It is proved that linear structures can be constructed by means of metrics on translation spaces under certain conditions. The concept of sub-normed Z-linear spaces is introduced and it is clarified that a translation space is a sub-normed Z-linear space if and only if its translation group is Abelian. Analogues the bounded linear operator theorem, the Hahn-Banach theorem and the resonance theorem are established in sub-normed Z-linear space.

In this paper give an effective method to calculate the Hausdorff measure of Marion sets on straight line and show how to calculate the Hausdorff measures of fractals on straight line with some examples.

Let A be an n x n Boolean matrix with period p, 1≤i≤n. The generalized index of convergence k(A, i) of A is the least nonnegative integers k such that i rows of Ak and Ak+p are mutually equal. We characterize the extremal matrices of generalized indices of convergence of the n×n Boolean matrices with exactly d (1≤d≤n) non-zero diagonal entries.

This paper investigates the existence of positive solutions of singular su-perlinear boundary value problems of fourth order ordinary differential equations. A necessary and sufficient condition for the existence of C2[0,1] positive solutions as well as C3[0,1] positive solutions is given by means of the fixed point theorems on cones.

In this paper, we discuss the existence of positive solutions for a singular boundary value systems with p-Laplacian (φp(x'))' + α1　(t)f(x(t),y(t)) = 0, (φp(y'))' +α2(t)g(x(t),y(t)) = 0, x(0) - β1x'(0) = 0, x(1) +δx'(1) = 0, y(0) - β2y'(0) = 0, y(1) + δ2y'(1) = 0, where φp(x) = |x|p-2x, p > 1. By using the fixed-point index theorem, the sufficient conditions of the existence of one or multiple positive solutions for the boundary value systems under some conditions are established. These results can be applied to the study of the existence of the radial solutions for the elliptic boundary value systems.

This paper investigates the relationship between the multivariate Stancu polynomials and the modulus of continuity. First, it is shown that the property of modulus of continuity is preserved by the multivariate Stancu polynomials and hence a corresponding one to univariate Bernstein polynomials is generalized. Then, by means of the Ditzian-Totik's modulus of continuity, the estimates of upper and lower bounds of rate for the polynomials approximating the continuous function are given. A necessary and sufficient condition on the smoothness of function is also obtained from the estimates so that the rate of approximation is O(n-a) (0 < a < 1).

In this paper, a one-dimensional perturbed neutral delay difference equation of the form: △(x(n) - cx(n - k)) = f(n,xn) + g(n,xn), n ∈ Z+ is considered. It is shown that under some conditions, the zero solution of the above equation is uniformly asymptotically stable provided that the zero solution of the corresponding equationwithout perturbation: △(x(n) - cx(n - k)) = f(n,xn), n∈Z+ is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.

In this paper, a new characterization of eventually norm continuous semigroups on Hilbert space is given. By using the resolvent of the generator, we give the sufficient and necessary conditions for the semigroups to be being eventually norm continuous.

A subgroup H of a group G is said to be semipermutable if it is permutable with every subgroup K of G with (|H|, |K|) = 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. In this paper, we extend and improve some of Asaad and Pingchao Wang's results by examining the influence of s-semipermutability on the structure of groups.

The relationship between the tangent cone, the tangent manifold and the tangent space of a convex a-body and the subdifferential of the Minkowski functional of the convex α-body is investigated. For each algebraic boundary point of a convex α-body, there exists a quasi-direct sum decomposition such that in the algebraic sense the boundary point is both a subspace smooth point and a quasi-complementary subspace strictly extreme point. The general results obtained are efficiently applied to the theory of polyhedral normed spaces.

In this paper, existence of the multiple solutions for a kind of superlinear operator equations is discussed via two coupled parallel sub-super solutions. The results obtained in this paper essentially improve the famous Amann three-solution theorem. As applications, a second order system of differential equations is considered.

In this paper, the techniques of partial order theory are used to study the solvability of a class of operator equations Lx = NX, where neither L nor N need to be continuous. Existence and multiplicity results are obtained in complete metric space and Banach space respectively, and then applied to a periodical boundary value problem of an intergro-differential equation.

