In this paper, we consider the generalized variational inequality GVI(F, g,C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI(F, g,C). Strong convergence results are established and applications to constrained generalized pseudo–inverse are included.

 In this paper, we interpret Massey products in terms of realizations (twitsting cochains) of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.

 In this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact non–negative curved manifolds admit (complete) metrics with non–negative curvature.

In this part II the fundamental inequality of the third order general regular variation is proved and the second order Edgeworth expansion of the distribution of the extreme values is discussed.

  The symmetry of singular Hamiltonian differential operators is proved under the standard "definiteness condition", which is strictly weaker than the densely definite condition used by A. M. Krall. Meanwhile, some properties of deficiency indices are given.

The purpose of this paper is to investigate the solutions of refinement equations of the form

The main purpose of this paper is to use the Fourier expansion for character sums and the mean value theorem of Dirichlet L–functions to study the asymptotic property of the difference between a D. H. Lehmer number and its inverse modulo p (an odd prime). A interesting mean square value formula is also given.

 Let be the maxiamal multilinear Bochner–Riesz operators generated by Bochner– Riesz operators and D ^α A ∈ Lipβ(|α| = m). The continuity of the operator on some Hardy and Herz type Hardy is obtained.

 In this paper, by the KAM method, under weaker small denominator conditions and nondegeneracy conditions, we prove a positive measure reducibility for quasi-periodic linear systems close to constant: X? = (A(λ) + F(?, λ))X, where the parameter λ ∈ (a, b), ω is a fixed Diophantine vector, which is a generalization of Jorba & Simó’s positive measure reducibility result.

Two boundary value problems are investigated for an over–determined elliptic system with several complex variables in polydisc. Necessary and sufficient conditions for the existence of finitely many linearly independent solutions and finitely many solvability conditions are derived. Moreover, the boundary value problem for any number of complex variables is treated in a unified way and the essential difference between the case of one complex variable and that of several complex variables is revealed.

 We consider the g–function related to a class of radial functions which gives a characterization of the L ^p –norm of a function on the Heisenberg group.

 For 1 < p ≤ 2, an L ^p -gradient estimate for a symmetric Markov semigroup is derived in a general framework, i. e. , where Γ is a carré du champ operator. As a simple application we prove that Γ^1/2((I-L)^-α) is a bounded operator from L ^p to L ^p provided that 1 < p < 2 and . For any 1 < p < 2, q > 2 and , there exist two positive constants c _q,α,C _p,α such that ∥Df∥ _p ≤ C _p,α∥(I - L)^α f∥ _p , c _q,α∥(I - L)^1-α f∥ _q ≤ ∥Df∥ _q + ∥f∥q, where D is the Malliavin gradient ([2]) and L the Ornstein–Uhlenbeck operator.

In this paper, the authors study the boundedness properties of generated by the function b ∈ Lip _β (?ⁿ)(0 < β ≤ 1/m) and the Marcinkiewicz integrals operator μ _Ω . The boundednesses are established on the Hardy type spaces and the Herz–Hardy type spaces .

Let (M, ω) be a closed symplectic 2n–dimensional manifold. Donaldson in his paper showed that there exist 2m–dimensional symplectic submanifolds (V ^{2 m} , ω) of (M,ω), 1 ≤ m ≤ n – 1, with (m – 1)–equivalent inclusions. On the basis of this fact we obtain isomorphic relations between kernel of Lefschetz map of M and kernels of Lefschetz maps of Donaldson submanifolds V ^{2 m} , 2 ≤ m ≤ n−1. Then, using this relation, we show that the flux group of M is discrete if the action of π₁(M) on π₂(M) is trivial and there exists a retraction r : M → V , where V is a 4–dimensional Donaldson submanifold. And, in the symplectically aspherical case, we investigate the flux groups of the manifolds.

In this paper, we offer a graded equivalence between the quotient categories defined by any graded Morita–Takeuchi context via certain modifications of the graded cotensor functors. As a consequence, we show a commutative diagram that establish the relation between the closed objects of the categories gr ^C and M ^C , where C is a graded coalgebra.

