Semilattices of left reversible semigroups

Let S be a cancellative semigroup which is a semilattice of left reversible semigroups S, . This article studies the relationship between the group of quotients G of S and the groups of quotients G of S, . It is shown that G is the maximum group homomorphic image of an inverse semigroup which is a semilattice of groups G (up to isomorphism).The technique used here which involves the use of Ore's quotients also applies to the study of the maximum group homomorphic image of a semigroup which is a semilattice of inverse semigroups.     

Infinitesimal Stokes’ Formula for Higher-Order de Rham Complexes

A higher-order de Rham complex dR [14] is associated with a commutative algebra A and a sequence of positive integers = (₁₂... It is called regular if is nondecreasing. We extend the algebraic definitions of the Lie derivative and interior product with respect to a derivation of A, to higher-order differential forms. These allow us to prove a generalization of the infinitesimal Stokes formula (also known as the Cartan homotopy formula) for higher regular de Rham complexes. In particular, this implies the homotopy invariance property of higher regular de Rham cohomologies for differentiable manifolds.     

The Choquet-Deny convolution equation μ=μ*σ for probability measures on Abelian semigroups

In this note, we characterize the regular probability measures satisfying the Choquet-Deny convolution equation =* on Abelian topological semigroups for a given probability measure .     

Matrix semigroups with commutable rank

We focus on matrix semigroups (and algebras) on which rank is commutable [ rank(AB)= rank(BA)]. It is shown that in a number of cases (for example, in dimensions less than 6), but not always, commutativity of rank entails permutability of rank [rank(A₁A₂... A_n)=rank(A₍₁₎A₍₂₎... A_(n))]. It is shown that a commutable-rank semigroup has a natural decomposition as a semilattice of semigroups that have a simpler structure. While it is still unknown whether commutativity of rank entails permutability of rank for algebras, the question is reduced to the case of algebras of nilpotents.     

A characterization of subgroups of the orthogonal group

LetG be a subgroup of the general linear group GL_n(K), where charK 2. Put Kⁿ =V. AssumeG is generated by the setS of all elements inG for which dimV( – 1) = 1, and suppose ²=1_V for each inS. If {V(–1)¦S} contains a simplex, if – 1_V G, and if inG is a product of dim v(–1) elements inS wheneverV(–1) is not contained in the kernel of–1, thenG is a subgroup of an orthogonal group.This research was supported in part by NSERC Canada grant A7251.To Helmut Mäurer on his 60th birthday     

Large Deviations of a Storage Process with Fractional Brownian Motion as Input

We study probabilities of large extremes of the storage process Y(t) = sup _t (X() - X(t) - c( - t)), where X(t) is the fractional Brownian motion. We derive asymptotic behavior of the maximum tail distribution for the process on fixed or slowly increased intervals by a reduction the problem to a large extremes problem for a Gaussian field.     

Homotopy stability of orthogonal spaces in commutative banach algebras

We study the concept of real stable rank for a complex commutative Banach algebra A (rsr A). It is shown that this invariant has a behaviour completely analogous to the classical Bass stable rank; in particular, we establish a precise relation between both invariants.Further, we use this machinery to show that the connected components of the orthogonal spaces of A, O _k ⁿ (A)={ A ^k×n:^t=id_k×k}, stabilize when k and n increase in a way that depends on rsr A Finally, we prove an analogous stabilization result for the homotopy classes of maps from the j-sphere into O _k ⁿ (A), where j is an arbitrary positive integer.Research supported by a grant of the CONICET, Argentina.     

Self-intersections of 1-dimensional random walks

Summary Consider a random walk S _n on the integers, where the steps _i have mean 0 and variance ². Let T be the time of first self-intersection of the random walk. It is shown that, as , T grows at rate ^2/3. More precisely, T^2/3 has a non-degenerate limit distribution which can be described in terms of Brownian motion local time.Research supported by National Science Foundation Grant MCS80-02698.     

Biordered Sets of Regular Semigroups with Inverse Transversals1

In a regular semigroup S, an inverse subsemigroup S° of S is called an inverse transversal of S if S° contains a unique inverse x° of each element x of S. An inverse transversal S° of S is called a Q-inverse transversal of S if S° is a quasi-ideal of S.If S is a regular semigroup with set of idempotents E then E is a biordered set. T.E. Hall obtained a fundamental regular semigroup T_E from the subsemigroup E which is generated by the set of idempotents of a regular semigroup. K.S.S. Nambooripad constructed a fundamental regular semigroup by a regular biordered set abstractly. In this paper, we discuss the properties of the biordered sets of regular semigroups with inverse transversals. This kind of regular biordered sets is called IT-biordered sets. We also describe the fundamental regular semigroup T_E when E is an IT-biordered set. In the sequel, we give the construction of an IT-biordered set by a left regular IT-biordered set and a right regular IT-biordered set.This project has been supported by the Provincial Natural Science Foundation of Guangdong Province, PR China     

Semilattices of proper inverse semigroups

Given a semilattice Y of inverse semigroups S, there corresponds a semilattice Y of groups G in a natural way. This correspondence is used to study semilattices of proper inverse semigroups. In paticular, it is shown that if S is a semilattice of proper inverse semigroups, then there exists a minimum semilattice congruence such that each -class is proper and there exists a maximum semilattice congruence such that each -class is proper.     

