[ES] El empleo de organismos vivos como biofactorías ha ganado una atención significativa en la industria debido a la creciente demanda de sistemas de producción sostenibles y la escasez de recursos. Entre sus muchas aplicaciones, las biofactorías pueden ser diseñadas para producir feromonas de insectos, las cuales sirven como alternativa ecológica a los pesticidas para el control de plagas en la agricultura. Como prueba de concepto, en esta tesis doctoral se caracterizaron plantas de Nicotiana benthamiana modificadas genéticamente con una ruta multigénica para producir las feromonas de polillas (Z)-11-hexadecenol (Z11-16OH) y (Z)-11-hexadecenil acetato (Z11-16OAc). Las plantas resultantes produjeron cantidades moderadas de ambas feromonas (111.4 µg g-1 FW y 11.8 µg g-1 FW para Z11-16OH y Z11-16OAc, respectivamente), y tasas de emisión diarias de ~10 ng g-1 FW para cada feromona. La producción de feromonas afectó negativamente al desarrollo de las plantas, probablemente debido a la sustancial carga metabólica y posible toxicidad de estos productos. Una estrategia para superar estas limitaciones es diseñar un sistema de expresión condicional que permita a las plantas crecer con normalidad antes de inducir la producción de feromonas. Para ello desarrollamos un conjunto de promotores sintéticos personalizables, llamados GB_SynP, activables con dCasEV2.1, un activador transcripcional potente y programable desarrollado recientemente para la inducción de genes en plantas. Los promotores GB_SynP permitieron una regulación precisa de los transgenes, con unos niveles de transcripción robustos y modulables en el estado "encendido" (presencia de dCasEV2.1 y la correspondiente guía de ARN), y una expresión mínima en el estado "apagado". Para implementar el sistema de producción condicional de feromonas en plantas se generó una nueva ruta multigénica para la biosíntesis de feromonas de polilla bajo el control de los promotores GB_SynP. Paralelamente, el activador dCasEV2.1 se reguló transcripcionalmente mediante el módulo CUP2:GAL4 sensible a sulfato de cobre, un inductor químico ampliamente utilizado en la agricultura. La funcionalidad del sistema se probó mediante expresión transitoria en N. benthamiana, resultando en unos rendimientos en el estado "encendido" de 32.7 µg g-1 FW y 25 µg g-1 FW para Z11-16OH y Z11-16OAc, respectivamente, y unos niveles insignificantes en ausencia de cobre. Sin embargo, la expresión en estable de esta ruta en N. benthamiana produjo unos niveles de expresión de los transgenes significativamente menores y una marcada disminución en la producción de feromonas. Esto supone que el sistema en su forma actual resulte inviable como biofactoría de feromonas en términos prácticos. La optimización de este sistema debe centrarse en mejorar la cascada de activación, en el uso de especies de plantas alternativas con mayor biomasa, y/o en incrementar las tasas de emisión en planta. Como alternativa a la producción de feromonas en plantas, la intercambiabilidad de piezas génicas entre plantas y hongos filamentosos puede aprovecharse para crear biofactorías fúngicas de feromonas. En este sentido, nuestro grupo adaptó previamente el sistema GoldenBraid a hongos filamentosos, llamado FungalBraid. En esta tesis ampliamos la colección de FungalBraid incorporando 27 piezas nuevas que incluyen diferentes marcadores de selección y promotores constitutivos e inducibles, los cuales se caracterizaron funcionalmente en Penicillium digitatum y P. chrysogenum. Además, se expresaron con éxito los promotores GB_SynP en P. digitatum, en combinación con el sistema de dCas9 activadora contenido en el vector pAMA18. Aunque los niveles de expresión de GB_SynP en hongos filamentosos fueron menores que los observados previamente en plantas, ésta y otras herramientas disponibles en la colección FungalBraid pueden utilizarse en el futuro para el desarrollo de biofactorías fúngicas que produzcan feromonas de insectos y otras biomoléculas de alto valor. / [CA] L'ús d'organismes vius com biofàbriques ha guanyat una atenció significativa a la indústria a causa de la creixent demanda de sistemes de producció sostenible i l'escassetat de recursos. Entre les seues moltes aplicacions, les biofàbriques poden ser dissenyades per a produir feromones d'insectes, les quals serveixen com a alternativa ecològica als pesticides per al control de plagues a