On Random Planar Curves and Their Scaling Limits

Planar curves arise naturally as interfaces between two regions of the plane. An important part of statistical physics is the study of lattice models. This thesis is about the interfaces of 2D lattice models. The scaling limit is an infinite system limit which is taken by letting the lattice mesh decrease to zero. At criticality, the scaling limit of an interface is one of the SLE curves (Schramm-Loewner evolution), introduced by Oded Schramm. This family of random curves is parametrized by a real variable, which determines the universality class of the model. The first and the second paper of this thesis study properties of SLEs. They contain two different methods to study the whole SLE curve, which is, in fact, the most interesting object from the statistical physics point of view. These methods are applied to study two symmetries of SLE: reversibility and duality. The first paper uses an algebraic method and a representation of the Virasoro algebra to find common martingales to different processes, and that way, to confirm the symmetries for polynomial expected values of natural SLE data. In the second paper, a recursion is obtained for the same kind of expected values. The recursion is based on stationarity of the law of the whole SLE curve under a SLE induced flow. The third paper deals with one of the most central questions of the field and provides a framework of estimates for describing 2D scaling limits by SLE curves. In particular, it is shown that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical physics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces.

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.

Application of ultraspherical polynomials to forced oscillations of a third-order non-linear system

In this paper the method of ultraspherical polynomial approximation is applied to study the steady-state response in forced oscillations of a third-order non-linear system. The non-linear function is expanded in ultraspherical polynomials and the expansion is restricted to the linear term. The equation for the response curve is obtained by using the linearized equation and the results are presented graphically. The agreement between the approximate solution and the analog computer solution is satisfactory. The problem of stability is not dealt with in this paper.

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.

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.

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.

Optimisation Problems in Wireless Sensor Networks : Local Algorithms and Local Graphs

This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs.

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.

A new type of high-order elements based on the mesh-free interpolations

We propose a new type of high-order elements that incorporates the mesh-free Galerkin formulations into the framework of finite element method. Traditional polynomial interpolation is replaced by mesh-free interpolations in the present high-order elements, and the strain smoothing technique is used for integration of the governing equations based on smoothing cells. The properties of high-order elements, which are influenced by the basis function of mesh-free interpolations and boundary nodes, are discussed through numerical examples. It can be found that the basis function has significant influence on the computational accuracy and upper-lower bounds of energy norm, when the strain smoothing technique retains the softening phenomenon. This new type of high-order elements shows good performance when quadratic basis functions are used in the mesh-free interpolations and present elements prove advantageous in adaptive mesh and nodes refinement schemes. Furthermore, it shows less sensitive to the quality of element because it uses the mesh-free interpolations and obeys the Weakened Weak (W2) formulation as introduced in [3, 5].

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.

Transient response of coupled non-linear non-conservative systems

This paper deals with an approximate method of analysis of non-linear, non-conservative systems of two degrees of freedom. The approximate equations for amplitude and phase are obtained by a generalized averaging technique based on the ultraspherical polynomial approximation. The method is illustrated by an example of a spring-mass-damper system.

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.