一类具有高阶转向点的二次问题的奇摄动

研究带有高阶转向点的二阶非线性微分方程的边值问题εy″=f(t)y′2+g(t,y)y(a,ε)=A,y(b,ε)=B的奇异摄动现象.在一定的条件下,得到了摄动解关于退化解的渐近性质及误差估计.     

Torelli Theorem for the Moduli Spaces of Rank 2 Quadratic Pairs

Let X be a smooth projective complex curve. We prove that a Torelli type theorem holds, under certain conditions, for the moduli space of α-polystable quadratic pairs on X of rank 2.     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

A Note on 3-Divisibility of Class Number of Quadratic Field*

In this paper, the authors show that there exists infinitely many family of pairs of quadratic fields Q(D^1/2) and Q((D+n)(1/2)) with D, n∈Z whose class numbers are both divisible by 3.     

Singular cycles of vector fields on regular parts of the boundary of Morse-Smale systems

At the boundary of the class of Morse-Smale vector fields there are vector fields whose unique degenerate phenomena is a singular cycle. We first characterize and classify all singular cycles which contains only one degeneracy (thesimple singular cycles: ssc). Each of these cycles defines a condimension one submanifold of vector fields. For some ssc its codimension one submanifold is a regular part of the boundary of the Morse-Smale systems. We characterize those ssc that defines this type of submanifold. Our ambient space isn dimensional,n2.Supported by Fondecyt, Proyecto 1930863.     

二次李三系的分解定理

有限维李三系称为二次的,如果它容许一个非退化的不变对称双线性型.无论是李三系分解为不可分解理想的直和,还是二次李三系分解为不可分解非退化理想的正交直和,证明了这两类分解在同构意义下都是唯一的.     

Quadratic prediction and quadratic sufficiency in finite populations

The problem of quadratic prediction for population quadratic quantities in finite populations has been considered in the literature. In this paper, we mainly aim at extending the ordinary quadratic prediction problems to a general case, and derive the representations of the two essentially unique optimal predictors: one is an optimal invariant quadratic unbiased predictor, and the other is an optimal invariant quadratic (potentially) biased predictor. Further, we show that the two predictors are nonnegative and reasonable by considering an extreme situation, and apply resulting conclusions to a special model with a compound symmetric variance matrix. In addition, we propose a notion of quadratic sufficiency with regard to the optimal prediction problems by employing materials derived in the first part, and investigate corresponding characterizations in detail.     

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits.More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x~2+ y~2+ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincar′e disc.     

Geometry of quadratic differential systems in the neighborhood of infinity

In this article we give a complete global classification of the class QS_ess of planar, essentially quadratic differential systems (i.e. defined by relatively prime polynomials and whose points at infinity are not all singular), according to their topological behavior in the vicinity of infinity. This class depends on 12 parameters but due to the action of the affine group and re-scaling of time, the family actually depends on five parameters. Our classification theorem (Theorem 7.1) gives us a complete dictionary connecting very simple integer-valued invariants which encode the geometry of the systems in the vicinity of infinity, with algebraic invariants and comitants which are a powerful tool for computer algebra computations helpful in the route to obtain the full topological classification of the class QS of all quadratic differential systems.     

Quadratic interior-point methods in statistical disclosure control

The safe dissemination of statistical tabular data is one of the main concerns of National Statistical Institutes (NSIs). Although each cell of the tables is made up of the aggregated information of several individuals, the statistical confidentiality can be violated. NSIs must guarantee that no individual information can be derived from the released tables. One widely used type of methods to reduce the disclosure risk is based on the perturbation of the cell values. We consider a new controlled perturbation method which, given a set of tables to be protected, finds the closest safe ones - thus reducing the information loss while preserving confidentiality. This approach means solving a quadratic optimization problem with a much larger number of variables than constraints. Real instances can provide problems with millions of variables. We show that interior-point methods are an effective choice for that model, and, also, that specialized algorithms which exploit the problem structure can be faster than state-of-the art general solvers. Computational results are presented for instances of up to 1000000 variables.AMS Subject Classification: 90C06, 90C20, 90C51, 90C90Jordi Castro: Partially supported by the EU IST-2000-25069 CASC project and by the Spanish MCyT project TIC2003-00997.     

On the Limit Cycles of Quadratic Differential Systems

In this paper we give the necessary and sufficient conditions for all finite critical points of quadratic differential systems to be weak foci, and solve an open problem proposed by Yanquian Ye. Received January 11, 1999, Revised October 10, 2000, Accepted March 5, 2001     

Closure method and asymptotic expansions for linear stochastic systems

A closure procedure for the hierarchy of moment equations related to linear systems of ordinary differential equations with a random parametric excitation is introduced. A generalization of Pringsheim's theorem for continued fractions is used in a proof of the procedure convergence. The boundary function method for singular perturbation problems is applied to obtain asymptotic expansions for the moments of the solutions of such systems.     

二次李超三系的分解定理

讨论了二次李超三系的唯一分解,得到了二次李超三系分解为非退化不可约阶化理想的分解定理.     

向量二阶非线性积分微分方程边值问题的奇摄动

该文研究向量二阶非线性积分微分方程边值问题的奇摄动, 在适当的条件下利用对角化方法证明了解的存在性, 构造出解的渐近展式并给出余项的一致有效的估计.     

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields K=k(D)over k=Fq(T)．For five series of real quadratic function fields K,the bounds of h(D)are given more explicitly,e．g.,if D=F^2 C．then h(D)≥degF／degP;if D=(SG)^2 cS．then h(D)≥degS／degP;if D=(A^m a)^2 A,then h(D)≥degA／degP,where P is an irreducible polynomial splitting in K,c∈Fq．In addition,three types of quadratic function fields K are found to have ideal class numbers bigger than one．     

Quadratic variation of martingales in Riesz spaces

We derive quadratic variation inequalities for discrete-time martingales, sub- and supermartingales in the measure-free setting of Riesz spaces. Our main result is a Riesz space analogue of Austin?s sample function theorem, on convergence of the quadratic variation processes of martingales.     

Homoclinic Orbits of a Quadratic Isochronous System by the Perturbation-incremental Method

In this paper, the perturbation-incremental method is presented for the analysis of a quadratic isochronous system. This method combines the remarkable characteristics of the perturbation method and the incremental method. The first step is the perturbation method. Assume that the parameter $\lambda$ is small, i.e. $\lambda\approx0$, the initial expression of the homoclinic orbit is obtained. The second step is the parameter incremental method. By extending the solution corresponding to small parameters to large parameters, we can get the analytical-expressions of homoclinic orbits.     

Quadratic assignment problems on series-parallel digraphs

Christofides has shown that the quadratic assignment problem (QAP) on isomorphic trees is solvable in polynomial time by dynamic programming. We generalize the approach to minimal series-parallel digraphs and show that this problem is equivalent to the general QAP, thus NP-hard. However the restriction to the subclass of series-parallel digraphs not containing bipartite subgraphs again is solvable in polynomial time. An algorithm for this class is given.This work was financially supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, Projekt S32/01.     

On the Conditions of a Center and General Integrals of Quadratic Differential Systems

We give the general integral of quadratic differential system with center in two cases under the Chinese classification. Project supported by the National Natural Science Foundation of China and Natural Science Foundation of Jiangsu Province Education Commission     

二次微分系统的中心条件和通积分

本文给出在中国分类之下二次微分系统有中心时两种情况的通积分.