In this thesis we demonstrate the algorithmic usefulness of the so-called Mal'cev correspondence for computations with infinite polycyclic groups. This correspondence between Q-powered nilpotent groups and rational nilpotent Lie algebras was discovered by Anatoly Mal'cev in 1951. We show how the Mal'cev correspondence can be realized on a computer. We explore two possibilities for this purpose and compare them: the first one uses matrix embeddings and the second the Baker-Campbell-Hausdorff formula. Then, we describe a new collection algorithm for polycyclically presented groups, which we call Mal'cev collection. Algorithms for collection lie at the heart of most methods dealing with polycyclically presented groups. The current state of the art is "collection from the left" as recently studied by Gebhardt, Leedham-Green/Soicher and Vaughan-Lee. Mal'cev collection is in some cases dramatically faster than collection from the left, while using less memory. Further, we explore how the Mal'cev correspondence can be used to describe symbolically the collection process in polycyclically presented groups. In particular, we describe an algorithm that computes the collection functions for splittable polycyclic groups. This algorithm is based on work by du Sautoy. We apply it to the computation of pro-p-completions of polycyclic groups. Finally we describe a practical algorithm for testing polycyclicity of finitely generated rational matrix groups. Previously, not only did no such method exist but it was not clear whether this question was decidable at all. Most of the methods described in this thesis are implemented in the computer algebra system GAP and publicly available as part of the GAP packages Guarana and Polenta. Reports on the implementation including runtimes for some examples are given at the appropriate places.

Made available in DSpace on 2016-04-27T17:27:08Z (GMT). No. of bitstreams: 1 Jesuino Junior Pires.pdf: 1325069 bytes, checksum: bb85f406de727e9479fc7148c6297230 (MD5) Previous issue date: 2014-09-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work aims to investigate the distinction between content and object of presentation as presented by Kasimir Twardowski in his opuscule Zur Lehre vom Inhalt und Gegenstand der Vorstellungen. Our object of study is the distinction between the content and the object of Twardowski before Brentano and Höfler. This research is characterized as a literature and the procedures are meant for reconstitution, analysis and reflection on the topic. The text is structured in three parts: the first part deals with some points developed by Brentano in his work Psychologie vom Empirischen Stankpunkt, especially with regard to the distinction between psychic and physical phenomena, and, consequently, on his assertion that the whole psychic phenomenon there is something given as immanent object. The second part presents the distinction between content and object, and the particularities of Twardowski before Höfler and Brentano. The main criticism of Twardowski to Brentano is that, like the term presentation , the term presented is also ambiguous and what is meant by immanent object , is in fact the content of representation. For this reason, it is proposed a threefold distinction of psychic phenomenon: act, content and object. This award culminates in the formation of one theory of objects in general, advocated by Twardowski. In the last part of the text some problematic issues are identified regarding the Twardowski s theory mainly on the notion of content and the concept of reality and possibility. Thus, we can infer that the particularities of Twardowski s distinction must be evidenced, firstly by his conception of content as a mediating body between the act and the object of representation; secondly, because this distinction has its linguistic counterpart from the functions name; and finally, through different logical functions performed by the adjective the presented / O presente trabalho tem por objetivo investigar a distinção entre conteúdo e objeto de representação tal como apresentada por Kasimir Twardowski em seu opúsculo Zur Lehre vom Inhalt und Gegenstand der Vorstellungen. Nosso objeto de estudo é a distinção entre conteúdo e objeto de Twardowski perante Brentano e Höfler. Esta pesquisa se caracteriza como uma pesquisa bibliográfica e os procedimentos são voltados para a reconstituição, análise e reflexão sobre o tema. O texto está estruturado em três partes: a primeira parte trata de alguns pontos desenvolvidos por Brentano em sua obra Psychologie vom Empirischen Standpunkt, principalmente no que diz respeito à distinção entre fenômenos psíquicos e fenômenos físicos e, consequentemente, sobre sua afirmação de que a todo fenômeno psíquico há algo dado como objeto imanente. A segunda parte apresenta a distinção entre conteúdo e objeto e as particularidades de Twardowski perante Höfler e Brentano. A principal crítica de Twardowski a Brentano é que, assim como o termo representação , o termo representado também é ambíguo e aquilo que se entende por objeto imanente é, em realidade, o conteúdo de representação. Propõe-se, assim, uma tripla distinção do fenômeno psíquico em: ato, conteúdo e objeto. Esta distinção culmina na constituição de uma teoria dos objetos em geral defendida por Twardowski. Na última parte do texto algumas questões problemáticas são apontadas no tocante à teoria de Twardowski, principalmente sobre a noção de conteúdo e sobre o conceito de realidade e possibilidade. Inferimos assim, que as particularidades da distinção de Twardowski podem ser evidenciadas: primeiro, por sua concepção de conteúdo como uma instância mediadora entre o ato e o objeto de representação; segundo, porque esta distinção tem sua contraparte linguística a partir das funções do nome; e terceiro, pelas diferentes funções lógicas exercidas pelo adjetivo o representado

