The theory of harmonic maps has been developed since the 1960's (see [2]). In recent years, some authors discussed the harmonicity of "homogeneous" maps between Riemannian homogeneous spaces using the theory of Lie groups. Let G and G' be compact Lie groups, H and H' their closed subgroups respectively. Assume that a homomorphism θ: G→G' maps H into H'; then there exists an induced map f_θ: G/H→G'/H'. M.A. Guest gave a necessary and sufficient condition for such a map to be harmonic, when G/H and G'/H' are generalized flag manifolds, H=T is a maximal torus and G' is a unitary group; and he gave some interesting examples (see [3]). We generalize his results to the case of general generalized flag manifolds G/H, i.e. H is a centralizer of a torus, and give some new examples of harmonic maps.

In [6], D. Fux determined the structures of cohomology of formal vector fields Lie algebras with coefficients in trivial modules or dual adjoint modules. We determine the structure of H¹ (L, V), where L=W (n), S(n) or H(n) is an infinite dimensional Lie algebra of characteristic 0 and V is a graded L-module V₀ (mixed product) or a graded irreducible L-module.

In [6], D. Fux determined the structures of cohomology of formal vector fields Lie algebras with coefficients in trivial modules or dual adjoint modules. We determine the structure of H¹ (L, V), where L=W (n), S(n) or H(n) is an infinite dimensional Lie algebra of characteristic 0 and V is a graded L-module V₀ (mixed product) or a graded irreducible L-module.

In this paper the thought on the uniform convergence of an empirical Bayes estimator or linear empirical Bayes (l.e.B.) estimator is advanced. Under two different models the l.e.B. estimators of the parameter are constructed respectively. It is proved that the convergence rates and uniform convergence rates of these l.e.B. estimators are all one with respect to the corresponding prior families. It is shown that the uniform convergence rate one is the best, under mild assumptions imposed on the conditional density of the sample.

In this paper, we construct a double indexing family of reducible SU(2)-connctions over S²×S² in a geometric way. By separation of variables, we compute the spectrum of the Hessian of the Yang-Mills functional at these connections. Using the implicit function theorem, we prove that these connections are isolated non-minimal solutions to the Yang-Mills equations on S²×S².

Under the constraint determined by a relation (a_n, b_n)^T={f(Φ)}_n between the reflectionless potentials and the eigenfunctions of the general discrete Schrödinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution variety N of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented. f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.

We consider the nonlinear periodic differential equation

where a(t) and p(t) are continuous and 1-periodic, β is a positive constant. The purpose of this paper is to prove that all solutions of the above-mentioned equation are bounded for t∈R and there are infinitely many quasi-periodic solutions and an infinity of periodic solutions of minimal period m, for each positive integer m.

In this paper, we show that under a certain technical condition, if a space has no 2-torsion, then either Sq²ⁿ x≠0 or there exists some y with Sq²ⁿy=Sq²ⁿ⁺¹ x, if for some n≥3 Sq²ⁿ⁺¹ x≠0. The proof uses relations between Steenrod operations and operations in connective real K-Theory.

Let f:Mⁿ→N^v(n,p)+m be a map.Suppose that m=n-1 or n<4.We obtain the necessary and sufflcient conditions forf to be homotopic to apth order immersion.Some concrete pth order immersion results of RPⁿ are also proved.

Consider a pair of genuinely nonlinear strictly hyperbolic conservation laws U_t+F(U)_x=0 with initial data U(O,X)=U_o(X).Suppose that the initial data U_o(X)=U₁(X)+U₂(X),where U₁(X) will issue rarefaction waves only,U₂(X) has any finite total variation and sufficiently small deviation.We prove that the Cauchy problem has a global solution.

In this paper,we consider a class of quasilinear elliptic eigenvalue problems with limiting nonlinearity.First,we use the concentration-compactness principle to get the existence of a minimum u∈H₀¹(ω,R^N) of the minimization problem

then we apply the reverse Hölder inequality to prove that u∈L^∞(ω,R^N).

