Kernel estimates and -spectral independence of generators of -semigroups

After the appearance of W. Arendt's result that “Gaussian estimate of a semigroup implies the L^p-spectral independence of the generator,” various generalizations have been obtained. This paper shows that a certain kernel estimate of a semigroup implies the L^p-spectral independence of the generator, generalizing the case of upper Gaussian estimate and “Gaussian estimate of order α(0,1] [S. Miyajima, H. Shindoh, Gaussian estimates of order α and L^p-spectral independence of generators of C₀-semigroups, Positivity 11 (1) (2007) 15–39], Definition 3.1.” The proof uses S. Karrmann's result about the L^p-spectral independence and B.A. Barnes' theorem about the spectrum of integral operators. As an application, the L^p-spectral independence of −[(−Δ)^α+V] (α(0,1]) for a suitable V is proved with the help of a recent result by V. Liskevich, H. Vogt and J. Voigt [V. Liskevich, H. Vogt, J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006) 245–277].     

Algebraic aspects of an ordered band arising in nonlinear signal processing

For any positive integers M and N we define a selfdually ordered band ℬ(M,N) of cardinality and ask whether or not it is lattice-ordered. The origin of ℬ(M,N) in nonlinear signal processing is outlined.     

Frequency modules and nonexistence of quasi-periodic solutions of nonlinear evolution equations

This paper gives lower estimates for the frequency modules of almost periodic solutions to equations of the form , where A generates a strongly continuous semigroup in a Banach space , F(t,x) is 2π-periodic in t and continuous in (t,x), and f is almost periodic. We show that the frequency module ℳ(u) of any almost periodic mild solution u of (*) and the frequency module ℳ(f) of f satisfy the estimate e ^{2π iℳ(f)}⊂e ^{2π iℳ(u)}. If F is independent of t, then the estimate can be improved: ℳ(f)⊂ℳ(u). Applications to the nonexistence of quasi-periodic solutions are also given.     

Projectively condensed semigroups,generalized completely regular semigroups and projective orthomonoids

The class of projectively condensed semigroups is a quasivariety of unary semigroups, the class of projective orthomonoids is a subquasivariety of . Some well-known classes of generalized completely regular semigroups will be regarded as subquasivarieties of . We give the structure semilattice composition and the standard representation of projective orthomonoids, and then obtain the structure theorems of various generalized orthogroups. Partially supported by a UGC (HK) grant #2060123 (04-05).     

半群上的Green(e)-关系

半群上的Green(e)-关系是半群上通常Green's关系的一种推广.借助半群的左(右)S-系及半群的双系深入研究了Green(e)-关系的代数性质,证明了每个H(e)-类R(e)e∩L(e)f为一个强无挠的(H(e)e,H(e)f)-双系,其中e,f为幂等元,并给出了每个含幂等元的D(e)-类的代数结构.     

On Robustness of Orbit Spaces for Partially Hyperbolic Endomorphisms

In this paper, the robustness of the orbit structure is investigated for a partially hyperbolic endomorphism f on a compact manifold M. It is first proved that the dynamical structure of its orbit space(the inverse limit space) M~f of f is topologically quasi-stable under C~0-small perturbations in the following sense: For any covering endomorphism g C~0-close to f, there is a continuous map φ from M~g to Multiply form -∞ to ∞ M such that for any {y_i }_(i∈Z) ∈φ(M~g), y_(i+1) and f(y_i) differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense: For any pseudo-orbit {x_i }_(i∈Z),there is a sequence of points {y_i }_(i∈Z) tracing it, in which y_(i+1) is obtained from f(y_i) by a motion along the center direction.     

On attractors and stability for semigroup actions and control systems

The present paper is dedicated to the study of various aspects of attraction and stability for semigroup actions on topological spaces. The main purpose is to present the connections among the distinct notions of attractors and stable sets. The concept of Conley attractor is also investigated and related to the other notions of attractors. All the results are applied to the theory of control systems.     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

A Note on a Class of Strongly Coupled Semilinear Reaction-diffusion Equations

A class of strongly coupled semilinear reaction-diffusion equations is discussed. The FitzHugh-Nagumo equation in [1] is only an example. The results that strongly coupled term generates an analytic semigroup extended the results in [4]. The results of the existence and uniqueness of solution and the stability of zero solution improved that in [1]. This paper also discusses the existence of periodic solution.     

紧拓扑半群上概率测度的简单半群及其支撑集

本文讨论一类拓扑半群上概率测度的极限性质．首先在紧半群上研究测度的简单半群和它的支撑集的相依关系;然后讨论测度的卷积幂ｕｎ收敛到Ｈａａｒｒ测度的充要条件．