On contractive cyclic fuzzy maps in metric spaces and some related results on fuzzy best proximity points and fuzzy fixed points

This paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.

Approximate Best Proximity Points of Cyclic Self-maps in Metric Spaces and Cyclic Asymptotic Regularity

3rd International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE) Madrid, AUG 28-31, 2014 / editado por Vagenas, EC; Vlachos, DS; Bastos, C; Hofer, T; Kominis, Y; Kosmas, O; LeLay, G; DePadova, P; Rode, B; Suraud, E; Varga, K

Weak Compatibility and Common Fixed Point Theorems for A-Contractive and E-expansive Maps in Uniform Spaces

2000 Mathematics Subject Classification: 47H10, 54E15.

Components of error of maps from remotely sensed images

The concept of a “true” ground-truth map is introduced, from which the inaccuracy/error of any production map may be measured. A partition of the mapped region is defined in terms of the “residual rectification” transformation. Geometric RMS-type and Geometric Distortion error criteria are defined as well as a map mis-classification error criterion (the latter for hard and fuzzy produc-tion maps). The total map error is defined to be the sum (over each set of the map partition men-tioned above) of these three error components integrated over each set of the partition.

Study on Fuzzy Ordered FuzzyTopological Spaces

In this study we combine the notions of fuzzy order and fuzzy topology of Chang and define fuzzy ordered fuzzy topological space. Its various properties are analysed. Product, quotient, union and intersection of fuzzy orders are introduced. Besides, fuzzy order preserving maps and various fuzzy completeness are investigated. Finally an attempt is made to study the notion of generalized fuzzy ordered fuzzy topological space by considering fuzzy order defined on a fuzzy subset.

Concept maps as a tool to analyze the ideas of teacher training students regarding what to teach on human nutrition in Primary Education

Concept maps are a technique used to obtain a visual representation of a person's ideas about a concept or a set of related concepts. Specifically, in this paper, through a qualitative methodology, we analyze the concept maps proposed by 52 groups of teacher training students in order to find out the characteristics of the maps and the degree of adequacy of the contents with regard to the teaching of human nutrition in the 3rd cycle of primary education. The participants were enrolled in the Teacher Training Degree majoring in Primary Education, and the data collection was carried out through a training activity under the theme of what to teach about Science in Primary School? The results show that the maps are a useful tool for working in teacher education as they allow organizing, synthesizing, and communicating what students know. Moreover, through this work, it has been possible to see that future teachers have acceptable skills for representing the concepts/ideas in a concept map, although the level of adequacy of concepts/ideas about human nutrition and its relations is usually medium or low. These results are a wake-up call for teacher training, both initial and ongoing, because they shows the inability to change priorities as far as the selection of content is concerned.

Gestão Ambiental apoiada por SIG na estrada da Colônia, Ilha Grande (RJ): uma visão do futuro.

Os SIG Sistemas de Informação Geográfica vêm sendo cada vez mais estudados como ferramentas facilitadoras de análises territoriais com o objetivo de subsidiar a gestão ambiental. A Ilha Grande, que pertence ao município de Angra dos Reis, localiza-se na baía de Ilha Grande no sul do estado do Rio de Janeiro e constitui-se no recorte espacial de análise. Apresenta uma dinâmica ambiental complexa que se sobrepõem principalmente aos usos de proteção ambiental e de atividade turística em uma porção do território em que as normatizações legais são difíceis de serem aplicadas, pois são reflexos de interesses que se manifestam em três esferas do poder a municipal, a estadual e a federal. O objetivo principal desta pesquisa é a realização do processamento digital de imagem para auxiliar a gestão territorial da Ilha Grande. Em foco, a estrada Abraão - Dois Rios, que liga Abraão (local de desembarque dos turistas, principal núcleo da Ilha) a Dois Rios (local de visitação por estudantes e pesquisadores, núcleo que abrigava o presídio, atualmente abriga sede do centro de pesquisa e museu da Universidade do Estado do Rio de Janeiro), ambos protegidos por diferentes categorias de unidades de conservação. A metodologia fundamenta-se no processamento digital de imagem através da segmentação e da classificação supervisionada por pixel e por região. O processamento deu-se a partir da segmentação (divisão de uma imagem digital em múltiplas regiões ou objetos, para simplificar e/ou mudar a representação de uma imagem) e dos processos de classificações de imagem, com a utilização de classificação por pixel e classificação por regiões (com a utilização do algoritmo Bhattacharya). As segmentações e classificações foram processadas no sistema computacional SPRING versão 5.1.7 e têm como objetivo auxiliar na análise de uso da Terra e projetar cenários a partir da identificação dos pontos focais de fragilidade encontrados ao longo da estrada Abraão-Dois Rios, propensos a ocorrências de movimentos de massa e que potencializam o efeito de borda da floresta e os impactos ambientais. A metodologia utilizada baseou-se em análise de campo e comparações de tecnologias de classificação de imagens. Essa estrada eixo de ligação entre os dois núcleos tem significativa importância na história da Ilha, nela circulam veículos, pesados e leves, de serviço, pedestres e turistas. Como resultados da presente foram gerados os mapas de classificação por pixel, os mapas de classificação por região, o mapa fuzzy com a intersecção dos mapas de classificação supervisionada por região e os mapas com os locais coletados em campo onde são verificadas ocorrências de movimentos de massa nas imagens ALOS, 2000, IKONOS, 2003 e ortofotografias, 2006. Esses mapas buscam servir de apoio à tomada de decisões por parte dos órgãos locais responsáveis.