In this paper we give a method for solving the functional equations arising from the differential embedding problem.We also obtain the conditions for embedding one-dimensional diffeomorphisms into differential flows.

In this paper,we study a two-parameter family of systems E_c in which E₀ as a contour consisting of a saddle point and two periodic motions of saddle type,i.e.,the situation is similar to that described by Lorenz equations for parameters b=8/3,σ=10,r=r_l=24.06,and get some results concerning bifurcation phenomenon and dynamical behavior of the orbits of E_c in a small neighborhood of the contour for |ε| near zero.Thus,under a few natural assumptions which are verified numerically,we can explain some numerical results of Lorenz equations for parameters near the above values in a mathematically precise way,which is different from the methods of J.Guckenheimer et al.^([3],[4]),by considering Lorenz equation as a one-or two-dimensional map.

In this paper the "residual complex" is defined when a group and its subgroups act on a complex.With its aid a homological spectral sequence of group products is given.And the author makes a concentrated study of the structure of the residual complex and proves that it becomes a clear "step complex" if the group can be expressed as an amalgamated free product of its subgroups.

This paper considers the problem of the existence and uniqueness of the connecting orbit for a cooperative vector field F in the retangular box B.Suppose that the origin O and the point P with positive components are vertices of B,only containing the equilibria O and P,and that the Jacobian matrices of F at O and P are irreducible.Then there is a unique orbit of F contained inB which joins O and P.

In this paper it will be shown that the spectrum of every unicellular unilateral weighted shift operator on a symmetric Banach space is the singleton set {0}.From this,we give an affirmative answer to Rosenthal-Shields' problem.

In the present paper we establish a sharp asymptotic formula for

with an error term (σ is a suitable positive constant) uniformly inq and T,which improves the best result hitherto given.

Suppose X and the coefficients A₁,…,A_m are n×n matrices.Let B be an mn×mn matrix as in (7).Let J be the Jordan canonical matrix of B and B=PJP^-1.Let E_i denote the i×i unit matrix.V is defined by dV/dt=JV and V(t=0)=E_mn.Then Z=PV satisfies dZ/dt=BZ.P^* is a matrix which consists of the first n rows of P.The author proves: There is a solution of (1)↔there are an mn×n matrix C,an n×n matrix Q and an n×n function matrix N such that P^*VC=QN,where det Q≠0 and N is defined by N(t=0)=E_n and dN/dt=RN,in which R is an n×n Jordan canonical matrix.There are three cases regarding the solutions of (1): No solution,finite k solutions,1<k<C_n^m,and infinite solutions which contain j parameters,1<-j<-mn².

For infinite discrete groups,Effros introduced the notion of inner amenability which gives a new classification of discrete groups.The inner amenability is a considerably weaker condition than amenability,but closely related to the quite deep property Γ of groups.In this paper the author investigates the structures of inner amenable groups by theoretical set theory.A sequence of characterizations of inner amenable groups is given here by developing the well-known Folner's conditions for amenable locally compact groups.

To estimate the exponent of a regularly varying d.f.F,the asymptotic behaviour of Hill's estimator has been extensively discussed.Under the assumption that the d.f.F is continuous,we obtain the large deviation theorem for Hill's estimator.

This paper deals with the closedness of generalized vertex operators defined by the central binomial coefficient under Lie bracket.The formula of Lie product of generalized vertex operators has also been obtained.

In this paper,we give a necessary and sufficient condition for the one-parameter families of diffeomorphisms on S¹ to be stable and a necessary condition for the multi-parameter families to be stable; and,moreover,we prove that phase-locking is a generic property of the one-parameter families of diffeomorphisms on S¹.We also get a necessary and sufficient condition of phase-locking for the one-parameter families of integral diffeomorphisms on S¹ which strengthens a result in [2].

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem
(A)
where L is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation of u^p at infinity.The existence of solutions to (A) is strongly dependent on the behaviour of a_ij (x),b(x) and f(x,u).For example,for any bounded smooth domain Ω,we have such that Lu=u^p possesses a positive solution in H₀¹(Ω).

