Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition for a homomorphism φ : Hⁿ (L;Z) → Hⁿ (M;Z) to be realized by a map f : M → L of degree k for closed (n-1)-connected 2n-manifolds M and L, n > 1. A corollary is that each (n-1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 1.In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms; in particular, there is a map f : M → L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M.

In this paper,we are concerned with the existence of invariant curves of reversible mappings. A variant of the classical small twist theorem is given.

In this paper, we establish the relationship between Hausdorff measures and Bessel capacities on any nilpotent stratified Lie group G (i. e., Carnot group). In particular, as a special corollary of our much more general results, we have the following theorem (see Theorems A and E in Section 1):Let Q be the homogeneous dimension of G. Given any set E ⊂ G, B_α,p (E) = 0 implies H^{Q-αp+ ε}(E) = 0 for all ε > 0. On the other hand, H^Q-αp(E) < ∞ implies B_α,p (E) = 0. Consequently, given any set E ⊂ G of Hausdorff dimension Q-d, where 0 < d < Q, B_α,p (E) = 0 holds if and only if αp ≤ d.A version of O. Frostman's theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4).

The simplest NMS flow is a polar flow formed by an attractive periodic orbit and a repulsive periodic orbit as limit sets.In this paper we show that the only orientable,simple,compact, 3-dimensional manifolds without boundary that admit an NMS flow with none or one saddle periodic orbit are lens spaces. We also see that when a fattened round handle is a connected sum of tori,the corresponding flow is also a trivial connected sum of flows.

In this paper,some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved.The results of this paper reduce to those of Pachpatte(1998,J.Math.Anal.Appl. 226,166-179) and Zhao and Debnath(2001,J.Math.Anal.Appl.262,411-418).

In this paper,some feasibly suffcient conditions are obtained for the global asymptotic stability of a positive steady state of a predator-prey system with stage structure for the predator by using the theory of competitive systems,compound matrices and stability of periodic orbits,and then the work of Wang [4] is improved.

In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau's Liouville theorem on bounded harmonic functions.

This paper concerns the problem of average σ-width of Sobolev-Wiener classes , and Besov-Wiener classes in the metric L q (R d ) for 1 ≤ q ≤ p ≤ ∞. The weak asymptotic results concerning the average linear widths, the average Bernstein widths and the infinite-dimensional Gel'fand widths are obtained, respectively.

The authors establish the approximations to the distribution of M-estimates in a linear model by the bootstrap and the linear representation of bootstrap M-estimation, and prove that the approximation is valid in probability 1. A simulation is made to show the effects of bootstrap approximation, randomly weighted approximation and normal approximation.

Let U be a flat right R-module and N an infinite cardinal number. A left R-module M is said to be (N, U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (N, U)-finitely presented in σ[M]. It is proved under some additional conditions that a left R-module M is (N, U)-coherent if and only if is M-flat as a right R-module if and only if the (N, U)-coherent dimension of M is equal to zero. We also give some characterizations of left (N, U)-coherent dimension of rings and show that the left N-coherent dimension of a ring R is the supremum of (N, U)-coherent dimensions of R for all flat right R-modules U.

This paper is concerned with the initial boundary-value problem for the following nonlinear evolution equation:
.
Under certain conditions on the initial data and the function g(s), we study the existence and nonexistence of global solution for this equation. The blow-up solution and the blow-up time are also investigated.

In this paper, we study the ring #(D,B) and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all (B,D)-Hopf modules _BM^D. Cai and Chen have proved this result in the case B = D = A. Secondly they have proved that if A has a nonzero left integral then A#A^*rat is a dense subring of End_k (A). We prove that #(A,A) is a dense subring of End_k (Q), where Q is a certain subspace of #(A,A) under the condition that the antipode is bijective (see Theorem 18). This condition is weaker than the condition that A has a nonzero integral. It is well known the antipode is bijective in case A has a nonzero integral. Furthermore if A has nonzero left integral, Q can be chosen to be A (see Corollary 19) and #(A,A) is both left and right primitive. Thus A#A^*rat ⊆ #(A,A) ≈ End_k (A). Moreover we prove that the left singular ideal of the ring #(A,A) is zero. A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional, namely the ring #(A,A) has a finite uniform dimension.