Integração de dados em ambiente SIG para mapeamento de favorabilidade mineral de ouro na ilha cristalina de Rivera (Uruguai)

Guias para exploração mineral são normalmente baseados em modelos conceituais de depósitos. Esses guias são, normalmente, baseados na experiência dos geólogos, em dados descritivos e em dados genéticos. Modelamentos numéricos, probabilísticos e não probabilísticos, para estimar a ocorrência de depósitos minerais é um novo procedimento que vem a cada dia aumentando sua utilização e aceitação pela comunidade geológica. Essa tese utiliza recentes metodologias para a geração de mapas de favorablidade mineral. A denominada Ilha Cristalina de Rivera, uma janela erosional da Bacia do Paraná, situada na porção norte do Uruguai, foi escolhida como estudo de caso para a aplicação das metodologias. A construção dos mapas de favorabilidade mineral foi feita com base nos seguintes tipos de dados, informações e resultados de prospecção: 1) imagens orbitais; 2) prospecção geoquimica; 3) prospecção aerogeofísica; 4) mapeamento geo-estrutural e 5) altimetria. Essas informacões foram selecionadas e processadas com base em um modelo de depósito mineral (modelo conceitual), desenvolvido com base na Mina de Ouro San Gregorio. O modelo conceitual (modelo San Gregorio), incluiu características descritivas e genéticas da Mina San Gregorio, a qual abrange os elementos característicos significativos das demais ocorrências minerais conhecidas na Ilha Cristalina de Rivera. A geração dos mapas de favorabilidade mineral envolveu a construção de um banco de dados, o processamento dos dados, e a integração dos dados. As etapas de construção e processamento dos dados, compreenderam a coleta, a seleção e o tratamento dos dados de maneira a constituírem os denominados Planos de Informação. Esses Planos de Informação foram gerados e processados organizadamente em agrupamentos, de modo a constituírem os Fatores de Integração para o mapeamento de favorabilidade mineral na Ilha Cristalina de Rivera. Os dados foram integrados por meio da utilização de duas diferentes metodologias: 1) Pesos de Evidência (dirigida pelos dados) e 2) Lógica Difusa (dirigida pelo conhecimento). Os mapas de favorabilidade mineral resultantes da implementação das duas metodologias de integração foram primeiramente analisados e interpretados de maneira individual. Após foi feita uma análise comparativa entre os resultados. As duas metodologias xxiv obtiveram sucesso em identificar, como áreas de alta favorabilidade, as áreas mineralizadas conhecidas, além de outras áreas ainda não trabalhadas. Os mapas de favorabilidade mineral resultantes das duas metodologias mostraram-se coincidentes em relação as áreas de mais alta favorabilidade. A metodologia Pesos de Evidência apresentou o mapa de favorabilidade mineral mais conservador em termos de extensão areal, porém mais otimista em termos de valores de favorabilidade em comparação aos mapas de favorabilidade mineral resultantes da implementação da metodologia Lógica Difusa. Novos alvos para exploração mineral foram identificados e deverão ser objeto de investigação em detalhe.

Fixed and Best Proximity Points of Cyclic Jointly Accretive and Contractive Self-Mappings

p(>= 2)-cyclic and contractive self-mappings on a set of subsets of a metric space which are simultaneously accretive on the whole metric space are investigated. The joint fulfilment of the p-cyclic contractiveness and accretive properties is formulated as well as potential relationships with cyclic self-mappings in order to be Kannan self-mappings. The existence and uniqueness of best proximity points and fixed points is also investigated as well as some related properties of composed self-mappings from the union of any two adjacent subsets, belonging to the initial set of subsets, to themselves.