l'agricultura. Com a prova d'aquest concepte, en aquesta tesi doctoral es van caracteritzar plantes de Nicotiana benthamiana modificades genèticament plantes de amb una ruta multigènica per a produir les feromones d'arnes (Z)-11-hexadecenol (Z11-16OH) i (Z)-11-hexadecenil acetat (Z11-16OAc). Les plantes resultants van produir quantitats moderades de totes dues feromones (111.4 µg g-1 FW i 11.8 µg g-1 FW per a Z11-16OH i Z11-16OAc, respectivament), i taxes d'emissió diàries d'aproximadament 10 ng g-1 FW per a cada feromona. La producció de feromones en aquestes plantes va afectar negativament el seu desenvolupament, probablement a causa de la substancial càrrega metabòlica i possible toxicitat d'aquests productes. Una estratègia per superar aquestes limitacions és dissenyar un sistema d'expressió condicional que permeta a les plantes créixer amb normalitat abans d'induir la producció de feromones. Per això hem desenvolupat un conjunt de promotors sintètics personalitzables, anomenats GB_SynP, activables amb dCasEV2.1, un activador transcripcional potent i programable desenvolupat recentment per a la inducció de gens en plantes. Els promotors GB_SynP van permetre una regulació precisa des transgens, amb uns nivells de transcripció robustos i modulables a l'estat "encès" (presència de dCasEV2.1 i la corresponent guia d'ARN), i una expressió mínima a l'estat "apagat". Per implementar el sistema de producció condicional de feromones en plantes es va generar una nova ruta multigènica per a la biosíntesi de feromones d'arna sota el control dels promotors GB_SynP. Paral·lelament, l'activador dCasEV2.1 es va regular transcripcionalment al mòdul CUP2:GAL4 sensible al sulfat de coure, un inductor químic àmpliament utilitzat en l'agricultura. La funcionalitat del sistema es va provar mitjançant expressió transitòria en N. benthamiana, resultant en uns rendiments a l'estat "encès" de 32.7 g-1 FW i 25 µg g-1 FW per a Z11-16OH i Z11-16OAc, respectivament, i uns nivells insignificants en absència de coure. No obstant això, l'expressió estable d'aquesta ruta a N. benthamiana va produir uns nivells d'expressió dels transgens significativament menors i una marcada disminució en la producció de feromones. Això suposa que el sistema en la seua forma actual resulte inviable com a biofàbrica de feromones en termes pràctics. L'optimització d'aquest sistema ha de centrar-se en millorar la cascada d'activació, en l'ús d'espècies de plantes alternatives amb major biomassa, i/o en incrementar les taxes d'emissió a la planta. Com a alternativa a la producció de feromones en plantes, la intercanviabilitat de peces gèniques entre plantes i fongs filamentosos pot aprofitar-se per crear biofàbriques fúngiques de feromones. En aquest sentit, el nostre grup va adaptar prèviament el sistema GoldenBraid a fongs filamentosos, anomenat FungalBraid. En aquesta tesi, vam ampliar la col·lecció de FungalBraid incorporant 27 peces noves que inclouen diferents marcadors de selecció i promotors constitutius i induïbles, els quals es van caracteritzar funcionalment a Penicillium digitatum i P. chrysogenum. A més, es van expressar amb èxit els promotors GB_SynP en P. digitatum, en combinació amb el sistema de dCas9 activadora contingut en el vector pAMA18. Encara que els nivells d'expressió de GB_SynP en fongs filamentosos van ser menors que els observats prèviament en plantes, aquesta i altres eines disponibles a la col·lecció FungalBraid poden utilitzar-se en el futur per al desenvolupament de biofàbriques fúngiques que produeixin feromones d'insectes i altres biomolècules de gran valor. / [EN] The use of living organisms as biofactories have gained significant attention in the industry due to the increasing demand for sustainable production systems and the shortage of resources. Among their many applications, biofactories can be engineered to produce insect pheromones, which serve as eco-friendly alternatives to pesticides for pest management in agriculture. As a proof of concept, in this thesis we characterized Nicotiana benthamiana plants engineered with a multigene pathway to produce the moth pheromones (Z)-11-hexadecenol (Z11-16OH) and (Z)-11-hexadecenyl acetate (Z11-16OAc). The resulting transgenic plants produced modest amounts of both pheromones (111.4 µg g-1 FW and 11.8 µg g-1 FW for Z11-16OH and Z11-16OAc, respectively), and daily emission rates of ~10 ng