In the complex n-dimensional projective space CPⁿ, let λ_p(=4 p(p+n)) be the eigen value of the Laplace-Beltrami operator and H_p be the space of all eigen functions of eigen value λ_p. The reproducing kernel h_p(z, w) of H_p is constructed explicitly in this paper, and a system of complete orthogohal functions of H_p is constructed from h_p(z,w)(p=1,2, …).

This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthermore, in the compact case or for birth-death processes or half-line diffusions, the rate is shown to be equal to the spectral gap.

In this paper, general modular theorems are obtained for meromorphic functions and their derivatives. The related criteria for normality of families of meromorphic functions are proved.

In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations:

In the present paper we consider the small random perturbations of one-dimensional diffusion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistical estimates of the exit times as the perturbations tend to zero.

In this paper we estimate the solutions of homogeneous linear system of differential equations with unbounded coefficients on the real line R. We also give a necessary and sufficient condition in order that the linear differential operator with unbounded coefficients has a bounded inverse in the scalar case.

This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.

Let f(X) be an additive form defined by

where a_#em/em#≠0 is integer, #em/em#=1,2…, s. In 1979, Schmidt proved that if ε>0 then there is a large constant C(k,ε) such that for s>C(k,ε) the equation f(X)=0 has a nontrivial, integer solution in σ₁, σ₂, …, σ_s, x₁, x₂, …, x_s satisfying

Schmidt did not estimate this constant C(k,∈) since it would be extremely large. In this paper, we prove the following result.

Let 0<p≤1<q<0, and w₁, w₂∈A₁ (Muckenhoupt-class). In this paper the authors prove that the strongly singular convolution operators are bounded from the homogeneous weighted Herz-type Hardy spaces HK_q^{α, p}(w₁; w₂) to the homogeneous weighted Herz spaces K_q^{α, p}(w₁; w₂), provided α=n(1-1/q). Moreover, the boundedness of these operators on the non-homogeneous weighted Herz-type Hardy spaces HK_q^{α, p} (w₁; w₂) is also investigated.

A series of convex functions on a Banach space is studied. The generalizations of the Moreau-Rockafellar theorem are given. As its application, the partial generalization of the Kuhn-Tucker theorem is obtained.

In this paper, we point out that most results on abelian (complex) W^*-algebras hold in the real case. Of course, there are differences in homeomorphisms of period 2. Moreover, an abelian real Von Neumann algebra not containing any minimal projection on a separable real Hilbert space is * isomorphic to L_τ^∞ ([0, 1]) (all real functions in L^∞([0, 1])), or L^∞([0, 1]) (as a real W^*-algebra), or L_τ^∞ ([0, 1])⊕L^∞ ([0, 1]) (as a real W^*-algebra), and it is different from the complex case.

Here we show that there exists a greatest fixed point separating congruence denoted by Ψ on a double MS-algebra (L;^0,+). Some properties of Ψ under certain conditions on fixed point complete and fixed point distributive have also been described.

In this paper, we will prove that every derivation of completely distributive subspace lattice (CDS) algebras on Banach space is automatically continuous. This is new even in the Hilbert space case. As an application of this result, we obtain that every additive derivation of nest algebras on Banach spaces is inne. We will also prove that every isomorphism between nest algebras on Banach space is automatically continuous, and in addition, is spatial.

Let {X, X_n; n≥1} be a sequence of i.i.d.r.v.s with EX=0 and EX²=σ²(0 < σ < ∞). we obtain some sufficient and necessary conditions for

to hold, get the widest range of k's and answer a question of Hanson and Russo (1983).

This paper deals with a class of nonlinear boundary value problems which appears in the study of models of flows through porous media. Existence results of asymptotic bifurcation and continua are reported both for operator equations and for boundary value problems.

In this paper, we completely solve the embedding problem of simple directed triple systems by proving that the necessary conditions for the embeddings of directed triple systems are also sufficient.

Let f be a holomorphic self-map of the punctured plane C^*=C\{0} with essentially singular points 0 and ∞. In this note, we discuss the sets I₀(f)={z∈C^*: fⁿ(z)→0, n→∞} and I_∞(f)={z∈C^*: fⁿ(z)→0, n→∞}. We try to find the relation between I₀(f), I_∞(t) and J(f). It is proved that both the boundary of I₀(f) and the boundary of I_∞(f) equal to J(f), I₀(f)∩J(f)≠θ and I_∞(f)∩J(f)≠θ. As a consequence of these results, we find both I₀(f) and I_∞(f) are not doubly-bounded.

By the use of Moser iteration and Campanato space estimate, the L^∞ and C^α regularity estimate of the gradient of solutions of nonlinear parabolic system ∂uⁱ/∂t-▽·(g|▽u|▽uⁱ)=0 with nonstandard growth conditions are obtained under the natural structure constraints.

In this paper, a distortion theorem and some equicontinuity and compactness theorems are obtained for the homeomorphism f with the sphere dilatation H(x,f)∈L_loc(D).

Let {E_i:i∈I} be a family of Archimedean Riesz spaces. The Riesz product space is denoted by ∏_i∈I E_i. The main result in this paper is the following conclusion: There exists a completely regular Hausdorff space X such that ∏_i∈I E_i is Riesz isomorphic to C(X) if and only if for everyem∈I there exists a completely regular Hausdorff space X_i such that E_i is Riesz isomorphic to C(X_i).

