We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian groups.

Given an action of a countable group with a quasi-invariant measure, there exists a multiplicative group in (0, ∞), called the ratio set of the group action, which in a sense describes the values of the Radon-Nikodym derivative. The main purpose of this thesis is to find the ratio set of the action of a finitely generated free group Ƒ on its topological boundary ∂Ƒ (the set of infinite words) for a certain natural class of quasi-invariant boundary measures. -- In Section 1, we focus on the general ergodic theory of equivalence relations. We outline the set-up, borrow from [1], [4] the definitions of the central notions of the theory, including counting measures (Proposition 1.8), quasi-invariance (Definition 1.6), Radon-Nikodym cocycle (Definition 1.15) and raio set (Definition 1.19), and illustrate them on the example of the orbit equivalence relation of a Markov shift (Definition 1.22). We also introduce the principal object: the boundary action of a finitely generated free group (see Section 1.2). -- In Section 2, we define the class of multiplicative Markov measures (Definition 2.1). These are the measures on a topological Markov chain entirely determined just by an initial (base) distribution and the admissibility matrix; the transition probabilities are then just the normalized restrictions of the base distribution onto the set of admissible transitions (see [7]). In the case of the free group, its boundary has a natural structure of a topological Markov chain (determined by the irreducibility condition from the definition of a free group: consecutive letters should not cancel each other), and in this case, we show that the multiplicative Markov measures are precisely the ones for which the Radon-Nikodym cocycle is a product cocycle (i.e. a cocycle whose potential only depends on the first letter of the input; see Definition 2.8). The final result of this section is an explicit description of the ratio set of the boundary action with respect to multiplicative Markov measures. -- In Section 3, given a probability measure 𝜇 on the set of free generators and their inverses, the definition of the associated nearest neighbor random walk is given. According to Furstenberg's Theorem (proof provided in Appendix), in this random walk, sample paths converge almost surely to a random boundary point, and the resulting limit distribution on the boundary of the free group is called the harmonic measure of the random walk (see Section 3.1). We show that the harmonic measure is a multiplicative measure (Theorem 3.3), and therefore the results of Section 2 allow us to describe the ratio set of the harmonic measure (Theorem 3.5). A significant role in these considerations is played by the passage probabilities of the random walk (given a group element, the probability that it is ever visited by a random walk). Since the harmonic measure is multiplicative, its potential only depends on the first letter, and this dependence actually amounts to taking the inverse of the corresponding passage probability (Proposition 2.9, Remark 2.10). Finally, we establish a one-to-one correspondence between three families of numbers indexed by the alphabet of the free group and subject to natural conditions; these are the step distributions of the random walk, the base of the harmonic measure (which is multiplicative Markov) and the family of passage probabilities (Theorem 3.6). -- In Section 4, we discuss another method for finding the ratio set of the harmonic measure based on using Martin theory (see [2]). -- In the Appendix, we prove Furstenberg's Theorem, a result used for defining the harmonic measure in Section 3. Actually, it is applicable not only for the nearest neighbor random walk (i.e. not only when the probability measure 𝜇 is supported on the alphabet set) but also the more general case where the support of the step distribution generates the free group. Moreover, in addition to the existence it also characterizes the harmonic measure as the unique 𝜇-stationary measure on the boundary

