# PhD Theses (Mathematics)

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Item Ulam-Hyers and Lyapunov Stability for Some Classes of Fractional Differential Equations and Difference Equations(2024) Shankar, MatapStudying 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.Item Local Geometry of Curve Graphs of Closed Surfaces(2024) Mahanta, KuwariLet 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 .Item Some Spaces of Holomorphic Functions and Their Applications(2023) Bhardwaj, Arun KumarIn 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.Item Paley and Peisert graphs over finite fields, and their generalizations(2023) Bhowmik, AnwitaThis 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.Item On the bipartite distance matrix and the bipartite Laplacian matrix(2021) Jana, RakeshThe 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.Item Weak Galerkin Finite Element Methods for Time Dependent Problems on Polygonal Meshes(2022) Kumar, NareshIn 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.Item Nearly Invariant Subspaces with Finite Defect in Vector Valued Hardy Spaces and its Applications(2023) Das, SomaIn 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.Item Trace Formulas and Finite Dimensional Approximations(2023) Pradhan, ChandanThe 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.Item Risk-Sensitive Stochastic Control and Games(2023) Golui, SubrataThis thesis considers risk-sensitive stochastic control and game problems on countable/Borel state space for discrete/continuous-time Markov decision processes (MDPs) under certain Lyapunov conditions. Here, infinite horizon control/game problems are analyzed with various cost criteria. The controllers can take action in discrete/continuous time from their admissible strategies.Item Higher Order Compact Explicit Jump Immersed Interface Methods for Incompressible Viscous Flows: Application and Development(2023) Singhal, Raghav"This study is primarily focused on the development of explicit jump high-order compact finite difference immersed interface approaches for the purpose of solving incompressible viscous flows that are governed by the Navier-Stokes (N-S) equation on uniform and non-uniform grids on a Cartesian mesh. In all, three basic schemes have been developed in the process: one for elliptic problems and the steady state of N-S equations with discontinuities in the solutions, source terms, and coefficients across the interface; the next one is the transient counterpart of the previously developed one uniform grids; and lastly, a discrete level-set approach on non-uniform grids with complex interfaces. The overall accuracy of the scheme is four in space and two in time. Throughout the whole physical domain, a nine-point compact stencil is maintained by incorporating the jump conditions into the right-hand side of the matrix equation Ax = b resulting from discretization of the concerned equations. We use the streamfunction- vorticity ( - ) formulation of the N-S equation, and the jump conditions for and at the irregular point across the interface are taken into account by using a new method based on Lagrangian interpolation.Item Integral Mixed Cayley Graphs(2023) Kadyan, MonuIn spectral graph theory, mixed graphs are crucial because they give a structure in which directed and undirected edges may coexist. If every edge of a mixed graph is undirected, then such a graph is known as an undirected graph.Item Stochastic Control Problems with Probability and Risk-sensitive Criteria(2023) Bhabak, ArnabIn this thesis we consider stochastic control problems with probability and risk-sensitive criterion. We consider both single and multi controller problems. Under probability criterion we first consider a zero-sum game with semi-Markov state process. We consider a general state and finite action spaces. Under suitable assumptions, we establish the existence of value of the game and also characterize it through an optimality equation. In the process we also prescribe a saddle point equilibrium. Next we consider a zero-sum game with probability criterion for continuous time Markov chains. We consider denumerable state space and unbounded transition rates. Again under suitable assumptions, we show the existence of value of the game and also characterize it as the unique solution of a pair of Shapley equations. We also establish the existence of a randomized stationary saddle point equilibrium. In the risk-sensitive setup we consider a single controller problem with semi-Markov state process. The state space is assumed to be discrete. In place of the classical risk-sensitive utility function, which is the exponential function, we consider general utility functions. The optimization criteria also contains a discount factor. We investigate random finite horizon and infinite horizon problems. Using a state augmentation technique we characterize the value functions and also prescribe optimal controls. We then consider risk-sensitive game problems. We study zero and non-zero sum risk-sensitive average criterion games for semi-Markov processes with a finite state space. For the zero-sum case, under suitable assumptions we show that the game has a value. We also establish the existence of a stationary saddle point equilibrium. For the non-zero sum case, under suitable assumptions we establish the existence of a stationary Nash equilibrium. Finally, we also consider a partially observable model. More specifically, we investigate partially observable zero sum games where the state process is a discrete time Markov chain. We consider a general utility function in the optimization criterion. We show the existence of value for both finite and infinite horizon games and also establish the existence of optimal polices. The main step involves converting the partially observable game into a completely observable game which also keeps track of the total discounted accumulated reward/cost.Item Adaptive Finite Element Methods for Parabolic Interface Problems(2021) Ray, TanushreeThe main objective of this thesis is to study adaptive nite element methods (AFEMs) for parabolic interface problems in a bounded convex polygonal domain in R2. Interface problems arise in a wide variety of applications in science and engineering such as material sciences and fluid dynamics when two or more distinct materials or