etd@IISc Collection:
http://hdl.handle.net/2005/41
Fri, 13 Oct 2017 21:36:42 GMT2017-10-13T21:36:42ZThe Channel Imagehttp://etd.ncsi.iisc.ernet.in:80/retrieve/46/mathematics.jpg
http://hdl.handle.net/2005/41
On The Structure of Proper Holomorphic Mappings
http://hdl.handle.net/2005/2695
Title: On The Structure of Proper Holomorphic Mappings
Authors: Jaikrishnan, J
Abstract: The aim of this dissertation is to give explicit descriptions of the set of proper holomorphic mappings between two complex manifolds with reasonable restrictions on the domain and target spaces. Without any restrictions, this problem is intractable even when posed for do-mains in . We give partial results for special classes of manifolds. We study, broadly, two types of structure results:
Descriptive. The first result of this thesis is a structure theorem for finite proper holomorphic mappings between products of connected, hyperbolic open subsets of compact Riemann surfaces. A special case of our result follows from the techniques used in a classical result due to Remmert and Stein, adapted to the above setting. However, the presence of factors that have no boundary or boundaries that consist of a discrete set of points necessitates the use of techniques that are quite divergent from those used by Remmert and Stein. We make use of a finiteness theorem of Imayoshi to deal with these factors.
Rigidity. A famous theorem of H. Alexander proves the non-existence of non-injective proper holomorphic self-maps of the unit ball in . ,n >1. Several extensions of this result for various classes of domains have been established since the appearance of Alexander’s result, and it is conjectured that the result is true for all bounded domains in . , n > 1, whose boundary is C2-smooth. This conjecture is still very far from being settled. Our first rigidity result establishes the non-existence of non-injective proper holomorphic self-maps of bounded, balanced pseudo convex domains of finite type (in the sense of D’Angelo) in ,n >1. This generalizes a result in 2, by Coupet, Pan and Sukhov, to higher dimensions. As in Coupet–Pan–Sukhov, the aforementioned domains need not have real-analytic boundaries. However, in higher dimensions, several aspects of their argument do not work. Instead, we exploit the circular symmetry and a recent result in complex dynamics by Opshtein.
Our next rigidity result is for bounded symmetric domains. We prove that a proper holomorphic map between two non-planar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. Furthermore, our proof of this result does not rely on the fine structure (in the sense of Wolf et al.) of bounded symmetric domains. Thus, we are able to apply our techniques to more general classes of domains. We illustrate this by proving a rigidity result for certain convex balanced domains whose automorphism groups are assumed to only be non-compact. For bounded symmetric domains, our key tool is that of Jordan triple systems, which is used to describe the boundary geometry.Wed, 27 Sep 2017 18:30:00 GMThttp://hdl.handle.net/2005/26952017-09-27T18:30:00ZDevelopment Of NMR Methods For Metabolomics And Protein Resonance Assignments
http://hdl.handle.net/2005/2633
Title: Development Of NMR Methods For Metabolomics And Protein Resonance Assignments
Authors: Dubey, Abhinav
Abstract: Nuclear Magnetic Resonance (NMR) spectroscopy is a quantitative, non-invasive and non-destructive technique useful in biological studies. By manipulating the magnetization of nuclei with non-zero spin, NMR gives insights into atomic level details. Application of NMR as a tool for discovering structure, understanding dynamics of bio-molecules such as proteins, metabolites, DNA, RNA and their interactions constitutes the field of bio-molecular NMR. In this thesis, new methods for rapid data analysis of NMR spectrum of proteins and metabolites are proposed.
The first computational method, PROMEB (Pattern Recognition Based Assignment in Metabolomics) is useful for the identification and assignments of metabolites. This is an important step in metabolomics and is necessary for the discovery of new biomarkers. In NMR spectroscopy based studies, the conventional approach involves a database search, wherein chemical shifts are assigned to specific metabolites by use of a tolerance limit. This is inefficient because deviation in chemical shifts associated with pH or temperature variations, as well as missing peaks, impairs a robust comparison with the database. These drawbacks are overcome in PROMEB, which is a method based on matching the pattern of peaks of a metabolite in 2D [13C, 1H] HSQC NMR spectrum, rather than conventionally used absolute tolerance thresholds. A high success rate is obtained even in the presence of large chemical shift deviations such as 0.5 ppm in 1H and 3 ppm in 13C and missing peaks (up to 50%), compared to nearly no assignments obtained under these conditions with existing methods that employ a direct database search approach. The pattern recognition approach thus helps in identification and assignment of metabolites in-dependent of the pH, temperature, and ionic strength used, thereby obviating the need for spectral calibration with internal or external standards.
Another computational method, ChemSMP(Chemical Shifts to Metabolic Path-ways), is described which facilitates the identification of metabolic pathways from a single two dimensional (2D) NMR spectrum. Typically in other approaches, this is done after relevant metabolites are identified to allow their mapping onto specific metabolic pathways. This task is daunting due to the complex nature of cellular processes and the difficulty in establishing the identity of individual metabolites. ChemSMP uses a novel indexing and scoring system comprised of a uniqueness
score and a coverage score. Benchmarks show that ChemSMP has a positive prediction rate of > 90% in the presence of decluttered data and can sustain the same at 60 − 70% even in the presence of noise, such as deletions of peaks and chemical shift deviations. The method tested on NMR data acquired for a mixture of 20 amino acids shows a success rate of 93% in correct recovery of metabolic pathways.