In this paper, characteristic condition is given for a generalized steepest descent method to converge to the zeros of quasi-accretive locally bounded operators defined on proper subsets of a uniformly smooth real Banach space. These results extend and unify the corresponding ones by Xu Zhongben and Jiang Yaolin, and others.

In this paper, by studying the factorization of adjoint polynomials of Γ-class graphs, we prove that structure characteristics of the chromatically equivalent graphs and chromatically non-uniqueness of their complement graphs.

Let A0,A1,…,An-1be entire functions with finite order, which are not all polynomials. In this paper we study the angular distribution as well as the removable sets of zeors of solutions of differential equations f(n) + An-1f(n-1) +…+ A0f = 0.

In this paper, a subalgpbra A2 of the Lie algebra A2 is constructed, which gives a corresponding loop algebra A2 by properly choosing the gradation of the basis elements. It follows that an isospectral problem is established. Furthermore, a new Liouville integrable Hamiltonian hierarchy of equations is obtaind by the use of Tu scheme, from which the hierarchy reduces to a nonlinear evolution equation of rational fractal form. By making use of a matrix transformation, a subalgebra A2 of the Lie algebra A1 is presented, which possesses the same communicative operations of basis elements as those in A2. Again we expand the Lie algebra A1 into a high-dimensional loop algebra G, and a type of expanding integrable system of the hierarchy obtained above is worked out.

Let a be a positive integer. Let D = 3a2 + 1 and p = 4a2 + 1, where p is a prime. In this paper we prove that if a is not a multiple of 4, then the equation x2 + Dm = pn has exactly two solutions (x, m, n) = (a, 1,1) and (8a3 + 3a, 1,3), except for (D,p) = (4,5), in which case the equation has exactly three solutions (x,m, n) = (1,1,1), (3,2,2) and (11,1,3).

The estimator σ_n~2 of error variance in semiparametric regression model is given by using the least squares method and local linear smoothers. It is provend that the Berry-Esseen bound of σ_n~2 can attain the accuracy of O(n-1/2) under suitable conditions.

This paper uses the method of upper and lower solutions and the monotone iterative technique to investigate the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive functional differential equations.

In this paper, the exchange rings R whose primitive factor rings are Artinian is studied. The K0-groups of R is characterized and it is proved that the uniqueness of n-th roots always holds in the category of finitely generated projective R-modules.

We present an upper and a lower bound for the spectral radius of non-negative matrices. Then we give the bounds for the spectral radius of digraphs.

Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given, among them all the prime ideals of a Dubrovin valuation skew group ring are characterised.

In this paper, the authors study the continuity properties of higher order commutators generated by the homogeneous fractional integral and BMO functions on certain Hardy spaces, weak Hardy spaces and Herz-type Hardy spaces.

It is proved that every proper holomorphic self-mapping of some kinds of Generalized Hartogs Triangles is an automorphism, and its explicit expression is given.

Motivated from the study on logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examined.

In this paper, the concepts of quasi Z-continuous domains and Z-meet continuous domains are introduced, and some relations between Z-continuous domains, quasi Z-continuous domains, Z-meet continuous domains and the Hausdorff separation of Z-Lawson topology are investigated. It is proved that if a subset system Z satisfies certain conditions and P is a quasi Z-qcontinuous domain, then the Z-way below relation <

The existence of periodic solutions of neutral periodic differential periodic system with infinite delay is considered. Using the exponential dichotomy and Krasnoselskii fixed point theorem, some sufficient conditions to guarantee the existence of T-periodic solution for the system are established.

This article is based on the quantifier elimination of the theory of complete binary trees. The countable atomic model and saturated model are presented. We also calculate the CB rank of the types in one and two variables. Finally, the geometric meaning of CB rank in this theory is found.

In this paper, the concept of quasi-linear projector is introduced in Banach space, and a unified definition for homogeneous generalized inverse of linear operators in Banach space is thus given. The homogeneous generalized inverses contain the bounded linear generalized invenses and the single valued metric generalized inverses. A unified criterion for the existence of the homogeneous generalized inverse of a linear operaor in Banach space is also given.

