The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and measures. In particular, we show that these sets and measures are natural generalizations of the well known self-similar sets and measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary) self-similar sets. In the third chapter we investigate the L {q} spectra and the Renyi dimensions of inhomogeneous self-similar measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case. Namely, we show that inhomogeneous self-similar measures may have phase transitions which is in sharp contrast to the behaviour of the L {q} spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra of inhomogeneous self-similar measures. We show that the multifractal spectra of inhomogeneous self-similar measures may be non-concave which is again in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we present a number of applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of inhomogeneous self-similar measures and again we present a number of applications of our results, in particular to non-linear self-similar measures.

本研究的目的在於結合部份計分模式(partial credit model, PCM)與模糊集合論(fuzzy set theory)，提出評定量表的不同計分方式：模糊部份計分法(fuzzy partial credit scaling, FPCS)。FPCS是根據 PCM 所估計出的梯度參數(step parameters)來建構三角形模糊數，三角形模糊數代表選擇某個特定選項的受試者的能力分配情形。接著，利用中心法(center of gravity method) 將三角形模糊數解模糊化為純量。最後，利用隸屬度當作權重，計算個別受試者的模糊觀察分數，並且用模糊觀察分數當作量表的總分。 本研究採用貝克憂鬱量表(Beck Depression Inventory-II, BDI)中文版為研究工具。本研究的樣本分為憂鬱症病患與非憂鬱症的一般大學生兩大類。240位憂鬱症病患樣本是由台北市立和平醫院精神科門診募集而來；321位大學生則以便利抽樣的方式募集而來。 為了驗証FPCS的有效性，本研究進行三個子研究，來比較FPCS與傳統計分法在信度、效度、集群分析的分類正確性。 子研究一探討FPCS的信度。本研究以Cronbach alpha係數來衡量量表的內部一致性，並且以結構方程式模式(structure equation modeling)進行驗證性因素分析所估計的各試題的變異數被潛在構念解釋的比例當作信度的指標。由研究結果顯示，以量表整體而言，FPCS計分的結果得到較高的內部一致性；以各題而言，量表各試題的變異數被潛在構念解釋的百分比高於傳統的原始分數。此結果顯示FPCS的計分方式可以降低測量誤差，提升信度。 子研究二探討FPCS的效度，本研究以精神科醫師的診斷當作效標，分別以FPCS與原始分數兩種不同的計分法當作自變項，以預測效度當作效度的指標。首先，將是否罹患憂鬱症編碼為二元變數，不同計分法所得到的量表分數當作自變數，進行Logistic迴歸分析。研究結果顯示，相較於原始分數，FPCS預測罹患憂鬱症的正確率由 74.8% 提升到 77.2%。接下來，依照所有樣本的憂鬱程度，區分為一般樣本、憂鬱症且緩解、憂鬱症無緩解三類，進行區別分析。研究結果顯示，相較於原始分數，FPCS分類正確率由 71.2% 提升到 80.7%。上述的研究結果顯示，FPCS具有較高的效度，可以降低誤判憂鬱症的機率。 子研究三比較模糊集群分析(fuzzy c-means, FCM)與傳統明確邏輯的集群分析。首先利用分群效度(clustering validity)指標，決定群數為三群。並以此結果，指定模糊集群、Wald法、k-means法之群數。為了比較分類的效果，將模糊集群之樣本，指定給獲得最大隸屬度之集群。並且以醫師的診斷的憂鬱程度當作評估分類結果之標準。研究結果顯示，相較於傳統明確邏輯的集群分析(Wald法、k-means法)，模糊集群分析得到分群結果，與醫師的診斷的結果有最高的相關。結果顯示模糊集群分析更能夠忠實的反映資料結構。 整體而言，相較於原始分數，FPCS有較高的信度、效度、分類正確性。此實証性研究結果支持了模糊集合論應用於心理學研究的可行性；多值的模糊邏輯比二值明確邏輯更能夠正確反映出人類的思維。 / The aim of this study was to propose and validate the new scaling method, fuzzy partial credit scaling (FPCS), which combines fuzzy set theory with the partial credit model (PCM) to score rating scales. To achieve this goal, the Chinese version of BDI (Beck Depression Inventory-II) was administrated to a depressed sample of patients and a non-depressed sample. The depressed sample consisted of 240 outpatients who were diagnosed as depressed by a psychiatric doctor, while 321 undergraduate students were recruited for the nondepressed sample. In FPCS, triangular fuzzy numbers were generated by step parameters to characterize distributions of each alternative value. Next, the center of gravity (COG) method was applied to “de-fuzzify” the fuzzy number into a scalar. Then, the “observed fuzzy scores” defined in FPCS were calculated as the sums of fuzzy number values weighted by membership degrees for the following analysis. Three studies were performed to compare the differences in reliability, validity and clustering precision between the raw score and FPCS. In Study One, the reliability issue of FPCS was discussed. The results of confirmatory factor analysis demonstrate that the BDI reliability was higher in FCPS than in raw