"... it will be the purpose of this paper to search the writings of Classicism as exemplified by Smith, Say, Ricardo, Senior, Mill and Marshall, and the works of Veblen for their theories of monopoly. By a contrast and comparison of these theories, it is hoped that the information developed may become the basis for checking any hidden holdovers from these two schools and, by showing their origin, to develop some understanding of a singularly complex problem of the modern world."--leaf 2

This dissertation modifies the model of Yang and Ng (1995) to investigate the condition of the emergence of the firm from a specialized exchange economy. It is assumed in this dissertation that there are fixed costs involved in the operation of a firm. After taking into account of this factor, I re-examine its effects on the division of labor and the structure of firm following Yang and Ng¡¦s framework. The model adopts an inframarginal framework to analyze the subject, in which a firm demonstrates diminishing returns, while both the final labor input and the intermediate labor input demonstrates increasing returns defined upon individuals. However, it is assumed that only the final labor input has the economies of specialization. It is argued in this dissertation that the existence of fixed learning costs may stimulate the economy to undergo structural changes if suitable conditions are met, which are largely related to relative market efficiency between markets.

SOUSA, Leandro Araujo de. Análise psicométrica dos itens de educação física do Exame Nacional do Ensino Médio (ENEM) via Teoria Clássica dos Testes. 2017. 69f. – Dissertação (Mestrado) – Universidade Federal do Ceará, Programa de Pós-graduação em Educação Brasileira, Fortaleza (CE), 2017. / Submitted by Gustavo Daher (gdaherufc@hotmail.com) on 2017-03-17T13:08:49Z No. of bitstreams: 1 2017_dis_lasousa.pdf: 765171 bytes, checksum: 136d56a7eae5ae977bedd331d9f77052 (MD5) / Approved for entry into archive by Márcia Araújo (marcia_m_bezerra@yahoo.com.br) on 2017-03-20T12:22:25Z (GMT) No. of bitstreams: 1 2017_dis_lasousa.pdf: 765171 bytes, checksum: 136d56a7eae5ae977bedd331d9f77052 (MD5) / Made available in DSpace on 2017-03-20T12:22:25Z (GMT). No. of bitstreams: 1 2017_dis_lasousa.pdf: 765171 bytes, checksum: 136d56a7eae5ae977bedd331d9f77052 (MD5) Previous issue date: 2017 / In recent years the importance of large-scale evaluations in the Brazilian context has grown, with emphasis in this scenario on the National High School Examination (ENEM). With its reformulation in 2009, the skills and abilities of the Physical Education have been inserted in the reference matrix of this exam. In that same year, the method of analysis of the results, based on the Classical Tests Theory (CTT), was also changed, using the Item Response Theory (IRT), under justification of being more adequate to allow the comparability of the Results. With this, this research aimed to analyze the Physical Education items of ENEM from the years 2009 to 2014 from the CTT. For that, we used the microdata of the exam provided by the National Institute of Studies and Educational Research Anísio Teixeira (INEP). The following metric parameters were analyzed: validity, reliability, difficulty and discrimination. SPSS software version 20.0 was used as a resource. The items presented good correlation values and adequacy of the item sample. They presented scores of commonality and factorial loads inadequate for the composition of the test. The Exploratory Factor Analysis presented low explanation of the variance considering only one factor, even the scree plot indicating that the test is unidimensionality. The reliability values of the test were good, with no influence of physical education items. The difficulty and discrimination presented values acceptable in almost every year. However, in 2014 the test did not present unidimensionality, considering the explained variance, as well as in the graphic analysis. This year, the items presented high difficulty and low discrimination. Thus, it is concluded that the Language and Codes tests of the ENEM presented difficulties in proving the unidimensionality, although it presented good accuracy, with the exception of 2014 and some Physical Education items of the exam did not present adequate parameters. Such factors may compromise the validity of the measure and consequently the results of such examination. / Nos últimos anos tem crescido a importância das avaliações em larga escala no contexto brasileiro, com destaque nesse cenário o Exame Nacional do Ensino Médio (ENEM). Com sua reformulação em 