A relação com a teoria das cordas renovou o interesse nas teorias quânticas de campo formuladas num espaço-tempo não-comutativo. O principal aspecto dessas teorias é o assim chamado "mecanismo UV/IR", segundo o qual divergências ultravioletas são parcialmente convertidas em infravermelhas. Para certos modelos, estas singularidades infravermelhas originadas do mecanismo UV/IR podem inviabilizar a solução perturbativa da teoria de campos. A questão principal, portanto, é encontrar teorias que sejam consistentes quando formuladas num espaço-tempo não-comutativo, sendo os modelos supersimétricos particularmente promissores neste sentido. Neste trabalho, examinamos as teorias de calibre supersimétricas Abelianas (NCSQED) e não-Abelianas com grupo de calibre U(N) (NCSYM) formuladas num espaço-tempo não-comutativo de quatro dimensões. Emambos os casos, calculamos as funções de vértice utilizando o formalismo covariante de supercampos que é tornado completamente operacional. Consideramos tanto as teorias N = 1 quanto as com supersimetria estendida. Mostramos rigorosamente que, a um laço da teoria de perturbações, estes modelos são livres de singularidades infravermelhas UV/IR não-integráveis. Para a função de dois pontos da NCSQED esta afirmação vale em qualquer calibre, ao passo que, para a função de três pontos, as singularidades infravermelhas UV/IR perigosas se anulam num calibre particular. Já para a NCSYM, demonstramos que as correções quânticas às funções de vértice de dois e três pontos não apresentam os efeitos indesejáveis do mecanismo UV/IR graças a certas relações envolvendo traços dos geradores do grupo de calibre que, surpreendentemente, são satisfeitas apenas na representação fundamental do grupoU (N). Como esperado, a função de dois pontos é também finita na teoria N = 4.

A relação com a teoria das cordas renovou o interesse nas teorias quânticas de campo formuladas num espaço-tempo não-comutativo. O principal aspecto dessas teorias é o assim chamado "mecanismo UV/IR", segundo o qual divergências ultravioletas são parcialmente convertidas em infravermelhas. Para certos modelos, estas singularidades infravermelhas originadas do mecanismo UV/IR podem inviabilizar a solução perturbativa da teoria de campos. A questão principal, portanto, é encontrar teorias que sejam consistentes quando formuladas num espaço-tempo não-comutativo, sendo os modelos supersimétricos particularmente promissores neste sentido. Neste trabalho, examinamos as teorias de calibre supersimétricas Abelianas (NCSQED) e não-Abelianas com grupo de calibre U(N) (NCSYM) formuladas num espaço-tempo não-comutativo de quatro dimensões. Emambos os casos, calculamos as funções de vértice utilizando o formalismo covariante de supercampos que é tornado completamente operacional. Consideramos tanto as teorias N = 1 quanto as com supersimetria estendida. Mostramos rigorosamente que, a um laço da teoria de perturbações, estes modelos são livres de singularidades infravermelhas UV/IR não-integráveis. Para a função de dois pontos da NCSQED esta afirmação vale em qualquer calibre, ao passo que, para a função de três pontos, as singularidades infravermelhas UV/IR perigosas se anulam num calibre particular. Já para a NCSYM, demonstramos que as correções quânticas às funções de vértice de dois e três pontos não apresentam os efeitos indesejáveis do mecanismo UV/IR graças a certas relações envolvendo traços dos geradores do grupo de calibre que, surpreendentemente, são satisfeitas apenas na representação fundamental do grupoU (N). Como esperado, a função de dois pontos é também finita na teoria N = 4.

The thesis deals with technology of hot-rolled profiles, describing total summary of principles and conditions this process is based on. There is a draft of a calibration of profile, the application of which is subsequently verified through a numerical simulation. Futher in the thesis also was verified the mechanical properties of applied construction steel S355J2 through the up-setting test and compared the results with the used material model in the simulation software Simufact Forming. Then, the test rolling of the calibrated profile was carried out. As a next step, the dimensions of the rolled bars were compared with the results of the simulation, on the basis of which the applicability and suitability of the simulation in the future development of other profiles was verified and evaluated.