scoring. That is, compared with raw scoring, scoring via FPCS produced fewer measurement errors, meaning that more variances in an item of BDI were explained by depression. In Study Two, the predictive validity issue of FPCS was investigated. First, logistic regression analysis was used to predict the odds of suffering depression based on FPCS and the raw scores. The analytical results showed that, via FPCS, the probability of correct classification of depressed and non-depressed was raised from 74.8% to 77.2%. Next, discrimination analysis was performed to classify the subjects according to the severity of depression into three categories: non-depression, depression with remission and depression without remission. The analytical results exhibited that, via FPCS, the probability of correct classification of severity of depression was raised from 71.2% to 80.7%. These two statistical analyses consistently show that FPCS exhibited higher predictive validity than did the raw score. That is, BDI scoring via FPCS makes more accuracy predictions for depression than raw score. In Study Three, fuzzy c-means (FCM) clustering was applied to partition the sample according to severity of depression. To examine explore whether fuzzy-based clustering methods uncover the information inherent in the latent structure more accurately than crisp clustering, FCM, Wald’s method, and k-means method were performed. The analytical results reveal that the association between the original and classified membership generated by FCM was stronger than that of the Wald and k-means methods. Hence, FCM revealed the data structure most accurately. Overall, FPCS has been consistently shown to be superior to raw scoring in terms of reliability, validity, and clustering accuracy. This study has empirically shown that fuzzy set theory is applicable to psychological research.

We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.

O principal objetivo deste trabalho é estudar diversas aplicações de combinatória infinita em espaços de funções contínuas, definidas em espaços compactos Hausdorff. Usando combinatória infinita para uma álgebra de Boole, por meio da dualidade de Stone, obtemos um espaço compacto Hausdorff. Com certas propriedades na álgebra de Boole é possível analisar propriedades analíticas no espaço de funções contínuas definidas em tal espaço. Especificamente, analisamos a propriedade de Grothendieck. Também analisamos a relação entre o espaço de funções contínuas e o espaço compacto Hausdorff sobre o qual é definido. Apresentamos um resultado que permite obter diversos resultados conhecidos de uma maneira uniforme (só usando fatos de topologia e teoria de conjuntos), dotando o espaço de funções contínuas de uma ordem peculiar. Finalmente, estudamos um pouco de jogos topológicos mediante diversos exemplos. / The main purpose of this work is to study some infinite combinatorics applications in spaces of continuous functions, defined in Hausdorff compact spaces. Using infinite combinatorics in Boolean algebras, through Stone duality, we obtain a compact Hausdorff space. With certain properties in Boolean algebras it is possible to analyze analytic properties in the space of continuous functions defined in such space. Specifically, we analyze the Grothendieck property. We also analyze the relationship between the space of continuous functions and the compact Hausdorff space on which it is defined. We present a result that allows to obtain several known results in a uniform way (only using facts of topology and set theory), giving the space of continuous functions a peculiar order. Finally, we study some topological games through several examples.

