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.

Topological stability through tame retractions

A smooth map is said to be stable if small perturbations of the map only differ from the original one by a smooth change of coordinates. Smoothly stable maps are generic among the proper maps between given source and target manifolds when the source and target dimensions belong to the so-called nice dimensions, but outside this range of dimensions, smooth maps cannot generally be approximated by stable maps. This leads to the definition of topologically stable maps, where the smooth coordinate changes are replaced with homeomorphisms. The topologically stable maps are generic among proper maps for any dimensions of source and target. The purpose of this thesis is to investigate methods for proving topological stability by constructing extremely tame (E-tame) retractions onto the map in question from one of its smoothly stable unfoldings. In particular, we investigate how to use E-tame retractions from stable unfoldings to find topologically ministable unfoldings for certain weighted homogeneous maps or germs. Our first results are concerned with the construction of E-tame retractions and their relation to topological stability. We study how to construct the E-tame retractions from partial or local information, and these results form our toolbox for the main constructions. In the next chapter we study the group of right-left equivalences leaving a given multigerm f invariant, and show that when the multigerm is finitely determined, the group has a maximal compact subgroup and that the corresponding quotient is contractible. This means, essentially, that the group can be replaced with a compact Lie group of symmetries without much loss of information. We also show how to split the group into a product whose components only depend on the monogerm components of f. In the final chapter we investigate representatives of the E- and Z-series of singularities, discuss their instability and use our tools to construct E-tame retractions for some of them. The construction is based on describing the geometry of the set of points where the map is not smoothly stable, discovering that by using induction and our constructional tools, we already know how to construct local E-tame retractions along the set. The local solutions can then be glued together using our knowledge about the symmetry group of the local germs. We also discuss how to generalize our method to the whole E- and Z- series.

Optical and Physical Properties of Atmospheric Aerosols over the Bay of Bengal during ICARB

Simultaneous and collocated measurements of total and hemispherical backscattering coefficients (σ and β, respectively) at three wavelengths, mass size distributions, and columnar spectral aerosol optical depth (AOD) were made onboard an extensive cruise experiment covering, for the first time, the entire Bay of Bengal (BoB) and northern Indian Ocean. The results are synthesized to understand the optical properties of aerosols in the marine atmospheric boundary layer and their dependence on the size distribution. The observations revealed distinct spatial and spectral variations of all the aerosol parameters over the BoB and the presence of strong latitudinal gradients. The size distributions varied spatially, with the majority of accumulation modes decreasing from north to south. The scattering coefficient decreased from very high values (resembling those reported for continental/urban locations) in the northern BoB to very low values seen over near-pristine environments in the southeastern BoB. The average mass scattering efficiency of BoB aerosols was found to be 2.66 ± 0.1 m2 g−1 at 550 nm. The spectral dependence of columnar AOD deviated significantly from that of the scattering coefficients in the northern BoB, implying vertical heterogeneity in the aerosol type in that region. However, a more homogeneous scenario was observed in the southern BoB. Simultaneous lidar and in situ measurements onboard an aircraft over the ocean revealed the presence of elevated aerosol layers of enhanced extinction at altitudes of 1 to 3 km with an offshore extent of a few hundred kilometers. Back-trajectory analyses showed these layers to be associated with advection from west Asia and western India. The large spatial variations and vertical heterogeneity in aerosol properties, revealed by the present study, need to be included in the regional radiative forcing over the Bay of Bengal.

