This thesis is about double product integrals with pseudo rotational generator, and aims to exhibit them as unitary implementors of Bogolubov transformations. We further introduce these concepts in this abstract and describe their roles in the thesis's chapters. The notion of product integral, (simple product integral, not double) is not a new one, but is unfamiliar to many a mathematician. Product integrals were first investigated by Volterra in the nineteenth century. Though often regarded as merely a notation for solutions of differential equations, they provide a priori a multiplicative analogue of the additive integration theories of Riemann, Stieltjes and Lebesgue. See Slavik [2007] for a historical overview of the subject. Extensions of the theory of product integrals to multiplicative versions of Ito and especially quantum Ito calculus were first studied by Hudson, Ion and Parthasarathy in the 1980's, Hudson et al. [1982]. The first developments of double product integrals was a theory of an algebraic kind developed by Hudson and Pulmannova motivated by the study of the solution of the quantum Yang-Baxter equation by the construction of quantum groups, see Hudson and Pulmaanova [2005]. This was a purely algebraic theory based on formal power series in a formal parameter. However, there also exists a developing analytic theory of double product integral. This thesis contributes to this analytic theory. The first papers in that direction are Hudson [2005b] and Hudson and Jones [2012]. Other motivations include quantum extension of Girsanov's theorem and hence a quantum version of the Black-Scholes model in finance. They may also provide a general model for causal interactions in noisy environments in quantum physics. From a different direction "causal" double products, (see Hudson [2005b]), have become of interest in connection with quantum versions of the Levy area, and in particular quantum Levy area formula (Hudson [2011] and Chen and Hudson [2013]) for its characteristic function. There is a close association of causal double products with the double products of rectangular type (Hudson and Jones [2012] pp 3). For this reason it is of interest to study "forwardforward" rectangular double products. In the first chapter we give our notation which will be used in the following chapters and we introduce some simple double products and show heuristically that they are the solution of two different quantum stochastic differential equations. For each example the order in which the products are taken is shown to be unimportant; either calculation gives the same answer. This is in fact a consequence of the so called multiplicative Fubini Theorem Hudson and Pulmaanova [2005]. In Chapter two we formally introduce the notion of product integral as a solution of two particular quantum stochastic differential equations. In Chapter three we introduce the Fock representation of the canonical commutation relations, and discuss the Stone-von Neumann uniqueness theorem. We define the notion of Bogolubov transformation (often called a symplectic automorphism, see Parthasarathy [1992] for example), implementation of these transformations by an implementor (a unitary operator) and introduce Shale's theorem which will be relevant to the following chapters. For an alternative coverage of Shale's Theorem, symplectic automorphism and their implementors see Derezinski [2003]. In Chapter four we study double product integrals of the pseudo rotational type. This is in contrast to double product integrals of the rotational type that have been studied in (Hudson and Jones [2012] and Hudson [2005b]). The notation of the product integral is suggestive of a natural discretisation scheme where the infinitesimals are replaced by discrete increments i.e. discretised creation and annihilation operators of quantum mechanics. Because of a weak commutativity condition, between the discretised creation and annihilation operators corresponding on different subintervals of R, the order of the factors of the product are unimportant (Hudson [2005a]), and hence the discrete product is well defined; we call this result the discrete multiplicative Fubini Theorem. It is also the case that the order in which the products are taken in the continuous (non-discretised case) does not matter (Hudson [2005a], Hudson and Jones [2012]). The resulting discrete double product is shown to be the implementor (a unitary operator) of a Bogolubov transformation acting on discretised creation and annihilation operators (Bogolubov transformations are invertible real linear operators on a Hilbert space that preserve the imaginary part of the inner product, but here we may regard them equivalently as liner transformations acting directly on creation and annihilations operators but preserving adjointness and commutation relations). Unitary operators on the same Hilbert space are a subgroup of the group of Bogolubov transformations. Essentially Bogolubov transformations are used to construct new canonical pairs from old ones (In the literature Bogolubov transformations are often called symplectic automorphisms). The aforementioned Bogolubov transformation (acting on the discretised creation and annihilation operators) can be embedded into the space L2(R+) L2(R+) and limits can be taken resulting in a limiting Bogolubov transformation in the space L2(R+) L2(R+). It has also been shown that the resulting family of Bogolubov transformation has three important properties, namely bi-evolution, shift covariance and time-reversal covariance, see (Hudson [2007]) for a detailed description of these properties. Subsequently we show rigorously that this transformation really is a Bogolubov