In this paper internal characterization on certain quotient images of locally separable metric spaces are discussed. We obtain some descriptions of quotient s-images, pseudo-open s-images, quotient compact images and closed images of locally separable metric spaces, and establish some relations between these and certain quotient images of metric spaces by the local separability of suitable subspaces.

Supposing the smooth involution of the DOLD manifold P(1,2l) satisfies the following condition: the fiberation π:P(1,2l)×_T×(-1)S^∞→RP(∞) is totally nonhomologous to zero (cf. [1, p373]), this paper determines the classification of smooth involution on the DOLD manifold P(1,2l) totally.

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system. We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents. Some examples of such surfaces are given in this article.

In this paper, it is proved that the Mori constant, the Hölder coefficient of the family QC_K (B) of K-quasiconformal self-mappings of the unit disk B with the origin fixed, is at most 46^1-1/K. It is also shown that the Hölder coefficient of the restriction of a map f∈QC_K(B) to the disk |z|≤ sin 25.7° is at most 16^1-1/K.

In this paper, the authors first establish some new real-variable characterizations of Herz-type Hardy spaces and , where ω₁,ω₃ ∈ A₁-weight, 1<q<∞,n(1-1/q)≤α<∞ and 0<p<∞. Then, using these new characterizations, they investigate the convergence of a bounded set in these spaces, and study the boundedness of some potential operators on these spaces.

We define a new topology on the space of strong types on a subset A of a model of a given theory and prove that either . We also deduce an open map theorem.

The author first investigates the limit cycles bifurcating from a center for general two dimensional systems, and then proves the conjecture that any unfolding of the cusp of order n has at most n-1 limit cycles.

Let H be a Hopf algebra,H₁ be a sub-Hopf algebra of H, H₂ be the quotient Hopt algebra of H modular H₁. This paper gives a simplified complex by defining a new base for the cobar complex and proves that the cobar complex of H has the same cohomology algebra with a twisted product of the cobar complexes of H₁ and H₂.

In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle Ω, where Ω is a bounded open subset of IRⁿ (n≥2) with non-smooth boundary ∂Ω and connected complement Ω_e =IRⁿ \Ω. We can prove that if Ω satisfies a certain geometrical condition, then , where depending only on n, and |·|_j (j/ = n-1, n) is aj-dimensional Lebesgue measure.

In this paper, we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space, such that for each point of the set the corresponding diffeomorphism possesses strange attractor. For some families of one-dimensional mapping satisfying certain transversality condition, we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor. Furthermore, we study the dynamics of this type of strange attractor.

Let S be a semigroup with identity and zero, the category of finitely generated projective S-systems. In this paper, the Whitehead group K₁ S of S is defined to be K₀ Ω , where Ω is the loop category of , and the structure of K₁ S is given.

This paper considers a class of fourth order nonlinear difference equations Δ²(r_n Δ²(y_n ) + f(n,y_n )=0,n ∈N(n₀) where f(n, y) may be classified as superlinear, sublinear, strongly superlinear and strongly sublinear. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. In strongly superlinear and strongly sublinear cases, sufficient conditions are given for all solutions to be oscillatory.

By using diffusion process with absorbing boundary, some lower bounds are obtained for the first Dirichlet eigenvalue of operator Δ+∇h on a non-compact complete Riemannian manifold. The resulting estimates contain McKean's estimate for ∇h=0. Moreover, the first Dirichlet eigenvalue for elliptic operators on R^d and the first mixed eigenvalue are also studied. Some examples show that our estimates can be sharp even for ∇h≠0.

Some people try to construct an orthonormal wavelet such that the corresponding scaling function φ(t) has the cardinal property,i.e. φ(n)= σ_n0, since such wavelets have many good applications. Unfortunately it is impossible to do so, except for a trivial case^[1]. In this work, a family of non-orthogonal cardinal wavelets with compact support is constructed and their duals are investigated.

If H and G are the transfer function matrix and the free response matrix of a linear finite automaton, then G is called a free response matrix matched to H. Given H, let G(H) be the set of all possible free response matrices matched to H. A classification and an enumeration on the set G(H) are given in this paper.

Let G=H^p (H_kⁿ ) be the (2n+1)-dimensional Heisenberg group over local field K. In this paper we prove some theorems about convolution operators on H^p (G) and vector-valued Hardy spaces. As an example, the distribution for some φ∈S(G), ξ φ=0 is a ramified 0-type kernel. These results can be applied to characterize H^p (G) spaces by square functions.

In this paper it is proved that any automorphism of an infinite dimensional Lie algebra of Cartan type over an algebraically closed field of characteristic p>2 can be induced from an automorphism of the associated divided power algebra.

In this paper we study the pointed representations of the Virasoro algebra. We show that unitary irreducible pointed representations of the Virasoro algebra are Harish-Chandra representations, thus they either are of highest or lowest weights or have all weight spaces of dimension1. Further, we prove that unitary irreducible weight representations of Virasoro superalgebras are either of highest weights or of lowest weights, hence they are also Harish-Chandra representations.

Let X be a compact metric space and let f: X→X be an Anosov map,i.e., an expansive selfmap with the pseudoorbit tracing property (abbr. POTP) (see Lemma 1). If Nn(f) denotes the number of fixed points of fⁿ which we name here the n-periodic number then we prove in the case asn tends to infinity that n^M ≤N_n(f)≤Hⁿ, whereM andH are two positive integers.

For all minimal immersions of the 2-sphere S² into the unit 2m-sphere S^2m with area 2π[m(m+1)+2], we are able to give an explicit representation by some real parameters. This representation is rather important in the study of minimal 2-spheres in S^2m because the parameters concerned are independent of each other. Particular examples are given and a classification theorem is also obtained.

