i-manager's Journal on Mathematics (JMAT)


Volume 11 Issue 2 July - December 2022

Research Paper

On Kolmogorov Complexity of Unitary Transformations in Quantum Computing

Alexei Kaltchenko*
Wilfrid Laurier University, Waterloo, Ontario, Canada.
Kaltchenko, A. (2022). On Kolmogorov Complexity of Unitary Transformations in Quantum Computing. i-manager’s Journal on Mathematics, 11(2), 1-7. https://doi.org/10.26634/jmat.11.2.19190

Abstract

We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the context of quantum computing, it corresponds to the least possible amount of data to define and describe a quantum circuit or quantum computer program. Our Kolmogorov complexity of unitary transformation is built upon Kolmogorov "qubit complexity" of Berthiaume, W. Van Dam and S. Laplante via mapping from unitary transformations to unnormalized density operators, which are subsequently "purified" into unnormalized vectors in Hilbert space. We discuss the optimality of our notion of Kolmogorov complexity in a broad sense.

Research Paper

A Second Derivative Block Method Derived from a Family of Modified Backward Differentiation Formula (BDF) Type for Solving Stiff Ordinary Differential Equations

Kaze Atsi* , Lydia Adiku**, Namuma Yarima***, G. M. Kumleng****
* Department of Mathematics, Federal University Gashua, Yobe State, Nigeria.
**-*** Department of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.
**** Department of Mathematics, University of Jos, Plateau State, Nigeria.
Atsi, K., Adiku, L., Yarima, N., and Kumleng, G. M. (2022). A Second Derivative Block Method Derived from a Family of Modified Backward Differentiation Formula (BDF) Type for Solving Stiff Ordinary Differential Equations. i-manager’s Journal on Mathematics, 11(2), 8-12. https://doi.org/10.26634/jmat.11.2.19206

Abstract

In this work, a second derivative block method derived from a family of modified backward differentiation formula (bdf) type for solving stiff ordinary differential equations has been constructed. Choosing a step number, k = 4, four discrete methods with uniform order 7 are obtained using the multistep collocation approach. The stability properties of the new method have been established. The solutions of two problems have been computed and compared with the corresponding exact and other existing solutions. Solutions are presented on graphs and the associated absolute errors are compared in tables.

Research Paper

Equiprime Ideals and Equiprime Semimodules in Boolean Like Semirings

Bhagavathula Venkata Narayana Murthy*
Department of Mathematics, Maharaj Vijayaram Gajapathi Raj College of Engineering Chintalavalsa, Vizianagaram, Andhra Pradesh, India.
Murthy, B. V. N. (2022). Equiprime Ideals and Equiprime Semimodules in Boolean Like Semirings. i-manager’s Journal on Mathematics, 11(2), 13-18. https://doi.org/10.26634/jmat.11.2.18889

Abstract

In this paper, the concepts of equiprime ideals and equiprime semi-modules in Boolean like semirings (now onwards denote ℜB) are introduced and illustrated with examples. It is established that if Ҏ is the equiprime ℜB–Ideal of the ℜB –module Ń then (Ҏ: Ń) is equiprime ideal of ℜB. Also it is approved that if Ń is an equiprime ℜB–module then (0: Ń) is an equiprime ideal of ℜB. Also, an important characterization of the equiprime ideal of ℜB in the semimodule structure over ℜB is put forward. Using this result, it can be proved that ℜB is equiprime if there exists a faithful equiprime ℜB–module Ń. Finally, it is proved that if Ń is an equiprime ℜB–module and Ϯ is an invariant subgroup of ℜB such that Ϯ is not contained in (0: Ń), then Ń is an equiprime Ϯ–module.

Mathematics Subject Classification: 16Y30, 16Y80.

Research Paper

Numerical Solution of Temperature Profile in Annulus

Reshu Gupta*
Applied Science Cluster, University of Petroleum and Energy Studies, Dehradun, India.
Gupta, R. (2022). Numerical Solution of Temperature Profile in Annulus. i-manager’s Journal on Mathematics, 11(2), 19-25. https://doi.org/10.26634/jmat.11.2.19035

Abstract

The model of temperature profile in the flow of non-Newtonian second-order fluid, flowing in an annulus, is investigated. The boundary of annulus is considered porous. Appropriate similarity transformation is used to convert non-linear PDEs into non-linear ODEs. The obtained differential equations are solved numerically. The nature of the temperature profile is presented graphically for the various physical parameters.

Review Paper

Mathematical Modelling of EOR Methods

Tirumala Rao Kotini* , Aman Singh**
*-** Energy Cluster, School of Engineering, UPES, Bidholi, Dehradun, Uttarakhand, India.
Kotni, T. R., and Singh, A. (2022). Mathematical Modelling of EOR Methods. i-manager’s Journal on Mathematics, 11(2), 26-34. https://doi.org/10.26634/jmat.11.2.19034

Abstract

The choice of appropriate and affordable procedures to boost oil recovery is usually recognized as one of the main challenges in reservoir development due to the huge demand for crude oil. Reservoir flow simulators are valuable tools for understanding and forecasting fluid flow in complex systems. The goal of this study is to run a mathematical model to evaluate the performance of various oil recovery methods, as well as to validate the model's accuracy with simulated field data. Thereby, the results of this developed model indicate that the model is approximately matched with the simulated field data. Enhanced oil recovery typically refers to chemical, miscible, thermal, and microbial processes. A system of nonlinear partial differential equations composed of Darcy's and mass conservation equations governs the model. The system is then numerically solved using the IMPEC (Implicit Pressure and Explicit Concentration) scheme by a finite difference method. We chose this approach because the experimental approaches are not only time consuming, but also costly. As a result, mathematical models could aid in the understanding of a reservoir and how such processes can be optimized to maximize oil recovery while lowering production costs. This paper provides a brief overview of mathematical modelling of various enhanced oil recovery methods, focusing on developing a generalized framework and describing some of the key challenges and opportunities.