波网络系统的适定性和正则性

研究了一波网络系统,在此网络的两个外部节点处施加控制并放置同位观测器,通过选择合适状态空间,将系统方程转化为抽象线性系统的形式.利用能量乘子法验证观测算子是允许的,通过对偶性原理得到控制算子也是允许的.求出传递函数后并验证传递函数在某个右半平面一致有界,从而说明系统是适定的,再求出传递函数在实轴上正方向的无穷远处的极限,找到直接传输算子,进而说明系统的正则性.     

Work-preserving emulations of shuffle-exchange networks: An analysis of the complex plane diagram

In this paper we show that for each n, the order-n shuffle-exchange network can be emulated by an n-node linear processor array or an n²-node mesh in a work-preserving manner. An emulation of a computation on a guest network is work-preserving on a host network , if the time-processor products are equal to within a constant factor. To achieve this result we demonstrate a uniform many-to-one embedding of the nodes of a shuffle-exchange network into a linear array. We then give a simple, deterministic routing algorithm on the linear array which schedules the communication of messages necessary to achieve the emulation within the required time bounds. The analysis of the algorithm relies on certain statistical properties of the complex plane diagram of the shuffle-exchange network.     

Fluctuations of deformed Wigner random matrices

Let X_n be a standard real symmetric (complex Hermitian) Wigner matrix, y₁, y₂, . . . , y_n a sequence of independent real random variables independent of X_n. Consider the deformed Wigner matrix H_n,α = n^-1/2X_n + n^-α/2 diag (y₁, . . . , y_n), where 0<α<1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform m_n,α(z) converges in probability to the corresponding Stieltjes transform m(z). In this paper, we shall give the asymptotic estimate for the expectation Em_n,α(z) and varianceVar(m_n,α(z)), and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein’s equation and its generalization which naturally leads to a certain recursive equation.     

The Existence and Stability of Asymmetric Spike Patterns for the Schnakenberg Model

Asymmetric spike patterns are constructed for the two-component Schnakenburg reaction–diffusion system in the singularly perturbed limit of a small diffusivity of one of the components. For a pattern with k spikes, the construction yields   k ₁  spikes that have a common small amplitude and   k ₂= k − k ₁  spikes that have a common large amplitude. A k -spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. Explicit conditions for the existence and linear stability of these asymmetric spike patterns are determined using a combination of asymptotic techniques and spectral properties associated with a certain nonlocal eigenvalue problem. These asymmetric solutions are found to bifurcate from symmetric spike patterns at certain critical values of the parameters. Two interesting conclusions are that asymmetric patterns can exist for a reaction–diffusion system with spatially homogeneous coefficients under Neumann boundary conditions and that these solutions can be linearly stable on an O (1) time scale.     

某四阶微分算子的一个正则性定理

本文给出了某带参数λ的四阶微分算子B_λ的一个正则性定理,由此我们分别获得了关于该算子在“非线性情形”的二个同胚类及在“线性情形”的三个线性同构类。这对描述某飞行器在其运行过程中某些类双向稳定性质是有用与方便的。     

Completely J —positive linear systems of finite order

Completely J — positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J — positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J — symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J — symmetric realizations are discussed.     

New enumerative results on two-dimensional directed animals

We list several open problems concerning the enumeration of directed animals on two-dimensional lattices. We show that most of these problems are special cases of two central problems: calculating the position generating function and the perimeter and area generating function for square lattice animals.

We propose a possible direction for solving these two problems: we extend Dhar's correspondence between hard particle gas models and enumeration of animals according to the area, and show that each of the main two generating functions is, essentially, the density of a one-dimensional gas model given by the stationary distribution of a probabilistic transition.

We are able to compute the density of certain stationary distributions. We thus obtain new bivariate generating functions for directed animals on the square and triangular lattices. We derive from these results the generating functions for animals on the decorated square and triangular lattices, as well as the average number of loops in directed animals.     

On the pointwise limits of bivariate lagrange projectors

A linear algebra proof is given of the fact that the nullspace of a finite-rank linear projector, on polynomials in two complex variables, is an ideal if and only if the projector is the bounded pointwise limit of Lagrange projectors, i.e., projectors whose nullspace is a radical ideal, i.e., the set of all polynomials that vanish on a certain given finite set. A characterization of such projectors is also given in the real case. More generally, a characterization is given of those finite-rank linear projectors, on polynomials in d complex variables, with nullspace an ideal that are the bounded pointwise limit of Lagrange projectors. The characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences.     