We also prove the existence of nonradial solutions to the problem -Δu=f(|x|,u) in Ω,u>0 in Ω u=0 on ∂Ω,Ω=B(0,1).for a class of f(r,u).

A coincidence theorem for a single-valued mapping in [3] is generalized as one for a strongly decomposable multivalued mapping.

As we know,B.Sz-Nagy and C.Foins studied systematically contractions on Hilbert spaces and developed the harmonic analysis theory of operators on Hilbert spaces.Since 1950s,people paid great attention to the study of contractions on π_k spaces.Only a few results have been obtained until today; in particular,the spectral theory of contractions on π_k Spaces and corresponding harmonic analysis theory have left still unexplored.This paper,as a continuation of [1],[2],[6],in which the authors after discussing some problems such as the negative invariant subspaces and unitary dilations of contractions on complete spaces with indefinite metrics,establish the triangle model of contractions on π_k spaces and furthermore,apply the triangle model to the study of spectral theory of contractions on π_k spaces,which is essential to the harmonic analysis of operators on π_k spaces.

In this paper we give the volume of tubes around complex submanifolds in complex space form in terms of the technique of Jacobi fields.

We construct an oracle A such that .So the polynomial time hierarchy is separated from the polynomial time probabilistic complexity class in relativization.

In his report at ICM,Smale posed a problem whether the inequality α(t,ψ_d )≤1 holds for allt∈(0,1),where .This paper gives a negative answer.In addition,elementary proofs are given for the theorems on the tractability of random algorithm and on the average area of approximate zeros.Meanwhile,discussion about the algorithm of the global Newton's method is devoted in this paper.

In this paper,we study the uniformly asymptotic stability of nonautonomous retarded difference-differential equations by constructing Lyapunov functionals.We conclude that the retarded difference-differential equation is uniformly asymptotically stable under the strong diagonal dominance.

Let f:(X,A)→(X,A) be an extension of a given map ψ:A→A,where (X,A) is a pair of compact polyhedra.We shall introduce a special Nielsen number,SN(f|ψ),which is a lower bound for the number of fixed points on X-A for all extensions in the homotopy class of f.It is shown that for many space pairs this lower bound is the best possible one,and that it can be realized without the by-passing condition.

We prove in this paper the C^∞ regularity for a "very strict" local minimum of class C_loc^ρ,ρ>3,of functionals with genuine degenerate quasiconvex integrand depending on a vector-valued function u.Such a minimum satisfies the condition: for all x∈Ω,there exists a neighbourhood K(x) of x in Ω and C₁ (x)>0,C₂ (x)>0,1≥ε(x)>0,such that for all real Ψ∈c₀^∞ (K).

In this paper we discuss the concept ‘generalized exponential dichotomy' and give the existence of C^k invariant manifolds for abstract nonautonomous differential equations in Banach or Hilbert spaces.Also we give a classification of invariant manifolds and an estimate of the locality radius of invariant manifolds.

We define the analyticity of Wiener functionals and study its properties and applications to oscillatory Wiener functionals.

Let A,B be unital C^*-algebras,D_A¹ the set of all completely positive maps ψ from A to M_n (C),with Tr ψI)≤1(n≥3).If Ψ is an α-invariant affine homeomorphism between D_A ¹ and D_B¹ with Ψ (0)=0,then A is ^*-isomorphic to B.Obtained results can be viewed as non-commutative Kadison-Shultz theorems.

This paper provides two kinds of forbidden configurations for the rectilinear O-embeddability of triangle free planar graphs.Meanwhile,the characterizations of the O-embeddability for outerplanar graphs and Halin graphs are found.

We prove the existance of a kind of singular directions concerning the differential polynomials .

In this paper we apply the Malliavin calculus for two-parameter Wiener functionals to show that the solutions of stochastic differential equations in plane have a smooth density under the restricted Hörmander's condition.This answers a question mentioned by Nualart and Sanz in [3].