Recursos de colección

ETD at Indian Institute of Science (2.439 recursos)

Repository of Theses and Dissertations of Indian Institute of Science, Bangalore, India. The repository has been developed to capture, disseminate and preserve research theses of Indian Institute of Science.

Mathematics (math)

Mostrando recursos 1 - 20 de 49

  1. Grothendieck Inequality

    Ray, Samya Kumar
    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...

  2. Analyzing Credit Risk Models In A Regime Switching Market

    Banerjee, Tamal
    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)...

  3. Riesz Transforms Associated With Heisenberg Groups And Grushin Operators

    Sanjay, P K
    We characterise the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions. Next we study the Riesz transforms associated to the Grushin operator G = - Δ - |x|2@t2 on Rn+1. We prove that both the first order and higher order Riesz...

  4. Rigidity And Regularity Of Holomorphic Mappings

    Balakumar, G P
    We deal with two themes that are illustrative of the rigidity and regularity of holomorphic mappings. The first one concerns the regularity of continuous CR mappings between smooth pseudo convex, finite type hypersurfaces which is a well studied subject for it is linked with the problem of studying the boundary behaviour of proper holomorphic mappings between domains bounded by such hypersurfaces. More specifically, we study the regularity of Lipschitz CR mappings from an h-extendible(or semi-regular) hypersurface in Cn .Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from...

  5. Ricci Flow And Isotropic Curvature

    Gururaja, H A
    This thesis consists of two parts. In the first part, we study certain Ricci flow invariant nonnegative curvature conditions as given by B. Wilking. We begin by proving that any such nonnegative curvature implies nonnegative isotropic curvature in the Riemannian case and nonnegative orthogonal bisectional curvature in the K¨ahler case. For any closed AdSO(n,C) invariant subset S so(n, C) we consider the notion of positive curvature on S, which we call positive S- curvature. We show that the class of all such subsets can be naturally divided into two subclasses: The first subclass consists of those sets S for which...

  6. Fourier Analysis On Number Fields And The Global Zeta Functions

    Fernandes, Jonathan
    The study of zeta functions is one of the primary aspects of modern number theory. Hecke was the first to prove that the Dedekind zeta function of any algebraic number field has an analytic continuation over the whole plane and satisfies a simple functional equation. He soon realized that his method would work, not only for Dedekind zeta functions and L–series, but also for a zeta function formed with a new type of ideal character which, for principal ideals depends not only on the residue class of the number(representing the principal ideal) modulo the conductor, but also on the position...

  7. Infinitely Divisible Metrics, Curvature Inequalities And Curvature Formulae

    Keshari, Dinesh Kumar
    The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples of operators in the Cowen-Douglas class. Secondly, we obtain...

  8. Analytic Continuation In Several Complex Variables

    Biswas, Chandan
    We wish to study those domains in Cn,for n ≥ 2, the so-called domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We demonstrate that this study is radically different from that of domains in C by discussing some examples of special types of domains in Cn , n ≥2, such that every function holomorphic on them extends to strictly larger domains. Given a domain in Cn , n ≥ 2, we wish to construct the maximal domain of existence for the holomorphic functions defined on the given domain....

  9. Function Theory On Non-Compact Riemann Surfaces

    Philip, Eliza
    The theory of Riemann surfaces is quite old, consequently it is well developed. Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions. Such multi-valued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. The theory splits in two parts; the compact and the non-compact case. The function theory developed on these cases are quite dissimilar. The main difficulty one encounters in the compact case is the scarcity of global holomorphic functions, which limits one’s study to meromorphic...

  10. Matchings Between Point Processes

    Jana, Indrajit

  11. Vector Bundles Over Hypersurfaces Of Projective Varieties

    Tripathi, Amit
    In this thesis we study some questions related to vector bundles over hypersurfaces. More precisely, for hypersurfaces of dimension ≥ 2, we study the extension problem of vector bundles. We find some cohomological conditions under which a vector bundle over an ample divisor of non-singular projective variety, extends as a vector bundle to an open set containing that ample divisor. Our method is to follow the general Groethendieck-Lefschetz theory by showing that a vector bundle extension exists over various thickenings of the ample divisor. For vector bundles of rank > 1, we find two separate cohomological conditions on vector bundles...

  12. The Role Of Potential Theory In Complex Dynamics

    Bandyopadhyay, Choiti
    Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. In this text, our main goal will be to gain a deeper understanding towards complex dynamics, the study of dynamical systems defined by the iteration of analytic functions, using the tools and techniques of potential theory. We will restrict ourselves to holomorphic polynomials in C. At first, we will discuss briefly about harmonic and subharmonic functions. In course, potential theory will repay its debt to complex analysis in the form of some...

  13. On The Role Of The Bargmann Transform In Uncertainty Principles

    Garg, Rahul

  14. Joint Eigenfunctions On The Heisenberg Group And Support Theorems On Rn

    Samanta, Amit
    This work is concerned with two different problems in harmonic analysis, one on the Heisenberg group and other on Rn, as described in the following two paragraphs respectively. Let Hn be the (2n + 1)-dimensional Heisenberg group, and let K be a compact subgroup of U(n), such that (K, Hn) is a Gelfand pair. Also assume that the K-action on Cn is polar. We prove a Hecke-Bochner identity associated to the Gelfand pair (K, Hn). For the special case K = U(n), this was proved by Geller, giving a formula for the Weyl transform of a function f of the...

  15. Relative Symplectic Caps, Fibered Knots And 4-Genus

    Kulkarni, Dheeraj
    The 4-genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4-genus and related invariants of homology classes is the Thom conjecture, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsv´ath-Szab´o, which say that closed symplectic surfaces minimize genus. In this thesis, we prove a relative version of the symplectic capping theorem. More precisely, suppose (X, ω) is a symplectic 4-manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. We show...

  16. Curvature Calculations Of The Operators In Cowen-Douglas Class

    Deb, Prahllad
    In a foundational paper “Operators Possesing an Open Set of Eigenvalues” written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space ‘H possessing an open set Ω C of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and vice-versa. Fortunately,...

  17. Segal-Bargmann Transform And Paley Wiener Theorems On Motion Groups

    Sen, Suparna

  18. A Study Of The Metric Induced By The Robin Function

    Borah, Diganta
    Let D be a smoothly bounded domain in Cn , n> 1. For each point p _ D, we have the Green function G(z, p) associated to the standard sum-of-squares Laplacian Δ with pole at p and the Robin constant __ Λ(p) = lim G(z, p) −|z − p−2n+2 z→p | at p. The function p _→ Λ(p) is called the Robin function for D. Levenberg and Yamaguchi had proved that if D is a C∞-smoothly bounded pseudoconvex domain, then the function log(−Λ) is a real analytic, strictly plurisubharmonic exhaustion function for D and thus induces a metric ds2 =...

  19. Some Mixed Boundary Value Problems Arising In Viscous Flow Theory

    Manna, Durga Pada

  20. Dilations, Functoinal Model And A Complete Unitary Invariant Of A r-contraction.

    Pal, Sourav
    A pair of commuting bounded operators (S, P) for which the set r = {(z 1 +z 2,z 1z 2) : |z 1| ≤1, |z 2| ≤1} C2 is a spectral set, is called a r-contraction in the literature. For a contraction P and a bounded commutant S of P, we seek a solution of the operator equation S –S*P = (I –P*P)½ X(I –P*P)½ where X is a bounded operator on Ran(I – P*P)½ with numerical radius of X being not greater than 1. We show the existence and uniqueness of solution to the operator equation above when (S,P) is a...

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