Existence of Generalized Heteroclinic Solutions of the Coupled Schrdinger System under a Small Perturbation

The following coupled Schrdinger system with a small perturbation uxx + u- u3+ βuv2+ f(, u, ux, v, vx) = 0 in R,vxx- v + v3+ βu2v + g(, u, ux, v, vx) = 0 in R is considered, where β and are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution(called the generalized heteroclinic solution thereafter).     

带有广义导子的素环

The concept of derivations and generalized inner derivations has been generalized as an additive function δ: R→ R satisfying δ(xy) = δ(x)y xd(y) for all x,y∈R,where d is a derivation on R.Such a function δis called a generalized derivation.Suppose that U is a Lie ideal of R such that u2 ∈ U for all u ∈U.In this paper,we prove that U(C)Z(R) when one of the following holds:(1)δ([u,v]) = uov (2)δ([u,v]) uov=O(3)δ(uov) =[u,v](4)δ(uov) [u,v]= O for all u,v ∈U.     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.     

Existence and bifurcation of solutions for a double coupled system of Schrdinger equations

Consider the following system of double coupled Schr¨odinger equations arising from Bose-Einstein condensates etc.,-△u+u=μ1u3+βuv2-κv,-△v+v=μ2v3+βu2v-κu,u≠0,v≠0 and u,v∈H1(RN),whereμ1,μ2are positive and fixed;κandβare linear and nonlinear coupling parameters respectively.We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system.Then using the positive and non-degenerate solution to the scalar equation-△ω+ω=ω3,ω∈H1r(RN),we construct a synchronized solution branch to prove that forβin certain range and fixed,there exist a series of bifurcations in product space R×H1r(RN)×H1r(RN)with parameter κ.     

一类优美树

§1. Preliminary Lemma Using A to denote the cardinal number of a finite set A, we do not explain anymore what are similar to A in this paper. For a given tree T(V,E), if there is a labeling f of its vertices, which satisfiesf［V(T)］={f(u)u∈V(T)}={0,1,2,…,E(T)},lf［E(T)］={lf(uv)=f(u)-f(v), uv∈E(T)}={1,2,…,E(T)},then f is called a graceful labeling of T, and T is called a graceful tree. lf denotes the labeling of edges that is derived from f. In the 1960s, RingedKotzing and A. R…     

方程u_(tt)=u_(xxt)+f(u_x)_x初边值问题的差分法

The finite difference method is considered for the followinginitial-boundary-value problem: arrayllutt=uxxt+f(ux)x, & (x,t) QT, u(x,0) =(x), & x ［0,1］, ut(x,0) = (x), & x ［0,1］, u(0,t) =u(1,t) =0, & t ［0,T］,array. where f(s),(x) and (x) are given functions;QT=［0,1］ ［0,T］. The convergence of the finite difference schemesis verified by discrete functional analysis methods and prior estimationtechniques.     

Almost homomorphisms between unital <Emphasis Type="Italic">C</Emphasis>*-algebras: A fixed point approach

Let A , B be two unital C*-algebras. By using fixed pint methods, we prove that every almost unital almost linear mapping h : A → B which satisfies h(2 n uy) = h(2 n u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, … , is a homomorphism. Also, we establish the generalized Hyers-Ulam-Rassias stability of *-homomorphisms on unital C*-algebras.     

Invariant sets and solutions to the generalized thin film equation

The invariant sets and the solutions of the 1 2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set E0 = {u : ux = vxF(u), uy = vyF(u)}, where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the l l-dimensional nonlinear evolution equations.     

A concentration behavior for semilinear elliptic systems with indefinite weight

Consider the Schrdinger system{-Δu+V1,nu=αQn(x)︱u︱α-2u︱v︱β,-Δv+V2,nv=βQn(x)︱u︱α︱v︱β-2v,u,v∈H10(Ω) where ΩR~N,α,β 1,α + β 2* and the spectrum σ(-△ + V_(i,n))(0,+∞),i = 1,2;Q_n is a bounded function and is positive in a region contained in Ω and negative outside.Moreover,the sets{Q_n 0} shrink to a point x_0∈Ω as n→+∞.We obtain the concentration phenomenon.Precisely,we first show that the system has a nontrivial solution(u_n,v_n) corresponding to Q_n,then we prove that the sequences(u_n) and(v_n) concentrate at x_0 with respect to the H~1-norm.Moreover,if the sets {Q_n 0} shrink to finite points and(u_n,v_n) is a ground state solution,then we must have that both u_n and v_n concentrate at exactly one of these points.Surprisingly,the concentration of u_n and v_n occurs at the same point.Hence,we generalize the results due to Ackermann and Szulkin.     