g-1 FW for each pheromone. Pheromone production in these plants significantly affected their fitness, likely due to the substantial metabolic burden and possible toxicity of lipid-derived products. One strategy to address these developmental abnormalities consists of engineering conditional transgene expression systems, thus allowing plants to grow normally before inducing the production of pheromones. To achieve this goal, in this thesis we developed a set of customizable synthetic promoters called GB_SynP, which can be activated by dCasEV2.1, a strong programable transcriptional activator recently developed for plant gene regulation. These GB_SynP promoters enabled tight regulation of single and multiple transgenes, with robust and tunable transcription levels in the ON state (presence of dCasEV2.1 loaded with the corresponding gRNA), and minimal or undetectable expression in the OFF state. To implement a conditional expression system for pheromone production in plants, a newly engineered multigene pathway for the biosynthesis of moth pheromones was constructed under the control of GB_SynP promoters. In parallel, the dCasEV2.1 activator was transcriptionally regulated with the CUP2:GAL4 sensor for copper sulphate, an agronomically-compatible chemical trigger. The functionality of this system was tested transiently in N. benthamiana, resulting in estimated yields of 32.7 µg g-1 FW and 25 µg g-1 FW for Z11-16OH and Z11-16OAc respectively in the ON state, and negligible levels in the absence of copper. However, stable transformation of the same copper-regulated pheromone pathway in N. benthamiana plants resulted in significantly lower transgene expression levels, which translated into a great reduction of pheromone yields. This makes the system in its current form a non-viable pheromone biofactory in practical terms. Further optimization should focus on the improvement of the activation cascade, the use of alternative plant hosts with more biomass, and/or the enhancement of emission rates in planta. As an alternative to pheromone production in plants, the interchangeability of DNA parts between plants and filamentous fungi could also be exploited to create fungal biofactories for pheromone production. In this regard, our research group previously adapted the GoldenBraid system for filamentous fungi, which we named FungalBraid. In this thesis, we expanded the FungalBraid collection by incorporating 27 new DNA parts, including different selection markers and several constitutive and inducible promoters, all of which were functionally characterized in Penicillium digitatum and P. chrysogenum. Furthermore, we successfully expressed the GB_SynP promoters developed for plants in P. digitatum, in combination with the non-integrative pAMA18-derived vector for the expression of a dCas9-based activator. Although further optimization of GB_SynP in filamentous fungi is required, as expression levels were lower than those previously observed in plants, this and the other tools available in the FungalBraid collection can be effectively employed in the future for the development of fungal biofactories that produce insect pheromones and other high value biomolecules. / Este trabajo ha sido financiado mediante la Ayuda para la Formación de Profesorado Universitario FPU18/02019 (Ministerio de Educación, Cultura y Deporte), así como por el proyecto europeo SUSPHIRE (PCI2018-092893, Era- CoBiotech), y los proyectos de Plan Nacional I+D PID2019-108203RB-100 y PID2021-125858OB-100 (Ministerio de Ciencia e Innovación). / Moreno Giménez, E. (2023). Synthetic biology tools for production of insect pheromones in plants and filamentous fungi [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/201180

