Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Passive vector turbulence

This thesis consists of three articles on passive vector fields in turbulence. The vector fields interact with a turbulent velocity field, which is described by the Kraichnan model. The effect of the Kraichnan model on the passive vectors is studied via an equation for the pair correlation function and its solutions. The first paper is concerned with the passive magnetohydrodynamic equations. Emphasis is placed on the so called "dynamo effect", which in the present context is understood as an unbounded growth of the pair correlation function. The exact analytical conditions for such growth are found in the cases of zero and infinite Prandtl numbers. The second paper contains an extensive study of a number of passive vector models. Emphasis is now on the properties of the (assumed) steady state, namely anomalous scaling, anisotropy and small and large scale behavior with different types of forcing or stirring. The third paper is in many ways a completion to the previous one in its study of the steady state existence problem. Conditions for the existence of the steady state are found in terms of the spatial roughness parameter of the turbulent velocity field.

Two Stage Transform Vector Quantization of LSFs for Wideband Speech Coding

We investigate the use of a two stage transform vector quantizer (TSTVQ) for coding of line spectral frequency (LSF) parameters in wideband speech coding. The first stage quantizer of TSTVQ, provides better matching of source distribution and the second stage quantizer provides additional coding gain through using an individual cluster specific decorrelating transform and variance normalization. Further coding gain is shown to be achieved by exploiting the slow time-varying nature of speech spectra and thus using inter-frame cluster continuity (ICC) property in the first stage of TSTVQ method. The proposed method saves 3-4 bits and reduces the computational complexity by 58-66%, compared to the traditional split vector quantizer (SVQ), but at the expense of 1.5-2.5 times of memory.

From actin monomers to bundles: The roles of twinfilin and α-actinin in Drosophila melanogaster development

The actin cytoskeleton is essential for a large variety of cell biological processes. Actin exists in either a monomeric or a filamentous form, and it is very important for many cellular functions that the local balance between these two actin populations is properly regulated. A large number of proteins participate in the regulation of actin dynamics in the cell, and twinfilin, one of the proteins examined in this thesis, belongs to this category. The second level of regulation involves proteins that crosslink or bundle actin filaments, thereby providing the cell with a certain shape. α-Actinin, the second protein studied, mainly acts as an actin crosslinking protein. Both proteins are conserved in organisms ranging from yeast to mammals. In this thesis, the roles of twinfilin and α-actinin in development were examined using Drosophila melanogaster as a model organism. Twinfilin is an actin monomer binding protein that is structurally related to cofilin. In vitro, twinfilin reduces actin polymerisation by sequestering actin monomers. The Drosophila twinfilin (twf) gene was identified and found to encode a protein functionally similar to yeast and mammalian twinfilins. A strong hypomorphic twf mutation was identified, and flies homozygous for this allele were viable and fertile. The adult twf mutant flies displayed reduced viability, a rough eye phenotype and severely malformed bristles. The shape of the adult bristle is determined by the actin bundles that are regularly spaced around the perimeter of the developing pupal bristles. Examination of the twf pupal bristles revealed an increased level of filamentous actin, which in turn resulted in splitting and displacement of the actin bundles. The bristle defect was rescued by twf overexpression in developing bristles. The Twinfilin protein was localised at sites of actin filament assembly, where it was required to limit actin polymerisation. A genetic interaction between twinfilin and twinstar (the gene encoding Cofilin) was detected, consistent with the model predicting that both proteins act to limit the amount of filamentous actin. α-Actinin has been implicated in several diverse cell biological processes. In Drosophila, the only function for α-actinin yet known is in the organisation of the muscle sarcomere. Muscle and non-muscle cells utilise different α-actinin isoforms, which in Drosophila are produced by alternative splicing of a single gene. In this work, novel α-actinin deletion alleles, including ActnΔ233, were generated, which specifically disrupted the transcript encoding the non-muscle α-actinin isoform. Nevertheless, ActnΔ233 homozygous mutant flies were viable and fertile with no obvious defects. By comparing α-actinin protein distribution in wild type and ActnΔ233 mutant animals, it could be concluded that non-muscle α-actinin is the only isoform expressed in young embryos, in the embryonic central nervous system and in various actin-rich structures of the ovarian germline cells. In the ActnΔ233 mutant, α-actinin was detected not only in muscle tissue, but also in embryonic epidermal cells and in certain follicle cell populations in the ovaries. The population of α-actinin protein present in non-muscle cells of the ActnΔ233 mutant is referred to as FC-α-actinin (Follicle Cell). The follicular epithelium in the Drosophila ovary is a well characterised model system for studies on patterning and morphogenesis. Therefore, α-actinin expression, regulation and function in this tissue were further analysed. Examination of the α-actinin localisation pattern revealed that the basal actin fibres of the main body follicle cells underwent an organised remodelling during the final stages of oogenesis. This involved the assembly of a transient adhesion site in the posterior of the cell, in which α-actinin and Enabled (Ena) accumulated. Follicle cells genetically manipulated to lack all α-actinin isoforms failed to remodel their cytoskeleton and translocate Ena to the posterior of the cell, while the actin fibres as such were not affected. Neither was epithelial morphogenesis disrupted. The reorganisation of the basal actin cytoskeleton was also disturbed following ectopic expression of Decapentaplegic (Dpp) or as a result of a heat shock. At late oogenesis, the main body follicle cells express both non-muscle α-actinin and FC-α-actinin, while the dorsal anterior follicle cells express only non-muscle α-actinin. The dorsal anterior cells are patterned by the Dpp and Epidermal growth factor receptor (EGFR) signalling pathways, and they will ultimately secrete the dorsal appendages of the egg. Experiments involving ectopic activation of EGFR and Dpp signalling showed that FC-α-actinin is negatively regulated by combined EGFR and Dpp signalling. Ubiquitous overexpression of the adult muscle-specific α-actinin isoform induced the formation of aberrant actin bundles in migrating follicle cells that did not normally express FC-α-actinin, provided that the EGFR signalling pathway was activated in the cells. Taken together, this work contributes new data to our knowledge of α-actinin function and regulation in Drosophila. The cytoskeletal remodelling shown to depend on α-actinin function provides the first evidence that α-actinin has a role in the organisation of the cytoskeleton in a non-muscle tissue. Furthermore, the cytoskeletal remodelling constitutes a previously undescribed morphogenetic event, which may provide us with a model system for in vivo studies on adhesion dynamics in Drosophila.

