# PhD Theses (Mathematics)

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Item Geometric analysis of spectral stability of matrices and operators(2001) Bora, ShreemayeeIn this thesis, an attempt is made to undertake a systematic analysis of the sensitivity of eigen systems in the natural geometric framework of the spectral portraits of the matrices. The e-spectra and the spectral portraits are shown to be efficient graphical tools for sensitivity analysis of eigenvalues and spectral decomposition of matrices. The notion of e-spectra is also shown to be an appropriate logical setting for spectral analysis of matrices which are known only up to a given accuracy. The geometric separation of eigenvalues of a matrix A which can be read off from the e-spectra of A is shown to be an appropriate measure of sensitivity of eigenvalues and spectral decompositions. For the l 2–norm, a characterization of the sensitivity of spectral decompositions is provided and in the process a problem raised by Demmel[13] is solved. Sufficient conditions are obtained for the stability of spectral decompositions with respect to operator norms. Several bounds on the magnitude of the perturbations which ensure stability are also derived. Under appropriate assumptions, a conjecture of Demmel[13] on the separation of matrices is also settled.Item HOC Schemes for Incompressible Viscous Flows: Application and Development(2001) Kalita, Jiten Ch.This work is concerned with higher-order compact (HOC) schemes for convection-diffusion equations in general and incompressible viscos flows in particular. A fully compact and fully higher-order accurate scheme is developed for the standard buoyancy driven square cavity problem through O(h4) compact approximation of the derivative source term and zero-gradient temperature boundary conditions of identical accuracy at the adiabatic walls. The scheme produces highly accurate results even for very high laminar Rayleigh numbers for which no other HOC results are seen. ...Item Expansioins of C-Sets and D-Sets Having Jordan Automorphism Groups and Some Related Questions(2002) Ahmed, ShabeenaThis thesis is about the interplay between permutation groups and some relational structures. Special emphasis is on a class of permutation groups called Jordan groups i which, in some non-degenerated way, the pointwise stabiliser of a subset is transitive on the complement................................Item Certain Aspects of Spectra of Unicyclic Graphs(2006) Nath, MilanThis thesis aims at filling some conspicuous gaps in the study of spectra of unicyclic graphs, and answering some recent questions on relations between the structure of a unicyclic graph and the spectrum of its adjacency matrix...Item On the spectra and the Laplacian spectra of graphs(2006) Barik, SasmitaThroughout all graphs are assumed to be simple. Let A(G) and L(G) denote the adjacency and the Laplacian matrix corresponding to a graph G, respectively. The second smallest eigen- value of L(G) is called the algebraic connectivity of G and is denoted by a(G). A corresponding eigenvector is called a Fiedler vector of G. The study of spectral integral variations in graphs has been a subject of interest in the past few years (see Fan [21, 22, 23], Kirkland [46] and So [65]). We say that the spectral integral variation occurs in one place by adding an edge e 2 G if (i) L(G) and L(G + e) have exactly n-1 eigenvalues in common and (ii) if उ is the other eigenvalue of L(G), then उ+2 is the other eigenvalue of L(G + e). We supply a characterization of connected graphs in which spectral integral variation occurs in one place by adding an edge where the changed eigenvalue is the algebraic connectivity. The results proved for that purpose suggest methods to construct graphs for which such variation occurs and also to construct graphs for which such variation never occurs. An argument showing that such a variation can never occur for the Laplacian spectral radius is supplied. Graphs with integer Laplacian spectrum has been a subject of study for many researchers, see, for example, Grone and Merris [34] and Grone, Merris and Sunder [33]. Probably the most common example is the star on n >= 3 vertices with Laplacian eigenvalues 0, 1 (multiplicity n - 2), and n. The characterization of trees with 1 as a Laplacian eigenvalue is rather difficult. It is known that the star is the only tree with algebraic connectivity 1. We take up the problem of characterizing trees that have 1 as the third smallest Laplacian eigenvalue. ...Item Higher order compact schemes for incompressible viscous flows on geometries beyond rectangular(2006) Pandit, Swapan KumarIn this dissertation, we have proposed a new class of higher order compact (HOC) finite difference schemes for solving the two-dimensional (2D) incompressible viscous flows through geometries beyond rectangular. The proposed schemes are developed for both steady-state and transient flows which are governed by the Navier-Stokes (N-S) equations. All these schemes are fourth order accurate in space while the ones for the transient flows are implicit, and first or second order accurate in time depend- ing on the choice of a weighted average parameter. We have employed these