Dissertations / Theses on the topic 'Descrete Mathematics'

This study establishes a framework for the practice and the acquisition of mathematical knowledge. The natures of mathematics and rituals/ritual-like activities are examined compared and contrasted. Using a four-fold typology of core features, surface features, content features and functions of mathematics it is established that the nature of mathematics, its practice and the acquisition is typologically similar to that of rituals/ ritual-like activities. The practice of mathematics and its acquisition can hence be metaphorically compared to that of rituals/ritual-like activities and be enriched by the latter. A case study was conducted using the ritual metaphor at two levels to introduce and teach a topic within the current year eleven West Australian Geometry and Trigonometry course. In the first level, instructional materials were written using a ritual-like mentor-exemplar, exposition, replicate and extrapolate model (through the use of specially organised examples and exercises) based on the approaches of several mathematics text book authors as they attempted to introduce a topic new to the West Australian mathematics curriculum.In the second level, the classroom instruction was organised using a ritual-like pattern with direct exemplar mentoring and exposition by the teacher followed by replication and extrapolation from the students. Embedded within this ritual-like process was the personal (and communal) engagement with each student vis-a-vis the establishment of the relationships between the referent concepts, procedures and skills. This resulted in the emergence of solution behaviours appropriate to specific tasks imitating and extrapolating the mentored solution behaviours of the teacher. In determining the extent to which the instruction, mentoring and acquisition was successful, each student's solution 'behaviour was compared "topographically" with the expected solution behaviour for the task at various critical points to determine the degree of congruence. Marks were allocated for congruence (or removed for incongruence), hence a percentage of congruence was established. The ritual-like model for the teaching and acquisition of mathematical knowledge required agreement with all stake-holders as to the purpose of the activity, expert knowledge on the part of the teacher, and within a classroom context requires students to possess similar levels of prerequisite mathematical knowledge.This agreement and the presence of an expert practitioner, provides the affirmation and security that is inherent in the practice of rituals. The study concluded that there is evidence to suggest that some aspects of mathematical ability are wired into the cognitive structures of human beings providing support to the hypothesis that some aspects of mathematics are discovered rather than created. The physical origin of mathematical abilities and activities was one of the factors used in this study to establish an isomorphism between the nature and practice of mathematics with that of rituals. This isomorphism provides the teaching and learning of mathematics with a more robust framework that is more attuned to the social nature of human beings. The ritual metaphor for the teaching and learning of mathematics can then be used as a framework to determine the relative adequacies of mathematics curricula, mathematics textbooks and teaching approaches.

This study establishes a framework for the practice and the acquisition of mathematical knowledge. The natures of mathematics and rituals/ritual-like activities are examined compared and contrasted. Using a four-fold typology of core features, surface features, content features and functions of mathematics it is established that the nature of mathematics, its practice and the acquisition is typologically similar to that of rituals/ ritual-like activities. The practice of mathematics and its acquisition can hence be metaphorically compared to that of rituals/ritual-like activities and be enriched by the latter. A case study was conducted using the ritual metaphor at two levels to introduce and teach a topic within the current year eleven West Australian Geometry and Trigonometry course. In the first level, instructional materials were written using a ritual-like mentor-exemplar, exposition, replicate and extrapolate model (through the use of specially organised examples and exercises) based on the approaches of several mathematics text book authors as they attempted to introduce a topic new to the West Australian mathematics curriculum.
In the second level, the classroom instruction was organised using a ritual-like pattern with direct exemplar mentoring and exposition by the teacher followed by replication and extrapolation from the students. Embedded within this ritual-like process was the personal (and communal) engagement with each student vis-a-vis the establishment of the relationships between the referent concepts, procedures and skills. This resulted in the emergence of solution behaviours appropriate to specific tasks imitating and extrapolating the mentored solution behaviours of the teacher. In determining the extent to which the instruction, mentoring and acquisition was successful, each student's solution 'behaviour was compared "topographically" with the expected solution behaviour for the task at various critical points to determine the degree of congruence. Marks were allocated for congruence (or removed for incongruence), hence a percentage of congruence was established. The ritual-like model for the teaching and acquisition of mathematical knowledge required agreement with all stake-holders as to the purpose of the activity, expert knowledge on the part of the teacher, and within a classroom context requires students to possess similar levels of prerequisite mathematical knowledge.
