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Scaling Limits Of Some Random Interface Models., Biltu Dan Dr.
Scaling Limits Of Some Random Interface Models., Biltu Dan Dr.
Doctoral Theses
In this thesis, we study some probabilistic models of random interfaces. Interfaces between different phases have been topic of considerable interest in statistical physics. These interfaces are described by a family of random variables, indexed by the ddimensional integer lattice, which are considered as a height configuration, namely they indicate the height of the interface above a reference hyperplane. The models are defined in terms of an energy function (Hamiltonian), which defines a Gibbs measure on the set of height configurations. More formally, letÏ• = {Ï•x}x∈Z dbe a collection of real numbers indexed by the d-dimensional integer lattice Z d. …
On Tests Of Independence Among Multiplerandom Vectors Of Arbitrary Dimensions., Angshuman Roy Dr.
On Tests Of Independence Among Multiplerandom Vectors Of Arbitrary Dimensions., Angshuman Roy Dr.
Doctoral Theses
Measures of dependence among several random vectors and associated tests of independence play a major role in different statistical applications. Blind source separation or independent component analysis (see, e.g., Hyv¨arinen et al., 2001; Shen et al., 2009), feature selection and feature extraction (see, e.g., Li et al., 2012), detection of serial correlation in time series (see, e.g., Ghoudi et al., 2001) and finding the causal relationships among the variables (see, e.g., Chakraborty and Zhang, 2019) are some examples of their wide-spread applications. Tests of independence has vast applications in other areas of sciences as well. For instance, to characterize the …
Characterization Of Eigenfunctions Of The Laplace-Beltrami Operator Through Radial Averages On Rank One Symmetric Spaces., Muna Naik Dr.
Characterization Of Eigenfunctions Of The Laplace-Beltrami Operator Through Radial Averages On Rank One Symmetric Spaces., Muna Naik Dr.
Doctoral Theses
Let X be a rank one Riemannian symmetric space of noncompact type and ∆ be the Laplace–Beltrami operator of X. The space X can be identified with the quotient space G/K where G is a connected noncompact semisimple Lie group of real rank one with finite centre and K is a maximal compact subgroup of G. Thus G acts naturally on X by left translations. Through this identification, a function or measure on X is radial (i.e. depends only on the distance from eK), when it is invariant under the left-action of K. We consider right-convolution operators Θ on functions …
A Study Of Operators On The Discrete Analogue Of Hardy Spaces On Homogeneous Trees And On Other Structures., P. Muthukumar Dr.
A Study Of Operators On The Discrete Analogue Of Hardy Spaces On Homogeneous Trees And On Other Structures., P. Muthukumar Dr.
Doctoral Theses
In analytic function theory, the study of multiplication and composition operators has a rich structure for various analytic function spaces of the unit disk D = {z ∈ C : |z| < 1} such as the Hardy spaces Hp, the Bergman spaces Ap and the Bloch space B. This theory connects the operator theoretic properties such as boundedness, compactness, spectrum, invertibility, isometry with that of the function theoretic properties of the inducing map (symbol) such as bijectivity, boundary behaviour and vise versa In Chapter 2, we define discrete analogue of generalized Hardy spaces (Tp) and their separable subspaces (Tp,0) on a homogenous rooted tree and study some of their properties such as completeness, inclusion relations with other spaces, separability and growth estimate for functions in these spaces and their consequences. In Chapter 3, we obtain equivalent conditions for multiplication operators Mψ on Tp and Tp,0 to be bounded and compact. Furthermore, we discuss point spectrum, approximate point spectrum and spectrum of multiplication operators and discuss when a multiplication operator is an isometry. In Chapter 4, we give an equivalent conditions for the composition operator Cφ to be bounded on Tp and on Tp,0 spaces and compute their operator norms. We have considered the composition operators induced by special symbols such as univalent and multivalent maps and automorphism of a homogenous tree. We also characterize invertible composition operators and isometric composition operators on Tp and on Tp,0 spaces. Also, we discuss the compactness of Cφ on Tp spaces and finally we prove that there are no compact composition operators on Tp,0 spaces. In Chapter 5, we consider the composition operators on the Hardy-Dirichlet space H2, the space of Dirichlet series with square summable coefficients. By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on H2 , for the affine-like inducing symbol ϕ(s) = c1 + cqq −s , where q ≥ 2 is a fixed integer. We also give an estimate for approximation numbers of a composition operators in our H2 setting. In Chapter 6, we study the weighted composition operators preserving the class Pα. Some of its consequences and examples of certain special cases are presented. Furthermore, we discuss about the fixed points of weighted composition operators.
