title: Bounded Eigenvalues of Fully Clamped and Completely Free Rectangular Plates

authors: Mochida, Yusuke

subjects: vibration

abstract: Exact solution to the vibration of rectangular plates is available only for plates with two opposite edges subject to simply supported conditions. Otherwise, they are analysed by using approximate methods. There are several approximate methods to conduct a vibration analysis, such as the Rayleigh-Ritz method, the Finite Element Method, the Finite Difference Method, and the Superposition Method. The Rayleigh-Ritz method and the finite element method give upper bound results for the natural frequencies of plates. However, there is a disadvantage in using this method in that the error due to discretisation cannot be calculated easily. Therefore, it would be good to find a suitable method that gives lower bound results for the natural frequencies to complement the results from the Rayleigh-Ritz method. The superposition method is also a convenient and efficient method but it gives lower bound solution only in some cases. Whether it gives upper bound or lower bound results for the natural frequencies depends on the boundary conditions. It is also known that the finite difference method always gives lower bound results. This thesis presents bounded eigenvalues, which are dimensionless form of natural frequencies, calculated using the superposition method and the finite difference method. All computations were done using the MATLAB software package. The convergence tests show that the superposition method gives a lower bound for the eigenvalues of fully clamped plates, and an upper bound for the completely free plates. It is also shown that the finite difference method gives a lower bound for the eigenvalues of completely free plates. Finally, the upper bounds and lower bounds for the eigenvalues of fully clamped and completely free plates are given.

text: Bounded Eigenvalues of Fully Clamped and Completely Free Rectangular Plates Exact solution to the vibration of rectangular plates is available only for plates with two opposite edges subject to simply supported conditions. Otherwise, they are analysed by using approximate methods. There are several approximate methods to conduct a vibration analysis, such as the Rayleigh-Ritz method, the Finite Element Method, the Finite Difference Method, and the Superposition Method. The Rayleigh-Ritz method and the finite element method give upper bound results for the natural frequencies of plates. However, there is a disadvantage in using this method in that the error due to discretisation cannot be calculated easily. Therefore, it would be good to find a suitable method that gives lower bound results for the natural frequencies to complement the results from the Rayleigh-Ritz method. The superposition method is also a convenient and efficient method but it gives lower bound solution only in some cases. Whether it gives upper bound or lower bound results for the natural frequencies depends on the boundary conditions. It is also known that the finite difference method always gives lower bound results. This thesis presents bounded eigenvalues, which are dimensionless form of natural frequencies, calculated using the superposition method and the finite difference method. All computations were done using the MATLAB software package. The convergence tests show that the superposition method gives a lower bound for the eigenvalues of fully clamped plates, and an upper bound for the completely free plates. It is also shown that the finite difference method gives a lower bound for the eigenvalues of completely free plates. Finally, the upper bounds and lower bounds for the eigenvalues of fully clamped and completely free plates are given.

year: 2007

title: Scoping the meaning of 'critical' in mathematical thinking for Initial Teacher Education

