Paper's Title:

A New Relaxed Complex-valued b-metric Type and Fixed Point Results

Author(s):

P. Singh, V. Singh and T. C. M. Jele

Department of Mathematics, University of KwaZulu-Natal,
Private Bag X54001, Durban,
South Africa.
E-mail: singhp@ukzn.ac.za
singhv@ukzn.ac.za
thokozani.jele@nwu.ac.za

Abstract:

In this paper, we study the existence and uniqueness of fixed point in complex valued b-metric spaces and introduce a new relaxed α, β Complex-valued b-metric type by relaxing the triangle inequality and determine whether the fixed point theorems are applicable in these spaces.

Paper's Title:

A Generalization of a Partial b-metric and Fixed Point Theorems

Author(s):

Pravin Singh and Virath Singh

Department of Mathematics, University of KwaZulu-Natal,
Private Bag X54001, Durban,
South Africa.
E-mail: singhp@ukzn.ac.za
singhv@ukzn.ac.za

Abstract:

The purpose of this paper is to introduce the concept of a Partial α, β b-metric as a generalization of a partial b-metric and prove theorems for some contractive type mapping.

Paper's Title:

Oblique Projectors from the Simpson Discrete Fourier Transformation Matrix

Author(s):

P. Singh and V. Singh

School of Mathematics, Computer Science and Statistics,
University of Kwazulu-Natal,
Private Bag X54001, Durban 4001,
South Africa.
E-mail: singhp@ukzn.ac.za, singhv@ukzn.ac.za

Abstract:

In this paper we examine the projectors of the Simpson Discrete Fourier Transform matrix of dimension two modulus four and show how they decompose the complex vector space into a direct sum of oblique eigenspaces. These projection operators are used to define a Simpson Discrete Fractional Fourier Transform (SDFRFT).

Paper's Title:

A New Relaxed b-metric Type and Fixed Point Results

Author(s):

P. Singh, V. Singh and Thokozani Cyprian Martin Jele

Department of Mathematics,
University of KwaZulu-Natal,
Private Bag X54001, Durban,
South Africa.
E-mail: singhp@ukzn.ac.za, singhv@ukzn.ac.za, thokozani.jele@nwu.ac.za

Abstract:

The purpose of this paper is to introduce a new relaxed α, β b-metric type by relaxing the triangle inequality. We investigate the effect that this generalization has on fixed point theorems.

Paper's Title:

Introducing the Picard-S3 Iteration for Fixed Points

Author(s):

Pravin Singh, Virath Singh and Shivani Singh

University of KwaZulu-Natal,
School of Mathematics Statistics and Computer Sciences,
Private Bag X54001,
Durban, 4000
South Africa.

Unisa,
Department of Decision Sciences,
PO Box 392,
Pretoria, 0003
South Africa.
E-mail: singhprook@gmail.com
singhv@ukzn.ac.za
shivstarsingh@gmail.com

Abstract:

In this paper we introduce a three step iteration method and show that it can be used to approximate the fixed point of a weak contraction mapping. Furthermore, we prove that this scheme is equivalent to the Mann iterative scheme. A comparison is made with other three step iterative methods by examining the speed of convergence. Results are presented in tables to support our conclusion.

Paper's Title:

Results on Bounds of the Spectrum of Positive Definite Matrices by Projections

Author(s):

P. Singh, S. Singh, V. Singh

University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.
E-mail: singhp@ukzn.ac.za

University of South Africa, Department of Decision Sciences,
PO Box 392, Pretoria, 0003,
South Africa.
E-mail: singhs2@unisa.ac.za

University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.
E-mail: singhv@ukzn.ac.za

Abstract:

In this paper, we develop further the theory of trace bounds and show that in some sense that the earlier bounds obtained by various authors on the spectrum of symmetric positive definite matrices are optimal. Our approach is by considering projection operators, from which several mathematical relationships may be derived. Also criteria for positive lower bounds are derived.

Paper's Title:

Eigenvalue Bounds based on Projections

Author(s):

Pravin Singh, Shivani Singh, Virath Singh

University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.

University of South Africa,
Department of Decision Sciences, PO Box 392,
 Pretoria,0003,
South Africa.

University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.

E-mail: singhp@ukzn.ac.za, singhs2@unisa.ac.za, singhv@ukzn.ac.za
 

Abstract:

In this paper, we derive expressions for the bounds of the extremal eigenvalues of positive definite matrices. Our approach is to use a symmetric projection operator onto an n-2 dimensional subspace of the real space of n tuples. These bounds are based on traces of the matrix and its powers. They are relatively easy and inexpensive to compute.

Paper's Title:

Recursive Bounds for the Eigenvalues of Symmetric Positive Definite Matrices

Author(s):

Pravin Singh, Shivani Singh, Virath Singh

University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.

University of South Africa,
Department of Decision Sciences, PO Box 392,
 Pretoria,0003,
South Africa.

University of KwaZulu-Natal,
Private Bag X54001,
Durban, 4001,
South Africa.

E-mail: singhp@ukzn.ac.za, singhs2@unisa.ac.za, singhv@ukzn.ac.za
 

Abstract:

In this paper, we bound the extremal eigenvalues of a positive definite real symmetric matrix by considering a part of the characteristic equation in the region of the smallest and largest eigenvalues. An expansion around these values leads to a sequence of monotonic functions, whose zeros coincide with the extremal zeros of associated polynomials. The latter is shown to yield bounds that are fairly accurate.

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