O principal objetivo deste trabalho é estudar diversas aplicações de combinatória infinita em espaços de funções contínuas, definidas em espaços compactos Hausdorff. Usando combinatória infinita para uma álgebra de Boole, por meio da dualidade de Stone, obtemos um espaço compacto Hausdorff. Com certas propriedades na álgebra de Boole é possível analisar propriedades analíticas no espaço de funções contínuas definidas em tal espaço. Especificamente, analisamos a propriedade de Grothendieck. Também analisamos a relação entre o espaço de funções contínuas e o espaço compacto Hausdorff sobre o qual é definido. Apresentamos um resultado que permite obter diversos resultados conhecidos de uma maneira uniforme (só usando fatos de topologia e teoria de conjuntos), dotando o espaço de funções contínuas de uma ordem peculiar. Finalmente, estudamos um pouco de jogos topológicos mediante diversos exemplos. / The main purpose of this work is to study some infinite combinatorics applications in spaces of continuous functions, defined in Hausdorff compact spaces. Using infinite combinatorics in Boolean algebras, through Stone duality, we obtain a compact Hausdorff space. With certain properties in Boolean algebras it is possible to analyze analytic properties in the space of continuous functions defined in such space. Specifically, we analyze the Grothendieck property. We also analyze the relationship between the space of continuous functions and the compact Hausdorff space on which it is defined. We present a result that allows to obtain several known results in a uniform way (only using facts of topology and set theory), giving the space of continuous functions a peculiar order. Finally, we study some topological games through several examples.

Made available in DSpace on 2016-04-27T16:57:50Z (GMT). No. of bitstreams: 1 EDM - Michaela C Schon.pdf: 1931458 bytes, checksum: 5cde0886ff87d5dafb588e52ab96ed50 (MD5) Previous issue date: 2006-11-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This paper aimed the realization of a study concerning the philosophical epistemology of the concept of number, in which it still makes sense to ask: What is number? In this perspective, we have assumed as problematic the philosophical duality of the conceptualizations of numbers, according to Axiomatic (proposed by Peano) e by the Set Theory and Logics (proposed by Russell), being the Conceptualization of Number the problem of this research, concerning the possibility of introducing an ultimate definition to this concept. The focus of this research is in the polemics that exists about the number introduced by Russell (1872-1970) contrary to Piano s (1858-1932), taking as a basis Otte s criticism, introduced in the article: B. Russell Introduction to Mathematical Philosophy , 2001. The research was developed using, as a reference, the sense of Complementarity, as well as using proper qualitative methodological research procedures. As a conclusion, we are able to claim that numbers are: on one hand, characteristics of certain classes and, on the other hand, operative concepts. This way, the existence of polemics between philosophers like Frege and Russell, who have favored predicative aspects, that is, they define number in terms of cardinality and, others like Grassmann, Dedekind and Peano who have highlighted the ordinal numbers, justify Otto s proposition of complementarity between the approaches. The possibility of having cognitive and didactical consequences on the teaching in the use of one or another approach of conceptualization of the number or both, as Otte intends, makes this study a contribution to Mathematical Education / Este trabalho objetivou realizar um estudo sobre a epistemologia filosófica do conceito de número, na qual ainda faz sentido o questionamento: O que é número? Nesta perspectiva, assumiu-se como problemática a dualidade filosófica das conceituações de número, sustentadas pela Axiomática (proposta por Peano) e pela Teoria dos Conjuntos e Lógica (proposta por Russell), sendo o problema de pesquisa a Conceituação de Número frente a essa dualidade e à possibilidade de ser apresentada uma definição em definitivo ao conceito de número. O foco da presente pesquisa está na polêmica existente entre a concepção de número apresentada por Russell (1872-1970) contraposta à de Peano (1858-1932), tomando-se por base as críticas de Otte, apresentadas no artigo: B. Russell Introduction to Mathematical Philosophy , de 2001. A pesquisa desenvolveu-se tendo por referência a noção de Complementaridade, tendo sido utilizados procedimentos metodológicos adequados às pesquisas qualitativas. Como conclusão pode-se afirmar que os números são: por um lado, características de certas classes e, por outro, conceitos operativos. Deste modo, a existência da polêmica entre filósofos como Frege e Russell, que favoreceram os aspectos predicativos, isto é, definem os números em termos de cardinalidade e, outros como Grassmann, Dedekind e Peano que destacam os números ordinais, justifica a proposição de Otte da complementaridade entre as abordagens. A possibilidade de existirem conseqüências cognitivas e didáticas na utilização no ensino de uma ou outra abordagem da conceituação de número ou de ambas como pretende Otte torna, este estudo, uma contribuição para a Educação Matemática

