<p>On the category of representations of a given quiver we define a tensor product point-wise and arrow-wise. The corresponding Clebsch-Gordan problem of how the tensor product of indecomposable representations decomposes into a direct sum of indecomposable representations is the topic of this thesis.</p><p>The choice of tensor product is motivated by an investigation of possible ways to modify the classical tensor product from group representation theory to the case of quiver representations. It turns out that all of them yield tensor products which essentially are the same as the point-wise tensor product.</p><p>We solve the Clebsch-Gordan problem for all Dynkin quivers of type A, D and E₆, and provide explicit descriptions of their respective representation rings. Furthermore, we investigate how the tensor product interacts with Galois coverings. The results obtained are used to solve the Clebsch-Gordan problem for all extended Dynkin quivers of type Ã_n and the double loop quiver with relations βα=αβ=αⁿ=βⁿ=0.</p>

Os P-splines de Eilers e Marx (1996) são métodos de suavização que é uma combinação de bases B-splines e uma penalização discreta sobre os coeficientes das bases utilizados para suavizar dados normais e não normais em uma ou mais dimensões, no caso de várias dimensões utiliza-se como suavização o produto tensor dos P-splines. Também os P-splines são utilizados como representação de modelos mistos Currie et al. (2006) pela presença de características tais como: efeitos fixos, efeitos aleatórios, correlação espacial ou temporal e utilizados em modelos mais generalizados tais como os modelos mistos lineares generalizados e modelos mistos aditivos generalizados. Neste trabalho apresentou-se toda a abordagem, metodologia e descrição dos P-splines como modelos mistos e como componentes das estruturas suavizadoras de variáveis unidimensionais e bidimensionais dos modelos mistos aditivos generalizados, mostrando essa abordagem e propondo seu uso em uma aplicação no comportamento dos níveis médios da produção de cana-de-açúcar sob a influência das alterações das variáveis climáticas como temperatura e precipitação, que foram medidos ao longo de 10 anos em cada mesorregião do Estado de São Paulo. O motivo de usar essa abordagem como método de suavização é que muitas vezes não é conhecido a tendência dessas covariáveis climáticas mas sabe-se que elas influenciam diretamente sobre a variável resposta. Além de permitir essa abordagem inclusão de efeitos fixos e aleatórios nos modelos a serem propostos, permitirá a inclusão do processo autoregressivo AR(1) como estrutura de correlação nos resíduos. / P-splines of Eilers e Marx (1996) are methods of smoothing that is a combination of B-splines bases and penalty the coefficients of the bases used to smooth normal and non-normal data in one or more dimensions; in the case of several dimensions it is used as smoothing the tensor product of the P-splines. Also the P-splines are used as representation of mixed models Currie et al. (2006) by the presence of characteristics such as: fixed effects, random effects, spatial or temporal correlation and used in more generalized models such as generalized linear mixed models and generalized additive mixed models. In this work the whole approach, methodology and description of the P-splines as mixed models and as components of the smoothing structures of one-dimensional and two-dimensional variables of generalized additive mixed models were presented, showing this approach and proposing its application in the behavior of the average levels of sugarcane production, which is influenced by changes in climatic variables such as temperature and precipitation , which were measured over 10 years in each mesoregion of the state of São Paulo. The reason for using this approach as a smoothing method is that the tendency of these climate covariables is not know for the most part, but is known that they influence directly the response variable, besides allowing this approach to include fixed and random effects in the models to be proposed, will allow the inclusion of the autoregressive process AR(1) as a correlation structure in the residuos.

<p>The study of symmetries is an essential tool in modern physics. The analysis of symmetries is often carried out in the form of Lie algebras and their representations. Knowing the representation theory of a Lie algebra includes knowing how tensor products of representations behave. In this thesis two methods to study and decompose tensor products of representations of non-compact Lie algebras are presented and applied to sl(2,R). We focus on products containing non-unitary representations, especially the product of a unitary highest weight representation and a non-unitary finite dimensional. Such products are not necessarily decomposable. Following the theory of B. Kostant we use infinitesimal characters to show that this kind of tensor product is fully reducible iff the sum of the highest weights in the two modules is not a positive integer or zero. The same result is obtained by looking for an invariant coupling between the product module and the contragredient module of some possible submodule. This is done in the formulation by Barut & Fronsdal. From the latter method we also obtain a basis for the submodules consisting of vectors from the product module. The described methods could be used to study more complicated semisimple Lie algebras.</p>

