ON THE SPECTRAL CHARACTERIZATION OF A STRONGLY CONTINUOUS PERTURBED SEMIGROUP

In this paper, we get a sufficient condition to estimate the essential type of a strongly continuous sendgroup in a Banach space, using oh condition we prove that the essential type of the strongly continuous semigroup don't increase under the A-smoothing perturbation and establish the spectral mapping theorem for the asymptotic parts of the spectrum of generator and semgroup.

Characteristic conditions-for strongly continuous groups of linear operators on a Hilbert space

IN 1932, Stone [1] gave the first result concerning C0 semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C0 group of unitary operators on a Hilbert space if and only if it is skew-adjoint. This result has been applied extensively to linear partial differential equations （PDEs） with a law of conservation. Hille [2] then discovered the generation theorem of the Hille-Yosida type for a C0 group on a Banach space. There are also some conditions under which a C0 semigroup on a Banach space can be embedded in a C0 group. We refer the readers to ref. [3] for details of the results mentioned above.

Rothe’s Fixed Point Theorem and the Controllability of the Benjamin-Bona-Mahony Equation with Impulses and Delay

For many control systems in real life, impulses and delays are intrinsic phenomena that do not modify their controllability. So we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system do not destroy its controllability. There are many practical examples of impulsive control systems with delays, such as a chemical reactor system, a financial system with two state variables, the amount of money in a market and the savings rate of a central bank, and the growth of a population diffusing throughout its habitat modeled by a reaction-diffusion equation. In this paper we apply the Rothe’s Fixed Point Theorem to prove the interior approximate controllability of the following Benjamin Bona-Mohany(BBM) type equation with impulses and delay where and are constants, Ω is a domain in , ω is an open non-empty subset of Ω , denotes the characteristic function of the set ω , the distributed control , are continuous functions and the nonlinear functions are smooth enough functions satisfying some additional conditions.

The Strong Continuity Hypothesis: Evidence from Arabic-speaking children data

This paper examines the acquisition of subject-verb agreement inflections in the natural speech corpus of two mono-lingual children speaking Yemeni Ibbi Arabic(YIA)between 2 and 3 years old.The two children are Ibrahim and Wala(between the age of 2;1 and 2;10)with Mean Lenghth of Utterance(MLU)range of 2.72 to 3.23 for Ibrahim and 2.9 to 3.27 for Wala.YIA,as a variety of Arabic,has rich and complex morphological system with a fusional type.Verbs are inflected with tense and agreement.Each verbal inflection is marked for person,number,and gender agreement.However,this paper attempts to explore how agreement forms are acquired by YIA-speaking children and examines when YIA children distinguish between first,second,and third person agreement,singular and plural,masculine and feminine agreement forms.The paper argues that agreement inflections(person,number,gender)are available to children early,thereby supporting the Strong Continuity Hypothesis(Lust,1999).Moreover,the results give evidence to Wexler’s Hypothesis(1998),Very Early Knowledge of Inflection(VEKI),which says that children know the grammatical and phonological properties of inflections in a language in the earliest stages when they enter the two-word stage.Similarly,this study tests Hoekstra and Hyams’(1995)Early Morpho-syntactic Convergence(EMC)which proposed that children acquire the specifics of inflections of the target language at an early stage.

SPECTRA OF COMPOSITION GROUPS ON THE WEIGHTED DIRICHLET SPACE OF THE UPPER HALF-PLANE

We prove that the group of weighted composition operators induced by continuous automorphism groups of the upper half plane U is strongly continuous on the weighted Dirichlet space of U,Dα(U).Further,we investigate when they are isometries on Dα(U).In each case,we determine the semigroup properties while in the case that the induced composition group is an isometry,we apply similarity theory to determine the spectral properties of the group.

ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

Suppose that X is a right process which is associated with a semi-Dirichlet form （ε, D（ε）） on L2（E; m）. Let J be the jumping measure of （ε, D（ε）） satisfying J（E x E- d） 〈 ∞. Let u E D（ε）b ：= D（ε） N L（E; m）, we have the following Pukushima＇s decomposition u（Xt）-u（X0） --- Mut ＋ Nut. Define Pu f（x） = Ex[eNT f（Xt）]. Let Qu（f,g） = ε（f,g）＋ε（u, fg） for f, g E D（ε）b. In the first part, under some assumptions we show that （Qu, D（ε）b） is lower semi-bounded if and only if there exists a constant a0 〉 0 such that /Put/2 ≤eaot for every t 〉 0. If one of these assertions holds, then （Put〉0is strongly continuous on L2（E;m）. If X is equipped with a differential structure, then under some other assumptions, these conclusions remain valid without assuming J（E x E - d） 〈 ∞. Some examples are also given in this part. Let At be a local continuous additive functional with zero quadratic variation. In the second part, we get the representation of At and give two sufficient conditions for PAf（x） = Ex[eAtf（Xt）] to be strongly continuous.

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.

Feller property and exponential ergodicity of diffusion processes with state-dependent switching

In this paper we consider the Feller property and the exponential ergodicity for general diffusion processes with state-dependent switching. We prove their Feller continuity by means of intro- ducing some auxiliary processes and by making use of the Radon-Nikodym derivatives. Furthermore, we also prove their strong Feller continuity and their exponential ergodicity under some reasonable conditions.