图的笛卡儿积的测地数

对于图G内的任意两点u和v，u-v测地线是指u和v之间的最短路．I（u，v）表示位于u-v测地线上所有点的集合，对于．S∈V（G），I（S）表示所有I（u，v）的并，这里“u,v∈．S．G的测地数g（G）是使I（S）=V（G）的点集．S的最小基数．在这篇文章，我们研究G×K3的测地数和g（G）与g（G×K3）相等的充分必要条件，还给出了T×Km和Cn×Km的测地数，这里T是树．     

一阶非线性周期方程的奇异点方法

本文应用奇异点理论,在g(x)为凹(凸)型函数时,给出周期系统(?)+a(t)g(x)=h(t)整体等价于Whitney意义下的尖点映射的结果.精确地说,算子Fx(t)=(?)+a(t)g(x(t))的奇异值集F(∑)为单连通超曲面并且将C[0,1]分成两个连通分支A1和A3,使得:(1)对周期为1的连续函数p(t)∈A1有唯一解.(2)对周期为1的连续函数p(t)∈A3恰有三个周期解.进一步,尖点集C的像集F(C)是C[0,1]中的,余维数等于2的子流形.对p∈F(C)有唯一解,而对p(t)∈F(∑)\F(C)恰有两个周期解.     

Generalizations of the notion of hyperbolicity

We describe several generalizations of the classical notion of hyperbolicity for a sequence of linear mappings. It is shown that the following three statements are equivalent: (i) the corresponding linear non-homogeneous system has a bounded solution for any bounded nonhomogeneity, (ii) the sequence has a (C, λ)-structure, (iii) the sequence is piecewise hyperbolic with long enough intervals of hyperbolicity.     

一类具时滞耗散型Duffing方程的周期解

利用Mawhin重合度理论研究了一类耗散型时滞Duffing方程ax″+f[x′(t-τ1(t))]+cx+g(x(t-τ2(t)))=p(t)周期解的存在性,得到了该方程2π周期解存在的充分性定理.     

The properties of solutions of a class of isoperimetric problems of stability optimization

A class of isoperimetric problems of stability optimization is considered. These arise, for example, when maximizing the Euler force in the destabilization of a column (rod) of varying cross-section and given volume (Lagrange's problem). It is well known that an extremum which depends on the form of the boundary conditions can be achieved for both simple and double eigenvalues. A class of problems is identified for which a global maximum is found at a simple eigenvalue. The possibility of achieving a local extremum for the first (simple) eigenvalue at stationary points is analysed qualitatively in terms of the parameter values and the form of the boundary conditions.     

The Evolution of Linearized Perturbations of Parallel Flows

The equations of an incompressible fluid are linearized for small perturbations of a basic parallel flow. The initial-value problem is then posed by use of Fourier transforms in space. Previous results are systematized in a general framework and used to solve a series of problems for prototypical examples of basic shear flow and of initial disturbance. The prototypes of shear flow are (a) plane Couette flow bounded by rigid parallel walls, (b) plane Couette flow bounded by rigid walls at constant pressure, (c) unbounded two-layer flow with linear velocity profile in each layer, (d) a piecewise linear profile of a boundary layer on a rigid wall. The prototypes of initial perturbation are the fundamental ones: (i) a point source of the field of the transverse velocity (represented by delta functions), (ii) an unbounded sinusoidal field of the transverse velocity, (iii) a point source of the lateral component of vorticity, (iv) a sinusoidal field of the lateral vorticity. Detailed solutions for an inviscid fluid are presented, but the problem for a viscous fluid is only broached.     