The third method developed is a new approach for rapid resonance assignments in proteins based on amino acid selective unlabeling. The method involves choosing a set of multiple amino acid types for selective unlabeling and identifying specific tripeptides surrounding the labeled residues from specific 2D NMR spectra in a combinatorial manner. The methodology directly yields sequence specific resonance assignments, without requiring a contiguously assigned stretch of amino acid residues to be linked, and is applicable to deuterated proteins.
The fourth method involves a simple approach to rapidly identify amino acid types in proteins from a 2D NMR spectrum. The method is based on the fact that 13Cβ chemical shifts of different amino acid types fall in distinct spectral regions. By evolving the 13C chemical shifts in the conventional HNCACB or HN(CO)CACB type experiment for a single specified delay period, the phase of the cross peaks of different amino acid residues are modulated depending on their 13Cβ chemical shift values. Following this specified evolution period, the 2D HN projections of these experiments are acquired. The 13C evolution period can be chosen such that all residues belonging to a given set of amino acid types have the same phase pattern (positive or negative) facilitating their identification. This approach does not re-quire the preparation of any additional samples, involves the analysis of 2D [15N,1H] HSQC-type spectra obtained from the routinely used triple resonance experiments with minor modifications, and is applicable to deuterated proteins.
Finally, the practical application of these methods for laboratory research is presented. PROMEB and ChemSMP is used to study cancer cell metabolism in previously unexplored oncogenic cell line. PROMEB helped in assigning a differential metabolite present at high concentration in cancer cell line compared to control non-cancerous cell line. ChemSMP revealed active metabolic pathways responsible for regulating energy homeostasis of cancer cells which were previously reported in literature.
The two methods developed for rapid protein resonance assignments can be used in applications such as identifying active-site residues involved in ligand binding, phosphorylation, or protein-protein interactions. The phase modulated experiments will be useful for quick assignment of signals that shift during ligand binding or in combination with selective labeling/unlabeling approaches for identification of amino acid types to aid the sequential assignment process. Both the methodology was applied to two proteins: Ubiquitin (8 kDa) and L-IGFBP2 an intrinsically disordered protein (12 kDa), for demonstrating rapid resonance assignment using only set of 2D NMR experiments.Wed, 24 May 2017 18:30:00 GMThttp://hdl.handle.net/2005/26332017-05-24T18:30:00ZGrothendieck Inequality
http://hdl.handle.net/2005/2540
Title: Grothendieck Inequality
Authors: Ray, Samya Kumar
Abstract: Grothendieck published an extraordinary paper entitled ”Resume de la theorie metrique des pro¬duits tensoriels topologiques” in 1953. The main result of this paper is the inequality which is commonly known as Grothendieck Inequality.
Following Kirivine, in this article, we give the proof of Grothendieck Inequality. We refor¬mulate it in different forms. We also investigate the famous Grothendieck constant KG. The Grothendieck constant was achieved by taking supremum over a special class of matrices. But our attempt will be to investigate it, considering a smaller class of matrices, namely only the positive definite matrices in this class. Actually we want to use it to get a counterexample of Matsaev’s conjecture, which was proved to be right by Von Neumann in some specific cases.
In chapter 1, we shall state and prove the Grothendieck Inequality. In chapter 2, we shall introduce tensor product of vector spaces and different tensor norms. In chapter 3, we shall formulate Grothendieck Inequality in different forms and use the notion of tensor norms for its equivalent formation .In the last chapteri.ein chapter4we shall investigate on the Grothendieck constant.Sun, 19 Jun 2016 18:30:00 GMThttp://hdl.handle.net/2005/25402016-06-19T18:30:00ZAnalyzing Credit Risk Models In A Regime Switching Market
http://hdl.handle.net/2005/2517
Title: Analyzing Credit Risk Models In A Regime Switching Market
Authors: Banerjee, Tamal
Abstract: Recently, the financial world witnessed a series of major defaults by several institutions and investment banks. Therefore, it is not at all surprising that credit risk analysis have turned out to be one of the most important aspect among the finance community. As credit derivatives are long term instruments, it is affected by the changes in the market conditions. Thus, it is a appropriate to take into consideration the effects of the market economy. This thesis addresses some of the important issues in credit risk analysis in a regime switching market. The main contribution in this thesis are the followings:
(1) We determine the price of default able bonds in a regime switching market for structural models with European type payoff. We use the method of quadratic hedging and minimal martingale measure to determine the defaultble bond prices. We also obtain hedging strategies and the corresponding residual risks in these models. The defaultable bond prices are obtained as solution to a system of PDEs (partial differential equations) with appropriate terminal and boundary conditions. We show the existence and uniqueness of the system of PDEs on an appropriate domain.
(2) We carry out a similar analysis in a regime switching market for the reduced form models. We extend some of the existing models in the literature for correlated default timings. We price single-name and multi-name credit derivatives using our regime switching models. The prices are obtained as solution to a system of ODEs(ordinary differential equations) with appropriate terminal conditions.
(3) The price of the credit derivatives in our regime switching models are obtained as solutions to a system of ODEs/PDEs subject to appropriate terminal and boundary conditions. We solve these ODEs/PDEs numerically and compare the relative behavior of the credit derivative prices with and without regime switching. We observe higher spread in our regime switching models. This resolves the low spread discrepancy that were prevalent in the classical structural models. We show further applications of our model by capturing important phenomena that arises frequently in the financial market. For instance, we model the business cycle, tight liquidity situations and the effects of firm restructuring. We indicate how our models may be extended to price various other credit derivatives.Sun, 24 Apr 2016 18:30:00 GMThttp://hdl.handle.net/2005/25172016-04-24T18:30:00Z