In this paper the algebraic structures of phase spaces are investigated. The phase spaces with hierarchy are defined and a property of the least subspaces of a phase space is given. According to these results, some characteristics of linear tautologies are discussed.

In this paper, by the method of stochastic linear programming and some skill in s-space, we give a proof for the existence of the optimal measurable coupling of transition probability.

In this paper, we establish some inequalities involving two n-dimensional simplexes in Euclidean space En, which are generalizations of the well-known Pedoe inequality, and prove some applications thereof.

Let {Xn, n≥0} be a sequence of measurable functions taking their values in the alphabet S = {1,2,…, N}. Let P,Q be two probability measures on the measurable space, such that {Xn,n ≥ 0} is Markovian under Q, Let h(P \ Q) = limsupn-1 log[P(X0,…, Xn)/Q(X0,…, Xn)} be the sample divergence-rate distance n→∞ of P relative to Q. In this paper, a class of small deviations theorems for the averages of the functions of two variables of an arbitrary information sources are discussed by using the concept h(P \ Q), and, as a corollary, a small deviations theorem for the entropy densities of arbitrary information sources is obtained. Finally, an extension of Shannon-McMillan Theorem on the case of nonhomogeneous Markov information sources is given.

Suppose D is a nonempty regular open set in Rd (d≥ 3). This paper gives the usual formula of the nonegative bounded solution for the Dirichlet problem for one class of nonlinear equations by using super Brownian motion. Under some conditions, this paper proves the existence and the uniqueness for the Dirichlet problem have fixed limit at infinity and obtain the probabilistic formula and extended some results in [4].

We give a sufficient condition, for a densely defined operator A, generating a contract C-semigroup, which is also necessary if C is a isometric operator, this generalize the Lumer-Phillips theorem; moreover, we use this result to describe the generators of isometric C-semigroups.

Consider the impulsive delay differential equation We obtain a global existence and uniqueness theorem of solutions of Eq.(E). General impulsive perturbed conditions are also obtained for the oscillatory or nonoscillatory behavior of all solutions when the corresponding linear delay differential equations has same behavior.

We study the existence of hamonic maps with free boundaries from disk into Riemannian manifolds. By using the perturbation technique of Sacks-Uhlenbeck, and the estimate of morse index of hamonic maps, we obtain some existence results.

In the paper "On theta pairs for a maximal subgroup" (Proc. Amer. Math. Soc., V109N3(1990)), N.P. Mukherjee and P.Bhattacharya denned the concept of θ-pairs associated to a maximal subgroup of a finite group, studied the influence of θ-pairs on the structure of the group itself and obtained several results. In this paper we study first the further properties of θ-pairs, then improve in essence some of the main results in Mukherjee and Bhattacharya's paper by using these properties and deal with some new conditions which are necessary and sufficient for a group to be solvable or nilpotent.

In this paper, we give the duality properties of relative Hopf modules. At first, we prove that the duality modules of relative Hopf modules are still relative Hopf modules. Then, we give the basic structure theorem and Maschke's theorem to the duality relative Hopf modules of relative Hopf modules.

This paper gives some sufficient conditions which ganrantee the existence and uniqueness and stability and instability of periodic solutions and almost periodic solutions and the estimate on the existence bound of the periodic solutions for some differential equations. Our results extend the main results in [1] and the relative results in [2-6].

This article apply the idea of the uncertainty principle to consider the uniqueness of the Cauchy problems for second order elliptic operators with non-smooth characteristics. We first construct a decomposition of function and study the Carleman estimate in the micro-local sense, then we prove the uniqueness of the Cauchy problems in more general cases.

In this paper, sufficient conditions are obtained for the oscillation of all solutions of the homogeneous Neumann, Dirichlet and Robin's boundary value problems associated with the hyperbolic system for i = 1, 2, ???, m, (x, t) ? ft x (0, oo), $1 C Rn, where A denotes the Laplacian in E".

For Szasz-Mirakian operator Sn, Bernstein operator Bn and Baskakov operators Vn, we prove that there exists a constant C > 0 such that wψ12 (f; 1/n~1/n)≤ and and hold for all where and denote Ditzian-Totik moduli of smoothness.

