Let R be a domains. In this paper we prove that wgl(R)2 if and only if every w-module over R is flat, and if and only if every finite type w-module over R is projective.

The nonlinear approximation problems with generalized weights are investigated. The maximal families of Kolmogorov and alternation type are characterized.Results on uniqueness elements and applications are also given.

In this paper the sufficient and necessary condition that holomorphic mappings are starlike on bounded starlike circle domain is discussed.

In this paper, we give a characterization of Bergman type spaces, and the boundedness of the Cesaro operator on Bergman type spaces.

Let Fq be a finite field with q=2t elements. In this paper, using singular symplectic and singular pseudo-symplactic geometry over Fq, we have given the sufficient and necessary condition of there exists solutions and the number of solutions X of rank k and solutions X to the equation XAXT=B over Fq. when A and B are symmetric matrices. Finally. we have obtained simple representations of enumerational formulas by q-hypergeometric series.

In this paper,we obtain a lower bound estimation about the Hausdorff dimension of the graph of a fractal interpolation function with the interpolation points

A two-dimensional hyperbolic system of nonlinear conservation laws was considered for any piecewise constant initial data having two discontinuity rays with the origin as vertex. One kind of new waves, which was labelled as the Dirac-contact wave, appeared in the solution. The entropy conditions for the Dirac-contact waves were given. The solutions on the Dirac-contact waves can be viewed as the bounded linear functionals on C(R2×R+).

In this paper the concept of sequential networks is introduced, the relations among that and the concepts of sequential quasi-bases and cs*-networks are established,and some new characterizations of the certain quotient images of metric spaces are obtained.

In this paper, the problem on limit cycles of Type I equation of quadratic differential systems dx/dt=-y+δx+lx2+mxy+ny2, dx/dt=x has been researched.It obtained the existential condition of limit cycles on large range when l≠0.

A new dual space notion of a Banach space, named Lipschitz dual space,is introduced, and within the new introduced space framework, the concept of the Lipschitz dual operator of a nonlinear Lipschitz operator is further defined. It is proved that the Lipschitz dual space of any Banach space E is an ordinary dual space of a certain Banach space containing E in the isometric embeding sense, and that the Lipschitz dual operator of any nonlinear Lipscitz operator becomes linear and bounded.By means of these findings, a lot of important results in linear analysis and theorems on linear operators are generalized to nonlinear cases. Thereby, a completely new way to generalize the linear operator theory to the nonlinear case is developed. As examples,several new mean ergodic theorems of the uniformly Lipschitz operators are proved.

In the category of general group-graded rings and graded-preserving ring homomorphisms, the characterizations of classical graded radical NG, LG, gG, rG and KG by means of elements are presented, the constructions of Nref, Lref, gref, rref and Kref are investigated and the relations between these graded radicals and their corresponding reflected radicals are discussed by introducing the weakly graded radicals.

Let P=NAM be the minimal parabolic subgroup of SU(n+1,1), which can be regarded as the affine automorphism group of Hn. According to the admissible condition in [1], we give the zonal wavelets on Heisenberg group Hn. Using the wavelet transform, we get the other decomposition of orthogonal direct sum of Li (Un+1, dμl(z, t, ρ) ). Moreover, the orthogonal decomposition of L2(P, dμl (z,t, ρ, u) )is given.

In this paper, N always denotes continuous nest, and T(N) = {T∈L(H),TMM: M∈N}. We describe the spectral picture of some strongly irreducible operators in T(N). and characterize the closure of the unitary orbit of the strongly irreducible operators in T(N).

Let X and Y be Banach spaces. We show that a spectrum-preserving surjective multiplicative map from B(X) to B(Y) is of the form (T) = ATA-1 for an isomorphism A of X onto Y.

Let. In this paper it is proved that there exists N(c)>0 such that for any N>N(c), the inequalityis solvable in prime numbers p1, p2, p3, where c1 is an absolute positive constant.

Let f be a continuous map from compact metric space onto itself. In this paper, for f has completely positive entropy, we get a necessary condition and some equivalent conditions when f has the pseudo-orbit-tracing property.