For a vertex set {u₁,u₂,…,u_k} of a graph G withn vertices, let

In this paper, we shall prove that if G is 3-connected and NC₄≥3n, then G is either a hamiltonian or Petersen graph. This generalizes some results on the neighborhood union conditions for hamiltonian graphs.

In the present paper sufficient conditions are obtained for the stability of the zero solution of a nonlinear delay differential equation caused by impulses.

In this paper, we will discuss some properties of the (n, m)-spherical functions on the Lie group G = SL(2,R), and obtain the decomposition of f in C_c⁴ (G) into these functions. Also we give the Fourier inversion formula for the (n, m)-spherical functions in C_c³ (G).

In this paper, we use the canonical forms of homogeneous polynomials of degree 3 to study the global properties of cubic systems
(0, 1)
Where P₃ and Q₃ are homogeneous polynomials of degree 3 in x, y. Through this work, we draw an overall outline of such systems.

Around 1994, Erdos et al. abstracted from their work the following problem: "Given ten points A_ij, 1≤i≤j≤5, on a plane and no three of them being collinear, if there are five points A_k, 1≤k≤5, on the plane, including points at infinity, with at least two points distinct, such that A_i, A_j, A_ij are collinear, where 1≤i≤j≤5, is it true that there are only finitely many such A_k's?" Erdos et al. Obtained the result that generally there are at most 49 groups of such A_k's. In this paper, using Clifford algebra and Wu's method, we obtain the results that generally there are at most 6 such groups of A_k's.

In Section 1 of this paper, we investigate the finitely presented dimension of an essential extension for a module, and obtain results concerning an essential extension of a torsion-free module. We partially answer the question: When is an essential extension of a finitely presented module (an almost finitely presented module) also finitely presented (almost finitely presented)? In Section 2, we study the C-excellent extensions and the finitely presented dimensions. We obtain some results on the homological dimensions of matrix rings and group rings.

We prove that two parameter smooth continuous martingales have ∞-modification and establish a Doob's inequality in terms of (p, r)-capacity for two parameter smooth martingales.

We consider the scattering problem for the Hartree equation with potential |x|^-1 in a space of dimension n≥2. We prove the existence of H^m -modified wave operator for Hartree equation on a dense set of a neighborhood of zero in H^m (Rⁿ), meanwhile, we obtain also the global existence for the Cauchy problem of Hartree equation in a space of dimension n≥2.

Let Δ(x) be the error term in the asymptotic formula for the counting function of square-full integers. In the present paper it is proved that , which improves on the exponent 5/33 obtained by X. D. CAO.

In this paper, we prove that a composition operator on H^p (B) is Fredholm if and only if it is invertible if and only if its symbol is an automorphism on B, and give the representation of the spectra of a class of composition operators. In addition, using composition operator, we discuss intertwining Toeplitz operators.

Two theorems about the distribution and moments of the Fourier coefficients of cusp forms are proved in this paper.

An existence theorem for the solution to the equation

is given by means of variational method where b(x)→∞, as |x|→∞ and f(x, s) has linear growth in s at infinity and sublinear growth in s at zero. For a special case, some multiplicity result is proved.

The boundedness on weighted local Hardy spaces h_w^1,p of the oscillatory singular integral

is considered when Q(x, y)=P(x-y) for some real-valued polynomial P with its degree not less than two. Also a sufficient and necessary condition on polynomial Q on Rⁿ×Rⁿ such that T maps h_w^1,p to the weighted integrable function space L_w¹ is found.

For finite rank operators in a commutative subspace lattice algebra algL we introduce the concept of correlation matrices, basing on which we prove that a finite rank operator in algL can be written as a finite sum of rank-one operators in algL, if it has only finitely many different correlation matrices. Thus we can recapture the results of J.R. Ringrose, A. Hopenwasser and R.Moore as corollaries of our theorems.

We study overdetermined systems of first order partial differential equations with singular solutions. The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.

The space of continuous maps from a topological space X to topological space Y is denoted by C(X,Y) with the compact-open topology. In this paper we prove that C(X,Y) is an absolute retract if X is a locally compact separable metric space and Y a convex set in a Banach space. From the above fact we know that C(X,Y) is homomorphic to Hilbert space l₂ if X is a locally compact separable metric space and Y a separable Banach space; in particular, C(Rⁿ,R^m) is homomorphic to Hilbert space l₂.

This paper is devoted to asymptotic formulae for functions related with the spectrum of the negative Laplacian in two and three dimensional bounded simply connected domains with impedance boundary conditions, where the impedances are assumed to be discontinuous functions. Moreover, asymptotic expressions for the difference of eigenvalues related to the impedance problems with different impedances are derived. Further results may be obtained.

Relative extreme values are defined by the supremum and minimum of a general jump process before its first time quitting from some state set, and relative extremum-times are defined by the first times reaching relative extreme values. The main objective of this paper is to find out the exact distributions and moments of them as the maximum of the set is up or equal to the process initial state. As especial cases, these results are applied to a general birth-death process and generalized birth-death processes.

By the using of determinantal varieties from moduli algebras of hypersurface singularites the relation of the deformation of hypersurface singularities and the deformation of their moduli algebras is studied. For a type of hypersurface singularities a weak Torelli type result is proved. This weak Torelli type result showes that for families of hypersurface singularities the moduli algebras can be used to distinguish the complex structures of singularities at least in some weak sence.

We study a system (D)x'=F(t,x_t) of functional differential equations, together with a scalar equation (S)x'=-a(t)f(x)+b(t)g(x(t-h)) as well as perturbed forms. A Liapunov functional is constructed which has a derivative of a nature that has been widely discussed in the literature. On the basis of this example we prove results for (D) on asymptotic stability and equi-boundedness.