Under some additional conditions, we prove a conjecture on fixed point indices of positive mappings in cones proposed by Dancer E. N. and give an example to illustrate the application of the theory.

We introduce meet-function and join-function on a subset of a poset and use them to give some explicit expressions of Smith's determinant.

Let ^Fn be the Kaplan-Meier estimator of distribution function F. Let J(-) be measureable real-valued function. In this paper, U-statistic representation for Kaplan-Meier L-estimator, T(^Fn) = (?)xJ(^Fn(x))d^Fn(x), is derived. Furthermore, the representation is also used to establish a Berry-Essen inequality for T(^Fn).

This paper gives the exact Hausdorff of measures a class of homogeneous Cantor sets.

We prove the weighted integral inequalities for differential forms, which can be considered as generalizations of the classical Poincare-type inequality.

This paper discusses the local and global centers of the generalized Lienard system, and gives two theorems which extend and improve on the related results of [1-3]. Moreover, the lemmas in this paper are also useful to the study of oscillation, stability of solution and existence of limit cycles of the generalized Lienard system.

Let M be a submanifold without umbilic point on unit sphere; its Moebius form Φ vanishes, in this paper the pinching theorems of sectional curvature about Moebius metric are obtained.

In this paper, the existence of periodic solutions for high order Duffing equations is studied by using degree theory and some known results are improved to some extent.

In this note by using an idea of Moscovici and the techniques of asymptotic expansions for heat kernels, an entire cyclic Chern character formula is given for a Dirac operator denned on a Clifford module.

The authors investigate the approximation of non-periodic functions in by a kind of s-dimensional network operator constructed by orthogonal algebraic polynomial. The upper bound of approximation is achieved and, in particular, the upper bound of approximation of the derivatives of the target functions by the corresponding derivatives of the translate and neural network operators is discussed.

In this article, we study the reqularity of the attractor in Cauchy problem for a system of nonlinear strain wave equation coupled with nonlinear Schrodinger equation. Existence of the global attractor A0 in E0 and A0 is actally equal to the global strong attractor A1 in E1 are obtained by a new operator decomposition technique.

We obtain explicit formulas of the Bergman kernels and holomorphic automorphism groups on HUA domains of the fourth type. And we also obtain of the explicit formula completely orthonormal system on semi-Reinhardt domain.

The method of quantifier elimination has been one of the powerful tools in the computer science and algebraic model theory. In this article, we deal with the theory of complete binary trees. After givling a set of fomulas as the Basic Formulas, we use the layout formulas and finite covering to prove that the theory of complete binary trees admits quantifier elimination.

We call a Cayley digraph (?)= Cay(G, S) normal for G if GR, the right regular representation of G, is a normal subgroup of the full automorphism group Aut(?) of (?). In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p2 (p odd prime). As a result, a family of nonnormal Caylay digraphs is found.

We prove the existence of quasiperiodic solutions and Lagrange stability for a class of differential equations with jumping nonlinearity , where a, b > 0, p(t) and is an unbounded function (see [1-19]).

This paper gives a formula of integral for the rectifiable curves in the plane by the Hausdorff fractional derivation and integral.

In this paper, we investigate multiplicative perturbation of the regularized cosine operator functions. We obtain some multiplicative perturbation theorems for the regularized cosine operator functions without the density of the range of the regularizing operator C.

In this paper, we characterize the linear surjective isometrics between the preannihilators of weakly closed T(N)-modules. Let U and W be the weakly closed T(N)-modules determined by the left order homomorphisms N → N and N→N respectively, and let U(?) and W(?) be the preannihilators of U and W on a Hilbert space H respectively. We prove that the onto isometrics between the spaces U(?) and W(?) are of the form A→U1AV1* or A → U2A*V2, where Ui and Vi (i = 1,2) are unitary operators, if (0)* = (0)# = (0), dim(0)+ ≠ 1 and min{dim(H(?)H), dim(H(?) H)} ≥ 2.

Let H be a finite dimensional Hopf algebra acting on an algebra yl, and let AH denote the invariant subring of A and A#H the smash product. Assume that A/AH is a semisimple extension and A is a flat right AH-module. It is shown that if H* is unimodular and A has an element of trace one then WD(AH) = WD(A) = WD(A#H). Moreover, if A is a projective left and right AH-module, then LD(AH) = LD(A) = LD(A#H).

A regularity result of a class of elliptic equations with singular lower term is obtained. The form of the equation is: Lu + vu =u, where L is a second order divergence elliptic operator, v belongs to some Morrey space and u, is a nonnegative Radon measure.

In this article, we shall give the necessary and suffcient conditions for a conformally immersed surface f : M →R2+p to be G*-deformable. After introducing the concept of adjoint form for /, the following results are proved: (1) If / is G*-deformable, then the normal bundle of / is flat and the adjoint form is closed; (2) When M is simply connected, then / is G*-deformable if it has flat normal bundle and closed adjoint form.

In this paper, we firstly discuss the existence, uniqueness and non-symmetric iterative approximation of solution for a class of systems of mixed monotone operator equations, and obtain some new fixed point theorems of mixed monotone operators, increasing operators and decreasing operators which have no continuous and compact conditions. Secondly, we study the fixed point of mixed monotone operators which have a-concave and -a-convex conditions while the operators also have no continuous and compact conditions, and get a new result. Finally, we apply the new results presented in this paper to Hammerstein integral equations on RN (see [1-12]).

The positively steady state, Hopf bifurcation and existence of periodic solution of 3-D Oregonator are analysed. The topological equivalence of 3-D Oregonator to its 2-D simplified model is proved, which testifies that the investigation of 2-D model for the 3-D Oregonator is valid.

