Applications of complex functions on problems in Banach algebras
- Authors: Snyman, Jandré Dillon
- Date: 2019
- Subjects: Banach algebras , Functions of several complex variables
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10948/48643 , vital:41055
- Description: In this dissertation, we provide applications of complex function theory to problems in Banach algebras. We discuss the structure of analytic functions f : D → A, where D is a domain of C and A is a Banach algebra as given by Aupetit in [3]: either the set {λ ∈ D : Sp(f(λ)) is finite} is of capacity zero, or there exists an integer n such that Sp(f(λ)) has exactly n elements, for every λ, except on a closed, discrete set of capacity zero, where the spectrum has at most n−1 elements. This deep result, which describes the structure of Sp(f(λ)) for all λ ∈ D, relies heavily on subharmonic techniques, which are also included in the dissertation. Let A and B be Banach algebras. A linear mapping φ : A → B is called a Jordan homomorphism if and only if φ(xy + yx) = φ(x)φ(y) + φ(y)φ(x) for every x, y ∈ A. This is equivalent to saying that φ(x 2 ) = φ(x) 2 for every x ∈ A. The following problem, due to I. Kaplansky, is still unsolved for the general Banach algebra case: Let A and B be unital Banach algebras and φ : A → B a unital, invertibility preserving linear mapping. Under what conditions of A and B is φ a Jordan homomorphism? The author’s honours project [24] served as an exposition of the GleasonKahane-Żelazko Theorem [7, Theorem 4], which provides an answer to Kaplansky’s problem in the case where B = C. In this dissertation we look at other special cases of Kaplansky’s problem, such as the case where A and B are von Neumann algberas, as solved by Aupetit [4, Theorem 1.3] and remark that his result holds for the more general case where A is any C ∗ -algebra that has the property that every self-adjoint element is the limit of a sequence of linear combinations of orthogonal idempotents in A, and B is a semi-simple Banach algebra. This result relies heavily on complex function theory, spectral theory and holomorphic functional calculus. We also provide detailed expositions of the work of Taylor [27] in which an operator calculus on undounded, closed linear operators is developed, and the work of Allan [1] in which a holomorphic functional calculus is defined for locally convex algebras.
- Full Text:
- Date Issued: 2019
- Authors: Snyman, Jandré Dillon
- Date: 2019
- Subjects: Banach algebras , Functions of several complex variables
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10948/48643 , vital:41055
- Description: In this dissertation, we provide applications of complex function theory to problems in Banach algebras. We discuss the structure of analytic functions f : D → A, where D is a domain of C and A is a Banach algebra as given by Aupetit in [3]: either the set {λ ∈ D : Sp(f(λ)) is finite} is of capacity zero, or there exists an integer n such that Sp(f(λ)) has exactly n elements, for every λ, except on a closed, discrete set of capacity zero, where the spectrum has at most n−1 elements. This deep result, which describes the structure of Sp(f(λ)) for all λ ∈ D, relies heavily on subharmonic techniques, which are also included in the dissertation. Let A and B be Banach algebras. A linear mapping φ : A → B is called a Jordan homomorphism if and only if φ(xy + yx) = φ(x)φ(y) + φ(y)φ(x) for every x, y ∈ A. This is equivalent to saying that φ(x 2 ) = φ(x) 2 for every x ∈ A. The following problem, due to I. Kaplansky, is still unsolved for the general Banach algebra case: Let A and B be unital Banach algebras and φ : A → B a unital, invertibility preserving linear mapping. Under what conditions of A and B is φ a Jordan homomorphism? The author’s honours project [24] served as an exposition of the GleasonKahane-Żelazko Theorem [7, Theorem 4], which provides an answer to Kaplansky’s problem in the case where B = C. In this dissertation we look at other special cases of Kaplansky’s problem, such as the case where A and B are von Neumann algberas, as solved by Aupetit [4, Theorem 1.3] and remark that his result holds for the more general case where A is any C ∗ -algebra that has the property that every self-adjoint element is the limit of a sequence of linear combinations of orthogonal idempotents in A, and B is a semi-simple Banach algebra. This result relies heavily on complex function theory, spectral theory and holomorphic functional calculus. We also provide detailed expositions of the work of Taylor [27] in which an operator calculus on undounded, closed linear operators is developed, and the work of Allan [1] in which a holomorphic functional calculus is defined for locally convex algebras.