On Padé-type model order reduction of J-Hermitian linear dynamical systems

A simple, yet powerful approach to model order reduction of large-scale linear dynamical systems is to employ projection onto block Krylov subspaces. The transfer functions of the resulting reduced-order models of such projection methods can be characterized as Padé-type approximants of the transfer function of the original large-scale system. If the original system exhibits certain symmetries, then the reduced-order models are considerably more accurate than the theory for general systems predicts. In this paper, the framework of J-Hermitian linear dynamical systems is used to establish a general result about this higher accuracy. In particular, it is shown that in the case of J-Hermitian linear dynamical systems, the reduced-order transfer functions match twice as many Taylor coefficients of the original transfer function as in the general case. An application to the SPRIM algorithm for order reduction of general RCL electrical networks is discussed.     

Piecewise linear test functions for stability and instability of queueing networks

We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient. This revised version was published online in June 2006 with corrections to the Cover Date.     

The dynamic stability of an elastic column

Lyapunov's second method is used to investigate the stability of the rectilinear equilibrium modes of a non-linearly elastic thin rod (column) compressed at its end. Stability here is implied relative to certain integral characteristics, of the type of norms in Sobolev spaces; the analysis is carried out for all values of the problem parameter except the bifurcation values.

The realm of problems connected with the Lagrange-Dirichlet equilibrium stability theorem and its converse involves specific difficulties when considered in the infinite-dimensional case: stability in infinite-dimensional systems is investigated relative to certain integral characteristics such as norms /1/, and as the latter may be chosen with a certain degree of arbitrariness, different choices may result in different stability results. On the other hand, there is no relaxation of any of the difficulties encountered in the case of a finite number of degrees of freedom.

We shall consider a certain natural mechanical system with a finite number of degrees of freedom. If the first non-trivial form of the potential energy expansion is positive-definite, the equilibrium position is stable. A similar statement has been proved for infinitely many dimensions as well /1–3/, using Lyapunov's direct method, and the total energy may play the role of the Lyapunov function.

The situation with respect to instability is more complex. In the finite-dimensional case, if the first non-trivial form of the potential energy expansion may take negative values, instability may be demonstrated in many cases by means of a function proposed by Chetayev in /4/. A general theorem has been proved /1/ for instability in infinitely many dimensions, relying on an analogue of Chetayev's function. Such functions have also been used /5, 6/ to prove the instability of equilibrium in specific linear systems with an infinite number of degrees of freedom.

However, Chetayev's functions /4/ are not suitable tools to prove the instability of equilibrium in most non-linear systems. Another “Chetayev function”, which is actually a perturbed form of Chetayev's original function from /4/, has been proposed /7/, and it has been used to prove instability when the equilibrium position is an isolated critical point of the first non-trivial form of the potential energy expansion.

The majority of problems concerning the onset of instability of equilibrium configurations of elastic systems have been considered from a quasistatic point of view (see, e.g., /8, 9/). Problems of elastic stability and instability were considered in a dynamical setting in /2, 5/, where stability was investigated by Lyapunov's direct method. However, most of the results obtained in this branch of the field concern linear systems, and there are extremely few publications dealing with the onset of instability in non-linear elastic systems using Lyapunov's direct method. This is because in an unstable elastic system the quadratic part of the potential energy may change sign, and therefore the analogues of Chetayev's function from /4/ are not usually suitable for solving these problems. Dynamic instability has been studied or a specific non-linearly elastic system /10/, with the fact of instability established by using an analogue of the Chetayev function from /7/.

This paper presents one more example of a study of dynamic instability crried out for a non-linearly elastic system by Lyapunov's direct method.     

State space collapse and stability of queueing networks

We study the stability of subcritical multi-class queueing networks with feedback allowed and a work-conserving head-of-the-line service discipline. Assuming that the fluid limit model associated to the queueing network satisfies a state space collapse condition, we show that the queueing network is stable provided that any solution of an associated linear Skorokhod problem is attracted to the origin in finite time. We also give sufficient conditions ensuring this attraction in terms of the reflection matrix of the Skorokhod problem, by using an adequate Lyapunov function. State space collapse establishes that the fluid limit of the queue process can be expressed in terms of the fluid limit of the workload process by means of a lifting matrix.     

