Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 99-101). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 99-101). Also available on the Internet.

Inaug.-Diss.- Bonn. / Includes bibliographical references (p. 58).

Inaug.-Diss.- Bonn. / Includes bibliographical references (p. 58).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf 63).

Sidman and Tailby (1982) established procedures to analyze the nature of stimulus to stimulus relations established by conditional discriminations. Their research describes specific behavioral tests to determine the establishment of properties that define the relations of equivalence. An equivalence relation requires the demonstration of three conditional relations: reflexivity, symmetry, and transitivity. The equivalence stimulus paradigm provides a method to account for novel responding. The research suggests that equivalence relations provide a more efficient and effective approach to the assessment, analysis, and instruction of skills. The present research examined the effectiveness of the formation of an equivalence class in teaching young children coin value relations. The second aspect of the study was to determine if there was a relationship between equivalence class formation and generalization of the skills established to other settings. Five children, 4- and 5-years old, were selected to participate in the study based on their lack of skills in the area of coin values and purchasing an item with dimes or quarters equaling fifth cents. The experimental task was presented on a Macintosh computer with HyperCard programming. The experimental stimuli consisted of pictures of dimes, quarters, and Hershey candy bars presented in match-to-sample procedures. Two conditional discriminations were taught (if A then B and if B then C.). The formation of an equivalence class was evaluated by if C then A. Generalization across settings was tested after the formation of an equivalence class by having the children purchase a Hershey candy bar with dimes at a play store. A multiple baseline experimentaldesign was used to demonstrate a functional relationship between the formation of an equivalence class and generalization of skills across settings. The present research provides supportive evidence that coin value relations can be taught to young children using equivalence procedures. The study also demonstrated generalization of novel, untaught stimuli across settings, after the formation of an equivalence class. A posttest on generalization across settings was conducted 3 months after the study. Long-term stability of equivalence relations was demonstrated by three of the subjects. / Department of Special Education

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 .

Let X be a Polish space and Q a measurable partition of X into Gδ equivalence classes. In 1978, S. M. Srivastava proved the existence of a Borel cross section for Q. He asked whether more can be concluded in case each equivalence class is uncountable. This question is answered here in the affirmative. The main result of the author is a proof that shows the existence of a Castaing Representation for Q.

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2011. / Includes bibliography. / ENGLISH ABSTRACT: Latimer & MacDuffee showed in 1933 that there is a one-to-one correspondence between equivalence classes of matrices with a given minimum polynomial and equivalence classes of ideals of a certain ring. In the case where the matrices are taken over the integers, Behn and Van der Merwe developed an algorithm in 2002 to produce a representative in each equivalence class. We extend this algorithm to matrices taken over the ring Fq[T] of polynomials over a finite field and prove a modified version of the Latimer-MacDuffee theorem which holds for proper equivalence classes of matrices. / AFRIKAANSE OPSOMMING: Latimer & MacDuffee het in 1933 bewys dat daar 'n een-tot-een korrespondensie is tussen ekwivalensieklasse van matrikse met 'n gegewe minimumpolinoom en ekwivalensieklasse van ideale van 'n sekere ring. In die geval waar die matrikse heeltallige inskrywings het, het Behn en Van der Merwe in 2002 'n algoritme ontwikkel om verteenwoordigers in elke ekwivalensieklas voort te bring. Ons brei hierdie algoritme uit na die geval van matrikse met inskrywings in die ring Fq[T] van polinome oor 'n eindige liggaam en ons bewys 'n gewysigde weergawe van die Latimer-MacDuffee stelling wat geld vir klasse van streng ekwivalente matrikse.

