One of today's most serious social, economical and environmental problems is traffic congestion. In addition to the financial cost of the problem, the number of traffic related injuries and casualties is very high. A recently considered approach to increase safety while reducing congestion and improving driving conditions is Automated Highway Systems (AHS). The AHS will evolve from the present highway system to an intelligent vehicle/highway system that will incorporate communication, vehicle control and traffic management techniques to provide safe, fast and more efficient surface transportation. A key factor in AHS deployment is intelligent vehicle control. While the technology to safely maneuver the vehicles exists, the problem of making intelligent decisions to improve a single vehicle's travel time and safety while optimizing the overall traffic flow is still a stumbling block. We propose an artificial intelligence technique called stochastic learning automata to design an intelligent vehicle path controller. Using the information obtained by on-board sensors and local communication modules, two automata are capable of learning the best possible (lateral and longitudinal) actions to avoid collisions. This learning method is capable of adapting to the automata environment resulting from unmodeled physical environment. Simulations for simultaneous lateral and longitudinal control of an autonomous vehicle provide encouraging results. Although the learning approach taken is capable of providing a safe decision, optimization of the overall traffic flow is also possible by studying the interaction of the vehicles. The design of the adaptive vehicle path planner based on local information is then carried onto the interaction of multiple intelligent vehicles. By analyzing the situations consisting of conflicting desired vehicle paths, we extend our design by additional decision structures. The analysis of the situations and the design of the additional structures are made possible by the study of the interacting reward-penalty mechanisms in individual vehicles. The definition of the physical environment of a vehicle as a series of discrete state transitions associated with a "stationary automata environment" is the key to this analysis and to the design of the intelligent vehicle path controller. This work was supported in part by the Center for Transportation Research and Virginia DOT under Smart Road project, by General Motors ITS Fellowship program, and by Naval Research Laboratory under grant no. N000114-93-1-G022. / Ph. D.