O objetivo principal desta pesquisa é o aprimoramento de ferramentas teóricas desenvolvidas para a análise harmônica de obras de vanguarda do início do século XX e afins (incluindo algumas obras de Villa-Lobos), destacando a utilização da simetria intervalar como fator de coerência em amostras de músicas deste período e correlacionadas. Observaremos principalmente aspectos relacionados a presença de padrões intervalares simétricos por reflexão inerentes ao sistema cromático. Procurando estabelecer um arcabouço teórico consistente que contemplasse as demandas circunscritas nas músicas pós-tonais analisadas, buscamos entender a importância da simetria entre alturas em níveis estruturais mais profundos supondo que a proporcionalidade intervalar pudesse corroborar no delineamento harmônico das obras escolhidas. Nossa proposta de estudo é calcada em fundamentos e conceitos desenvolvidos por teóricos ligados às pesquisas sobre a teoria dos conjuntos, com destaque para os textos de Joseph Straus (2005). Em outra esfera, elencamos ferramentas teóricas de uma segunda abordagem metodológica que cada vez mais ganha destaque entre musicólogos que atuam no campo da análise de obras pós-tonais - a teoria neorriemanniana - em particular observando os conceitos apresentados por David Lewin (1982; 1987) e os seus desdobramentos discutidos por Richard Cohn (1998; 2012), autor em torno do qual gravita a vertente secundária conhecida como teoria transformacional. A partir da intersecção entre estas duas correntes teóricas pós-tonais escolhidas, desenvolvemos uma terceira proposta metodológica aparentemente inédita - que chamaremos neste trabalho de teoria da inversão - um desdobramento decorrente do aperfeiçoamento de conceitos da teoria neorriemanniana de David Lewin e Brian Hyer que envolvem a reflexão intervalar de conjuntos de alturas, parâmetros não contemplados pela recente teoria transformacional de Dmitri Tymoczko (2007; 2011) e Richard Cohn (1998; 2012). No âmbito desta proposição, seguindo a suposição levantada por Robert Morris (2007) de que existem aspectos fundamentais em comum entre as principais correntes teóricas dedicadas à música pós-tonal, exploramos alguns apontamentos deste autor que nos direcionaram à apropriação de ferramentas pertencentes à teoria dos grupos - campo de conhecimento oriundo da matemática, especializada no estudo de simetria utilizando estruturas algébricas conhecidas como matrizes. Deste processo surge a descoberta de um conglomerado de classes de conjuntos que podem ser alinhados em uma mesma rede de projeções por inversão. Avaliando os aspectos inerentes a este sistema apresentado, identificamos a construção de toda uma estrutura em disposição espelhada, revelando a existência de uma simetria transversal que abrange um grande número de conjuntos de alturas inerentes ao universo das doze alturas, confirmando a hipótese levantada por Robert Morris. Verificamos ainda outras correlações entre os conjuntos correspondentes presentes nesta rede de projeções por inversão - relações por multiplicação pelo fator M5 e M7 e invariância entre entradas de vetores intervalares (RAHN, 1980; OLIVEIRA, 2007) - que corroboram a constatação desta dimensão simétrica envolvendo o campo harmônico cromático. Outra proposta neste trabalho foi a ampliação na gama de possibilidades de utilização de redes de alturas (Tonnetze) - ferramenta emblemática da teoria neorriemanniana - apresentando outras opções de conjuntos para os desdobramentos por inversões, indo além dos convencionais conjuntos 3-11 (tricorde Maior e menor) e 4-27 (tetracordes Maior 7 e meio diminuto) recorrentes em formatações tradicionais. Seguindo neste propósito, desenvolvemos aprofundamentos abrangendo a remota rede de alturas de Euler com o tetracorde 4-20 e o Tonnetz tridimensional de Gollin (1998), alinhando esta pesquisa também aos resultados encontrados por Henri Pousseur ([1968], 2009) em suas \"redes harmônicas\" e aos conceitos desenvolvidos por George Perle (1977) e sua \"teoria dos ciclos intervalares\". / The main objective of this research is to improve the theoretical tools developed for the harmonic analysis of the early works of the early 20th century and related works (including some works by Villa-Lobos), highlighting the use of interval symmetry as a coherence factor in samples of pieces from this period and