Editor(s)

Dr. Luigi Giacomo Rodino
Professor,
Department of Mathematics, University of Turin, Italy.

ISBN 978-93-91595-24-1 (Print)
ISBN 978-93-91595-32-6 (eBook)
DOI: 10.9734/bpi/ctmcs/v7

This book covers key areas of mathematics and computer science. The contributions by the authors include Thomas equation, generalised tanh method, travelling wave hypothesis method, exact solution, solitary wave solution, mean graph, near-mean graph, direct product, Cartesian product, decomposition of graph, continuous monotonic decomposition, arithmetic decomposition, odd star decomposition, spZc –closed, strongly spZc-connected space, spZc-continuous, spZc-continuum, Lyness’ equations, periodic solution, Age-structured model, BLL- model, age-dependent harvesting, modified Leslie matrix, b-metric space, b2-metric space, weakly compatible, contractive conditions, computable real numbers, computable real functions, algebra of real functions, classification of real functions, extended rectangular B- metric spaces, almost alpha-Suzuki generalized rational type contraction conditions, alpha-Suzuki Ciric type contractions, common fixed points, transfinite cardinals, aleph zero, power set, twin prime conjecture, binary conversion, decimal conversion, decimal to binary conversion, digital system, binary system, Hankel-Schwartz transformation, bounded linear operator, generalized transformation, distributions, Bessel function, logistics modeling, estimation of plant equipment, assessing equipment capacity, optimization of plant layout, transport modeling, complete graphs, corona, graph operations, join, weakly connected dominating sets, weakly connected domination number, informal learning, E-learning, problem-based learning, social learning, cooperation, mentoring, degenerate transformation, discrete transitions, nonidentical and identical relations, non-integrable manifolds, skew-symmetric forms. This book contains various materials suitable for students, researchers and academicians in the field of mathematics and computer science.

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### Chapters

##### Study on Analytic Solutions of the Thomas Equation by Generalized Tanh and Travelling Wave Hypothesis Methods

K. S. Al-Ghafri

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 1-7
https://doi.org/10.9734/bpi/ctmcs/v7/3632F

Analytical solutions to the Thomas equation are studied. The Thomas equation is one of the important equations in physical sciences. The wave transformation is used to transform the Thomas equation from a partial differential equation (PDE) to an ordinary differential equation (ODE). Exact solutions for the Thomas equation are obtained using both the generalised tanh and the travelling wave hypothesis methods.

##### Study of Mean Labelings on Product Graphs

Teena Liza John, T. K. Mathew Varkey

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 8-16
https://doi.org/10.9734/bpi/ctmcs/v7/3317F

Let G be a(p,q) graph and let: $$V(G) \rightarrow$$ {O,1,…,q} be an injection. Then G is said to have a mean labeling if for each edge uv, there exists an induced injective map $$f^*:E(G) \rightarrow$$ {1,2,…,q} defined by

$$f^* (uv)= {f(u) + f(v) \over 2}$$ if $$f(u)+f(v)$$ is even

$$= {f(u) + f(v) + 1 \over 2}$$ if $$f(u)+f(v)$$ is odd

The graph G is said to be a near mean graph if the injective map f : V(G)$$\rightarrow$${1,2,…,q-1,q+1} induces $$f^*:E(G)\rightarrow$${1,2,…,q} which is also injective, defined as above. In this paper , We looked at the direct product of paths and augmented star graphs for their mean labelings and the Cartesian product of Pn and K4 for its near- meanness in labelings.

##### Arithmetic Odd Star Decomposition of Graphs

E. Ebin Raja Merly

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 17-23
https://doi.org/10.9734/bpi/ctmcs/v7/3469F

Let G = (V, E) be a simple connected graph with p vertices and q edges. If G1, G2, …., Gn are connected edge disjoint subgraphs of G with E(G) = E(G1) $$\cup$$E(G2)$$\cup$$….$$\cup$$E(Gn), then (G1, G2, ……, Gn) is said to be a decomposition of G. A decomposition (G1, G2, ……, Gn) of G is said to be continuous monotonic decomposition if each Gi is connected and |E(Gi)| = i for every i = 1, 2, …., n. [1]. In this chapter, we introduced the concept Arithmetic odd star decomposition. A decomposition (G1, G2, ……, Gn) of G is said be an Arithmetic decomposition if |E(Gi) = a + (i-1)d, for every I = 1, 2 ……, n and a, d $$\in$$ Z. Clearly q=$$\frac{n}{2}$$[2a + (n - 1) d]. If a = 1 and d = 1, then q = $$\frac{n(n+1)}{2}$$. In this chapter, we study the graphs with a = 1 and d = 2. Then q = n2. That is, the number of edges of G is a perfect square.