Lyapunov numbers of markov solutions of linear stochastic systems

All nontrivial solutions of x = A(t)x grow exponentially with rate X(x,w)e{A1,...,Xr}, A a (strictly) stationary matrix process. Projecting x to the unit sphere one obtains for each of the Lyapunov exponents Xt a solution xt with stationary angle st. Now if A is a Markov process one can restrict oneself to Markov solutions, i.e., (x, A) shall be a (joint) Markov process (wich is a restriction on the inital conditions). We prove that whenever there is a Markov solution x with Lyapunov number X then there is another Markov solution with a stationary angle (or equivalently: an invariant measure for the transition probabilities of (s, A)) with the same Lyapunov number. This has some consequences, e.g., for the uniqueness of the Lyapunov numbers     

亚纯函数的差分多项式

假设函数f(z)是亚纯函数,H(z,f)是关于f(z)的差分多项式,s(z)是关于f(z)的小函数,考察了差分多项式f(z)~nH(z,f)-s(z)的零点分布问题.首先得到了差分多项式f(z)~nH(z,f)-s(z)的零点计数函数和函数f(z)的特征函数以及极点计数函数之间的一些不等式估计,再根据这些不等式,建立了Hayman关于亚纯函数的一个经典结果的差分模拟.     

A parallel algorithm of multi-variant evolutionary synthesis of nonlinear models

A parallel algorithm has been proposed for solving the problem of construction of nonlinear models (mathematical expressions, functions, algorithms, and programs) using given experimental data, set of variables, basic functions and operations. The designed algorithm of multivariant evolutionary synthesis of nonlinear models includes linear representation of a chromosome, modular operations in decoding of a genotype into a phenotype for interpreting a chromosome as a sequence of instructions, and a multi-variant method for presenting a set of models (expressions) using a single chromosome. A sequential version of the algorithm is compared with a standard genetic programming (GP) algorithm and a Cartesian genetic programming (CGP) one. The algorithm proposed was shown to excel the GP and CGP algorithms both in the time required for search for a solution (more than by an order of magnitude in most cases) and in the probability of finding a given function (model). Experiments have been carried out on parallel supercomputer systems, and estimates of the efficiency of the parallel algorithm offered have been obtained; the estimates demonstrate linear acceleration and scalability.     

一类滞后差分方程解的渐近性

考虑时滞差分方程xn－xn-1＝F（－f（xn）＋g（xn－k）），这里k是正整数，F，f，g是R→R的连续函数，F和f在R上单调增加，且对所有的u≠0，uF（u）＞0.我们证明了如果对所有的y∈R，有f（y）≥g（y）（f（y）≤g（y）），则方程的每个解趋于一个常数或－∞（∞）．进一步，如果对所有的y∈R，有f（y）≡g（y）；则方程的每个解当n→∞时趋于常数．     

A few applications of negative- type inequalities

Applying the negative-type inequalities for square Euclidean distance, we present (1) a parallelatope theorem (a generalization of the parallelogram theorem), (2) a short proof of Rankin’s theorem for the maximum number of dispersed points on a sphere, and (3) a proof of impossibility of a certain geometric embedding for some graphs.     

Community Structures of Networks

We present an approach to studying the community structures of networks by using linear programming (LP). Starting with a network in terms of (a) a collection of nodes and (b) a collection of edges connecting some of these nodes, we use a new LP-based method for simultaneously (i) finding, at minimal cost, a second edge set by deleting existing and inserting additional edges so that the network becomes a disjoint union of cliques and (ii) appropriately calibrating the costs for doing so. We provide examples that suggest that, in practice, this approach provides a surprisingly good strategy for detecting community structures in given networks.      

环上矩阵广义Moore-Penrose逆的存在性

讨论带有对合反自同构*有单位元的结合环R上矩阵的广义Moore-Penrose 逆,给出了环R上矩阵的广义Moore-Penrose逆存在的几个充要条件．特别,得到了环 R上矩阵A的关于M和N的广义Moore-Penrose逆存在的充要条件是A有分解A= GDH,其中D2=D,(MD)*=MD,(GD)*MGD+M(I-D)和DHN-1(DH)*+ (I-D)M-1均可逆．     

Non-wandering sets of the powers of dendrite maps

Let(X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f)and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold:(1) If x ∈Ω(f)-Ω(f~n) for some n ≥ 2,then x ∈ EP(f).(2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = ?0(the cardinal number of the set of positive integers).     