2009, competências e habilidades da área de Educação Física têm sido inseridas na matriz de referência desse exame. Nesse mesmo ano é alterado também o método de análise dos resultados, realizado a partir da Teoria Clássica dos Testes (TCT), passando a ser utilizada a Teoria de Resposta ao Item (TRI), sob justificativa de ser mais adequada por permitir a comparabilidade dos resultados. Com isso, esta pesquisa objetivou analisar os itens de Educação Física do ENEM dos anos de 2009 a 2014 a partir da TCT. Para tanto, utilizou-se os microdados do exame disponibilizados pelo Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira (INEP). Foram analisados os seguintes parâmetros métricos: validade, fidedignidade, dificuldade e discriminação. Utilizou-se como recurso o software SPSS, versão 20.0. Os itens apresentaram bons valores de correlação e adequação da amostra de itens. Apresentaram escores de comunalidade e cargas fatoriais inadequados para composição da prova. A Análise Fatorial Exploratória apresentou baixa explicação da variância considerando apenas um fator, mesmo a análise gráfica (scree plot) indicando a unidimensionalidade do teste. Os valores de fidedignidade da prova foram bons, não havendo influência dos itens de Educação Física. A dificuldade e discriminação apresentaram valores aceitáveis em quase todos os anos. No entanto, em 2014 a prova não apresentou unidimensionalidade, considerando a variância explicada, bem como na análise gráfica. Neste ano, os itens apresentaram alta dificuldade e baixa discriminação. Dessa forma, conclui-se que as provas de Linguagens e Códigos do ENEM apresentaram dificuldades de comprovação da unidimensionalidade, embora, tenha apresentado boa precisão, com exceção de 2014 e alguns itens de Educação Física do exame não apresentaram parâmetros adequados. Tais fatores podem comprometer a validade da medida e consequentemente dos resultados desse exame.

nÃo hà / Nos Ãltimos anos tem crescido a importÃncia das avaliaÃÃes em larga escala no contexto brasileiro, com destaque nesse cenÃrio o Exame Nacional do Ensino MÃdio (ENEM). Com sua reformulaÃÃo em 2009, competÃncias e habilidades da Ãrea de EducaÃÃo FÃsica tÃm sido inseridas na matriz de referÃncia desse exame. Nesse mesmo ano à alterado tambÃm o mÃtodo de anÃlise dos resultados, realizado a partir da Teoria ClÃssica dos Testes (TCT), passando a ser utilizada a Teoria de Resposta ao Item (TRI), sob justificativa de ser mais adequada por permitir a comparabilidade dos resultados. Com isso, esta pesquisa objetivou analisar os itens de EducaÃÃo FÃsica do ENEM dos anos de 2009 a 2014 a partir da TCT. Para tanto, utilizou-se os microdados do exame disponibilizados pelo Instituto Nacional de Estudos e Pesquisas Educacionais AnÃsio Teixeira (INEP). Foram analisados os seguintes parÃmetros mÃtricos: validade, fidedignidade, dificuldade e discriminaÃÃo. Utilizou-se como recurso o software SPSS, versÃo 20.0. Os itens apresentaram bons valores de correlaÃÃo e adequaÃÃo da amostra de itens. Apresentaram escores de comunalidade e cargas fatoriais inadequados para composiÃÃo da prova. A AnÃlise Fatorial ExploratÃria apresentou baixa explicaÃÃo da variÃncia considerando apenas um fator, mesmo a anÃlise grÃfica (scree plot) indicando a unidimensionalidade do teste. Os valores de fidedignidade da prova foram bons, nÃo havendo influÃncia dos itens de EducaÃÃo FÃsica. A dificuldade e discriminaÃÃo apresentaram valores aceitÃveis em quase todos os anos. No entanto, em 2014 a prova nÃo apresentou unidimensionalidade, considerando a variÃncia explicada, bem como na anÃlise grÃfica. Neste ano, os itens apresentaram alta dificuldade e baixa discriminaÃÃo. Dessa forma, conclui-se que as provas de Linguagens e CÃdigos do ENEM apresentaram dificuldades de comprovaÃÃo da unidimensionalidade, embora, tenha apresentado boa precisÃo, com exceÃÃo de 2014 e alguns itens de EducaÃÃo FÃsica do exame nÃo apresentaram parÃmetros adequados. Tais fatores podem comprometer a validade da medida e consequentemente dos resultados desse exame. / In recent years the importance of large-scale evaluations in the Brazilian context has grown, with emphasis in this scenario on the National High School Examination (ENEM). With its reformulation in 2009, the skills and abilities of the Physical Education have been inserted in the reference matrix of this exam. In that same year, the method of analysis of the results, based on the Classical Tests Theory (CTT), was also changed, using the Item Response Theory (IRT), under justification of being more adequate to allow the comparability of the Results. With