PhD Theses (Mathematics)Department of Mathematicshttps://gyan.iitg.ac.in/handle/IITG2023/MatPhd2024-07-25T15:08:16Z2024-07-25T15:08:16Z1231On nonlocal purely critical and supercritical problems and superlinear semipositone problemsKumar, Uttamhttps://gyan.iitg.ac.in/handle/123456789/26342024-06-07T11:02:05Z2022-01-01T00:00:00Zdc.title: On nonlocal purely critical and supercritical problems and superlinear semipositone problems
dc.contributor.author: Kumar, Uttam
dc.description.abstract: The main objective of this thesis is to examine purely critical and supercritical exponent problems involving nonlocal operator in the symmetric domain. The nonlocal superlinear semipositone problem is also investigated.
dc.description: Supervisor: Tiwari, Sweta
2022-01-01T00:00:00ZOn the Darboux Polynomials and Simplicity of Polynomial DerivationsKesarwamy, Ashish Kumarhttps://gyan.iitg.ac.in/handle/123456789/26152024-05-17T10:27:41Z2024-01-01T00:00:00Zdc.title: On the Darboux Polynomials and Simplicity of Polynomial Derivations
dc.contributor.author: Kesarwamy, Ashish Kumar
dc.description.abstract: Derivations and Darboux polynomials are useful methods to study polynomial or rational differential system. In the vocabulary of differential algebra, Darboux polynomials coincides with generators of polynomial differential ideals, that is f∈k [ x1 ,…, xn] is a Darboux polynomial iff f ≠ 0 and the ideal ( f ) is differential. The present thesis studies certain classes of derivations having no Darboux polynomials with monomial cofactor. Another important notion in commutative algebra, simple derivations has also been studied in this thesis. Simple derivationsplay an important role in numerous problems. This thesis studies certain classes of derivations that are simple. In Chapter-1, we talk about definitions, notations and basic facts. In Chapter-2, we study a class of derivations having no Darboux polynomial with monomial cofactor. In Chapter-3, we study simplicity of certain polynomial derviations in the polynomial ring k [ x , y ]. In Chapter-4, we generalize the results obtained in Chapter-3 in more than two variables, i.e., we talk about few classes of simple derivations of the polynomial ring k [ x1 ,…, xn ]. In this Chapter- 5, We pose some problems arising out of the work carried out in this thesis.
dc.description: Supervisor: Wagh, Vinay Vilas
2024-01-01T00:00:00ZUlam-Hyers and Lyapunov Stability for Some Classes of Fractional Differential Equations and Difference EquationsShankar, Mataphttps://gyan.iitg.ac.in/handle/123456789/25452024-02-05T09:59:57Z2024-01-01T00:00:00Zdc.title: Ulam-Hyers and Lyapunov Stability for Some Classes of Fractional Differential Equations and Difference Equations
dc.contributor.author: Shankar, Matap
dc.description.abstract: Studying the behavior of a dynamical system and the dependency of its solution on the initial state or initial condition began in the late 1880s. Describing the dynamics of dynamical systems as a function of time on the state space can generate a differential equation. Thus, the theory of dynamical systems may be said to be a special and important topic in the theory of differential equations. It falls under the qualitative theory which is mainly concerned with properties which are not quantified. The study of qualitative theory leads to a better understanding of the dynamical systems.
dc.description: Supervisor: Bora, Swaroop Nandan
2024-01-01T00:00:00ZLocal Geometry of Curve Graphs of Closed SurfacesMahanta, Kuwarihttps://gyan.iitg.ac.in/handle/123456789/25442024-02-05T09:59:43Z2024-01-01T00:00:00Zdc.title: Local Geometry of Curve Graphs of Closed Surfaces
dc.contributor.author: Mahanta, Kuwari
dc.description.abstract: Let Sg denote a closed, orientable surface of genus g 2. Let C(Sg) be the associated curve graph and d be the associated path metric. Let and be curves on Sg and T ( ) be the Dehn twist of about .