fluids with different conductivities or densities or diffusion are interacting across the interface. Due to the discontinuity of the coefficients along the interface, the analytic solutions are rarely available for the interface problems. Therefore, the numerical approximation is the only way to proceed with such problems. Even if the solution is smooth in each individual subdomain, the global regularity of the solution of such problem is very low. As a result, it is very challenging to achieve higher order accuracy in the nite element method (FEM). Therefore, much attention has been paid in the recent years to the study both theory and numerics of time-dependent interface problems. It is known that AFEMs are widely used numerical techniques to enhance the accuracy and effciency of the nite element method. The key to the success of AFEMs relies on the a posteriori error analysis, which provides error indicators for the design of adaptive algorithms. The adaptive method reduces the computational efforts and ensures higher density nodes in a particular area of the given domain where the solution is very diffcult to approximate.Item Discretization in Space and Time of Subdiffusion Equations with Memory(2022) Mahata, ShantiramThe main objective of this thesis is to investigate the numerical analysis of nite element method for the subdi usion equations with memory. Both smooth and singular kernels are considered covering smooth and nonsmooth initial data.Item Localizing Spectra and Pseudospectra of Matrices and Matrix Polynomials(2022) Roy, NanditaThis thesis considers different aspects of the study of sets that localize spectra and pseudospectra of matrices and matrix polynomials. Given a block upper triangular matrix, the literature has several results for outer approximations of the pseudospec- tra of the matrix by the pseudospectra of the diagonal blocks. The thesis provides inner approximations of the pseudopectra of a block upper triangular matrix in terms of pseudospectra of the diagonal blocks. Next the definitions of block Geršgorin sets, block Brualdi sets, block minimal Geršgorin sets and permuted pointwise minimal Geršgorin sets are extended to matrix polynomials in homogeneous form. The use of the homogeneous form is justified by its unified treatment of both the finite and infinite eigenvalues. Many properties of these sets are derived and efficient numerical methods for plotting the sets that can compare favourably with some existing meth- ods under certain conditions, are proposed. In particular, the proposed methods may be used to plot all eigenvalue localization sets for matrix polynomials without laying a grid on any part of the complex plane. Eigenvalue problems associated with the quadratic matrix polynomials have a wide range of applications. Localizations of eigenvalues of such polynomials are proposed via block Geršgorin sets that arise from some special linearizations. Several properties of these sets are derived and used to obtain some easily computable bounds on the eigenvalues of the matrix polynomial. They are also used to obtain sufficient conditions on the coefficient matrices of the polynomial for its eigenvalues to lie in particular regions of the complex plane that are important from the point of view of applications. The results also include various structured matrix polynomials. As an outcome of the analysis, several upper bounds on solutions of important distance problems associated with structured and unstructured quadratic matrix polynomials are derived for various choices of norms. In all cases, numerical experiments are performed to illustrate the results.Item Some classical problems in harmonie analysis(2023) Mondal, Shyam SwarupThis thesis focuses on certain classical problems in harmonic analysis in connection with mathematical physics. We begin with the Fourier analysis on the Euciidean space, discuss some well known results, basic definitions, and review of recent developments that motivates us to consider the problems discussed in the thesis. We prove a restriction theorem for the Fourier-Hermite transform and obtain a Strichartz estimate for systems of orthonormal fi.rnctions associated with the Hermite operator H : -L, + lrl' on R.' for the range I I q < ffi as an application. Besides, we show an optimal behavior of the constant in the Strichartz estimate as limit of a large number of functions.Item On the penetration and distribution of drugs into biological tissues: A multiscale approach(2022) Yadav, Kuldeep SinghIn this thesis, a finite volume heterogeneous multiscale method (FV-HMM) is propounded to study drug transport into biological tissues by considering cell scale heterogeneity. The partition coefficient is incorporated in the diffusionbased drug transport model as the first objective. A new upscaling technique is devised to evaluate the effective drug transport at the macro level. Next, the FV-HMM is improvised (to FVHMM-p) to incorporate the passive diffusion across the cell membrane. The permeable cell membrane treated in the microscale model incorporates the solute diffusivity, membrane thickness, and partition coefficient. For the microscale model simulation, a novel permeable interface method (PIM) based on the central-type finite difference discretization is developed. Further, the FVHMM-p is reconstructed to investigate the effects of biological cell orientation on the penetration and distribution of a drug in tissues. The simulation results reveal that the biological cell orientation is an important factor, which can potentially affect the drug penetration and distribution in the tissues. In the next objective, the FVHMM-p incorporates the fluid flow and drug metabolism effects on drug transport. On treating the tissue as a porous medium, Darcy’s law is used for fluid flow, and the drug metabolism is calculated using the Michaelis-Menten equation. It is observed that the particles of sizes 10 and 100 nm can penetrate the tissue in fluid flow regions. Furthermore, local sensitivity analysis is also performed to determine the model response to the input parameters. It is observed that the parameters such as fluid velocity, extracellular diffusivity, and microscale domain size are the most sensitive to the model outcome. Finally, the last multiscale model is employed to study the tissue penetration and distribution