this, this research aimed to analyze the Physical Education items of ENEM from the years 2009 to 2014 from the CTT. For that, we used the microdata of the exam provided by the National Institute of Studies and Educational Research AnÃsio Teixeira (INEP). The following metric parameters were analyzed: validity, reliability, difficulty and discrimination. SPSS software version 20.0 was used as a resource. The items presented good correlation values and adequacy of the item sample. They presented scores of commonality and factorial loads inadequate for the composition of the test. The Exploratory Factor Analysis presented low explanation of the variance considering only one factor, even the scree plot indicating that the test is unidimensionality. The reliability values of the test were good, with no influence of physical education items. The difficulty and discrimination presented values acceptable in almost every year. However, in 2014 the test did not present unidimensionality, considering the explained variance, as well as in the graphic analysis. This year, the items presented high difficulty and low discrimination. Thus, it is concluded that the Language and Codes tests of the ENEM presented difficulties in proving the unidimensionality, although it presented good accuracy, with the exception of 2014 and some Physical Education items of the exam did not present adequate parameters. Such factors may compromise the validity of the measure and consequently the results of such examination.

Cette thèse montre que l’on peut trouver dans la théorie classique contemporaine deux conceptions du salaire selon que l’évaluation du travail salarié est le résultat d’une lutte pour le partage de la valeur ajoutée (salaire exogène) ou du jeu concurrentiel sur le marché du travail (salaire endogène). Sur cette base, nous soutenons que l’approche classique, dont les antécédents historiques reposent sur les œuvres de David Ricardo (1772-1823) et d’un auteur méconnu, Robert Torrens (1780-1864), est susceptible de rendre compte : 1) du partage de la valeur ajoutée et de son influence sur les prix relatifs dans les économies à l’équilibre et en déséquilibre, 2) du statut global des salariés, c'est-à-dire à la fois dans la production et dans les échanges, 3) de la condition d’existence du profit défini comme la rémunération d’une pratique spécifiquement capitaliste. Le chapitre I reconsidère les arguments qui font de Torrens l’antécédent historique du concept de salaire exogène. Le chapitre II expose sa théorie des prix, sur la base de laquelle le chapitre III étudie la relation entre les variables de répartition et les prix relatifs dans les économies à l’équilibre et en déséquilibre. Le chapitre IV analyse la répartition des biens de consommation finale entre les membres de la société (salariés et capitalistes). Le chapitre V dégage les éléments centraux de la théorie physique du salaire de Torrens. Par rapport au corpus théorique actuel issu des travaux de Piero Sraffa, cette théorie se singularise par un salaire réel déterminé par la concurrence sur le marché du travail et une expression du taux de profit en termes d’un surtravail, non assimilable à du travail non payé. / The dissertation shows that contemporary classical theory relies on two representations of the idea of ‘wages’. Those representations depend on the way labour is valued. If wages are the result of a struggle for added value, they are seen as exogenous. On the other hand, if wages are the result of competition in labour markets, they are seen as endogenous. The classical theory, which is based on Ricardo and Torrens’ seminal works allows to understand: 1) the distribution of added value, and its effects on relative prices in economies both in equilibrium and disequilibrium, 2) the position of wage-earners in the processes of production and exchange, 3) the source of profits defined as compensation of specifically capitalist practice. Chapter I discusses Torrens’ contribution to the classical idea of exogenous wages. Chapter II presents Torrens’ theory of prices. This theory is then used in chapter III to analyse the relationships between distribution variables and relative prices in economies in equilibrium or in disequilibrium. Chapter IV focuses on the distribution of final goods among wage earners and capitalists. The last chapter (V) underlines the main features of Torrens’ theory of wages. In comparison with Sraffa’s work, this theory proposes first that real wages are fixed on labour market, and then that the rate of profit can be defined as a surplus labour, but not as an unpaid labour.

