On the algebraic structure of factorized S-matrices

This thesis investigates the algebraic structure of certain quantum field theories in one space and one time dimension. These theories are integrable - essentially, highly constrained and therefore soluble. Thus, instead of having to use perturbative techniques, it is possible to conjecture their exact 5-matrices, which have the property that they are factorized into two-particle 5-matrices. In particular, there are two types of such theory: in one, scattering is purely elastic, whilst in the other, there is additional structure dictated by the Yang-Baxter equation. This thesis explores the algebraic structure of the latter and its links with the former. We begin, in chapter one, with an informal summary of the development of the subject, followed by a more mathematical exposition in chapter two. Chapter three constructs explicitly some exact factorized 5-matrices with Yang-Baxter structure, and comments on their features, both intrinsic and in relation to purely elastic 5-matrices. In particular, there is an unexplained close correspondence between the mass spectra and particle fusings in the two types of theory. The next three chapters attempt to shed some light on these features. Chapter four constructs similar 5-matrices, but based on quantum-deformed algebras rather than classical algebras. In chapter five we describe the structure of the 5-matrices when the particles they describe transform in irreducible representations of classical algebras. This leads us to consider the Yangian algebra, the representation theory of which underlies Yang-Baxter dependent 5-matrices, and which we therefore review briefly. We begin chapter six by reviewing the work which shows that the Yangian is also the charge algebra of the integrable quantum field theory, and subsequently show that the Yangian is also to a great extent present in the corresponding classical theory. We conclude with a brief seventh chapter describing the outlook for further research, followed by appendices containing respectively details of the Lagrangians of some integrable quantum field theories, a continuum formulation of the quantum inverse problem, explicit expressions for some of the R-matrices computed in the text, and a summary of known solutions of the Yang-Baxter equation

Toward a mathematical theory of perception

A new technique for the modelling of perceptual systems called formal modelling is developed. This technique begins with qualitative observations about the perceptual system, the so-called perceptual symmetries, to obtain through mathematical analysis certain model structures which may then be calibrated by experiment. The analysis proceeds in two different ways depending upon the choice of linear or nonlinear models. For the linear case, the analysis proceeds through the methods of unitary representation theory. It begins with a unitary group representation on the image space and produces what we have called the fundamental structure theorem. For the nonlinear case, the analysis makes essential use of infinite-dimensional manifold theory. It begins with a Lie group action on an image manifold and produces the fundamental structure formula. These techniques will be used to study the brightness perception mechanism of the human visual system. Several visual groups are defined and their corresponding structures for visual system models are obtained. A new transform called the Mandala transform will be deduced from a certain visual group and its implications for image processing will be discussed. Several new phenomena of brightness perception will be presented. New facts about the Mach band illusion along with new adaptation phenomena will be presented. Also a new visual illusion will be presented. A visual model based on the above techniques will be presented. It will also be shown how use of statistical estimation theory can be made in the study of contrast adaptation. Furthermore, a mathematical interpretation of unconscious inference and a simple explanation of the Tolhurst effect without mutual channel inhibition will be given. Finally, image processing algorithms suggested by the model will be used to process a real-world image for enhancement and for "form" and texture extraction

Residential patterns in the nineteenth century city : Kingston upon Hull, 1851

Studies of residential patterns have tended to concentrate on cities in modern societies at a similar stage of advanced industrial development. Those studies which have been carried out in less advanced societies, however, suggest that the forces behind residential differentiation vary with the nature of society itself. The three factors of social rank, family status and migrant status have been identified as major dimensions of differentiation within cities, but at a less advanced stage of development these factors are often measured in terms of different criteria, and show differing degrees of interdependence, particularly between the social rank and family status axes.Nineteenth century Britain presents an interesting example of a society in the transition stage from a pre-industrial to a modern form of organisation. Available evidence suggests the importance of a social rank criterion based on subjective rather than purely economic definitions of social status, and the differing economic circumstances between strata suggest possible links between family status and social rank. Using Hull as a case study, and the 1851 census enumerators' books as a source of data, factor analysis techniques have been used to try to define this pattern of differentiation more precisely.The main dimensions of residential differentiation are shown to be consistent with the patterns found elsewhere, although the composition of these factors contrasts markedly with the twentieth century situation, due to the specific conditions of the period. Social rank, in particular, illustrates the dichotomy within society between employers and the employed, and migrant status reflects the specific situation of Irish immigrants. An oblique solution supports the idea that social rank and family status show a marked degree of interdependence in this context. The results have clear implications for the study of nineteenth century society, and also contribute to a general theory of urban residential patterns