Campanato space is widely used to study partial differential equation. Thus in this paper we generalize Morrey space and Campanato space, and study the properties of the introduced generalized Morrey space and generalized Campanato space. In another paper, we obtain the regularity for nonpower growth elliptic partial differential equation by means of generalized Campanato space.

This paper considers the following oscillatory singular integrals: Tf(x)= P.V. ∫Rn eip(x-y)K(x - y)f(x)dy, where K(y) = Ω(y)/||y||n and ∫Sn-1Ω(y)dσ(y) = 0. For P which satisfies certain convex condition andΩ∈ Lq(Sn-1) (q > 1), we prove that T is bounded on Lp(Rn), p > 1.

In this paper we study the relationships of inequalities of Hardy martingales with values in complex quasi-Banach space and analytic uniform convexity of the space, prove weak (1,1), strong (p,p) and convex φ-function inequalities of q-mean-square function of Hardy martingales, and using them give characterizations of analytic q-uniform convexifiablity of complex space. In the same time we characterize the same property by using Banach-space-valued analytic functions and Brown motion in unit disk of complex plane.

The following problem of Heilbronn type is considered. Let S Rk be a set consists of n points A1, A2,…,An, d(S)=min{AiAj|1≤i

Let F be a field, R an F-algebra with an involution, if any matrix over R has a singular value decomposition (SVD for short), then R is said to have the SVD property. This paper shows that: R is an F-algebra having the SVD property if and only if R is isomorphic to either R+ or R+(-1~1/-1) or the quaternion field (-1,-1/R+) over R+, where R+ is the set of all symmetric elements of R, and R+ is a Galois order closed field.

In this paper, we introduce the notion of Wn-modules and describe non-commutative coherent rings with finite FP-selfinjective dimension. The obtained results generalize the works of Stentrom and Bass. Finally we apply these results to extension closed categories of modules.

In this paper, by introducing the notion of a strongly real Hilbert ring, we show that (1) A ring A is a strqngly real Hilbert ring if and only if A[X] is a real Hilbert ring; and (2) A ring A is a strongly real Hilbert ring if and only if every real maximal ideal of A[X]contracts in A to a real maximal ideal. Thereby, the two main results listed in reference [1] are negated. Moreover, the so-called strictly real Hilbert rings are investigated; these rings are of significance for the discussion of the semialgebraic Nullstellensatz.

This paper first generalizes Ekeland's variational principle and then considers the relation between (P.S.) condition and coercivity of functional.

In this paper, we provide nonlinear ergodic theorems and weak convergence theorems for commutative semigroups of asymptotically nonexpansive type mappings iu uniformly convex Banach spaces that satisfy opial's condition or have a Frechet differentiable norm.

An answer to the problem formulated by W. Fenchel[1] is obtained. A global property of spherical curves is given and applied to generalize Jacobi theorem on closed space curves.

In the present paper we establish some surjectivity theorems for the operators A and C, where A is m-accretive and C is compact. Our results generalize and improve the main results of Zhu[5].

In this paper, by using the theory of exponential dichotomies and functional analystic method, we investigate the degenerate heteroclinic bifurcation of the systems whose unperturbed equations have not heteroclinic manifold. The sufficient condition assuring the existence of transversal heteroclinic orbits is given by making use of Melnikov vector.

Let M be a Γ-ring in the sense of Nobusawa.The ring M2=(RΓML) was defined by Kyuno. In this paper, first we characterise primitive ideals and the Jacobson radical of M2. Next, we introduce a new class of Γ-rings in which, for every homomorphic image, the Jacobson radical equals the simplicial radical. We shall call this the class of PM-Γ-rings. Relationships between PM properties of Γ-ring M and the corresponding properties of Γn,m-ring Mm,n, the right operator ring R of Γ-ring M, M-ring Γ and the ring M2 are established. Finally, we give a general version of Jacobson properties for Γ-rings, which includes also Jacobson property, Brown-McCoy property and PM property.

This paper deals with a kind of superlinear boundary value problems with singularity. We prove continua to the solution set exist. Our corollary improves the main results of [1] to some extent, and we drop their main condition off.

This paper discussed the complex measure dμ= ψdv where the complex valued function ψ∈ Lloc1 and some properties of conditional expectations and matin-gales with respect to the measure dμ. Then it proved that the maximal operators of the martingales are week (p, p) bounded and strong (p, p) bounded if ψsatisfies bp+(K) condition and that the square operators of the martingales are L2-bounded if ψ satisfies b∞+(K) ∩ a1(K) condition.