Submitted by Liliane Ferreira (ljuvencia30@gmail.com) on 2018-08-21T12:44:31Z No. of bitstreams: 2 Dissertação - Lucas Elias Vieira - 2018.pdf: 1438523 bytes, checksum: 81df75730e0ab41f69b4619af81cd7a2 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-08-22T12:42:43Z (GMT) No. of bitstreams: 2 Dissertação - Lucas Elias Vieira - 2018.pdf: 1438523 bytes, checksum: 81df75730e0ab41f69b4619af81cd7a2 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-08-22T12:42:43Z (GMT). No. of bitstreams: 2 Dissertação - Lucas Elias Vieira - 2018.pdf: 1438523 bytes, checksum: 81df75730e0ab41f69b4619af81cd7a2 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-07-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we study topological defects in classical field theory aiming applications in cosmology. Topological and non-topological solutions were investigated in two dimensions, where we also analyzed the stability of some of these. We consider models of one and two real scalar fields, whose solutions were obtained via quadrature, BPS and orbit methods. The stability of these solutions also were investigated in some cases. In order to apply scalar field theory in cosmology, we revisit some topics of general relativity, such that the principle of equivalence, Einstein's field equations and Schwarzschild's solution for Einstein's equation. We also present some topics in cosmology, relevant to the present work, such that the cosmological principle, Hubble's law and cosmological inflation. We discuss the solutions of the Einstein's equations in the FRW metric in the so-called Hot Big Bang model-the universe composed of matter and radiation. In order to solve some problems of initial conditions of the universe, we introduce the inflationary theory, which supposes that the primordial universe was underwent an accelerated expansion guided by a scalar field. In sequence, we introduce the standard cosmological model, which states that the universe is in accelerated expansion driving by a cosmological constant, named dark energy, which constitutes about seventy percent of the universe energy. Finally, we present a first order formalism for cosmology and discuss two model describing the standard and tachyonics dynamics for scalar field that represent the dark energy. / Neste trabalho foram abordadas soluções do tipo defeitos topológicos em teoria clássica de campos. Foram investigadas soluções topológicas e não topológicas em duas dimensões, onde também analisamos a estabilidade de algumas destas. Consideramos modelos de um e dois campos escalares reais, onde utilizamos os métodos de quadratura, BPS e das órbitas para obter soluções. Com o intuito de aplicar a teoria de campos escalares em cosmologia, introduzimos a teoria da relatividade geral, falando sobre o princípio da equivalência, equações de campo de Einstein e solução de Schwarzschild para as equações de Einstein. Abordamos alguns tópicos em cosmologia, onde enunciamos o princípio cosmológico, lei de Hubble e inflação cosmológica. Resolvemos as equações de Einstein na métrica FRW considerando o universo composto de matéria e radiação denominado modelo Big Bang quente. Com o objetivo de resolver alguns problemas de condições iniciais do universo, introduzimos a teoria inflacionária, a qual propõe que o universo primordial sofreu uma expansão acelerada guiada por um campo escalar. Após mostrar como a teoria da inflação corrige tais problemas, apresentamos o modelo cosmológico padrão, o qual afirma que o universo está em expansão acelerada e utiliza uma constante cosmológica como guia da expansão, tal constante neste cenário representa a denominada energia escura que compõe aproximadamente setenta por cento do universo. Finalmente, nós apresentamos o formalismo de primeira ordem para cosmologia e discutimos dois modelos descrevendo dinâmicas padrão e taquiônica para o campo escalar, representando a energia escura.