Dans ce mémoire, on s'intéresse à l'action du groupe des transformations affines et des homothéties sur l'axe du temps des systèmes différentiels quadratiques à foyer faible d'ordre trois, dans le plan. Ces systèmes sont importants dans le cadre du seizième problème d'Hilbert. Le diagramme de bifurcation a été produit à l'aide de la forme normale de Li dans des travaux de Andronova [2] et Artès et Llibre [4], sans utiliser le plan projectif comme espace des paramètres ni de méthodes globales. Dans [7], Llibre et Schlomiuk ont utilisé le plan projectif comme espace des paramètres et des notions à caractère géométrique global (invariants affines et topologiques). Ce diagramme contient 18 portraits de phase et certains de ces portraits sont répétés dans des parties distinctes du diagramme. Ceci nous mène à poser la question suivante : existe-t-il des systèmes distincts, correspondant à des valeurs distinctes de paramètres, se trouvant sur la même orbite par rapport à l'action du groupe? Dans ce mémoire, on prouve un résultat original : l'action du groupe n'est pas triviale sur la forme de Li (théorème 3.1), ni sur la forme normale de Bautin (théorème 4.1). En utilisant le deuxième résultat, on construit l'espace topologique quotient des systèmes quadratiques à foyer faible d'ordre trois par rapport à l'action de ce groupe. / We are interessed here in the action of the group of affine transformations and time homotheties on quadratic differential systems which have a weak focus of third order. These systems are important for Hilbert sixteenth problem. The bifurcation diagram was produced using Li's normal form in the articles of Andronova [2], and Artès and Llibre [4], without using the projective plane as parameter space, and without using global methods. In [7], Llibre and Schlomiuk used the projective plane as parameter space and global geometric methods (affine and topological invariants). This diagram contains 18 phase portraits and some of these portraits are repeated in distinct parts of the diagram. This led us to ask the following question : do there exist distinct differential systems, corresponding to distinct values of the parameter, which are on the same orbit of the group action? In this master's thesis, we prove an original result : the action of the group is not trivial on Li's normal form (theorem 3.1), neither is it trivial on Bautin's normal form (theorem 4.1). Using the second result, we construct the quotient topological space of these systems with a weak focus of third order, with respect to the group action.

Šiame darbe nagrinėjamas grupės ieškinio teisinis institutas kaip priemonė ginti viešuosius ir privačius interesus. Pirmoje dalyje siekiama atskleisti teisinio intereso sąvoką, viešojo ir privataus interesų sampratas, bei analizuojame galimybes šiuos interesus apginti pasitelkus grupės ieškinio institutą. Antrojoje dalyje nagrinėjame klasikinio ir kontinentinio grupės ieškinio instituto pagrindinius požymius bei apžvelgiame grupės ieškinio instituto ypatumus Lietuvos Respublikos civiliniame procese. Trečiojoje dalyje apžvelgiami pagrindiniai grupės ieškinio turinio reikalavimai, jo priėmimo sąlygos bei priimamo teismo sprendimo ypatumai. Magistro darbo tikslas pateikti mokslinį grupės ieškinio instituto vertinimą ginant pažeistus viešuosius ir privačius interesus. Daugelis užsienio šalių, savo įstatymuose įtvirtindamos grupės ieškinio institutą, visada jį lygina su Jungtinėse Amerikos Valstijose egzistuojančiu modeliu, todėl darbe daugiausia remsimės šios šalies civilinio proceso įstatymais. Nagrinėdami teisės literatūrą sutiksime teisės mokslininkų poziciją, kad grupės ieškiniai Europoje neegzistuoja, nes, anot jų, grupės ieškiniais vadintini tik tokie ieškiniai, kuriuose reikalavimus reiškia grupės nariai. Magistro darbe analizuosime šią nuomonę. Grupės ieškinio paplitimui Europos šalyse didžiausią įtaką padarė viešojo intereso bei kolektyvinių vartotojų interesų gynimo paieška. Todėl išsamiau panagrinėsime pačią teisinio intereso sąvoką, viešojo ir... [toliau žr. visą tekstą] / Legal institute of group action, as a measure of protecting public and private interests, is tackled in this work. The first part includes the explanation of legal interest, the concepts of public and private interests, as well as the analysis of possibilities of protection of these interests, using the institute of group action. The second part explains the main features of classic and continental institute of group action, as well as the peculiarities of the institute of group action in the Lawsuit of the Republic of Lithuania. The third part comprises of the main content requirements to the group action, the conditions of its acceptation and the peculiarities of awarded judgment. The goal of this Master’s Thesis – to present scientific evaluation of the institute of group action, while protecting violated public and private interests. Various foreign countries consolidating the institute of group action in their Law, compare it to the model existing in the United States of America, therefore our thesis will mainly be based on the Lawsuit of the United States of America. While tackling the legal literature, we will meet the position of legal scientists, claiming the group action do not exist in Europe, as group actions may only be called the claims when the claims are represented by the group members. We will analyze this opinion in our Master’s Thesis. The search for protection of public interest and collective interests of the users had a major... [to full text]

