BIFURCATION OF SINGULAR POINTS NEAR A DOUBLE FOLD POINT INZ_2 -SYMMETRIC NONLINEAR EQUATIONS

We consider the bifurcation of singular points near a double fold point in Z2 -symmetric nonlinear equations with two parameters,where the linearization has a two dimensional null space spanned by a symmetric null vector and an ami-symmetric null vector. In particular, we show the existence of a turning point path and a pitchfork point path passing ihrough the double fold point and they are the only singular points nearby. Their nondegeneracy is confirmed. A supporting numerical example is also provided. The main tools for our analysis as well as the compulation are some extended systems.     

BIFURCATION ANALYSIS AND COMPUTATION OF DOUBLE TAKENS-BOGDANOV POINT IN Z2-EQUIVARIABLE NONLINEAR EQUATIONS

The paper deals with the computation and bifurcation analysis of double Takens-Bogdanov point (u~0, Λ~0) (in short, DTB point) in the Z_2-equivariable nonlinear equation f(u,Λ)=0,f:U×R~4→ V, where U and V are Banach spaces, parameters A∈R~4. At (u~0,Λ~0) , the null space of f_u~0 has geometric multiplicity 2 and algebraic multiplicity 4. Firstly a regular extended system for computing DTB point is proposed. Secondly, it is proved that there arc four branches of singular points bifurcated from DTB point: two paths of STB points, two paths of TB-Hopf points. Finally, the numerical results of one dimensional Brusselator equations are given to show the effectiveness of our theory and method.     

MULTIPLE POINTS OF IMMERSIONS OF M~6■R~7

In this article, we determine the multiple point set of immersions of 6-dimensional manifolds into 7-dimensional Euclidean space R7. The method is to evaluate the characteristic numbers of these manifolds. By a certain method, these numbers can be read of from the Hurewicz image ofh : π7QMO(1) → H7QMO(1).In particular, we show that for any immersion of P 6 in R7 the double, 4-fold and 6-fold point manifolds are cobordant to boundaries and triple point manifolds cobordant to P 4, 5-fold point manifolds cobordant to P 2 and 7-fold point manifold are odd number.     

BIFURCATION PROBLEMS OF MULTIPARAMETER AND MULTIBRANCH AND THEIR APPLICATIONS

This paper deals with the bifurcation problems of multiparameter and multibranch and their approximations. It derives general approximation results of solution branches in the neighbourhood of a singular (limit, turning or multibranch) point, and shows that turning and multibranch points are bifurcation points. The abstract theories are applied to a mixed finite element approximation of stationary Navier-Stokes equation.     

THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

For two rational quadratic B-spline curves with same control vertexes, the cross ratio of four eollinear points are represented; which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the twocurves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Beeier curves, the value is generally related with the loeation of the ray, and the necessary and sufficient condition o5 the ratio being independent of the ray‘s loeation is showed. Alsn another cross ratio o5 the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initlal vertex intersects respectivdy with the curve segmentt the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only whh the ray‘s location, butnot with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.     

Heteroclinic orbits and heteroclinic chains for a discrete Hamiltonian system

In the present work we prove some existence results of heteroclinic orbits and heteroclinic chains for a second order discrete Hamiltonian system of the form Δ2q(t-1)+V(q(t))=0,t∈Z.The methods we use are variational in nature.Our results show that under general conditions,for each maximum point β of V,the above system possesses multiple heteroclinic orbits joining β and some other maximum points of V.We also prove that for any pair of distinct maximum points η and ξ of V,there exists at least one heteroclinic chain from η to ξ.     

无界区域上Cauchy型奇异积分算子的性质

On the basis of introducing the modified Cauchy kernel, we discuss the Hoelder continuity of the Cauchy-type singular integral operator on unbounded domains for regular functions by dividing into the following three cases： two points are on the boundary of region; one point is on the boundary and another point is in the interior（or exterior） of the region; two points are in the interior （or exterior） of the region.     

Von Neumann Regularity and Quadratic Conorms in JB^＊-triples and C^＊-algebras

We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.     

