There has been much interest in algebras which have a basis consisting of diagrams, which are multiplied in some natural diagrammatic way. Examples of these so-called diagram algebras include the partition, Brauer and Temperley-Lieb algebras. These three examples all have the property that the product of two diagram basis elements is always a scalar multiple of another basis element. Motivated by this observation, we find that these algebras are examples of twisted semigroup algebras. Such algebras are an obvious extension of twisted group algebras, which arise naturally in various contexts; examples include the complex numbers and the quaternions, considered as algebras over the real numbers. The concept of a cellular algebra was introduced in a famous paper of Graham and Lehrer; an algebra is called cellular if it has a basis of a certain form, in which case the general theory of cellular algebras allows us to easily derive information about the semisimplicity of the algebra and about its representation theory, even in the non-semisimple case. Many diagram algebras (including the above three examples) are known to be cellular. The aim of this thesis is to deduce the cellularity of these examples (and others) by proving a general result about the cellularity of twisted semigroup algebras. This will extend a recent result of East. In Chapters 2 and 3 we discuss semigroup theory and twisted semigroup algebras, and realise the above three examples as twisted semigroup algebras. Chapters 4 to 7 detail and extend slightly the theory of cellular algebras. In Chapter 8 we state and prove the main theorem, which shows that certain twisted semigroup algebras are cellular. Under the assumptions of the main theorem, we explore the cell representations of twisted semigroup algebras in Chapter 9. Finally in Chapter 10, we apply the theorem to various examples, including the three diagram algebras mentioned above.

PhD / None.

This dissertation will seek to explore if the properties of some of the key results on semigroups and their compacti cations under the operation of addition also apply under the operation of multiplication. Consider- able emphasis will be placed on the semigroup N of the set of natural numbers and its compacti cation N. Furthermore, the dissertation will discuss the smallest ideal of a semi- group and highlight some of its fundamental properties. / Mathematics / M. Sc. (Mathematics)

No description available.

In this paper we present an exponential stability result for a diffusion equation arising from dumbbell models for polymer flow. Using the methods of semigroup theory, we show that the semigroup U(t) associated with the diffusion equation is well defined and that all solutions converge exponentially to an equilibrium solution. Both finitely and infinitely extensible dumbbell models are considered. The main tool in establishing stability is the proof of compactness of the semigroup. / Ph. D.

This dissertation is devoted to investigating the time dependent neutron transport equations with reflecting boundary conditions. Two typical geometries --- slab geometry and spherical geometry --- are considered in the setting of <I>L^p</I> including <I>L^1</I>. Some aspects of the spectral properties of the transport operator <I>A</I> and the strongly continuous semigroup <I>T(t)</I> generated by <I>A</I> are studied. It is shown under fairly general assumptions that the accumulation points of { m Pas}(A):=sigma (A) cap { lambda :{ m Re}lambda > -lambda^{ast} }, if they exist, could only appear on the line { m Re}lambda =-lambda^{ast}, where lambda^{ast} is the essential infimum of the total collision frequency. The spectrum of <I>T(t)</I> outside the disk {lambda : |lambda| leq exp (-lambda^{ast} t)} consists of isolated eigenvalues of <I>T(t)</I> with finite algebraic multiplicity, and the accumulation points of sigma (T(t)) igcap{ lambda : |lambda| > exp (-lambda^{ast} t)}, if they exist, could only appear on the circle {lambda :|lambda| =exp (-lambda^{ast} t)}. Consequently, the asymptotic behavior of the time dependent solution is obtained. / Ph. D.

<p>Definisane su neke klase polugrupa koje su uop&scaron;tenja tzv. antiinverznih polugrupa. Za neke op&scaron;te klase polugrupa nađene su baze u smislu Ljapina. Dati su identiteti algebri koje se potapaju u polumreže.</p> / <p>Some new classes of semigroups are defined which are generalizations of so called antiinverse semigroups. The Lyapin bases of some well known classes of semigroups are determined. The identities of the variety of subalgebras of semilattices are determined.</p>

An integral domain D is weakly integrally closed if whenever there is an element x in the quotient field of D and a nonzero finitely generated ideal J of D such that xJ J2, then x is in D. We define weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of finitely many 0's and 1's is forbidden if whenever the characteristic binary string of a numerical monoid M contains F, then M is not weakly integrally closed. Any stretch of the pattern 11011 is forbidden. A numerical monoid M is weakly integrally closed if and only if it has a forbidden pattern. For every finite set S of forbidden patterns, there exists a monoid that is not weakly integrally closed and that contains no stretch of a pattern in S. It is shown that particular monoid algebras are weakly integrally closed. / by Mary E. Hopkins. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

O propósito deste trabalho é estudar a existência de solução periódica para problemas de oscilação não linear de barras submetidas a forças periódicas. Estudaremos concretamente dois problemas, que serão interpretados como equações diferenciais abstratas de segunda ordem cuja classe foi considerada em Ceron e Lopes [1]. Para garantir a existência de solução periódica dos problemas considerados, mostraremos que a aplicação de Poincaré S é limitada dissipativa e alfa-contração. Isso garante a existência de um atrator invariante compacto e a existência de um ponto fixo de S, o que é equivalente a existência da solução periódica. / Our aim in this work is to study the existence of periodic solution to oscillation in nonlinear problems of beams submitted to periodic forcing. We will study concretely two problems, which can be interpreted as an abstract second order diferential equation studied by Ceron and Lopes [1]. Our intention is to prove the existence of periodic solution to these problems. To this end, we will show that the Poincaré map S is uniform ultimately bounded and alpha-contraction. Thus we have the existence of invariant compact attractor, therefore S have a fixed point, which is equivalent the existence of a periodic solution.

We consider the flow of an ideal gas with internal friction and heat conduction in a layer between a fixed plane and an upper free boundary. We describe the top free surface as the graph of a time dependent function. This forces us to exclude breaking waves on the surface. For this and other reasons we need to confine ourselves to flow close to a motionless equilibrium state which is fairly easy to compute. The full equations of motion, in contrast to that, are quite difficult to solve. As we are close to an equilibrium, a linear system of equations can be used to approximate the behavior of the nonlinear system. Analytic, strongly continuous semigroups defined on a suitable Banach space X are used to determine the behavior of the linear problem. A strongly continuous semigroup is a family of bounded linear operators {T(t)} on X where 0 ≤ t < infinity satisfying the following conditions. 1. T(s+t)=T(s)T(t) for all s,t ≥ 0 2. T(0)=E, the identity mapping. 3. For each x ∈ X, T(t)x is continuous in t on [0,infinity). Then there exists an operator A known as the infinitesimal generator of such that T(t)=exp (tA). Thus, an analytic semigroup can be viewed as a generalization of the exponential function. Some estimates about the decay rates are derived using this theory. We then prove the existence of long term solutions for small initial values. It ought to be emphasized that the decay is not an exponential one which engenders significant difficulties in the transition to nonlinear stability.