Necessary and sufficient conditions for conservativeness of dynamical semigroups

Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let be a Hilbert space andA a von Neumann algebra. A dynamical semigroup P_t is a -weakly continuous one-parameter semigroup of completely positive maps ofA into itself. A semigroup P_t possessing the property of preserving the identityIA is said to be conservative and its infinitesimal operator L[·] is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation P=L[P] has no solutions inA ₊. Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra of (R ⁿ ) yields a series of new regularity conditions for both diffusion and jump processes.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 149–184, 1990.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

Messbare Mengen von Massen und Inhalten

Let X be a completely regular space. The customary -field is the coarsest -field on the space of Bairemeasures on X which makes (A) measurable for any Baire set A. We compare the customary -field with the Baire and Borel -field induced by the weak* topology which lies on the dual space C(X). In (2.3) it is shown that the customary -field is just the Baire -field. In part 3 necessary and sufficient conditions are given under which the set of -smooth measures is measurable with respect to the Borel -field which lies on the positive cone of the space of finitely additive, regular measures C(X). Finally, a decomposition theorem for generalized kernels is proved. The -smooth part of a generalized kernel is a kernel again if certain conditions are fulfilled.     

Stationary Values of Sectional Curvature in Grassmann Manifolds of Bivectors

In the Grassmann manifold G _2,n ⁺ of bivectors (n4), the sectional curvature K() in the direction of a 2-plane takes values in [0,2]. All stationary values a of K() such that the gradient K|₌₀ vanishes for at least one ₀ K^–1(a) are found. The values are {0,1,2} for n=4, {0,1/5,1,2} for n=5, and {0,1/5,1/2,1,2} for n6. Bibliography: 7 titles.     

Extending partial isomorphisms of graphs

Theorem Let X be a finite graph. Then there exists a finite graph Z containing X as an induced subgraphs, such that every isomorphism between induced subgraphs of X extends to an automorphism of Z.The graphZ may be required to be edge-transitive. The result implies that for anyn, there exists a notion of a genericn-tuple of automorphism of the Rado graphR (the random countable graph): for almost all automorphism ₁,..., _n and ₁ ofR (in the sense of Baire category), (R,₁,..., _n ), (R,₁,..., _n ). The problem arose in a recent paper of Hodges, Hodgkinson, Lascar and Shelah, where the theorem is used to prove the small index property forR.Work supported by a Sloan Fellowship and by NSF grant DMS-8903378.     

Group automorphisms with finite entropy

A compact metrie abelian group X with the normalized Haar measure is a Lebesgue probability space. A group automorphism ofX is an invertible measure preserving transformation of the probability space. This paper is to show that if the entropy of is finite, then there exist totally disconnected subgroupsH andN, a finite-dimensional subgroupS and a subgroupT satisfying the conditions: (i)H, N, S andT are strictly -invariant, (ii)N=HST, (iii)h(| _N )=0, (iv) ifS/N is non-trivial then it is a finite-dimensional solenoidal group with condition (**) (see the definition in §1), (v) ifT/N is non-trivial then it is connected and locally connected, such thatX/N splits into a direct sumX/N=H/NS/NT/N. This result characterizes the structure of finite entropy automorphisms.     

A characterization of plane Lorentz transformations

Denote (x_i,y_i=ct_i), i=1,2, by X_i and (x₂–x₁)²–(y₂–y₁)² by F(X₁,X₂). Then our result is the following: Given a fixed real number 0 and given a bijection of M=IR² such that F(X₁,X₂) = iff F(X _in1 ^su , _in2 ^su ) =p for all X₁, X₂ M. Then must be a Lorentz transformation (time reversal and inhomogeneity included).     

Abundant Left C-lpp Proper Semigroups

The aim of this paper is to study a class of left abundant semigroups, so-called abundant left C-lpp proper semigroups including left type A proper semigroups and right inverse proper semigroups as its subclasses. A structure theorem similar to McAlisters for inverse proper semigroups is obtained. As its application, it is verified that any abundant left C-lpp proper semigroup can be embedded into a semidirect product of a left regular band by a cancellative monoid.AMS Subject Classification (1991): 20M10Suppoted by the Foundation of Yunnan University and also by the Director Foundation of Yunnan Province.     

Universal measurability of the identity mapping of a Banach space in certain topologies

If X is a Banach space and X is its conjugate, then a subset Y of X is called madmissible for X if a) the topology (X, Y) is Hausdorff, b) the identity embedding of (X, (X, Y)) into X is universally measurable (Ref. Zh. Mat., 1975, 8B 75 8K). If X is separable, then the existence of an m-admissible set is well known. In this note it is shown that there exist nonseparable X having separable m-admissible sets. The properties of spaces with separable m-admissible sets are considered. It is proved, in particular, that a separable normalizing subset Y of X is m-admissible for X if and only if every (X, Y)-compact set is separable in X.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 305–314, February, 1978.     

Metric and Complete Metric σ-Frames

A -frame is a lattice in which countable joins exist and binary meets distribute over countable joins. In this paper, the category MFrm, of metric -frames, is introduced, and it is shown to be equivalent to the category MLFrm _u, of metric Lindelöf frames.Finally, it is shown that the complete metric -frames are exactly the cozero parts of complete metric Lindelöf frames.