A motivational problem for group homology is a conjecture of Quillen that states, as reformulated by Anton, that the second homology of the general linear group over R = Z[1/p; ζp], for p an odd prime, is isomorphic to the second homology of the group of units of R, where the homology calculations are over the field of order p. By considering the group extension spectral sequence applied to the short exact sequence 1 → SL2 → GL2 → GL1 → 1 we show that the calculation of the homology of SL2 gives information about this conjecture. We also present a series of algorithms that finds an upper bound on the second homology group of a finitely-presented group. In particular, given a finitely-presented group G, Hopf's formula expresses the second integral homology of G in terms of generators and relators; the algorithms exploit Hopf's formula to estimate H2(G; k), with coefficients in a finite field k. We conclude with sample calculations using the algorithms.

We investigate various relationships between categories of functors. The major examples are given by extending some duality to a larger structure, such as an adjunction or a recollement of abelian categories. We prove a theorem which provides a method of constructing recollements which uses 0-th derived functors. We will show that the hypotheses of this theorem are very commonly satisfied by giving many examples. In our most important example we show that the well-known Auslander-Gruson-Jensen equivalence extends to a recollement. We show that two recollements, both arising from different characterisations of purity, are strongly related to each other via a commutative diagram. This provides a structural explanation for the equivalence between two functorial characterisations of purity for modules. We show that the Auslander-Reiten formulas are a consequence of this commutative diagram. We define and characterise the contravariant functors which arise from a pp-pair. When working over an artin algebra, this provides a contravariant analogue of the well-known relationship between pp-pairs and covariant functors. We show that some of these results can be generalised to studying contravariant functors on locally finitely presented categories whose category of finitely presented objects is a dualising variety.

In this thesis we prove the following: The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup. The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type. The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem.

Summary Speaking about peculiarities of Cechovas creation in scientific literature, the two main features should be pointed out, i.e, laconic depiction and writer’s ability to present three periods of a character life in a short story. In most of his works the action is short, but it reveals a long period of the hero’s life. Taking this feature into consideration we have the aim to analyze five stories and to prove that short stories may contain the past, present and future prospects are presented. At the same time the content highlights plenty of hero’s life moments, which are of great importance. The attention was paid on the open endings. Cechovas manages to end his stories in a particular way, that for the reader the future of all the heroes in the story is clear.

In the presented work we focus on applications of decision problems from combinatorial group theory. Namely we analyse the Shpilrain-Zapata pro- tocol. We give formal proof that small cancellation groups are good platform for the protocol because the word problem is solvable in linear time and they are generic. We also analyse the complexity of the brute force attack on the protocol and show that in a theoretical way the protocol is immune to attack by adversary with arbitrary computing power.