In this note we prove that if R is a strong Mori domain with t–dim R = n and with countably many prime v–ideals, then there is a chain of rings between R and R ^w

such that each R _i is also a strong Mori domain and t–dim Rk = n − k + 1 for k = 1, 2, . . . , n.

In this paper, we study the boundary dilatation of quasiconformal mappings in the unit disc. By using Strebel mapping by heights theory we show that an asymptotical Hamilton sequence is determined by a quasisymmetric function

In this paper, we discuss the moving-average process , where {α _i ;-∞ < i < ∞} is a doubly infinite sequence of identically distributed φ-mixing or negatively associated random variables with mean zeros and finite variances, {α _i ;-∞ < i < ∞} is an absolutely summable sequence of real numbers. Set . Suppose that . We prove that for any ,

, and if ,

where is a Gamma function and μ^(2δ+2) stands for the (2δ + 2)-th absolute moment of the standard normal distribution.

 Kernel theorems are established for Banach space–valued multilinear mappings. A moment characterization theorem for Banach space–valued generalized functionals of white noise is proved by using the above kernel theorems. A necessary and sufficient condition in terms of moments is given for sequences of Banach space–valued generalized functionals of white noise to converge strongly. The integration is also discussed of functions valued in the space of Banach space–valued generalized functionals.

When A ∈ B(H) and B ∈ B(K) are given, we denote by M _C an operator acting on the Hilbert space H ⊕ K of the form In this paper, first we give the necessary and sufficient condition for M _C to be an upper semi-Fredholm (lower semi–Fredholm, or Fredholm) operator for some C ∈ B(K,H). In addition, let (A) ={λ ∈ ? : A − λI is not an upper semi-Fredholm operator} be the upper semi–Fredholm spectrum of A ∈ B(H) and let (A) = {λ ∈ ? : A − λI is not a lower semi–Fredholm operator} be the lower semi–Fredholm spectrum of A. We show that the passage from is accomplished by removing certain open subsets of from the former, that is, there is an equality

where is the union of certain of the holes in which happen to be subsets of Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl's theorem, Browder's theorem, a–Weyl's theorem and a–Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.

 Let X be an infinite–-dimensional complex Banach space and denote by ?(X) the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from ?(X) onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional φ on ?(X) such that either Φ(T) = cATA ⁻¹ + φ(T)I for all T, or Φ(T) = cAT*A ⁻¹ + φ(T)I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite–dimensional Hilbert space, the above similarity invariant additive functional φ is always zero.

 We prove the p–adic transcendence and p–adic transcendence measures for the values of some Mahler type functions.

We continue the work initiated in [1] (Second order nonlinear evolution inclusions I: Existence and relaxation results. Acta Mathematics Science, English Series, 21(5), 977-966 (2005)) and study the structural properties of the solution set of second order evolution inclusions which are defined in the analytic framework of the evolution triple. For the convex problem we show that the solution set is compact R _δ , while for the nonconvex problem we show that it is path connected. Also we show that the solution set is closed only if the multivalued nonlinearity is convex valued. Finally we illustrate the results by considering a nonlinear hyperbolic problem with discontinuities.

Let X be a metrizable space and let ?:? × X → X be a continuous flow on X. For any given {φt}–invariant Borel probability measure, this paper presents a {? _t }–invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the continuous flow (X, {? _t }). The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non–uniformly hyperbolic systems and skew–product flows.

The abc–conjecture for the ring of integers states that, for every ε > 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)^{1 + ε} with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m.

 In this paper we shall study the solvability of discontinuous functional equations, and apply the so-obtained results to discontinuous implicit initial value problems in ordered Banach spaces. The proofs are based on fixed point results in ordered spaces proved recently by the author. A concrete example is solved to demonstrate the obtained results.

 We introduce and discuss the notion of a naturally full functor. The definition is similar to the definition of a separable functor; a naturally full functor is a functorial version of a full functor, while a separable functor is a functorial version of a faithful functor. We study the general properties of naturally full functors. We also discuss when functors between module categories and between categories of comodules over a coring are naturally full.