Made available in DSpace on 2016-06-02T20:30:50Z (GMT). No. of bitstreams: 1 3021.pdf: 1554861 bytes, checksum: 79bd6fbda8332922bee3c9d6c05f7a59 (MD5) Previous issue date: 2010-03-08 / Universidade Federal de Minas Gerais / The paradigm of stimulus equivalence has been widely studied, with different populations. Several studies have investigated the variables that can influence the formation and maintenance of equivalent stimulus classes. Regarding to formation, some authors point that the structure of train CaN and the arrangement simple to complex can facilitate this formation. Regarding to maintenance, some authors propose, for example, that classes with higher number of stimuli are more stable and are easier to be remembered over time. Despite the diversity of populations studied with procedures derived from the paradigm of stimulus equivalence, there is still a lack of studies with the elderly population. In order to enlarge the knowledge about the generality of the phenomenon of formation and maintenance of equivalent classes stimulus, this study proposed to investigate the formation of equivalent classes of different sizes and the role of this size in the maintenance of the classes with participants between 60 and 75 years. The procedure of matching-to-sample was used to teach three classes of equivalent stimuli composed of three or six stimulus (Condition 1 and Condition 2). The experimental procedure consisted of a training phase of baseline relations and testing of emerging relations, and six weeks after the first phase, a test was done with the emerging and learned relations to check the maintenance of classes formed. All participants showed the emergence of classes of equivalent stimulus. Three participants showed immediate emergency and five delayed emergency. All the participants who went through Condition 1 had percentages of correct responses below 90% in the test done after six weeks, while participants who underwent Condition 2 had hit percentages above 90%. Since most participants from Condition 2 showed accuracy right in the beginning of the test, it was not possible to evaluate the recovery of the relations throughout the test made six weeks later. This performance showed that the size of the class may be a relevant variant that influences the stability of the class over time, since the participants in the condition that trained with larger classes had better results on this test. This paper presents contributions to the studies of stimulus equivalence, since studies with this population are rare, and to understanding the factors involved in the formation and maintenance of stimulus equivalence. / O paradigma da equivalência de estímulos tem sido amplamente estudado, com diferentes populações. Diversas pesquisas têm investigado as variáveis que podem influenciar na formação e na manutenção de classes de estímulos equivalentes. Em relação à formação, autores apontam que a estrutura de treino CaN e o arranjo do simples para o complexo podem facilitar essa formação. Em relação à manutenção, autores propõem, por exemplo, que classes com maior número de estímulos são mais estáveis ao longo do tempo e são recuperadas mais facilmente. Apesar da diversidade de populações estudadas com procedimentos oriundos do paradigma de equivalência de estímulos, existe ainda uma carência de estudos com a população idosa. Com o objetivo de ampliar o conhecimento sobre a generalidade do fenômeno de formação e manutenção de classes de estímulos equivalentes, o presente estudo propôs verificar a formação de classes de estímulos equivalentes de diferentes tamanhos e o papel deste tamanho na manutenção das mesmas com participantes idosos entre 60 e 75 anos. O procedimento de emparelhamento com o modelo foi utilizado para ensinar três classes de estímulos equivalentes compostas por três ou seis estímulos (Condição 1 e Condição 2). O procedimento experimental consistiu em uma fase de treino de relações de linha de base e teste de relações emergentes seguida, seis semanas depois, de um teste com as relações aprendidas e emergentes a fim de verificar a manutenção das classes formadas. Todos os participantes mostraram a emergência de classes de equivalência. Três participantes mostraram emergência imediata e cinco, emergência atrasada. Em geral, os participantes que passaram pela Condição 1 apresentaram porcentagens de acertos abaixo de 90% no teste feito após seis semanas, enquanto os participantes que passaram pela Condição 2 apresentaram porcentagens de acerto acima de 90%. Uma vez que a maior parte dos participantes da Condição 2 apresentaram porcentagens de acertos muito altas desde o inicio do teste, não foi possível avaliar a recuperação das relações ao longo do teste feito após seis semanas. Esse desempenho evidenciou que o número de estímulos em um classe pode ser uma variável que influencia na estabilidade das classes ao longo do tempo, uma vez que os participantes da condição que treinaram com classes maiores tiveram resultados melhores neste teste. O trabalho apresenta contribuições para os estudos em equivalência de estímulos, uma vez que estudos com população dessa idade são escassos, para a compreensão dos fatores envolvidos na formação e manutenção das classes de estímulos equivalentes.