Let where a(x) and γ(x) are positive bounded integrable functions in D and We first establish a limit theorem of the super-Brownian motion X with parameters Then in Section 3 and 4, we study the structure of the set of all positive bounded solutions of the differential equation in a domain D (bounded or unbounded). All positive bounded solutions with Dirichlet boundary condition or stochastic Dirichlet boundary condition are represented in terms of the super-Brownian motion X.

In this paper, we consider existence of positive solutions for singular boundary value problem by a fixed point theorem in cones. At last, we will prove a brief theorem on existence of positive solutions for singular problem in decreasing condition.

In this paper, we improve the integrability of the derivatives of very weak solutions for a class of nonlinear elliptic systems (1.1). which satisfy. the controllable growth conditions (1.2)-(1.4).

The purpose of this paper is to establish a theorem on structure of ordered fields satisfying the infinite-dimensional real Nullstellensatz. For the preliminaries, in the first part of this paper, a class of ordered groups, which have no gap of rank d for an infinite cardinal number d, is investigated. With the aid of Hahn Embeding Theorem,we give the structure of ordered groups without gap of rank d for an infinite cardinal numberd.

Taking the concept of L*-Lindenbaum algebra as background, this paper introduced the concepts of implication lattice and regular implication lattice and discussed their basic properties. On the other hand, this paper introduced the concept of fuzzy implication space that is generallization of the concept of zero-dimensional space in point set topology. The representation theorem of regular implication lattices by means of fuzzy implication spaces had been established, that can be seen as, in certain sense, a generalization of the famous Stone’s representation theorem.

In this paper, we give sufficient and necessary conditions for operator ma trix to generate an exponentially bounded semigroup, where A1 and A0 are the coefficent operators of the complete sedond order abstract Cauchy problem (ACP2). we also give the relationship between the fact that generates an exponentially bounded semigroup and the C-wellposedness of semigroup and the C-wellposedness of (ACP2). Finally, we apply these,esult and those in [3] to discuss the C-wellposedness of a kind of (ACP2) with application background.

It is proved that norm continuous linear mappings of subalgebras of AF C*-algebras which preserve idempotents are Jordan homomorphisms, and that local derivation from subalgebra A of AF C*-algebra E into an arbitrary normed E-bimodule M is derivation, which generalizes the result of Crist about local derivation.

In this paper, hi-injectors are defined. Some new representations of injectors,flatjectors, perfect injectors and coflatjectors are given. Endomorphism ring S of finitely generated modules over R and some relations between ring R and ring S are discussed.

In this paper, we will study*-regular semigroups from a universal perspective. First. we introduce the concept of a*-relational morphism and discuss some properties of*-relational morphisms. Then we describe the varieties of*-regular semigroups generated by Mal’cev products of varieties of*-regular semigroups in terms of *-relational morphisms. Last, using this description, we determine the varieties of the forms (uog), (uoReB) and (uoReg).

In this paper, an extrapolation theorem of interpolation spaces by approximation is established on a quasi-norm Abelian group. The results generalizes the extrapolation theorem by the best approximation of a family of normal approximation sets, which was found by Devore R. The conditions here is weakened. Moreover. the similar theorem by operator approximation can be set up by our results.

In this paper, we prove that a quadratic system with a weak focus and weak pairing infinitehyperbolic saddles has at most one limit cycle, and that if this system has a weak infinite homoclinic loop, then its stability is contrary to that of the singular point surrounded by it, and this system has no limit cycle in the region formed by it.

For a 2 by 2 operator partial matrix of the form with A, C and D given operators acting on suitable separable complex Hilbert spaces, the intersection and the union of the spectra of all completions are completely characterized.are completely characterized.

In this article, preservation of hall, class, bound and rank in stability theory of module is discussed. The following results are proved: 1) NR is the hall of A in MR if and only if NS-1R is the hall of A in MS-1R; 2) β(p) =β(q) if and only if β(pR)=β(qR), 3) cl(p) =cl(q) if and only if cl(pR)=cl(qR), 4) U(P)=U(pR). Some conditions which ensure preservation of hall, class, bound and rank between MR and its fracted module (S-1 M)s-1 R are also given.