A composition algebra is a non-zero algebra endowed with a strictly non-degenerate, multiplicative quadratic form. Finite-dimensional composition algebras exist only in dimension 1, 2, 4 and 8 and are in general not associative or unital. Over the real numbers, such algebras are division algebras if and only if they are absolute valued, i.e. equipped with a multiplicative norm. The problem of classifying all absolute valued algebras and, more generally, all composition algebras of finite dimension remains unsolved. In dimension eight, this is related to the triality phenomenon. We approach this problem using a categorical language and tools from representation theory and the theory of algebraic groups. We begin by considering the category of absolute valued algebras of dimension at most four. In Paper I we determine the morphisms of this category completely, and describe their irreducibility and behaviour under the actions of the automorphism groups of the algebras. We then consider the category of eight-dimensional absolute valued algebras, for which we provide a description in Paper II in terms of a group action involving triality. Then we establish general criteria for subcategories of group action groupoids to be full, and applying this to the present setting, we obtain hitherto unstudied subcategories determined by reflections. The reflection approach is further systematized in Paper III, where we obtain a coproduct decomposition of the category of finite-dimensional absolute valued algebras into blocks, for several of which the classification problem does not involve triality. We study these in detail, reducing the problem to that of certain group actions, which we express geometrically. In Paper IV, we use representation theory of Lie algebras to completely classify all finite-dimensional absolute valued algebras having a non-abelian derivation algebra. Introducing the notion of quasi-descriptions, we reduce the problem to the study of actions of rotation groups on products of spheres. We conclude by considering composition algebras over arbitrary fields of characteristic not two in Paper V. We establish an equivalence of categories between the category of eight-dimensional composition algebras with a given quadratic form and a groupoid arising from a group action on certain pairs of outer automorphisms of affine group schemes

>Magister Scientiae - MSc / The groups we consider in this study belong to the class Xo of all finitely generated groups with finite commutator subgroups. We shall eventually narrow down to the groups of the form T)<lw zn for some nE N and some finite abelian group T. For a Xo-group H, we study the non-cancellation set, X(H), which is defined to be the set of all isomorphism classes of groups K such that H x Z ~ K x Z. For Xo-groups H, on X(H) there is an abelian group structure [38], defined in terms of embeddings of K into H, for groups K of which the isomorphism classes belong to X(H). If H is a nilpotent Xo-group, then the group X(H) is the same as the Hilton-Mislin (see [10]) genus group Q(H) of H. A number of calculations of such Hilton-Mislin genus groups can be found in the literature, and in particular there is a very nice calculation in article [11] of Hilton and Scevenels. The main aim of this thesis is to compute non-cancellation (or genus) groups of special types of .Xo-groups such as mentioned above. The groups in question can in fact be considered to be direct products of metacyclic groups, very much as in [11]. We shall make extensive use of the methods developed in [30] and employ computer algebra packages to compute determinants of endomorphisms of finite groups.

In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern, we show how to find the equation determining the spectrum. We give two length 4 applications, and a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.

