The aim of this thesis is to present efficient (strongly polynomial) methods and algorithms for problems in max­algebra when certain matrices have special entries or are structured. First, we describe all solutions to a one-sided parametrised system. Next, we consider special cases of two-sided systems of equations/inequalities. Usually, we describe a set of generators of all solutions but sometimes we are satisfied with finding a non-trivial solution or being able to say something meaningful about a non-trivial solution should it exist. We look at special cases of the generalised eigenproblem, describing the full spectrum usually. Finally, we prove some results on 2x2 matrix roots and generalise these results to a class of nxn matrices. Main results include: a description of all solutions to the two-dimensional generalised eigenproblem; observations about a non-trivial solution (should it exist) to essential/minimally active two-sided systems of equations; the full spectrum of the generalised eigenproblem when one of the matrices is an outer-product; the unique candidate for the generalised eigenproblem when the difference of two matrices is symmetric and has a saddle point and finally we explicitly say when a 2x2 matrix has a kth root for a fixed positive integer k.

Let X be a compact connected Riemann surface of genus g and let f∶ X →P1 be a meromorphic function of degree n. Classes of such covers are in one to one correspondence with the primitive systems, which are tuples of elements (x1,x2,⋯,xr) in the symmetric group Sn taken up to conjugation and the action of the braid group, such that x1.x2.⋯.xr=1 and G=〈x1,x2,⋯,xr〉 is a primitive subgroup G of Sn. This thesis is a contribution to the classification of primitive genus g ≤ 2 systems of sporadic almost simple groups.

We employ the method of asymptotic coordinate expansions in time and space to determine the detailed structure of the solution to the linearised dambreak problem at the initial stage, in the far fields and at large time. We consider the situation where an inclined dam separates a horizontal layer of incompressible and inviscid fluid from a shallower horizontal layer of the fluid. The fluid is initially at rest, sits on a horizontal, impermeable base, and is bounded above by a free surface. We consider the linearised dambreak problem, which corresponds to a dam with a small step height and slope. We formulate the problem for the free surface and fluid velocity potential, to which the exact solution is found via the theory of complex Fourier transforms. This gives the free surface and fluid velocity potential in complex Fourier integral form. We examine the detailed asymptotic form of the exact solution for the free surface at the initial stage, in the far field and at large time. The asymptotic approximations are then compared to a numerical evaluation of the exact solution for the free surface, and to the case where the free surface is described by the linearised shallow water theory.

The aim of this thesis is to study local and global invariants in representation theory of finite groups using the (restricted) Lie algebra structure of the first degree of Hochschild cohomology of a block algebra B as a main tool. This lead to two directions: In the first part we investigate the global approach. In particular, we prove the compatibility of the p-power map under stable equivalence of Morita type of subclasses of the first Hochschild cohomology represented by integrable derivations. Further results in this aspect include an example showing that the p-power map cannot generally be expressed in terms of the BV operator. We also study some properties of r-integrable derivations and we provide a family of examples given by the quantum complete intersections where all the derivations are r-integrable. In the second part our attention is focused on the local invariants. More precisely, we fully characterise blocks B with unique isomorphism class of simple modules such that the first degree Hochschild cohomology HH1(B) is a simple as Lie algebra. In this case we prove that B is a nilpotent block with an elementary abelian defect group P of order at least 3 and HH1(B) is isomorphic to the Witt algebra HH1(kP).