Understanding fractions is fundamental for expanding number knowledge from the whole number system to the rational number system. According to the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (2000), learning fractions is an important mathematical goal for students in grades three through five in the U.S. Moreover, the NCTM suggests that fraction instruction start in Pre-K and continue through 8th grade. At the same time, the Common Core State Standards for Mathematics (CCSSM) suggests that fraction instruction should occur from Grade 3 to 7. In contrast to the time spent on learning fractions in the U.S., students in China spend a relatively short time learning fractions (Zhang and Siegler, 2022). According to the Chinese national curriculum standards, the Chinese National Mathematics Curriculum Standards (CNMCS) for five-four system, the fundamental fraction concepts are taught in grades 3 and 5 only. However, Chinese students continue to have higher performance on fraction items in international assessments when compared with American students (Fan and Zhu, 2004). Consequently, over the last several years, researchers have investigated subject content knowledge and pedagogical content knowledge of Chinese in-service teachers and pre-service teachers via fraction division (e.g., Li and Huang, 2008; Ma, 1999). There are also studies exploring Chinese written curricula of fraction division (e. g., Li, Zhang, and Ma, 2009). Recently, a quantitative study from Norton, Wilkins, and Xu (2018) investigated the process of Chinese students' construction of fraction knowledge through the lens of fraction schemes, a model established by western scholars Steffe (2002) and his colleague Olive (Steffe and Olive, 2010). However, there is a lack of qualitative research that attempts to use fraction schemes as an explanatory framework to interpret the process of Chinese students' construction of fraction knowledge. The main purpose of this study was to investigate Chinese students' understanding of the fundamental fraction knowledge in terms of their understanding of the "fraction unit," referred to as a "unit fraction" in the U.S., using Steffe and Olive's (2010) fraction schemes as the conceptual framework. A sequential mixed methods design was used in this study. The design included two consecutive phases, namely a quantitative phase followed by a qualitative phase (Creswell and Plano Clark, 2011). During the quantitative phase, five hundred and thirty-four Chinese fourth and fifth grade students were administered an assessment. The quantitative data was first analyzed using a Cochran's Q test to determine if the Chinese participants in this study follow the same progression of fraction schemes as their American peers. Results indicate that the development of fractional schemes among Chinese 4th and 5th grade participants in this study is similar to their U.S. counterparts and the Chinese participants in Norton et al.'s (2018) study regardless of the curricula differences across countries or areas in the same country, the textbook differences, and the language differences. Next, two different analysis of variances (ANOVA), a three-way mixed ANOVA and a two-way repeated measures ANOVA were conducted. The three-way mixed ANOVA was used to inform the researcher as to the fraction schemes these students had constructed before the concept of fraction unit is formally introduced and after the concept of fraction unit is formally introduced. The results showed that the fraction knowledge of the students in this study developed from 4th grade to 5th grade. The analysis of clinical interview data confirmed this conclusion. The two-way repeated measures ANOVA was used to determine which model (i.e., linear, circular, or rectangular) is more or less problematic for Chinese students when solving fraction tasks. The results suggest that generally students' performance on linear model tasks was better than their performance on circular model tasks, but there was no statistically significant difference between performance on circular model and its corresponding rectangular model tasks. The results from the quantitative analyses were also used to screen students to form groups based on their highest available fraction scheme for a clinical interview in the second phase, the qualitative phase. In the qualitative phase, a clinical interview using a think-aloud method was used to gain insight into the role of students' conceptual understanding of the fraction unit in their construction of fraction knowledge. In this phase, students were asked to solve the tasks in the clinical interview protocol using the think aloud method. Two main findings were revealed analyzing the clinical interview data. First, a conceptual understanding of fraction units as well as a conceptual understanding of a unit whole play a critical role in the construction of Chinese students' fraction knowledge. Second, the lack of the understanding of a fraction unit as an iterable unit may be one of the reasons that obstructs students move from part-whole concept of fractions to the measurement concept of fractions. This study also demonstrates that a conceptual understanding of fraction units and the unit whole are a necessary condition for constructing of a conceptual understanding of fraction knowledge. Thus, implications of this study suggest that teachers not only should help students build a conceptual understanding of fraction units, but also need to confirm that students have constructed the concept of what the unit whole is before asking students to identify the fraction units for the referent whole. On the other hand, the tasks used in the present study only include continuous but not discrete wholes. Therefore, future research may focus on investigating how students identify fraction units and in what way the iterating operation could be used when students encounter a discrete whole. / Doctor of Philosophy / Understanding fractions is fundamental for expanding number knowledge from the whole number system to the rational number system. According to the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics (2000), learning fractions is an important mathematical goal for students in grades three through five in the U.S. At the same time, the Common Core State Standards for Mathematics (CCSSM), suggests that fraction instruction should occur from Grade 3 to 7. In contrast to the time spent on learning fractions in the U.S., students in China spend a relatively short time learning fractions (Zhang and Siegler, 2022). According to the Chinese national curriculum standards, the Chinese National Mathematics Curriculum Standards (CNMCS) for five-four system, the fundamental fraction concepts are taught in grades 3 and 5 only. However, Chinese students continue to have higher performance on fraction items in international assessments when compared with American students (Fan and Zhu, 2004). Consequently, over the last several years, researchers have investigated fraction knowledge of Chinese in-service teachers and pre-service teachers via fraction division (e.g., Li and Huang, 2008; Ma, 1999). There are also studies exploring Chinese written curricula of fraction division (e. g., Li, Zhang, and Ma, 2009). Recently, Norton, Wilkins, and Xu (2018) collected and analyzed numerical data from Chinese students and investigated the process of how Chinese students learn fraction knowledge through a model established by western scholars Steffe (2002) and his colleague Olive (Steffe and Olive, 2010). However, there is a lack of research study that attempts to seek an in-depth understanding of how Chinese students learn their fraction knowledge. This study used both numerical data and data gathering from interviewing 29 4th and 5th grade Chinese students. It aimed to investigate Chinese students' understanding of the fundamental fraction knowledge in terms of their understanding of the "fraction unit," referred to as a "unit fraction" in the U.S., using Steffe and Olive's (2010) fraction schemes as the conceptual framework. This study demonstrates that a comprehensive and practical understanding of fraction units and the whole of a given fraction are a necessary condition for building a comprehensive understanding of fraction knowledge. The implications of this study suggest that teachers not only should help students build a comprehensive understanding of fraction units, but also need to confirm that students have built the concept of what the whole of a given fraction is before asking students to identify the fraction units for the referent whole.