For two operators acting on an infinite dimensional Hilbert space, it is interesting to investigate whether the essential similarity is equivalent to the similarity in the Calkin algebra. In this paper we introduce an operator class ES((?)), and show that, for two operators in ES((?)), the similarities above are the same. Moreover, we show that ES((?)) is dense in B((?)) in norm topology.

In this paper, the existence of positive solutions is studied for higher order boundary value problems with a p-Laplacian operator, by useing the method of the compression and expansion of cone.

In this paper, we compute, by using cone theory, the topological degree of a class of completely continuous fields where the condition on the lower boundedness of nonlinear operators is sharply weakened. Therefore our results substantially improve and generalize the existing ones in the literature. Finally we use our abstract results to establish the existence of nontrivial solutions for superlinear Hammerstein integral equations.

The paper is concerned with the existence of solutions for the following multi-point boundary value prolem x"(t) = f(t,x(t), x'(t)) + e(t), t ∈(0,1); x'(0) = x'(ξ), x(1) =sum from i=1 to m-3 βix(ηi) where βi ∈ R, rum form i=1 to m-3βi= 1, 0 <η1<η2 <... <ηm-3 < 1, 0 <ξ< 1, and rum form i=1 to m-3βiηi=1. This is the case dim Ker L = 2. When the βi's have different signs, we prove existence results for the m-point boundary value problem at resonance by the use of Mawhin's coincidence degree theory. Since all the existence results obtained in previous papers are for the case dim Ker L=1, our work is new.

we prove that the converse proposition of the famous Lomonosov lemma is true and obtain a new necessary and sufficient condition for the existence of a nontrivial closed invariant subspace for a subalgebra (?) of (?)(X) in a Banach space X.

In this paper, we obtain some existence results for a class of abstract equilibrium problem; our results give answers to the open problem raised by Isac, Sehgal and Singh. Furthermore, we study the equilibrium problems with lower and upper bounds under restricted conditions, and obtain more results. Finally, we generalize these open problems to the case of two spaces.

Let (?) be a subalgebra with the unit operator I in B(H), where H is a real Hilbert space. We say that a linear mapping (?) from (?) into B(H) is a generalized Jordan*-left derivation if (?)(T2) = T*(?)(T) + (?)(T)T - T*(?)(I)T for any T ∈(?); (?) is called a generalized*-left kernel-into-range preserving mapping if (?)(T)(ker(T)) (?) ran(T*). In this paper, we show the following main result. If a generalized*-left preserving kernel-into-range mapping on real nest algebras is continuous in the weak operator topology, then (?) is a generalized*-left inner derivation, i.e. (?)(T) = T*A + BT for some A, B∈B(H); In particular, (?) is also a generalized Jordan*-left derivation.

This paper is concerned with the Cauchy problem for a class of nonlinear Klein-Gordon equations with several competing potential functions. In terms of the characteristics of the ground state, the sharp conditions for blowing up and global existence are derived out by applying the potential well argument and the concavity method. And the question that how small the initial data are, the global solutions exist is answered.

A family of finite-dimensional modular Lie superalgebras of Cartan types HO is constructed in this paper. They are proved to be simple. By establishing dimension formulas of Lie superalgebras of Cartan-type W, H, K, S and HO, we study the isomorphism relations between HO and W, S, H and K.

In this paper, we characterize the structure of (?)*-covered regular semigroups and classify weakly covered regular semigroups and covered regular semigroups. By using the main results, we also obtain the structural characterizations for (?)-covered completely regular semigroups, weakly covered completely regular semigroups and covered completely regular semigroups.

The relationship between weak vector variational inequality with a mapping from a topological vector space to continuous linear mapping space and its scalarization is discussed. A sufficient condition is presented on the connectedness of solution set for the weak vector variational inequality in a topological vector space. The conceptions of weakly and strongly uniformly continuous are introduced. The result on connectedness is derived by discussing the properties of set-valued mapping induced by solution set of a scalar variational inequality related to the weak vector variational inequality.