Soliton solutions of noncommutative integrable systems

This thesis is concerned with solutions of noncommutative integrable systems where the noncommutativity arises through the dependent variables in either the hierarchy or Lax pair generating the equation. Both Chapters 1 and 2 are entirely made up of background material and contain no new material. Furthermore, these chapters are concerned with commutative equations. Chapter 1 outlines some of the basic concepts of integrable systems including historical attempts at finding solutions of the KdV equation, the Lax method and Hirota's direct method for finding multi-soliton solutions of an integrable system. Chapter 2 extends the ideas in Chapter 1 from equations of one spatial dimension to equations of two spatial dimensions, namely the KP and mKP equations. Chapter 2 also covers the concepts of hierarchies and Darboux transformations. The Darboux transformations are iterated to give multi-soliton solutions of the KP and mKP equations. Furthermore, this chapter shows that multi-soliton solutions can be expressed as two types of determinant: the Wronskian and the Grammian. These determinantal solutions are then verified directly. In Chapter 3, the ideas detailed in the preceding chapters are extended to the noncommutative setting. We begin by outlining some known material on quasideterminants, a noncommutative KP hierarchy containing a noncommutative KP equation, and also two families of solutions. The two families of solutions are obtained from Darboux transformations and can be expressed as quasideterminants. One family of solutions is termed ``quasiwronskian'' and the other ``quasigrammian'' as both reduce to Wronskian and Grammian determinants when their entries commute. Both families of solutions are then verified directly. The remainder of Chapter 3 is original material, based on joint work with Claire Gilson and Jon Nimmo. Building on some known results, the solutions obtained from the Darboux transformations are specified as matrices. These solutions have interesting interaction properties not found in the commutative setting. We therefore show various plots of the solutions illustrating these properties. In Chapter 4, we repeat all of the work of Chapter 3 for a noncommutative mKP equation. The material in this chapter is again based on joint work with Claire Gilson and Jon Nimmo and is mainly original. The original material in Chapters 3 and 4 appears in \cite{gilson:nimmo:sooman2008} and in \cite{gilson:nimmo:sooman2009}. Chapter 5 builds on the work of Chapters 3 and 4 and is concerned with exponentially localised structures called dromions, which are obtained by taking the determinant of the matrix solutions of the noncommutative KP and mKP equations. For both equations, we look at a three-dromion structure from which we then perform a detailed asymptotic analysis. This aymptotic forms show interesting interaction properties which are demonstrated by various plots. This chapter is entirely the author's own work. Chapter 6 presents a summary and conclusions of the thesis

Soliton solutions of noncommutative integrable systems

This thesis is concerned with solutions of noncommutative integrable systems where the noncommutativity arises through the dependent variables in either the hierarchy or Lax pair generating the equation. Both Chapters 1 and 2 are entirely made up of background material and contain no new material. Furthermore, these chapters are concerned with commutative equations. Chapter 1 outlines some of the basic concepts of integrable systems including historical attempts at finding solutions of the KdV equation, the Lax method and Hirota's direct method for finding multi-soliton solutions of an integrable system. Chapter 2 extends the ideas in Chapter 1 from equations of one spatial dimension to equations of two spatial dimensions, namely the KP and mKP equations. Chapter 2 also covers the concepts of hierarchies and Darboux transformations. The Darboux transformations are iterated to give multi-soliton solutions of the KP and mKP equations. Furthermore, this chapter shows that multi-soliton solutions can be expressed as two types of determinant: the Wronskian and the Grammian. These determinantal solutions are then verified directly. In Chapter 3, the ideas detailed in the preceding chapters are extended to the noncommutative setting. We begin by outlining some known material on quasideterminants, a noncommutative KP hierarchy containing a noncommutative KP equation, and also two families of solutions. The two families of solutions are obtained from Darboux transformations and can be expressed as quasideterminants. One family of solutions is termed ``quasiwronskian'' and the other ``quasigrammian'' as both reduce to Wronskian and Grammian determinants when their entries commute. Both families of solutions are then verified directly. The remainder of Chapter 3 is original material, based on joint work with Claire Gilson and Jon Nimmo. Building on some known results, the solutions obtained from the Darboux transformations are specified as matrices. These solutions have interesting interaction properties not found in the commutative setting. We therefore show various plots of the solutions illustrating these properties. In Chapter 4, we repeat all of the work of Chapter 3 for a noncommutative mKP equation. The material in this chapter is again based on joint work with Claire Gilson and Jon Nimmo and is mainly original. The original material in Chapters 3 and 4 appears in \cite{gilson:nimmo:sooman2008} and in \cite{gilson:nimmo:sooman2009}. Chapter 5 builds on the work of Chapters 3 and 4 and is concerned with exponentially localised structures called dromions, which are obtained by taking the determinant of the matrix solutions of the noncommutative KP and mKP equations. For both equations, we look at a three-dromion structure from which we then perform a detailed asymptotic analysis. This aymptotic forms show interesting interaction properties which are demonstrated by various plots. This chapter is entirely the author's own work. Chapter 6 presents a summary and conclusions of the thesis.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Long term performance of gravel base course layers in asphalt