schemes not only on problems having analytical solutions to verify their order of accuracy, ef- ficiency, effectiveness and robustness but also to explore new flow phenomena of some other complicated problems, namely, the lid-driven cavity flow, channel flow with forward and backward constriction, flow in a lateral and symmetric dilated channels, flow through constricted tube etc. They are seen to efficiently capture both steady- state and transient solutions with Dirichlet as well as Neumann boundary conditions. Apart from including all the features of existing HOC schemes, the formulation has the added advantage of capturing transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variation. We present both quali- tative and quantitative results produced by our schemes on relatively coarser grid for all the test cases and compare them with theoretical predictions, analytical as well as established numerical and experimental results available in the literature. Excellent agreement is obtained in all the cases..Item Finite element methods for elliptic and parabolic interface problems(2006) Deka, BhupenThe main objective of this thesis is to study the convergence of finite element solutions to the exact solutions of elliptic and parabolic interface problems by means of classical finite element method. Due to low global regularity of the true solution it is difficult to apply the classical finite element analysis to obtain optimal order of convergence for interface problems (cf. [3, 14]). The emphasis is on the theoretical aspects of such methods. In order to maintain the best possible convergence rate, a finite element discretization is proposed and analyzed for both elliptic and parabolic interface problems. More precisely, we have shown that the finite element solution converges to the exact solution at an optimal rate in L2 and H1 norms if the grid lines coincide with the actual interface by allowing interface triangles to be curved triangles. further, if the grid lines form an ...Item Reflection and transmission of surface water waves by undulating bottom topography(2006) Martha, Subash ChandraThe objective of this thesis is to investigate the scattering of a train of small amplitude harmonic surface water waves by small undulation of a sea-bed for both normal and oblique incidence. In this study of scattering, mixed boundary value problems are set up for the determi- nation of a velocity potential where the governing partial differential equation happens to be Laplace's equation in two dimensions for normal incidence and in three dimensions for oblique incidence within the fluid with a mixed boundary condition on the free surface and a condition on the bottom boundary. As the fluid domain extends to infinity, a far-field condition or an infinity condition arises to ensure uniqueness of the problem. Applying a perturbation analysis, which involves a small parameter Epsilon present in the representation of the small undulation of the sea-bed, directly to the boundary value problem the original problem is reduced to a simpler boundary value problem for the first order correction of the potential..Item Pseudospectra of Matrix Pencils and their applications in perturbation analysis of Eigenvalues and Eigendecompositions(2007) Ahmad, Sk. SafiqueThe main theme of the thesis revolves around pseudospectra of matrix pencils and matrix polynomials and their applications in perturbation theory...Item Dynamics of Certaie Transcendental Entireand Meromorphic Functions(2007) Nayak, TarakaantaLet f : C ---- C = C U {00} be a non-constant transcendendental entrie or meromorphic function. The function f Xn ,the n-times composition of f is called the n-th it-crate of f . The Fatou set of the function f, ....Item On the Convergence of H1-Galerkin Mixed Finite Element Method for Parabolic Problems(2008) Tripathy, MadhusmitaThe purpose of the present work is to study the convergence of H1-Galerkin mixed Dnite element method for the linear parabolic partial diDerential equations. The emphasis is on the theoretical aspects of such methods. An attempt has been made in this thesis to study the error analysis for the semidiscrete and fully discrete schemes with lesser regularity assumptions on the initial data. More precisely, for homogeneous parabolic problem an energy technique is used to obtain error estimates of order O(h2tD1=2) with positive time in the L2-norm for both the solution and the Dux when the given initial data is in H2(D)\H1 0 (D). Further, a parabolic duality argument is used to obtain optimal order error estimates of order O(h2tD1) for both the solution and its Dux when the given initial function is only in H1 0 (D). Analogous results are shown to hold for two dimensional parabolic problems. Since the smoothing property of the exact solution plays a signiDcant role in the study of error analysis in the semidiscrete case we, therefore, Drst investigate the smoothing property of the exact solution of this problem using energy arguments. Based on backward Euler method, a fully discrete scheme is analyzed for one-dimensional homogeneous parabolic problems and almost optimal order error bounds are established. Optimal order error estimate is the best that one can get