Maximal domains of quasi-convexity and pseudo-convexity for quadratic functions

We study quasi-convex and pseudo-convex quadratic functions on solid convex sets. This generalizes Martos' results in [12] and [13] by using Koecher's results in [8].This research was supported by Hydro—Quebec; University of Montreal; Office of Naval Research, Contract N-00014-47-A0112-0011; National Science Foundation, Grant GP 25738.     

局部紧H半群上概率测度卷积幂的弱收敛性

本文讨论局部紧Ｈ半概率测度卷积幂的弱收敛性,将紧群上的Ｋａｗａｄａ－Ｉｔｏ型结果以某种相应的形式建立到局部紧Ｈ半群,由于紧半群上的概率测度卷积幂序列必为 ,所以,不仅Ｋａｗａｄａ－Ｉｔｏ经典结果是本文的特例,而且「１」中的定理和「２」中的定理２都可以作为本文定理的推论     

Uniform semigroups and fixed point properties

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group. Research Supported by NSERC of Canada Grant No. A8227.     

E-free objects and e-locality for completely regular semigroups

We prove that the e-variety CR(H), of all completely regular semigroups whose subgroups belong to some group variety H, is e-local; that is, every regular, locally completely regular semigroupoid [with subgroups fromH] divides a completely regular semigroup [with subgroups from H], in a ‘regular’ way. In a future paper with P.G. Trotter, this theorem will be applied to semidirect products of e-varieties and to e-free E-solid regular semigroups. A key role in the proof is played by the e-free semigroups in the e-variety CR(H). We provide a solution to the ‘word problem’ in these semigroups, in the style of that for free completely regular semigroups given by Kadourek and Polàk. The solution is derived from the author's work on free products of completely regular semigroups. Communicated by F. Pastijn The author is indebted to the Australian Research Council and to National Science Foundation grant INT-8913404 for their support of this research.     

Two-Parameter Semigroups

In this paper we study compact semigroups that can be written as a product of two one-parameter semigroups. We show that such semigroups admit locally a unique factorization and can locally be embedded into a two-dimensional Lie group. The latter fact allows us to determine their local structure. We show how these results provide generalizations of a number of earlier results of this type.     

The ergodic theorem for Markov processes

Most of the material in Sections 4-5-6-8-11 has been published in [4]-[10]. We shall deal with the asymptotical behavior of the iterates of a Markov transition function. Our aim is to generalize the results about the ‘cyclic’ convergence of the iterates of a Markov matrix. Throughout the paper functional analytic methods are used and not probabilistic arguments. The report is self contained, modulo standart results from functional analysis, except for the decomposition into conservative and dissipative parts. Also we assume the existence of an invariant σ finite measure on the conservative part. This has been proved, under some restrictions, by several authors using probabilistic methods. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR) United States Air Force.     

Some subdirectly irreducible idempotent semigroups

Subdirectly irreducible idempotent semigroups were characterized in [3], and in that paper, their connection with the various equational classes of idempotent semigroups was discussed. All these results are in terms of identities, so that examples of subdirectly irreducibles in the equational classes are explicitly known only for small classes. It is easy to show from general considerations (see the last section of the present paper) that every proper equational subclass of the class of idempotent semigroups is generated (as an equational class) by one or two subdirectly irreducibles. In this paper we give an example of a subdirectly irreducible for each join irreducible equational class of idempotent semigroups, which generates the class. This list, together with known results, gives explicit examples of one or two finite subdirectly irreducibles which generate the various equational classes. Research supported by the National Research Council of Canada.     

On semigroups in which every Rees one-sided congruence is a congruence

The purpose of this paper is to examine the structure of those semigroups which satisfy one or both of the following conditions: A_r(A_ℓ): The Rees right (left) congruence associated with any right (left) ideal is a congruence. The conditions A_r and A_ℓ are generalizations of commutativity for semigroups. This paper is a continuation of the work of Oehmke [5] and Jordan [4] on H-semigroups (H for hamiltonian, a semigroup is called an H-semigroup if every one-sided congruence is a two-sided congruence). In fact the results of section 2 of Oehmke [5] are proved here under the condition A_r and/or A_ℓ and not the stronger hamiltonian condition. Section 1 of this paper is essentially a summary of the known results of Oehmke. In section 2 we examine the structure of irreducible semigroups satisfying the condition A_r and/or A_ℓ. In particular we determine all regular (torsion) irreducible semigroups satisfying both the conditions A_r and A_ℓ. This research has been supported by Grant A7877 of the National Research Council of Canada.     

