Problems of Lifts in Symplectic Geometry

Let(M,ω)be a symplectic manifold.In this paper,the authors consider the notions of musical(bemolle and diesis)isomorphisms ω~b:T M→T~*M and ω~?:T~*M→TM between tangent and cotangent bundles.The authors prove that the complete lifts of symplectic vector field to tangent and cotangent bundles is ω~b-related.As consequence of analyze of connections between the complete lift ~cω_(T M )of symplectic 2-form ω to tangent bundle and the natural symplectic 2-form dp on cotangent bundle,the authors proved that dp is a pullback o f~cω_(TM)by ω~?.Also,the authors investigate the complete lift ~cφ_T~*_M )of almost complex structure φ to cotangent bundle and prove that it is a transform by ω~?of complete lift~cφ_(T M )to tangent bundle if the triple(M,ω,φ)is an almost holomorphic A-manifold.The transform of complete lifts of vector-valued 2-form is also studied.     

Property T＊＊ for C＊-algebras

Extending the notion of property T of finite von Neumann algebras to general von Neumann algebras, we define and study in this paper property T** for (possibly non-unital) C* -algebras. We obtain several results of property T** parallel to those of property T for unital C* -algebras. Moreover, we show that a discrete group Γ has property T if and only if the group C* -algebra Cr* (Γ) (or equivalently, the reduced group C* -algebra Cr* (Γ)) has property T**. We also show that the compact operators K(l2 ) has property T** but c0 does not have property T**.     

A Survey of the Homotopy Properties of Inclusion of Certain Types of Configuration Spaces into the Cartesian Product

Let X be a topological space.In this survey the authors consider several types of configuration spaces,namely,the classical(usual)configuration spaces F_n(X)and D_n(X),the orbit configuration spaces F_n~G(X)and F_n~G(X)/S_nwith respect to a free action of a group G on X,and the graph configuration spaces F_n~Γ(X)and F_n~Γ(X)/H,whereΓis a graph and H is a suitable subgroup of the symmetric group S_n.The ordered configuration spaces F_n(X),F_n~G(X),F_n~Γ(X)are all subsets of the n-fold Cartesian product ∏_1~nX of X with itself,and satisfy F_n~G(X)?F_n(X)?F_n~Γ(X)?∏_1~nX.If A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of view.The principal results known in the literature concern the usual configuration spaces.The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusionι:A-→∏_1~nX,the homotopy type of the homotopy fibre I_ιof the mapιvia certain constructions on various spaces that depend on X,and the long exact sequence in homotopy of the fibration involving I_ιand arising from the inclusionι.In this respect,if X is either a surface without boundary,in particular if X is the 2-sphere or the real projective plane,or a space whose universal covering is contractible,or an orbit space S~k/Gof the k-dimensional sphere by a free action of a Lie group G,the authors present recent results obtained by themselves for the first case,and in collaboration with Golasi′nski for the second and third cases.The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest.In order to motivate various questions,for the remaining types of configuration spaces,a few of their basic properties are described and proved.A list of open questions and problems is given at the end of the paper.     

SPECTRAL PROPERTIES OF POPULATION OPERATOR AND ASYMTOTIC BEHAVIOUR OF POPULATION SEMIGROUP

In this paper we discuss the spectral properties of the Population operator, prove that the population operator has only one real eigenvalue γ_0, which is greater that real parts of other eigenvalues,and find the corresponding relation between γ_0 and the critical fertility rate β_(cr). We also study the existence and asymptotic behaviour of the semigroup for the population system, then come to the conclusion about the stability of the population system,including the existene of the steaby population state in the critical case of the fertility rate.These are all the new results of the quantitative population theory.     

COMBINATION OF GLOBAL AND LOCAL APPROXIMATION SCHEMES FOR HARMONIC MAPS INTO SPHERES

It is well understood that a good way to discretize a pointwise length constraint in partial differential equations or variational problems is to impose it at the nodes of a triangulation that defines a lowest order finite element space. This article pursues this approach and discusses the iterative solution of the resulting discrete nonlinear system of equations for a simple model problem which defines harmonic maps into spheres. An iterative scheme that is globally convergent and energy decreasing is combined with a locally rapidly convergent approximation scheme. An explicit example proves that the local approach alone may lead to ill-posed problems; numerical experiments show that it may diverge or lead to highly irregular solutions with large energy if the starting value is not chosen carefully. The combination of the global and local method defines a reliable algorithm that performs very efficiently in practice and provides numerical approximations with low energy.     

