Let S^1.1 denote the submanifold of Lorentz space R^2.1,which is composed of all points l with l²=1 and let T^1.1= R^1.1/Z×Z.In this paper we study the existence of nontrivial harmonic maps from T^1.1 to S^1.1 and H²,and construct a harmonic map for any homotopy class of maps from T² to S^1.1.

The differential equation
in considered,where θ is a real parameter.We prove that Hopf bifurcation occurs at the nonzero foci and is always subcritical for any A in a subregion in the complex plane.

In this paper,we introduce a Z_p index theory.For any given positive integer p,we introduce a subset E_p of positive integers and define a family of index mappings σ_n,∀_n∈E_p.We prove that this index theory possesses the similar properties as Z₂ and S¹ index theories do.In particular,by means of a Z_p Borsuk-Ulam theorem given in one of our recent papers we prove that under some suitable conditions this theory also possesses dimensional property which is important in applications.As a simple application,we study the bifurcation problem of periodic solutions of nonautonomous Hamiltonian systems.

In the present paper we establish a criterion of algebraic independence of complex numbers θ₁,...,θ_n over a field K⊂C of finite transcendence type using a sequence of nonzero polynomials in several variables with integral coefficients,which satisfy simultaneously certain upper and lower estimates in different orders of magnitude at the point (ω₁,...,ω_q,θ₁,...,θ_n),where {ω₁,...,ω_q} is a transcendence basis of K over C.

Harry-Dym's equation (HD)r_t=(r^-1/2)_xxx is well-known for ies cusp soliton solutions.In this paper,relatoions are revealed HD and a completely integrable Hamkltonian system in Liouvile sense given by

in the symplectic manifold (R^2N,dp∧dp).Here <ξ,η〉 is the standard inner product in R^N.

In this paper,we prove that if ZFC is consistent,then so are the following theories:

where MA denotes Martin's axiom.KT(ω₂) the statement:"There exists an ω₂-Kurepa tree",and SOCA,SOCA1,OCA and ISA are axioms introduced in [1].

In this paper we discuss the completions (G,τ) of a commutative l-group G with respect to the intrinsic topologies τ.We give some conditions under which τ is the intrinsic topology of the same type on G as τ and give the relations between these completions.

In this paper,we establish a sharp inequality of the gradient of energy density.We use it in studying stability of domain and pinching of energy.And we get the sharp conclusion respectly.In addition,we connect the existence of non-constant totally geodesic maps with the construction of manifolds on topology and geometry.

In this paper the limit points of the Kaplan-Meier estimator is discussed.We use the method of strong approximation to get the unit ball of the reproducing kernel Hilbert space.

Let F=(f_ij)be an n×n matrix of H^∞entries,and define

This type of operator plays an important role in Cowen-Douglas theory.We call it the imbedded operator with symbol F.In the paper [L],we have already shown that S_F is a compact pertubation of T_z⊕ ^* I_n by BDF Theorem.In this paper,we begin to investigate the compact part of S_F.We give a practical method to calculate this compact part when n=1 and F is any finite Blaschke product.

This paper considers geodesic triangies in a Riemannian manifold M.First we imbed the set Σ of geodesic triangles in M into a big space E,then find some equations in E satisfied by tangent vectors of Σ.Finally we give an application of the result.

To each curve singularity there associates a tree with suitably defined weights and special paths.These combinatorial data reflect the topological properties of the singularity.It is shown that such a curve can be specified by several axioms,so that every tree satisfying these axioms arises from a curve singularity.

It is proved that the modules in any component of the AR-quiver of a wild hereditary Artin algebra are uniquely determined by their composition factors.

Resonances of Schrödinger operators are closely related to the poles of certain meromorphic extension of the resolvent. In this paper, we study the resonances in Stark effect, generated by negative eigenvalues. We give precise location for the resonances and prove in particular that the widths of resonances are exponentially small and can be measured by some Agmon's distance.

For any polynomial p(z)=a₀zⁿ+a|z^n-1+…+a_n, a ₀≠0, n≥2,F is the Julia set and μ^* is the equilibrium distribution on F. Hans Brolin ^[1] proved that γ (F)>0, and S_μ^* =F. Up to now, we know nothing about rational functions. The aim of this paper is to discuss the case of rational functions.

Paracommutator was first introduced and studied by Janson and Peetre in 1985. They mainly discussed its boundedness and Schatten-von Neumann properties on L²-space. In this paper we study the boundedness of paracommutator on L^p-spaces and obtain some useful criteria.

It is shown that the number of square-full numbers in the interval is asymptotically equal to for every θ in the range 1/6>θ≥0.14254, which extends P.Shiu's range 1/6>θ≥0.1526.

In this paper,utilizing equivariant Mose theory for isolated critical orbits developed in [16] we study the versions of Morse identities under group actions,i.e.the global property of equivariant Morse theory. Combining the concept of critical multiplicity we give a lower bound of the critical multiplicities for a class of invariant functionals.As applications we study the bifuractions of a class of equivariant potential operators and generalize some results due to Benci and Pacella, Chang Rabinowitz, etc.

Suppose that X_l,…, X_n are samples drawn from a population with density function f andf_n(x)=f_n (x;X_l,…, X_n is an estimate off(x), Denote by m_nr=∫|f_n(x)-f(n)|^rdx and M_nr=E(m_nr) the Integrated r-th Order Error and Mean Integrated r-th Order Error of f_n for some r-1 (when r=2,they are the familiar and widely studied ISE and MISE), In this paper the same necessary and sufficient condition for and a.s. is obtained when f_n(x) is the ordinary histogram estimator.