Ball Remotality In Banach Spaces And Related Topics, Tanmoy Paul Dr.
Ball Remotality In Banach Spaces And Related Topics, Tanmoy Paul Dr.
Doctoral Theses
In this work we aim to study Ball Remotality and densely Ball Remotality of subspaces in Banach spaces. We study this property in many classical spaces of type c0, c,\\â„“p and C(K) where K is a compact Hausdorff space. The said problem also discussed for Banach spaces when considered as a subspace in its bidual. It is observed M-ideals in C(K) are densely ball remotal. It is shown that a particular type of M-ideal in A(K) where K is a Choquet simplex is densely ball remotal.
Certain Pattern Recognition Tasks Using Genetic Programming., Durga Muni Dr.
Certain Pattern Recognition Tasks Using Genetic Programming., Durga Muni Dr.
Doctoral Theses
No abstract provided.
Spectral Analysis And Synthesis For Radial Sections Of Homogenous Vector Bundles On Certain Noncompath Riemannian Symmetric Spaces., Sanjoy Pusti Dr.
Spectral Analysis And Synthesis For Radial Sections Of Homogenous Vector Bundles On Certain Noncompath Riemannian Symmetric Spaces., Sanjoy Pusti Dr.
Doctoral Theses
We consider two classical theorems of real analysis which deals with translation invariant subspaces of integrable and smooth functions on R respectively. The first one is a theorem of Norbert Wiener [63] which states that if the Fourier transform of a function f ∈ L 1 (R) has no real zeros then the finite linear combinations of translations f(x − a) of f with complex coefficients form a dense subspace in L 1 (R), equivalently, span{g ∗ f | g ∈ L 1 (R)} is dense in L 1 (R). This theorem is well known as the Wiener-Tauberian Theorem (WTT). …
Intersection Properties Of Balls In Banach Spaces And Related Topics., Sudipta Dutta Dr.
Intersection Properties Of Balls In Banach Spaces And Related Topics., Sudipta Dutta Dr.
Doctoral Theses
In the first part of this chapter, we explain in general terms the background and the main theme of this thesis and provide a chapter-wise summary of its principal results. In the second part, we introduce some notations and preliminaries that will be used in the subsequent chapters.As a prototype of the properties we will study in this thesis, let us call a closed linear subspace Y of a Banach space X a (P)-subspace of X if Y has a certain property P as a subspace of X. If a Banach space X, in its canonical embedding, is a (P)-subspace …
Static Interconnection Networks And Parallel Algorithms For Efficient Problem Solving., T. Krishnan Dr.
Static Interconnection Networks And Parallel Algorithms For Efficient Problem Solving., T. Krishnan Dr.
Doctoral Theses
There is nothing more difficult to take in hand, more perilous to conduct, or more uncertain in its success, than to take the lead in the introduction of a new order of things.
Fourier Transforms Of Very Rapidly Decreasing Functions On Certain Lie Groups., M. Sundari Dr.
Fourier Transforms Of Very Rapidly Decreasing Functions On Certain Lie Groups., M. Sundari Dr.
Doctoral Theses
Recall that for a function f ϵ L1(Rn ), its Fourier transform fÌ‚ is definedby: fÌ‚ (ƹ) = ʃ Rnf(x)ei(ƹ,x)dx ( 0.1.1)where (.,.) denotes the standard inner product on Rn and dr the Lebesgue measure on Rn. A celebrated theorem of L. Schwartz asserts that a function f on Rn is rapidly decreasing (or in the Schwartz class ) if and only if its Fourier transform is rapidly decreasing . In sharp contrast to Schwartz s theorem, is a result due to Hardy ([18) which says that ʃ and fÌ‚ cannot both be very rapidly decreasing . More precisely, if …
On Some Problems Related To Hermite And Laguerre Expansions., P. K. Ratnakumar Dr.
On Some Problems Related To Hermite And Laguerre Expansions., P. K. Ratnakumar Dr.
Doctoral Theses
The first three chapters of this thesis are concerned with the spherical means associated to the Hermite and Laguerre expansions. The study of spherical means has a very long history. The classic work of F. John deals with various applications of the spherical means to the theory of partial differential equations. They entered Fourier analysis with the celebrated theorem of E. Stein on spherical analogue of the Lebesgue differentiation theorem. Ever since they have appcared again and again in several areas of analysis like integral geometry, inversion of Fourier transforms and related arcas.
Some Limit Theorem On Conditional U-Statistics And Censored Data Non Parametric Regression., Arusharka Sen Dr.