Extension of Floquet's theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

A New Approach to Synchronization Analysis of Linearly Coupled Map Lattices

In this paper, a new approach to analyze synchronization of linearly coupled map lattices (LCMLs) is presented. A reference vector x(t) is introduced as the projection of the trajectory of the coupled system on the synchronization manifold. The stability analysis of the synchronization manifold can be regarded as investigating the difference between the trajectory and the projection. By this method, some criteria are given for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to the eigenvalue "0" of the coupling matrix play key roles in the stability of synchronization manifold for the coupled system. Moreover, it is revealed that the stability of synchronization manifold for the coupled system is different from the stability for dynamical system in usual sense. That is, the solution of the coupled system does not converge to a certain knowable s(t) satisfying s(t 1) = f(s(t)) but to the reference vector on the synchronization manifold, which in fact is a certain weighted average of each xi(t) for i = 1, ... ,m, but not a solution s(t) satisfying s(t 1) = f(s(t)).     

Chaotic invariant sets of a delayed discrete neural network of two non-identical neurons

In this paper, we show that a delayed discrete Hopfield neural network of two nonidentical neurons with no self-connections can demonstrate chaotic behavior in a region away from the origin. To this end, we first transform the model, by a novel way, into an equivalent system which enjoys some nice properties. Then, we identify a chaotic invariant set for this system and show that the system within this set is topologically conjugate to the full shift map on two symbols. This confirms chaos in the sense of Devaney. Our main result is complementary to the results in Kaslik and Balint (2008) and Huang and Zou (2005), where it was shown that chaos may occur in neighborhoods of the origin for the same system. We also present some numeric simulations to demonstrate our theoretical results.     

A COMPREHENSIVE PROOF OF THE GREENBERGER-HORNE-ZEILINGER THEOREM FOR THE FOUR-QUBIT SYSTEM

Greenberger-Horne-Zeilinger(GHZ)theorem asserts that there is a set of mutually commuting nonlocal observables with a common eigenstate on which those ob- servables assume values that refute the attempt to assign values only required to have them by the local realism of Einstein,Podolsky,and Rosen(EPR).It is known that for a three-qubit system.there is only one form of the GHZ-Mermin-like argument with equiva- lence up to a local unitary transformation,which is exactly Mermin's version of the GHZ theorem.This article for a four-qubit system,which was originally studied by GHZ,the authors show that there are nine distinct forms of the GHZ-Mermin-like argument.The proof is obtained using certain geometric invariants to characterize the sets of mutually commuting nonlocal spin observables on the four-qubit system.It is proved that there are at most nine elements(except for a different sign)in a set of mutually commuting nonlocal spin observables in the four-qubit system,and each GHZ-Mermin-like argument involves a set of at least five mutually commuting four-qubit nonlocal spin observables with a GHZ state as a common eigenstate in GHZ's theorem.Therefore,we present a complete construction of the GHZ theorem for the four-qubit system.     

Dense 3-uniform hypergraphs containing a large clique

An r-uniform graph C is dense if and only if every proper subgraph G' of G satisfies λ(G') λ(G).,where λ(G) is the Lagrangian of a hypergraph G. In 1980's, Sidorenko showed that π(F), the Turán density of an γ-uniform hypergraph F is r! multiplying the supremum of the Lagrangians of all dense F-hom-free γ-uniform hypergraphs. This connection has been applied in the estimating Turán density of hypergraphs. When γ=2 the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However,when r ≥ 3, it becomes much harder to estimate the Lagrangians of γ-uniform hypergraphs and to characterize the structure of all dense γ-uniform graphs. The main goal of this note is to give some sufficient conditions for3-uniform graphs with given substructures to be dense. For example, if G is a 3-graph with vertex set [t] and m edges containing [t-1]~(3),then G is dense if and only if m≥{t-2 3)+(t-2 2)+1. We also give a sufficient condition on the number of edges for a 3-uniform hypergraph containing a large clique minus 1 or 2 edges to be dense.     

Operator Equations and Duality Mappings in Sobolev Spaces with Variable Exponents

After studying in a previous work the smoothness of the space UΓ0={u∈W1,p(·)(Ω);u=0 on Γ0 Γ=Ω},where dΓ-measΓ0>0,with p(·)∈C(Ω)and p(x)>1 for all x∈Ω,the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space.The results obtained in this direction are used for proving existence results for operator equations having the form Ju=Nfu,where J is a duality mapping on UΓ0 corresponding to the gauge function,and Nf is the Nemytskij operator generated by a Carath′eodory function f satisfying an appropriate growth condition ensuring that Nf may be viewed as acting from UΓ0 into its dual.     