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-10T14:24:23Z No. of bitstreams: 1 denisemendoncademelo.pdf: 507802 bytes, checksum: c3b5542230bfc09c6418bdaf6d4288e2 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-04-11T11:35:29Z (GMT) No. of bitstreams: 1 denisemendoncademelo.pdf: 507802 bytes, checksum: c3b5542230bfc09c6418bdaf6d4288e2 (MD5) / Made available in DSpace on 2017-04-11T11:35:29Z (GMT). No. of bitstreams: 1 denisemendoncademelo.pdf: 507802 bytes, checksum: c3b5542230bfc09c6418bdaf6d4288e2 (MD5) Previous issue date: 2016-12-07 / O Mini Exame do Estado Mental (MEEM) é o instrumento de rastreio do status cognitivo mais utilizado no mundo. Apesar de sua relevância, suas propriedades psicométricas não têm sido suficientemente analisadas no Brasil. O objetivo principal desta tese foi analisar as propriedades psicométricas da versão brasileira dessa medida proposta por Brucki et al (2003) com Teoria Clássica dos Testes e com Teoria de Resposta ao Item (TRI), enfatizando a escolaridade. Para tanto, foram desenvolvidos dois estudos. Antes, porém, realizou-se uma revisão sistemática das pesquisas empíricas indexadas na base de dados SciELO que utilizaram o MEEM com idosos no Brasil (N=74), que identificou 11 versões do MEEM. Mais da metade desses estudos adotou a escolaridade para estabelecer pontos de corte. Constatou-se que as propriedades psicométricas dessa medida têm sido pouco investigadas no país e, consequentemente, faltam padronização e evidências de validade. O segundo estudo teve como objetivo obter evidências de validade baseadas na estrutura interna para essa medida por meio de análise de componentes principais (ACP) e de análise fatorial confirmatória (AFC). Uma amostra de 2734 idosos participantes do estudo FIBRA Unicamp foi pesquisada. Na ACP, uma solução com cinco componentes foi a mais adequada. Quatro modelos de estrutura interna foram testados com AFC. Dois modelos com estrutura multidimensional de cinco fatores e um fator de segunda ordem apresentaram índices de ajuste satisfatórios, o que não ocorreu com o modelo unidimensional. Assim, há evidências de que se trata de uma medida multidimensional. O terceiro estudo apresentou uma análise com TRI para avaliar o funcionamento diferencial dos itens (DIF), a dificuldade e a discriminação dos itens. Constatou-se que quatro itens do sete seriado foram os mais difíceis e os itens nove, 13, 22 e 23, os mais fáceis. O nível de habilidade das pessoas foi maior do que o nível de dificuldade dos itens. Observou-se DIF para escolaridade (27 itens), sexo (18 itens) e idade (16 itens). Conclui-se que o uso do MEEM com idosos da comunidade deve ser cauteloso devido ao grande número de itens enviesados, principalmente pela escolaridade. Em suma, a versão do MEEM analisada apresentou evidências de validade baseadas na estrutura interna, e é fortemente influenciada pela escolaridade. / The Mini Mental State Examination (MMSE) is the screening tool of cognitive status most used in the world. Despite its relevance, its psychometric properties have not been sufficiently analyzed in Brazil. The main purpose of this thesis was to analyze the psychometric properties of a Brazilian version of this measure suggested by Brucki et al (2003) with Classical Theory of Tests and Item Response Theory (IRT), emphasizing education. For this, two studies were developed. First, however, there was a systematic review of empirical research indexed in SciELO database that used the MMSE with elderly (N = 74), which identified 11 versions of the MMSE. More than half of these studies adopted the school to establish cutoffs. It was found that the psychometric properties of this measure have been little investigated in the country and, consequently, lack standardization and evidence of validity. The second study aimed to obtain evidence of validity based on the internal structure for this measure by principal component analysis (PCA) and confirmatory factor analysis (CFA). A sample of 2734 elderly participants of the FIBRA Unicamp study was researched. In the PCA, a solution with five components was the most appropriate. Four internal structure models were tested with CFA. Two models with multidimensional structure of five factors and a factor of second order showed satisfactory fit indices, which did not happen with the unidimensional model. Therefore, there is evidence that it is a multidimensional measurement. The third study presents an analysis with IRT to evaluate the differential item functioning (DIF), difficulty and discrimination of the items. It was found that four items of serial seven were the most difficult and the items 9, 13, 22 and 23 were the easiest ones. The skill level of the people was higher than the level of difficulty of the items. There was DIF for education (27 items), sex (18 items) and age (16 items). We conclude that the use of MMSE with community dwelling elderly should be cautious due to the large number of biased items, especially for education. In conclusion, the version of the MMSE analyzed the version of MMSE analyzed obtained validity evidences regarding the internal structure, and is strongly influenced by education.