correlated. We will mainly observe aspects related to the presence of symmetrical interval patterns by reflection inherent to the chromatic system. We will mainly observe aspects related to the presence of symmetrical interval patterns by reflection inherent to the chromatic system. In order to establish a consistent theoretical framework that contemplates the circumscribed demands in the analyzed post-tonal pieces, we sought to understand the importance of symmetry by comparing pitches at deeper structural levels, assuming that the interval proportionality could corroborate the harmonic delineation of the chosen works. Our proposal is based on fundamentals and concepts developed by theorists related to research on pitch-class set theory, especially the texts of Joseph Straus (2005). Our study proposal is based on fundamentals and concepts developed by theorists related to research on pitch-class set theory, especially the texts of Joseph Straus (2005). In another sphere, we have ellipped theoretical tools of a second methodological approach that is increasingly prominent among musicologists working in the field of post-tonal analysis - neo-Riemannian theory - in particular, observing the concepts presented by David Lewin (1982, 1987) and its ramifications discussed by Richard Cohn (1998, 2012), author around which gravitates the secondary slope known as transformational theory. From the intersection between these two post-tonal theoretical currents chosen, we have developed a third methodological proposal that is apparently unpublished - which we will call inversional pitch-class set theory - an unfolding resulting from the refinement of David Lewin and Brian Hyer\'s concepts of neo-Riemannian theory involving interval analysis of sets of pitch-class sets, parameters not contemplated by the recent transformational theory of Dmitri Tymoczko (2007; 2011) and Richard Cohn (1998, 2012). In this context, following the assumption made by Robert Morris (2007) that there are fundamental aspects in common among the main theoretical currents dedicated to post-tonal music, we explore some notes of this author that have directed us to the appropriation of tools belonging to the theory of groups - field of knowledge from mathematics, specialized in the study of symmetry using algebraic structures known as matrices. From this process comes the discovery of a conglomeration of pitch-class sets that can be aligned in the same inversional pitch-class set network. Evaluating the inherent aspects of this system, we identified the construction of a whole structure in a mirrored arrangement, revealing the existence of a transversal symmetry that covers a substantial number of pitch-class sets inherent to the universe of the twelve pitches, confirming the hypothesis raised by Robert Morris. We also verified other correlations between the corresponding sets in this inversional pitch-class set network - relations by multiplication by the factor M5 and M7 and invariance between interval vectors (RAHN, 1980; OLIVEIRA, 2007) - which corroborate the observation of this symmetrical dimension involving the chromatic harmonic field. Another proposal in this work was the expansion of the range of possibilities of use of pitch-classes networks (Tonnetze) - emblematic tool of the neo-Riemannian theory - presenting other options of sets for the inversion unfolding, going beyond the conventional sets 3-11 (Major and minor chord) and 4-27 (Major 7th and half-diminished 7th chords) recurrent in traditional formatting. Following this, we developed deepening studies covering the remote pitch-class network of Euler with the tetrachord 4-20 and Gollin\'s three-dimensional Tonnetz (1998), aligning this research also with the results found by Henri Pousseur ([1968], 2009) about his harmonic networks and the concepts developed by George Perle (1977) and his \"interval cycles theory\".

L'application de méthodes algébriques en musique représente une démarche récente dans la recherche musicale. Une réflexion historique sur l'émergence du concept de structure algébrique en musique met en évidence la place centrale occupée par trois compositeurs/théoriciens du XXe siècle : Milton Babbitt, Iannis Xenakis et Anatol Vieru. À partir de leurs propositions théoriques, cette étude développe une réflexion approfondie sur la notion de théorie musicale dans ses applications aussi bien analytiques que compositionnelles. Elle offre également une formalisation algébrique de la Set Theory et de ses développements transformationnels tout en discutant les rapports entre la tradition analytique américaine et la démarche théorique formelle en Europe. Les concepts abordés permettent de définir la place d'une démarche computationnelle en musicologie et ouvrent des questions philosophiques sur le rapport entre mathématiques et musique.