The study of symmetries is an essential tool in modern physics. The analysis of symmetries is often carried out in the form of Lie algebras and their representations. Knowing the representation theory of a Lie algebra includes knowing how tensor products of representations behave. In this thesis two methods to study and decompose tensor products of representations of non-compact Lie algebras are presented and applied to sl(2,R). We focus on products containing non-unitary representations, especially the product of a unitary highest weight representation and a non-unitary finite dimensional. Such products are not necessarily decomposable. Following the theory of B. Kostant we use infinitesimal characters to show that this kind of tensor product is fully reducible iff the sum of the highest weights in the two modules is not a positive integer or zero. The same result is obtained by looking for an invariant coupling between the product module and the contragredient module of some possible submodule. This is done in the formulation by Barut &amp; Fronsdal. From the latter method we also obtain a basis for the submodules consisting of vectors from the product module. The described methods could be used to study more complicated semisimple Lie algebras.

Os P-splines de Eilers e Marx (1996) são métodos de suavização que é uma combinação de bases B-splines e uma penalização discreta sobre os coeficientes das bases utilizados para suavizar dados normais e não normais em uma ou mais dimensões, no caso de várias dimensões utiliza-se como suavização o produto tensor dos P-splines. Também os P-splines são utilizados como representação de modelos mistos Currie et al. (2006) pela presença de características tais como: efeitos fixos, efeitos aleatórios, correlação espacial ou temporal e utilizados em modelos mais generalizados tais como os modelos mistos lineares generalizados e modelos mistos aditivos generalizados. Neste trabalho apresentou-se toda a abordagem, metodologia e descrição dos P-splines como modelos mistos e como componentes das estruturas suavizadoras de variáveis unidimensionais e bidimensionais dos modelos mistos aditivos generalizados, mostrando essa abordagem e propondo seu uso em uma aplicação no comportamento dos níveis médios da produção de cana-de-açúcar sob a influência das alterações das variáveis climáticas como temperatura e precipitação, que foram medidos ao longo de 10 anos em cada mesorregião do Estado de São Paulo. O motivo de usar essa abordagem como método de suavização é que muitas vezes não é conhecido a tendência dessas covariáveis climáticas mas sabe-se que elas influenciam diretamente sobre a variável resposta. Além de permitir essa abordagem inclusão de efeitos fixos e aleatórios nos modelos a serem propostos, permitirá a inclusão do processo autoregressivo AR(1) como estrutura de correlação nos resíduos. / P-splines of Eilers e Marx (1996) are methods of smoothing that is a combination of B-splines bases and penalty the coefficients of the bases used to smooth normal and non-normal data in one or more dimensions; in the case of several dimensions it is used as smoothing the tensor product of the P-splines. Also the P-splines are used as representation of mixed models Currie et al. (2006) by the presence of characteristics such as: fixed effects, random effects, spatial or temporal correlation and used in more generalized models such as generalized linear mixed models and generalized additive mixed models. In this work the whole approach, methodology and description of the P-splines as mixed models and as components of the smoothing structures of one-dimensional and two-dimensional variables of generalized additive mixed models were presented, showing this approach and proposing its application in the behavior of the average levels of sugarcane production, which is influenced by changes in climatic variables such as temperature and precipitation , which were measured over 10 years in each mesoregion of the state of São Paulo. The reason for using this approach as a smoothing method is that the tendency of these climate covariables is not know for the most part, but is known that they influence directly the response variable, besides allowing this approach to include fixed and random effects in the models to be proposed, will allow the inclusion of the autoregressive process AR(1) as a correlation structure in the residuos.

