CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Na tentativa de facilitar o diagnÃstico dos diversos fatores que afetam a qualidade da Ãgua e antever possÃveis impactos futuros sobre o meio ambiente , sÃo adotadas aÃÃes que racionalize m o uso da Ãgua a partir da otimizaÃÃo de processos naturais ou tecnolÃgicos. A modelagem matemÃtica à um exemplo disso e, em conjunto com a Teoria Fuzzy , que permite fazer a anÃlise dos resultados sem necessidade de significativos bancos de dados, pode - se estabelecer o risco como indicador de contaminaÃÃo das Ãguas de rios, sendo de valor prÃtico na tomada de decisÃo e concessÃo de outorga de lanÃamentos. Neste estudo, foi desenvolvido um modelo matemÃtico aplicado Ãs equaÃÃes completas de Streeter - Phelps utilizando a Teoria dos nÃmeros Fuzzy , a fim de analisar o risco de contaminaÃÃo de um curso d'Ãgua que recebe agentes poluentes de mÃltiplas fontes de lanÃamento. Pelas simulaÃÃes do modelo, foram analisados diferentes cenÃrios, verificando a influÃncia d os seus parÃmetros, bem como o lanÃamento de fontes poluidoras pontuais e difusas, nos percentuais de risco. De acordo com os resultados, observou - se que a quantidade de carga lanÃada tem influÃncia no tempo de diluiÃÃo desta massa no sistema, de forma que , para maiores valores de lanÃamento, o tempo de diluiÃÃo à menor, favorecendo os processos de decaimento e formaÃÃo da camada bentÃnica; em relaÃÃo Ãs reaÃÃes fÃsicas, quÃmicas e biolÃgicas, verifica - se que os processos de sedimentaÃÃo, fotossÃntese e res piraÃÃo, para os dados mÃdios encontrados em literatura, tem pequena influÃncia no comportamento das curvas de concentraÃÃo de OD e curvas de risco, enquanto que o processo de nitrificaÃÃo tem forte influÃncia; jà a temperatura desempenha um significativo papel no comportamento do OD, onde, para valores maiores, maior serà o dÃficit OD e, em consequÃncia, aumento dos percentuais de risco. Por fim, o modelo desenvolvido como proposta de facilitar a tomada de decisÃo no controle de lanÃamento de efluentes em rios mostrou - se uma alternativa viÃvel e de valor prÃtico de anÃlise, jà que os objetivos foram alcanÃados / In an attempt to facilitate the diagnosis of the various factors that affect water quality and predict possible future impacts on the environment, actions to rationalize the use of water from the optimization of natural and technological processes are adopted. Mathematical modeling is one example and, together with Fuzzy Theory, which allows the analysis of the results without the need for significant databases, one can establish the risk as an indicator of contamination of rivers, and of practical value in decision making and allocation of grant releases. In this study, the full Streeter-Phelps equations, using the Fuzzy set Theory, was applied, in order to analyze the risk of contamination of a watercourse that receives multiple sources release pollutants. Through the model simulations, different scenarios were analyzed, and the influence of its parameters as well as the launch point and nonpoint pollution sources, in the calculation of the risk. According to the results, it was observed that the amount of discharge released influences the time of the mass dilution in the system, so that for higher values of launch, the dilution time is less favoring the formation and decay processes of benthic layer; regarding the physical, chemical and biological reactions, it appears that sedimentation processes, photosynthesis and respiration, concerning with the average data found in literature, have little influence on the behavior of the curves of DO concentration curves and risk, while the nitrification process has a strong influence; with respect to the temperature, the results showed that it plays a significant role in the behavior of DO, where, for larger values of it, the higher the DO deficit and, consequently, increase in the risk. Finally, the model developed as a proposal to facilitate the decision making in the control of discharge of effluents into rivers proved to be a viable and practical analytical alternative way, since the goals were achieved.

In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .

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.

We compute the partial isomorphism rank, in the sense Scott and Karp, of a pair of ordinal structures using an Ehrenfeucht-Fraisse game. A complete formula is proven by induction given any two arbitrary ordinals written in Cantor normal form.

In this thesis, I survey 11 approximation algorithms for unweighted set cover problem. I have also implemented the three algorithms and created a software library that stores the code I have written. The algorithms I survey are: 1. Johnson's standard greedy; 2. f-frequency greedy; 3. Goldsmidt, Hochbaum and Yu's modified greedy; 4. Halldorsson's local optimization; 5. Dur and Furer semi local optimization; 6. Asaf Levin's improvement to Dur and Furer; 7. Simple rounding; 8. Randomized rounding; 9. LP duality; 10. Primal-dual schema; and 11. Network flow technique. Most of the algorithms surveyed are refinements of standard greedy algorithm.

In the study of sets of points certain sets are found to be negligible, especially when applied to the theory of functions. The purpose of this paper is to discuss three of these "negligible" types, namely, exhaustible sets, denumerable sets, and sets of Lebesgue measure zero. We will present a complete existential theory in q-space for the three set properties mentioned above, followed by a more restricted discussion in the linear continuum by use of interval properties.

Rough set theory (RST) is concerned with the formal approximation of crisp sets and is a mathematical tool which deals with vagueness and uncertainty. RST can be integrated into machine learning and can be used to forecast predictions as well as to determine the causal interpretations for a particular data set. The work performed in this research is concerned with using various optimisation techniques to granulise the rough set input partitions in order to achieve the highest forecasting accuracy produced by the rough set. The forecasting accuracy is measured by using the area under the curve (AUC) of the receiver operating characteristic (ROC) curve. The four optimisation techniques used are genetic algorithm, particle swarm optimisation, hill climbing and simulated annealing. This newly proposed method is tested on two data sets, namely, the human immunodeficiency virus (HIV) data set and the militarised interstate dispute (MID) data set. The results obtained from this granulisation method are compared to two previous static granulisation methods, namely, equal-width-bin and equal-frequency-bin partitioning. The results conclude that all of the proposed optimised methods produce higher forecasting accuracies than that of the two static methods. In the case of the HIV data set, the hill climbing approach produced the highest accuracy, an accuracy of 69.02% is achieved in a time of 12624 minutes. For the MID data, the genetic algorithm approach produced the highest accuracy. The accuracy achieved is 95.82% in a time of 420 minutes. The rules generated from the rough set are linguistic and easy-to-interpret, but this does come at the expense of the accuracy lost in the discretisation process where the granularity of the variables are decreased.

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