- Full Text:
- Date Issued: 2019
Rethinking biokinetics: a philosophical critique concerning the roles and responsibilities of practitioners and patients
- Authors: Greene, Mark A
- Date: 2016
- Subjects: Holistic education -- South Africa Physical education and training -- South Africa -- History Exercise therapy -- South Africa
- Language: English
- Type: Thesis , Masters , MA
- Identifier: http://hdl.handle.net/10948/10381 , vital:26743
- Description: The specialist responsible for providing prescribed exercise as a form of medicine to the South African population is the biokineticist. Biokinetics is a relatively new profession developed in response to empirical research supporting exercise as medicine. Although the role of the biokineticist is clearly defined by the Health Professions Council of South Africa (HPCSA), there is still confusion concerning this profession and its services. At present, the dominant medicalscientific and capitalist-economic discourses offer a narrowly mechanistic and instrumental understanding of the human body and its treatment, which contributes to this confusion. This dissertation proposes a new point of departure, enriched by historical and philosophical understandings of the human body, from which the biokineticist can begin to redefine him/herself. The study aims to shift biokinetics as a practice away from the reductionist and dehumanising influences of instrumentalist, scientific and neoliberal capitalist-economic discourses, and to restore to biokineticists, and their clients, the ability they need to interact as relatively autonomous individuals. The study draws on the ideas of Michel Foucault, specifically on his notions for ‘discipline’, ‘docility’, and ‘the care of the self’ (1991; 2005), as well as Gilles Deleuze and Felix Guattari and their ideas of ‘becoming’ and ‘rhizomatic thinking’ (1983; 1987); ultimately presenting a philosophically enriched, holistic representation of the human body. From here recommendations for best practice in contemporary biokinetics are suggested that encourage interactions and connections between the professional and his/her patients, which move beyond the mere physiological interpretations currently dominating health discourse.
- Full Text:
- Date Issued: 2016
- Authors: Greene, Mark A
- Date: 2016
- Subjects: Holistic education -- South Africa Physical education and training -- South Africa -- History Exercise therapy -- South Africa
- Language: English
- Type: Thesis , Masters , MA
- Identifier: http://hdl.handle.net/10948/10381 , vital:26743
- Description: The specialist responsible for providing prescribed exercise as a form of medicine to the South African population is the biokineticist. Biokinetics is a relatively new profession developed in response to empirical research supporting exercise as medicine. Although the role of the biokineticist is clearly defined by the Health Professions Council of South Africa (HPCSA), there is still confusion concerning this profession and its services. At present, the dominant medicalscientific and capitalist-economic discourses offer a narrowly mechanistic and instrumental understanding of the human body and its treatment, which contributes to this confusion. This dissertation proposes a new point of departure, enriched by historical and philosophical understandings of the human body, from which the biokineticist can begin to redefine him/herself. The study aims to shift biokinetics as a practice away from the reductionist and dehumanising influences of instrumentalist, scientific and neoliberal capitalist-economic discourses, and to restore to biokineticists, and their clients, the ability they need to interact as relatively autonomous individuals. The study draws on the ideas of Michel Foucault, specifically on his notions for ‘discipline’, ‘docility’, and ‘the care of the self’ (1991; 2005), as well as Gilles Deleuze and Felix Guattari and their ideas of ‘becoming’ and ‘rhizomatic thinking’ (1983; 1987); ultimately presenting a philosophically enriched, holistic representation of the human body. From here recommendations for best practice in contemporary biokinetics are suggested that encourage interactions and connections between the professional and his/her patients, which move beyond the mere physiological interpretations currently dominating health discourse.