The study of an evolutionary process has traditionally considered a population with a homogeneous structure where each pair of individuals is equally likely to interact with one another. Later studies have considered heterogeneous structures implemented using evolutionary graph theory, and other studies have considered group interactions of fixed size. This work builds upon these later studies by implementing a set of evolutionary dynamics that can be used to study more complex evolutionary processes consisting of a population with a heterogeneous structure where individuals interact in groups of varying size. This research begins by analytically studying simple evolutionary processes using a set of standard evolutionary dynamics. Results are derived that identify the structures for which an evolutionary process is identical to a Moran process, which has a homogeneous population structure, for each of the evolutionary dynamics. These results form a basis for the work that follows by providing a better understanding of evolutionary dynamics. Before considering more complex evolutionary processes, a class of multiplayer games called social dilemmas are defined for variable group sizes. The two main types of social dilemmas are identified, namely public goods dilemmas and commons dilemmas, and examples of each type of dilemma are given whose characteristics are visually illustrated. More complex evolutionary processes are then studied based on the framework of Broom-Rychtář that provides the mathematical tools to model group interactions in mobile individuals. First, the evolutionary dynamics that can be used within this framework are developed. The updated version of the framework is then used to demonstrate how it can applied to study various kinds of behaviour in an evolutionary setting. The first application is the territorial raider model. It considers territorial behaviour where each individual has their own territory that overlaps with those of other individuals. Interactions take place between groups of individuals when they meet in the overlapping parts of their territories. Two kinds of social dilemmas are studied in this model: a multiplayer hawk-dove game and a multiplayer public goods game. It is shown that the temperature, which measures how often an individual is likely to be replaced, plays an important role in determining the success of a given strategy. A generalized version of the territorial raider model is also considered where subpopulations rather than individuals share the same territory. A multiplayer public goods game is used to study the evolution of cooperation, which is a suboptimal strategy at the individual level but an optimal strategy at the group level. The structure and dynamics are shown to be critical in the evolution of cooperation where an extension of the temperature, called the subpopulation temperature, dictates the relative success of cooperators. Finally, a model where individual move base upon their previous interactions is considered called the Markov movement model. A multiplayer public goods game is used to study the evolution of cooperation. It is shown that cooperators can benefit by staying with one another provided that there is a movement cost that slows down their competitors, the defectors. In this case, the dynamics play a less critical role in the evolution of cooperation.

In this thesis we study saturated fusion systems on ρ-groups having sectional rank 3, for ρ odd. We obtain a complete classification of simple fusion systems on p-groups having sectional rank 3 for ρ ≥ 5, exhibiting a new simple exotic fusion system on a 7-group of order 7^5. We introduce the notion of pearls, defined as essential subgroups isomorphic to the groups C_ρ X_ρ and ρ₊¹⁺² (for odd), and we illustrate some properties of fusion systems involving pearls. As for ρ = 3, we determine the isomorphism type of a certain section of the 3-groups considered.

Let \(G\) be a split finite group of Lie type defined over F\(_q\), where \(q\)=\(p\)\(^e\) is a prime power and \(p\) is not a very bad prime for \(G\). Let \(U\) be a Sylow \(p\)-subgroup of \(G\). In this thesis, we provide a full parametrization of the set Irr(\(U\)) of irreducible characters of \(U\) when \(G\) is of rank 5 or less. In particular, for every character χ ∈ Irr(\(U\)) we determine an abelian subquotient of \(U\) such that χ is obtained by an inflation, followed by an induction of a linear character of this subquotient. The characters are given in most cases as the output of algorithm that has been implemented in the computer system GAP, whose validity is proved in this thesis using classical results in representation theory and properties of the root system associated to \(G\). We also develop a method to determine a parametrization of the remaining irreducible characters, which applies for every split finite group of Lie type of rank at most 5, and lays the groundwork to provide such a parametrization in rank 6 and higher.

A una superficie proyectiva X cualquiera se le pueden asociar una serie de conos convexos (cono de curvas, cono semiamplio y cono característico) que proporcionan información sobre la geometría de la superficie. En esta memoria se hace un estudio del cono de curvas asociado a una superficie proyectiva racional y regular. Más concretamente, se establecen condiciones que implican la poliedricidad de dicho cono. Estas condiciones son de dos tipos: unas que dependen de la existencia de determinados divisores efectivos, y otras que dependen únicamente de la obtención de la superficie a partir de una superficie relativamente minimal (que puede ser el plano proyectivo o una superficie de Hirzebruch). La poliedricidad del cono de curvas tiene importantes implicaciones geométricas, como el hecho de que el número de morfismos proyectivos con fibras conexas de X a otra variedad (contracciones) es finito, y también que el número de (-1)-curvas de X (es decir, de curvas no singulares, racionales y de auto-intersección ­1) es finito.