It is very importent in algebraic semigroup theory to characterize congruencesand congruence extensions on semigroups (see [1-5]). In this paper, we describe theclass of orthogroups by virtue of ways in [6] that investigating the normality and thevariance for the congruences oil bands and congruence extension (Theorem 2.6), andobtain the sufficient and necessary conditions for the class of orthodox semigroups thatall congruences on whose idempotent bands extend to these orthodox semigroups.

In this paper, we cosider the neutral higher dimensional periodic system:Where (t, x) ∈ R ×Rn, the n×n matrix A(t, x) is continuous, xt ∈C([-r, 0], Rn),xt(9) = x(t + θ), 0 E [-r, 0], denote C = C([-r, 0], Rn), f. R ×C→ Rn is continuous,furthermore A(t + T, x) = A(t, x)∈(t,x) E R ×Rn, f(t + T, ) = f(t,),(t，) ER × C, T, r ＞ 0, c ∈R. Using fixed point method, we get some sufficient conditionsto guarantee the existence and uniqueness of T-periodic solutions for the system. Thecorresponding results in [1-3] are extended and improved.

In this paper, a class of functionals of Kaplan-Meter estimator are investigated. Counting process martingale methods are used to show the asymptotic normality, and establish a mean square error inequality and a probability inequality ofthem without the assumption that F, G are continuous, where F, G are survival timedistribution and censoring time distribution respectively.

We study the conformal deformation for prescribing scalar curvature function S on Cartan-Hadamard manifold Mm (n 3) with strongly negative curvature.By employing the super-subsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for S near infinity so thatthe problem of complete conformal deformal is solvable. In the more general cases,we prove an asymptotic estimation oil the solutions of the conformal scalar curvatureequation.

For any nonempty closed set E in the unit circle, any positive number A andany real number μ (0 ≤μ≤λ), we construct a meromorphic function f(z) of orderA in the unit disc, which takes the set {L(θ)|θ∈E} as its Borel radii set and the set{L(θ)|θ| E} as its non-Julia radii set. And the order of its counting function N(r, f)Is μ.

In this paper,we prove that the plane quadratic system has at most one limitcycle surrounding a 2nd-order weak focus, and that if the product of the second andthird focus value of the weak focus is positive definite, then the system has no limitcycle surrounding the 2nd-order weak focus.

We discuss the consistency of the weighted kernel estimators of nonparametric regression function with censored data.Some more weakly sufficient conditions forstrong consistency are given. These results greatly improve the current ones.

In this paper, the convergence of solutions of cooperative systems is studied.The basic assumption is that the Jacobian matrices are uniformly stable in Withthis assumption, we provide the complete classification on the global behavior of suchsystems. The main results are the following. If the system has a positive steadystate then it is globally asymptotically stable in Int A sufficient and necessarycondition for a nonnegative steady state of the system to be globally asymptoticallystable is presented.

In the second part of this paper, we investigate the structure of ordered fieldssatisfying the infinite-dimensional real Nullstellensatz. By embedding these fields intothe associated fields of power series, we prove that an ordered field satisfies the infinitedimensional real Nullstellensatz if and only if its real closure is isomorphic to somesubfield of , which has the infinite-dimensional approximation property, where is a divisible ordered group without gap, and is the field of power series withcoefficients in the field of real numbers and exponents in .

From point view of Thurston we define and discuss the Schwarzian derivativeand Schwarzian derivative of high order of holomorphic mappings on domains of thematrix space Cm×n (m n) by using the approximation of second order. We provethat if the Schwarzian derivatives exist,then they are the invariant up to similarityunder the group of holomorphic auomophism of the compact dual space of RI(m,n)Also we prove that the Schwarzian in this paper is same as the Schwarzian in [1-4]Schwarzian of holomorphic mappings on domains of Cn

In this paper, the decreasing operator with convexity or concavity is studied,and existence, uniqueness and iteration of the fixed point on decreasing operator areobtained. Finaly, some applications to various equations are also done.