Some Notes on a Minimization Problem

Let X[a,b] be a compact set containing at least n+1 points and Kan n-dimensional Haar subspace in c[a,b]. Let F(x,y) be a nonnegativefunction, defined on X×(-∞,∞), satisfying ‖F(·,p)‖<∞ with the L_∞norm forsome∈K, where F(x,p)≡F(x,p(x)). The minimization problem discussed in this paper is to find an elementp∈K such that ‖F(·,p)‖=inf ‖F(·,q)‖, such an element p(if any) is saidto be a minimum to F in K~(q∈K). The author in [1,2] studied this problem and has given the main theoremsin the Cbebyshev theory under the following assumptions: (A) lim F(x,y)=∞, x∈X; (B) lim F(x,u)=F(x,y), x∈X,y; (C)lim F(u,υ)=F(x,y),x∈X,y; (D) For each x∈X there existtwo real numbers f~-(x) and f~+(x),f~-(x)f~+(x). such that F(x,y) is strictlydecreasing with respect to y on (-∞,f~-(x)] and strictly increasing on [f~+(x),∞), and F(x,y)=F(x):=inf F(x,υ) on [f~-(x),f~+(x)]. Denote f_1(x)=inf{y:F(x,y)‖F~*‖},f_2(x)=sup{y:F(x,) ‖F‖},f_1(x)=lim f_1(u),f_2(x)=lim f_2(u), G=(q∈K: f_1qf_2}.For pεK set X_p={     

关于弹性半无限体表面的稳定性

本文综合报道有关弹性半无限体表面稳定性的若干工作.对于不可压缩弹性半无限体,概述在双向受载下自由表面的失稳分析,给出失稳的临界条件.对于可压缩弹性材料情况,分析了由标准材料组成的半无限体的表面轴对称失稳,得到失稳临界参数对于材料参数的依赖关系.     

Sets of finite perimeter and the Hausdorff-Gauss measure on the Wiener space

In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the one-codimensional Hausdorff measure restricted on the reduced boundary and/or the measure-theoretic boundary, which may be strictly smaller than the topological boundary. In this paper, we discuss the counterpart of this measure in the abstract Wiener space, which is a typical infinite-dimensional space. We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter. The formula is presented in terms of the integration with respect to the one-codimensional Hausdorff-Gauss measure restricted on the measure-theoretic boundary.     

Complex structures in the Nash-Moser category

Working in the Nash-Moser category, it is shown that the harmonic and holomorphic differentials and the Weierstrass points on a closed Riemann surface depend smoothly on the complex structure. It is also shown that the space of complex structures on any compact surface forms a principal bundle over the Teichmüller space and hence that the uniformization maps of the closed disk and the sphere depend smoothly on the complex structure.     

Broken hyperbolic structures and affine foliations on surfaces

In this paper, we introduce two new kinds of structures on a non-compact surface: broken hyperbolic structures and broken measured foliations. The space of broken hyperbolic structures contains the Teichmüller space of the surface as a subspace. The space of broken measured foliations is naturally identified with the space of affine foliations of the surface. We describe a topology on the union of the space of broken hyperbolic structures and of the space of broken measured foliations which generalizes Thurston's compactification of Teichmüller space.     

Flat Partial Connections on a Three Manifold Equipped with a Codimension One Foliation

Given a compact connected oriented three manifold, equipped with a codimension one foliation, such that the Bott connection on the normal bundle is flat, a 2-form on the space parametrizing flat partial connections on it has been constructed. This form is closed. In the special case where the foliated three manifold is a surface bundle over the circle, this 2-form is identified with a certain 2-form on the parameter space for a class of paths in the representation space for the surface group. The 2-form, in question, on the parameter space for paths is constructed from the natural symplectic form on the representation space for a surface group.     

Saddle surfaces in singular spaces

The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel' on compactness of saddle surfaces in a Euclidean space.

     

The Picard group of the moduli of G-bundles on a curve

Let G be a complex semi-simple group, and X a compact Riemann surface. The moduli space of principal G-bundles on X, and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G₂ coarse moduli space and the moduli stack).     

A criterion for straightening of a lipschitz surface in the Lizorkin-Triebel sense. I

We study the conditions when the trace of a Lizorkin-Triebel space on a Lipschitz surface coincides with the trace of this space on a hyperplane. A criterion in terms of a dyadic weighted inequality is found for a wide range of indices.     

LIFE SPAN OF A SMOOTH SOLUTION FOR THE SURFACE DIFFUSION FLOW

Consider the motion of immersed hypersurfaces driven by surface diffusion flow and give an lower bound on the life span of a smooth immersed solution, which depends only on how much the curvature of the initial surface is concentrated in space.     

Affine foliations and covering hyperbolic structures

We describe the relationship between closed affine laminations in a punctured surface and some associated hyperbolic structures on certain covers of the punctured surface, which we call covering hyperbolic structures. Further, in analogy with the theory of William Thurston relating the Teichmüller space of a surface to the projective lamination space, we describe a space with points representing affine laminations in a given surface and with other points representing the associated covering hyperbolic structures. Received: 27 March 2000 / Revised version: 10 January 2001