Construction and regularity of transition functions on polish spaces under measurability conditions

This paper concerns the construction and regularity of a transition (probability) function of a non-homogeneous continuous-time Markov process with given transition rates and a general state space. Motivating from a lot of restriction in applications of a transition function with continuous (in t≥0) and conservative transition rates q(t, x, Λ), we consider the case that q(t,x,Λ) are only required to satisfy a mild measurability (in t≥0) condition, which is a generalization of the continuity condition. Under the measurability condition we construct a transition function with the given transition rates, provide a necessary and sufficient condition for it to be regular, and further obtain some interesting additional results.     

Boundedness from below of composition operators on the Bloch spaces

This paper proves that a necessary condition for a composition operator on the Bloch space of the unit disk to be bounded below given by Ghatage, Yan and Zheng is also sufficient. Given furthermore are a sufficient condition and a necessary condition of the boundedness from below for a composition operator on the Bloch space of a unit ball with dimensions bigger than one.     

Quasi-d-Koszul Modules and Applications

Let R be a Noetherian semiperfect algebra. A necessary and sufficient condition for a finitely generated R-module to be quasi-d-Koszul is given, which improves one of the main results in [1]. The authors also give a necessary and sufficient condition for the Minimal Horseshoe Lemma holding in mod（R）. As an application, it is proved that the ＂Minimal Horseshoe Lemma＂ is true in the category of quasi-d-Koszul modules under certain conditions.     

THE FINITE INVARIANT MEASURES AND THE REVERSIBLE MEASURES OF THE DIFFUSIONS WITH THE VENTZEL''''S BOUNDARY CONDITIONS IN THE CIRCULAR DISC

For a given rotation-invariant operator Ω in the unit disc, the necessary condition for thef xistence of finite invariant measures of the strong Markov transition functions determined by Ω with the Ventzel's boundary conditions of finite rank is established If only rotation-invariant transition functions are considered, then the necessary and sufficient conditions for a given measure to be invariant and to be reversible, and all the cor6esponding boundary conditions for posseusiug these properties have been found explicitly in this paper.     

The spectrum sequences of periodic frame multiresolution analysis

The periodic multiresolution analysis （PMRA） and the periodic frame multiresolution analysis （PFMRA） provide a general recipe for the construction of periodic wavelets and periodic wavelet frames, respectively. This paper addresses PFMRAs by the introduction of the notion of spectrum sequence. In terms of spectrum sequences, the scaling function sequences generating a normalized PFMRA are characterized; a characterization of the spectrum sequences of PFMRAs is obtained, which provides a method to construct PFMRAs since its proof is constructive; a necessary and sufficient condition for a PFMRA to admit a single wavelet frame sequence is obtained; a necessary and sufficient condition for a PFMRA to be contained in a given PMRA is also obtained. What is more, it is proved that an arbitrary PFMRA must be contained in some PMRA. In the meanwhile, some examples are provided to illustrate the general theory.     

A note on characterization of Hardy space <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>

An all-round answer is given to the following question: "Find a necessary and sufficient size condition for an integrable compactly supported function on Rn with mean value zero to be in the Hardy space H1." Stefanov answers it only for n = 1. Equivalent answer is also given for n = 1.     

Projection on Segre varieties and determination of holomorphic mappings between real submanifolds Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given.     

GLOBAL WEAK SHARP MINIMA AND COMPLETENESS OF METRIC SPACE

A sufficient condition on the existence of a global weak sharp minima for general function in metric space is established. A characterization for convex function to have global weak sharp minima is also presented, which generalized Burke and Ferris‘ result to infinite dimensional space. A characterization of the completeness of a metric space is given by the existence of global weak sharp minima.     

PM环的扩张

A ring is called PM-ring if every primitive ideal is a intersection of its some maximal ideals.In this paper,the property of PM-ring is discussed and the necessary and sufficient condition are given for the polynomial,centralizing,strongly normal extensions of PM-ring to be PM-rings.We also get some applications of the extension of PM-ring in group ring.     

The Partial Stability for Linear Discrete Systems

In this paper, we study the partial stability of linear discrete systems by means of Liapunov's functions of quadratic form. We obtain a necessary and sufficient condition for the system being stable with respect to part of variables and generalize Liapunov's equation to the partial stability of linear discrete systems. A method of constructing Liapunov's function of quadratic form for the stability of the systems is given.     