Subspace intersection method of high-resolution bearing estimation in shallow ocean using acoustic vector sensors

The subspace intersection method (SIM) provides unbiased bearing estimates of multiple acoustic sources in a range-independent shallow ocean using a one-dimensional search without prior knowledge of source ranges and depths. The original formulation of this method is based on deployment of a horizontal linear array of hydrophones which measure acoustic pressure. In this paper, we extend SIM to an array of acoustic vector sensors which measure pressure as well as all components of particle velocity. Use of vector sensors reduces the minimum number of sensors required by a factor of 4, and also eliminates the constraint that the intersensor spacing should not exceed half wavelength. The additional information provided by the vector sensors leads to performance enhancement in the form of lower estimation error and higher resolution.

Prediction of Candidate Primary Immunodeficiency Disease Genes Using a Support Vector Machine Learning Approach

Screening and early identification of primary immunodeficiency disease (PID) genes is a major challenge for physicians. Many resources have catalogued molecular alterations in known PID genes along with their associated clinical and immunological phenotypes. However, these resources do not assist in identifying candidate PID genes. We have recently developed a platform designated Resource of Asian PDIs, which hosts information pertaining to molecular alterations, protein-protein interaction networks, mouse studies and microarray gene expression profiling of all known PID genes. Using this resource as a discovery tool, we describe the development of an algorithm for prediction of candidate PID genes. Using a support vector machine learning approach, we have predicted 1442 candidate PID genes using 69 binary features of 148 known PID genes and 3162 non-PID genes as a training data set. The power of this approach is illustrated by the fact that six of the predicted genes have recently been experimentally confirmed to be PID genes. The remaining genes in this predicted data set represent attractive candidates for testing in patients where the etiology cannot be ascribed to any of the known PID genes.

Holomorphic Sobolev spaces, Hermite and special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

The images of Hermite and Laguerre-Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterized. These are used to characterize the image of Schwartz class of rapidly decreasing functions f on R-n and C-n under these semigroups. The image of the space of tempered distributions is also considered and a Paley-Wiener theorem for the windowed (short-time) Fourier transform is proved.

Gaussian mixture model based switched split vector quantization of LSF parameters

We address the issue of rate-distortion (R/D) performance optimality of the recently proposed switched split vector quantization (SSVQ) method. The distribution of the source is modeled using Gaussian mixture density and thus, the non-parametric SSVQ is analyzed in a parametric model based framework for achieving optimum R/D performance. Using high rate quantization theory, we derive the optimum bit allocation formulae for the intra-cluster split vector quantizer (SVQ) and the inter-cluster switching. For the wide-band speech line spectrum frequency (LSF) parameter quantization, it is shown that the Gaussian mixture model (GMM) based parametric SSVQ method provides 1 bit/vector advantage over the non-parametric SSVQ method.

Computationally efficient optimum weighting function for vector quantization of LSF parameters

We propose a new weighting function which is computationally simple and an approximation to the theoretically derived optimum weighting function shown in the literature. The proposed weighting function is perceptually motivated and provides improved vector quantization performance compared to several weighting functions proposed so far, for line spectrum frequency (LSF) parameter quantization of both clean and noisy speech data.

Cluster based training for scaling non-linear support vector machines

Support Vector Machines(SVMs) are hyperplane classifiers defined in a kernel induced feature space. The data size dependent training time complexity of SVMs usually prohibits its use in applications involving more than a few thousands of data points. In this paper we propose a novel kernel based incremental data clustering approach and its use for scaling Non-linear Support Vector Machines to handle large data sets. The clustering method introduced can find cluster abstractions of the training data in a kernel induced feature space. These cluster abstractions are then used for selective sampling based training of Support Vector Machines to reduce the training time without compromising the generalization performance. Experiments done with real world datasets show that this approach gives good generalization performance at reasonable computational expense.

OCR Prediction Using Support Vector Machine Based on Piezocone Data

The determination of the overconsolidation ratio (OCR) of clay deposits is an important task in geotechnical engineering practice. This paper examines the potential of a support vector machine (SVM) for predicting the OCR of clays from piezocone penetration test data. SVM is a statistical learning theory based on a structural risk minimization principle that minimizes both error and weight terms. The five input variables used for the SVM model for prediction of OCR are the corrected cone resistance (qt), vertical total stress (sigmav), hydrostatic pore pressure (u0), pore pressure at the cone tip (u1), and the pore pressure just above the cone base (u2). Sensitivity analysis has been performed to investigate the relative importance of each of the input parameters. From the sensitivity analysis, it is clear that qt=primary in situ data influenced by OCR followed by sigmav, u0, u2, and u1. Comparison between SVM and some of the traditional interpretation methods is also presented. The results of this study have shown that the SVM approach has the potential to be a practical tool for determination of OCR.