transformation. We remark that these transformations are Hilbert-Schmidt perturbations of the identity map and satisfy a criterion specified by Shale's theorem. By Shale's theorem we then know that each Bogolubov transformation is implemented in the Fock representation of the CCR. We also compute the constituent kernels of the integral operators making up the Hilbert-Schmidt operators involved in the Bogolubov transformations, and show that the order in which the approximating discrete products are taken has no bearing on the final Bogolubov transformation got by the limiting procedure, as would be expected from the multiplicative Fubini Theorem. In Chapter five we generalise the canonical form of the double product studied in Chapter four by the use of gauge transformations. We show that all the theory of Chapter four carries over to these generalised double product integrals. This is because there is unitary equivalence between the Bogolubov transformation got from the generalised canonical form of the double product and the corresponding original one. In Chapter six we make progress towards showing that a system of implementors of this family of Bogolubov transformations can be found which inherits properties of the original family such as being a bi-evolution and being covariant under shifts. We make use of Shales theorem (Parthasarathy [1992] and Derezinski [2003]). More specifically, Shale's theorem ensures that each Bogolubov transformation of our system is implemented by a unitary operator which is unique to with multiplicaiton by a scalar of modulus 1. We expect that there is a unique system of implementors, which is a bi-evolution, shift covariant, and time reversal covariant (i.e. which inherits the properties of the corresponding system of Bogolubov transformation). This is partly on-going research. We also expect the implementor of the Bogolubov transformation to be the original double product. In Evans [1988], Evan's showed that the the implementor of a Bogolubov transformation in the simple product case is indeed the simple product. If given more time it might be possible to adapt Evan's result to the double product case.

Motivation: Recently, with urbanisation becoming an increasing trend, the African population have been introduced to the westernised lifestyle. This contributed to severe health implications and a rapid increase in cardiovascular morbidity and mortality in the African population. In South Africa and other sub–Saharan African countries, the prevalence of cardiovascular disease is increasing rapidly. The African population is thus regarded as a high risk group, and we deem it necessary and important to investigate additional possible cardiovascular risk markers in the attempt to improve the diagnosis of cardiovascular diseases and the treatment thereof. We investigate double product as a possible cardiovascular risk marker in African and Caucasian men from South Africa. Recent studies have suggested that increased double product values might be an independent predictor of cardiovascular morbidity and mortality. However, investigations on double product and endorgan damage are limited in this population group. The strength of associations between double product and various measurements of end–organ damage, including intima–media thickness, left ventricular hypertrophy and albumin–to–creatinine ratio, are compared with the associations between the more traditional risk factor, systolic blood pressure, and the measurements of end–organ damage. Aim: The aim of this study is to investigate the usability of double product as a possible cardiovascular risk marker in African and Caucasian men from South Africa. Methodology: The manuscript presented in Chapter 3 made use of the cross–sectional SABPA (Sympathetic Activity and Ambulatory Blood Pressure in Africans) study, where 101 urbanised African and 101 Caucasian male school teachers from the North West Province of South Africa were recruited. Anthropometric measurements were taken in triplicate using standard methods. The cardiovascular measurements included 24–hour ambulatory blood pressure and electrocardiogram measurements. Double products were calculated as the product of 24–hour, daytime and night–time mean systolic blood pressure and mean heart rate values. The carotid intima–media thickness was also obtained following the prescribed protocols. A registered nurse collected blood samples and the biochemical measurements were performed by independent laboratories, blinded to the subject’s cardiovascular profile. Means and proportions were compared by standard t–tests and the x2 –tests, respectively. The association between markers of end–organ damage and double product were investigated using single, partial and multiple regression analyses. All p–values <= 0.05 were deemed significant. Results and conclusions: Results from this study showed that Africans displayed significantly higher systolic blood pressure values, heart rate values and subsequent double product values compared to the Caucasians. Despite these significant differences, double product only showed borderline significant correlations with the markers of end–organ damage in African men, while no correlations were evident in Caucasian men. In African men, systolic blood pressure displayed stronger and significant correlations with intimamedia thickness, left ventricular hypertrophy, and albumin–to–creatinine ratio than double product. These findings suggest that double product may not be a good marker of increased risk for end–organ damage and subsequent cardiovascular–related mortality. / Thesis (M.Sc. (Physiology))--North-West University, Potchefstroom Campus, 2011.