A tutela dos interesses metaindividuais em juízo e a busca de mecanismos que assegurem que o processo seja instrumento para o acesso substancial à justiça inserem-se no contexto das ondas renovatórias do processo civil. Embora o movimento em questão situe-se cronologicamente na década de 1960, o ordenamento jurídico pátrio ainda não fez fluir simultaneamente as três ondas. Os denominados interesses metaindividuais abrangem tanto os interesses essencialmente coletivos, nos quais se incluem os difusos e coletivos, quanto os interesses doutrinariamente denominados de acidentalmente coletivos, correspondentes, nos termos do microssistema processual coletivo, ao conceito de interesse individual homogêneo. No Brasil, a tutela jurisdicional de interesses metaindividuais pode se dar através do processo coletivo e, também, por meio do processo civil tradicional, caso apresentem viés estritamente individual. A dificuldade de enquadramento de determinados interesses dentro dos conceitos legais dados pelo microssistema processual coletivo é recorrente no cotidiano jurisprudencial. O processo civil vigente não apresenta mecanismo normativo apto a evitar a concomitância entre ações individuais e coletivas sobre uma mesma questão jurídica, razão pela qual o sistema convive com demandas de massa repetitivamente levadas ao Poder Judiciário. O processo coletivo, no entanto, não é a única alternativa processual existente para o manejo racional de processos repetitivos: as denominadas ações de grupo prestam-se à resolução isonômica de tais demandas sem que, para tanto, sejam necessárias ficções representativas e grandes debates acerca da legitimidade. O presente trabalho presta-se à análise de tais possibilidades, tendo por norte a otimização do alcance subjetivo da decisão judicial que trate de interesse metaindividual. Encontra-se em trâmite na Câmara dos Deputados o projeto do Novo Código de Processo Civil, que traz, dentre as inovações destinadas ao tratamento de demandas de massa, o incidente de resolução de demandas repetitivas, a centralização de processos repetitivos e a conversão da ação individual em coletiva, mecanismos estes que integrarão o objeto do presente estudo. / The judicial protection of group rights and the search for mechanisms that ensure that civil procedure is actually an instrument for substantial access to justice fall into the context of the waves of reform. Although such motion is chronologically situated in the 1960s, Brazilian Law has not yet fulfilled the task of simultaneously flowing the three waves. The so called group rights refer, in Brazilian Law, both essencially and occasionally group rights. The latter refers, in Brazilian Civil Procedure, to the concept of homogeneous individual rights. In Brazil, the judicial protection of group rights can be led through class actions or individual suits, as long as the right involved has a strictly individual aspect. The difficulty of framing some rights into the legal concepts is iterant in Brazilian Courts daily routine. The current Civil Procedure does not have tools able to avoid the concomitance between individual suits and class actions that refer to the same legal issue. For that reason, the judiciary deals with repetitive suits. Class actions, however, are not the only procedural alternative to rationally dealing with repetitive suits: group actions also aim to adequately resolving suits, but without involving issues as adequacy of representation or further concerns related to standing to sue. This research is dedicated to the analysis of such possibilities, having in mind the optimization of the subjective range of judicial decisions that refer to group rights. The Project of the new Brazilian Civil Procedure Code, currently at the Brazilian Chamber of Deputies, brings, among the innovations that intend to deal with repetitive suits, the incident of resolution of repetitive suits, the centralization of repetitive suits and the possibility of convertion of individual suits into class actions. All of these innovations will also be analised.