authors: Furness, Jane Amanda

abstract: Current strong emphasis on literacy and numeracy in New Zealand educational policy, as elsewhere, reverberates in different ways in institutions charged with children's and adults' learning. A common response is to locate literacy and numeracy centrally in programmes aimed at preparing children for and enhancing adult participation in 21st century life and work. One place these agendas overlap is in preservice teacher education or Initial Teacher Education (ITE). Preservice teachers enter their ITE programmes as adult learners engaged in building literacy and numeracy knowledge and skills for teaching across the curriculum, working with student achievement data and administration, aspects of which may also be useful in their wider lives. These tasks also involve making judgements which require critical thinking, another key policy focus in New Zealand and internationally for children's and adults' learning. As part of a project in which a group of researchers are exploring mathematical thinking and reasoning in initial teacher education (MARKITE) the researchers included critical awareness as one of three dimensions in their definition of mathematical thinking. As the project has progressed our interest in the notion of criticality in mathematics education and its implications for mathematical thinking in the context of the needs of 2 1st century citizens has grown. We have set out in this paper to illuminate what 'critical' in mathematical thinking could mean and what we mean by it in our project. Our aim in this reflective undertaking is to shed light on options and elaborate why we believe developing criticality is important in initial teacher education.
The primary mode of inquiry was a literature search, covering critical theory and critical literacy, critical mathematics and various definitions of mathematics education (critical numeracy, data literacy and quantitative literacy). Next we reviewed current thinking on the context in which ITE takes place, in particular the needs of democratic citizens in the 2 1st century. Ideas gleaned from these documents have been analysed against the Key Competencies Framework and other policy and implementation documents to consider how what is suggested as important in the Mathematics and Statistics learning area (MOE, 2007) provides grounds for the development of criticality in the identification and use of mathematical thinking across an ITE programme. In this examination we were cognisant of the increasing demand for (quantitative) data use in schools.
Our investigations are drawing us towards conclusions that favour quite classic critical theory applications (Freire, 1974), which we see also reflected in current works which explore the link between mathematics and the globally experienced social and environmental conditions of the present and the future (e.g. Atweh & Brady, 2009; Greer & Skovsmose, 2012). Mathematics is deeply involved in framing understanding and solving social, economical and environmental problems. To date we have found very little literature that focuses on the nature of and how to develop student teachers' mathematical thinking beyond a focus on curriculum mathematics. In the National Standards, we find the stem for criteria include a focus on context and problem solving. This creates the potential for critical engagement with the mathematical ideas embedded in real world problems and solutions. However the exemplifications of the standards do not encompass pressing social issues. 
This work brings together current thinking about criticality and critical mathematics/mathematics education in a first attempt to put forward a useful understanding of critical mathematical thinking for ITE. It also brings to the attention of educators the role of mathematics in critical democratic citizenship, alongside the more commonplace focus on critical literacy. The kinds of critical mathematical thinking we have become more specifically interested in are those that enable the inclusion of different culturally based world views am actions, equitable participation and outcomes, and so, justice for all citizens. We propose that the develop in preservice teachers of capacity for critical mathematical thinking is a crucial component of ITE, especially in these times of significant global change which mathematics plays a central yet often hidden re-Critical mathematical thinking holds the potential to facilitate teachers to work actively in pursuit of an inclusive, socially just world.

text: Scoping the meaning of 'critical' in mathematical thinking for Initial Teacher Education Current strong emphasis on literacy and numeracy in New Zealand educational policy, as elsewhere, reverberates in different ways in institutions charged with children's and adults' learning. A common response is to locate literacy and numeracy centrally in programmes aimed at preparing children for and enhancing adult participation in 21st century life and work. One place these agendas overlap is in preservice teacher education or Initial Teacher Education (ITE). Preservice teachers enter their ITE programmes as adult learners engaged in building literacy and numeracy knowledge and skills for teaching across the curriculum, working with student achievement data and administration, aspects of which may also be useful in their wider lives. These tasks also involve making judgements which require critical thinking, another key policy focus in New Zealand and internationally for children's and adults' learning. As part of a project in which a group of researchers are exploring mathematical thinking and reasoning in initial teacher education (MARKITE) the researchers included critical awareness as one of three dimensions in their definition of mathematical thinking. As the project has progressed our interest in the notion of criticality in mathematics education and its implications for mathematical thinking in the context of the needs of 2 1st century citizens has grown. We have set out in this paper to illuminate what 'critical' in mathematical thinking could mean and what we mean by it in our project. Our aim in this reflective undertaking is to shed light on options and elaborate why we believe developing criticality is important in initial teacher education.
The primary mode of inquiry was a literature search, covering critical theory and critical literacy, critical mathematics and various definitions of mathematics education (critical numeracy, data literacy and quantitative literacy). Next we reviewed current thinking on the context in which ITE takes place, in particular the needs of democratic citizens in the 2 1st century. Ideas gleaned from these documents have been analysed against the Key Competencies Framework and other policy and implementation documents to consider how what is suggested as important in the Mathematics and Statistics learning area (MOE, 2007) provides grounds for the development of criticality in the identification and use of mathematical thinking across an ITE programme. In this examination we were cognisant of the increasing demand for (quantitative) data use in schools.
Our investigations are drawing us towards conclusions that favour quite classic critical theory applications (Freire, 1974), which we see also reflected in current works which explore the link between mathematics and the globally experienced social and environmental conditions of the present and the future (e.g. Atweh & Brady, 2009; Greer & Skovsmose, 2012). Mathematics is deeply involved in framing understanding and solving social, economical and environmental problems. To date we have found very little literature that focuses on the nature of and how to develop student teachers' mathematical thinking beyond a focus on curriculum mathematics. In the National Standards, we find the stem for criteria include a focus on context and problem solving. This creates the potential for critical engagement with the mathematical ideas embedded in real world problems and solutions. However the exemplifications of the standards do not encompass pressing social issues. 
This work brings together current thinking about criticality and critical mathematics/mathematics education in a first attempt to put forward a useful understanding of critical mathematical thinking for ITE. It also brings to the attention of educators the role of mathematics in critical democratic citizenship, alongside the more commonplace focus on critical literacy. The kinds of critical mathematical thinking we have become more specifically interested in are those that enable the inclusion of different culturally based world views am actions, equitable participation and outcomes, and so, justice for all citizens. We propose that the develop in preservice teachers of capacity for critical mathematical thinking is a crucial component of ITE, especially in these times of significant global change which mathematics plays a central yet often hidden re-Critical mathematical thinking holds the potential to facilitate teachers to work actively in pursuit of an inclusive, socially just world.