Metodología para estimar un modelo matemático desarrollado y validado a partir de la información obtenida en un proceso, para explicar y predecir la variación observable en una característica de calidad así como un modelo que explique el coste de la variación. Los modelos se obtienen utilizando herramientas de control estadístico y los conceptos de costes de calidad y su combinación permite identificar las actuaciones más eficaces para la mejora del proceso y predecir su resultado. La aplicación práctica se realiza en un proceso de fabriación de baldosas cerámicas para reducir la variación dimensional.

Motivation In the black population of South Africa, cardiovascular disease (CVD) is rapidly increasing due to urbanisation. Stroke is usually accompanied by a prothrombotic haemostatic profile. Changing lifestyle factors that accompany the urbanisation process could have a negative impact on the haemostatic profile of black South Africans. Elevated levels of pro-coagulant factors, von Willebrand factor (vWF), fibrinogen and fibrin D-dimer have been reported in the black population, which could increase the black population’s susceptibility to CVD. However, low levels of plasminogen activator inhibitor-1 (PAI-1) previously reported in the black population could contribute towards a pro-fibrinolytic state, which may counteract the hypercoagulant state. This may have a beneficial effect on the haemostatic profile of the black population. More investigation into the haemostatic profile of black South Africans is therefore needed to determine if an altered haemostatic profile exists in this group, and if so, to what extent these alterations may relate to cardiovascular dysfunction. This study included markers of both the coagulation (vWF, fibrinogen, fibrin D-dimer) and fibrinolytic (PAI-1, fibrin D-dimer and fibrinolytic potential) systems in an attempt to investigate the haemostatic profile of the black population of South Africa, and for comparison purposes that of the white population as well. The relationship of these markers’ with selected markers of cardiovascular function was also examined to determine if they could possibly contribute to an increase in cardiovascular risk, especially in the black population. Aims The aims of this study were to first compare coagulation and fibrinolysis markers in the black and white populations of South Africa. Furthermore, to determine if associations exist between the selected components of the haemostatic system and markers of cardiovascular function, especially in the black population of South Africa, who tends to be at a higher cardiovascular risk due to altered metabolic and haemostatic profiles. Methodology The Sympathetic activity and Ambulatory Blood Pressure in Africans (SABPA) study was a prospective cohort study that consisted of 409 participants at baseline (2008-2009) that were equally distributed according to both ethnicity (200 black; 209 white) and gender (black, 101 men, 99 women; white, 101 men, 108 women). At follow-up (2011/2012) the cohort totalled 359 participants (170 black, 88 men and 82 women; 189 white, 93 men and 96 women). Data from baseline measurements were used for the first two manuscripts (chapters 2 and 3), while followup data was used for the third manuscript (chapter 4). vWF, fibrinogen, PAI-1, fibrin D-dimer, CLT, serum peroxides, glutathione, glutathione peroxidase and reductase activity were determined, and ambulatory blood pressure and the retinal vessel calibres were measured. The groups were stratified by ethnicity as specified by statistical interaction terms. T-tests and chi-square tests were used to compare means and proportions, respectively. Pearson and partial regression analyses were used to determine correlations between the components of the haemostatic system and cardiovascular function markers. This was followed by multiple linear regression analyses to investigate whether independent associations exist between the variables in both ethnic groups. P-values ≤0.050 were deemed significant. Results and conclusion of each manuscript The first manuscript (chapter 2) compares the haemostatic profiles of the black and white population to determine whether ambulatory blood pressure is related to components of the haemostatic system. The black participants displayed a prothrombotic profile with significantly higher vWF, fibrinogen, PAI-1, fibrin D-dimer and a longer CLT than their white counterparts. Furthermore, partial and multiple linear regression analyses showed a positive association of systolic and diastolic blood pressure with fibrin D-dimer in the black population, while a negative association existed between ambulatory blood pressure and CLT in the white population. These associations suggest that fibrin D-dimer may contribute, at least in part, to the high prevalence of hypertension in the black population. The second manuscript (chapter 3) determined associations between markers of the haemostatic and oxidant-antioxidant systems in the black and white populations. In addition to the prothrombotic profile that exists in the black population, this group also had significantly higher serum peroxides (oxidative stress) and lower glutathione peroxidase activity (antioxidant) levels. Multiple linear regression analyses indicated positive associations between fibrinogen and serum peroxides in both populations. In the white population, an additional positive association was found between serum peroxide and CLT. In the black population, vWF and CLT were negatively associated with GPx activity. The results