Starting with the work of Volker Strassen, algorithms for matrix multiplication have been developed which are time complexity-wise more efficient than the standard algorithm from the definition of multiplication. The general method of the developments has been viewing the bilinear mapping that matrix multiplication is as a three-dimensional tensor, where there is an exact correspondence between time complexity of the multiplication algorithm and tensor rank. The latter can be seen as a generalisation of matrix rank, being the minimum number of terms a tensor can be decomposed as. However, in contrast to matrix rank there is no general method of computing tensor ranks, with many values being unknown for important three-dimensional tensors. To further improve the theoretical bounds of the time complexity of matrix multiplication, support rank of tensors has been introduced, which is the lowest rank of tensors with the same support in some basis. The goal of this master's thesis has been to go through the history of faster matrix multiplication, as well as specifically examining the properties of support rank for general tensors. In regards to the latter, a complete classification of rank structures of support classes is made for the smallest non-degenerate tensor product space in three dimensions. From this, the size of a support can be seen affecting the pool of possible ranks within a support class. At the same time, there is in general no symmetry with regards to support size occurring in the rank structures of the support classes, despite there existing a symmetry and bijection between mirrored supports. Discussions about how to classify support rank structures for larger tensor product spaces are also included. / Från och med forskning gjord av Volker Strassen har flera algoritmer för matrismultiplikation utvecklats som är effektivare visavi tidskomplexitet än standardalgoritmen som utgår från defintionen av multiplikation. Generellt sett har metoden varit att se den bilinjära avbildningen som matrismultiplikation är som en tredimensionell tensor. Där används att det finns en exakt korrespondens mellan multiplikationsalgoritmens tidskomplexitet och tensorrang. Det sistnämnda är ett slags generalisering av matrisrang, och är minsta antalet termer en tensor kan skrivas som. Till skillnad frpn matrisrang finns ingen allmän metod för att beräkna tensorrang, och många värden är okända även för välstuderade och viktiga tensorer. För att hitta fler övre begränsningar på matrismultiplikations tidskomplexitet har stödrang av tensorer införts, som är den lägsta rangen bland tensor med samma stöd i en viss bas. Målet med detta examensarbete har varit att göra en genomgång av historien om snabbare matrismultiplikation, samt att specifikt undersöka egenskaper av stödrang för allmänna tredimensionella tensorer. För det sistnämnda görs en fullständig klassificering av rangstrukturer bland stödklasser för den minsta icke-degenererade tensorprodukten av tre vektorrum. Slutsatser är bl.a. att storleken av ett stöd kan ses påverka antalet möjliga ranger inom en stödklass. Samtidigt finns i allmänhet ingen symmetri med avseende på stödstorlek i stödklassernas rangstrukturer. Detta trots att det finns en symmetri och bijektion mellan speglade stöd. I arbetet ingår även en diskussion om hur stödrangstrukturer skulle kunna klassificeras för större tensorprodukter.

Title: Homotopic structures in algebra, geometry and mathematical physics Author: Eva Černohorská Department: Mathematical Institute of Charles University Supervisor: RNDr. Martin Markl, DrSc., Institute of Mathematics of the Academy of Sciences of the Czech Republic, Mathematical Institute of Charles University Abstract: The aim of this thesis was to generalize the result that associative algebras on finite dimensional vector spaces can be described using differentials on free algebras. This result is limited by the duality of vector spaces. If we assume that the underlying space has a linear topology, then we can use the duality between discrete and linearly compact (profinite) vector spaces. To generalize the notion of an algebra, we need to recall the completed tensor product on linear vector spaces. Since this topics does not seem to be sufficiently covered by the literature, this thesis could serve also as a comprehensive text on linear vector spaces and their completed tensor products. We prove that also A∞ structures on linearly compact vector spaces could be represented by differentials on a free algebra. Keywords: Strongly homotopy associative algebra, linear topological vector space, Pontryagin duality, completed tensor product, differential

[pt] Um operador definido em um espaço de Hilbert é uniformemente estável se ele converge na topologia da norma para o operador nulo. O problema de Estabilidade Multiplicativa investiga quais são as classes de operadores que estabilizam uniformemente o operador original por uma perturbação multiplicativa. Neste trabalho colocamos este problema no contexto de produto tensorial e investigamos quais as classes que estabilizam multiplicativamente Contrações Fortemente Estáveis sob uma perturbação compacta. Em particular, apresentamos uma solução para o Problema de Estabilidade Multiplicativa para Contrações Fortemente Estáveis. / [en] An operator on a Hilbert space is uniformly stable if it converges to the null operator on the norm topology. The Multiplicative Stabilization Problem investigates which operators classes uniformly stabilize de original operator under multiplicative perturbation. This work consider the previous problem under the tensor product framework and investigates which operators classes multiplicative stabilize Strongly Stable Contraction under compact perturbations. We have established a solution to the Multiplicative Stabilization Problem for Strongly Stable Contractions.