Some Limit Theorem On Conditional U-Statistics And Censored Data Non Parametric Regression., Arusharka Sen Dr.
Doctoral Theses
In Statistics, a classical problem is that of estimating the regression function which is defined as m{t) := E(Y|X = ), te R, for two random variables X and Y such that EY < 0o. The estimators are constructed iased on a sample {(Xi, Yi.)}, 1sis n,n 2 1, from the distribution of (X, Y). Throughout this thesis, we assume X and Y to be real-valued for the sake of convenience. The classical approach to this problem is to assume a parametrized, polynomial form for nt-), i.e., m(t) := Bo + E-1 P,ti, p 21, and obtain estimates of the unknown paraineters Bo, Bj,, 1sjsp. Later, with the development of techıniques for non-parametrie density estimation, it was sought to extend these techniques to regression estimation. Heuristically, the two problems can be seen to be related as follows : let fi(-) be the marginal density of X and note that E1(X S x) = h(t)dt, z € R, whereas EY 1(X Sx) = m(t)fi(t)dt, x E MR. (1.0.2) In other wordds, (1.0.1) can be looked upon as a special case of (1.0.2), with Y = 1. + similarity, as we shall see later on, has been the underlying theme in Chapters 2 and 4 of the present work.) The following non-parametric regression estimator was proposed independently by Nadaraya (1964) and Watson (1964): "(): := m.(Y,)/m.(1, ), te R, (1.0.3) where m,(Y, t) = (nan)- E-, Y;K((t - X:)/an). (1.0.4) m,(1,1) (na,)-E, K((I - X)/a,). Here K(), the so-called kernel function, is chosen to satiafy various analytical conditions (typically, K(-) is taken to be a density function), and a, 1 0 are the bandwidths which go to zero sufficiently slowly (e.g., na,0o as n00) in order to ensure consistency of the estimator mW (). The intuition behind such an estimator is that m,(Y,) is an estimator of mt-)fi() while m,(1,) cstimates the density fa(-). See Prakasa Rao (1983), Chapters 1-4. for an introduction to non-parametric density and regression estimation. Now, m(t) is a functional of the conditional distribution of Y, given X = t. A natu- ral generalisation of the regression estimation problem seems to be the estimation of the following functionals: mh(t1,....tk) := E{h(Y1,.....Yk) | X1, = t1.,Xk. = tk), (t....) € R*, k 2 1, (1.0.5) where h: R*- R is such that Elh(Y...., Y) < 0. A similar generalisation led Hoelfding (1948) from the sample mcan to the theory of so-called U-statistics, in the uncondilional set-up. The estimation of (1.0.5) were considerexl, for the first time in published form, in Stute (1991) where the following conditional U-statistics were proposed as estimators;where Fn(-) := n-1E, 1(Xi; < ) denotes the empirical distribution function (c.d.f Bochynek discussed the asymptotic normality of conditional U- and V-statistics and pei formed simulation studies on them. Stute (1991) established weak and strong pointwis consistency and asymptotic normality of U(t). Liero (1991) studied uniform strong con sistency of conditional U-statistics and established asymptotic normality of the integrate squared error (ISE) statistic:for suitable A c R* and weight function w(-). We quote the following examples to illustrate the possible use of conditional U-statistics See Stute (1991) and Bochynek (1987) for other examples. Throughout this thesis, our set up will be as foliows: {(Xn, Yn)}n>ı is a bi-variate i.i.d sequence, with (X1, Y1) having join density f(,-) and X, having marginal density fi(-). Consequently,
On Some Measurement Problem In Economics., Nachiketa Chattopadhyay Dr.
On Some Measurement Problem In Economics., Nachiketa Chattopadhyay Dr.
Doctoral Theses
In many branches of economie theory, quantitative relationships play a major role. Translating actions of individuals or groups of individuals, in the various soci- etal processes of production, distribution or consumption, into measurable or quan- tifiable notions has been one of the most important problems. When we take the society as a whole, diverse economic activities of ita memhers need to be collectively presented and it has given rise to interesting aggregation problems. Economists have tried to solve these problems by constructing social index numbers. Simi- larly, quantifying actions of individual units (for example, a firm /a consumer) has brought …
Contributions To The Neyman-Scott And Mixture Problem., Joydeep Bhanja Dr.
Contributions To The Neyman-Scott And Mixture Problem., Joydeep Bhanja Dr.