For any integers a₁,a₂,a₃,a₄ and c with a₁a₂a₃a₄0(modp), this paper shows that there exists a solution X=(x₁,x₂,x₃,x₄)∈Z⁴ of the congruence a₁x₁²+a₂x₂²+a₃x₃²+a₄x₄²≡c(modp) such that

A nonlinear hinged extensible elastic body equation with strong structural damping and Balakrishnan-Taylor damping of full exponent is studied as a general model for large space structures of higher dimensions. In this paper, the absorbing sets and flat inertial manifold are obtained for this nonlinear body equation. The control spillover problem associated with the stabilization of this equation is resolved by constructing a linear finite dimensional feedback, control based on the existence of inertial manifolds of the uncontrolled equation. Moreover, the results obtained are robust with respect to the uncertainty in structural parameters.

In this paper, a class of functionals of Kaplan-Meier estimator is investigated. Counting process martingale methods are used to show the asymptotic normality, and we establish a mean square error inequality and a probability inequality of them without the assumption that F, G are continuous, where,F, G are survival time distribution and censoring time distribution respectively.

In this paper, we study composition operators on a Banach space of analytic functions, denoted by X, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman space L_α¹(D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range.

The differentiability of a norm of a Banach space may be characterized by its unit sphere. This paper generalizes these geometric conditions of norm's differentiability to the case of a regular locally Lipschitz function.

Let E(X)=|{N≤X;N≠p₁²+p₂³+p₃⁴+p₄⁵ for any primesp_i}|. It is proved in this paper that there exists a positive constant δ>0 such that which improves a result of prachar.

Suppose that {α_k}_k^∞=-∞ is a Lacunary sequance of positive numbers satisfying inf and that Ω(y') is a function in the Besov space B₁^0,1 (S^n-1) where S^n-1 is the unit sphere on Rⁿ(n≥2). We prove that if ∫S^n-1Ω(y')dσ(y') then the discrete singular integral operator

and the associated maximal operator

are both bounded in the space L²(Rⁿ)

The theorems in this paper improve a result by Duoandikoetxea and Rubio de Francia^[1] in the L² case.

Existence and uniqueness results are obtained for positive radial solutions of a class of quasilinear elliptic equations in aN-ball or an annulus without monotone assumptions on the nonlinear term f. It is also proved that there is no non-radial positive solution.

In this paper, we study the spectral asymptotics for connected fractal domains and Weyl-Berry conjecture. We prove, for some special connected fractal domains, the sharp estimate for second term of counting function asymptotics, which implies that the weak form of the Weyl-Berry conjecture holds for the case. Finally, we also study a naturally connected fractal domain, and we prove, in this case, the weak Weyl-Berry conjecture holds as well.

We confirm the Vaught's conjecture and obtain a complete classification for the theories of unitary relations.

We establish integral formulas of Minkowski's type for compact spacelike hypersurfaces in de sitter space S₁ⁿ⁺¹ (1) and give their applications to the case of constant r-th mean curvature (r=1,2,…,n-1). When r=1 we recover Montiel's result.

In this paper,using the Guichardet space technique,the relationship between Fermion quantum stochastic calculus and non-causal calculus in Segal space L2(H) is discussed,and an anticipating quantum stochastic calculus is naturally given.

By the use of approximation method a general criterion of algebraic independence of complex numbers is established.As its application the algebraic independence of values of certain gap series at algebraic and transcendental points is given.

In this paper,under some fairly general conditions,a first-order Edgeworth expansion for the standardized statistic of β in partial linear models is given,then a non-residual type of consistent estimation for the error variance is constructed,and finally an Edgeworth expansion for the corresponding studentized version is presented.

We start from a super-Brownian motion with the branching mechanism presented by Dawson and Vinogradov.Its behaviour near extinction is studied in this paper,and the main result is that the diameter of the support tends to zero almost surely at the time of extinction.

K.A.Hardie and K.H.Kamps investigated the track homotopy category H_B over a fixed space B ([1]).They have introduced two pairs of adjoint functors:P_B——|N_B and m_*——|m*,where P_B:H_B→HB,and m_*:H_A→H_B for a fixed map m:A→B.We have introduced a split fibration of categories L:H_b→H_B and proved L J,J——|L in [2].This paper first extends P_B——|N_B to for any fixed map b:B→B.Moreover we also extend these results to obtain two pairs of adjoint functors involving track homotopy categories H_b and H b where H b is the dual of H_b.One of our results is N_b——|P_b.This differs from P_B——|N_B.

In this paper,we shall introduce the concept of the Bessel (Riesz) potential Köthe function spaces Xs and give some dual estimates for a class of operators determined by a semi-group in the spaces Lq (-T,T;Xs) (Lq(-T,T;)).Moreover,some time-space Lp -Lp' estimates for the semi-group exp(emt(-Δ)m/2) and the operator A:=∫₀t exp(i(t-τ)(-Δ)m/2)·dτ in the Lebesgue-Besov spaces Lq(-T,T;) are given.On the basis of these results,in a subsequent paper we shall present some further applications to a class of nonlinear wave equations.

In this paper,a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems.Then the Poincarémap is established to solve various problems in homoclinic bifurcations with codimension one or two.Bifurcation diagrams and bifurcation curves are given.

Nonlinear integral and discrete inequalities are obtained,which are related to some recent results of B.G.Pachpatte in [1] given therein as generalizations of Ou-Iang's integral inequality [2].As special cases,some new inequalities with the power nonlinearity are derived from the main results.To show the contribution of our results,boundedness of solutions to certain nonlinear difference equation is also considered.