En aquesta tesi interdisciplinar desenvolupem eines algebraiques per a problemes en filogenètica i genòmica. Per estudiar l'evolució molecular de les espècies sovint s'usen models evolutius estocàstics. L'evolució es representa en un arbre (anomenat filogenètic) on les espècies actuals corresponen a fulles de l'arbre i els nodes interiors corresponen a ancestres comuns a elles. La longitud d'una branca de l'arbre representa la quantitat de mutacions que han ocorregut entre les dues espècies adjacents a la branca. Llavors l'evolució de seqüències d'ADN en aquestes espècies es modelitza amb un procés Markov ocult al llarg de l'arbre. Si el procés de Markov se suposa a temps continu, normalment s'assumeix que també és homogeni i, en tal cas, els paràmetres del model són les entrades d'una raó de mutació instantània i les longituds de les branques. Si el procés de Markov és a temps discret, llavors els paràmetres del model són les probabilitats condicionades de substitució de nucleòtids al llarg de l'arbre i no hi ha cap hipòtesi d'homogeneïtat. Aquests últims són els tipus de models que considerem en aquesta tesi i són, per tant, més generals que els de temps continu. Des d'aquesta perspectiva s'estudien els problemes més bàsics de la filogenètica: donat un conjunt de seqüències d'ADN, com decidim quin és el model evolutiu més adequat? com inferim de forma eficient els paràmetres del model? I fins i tot, tal i com també hem provat en aquesta tesi, és possible que les espècies no hagin evolucionat seguint un sol arbre sinó una mescla d'arbres i llavors cal abordar aquestes preguntes en aquest cas més general. Per a models evolutius a temps continu i homogenis, s'ha proposat solucions diverses a aquestes preguntes al llarg de les últimes dècades. En aquesta tesi resolem aquests dos problemes per a models evolutius a temps discret usant tècniques algebraiques provinents d'àlgebra lineal, teoria de grups, geometria algebraica i estadística algebraica. A més a més, la nostra solució per al primer problema és vàlida també per a mescles filogenètiques. Hem fet tests dels mètodes proposats en aquesta tesi sobre dades simulades i dades reals del projectes ENCODE (Encyclopedia Of DNA Elements). Per tal de provar els nostres mètodes hem donat algoritmes per a generar seqüències evolucionant sota un model a temps discret amb un nombre esperat de mutacions prefixat. I així mateix, hem demostrat que aquests algorismes generen totes les seqüències possibles (per la majoria de models). Els tests sobre dades simulades mostren que els mètodes proposats són molt acurats i els resultats sobre dades reals permeten corroborar hipòtesis prèviament formulades. Tots els mètodes proposats en aquesta tesi han estat implementats per a un nombre arbitrari d'espècies i estan disponibles públicament. / In this thesis we develop interdisciplinary algebraic tools for genomic and phylogenetic problems. To study the molecular evolution of species one often uses stochastic evolutionary models. The evolution is represented in a tree (called phylogenetic tree) whose leaves represent current species and whose internal nodes correspond to their common ancestors. The length of a branch of the tree represents the number of mutations that have occurred between the two species adjacent to the branch. Then ,the evolution of DNA sequences in these species is modeled with a hidden Markov process along the tree. If the Markov process is assumed to be continuous in time, it is usually assumed homogeneous as well and, if so, the model parameters are the instantaneous rate of mutation and the lengths of the branches. If the Markov process is discrete in time, then the model parameters are the conditional probabilities of nucleotide substitution along the tree and there is no assumption of homogeneity. The latter are the types of models we consider in this thesis and are therefore more general than the homogeneous continuous ones. From this perspective we study the basic problems of phylogenetics: Given a set of DNA sequences, what is the evolutionary model that best fits the data? how can we efficiently infer the model parameters? Also, as we also checked in this thesis, it is possible that species have not evolved along a single tree but a mixture of trees so that we need to address these questions in this more general case. For continuous-time, homogeneous, evolutionary models, several solutions to these questions have been proposed during the last decades. In this thesis we solve these two problems for discrete-time evolutionary models, using algebraic techniques from linear algebra, group theory, algebraic geometry and algebraic statistics. In addition, our solution to the first problem is also valid for phylogenetic mixtures. We have made tests of the methods proposed in this thesis on simulated and real data from ENCODE Project (Encyclopedia Of DNA Elements). To test our methods, we also provide algorithms to generate sequences evolving under discrete-time models with a given expected number of mutations. Even more, we have proved that these algorithms generate all possible sequences (for most models). Tests on simulated data show that the methods are very accurate and our results on real data confirm hypotheses previously formulated. All the methods in this thesis have been implemented for an arbitrary number of species and are publicly available.