Doctoral Theses
Neyman and Scott (1948) were the first to point out that the method of maximum likelihood fails to provide elfficient estimates when the number of parameters grows with the sample size n. Consider the following examples introduced by them:Ezample 1.1 Let { Xi } be a sequence of independent random vectors in I", components Xij of Xi being independent normal with mean u, and variance o2 Ilere o2 is the parameter of interest. It is casy to sce that the maxinum likelihood estimate for o2 is not even consistent. It is also known (see Lindsay (1980), Pfanzagl (1982), van der …
The Mazur Intersection Property In Banach Spaces And Related Topics., Pradipta Bandyopadhyaya Dr.
The Mazur Intersection Property In Banach Spaces And Related Topics., Pradipta Bandyopadhyaya Dr.
Doctoral Theses
In the first part of this chapter we explain in general terms the main theme of this thesis and provide a chapter wise summary of its principal results. The second part recapitulates some of the known notions and results used in the subsequent chapters. The numbers given in parentheses correspond to those in the list of references on page 70.S. Mazur (40] was the first to consider the following smoothness property in normed linear spaces, called the Mazur Intersection Property (MIP), or, more briefly, the Property (1):Every closed bounded convex set is the intersection of closed balls containing it.He showed …
Some Aspects Of Majorization And Their Applications In Statistics., Subir Kumar Bhandari Dr.
Some Aspects Of Majorization And Their Applications In Statistics., Subir Kumar Bhandari Dr.
Doctoral Theses
The works in this dissertation are primarily baset ur. difterent concepts of majorization and the results thereof. True first part of this dissertation is a study on dirfferent concepts of uni variate and multivariate majorization. The latter part of this dissertation includes studies on some problems in Sample Survey, and problems relating to ranking and selection with the use of some results, old and new, in majorization.The concept of univariate majorization has been consi- dered by economists in relation to Lorenz ourve, as well as by mathematicians and statisticians, especially in the field of reliability. It appears that the results …
On A Class Of Stochastic Approximation -Type Parameter-Learning Algorithms For Pattern Recognition., Amita Pal Dr.
On A Class Of Stochastic Approximation -Type Parameter-Learning Algorithms For Pattern Recognition., Amita Pal Dr.
Doctoral Theses
The first tank can involve one or more subtasks. For instance, it may require the design of a classifier on the basis of whatever prior knowledge there is of the feature space, or given the design, to estimate efficiently the parameters of the classifier. The latter might involve the estimation of the density function itself if very little is known about the class-conditional fenture distribution, or it may necessitate the entimation of the parameters of the fenture distribution, if one can assume it to have some known form. It may also involve estimating the boundaries of the classes, if even …
Theory Of Estimation In Algebraic And Analytic Exponential Families With Applications To Variance Components Models., Krishnan Unni Dr.
Theory Of Estimation In Algebraic And Analytic Exponential Families With Applications To Variance Components Models., Krishnan Unni Dr.
Doctoral Theses
There are many important statistical problems of the following kind. The family of probability measures O is parametrized by a vector parame ter n varying in a q-dimen- sional domain. P can be represented as an exponential family of probability distributions with k canonical para- me ters where k is greater than q. The canonical parameters do not vary in a domain in R, but are restricted by polyno- mial or analytic equations. They vary on a curved surface defined by the polynomial or analytic equations within the natural parame ter space of the exponential family. The present work is …
Contribution To The Ergodic Theory., S. Natarajan Dr.
Contribution To The Ergodic Theory., S. Natarajan Dr.
Doctoral Theses
Ergodic theory is chiefly concerned with the study of transformations on a measure space which preserve the measure. Interesting classes of such transformations are the classes of ergodic, weakly mixing and mixing transformations. The bulk of this thesis is devoted to the study of a closely related family of transformations called the we akly stable transformations; these are more general than the weakly mäxing transformations.This thesis is divided into three parts. Part I contains preliminary ideas, notations and some results on the invertibility and continuity properties of transition functions (Chapters 1 and 2). In Chapter 3 we collect known results …
Threshold Order Of A Switching Function And Stochastic Dependence Of Inputs., T. Krishnan Dr.
Threshold Order Of A Switching Function And Stochastic Dependence Of Inputs., T. Krishnan Dr.
Doctoral Theses
This Ä‘insertation is being submitted to the Indian Statiatical Inatitute, in support of the author's application for the degree of Doctor of Philosophy. This diasertation enbodien the work carried out by the author at the Indian Statistieal Institute under the aupervision of frofensor C. R. Rso. le is grate ful to Professor Rao for his constant encouragenent and inspi- ration and for the facilities provided at the Institute.The main part of the results of Chapter 5 have already been ublished (20). Zhe resulte of Chapters 2 and 3 and of Chapter 4 &e being reviewed for publie cation. ReferRes of …