Fuzzy ARTMAP: A Neural Network Architecture for Incremental Supervised Learning of Analog Multidimensional Maps

A new neural network architecture is introduced for incremental supervised learning of recognition categories and multidimensional maps in response to arbitrary sequences of analog or binary input vectors. The architecture, called Fuzzy ARTMAP, achieves a synthesis of fuzzy logic and Adaptive Resonance Theory (ART) neural networks by exploiting a close formal similarity between the computations of fuzzy subsethood and ART category choice, resonance, and learning. Fuzzy ARTMAP also realizes a new Minimax Learning Rule that conjointly minimizes predictive error and maximizes code compression, or generalization. This is achieved by a match tracking process that increases the ART vigilance parameter by the minimum amount needed to correct a predictive error. As a result, the system automatically learns a minimal number of recognition categories, or "hidden units", to met accuracy criteria. Category proliferation is prevented by normalizing input vectors at a preprocessing stage. A normalization procedure called complement coding leads to a symmetric theory in which the MIN operator (Λ) and the MAX operator (v) of fuzzy logic play complementary roles. Complement coding uses on-cells and off-cells to represent the input pattern, and preserves individual feature amplitudes while normalizing the total on-cell/off-cell vector. Learning is stable because all adaptive weights can only decrease in time. Decreasing weights correspond to increasing sizes of category "boxes". Smaller vigilance values lead to larger category boxes. Improved prediction is achieved by training the system several times using different orderings of the input set. This voting strategy can also be used to assign probability estimates to competing predictions given small, noisy, or incomplete training sets. Four classes of simulations illustrate Fuzzy ARTMAP performance as compared to benchmark back propagation and genetic algorithm systems. These simulations include (i) finding points inside vs. outside a circle; (ii) learning to tell two spirals apart; (iii) incremental approximation of a piecewise continuous function; and (iv) a letter recognition database. The Fuzzy ARTMAP system is also compared to Salzberg's NGE system and to Simpson's FMMC system.

An improved lesion detection approach based on similarity measurement between fuzzy intensity segmentation and spatial probability maps

The application of automatic segmentation methods in lesion detection is desirable. However, such methods are restricted by intensity similarities between lesioned and healthy brain tissue. Using multi-spectral magnetic resonance imaging (MRI) modalities may overcome this problem but it is not always practicable. In this article, a lesion detection approach requiring a single MRI modality is presented, which is an improved method based on a recent publication. This new method assumes that a low similarity should be found in the regions of lesions when the likeness between an intensity based fuzzy segmentation and a location based tissue probabilities is measured. The usage of a normalized similarity measurement enables the current method to fine-tune the threshold for lesion detection, thus maximizing the possibility of reaching high detection accuracy. Importantly, an extra cleaning step is included in the current approach which removes enlarged ventricles from detected lesions. The performance investigation using simulated lesions demonstrated that not only the majority of lesions were well detected but also normal tissues were identified effectively. Tests on images acquired in stroke patients further confirmed the strength of the method in lesion detection. When compared with the previous version, the current approach showed a higher sensitivity in detecting small lesions and had less false positives around the ventricle and the edge of the brain

The Meaningful Fractal Fuzzy Dimension Applied to the Design of Self Organizing Maps

This paper presents the principal results of a detailed study about the use of the Meaningful Fractal Fuzzy Dimension measure in the problem in determining adequately the topological dimension of output space of a Self-Organizing Map. This fractal measure is conceived by combining the Fractals Theory and Fuzzy Approximate Reasoning. In this work this measure was applied on the dataset in order to obtain a priori knowledge, which is used to support the decision making about the SOM output space design. Several maps were designed with this approach and their evaluations are discussed here.

A Conceptual Model to Combine Creativity Techniques with Fuzzy Cognitive Maps for Enhanced Knowledge Management

This chapter introduces a conceptual model to combine creativity techniques with fuzzy cognitive maps (FCMs) and aims to support knowledge management methods by improving expert knowledge acquisition and aggregation. The aim of the conceptual model is to represent acquired knowledge in a manner that is as computer-understandable as possible with the intention of developing automated reasoning in the future as part of intelligent information systems. The formal represented knowledge thus may provide businesses with intelligent information integration. To this end, we introduce and evaluate various creativity techniques with a list of attributes to define the most suitable to combine with FCMs. This proposed combination enables enhanced knowledge management through the acquisition and representation of expert knowledge with FCMs. Our evaluation indicates that the creativity technique known as mind mapping is the most suitable technique in our set. Finally, a scenario from stakeholder management demonstrates the combination of mind mapping with FCMs as an integrated system.