Browsing by Author "Singh, Virath Sewnath."

Now showing 1 - 7 of 7

Bounds on the extremal eigenvalues of positive definite matrices.
(2018) Jele, Thokozani Cyprian Martin.; Singh, Virath Sewnath.; Singh, Pravin.
The minimum and maximum eigenvalues of a positive de nite matrix are crucial to determining the condition number of linear systems. These can be bounded below and above respectively using the Gershgorin circle theorem. Here we seek upper bounds for the minimum eigenvalue and lower bounds for the maximum eigenvalue. Intervals containing the extremal eigenvalues are obtained for the special case of Toeplitz matrices. The theory of quadratic forms is discussed in detail as it is fundamental in obtaining these bounds.
A discrete Fourier transform based on Simpson's rule.
(2013) Sibandze, Dan Behlule.; Singh, Pravin.; Singh, Virath Sewnath.
Fourier transforms are mathematical operations which play a vital role in the analysis of mathematical models for problems originating from a broad spectrum of elds. In this thesis, we formulate a discrete transform based on Simpson's quadrature for 4m + 2 quadrature nodes and analyse its various properties and provide detailed proofs thereof. In addition, we make applications to encryption and watermarking in the frequency domain.
A discrete Hartley transform based on Simpson's rule.
(2015) Ramsunder, Ashai.; Singh, Pravin.; Singh, Virath Sewnath.
The Discrete Hartley Transform and Discrete Fourier Transform are classical transfor- mations designed for e cient computations in the frequency domain. We introduce a relatively new transformation based on the existing Discrete Hartley Transform by applying Simpson's quadrature for N = 4m + 2 quadrature nodes. The majority of our investigation involves exploring the mathematical properties satis ed by our newly derived transformation. We formulate the convolution and cross correlation properties both in the real and frequency domain. An intensive spectral analysis is performed to ascertain the multiplicities of the eigenvalues corresponding to the transformation matrix.
Eigenvalue bounds for matrices.
(2018) Khambule, Pretty Nombuyiselo.; Singh, Pravin.; Singh, Virath Sewnath.
Eigenvalues are characteristic of linear operators. Once the spectrum of a matrix is known then its Jordan Canonical form can be determined which simplifies the un- derstanding of the matrix. For large matrices and spectral analysis sometimes it is only necessary to know the eigenvalues of smallest and largest absolute values. Hence we consider various strategies of bounding the spectrum in the complex plane. Such bounds may be numerically improved by various algorithms. The minimal and maximal eigenvalues are crucial to determine the condition number of linear systems.
Fixed point theory in various generalized metric-type spaces.
(2022) Jele, Thokozani Cyprian Martin.; Singh, Virath Sewnath.; Singh, Pravin.
In the theory of fixed points, there are numerous articles dealing with generalization of the basic Banach contraction mapping principle. There has been two lines of approach. The first one is concerned with generalizations of the contractive conditions on the mapping space. The other line of investigation deals with various generalizations of the metric spaces and the results that can be obtained in these new frameworks, referred to as metric-type spaces. In this thesis, we elected for the latter approach by providing a more general framework for a b-metric space , G-metric space and S-metric space. In this thesis, we proved that these new metric-type spaces equipped with various contractions type mappings have unique fixed points and provide numerous examples of each metric-type spaced mentioned.
A new approach to ill-posed evolution equations : C-regularized and B- bounded semigroups.
(2001) Singh, Virath Sewnath.; Banasiak, Jacek.
The theory of semigroups of linear operators forms an integral part of Functional Analysis with substantial applications to many fields of the natural sciences. In this study we are concerned with the application to equations of mathematical physics. The theory of semigroups of bounded linear operators is closely related to the solvability of evolution equations in Banach spaces that model time dependent processes in nature. Well-posed evolution problems give rise to a semigroup of bounded linear operators. However, in many important and interesting cases the problem is ill-posed making it inaccessible to the classical semigroup theory. One way of dealing with this problem is to generalize the theory of semigroups. In this thesis we give an outline of the theory of two such generalizations, namely, C-regularized semigroups and B-bounded semigroups, with the inter-relations between them and show a number of applications to ill-posed problems.
Solutions of the Volterra integro-differential equation.
(2018) Ali, Yusuf.; Singh, Virath Sewnath.; Singh, Pravind Sewsanker.; Narain, Rivendra Basanth
Integro-di erential equations has found extensive applications in the eld of engineering, sciences and mathematical modelling of various physical and biological phenomena. In this thesis we focus on the Volterra type integro-di erential equation which has been used to model biological species co-existing, heat di usion, electromagnetic theory etc. In recent years much research has focused on nding approximate solutions of the integro-di erential equation by polynomial methods, speci cally focusing on the Lagrange collocation and piecewise cubic Hermite collocation methods. A further aspect to the thesis will be on analytical methods, mainly the applications of Lie group theory to the Volterra type equation. Lie group theory is one of the most powerful methods applied to obtain solutions of di erential equations. We will present the linear independent symmetries of the Volterra type equation of the rst and second kind. In addition, we shall apply the Laplace transform and it's inverse to determine general solutions for selected forms of kernel, speci cally those with convolution integrands.