In this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute Auslander-Reiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and Ringel-Tachikawa Theorem which states that for an artinian ring R of finite representation type, each R-module is a direct sum of finite-length indecomposable R-modules. In cases where this applies, the indecomposable objects obtained in the Auslander-Reiten quiver give the building blocks for the objects in the category. We also briefly discuss in which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise Calabi-Yau dimension of such a category. / by Audrey Moore. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

Word embeddings are a powerful approach for capturing semantic similarity among terms in a vocabulary. Exponential family embeddings extend the idea of word embeddings to other types of high-dimensional data. Exponential family embeddings have three ingredients; embeddings as latent variables, a predefined conditioning set for each observation called the context and a conditional likelihood from the exponential family. The embeddings are inferred with a scalable algorithm. This thesis highlights three advantages of the exponential family embeddings model class: (A) The approximations used for existing methods such as word2vec can be understood as a biased stochastic gradients procedure on a specific type of exponential family embedding model --- the Bernoulli embedding. (B) By choosing different likelihoods from the exponential family we can generalize the task of learning distributed representations to different application domains. For example, we can learn embeddings of grocery items from shopping data, embeddings of movies from click data, or embeddings of neurons from recordings of zebrafish brains. On all three applications, we find exponential family embedding models to be more effective than other types of dimensionality reduction. They better reconstruct held-out data and find interesting qualitative structure. (C) Finally, the probabilistic modeling perspective allows us to incorporate structure and domain knowledge in the embedding space. We develop models for studying how language varies over time, differs between related groups of data, and how word usage differs between languages. Key to the success of these methods is that the embeddings share statistical information through hierarchical priors or neural networks. We demonstrate the benefits of this approach in empirical studies of Senate speeches, scientific abstracts, and shopping baskets.

This thesis is concerned with the Regular Inverse Galois Problem for p-groups over fields of characteristic unequal to p. Building upon results of Saltman, Dentzer characterized a class of finite groups that are automatically realized over every field, and proceeded to show that every group of order dividing p⁴ belongs to this class. We extend this result to include groups of order p⁵, provided that the base field k contains the p³-th roots of unity. The proof involves reducing to certain Brauer embedding problems defined over the rational function field k(x). Through explicit computation, we describe the cohomological obstructions to these embedding problems. Then by applying results about the Brauer group of a Dedekind domain, we show that they all possess solutions. / text