We prove the boundedness from L _p (T ²) to itself, 1 < p < ∞, of highly oscillatory singular integrals Sf(x, y) presenting singularities of the kind of the double Hilbert transform on a non–rectangular domain of integration, roughly speaking, defined by |y'| > |x'|, and presenting phases λ(Ax+By) with 0 ≤ A, B ≤ 1 and λ ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A, B and λ involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms (of left A–modules) from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.

Finite dimensional modules over Weil algebras are investigated and corresponding gauge bundle functors, from the category of vector bundles into the category of fibered manifolds, are determined. The equivalence of the two definitions of gauge Weil functors is proved and a number of geometric examples is presented, including a new description of vertical Weil bundles.

 In this paper, we shall present a short and simple proof on the isometric linear extension problem of into–isometries between two unit spheres of atomic abstract L ^p –spaces (0 < p < ∞).

 When an independent estimate of covariance matrix is available, we often prefer two–stage estimate (TSE). Expressions of exact covariance matrix of the TSE obtained by using all and some covariables in covariance adjustment approach are given, and a necessary and sufficient condition for the TSE to be superior to the least square estimate and related large sample test is also established. Furthermore the TSE, by using some covariables, is expressed as weighted least square estimate. Basing on this fact, a necessary and sufficient condition for the TSE by using some covariables to be superior to the TSE by using all covariables is obtained. These results give us some insight into the selection of covariables in the TSE and its application.

For any element a in a generalized 2 ⁿ –dimensional Clifford algebra ? _n ( ) over an arbitrary field of characteristic not equal to two, it is shown that there exits a universal invertible matrix P _n over ? _n ( ) such that , where ?(a) is a matrix representation of a over and D _a is a diagonal matrix consisting of a or its conjugate.

Let (X, (X), be a probability space with σ-algebra , (X) and probability measure The set V in is called P-admissible, provided that for any positive integer n and positive-measure set V _n ∈ contained in V , there exists a Z _n ∈ such that Z _n ⊂ V _n and 0 < (Z _n ) < 1/n. Let T be an ergodic automorphism of (X, preserving and A belong to the space of linear measurable symplectic cocycles

In this paper, we first define a doubly transitive resolvable idempotent quasigroup (DTRIQ), and show that a DTRIQ of order v exists if and only if v ≡ 0 (mod 3) and v ? 2 (mod 4). Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang (J. Combin. Designs, 4 (1996), 301–321). As an application, we obtain an LRDTS(4 · 3 ⁿ ) for any integer n ≥ 1, which provides an infinite family of even orders.

For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S^k . We show that the singular set of stable-stationary harmonic maps from B⁵ to S³ is the union of finitely many isolated singular points and finitely many Hölder continuous curves. We also discuss the minimization problem among continuous maps from Bⁿ to S².

A construction of Λ-adic modular forms from p-adic modular symbols is described. It shows that each Λ linear map satisfying some certain conditions from the module of p-adic modular symbols to Λ corresponds to a Λ-adic modular form.

So far the study of exponential bounds of an empirical process has been restricted to a bounded index class of functions. The case of an unbounded index class of functions is now studied on the basis of a new symmetrization idea and a new method of truncating the original probability space; the exponential bounds of the tail probabilities for the supremum of the empirical process over an unbounded class of functions are obtained. The exponential bounds can be used to establish laws of the logarithm for the empirical processes over unbounded classes of functions.

In this paper, we treat a class of non-standard commutators with higher order remainders in the Lipschitz spaces and give (L^p , L^q ), (H^p , L^q ) boundedness and the boundedness in the Triebel-Lizorkin spaces. Our results give simplified proofs of the recent works by Chen, and extend his result.

This paper is concerned with the standing wave in the inhomogeneous nonlinear Klein- Gordon equations with critical exponent. Firstly, we obtain the existence of standing waves associated with the ground states by using variational calculus as well as a compactness lemma. Next, we establish some sharp conditions for global existence in terms of the characteristics of the ground state. Then, we show that how small the initial data are for the global solutions to exist. Finally, we prove the instability of the standing wave by combining the former results.