In 2010 high harmonic generation (HHG) in solids was first observed where high order harmonics of a strong laser field's frequency were observed. HHG in solids is now a rapidly developing field that allows for exciting applications like fully solid state attosecond XUV sources and new ultrafast resolution imaging techniques for quantum dynamics in solids. HHG in solids has been explained by two mechanisms: an interband mechanism, due to polarization associated with separate energy bands, and an intraband mechanism that results from nonlinearities and population changes associated with each individual band. While interband HHG has been seen in wide bandwidth semiconductors, intraband HHG has been observed in narrow bandwidth dielectrics. There has not yet been an explanation of the alternation of mechanisms with material differences. The main goal of this thesis is to attempt to provide a better understanding of the most important mechanisms and where they prevail. Although numerical modelling of HHG requires consideration of multiple energy bands, a two-band model consisting only of a valence band and a single conduction band can explain the most important mechanisms. This model requires a given material's band gap between its valence and conduction bands as well as dipole matrix elements between the bands. In this thesis we follow the Kronig-Penney model to develop a 1D delta-function potential model of solids to obtain these properties required of the two-band model. We implement this in a Wannier quasi-classical (WQC) model of interband HHG in semiconductors that explains the dominant dynamics leading to such through quasi-classical real space electron-hole pair trajectories. Although HHG in solids can be explained to be the result of a resonant process in which an electron-hole pair is generated in the first step, there are also virtual transition processes that lack consideration. These processes do not conserve energy and correspond to transitions to conduction bands resulting from field induced distortions of the ground state. We use methodology introduced by Keldysh for optical field ionization of atoms and solids along with the 1D delta-function potential model to quantify how both resonant and virtual transitions lead to HHG in solids for wide and low bandwidth solids.

Le but de cette thèse était de modéliser la réponse mécanique de nanotubes de carbone à des champs électriques. Nous avons commencé par utiliser le potentiel AIREBO dans des simulations de dynamique moléculaire afin d'étudier l'élasticité non-linéaire et la limite de déformation en torsion de divers nanotubes, en fonction de leur longueur, rayon et chiralité. Nous trouvons notamment que le module d'Young effectif des tubes décroît d'autant plus vite que la chiralité est faible. D'autre part, nous montrons que la limite de l'énergie stockable par atome lors de la torsion d'un tube est d'autant plus grande que le diamètre est petit.<br><br>Nous modélisons ensuite, de façon atomistique, la distribution surfacique de charge électrique sur des nanotubes de carbone possédant une charge nette. Nous retrouvons notamment l'effet de pointe classique avec un très bon accord quantitatif avec des résultats expérimentaux obtenus par microscopie à force électrostatique.<br><br>Par combinaison des méthodes utilisées dans les études précédentes, nous simulons la déflection de nanotubes semi-conducteurs et métalliques par un champ électrique extérieur, dans une configuration de type interrupteur moléculaire. L'effet des caractéristiques géométriques des tubes et du champ sur cette déflection ont été systématiquement étudiés.<br><br>En outre, nous avons vu que des simulations de dynamique moléculaire avec le potentiel AIREBO permettent de retrouver quantitativement les énergies expérimentales d'adsorption du benzène, du naphtalène et d'anthracène sur le graphite. Ce type de simulation nous permet d'avancer sur la voie de la compréhension de la sélectivité de l'adsorption de certaines molécules surfactantes à plusieurs cycles benzéniques sur des nanotubes de chiralité donnée.

After giving a brief background on the subject we introduce in section two the Phase-Integral Method of Fröman &amp; Fröman in terms of the platform function of Yngve and Thidé. In section three we derive a different form of the radial Bohr-Sommerfeld condition in terms of the apsidal angle of the corresponding classical motion. Using the derived expression, we then show how easily one can calculate the exact energy eigenvalues of the hydrogen atom and the isotropic three-dimensional harmonic oscillator, we also derive an expression for higher order quantization condition. In section four we derive an expression for the angular frequencies of a restricted (0≤φ≤β) soap bubble and also give a proposition concerning the parameters l and m of the associated Legendre differential equation. / Vi använder Fröman &amp; Frömans Fas-Integral Metod tillsammans med Yngve &amp; Thidés plattformfunktion för att härleda kvantiseringsvilkoret för högre ordningar. I sektion tre skriver vi Bohr-Sommerfelds kvantiseringsvillkor på ett annorlunda sätt med hjälp av den så kallade apsidvinkeln (definierad i samma sektion) för motsvarande klassiska rörelse, vi visar också hur mycket detta underlättar beräkningar av energiegenvärden för väteatomen och den isotropa tredimensionella harmoniska oscillatorn. I sektion fyra tittar vi på en såpbubbla begränsad till området 0≤φ≤β för vilket vi härleder ett uttryck för dess (vinkel)egenfrekvenser. Här ger vi också en proposition angående parametrarna l och m tillhörande den associerade Legendreekvationen.