This thesis deals with finite groups. More precisely, it studies formations,which are classes of groups closed under taking homomorphic images andsubdirect products. The concept of X-local formation, where X is a classof simple groups, generalises the concepts of local formation and Baer-localformation. It was introduced by F¨orster with the aim of generalising thewell-known theorems of Gasch¨utz-Lubeseder-Schmid and Baer. The first onestates that a formation is saturated if and only if it is local. The second onecharacterises Baer-local formations as the solubly saturated ones. F¨orsterintroduced a Frattini-like subgroup associated with the class X and whichallowed him to introduce the concept of X-saturation. In the first chapterof the thesis a new Frattini-like subgroup is defined. This subgroup allowsus to give a more natural generalisation of the above-mentioned theorems.Moreover, X-local formations are characterised as the classes of groups withgeneralised central chief factors. In the second chapter the relation betweenthis family of formations and the one of !-saturated formations, where ! isa set of primes, is studied. The third chapter is devoted to analyse someproblems on the product of two formations. A classical result states that theproduct of two saturated formations is a saturated formation. This resultis not true in general for solubly saturated formations. We give conditionson two X-saturated formations to ensure that the product is an X-saturatedformation. As a corollary we obtain a result for products of Baer-localformations. The starting point of chapter four is a question posed by Skibain the Kourovka Notebook dealing Baer-local formations generated by agroup. We study a more general question by using the concept of X-localformation. Moreover, we give a complete description of the factorisations ofan X-local formation generated by a group. / La tesis se enmarca dentro de la teor´ýa de grupos finitos. M´as concretamente,se centra en el estudio de las formaciones de grupos, que son lasclases de grupos cerradas para cocientes y para productos subdirectos. Elconcepto de formaci´on X-local, donde X es una clase de grupos simples,generaliza los conceptos de formaci´on local y formaci´on de Baer. Fue introducidopor F¨orster con la idea de generalizar los conocidos teoremas deGasch¨utz-Lubeseder-Schmid y de Baer. El primero de ellos afirma que unaformaci´on es saturada si, y s´olo si, es local. El segundo caracteriza las formacionesBaer-locales como las resolublemente saturadas. Para ello F¨orsterintrodujo un subgrupo de tipo Frattini asociado a la clase X del cual surgeel concepto de X-saturaci´on. En el primer cap´ýtulo de la tesis se define unnuevo subgrupo mediante el que se consigue una generalizaci´on m´as satisfactoriade los teoremas mencionados. Tambi´en se caracterizan las formacionesX-locales como las clases de grupos con factores principales generalizados.En el segundo cap´ýtulo se estudia la relaci´on entre esta familia de formacionesy la de las formaciones w-saturadas, donde w es un conjunto de n´umerosprimos. En la tesis tambi´en se abordan problemas relativos al producto dedos formaciones. Un resultado cl´asico de teor´ýa de clases de grupos afirmaque el producto de dos formaciones saturadas es una formaci´on saturada.Este resultado no es cierto en general para formaciones resolublemente saturadas.En el cap´ýtulo tercero se presentan condiciones necesarias y suficientespara que el producto de dos formaciones X-saturadas sea una formaci´on Xsaturada.Como consecuencia, cuando X es la clase de los grupos simplesabelianos, obtenemos un resultado para productos de formaciones resolublementesaturadas. El punto de partida del cap´ýtulo cuarto es una preguntade Skiba que aparece en la edici´on de 1992 del Kourovka Notebook sobreformaciones Baer-locales generadas por un grupo. Nosotros abordamos unacuesti´on m´as general usando formaciones X-locales. Adem´as, damos una descripci´on completa de las factorizaciones de una formaci´on X-local generadapor un grupo.