Recently, Dawson and Feinstein showed that a Banach algebra integral extension B of a commutative Banach algebra A of topological stable rank one is again of topological stable rank one. In this thesis, we provide a partial converse to this statement: If an Arens-Hoffman extension A® of a commutative C*-algebra A has topological stable rank one then A has topological stable rank one.

This dissertation examines subfunctors of Ext relative to covering (enveloping) classes and the theory of covering (enveloping) ideals. The notion of covers and envelopes by modules was introduced independently by Auslander-Smalø and Enochs and has proven to be beneficial for module theory as well as for representation theory. The first few chapters examine the subfunctors of Ext and their properties. It is showed how the class of precoverings give us subfunctors of Ext. Furthermore, the characterization of these subfunctors and some examples are given. In the latter chapters ideals, the subfunctors of Hom, are investigated. The definition of cover and envelope carry over to the ideals naturally. Classical conditions for existence theorems for covers led to similar approaches in the ideal case. Even though some theorems such as Salce’s Lemma were proven to extend to ideals, most of the theorems do not directly apply to the new case. It is showed how Eklof & Trlifaj’s result can partially be extended to the ideals generated by a set. In that case, one also obtains a significant result about the orthogonal complement of the ideal. We relate the existence theorems for covering ideals of morphisms by identifying the morphisms with objects in A2 (which is the category of all representations of 2-quiver by R-modules) and obtain a sufficient condition for the existence of covering ideals in a more general setting. We finish with applying this result to the class of phantom morphisms.

Konstanz, Univ., Diplomarb., 2003.

abstract: Finding the optimal solution to a problem with an enormous search space can be challenging. Unless a combinatorial construction technique is found that also guarantees the optimality of the resulting solution, this could be an infeasible task. If such a technique is unavailable, different heuristic methods are generally used to improve the upper bound on the size of the optimal solution. This dissertation presents an alternative method which can be used to improve a solution to a problem rather than construct a solution from scratch. Necessity analysis, which is the key to this approach, is the process of analyzing the necessity of each element in a solution. The post-optimization algorithm presented here utilizes the result of the necessity analysis to improve the quality of the solution by eliminating unnecessary objects from the solution. While this technique could potentially be applied to different domains, this dissertation focuses on k-restriction problems, where a solution to the problem can be presented as an array. A scalable post-optimization algorithm for covering arrays is described, which starts from a valid solution and performs necessity analysis to iteratively improve the quality of the solution. It is shown that not only can this technique improve upon the previously best known results, it can also be added as a refinement step to any construction technique and in most cases further improvements are expected. The post-optimization algorithm is then modified to accommodate every k-restriction problem; and this generic algorithm can be used as a starting point to create a reasonable sized solution for any such problem. This generic algorithm is then further refined for hash family problems, by adding a conflict graph analysis to the necessity analysis phase. By recoloring the conflict graphs a new degree of flexibility is explored, which can further improve the quality of the solution. / Dissertation/Thesis / Ph.D. Computer Science 2011