This agreement and the presence of an expert practitioner, provides the affirmation and security that is inherent in the practice of rituals. The study concluded that there is evidence to suggest that some aspects of mathematical ability are wired into the cognitive structures of human beings providing support to the hypothesis that some aspects of mathematics are discovered rather than created. The physical origin of mathematical abilities and activities was one of the factors used in this study to establish an isomorphism between the nature and practice of mathematics with that of rituals. This isomorphism provides the teaching and learning of mathematics with a more robust framework that is more attuned to the social nature of human beings. The ritual metaphor for the teaching and learning of mathematics can then be used as a framework to determine the relative adequacies of mathematics curricula, mathematics textbooks and teaching approaches.

This work is primarily aimed at determining the effect that an immunisation policy Is likely to have on the incidence of Haemophllus influenzae Type B (HIB) and systematic HIB in Western Australia. There was a significant effort made to collect data pertinent to the estimation of parameter values but since HIB has only been a notifiable disease since 1992, there was a distinct lack of relevant data available. Private communication with individual’s such as Dr Jeffrey Hanna and Dr Beryl Wild resulted in practical information being obtained that was used to estimate certain parameters. The deterministic mathematical models within the thesis are extensions of existing ideas tailored to suit the needs of this thesis. Chapter one is a basic introduction to the pursuit of modelling infectious diseases with a brief description of basic epidemiology concepts. It also shows that even simple models may not deliver analytical results. Chapter two extends a model used by Angela Mclean and allows some analytical results to be obtained by first simplifying the model and then solving using standard methods to give the equilibrium distributions for the proportions of people in each state within the model

Educational Administration
Ed.D.
Some students do not learn mathematics even though they have both the potential and ability to learn math. This problem typically diminishes opportunities for students who are already marginalized by society. Educators, educational administrators, education policy makers, and the education community have been aware of the significant disparities in mathematics and science achievement between Asian/Pacific Islanders and Caucasians and underrepresented minority groups. If we are to understand students and to alter their motivational patterns and attitudes, continued research in the area of student motivation and attitude is essential. This case study provides a detailed examination of a 10th grade geometry class located in an urban magnet public high school with 95% minority students. The primary purpose was to learn how students perceive and describe their mathematical learning experiences. The secondary purpose was to determine the factors that influenced on students' motivation, attitudes, or perceptions of their mathematical learning experiences. Students described not only their perceptions and attitudes in light of their actual degree of success, but also the impact of their mathematics teacher's pedagogy. Using qualitative methods, this study suggests the potential of some factors that mathematics educators, educational administrators, or policy makers should consider in order to explain why and how some students do not learn mathematics, even though they have the ability to learn it. The researcher analyzes data from surveys, interviews, and classroom observation. There are seven emergent themes--three themes which arose as influencing students' attitudes: (1) family background, (2) teacher's beliefs and attitudes, and (3) the concept of success as a turning point and four themes which had been anticipated as potentially explanatory, but ultimately were not: (1) student initial attitude, (2) gender, (3) ethnicity, and (4) teacher's pedagogy alone. Furthermore, the data indicate that the classic stereotypes about how gender and/or ethnicity influence the mathematics achievement gap in the U.S. may not apply in settings where all students receive appropriate support and the educational environment is conducive to learning mathematics. Moreover, the data indicate that the focus on content knowledge in determining who is a highly qualified teacher in the No Child Left Behind Act of 2001 may need to be examined further. This study will be of value to educators in the design and understanding of interventions to enhance achievement in high school mathematics.
Temple University--Theses

Modelling is a powerful method for understanding complex systems, which works by simplifying them to their most essential components. The choice of the components is driven by the aspects studied. The tool chosen to perform this task will determine what can be modelled, the maximum number of components which can be represented, as well as the analyses which can be performed on the system. Performance Evaluation Process Algebra (PEPA) was initially developed to tackle computer systems issues. Nevertheless, it possesses some interesting properties which could be exploited for the study of epidemiological systems. PEPA's main advantage resides in its capacity to change scale: the assumptions and parameter values describe the behaviour of a single individual, while the resulting model provides information on the population behaviour. Additionally, stochasticity and continuous time have already proven to be useful features in epidemiology. While each of these features is already available in other tools, to find all three combined in a single tool is novel, and PEPA is proposed as a useful addition to the epidemiologist's toolbox. Moreover, an algorithm has been developed which allows converting a PEPA model into a system of Ordinary Differential Equations (ODEs). This provides access to countless additional software and theoretical analysis methods which enable the epidemiologist to gain further insight into the model. Finally, most existing tools require a deep understanding of the logic they are based on and the resulting model can be difficult to read and modify. PEPA's grammar, on the other hand, is easy to understand since it is based on few, yet powerful concepts. This makes it a very accessible formalism for any epidemiologist. The objective of this thesis is to determine precisely PEPA's ability to describe epidemiological systems, as well as extend the formalism when required. This involved modelling two systems: the bubonic plague in prairie dogs, and measles in England and Wales. These models were chosen as they exhibit a good range of typical features, allowing to thoroughly test PEPA. All features required in each of these models have been analysed in detail, and a solution has been provided for representing each of these features. While some of them could be expressed in a straightforward manner, PEPA did not provide the tools to express others. In those cases, we determined methods to approach the desired behaviour, and the limitations of said methods were carefully analysed. In the case of models with a structured population, PEPA was extended to simplify their expression and facilitate the writing process of the PEPA model. The work also required the development of an algorithm to derive ODEs adapted to the type of models encountered. Finally, the PEPAdum software was developed to assist the modeller in the generation and analysis of PEPA models, by simplifying the process of writing a PEPA model with compartments, performing the average of stochastic simulations and deriving and explicitly providing the ODEs using the Stirling Amendment.

