Using a kind of Mayer-Vietoris principle for the symplectic Floer homology of knots, we compute the symplectic Floer homology of the square knot and granny knots.

The problem of deciding whether a graph manifold is finitely covered by a surface bundle over the circle is discussed in this paper. A necessary and sufficient condition in term of the solutions of a certain matrix equation is obtained, as well as a necessary condition which is easy to compute. Our results sharpen and extend the earlier results of J. Leucke-Y. Wu, W. Neumann, and S. Wang-F. Yu in this topic.

We provide an elementary formula for the non-integer powers of the Laplace operator in the Euclidean spaces. Such formulas are helpful in establishing the elliptic nature of some pseudo-differential operators arising from the study of energies of knotted loops in space, and possibly of embedded submanifolds in a Riemannian manifold.

We study all the possible weak limits of a minimizing sequence, for p-energy functionals, consisting of continuous maps between Riemannian manifolds subject to a Dirichlet boundary condition or a homotopy condition. We show that if p is not an integer, then any such weak limit is a strong limit and, in particular, a stationary p-harmonic map which is C^1,α continuous away from a closed subset of the Hausdorff dimension ≤ n-[p]-1. If p is an integer, then any such weak limit is a weakly p-harmonic map along with a (n-p)-rectifiable Radon measure μ. Moreover, the limiting map is C^1,α continuous away from a closed subset Σ=spt μ ∪ S with H^n-p (S)=0. Finally, we discuss the possible varifolds type theory for Sobolev mappings.

In this paper, we will prove that Floer homology equipped with either the intrinsic or exterior product is isomorphic to quantum homology as a ring. We will also prove that GW-invariants in Floer homology and quantum homology are equivalent.

Let G be a finite group and k a field of characteristic p > 0. In this paper we consider the support variety for the cohomology module Ext_kG^* (M, N) where M and N are kG-modules. It is the subvariety of the maximal ideal spectrum of H*(G, k) of the annihilator of the cohomology module. For modules in the principal block we show that that the variety is contained in the intersections of the varieties of M and N and the difference between the that intersection and the support variety of the cohomology module is contained in the group theoretic nucleus. For other blocks a new nucleus is defined and a similar theorem is proven. However in the case of modules in a nonprincipal block several new difficulties are highlighted by some examples.

Let be a compact complex manifold of complex dimension two with a smooth Kähler 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 the existence of a metric on E'=E _\D (compatible with the parabolic structure) which is Hermitian-Einstein with respect to the restriction of the Kähler metric to \D. A converse is also proved.

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L²(R) initial data.

In this paper we have completely determined: (1) all almost simple groups which act 2-transitively on one of their sets of Sylow p-subgroups. (2) all non-abelian simple groups T whose automorphism group acts 2-transitively on one of the sets of Sylow p-subgroups of T. (3) all finite groups which are 2-transitive on all their sets of Sylow subgroups.

In this paper, automorphisms of the algebra of q-difference operators, as an associative algebra for arbitrary q and as a Lie algebra for q being not a root of unity, are determined.

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformly C³ boundary, under the assumption that‖a‖_L ²(Θ)+‖f‖_L ¹(0,∞;L²(Θ)) or‖▽a‖_L ²(Θ)+‖f|_L ²(0,∞;L²(Θ)) small or viscosityv large. Here a is a given initial velocity and f is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed.

In this paper the asymptotic behavior of Gibbs function for a class of M-band wavelet expansions is given. In particular, the Daubechies' wavelets are included in this class.

In this paper we construct models obtained by suitably combining Brownian motions and telegraphs in such a way that their transition functions satisfy higher-order parabolic or hyperbolic equations of different types.Equations with time-varying coefficients are also derived by considering processes endowed either with drift or with suitable modifications of their structure.Finally the distribution of the maximum of the iterated Brownian motion (along with some other properties) is presented.

Special generators of the unoriented cobordism ring MO_* are constructed to determine the groups J_n,κ^r of n-dimensional cobordism classes in MO_n containing a representative Mⁿ admitting a (Z₂)^k -action with fixed point set of constant codimension.

In this paper, we prove that under the F₄ conditions, any L log⁺ L bounded two-parameter Banach spece valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if the F₄ condition is replaced by the weaker local F₄ condition.

Two families of Liapunov functions are employed to study the global stability and boundedness of functional differential systems. New stability and boundedness theorems are obtained. Applications of these theorems to some nonlinear differential systems with infinite delay are discussed.

We consider the convolution transforms of measures on R^d defined by some approximate identity. We shall establish some relations between the irregular boundary properties of the convolution function and the local Lipschitz exponent of the measure. In particular, the results can be applied to the Poisson and Gauss-Weierstrass kernels. We can then obtain some singular boundary behavior of positive harmonic or parabolic functions on R₊^d+1 by multifractal analysis of measures.