Made available in DSpace on 2014-06-11T19:34:57Z (GMT). No. of bitstreams: 0 Previous issue date: 2003-05-23Bitstream added on 2014-06-13T19:23:59Z : No. of bitstreams: 1 torres_jlr_dr_jabo.pdf: 417695 bytes, checksum: 1a1abee9864a748c29b85912bea72f67 (MD5) / O estabelecimento de culturas de cobertura para formação e manutenção dos resíduos culturais na superfície do solo, principalmente nas regiões de cerrado, tem encontrado alguns obstáculos, pois as condições climáticas nestas regiões favorecem a decomposição destes resíduos vegetais. A implantação do plantio direto nestas áreas tem crescido exponencialmente, porém tem sido utilizado como base em dados gerados em outras regiões do País, em outras condições climáticas. O presente estudo teve como objetivo principal estudar as plantas de cobertura mais utilizadas na região, avaliando o tempo de decomposição dos restos culturais, o acúmulo e liberação de nitrogênio por um período de até 320 dias, associando estes dados a produtividade das culturas. Também, fez-se o monitoramento da temperatura e umidade do solo, semanalmente, durante o período de janeiro a junho de 2000 nas profundidades de 0 a 5 e 5 a 10 cm. Após dois anos agrícolas de implantação do experimento, fez-se uma avaliação da influência destas coberturas em algumas propriedades físicas do solo da área. O delineamento experimental utilizado foi de blocos ao acaso, com parcelas subdivididas e quatro repetições. Nas parcelas principais, os tratamentos utilizados constaram de sete tipos de coberturas: milheto (Pennisetum americanum sin. tiphoydes), braquiária (Brachiaria brizantha), sorgo forrageiro (Sorghum bicolor L. Moench), guandu (Cajanus cajan (L.) Millsp), crotalária juncea (Crotalarea juncea) e aveia preta (Avena strigosa Schreb), pousio e área sob sistema de plantio convencional (testemunha). Nas subparcelas, após a dessecação das coberturas, plantou-se milho e soja, sendo estas culturas rotacionadas no segundo ano... . / The establishment of covering cultures for formation and maintenance of the cultural residues in the soil surface, mainly in the cerrado areas, has been finding some obstacles, because the climatic conditions in these areas favor the decomposition of these vegetable residues. The no tillage system implantation in these areas, has been increasing exponentially, however it has been used as base, the data generated in other areas of the country, in other climatic conditions. The present study had as main objective to study the covering plants more used in this area, evaluating the time of decomposition of cultural remains, the accumulation and liberation of nitrogen for a period up to 320 days, associating these data to the productivity of the cultures. Also, it was made the observation the soil of temperature and humidity, weekly, during the period of January to June 2000, in the depths of 0 - 5 and 5 - 10 cm. After two agricultural years, it was made an evaluation of the influence of the covering plants in some physical soil properties in the area. The experimental design was random blocks, with subdivided plots and four repetitions. The treatments used consisted in eight covering types: pearl millet (Pennisetum americanum sin. tiphoydes), braquiária (Brachiaria brizantha), sorgun (Sorghum bicolor L. Moench), pigeonpea (Cajanus cajan (L.) Millsp), sunn hemp (Crotalárea juncea) and black oat (Avena strigosa Schreb), fallow land, area in conventional system. In the subplots, after the covering plants dessecation, it was planted corn and soybean, and these cultures were rotated in the second year. Among the appraised coverings it was verified that millet and the sunn hemp were the cultures that presented the best score in dry mass production. Nitrogen accumulation and liberation in the appraised periods... (Complete abstract, click eletronic address below).

abstract: In software testing, components are tested individually to make sure each performs as expected. The next step is to confirm that two or more components are able to work together. This stage of testing is often difficult because there can be numerous configurations between just two components. Covering arrays are one way to ensure a set of tests will cover every possible configuration at least once. However, on systems with many settings, it is computationally intensive to run every possible test. Test prioritization methods can identify tests of greater importance. This concept of test prioritization can help determine which tests can be removed with minimal impact to the overall testing of the system. This thesis presents three algorithms that generate covering arrays that test the interaction of every two components at least twice. These algorithms extend the functionality of an established greedy test prioritization method to ensure important components are selected in earlier tests. The algorithms are tested on various inputs and the results reveal that on average, the resulting covering arrays are two-fifths to one-half times smaller than a covering array generated through brute force. / Dissertation/Thesis / Masters Thesis Computer Science 2015

