For every positive, decreasing, summable sequence $a=(a_i)$, we can construct a Cantor set $C_a$ associated with $a$. These Cantor sets are not necessarily self-similar. Their dimensional properties and measures have been studied in terms of the sequence $a$. In this thesis, we extend these results to a more general collection of Cantor sets. We study their Hausdorff and packing measures, and compare the size of Cantor sets with the more refined notion of dimension partitions. The properties of these Cantor sets in relation to the collection of cut-out sets are then considered. The multifractal spectrum of $\mathbf{p}$-Cantor measures on these Cantor sets are also computed. We then focus on the special case of homogeneous Cantor sets and obtain a more accurate estimate of their exact measures. Finally, we prove the $L^p$-improving property of the $\mathbf{p}$-Cantor measure on a homogeneous Cantor set as a convolution operator.

The focus of this research is on a tolerance space-based approach to image analysis and correspondence. The problem considered in this thesis is one of extracting perceptually relevant information from groups of objects based on their descriptions. Object descriptions are represented by feature vectors containing probe function values in a manner similar to feature extraction in pattern classification theory. The motivation behind this work is the synthesizing of human perception of nearness for improvement of image processing systems. In these systems, the desired output is similar to the output of a human performing the same task. Thus, it is important to have systems that accurately model human perception. Near set theory provides a framework for measuring the similarity of objects based on features that describe them in much the same way that humans perceive the similarity of objects. In this thesis, near set theory is presented and advanced, and work is presented toward a near set approach to performing content-based image retrieval. Furthermore, results are given based on these new techniques and future work is presented. The contributions of this thesis are: the introduction of a nearness measure to determine the degree that near sets resemble each other; a systematic approach to finding tolerance classes, together with proofs demonstrating that the proposed approach will find all tolerance classes on a set of objects; an approach to applying near set theory to images; the application of near set theory to the problem of content-based image retrieval; demonstration that near set theory is well suited to solving problems in which the outcome is similar to that of human perception; two other near set measures, one based on Hausdorff distance, the other based on Hamming distance.

[Truncated abstract] Fuzzy inference systems (FIS) are information processing systems using fuzzy logic mechanism to represent the human reasoning process and to make decisions based on uncertain, imprecise environments in our daily lives. Since the introduction of fuzzy set theory, fuzzy inference systems have been widely used mainly for system modeling, industrial plant control for a variety of practical applications, and also other decisionmaking purposes; advanced data analysis in medical research, risk management in business, stock market prediction in finance, data analysis in bioinformatics, and so on. Many approaches have been proposed to address the issue of automatic generation of membership functions and rules with the corresponding subsequent adjustment of them towards more satisfactory system performance. Because one of the most important factors for building high quality of FIS is the generation of the knowledge base of it, which consists of membership functions, fuzzy rules, fuzzy logic operators and other components for fuzzy calculations. The design of FIS comes from either the experience of human experts in the corresponding field of research or input and output data observations collected from operations of systems. Therefore, it is crucial to generate high quality FIS from a highly reliable design scheme to model the desired system process best. Furthermore, due to a lack of a learning property of fuzzy systems themselves most of the suggested schemes incorporate hybridization techniques towards better performance within a fuzzy system framework. ... This systematic enhancement is required to update the FIS in order to produce flexible and robust fuzzy systems for unexpected unknown inputs from real-world environments. This thesis proposes a general framework of Adaptive T-S (Takagi-Sugeno) type Rough-Fuzzy Inference Systems (ARFIS) for a variety of practical applications in order to resolve the problems mentioned above in the context of a Rough-Fuzzy hybridization scheme. Rough set theory is employed to effectively reduce the number of attributes that pertain to input variables and obtain a minimal set of decision rules based on input and output data sets. The generated rules are examined by checking their validity to use them as T-S type fuzzy rules. Using its excellent advantages in modeling non-linear systems, the T-S type fuzzy model is chosen to perform the fuzzy inference process. A T-S type fuzzy inference system is constructed by an automatic generation of membership functions and rules by the Fuzzy C-Means (FCM) clustering algorithm and the rough set approach, respectively. The generated T-S type rough-fuzzy inference system is then adjusted by the least-squares method and a conjugate gradient descent algorithm towards better performance within a fuzzy system framework. To show the viability of the proposed framework of ARFIS, the performance of ARFIS is compared with other existing approaches in a variety of practical applications; pattern classification, face recognition, and mobile robot navigation. The results are very satisfactory and competitive, and suggest the ARFIS is a suitable new framework for fuzzy inference systems by showing a better system performance with less number of attributes and rules in each application.