between the exact solution and its numerical approximation when measured globally on the computational domain. But, there are places (points or lines) in the computational domain where the approximate solution is more closer to the exact solution than what is predicted by the global error estimates. It would be advantageous to make use of those points or lines in the modelling process. Therefore, we study superconvergence phenomenon for the semidiscrete H1-Galerkin mixed Dnite element method for parabolic problems. A new approximate solution for the Dux with superconvergence of order O(hk+3) is realized via a postprocessing technique, where k D 1 is the order of the approximating polynomials employed in the Raviart-Thomas element. A priori error estimates can give asymptotic rates of convergence as the mesh parameter goes to zero, but often can not provide much practical information about the actual errors encountered on a given mesh. The question of quantifying the error brings attention to a posteriori estimates. A posteriori error estimators are computable quantities which bound the errors or approximate the errors by computed numerical solution and input data of the problem. Further, to guarantee a good convergence behavior of the discrete solution, one needs to apply a reDnement algorithm based on a posteriori error estimates. Therefore, we study a posteriori error analysis for the semidiscrete and fully discrete H1-Galerkin mixed Dnite element method for parabolic problems. The estimators are derived based on a residual approach...Item Backward Perturbation and Sensitivity analysis of Structured polynomial Eigenomial Eigenvalue Problem(2008) Adhikari, BibhasThe main theme of the thesis is structured perturbation and sensitivity analysis of structured polynomial eigenvalue problem. Structured mapping problem naturally arises when analyzing structured backward per- turbation of structured eigenvalue problem. Given two matrices X and B of same size, the structured mapping problem requires to Dnd a \structured"" matrix A; if any, having the small- est norm such that AX = B: We provide a complete solution of structured mapping problem. More generally, we provide a complete solution of the structured inverse least-squared problem (SILSP): min A kAX D BkF ; where the minimum is taken over \structured"" matrices. As a consequence of structured map- ping problem, we determine structured backward errors of approximate invariant subspaces of structured matrices. We also analyze structured pseudospectra of structured matrices. Next, we undertake a detailed structured backward perturbation analysis of structured ma- trix polynomials and derive explicit computable expressions for structured backward errors of approximate eigenelements. We analyze structured pseudospectra of structured matrix poly- nomials and establish a partial equality between unstructured and structured pseudospectra, which plays an important role in solving certain distance problems associated with structured polynomials. We also derive relatively simple expressions for structured condition numbers of simple eigenvalues of structured matrix polynomials, which play an important role in ana- lyzing sensitivity of eigenvalues of structured polynomial eigenvalue problem. Generally, a polynomial eigenvalue problem is \linearized"" Drst and then solved by a back- ward stable algorithm. However, the eigenvalues of the resulting linear problem is usually more sensitive to perturbation than the original problem. Moreover, a polynomial admits inDnitely many linearizations. The same holds true for structured polynomials as well. Therefore, for computational purposes, it is of paramount importance to identify potential structured lin- earizations which are as well conditioned as possible. With the help of structured backward perturbation analysis and structured condition numbers of eigenvalues, we identify \good"" structured linearizations which guarantee almost as accurate solutions as that of the original polynomial eigenvalue problem..Item Some New Directions in Hoc Methodology:Tackling circular Geometries(2009) Ray, Rajendra KumarThe present work is mainly deals with the development of a class of higher-order com- pact (HOC) Dnite di Derence formulations to tackle the circular geometries both for the continuous and discontinuous cases. Depending upon this, the contains of the present work can be divided into two parts. The Drst part concerned with the development of HOC schemes for convection-diDusion equations in general and incompressible viscous Dows in particular on nonuniform polar coordinate system. The basic diDerence be- tween the proposed scheme and the earlier HOC schemes is that the proposed schemes are able to handle variable coeDcients of the second order derivatives while the previ- ous schemes could deal only with unit diDusion coeDcients on cartesian or cylindrical polar coordinate on uniform grid. A fourth order accurate HOC scheme for the steady state convection-diDusion equations on non-uniform polar grid has been developed Drst. The scheme produces highly accurate results even in coarser grids for diDerent Duid Dow problems. An HOC treatment for the streamfunction-vorticity (D-!) formulation