Topological regular semigroups and topological inductive groupoids

The theory of topological semigroups has emerged as a separate discipline during the fifties (cf. Hofmann and Mostert [5] and Carruth, Koch and Hildebrant [3]), but the idea of topologically consistent regularity in such semigroups does not seem to have gained much attention. In this paper topological regular semigroups are introduced where the regularity is topologically consistent. The topological version of the structure theorem for regular semigroups given by Nambooripad [7] is provided here. Some special cases are also discussed. The necessary terminology on topological groupoids are available in Brown and Hardy [1] and Mackenzie [6].     

Continuous state branching semigroups

Summary This paper is concerned with Markov processes with continuous creation where the phase space is a general separable compact metric space. The transition probabilities for such a process determine a semigroup of operators acting on a function space over the collection of bounded Borel measures on the phase space. Such a semigroup is characterized by a particular convolution condition and is called a continuous state branching semigroup. A connection is established between continuous state branching semigroups and certain semigroups of nonlinear operators and then this connection is exploited to establish an existence theorem for the former.Research associated with a project in probability at Princeton University supported by the Office of Army Research.     

纯正半群上的同余扩张(一)

刻划半群上的同余及其扩张是半群的代数理论中的一个非常重要的课题．本文讨论了带上的同余的正规性和不变性以及在其Ｈａｌｌ半群上的扩张，从同余扩张的角度刻划了带上的同余的性质，给出了扩张的极大、极小同余的描述．     

Compact totally disconnected k-divisible semigroups

One of the early results [5] regarding divisibility in semigroups states that no finite non-degenerate group is divisible. A sequel to this (which in view of well-known results on compact semigroups is a generalization) is that a compact semigroup is divisible if and only if each component is a divisible subsemigroup [2]. Consequently, a finite semigroup is divisible if and only if it is an idempotent semigroup. However, it is of some interest to know which finite semigroups are k-divisible for a given positive integerk≥2. In this note we present a complete characterization of finitek-divisible semigroups, and use this along with a result of K. Numakura [8], to characterize compact totally disconnected k-divisible semigroups     

Quadratic programming with one negative eigenvalue is NP-hard

We show that the problem of minimizing a concave quadratic function with one concave direction is NP-hard. This result can be interpreted as an attempt to understand exactly what makes nonconvex quadratic programming problems hard. Sahni in 1974 [8] showed that quadratic programming with a negative definite quadratic term (n negative eigenvalues) is NP-hard, whereas Kozlov, Tarasov and Haijan [2] showed in 1979 that the ellipsoid algorithm solves the convex quadratic problem (no negative eigenvalues) in polynomial time. This report shows that even one negative eigenvalue makes the problem NP-hard.This author's work supported by the Applied Mathematical Sciences Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013. A000 and in part by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS 8920550.     

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.     

The kernel of the wreath product of semigroups

All wreath products, in the general sense, of two semigroups with completely simple kernels have completely simple kernels. Presented here are computational characterizations of the primitive idempotents of such a wreath and of the identities and inverses in subgroups of its kernel. Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant AF AFOSR 924-67.     

Semilattice decompositions of semigroups

The purpose of this paper is to develop a general theory of semilattice decompositions of semigroups from the point of view of obtaining theorems of the type: A semigroup S has propertyD if and only if S is a semilattice of semigroups having property β. As such we are able to extend the theories of Clifford [3], Andersen [1], Croisot [5], Tamura and Kimura [14], Petrich [9], Chrislock [2], Tamura and Shafer [15], Iyengar [7] and Weissglass and the author [10]. The root of our whole theory is Tamura's semilattice decomposition theorem [12, 13]. Of this, we give a new proof. The results of this paper were obtained by the author between January and July of 1971, while an undergraduate at the University of California, Santa Barbara.     

Representations of involutive semigroups

Involutive [topological] semigroups are semigroups endowed with an involutive antiautomorphism. A representation of such a semigroup means an involution preserving [weakly continuous] morphism into the algebra of bounded operators on a Hilbert space. We develop a representation theory of involutive [topological] semigroups based on positive definite functions on them. We do not generally assume the existence of an identity element. This makes the proofs of some results harder, but most results hold in this general setting. The author thanks the referee for many constructive suggestions to improve the exposition of the paper.     

Shift invariant manifolds and nonlinear analytic function theory

The ultimate goal of our campaign is to generalize a substantial collection of results in classical complex variables to highly nonlinear situations. In [BH1] and subsequent works (c. f. [BGR],[H]) it was shown how an extension of the classical Beurling-Lax-Halmos theorem to Hilbert spasces with a signed bilinear form (indefinite metric) could be regarded as the key to many theorems in complex analysis. Thus our approach to the nonlinear case is to first extend our indefinite metric Beurling-Lax-Halmos theory to nonlinear situations that is the subject of this article.Supported in part by the Air Force Office of Scientific Research, the National Science Foundation and the Office of Naval Research.