The purpose with this research paper is to examine the students’ shown knowledge in geometry, with a focus on construction and its concepts, and the educational value and teaching the students got in this area. The students’ homes are used as a starting-point. The students shall, from a self-made drawing of their home and a photograph of it, describe what their home looks like. In this paper, the mathematical concepts the students used will be analyzed and compared with the education they received. The analytical framework is based on Van Hieles levels of knowledge and Blooms Taxonomy. The study was done at a Secondary School in Kenya. Four students were selected and interviewed. The lesson observations were made with the purpose to get an understanding for how the education for these students look like and to get examples on how the teaching is conducted for these students. Finally, interviews with the teachers were carried out. The students show a good knowledge in the national exams. However, the study shows that when the students are supposed to use this particular knowledge outside of the classroom, the students experience difficulties. Mostly, the students encounter problems when they are supposed to estimate measurements. Furthermore, they lack the ability to compare scales. The research also shows that the education for these students is monotone and much time during the lessons is spend either with a teacher lecturing in front of the board or students working with examples in the textbook. According to the Variation Theory, the knowledge of the students should deepen if the objects of learning are varying. This variation is not something the students receive in the present situation.
Syftet är att undersöka några gymnasieelevers visade kunskaper i geometri med fokus på konstruktion och begreppsanvändning samt den undervisning som erbjuds eleverna inom området. Elevernas hem används som utgångspunkt. Eleverna ska utifrån en teckning, som de själva ritat, och ett fotografi beskriva hemmet. De matematiska begrepp som eleverna använder analyseras. Analysverktyget bygger på van Hieles kvalitativa kunskapsnivåer och Blooms Taxonomi. Undersökningen genomfördes på en gymnasieskola i Kenya. Fyra utvalda elever intervjuades. Lektionsobservationer genomfördes i syfte att få förståelse för hur elevernas undervisningssituation ser ut och få exempel på hur undervisningen bedrivs. Slutligen intervjuades två av elevernas lärare. Eleverna har goda kunskaper på nationella prov men undersökningen visar att när dessa kunskaper skall överföras till något utanför lektionssalen stöter eleverna på problem. De har svårt att uppskatta längdenheter och svårt att jämföra skala. Det kommer också fram att deras undervisning är ganska monoton. Mycket tid läggs till att läraren undervisar eleverna framme vid tavlan eller att eleverna jobbar med uppgifter i sin övningsbok. Enligt variationsteorin, som beskrivs i arbetet, skulle elevernas kunskaper ges möjlighet att fördjupas om de geometriska objekt som skall förstås varieras. Denna variation erbjuds inte eleverna i nuläget.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
Apresenta-se um modelo de equações diferenciais ordinárias que descreve a transmissão de dengue em uma população humana e de mosquitos quando há circulação de dois sorotipos de vírus. Resultados analíticos e numéricos para os pontos de equilíbrio deste modelo, e o estudo da estabilidade dos mesmos são obtidos. Faz-se uma aproximação de estado quase-estacionário para a população de mosquito, com o objetivo de estudar e comparar a dinâmica da transmissão da dengue em redes de diferentes topologias. O modelo de transmissão através de redes complexas considera diferentes graus de conectividade entre os indivíduos da população e por isso representa melhor as interações sociais. Observa-se que a dinâmica da transmissão da dengue depende fortemente da topologia da rede e do número médio de conexões, portanto medidas de controle da doença devem ter um impacto diferente dada a diversidade das conexões entre os indivíduos de uma população
We present a model of ordinary differential equations to describe the dengue transmission in a human and a mosquito populations when there are two serotypes of circulating virus. Analytic and numeric results to the equilibruim points of this model, and the study of the stability of this points were obtained. We assume the quasi-steady state approach to the mosquito population, in order to study and compare the dynamics of transmission of two serotypes of dengue virus in networks with different topologies. We consider the transmission model through complex networks with different degrees of conectivity among the individuals and, thus, it provides a better representation of the social interations. We observe that the transmission dynamics of dengue depends strongly on the network topology and the mean number of conections, thus the control measures must have a different impact given the diversity of conections among the individuals on the population