abstract: Exhaustive testing is generally infeasible except in the smallest of systems. Research has shown that testing the interactions among fewer (up to 6) components is generally sufficient while retaining the capability to detect up to 99% of defects. This leads to a substantial decrease in the number of tests. Covering arrays are combinatorial objects that guarantee that every interaction is tested at least once. In the absence of direct constructions, forming small covering arrays is generally an expensive computational task. Algorithms to generate covering arrays have been extensively studied yet no single algorithm provides the smallest solution. More recently research has been directed towards a new technique called post-optimization. These algorithms take an existing covering array and attempt to reduce its size. This thesis presents a new idea for post-optimization by representing covering arrays as graphs. Some properties of these graphs are established and the results are contrasted with existing post-optimization algorithms. The idea is then generalized to close variants of covering arrays with surprising results which in some cases reduce the size by 30%. Applications of the method to generation and test prioritization are studied and some interesting results are reported. / Dissertation/Thesis / Masters Thesis Computer Science 2015

The problems we study in this thesis are all related to covering arrays. Covering arrays are combinatorial designs, widely used as templates for efficient interaction-testing suites. They have connections to many areas including extremal set theory, design theory, and graph theory. We define and study several generalizations of covering arrays, and we develop a method which produces an infinite family of LYM inequalities for graph-intersecting collections. A common theme throughout is the dependence of these problems on graphs. Our main contribution is an extremal method yielding LYM inequalities for $H$-intersecting collections, for every undirected graph $H$. Briefly, an $H$-intersecting collection is a collection of packings (or partitions) of an $n$-set in which the classes of every two distinct packings in the collection intersect according to the edges of $H$. We define ``$F$-following" collections which, by definition, satisfy a LYM-like inequality that depends on the arcs of a ``follow" digraph $F$ and a permutation-counting technique. We fully characterize the correspondence between ``$F$-following" and ``$H$-intersecting" collections. This enables us to apply our inequalities to $H$-intersecting collections. For each graph $H$, the corresponding inequality inherently bounds the maximum number of columns in a covering array with alphabet graph $H$. We use this feature to derive bounds for covering arrays with the alphabet graphs $S_3$ (the star on three vertices) and $\kvloop{3}$ ($K_3$ with loops). The latter improves a known bound for classical covering arrays of strength two. We define covering arrays on column graphs and alphabet graphs which generalize covering arrays on graphs. The column graph encodes which pairs of columns must be $H$-intersecting, where $H$ is a given alphabet graph. Optimizing covering arrays on column graphs and alphabet graphs is equivalent to a graph-homomorphism problem to a suitable family of targets which generalize qualitative independence graphs. When $H$ is the two-vertex tournament, we give constructions and bounds for covering arrays on directed column graphs. FOR arrays are the broadest generalization of covering arrays that we consider. We define FOR arrays to encompass testing applications where constraints must be considered, leading to forbidden, optional, and required interactions of any strength. We model these testing problems using a hypergraph. We investigate the existence of FOR arrays, the compatibility of their required interactions, critical systems, and binary relational systems that model the problem using homomorphisms.

Let L be a discrete subgroup of \mathbb{R}^n under addition. Let D be a finite set of points including the origin. These two sets will define a multilattice of \mathbb{R}^n. We explore how to generate a periodic covering of the space \mathbb{R}^n based on L and $D$. Additionally, we explore the problem of covering when we restrict ourselves to covering \mathbb{R}^n using only dilations of the right regular simplex in our covering. We show that using a set D= {0,d} to define our multilattice the minimum covering density is 5-\sqrt{13}. Furthermore, we show that when we allow for an arbitrary number of displacements, we may get arbitrarily close to a covering density of 1.

Most of the research on domination focuses on vertices dominating other vertices. In this paper we consider vertexedge domination where a vertex dominates the edges incident to it as well as the edges adjacent to these incident edges. The minimum cardinality of a vertex-edge dominating set of a graph G is the vertex-edge domination number γve(G). We present bounds on γve(G) and relationships between γve(G) and other domination related parameters. Since any ordinary dominating set is also a vertex-edge dominating set, it follows that γve(G) is bounded above by the domination number of G. Our main result characterizes the trees having equal domination and vertex-edge domination numbers.