Middleware for Mobile Sensing Applications in Urban Environments

Sensor networks represent an attractive tool to observe the physical world. Networks of tiny sensors can be used to detect a fire in a forest, to monitor the level of pollution in a river, or to check on the structural integrity of a bridge. Application-specific deployments of static-sensor networks have been widely investigated. Commonly, these networks involve a centralized data-collection point and no sharing of data outside the organization that owns it. Although this approach can accommodate many application scenarios, it significantly deviates from the pervasive computing vision of ubiquitous sensing where user applications seamlessly access anytime, anywhere data produced by sensors embedded in the surroundings. With the ubiquity and ever-increasing capabilities of mobile devices, urban environments can help give substance to the ubiquitous sensing vision through Urbanets, spontaneously created urban networks. Urbanets consist of mobile multi-sensor devices, such as smart phones and vehicular systems, public sensor networks deployed by municipalities, and individual sensors incorporated in buildings, roads, or daily artifacts. My thesis is that "multi-sensor mobile devices can be successfully programmed to become the underpinning elements of an open, infrastructure-less, distributed sensing platform that can bring sensor data out of their traditional close-loop networks into everyday urban applications". Urbanets can support a variety of services ranging from emergency and surveillance to tourist guidance and entertainment. For instance, cars can be used to provide traffic information services to alert drivers to upcoming traffic jams, and phones to provide shopping recommender services to inform users of special offers at the mall. Urbanets cannot be programmed using traditional distributed computing models, which assume underlying networks with functionally homogeneous nodes, stable configurations, and known delays. Conversely, Urbanets have functionally heterogeneous nodes, volatile configurations, and unknown delays. Instead, solutions developed for sensor networks and mobile ad hoc networks can be leveraged to provide novel architectures that address Urbanet-specific requirements, while providing useful abstractions that hide the network complexity from the programmer. This dissertation presents two middleware architectures that can support mobile sensing applications in Urbanets. Contory offers a declarative programming model that views Urbanets as a distributed sensor database and exposes an SQL-like interface to developers. Context-aware Migratory Services provides a client-server paradigm, where services are capable of migrating to different nodes in the network in order to maintain a continuous and semantically correct interaction with clients. Compared to previous approaches to supporting mobile sensing urban applications, our architectures are entirely distributed and do not assume constant availability of Internet connectivity. In addition, they allow on-demand collection of sensor data with the accuracy and at the frequency required by every application. These architectures have been implemented in Java and tested on smart phones. They have proved successful in supporting several prototype applications and experimental results obtained in ad hoc networks of phones have demonstrated their feasibility with reasonable performance in terms of latency, memory, and energy consumption.

Problems and Algorithms for Sequence Segmentations

The analysis of sequential data is required in many diverse areas such as telecommunications, stock market analysis, and bioinformatics. A basic problem related to the analysis of sequential data is the sequence segmentation problem. A sequence segmentation is a partition of the sequence into a number of non-overlapping segments that cover all data points, such that each segment is as homogeneous as possible. This problem can be solved optimally using a standard dynamic programming algorithm. In the first part of the thesis, we present a new approximation algorithm for the sequence segmentation problem. This algorithm has smaller running time than the optimal dynamic programming algorithm, while it has bounded approximation ratio. The basic idea is to divide the input sequence into subsequences, solve the problem optimally in each subsequence, and then appropriately combine the solutions to the subproblems into one final solution. In the second part of the thesis, we study alternative segmentation models that are devised to better fit the data. More specifically, we focus on clustered segmentations and segmentations with rearrangements. While in the standard segmentation of a multidimensional sequence all dimensions share the same segment boundaries, in a clustered segmentation the multidimensional sequence is segmented in such a way that dimensions are allowed to form clusters. Each cluster of dimensions is then segmented separately. We formally define the problem of clustered segmentations and we experimentally show that segmenting sequences using this segmentation model, leads to solutions with smaller error for the same model cost. Segmentation with rearrangements is a novel variation to the segmentation problem: in addition to partitioning the sequence we also seek to apply a limited amount of reordering, so that the overall representation error is minimized. We formulate the problem of segmentation with rearrangements and we show that it is an NP-hard problem to solve or even to approximate. We devise effective algorithms for the proposed problem, combining ideas from dynamic programming and outlier detection algorithms in sequences. In the final part of the thesis, we discuss the problem of aggregating results of segmentation algorithms on the same set of data points. In this case, we are interested in producing a partitioning of the data that agrees as much as possible with the input partitions. We show that this problem can be solved optimally in polynomial time using dynamic programming. Furthermore, we show that not all data points are candidates for segment boundaries in the optimal solution.

Mining Sequential Data – in Search of Segmental Structures

Segmentation is a data mining technique yielding simplified representations of sequences of ordered points. A sequence is divided into some number of homogeneous blocks, and all points within a segment are described by a single value. The focus in this thesis is on piecewise-constant segments, where the most likely description for each segment and the most likely segmentation into some number of blocks can be computed efficiently. Representing sequences as segmentations is useful in, e.g., storage and indexing tasks in sequence databases, and segmentation can be used as a tool in learning about the structure of a given sequence. The discussion in this thesis begins with basic questions related to segmentation analysis, such as choosing the number of segments, and evaluating the obtained segmentations. Standard model selection techniques are shown to perform well for the sequence segmentation task. Segmentation evaluation is proposed with respect to a known segmentation structure. Applying segmentation on certain features of a sequence is shown to yield segmentations that are significantly close to the known underlying structure. Two extensions to the basic segmentation framework are introduced: unimodal segmentation and basis segmentation. The former is concerned with segmentations where the segment descriptions first increase and then decrease, and the latter with the interplay between different dimensions and segments in the sequence. These problems are formally defined and algorithms for solving them are provided and analyzed. Practical applications for segmentation techniques include time series and data stream analysis, text analysis, and biological sequence analysis. In this thesis segmentation applications are demonstrated in analyzing genomic sequences.