In this paper, we consider the solution of the biharmonic equation using Adini noncon-forming finite element, and report new results of the asymptotic error expansions of the interpolation error functionals and nonconforming remainder. These expansions are used to develop two extrapolation formulas and a series of sharp error estimates. Finally, the numerical results have verified the extrapolation theory.

Let G be a non-empty closed (resp. bounded closed) boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X. Let K(X) denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let denote the closure of the set . We prove that the set of all , such that the minimization (resp. maximization) problem min(A,G) (resp. max(A,G)) is well posed, contains a dense G δ-subset of , thus extending the recent results due to Blasi, Myjak and Papini and Li.

In two centuries ago, Ming Autu discovered the famous Catalan numbers while he tried to expand the function sin(2px) as power series of sin(x) for the case p = 1, 2, 3. Very recently, P. J. Larcombe shows that for any p, sin(2px) can always be expressed as an infinite power series of sin(x) involving precise combinations of Catalan numbers as part of all but the initial p terms and gave all expansions for the case p = 4, 5. The present paper presents the desired expansion for arbitrary integer p.

We prove that there are non-recursive r.e. sets A and C with A < _T C such that for every set .

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields over k = F q (T). For five series of real quadratic function fields K, the bounds of h(D) are given more explicitly, e. g., if D = F² + c, then h(D) ≥ degF/degP; if D = (SG)² + cS, then h(D) ≥ degS/degP; if D = (A m + a)² + A, then h(D) ≥ degA/degP, where P is an irreducible polynomial splitting in K, c ∈ F q. In addition, three types of quadratic function fields K are found to have ideal class numbers bigger than one.

The main purpose of this paper is to use the estimate for character sums and the method of trigonometric sums to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of the Gauss sums, and give a sharper asymptotic formula.

In this paper, two new functions are introduced to depict the Jamison weighted sum of random variables instead using the common methods, their properties and relationships are system-atically discussed. We also analysed the implication of the conditions in previous papers. Then we apply these consequences to B-valued random variables, and greatly improve the original results of the strong convergence of the general Jamison weighted sum. Furthermore, our discussions are useful to the corresponding questions of real-valued random variables.

In this work, Rademacher's two questions about Dedekind sums are applied to the Hardy sums s₂(h, k) and s₃(h, k). Some relations between these sums are obtained and some inequalities are given.

For a locally compact group G, L¹(G) is its group algebra and L^∞ (G) is the dual of L¹ (G). Lau has studied the bounded linear operators T : L^∞ (G) → L^∞ (G) which commute with convolutions and translations. For a subspace H of L^∞ (G), we know that M (L^∞ (G), H), the Banach algebra of all bounded linear operators on L^∞ (G) into H which commute with convolutions, has been studied by Pym and Lau. In this paper, we generalize these problems to L(K), the dual of a hypergroup algebra L(K) in a very general setting, i.e. we do not assume that K> admits a Haar measure. It should be noted that these algebras include not only the group algebra L¹ (G) but also most of the semigroup algebras. Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however, we succeed in getting some interesting results.

The asymptotic expansion for small |t| of the trace of the wave kernel , where and are the eigenvalues of the negative Laplacian in the (x 2,x 2)-plane, is studied for a multi-connected vibrating membrane Ω in R² surrounded by simply connected bounded domains Ω_j with smooth boundaries ∂Ω_j (j = 1,..., n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Γ_i(i = 1+k_j-1,…, k_j) of the boundaries ∂Ω_j are considered, such that and k₀ = 0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e. g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel for small |t|.

In this paper, we construct first a new concrete example of asymmetric convex compact C^1,1-hypersurfaces in R²ⁿ possessing precisely n closed characteristics. Then we prove multiplicity results on the closed characteristics on convex compact hypersurfaces in R²ⁿ pinched by not necessarily symmetric convex compact hypersurfaces.

In this paper we discuss maximal attractors of the m-dimensional Cahn-Hilliard System in the product spaces (L²(Ω))^m and (H²(Ω))^m in terms of D. Henry's general theory and from the viewpoint of compactness and absorptivity of semigroups as R. Temam did. After giving the existence and uniqueness of global solutions, we technically restrict our discussion to some subspaces, give estimates with a new graph norm, and obtain the existence of maximal attractors and some properties of them.

We establish a criterion for the uniqueness of periodic solutions for a class of second-order equations. We also give an application to a polynomial system and corrections to some known results.