The automatic extraction of knowledge from the world by a robotic system as human beings interpret their environment through their senses is still an unsolved task in Artificial Intelligence. A robotic agent is in contact with the world through its sensors and other electronic components which obtain and process mainly numerical information. Sonar, infrared and laser sensors obtain distance information. Webcams obtain digital images that are represented internally as matrices of red, blue and green (RGB) colour coordinate values. All this numerical values obtained from the environment need a later interpretation in order to provide the knowledge required by the robotic agent in order to carry out a task. Similarly, light wavelengths with specific amplitude are captured by cone cells of human eyes obtaining also stimulus without meaning. However, the information that human beings can describe and remember from what they see is expressed using words, that is qualitatively. The exact process carried out after our eyes perceive light wavelengths and our brain interpret them is quite unknown. However, a real fact in human cognition is that people go beyond the purely perceptual experience to classify things as members of categories and attach linguistic labels to them. As the information provided by all the electronic components incorporated in a robotic agent is numerical, the approaches that first appeared in the literature giving an interpretation of this information followed a mathematical trend. In this thesis, this problem is addressed from the other side, its main aim is to process these numerical data in order to obtain qualitative information as human beings can do. The research work done in this thesis tries to narrow the gap between the acquisition of low level information by robot sensors and the need of obtaining high level or qualitative information for enhancing human-machine communication and for applying logical reasoning processes based on concepts. Moreover, qualitative concepts can be added a meaning by relating them to others. They can be used for reasoning applying qualitative models that have been developed in the last twenty years for describing and interpreting metrical and mathematical concepts such as orientation, distance, velocity, acceleration, and so on. And they can be also understood by human-users both written and read aloud. The first contributions presented are the definition of a method for obtaining fuzzy distance patterns (which include qualitative distances such as ‘near’, far’, ‘very far’ and so on) from the data obtained by any kind of distance sensors incorporated in a mobile robot and the definition of a factor to measure the dissimilarity between those fuzzy patterns. Both have been applied to the integration of the distances obtained by the sonar and laser distance sensors incorporated in a Pioneer 2 dx mobile robot and, as a result, special obstacles have been detected as ‘glass window’, ‘mirror’, and so on. Moreover, the fuzzy distance patterns provided have been also defuzzified in order to obtain a smooth robot speed and used to classify orientation reference systems into ‘open’ (it defines an open space to be explored) or ‘closed’. The second contribution presented is the definition of a model for qualitative image description (QID) by applying the new defined models for qualitative shape and colour description and the topology model by Egenhofer and Al-Taha [1992] and the orientation models by Hernández [1991] and Freksa [1992]. This model can qualitatively describe any kind of digital image and is independent of the image segmentation method used. The QID model have been tested in two scenarios in robotics: (i) the description of digital images captured by the camera of a Pioneer 2 dx mobile robot and (ii) the description of digital images of tile mosaics taken by an industrial camera located on a platform used by a robot arm to assemble tile mosaics. In order to provide a formal and explicit meaning to the qualitative description of the images generated, a Description Logic (DL) based ontology has been designed and presented as the third contribution. Our approach can automatically process any random image and obtain a set of DL-axioms that describe it visually and spatially. And objects included in the images are classified according to the ontology schema using a DL reasoner. Tests have been carried out using digital images captured by a webcam incorporated in a Pioneer 2 dx mobile robot. The images taken correspond to the corridors of a building at University Jaume I and objects with them have been classified into ‘walls’, ‘floor’, ‘office doors’ and ‘fire extinguishers’ under different illumination conditions and from different observer viewpoints. The final contribution is the definition of a similarity measure between qualitative descriptions of shape, colour, topology and orientation. And the integration of those measures into the definition of a general similarity measure between two qualitative descriptions of images. These similarity measures have been applied to: (i) extract objects with similar shapes from the MPEG7 CE Shape-1 library; (ii) assemble tile mosaics by qualitative shape and colour similarity matching; (iii) compare images of tile compositions; and (iv) compare images of natural landmarks in a mobile robot world for their recognition. The contributions made in this thesis are only a small step forward in the direction of enhancing robot knowledge acquisition from the world. And it is also written with the aim of inspiring others in their research, so that bigger contributions can be achieved in the future which can improve the life quality of our society.