FAPESP: 2013/01552-7

Orientador: Cláudia Pio Ferreira
Banca: Fernanado Luiz Pio dos Santos
Banca: Suani Tavares Rubim de Pinho
Resumo: Apresenta-se um modelo de equações diferenciais ordinárias que descreve a transmissão de dengue em uma população humana e de mosquitos quando há circulação de dois sorotipos de vírus. Resultados analíticos e numéricos para os pontos de equilíbrio deste modelo, e o estudo da estabilidade dos mesmos são obtidos. Faz-se uma aproximação de estado quase-estacionário para a população de mosquito, com o objetivo de estudar e comparar a dinâmica da transmissão da dengue em redes de diferentes topologias. O modelo de transmissão através de redes complexas considera diferentes graus de conectividade entre os indivíduos da população e por isso representa melhor as interações sociais. Observa-se que a dinâmica da transmissão da dengue depende fortemente da topologia da rede e do número médio de conexões, portanto medidas de controle da doença devem ter um impacto diferente dada a diversidade das conexões entre os indivíduos de uma população
Abstract: We present a model of ordinary differential equations to describe the dengue transmission in a human and a mosquito populations when there are two serotypes of circulating virus. Analytic and numeric results to the equilibruim points of this model, and the study of the stability of this points were obtained. We assume the quasi-steady state approach to the mosquito population, in order to study and compare the dynamics of transmission of two serotypes of dengue virus in networks with different topologies. We consider the transmission model through complex networks with different degrees of conectivity among the individuals and, thus, it provides a better representation of the social interations. We observe that the transmission dynamics of dengue depends strongly on the network topology and the mean number of conections, thus the control measures must have a different impact given the diversity of conections among the individuals on the population
Mestre

Thesis (MScAgric)--Stellenbosch University, 2013.
ENGLISH ABSTRACT: The first study (Chapter 3) evaluated the growth response of ostrich chicks on diets containing three different levels of protein and amino acids. Linear and nonlinear models were fitted to the data and compared by using Akaike’s information criterion (AIC). The linear polynomial of the third degree had the lowest AIC value for all three treatments thus making it the most suitable model for the data. Significant differences were found between treatments for growth data. The results from this study can aid in describing the growth of ostriches subjected to assumed optimum feeding conditions. In the second study (Chapter 4), a range of diets was formulated for the five growth stages of ostriches (pre-starter, starter, grower, finisher and maintenance) according to their nutrient requirements. The diets were diluted with wheat straw. Three dilution levels (0%, 10% and 20%) were used for the pre-starter and starter phases, five dilution levels (0%, 15%, 30%, 45% and 60%) were used for the grower and the finisher phases, and five dilution levels (0%, 20%, 40%, 60% and 80%) were used for the maintenance phase. Weekly intake data were collected throughout each phase. Feed bulk restricted intake by 21% and 52% at the 10% and 20% dilution level, respectively (P < 0.05) in the pre-starter phase, whereas intake was not restricted during the starter phase(P > 0.05). Intake was constrained by 39% and 42% at the 45% and 60% dilution levels in the grower phase, respectively (P < 0.05), and by 17% and 39% at the 45% and 60% dilution levels (P < 0.05) in the finisher phase, respectively. Feed bulk restricted intake by 60% and 69% for the 60% and 80% dilution levels (P < 0.05), respectively, in the maintenance phase. Defining the bulk density that will constrain feed intake, as established in this study, will aid in least-cost feed formulations, feed intake modelling and growth predictions. In the third study (Chapter 5) the effect of three different dietary protein (with a specific associated amino acid content) concentrations on certain production parameters in growing ostriches were investigated. Significant differences were found for the final live weight of birds, cold carcass weight, thigh weight as well as for most of the weighed muscles at slaughter (350 days old). Concerning the growth and feed related parameters, only average daily gain (ADG) was influenced by dietary treatment (P < 0.05). Results indicated that birds on the diet with the medium protein performed optimally. One exception is the starter phase (26 – 47 kg) where chicks on the high protein diet outperformed those on the medium protein diet. In the fourth study (Chapter 6) the effects of different dietary energy concentrations on ostrich production parameters were examined in two different trials. The first trial included measurements from the pre-starter phase through the starter phase until the grower phase. The second trial was based on the finisher phase per se. Overall dietary levels provided in the pre-starter, starter and grower phases indicated better growth, FCR, skin size and grade, thigh weight, live weight, and carcass weight for the birds fed the medium energy diet. Dietary energy levels provided during the finisher phase indicated that the energy level above the medium level used improved growth rate and tanned skin size. The gender of the birds influenced carcass weight, growth rate, and certain feather parameters (P < 0.05). In the fifth study (Chapter 7) the effect of feather clipping at six to eight months of age on the production parameters of ostrich chicks were investigated. The study was conducted in three different trials. In each of the trials the feathers of half the amount of birds were clipped at six to eight months of age. Significant differences (P < 0.05) were found for the feed conversion ratio (FCR), the average daily gain (ADG) and for the quantity of valuable feathers. Results indicated that the growth rate and FCR was better for the birds which had their feathers clipped at six to eight months of age. Results also showed that the quantity of feathers with commercial value were significantly higher for the clipped group. This study showed that there may be an advantage for ostrich producers concerning the harvesting of feathers at six to eight months of age. The work in this thesis is a follow up on the framework set by Kritzinger (2011) and is part of the same project. Most of the results obtained in these studies will be incorporated in to the mathematical optimisation model of Gous and Brand (2008) for more accurate predictions concerning feed intake and other production parameters that may lower feeding costs.
AFRIKAANSE OPSOMMING: Die eerste studie (Hoofstuk 3) evalueer die groei van volstruiskuikens op diëte met drie verskillende vlakke van proteïene en aminosure. Liniêre en nie-liniêre regressiemodelle is op die data gepas en met Akaike se inligting kriterium (AIC) vergelyk. Die liniêre polinoom van die derde graad het die laagste AIC waarde vir al drie behandelings gehad. Daarom is die voorspellings van hierdie model gebruik om die groeidata te interpreteer. Beduidende verskille tussen behandelings vir groeidata (P < 0.05) is gevind. Die resultate van hierdie studie kan help met die beskrywing van die groei van volstruise, onderworpe aan aangeneemde optimale voedingsbehoeftes. In die tweede studie (Hoofstuk 4) is 'n verskeidenheid diëte geformuleer vir die vyf groeistadiums van volstruise (voor-aanvangs, aanvangs, groei, afronding en onderhoud) volgens hul voedingsbehoeftes. Die diëte is verdun met koringstrooi. Drie verdunningsvlakke (0%, 10% en 20%) is gebruik vir die voor-aanvangs- en aanvangsfase, vyf verdunningvlakke (0%, 15%, 30%, 45% en 60%) is gebruik vir die groei- en die afrondingsfase en vyf verdunningsvlakke (0%, 20%, 40%, 60% en 80%) is gebruik vir die onderhoudsfase. Weeklikse inname-data is ingesamel gedurende elke fase. In die voor-aanvangsfase het voerlywigheid (verhoging van ruvesel) inname beperk met 21% en 52% vir die 10% en 20% verdunningsvlakke (P < 0.05) onderskeidelik, terwyl inname nie beperk is gedurende die aanvangsfase nie (P > 0.05). Inname is beperk met 39% en 42% op die 45% en 60% verdunningsvlakke in die groeifase (P < 0.05) onderskeidelik, en met 17% en 39% op die 45% en 60% verdunningsvlakke in die afrondingsfase (P < 0.05), onderskeidelik. Voerdigtheid het inname beperk met 60% en 69% vir die 60% en 80% verdunningsvlakke, onderskeidelik, in die onderhoudsfase (P < 0.05). Die definiëring van die digtheid of ruvoerinhoud van voer wat inname beperk, soos in die studie bepaal, sal help met die optimering van voerformulasies, voerinname-modellering en groeivoorspellings. In die derde studie (Hoofstuk 5) is die effek van drie verskillende dieëtproteïenkonsentrasies (met 'n spesifieke gepaardgaande aminosuurinhoud) op sekere produksieparameters in die groei van volstruise ondersoek. Beduidende verskille is gevind vir die finale lewende gewig, koue karkasmassa, boudgewig sowel as vir die meeste van die geweegde spiere van voëls op slagouderdom (350 dae oud). Met betrekking tot die groei en voedingsverwante parameters, is slegs die gemiddelde daaglikse toename (GDT) beïnvloed deur die dieet (P < 0.05). Resultate het aangedui dat voëls op die medium-proteïendieet optimaal presteer. Een uitsondering is die aanvangsfase (26 – 47 kg), waar kuikens op die hoë-proteïendieet beter gevaar het as die voëls wat die medium-proteïendieet ontvang het. In die vierde studie (Hoofstuk 6) is die invloed van verskillende dieet-energiekonsentrasies op volstruis-produksieparameters in twee verskillende proewe ondersoek. Die eerste proef het gestrek vanaf die voor-aanvangsfase, deur die aanvangsfase tot en met die einde van die groeifase. Die tweede proef is gedoen vir die afrondingsfase. In die voor-aanvangs-, aanvangs- en groeifase is beter groei, voeromsetverhouding (VOV), velgrootte en -graad, boudgewig, lewende gewig en karkasgewig verkry vir die voëls wat die standaard-energie dieet ontvang het (P < 0.05). Dieet-energievlakke wat tydens die afrondingsfase fase verskaf is, het aangedui dat die energievlak bo die medium-vlak verbeterde groeitempo en gelooide velgrootte tot gevolg het (P < 0.05). Die geslag van die voëls het ’n invloed gehad op karkasgewig, groei, en sekere veerparameters. In die vyfde studie (Hoofstuk 7) is die effek van die knip van vere, op die ouderdom van ses tot agt maande, op die produksieparameters van volstruiskuikens ondersoek. Die studie is uitgevoer in drie verskillende proewe. In elk van die proewe is die vere van die helfte van die hoeveelheid voëls geknip op ses tot agt maande ouderdom. Beduidende verskille is gevind vir die VOV, die gemiddelde daaglikse toename (GDT) en vir die hoeveelheid waardevolle vere (P < 0.05). Die groeitempo en VOV was beter vir die voëls waarvan die vere op ses tot agt maande ouderdom geknip is (P < 0.05). Resultate het ook getoon dat die hoeveelheid waardevolle vere aansienlik hoër was vir die groep waarvan die vere op ses tot agt maande ouderdom geknip is (P < 0.05). Hierdie studie het getoon dat daar 'n voordeel mag wees vir volstruisprodusente indien vere geknip word op die ouderdom van ses tot agt maande. Die werk in hierdie tesis volg op die raamwerk van Kritzinger (2011) en was deel van dieselfde projek. Die meeste van die resultate wat verkry is in die studies sal in die wiskundige optimeringsmodel van Gous en Brand (2008) geïnkorporeer word vir meer akkurate voorspellings van voerinname en produksieparameters wat die voerkostes kan verlaag.

The problems of estimation and prediction for statistical models based on progressively type-II censored sample play a crucial role in various areas of research such as reliability theory, survival analysis and statistics. In this thesis, statistical inferences for five distributions are considered under progressively type-II censored sample. For generalized Rayleigh, gamma-mixed Rayleigh, log-logistic and generalized Fréchet distributions, the estimators have been obtained for the unknown parameters, reliability and hazard rate functions. For the case of exponentiated Gumbel type-II distribution, the estimands are proposed for the model parameters. Various estimates are proposed in this thesis. The maximum likelihood estimates are obtained. These are not of closed-form. Thus, Newton-Raphson method, expectation-maximization and stochastic expectation-maximization algorithms are used to compute the maximum likelihood estimates. Sufficient conditions for the existence and uniqueness of the maximum likelihood estimates are obtained for the exponentiated Gumbel type-II distribution. For each problem, Bayes estimates are derived with respect to various symmetric and asymmetric loss functions. The priors are considered as independent gamma distributions for the purpose of Bayesian estimation. It is observed that the Bayes estimates are not of closed-form. So, approximation techniques such as Lindley's method, importance sampling method and Metropolis-Hastings algorithm are employed. The approximate confidence intervals are constructed using the normal approximation of the maximum likelihood estimates, normal approximation of the log-transformed maximum likelihood estimates and two bootstrap procedures. Highest posterior density credible intervals are introduced. In addition, the problem of Bayesian prediction and interval estimation is studied for gamma-mixed Rayleigh and generalized Fréchet distributions. In this purpose, one- and two-sample prediction problems are studied. To observe the performance of the proposed estimates, a detailed simulation is conducted using R software. The performance of the maximum likelihood and Bayes estimates is observed based on the average values and mean squared errors. The interval estimates are compared with respect to the average lengths and coverage probabilities. For every problem, real datasets are considered and analysed for illustrative purposes.