dc.description: Supervisor: Palaparthi, Sreekrishna
2024-01-01T00:00:00ZSome Spaces of Holomorphic Functions and Their ApplicationsBhardwaj, Arun Kumarhttps://gyan.iitg.ac.in/handle/123456789/25432024-02-05T09:59:28Z2023-01-01T00:00:00Zdc.title: Some Spaces of Holomorphic Functions and Their Applications
dc.contributor.author: Bhardwaj, Arun Kumar
dc.description.abstract: In this dissertation, we consider several problems in complex analysis. In the first part, we study about an explicit formula for the Hilbert transform. The celebrated integral transforms such as Fourier transform, Laplace transform, and Hilbert transform have tremendous applications in various branches of science and engineering. However, unlike to Fourier or Laplace transform, very few functions have an explicit formula for their Hilbert transforms. In this dissertation, we obtain an explicit formula for the Hilbert transform of log|f|, for the function f in Nevanlinna class having continuous extension to the real line. This family is the largest possible for which such a formula for the Hilbert transform of log|f| can be obtained. The formula is very general and implies several previously known results.
dc.description: Supervisor: Srivastava, Rajesh Kumar
2023-01-01T00:00:00ZPaley and Peisert graphs over finite fields, and their generalizationsBhowmik, Anwitahttps://gyan.iitg.ac.in/handle/123456789/25152024-01-10T11:45:22Z2023-01-01T00:00:00Zdc.title: Paley and Peisert graphs over finite fields, and their generalizations
dc.contributor.author: Bhowmik, Anwita
dc.description.abstract: This thesis is mainly devoted to the computation of the number of cliques of certain Cayley graphs, namely the Paley- type graphs, Peisert graphs and Peisert-like graphs. Barring the case of the Peisert graphs, the focus is on the number of cliques of orders three (triangles) and four. Let q be a prime power such that q 1 (mod 4). The Paley graph of order q is the graph with vertex set as the nite eld Fq and edges de ned as, ab is an edge if and only if a b is a non-zero square in Fq. The rst part of this thesis involves de ning a generalization of the Paley graph, called the Paley-type graph on the commutative ring Zn for certain values of n, precisely n = 2sp 1 1 p k k , where s = 0 or 1, i 1, where the distinct primes pi satisfy pi 1 (mod 4) for all i = 1; : : : ; k. For such n, we de ne the graph with vertex set Zn and edges de ned as, ab is an edge if and only if a b is a square in the set of units of Zn. We look at some properties of this graph. For primes p 1 (mod 4), Evans, Pulham and Sheehan computed the number of complete subgraphs of order four in the Paley graph. Recently, Dawsey and McCarthy found the number of triangles and complete subgraphs of order four in the generalized Paley graph of prime power order. We nd the number of triangles and complete subgraphs of order four in the Paley-type graph successively for n = p (p 1 (mod 4) being a prime and 1) and for general n, using character sums and combinatorial methods.
dc.description: Supervisor: Barman, Rupam
2023-01-01T00:00:00ZOn the bipartite distance matrix and the bipartite Laplacian matrixJana, Rakeshhttps://gyan.iitg.ac.in/handle/123456789/24752024-01-16T11:15:19Z2021-01-01T00:00:00Zdc.title: On the bipartite distance matrix and the bipartite Laplacian matrix
dc.contributor.author: Jana, Rakesh
dc.description.abstract: The study of the properties of graphs via matrices is a widely studied subject that ties together two seemingly unrelated branches of mathematics; graph theory and linear algebra. Graham and Pollak in 1971 proved a remarkable result which tells that the determinant of the distance matrix of a tree only depends on the number of
vertices in the tree. This impressive result created a lot of interest among the researchers. Since then many generalizations have been proposed in order to understand the distance matrix better. Yet, the understanding seems
to be far from complete. We present one such point of view here showing how many more combinatorial objects are linked together.
