Contributions to the theory of O-simple inverse semigroups.

The structure of O-bisimple and bisimple inverse semigroups has been extensively studied and established by Clifford, Reilly, Warne, Munn and McAlister. The initial work was done by Clifford in [0] on bisimple inverse semigroups with an identity and this was generalised by Reilly and Clifford in [13]. In [5] McAlister has produced a structure theorem for O-bisimple inverse semigroups in terms of groups and semilattices which can be specialised to give most of the previously known results in this area. These include the result of Munn described in [10] and the result of Reilly in [12], which deals with bisimple inverse semigroups whose semilattices are order isomorphic to the non negative integers with the reverse of the natural ordering, i.e. semilattices which are w-chains. Warne has made a study of those bisimple inverse semigroups whose semilattices are order isomorphic to the integers with the reverse of the natural ordering and has obtained in [14] a structure for these which ties closely with [12]

A class of abstract quasi-linear evolution equations of second order

In this paper we study the abstract quasi-linear evolution equation of second order formula here in a general banach space z. it is well-known that the abstract quasi-linear theory due to kato [10, 11] is widely applicable to quasi-linear partial differential equations of second order and that his theory is based on the theory of semigroups of class (C0). (for example, see the work of hughes et al. [9] and heard [8].) however, even in the special case where a (t,w, v) = a is independent of (t, w, v), it is found in [2] and [14] that there exist linear partial differential equations of second order for which cauchy problems are not solvable by the theory of semigroups of class (C0) but fit into the mould of well-posed problems where the solution and its derivative depend continuously on the initial data if the initial condition is measured in the graph norm of a suitable power of a. (see also work by krein and khazan [13] and fattorini [6, chapter 8].) this kind of cauchy problem has recently been studied extensively, using the theory of integrated semigroups or regularized semigroups. the theory of integrated semigroups was studied intensively by arendt [1] and that of regularized semigroups was initiated by da prato [3] and renewed by davies and pang [4]. for the theory of regularized semigroups we refer the reader to [5] and [16]. (u(t),v(t))' = Ãu(t)(u(t),v(t)) for t&#8712;[0,T] and (u(0),v(0)) = (&#966;,&#968;) in a suitable Banach space X, where for each solution w of equation (1.1) the matrix operator Aw(t) in X is defined by Aw(t)(u,v)=(v,A(t,w(t),w'(t)) u). We are here interested in studying the case where each matrix operator Aw(t) is the (complete infinitesimal) generator of a regularized semigroup on X. In Section 3 we set up basic hypotheses on the operators appearing in equation (1.1), and prove a fundamental existence and uniqueness theorem (Theorem 3.6) for the Cauchy problem (1.1). The proof is based on the theory of regularized evolution operators developed by the author [15], and a method of successive approximations proposed by Kobayasi and Sanekata [12] is applied to construct a unique twice continuously differentiable function u satisfying equation (1.1). Our formulation includes the abstract quasi-linear wave equation of Kirchhoff type u&#34;(t)+­m(|A1/2u(t)|2)Au(t)=0 (1.2) in a real Hilbert space H, where A is a nonnegative selfadjoint operator in H. Section 4 presents a regularized semigroup theoretical approach to the local solvability of equation (1.2) in the `degenerate case' where the function m(r) has zeros (Theorems 4.1 and 4.2), by using the result obtained in Section 3. In Section 2 we summarize some results on the generation of a regularized evolution operator associated with the linearized equation of (1.1), under the `regularized stability ' condition, and show that the family of matrix operators used to solve the linearized equation (1.2) satisfies the regularized stability condition. This fact will be useful for our arguments in Section 4.</p

Relaxation in time elapsed neuron network models in the weak connectivity regime

In order to describe the firing activity of a homogenous assembly of neurons, we consider time elapsed models, which give mathematical descriptions of the probability density of neurons structured by the distribution of times elapsed since the last discharge. Under general assumption on the firing rate and the delay distribution, we prove the uniqueness of the steady state and its nonlinear exponential stability in the weak connectivity regime. The result generalizes some similar results obtained in [10] in the case without delay. Our approach uses the spectral analysis theory for semigroups in Banach spaces developed recently by the first author and collaborators

Convoluted CC-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines

Convoluted $C$-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated $C$-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted $C$-cosine functions and analytic convoluted $C$-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convoluted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator $(-\Delta)^{2^{n}},$ $n\in {\mathbb N},$ acting on $L^{2}[0,\pi]$ with appropriate boundary conditions, generates an exponentially bounded $K_{n}$-convoluted cosine function, and consequently, an exponentially bounded analytic $K_{n+1}$-convoluted semigroup of angle $\frac{\pi}{2},$ for suitable exponentially bounded kernels $K_{n}$ and $K_{n+1}.

The number of nilpotent semigroups of degree 3

A semigroup is \emph{nilpotent} of degree 3 if it has a zero, every product of 3 elements equals the zero, and some product of 2 elements is non-zero. It is part of the folklore of semigroup theory that almost all finite semigroups are nilpotent of degree 3. We give formulae for the number of nilpotent semigroups of degree 3 with $n\in\N$ elements up to equality, isomorphism, and isomorphism or anti-isomorphism. Likewise, we give formulae for the number of nilpotent commutative semigroups with $n$ elements up to equality and up to isomorphism

Extending the Ehresmann-Schein-Nambooripad Theorem

We extend the `join-premorphisms' part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the `morphisms' part. However, it is so-called `meet-premorphisms' which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for meet-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.Comment: 23 pages; final section on Szendrei expansions removed; further reordering of materia