present efficient algorithms for computing theX – Y convex hulls of anX – Y polygon and of a set ofX – Y polygons under various conditions. Unlike convex hulls in the Euclidean plane, theX – Y convex hull of a set ofX – Y polygons may not exist. The condition under which theX – Y convex hull exists is given and an algorithm for testing if the given set ofX – Y polygons satisfies the condition is also presented.     

Efimov's Effect in a Model of Perturbation Theory of the Essential Spectrum

A model operator similar to the energy operator of a system with a nonconserved number of particles is studied. The essential spectrum of the operator is described, and under some natural conditions on the parameters it is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum.     

Spectrum of a Model Operator in the Perturbation Theory of the Essential Spectrum

We consider a model operator acting in a subspace of a Fock space and obtain a symmetrized analogue of the Faddeev equation. For the operator considered, we describe the position and the structure of its essential spectrum.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 544–554, September, 2005.     

Effective Dynamics of a Magnetic Vortex in a Local Potential

We study a simplified mean field model of superconductor dynamics in the presence of impurities or for variable superconductor depth. This model is given by the gradient-flow version of the Ginzburg-Landau equations (Gorkov-Eliashberg equations) with an addition of a potential term. We find a dynamical law of motion of the vortex center, involving the potential, such that for datum close to a (static) magnetic vortex the solution is close, for all times, to a magnetic vortex whose center obeys this law.     

Large quartic groups on translation planes,II—Even order: Characterization of the Ott‐Schaeffer planes

A general theory of collineation groups generated by quartic groups of even order is considered. Applications are given to collineation groups generated by ‘large’ quartic groups. © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 195–210, 2005.     

Field Theory Approach to the Dynamics of a Continuous Medium: A Perturbation Theory

We propose a perturbation theory that allows solving the equations of motion for the displacement vector in the body of the Earth in the framework of the linear theory of elasticity. We show that tectonic processes are primarily determined by tidal actions. We analyze the tidal effects in the Earth–Moon–Sun system.     

周期扰动下卫星运动系统的动力学性质

本文运用Melnikov方法对平面卫星运动系统在周期扰动下所表现出来的动力学性质进行了探讨.首先运用次谐Melnikov方法给出了卫星轨道在周期扰动下存在次谐周期轨道的条件,并进一步运用同宿.Melnikov方法证实了该系统存在Smale马蹄意义下的混沌性质.     

随机参数作用下参激双势阱Duffing系统的随机动力学行为分析

基于正交多项式逼近理论,研究了在不同随机参数作用下参激双势阱Duffing系统的随机动力学行为.首先,借助Poincaré(庞加莱)截面分析系统的复杂动力学行为;其次,分别针对系统非线性项系数和阻尼项系数为随机参数的情况,运用正交多项式逼近法,将随机参数Duffing系统转化为与之等价的确定性扩阶系统,并证明其有效性;最后,运用等价确定性扩阶系统的集合平均响应,揭示随机系统的动力学特性,以及随机变量强度变化对系统产生的影响.数值结果表明,对于多吸引子共存情形,参激双势阱Duffing系统在随机非线性项系数影响下,其动力学行为较为稳定,共存吸引子与确定性情形保持一致;而当阻尼系数为随机参数时,随着随机变量强度的增加,部分共存吸引子将发生分岔现象.     

一个呼吸动力学模型的全局吸引性

研究了一个呼吸动力学时滞微分方程模型.x(t)=1-αx(t)xn(t-τ)1+xn(t-τ).通过利用一种映射方法得到了该系统平衡点全局吸引的充分条件.所得结论优于已有的结果.     

First Order Perturbation and Local Stability of Parametrized Systems

A problem frequently encountered in geometric constraint solving and related settings is to ascertain sensitivity of solutions arising from a well constrained input configuration. This is important for tolerancing and motion planning, for example. An example would be determining lines simultaneously tangent to four given spheres (which originates as a line-of-sight problem); how much does a perturbation of the input affect the positioning of these lines. Once translated to an algebraic setting one has a system of polynomial equations with some coefficients parametrized, and wants to determine the solutions and a good approximation of their sensitivity to changes in the parameters. We will compute this sensitivity from first order changes in an appropriate Gröbner basis. We demonstrate the applicability on several examples. We also discuss a more global form of stability, wherein one wants to know about perturbations that might change the character of the solution space e.g. by having fewer than the generic number of solutions.     

Perturbation determinants in Banach spaces — with an application to eigenvalue estimates for perturbed operators

In the first part of this paper we provide a self‐contained introduction to (regularized) perturbation determinants for operators in Banach spaces. In the second part, we use these determinants to derive new bounds on the discrete eigenvalues of compactly perturbed operators, broadly extending some recent results by Demuth et al. In addition, we also establish new bounds on the discrete eigenvalues of generators of C₀‐semigroups.     

营养基被污染且脉冲扰动的时滞Chemostat模型分析(英文)

研究营养基被污染且脉冲扰动的时滞Chemostat模型.利用离散动力系统频闪映射,得到了微生物种群灭绝周期解,且它是全局吸引的;利用时滞脉冲微分方程理论,得到了系统持久的条件.结论提示了时滞增长反应对Chemostat的产量起着重要的作用.     

一类高维生物数学模型小扰动下的Hopf分支

讨论一类六维生物数学模型,在［１,２］的基础上,进一步考虑生物种群之间的相互作用,引进扰动项,研究了中心流形及Ｈｏｐｆ分支问题．利用计算机代数符号软件计算了中心流形,得到了产生Ｈｏｐｆ分支的判定条件,并给出了数值算例,所用机器证明的思想方法可广泛运用于类似复杂生物数学模型的研究．     

Justification of a Perturbation Approximation of the Klein–Gordon Equation

Consider the nonlinear wave equation
u_tt − γ ² u_xx + f(u) = 0
with the initial conditions
u ( x ,0) = εφ ( x ), u _t( x ,0) = εψ ( x ),
where f ( u ) is either of the form f ( u )= c ² u −σ u ^{2 s +1}, s =1, 2,…, or an odd smooth function with f '(0)>0 and | f '( u )|≤ C ₀².The initial data φ( x )∈ C ² and ψ( x )∈ C ¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u ( x ,  t ; ɛ), and prove its boundedness in x ∈ R and t >0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u ( x ,  t ; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ³ uniformly for x ∈ R and 0≤ t ≤ T /ɛ², as long as ɛ is sufficiently small, T being an arbitrary positive number.     

Spectrum of the One-dimensional Schrödinger Operator With a Periodic Potential Subjected to a Local Dilative Perturbation

We study the spectrum of the one-dimensional Schr?dinger operator with a potential, whose periodicity is violated via a local dilation. We obtain conditions under which this violation preserves the essential spectrum of the Schr?dinger operator and an infinite number of isolated eigenvalues appear in a gap of the essential spectrum. We show that the considered perturbation of the periodic potential is not relative compact in general.     

Dynamics of a Mutualistic Model with Diffusion

This article is concerned with a system of semilinear parabolic equations with no-flux boundary condition in a mutualistic ecological model. Stability result of the equilibrium about relevant ODE problem is proved by discussing its Jacobian matrix, we give two priori estimates and prove that the model is permanent when ε_1+ ε_2 ≠ 0. Moreover sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the model are obtained. Nonexistence of nonconstant positive steady states of the model is also given. When ε_1+ε_2= 0, grow up property is derived if the geometric mean of the interaction coefficients is greater than 1(α_1α_2 1),while if the geometric mean of the interaction coefficients is less than 1(α_1α_2 1), there exists a global solution. Finally, numerical simulations are given.