Group delay between two univariate time series is a measure, in units of time, of how one series leads or lags the other at specific frequencies. The only published method of estimating group delay is Hannan and Thomson (1973); however, their method is highly asymptotic and does not allow inference to be performed on the group delay parameter in finite samples. In fact, spectral analysis in general does not allow for small sample inference which is a difficulty with the frequency domain approach to time series analysis. The reason that no statistical inference may be performed in small samples is the fact that distribution theory for spectral estimates is highly asymptotic and one can never be certain in a particular application what finite sample size is required to justify the asymptotic result. In the dissertation the asymptotic distribution theory is circumvented by use of the Box-Cox power transformation on the observed sample phase function. Once transformed, it is assumed that the sample phase is approximately normally distributed and the relationship between phase and frequency is modelled by a simple linear regression model. ln order to estimate group delay it is necessary to inversely transform the predicted values to the original scale of measurement and this is done by expanding the inverse Box-Cox transformation function in a Taylor Series expansion. The group delay estimates are generated by using the derivative of the Taylor Series expansion for phase. The UMVUE property comes from the fact that the Taylor Series expansions are functions of complete, sufficient statistics from the transformed domain and the Lehmann-Scheffe' result (1950) is invoked to justify the UMVUE property. / Ph. D.

This thesis introduces models for pricing credit default swaps (CDS) and evaluating the counterparty risk when buying a CDS in the over-the-counter (OTC) market from a counterpart subjected to default risk. Rather than assuming that the default of the referencing firm of the CDS is independent of the trading parties in the CDS, this thesis proposes models that capture the default correlation amongst the three parties involved in the trade, namely the referencing firm, the buyer and the seller. We investigate how the counterparty risk that CDS buyers face can be affected by default correlation and how their balance sheet could be influenced by the changes in counterparty risk. The correlation of corporate default events has been frequently observed in credit markets due to the close business relationships of certain firms in the economy. One of the many mathematical approaches to model that correlation is default contagion. We propose an innovative model of default contagion which provides more flexibility by allowing the affected firm to recover from a default contagion event. We give a detailed derivation of the partial differential equations (PDE) for valuing both the CDS and the credit value adjustment (CVA). Numerical techniques are exploited to solve these PDEs. We compare our model against other models from the literature when measuring the CVA of an OTC CDS when the default risk of the referencing firm and the CDS seller is correlated. Further, the model is extended to incorporate economy-wide events that will damage all firms' credit at the same time-this is another kind of default correlation. Advanced numerical techniques are proposed to solve the resulting partial-integro differential equations (PIDE). We focus on investigating the different role of default contagion and economy-wide events have in terms of shaping the default correlation and counterparty risk. We complete the study by extending the model to include bilateral counterparty risk, which considers the default of the buyer and the correlation among the three parties. Again, our extension leads to a higher-dimensional problem that we must tackle with hybrid numerical schemes. The CVA and debit value adjustment (DVA) are analysed in detail and we are able to value the profit and loss to the investor's balance sheet due to CVA and DVA profit and loss under different market circumstances including default contagion.

Mobile sinks are vital in many wireless sensor applications for efficient data collection, data querying, and localized sensor reprogramming. Mobile sinks prolong the lifetime of a sensor network. However, when sensor networks with mobile sinks are deployed in a hostile environment, security became a critical issue. They become exposed to varieties of malicious attacks. Thus, anti threats schemes and security services, such as mobile sink?s authentication and pairwise key establishment, are essential components for the secure operation of such networks. Due to the sensors, limited resources designing efficient security schemes with low communication overhead to secure communication links between sensors and MS (Mobile Sink) is not a trivial task. In addition to the sensors limited resources, sink mobility required frequent exchange of cryptography information between the sensors and MS each time the MS updates its location which imposes extra communication overhead on the sensors. In this dissertation, we consider a number of security schemes for WSN (wireless sensor network) with MS. The schemes offer high network?s resiliency and low communication overhead against nodes capture, MS replication and wormhole attacks. We propose two schemes based on the polynomial pool scheme for tolerating nodes capture: the probabilistic generation key pre-distribution scheme combined with polynomial pool scheme, and the Q-composite generation key scheme combined with polynomial pool scheme. The schemes ensure low communication overhead and high resiliency. For anti MS replication attack scheme, we propose the multiple polynomial pools scheme that provide much higher resiliency to MS replication attack as compared to the single polynomial pool approach. Furthermore, to improve the network resiliency against wormhole attack, two defensive mechanisms were developed according to the MS mobility type. In the first technique, MS uses controlled mobility. We investigate the problem of using a single authentication code by sensors network to verify the source of MS beacons, and then we develop a defensive approach that divide the sensor network into different authentication code?s grids. In the second technique, random mobility is used by MS. We explore the use of different communication channels available in the sensor hardware combined with polynomial pool scheme.