Recursion is a fundamental tool of mathematics used to define, construct, and analyze mathematical objects. This work employs induction, sieving, inversion, and other recursive methods to solve a variety of problems in the areas of algebraic number theory, topological and combinatorial graph theory, and analytic probability and statistics. A common theme of recursively defined functions, weighted sums, and cross-referencing sequences arises in all three contexts, and supplemented by sieving methods, generating functions, asymptotics, and heuristic algorithms. In the area of number theory, this work generalizes the sieve of Eratosthenes to a sequence of polynomial values called polynomial-value sieving. In the case of quadratics, the method of polynomial-value sieving may be characterized briefly as a product presentation of two binary quadratic forms. Polynomials for which the polynomial-value sieving yields all possible integer factorizations of the polynomial values are called recursively-factorable. The Euler and Legendre prime producing polynomials of the form n2+n+p and 2n2+p, respectively, and Landau's n2+1 are shown to be recursively-factorable. Integer factorizations realized by the polynomial-value sieving method, applied to quadratic functions, are in direct correspondence with the lattice point solutions (X,Y) of the conic sections aX2+bXY +cY2+X-nY=0. The factorization structure of the underlying quadratic polynomial is shown to have geometric properties in the space of the associated lattice point solutions of these conic sections. In the area of combinatorial graph theory, this work considers two topological structures that are used to model the process of homologous genetic recombination: assembly graphs and chord diagrams. The result of a homologous recombination can be recorded as a sequence of signed permutations called a micronuclear arrangement. In the assembly graph model, each micronuclear arrangement corresponds to a directed Hamiltonian polygonal path within a directed assembly graph. Starting from a given assembly graph, we construct all the associated micronuclear arrangements. Another way of modeling genetic rearrangement is to represent precursor and product genes as a sequence of blocks which form arcs of a circle. Associating matching blocks in the precursor and product gene with chords produces a chord diagram. The braid index of a chord diagram can be used to measure the scope of interaction between the crossings of the chords. We augment the brute force algorithm for computing the braid index to utilize a divide and conquer strategy. Both assembly graphs and chord diagrams are closely associated with double occurrence words, so we classify and enumerate the double occurrence words based on several notions of irreducibility. In the area of analytic probability, moments abstractly describe the shape of a probability distribution. Over the years, numerous varieties of moments such as central moments, factorial moments, and cumulants have been developed to assist in statistical analysis. We use inversion formulas to compute high order moments of various types for common probability distributions, and show how the successive ratios of moments can be used for distribution and parameter fitting. We consider examples for both simulated binomial data and the probability distribution affiliated with the braid index counting sequence. Finally we consider a sequence of multiparameter binomial sums which shares similar properties with the moment sequences generated by the binomial and beta-binomial distributions. This sequence of sums behaves asymptotically like the high order moments of the beta distribution, and has completely monotonic properties.

In this project the objective is to understand how to design a braiding machine capable of automated production of Omnifibre in a research environment. Automated production of Omnifibre is the key issue for the researchers as they want to increase the weaveabilty of the fibers and make it more suitable for use in active textiles. To achieve the necessary knowledge when designing a braiding machine, an extensive literature study was performed which focused on braids, braiding machines, and CNC manufacturing. An Interview with a researcher and with a manufacturing expert was conducted. Simulations of different braiding machine configurations were performed in TexMind braiding machine configurator. Solidworks was used to estimate the size of the braiding machine. A large amount of the machine’s parts were manufactured on a CNC mill and lathe to test manufacturability and to aid in designing optimal subsystems. The result is a proposed design for a braiding machine in the form of a 3-D model and a partially completed prototype used for testing and design evaluation. The conclusion is that Omnifibre is much like other ultra-fine braided threads, and the research on its applicability has a big impact on the braiding machine’s design, especially on flexibility in thread material and braiding patterns.

