[EN] The purpose of the Ph.D. Thesis "Dynamics of strongly continuous semigroups associated to certain differential equations'' is to analyse, from the point of view of functional analysis, the dynamics of solutions of linear evolution equations. These solutions can be represented by a strongly continuous semigroup on an infinite-dimensional Banach space. The aim of our research is to provide global conditions for chaos, in the sense of Devaney, and stability properties of strongly continuous semigroups which are solutions of linear evolution equations. This work is composed of three principal chapters. Chapter 0 is introductory and defines basic terminology and notation used, besides presenting the basic results that we will use throughout this thesis. Chapters 1 and 2 describe, in general way, a strongly continuous semigroup induced by a semiflow in Lebesgue and Sobolev spaces which is a solution of a linear first order partial differential equation. Moreover, some characterizations of the main dynamical properties, including hypercyclicity, mixing, weakly mixing, chaos and stability are given along these chapters. Chapter 3 describes the dynamical properties of a difference equation based on the so-called birth-and-death model and analyses the conditions previously proven for this model improving them by employing a different strategy. The goal of this thesis is to characterize dynamical properties of these kind of strongly continuous semigroups in a general way, whenever possible, and to extend these results to another spaces. Along this memory, these findings are compared with the previous ones given by many authors in recent years. / [ES] La presente memoria "Dinámica de semigrupos fuertemente continuos asociadas a ciertas ecuaciones diferenciales'' es analizar, desde el punto de vista del análisis funcional, la dinámica de las soluciones de ecuaciones de evolución lineales. Estas soluciones pueden ser representadas por semigrupos fuertemente continuos en espacios de Banach de dimensión infinita. El objetivo de nuestra investigación es proporcionar condiciones globales para obtener caos, en el sentido de Devaney, y propiedades de estabilidad de semigrupos fuertemente continuos, los cuales son soluciones de ecuaciones de evolución lineales. Este trabajo está compuesto de tres capítulos principales. El Capítulo 0 es introductorio y define la terminología básica y notación usada, además de presentar los resultados básicos que usaremos a lo largo de esta tesis. Los Capítulos 1 y 2 describen, de forma general, un semigrupo fuertemente continuo inducido por un semiflujo en espacios de Lebesgue y en espacios de Sobolev, los cuales son solución de una ecuación diferencial lineal en derivadas parciales de primer orden. Además, algunas caracterizaciones de las principales propiedades dinámicas, incluyendo hiperciclicidad, mezclante, débil mezclante, caos y estabilidad, se obtienen a lo largo de estos capítulos. El Capítulo 3 describe las propiedades dinámicas de una ecuación en diferencias basada en el llamado modelo de nacimiento-muerte y analiza las condiciones previamente probadas para este modelo, mejorándolas empleando una estrategia diferente. La finalidad de esta tesis es caracterizar propiedades dinámicas para este tipo de semigrupos fuertemente continuos de forma general, cuando sea posible, y extender estos resultados a otros espacios. A lo largo de esta memoria, estos resultados son comparados con los resultados previos dados por varios autores en años recientes. / [CA] La present memòria "Dinàmica de semigrups fortament continus associats a certes equacions diferencials'' és analitzar, des del punt de vista de l'anàlisi funcional, la dinàmica de les solucions d'equacions d'evolució lineals. Aquestes solucions poden ser representades per semigrups fortament continus en espais de Banach de dimensió infinita. L'objectiu de la nostra investigació es proporcionar condicions globals per obtenir caos, en el sentit de Devaney, i propietats d'estabilitat de semigrups fortament continus, els quals són solucions d'equacions d'evolució lineals. Aquest treball està compost de tres capítols principals. El Capítol 0 és introductori i defineix la terminologia bàsica i notació utilitzada, a més de presentar els resultats bàsics que utilitzarem al llarg d'aquesta tesi. Els Capítols 1 i 2 descriuen, de forma general, un semigrup fortament continu induït per un semiflux en espais de Lebesgue i en espais de Sobolev, els quals són solució d'una equació diferencial lineal en derivades parcials de primer ordre. A més, algunes caracteritzacions de les principals propietats dinàmiques, incloent-hi hiperciclicitat, mesclant, dèbil mesclant, caos i estabilitat, s'obtenen al llarg d'aquests capítols. El Capítol 3 descrivís les propietats dinàmiques d'una equació en diferències basada en el model de naixement-mort i analitza les condicions prèviament provades per aquest model, millorant-les utilitzant una estratègia diferent. La finalitat d'aquesta tesi és caracteritzar propietats dinàmiques d'aquest tipus de semigrups fortament continus de forma general, quan siga possible, i estendre aquests resultats a altres espais. Al llarg d'aquesta memòria, aquests resultats són comparats amb els resultats previs obtinguts per diversos autors en anys recents. / Aroza Benlloch, J. (2015). Dynamics of strongly continuous semigroups associated to certain differential equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/57186