Environmental stewardship schemes are an important driver of biodiversity and habitat improvement throughout England, with the provision of funding to land managers to deliver effective land management that will benefit wildlife, habitats, natural resources and the population. Participation in agri-environment schemes is voluntary and much is done to encourage scheme participation. Dartmoor is a designated landscape, a National Park, and a Special Area of Conservation and encompasses areas of Sites of Special Scientific Interest (SSSI). Dartmoor is a farmed landscape, with the area divided into 92 common land units, over which a diversity of common rights are exercised. At present, the majority of Dartmoor Commons are managed by Environmental agreements, protecting the habitat and the SSSI’s. There is a demand for Dartmoor to be a recreational area, an environmental jewel, and a farmed landscape. Protection of this landscape requires the amalgamation of various organisations and individuals. Research indicates the benefits of group agri-environment schemes, of a landscape-scale approach to the improvement of habitats and the provision of wildlife corridors, crossing the boundaries of land ownership. vi The research considers the factors associated with group agri-environment schemes, where it is not necessarily like-minded individuals who come together because of a common cause and a shared vision. The issue surrounding common rights results in persons coming together because they have common rights and not necessarily a common view. A combination of interview responses and questionnaire data has been pulled together to ascertain the factors affecting agri-environment scheme participation on Dartmoor. The data reflects on the impacts of group agri-environment schemes on the commons of Dartmoor. The question remains as to the voluntary nature of group agri-environment schemes on Dartmoor. Hardin (1968) recognised the impact of one commoner’s decision had on another. Dartmoor agri-environment schemes require participation from the majority of commoners; therefore an individual’s action has a consequence. This research aims to investigate the impacts of agri-environment schemes on the commons of Dartmoor.

Stochastic differential equations (SDE’s) are used to describe systems which are influenced by randomness. Here, randomness is modelled as some external source interacting with the system, thus ensuring that the stochastic differential equation provides a more realistic mathematical model of the system under investigation than deterministic differential equations. The behaviour of the physical system can often be described by probability and thus understanding the theory of SDE’s requires the familiarity of advanced probability theory and stochastic processes. SDE’s have found applications in chemistry, physical and engineering sciences, microelectronics and economics. But recently, there has been an increase in the use of SDE’s in other areas like social sciences, computational biology and finance. In modern financial practice, asset prices are modelled by means of stochastic processes. Thus, continuous-time stochastic calculus plays a central role in financial modelling. The theory and application of interest rate modelling is one of the most important areas of modern finance. For example, SDE’s are used to price bonds and to explain the term structure of interest rates. Commonly used models include the Cox-Ingersoll-Ross model; the Hull-White model; and Heath-Jarrow-Morton model. Since there has been an expansion in the range and volume of interest rate related products being traded in the international financial markets in the past decade, it has become important for investment banks, other financial institutions, government and corporate treasury offices to require ever more accurate, objective and scientific forms for the pricing, hedging and general risk management of the resulting positions. Similar to ordinary differential equations, many SDE’s that appear in practical applications cannot be solved explicitly and therefore require the use of numerical methods. For example, to price an American put option, one requires the numerical solution of a free-boundary partial differential equation. There are various approaches to solving SDE’s numerically. Monte Carlo methods could be used whereby the physical system is simulated directly using a sequence of random numbers. Another method involves the discretisation of both the time and space variables. However, the most efficient and widely applicable approach to solving SDE’s involves the discretisation of the time variable only and thus generating approximate values of the sample paths at the discretisation times. This paper highlights some of the various numerical methods that can be used to solve stochastic differential equations. These numerical methods are based on the simulation of sample paths of time discrete approximations. It also highlights how these methods can be derived from the Taylor expansion of the SDE, thus providing opportunities to derive more advanced numerical schemes. / Dissertation (MSc (Mathematics of Finance))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / MSc / unrestricted

The Internet of Bodies (IoB) is a wireless network of on-body or in-body commu- nication formed by wearable, ingestible, injectable, and implantable smart devices. The vast majority of on-body communications, is typically required to be within <5 cm vicinity of the human body. The radiative nature of currently used RF devices leads to wasted energy that is radiated in unneeded off-body directions. Consequently, it degrades energy efficiency, introduces co-existence and interference problems, and imposes security threats on sensitive data. As an alternative, the capacitive body channel communication (BCC) couples the signal (between 10 kHz-100 MHZ) to the human body, which is more conductive than air. Hence, it provides lower loss, bet- ter privacy and confidentiality, and nJ/bit to pJ/bit energy efficiency. Accordingly, our work investigates orthogonal and non-orthogonal capacitive body channel access schemes for ultralow-power IoB networks with or without cooperation. We derive the closed-form optimal power allocation for uplink and downlink transmissions and the maximum number of IoB nodes satisfying a reliable and feasible network for non- cooperative schemes. The cooperative schemes necessitate joint optimization of both power and phase time allocations. We achieve this by using the Golden-Section search algorithm to minimize the power consumption in both phases.