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (MSc)-- Stellenbosch University, 2003. / ENGLISH ABSTRACT: This thesis implements and evaluates two prediction techniques used to forecast deterministic chaotic time series. For a large number of such techniques, the reconstruction of the phase space attractor associated with the time series is required. Embedding is presented as the means of reconstructing the attractor from limited data. Methods for obtaining the minimal embedding dimension and optimal time delay from the false neighbour heuristic and average mutual information method are discussed. The first prediction algorithm that is discussed is based on work by Sauer, which includes the implementation of the singular value decomposition on data obtained from the embedding of the time series being predicted. The second prediction algorithm is based on neural networks. A specific architecture, suited to the prediction of deterministic chaotic time series, namely the time dependent neural network architecture is discussed and implemented. Adaptations to the back propagation training algorithm for use with the time dependent neural networks are also presented. Both algorithms are evaluated by means of predictions made for the well-known Lorenz time series. Different embedding and algorithm-specific parameters are used to obtain predicted time series. Actual values corresponding to the predictions are obtained from Lorenz time series, which aid in evaluating the prediction accuracies. The predicted time series are evaluated in terms of two criteria, prediction accuracy and qualitative behavioural accuracy. Behavioural accuracy refers to the ability of the algorithm to simulate qualitative features of the time series being predicted. It is shown that for both algorithms the choice of the embedding dimension greater than the minimum embedding dimension, obtained from the false neighbour heuristic, produces greater prediction accuracy. For the neural network algorithm, values of the embedding dimension greater than the minimum embedding dimension satisfy the behavioural criterion adequately, as expected. Sauer's algorithm has the greatest behavioural accuracy for embedding dimensions smaller than the minimal embedding dimension. In terms of the time delay, it is shown that both algorithms have the greatest prediction accuracy for values of the time delay in a small interval around the optimal time delay. The neural network algorithm is shown to have the greatest behavioural accuracy for time delay close to the optimal time delay and Sauer's algorithm has the best behavioural accuracy for small values of the time delay. Matlab code is presented for both algorithms. / AFRIKAANSE OPSOMMING: In hierdie tesis word twee voorspellings-tegnieke geskik vir voorspelling van deterministiese chaotiese tydreekse ge"implementeer en geevalueer. Vir sulke tegnieke word die rekonstruksie van die aantrekker in fase-ruimte geassosieer met die tydreeks gewoonlik vereis. Inbedmetodes word aangebied as 'n manier om die aantrekker te rekonstrueer uit beperkte data. Metodes om die minimum inbed-dimensie te bereken uit gemiddelde wedersydse inligting sowel as die optimale tydsvertraging te bereken uit vals-buurpunt-heuristiek, word bespreek. Die eerste voorspellingsalgoritme wat bespreek word is gebaseer op 'n tegniek van Sauer. Hierdie algoritme maak gebruik van die implementering van singulierwaarde-ontbinding van die ingebedde tydreeks wat voorspel word. Die tweede voorspellingsalgoritme is gebaseer op neurale netwerke. 'n Spesifieke netwerkargitektuur geskik vir deterministiese chaotiese tydreekse, naamlik die tydafhanklike neurale netwerk argitektuur word bespreek en ge"implementeer. 'n Modifikasie van die terugprapagerende leer-algoritme vir gebruik met die tydafhanklike neurale netwerk word ook aangebied. Albei algoritmes word geevalueer deur voorspellings te maak vir die bekende Lorenz tydreeks. Verskeie inbed parameters en ander algoritme-spesifieke parameters word gebruik om die voorspelling te maak. Die werklike waardes vanuit die Lorentz tydreeks word gebruik om die voorspellings te evalueer en om voorspellingsakkuraatheid te bepaal. Die voorspelde tydreekse word geevalueer op grand van twee kriteria, naamlik voorspellingsakkuraatheid, en kwalitatiewe gedragsakkuraatheid. Gedragsakkuraatheid verwys na die vermoe van die algoritme om die kwalitatiewe eienskappe van die tydreeks korrek te simuleer. Daar word aangetoon dat vir beide algoritmes die keuse van inbed-dimensie grater as die minimum inbeddimensie soos bereken uit die vals-buurpunt-heuristiek, grater akkuraatheid gee. Vir die neurale netwerkalgoritme gee 'n inbed-dimensie grater as die minimum inbed-dimensie ook betel' gedragsakkuraatheid soos verwag. Vir Sauer se algoritme, egter, word betel' gedragsakkuraatheid gevind vir 'n inbed-dimensie kleiner as die minimale inbed-dimensie. In terme van tydsvertraging word dit aangetoon dat vir beide algoritmes die grootste voorspellingsakkuraatheid verkry word by tydvertragings in 'n interval rondom die optimale tydsvetraging. Daar word ook aangetoon dat die neurale netwerk-algoritme die beste gedragsakkuraatheid gee vir tydsvertragings naby aan die optimale tydsvertraging, terwyl Sauer se algoritme betel' gedragsakkuraatheid gee by kleineI' waardes van die tydsvertraging. Die Matlab kode van beide algoritmes word ook aangebied.

In this thesis we study two different topics, both in the context of semigroup constructions. The first is the investigation of an embedding problem, specifically the problem of whether it is possible to embed any given finitely presentable semigroup into a D-simple finitely presentable semigroup. We consider some well-known semigroup constructions, investigating their properties to determine whether they might prove useful for finding a solution to our problem. We carry out a more detailed study into a more complicated semigroup construction, the Byleen extension, which has been used to solve several other embedding problems. We prove several results regarding the structure of this extension, finding necessary and sufficient conditions for an extension to be D-simple and a very strong necessary condition for an extension to be finitely presentable. The second topic covered in this thesis is relative rank, specifically the sequence obtained by taking the rank of incremental direct powers of a given semigroup modulo the diagonal subsemigroup. We investigate the relative rank sequences of infinite Cartesian products of groups and of semigroups. We characterise all semigroups for which the relative rank sequence of an infinite Cartesian product is finite, and show that if the sequence is finite then it is bounded above by a logarithmic function. We will find sufficient conditions for the relative rank sequence of an infinite Cartesian product to be logarithmic, and sufficient conditions for it to be constant. Chapter 4 ends with the introduction of a new topic, relative presentability, which follows naturally from the topic of relative rank.