- Full Text:
- Date Issued: 2016
The community structure and feeding ecology of the ichthyofauna in the Mngazana and Mngazi estuaries, Port St. Johns, South Africa
- Authors: Grant, Walter Lawrence
- Date: 2007
- Subjects: Estuarine fishes -- South Africa -- Eastern Cape , Fish communities -- South Africa -- Eastern Cape , Estuaries -- South Africa -- Eastern Cape
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:10713 , http://hdl.handle.net/10948/513 , http://hdl.handle.net/10948/d1012017 , Estuarine fishes -- South Africa -- Eastern Cape , Fish communities -- South Africa -- Eastern Cape , Estuaries -- South Africa -- Eastern Cape
- Description: The Mngazana and Mngazi estuaries are situated just south of Port St. Johns close to the biogeographical boundary between the warm temperate and subtropical zones along the east coast of Southern Africa. The Mngazana estuary is a permanently open estuary with three species of mangrove. The Mngazi estuary is a temporarily open/closed system. The ichthyofauna was sampled with seine nets and gillnets during June 2005, August 2005, November 2005, February 2006 and April 2006. The fish community structure was significantly different between the two estuaries with the differences being especially noticeable in the larger fish sampled with gillnets. There were significant differences in the fish community sampled with the seine net on both spatial and temporal scales in both of the estuaries. Certain warm temperate species were less abundant and certain subtropical species were more abundant at the Mngazana estuary than they were during a study conducted in 1975. Stomach content analysis revealed that most species had similar diets in the two estuaries with the only differences occurring among benthic feeding species. The diet of sub-adult piscivores comprised of a large proportion of penaeid prawns while the juveniles feed mainly on mysids. The zooplanktivorous fish feed mainly on copepods and mysids with the proportions varying between the species. Benthic feeding fishes showed the greatest variability in diet with plant material, detritus, crabs and copepods being the main food sources. Stable isotopes revealed that particulate organic matter and microphytobenthos were the most important carbon sources to fish in the Mngazi estuary. In the Mngazana estuary a combination of mangrove material, detritus, particulate organic matter and Zostera material were the most important carbon sources. The importance of each of these sources did, however, vary greatly between the different trophic guilds of the fish in the Mngazana estuary. The distribution of certain fish species was investigated in relation to physical parameters (temperature, salinity, turbidity and depth) and biological parameters (prey and predators) in the environment. It was found that most small fish (< 100 mm SL) were strongly correlated with turbidity where-as they showed weak correlations with their prey. Sub-adult piscivores fell into two groups, i.e. those that could tolerate turbid water and those that were only found in clear water. The group that also occurred in turbid water showed very strong correlations with their prey but the clear water group did not. It would appear that the small and thus vulnerable fish seemed to occur in areas that offered them protection rather than areas where their food supply is most abundant, where-as the piscivores were more influenced by prey distribution, as long as the physical conditions were tolerable to them.
- Full Text:
- Date Issued: 2007
- Authors: Grant, Walter Lawrence
- Date: 2007
- Subjects: Estuarine fishes -- South Africa -- Eastern Cape , Fish communities -- South Africa -- Eastern Cape , Estuaries -- South Africa -- Eastern Cape
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:10713 , http://hdl.handle.net/10948/513 , http://hdl.handle.net/10948/d1012017 , Estuarine fishes -- South Africa -- Eastern Cape , Fish communities -- South Africa -- Eastern Cape , Estuaries -- South Africa -- Eastern Cape
- Description: The Mngazana and Mngazi estuaries are situated just south of Port St. Johns close to the biogeographical boundary between the warm temperate and subtropical zones along the east coast of Southern Africa. The Mngazana estuary is a permanently open estuary with three species of mangrove. The Mngazi estuary is a temporarily open/closed system. The ichthyofauna was sampled with seine nets and gillnets during June 2005, August 2005, November 2005, February 2006 and April 2006. The fish community structure was significantly different between the two estuaries with the differences being especially noticeable in the larger fish sampled with gillnets. There were significant differences in the fish community sampled with the seine net on both spatial and temporal scales in both of the estuaries. Certain warm temperate species were less abundant and certain subtropical species were more abundant at the Mngazana estuary than they were during a study conducted in 1975. Stomach content analysis revealed that most species had similar diets in the two estuaries with the only differences occurring among benthic feeding species. The diet of sub-adult piscivores comprised of a large proportion of penaeid prawns while the juveniles feed mainly on mysids. The zooplanktivorous fish feed mainly on copepods and mysids with the proportions varying between the species. Benthic feeding fishes showed the greatest variability in diet with plant material, detritus, crabs and copepods being the main food sources. Stable isotopes revealed that particulate organic matter and microphytobenthos were the most important carbon sources to fish in the Mngazi estuary. In the Mngazana estuary a combination of mangrove material, detritus, particulate organic matter and Zostera material were the most important carbon sources. The importance of each of these sources did, however, vary greatly between the different trophic guilds of the fish in the Mngazana estuary. The distribution of certain fish species was investigated in relation to physical parameters (temperature, salinity, turbidity and depth) and biological parameters (prey and predators) in the environment. It was found that most small fish (< 100 mm SL) were strongly correlated with turbidity where-as they showed weak correlations with their prey. Sub-adult piscivores fell into two groups, i.e. those that could tolerate turbid water and those that were only found in clear water. The group that also occurred in turbid water showed very strong correlations with their prey but the clear water group did not. It would appear that the small and thus vulnerable fish seemed to occur in areas that offered them protection rather than areas where their food supply is most abundant, where-as the piscivores were more influenced by prey distribution, as long as the physical conditions were tolerable to them.
- Full Text:
- Date Issued: 2007
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