suggest that there are ethnic-specific relationships between the haemostatic and oxidant-antioxidant systems. The third manuscript (chapter 4) investigated the relationships between the retinal vessel calibres and components of the haemostatic system in the black and white population. The investigation focussed specifically on arteriolar diameters in the lower median, since a narrow arteriolar diameter is known to be associated with elevated blood pressure. In both ethnic groups, a narrower arteriolar calibre was accompanied by narrower venular calibres. Independent positive associations were found between the central retinal vein equivalent (CRVE) and fibrinogen in the black population, as well as vWF and CLT in the white population. In addition, independent negative associations were found between the central retinal artery equivalent and CLT in the black population and with vWF in the white population. The results suggest that haemostatic alterations are linked to early vascular changes that may differ between ethnicities. General conclusion Ethnic-specific relationships between the components of the haemostatic system and measures of cardiovascular function are evident. The prothrombotic profile that is observed in the black population, together with the adverse associations of the haemostatic components with blood pressure, a compromised oxidant-antioxidant profile, and retinal vessel calibres may contribute, at least in part, to the high cardiovascular and cerebrovascular risk evident in this population group. / PhD (Physiology), North-West University, Potchefstroom Campus, 2015

Motivation In the black population of South Africa, cardiovascular disease (CVD) is rapidly increasing due to urbanisation. Stroke is usually accompanied by a prothrombotic haemostatic profile. Changing lifestyle factors that accompany the urbanisation process could have a negative impact on the haemostatic profile of black South Africans. Elevated levels of pro-coagulant factors, von Willebrand factor (vWF), fibrinogen and fibrin D-dimer have been reported in the black population, which could increase the black population’s susceptibility to CVD. However, low levels of plasminogen activator inhibitor-1 (PAI-1) previously reported in the black population could contribute towards a pro-fibrinolytic state, which may counteract the hypercoagulant state. This may have a beneficial effect on the haemostatic profile of the black population. More investigation into the haemostatic profile of black South Africans is therefore needed to determine if an altered haemostatic profile exists in this group, and if so, to what extent these alterations may relate to cardiovascular dysfunction. This study included markers of both the coagulation (vWF, fibrinogen, fibrin D-dimer) and fibrinolytic (PAI-1, fibrin D-dimer and fibrinolytic potential) systems in an attempt to investigate the haemostatic profile of the black population of South Africa, and for comparison purposes that of the white population as well. The relationship of these markers’ with selected markers of cardiovascular function was also examined to determine if they could possibly contribute to an increase in cardiovascular risk, especially in the black population. Aims The aims of this study were to first compare coagulation and fibrinolysis markers in the black and white populations of South Africa. Furthermore, to determine if associations exist between the selected components of the haemostatic system and markers of cardiovascular function, especially in the black population of South Africa, who tends to be at a higher cardiovascular risk due to altered metabolic and haemostatic profiles. Methodology The Sympathetic activity and Ambulatory Blood Pressure in Africans (SABPA) study was a prospective cohort study that consisted of 409 participants at baseline (2008-2009) that were equally distributed according to both ethnicity (200 black; 209 white) and gender (black, 101 men, 99 women; white, 101 men, 108 women). At follow-up (2011/2012) the cohort totalled 359 participants (170 black, 88 men and 82 women; 189 white, 93 men and 96 women). Data from baseline measurements were used for the first two manuscripts (chapters 2 and 3), while followup data was used for the third manuscript (chapter 4). vWF, fibrinogen, PAI-1, fibrin D-dimer, CLT, serum peroxides, glutathione, glutathione peroxidase and reductase activity were determined, and ambulatory blood pressure and the retinal vessel calibres were measured. The groups were stratified by ethnicity as specified by statistical interaction terms. T-tests and chi-square tests were used to compare means and proportions, respectively. Pearson and partial regression analyses were used to determine correlations between the components of the haemostatic system and cardiovascular function markers. This was followed by multiple linear regression analyses to investigate whether independent associations exist between the variables in both ethnic groups. P-values ≤0.050 were deemed significant. Results and conclusion of each manuscript The first manuscript (chapter 2) compares the haemostatic profiles of the black and white population to determine whether ambulatory blood pressure is related to components of the haemostatic system. The black participants displayed a prothrombotic profile with significantly higher vWF, fibrinogen, PAI-1, fibrin D-dimer and a longer CLT than their white counterparts. Furthermore, partial and multiple linear regression analyses showed a positive association of systolic and diastolic blood