Rapid growth in population along with positive trends in global economy over the past several decades has significantly contributed to an increased demand for various infrastructure needs worldwide. For this reason, the focus of this thesis has been directed towards extending the mechanics of unsaturated soils, which is an emerging geotechnical engineering field to investigate the behavior of two key infrastructure systems, namely pile foundations and energy pipeline systems. The mechanism of soil-pile foundations and soil-pipeline systems interaction behavior has several similarities. Both these infrastructure facilities require comprehensive understanding of the soil-structure interaction mechanism. Reliable estimation of mechanical properties of both the soil and the soil-structure interface is required for the rational interpretation the load-displacement behavior of pile foundations and pipeline systems. Currently, the design of systems is predominantly based on design codes and guidelines that use empirical procedures or employ the principles of saturated soil mechanics. In many scenarios, pile foundations extend either totally or partly in unsaturated soils as the groundwater table level in many regions is at a greater depth. Such scenarios are commonly encountered in semi-arid and arid regions of the world. In addition, pipeline systems are typically buried at shallow depths in unsaturated soil strata, which are susceptible to wetting and drying, freezing and thawing cycles or both, due to seasonal environmental changes. Capillary stress or matric suction in the unsaturated zone increases the effective stress contribution towards the shear strength and stiffness of soil and soil-structure interface. Extending saturated soil mechanics to design or analyze such structures may lead to erroneous estimation of pile foundation carrying capacity or loads transferred on pipeline body from the surrounding unsaturated soil. Experimental, analytical and numerical investigations were undertaken to study the behavior of single pile, pile group, and pipeline systems in saturated and unsaturated sands under static loading. The experimental program includes 40 single model pile and 2×2 pile group, and six prototype pipeline tests under saturated and unsaturated condition. The results of the experimental studies suggest that matric suction has significant contribution towards the mechanical behavior of both pile foundation and pipeline system. The axial load carrying capacity of single pile and pile group increased approximately 2 to 2.5 times and the settlement reduced significantly compared to saturated condition. The influence of matric suction towards a single pile is significantly different in comparison to pile group behavior. The cumulative influence of matric suction and stress overlap of pile group behavior in sandy soils result in erroneous estimation of pile group capacity, if principles of saturated soil mechanics are extended. Group action plays major role in changing the moisture regime under the pile group leading to incompatible stress state condition in comparison to single pile behavior. On the other hand, the peak axial load on the pipe is almost 2.5 folds greater in unsaturated sand that undergoes much less displacement in comparison to saturated condition. Such an increase in the external axial forces may jeopardize the integrity of energy pipeline systems and requires careful reevaluation of existing design models extending the principles of unsaturated soil mechanics. Two analytical design models to estimate the axial force exerted on pipeline body were proposed. The proposed models take account of matric suction effect and soil dilatancy and provide smooth transition from unsaturated to saturated condition. These models were developed since measurement of the unsaturated soil and interface shear strength and stiffness properties need extensive equipment that require services of trained professional, which are expensive and time consuming. The models utilize the saturated soil shear strength parameters and soil-water characteristic curve (SWCC) to predict the mechanical behavior of the structure in saturated and unsaturated cohesionless soils. The prototype pipeline experimental results were used to verify the proposed models. The predicted axial force on pipeline using the proposed models agrees well with the measured behavior under both saturated and unsaturated conditions. Moreover, numerical techniques were proposed to investigate the behavior of pile foundation and pipeline system in saturated and unsaturated sand. The proposed methodology can be used with different commercially available software programs. Two finite element analysis programs were used in this study; namely, PLAXIS 2D (2012) to simulate soil-pile foundation behavior and SIGMA/W (2012) to simulate soil-pipeline system behavior. The proposed techniques require the information of unsaturated shear strength and stiffness, which can be derived from saturated soil properties and the SWCC. The model was verified using pile and pipeline test results from this study and other research studies from the published literature. There is a good agreement between the measured behavior and the predicted behavior for both the saturated and unsaturated conditions. The methodology was further extended to investigate the behavior of rigid and flexible pipelines buried in Indian Head till (IHT) during nearby soil excavation activity. The simulation results suggest that excavation can be extended safely without excessive deformation to several meters without the need for supporting system under unsaturated condition. The studies summarized in the thesis provide evidence that the principles of saturated soil mechanics underestimate the pile foundations carrying capacity as well as the axial force exerted on pipelines in unsaturated soils. Such approaches lead to both uneconomical pile foundation and unsafe pipeline systems designs. For this reason, the pile and pile group carrying capacity and pipeline axial force should be estimated taking into account the influence of matric suction as well as the dilatancy of the compacted sand. The experimental studies, testing techniques along with the analyses of test results and the proposed analytical and numerical models are useful for better understanding the pile foundation and buried pipeline behaviors under both saturated and unsaturated conditions. The proposed analytical and finite element models are promising for applying the mechanics of unsaturated soils into conventional geotechnical engineering practice using simple methods.

[pt] Temos como objetivo principal classificar, a menos de semi-conjugação, os grupos de homeomorfismos crescentes da reta, com no máximo N pontos fixos. Começando pelos casos já classificados, N igual a 0 pelo Teorema de Holder e N igual a 1 pelo Teorema de Solodov, nosso próximo passo será classificar o caso N igual a 2. Para isso, iremos introduzir a noção de Abertura de Órbita como uma semi-conjugação dos grupos de homeomorfismos. / [en] We have as main objective to classify, by semi-conjugation, the groups of increasing homeomorphisms of the line, with at most N fixed points. Starting with the cases already classified, N equal to 0 by Holder s Theorem and N equal to 1 by Solodov s Theorem, our next step will be to classify the case N equal to 2. For this, we will introduce the notion of Orbital Opening as a semiconjugation of the groups of homeomorphisms.