In this paper, a relation between hyperbolic Gauss map and the normal Gauss map of surfaces in a 3-dimensional hyperbolic space has been obtained. It is proved that generic surfaces are determined uniquely by the hyperbolic Gauss map and the mean curvature function. A linear elliptic equation of second order for the hyperbolic Gauss map has been derived. The author also obtains a solution for two kinds of nonlinear partial differential equations (systems) of third order for the hyperbolic Gauss map.

The purpose of this paper is to study a class of generalized nonlinear set-valued mixed quasi-variational inclusions which includes the known class of generalized set-valued variational inclusions, introduced and studied by Shang et al., as a special case. Using the technique of resolvent operators, we establish the equivalence between the generalized nonlinear set-valued mixed quasi-variational inclusions and the fixed point problems, where the resolvent operatorsJpA(.,x) are Lipschitz continuous with constantl/(l + cp), and also suggest some perturbed iterative algorithms. Further, we provide the convergence criteria for approximate solutions generated by the algorithms. The algorithms and results presented in this paper improve and generalize the corresponding algorithms and results of Shang et al.

We give some Liouville type theorems of harmonic maps which solve some free boundary problems. By using some technical energy estimates and choosing special parameters of variation of energy for harmonic maps, we get that the harmonic map of half simple manifolds into a Riemannian manifold with free boundaries must be constant.

Let Ω be a finite dimensional central algebra with an involutorial antiauto-morphism and char Ω≠2. Persymmetric and perskewsymmetric matrices over Ω are defined. Three systems of matrix equations over Ω[λ] are considered. Necessary and sufficient conditions for the existences of constant solutions with persymmmetric and perskewsymmetric constraints to the systems mentioned above are given. As a special case, some results dealing with the corresponding matrix equations are also presented.

In this paper it is shown that the problem of extensions of complete minimal sequences in separable Banach spaces has an affirmative answer. We give two examples in X = l∞ to show that corresponding results do not hold for minimal systems in nonseparable Banach spaces. In addition, a characterization of hereditarily indecomposable Banach spaces is given in terms of the properties of the extension of minimal sequences.

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate and singular semilinear parabolic equation ut - (xαux)x = ∫0af(u)dx in (0, a)×(0, T) with homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. It is also proved that the blow-up set is the whole domain, which differs from the local case. Furthermore, the blow-up rate is precisely determined for that in the special cases: f(u) = up, p > 1 and f(u) = eu.

In this paper, upper bounds for the fractal dimensions of invariant sets of C1-maps on smooth Riemannian manifolds are given in terms of the globe Lyapunov exponents.

In the present note an equivalent version of a criterion for the linear independence of a set of complex numbers over a number field by the author [1] (see Acta Math. Sinica, 1997, 40(5): 713-716, in Chinese) is given, and its application to the study of transcendency of certain power series and trigonometric series is also given.

One *-isomorphism of C*-algebras must be (completely) isometric map, but the inverse is not. In this paper, the result that a real completely isometric map can determine one *-isomorphism of C*-algebras is proved.

By using the partial order method and some new comparison results, the maximal or minimal solutions, the unique solution and the iterative approximation of solutions of the initial value problem for nonlinear second order integro-differential equations in Banach spaces are investigated. In this paper, we require only a lower solution or an upper solution and some weaker conditions presented here, and we extend and improve some recent results (see [1-10]).

Furuta showed that if A > B > 0, then for fixed p > 0 and r > 0, F(α) = (ArBαAr)p+2r/α+2r is decreasing for α > p. But he pointed out that this result does not remain valid for 0 < α < p and r > 0. In this paper, we show that (1) If -1/2 < r < 0 and p < -2r then F(α) is decreasing in [-p - 4r,+∞) and (-∞,p], and the two intervals can not be enlarged; (2) If -1/2 < r < 0 and p > -2r then F(α) is increasing in (-∞, -p - 4r] and [p,+∞), and the two intervals also can not be enlarged; (3) If r > 0, p > 0, then [p, +∞) is also the best monotonic interval of F(α).