A Boolean matrix is a matrix with elements from the Boolean semiring ({0, 1}, +, x), where the addition and multiplication are as usual with the exception that 1 + 1 = 1. In this thesis we study eight classes of monoids whose elements are Boolean matrices. Green's relations are five equivalence relations and three pre-orders which are defined on an arbitrary monoid M and describe much of its structure. In the monoids we consider the equivalence relations are uninteresting - and in most cases completely trivial - but the pre-orders are not and play a vital part in understanding the structure of the monoids. Each of the three pre-orders in each of the eight classes of monoids can be viewed as a computational decision problem: given two elements of the monoid, are they related by the pre-order? The main focus of this thesis is determining the computational complexity of each of these twenty-four decision problems, which we successfully do for all but one.

PhD / None.

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.

Let S and T be numerical semigroups and let k be a positive integer. We say that S is the quotient of T by k if an integer x belongs to S if and only if kx belongs to T. Given any integer k larger than 1 (resp., larger than 2), every numerical semigroup S is the quotient T/k of infinitely many symmetric (resp., pseudo-symmetric) numerical semigroups T by k. Related examples, probabilistic results, and applications to ring theory are shown. Given an arbitrary positive integer k, it is not true in general that every numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by k. In fact, a numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by each positive integer k larger than 1 if and only if S is itself of maximal embedding dimension. Nevertheless, for each numerical semigroup S, for all sufficiently large positive integers k, S is the quotient of a numerical semigroup of maximal embedding dimension by k. Related results and examples are also given.

A compact semigroup is, roughly, a semigroup compactification of a locally compact group if it contains a dense homomorphic image of the group. The theory of semigroup compactifications has been developed in connection with subalgebras of continuous bounded functions on locally compact groups. The Eberlein algebra of a locally compact group is defined to be the uniform closure of its Fourier-Stieltjes algebra. In this thesis, we study the semigroup compactification associated with the Eberlein algebra. It is called the Eberlein compactification and it can be constructed as the spectrum of the Eberlein algebra. The algebra of weakly almost periodic functions is one of the most important function spaces in the theory of topological semigroups. Both the weakly almost periodic functions and the associated weakly almost periodic compactification have been extensively studied since the 1930s. The Fourier-Stieltjes algebra, and hence its uniform closure, are subalgebras of the weakly almost periodic functions for any locally compact group. As a consequence, the Eberlein compactification is always a semitopological semigroup and a quotient of the weakly almost periodic compactification. We aim to study the structure and complexity of the Eberlein compactifications. In particular, we prove that for certain Abelian groups, weak^{*}-closed subsemigroups of L^{\infty}[0, 1] may be realized as quotients of their Eberlein compactifications, thus showing that both the Eberlein and weakly almost periodic compactifications are large and complicated in these situations. Moreover, we establish various extension results for the Eberlein algebra and Eberlein compactification and observe that levels of complexity of these structures mimic those of the weakly almost periodic ones. Finally, we investigate the structure of the Eberlein compactification for a certain class of non-Abelian, Heisenberg type locally compact groups and show that aspects of the structure of the Eberlein compactification can be relatively simple.

