In this paper, we study the existence of solutions for random second-order multivalued periodic problem, and prove the existence theorems under most common hypotheses for both the convex and nonconvex problems. An application to random periodic feedback control system is also considered.

In this paper, we introduce the concept of "Bounded generalized inverse module maps" and give an equivalent characterization. As application, we obtain a factor decomposition theorem of certain important bounded module maps, such that the famous Douglas decomposition theorem on Hilbert space becomes a special case.

The uniform 3-part Cantor set K(λ, 3) is a self-similar fractal set satisfying the open set condition. In this paper, the exact value of the s-dimensional Hausdorff centered measure of K(λ,3) is given by computing the upper spherical density of a probability measure μ at point x, where s = logλ1/3 is the Hausdorff dimension of K(λ,3).

For 0

<1, the class of p-w-hyponormal operators is introduced. This class contains all w-hyponormal operators. Certain properties of this class of operators are obtained. First, many properties that the w-hyponormal operators possess are shown to hold for the p-w-hyponormal operators; for example, if T is a p-w-hyponormal operator, then its spectral radius and norm are identical, and the nonzero points of its joint point spectum and point spectum are identical. Secondly, for some special p-w-hyponormal operators, we also prove that their squares are p-w-hyponormal operators; but in general, it is not true for all p-w-hyponormal operators. Lastty, we show that there is no p-w-hyponormal operator in an n-dimensional space unless it is a normal operator.

In this paper some functions of product and mapping properties of k-spaces with a point-countable k-network are discussed via special metric spaces Tω and Tω1. The relationships among the three classes of spaces, which are introduced respectively for solving the "the products of k-spaces question" posed by Michael in 1973, are analysed via Tω; on the other hand, some internal characterizations of closed images of locally separable metric spaces are studied via Tω1.

By applying the way of weigh coefficient and Fubini theorem, we give some new Hardy-type integral inequalities with some functions.

Let A be an associative algebra, T1,... ,Tn EEEEE A. Write T = (T1,..., Tn), ε(T) denotes Lie algebra generated by T. It is shown that if ε(T) is a nilpotent Lie algebra, it is finite-dimensional. A sufficient and necessary condition for ε(T) to be a finite-dimensional nilpotent Lie algebra is given. If A is considered as a Banach algebra B(X) of all bounded linear operators on a Banach space X, it is proved that if the Lie algebra generated by quasinilpotent T1, T2 is finite-dimensional nilpotent Lie algebra, T1T2,T1 + T2 are also quasinilpotent; if the Lie algebra generated by non-zero compact operator T1 and non-scalar value operator T2 is finite-dimensional, T2 has nontrivial hyperinvariant subspace. This generalizes Lomonosov's theorem on invariant subspace of bounded linear operators.

Let M be a hyperfinite factor of type Ⅱ1, D is a Cartan masa of M, T be a Cartan subalgebas of M with diagonal D which generates M. If Φ: T →T be an σ-weakly continuous (Banach) isometry, then Φ can be extended a isometry on M.If a map Φ: T→T satisfies that for every pair a, b ∈ T, there is a oooooooooo-weakly continuous isometry Φa,b on T such that Φa,b(a) = Φ(a), Φ>a,b(b) = Φ(b), then Φ is a linear isometry.

The formulae on the dimensions of Hochschild cohomological groups of dual extensions of hereditary algebras and a class of l-hereditary algebras are obtained ex-

Let H be a complex seperable Hilbert space and L(H) denote the collection of bounded linear operators on H. In this paper, we first show that the quotient algebra (?)'(T)/rad(?)'(T) is not always commutative for each T ∈ Bn(Ω). Then after giving a sufficient condition for (?)'(T)/rad.(?)'(T) to be commutative, we show that the set of ,Bbn(Ω) operators A, with (?)'(A)/rad(?)'(A) commutative, is dense in Bn(Ω). Similar results are also established for Bmn(Ω) operators.

Given an positive integer n, we define the Smarandache power functions SP(n) as follows: SP(n) = min{m : n|mm, m ∈ N}. In this paper, we study the mean value distribution property of SP(n), and obtain a sharper asymptotic formula for the mean value of the Smarandache power function.

If N is a factor of type Ⅱ1 and G is a finite group with a properly outer action a on N, then N is a maximal subfactor in N (?)α G when G is of prime order. Conversely, suppose that N (?) M is an inclusion of factors of type Ⅱ1 and M (?) M1 is the basic construction for N (?) M. If [M :N]= p is a prime number, N'∩M= CI, N'∩ M1 is abelian and N (?)M is of depth 2, then N is a maximal subfactor of M.

In this paper, we consider a third order nonlocal boundary value problem x'"(t) = f(t,x(t),x'(t),x"(t)), t ∈ (0,1), x(0) = 0, x"(0) = 0, z'(1) =(?)x'(s)dg(s), where f : [0,1] x R3 →R is a continuous function, g : [0,1]→[0,∞) is a nonde-creasing function with g(0) = 0. Under the resonance condition g(1) = 1, an existence result is given by using the coincidence degree theory.

