Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z0 be a subset of Z such than n∈Z0 implies n+1∈Z0. Denote the space of all compactly supported distributions by D', and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence Gn and Hn, n∈Z0, in D', define the corresponding nonstationary nonhomogeneous refinement equation
Φ_n = Hn ∗ Φ_n+1(A·) + G_n for all n ∈ Z₀, (*)
where Φn, n∈Z0, is in a bounded set of D'. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φn, n∈Z0, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations

where the matrices Sn and the vectors , n∈Z0, can be constructed explicitly from Hn and Gn respectively. The results above are still new even for stationary nonhomogeneous refinement equations.

This paper gives the Hausdorff measure of a class of homogeneous Cantor sets.

Rips conjectured that a non-clementary word hyperbolic group is cohopfian if and only if it is freely indecomposable. The results and examples in this paper show that cohopficity phenomenon in the case of word hyperbolic group with torsion is much more complicated than the conjecture. In particular, the cohopficity of such groups is not determined by the numbers of their ends and the cohopficity is not preserved by finite index subgroups. Our results and examples arise from Kleinian groups. Orbifold structures and orbifold maps are the new tools in our discussions.

For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and Hörmander. As an application of our result, we show that the solution of three-dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.

Let be a compact complex manifold of complex dimension two with a smooth Köhler metric and D a smooth divisor on . If E is a rank 2 holomorphic vector bundle on with a stable parabolic structure along D, we prove that there exista a Hermitian-Einstein metric on compatible with the parabolic structure, whose curvature is square integrable.

A multi-dimensional version of the duality principle of Sawyer type [1] is obtained whenever the corresponding weight satisfies some doubling property.

We use a semantical method of complete residuated lattice-valued logic to give a generalization of fuzzy topology as a partial answer to a problem by Roser and Turquette.

In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234-240) is true for 1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with int K≠Ø. This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractive maps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-selfmaps are proved under various well-known boundary conditions. Our results are generalizations and improvements of the recent results obtained by many authors.

In this paper, we discuss the connection between rigid graded selfduality over some noncommutative graded ring and the selfduality over its initial subring.

Let f∈C¹ (R², R²), f(0)=0. The Jacobian Conjecture states that if for any x∈R², the eigenvalues of the Jacobian matrix Df(x) have negative real parts, then the zero solution of the differential equation x=f(x) is globally asymptotically stable. In this paper we prove that the conjecture is true.

We prove a theorem on the commutativity of the PI ring and use it to generalize a famous theorem of Herstein.

We prove the uniqueness of infinite open cluster for high-density bond percolation on lattice Sierpinski Carpet; forthermore, an alternative proof of the existence of phase transition of the model is given. A rescaling technique is developed and used as the main tool of our proofs.

We call a Cayley digraph Γ=Cay(G, S) normal for G if G_R, the right regular representation of G, is a normal subgroup of the full automorphism group Aut(Γ) of Γ. In this paper we determine the normality of Cayley digraphs of valency 2 on nonabelian groups of order 2p² (p odd prime). As a result, a family of nonnormal Cayley digraphs is found.

Let λ₁, λ₂,..., λ₇ be real numbers satisfying λ_i≥1. In this paper, we prove there are integers x₁,..., x₇ such that the inequalities and hold simultaneously.

We introduce the meet-function and join-function on a subset of a poset and use them to give some explicit expressions of Smith's determinant.

Let be the Kaplan-Meier estimator of distribution function F. Let J(·) be a measurable real-valued function. In this paper, a U-statistic representation for the Kaplan-Meier L-estimator, T()=∫xJ((x))d(x), is derived. Furthermore, the representation is also used to establish a Berry-Essen inequality for .

Let Ω be a bounded Lipschitz domain.Define B_1,r^0,1(Ω)={f∈L¹(Ω):there is an F∈B₁^0,1(Rⁿ) such that F|_Ω=f} and B_1,z^0,1(Ω)={f∈B₁^0,1(Rⁿ):f=0 on (Rⁿ)\Ω In this paper,the authors establish the atomic decompositions of these spaces.As by-products,the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation on R₊ⁿ.

Recently much work has been devoted to nonlocal problems.However,very little has been accomplished in the literature for nonlocal initial problems with nonlinear boundary conditions.It is the purpose of this paper to prove the existence results for solutions to a semilinear parabolic PDE with linear homogeneous boundary conditions,and to other ones with nonlinear boundary conditions,provided the ordered upper and lower solutions are given.Semigroup,fractional order function spaces and generalized Poincaré operators play an important role in proving the existence of solutions.

In this paper,we study some geometrical and analytic properties of manifolds with non-negative sectional curvature at infinity.Then,we apply these results to the study of harmonic maps.

