Editor(s)

Dr. Manuel Alberto M. Ferreira
Professor,
Department of Mathematics, ISTA-School of Technology and Architecture,  Lisbon University,  Portugal.

ISBN 978-93-90768-00-4 (Print)
ISBN 978-93-90768-06-6 (eBook)
DOI: 10.9734/bpi/tpmcs/v7

This book covers key areas of mathematics and computer science. The contributions by the authors include susceptibility, impulse response function, Hilbert-transform, Fourier transform, conjugate Fourier transform, convolution, causal systems, non-causal systems, analyticity, Kramer-Kronigs relation, relaxation, retardation, epidemics model, hyper geometric function, inequalities, communicable diseases, Concept Lattice, formal concept, frequent pattern, association rules, landmark, Convection-diffusion-reaction equation, alternating-direction implicit scheme, traveling wave solutions, linear and nonlinear differential equations, best proximity point, Geraghty function, Geraghty contraction, series expansion, fluid models, intermolecular potential model, integral Equations, Timoshenko system, exponential stability, distributed delay, pattern recognizer, frame homomorphism, quotient frames, matrix inversion, matrix multiplication, recursive algorithm, matrix order expansion, matrix order condensation, Jackson operator, generalized modulus of smoothness, K-function. This book contains various materials suitable for students, researchers and academicians in the field of mathematics and computer science.

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### Chapters

##### Causality Principle- Simplified Deliberation and Explanation of Its Complex Mathematics

Shantanu Das

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 1-50
https://doi.org/10.9734/bpi/tpmcs/v7/6599D

Strike a bell, and then we hear the sound of its gong and not before striking. This is statement of Causality is Universal phenomena; and is a ‘matter of fact’- taken very casually. The information about theory of Causality is too scattered, and is not concise, and is lost is ‘theory of complex analysis’-making the understanding of this ‘matter-of-fact’ phenomena very complicated. In the available literatures authors use the mathematical formulas without explaining them thoroughly, and their practical utility seems missing; and gets lost in the complications. The formulas that are used do not carry out much detailed and elaborate steps that give readers jitters. The purpose of this Chapter and its deliberation with detailed derivations is to make stringent presentation of the Principles of Causality and develop the mathematics in a simplified way, and still make the purpose of applications in mind. A simple principle of nature that is: ‘the effect can only happen after the cause’, i.e. called causality has great mathematical treatment and development we term that as Kramer-Kronigs relation, analyticity, Titchmarsh principle etc. Like the statement- ‘a causal response function is analytic in upper-half of the complex plane’-sounds very complicated and abstract. Here in this Chapter we try to give elaborate treatment, on all the mathematical seemingly complicated and abstract statements and expressions.

Though in terms of ‘complex-analysis’ the Causality Principles looks very complicated and too abstract,  here in this Chapter we simplify the derivation of analyticity Kramer-Kronigs relations and obtain these expressions in time and frequency domains. We start from the basics of Impulse Response Function or Green’s Function and then we define the generalized susceptibility function. Thereafter we elaborate by use of Fourier transformation techniques the Kramer-Kronig relations in frequency domain and later also in time domain. This method gets applied to various fields, i.e. in impedance studies, in dielectric relaxation/retardation studies, in refractive index studies, in electric polarization studies, in magnetic systems studies, in stress-strain relaxation studies etc. Even we if we make an artificial material with negative permittivity and negative permeability (thus showing negative refraction), it should and must follow the mathematical tests of causality that is Kramer-Kronigs relation. In this Chapter the examples that we consider are for simple Debye systems, however, the theory and principles, which we deliberate can be extended to non-Debye systems-as well.  We are not sure about causality theory that is developed and discussed here if it can be applied to non-differentiable systems i.e. response function defined on fractal support? –perhaps a new formal mathematics needs to be developed in this regard. Our discussion is only for continuous and differentiable systems.

We make declaration- the contents of this Chapter are not new rather existed since 1930’s yet, the theory and its mathematics were difficult to grasp and also to teach. That is because information about the details are too scattered. This Chapter will help as teaching matter to the physics, engineering and mathematics students- and readers will find the explanations and detailed derivations useful in their research work.

##### Study on Some Inequalities of a Formula of Population Size due to Epidemics Model Problem

Hemant Kumar

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 51-56
https://doi.org/10.9734/bpi/tpmcs/v7/2800D

In the present chapter, we evaluate a formula of population size due to epidemics model problem for communicable diseases in which the daily contact rate is supposed to be varied with population size N (t) that is large so that it is considered as a continuous variable of time t. Then, we obtain some inequalities of above analytic formula of population size. We present an analysis to discuss the epidemics. These results may also be useful to plot some graphs on population dynamics of the system.

##### An Approach of Concept Lattice Theory in Data Mining and Its Applications

Pascal Sungu Ngoy, Kaninda Musumbu, Nathalie Wandji

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 57-75
https://doi.org/10.9734/bpi/tpmcs/v7/6864D

Concept lattice has been proven to be a very effective tool and architecture for data mining in general. It is widely used for data analysis and knowledge discovery and various concept lattice based approaches are used depending on the type of data. This extended version paper aims at presenting one application of the lattice theory in text mining and another one in image mining.