Representation theory, which encodes the elements of a group as linear operators on a vector space, has far-reaching implications in physics. Fundamental results in quantum physics emerge directly from the representations describing physical symmetries. We first examine the connections between specific representations and the principles of quantum mechanics. Then, we shift our focus to the braid group, which describes the algebraic structure of braids. We apply representations of the braid group to physical systems in order to investigate quasiparticles known as anyons. Finally, we obtain governing equations of anyonic systems to highlight the differences between braiding statistics and conventional Bose-Einstein/Fermi-Dirac statistics.

With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

Cette thèse traite la fabrication de pièces composites par le procédé « Resin Transert Molding » (RTM), appliquée à des tubes de protections thermiques assemblées dans des propulseurs de systèmes d’armes. Ces travaux ont pour objectif de démontrer la faisabilité d’utilisation de ce procédé pour la fabrication de ces pièces complexes. C’est le préformage, première étape du procédé de fabrication par RTM, qui est étudié dans le cadre de cette thèse. Cette étape est cruciale du point de vue de la faisabilité de l’étape d’injection qui la suit dans le procédé RTM mais aussi pour s’assurer de la qualité de la pièce composite finale obtenue. L’objectif des travaux de thèse est triple. Il faut tout d’abord développer le protocole de fabrication répétable adapté pour garantir l’obtention de préformes conformes. Ce protocole devra être viable du point de vue industriel. Pour cela, une démarche expérimentale a été mise en place. Un pilote de laboratoire puis un pilote industriel ont permis de comprendre et maitriser les phénomènes survenant en cours de préformage en faisant varier les paramètres procédé pour la fabrication de nombreux prototypes. Un modèle macroscopique prédictif de la forme globale des plis obtenus à partir des paramètres procédés a été développé à l’aide des observations expérimentales. Un modèle mésoscopique, à l’échelle de la maille élémentaire, a été écrit également. Il permet de prédire, à partir des données constitutives du matériau et d’une géométrie de pièce, la déformation de compaction et de cisaillement, modes de sollicitations prépondérants en cours de préformage, subie par le renfort en cours de la première étape du procédé de fabrication. Ces modèles mésoscopique et macroscopique couplés permettent le développement d’un outil global qui, de manière théorique et prédictive, assure la faisabilité d’une pièce de géométrie connue avec un matériau connu et fournit les paramètres « procédé » optimum pour assurer sa fabrication future. Les phénomènes de déformation en cisaillement et compaction apparaissant sur la tresse en cours de préformage sont donc identifiés et connus. Le procédé de fabrication est optimisé et l’outil prédictif permet d’envisager et tester en amont un changement de matériau, de géométrie de pièce à fabriquer ou de cahier descharges industriel. / This study deals with the manufacture of composite parts by the process "Resin Transert Molding" (RTM), applied to thermal protection tubes. This work aims to demonstrate the feasibility of using this method for the production of these complex parts. This study deals with the first step of the RTM process, the fiber performing. This is critical from the standpoint of the feasibility of injecting step that follows in the RTM process but also to ensure the quality of the final composite part obtained. The aim of the thesis is threefold. Must first develop the manufacturing protocol adapted to ensure repeatable obtaining preforms compliant. This protocol should be viable to the industrial point of view. For this purpose, an experimental approach was implemented. A pilot laboratory and an industrial pilot helped to understand and master the phenomena occurring during forming varying the process parameters for the production of many prototypes. A macroscopic model predictive of overall shape folds obtained from the process parameters has been developed with the experimental observations. A mesoscopic model, the scale of the unit cell was also writing. It can predict, based on the specifications of the material and part geometry, the deformation of compaction and shear stresses. These models mesoscopic and macroscopic allow the development of a global tool that, theoretically predictive and ensures the feasibility of a piece of known geometry with a known material parameters and provides the "process" to ensure its optimum manufacturing future. The phenomena of compaction and shear strain appearing on the braid during preforming are identified and known. The manufacturing process is optimized and the predictive tool allows to explore and test upstream change of material, part geometry in manufacturing or industrial specifications.

