Asymptotic expansions and convergence rates for Euler’s approximations of semigroups

We provide optimal bounds for errors in Euler’s approximations of semigroups in Banach algebras and of semigroups of operators in Banach spaces. Furthermore, we construct asymptotic expansions for such approximations with optimal bounds for remainder terms. The sizes of errors are controlled by smoothness properties of semigroups. In this paper we use Fourier–Laplace transforms and a reduction of the problem to the convergence rates and asymptotic expansions in the Law of Large Numbers. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09. This paper was written in 2004. In the interim, several related articles were published; let us mention [14, 13, 15].     

Profile empirical likelihood for parametric and semiparametric models

This paper introduces a profile empirical likelihood and a profile conditionally empirical likelihood to estimate the parameter of interest in the presence of nuisance parameters respectively for the parametric and semiparametric models. It is proven that these methods propose some efficient estimators of parameters of interest in the sense of least-favorable efficiency. Particularly, for the decomposable semiparametric models, an explicit representation for the estimator of parameter of interest is derived from the proposed nonparametric method. These new estimations are different from and more efficient than the existing estimations. Some examples and simulation studies are given to illustrate the theoretical results. The first author is supported by NNSF projects (10371059 and 10171051) of China. The second author is supported by a grant from The Research Grants Council of the Hong Kong Special Administrative Region, China (#HKU7060/04P). The third author is supported by the University Research Committee of the University of Hong Kong and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU7323/01M).     

Non-negative hereditary polynomials in a free *−algebra

We prove a non-negative-stellensatz and a null-stellensatz for a class of polynomials called hereditary polynomials in a free *-algebra.Partially supported by NSF, DARPA and Ford Motor Co.Partially supported by NSF grant DMS-0140112Partially supported by NSF grant DMS-0100367     

Medial permutable semigroups of the first kind

We say that a semigroup S is a permutable semigroup if the congruences of S commute with each other, that is, α○β=β○α is satisfied for all congruences α and β of S. A semigroup is called a medial semigroup if it satisfies the identity axyb=ayxb. The medial permutable semigroups were examined in Proc. Coll. Math. Soc. János Bolyai, vol. 39, pp. 21–39 (1981), where the medial semigroups of the first, the second and the third kind were characterized, respectively. In Atta Accad. Sci. Torino, I-Cl. Sci. Fis. Mat. Nat. 117, 355–368 (1983) a construction was given for medial permutable semigroups of the second [the third] kind. In the present paper we give a construction for medial permutable semigroups of the first kind. We prove that they can be obtained from non-archimedean commutative permutable semigroups (which were characterized in Semigroup Forum 10, 55–66, 1975). Research supported by the Hungarian NFSR grant No T042481 and No T043034.     

The structure of left C-rpp semigroups

This paper studies the class of left Clifford-rpp semigroups and investigates the structure of their semi-spined products and semilattice decompositions. These semigroups are generalizations of left Clifford semigroups and Clifford-rpp semigroups. We also discuss some special cases such as when a semilattice decomposition becomes a strong semilattice decomposition and a semi-spined product becomes a spined product. Communicated by Boris Schein This research is jointly supported by a grant of National Natural Science Foundation of China and a small project grant #200.600.380 of CUHK.     

Solvability of partial differential equations by meshless kernel methods

This paper first provides a common framework for partial differential equation problems in both strong and weak form by rewriting them as generalized interpolation problems. Then it is proven that any well-posed linear problem in strong or weak form can be solved by certain meshless kernel methods to any prescribed accuracy. The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101205). Robert Schaback’s research in Hong Kong was sponsored by DFG and City University of Hong Kong.     

On <Emphasis Type="Italic">U</Emphasis>-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W _U and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups. This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123)     

Critical Gevrey index for hypoellipticity of parabolic operators and Newton polygones

Summary In this paper we give geometrical expressions of the (non) hypoellipticity in Gevrey spaces of parabolic operators via Newton polygones. We also determine the critical Gevrey class for which the hypoellipticity holds.Partially supported by GNAFA, CNR, Italy.Partially supported by JSPS, Japan and a grant MM-410/94 with MES, Bulgaria.Partially supported by Chuo University special research fund.     