In this dissertation we study the algebraic properties of the intersection algebra of two ideals I and J in a Noetherian ring R. A major part of the dissertation is devoted to the finite generation of these algebras and developing methods of obtaining their generators when the algebra is finitely generated. We prove that the intersection algebra is a finitely generated R-algebra when R is a Unique Factorization Domain and the two ideals are principal, and use fans of cones to find the algebra generators. This is done in Chapter 2, which concludes with introducing a new class of algebras called fan algebras. Chapter 3 deals with the intersection algebra of principal monomial ideals in a polynomial ring, where the theory of semigroup rings and toric ideals can be used. A detailed investigation of the intersection algebra of the polynomial ring in one variable is obtained. The intersection algebra in this case is connected to semigroup rings associated to systems of linear diophantine equations with integer coefficients, introduced by Stanley. In Chapter 4, we present a method for obtaining the generators of the intersection algebra for arbitrary monomial ideals in the polynomial ring.

In this paper we discuss what is known so far about diagonal acts of monoids. The first results that will be discussed comprise an overview of some work done on determining whether or not the diagonal act can be finitely generated or cyclic when looking at specific classes of monoids. This has been a topic of interest to a handful of semigroup theorists over the past seven years. We then move on to discuss some results pertaining to flatness properties of diagonal acts. The theory of flatness properties of acts over monoids has been of major interest over the past two decades, but so far there are no papers published on this subject that relate specifically to diagonal acts. We attempt to shed some light on this topic as well as present some new problems.

In this paper we discuss what is known so far about diagonal acts of monoids. The first results that will be discussed comprise an overview of some work done on determining whether or not the diagonal act can be finitely generated or cyclic when looking at specific classes of monoids. This has been a topic of interest to a handful of semigroup theorists over the past seven years. We then move on to discuss some results pertaining to flatness properties of diagonal acts. The theory of flatness properties of acts over monoids has been of major interest over the past two decades, but so far there are no papers published on this subject that relate specifically to diagonal acts. We attempt to shed some light on this topic as well as present some new problems.

A compact semigroup is, roughly, a semigroup compactification of a locally compact group if it contains a dense homomorphic image of the group. The theory of semigroup compactifications has been developed in connection with subalgebras of continuous bounded functions on locally compact groups. The Eberlein algebra of a locally compact group is defined to be the uniform closure of its Fourier-Stieltjes algebra. In this thesis, we study the semigroup compactification associated with the Eberlein algebra. It is called the Eberlein compactification and it can be constructed as the spectrum of the Eberlein algebra. The algebra of weakly almost periodic functions is one of the most important function spaces in the theory of topological semigroups. Both the weakly almost periodic functions and the associated weakly almost periodic compactification have been extensively studied since the 1930s. The Fourier-Stieltjes algebra, and hence its uniform closure, are subalgebras of the weakly almost periodic functions for any locally compact group. As a consequence, the Eberlein compactification is always a semitopological semigroup and a quotient of the weakly almost periodic compactification. We aim to study the structure and complexity of the Eberlein compactifications. In particular, we prove that for certain Abelian groups, weak^{*}-closed subsemigroups of L^{\infty}[0, 1] may be realized as quotients of their Eberlein compactifications, thus showing that both the Eberlein and weakly almost periodic compactifications are large and complicated in these situations. Moreover, we establish various extension results for the Eberlein algebra and Eberlein compactification and observe that levels of complexity of these structures mimic those of the weakly almost periodic ones. Finally, we investigate the structure of the Eberlein compactification for a certain class of non-Abelian, Heisenberg type locally compact groups and show that aspects of the structure of the Eberlein compactification can be relatively simple.