pressure with fibrin D-dimer in the black population, while a negative association existed between ambulatory blood pressure and CLT in the white population. These associations suggest that fibrin D-dimer may contribute, at least in part, to the high prevalence of hypertension in the black population. The second manuscript (chapter 3) determined associations between markers of the haemostatic and oxidant-antioxidant systems in the black and white populations. In addition to the prothrombotic profile that exists in the black population, this group also had significantly higher serum peroxides (oxidative stress) and lower glutathione peroxidase activity (antioxidant) levels. Multiple linear regression analyses indicated positive associations between fibrinogen and serum peroxides in both populations. In the white population, an additional positive association was found between serum peroxide and CLT. In the black population, vWF and CLT were negatively associated with GPx activity. The results suggest that there are ethnic-specific relationships between the haemostatic and oxidant-antioxidant systems. The third manuscript (chapter 4) investigated the relationships between the retinal vessel calibres and components of the haemostatic system in the black and white population. The investigation focussed specifically on arteriolar diameters in the lower median, since a narrow arteriolar diameter is known to be associated with elevated blood pressure. In both ethnic groups, a narrower arteriolar calibre was accompanied by narrower venular calibres. Independent positive associations were found between the central retinal vein equivalent (CRVE) and fibrinogen in the black population, as well as vWF and CLT in the white population. In addition, independent negative associations were found between the central retinal artery equivalent and CLT in the black population and with vWF in the white population. The results suggest that haemostatic alterations are linked to early vascular changes that may differ between ethnicities. General conclusion Ethnic-specific relationships between the components of the haemostatic system and measures of cardiovascular function are evident. The prothrombotic profile that is observed in the black population, together with the adverse associations of the haemostatic components with blood pressure, a compromised oxidant-antioxidant profile, and retinal vessel calibres may contribute, at least in part, to the high cardiovascular and cerebrovascular risk evident in this population group. / PhD (Physiology), North-West University, Potchefstroom Campus, 2015

Este trabalho está dedicado a estudar a consistência global da dinâmica quântica de sistemas não-comutativos. Nosso ponto de partida é a teoria de sistemas vinculados, dado que esta provê uma descrição uni cada da dinâmica clássica e quântica para os modelos a serem investigados. Analisamos o problema relacionado com a existência da série de Born e unitariedade e focamos, na seqüência, na formulação funcional da dinâmica quântica dos sistemas não-comutativos. A compatibilidade entre as abordagens funcional e operatorial é substanciada de forma geral. Subseqüentemente, a transformada de Weyl generalizada de índice α é usada para implementar a de nição "via time-slicing" da integral de caminho no espaço de fase, o que nos permite calcular o correspondente propagador de Feynman. Como esperado, esta representação para o propagador de Feynman não é única, mas rotulada pelo parâmetro real α. Provamos que as contribuições dependentes de α desaparecem no limite quando o "slice" de tempo tende a zero, tal qual é requerido pela consistência da formulação. Esta prova é intrincada pois o Hamiltoniano envolve, necessariamente, produtos de operadores não comutantes. A anti-simetria da matriz que parametriza a não-comutatividade joga um papel fundamental no mecanismo de cancelamento dos termos dependentes de α. Por m, estudamos a implementação do processo formulado por Batalin, Fradkin e Tyutin (BFT), o qual permite transformar esses sistemas em uma teoria de calibre Abeliana exibindo apenas vínculos de primeira classe. A adequação da imersão BFT, como aplicada neste trabalho, é veri cada demonstrando que existe um mapeamento isomór co que conecta o modelo de segunda classe com o setor invariante de calibre da teoria de calibre. Como é sabido, a quantização funcional de uma teoria de calibre exige a eliminação da liberdade de calibre. Então, temos a nossa disposição um conjunto in nito de descrições alternativas para a mecânica quântica não-comutativa, uma para cada calibre. Estudamos as características relevantes deste in nito conjunto de correspondências. A quantização funcional da teoria de calibre é explicitamente realizada para dois calibres diferentes e os resultados comparados com o correspondente ao sistema de segunda classe. Dentro do quadro operatorial, a teoria de calibre é quantizada utilizando-se o método de Dirac. / This work is concerned with the global consistency of the quantum dynamics of noncommutative systems. Our point of departure is the theory of constrained systems, since it provides a uni ed description of the classical and quantum dynamics for the models under investigation. We then analise the problem concerned with the su cient conditions for the existence of the Born series and unitarity and turn, afterwards, into studying the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. Subsequently, the generalized Weyl transform of index α is used to implement the time-slice de nition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter α. We succeed in proving that the α-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian necessarily involves products of noncommuting operators. The antisymmetry of the matrix parameterizing the noncommutativity plays a key role in the cancelation mechanism of the α-dependent terms. Finally, we study the embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) which enables one to transform these noncommutative systems into an Abelian gauge theory exhibiting only rst class constraints. The appropriateness of the BFT embedding, as implemented in this work, is veri ed by showing that there exists a one to one mapping linking the second class model with the gauge invariant sector of the gauge theory. As is known, the functional quantization of a gauge theory calls for the elimination of its gauge freedom. Then, we have at our disposal an in nite set of alternative descriptions for noncommutative quantum mechanics, one for each gauge. We study the relevant features of this in nite set of correspondences. The functional quantization of the gauge theory is explicitly performed for two di erent gauges and the results compared with that corresponding to the second class system. Within the operator framework the gauge theory is quantized by using Dirac's method.

Este trabalho está dedicado a estudar a consistência global da dinâmica quântica de sistemas não-comutativos. Nosso ponto de partida é a teoria de sistemas vinculados, dado que esta provê uma descrição uni cada da dinâmica clássica e quântica para os modelos a serem investigados. Analisamos o problema relacionado com a existência da série de Born e unitariedade e focamos, na seqüência, na formulação funcional da dinâmica quântica dos sistemas não-comutativos. A compatibilidade entre as abordagens funcional e operatorial é substanciada de forma geral. Subseqüentemente, a transformada de Weyl generalizada de índice α é usada para implementar a de nição "via time-slicing" da integral de caminho no espaço de fase, o que nos permite calcular o correspondente propagador de Feynman. Como esperado, esta representação para o propagador de Feynman não é única, mas rotulada pelo parâmetro real α. Provamos que as contribuições dependentes de α desaparecem no limite quando o "slice" de tempo tende a zero, tal qual é requerido pela consistência da formulação. Esta prova é intrincada pois o Hamiltoniano envolve, necessariamente, produtos de operadores não comutantes. A anti-simetria da matriz que parametriza a não-comutatividade joga um papel fundamental no mecanismo de cancelamento dos termos dependentes de α. Por m, estudamos a implementação do processo formulado por Batalin, Fradkin e Tyutin (BFT), o qual permite transformar esses sistemas em uma teoria de calibre Abeliana exibindo apenas vínculos de primeira classe. A adequação da imersão BFT, como aplicada neste trabalho, é veri cada demonstrando que existe um mapeamento isomór co que conecta o modelo de segunda classe com o setor invariante de calibre da teoria de calibre. Como é sabido, a quantização funcional de uma teoria de calibre exige a eliminação da liberdade de calibre. Então, temos a nossa disposição um conjunto in nito de descrições alternativas para a mecânica quântica não-comutativa, uma para cada calibre. Estudamos as características relevantes deste in nito conjunto de correspondências. A quantização funcional da teoria de calibre é explicitamente realizada para dois calibres diferentes e os resultados comparados com o correspondente ao sistema de segunda classe. Dentro do quadro operatorial, a teoria de calibre é quantizada utilizando-se o método de Dirac. / This work is concerned with the global consistency of the quantum dynamics of noncommutative systems. Our point of departure is the theory of constrained systems, since it provides a uni ed description of the classical and quantum dynamics for the models under investigation. We then analise the problem concerned with the su cient conditions for the existence of the Born series and unitarity and turn, afterwards, into studying the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. Subsequently, the generalized Weyl transform of index α is used to implement the time-slice de nition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter α. We succeed in proving that the α-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian necessarily involves products of noncommuting operators. The antisymmetry of the matrix parameterizing the noncommutativity plays a key role in the cancelation mechanism of the α-dependent terms. Finally, we study the embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) which enables one to transform these noncommutative systems into an Abelian gauge theory exhibiting only rst class constraints. The appropriateness of the BFT embedding, as implemented in this work, is veri ed by showing that there exists a one to one mapping linking the second class model with the gauge invariant sector of the gauge theory. As is known, the functional quantization of a gauge theory calls for the elimination of its gauge freedom. Then, we have at our disposal an in nite set of alternative descriptions for noncommutative quantum mechanics, one for each gauge. We study the relevant features of this in nite set of correspondences. The functional quantization of the gauge theory is explicitly performed for two di erent gauges and the results compared with that corresponding to the second class system. Within the operator framework the gauge theory is quantized by using Dirac's method.