In this paper we develop a sequence Z₀,…,Z_n,…(n∈ω) of axiom systems for set theory, such that (1) the consistency of any system within the sequence is provable in its succeeding systems, (2) the first system in the sequence is Zermelo's system Z and the union of all systems in the sequence is just ZF.

Lattice-valued semicontinuous mappings play a basic and important role in solving the problems of L-fuzzy compactification theory, and make the previous work on weakly induced spaces and induced spaces determinatively generalized and strengthened. Moreover, we can describe the complete distributivity of lattices with them as well. In this paper, we give the mutually descriptive relation between lattice-valued semicontinuous mappings and the complete distributivity of lattices, and the construction theorems of open sets and closed sets in lattice-valued fully stratified spaces, weakly induced spaces and induced spaces (they are called S-spaces). Furthermore, we will investigate the structure of the co-topology of S-space, solve a series of interesting problems on product, N-compactness and metrization of S-spaces.

This is a sequel to our joint paper^[4] in which upper bound estimates for large deviations for Markov chains are studied. The purpose of this paper is to characterize the rate function of large deviations for jump processes. In particular, an explicit expression of the rate function is given in the case of the process being symmetrizable.

Suppose that f(x)∈C[-1,1], f(x) is convex in [0,1] and concave in [-1,0] with the new concept of regular convexity-turning points,we obtain the following estimates of Jackson type for coconvex approximation of f(x) by algebraic polynomials of degree ≤ n: .

We establish some new n-independent-variable discrete inequalities which are analogous to some Langenhop-Gollwitzer type integral inequalities obtained by the present author in J. Math. Anal. Appl., 109 (1985), 171-181. An application to hyperbolic summary-difference equations in n variables is also sketched.

In this paper we analyze the qualitative behaviour of the equation
,
where ε is a small parameter. We divide the interval of parameter b into four sets of subintervals. A, B, C and D. For b∈A, B or D, we discuss the different structures of the attractors of the equation and their stabilities. When chaotic phenomena appear, we also estimate the entropy. For b∈C, the set of bifurcation intervals, we analyze the bifurcating type and get a series of consequences from the results of Newhouse and Palis.

Under three kinds of constraints of potential for KdV hierarchy,the two systems, stemming from the spatial part and the time part of the Lax pair for KdV hierarchy, are shown to be naturally consistent. For the former system, the constants of the motion are presented and the Painlevé property is examined.

In this paper we construct a double Darboux Transformation for AKNS hierarchy and give its decomposition theorem. A remarkable characteristic of the double Darboux Transformation—incommutativity is proved.

Let f (m) be an irreducible quadratic polynomial with integral coefficients and positive leading coefficient. Under the assumption of Extended Riemann Hypothesis, we obtain new remainder terms in the upper bounds on primes represented by f(m) or f(p) which greatly improve Bantle's recent results. As an application, we obtain, in the second part of the paper, a new result on the lower bound of the least primes in arithmetic progressions with some difference.

We first study the Grassmannian manifold G_n (R^n+p)as a submanifold in Euclidean space Λⁿ (R^n+p). Then we give a local expression for each map from Riemannian manifold M toGⁿ (R^n+p) ⊂Λⁿ (R^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map from M to G_n (R^n+p). As a consequence of this formula we get a rigidity theorem.

We present in this note a simple proof of the Bott residue theorem in a slightly more general form.

In this paper, we consider the following question. For some given n-bordism class α and (n-k_i)-bordism classes β_n-ki,(1<k₁<...<k_i),. under what condition can we find a representation M of α and an involution T on M, such that the bordism class of the fixed point set of T is . This paper also gives some examples not satisfying the above property.

In this paper we shall give a new proof of the well-known theorem of Faith-Utumi ^[1]. Using our method we can show that every right order of K_n is a prime right Goldie ring, where K_n is the n×n-matrix ring over division ring K. Specially,D_n is a prime right Goldie ring, if D is a right order of K.

In this paper, we continue the discussion of the conjecture which says that infinitesimal Ⅱ-isometry of surface is infinitesimal Ⅰ-isometry, i.e., infinitesimally rigid. We have some invariants by means of which some integral formulas are worked out. As an application to these integral formulas, we get some results on infinitesimal Ⅱ-isometry of closed surface. The theorems proved are just more or less obvious generalizations of known results.

This paper is to give the necessary and sufficient conditions for a square integrable martingale array to converge stably, which improves the results given by Jeganathan, Brown & Eagleson and others.

The complete convergence theorem implies that starting from any initial distribution the one dimensional contact process converges to a limit as t→∞. In this paper we give a necessary and sufficient condition on the initial distribution for the convergence to occur with exponential rapidity.

In this paper, we study the Hankel operators belonging to E which denotes all of the essential Toeplitz operators, and show that the class of symbols of these Hankel operators is a Douglas algebra. Meanwhile, using the concept of essential commutant and some knowledge of the theory of function algebras, we solve a problem raised by Jos? Barria and P. R. Halmos in[1].

In this paper, we study the free boundary problem for degenerate parabolic equations (1.1)-(1.4). The existence of generalized solutions in BV₁, 1/2 is obtained by the means of parabolic regularization under certain restrictions. The uniqueness and regularity of generalized solutions are also discussed. In addition, a C^1+α smoothness for the free boundary is obtained in the parabolic case.