A technique to compute the colored Jones polynomials of satellite knots, illustrated by theWhitehead doubles of knots, is presented. Then the author proves the volume conjecture for Whitehead doubles of a family of torus knots and shows some interesting observations.

The aim of this work is to construct weak solutions for the three dimensional Vlasov-Poisson initial-boundary value problem with bounded electric field. The main ingredient consists of estimating the change in momentum along characteristics of regular electric fields inside bounded spatial domains. As direct consequences, the propagation of the momentum moments and the existence of weak solution satisfying the balance of totalenergy are obtained.

Compact Kähler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kähler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition $ \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} $, where X _j is a Calabi-Yau manifold, or a hyperKähler manifold, or X _j satisfies H ⁰(X _j , Ω ^p) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kähler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ∈ > 0, there exists a Kähler structure (J _∈, g _∈) on M such that the volume ${\text{Vol}}_{{g_{ \in } }} {\left( M \right)} < V$, the sectional curvature |K(g _∈)| < Λ², and the Ricci-tensor Ric(g _∈)> −∈g _∈, where V and Λ are two constants independent of ∈. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, $ \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} $, where X _i is a Calabi-Yau manifold, or a hyperKähler manifold, or X _i satisfies H ⁰(X _i , Ω ^p) = {0}, p > 0.

The classical Hardy theorem asserts that f and its Fourier transform $\ifmmode\expandafter\hat\else\expandafter\^\fi{f}$ can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU(1, 1) there are infinitely many “good” functions in the sense that f and its spherical Fourier transform $ \ifmmode\expandafter\tilde\else\expandafter\~\fi{f}$ both have good decay. In this paper, we shall characterize such functions on SU(1, 1).

The author studies the structure of solutions to the interface problems for second order linear elliptic partial differential equations in three space dimension. The set of singular points consists of some singular lines and some isolated singular points. It is proved that near a singular line or a singular point, each weak solution can be decomposed into two parts, a singular part and a regular part. The singular parts are some finite sum of particular solutions to some simpler equations, and the regular parts are bounded in some norms, which are slightly weaker than that in the Sobolev space H ².

In this paper, the stability and the Hopf bifurcation of small-world networks with time delay are studied. By analyzing the change of delay, we obtain several sufficient conditions on stable and unstable properties. When the delay passes a critical value, a Hopf bifurcation may appear. Furthermore, the direction and the stability of bifurcating periodic solutions are investigated by the normal form theory and the center manifold reduction. At last, by numerical simulations, we further illustrate the effectiveness of theorems in this paper.

Abstract In this paper, Lefschetz formulae for torus actions on p-adic groups are proven. These are similar to comparable formulae for real Lie groups. Applications lie in the realm of dynamical zeta functions.

The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form

u _t − Δ∅(u) = 0,

where ∅ ∈ C ¹(ℝ¹) is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of ∅'( · ) is permitted to have zero measure. This is an answer to an open problem in [13, p. 288].