The limit cycle bifurcations near a heteroclinic loop with two or three saddle points are studied. The uniqueness of limit cycles is proved uader a simple condition, and a necessary and sufficient condition for a limit cycle to exist is obtained. As an application to a codimension two bifurcation in a three dimensional system, the problem on the uniqueness of limit cycles near a heteroclinic loop of the deduced plane system is solved.

In this paper, we construct some even continuous solutions of the Feigen-baum Function Equation. For every 0 < λ < 1, we give out some C∞-even-solutions of this equation.

In this paper, we introduce the conceptions S∈ (Q) S∈ L(H) with σe(S) C σe(T) for every operator T quasisimilar to S, β(S) = {λ∈C : a neighbourhood U of λ such that (S -z) ○(V,H) is closed for every neighbour hood V of λ, V U}, characterize (Q) operators by means of β(S) and show that the family of (Q) operators includes many familiar operators (such as subdecompasable, M-hyponormal, semi-hyponormal operators, etc.) and is norm-dense in L(H).

A Markov chain in markovian environment is studied in this paper. For this process we prove a limit law for the recurrence to small cylindrical sets. Moreover, we give a sufficient condition for the process being φ-mixing, and give a exponential estimation of the probability for the process to return to small cylindrical sets.

This paper gives two moment inequalities for σ-mixing and (ψ-mixing sequences. In the study of weighted sums, they are more useful than ones abtained by Shao Qiman in [6] and [7]. Here they are applied to discuss the rate of strong convergence for the nonparametric recursive kernel estimators and the strong consistency for the nonparametric weighted kernel estimators of regression function. The obtained results are much better.

In this paper, higher dimensional periodic systems with delay of the form x'(t) = A(t,x(t))x(t) + f(t,x(t - r)) are considered, where the n x n matrix A(t, x) and the n-vector f(t,x) are continuous in (t,x) ∈ R × Rn, and A(t + T,x)-A(t,x), f(t + T, x) = f(t, x). Using fixed point method, some sufficient conditions to guarantee the existence and uniqueness of T periodic solution for the systems are obtained. The main results in [1-3] are extended and improved.

We give some conditions of the pointwise continuity of the M-P inverses Ax+ of Ax, when Ax is pointwise continuous. Then some applications to integral equations are given in 3.

The main result of this paper is: suppose that A(ζ,η) satisfies A0, A1, A2, A3, A4, then Tb is bounded on L2(K) if and only if b eBMO(K). We apply this result to prove that the commutator of fractional integration with b is bounded on L2(K) if and only if b∈I-s(BMO), (-1 < s < 0), the commutator of singular integral transform with b is bounded on L2(K) if and only if b∈BMO(K), and the commutator of multiplier with b is bounded on L2(K) if b∈BMO(K).

Let M be the ideal in Z[v] generated by v-1 and a odd prime p, U be a quantum algebra over A-Z[V]M with a symmetric Cartan matrix. Let k be algebraically closed field of characteristic zero. Consider a ring homomorphism A→k (v→ζ) and let Uk = U A k, Uk be infinitesimal quantum algebra of Uk,ζ be a primitive p-th root of 1. Let V and M be finite dimensional integrable Uk-modules. In this paper we show that Uk module isomorphisms V M ≌ M V when at least one of V or M is injective, or least one of V or M is trivial Uk module. Moreover, if V is indecomposable Uk module, M is indecomposable Uk module and M restricts to an indecomposable Uk module, then we show that V M is indecomposable Uk module. Finally, if V and M is Uk module isomorphisms and with unique simple submodule, we prove that the V and M also is Uk module isomorphisms, this shows Uk structure on indecomposable injective Uk module is unique.

In this paper, we study the structures of modules over group ring RG with KdimR ≤1, gl.dimR ≤ 2 or w.gl.dimR = 0. As applications, we give some characterizations of group rings with lower dimensional coefficient rings.

Comparison theorems for nonnegative splittings of bounded linear operators have been presented by Marek and Szyld. Without detailed proofs, they also gave the hypotheses which provide strict inequalities for the comparisons. In this note, several counter examples are given which show that all the hypotheses by Marek and Szyld, which provide strict inequalities, are not sufficient. Then the corrected comparison theorems are proposed.