##### Various Concepts of spZc-open Sets in Topology

R. M. Sivagama Sundari, A. P. Dhana Balan

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 24-31
https://doi.org/10.9734/bpi/ctmcs/v7/3342F

Connectedness is one of the principal property in the study of topological spaces. Weierstrass introduced the notion of arcwise connectedness. However Cantor introduced the notion of Connectedness which we are using today. Connectedness is mainly used to distinguish topological spaces. In this book chapter, we introduce the concept of spZc-connectedness and separate sets using Zc-open sets. Some properties and theorems using spZc-continuous, spZc-irresolute through spZc –open sets are also discussed. We also introduce the concept of  spZc-open set strongly spZc-connectedness , spZc-continuum using  the concept of spZc-open sets. Some properties and theorems using locally spZc-connected space are also discussed.

##### Study on Periodicity and Transformation of Difference Equations

Khalil I. T. Al-Dosary

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 32-41
https://doi.org/10.9734/bpi/ctmcs/v7/11651D

In this research we develop a method to construct periodic dierence equations with desired period. Also, we investigate the transformation of periodicity of dierence equations under invertible maps. These tech- niques are applied to generalize and extend some results that have been obtained recently. One of the interesting results is derivation of some generalized Lyness equations.

##### Study on Single Species One Sex Age-structured Fish Population Model with Restricted Harvesting

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 42-49
https://doi.org/10.9734/bpi/ctmcs/v7/10528D

A discrete age-structured Bernardelli, Lewis and Leslie (BLL) population model for a single species is formulated under age dependent harvesting condition. Here, after an initial period during which the growth of the species is not much, harvesting is performed. Its rate is proportional to the available bio-mass (number of species) of different age group population and decreases with the age of the species by considering monotonic decreasing catch per unit effort parameter. The modified Leslie matrix for the present models is derived. Stability of the system is studied from the ratio of the population densities with respect to times. This kind of study is very important in population biology as to get an idea about the future biomass and sustainability of the species.

##### Study on Fixed Point Theorems of Weakly Compatible Mappings in b2-Metric Space Satisfying Contractive Conditions

Thokchom Chhatrajit Singh, Yumnam Rohen Singh, K. Anthony Singh

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 50-64
https://doi.org/10.9734/bpi/ctmcs/v7/3281F

The concept of 2-metric space and b-metric space.  Mustafa et.  al. (Z. Mustafa. V, Paraneh, J. Razaei and  Z. Kadulberg: b2 -metric spaces and  some fixed point theorems, Fixed Point Theory  and  Applications 2014,  2014:144)  introduced b2 -metric space  are generalised. In this  paper, we prove  a common fixed point theorem for two pairs  of weakly  compatible mappings satisfying ($$\varphi$$, $$\psi$$) contractive condition in b2 -metric space.

##### The Set of Real Functions are Countable in Applied Mathematics, Algebra of the Functions and Their Classification

M. A. Malkov

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 65-71
https://doi.org/10.9734/bpi/ctmcs/v7/3406F

The set of real functions is countable since the functions must be computable, i.e. there must be an algorithm for computing them. But the set of algorithms is countable. Uncomputable functions are useless, they do not exist in applied mathematics. The set of computable real numbers is also countable. Uncomputable numbers are useless. The definition of algebra of computable real functions is given and a classification of subalgebras with one-element bases is constructed. This classification is a classification of functions too. Algebras with multielement bases are fictitious, they are useless for classification of functions. All infinite sequences of inclusions of subalgebras are constructed.

##### Common Fixed Points for a Pair of Selfmaps in Extended Rectangular B-Metric Spaces for Almost Alpha-Suzuki Non Linear Type Contractions

M. V. R. Kameswari

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 72-87
https://doi.org/10.9734/bpi/ctmcs/v7/3234F

In this paper, we define almost alpha-Suzuki generalized rational type contraction and almost alpha- Suzuki Ciric type contraction for a pair of self maps in extended rectangular B-metric spaces and prove the existence of common fixed points for these contractions. The presented results are supported by examples. Our results improve and generalize the many results in the literature. As a consequence, we applied our results to study the existence of common fixed point theorems equipped with a directed graph.