OLIVEIRA, Henrique Viana. Refinamento de Consultas em Lógicas de Descrição Utilizando Teoria dos Rough Sets. 2012. 111 f. : Dissertação (mestrado) - Universidade Federal do Ceará, Centro de Ciências, Departamento de Computação, Fortaleza-CE, 2012. / Submitted by guaracy araujo (guaraa3355@gmail.com) on 2016-07-01T17:23:02Z No. of bitstreams: 1 2012_dis_hvoliveira.pdf: 789598 bytes, checksum: d75ef093adc56cc930f52c1e486ead5a (MD5) / Approved for entry into archive by guaracy araujo (guaraa3355@gmail.com) on 2016-07-01T17:23:47Z (GMT) No. of bitstreams: 1 2012_dis_hvoliveira.pdf: 789598 bytes, checksum: d75ef093adc56cc930f52c1e486ead5a (MD5) / Made available in DSpace on 2016-07-01T17:23:47Z (GMT). No. of bitstreams: 1 2012_dis_hvoliveira.pdf: 789598 bytes, checksum: d75ef093adc56cc930f52c1e486ead5a (MD5) Previous issue date: 2012 / Query Refinement consists of methods that modify the terms of a consult aiming the change of its result obtained previously. Refinements can be done of several ways and different approaches can be applied to it. This work proposes to apply methods of Query Refinement based on Rough Set theory, using it as an alternative for the refinement problem. The proposed methods will be grounded in the languages of Description Logics, which are commonly used on problems involving knowledge bases or ontologies representation. Two extensions of Description Logics with the Rough Set theory are introduced in this dissertation. We will prove the complexity of satisfiability of these logics, as well as the complexities of the query refinement methods applied to these logics. Finally, we will show quality measures which will aid to choose the results of the refinements obtained. / Refinamento de consulta consiste de técnicas que modificam os termos de uma consulta com o objetivo de alterar os resultados obtidos inicialmente. Para a realização de tal fim, diversas abordagens podem ser aplicadas e diferentes tipos de refinamentos podem ser considerados. Este trabalho propõe aplicar a teoria dos Rough Sets como uma nova alternativa de solução para o problema. Através das noções presentes nessa teoria, iremos desenvolver técnicas que serão aplicadas nas linguagens de Lógicas de Descrição, que são comumente utilizadas em problemas de representação de bases de conhecimento ou ontologias. Além disso, introduziremos duas extensões de Lógicas de Descrição capazes de representar as operações da teoria dos Rough Sets. Provaremos os resultados de complexidade de decisão dessas duas lógicas, assim como os resultados de complexidade das técnicas de refinamentos desenvolvidas. Por fim, mostraremos métricas de qualidade que poderão ser usadas para melhorar o resultado dos refinamentos obtidos.

This master's thesis deals with extensions of TOPSIS method, which is one of methods for multi-criteria evaluation of alternatives. These extensions use theory of fuzzy sets (FS) and their futher extensions to interval-valued (IVFS), intuitionistic (IFS) and hesitant (HFS) fuzzy sets and their combinations (IVIFS, IVIHFS). Significant part of this thesis explains the principle of fuzzy sets and their generalizations. Descriptions of operators for aggregations of grades of membership has very important role here. Next, very short description of multi-criteria evaluation problems and detailed description of TOPSIS method are contained. The second half of this thesis is dedicated to four existing extensions of TOPSIS metod that uses theories of FS, IVFS, IVIFS and IVIHFS. Every method is illustrated with an example that shows principle of calculations. It illustrates new possibilities of the methods that use new sets theories and potential complications and deviations from the original TOPSIS method. At the end of this thesis, evaluation of usefulness of used approaches is mentioned.

Thesis focuses on disk arrays, where the goal is to design test scenarios to measure performance of disk array and use predictive analytics tools to train a model that will predict the selected performance parameter on a measured set of data. The implemented web application demonstrates the functionality of the trained model and shows estimate of the disk array performance.

In the exploration of chemical systems through quantum mechanics, accurate treatment of the electron wavefunction, and the related electron density, is fundamental to extracting information concerning properties of a system. This work examines challenges in achieving accurate chemical information through manipulation of the one-electron basis set.

For any integer k≥0, a set of vertices S of a graph G=(V,E) is k-cost-effective if for every v∈S,|N(v)∩(V∖S)|≥|N(v)∩S|+k. In this paper we study the minimum cardinality of a maximal k-cost-effective set and the maximum cardinality of a k-cost-effective set. We obtain Gallai-type results involving the k-cost-effective and global k-offensive alliance parameters, and we provide bounds on the maximum k-cost-effective number. Finally, we consider k-cost-effective sets that are also dominating. We show that computing the k-cost-effective domination number is NP-complete for bipartite graphs. Moreover, we note that not all trees have a k-cost-effective dominating set and give a constructive characterization of those that do.

Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances differ in different situations, such as, kinship and friendship (in the case of individuals), common economic interests (for both individuals and states), common political interests, and geographical proximity. This structure of alliances is not only prevalent in social networks, but it is also an important characteristic of similarity networks of natural and unnatural objects. (A similarity network defines the links between two objects based on their similarities). Discovery of such structure in a data set is called clustering or unsupervised learning and the ability to do it automatically is desirable for many applications in the areas of pattern recognition, computer vision, artificial intelligence, behavioral and social sciences, life sciences, earth sciences, medicine, and information theory. In this dissertation, we study a graph theoretical model of alliances where an alliance of the vertices of a graph is a set of vertices in the graph, such that every vertex in the set is adjacent to equal or more vertices inside the set than the vertices outside it. We study the problem of partitioning a graph into alliances and identify classes of graphs that have such a partition. We present results on the relationship between the existence of such a partition and other well known graph parameters, such as connectivity, subgraph structure, and degrees of vertices. We also present results on the computational complexity of finding such a partition. An alliance cover set is a set of vertices in a graph that contains at least one vertex from every alliance of the graph. The complement of an alliance cover set is an alliance free set, that is, a set that does not contain any alliance as a subset. We study the properties of these sets and present tight bounds on their cardinalities. In addition, we also characterize the graphs that can be partitioned into alliance free and alliance cover sets. Finally, we present an approximate algorithm to discover alliances in a given graph. At each step, the algorithm finds a partition of the vertices into two alliances such that the alliances are strongest among all such partitions. The strength of an alliance is defined as a real number p, such that every vertex in the alliance has at least p times more neighbors in the set than its total number of neighbors in the graph). We evaluate the performance of the proposed algorithm on standard data sets.

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