year: 2015

title: Telomere length in early childhood is associated with sex and ethnicity

authors: Ly, Kien

subjects: Science & Technology

abstract: Telomeres are repetitive DNA sequences at the end of chromosomes that function to protect chromosomes from degradation. Throughout the life course, telomere length decreases with age and is influenced by environmental factors and health conditions. This study aimed to determine the relative telomere lengths in a diverse cohort of about 4000 four-year-old children in New Zealand. Linear regression was used to investigate the relationship between telomere length, child gender, ethnicity, paternal age and deprivation. We observed substantial variation in telomere length according to sex and self-identified ethnicity. Telomere length was longer in females compared to males (coefficient of 0.042, 95% confidence interval (CI) 0.024–0.060). European children had shorter telomere than both the indigenous Māori (coefficient of 0.03, CI 0.007–0.055) and Pacific children (coefficient of 0.15, CI 0.12–0.18). The data suggest that telomere lengths are highly variable and variability between individuals arise from early age, influenced partly by sex and ethnicity. Longer telomeres in indigenous Māori and Pacific children may reflect the heritability of telomere length in genetically less complex populations. This study increases our understanding of telomere dynamics in young children since the majority of telomere studies are conducted in adults.

text: Telomere length in early childhood is associated with sex and ethnicity Telomeres are repetitive DNA sequences at the end of chromosomes that function to protect chromosomes from degradation. Throughout the life course, telomere length decreases with age and is influenced by environmental factors and health conditions. This study aimed to determine the relative telomere lengths in a diverse cohort of about 4000 four-year-old children in New Zealand. Linear regression was used to investigate the relationship between telomere length, child gender, ethnicity, paternal age and deprivation. We observed substantial variation in telomere length according to sex and self-identified ethnicity. Telomere length was longer in females compared to males (coefficient of 0.042, 95% confidence interval (CI) 0.024–0.060). European children had shorter telomere than both the indigenous Māori (coefficient of 0.03, CI 0.007–0.055) and Pacific children (coefficient of 0.15, CI 0.12–0.18). The data suggest that telomere lengths are highly variable and variability between individuals arise from early age, influenced partly by sex and ethnicity. Longer telomeres in indigenous Māori and Pacific children may reflect the heritability of telomere length in genetically less complex populations. This study increases our understanding of telomere dynamics in young children since the majority of telomere studies are conducted in adults.

year: 2019