This thesis presents the construction, the analysis and the verication of compact residual discretizations for the solution of conservation laws on unstructured meshes. The schemes considered belong to the class of residual distribution (RD) or fluctuation splitting (FS) schemes. The methodology presented relies on three main elements: design of compact linear first-order stable schemes for linear hyperbolic PDEs, a positivity preserving procedure mapping stable first-order linear schemes onto nonlinear second-order schemes with non-oscillatory shock capturing capabilities, and a conservative formulation enabling to extend the schemes to nonlinear CLs. These three design steps, and the underlying theoretical tools, are discussed in depth. The nonlinear RD schemes resulting from this construction are tested on a large set of problems involving the solution of scalar models, and systems of CLs. This extensive verification fills the gaps left open, where no theoretical analysis is possible. Numerical results are presented on the Euler equations of a perfect gas, on a two-phase flow model with highly nonlinear thermodynamics, and on the shallow-water equations. On irregular grids, the schemes proposed yield quite accurate and stable solutions even on very difficult computations. Direct comparisone show that these results are more accurate than the ones given by FV and WENO schemes. Moreover, our schemes have a compact nearest-neighbor stencil. This encourages to further develop our approach, toward the design of very high-order schemes, which would represent a very appealing alternative, both in terms of accuracy and efficiency, to now classical FV and ENO/WENO discretizations. These schemes might also be very competitive with respect to very high-order DG schemes.

La Simulation aux Grandes Echelles (SGE) est de plus en plus utilisée dans les processus de développement et la conception des réacteurs aéronautiques industriels. L'une des raisons pour ce besoin résulte dans la capacité de la SGE à fournir des informations instantanées d'un écoulement turbulent augmentant la quantité des prédictions de la composition des gaz d'échappement. Ce manuscrit de thèse aborde deux sujets récurrents de la SGE. D'une part, les schémas numériques pour la SGE nécessitent certaines propriétés, notamment une précision élevée avec une diffusivité faible pour ne pas nuire aux modèles de turbulence. Afin de répondre à ce pré requis, une famille de schémas d'intégration temporelle d'ordre élevée est proposée, permettant de modifier la diffusion numérique du schéma. D'autre part, la SGE étant intrinsèquement instationnaire, elle est très consommatrice en temps CPU. De plus, une géométrie complexe prend beaucoup de temps de simulation même avec les super calculateurs d'aujourd'hui. Dans le cas particulier d'intérêt et souvent rencontré dans les applications industrielles, l'approche bas-Mach est constitue une alternative intéressante permettant de réduire le coût et le temps de retour d'une simulation LES. L'impact et la comparaison des formalismes compressible et incompressible sont toutefois rarement quantifiés, ce qui est proposé dans ce travail pour une configuration représentative d'un brûleur swirlé industriel mesuré au CORIA / Large-Eddy Simulation (LES) becomes a more and more demanded tool to improve the design of aero-engines. The main reason for this request stems from the constraints imposed on the next generation low-emission engines at the industrial development level and the ability for LES to provide information on the instantaneous turbulent flow field which greatly contributes to improving the prediction of mixing and combustion thereby offering an improved prediction of the exhaust emission. The work presented in this thesis discusses two recurring issues of LES. For one, numerical schemes for LES require certain properties, i.e. low-diffusion schemes of high order of accuracy so as not to interfere with the turbulence models. To meet this purpose in the context of fully unstructured solvers, a new family of high-order time-integration schemes is proposed. With this class of schemes, the diffusion implied by the numerical scheme become adjustable and built-in. Second, since fully unsteady by nature, LES is very consuming in terms of CPU time. Even with today's supercomputers complex problems require long simulation times. Due to the low flow velocities often occurring in industrial applications, the use of a low-Mach number solver seems suitable and can lead to large reductions in CPU time if comparable to fully compressible solvers. The impact of the incompressibility assumption and the different nature of the numerical algorithms are rarely discussed. To partly answer the question, detailed comparisons are proposed for an experimental swirled configuration representative of a real burner that is simulated by LES using a fully explicit compressible solver and an incompressible solution developed at CORIA