Les méthodes de décomposition de domaine en espace ont prouvé leur utilité dans le cadre des architectures parallèles. Pour les problèmes d’évolution en temps, il est nécessaire d’introduire une dimension supplémentaire de parallélisme dans la direction du temps. Ceci peut alors être couplé avec des méthodes de type optimisé Schwarz waveform relaxation. Nous nous intéressons dans cette thèse aux méthodes directes de décomposition en temps. Nous en étudions particulièrement deux. Dans une première partie nous étudions la méthode de produit tensoriel, introduite par R. E. Lynch, J. R. Rice, et D. H. Thomas in 1963. Nous proposons une méthode d’optimisation des pas de temps, basée sur une étude d’erreur en variable de Fourier en temps. Nous menons cette étude sur les schémas d’Euler et de Newmark pour la discrétisation en temps de l’équation de la chaleur. Nous présentons ensuite des tests numériques établissant la validité de cette approche. Dans la seconde partie, nous étudions les méthodes dites de Bloc, introduites par Amodio et Brugnano en 1997. Nous comparons diverses implémentations de la méthode, basées sur différentes approximations de l’exponentielle de matrice. Nous traitons l’équation de la chaleur et l’équation des ondes, et montrons par une étude numérique bidimensionnelle la puissance de la méthode. / Domain decomposition methods in space applied to Partial Differential Equations (PDEs) expanded considerably thanks to their effectiveness (memory costs, calculation costs, better local conditioned problems) and this related to the development of massively parallel machines. Domain decomposition in space-time brings an extra dimension to this optimization. In this work, we study two different direct time-parallel methods for the resolution of Partial Differential Equations. The first part of this work is devoted to the Tensor-product space-time method introduced by R.E. Lynch, J. R. Rice, and D. H. Thomas in 1963. We analyze it in depth for Euler and Crank-Nicolson schemes in time applied to the heat equation. The method needs all time steps to be different, while accuracy is optimal when they are all equal (in the Euler case). Furthermore, when they are close to each other, the condition number of the linear problems involved becomes very big. We thus give for each scheme an algorithm to compute optimal time steps, and present numerical evidences of the quality of the method. The second part of this work deals with the numerical implementation of the Block method of Amodio and Brugnano presented in 1997 to solve the heat equation with Euler and Crank- Nicolson time schemes and the elasticity equation with Euler and Gear time schemes. Our implementation shows how the method is accurate and scalable.

Dans le premier chapitre, nous commençons par définir certains produits tensoriels et identifions leur dual. Nous donnons ensuite quelques propriétés des classes de Schatten. La fin du chapitre est dédiée à l’étude des espaces de Bochner à valeurs dans l'espace des opérateurs factorisables par un espace de Hilbert. Le deuxième chapitre est consacré aux multiplicateurs de Schur linéaires. Nous caractérisons les multiplicateurs bornés sur B(Lp, Lq) lorsque p est inférieur à q puis appliquons ce résultat pour obtenir de nouvelles relations d'inclusion entre espaces de multiplicateurs. Dans le troisième chapitre, nous caractérisons, au moyen de multiplicateurs de Schur linéaires, les multiplicateurs de Schur bilinéaires continus à valeurs dans l'espace des opérateurs à trace. Dans le quatrième chapitre, nous donnons divers résultats concernant les opérateurs intégraux multiples. En particulier, nous caractérisons les opérateurs intégraux triples à valeurs dans l'espace des opérateurs à trace puis nous donnons une condition nécessaire et suffisante pour qu'un opérateur intégral triple définisse une application complètement bornée sur le produit de Haagerup de l'espace des opérateurs compacts. Enfin, le cinquième chapitre est dédié à la résolution des problèmes de Peller. Nous commençons par étudier le lien entre opérateurs intégraux multiples et théorie de la perturbation pour le calcul fonctionnel des opérateurs autoadjoints pour finir par la construction de contre-exemples à ces problèmes. / In the first chapter, we define some tensor products and we identify their dual space. Then, we give some properties of Schatten classes. The end of the chapter is dedicated to the study of Bochner spaces valued in the space of operators that can be factorized by a Hilbert space.The second chapter is dedicated to linear Schur multipliers. We characterize bounded multipliers on B(Lp, Lq) when p is less than q and then apply this result to obtain new inclusion relationships among spaces of multipliers.In the third chapter, we characterize, by means of linear Schur multipliers, continuous bilinear Schur multipliers valued in the space of trace class operators. In the fourth chapter, we give several results concerning multiple operator integrals. In particular, we characterize triple operator integrals mapping valued in trace class operators and then we give a necessary and sufficient condition for a triple operator integral to define a completely bounded map on the Haagerup tensor product of compact operators. Finally, the fifth chapter is dedicated to the resolution of Peller's problems. We first study the connection between multiple operator integrals and perturbation theory for functional calculus of selfadjoint operators and we finish with the construction of counter-examples for those problems.