纯整Ehresmann半群的若干刻划

本文介绍纯整Ehresmann半群．纯整Ehresmann半群是一类特殊的U-半富足半群，我们给出了这类半群的若干刻划，并讨论了一些特殊的纯整Ehresmann半群．     

A note on the power semigroup of a completely simple semigroup

We prove the pseudovariety generated by power semigroups of completely simple semigroups is the semidirect product of the pseudovariety of block groups with the pseudovariety of right zero semigroups, and hence is decidable. This answers a question of Almeida from over 15 years ago. The author was supported in part by NSERC.     

正规Ehresmann型wrpp半群

众所周知,Clifford半群是正则半群类中的一类重要半群,本文定义正规 Ehresmann型wrpp半群,它是Clifford半群在wrpp半群类中的推广,给出了此类半群的若干刻划.     

Necessary and Sufficient Conditions for Weak No-Arbitrage in Securities Markets with Frictions

In this paper we consider a financial market model with frictions which include transaction costs, bid-ask spread and taxes. By using optimization, linear and nonlinear programming and convex programming techniques, several necessary and sufficient conditions are derived for the weak no-arbitrage. Some results on state prices are also provided. The results of this paper can provide at least some theoretical insight to the problem. This research was supported in part by a grant of the National Excellent Ph.D. Thesis Project of China (No. 200267), a grant of the National Natural Science Foundation of China (No. 10171115), a “Tenth Five-Year Plan” project of Ministry of Education of China (No. 01JA630009), a grant of the Natural Science Foundation of Guangdong Province (No. 011193), two grants from the Research Grants Council of Hong Kong (CityU 1081/02E, HKU 7139/01H), and a research grant of the University of Hong Kong.     

The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis. The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 10671025) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 102507). The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 102507) and the National Natural Science Foundation of China (No. 10671077).     

Global convoluted semigroups

Global exponentially bounded convoluted semigroups in Banach spaces are systematically treated with the help of Laplace transform. A perturbation theorem in this context is proved and some characterizations of the introduced class of analytic convoluted semigroups are obtained. Illustrative examples of generators of convoluted semigroups, including differential operators, are presented. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong stability of quasi-affine transforms of contraction semigroups and the steady-state riccati equation

This paper studies strong stability of the class of strongly continuous Hilbert space semigroups which are quasi-affine transforms of contraction semigroups. Sufficient conditions for such a semigroup to be approximately strong stable are given. Applications to the stabilizability problem of Hilbert space semigroups, using a feedback involving a solution of the steady-state Riccati equation, are made. Our key tool is the generalization of the LaSalle invariance principle by Hale.This research was supported by the United States Air Force Office of Scientific Research, Directorate of Mathematical and Information Sciences, under Grant No. AFOSR-79-0053D.     

Tight representations of semilattices and inverse semigroups

By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleanness present in the semilattice of idempotents of  . After observing that the Vagner–Preston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the regular tight representation. We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to continuous inverse semigroups, a class generalizing the E ^*-unitaries. Partially supported by CNPq.     

New higher-order equiaffine invariants

We introduce new affine invariants for smooth convex bodies. Some sharp affine isoperimetric inequalities are established for the new invariants. Partially supported by an NSERC grant and an FRDP grant. Partially supported by an NSF grant, an FRG-NSF grant and a BSF grant.     

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).     

Generic representations for the unitary group in three variables

We show that for the quasi-split unitary group in three variables every tempered packet of cuspidal automorphic representations contains a globally generic representation. Partially supported by a Minerva grant. Partially supported by NSF grant DMS 9619766 and 9988611. Partially supported by NSF grant DMS 9700950.