With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

The dynamics of a complex system can be understood by analyzing the phase space structure of that system. We apply geometric and statistical techniques to two Hamiltonian systems to find and exploit structure in the phase space that helps us get qualitative and quantitative results about the phase space transport. While the structure can be revealed by the study of invariant manifolds of fixed points and periodic orbits in the first system, there do not exist any fixed points (and hence invariant manifolds) in the second system. The use of statistical (or measure theoretic) and topological methods reveals the phase space structure even in the absence of fixed points or stable and unstable invariant manifolds. The first problem we study is the four-body problem in the context of a spacecraft in the presence of a planet and two of its moons, where we exploit the phase space structure of the problem to devise an intelligent control strategy to achieve mission objectives. We use a family of analytically derived controlled Keplerian Maps in the Patched-Three-Body framework to design fuel efficient trajectories with realistic flight times. These maps approximate the dynamics of the Planar Circular Restricted Three Body Problem (PCR3BP) and we patch solutions in two different PCR3BPs to form the desired trajectories in the four body system. The second problem we study concerns phase space mixing in a two-dimensional time dependent Stokes flow system. Topological analysis of the braiding of periodic points has been recently used to find lower bounds on the complexity of the flow via the Thurston-Nielsen classification theorem (TNCT). We extend this framework by demonstrating that in a perturbed system with no apparent periodic points, the almost-invariant sets computed using a transfer operator approach are the natural objects on which to pin the TNCT. / Ph. D.

A relação entre os grupos de tranças de superfícies e os grupos de homotopia das esferas é atualmente um tópico de bastante interesse. Nos últimos anos tem sido feitos avanços consideráveis no estudo desta relação no caso dos grupos de tranças de Artin com n cordas, denotado por Bn, da esfera e do plano projetivo. Nessa tese analisamos com detalhes as interações entre a teoria de tranças e a teoria de homotopia, e mostramos novos resultados que estabelecem conexões entre os grupos de homotopia da 2-esfera S2 e os grupos de tranças sobre qualquer superfície. No andamento deste trabalho, descobrimos uma conexão surpreendente dos grupos de tranças com os grupos cristalográficos e de Bieberbach: para n maior ou igual que 3, o grupo quociente Bn/[Pn, Pn] é um grupo cristalográfico que contém grupos de Bieberbach como subgrupos, onde Pn é o subgrupo de tranças puras de Bn. Com isto obtivemos uma formulação de um Teorema de Auslander e Kuranishi para 2-grupos finitos e exibimos variedades Riemannianas compactas planas que admitem difeomorfismo de Anosov e cujo grupo de holonomia é Z2k . Além disso, durante esta tese, detectamos e, quando possível, corrigimos algumas imprecisões em dois importantes artigos nessa área de estudo, escritos por J. Berrick, F. R. Cohen, Y. L. Wong e J. Wu (Jour. Amer. Math. Soc. - 2006) assim como por J. Y. Li e J.Wu (Proc. London Math. Soc. - 2009). / The relation between surface braid groups and homotopy groups of spheres is currently a subject of great interest. Considerable progress has been made in recent years in the study of these relations in the case of the n-string Artin braid groups, denoted by Bn, the sphere and the projective plane. In this thesis we analyse in detail the interactions between braid theory and homotopy theory, and we present new results that establish connections between the homotopy groups of the 2-sphere S2 and the braid groups of any surface. During the course of this work, we discovered an unexpected connection of braid groups with crystallographic and Bieberbach groups: for n greater or equal than 3, the quotient group Bn/[Pn, Pn] is a crystallographic group that contains Bieberbach groups as subgroups, where Pn is the pure braid subgroup of Bn. This enables us to obtain a formulation of a theorem of Auslander and Kuranishi for finite 2-groups, and to exhibit Riemannian compact flat manifolds that admit Anosov diffeomorphisms and whose holonomy group is Z2k. In addition, during the thesis, we have detected, and where possible, corrected some inaccuracies in two important papers in the area of study, by J. Berrick, F. R. Cohen, Y. L. Wong and J. Wu (Jour. Amer. Math. Soc. - 2006), and by J. Y. Li and J. Wu (Proc. London Math. Soc. - 2009).