Este trabalho está dedicado a estudar a consistência global da dinâmica quântica de sistemas não-comutativos. Nosso ponto de partida é a teoria de sistemas vinculados, dado que esta provê uma descrição uni cada da dinâmica clássica e quântica para os modelos a serem investigados. Analisamos o problema relacionado com a existência da série de Born e unitariedade e focamos, na seqüência, na formulação funcional da dinâmica quântica dos sistemas não-comutativos. A compatibilidade entre as abordagens funcional e operatorial é substanciada de forma geral. Subseqüentemente, a transformada de Weyl generalizada de índice α é usada para implementar a de nição "via time-slicing" da integral de caminho no espaço de fase, o que nos permite calcular o correspondente propagador de Feynman. Como esperado, esta representação para o propagador de Feynman não é única, mas rotulada pelo parâmetro real α. Provamos que as contribuições dependentes de α desaparecem no limite quando o "slice" de tempo tende a zero, tal qual é requerido pela consistência da formulação. Esta prova é intrincada pois o Hamiltoniano envolve, necessariamente, produtos de operadores não comutantes. A anti-simetria da matriz que parametriza a não-comutatividade joga um papel fundamental no mecanismo de cancelamento dos termos dependentes de α. Por m, estudamos a implementação do processo formulado por Batalin, Fradkin e Tyutin (BFT), o qual permite transformar esses sistemas em uma teoria de calibre Abeliana exibindo apenas vínculos de primeira classe. A adequação da imersão BFT, como aplicada neste trabalho, é veri cada demonstrando que existe um mapeamento isomór co que conecta o modelo de segunda classe com o setor invariante de calibre da teoria de calibre. Como é sabido, a quantização funcional de uma teoria de calibre exige a eliminação da liberdade de calibre. Então, temos a nossa disposição um conjunto in nito de descrições alternativas para a mecânica quântica não-comutativa, uma para cada calibre. Estudamos as características relevantes deste in nito conjunto de correspondências. A quantização funcional da teoria de calibre é explicitamente realizada para dois calibres diferentes e os resultados comparados com o correspondente ao sistema de segunda classe. Dentro do quadro operatorial, a teoria de calibre é quantizada utilizando-se o método de Dirac. / This work is concerned with the global consistency of the quantum dynamics of noncommutative systems. Our point of departure is the theory of constrained systems, since it provides a uni ed description of the classical and quantum dynamics for the models under investigation. We then analise the problem concerned with the su cient conditions for the existence of the Born series and unitarity and turn, afterwards, into studying the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. Subsequently, the generalized Weyl transform of index α is used to implement the time-slice de nition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter α. We succeed in proving that the α-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian necessarily involves products of noncommuting operators. The antisymmetry of the matrix parameterizing the noncommutativity plays a key role in the cancelation mechanism of the α-dependent terms. Finally, we study the embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) which enables one to transform these noncommutative systems into an Abelian gauge theory exhibiting only rst class constraints. The appropriateness of the BFT embedding, as implemented in this work, is veri ed by showing that there exists a one to one mapping linking the second class model with the gauge invariant sector of the gauge theory. As is known, the functional quantization of a gauge theory calls for the elimination of its gauge freedom. Then, we have at our disposal an in nite set of alternative descriptions for noncommutative quantum mechanics, one for each gauge. We study the relevant features of this in nite set of correspondences. The functional quantization of the gauge theory is explicitly performed for two di erent gauges and the results compared with that corresponding to the second class system. Within the operator framework the gauge theory is quantized by using Dirac's method.