Finite element analysis of curved anisotropic laminated hollow bars

Curved hollow bars of laminated anisotropic construction are used as structural members in many industries. They are used in order to save weight without loss of stiffness in comparison with solid sections. In this paper are presented the details of the development of the stiffness matrices of laminated anisotropic curved hollow bars under line member assumptions for two typical sections, circular and square. They are 16dof elements which make use of one-dimensional first-order Hermite interpolation polynomials for the description of assumed displacement state. Problems for which analytical or other solutions are available are first solved using these elements. Good agreement was found between the results. In order to show the capability of the element, application is made to carbon fibre reinforced plastic layered anisotropic curved hollow bars.

Nature of the thiamin-binding protein from chicken egg yolk

A simple, rapid and efficient procedure for the purification of thiamin-binding protein from chicken egg yolk was developed. The method involved removal, by exclusion, of lipoproteins from DEAE-cellulose and subsequent elution of water-soluble proteins held on the ion-exchanger with 1 M-NaCl, followed by treatment of the eluted protein fraction with an aqueous suspension of dextran/charcoal to generate apoprotein from the holoprotein. The resultant protein fraction was subjected to bioaffinity chromatography on thiamin pyrophosphate--AE (aminoethyl)-Sepharose. The protein eluted specifically with 10 microM-thiamin at pH 7.0, was homogeneous by the criteria of polyacrylamide-gel disc electrophoresis, had a mol.wt. of 38 000 +/- 2000 and was not a glycoprotein. The purified thiamin-binding protein specifically interacted with riboflavin-binding protein with no detectable deleterious affect on its (14C)thiamin-binding capacity. The protein bound [14C]thiamin with a molar ratio of 1.0, with dissociation constant (Kd) 0.41 microM. This protein-ligand interaction was inhibited by thiamin analogues and antagonists. The absorption spectrum of the protein in the presence of thiamin exhibited significant hypochromism at the 278 nm band, indicating the involvement of aromatic amino acid residues of the protein, during its binding to the ligand. The protein cross-reacted with the monospecific antiserum to egg-white thiamin-binding protein, showing thereby that thiamin-binding proteins present in chicken egg yolk and white are the products of the same structural gene.

Arginine Decarboxylase from Lathyrus sativus Seedlings Purification and Properties

Arginine decarboxylase which makes its appearance in Lathyrus sativus seedlings after 24 h of seed germination reaches its highest level around 5–7 days, the cotyledons containing about 60% of the total activity in the seedlings at day 5. The cytosol enzyme was purified 977-fold from whole seedlings by steps involving manganese chloride treatment, ammonium sulphate and acetone fractionations, positive adsorption on alumina C-γ gel, DEAE-Sephadex chromatography followed by preparative disc gel electrophoresis. The enzyme was shown to be homogeneous by electrophoretic and immunological criteria, had a molecular weight of 220000 and appears to be a hexamer with identical subunits. The optimal pH and temperature for the enzyme activity were 8.5 and 45 °C respectively. The enzyme follows typical Michaelis-Menten kinetics with a Km value of 1.73 mM for arginine. Though Mn2+ at lower concentrations stimulated the enzyme activity, there was no dependence of the enzyme on any metal for the activity. The arginine decarboxylase of L. sativus is a sulfhydryl enzyme. The data on co-factor requirement, inhibition by carbonyl reagents, reducing agents and pyridoxal phosphate inhibitors, and a partial reversal by pyridoxal phosphate of inhibition by pyridoxal · HCl suggests that pyridoxal 5'-phosphate is involved as a co-factor for the enzyme. The enzyme activity was inhibited competitively by various amines including the product agmatine. Highest inhibition was obtained with spermine and arcain. The substrate analogue, l-canavanine, homologue l-homoarginine and other basic amino acids like l-lysine and l-ornithine inhibited the enzyme activity competitively, homoarginine being the most effective in this respect.

Finite element analysis of laminated anisotropic beams of bimodulus materials

This paper presents finite element analysis of laminated anisotropic beams of bimodulus materials. The finite element has 16 d.o.f. and uses the displacement field in terms of first order Hermite interpolation polynomials. As the neutral axis position may change from point to point along the length of the beam, an iterative procedure is employed to determine the location of zero strain points along the length. Using this element some problems of laminated beams of bimodulus materials are solved for concentrated loads/moments perpendicular and parallel to the layering planes as well as combined loads.

Purification and properties of a DNA polymerase from Mycobacterium tuberculosis H37Rv

DNA polymerase has been purified approximately 2000-fold from Mycobacterium tuberculosis H37Rv. The purified preparation was homogeneous by electrophoretic criteria and has a molecular weight of 135 000. The purified enzyme resembles Escherichia coli polymerase I in its properties, being insensitive to sulfhydryl drugs and possessing 5′,3′-exonuclease activity in addition to polymerase and 3′,5′-exonuclease activities. However, it differs from the latter in its sensitivity to higher salt concentration and DNA intercalating agents such as 8-aminoquinoline. The polymerase exhibited maximal activity between 37–42°C and pH 8.8–9.5. The polymerase was stable for several months below 0°C. However, the 5′,3′-exonuclease activity was more labile. The effects of different metal ions, polyamines and drugs on the polymerase activity are presented.