##### Study on the Proof of the Existence of Transfinite Cardinals Strictly Smaller than $$\aleph_\circ$$ with an Ensuing Solution to the Twin Prime Conjecture

Karan Doshi

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 88-98
https://doi.org/10.9734/bpi/ctmcs/v7/3399F

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than , the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.

##### A New Method for Decimal - Binary Conversion: Interpret as Octal - Binary Conversion

Lianly Rompis

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 99-106
https://doi.org/10.9734/bpi/ctmcs/v7/3611F

In digital technology designs and digital machine operations normally people use binary number system instead of decimal number system. Digital devices work with digital technologies that use digital signals to communicate between devices. This is the reason why decimal to binary conversion becomes so important. By converting decimal into binary, it would be very easy to convert the binary number into octal or hexadecimal. So far the division by base-2 has been used as a standard method to convert decimal numbers into binary numbers. This research brings a different method for conducting decimal to binary conversion where this conversion could be done using octal number system. Octal number system helps interpret decimal into octal and the octal into binary. Literature studies, comparisons, analysis and evaluation have been made to derive the solution for this alternative new method. This method is reliable and flexible to use for education, learning process, even for programming.

##### Bounded Linearness of Distributional Generalized Hankel-Schwartz Type Transformations on $$L^{'}_{p,v}$$ Spaces

B. B. Waphare

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 107-113
https://doi.org/10.9734/bpi/ctmcs/v7/3436F

Two Hankel-Schwartz type transformations are defined in this study. On the Lp,v and $$L^{'}_{p,v}$$ spaces, the Hankel-Schwartz type transformations are defined. It is also demonstrated that the transformations defined by (1) and (2) are bounded linear operators of Lp,v. We also study the behaviour of transformations defined by (3) and (4) on the Lp,v spaces. It is also shown that the transformations defined by (3) and (4) are bounded linear operator of Lp,v into Lp,2p(3$$\alpha$$+$$\beta$$)-v and of L p,v into Lp,-v-2($$\alpha$$-$$\beta$$)p respectively. Finally, we proved that distributional generalised HankelSchwartz type transformation on $$L^{'}_{p,v}$$ spaces are bounded linear.

##### Describing the Role of Modeling and Simulation in Estimation of Plant Equipment Requirement and Layout Design

Prabhakar Sastri

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 114-124
https://doi.org/10.9734/bpi/ctmcs/v7/11656D

The objective of this study is to develop modeling tools which help design large plants. The method can be applied to a new plant as well as for expansion of an existing plant.

The first requirement is the facilities that need to be put in place to meet the business objectives. Once the objectives are clear and the major facilities decided the model is built to assess the infrastructure, facilities planning, ancillary requirements etc.

The first example is discussed to evidence the usefulness of the technique is that of the expansion of a steel plant from million 2.5 tons/year to 4 million tons/year.

The model was then being expanded to a second different steel plant facility with different infrastructure for its expansion of from million 4.6 million tons/year to 10 million tons/year.

The model is currently used to a third facility with different infrastructure. It is that of the expansion of a treatment facility. The facilities for this plant are very different from the first two. This just shows that the usefulness of the technique shows that it can be used in different production environments.

##### Weakly Connected Domination in Graphs Resulting from Binary Operations

Elsie P. Sandueta, Sergio R. Canoy Jr.

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 125-128
https://doi.org/10.9734/bpi/ctmcs/v7/3592F

Let G = ( V (G),E (G) ) be a connected undirected graph. The closed neighborhood of any vertex v $$\epsilon$$ V (G) is NG[v] = {u $$\epsilon$$ V (G) : uv $$\epsilon$$ E (G) } U {v}. For $$\subseteq$$ V  (G) , the closed neighborhood of C is N[C] = $$U_{(vec)}$$ NG [v]. A dominating set C $$\subseteq$$ V (G) is a weakly connected dominating set in G if the subgraph $$\langle$$C$$\rangle$$w = (NG[C],EW) weakly induced by C is connected, where EW is the set of all edges with at least one vertex in C. The weakly connected domination number $$\gamma$$w (G) of G is the minimum cardinality among all weakly connected dominating sets in G. In this paper, we characterized the weakly connected dominating sets in the Kr-gluing of complete graphs and corona of graphs. As con- sequences, we determined the weakly connected domination number of the aforementioned graphs. Weakly connected domination number in the join of graphs is also determined.