Let p>2 be a prime, c be an integer with (c,p)=1. For each integer a with 1≤a≤p-1, there exists one and only one b with 1≤b≤p-1 such that ab ≡ c mod p. Professor Lehmer asked us to find the number of cases in which a and b are of opposite parity. In this paper, we give two generalizations on this problem, and obtain two interesting hybrid mean value formulae.

In this paper, we study the Hardy-Littlewood inverse theorem about rotation modulus of continuity in Bergman spaces Bqp(0 < p < 1,q > 1). In an ordinary condition, we obtain a result as in Hp (0 < p < 1) spaces.

In this paper, we introduce a new class of partially power comparability and get a number of necessary and sufficient conditions for an exchange ideal to be a generalized cu-ideal. These results extend the corresponding theorems for exchange generalized cu-rings.

Let N≡ 5 (mod 24) be a sufficiently large positive integer. Assume GRH (the generalized Riemann hypothesis), then the equation N = n_1~2+ n_2~2+…+ n_2~5 has solutions, where ni, i = 1,2,..., 5 has a fixed number of prime factors. Puthermore, an asymptotic formula for the number of solutions is obtained.

In this article, we prove that the ternary Goladbach problem in arithmetic progression can be solved for almost all large positive moduli, where the moduli can be as large as AT1/6-ε.

Some existence results of positive fixed points for a convex operators is given by means of fixed point theorem of cone expansion and compression, then they are used to superlinear Hammerstein integral equations.

Consider the nonlinear semiparametric model with errors in covariables, we construct three empirical log-likelihood ratio statistics for the unknown parameter with the help of validation data. It is shown that the proposed statistics have the asymptotic chi-square distribution, and hence the confidence regions of the parameter can be constructed accordingly. In addition, the least squares estimator of unknown parameter is constructed, and its asymptotic behaviors are proved. A simulation study is carried out to compare the proposed methods with the least-squares method in terms of the confidence regions and its coverage probabilities.

Let {Xi,i≥1} be ρ-mixing sequences with identical distributions, and these distributions belong to domain of normal attraction with non-degenerate stable distribution. With probability one, we have limsupn→∞ . And we get several equivaent conditions, which not only generalize the obtained results equivalent to ρ- mixing sequences, but also improve them.

Some results of existence of positive solutions for the singular boundary value problems have been given, where the function f(t,x) may be singular at t = 0,1.

Let G be a connected finite graph. We consider three types of probability measures on Q, the set of subgraphs of G, which govern a random spanning tree, a random spanning forest, and a random connected subgraph respectively. Basing on the edge-negative association in uniform spanning tree , we construct a family of random spanning forests and random connected subgraphs on G which are edge-negative associated. Finally, we prove the edge-negative association property for uniform spanning forest in a special case of G.

The problem of the uniform dimensions for image sets and graph sets of N-parameter d-dimensional generalized a-stable process, which may not possess the uniform stochastic Holder condition, is investigated. The uniform Hausdorff dimension and the uniform Packing dimension for image sets of N-parameter d-dimensional generalized a-stable process are obtained, and the upper bounds of the uniform Hausdorff dimension and the uniform Packing dimension for graph sets are also given, the results contain the results of multi-parameter stable processes and generalized Brownian sheet.

The existence of positive solutions for boundary value problem of a nonlinear functional difference equation with p-Laplacian operator is investigated, where Φp(u) = |u|p-2u, p > 1, (?)(0) = 0, C+ = {(?) :(?) C, (?)(k)≥ 0, k ∈ [-(?),0]}. Sufficent conditions are obtained for the existence of at least one positive solution and two positive solutions.

In this paper we enrich Hua's result by proving that each sufficiently large odd integer N can be written as N = p_1~3 +…+ p_9~3,with |Pj - 3((N/9)~(1/2))|≤U = N1/3-2/555+ε, where Pj are primes. This result is obtained by an iterative method and a hybrid estimate for Dirichlet polynomial.

In this paper, we study the sum Tk,v(x, y) := ∑x

Firstly, we introduce the concept of triangular matrix ring of order n. Then, it is shown that the category mod Гn of finitely generated modules over a triangular matrix algebra Гn of order n is equivalent to the category Г(?)n. Moreover, we give the Jacobson radical of Гn, indecomposable projective objects in the category Гn(?) and the global dimension of Гn.

The asymptotic solution and connection formulae of Painleve equation are researched via a more direct method in this paper. The concrete method is: (1) To get the numerical solution first, then fit the oscillating asymptotic solution with the method of least squares; (2) According to the expression form of the oscillating asymptotic solution, the solution which depends on the relation between parameters is determined by means of harmonic balance method, that is, connection formulae. Use this kind of method to the general fifth Painleve equation, and let the parameters α,β,γ and δ satisfy some conditions. We find out the oscillating asymptotic solution and the connection formulae.