Purification and characterization of a thermostable glucoamylase from the thermophilic fungus Thermomyces lanuginosus.

Glucoamylase (1,4-alpha-D-glucan glucohydrolase, EC 3.2.1.3) was purified from the culture filtrates of the thermophilic fungus Thermomyces lanuginosus and was established to be homogeneous by a number of criteria. The enzyme was a glycoprotein with an average molecular weight of about 57 000 and a carbohydrate content of 10-12%. The enzyme hydrolysed successive glucose residues from the non-reducing ends of the starch molecule. It did not exhibit any glucosyltransferase activity. The enzyme appeared to hydrolyse maltotriose by the multi-chain mechanism. The enzyme was unable to hydrolyse 1,6-alpha-D-glucosidic linkages of isomaltose and dextran. It was optimally active at 70 degrees C. The enzyme exhibited increase in the Vmax. and decreased in Km values with increasing chain length of the substrate molecule. The enzyme was inhibited by the substrate analogue D-glucono-delta-lactone in a non-competitive manner. The enzyme inhibited remarkable resistance towards chemical and thermal denaturation.

The purification and properties of peroxidase in Mycobacterium tuberculosis H37Rv and its possible role in the mechanism of action of isonicotinic acid hydrazide

Peroxidase from Mycobacterium tuberculosis H37Rv was purified to homogeneity. The homogeneous protein exhibits catalase and Y (Youatt's)-enzyme activities in addition to peroxidase activity. Further confirmation that the three activities are due to a single enzyme was accomplished by other criteria, such as differential thermal inactivation, sensitivity to different inhibitors, and co-purification. The Y enzyme (peroxidase) was separated from NADase (NAD+ glycohydrolase) inhibitor by gel filtration on Sephadex G-200. The molecular weights of peroxidase and NADase inhibitor, as determined by gel filtration, are 240000 and 98000 respectively. The Y enzyme shows two Km values for both isoniazid (isonicotinic acid hydrazide) and NAD at low and high concentrations. Analysis of the data by Hill plots revealed that the enzyme has one binding site at lower substrate concentrations and more than one at higher substrate concentration. The enzyme contains 6g-atoms of iron/mol. Highly purified preparations of peroxidases from different sources catalyse the Y-enzyme reaction, suggesting that the nature of the reaction may be a peroxidatic oxidation of isoniazid. Moreover, the Y-enzyme reaction is enhanced by O2. Isoniazid-resistant mutants do not exhibit Y-enzyme, peroxidase or catalase activities, and do not take up isoniazid. The Y-enzyme reaction is therefore implicated in the uptake of the drug.

Study of third-order non-linear systems- Variation of parameters approach

In this paper, the transient response of a third-order non-linear system is obtained by first reducing the given third-order equation to three first-order equations by applying the method of variation of parameters. On the assumption that the variations of amplitude and phase are small, the functions are expanded in ultraspherical polynomials. The expansion is restricted to the constant term. The resulting equations are solved to obtain the response of the given third-order system. A numerical example is considered to illustrate the method. The results show that the agreement between the approximate and digital solution is good thus vindicating the approximation.

Studies on nucleotidases in plants Dimerization of the crystalline mung bean nucleotide pyrophosphatase by 5-adenosine monophosphate and the properties of the dimerized enzyme

The addition of AMP to the crystalline and homogeneous mung bean nucleotide pyrophosphatase [EC 3.6.1.9]altered its electrophoretic mobility. AMP was tightly bound to the enzyme and was not removed on passage through a column of Sephadex G-25 or on electrophoresis. The molecular weight of the native and AMP-modified enzymes were 65,000 and 136,000, respectively. The properties of the native enzyme such as the pH (9.4) and temperature (49 °C) optima, inhibition by EDTA, reversal of EDTA-inhibition by Zn2+ and Co2+, were not altered on dimerization by AMP. The AMP-modified enzyme had a linear time-course of reaction, unlike the native enzyme which exhibited a biphasic time-course of reaction. The AMP-modified enzyme was irreversibly denatured by urea. AMP concentrations larger than 100 μM inhibited linearly the activity of the AMP-modified enzyme. ADP and ATP inhibited the activity in a sigmoidal manner. Km and V of the native and AMP-modified enzymes were, 0.25 mImage and 0.58 mImage ; and 3.3 and 2.5, respectively.