Motivation: Recently, with urbanisation becoming an increasing trend, the African population have been introduced to the westernised lifestyle. This contributed to severe health implications and a rapid increase in cardiovascular morbidity and mortality in the African population. In South Africa and other sub–Saharan African countries, the prevalence of cardiovascular disease is increasing rapidly. The African population is thus regarded as a high risk group, and we deem it necessary and important to investigate additional possible cardiovascular risk markers in the attempt to improve the diagnosis of cardiovascular diseases and the treatment thereof. We investigate double product as a possible cardiovascular risk marker in African and Caucasian men from South Africa. Recent studies have suggested that increased double product values might be an independent predictor of cardiovascular morbidity and mortality. However, investigations on double product and endorgan damage are limited in this population group. The strength of associations between double product and various measurements of end–organ damage, including intima–media thickness, left ventricular hypertrophy and albumin–to–creatinine ratio, are compared with the associations between the more traditional risk factor, systolic blood pressure, and the measurements of end–organ damage. Aim: The aim of this study is to investigate the usability of double product as a possible cardiovascular risk marker in African and Caucasian men from South Africa. Methodology: The manuscript presented in Chapter 3 made use of the cross–sectional SABPA (Sympathetic Activity and Ambulatory Blood Pressure in Africans) study, where 101 urbanised African and 101 Caucasian male school teachers from the North West Province of South Africa were recruited. Anthropometric measurements were taken in triplicate using standard methods. The cardiovascular measurements included 24–hour ambulatory blood pressure and electrocardiogram measurements. Double products were calculated as the product of 24–hour, daytime and night–time mean systolic blood pressure and mean heart rate values. The carotid intima–media thickness was also obtained following the prescribed protocols. A registered nurse collected blood samples and the biochemical measurements were performed by independent laboratories, blinded to the subject’s cardiovascular profile. Means and proportions were compared by standard t–tests and the x2 –tests, respectively. The association between markers of end–organ damage and double product were investigated using single, partial and multiple regression analyses. All p–values <= 0.05 were deemed significant. Results and conclusions: Results from this study showed that Africans displayed significantly higher systolic blood pressure values, heart rate values and subsequent double product values compared to the Caucasians. Despite these significant differences, double product only showed borderline significant correlations with the markers of end–organ damage in African men, while no correlations were evident in Caucasian men. In African men, systolic blood pressure displayed stronger and significant correlations with intimamedia thickness, left ventricular hypertrophy, and albumin–to–creatinine ratio than double product. These findings suggest that double product may not be a good marker of increased risk for end–organ damage and subsequent cardiovascular–related mortality. / Thesis (M.Sc. (Physiology))--North-West University, Potchefstroom Campus, 2011.

Made available in DSpace on 2016-06-02T19:22:12Z (GMT). No. of bitstreams: 1 6350.pdf: 3538170 bytes, checksum: 7d36b58142cff837550ab8086a4637b7 (MD5) Previous issue date: 2014-11-07 / Universidade Federal de Sao Carlos / The Normal weight obesity (NWO) syndrome has been characterized on subjects with normal BMI and high body fat mass percentage (BF%>30 on with women) and is a risk factor for cardiometabolic dysregulation and cardiovascular mortality. The aim of this study was to evaluate whether circuit resistance training (CRT) improves body composition, heart size and functions, cardiometabolic parameters, and cardiorespiratory, cardiovascular and skeletal muscle fitness on women with NWO. Subjects/Methods: Data are mean [95%CI]. Twenty-nine white women participated in the study allocated to three groups: ten NWO-CRT (baseline: BMI=22.4 [21.4-23.3] kg/m2; BF%=44.5 [41.0-48.0]%) performed CRT during ten weeks; thirteen untrained NWO-control (baseline: BMI=21.7 [20.8-22.7] kg/m2; BF%=37.8 [34.6-41.1]%) and six non-obese (BMI=19.2 [17.9-20.6] kg/m2; BF%=23.6 [18.3-29.0]%). At baseline (all groups) and after ten weeks (NWO groups) performed: dualenergy- X-ray-absorptiometry, echocardiography, blood tests, arterial pressure, exercise testing, and total-overload-by-training-session (TOL) was calculated. Results: At baseline NWO-control showed almost double of body fat mass (BF) (22.41 [19.5-25.3] kg) than nonobese (11.88 [9.0-14.8] kg) (p=0.0001), and NWO-CRT had more BF than NWO-control (27.28 [23.9-30.6] kg) (p=0.0227). The NWO-CRT after training: reduced more than 8 kg of BF (p=0.000002); the BF% became lower than NWO-control (33.1 [30.1-36.0] < 37.0 [34.3- 39.6]%, p=0.0423) with 30% of NWO-CRT becoming non-obese; reduced 3 kg of trunk fat mass (p=0.000005); showed fasting glucose (72.8 [69.4-76.2] mg/dl) smaller than NWOcontrol (81.7 [78.6-84.8] mg/dl) (p=0.004) and non-obese (92.7 [86.6-98.8] mg/dl) (p=0.000003); increased TOL (5,087.5 [4,142.5-6,032.5] to 6 963.3 [6,226.4-7,700.2] rep.kg, p=0.0004); increased load at VO2peak (122.5 [106.8-138,2] to 137.5 [118.18-156.82] W, p=0.0051); reduced the double product/load at VO2peak ratio (277.4 [222.1-332.8] to 237.7 [194.2-281.2] mmHg.bpm/W, p=0.0015). The CRT increased left ventricular mass/body surface area ratio (84.29 [78.98-89.6] to 90.29 [81.45-99.12] g/m2, p=0.0215). Conclusions: CRT improves the body composition, cardiovascular and skeletal-muscle fitness and reduces fasting glucose without cardiorespiratory changes on normal weight obese women. / A síndrome de obesidade com peso normal tem sido caracterizada em indivíduos com índice de massa corporal normal e elevado percentual de massa gorda corporal (%MG), maior que 30% em mulheres, e é considerado um fator de risco para distúrbios metabólicos e mortalidade cardiovascular. Assim, o objetivo deste trabalho foi avaliar se um treinamento de circuito resistido melhora a composição corporal, parâmetros estruturais e funções cardíacas, parâmetros cardiometabólicos e capacidades cardiorrespiratória, cardiovascular e muscular esquelética em mulheres adultas sedentárias obesas de peso normal. Dados são mostrados em média [intervalo de confiança de 95%]. Vinte e nove mulheres participaram do estudo sendo alocadas em três grupos: dez mulheres inicialmente obesas de peso normal (IMC=22,4 [21,4-23,3] kg/m2; %MG=44,5 [41,0-48,0]%) que realizaram dez semanas de treinamento resistido em circuito (grupo MOPN-TRC); treze mulheres no grupo MOPN-controle (IMC=21,7 [20,8-22,7] kg/m2; %MG 37,8 [34,6-41,1]%) que permaneceram sedentárias nas dez semanas; e seis mulheres num grupo controle de nãoobesas sedentárias avaliadas (IMC=19,2 [17,9-20,6] kg/m2; %MG=23,6 [18,3-29,0]%) somente inicialmente como valores de referência geral. Inicialmente todos os grupos realizaram: avaliação a composição corporal por absortometria radiológica de dupla energia dual-energy-X-ray-absorptiometry (DXA), ecocardiografia com doppler, perfil lipídico glicemia de jejum, teste ergoespirométrico máximo em ciclo-ergômetro, pressão arterial e a sobrecarga total da sessão de treino foi calculada. Após dez semanas os dois grupos MOPN refizeram estas análises. Inicialmente o grupo MOPN-controle apresentava mais que o dobro de massa gorda corporal (MG) (22,41[19,5-25,3] kg) que as não-obesas (11,88 [9,0-14,8] kg) (p=0,0001), enquanto o grupo MOPN-TRC possuía MG (27,28 [23,9-30,6] kg) ainda maior MOPN-controle (p=0,0227). Após o treinamento as MOPN-TRC: reduziram mais de 8 kg de MG (p=0,000002); o %MG que era maior tornou-se menor que o das MOPN-controle (33,1 [30,1-36,0] < 37,0 [34,3-39,6]%, p=0,0423) e 30% das voluntárias tornaram-se não-obesas; reduziram 3 kg de massa gorda no tronco (p=0,000005); apresentaram glicemia de jejum (72,8 [69,4-76,2] mg/dl) menor que os grupos MOPN-controle (81,7 [78,6-84,8] mg/dl) (p=0,004) e não-obeso (92,7 [86,6-98,8] mg/dl) (p=0,000003); aumentou a sobrecarga total da sessão de treino (5.087,5 [4.142,5-6.032,5] para 6.963,3 [6.226,4-7.700,2] rep.kg, p=0,0004); aumentou a carga no VO2pico (122,5 [106,8-138,2] para 137,5 [118,18-156,82] W, p=0,0051); reduziu a taxa duplo produto/carga no VO2pico (277,4 [222,1-332,8] para 237,7 [194,2-281,2] mmHg.bpm/W, p=0,0015). Conclui-se que o treinamento resistido em circuito realizado melhora a composição corporal, as capacidades cardiovascular e muscular esquelética, enquanto reduz a glicemia de jejum sem alterar a capacidade cardiorrespiratória de mulheres obesas de peso normal.