Let f : X →Y be a continuous and surjective map. f is a sequence-covering map if whenever {yn} is a convergent sequence in Y there is a convergent sequence {xn} in X with each xn ∈ f-1(yn). f is a 1-sequence-covering map if for each y ∈ Y, there is x ∈ f-1(y) such that whenever {yn} is a sequence converging to y in y there is a sequence {xn} converging to x in X with each xn ∈ f-1(yn}. In this paper the structure of sequence-covering and closed maps of metric spaces is investigated, a problem posed by "Topology and its Applications" is negatively answered.

In this note, we obtain the sufficient and necessary condition of crossed product A×α G, where A is a maximal abelian *-subalgebra of B(H) and G is a discrete group, to be a factor without assuming that G acts freely on A.

We propose a selection procedure for selecting the best Logistic population compared with a control by employing the empirical Bayes approach. The rate of the convergence of proposed selection rule is given.

In this paper, we set up some operator properties of slant Toeplitz operators on Bergman space. We prove that if S is a linear operator on every Lap (1 < p < ∞), then 5 is compact on La2 if and only if S(z), the Berezin transform of S, satisfies S(z)→0 (z→(?)D). Our main result can be used to prove that if S is a finite sum of finite products of Toeplitz operators or slant Toeplitz operators, then S is compact if and only if S(z)→0 as z→(?)D.

In this paper, we study by using a topological method the boundary value problem for systems of nonlinear second order ordinary differential equations of the form We shall prove the existence and multiplicity of positive solutions of the above problem under some conditions.

In this paper, we introduce Lie invariant subspace of a linear mapping on a Banach algebra and give the general form of a linear map on a nest subalgebra of a Von Neumann algebra, under which the derivation space is invariant. We also introduce Lie derivation and Lie automorphisn and discuss the relationship between Lie derivations and semigroups of Lie automorphisms.

The classification of finite groups in which the multiplicity of each nonlinear irreducible character degree is equal to one is well known. For a solvable group G, we consider a more general case in this paper, namely the multiplicities of nonlinear irreducible character degrees of G are always co-prime to the group order |G|. In particular, we classify finite groups G of odd order in which the multiplicities of nonlinear irreducible character degrees are all less than 2p, where p is the minimal prune divisor of the group order |G|.

In this paper, by constructing a special cone, we investigate the existence of positive solutions of a class of nonlinear singular boundary value problems in Banach space.

In this paper, we obtain a sufficient and necessary condition for the normality of the product of two GO-spaces.

Letx0 = 1,xn+1 = - 1 and let {xk}kn=1 be the zeros of n-th Jacobi polynomial. In this paper, the sufficient and necessary conditions for the mean convergence of Hermite-Fejer interpolation operators based on the nodes {xk}k=0 n+1 established.

In this paper, any differential polynomial P(w) of a v-valued algebroid function w is proved to be a λ-valued (1 <λ< v) algebroid function. Furthermore, if F = F(w) assume P(w)/wn or P(w)·(wq + bq-1wq-1 +…+ b0)p, respectively with suitable positive integer n or p and q, then the following estimate for the growth of the characteristic function of w T(r,w) < 1/AN(r, 1/F-φ) + S(r,w) is proved to be valid for any small function φ((?)0) with respect to w, where A is a positive integer.

In this paper, fixed point theorems for u0-concave sequential contraction operators and a class of mixed monotone operators without the assumfotion of continuity or compactness are proved, separately. An application to integral equations is given.

In this paper, we discuss finite groups with automorphism groups of order 4pq, where p and q are distinct odd primes, and clarify their structures.

We are concerned with the following Dirichlet problem -△u(x)=f(x,u), x∈Ω, u∈h01(Ω),(P) where f(x,t) ∈ C(Ω×R), f(x,t)/t is nondecreasing in t ∈ R and tends to an L∞-function q(x) uniformly in x ∈Ω as t→+∞ (i.e., f(x,t) is asymptotically linear in t at infinity). In this case, Ambrosetti-Rabinowitz-type condition, that is, for someθ> 2, M > 0, 0 <θF(x,s) < f(x,s)s, for all |s| > M and x ∈Ω, (AR) is no longer true, where F(x, s) = ∫08 f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass theorem, that problem (P) has a positive solution under suitable conditions on f(x,t) and q(x). Our methods also work for the case where f(x,t) is superlinear in t at infinity, i.e., q(x) ≡+∞.