of the two-dimensional unsteady, incompressible, viscous Navier-Stokes equations on polar grid has been developed next, speciDcally designed for the motion past circular cylinder prob- lems. The scheme is second order accurate in time and at least third order accurate in space. The HOC treatment is also used to discretize the Neumann boundary conditions. The scheme is then used to solve the Dow past an impulsively started circular cylinder problem for a wild range of Reynolds numbers (Re) and to solve the Dow past rotating cylinder problems for wild range of both Re and rotation parameter (D). Present numer- ical results are then compared with the existing experimental and standard numerical results. In every case an excellent agreement has been found. In this process, some new properties have been found and some extended works have been carried out which have not been studied earlier. The second part of the present work deals with the development of Dnite diDerence algorithms which are obtained by clubbing the existing HOC methodology with a special treatment to tackle the immersed interfaces for problems having discontinuities along the circular interfaces. Firstly, a new methodology for numerically solving one-dimensional (1D) elliptic equations with discontinuous coeDcients, Duxes and singular source terms and the corresponding unsteady parabolic equations on nonuniform space grids have been developed. Stability and convergence analysis of the newly developed scheme have been carried out next. Then, this 1D idea has been extended for the 2D elliptic problems with same type of discontinuities. For both the 1D and 2D cases, numerous numerical studies on a number of problems have been done and compared present results with those obtained by well known methods. In all cases, our formulation is found to produce better results on relatively coarser grids..Item Water Wave Scattering by a Spherical Structure and an Undulating Bottom Topogrphy in a Two-Layer Fluid(2009) Mohapatra, SmrutiranjanThis thesis studies (i) the interaction of water waves with spherical geometries in a two-layer fluid of finite depth, which is covered by either a rigid flat structure or a very thin ice shelf; (ii) the scattering of water waves by different types of unevenness on the bottom surface under such situations. To solve the ice-covered problems, the common idealization of ice as a thin elastic plate, which is static in all but its flexural response, is followed. Furthermore, the assumptions of linear and time harmonic motions are considered. Firstly, the problem consisting of wave interaction with a spherical body submerged in either layer of the fluid is divided into two parts: one describing the scattering of waves by the fixed structure and the other describing the radiation of waves by the body into otherwise calm water. The radiation problem is further split into a number of parts, each of which corresponds to the body moving in a separate mode of motion. The physical problem involving radiation or scattering case, is reduced to a boundary value problem governed by a three-dimensional LaplaceDs equation for both the upper and the lower layers. The method of solution for both the fluids is based upon the multipole expansions technique. The solutions help in calculating the hydrodynamic forces acting on the spherical body for different modes of motion such as heave and sway. A number of observations are made for these motions with regard to different submersion depths. The multipole expansion method is found to be an extremely powerful method for solving radiation and scattering problems for submerged spheres. It eliminates the need to use large and cumbersome numerical packages for the solution of such problems. Secondly, the latter part of this thesis is solely devoted to the investigation of the scattering of a train of small amplitude harmonic water waves by small bottom undulation of an ocean, which consists of a two-layer fluid, for both normal and oblique incidences. Moreover, it is assumed that both the fluids are of finite depth and the upper fluid is covered by either a rigidinvolving the shape function which represents the bottom undulation. Different special forms of the shape functions are considered to compute the integrals explicitly for the reflection and transmission coefficients and the results are suitably presented graphically. Out of these shape functions, the particular case of a patch of sinusoidal ripples (with the same wave number or two different wave numbers) has considerable significance due to the ability of an undulating bed to reflect incident wave energy which is important in respect of both coastal protection and of possible ripple growth if the bed is erodable. For this ripple patch, in the case of a channel bed assumed bounded above by a rigid boundary, it is observed that if the bed wave number is twice the interface wave number then there is a resonant Bragg-type interaction between the interface waves and the bed forms as observed earlier in the literature. Moreover, in case of an ocean-bed covered by an ice-cover, it is observed that when the wave is obliquely incident on the ice-cover surface we always find energy transfer to the