This thesis illuminates on the study of some elliptic partial differential equations (PDEs) involving singularity and measure data or a power non-linearity. The thesis emphasises mostly on the non-local PDEs. The main objective is to obtain the existence, multiplicity and regularity of solutions to the problems considered in the thesis prescribed with certain Dirichlet boundary conditions. Some of the key techniques employed in the thesis to guarantee the existence of solutions are the weak convergence method, Schauder fixed point theorem, Brouwer degree theory, different variants of mountain pass theorem, concentration compactness lemma etc. The existence of infinitely many solutions is accomplished by applying the symmetric mountain pass theorem. The regularity of the solutions is established mostly by the Moser iteration techniques.

Maximal monotone operators play an important role in non-linear modern analysis. In this thesis, we focus on the most famous and significant open problem that is “Sumproblem” in monotone operator theory. First, we provide some powerful sufficient conditions for partial solutions of the Sum problem. The maximality of the sum of an ultramaximal monotone operator and a maximal monotone operator of type (D) with Br_ezis-Crandall-Pazy conditions in Grothendieck Banach spaces is established. Also, we work on Br_ezis-Haraux conditions in Grothendieck Banach spaces. Finally, a sufficient condition is provided for the convergence of a sequence of maximal monotone operators of type (D) in general Banach spaces. The representability of the lower limit of a sequence of maximal monotone operators of type (D) is discussed through their representative functions. Moreover, some conditions are given which guarantee the maximality of the lower limit of a sequence of maximal monotone operators of type (D).

Swirling, whirling and rotating flows have fascinated people for centuries and the fascination continues to this day. Atmospheric or oceanic flows, typhoons and tornadoes, and closer to our daily lives, bathtub vortices and stirring tea in a teacup, are all examples of the ubiquity of swirling flows at all scales in nature. Moreover, rotating flows are of crucial importance in a wide range of scientific and engineering applications. Description of such flows, in more than just observational details had to wait for the Navier-Stokes equations. Even than the equations were so difficult to solve, that it had to wait for the advent of computers and with them many numerical, analytical or semi-analytical techniques. This thesis is devoted to investigation of a few problems of convective heat and mass transfer related to rotating disk systems. Rotating disk systems are widely used to model the flow and heat transfer associated with the internal-air systems of gas turbines, where disks rotate close to a rotating or a stationary surface. In addition, rotating-disk systems are used in electro-chemistry (rotating-disk electrodes), bio and chemical reactors, transport engineering (automobile brakes), rotating-disk cleaners, etc. These flows are among those few problems in fluid dynamics for which Navier-Stokes equations admit an exact solution. In view of such theoretical and practical importance of rotating disk flows, this thesis focuses on obtaining approximate analytical solutions of equations governing the flow problems arising in five different configurations of the rotating disk systems, viz. (i) flow over single rotating disk (under the influence of partial slip), (ii) flow between two rotating disks, (iii) fluid impinging over a stretchable rotating disk, (iv) flow between two stretchable rotating disks, and (v) flow over a stretchable rotating disk (in a rotating frame of reference). In addition, two different non-Newtonian fluid models, namely, visco-inelastic Reiner-Rivlin fluid and viscoelastic second grade fluid, are considered and the corresponding steady, laminar flows are investigated in the above mentioned configurations. The equations governing these flows are fully coupled and highly nonlinear, which provides a level of complexity that it becomes difficult to get a closed form analytic approximation. In this thesis, we have adopted a popular and promising non-perturbation technique, called, the ‘Homotopy Analysis Method (HAM)’ to obtain approximate analytical solutions of the considered problems. HAM is a general approximate analytic approach used to obtain series solutions of nonlinear equations of different types, such as algebraic equations, ordinary differential equations, partial differential equations and differential integral equations. The reasons why HAM is used are, first, this method is valid no matter whether or not a nonlinear problem contains small or large physical parameters, which is an essential requirement for perturbation techniques; second, unlike other perturbation or non-perturbation methods, the HAM provides us a simple way to guarantee the convergence of solution series by means of ~-curves. In addition, HAM provides us with the freedom to choose proper base functions to approximate a given nonlinear problem. In the consequent chapters, results in the form of a convergent Taylor series are obtained using HAM. In some cases the results obtained by using HAM are compared with those obtained by using commonly used semi analytical techniques, namely, Adomian decomposition method (ADM) and Homotopy perturbation method (HPM) to demonstrate the efficiency of HAM. In addition, convergence of the obtained series solutions are also analyzed. The effects of various parameters, such as, Reynolds number, stretching parameter, non-Newtonian parameter etc. on the flow fields are discussed in detail. Moreover, streamlines are drawn to better visualize and understand the flow problems under consideration.