dc.description: Supervisor: Pati, Sukanta
2021-01-01T00:00:00ZWeak Galerkin Finite Element Methods for Time Dependent Problems on Polygonal MeshesKumar, Nareshhttps://gyan.iitg.ac.in/handle/123456789/24742024-01-16T11:15:02Z2022-01-01T00:00:00Zdc.title: Weak Galerkin Finite Element Methods for Time Dependent Problems on Polygonal Meshes
dc.contributor.author: Kumar, Naresh
dc.description.abstract: In this thesis, an attempt has been made to study the higher order of convergence for time-dependent problems. This thesis aims to design and analyze higher-order convergence of weak Galerkin finite element approximations to the true solutions for time-dependent problems on polygonal meshes. The mathematical analysis of higher-order convergence for time-dependent problems with polygonal meshes adds more challenges than one could imagine. First, describe a systematic numerical study on WG-FEMs for second-order linear parabolic problems by allowing polynomial approximations with various degrees for each local element. Convergence of both semidiscrete and fully discrete WG solutions is established. Here, we assume that the true solution satisfies full regularity assumptions. Next, we proceed to discuss the WG algorithm for the parabolic problems, when the solution u in L2(0, T; Hk+1(Ω))∩H1(0, T; Hk-1(Ω)). Such regularity holds where forcing function f in L2(0, T; Hk-1(Ω)) and initial function u0 in Hk(Ω). for some k≥1. Optimal order error estimates in L2(L2) and L2(H1) norms are shown to hold for the spatially discrete-continuous time and the discrete-time weak Galerkin finite element schemes. Further, we explore the L2 error convergence of weak Galerkin finite element approximations for a homogeneous parabolic equation with non-smooth initial data using polygonal meshes. Our next focus is to describe WG-FEMs for solving the wave equation. We propose both semidiscrete and fully discrete schemes to solve the second-order linear wave equation numerically. In our last problem, we designed and analyzed the WG-FEMs to approximate a general linear second-order hyperbolic equation with variable coefficients on polygonal meshes. The convergence analysis is carried out for the semidiscrete and fully discrete weak Galerkin approximations. The fully discrete scheme can be reinterpreted as an implicit second-order accurate Newmark scheme that is unconditionally stable.
dc.description: Supervisor: Deka, Bhupen
2022-01-01T00:00:00ZNearly Invariant Subspaces with Finite Defect in Vector Valued Hardy Spaces and its ApplicationsDas, Somahttps://gyan.iitg.ac.in/handle/123456789/24582024-01-16T11:14:13Z2023-01-01T00:00:00Zdc.title: Nearly Invariant Subspaces with Finite Defect in Vector Valued Hardy Spaces and its Applications
dc.contributor.author: Das, Soma
dc.description.abstract: In this dissertation, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector valued Hardy space. Using this characterization we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space. Moreover, as an application, we also identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases using our characterization in connection with nearly invariant subspaces of finite defect for the backward shift operator acting on the vector valued Hardy space.
dc.description: Supervisor: Chattopadhyay, Arup
2023-01-01T00:00:00ZTrace Formulas and Finite Dimensional ApproximationsPradhan, Chandanhttps://gyan.iitg.ac.in/handle/123456789/24552024-01-16T11:13:53Z2023-01-01T00:00:00Zdc.title: Trace Formulas and Finite Dimensional Approximations
dc.contributor.author: Pradhan, Chandan
dc.description.abstract: The dissertation gives a new proof of some existing second-order trace formulas, namely the Koplienko-Neidhardt trace formula for pair of unitaries in the multiplicative path, the Koplienko-Neidhardt trace formula for pair of contractions via linear path with one of them being normal. Our proofs are based on the idea of the finite-dimensional approximation method introduced by Voiculescu. As a consequence of our results and the Schaffer matrix unitary dilation, we obtained second-order trace formula for a class of pairs of contractions via linear path. Using a different setup of finite dimensional approximations, we extend the Koplienko-Neidhardt trace formula for a class of pairs of contractions via multiplicative path.
dc.description: Supervisor: Chattopadhyay, Arup
2023-01-01T00:00:00Z