interface, but for inter- facial incident waves there are parameter ranges for which no energy transfer to the ice-cover surface is possible. Problems related.Item Analytical Study on Solute Transportation in Streams with Transient Storage and Hyporheic Zones(2009) Kumar, AkhileshThis thesis presents the derivation of general analytical solutions of the transient stor- age model and also of the diDusive transfer model for the longitudinal solute transport in streams with transient storage and hyporheic zones for conservative and reactive solutes. These general analytical solutions are derived by means of Laplace trans- form. The transient storage model deals with the Drst order mass transfer between the main channel of stream and the storage zone, and the diDusive transfer model deals with the diDusive mass transfer of solute between the main channel of stream and the hyporheic zone. Parameters of the transient storage model and also of the diDusive transfer model are estimated for the Uvas Creek tracer experiment by using the large scale Newton reDexive method. The analytical results of these models for conservative solutes are compared with the observed data of the Uvas Creek tracer experiment for chloride concentration. Sensitivity analysis is performed in order to identify the critical parameters on solute concentrations. EDects of diDerent param- eters that represent physical, chemical and hydrological processes involved in the transport of solutes in streams are studied for hypothetical situations. Results are presented for conservative solutes considering step concentration-time proDle as the upstream boundary condition whereas in the case of reactive solutes, an instantaneous release of the solute is considered to be the upstream boundary condition..Item Performance Analysis of Working Vacation Queueing Models in Communication Systems(2010) Goswami, CosmikaThis thesis studies, both analytically and through numerical experiments, the performance of queueing models with a `working vacation' policy arising naturally in communication systems, especially in wavelength division multiplexing (WDM) networks. In a queueing system with this vacation policy, the server switches between vacation and non-vacation periods. Unlike in a classical vacation framework, this system serves customers even during vacation periods but at a lower service rate. We study some working vacation queueing models incorporating diDerent features of system characteristics with a view to assisting optimization and guiding the design of new generation systems. First, we consider a single-server queueing model with working vacations and corre- lated arrivals as network traDc is seldom similar and this often leads to system congestion and packet loss. The model is studied in discrete-time scale by constructing a quasi-birth- death (QBD) process. Since the matrix-geometric method gives an eDcient way to solve homogeneous QBDs, we use this method to analyze and study the performance of the correlated model. The analogous continuous-time model is also outlined. Next, we con- sider a Dnite-buDer model with this working vacation policy and correlated arrivals to lay the emphasis on the role of correlation in arrivals on customer loss probabilities. A multi- server model is presented next, where the servers obey asynchronous multiple working vacation policy and the formulated non-homogeneous QBD process is analyzed using the Dnite truncation method of approximation. A working vacation model with diDerent priority classes of customers is studied as priority based traDc can enhance network eDciency and ensure quality of service (QoS). Explicit expressions for system performance measures are obtained and also comparisons are made for diDerent classes of customers. Another important model considered is with retrial or repeated attempts of customers. This model, mostly seen in mobile networks, is analyzed to obtain closed-form solutions. Finally, a queue with impatient customers is studied with two types of working vacation policies, multiple and single working vacation policy, and comparisons are made to determine the most effective policy....Item Iwasawa Invariants of Elliptic Curves and p_ADIC Measures(2010) Barman, RupamThe central theme of our work is to investigate Iwasawa invariants associated with elliptic curves and p-adic measures. Iwasawa D- and D-invariants of an elliptic curve contain valuable information about the curve. On the other hand, p-adic L-functions over Q arise as D-transforms of certain p-adic measures, hence there is considerable interest in Iwasawa invariants of such measures and their D-transforms. Suppose that E1 and E2 are elliptic curves defined over Q. Let p be an odd prime where E1 and E2 have good ordinary reduction. Assume that E1[pi] D= E2[pi] as Galois modules for i = D(E1)+1. Also assume that both E1(Q)[p] and E2(Q)[p] are trivial. Under the above assumptions we prove that D(E1) = D(E2). Also, if E1[pi] D= E2[pi] as Galois modules for i = D(E1), then D(E1) D D(E2). This result is an extension of earlier works of Greenberg and Vatsal, who studied this problem for elliptic curves with D-invariants zero. We also illustrate our results through some numerical examples. We