##### A Goal Programming Approach to Rubber Plantation Planning in Tripura and Formulation of Mathematical Model

Manish Nandi

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 129-139
https://doi.org/10.9734/bpi/ctmcs/v7/3473F

Rubber has been identified as one of the thrust area in Tripura. Tripura is the second largest rubber producer in the country after Kerala. The rubber tree takes around six to seven years to mature and naturally production starts after that. Rubber plantations have two distinct stages in its economic life- the immature phase and the mature phase. Growing rubber trees is a high cost and long term investment i.e. the first phase is the age of all expenses and no return. The expenditures in the rubber plantations can be classified in two broad heads- Material Cost and Labor Cost. In this study, an attempt has been taken to formulate a strategic planning using the Goal Programming approach.

##### Engaging Small and Medium Sized Companies in Cooperation and New Forms of Learning

Steffi Engert

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 140-149
https://doi.org/10.9734/bpi/ctmcs/v7/8562D

Small and medium sized companies (SMEs) represent the backbone of the European economy. SMEs were not prepared for the consequences of Covid-19 and the national lockdowns caused by it. Many aspects within SMEs were affected by the crisis: Rapidly, good solutions for work and learning therefore have to be found. E-Learning through its flexibility and ease of access is seen also as an enabler of lifelong learning with its potential to transform how and when employees learn, and to act as a catalyst of change and integration. SMEs need to find new ways of cooperation, for example technology-enabled Communities of Practice (CoP). Another form of cooperation are crossing sectors, especially between SMEs, research and higher education institutions, but this form needs to be improved. To be able to survive the pandemic, be innovative and play an important role. , SMEs have to tackle change and develop their learning processes on a far-reaching scale. This paper takes a closer look at the situation of European SMEs and presents findings from recent European projects identifying the ongoing problems in the adoption of digital forms of learning and methods like problem-based learning. Covid-19 should not be a barrier for SMEs to increase sales because the SMEs actors can promote their business through digital marketing. This paper will try to answer questions like: How do SMEs utilize digital methods for business, learning, reskilling the workforce for digital marketing? Which kind of cooperation should help in this context? The answer is based on literature research and the work of the authors within national and European projects.

##### Nonidentical Relations of Skew-Symmetric Forms: Generation of Closed Exterior Forms. Discrete Transitions. Connection between Field-theory Equations and Nonidentical Relations

L. I. Petrova

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 150-166
https://doi.org/10.9734/bpi/ctmcs/v7/3369F

Nonidentical relations of skew-symmetric differential forms, which basis are non?integrable deforming manifolds follow from differential equations. From nonidentical relations closed exterior forms are obtained. The process of obtaining closed exterior forms describes the discrete transitions and the emergence of structures and observable formations such as waves, vertices, and turbulent pulsations.

It is shown that the field theory equations (by Schroedinger, Maxwell, Einstein and others) turns to be nonidentical relations, obtained from the mathematical physics equations for material media such as the cosmologic systems, the systems of charged particles and others.

##### End-to-end Secured Fiber Optic Communication Using a Novel Encryption Algorithm

Bosco Paul Alapatt

Current Topics on Mathematics and Computer Science Vol. 7, 5 August 2021, Page 167-176
https://doi.org/10.9734/bpi/ctmcs/v7/3184F

Fiber optic communication has grown in popularity in recent decades as a result of its several advantages over conventional cable communication. Fiber optic communication allows for a download speed of 100 Mb/s with minimal latency and a fast upload speed. Despite the fact that fiber optics can transfer enormous amounts of data swiftly over great distances, it is susceptible to a variety of security threats. It is simple to tap into the fiber optic cable. Fiber optic cable is a thin strand of glass or plastic which is used to transmit data from one place to other. A unique encryption method based on the Advanced Encryption Standard (AES) is used to provide end-to-end secure fiber optic communication. It employs a variable key cypher in which the original key is used to generate a sequence of AES sub keys, with feedback from previous cypher blocks, and each sub block is encrypted using multiple sub keys. The performance of the proposed encryption algorithm is simulated and examined. The simulation results suggest that the proposed method in fiber optic communication delivers strong end-to-end security.