Consider the differential equation (?) + a(t)g(x) = h(t), where a(t) and h(t) are 1-periodic functions such that a(f) does not change sign, and g is a concave-convex type function. By using the singularity method we obtain a complete geometric structure of 1-periodic solution, and the exact multiplicity results. More precisely, the image of singularities consists of codimensional 1 manifold that divides the C[0, 1] into two open sets A1,A3: (1) for h(t) A1, the equation has a unique periodic solution. (2) for h(t) ∈ A3, the equation has exactly three periodic solutions. (3) Moreover, the image F(C) of cusp singularities C is a codimensional 2 manifolds of ∈ [0,1] such that for h(t) ∈ F(C), the equation has a unique periodic solution, and for h(t) ∈ F(E)\F(C), the equation has exactly two periodic solutions.

In this paper we investigate the conformal Gauss map of submanifolds in sphere space and get, by Moebius invariants, the condition for this kind of map to be harmonic. In particular, we show that the conformal Gauss map of a 2-dimensional submanifold in sphere space is harmonic if and only if it is a Willmore submanifold.

As a generalization of the famous Bernstein polynomials, a class of broader polynomials called the multivariate Stancu polynomials denned on the normal simplex is introduced. With the best polynomial approximation as a metric, we prove the approximation property, estimate the approximation rate and characterize the approximation order of the Stancu polynomials to any continuous functions. An inverse theorem of approximation for the Stancu polynomials is also proved.

In this paper, the concept of expansive action of homeomorphism is generalized to the case of topological group, the expansiveness of a finitely generated topological group with discrete topology is studied and the following result is obtained: Z×Z can not acts expansively on the unit interval I, but the free product Z★Z can acts expansively on it.

In this paper the enumeration of rooted cubic strong c-nets is studied. The explicit expressions of this kind of maps with the size and the valency of the root-face as parameters are derived. Moreover, these results are also the same as those of rooted 4-connected triangulations which are the dual maps of rooted cubic strong c-nets. Here, some of the corresponding results in [1, 2] are thus simplified or generalized properly.

BHW theorem is about holomophic extensions for invariant holomorphic functions. In this paper a generalization of the theorem is given.

In this note we get the explicit formula of holomorphic sectional curvature on super-Cartan domains of the third type under the invariant Kalher metric. By using the completeness of its Bergman metric, we construct an invariant Kalher metric which is not less than the Bergman metric and prove that the holomorphic sectional curvature with this invariant Kalher metric is bounded from above by a negative constant, hence we obtain the comparison theorm for the Bergman and Kobayashi metrics on super-Cartan domains of the third type.

The distribution of zeros for differential polynomials fkf'+ a and fkQ[f] + P[f] has been studied in [4] and [1]. There are some difficulties in proving zeros distribution of differential polynomials because of the presence of poles. So the above results are only for entire functions. In this paper, we prove that for a transcendental meromorphic function f and two differential polynomials Q[f], P[f] of f, fkQ[f] + P[f] has infinitely many zeros outside the union of the disce which do not contain the poles of f,Q[f], P[f] and the zeros of P[f]. Where f satisfies δ(∞, f) >and P are weights of Q[f] and P[f]. The result improves that of [1] and [2,4,6].

Let a = |m(m4 - 10m2 + 5)|, b =5m4 - 10m2 + 1, c = m2 + 1, where m is an even integer. In this paper, we prove that if m ≥ 542, then the equation ax + by = cz has only one positive integer solution (x,y, z) = (2,2,5).

A generalized steepness lemma of projected gradient operator on a closed convex set is discussed in this paper. The global convergence and strong convergence of the projected gradient methods using several slacked step-size rules are proved.

Herstein generalized Jacobson's famous theorem on the commutativity of rings in 1955 as follows. Theorem A. if for every x, y ∈ R, there is a polynomial p(t) with integer coefficients, depending possibly on x and y, such that [x - x2p(x), y] = 0, then R is commutative. In this paper, we define a property Fk of the polynomial f(x1,X2,…, xn) and prove, on the bassing is property, a theorem on the commutativity of rings. Herstein's theorem is just the one case when n = 1 in our theorem.

Let u(x, t) = (SΩf)(x,t) be the solution of Cauchy problem of the general dispersive equation tu-iΩ(D)u = 0, u(x,0) = f(x), (x,t) ×Rn ∈R, and SΩ*f, SΩ**f be the local and global maximal operators. This paper gives some estimates about them.

The aim of this paper is to construct the fundamental solution of p-sub-Laplacian on the Heisenberg group and establish Hardy's inequalities by proving Pi-cone's identity on vector fields. Furthermore, Pohozaev's identities are given by using the noncommutative properties of vector fields. These results generalize those by Fol-land, Garofalo-Lanconelli, and some methods in this paper are new.

In this paper, we study some topics for harmonic maps from surfaces into complex Grassmann manifolds. We obtain an equivalent condition for harmonic maps to be '-irreducible or "-irreducible, and provide an explicit formula to calculate the isotropy order of harmonic maps.

Let u1, u2, … be independent random variables uniformly distributed on (0,1). This paper derives that for any n ≥ 1, the (n - 1)th moments of 1 - u1u2… uk are exponential complete Bell polynomials Yk(ζn(1), 1!ζn(2), 2!ζn(3), 3!ζn(4),…). Here the partial sums ζn(r) =∑j=1n/jr, r≥1, so the Riemann-Zeta function ζ(k) can be expanded as the series involving Stirling numbers of the first kind. Furthermore, all the 5-order sums of ζn(r) such as ∑n>1 Hn2/n3 have been given.

In this paper, we study the geometrical and analytic properties of manifolds with nonnegatively sectional curvature at infinity. Then, we apply these results to the study of harmonic maps.

We give general integral of quadratic differential system with center in two cases under the Chinese classification.