find a generalization of an existing result of Satoh, Kida and Childress that deals with p-adic measures on Zp to p-adic measures on Zn p for any n. Let O be the ring of integers of a finite extension of Qp, where p is a fixed odd prime. If D is a O-valued measure on Zn p , then it gives a power series in n variables with coefficients in O. Analogous to the case n = 1, we define Iwasawa invariants of such a power series for any n. Given a O-valued measure D on Zn p , one obtains a new measure D = P D12V D D D P Dn2V (D D (D1; D D D ; Dn))jUn on Un while defining the D-transform of D. Extending by 0, D gives a measure on Zn p . We obtain a relation between the Iwasawa invariants of the power series associated to D and the D-transform for any n D 1. We prove the relation by deriving certain p-adic properties of Mahler coefficients of the continuous functions fm(x) = Dux m D and fm1;DDD ;mn(x1; D D D ; xn) = Dux1 m1 D D D D Duxn mn D , where u is a topological generator of 1 + pZp..Item Upwind Based Numerical Methods for Time-Dependent singularly perturbed problems with boundary and interior layers(2010) Mukherjee, KaushikThis thesis provides some efficient numerical techniques for solving time-dependent singularly perturbed problems (SPPs) possessing boundary and interior layers. These types of problems are described by partial differential equations in which the highest spatial derivative is multiplied by an arbitrarily small parameter "", known as Dsingular perturbation parameterD. This leads to the occurrence of boundary (or interior) layers, which are basically thin regions in the neighbourhood of the boundary (or interior) of the domain, where the gradients of the solutions steepen as the perturbation parameter "" tends to zero. Due to this layer phenomena, it is a very difficult and challenging task to provide ""-uniform numerical methods for solving SPPs. The term D""-uniformD is meant to identify those numerical methods in which the approximate solution converges (measured in the supremum norm) independently with respect to the parameter "" to the corresponding exact solution of SPP . The purpose of this thesis is therefore to develop, analyze, improve and optimize the ""-uniform upwind based numerical methods for solving time-dependent singularly perturbed initial-boundary-value problems (IBVPs) with smooth and non-smooth data. This is accomplished by constructing spacial non-uniform meshes resolving boundary and interior layers. At first, a uniformly convergent hybrid numerical scheme is proposed and analyzed on a layer resolving piecewise-uniform Shishkin mesh for singularly perturbed one-dimensional parabolic convection-diffusion IBVP with a regular boundary layer as well as a class of parabolic convection-diffusion IBVPs with strong interior layers. The scheme utilizes a proper combination of the midpoint upwind scheme and the classical central difference scheme for the spatial discretization and the backward-Euler scheme for discretizing the time derivative. The analogous study of a similar kind of hybrid scheme is also made for a class of singularly perturbed mixed parabolic-elliptic IBVPs exhibiting both boundary and interior layers. Further, the efficiency of the hybrid scheme (proposed for 1D parabolic IBVP with smooth data ) is tested by extending it for solving two-dimensional singularly perturbed parabolic convection-diffusion IBVP on a spacial rectangular mesh, utilizing the Peaceman and Rachford method for the time discretization. In all the cases, the newly proposed hybrid schemes attain an almost second-order spatial accuracy. Moreover, a unified theory is derived to obtain an optimal order of convergence of the classical implicit upwind finite difference scheme on Shishkin-type meshes (including the piecewise-uniform Shishkin mesh and the Bakhalov-Shishkin mesh), for a class of singularly perturbed parabolic IBVPs exhibiting strong interior layers. Finally, a post-processing technique (Richardson extrapolation), which improves the accuracy of the standard upwind scheme, is analyzed on a piecewise-uniform Shishkin mesh for singularly perturbed parabolic convection-diffusion IBVP exhibiting a regular boundary layer...Item PERMUTATION POLYNOMIALS AND THEIR APPLICATIONS IN CRYPTOGRAPHY(2010) Singh, Rajesh PratapA polynomial over a finite ring R is called a permutation polynomial of R if it induces a bijection from R to R. Permutation polynomials over finite rings have several applications in combinatorics, coding theory and cryptography. For example, the RC6 block cipher uses the permutation polynomial x + 2x2 over the finite ring Z2n, where 2n is the word size of machine. In 2001, Rivest found an exact characterization of permutation polynomials over finite rings Z2n. However, using Rivest's technique it was di±cult to characterize permutation polynomials over finite rings Zm, for m = 3n; 5n. In this thesis, we present some methods to characterize all permutation polynomials over finite rings Zm for m = 2n; 3n; 5n. In addition, we produce a new class of permutation binomials over the finite fields Zp. Moreover, we show that every polynomial over finite ring Zpn can be expressed as a triangular map over Zn p . Using this representation, we