efficacy of chemotherapeutic agents, such as fluorouracil, carmustine, cisplatin, methotrexate, doxorubicin, and paclitaxel. The physical properties of drugs are incorporated in the model to understand the effects under different situations. It is observed that carmustine penetrates deeper into the tissue, followed by paclitaxel, methotrexate, fluorouracil, doxorubicin, while cisplatin penetrates least.Item Local properties of Richardson varieties in symplectic and orthogonal Grassmannians(2022) Ray, PapiIn a paper by Kodiyalam and Raghavan, they provided an explicit combinatorial description of the Hilbert function of the tangent cone at any point on a Schubert variety in the Grassmannian, by giving a certain “degree-preserving” bijection between a set of monomials defined by an initial ideal and a “standard monomial basis”. In this thesis, we have proved that this bijection is in fact a bounded RSK correspondence. As an application, we have proved that the bijection given in a paper of Ghorpade and Raghavan (for the symplectic Grassmannian) is also a bounded RSK correspondence. In the PhD thesis of Kreiman, he had given a bijection between the same two combinatorially defined sets as in the paper of Kodiyalam and Raghavan. In this thesis, we have proved that the bijection given in Kreiman’s thesis and the bijection given in the paper of Kodiyalam and Raghavan are equivalent. Using the above results, we have given an explicit Gröbner basis for the ideal of the tangent cone at any T -fixed point of a Richardson variety in the symplectic Grassmannian. In this thesis, we have also provided formulae for the multiplicity at any T -fixed point of a Richardson variety in the symplectic as well as the orthogonal Grassmannians; together with an interpretation of the multiplicity in terms of certain non-intersecting lattice paths.Item (A) Study of Some Parameters in Signed Graphs(2022) DeepakGraphs with positive or negative edges are called signed graphs. We denote a signed graph Σ by (G, ϕ), where G is called the underlying graph of Σ and ϕ is a function that assigns +1 or −1 to the edges of G. The set of negative edges in Σ is known as the signature of Σ. An unsigned graph can be realized as a signed graph in which all edges are positive. Switching Σ by a vertex v is to change the sign of each edge incident to v. Switching is a way of turning one signed graph into another. Two signed graphs are called switching equivalent if one can be obtained from the other by a sequence of switchings. Further, two signed graphs are said to be switching isomorphic to each other if one is isomorphic to a switching of the other. In Chapter 2 of the thesis, we classify the switching non-isomorphic signed graphs arising from K6, P3,1, P5,1, P7,1, and B(m, n) for m ≥ 3, n ≥ 1, where K6 is the complete graph on six vertices, Pn,k denotes the generalized Petersen graph and B(m, n) denotes the book graph consisting of n copies of the cycle Cmwith exactly one common edge. We also count the switching non-isomorphic signatures of size two in P2n+1,1 for n ≥ 1. We prove that the size of a minimum signature of P2n+1,1, up to switching, is at most n + 1.Item Mathematical Modelling of Drug Delivery in Tissue Cells by Electroporation(2022) Mondal, NilayElectroporation method is a useful tool for delivering drugs into various diseased tissues in the human body. In this thesis, mathematical models are developed to demonstrate how to deliver drugs into the diseased cells using tissue electroporation. A detailed study is carried out on reversible electroporation, thermal effects resulting from pulse application, and on drug elimination and metabolism. The present dissertation starts with a mathematical model of single cell electroporation for delivering drug into the cell. The model is able to capture non-homogeneous drug transport in the cell due to non-uniform cell membrane permeabilization. Several numerical experiments are conducted to understand the effects of electric field and drug permeability on drug uptake into the cell. Through investigation, the appropriate electric field and drug permeability are identified that lead to sufficient drug uptake into the cell. The second work contributes an improved mathematical model for drug delivery into the electroporated tissue that deals with both reversibly and irreversibly electroporated cells. The time-dependent mass transfer coefficient as a function of pore density is used to find the drug concentrations throughout reversibly and irreversibly electroporated cells as well as in the extracellular space. The effects of permeability of drug, electric field, and pulse period on drug concentrations in extracellular and intracellular regions are discussed. The threshold value of an electric field to initiate drug uptake is identified in this study. Special emphasis is also given on two cases of electroporation, drug dynamics during ongoing electroporation and drug dynamics after the electric pulse period is over. The next work focuses on the development of a mathematical model of drug delivery based on reversible tissue electroporation in order to treat all of the cells at the targeted site. In addition, the thermal effects on the tissue, which is an outcome of Joule heating, are also considered. The model optimizes the electroporation parameters for the required drug uptake into the cells with no thermal damage. This model can be used in clinical experiments to predict the drug uptake into the infected cells by controlling the model parameters according to the nature of infections. The subsequent work emphasizes the tissue boundary where the drug is injected as a point source. Drug loss from the tissue boundaries through extracellular space and its reaction on drug transport are examined. The effects of electric field on tissue conductivity is studied. Multiple pulses are applied to deliver a sufficient amount of drug into the targeted cells. The final work concentrates on the process of drug metabolism in the cells in our modified model. The main objective is to examine the combined effects of drug loss across tissue boundaries and drug metabolism within cells. A detailed analysis of the effects of metabolism parameters on cellular drug uptake is presented. The model is able to calculate the amount of drug metabolites in cells at the end of drug transport.