In this paper we consider a random evolution equation with small perturbations, and show how to construct coupled solutions to the equation. As applications, we prove the Feller continuity of the solutions and the existence and uniqueness of invariant measures. Furthermore, we establish a large deviations principle for the family of invariant measures as the perturbations tend to zero.

In this paper, the problem of absolute stability for general Lurie control systems with multi-nonlinear feedback terms is investigated. Prom the Lyapunov function approach, new necessary and sufficient conditions and some practical sufficient conditions of absolute stability are derived, which improve absolute stability for the set. A numerical example illustrates the effectiveness of the proposed results.

In the present note the algebraic independence of values of certain functions denned by infinite products at algebraic and transcendental points is given.

The purpose of this paper is to investigate the problem of approximating to the fixed points of uniformly Lipschitzian asymptotically pseudocontractive mappings in an arbitrary real Banach space by the modified Ishikawa iterative sequences with errors. Under the condition of removing the restriction limn→∞βn = 0 we prove that Professor Zhang Shisheng's results (Acta Math. Appl. Sinica, 2001, 24 (2): 236-241) remain true. On the other hand, we also extend his results to the case of the modified Ishikawa iterative sequences with errors.

In our paper, under the condition that u1(t) is increasing, and the vector function tu1(‖t‖) is Lipschitz continuous and other mild conditions, we show the recursive algorithms proposed are strongly consistent.

In this paper, we prove the reduction of codinemsion for the surfaces in Sn with vanishing Moebius form and flat Moebius normal bundle, and classify this sort of surfaces. Moreover, we also give a classification of the surfaces with vanishing Moebius form in Sn.

In this paper, the σ-semi-convexity is introduced and a general Ekeland's variational principle in locally convex spaces is given.

If T is a regular operator on a Banach space X, with finite dimensional intersection K(T) ∩ N(T) and K(T) closed, and if 5 is invertible, commutes with T and has sufficiently small norm, then T - S is an upper semi-Fredholm operator. If in addition K(T) + N(T) or R(T) + N(T) has a complementary subspace with finite dimension, then T ?S is a Fredholm operator.

Q spaces (or Qp, or Qα spaces), are studied heavily by many authors. In particular, Wu and Xie generalized Q spaces to the Qpα,q spaces in recently. In this paper, we show first the equivalence of some real variable characterization and wavelet characterization for general Q spaces. Our main idea is to use In dimensional wavelet to analyze the elements in Qpα,q on Rn. Furthermore, we construct the pre-dual spaces Ppα,q of Qpα'q, where Ppα,q are generated by atoms which are denned by wavelets.

In this paper, we will prove some new inequalities on sequences. As applications of our theorems, we introduce a new kind of iterative procedure, and give a proof of the convergence for it. Furthermore, we show that it is equivalent to the Ishikawa iterative sequence which is well known.

In this paper, we give the conditions of existence of positive almost periodic type solutions for some nonlinear delay integral equations.

Let f(z) be a nonlinear meromorphic function in the complex plane satisfying Ni(r,1/f) = S(r, f), and c be any nonzero complex number. Then we have T(r, f) < 11{N(r, f) +N(r, 1/f'-c} + S(r, f), where N1(r, 1/f) denotes the counting function of the simple zeros of the function f(z).

This paper discusses the Poincare bifurcation for a class of cubic Hamiltonian systems with a global center. We prove that its Poincare bifurcation can produce two limit cycles at most and may produce two limit cycles.

By employing the continuation theorem of coincidence degree theory developed by Mawhin, we study a kind of Rayleigh equation with a deviating argument as follows x11(t) + f(x'(t)) + g(x(t -?τ(t))) = p(t), and some new results on the existence of periodic solutions are obtained. Our work generalizes the knowen result (see [8]).

In this paper, a necessary and sufficient condition for an operator to be a bound-limited self-adjoint extension of a singluar Sturm-Liouville operator is obtained. As a consequence, all possible forms of the bound-limited self-adjoint extensions are characterized via complete classification of the self-adjoint boundary conditions. Also, this result can answer directly the equal cases of the inequalites among the minimal eigenvalues for singular Sturm-Liouville problems.