obtain su±cient conditions for a polynomial over Zpn to be a permutation polynomial, for any prime p. Next, we consider permutation polynomials over finite fields. Permutation polynomials over finite fields have been the subject of study for many years. There is a considerable interest in finding new classes of permutation polynomifials over finite fields. However, only a handful of specific classes of permutation polynomials are known so far and very few of the known classes have permutation polynomials commuting with one another. We find certain new classes of permutation polynomials over finite fields. Some of these classes are commutative. Multivariate public key cryptography is a branch of public key cryptography in which cryptosystems are based on the problem of solving nonlinear equations over finite fields. This problem is proven to be NP complete. MIC*, the first practical public key cryptosystem based on this problem, was proposed in 1988 by T. Matsumoto and H. Imai. This cryptosystem was more e±cient than RSA and ECC (Elliptic curve cryptosystems). Unfortunately, this cryptosystem was broken by Patarin. In 1996 Patarin gave a generalization of MIC* cryptosystem called HFE. However, in HFE the secret key computation was not as eficient as in the original MIC* cryptosystem. The basic instance of HFE was broken in 1999. In recent years, designing a public key cryptosystem based on the problem of solving system of nonlinear equations has been a challenging area of research. In this thesis, we have designed two eficient multivariate public key cryptosystems using permutation polynomials over finite fields. We have shown that these cryptosystems are secure against all the known attacks...Item Parameter-Uniform Numerical Methods for Singularly perturbed convection-deffusion boundary-value problems using Adaptive Grid(2010) Mohapatra, JugalThis thesis provides some efficient numerical techniques for solving singularly perturbed convectiondiffusion boundary-value problems exhibiting boundary layers. These singular perturbation problems (SPPs) are described by differential equations in which the highest derivative is multiplied by an arbitrarily small parameter "" known as Dsingular perturbation parameterD. This leads to the occurrence of boundary layers, which are basically thin regions in the neighbourhood of the boundary of the domain, where the gradients of the solutions steepen as the perturbation parameter "" tends to zero. Due to this layer phenomena, it is very difficult and also a challenging task to provide ""-uniform numerical methods i.e., methods in which the approximate solution converges to the exact solution independently with respect to the perturbation parameter, measured in the supremum norm. The convergence properties of some ""-uniform numerical methods are developed and analyzed in this thesis for solving SPPs using nonuniform grids. Especially two types of nonuniform grids are discussed here. They are the well-known piecewise-uniform Shishkin mesh and the newly developed adaptive grid which is based on the equidistribution of a strictly positive monitor function depending on the solution. These nonuniform grids are so chosen as to give a numerical solution that is uniformly accurate with respect to the singular perturbation parameter. This thesis consists of eight chapters. Chapter 1 contains the general introduction and it also provides the motivation and objective for solving SPPs. Chapter 2 presents a brief discussion on the generation of the Shishkin mesh and the adaptive grid and proposes an algorithm for the practical implementation of an adaptive remeshing strategy. A model convection-diffusion problem is solved using the upwind scheme on this adaptively generated grid formed by the arc-length monitor function. Here, the first-order accurate global solution via interpolation and the first-order approximation to the normalized flux are found out on this adaptively generated grid and also their uniform convergence analysis is carried out in the whole domain. The monitor function remains the same irrespective of the location of the boundary layer (on left or right) of the domain which shows the advantage of using such kind of adaptive grid over the Shishkin mesh. Again, the same monitor function is used in Chapter 3 for solving a model convection-diffusion problem with Robin boundary conditions and achieves the optimal rate of convergence in the discrete supremum norm. The adaptive grid idea is then extended for solving the singularly perturbed differential-difference equations with the delay and the shift terms, for the first time in the literature in Chapters 4 and 5. The error analysis is carried out for the classical upwind scheme on the adaptive grid and an optimal first-order convergence is obtained. Next, two higher-order methods are discussed for solving the singularly perturbed delay differential equations on the Shishkin mesh namely: the Richardson extrapolation technique (a postprocessing technique) and the defect-correction method in Chapters 6 and 7, respectively, which give almost second-order convergence improving the almost first-order convergence of the upwind scheme. Finally, Chapter 8 summarizes the results obtai...