MULTIPLE POINTS OF IMMERSIONS OF M~6■R~7

In this article, we determine the multiple point set of immersions of 6-dimensional manifolds into 7-dimensional Euclidean space R7. The method is to evaluate the characteristic numbers of these manifolds. By a certain method, these numbers can be read of from the Hurewicz image ofh : π7QMO(1) → H7QMO(1).In particular, we show that for any immersion of P 6 in R7 the double, 4-fold and 6-fold point manifolds are cobordant to boundaries and triple point manifolds cobordant to P 4, 5-fold point manifolds cobor...     

THE UNIQUENESS OF LIMIT CYCLE AND THE STRUCTURE OF CRITICAL POINT AT INFINITY FOR A CLASS OF CUBIC SYSTEM

A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.     

A PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interior-point method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.     

（Z2）^k-Actions with Fixed Point Set of Constant Codimension 2^k＋ 2

Let J.^r ,k denote the ideal in MO. of cobordism classes containing a representative that admits (Z2)^k-actions with a fixed point set of constant codimension r. In this paper we determine J,k^2k 2 and Ja,3^2^3 1.     

Shock wave solution for a class of nonlocal nonlinear singularly perturbed boundary value problems with turning point

The nonlocal nonlinear boundary value problem with turning point is considered. Leading the stretched variables the formal asymptotic solution of the original problem is constructed. And by using the theory of differential inequalities the existence and uniformly validity of solution for the original boundary value problems are proved. The contribution of this paper is that, the asymptotic behavior of solution is studied by using a simple and special method.     

THE LIMITING PROFILE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS WITH A SHRINKING SELF-FOCUSING CORE

In this paper,we consider the semilinear elliptic equation systems ■ where N≥3,a,∞> 1,α+β<2~*,2~*=2N/(N-2) and Qn are bounded given functions whose self-focusing cores {x∈R^N|Q_n(x)> 0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang [13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentra...     

Op en Lo op Saddle Point on Linear Quadratic Sto chastic Differential Games

In this paper, we deal with one kind of two-player zero-sum linear quadratic stochastic differential game problem. We give the existence of an open loop saddle point if and only if the lower and upper values exist.     

Conjugate Points on a Type of Khler Manifolds

We study conjugate points on a type of Khler manifolds, which are submanifolds of Grassmannian manifolds. And then we give the applications to the study of the index of geodesics and homotopy groups.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

Chamber Structure for Some Equivariant Relative Gromov–Witten Invariants of P~1 in Genus 0

In this paper, we study genus 0 equivariant relative Gromov–Witten invariants of P1 whose corresponding relative stable maps are totally ramified over one point. For fixed number of marked points, we show that such invariants are piecewise polynomials in some parameter space. The parameter space can then be divided into polynomial domains, called chambers. We determine the difference of polynomials between two neighbouring chambers. In some special chamber, which we called the totally negative chamber, we show that such a polynomial can be expressed in a simple way. The chamber structure here shares some similarities to that of double Hurwitz numbers.     

Symmetry Adapted Block Diagonalisation in Equivariant Steady State Bifurcation Problems and its Numerical Application

Considering steady state bifurcation of single parameter r-equivariant systemswhere Γ is a compact Lie group acting on X via a faithful orthogonal representation, we simultaneously block-diagonalise all jacobians of g along a solution path with respect to symmetry adapted subspaces and apply bifurcation groups to reduce symmetry breaking bifurcation points to simple bifurcation points. In generical situations this algorithm allows for numerically stable use of continuation methods and steady state bifurcation point detection and computation. The large scale decrease of computing efforts is demonstrated at computing the equilibria of a hexagonal lattice dome with 21 degrees of freedom, we present a great variety of solution arcs with symmetry from full D6 down to Z2 symmetry.     

AN EQUIVALENCE BETWEEN TWO ALGORITHMS FOR GENERAL QUADRATIC PROGRAMMING

In this paper,we demonstrate that the general quadratic programming algorithms of Fletcher,and Keller are equivalent in the sense that they generate identical sequences of points.Suchan equivalence of the two algorithms extends earlier results for convex quadratic programming dueto Best.Several computational results have verified this equivalence.