For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state,we prove that the first-order derivatives of ρ,υ blow-up at the blow-up time,while ρ,υ remain continuous.In particular,in the irrotational case we prove S.Alinhac's statement.

We give the general integral of quadratic differential system with center in two cases under the Chinese classification.

We study the Bloch constant for K-quasiconformal holomorphic mappings of the unit ball B of Cⁿ into Cⁿ.The final result we prove in this paper is:If f is a K-quasiconformal holomorphic mapping of B into Cⁿ such that det(f'(0))=1,then f(B) contains a schlicht ball of radius at least

where C_n>1 is a constant depending on n only,and C_n→√10 as n→∞.

By using the Taylor expansions of the solutions of Jacobi equations,we obtain the complete formulas for the volumes of tubes about curves in a Riemannian manifold.This unifies the known results and simplifies the computations involved in this direction.In the special case of surfaces,we also obtain the corresponding complete formulas which generalize the known results.

We prove the existence of quasiperiodic solutions and Lagrange stability for a class of differential equations with jumping nonlinearity ,where a,b>0,p(t) ∈C(R/2πZ) and φ:R→R is an unbounded function.

In this paper,we characterize the pointwise multiplier space M (D_τ,D_μ) of Dirichlet type spaces in the unit ball of Cⁿ for the values of τ,μ in three cases:(ⅰ) τ<0,μ<0,(ⅱ) τ<μ,(ⅲ) τ≥μ,τ>n,and construct two functions to show that M(D_τ)⊂D_τ properly if τ≤n and M(D_τ)⊂M(D_μ) properly if τ>μ and τ>n-1.

In this paper,by calculating the commutator subgroup of the unit group of finite path algebra κΔ and the unit group abelianized,we explicitly characterize the Κ₁ group of finite dimensional path algebra over an arbitrary field.

Suppose that f(x)=(f₁(x),...,f_r(x))^T,x∈R^d is a vector-valued function satisfying the refinement equation f(x)=∑_Λc_κf(2x-κ) with finite set Λ of Z^d and some r×r matricex c_κ.The requirements for f to have accuracy p are given in terms of the symbol function m(ξ).

We prove local weighted integral inequalities for differential forms.Then byusing the local results,we prove global weighted integral inequalities for differential forms in L_s(μ)-averaging domains and in John domains,respectively,which can be considered as generalizations of the classical Poincaré-type inequality.

In this note,we use the so-called microlocal energy method to give a characterization of the Gevrey-Sobolev wave front set ,which will be useful in the study of non-linear microlocal analysis in Gevrey classes.

In this note,we construct some new harmonic maps from R^m to H^m via symmetric methods.

In this paper,the average σ-Κ width of Sobolev-Wiener classes with mixed smoothness in L_q(R^d) is studied for 1<q≤p<∞,and the weak asymptotical behaviour of these quantities is obtained.

In this paper we consider the boundedness of all the solutions for the equation x"+ax⁺-bx^-=f(t) is a smooth 2π-periodic function,a and b are positive constants (a≠b).

Let F be an algebracially closed field of characteristic p>2,and L be the pⁿ-dimensional Zassenhaus algebra with the maximal invariant subalgebra L₀ and the standard filtration Then the number of isomorphism classes of simple L-modules is equal to that of simple L₀-modules,corresponding to an arbitrary character of L except when its height is biggest.As to the number corresponding to the exception there was an earlier result saying that it is not bigger than pⁿ.

This paper gives a formula of integral for the rectifiable curves in the plane by the Hausdorff fractional derivation and integral.

In this paper,a class of finitely supported orthogonal filters with interpolation and symmetry is constructed.

Basic properties of the Herz-type Hardy spaces HK_q^α,p,such as the boundedness of singular integral operators and the fractional integration operators,atomic decomposition,dense subspaces,etc.,are established in the full range 0<q<∞.

We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but different twisted Alexander polynomials are given.

In this paper, under the Lorentz gauge, the global existence, local dissipativity and attractors of the evolutionary superconductivity problem which is a parabolic-hyperbolic system are obtained.

In this paper, we prove that the forced pendulum-type equation +G_x(x,t)-p(t),where G(x, t) ∈ C¹(R²) with 1-periodicity in x satisfies the conditions: sup_(x,t)∈R² |G_x(x,t)| < +∞ and lim sup_t→∞{sup_x∈R |G_x(x,t)/t|}=0, possesses infinitely many unbounded solutions on a cylinder S¹×R for any almost periodic function p(t) with nonvanishing mean value.

We make improvements to some of the constants concerned with univalent functions with quasiconformal extensions that have appeared in the literature.