In the first approach, the notion of lattice theory has been applied by using one of its components mostly used in data mining, the formal concept analysis which has a powerful method, the association rule extraction which helps to find in a database patterns which appear frequently together.

In the second one, the use of the lattice theory for image sets characterization has been shown by using landmarks to enable a machine to automatically classify objects with respect to the image class they belong to. For text mining, association rules discovery, mostly uses formal concept analysis to analyze the relations between patterns which appear at the same time.

##### Investigation of Numerical Simulation and New Traveling Wave Solutions of Convection-Diffusion Equation with Reaction

M. A. Ashabrawy, E. E. El-Behadi

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 76-84
https://doi.org/10.9734/bpi/tpmcs/v7/6623D

Throughout this paper we will discuss and analyze of the convection-diffusion equation which included a reaction term. First, we will apply an alternating-direction implicit (ADI) scheme akin to that proposed by Polezhaev. Use of this implicit operator-splitting scheme allows the application of a tridiagonal Thomas solver to obtain the solution of steady convection-diffusion equation with reaction. According to the unsteady case, we will use the improved (G' /G) -expansion method to construct the traveling wave solutions, where (G' /G) satisfies a second order linear ordinary differential equation. In this paper we will explore new applications of this method to some special nonlinear convection-diffusion equation with reaction. The new type of exact travelling wave solution obtained in this paper might have significant impact on future researches.

##### Best Proximity Point for a New Class of Proximal Contraction

Anil Kumar

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 85-92
https://doi.org/10.9734/bpi/tpmcs/v7/7227D

Here, a new class of contraction namely generalized (a-u) h-f Geraghty proximal contraction is introduced. Then after, we proved a best proximity theorem in a complete metric space which ensure the existence of best proximity point for such contraction. Our result extends the best proximity point theorem obtained by Hamzehnejadi and Lashkaripour (Fixed Point theory and Applications, 2016) and other related fixed point theorems existing in the literature.

##### The Half way Series Expansion

Alpha Ibrahim Turay

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 93-108
https://doi.org/10.9734/bpi/tpmcs/v7/1792D

A new formula for mathematics is derived which gives the upper half of the series expansion of the expression (ay+b)2n , where n  is a natural number. A proof of the new formula is given followed by a simple example to test its accuracy. This formula is helpful whenever n is  large.

##### Nonexistence of Global Solutions of Some Nonlinear Ultra-Parabolic Equations on the Heisenberg Group

Lamairia Abd Elhakim, Haouam Kamel

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 109-124
https://doi.org/10.9734/bpi/tpmcs/v7/2386E

This chapter provides sufficient conditions for non existence Global weak solutions for non-local and non-linear equivalent equations on HN × (0,$$\infty$$) × (0,$$\infty$$), where HN is the Heisenberg group. Our method of proof relies on a suitable choice of a test function and the weak formulation approach of the sought for solutions.

##### Study on Non-Intrusive Context Aware Transactional Framework to Derive Business Insights on Big Data

Siva Chidambaram, P. E. Rubini, V. Sellam, S. Venkata Lakshmi

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 125-132
https://doi.org/10.9734/bpi/tpmcs/v7/6603D

To convert invisible, unstructured and time-sensitive machine data into information for decision making is a challenge. Tools available today handle only structured data. All the transaction data are getting captured without understanding its future relevance and usage. It leads to other big data analytics related issue in storing, archiving, processing, not bringing in relevant business insights to the business user. In this paper, we are proposing a context aware pattern methodology to filter relevant transaction data based on the preference of business. The complexity and cost factor involved in data management like storing, archiving, backup, recovery, etc. can be reduced by this framework.

##### Studies on Quotient Frames and Filters

Rajesh K. Thumbakara

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 133-138
https://doi.org/10.9734/bpi/tpmcs/v7/7070D

Frame theory is the study of topology based on its open set lattice and it was studied extensively by various authors. In this paper we introduce the notion of quotient frames using filters and study the relation between the filters of the given frame and the filters of the corresponding quotient frame.

##### Study on Matrix Inverse as by-Product of Determinant

Feng Cheng Chang

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 139-158
https://doi.org/10.9734/bpi/tpmcs/v7/2406E

The determinant of a given square matrix is obtained as the product of pivot elements evaluated via the iterative matrix order condensation. It follows as the by-product that the inverse of this matrix is then evaluated via the iterative matrix order expansion. The fast and straightforward basic iterative procedure involves only simple elementary arithmetical operations without any high mathematical process. Remarkably, the revised optimal iterative process will compute without failing the inverse of any square matrix within minutes, be it real or complex, singular or nonsingular, and amazingly enough even for size as huge as 999x999. The manually extended iteration process is also developed to shorten the iteration steps, if the calculation of small size inverse matrices is feasible.

##### Detailed Study on Characterization of Best Algebraic Approximation by a Generalized Modulus of Smoothness

Teodora Zapryanova, Diko Souroujon

Theory and Practice of Mathematics and Computer Science Vol. 7, 12 February 2021, Page 159-169
https://doi.org/10.9734/bpi/tpmcs/v7/7352D

The paper presents Jackson inequality and the corresponding inverse inequality for the best algebraic approximation in terms of the generalized moduli of smoothness. Moreover the equivalence between the generalized modulus and the Butzer-Stens modulus is shown.