This paper is devoted to the study of the Cauchy problem for the coupled system of the Schrödinger-KdV equations which describes the nonlinear dynamics of the one-dimensional Langmuir and ion-acoustic waves. Global well-posedness of the problem is established in the space H^κ×H^κ (κεZ⁺), the first and second components of which correspond to the electric field of the Langmuir oscillations and the low-frequency density perturbation respectively.

In this paper, we give a sufficient condition for double crossproductX ?A to beX ?_τ A for some skew pairingτ ifX ?A is a 2-cocycle deformation ofX ?A. Then we give a sufficient and necessary condition forX ?A to beX ?_τ A by using natural isomorphism terminology.

In this paper we obtain exact rates of uniform convergence for oscillation moduli and Lipschitz-1/2 moduli of PL-process and cumulative hazard process when the data are subject to left truncation and right censorship. Based on these results, the exact rates of uniform convergence for various types of density and hazard function estimators are derived.

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-function to study the distribution property of Dedekind sums, and to give a sharper mean value formula.

All the symplectic matrices possessing a fixed eigenvalueθ on the unit circle form a hypersurface in the real symplectic group Sp(2n). This paper is devoted to the study of the topological structures of this hypersurface and its complement in Sp(2n).

We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry.

Some Liouville type theorems for harmonic maps from K?hler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (exceptR_IV(2)) to any Riemannian manifold with finite energy has to be constant.

In this paper,we give a suffcient condition for double crossproduct X A to be X _τA for some skew pairingτif X A is a 2-cocycle deformation of X A.Then we give a suffcient and necessary condition for X A to be X _τA by using natural isomorphism terminology.

In this paper we obtain exact rates of uniform convergence for oscillation moduli and Lipschitz-1/2 moduli of PL-process and cumulative hazard process when the data are subject to left truncation and right censorship.Based on these results,the exact rates of uniform convergence for various types of density and hazard function estimators are derived.

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-function to study the distribution property of Dedekind sums,and to give a sharper mean value formula.

All the symplectic matrices possessing a fixed eigenvalue ω on the unit circle form a hypersurface in the real symplectic group Sp(2n).This paper is devoted to the study of the topological structures of this hypersurface and its complement in Sp(2n).

We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature,in the three-dimensional Euclidean space.We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind.They are determined implicitly by elliptic integrals of the third kind.Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry.

Some Liouville type theorems for harmonic maps from Kähler manifolds are obtained.The main result is to prove that a harmonic map from a bounded symmetric domain(except R_IV(2))to any Riemannian manifold with finite energy has to be constant.

We provide some more explicit formulae to facilitate the computation of Ohtsuki's rational invariants λ_n of integral homology 3-spheres extracted from Reshetikhin-Turaev SU(2) quantum invariants. Several interesting consequences will follow from our computation of λ₂. One of them says that λ₂ is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that λ₁ is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial.

A two-dimensional hyperbolic system of nonlinear conservation laws is considered for any piecewise constant initial data having two discontinuity rays with the origin as vertex. One kind of new waves, which is labeled the Dirac-contact wave, appears in the solution. The entropy conditions for the Dirac-contact waves are given. The solutions on the Dirac-contact waves can be viewed as the bounded linear functionals on C₀^∞ (R² ×R₊).

Necessary and sufficient conditions are obtained for operator partial 2 × 2 matrices to have an idempotent completion, and all such completions are parametrically represented.

For a regular semigroup with an inverse transversal, we have Saito's structure W(I,S^o, Λ, *, {α, β}). We represent congruences on this kind of semigroups by the so-called congruence assemblage which consist of congruences on the structure component parts I,S^o and Λ. The structure of images of this type of semigroups is also presented.

This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some sufficient conditions in terms of the isoperimetric inequalities and some necessary conditions for the inequalities are presented. The resulting conditions can be sharp qualitatively as illustrated by some examples. It turns out that for a probability measure, the Nash inequalities are much stronger than the Poincaré and the logarithmic Sobolev inequalities in the present context.

We study here minmal graph evolutions in the hyperbolic space and prove that there exists a unique smooth solution.

In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the product L₂ L₁ of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators.

Let 1<c<11/10. In the present paper it is proved that there exists a number N(c)>0 such that for each real number N>N(c) the inequality |p₁^c+p₂^c+p₃^c-N|-1/c(11/10-c)log^c₁N is solvable in prime numbers p₁,p₂,p₃, where c₁ is some absolute positive constant.

An inverse theorem for the best weighted polynomial approximation of a function in (S) is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation in , wherew_α is a Jacobi-type weight on S, 0<p ≤ ∞,S is a simplex and W_λ is a Freud weight. For Ditzian-Totik K-functionals on L_p(S), 1 ≤p ≤ ∞, we obtain a new equivalence expression.

Singular limit is investigated for reaction-diffusion equations with an additive noise in a bounded domain of R². The solution converges to one of the two stable phases {+1,-1} determined from the reaction term; accordingly a phase separation curve is generated in the limit. We shall derive a randomly perturbed motion by curvature for the dynamics of the phase separation curve.