This research investigated the performance of base layer aggregates in HMA pavements using laboratory tests (standard compaction, particle size analysis, Atterberg Limits, sodium sulfate soundness, Micro-Deval abrasion, absorption, specific gravity, and soaked CBR) on existing base layer materials as well as pavement surface visual and automated distress surveys. The purpose of this research was to investigate potential degradation of aggregate bases, strength variations over time, and the likely causes for both. Analysis of laboratory and field test results indicated significant variability in the properties and characteristics of base layer aggregate materials in various pavement test sections. Based on the results of the laboratory and field tests, the research team believes that the long-term performance of the base layer aggregates impacted the overall pavement performance of the corresponding test sections. While base aggregate materials in general did not exhibit severe degradation or disintegration – as demonstrated by laboratory tests – nor significant contamination from subgrade, the performance of such materials was lower compared with typical crushed stone materials

SENSING MECHANISM AND APPLICATION OF MECHANICAL STRAIN SENSOR: A MINI-REVIEW

This study reviews the potential of flexible strain sensors based on nanomaterials such as carbon nanotubes (CNTs), graphene, and metal nanowires (NWs). These nanomaterials have excellent flexibility, conductivity, and mechanical properties, which enable them to be integrated into clothing or attached to the skin for the real-time monitoring of various activities. However, the main challenge is balancing high stretchability and sensitivity. This paper explains the basic concept of strain sensors that can convert mechanical deformation into electrical signals. Moreover, this paper focuses on simple, flexible, and stretchable resistive and capacitive sensors. It also discusses the important factors in choosing materials and fabrication methods, emphasizing the crucial role of suitable polymers in high-performance strain sensing. This study reviews the fabrication processes, mechanisms, performance, and applications of stretchable strain sensors in detail. It analyzes key aspects, such as sensitivity, stretchability, linearity, response time, and durability. This review provides useful insights into the current status and prospects of stretchable strain sensors in wearable technology and human–machine interfaces

Socio-economic inequalities in women`s fruit and vegetable intakes: a multilevel study of individual, social and environmental mediators

Objective: This study employed a multilevel design to test the contribution of individual, social and environmental factors to mediating socio-economic status (SES) inequalities in fruit and vegetable consumption among women. Design: A cross-sectional survey was linked with objective environmental data. Setting: A community sample involving 45 neighbourhoods. Subjects: In total, 1347 women from 45 neighbourhoods provided survey data on their SES (highest education level), nutrition knowledge, health considerations related to food purchasing, and social support for healthy eating. These data were linked with objective environmental data on the density of supermarkets and fruit and vegetable outlets in local neighbourhoods. Results: Multilevel modelling showed that individual and social factors partly mediated, but did not completely explain, SES variations in fruit and vegetable consumption. Store density did not mediate the relationship of SES with fruit or vegetable consumption. Conclusions: Nutrition promotion interventions should focus on enhancing nutrition knowledge and health considerations underlying food purchasing in order to promote healthy eating, particularly among those who are socio-economically disadvantaged. Further investigation is required to identify additional potential mediators of SES&ndash;diet relationships, particularly at the environmental level.<br /

Matrix solutions of a noncommutative KP equation and a noncommutative mKP equation

Matrix solutions of a noncommutative KP and a noncommutative mKP equation which can be expressed as quasideterminants are discussed. In particular, we investigate interaction properties of two-soliton solutions.Comment: 2 figure

The relative influence of neighbourhood incivilities, cognitive social capital, club membership and individual characteristics on positive mental health

Previous research indicates that residents׳ perceptions of their neighbourhoods can have an adverse influence on their health and wellbeing over and above the influence of structural disadvantage. Contrary to most prior research, this study employed an indicator of positive wellbeing and assessed the impact of individual characteristics, perceived social and environmental incivilities, indicators of cognitive and structural social capital, and perceived safety. Analyses of data from a large regional UK representative study (n=8237; 69.64% response rate) found the most influential determinants of wellbeing were physical health problems, age, SES and cognitive social capital. Smaller, significant effects were also found for environmental and social incivilities, and for perceived safety. The effect of cognitive social capital was moderated by age, with a stronger effect found among those aged 65 years and over than among younger participants. Findings indicate that the promotion of positive mental health within communities may be facilitated by efforts to foster a greater sense of belonging among residents, and that older adults may benefit most from such efforts