Various engineering and science problems may transform into linear and nonlinear equations, in general. In recent decades, Artificial Neural Network (ANN) has emerged as one of the prominent mechanism for solving linear and nonlinear equations. Although linear and nonlinear equations may be solved by different known analytical and numerical methods but those are sometimes having different complexity to handle. Traditional numerical methods may sometimes fail to solve these equations due to the involvement of singularities or complexity of the function etc. Moreover, (for example) there may exist two closely positioned roots or due to discontinuity of the curve in the problems of root finding and then the known numerical methods may sometimes difficult to use. In case of linear system of equations, the traditional numerical methods sometimes fail if the system is not diagonally dominant, positive definite etc. In those cases, ANN based methods may be an alternative for solving the equations. In this regard, detail ANN procedure with various example problems related to transcendental, Diophantine and linear system of equations with their network architectures have been addressed here to demonstrate the proposed procedure. Further, solving linear and nonlinear eigenvalue problems are also challenging task. For example, dynamic analysis of structure without damping may transform into a linear eigenvalue problem and with damping it leads to a nonlinear eigenvalue problem. Linear eigenvalue problems are studied though by many authors, but nonlinear eigenvalue problems are not studied much. However, these methods (for both linear as well as nonlinear) may sometimes be problem dependent and difficult to handle. As such, in these cases, ANN may also be advantageous over the existing methods. Few examples of linear eigenvalue problems such as vibration analysis of spring mass system and multi-storey shear building have been investigated. On the other hand, two examples of overdamped spring mass systems have been examined to show the efficacy of the proposed method in case of nonlinear eigenvalue problem. It may be noted that parameters involved in the above systems may not be crisp (exact) always because of errors in experiment, measurement and observation. In that case, the problem leads to an uncertain system. In order to handle these uncertainties, recently researchers have introduced interval and/or fuzzy numbers in place of crisp ones. In these regards, various techniques have been developed by different authors but these are sometimes valid for certain (particular) type of problems only. These methods may have few drawbacks that include number of iterations, triangularisation etc. In this context, application problems such as static problems of structures lead to system of equations. As mentioned earlier that inclusion of uncertainty makes the problem as uncertain. Similarly, computation of the interval controls using pole placement technique in case of uncertain plant system reduces to interval linear system, which itself is a challenging problem. Moreover, dynamic problems lead to eigenvalue problems which may become more complicated due to the inclusion of uncertainty. Accordingly, ANN methods have been developed to handle the above problems with ease. Different example problems have been solved in this context to validate the proposed ANN technique.

博士
國立中正大學
化學所
97
Substantial evidence has indicated that oxidative stress can be a causative factor in the pathogenesis of Alzheimer''s diseases (AD) and that scavenge ROS level may usefully prevent the development or progress of AD, and that diets high in antioxidants offer the potential to lower the associated risks. Astaxanthine (ASTx) is a novel carotenoid nutraceutics obtained from the crustaceans and red yeasts, having potent antioxidant, photoprotectant, hepatodetoxicant and anti-inflammatory activities. Document concerning the anti-Alzheimer’s disease (anti-AD) and neuroprotective effect of ASTx is still lacking. We utilized the Abeta25-35-PC12 cell model to investigate its neuron protective effect. Results indicated that ASTx protected the damaging effects induced by Abeta25-35 through several pathways, i.e. by blocking the direct fatal effect of Abeta25-35 oligomers, stabilizing calcium homeostasis, suppressing ROS generation, dowregulating the ratio of caspase 3/procaspase 3, downregulating the cytokine pathway, TNFalfa/IL-1beta, increasing translocation of NFkB and taking the pathway through p-p38MAPK. Conclusively, ASTx has a merit to be used as a very potential neuron protective and anti-AD adjuvant therapy. Molecular physicobiochemical calculations indicated that the metallic ion binding to beta-amyloids (Abeta) may induce production of hydrogen peroxide, which triggers the Ca ion redistribution from the extracellular to the intracellular compartmentation, resulting in a transient membrane electropotential drop by at least 208.06mV. Moreover, using the Mark and Houwink empirical equation, we predicted the diffusible distances of all Abeta identities would be confined in a very tiny region within a radius less than 3.96*10-4cm in brain at 192 h after produced. Because of the inherent tendency of aggregation behaved by the Abeta, the maximum diffusion coefficient and inherent viscosity were 8.24*10-15 cm2s-1 and 87.24 cps for the 12 mers, the largest soluble form of Abeta.