Algorithms in time series

The use of finitely parametrized linear models such as ARMA models in analysing time series data has been extensively studied and in recent years there has been an increasing emphasis on the development of fast regression—based algorithms for the problem of model identification. In this thesis we investigate the statistical properties of pseudo—linear regression algorithms in the context of off-line and online (real-time) identification. A review of these procedures is presented in Part I in relation to the problem of identifying an appropriate ARMA model from observed time series data. Thus, criteria introduced by Akaike and Rissanen are important here to ensure a model of sufficient complexity is selected, based on the data. In chapter 1 we survey published results pertaining to the statistical properties of identification procedures in the off-line context and show there are important differences as concerns the asymptotic performance of certain parameter estimation algorithms. However, to effect the identification process in real-time recursive estimation algorithms are required. Furthermore, these procedures need to be adaptive to be applicable in practice. This is discussed in chapter 2. Technical results and limit theorems required for the theoretical analysis conducted in Part II are collated in chapter 3. Chapters 4 and 5 of Part II are therefore devoted to the detailed investigation of particular algorithms discussed in Part I. Chapter 4 deals with off-line parameter estimation algorithms and in chapter 5, the important idea of a Description Length Principle introduced by Rissanen, is examined in the context of the recursive estimation of autoregressions. Empirical evidence from simulation experiments are also reported in each chapter and in chapter 5, aspects of speech analysis are incorporated in the simulation study. The simulation results bear out the theory and the proofs of asymptotic results are given at the end of the chapter

Structure-function characterization and engineering of polysaccharides and antibodies with therapeutic activity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Biological Engineering, 2012.Cataloged from PDF version of thesis.Includes bibliographical references.Proteins and polysaccharides are of growing importance as a source for novel therapeutic compounds and target a range of diseases, from cancer to infections from pathogens. However, owing to their large and complex structures, they face a unique set of challenges, compared to small molecules, in their discovery and development as safe, efficacious drugs. Towards addressing these challenges, we describe in this thesis the implementation of structure-function relationship approaches to characterize and engineer polysaccharides and antibodies to improve their therapeutic profiles. The plant polysaccharide pectin, when modified, has demonstrated significant anticancer activity in animal models and small-scale clinical trials. Its development has been hampered, however, due to its complex structure and lack of structure-activity correlates. Using an integrated approach, we engineer a modified pectin that exhibits significant in vivo anticancer activity, which we link to specific structural attributes and cellular functional mechanisms. These results improve our structure-function understanding of anticancer modified pectin, an important step towards the clinical use of this complex polysaccharide. Applying what we learned from pectin, we develop an integrated framework to identify a contaminant in batches of heparin, a polysaccharide anticoagulant drug, associated with an outbreak of allergic-type reactions in 2007-2008. Employing orthogonal analytical approaches to overcome challenges of characterizing structurally complex pharmaceutical heparin, we determine that the structurally related glycan, oversulfated chondroitin sulfate, is the major contaminant. We link its presence to activation of the contact pathway, thereby establishing a structure-function understanding of contaminated heparin and improving the safety profile of this polysaccharide drug. Transitioning knowledge gained from the structure-function characterization of polysaccharides, we engineer, by structure-based design, a broad spectrum neutralizing antibody to dengue virus, which yearly infects more than 200 million people, causing approximately 21,000 deaths. We incorporate complementary approaches of energetics and empirical informatics methods to rationally redesign an existing antibody for greater breadth and potency, resulting in an engineered antibody with binding to all four virus serotypes and good in vitro potency. Overall, this thesis provides important insights into structure-function approaches through the use of complementary methods to characterize and engineer therapeutic polysaccharides and antibodies.by Luke Robinson.Ph.D

Perturbative and non-perturbative studies in low dimensional quantum field theory

A relevant perturbation of a conformal field theory (CFT) on the half-plane, by both a bulk and boundary operator, often leads to a massive theory with a particle description in terms of the bulk S-matrix and boundary reflection factor R. The link between the particle basis and the CFT in the bulk is usually made with the thermodynamic Bethe ansatz effective central charge C(_eff). This allows a conjectured S-matrix to be identified with a specific perturbed CFT. Less is known about the links between the reflection factors and conformal boundary conditions, but it has been proposed that an exact, off-critical version of Affleck and Ludwig's g-function could be used, analogously to C(_eff), to identify the physically realised reflection factors and to match them with the corresponding boundary conditions. In the first part of this thesis, this exact g-function is tested for the purely elastic scattering theories related to the ADET Lie algebras. Minimal reflection factors are given, and a method to incorporate a boundary parameter is proposed. This enables the prediction of several new flows between conformal boundary conditions to be made. The second part of this thesis concerns the three-parameter family of PT-symmetric Hamiltonians H(M,o,1) = p(^2) – (ix) (^2M) – α(ix) The positions where the eigenvalues merge and become complex correspond to quadratic and cubic exceptional points. The quasi-exact solvability of the models for M = 3 is exploited to exploreaway from M = 3 is investigated using both numerical and perturbative approaches