This paper, by using of windowed Fourier transform (WFT), gives a family of embedding operators , s.t. are reproducing subspaces (n = 0, Bargmann Space); and gives a reproducing kernel and an orthonormal basis (ONB) of T n L 2(R). Furthermore, it shows the orthogonal spaces decomposition of . Finally, by using the preceding results, it shows the eigenvalues and eigenfunctions of a class of localization operators associated with WFT, which extends the result of Daubechies in[1] and[6].

In this paper the small Hankel operators on the Dirichlet-type spaces D^p on the unit ball of Cⁿ are considered. A similar result to that of the one-dimensional setting is given, which characterizes the boundedness of the small Hankel operators on D^p.

Let ξ_i ∈ (0, 1) with 0 < ξ₁ < ξ₂ <… < ξ_m-2 < 1, a_i, b_i ∈[0,∞) with and . We consider the m-point boundary-value problem , We consider the m-point boundary-value problem

where f(x, y) ≥ -M, and M is a positive constant. We show the existence and multiplicity of positive solutions by applying the fixed point theorem in cones.

The main purpose of this paper is to use estimates for character sums and analytic methods to study the first power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and three asymptotic formulae are obtained.

For the generalized Beltrami system with two characteristic matrices, we deal with the regularity of its very weak solutions in the Sobolev class (1 < r < n). By changing the generalized Beltrami system into a class of a divergent elliptic system with nonhomogeneous items, we obtain that each of its very weak solutions is essentially a classical weak solution of a usual Sobolev class. Furthermore, we also establish a higher regularity of its weak solution if the regularity hypotheses of two characteristic matrices are improved.

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.

In this paper, we consider the generation and propagation of interfaces for p-Laplacian equations with the derivative of a bi-stable potential.

The weight hierarchy of a binary linear[n, k] code C is the sequence (d₁, d₂,…, d_k), where d r is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes and the possible weight hierarchies in each class is determined by finite projective geometries. The possible weight hierarchies in class A, B, C, D are determined in Part (Ⅰ). The possible weight hierarchies in class E, F, G, H, I are determined in Part (Ⅱ).

In this paper, the asymptotic behavior of generalized risk processes without any moment assumptions on the controlling process is described.

With Littlewood-Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces and Triebel-Lizorkin spaces ; but the structure of dual spaces of is very different from that of Besov spaces or that of Triebel-Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood-Paley analysis. Our main goal is to characterize in tent spaces with wavelets. By the way, some applications are given: (i) Triebel-Lizorkin spaces for p = ∞ defined by Littlewood-Paley analysis cannot serve as the dual spaces of Triebel-Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among and L 1 are studied.

In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of local solutions of quasilinear parabolic equation with critical Sobolev exponent and with lower energy initial value; we also describe the asymptotic behavior of global solutions with high energy initial value.

In this paper we prove that if T is a regular n-partite tournament with n ≥ 4, then each arc of T lies on a cycle whose vertices are from exactly k partite sets for k = 4, 5,…,n. Our result, in a sense, generalizes a theorem due to Alspach.

In this paper, we are first concerned with viscous approximations for the three-dimensional axisymmetric incompressible Euler equations. It is proved that the viscous approximations, which are the solutions of the corresponding Navier-Stokes equations, converge strongly in L²([0, T]; L_loc²(R³)) provided that they have strong convergence in the region away from the symmetry axis. This result has been proved by the authors for the approximate solutions generated by smoothing the initial data, with no restriction of the sign of the initial data. Then we discuss the decay rate for maximal vorticity function, which is established for both approximate solutions generated by smoothing the initial data and viscous approximations respectively. One sufficient condition to guarantee the strong convergence in the region away from the symmetry axis is given, and a decay rate for maximal vorticity function in the region away from the symmetry axis is obtained for non-negative initial vorticity.

In this paper we consider the commutators of fractional integrals

on the Besov spaces B_p^s,q,where b is a locally integrable function and 0<α<n. We first establish the equivalence between the boundedness of the commutators and the paraproduct of J. M. Bony. Then we obtain two conditions on the boundedness of the commutators. One of these conditions is necessary and the other is sufficient.

Let G be a graph which contains exactly one simple closed curve. We prove that a continuous map f: G→G has zero topological entropy if and only if there exist at most k≤[(Edg(G)+End(G)+3)/2] different odd numbers n₁,...,n_k such that Per(f) is contained in , where Edg(G) is the number of edges of G and End(G) is the number of end points of G.