一类推广的干摩擦问题周期解的存在性

为了得到干摩擦问题存在周期解的判断方法,利用微分包含理论对系统+k~2x+μSngp(t)+T(t)a.e.进行了讨论.在周期外力p(t)周期等于固有周期2π/R的情形下得到了周期解存在的充要条件,在周期外力p(t)周期不等于固有周期整数倍2nπ/R的情形下得到了周期解存在的充分条件.     

Optimal detection of a recurring tuple of reference fragments in a quasiperiodic sequence

An a posteriori (off-line) approach to solving the problem of maximum-likelihood detection of a recurring tuple containing reference fragments in a numerical quasiperiodic sequence is studied. The case is analyzed where (1) the total number of fragments in a sequence is unknown; (2) the index of a sequence term corresponding to the beginning of a fragment is a deterministic (not random) value; (3) a sequence distorted by additive uncorrelated Gaussian noise is available for observation. It is shown that the problem under consideration is reduced to testing a set of simple hypotheses about the mean of a random Gaussian vector. The cardinality of this totality grows exponentially as the vector dimension (i.e., the length of the sequence under study) increases. It is established that searching for a maximum-likelihood hypothesis is equivalent to finding arguments that yield a maximum for an auxiliary objective function. It is shown that maximizing the objective function reduces to solving a special optimization problem, which is proved to be solvable in polynomial time. An exact algorithm for solving this problem, which underlies the optimal (maximum-likelihood) detection algorithm for a recurring tuple, is substantiated. The kernel of the exact algorithm is an algorithm for solving a special (basic) optimization problem. Results of numerical simulations are presented.     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

一类上界型拟合优度检验统计量的精确分布

对于简单假设的拟合优度检验,Zhang (2002)构造出一类上界型检验.取不同的参数$\lambda$和不同的权函数$q(t)$,这类检验包含了Kolmogorov-Smirov检验, Berk and Jones(1979)检验等已有的上界型检验.文献中仅对极少数$\lambda$和$q(t)$所对应的检验给出了零假设下的精确分布.然而, 针对不同的问题, ``好'的检验是不同的,因此有必要对任意给定的$\lambda$和$q(t)$情况, 讨论该类检验.本文对任意给定的$\lambda$和$q(t)\equiv 1$情况,导出了相应上界型检验统计量在零假设下的精确分布. 当样本容量$n$较大时,精确分布的计算时间较长, 本文还通过模拟比较得到了在不同样本量下,应采用的计算方法. 最后, 给出一个实际例子对前述方法加以简单说明.     

Subgraphs of hypercubes and subdiagrams of boolean lattices

In this paper we extensively treat the following problems: When is a given graph a subgraph (resp. induced subgraph) of a hypercube and when is an ordered set a subdiagram (resp. induced subdiagram) of a Boolean lattice? We present characterizations for that in terms of suitable edge-colorings of the graphs and, for ordered sets, of their covering graphs.     

On the Wielandt Subgroup in a p-Group of Maximal Class

The Wielandt subgroup of a group G,denoted by w(G),is the intersection of the normalizers of all subnormal subgroups of G.In this paper,the authors show that for a p-group of maximal class G,either wi(G) = ζi(G) for all integer i or wi(G) = ζi+1(G) for every integer i,and w(G/K) = ζ(G/K) for every normal subgroup K in G with K = 1.Meanwhile,a necessary and suflcient condition for a regular p-group of maximal class satisfying w(G) = ζ2(G) is given.Finally,the authors prove that the power automorphism group PAut(G) is an elementary abelian p-group if G is a non-abelian pgroup with elementary ζ(G) ∩ 1(G).