Column continuous transition functions

A column continuous transition function is by definition a standard transition function P(t) whose every column is continuous for t?0 in the norm topology of bounded sequence space l_∞. We will prove that it has a stable q-matrix and that there exists a one-to-one relationship between column continuous transition functions and increasing integrated semigroups on l_∞. Using the theory of integrated semigroups, we give some necessary and sufficient conditions under which the minimal q-function is column continuous, in terms of its generator (of the Markov semigroup) as well as its q-matrix. Furthermore, we will construct all column continuous Q-functions for a conservative, single-exit and column bounded q-matrix Q. As applications, we find that many interesting continuous-time Markov chains (CTMCs), say Feller-Reuter-Riley processes, monotone processes, birth-death processes and branching processes, etc., have column continuity.     

随机单调Markov积分半群

讨论了Markov积分半群的单调性和转移函数的单调性的等价性,并得到最小的Q半群是单调的充要条件.     

Weighted norm inequalities for heat-diffusion Laguerre’s semigroups

We consider three systems of Laguerre functions and their corresponding heat diffusion semigroups. For the associate maximal operators, we give necessary and sufficient conditions in order to obtain strong type, weak type and restricted weak type (p,p), with respect to a power weight x ^δ, for 1 ≤ p ≤ ∞. We also obtain sufficient conditions for more general weights. Both authors were supported by grants from CONICET and Universidad Nacional del Litoral (Argentina).     

Stabilities of Hilbert space contraction semigroups revisited

A necessary and sufficient condition for exponential stability of Hilbert space contraction semigroups is obtained in terms of an inequality involving the dissipative norm associated with the generator of the semigroup and with the Hilbert space norm.     

Markov Integrated Semigroups and their Applications to Continuous-Time Markov Chains

A Markov integrated semigroup G(t) is by definition a weaklystar differentiable and increasing contraction integrated semigroup on l _∞. We obtain a generation theorem for such semigroups and find that they are not integrated C ₀-semigroups unless the generators are bounded. To link up with the continuous-time Markov chains (CTMCs), we show that there exists a one-to-one relationship between Markov integrated semigroups and transition functions. This gives a clear probability explanation of G(t): it is just the mean transition time, and allows us to define and to investigate its q-matrix. For a given q-matrix Q, we give a criterion for the minimal Q-function to be a Feller-Reuter-Riley (FRR) transition function, this criterion gives an answer to a long-time question raised by Reuter and Riley (1972). This research was supported by the China Postdoctoral Science Foundation (No.2005038326).     

Markov积分算子半群的限制及关于增加积分算子半群的生成

证明了转移函数是l∞的一个子空C^1上的正的压缩C0半群，其极小生成元恰好是Markov积分算子半群的生成元在C^1中的部分；Markov积分算子半群的生成元稠定的充分必要条件是q-矩阵Q一致有界；同时转移函数是Feller—Reuter—Riley的充要条件是Markov积分算子半群的生成元在Cn中的部分产生一个强连续半群．最后，在序Banach空间给出了增加的压缩积分算子半群的生成定理．     

Stabilizability of two classes of contraction semigroups

This paper studies the strong stabilizability of two classes of Hilbert space contraction semigroups: (i) strict contraction semigroups, which include those with strictly dissipative generators; and (ii) isometric or unitary semigroups. The former class is already weakly stable, while the latter is not strongly stable over the whole space. Our tool is the functional model of Hilbert space contractions; hence, strong stability of the semigroup is studied via stability of its cogenerator. It is shown that a strict contraction semigroup is, in general, not strongly stabilized by the feedback –B*, while an isometric or a unitary semigroup is strongly stabilized by the same feedback, providedB is not compact.     

A Characterisation of a Class of Semigroups with Locally Commuting Idempotents

McAlister proved that a necessary and sufficient condition for a regular semigroup S to be locally inverse is that it can be embedded as a quasi-ideal in a semigroup T which satisfies the following two conditions: (1) T = TeT, for some idempotent e; and (2) eTe is inverse. We generalise this result to the class of semigroups with local units in which all local submonoids have commuting idempotents.