Nous étudions un certain monoïde défini par une présentation, notée P, qui est un hybride de celles du monoïde de tresses infinies et du monoïde de Thompson. Pour cela, nous utilisons plusieurs approches. On décrit d’abord un système de réécriture convergent pour la présentation P, ce qui fournit en particulier une solution au problème de mots de P et rapproche le monoïde hybride du monoïde de Thompson. Puis, suivant le modèle du monoïde de tresses, on utilise la méthode du retournement de facteur pour analyser la relation de divisibilité à gauche, et montrer en particulier que le monoïde hybride admet la simplification et des ppcm à droite conditionnels. Ensuite, on étudie la combinatoire de Garside de l'hybride: pour chaque entier n, on introduit un élément ∆(n) comme ppcm à droite des (n−1) premiers atomes, et on étudie les diviseurs à gauche des éléments ∆(n), appelés éléments simples. Les principaux résultats sont les dénombrement des diviseurs à gauche de ∆(n) et la détermination effective des formes normales des éléments simples. On termine en construisant des représentations du monoïde hybride dans divers monoïdes, en particulier une représentation dans des matrices à coefficients polynômes de Laurent dont on conjecture qu’elle est fidèle. / We study a certain monoid specified by a presentation, denoted P, that is a hybrid of the classical presentation of the infinite braid monoid and of the presentation of Thompson’s monoid. To this end, we use several approaches. First, we describe a convergent rewrite system for P, which provides in particular a solution to the word problem, and makes the hybrid monoid reminiscent of Thompson’s monoid. Next, on the shape of the braid monoid, we use the factor reversing method to analyze the divisibility relation, and show in particular that the hybrid monoid admits cancellation and conditional right lcms. Then, we study Garside combinatorics of the hybrid: for every integer n, we introduce an element ∆(n) as the right lcm of the first (n−1) atoms, and one investigates the left divisors of the elements ∆(n), called simple elements. The main results are a counting of the left divisors of ∆(n) and a characterization of the normal forms of simple elements. We conclude with the construction of several representations of the hybrid monoid in various monoids, in particular a representation in a monoid of matrices whose entries are Laurent polynomials, which we conjecture could be faithful.

People's inability to cope with the demands of modern life, has led to a significant increase in the incidence of anxiety being experienced by people from all walks of life. People are suffering from anxiety without knowing the root cause of it and it was found from the anxiety cases studied that its origin can in many cases be traced to the birth experience where specific negative suggestions were imprinted on the subconscious. In this study the influence of the birth experience as the underlying cause of anxiety and the effect of it on the formation of the self-concept and self-actualisation have been explored. The research shows that a subconscious origin exists in the development of anxiety. In the research for this study, Medical Hypnoanalysis, which is based on a process of diagnosis and therapy, was used as therapeutic method with several clients suffering from anxiety symptoms. In this research the perceptions formed in the subconscious during the birth experience were examined and related to the anxiety experienced during childhood and later life. The case studies, their diagnoses and the follow-up therapeutic sessions relating to the birth experience were investigated and discussed. Educational Psychologists show a specific interest in and a tendency to get involved with, or implement Medical Hypnoanalysis in therapy. It became obvious from this study that Medical Hypnoanalysis supplements the needs of the Educational Psychologist as a tool in therapy. In this study guide lines have been presented to the Educational Psychologist for the treatment of anxiety originating from the birth trauma by regressing the client back to this traumatic experience. Suggestions on how to conduct the birth regression sessions have also been presented and clarified. Through the case studies the identification and removal of the highly charged emotional and negative beliefs that were responsible for the anxiety symptom have been demonstrated. The case studies have furthermore demonstrated the successful utilisation of positive and healing suggestions to achieve the therapeutic goals. The results of this study show that Medical Hypnoanalysis can be used effectively in the treatment of the root causes of anxiety. / Psychology of Education / D. Ed. (Psychology of Education)

In this thesis we study several properties of finitely presented groups, through the unifying paradigm of encoding sought-after group properties into presentations and detecting group properties from presentations, in the context of Geometric Group Theory. A group law is said to be detectable in power subgroups if, for all coprime m and n, a group G satisfies the law if and only if the power subgroups G(^m) and G(ⁿ) both satisfy the law. We prove that for all positive integers c, nilpotency of class at most c is detectable in power subgroups, as is the k-Engel law for k at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group W such that W(²) and W(³) are metabelian but W has derived length 3. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result. We construct a census of two-generator one-relator groups of relator length at most 9, with complete determination of isomorphism type, and verify a conjecture regarding conditions under which such groups are automatic. Furthermore, we introduce a family of one-relator groups and classify which of them act properly cocompactly on complete CAT(0) spaces; the non-CAT(0) examples are counterexamples to a variation on the aforementioned conjecture. For a subclass, we establish automaticity, which is needed for the census. The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has non-positive deficiency. For every prime p we construct finite p-groups of arbitrary negative deficiency, and thereby complete Kotschick's proposed classification of the integers which are deficiencies of Kähler groups. We explore variations and embellishments of our basic construction, which require subtle Schur multiplier computations, and we investigate the conditions on inputs to the construction that are necessary for success. A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3-manifolds have the same finite quotients. We determine examples of sizeable graphs, as required to construct finitely presented non-hyperbolic subgroups of hyperbolic groups, which have the fewest vertices possible modulo mild topological assumptions.