THE EIGENVALUE PROBLEM EQUIVALENT TO MULTIVARIATE POLYNOMIAL SYSTEM

In this paper, the authors proved that finding all solutions of a given multivariate polynomial system is equivalent to solving a relative joint eigenvalue problem(Theorem 1) and in some cases one can find all solutions of the given system from the eigenvalues and vectors of one matrix or matrix pencil (Theorem 2). Especially the situation that the ideal generated by the given system is 0-dimensional is discussed.     

INITIAL BOUNDARY VALUE PROBLEMS FOR QUASILINEAR HYPERBOLIC-PARABOLIC COUPLED SYSTEMS IN HIGHER DIMENSIONAL SPACES

The initial bounary value problem for quasilinear hyperbolic-parabolic coupled systemsin higher dimensional spaces,which arises in many mechanical problems is considered.Underthe assumptions that the hyperbolic part of the coupled system is a quasilinear symmetrichyperbolic system and the parabolic part is a quasilinear parabolic system of second orderand suitble asstunptions of smoothness and compatibiliy conditions,the existence anduniqueness of local smooth solution is proved in the cases that the boundary of domain isnoncharacteristic or uniformly characteristic with respect to the hyperbolic part.As an application,the existence and uniqueness of local smooth solution for the initialboundary problem of the radiation hydrodynamic system,as well as of the viscous compressiblehydrodynamic system,with solid wall boundary,is obtained.     

Grunwald-Wang theorem,an effective version

The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem,which asserts that given a family of local characters χvof K *vof exponent m, where v ∈ S for a finite set S of primes of K, there exists a global character χ of the idele class group CK of exponent m(unless some special case occurs, when it is 2m) whose local component at v is χv. The effectiveness problem for this theorem is to bound the norm N(χ) of the conductor of χ in terms of K, m, S and N(χv)(v ∈ S). The Kummer case(when K contains μm) is easy since it is almost an application of the Chinese remainder theorem. In this paper, we solve this problem completely in general case, and give three versions of bound, one is with GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an l-th power mod N. One also gets the least upper bound for N such that lr| φ(N) and p is not an l-th power mod N.Some part of this article is adopted(with some revision) from the unpublished thesis by Wang(2001).     

GLOBAL C~1 SOLUTION OF CAUCHY PROBLEM FOR TWO-DIMENSIONAL GAS DYNAMICS SYSTEM

Using the method of characteristic lines this paper considers the global C~1 solution of the Cauchy problem for two-dimensional gas dynamics system. When the initial data degenerate to the special case φ_0(x, y)=const, the global C~1 solution is obtained. For the case of isentropic exponent γ=1, a transformation about variables is introduced, which changes the system to a first order linear hyperbolic system with constant coefficients and the global C~1 solution is also obtained in this case when the initial data of the forms (φ_0(x, y), u_0(x, y), u_0(x, y))=(exp(w_(01) (c_1x d_1y) w_(02)(c_2x d_2y)), u_(01)(c_1x d_1y) u_(02)(c_2x d_2y), u_(01)(c_1x d_1y) u_(02)(c_2x d_2y)), where c_i and d_i(i=1, 2) are constants.     

Global well-posedness of the fractional Klein-Gordon-Schr¨odinger system with rough initial data

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R~(1+1). We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H~(s_1) and wave data in H~(s_2) × H~(s_2-1)for 3/4- α s_1≤0 and-1/2 s_2 3/2, where α is the fractional power of Laplacian which satisfies 3/4 α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 s_2 1/2 by using the conservation law for the L~2 norm of u.     

Local stable and unstable sets for positive entropy C;dynamical systems

For any C;diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy,a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent.The main line of our approach to this result is under the setting of topological dynamical systems,which is also applicable to infinite-dimensional C;dynamical systems.     

The product of the restrained domination numbers of a graph and its complement

Let G=(V,E) be a graph.A set S■V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S.The restrained domination number of G,denoted γr(G),is the smallest cardinality of a restrained dominating set of G.In this paper,we show that if G is a graph of order n≥4,then γr(G)γr(G)≤2n.We also characterize the graphs achieving the upper bound.     

On the Cauchy problem and peakons of a two-component Novikov system

We study a two-component Novikov system, which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity. The primary goal of this paper is to understand how multi-component equations, nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons. We establish the local well-posedness of the Cauchy problem in Besov spaces B_(p,r)~s with 1 p, r +∞, s max{1 + 1/p, 3/2} and Sobolev spaces Hs(R)with s 3/2, and the method is based on the estimates for transport equations and new invariant properties of the system. Furthermore, the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied. A blow-up criterion on solutions of the Cauchy problem is demonstrated. In addition, we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line, and the single-peaked solitons on the circle, which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.