Preserving Positive Realness Through Discretization

The importance of positive real transfer functions relies on the fact that they are associated with positive linear systems. Those systems possess the property that their input-output product time-integral, which is a measure of the total enerty, is nonnegative. Such a property can be also formulated in the discrete context. It is shown that a discrete positive real transfer function is obtained from a positive real continuous one of relative order zero being strictly stable poles via discretization by a sampler and zero-order hold device provided that the direct input-output transmission gain is sufficiently large. It is also proved that a discrete positive real transfer function may be obtained from a stable continuous one of relative order zero and high direct input-output gain which posses simple complex conjugate critically stable poles even in the case that this one is not positive real. For that purpose, the use of an appropriate phase-lag or phase lead compensating network for the continuous transfer function may be required to ensure positive realness of the discrete transfer function.     

奇异摄动问题在Bakhvalov-Shishkin 网格上的流线扩散有限元逼近

本文采用线性插值的流线扩散有限元在Bakhvalov-Shishkin网格上求解一维对流扩散型的奇异摄动问题. 在ε ≤ N^-1的前提下,可以得到,关于扰动参数ε 是一致收敛的. 在离散的SD范数下,其u-u_I的误差阶提高到N^-2,u-u_h的误差阶达到N^-2（lnN）^0.5. 最后,通过数值算例,验证了理论分析.     

一类新的广义割圆序列的线性复杂度及其自相关值

最近,丁存生基于新的割圆类(V_0,V_1)构造了循环码并研究了其性质.本文利用割圆类(V_0, V_1)构造了周期为pq的2阶二元序列,并计算了其自相关值、线性复杂度和极小多项式.     

垂直线性互补问题的一步全局线性和局部二次收敛光滑Newton法

基于凝聚函数,提出一个求解垂直线性互补问题的光滑Newton法.该算法具有以下优点:(i)每次迭代仅需解一个线性系统和实施一次线性搜索;(ⅱ)算法对垂直分块P0矩阵的线性互补问题有定义且迭代序列的每个聚点都是它的解.而且,对垂直分块P₀+R₀矩阵的线性互补问题,算法产生的迭代序列有界且其任一聚点都是它的解;(ⅲ)在无严格互补条件下证得算法即具有全局线性收敛性又具有局部二次收敛性.许多已存在的求解此问题的光滑Newton法都不具有性质(ⅲ).     

Spectral analysis and the Riemann hypothesis

The explicit formulas of Riemann and Guinand-Weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of Riemann. We recall Alain Connes’ spectral interpretation of the critical zeros of the Riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable Hilbert space. We then give a spectral interpretation of the zeros of the Dedekind zeta function of an algebraic number field K of degree n in an automorphic setting.

If K is a complex quadratic field, the torical forms are the functions defined on the modular surface X, such that the sum of this function over the “Gauss set” of K is zero, and Eisenstein series provide such torical forms.

In the case of a general number field, one can associate to K a maximal torus T of the general linear group G. The torical forms are the functions defined on the modular variety X associated to G, such that the integral over the subvariety induced by T is zero. Alternately, the torical forms are the functions which are orthogonal to orbital series on X.

We show here that the Riemann hypothesis is equivalent to certain conditions bearing on spaces of torical forms, constructed from Eisenstein series, the torical wave packets. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of critical zeros of the Dedekind zeta function of K.     

On n-widths of a Sobolev function class in Orlicz spaces

This paper considers the problem of n-widths of a Sobolev function class Ωr∞ determined by Pr(D) = Dσ∏lj =1(D2- t2jI) in Orlicz spaces. The relationship between the extreme value problem and width theory is revealed by using the methods of functional analysis. Particularly, as σ = 0 or σ = 1, the exact values of Kolmogorov's widths, Gelfand's widths, and linear widths are obtained respectively, and the related extremal subspaces and optimal linear operators are given.     

输送网络新模型——改进的K-H模型

引入了网络流中最小流函数的概念并对此进行了深入研究,给出了最小流函数s(λ_1,λ_2,…,λ_r)的分析表达式,建立了输送网络"发"收"叠加的新数学模型,该模型克服了传统的K-H模型中将输送源排除在网络系统之外的弊端,在输送网络的优化问题上具有极强的可操作性.     

高阶齐次线性微分方程解的零点

研究了一类高阶齐次线性微分方程解的零点收敛指数,并得到当方程的系数A_0为整函数,其泰勒展式为缺项级数,并且A_0起控制作用时,方程f~((k))+A_(k-2)f~((k-2))+…+A_1f′+A_0f=0的任意两个线性无关解f_1,f_2满足max{λ(f_1),λ(f_2)}=∞,其中λ(f)表示亚纯函数.f的零点收敛指数.