For a given negatively graded Lie superalgebra K-, the universal graded Lie superalgrbras U of type K- is defined and its existence is proved. By posing additional conditions on K-, other types of universal graded Lie superalgebras are defined and discussed. The concept of universal graded Lie superalgebras leads naturally to the graded Cartan type Lie superalgebras, and it is proved that the graded Cartan type Lie superalgerbras K(m,n,ωA),S(m,n) and H(m,n) can be characterized as certain universal graded Lie superalgebras.

In this paper, we present a method to solve system of algebraic equations or system of parametric algebraic equations by computing the normal zero decomposition of polynomial systems. For a system of algebraic equations, we can get a zero decomposition of triangular systems. We can get all the solutions of the algebraic system by computing the solutions of the triangular systems. For a system of parametric algebraic equations, we will give the conditions which the parametric should satisfy such that the algebraic system has solutions. Furthermore, the triangular systems in the normal decomposition still keep the triangular form for the specializations of the parameters. All the solutions of the parametric system can be got by computing the solutions of the triangular systems.

The structure of the ideal J*,k 2k+2k-1 of the unoriented cobordism ring MO* is determined in this paper.

By using of the contact equivalence of smooth mapping germs in singularity theory, this paper deals with the bifurcation problems with symmetry both on state variables and on parameters. The classiffication of multiparameter bifurcation problems with D3 symmetry on state variables and O(2) symmetry on bifurcation parameters up to condimension 3 is obtained, and the corresponding recognition conditions are given.

(M, T) denote the smooth involution on the r-dimention closed manifold M with fixed point set F. In this paper, the authors investage the bordism of the involution with F =∪i=1 mRPi(1)×HPi(n). Here HP(n) denote n-dimentional quaternionnic projective space.

In this paper, we present two types of the solvable Lie algebras with non degenerate invariant bilinear form. They are extended Heisenberg Lie algebra and 3-solvable non-degenerate Lie algebra. We determined the structure of a class of 3 solvable non-degenerate Lie algebra and its Derivation Lie algebras.

Each nonprincipal arithmetical ultrafilter p∈βω-ω is associated with a simple arithmetical model Np = {f(p) : f∈ω) N, and a positive real number is just an equivalence class of finite fractions made by arithmetical ultrafilters (instead of natural numbers), just like the ancient Greek's idea. It is well known that MA countable implies the statement (Q): (Q) If B■p(ω) has the sfip (strong finite intersection property) and |B| < 2ω, then there is a Q-point q  B. In this paper we will prove that the statement (Q) implies our Hypothesis: There exist nonprincipal arithmetical ultrafilters on ω.

In this paper, we have proved that L2 boundedness of commutators of BMO and Calderon-Zygmund singular integral operators with weak kernel, defined by [B,T]f = BTf-T(Bf) when B ∈ BMO, where T satisfies the following conditions: T(b1) ∈ BMO, T*(b2) ∈ BMO, b2Tb1 ∈ WBP for some accretive functions b1,b2.

In this paper, idempotents of factor von Neumann algebras is characterized by general Lie product; a characterization of idempotents of nest subalgebras in factor von Neumann algebras is obtained by Lie product. As its application, Lie isomorphisms of nest subalgebras in factor von Neumann algebras are studied, and it is shown that every Lie isomorphism between two nest subalgebras of a factor von Neumann algebra is either the sum of an isomorphism and a general trace or the sum of a negative anti-isomorphism and a general trace.

This note considers the local growth rate of variation γ(x,f) and local topological entropy h(x, f) at points x ∈ I for a continuous map f on a compact interval I such that the total variation Var (fn) is bounded for all n ≥ 0. We will show that γ(x,f) is always no less than h(x, f) and that the functions x →γ(x,f) and x → h(x, f), which map a point x to its local growth rate of variation and its local topological entropy respectively, are both upper semi-continuous. We also obtain a variational principle: the supremum of the local growth rate of variation and that of local topological entropies are equal to the global exponential growth rate γ(f) = limn→∞1/n ln Var (fn) of the total variations Var (fn) and the topological entropy h(f), respectively . When the map f :I→I is topologically transitive, we infer that the local growth rates of variation and local topological entropies functions are both constant on I° or on I° minus a fixed point. This is similar to the almost everywhere constancy of local entropy considered by Brin and Katok.

Some oscillation criteria are established by using Raccati transformation techniques for the following second order nonliner neutral difference equation Δ(pn(Δ(xn + φ(n,xτn))γ)) + qnfβ(xgn)=0, n=0,1,2,..., which extend some oscillation criteria and answer an open problem in [10,11], and also correct a theorem and its proof in [10].