## Articles in PressVolume 8, Issue 2 (2020)

 Authors : Chetan Kumar Sahu, S. Biswas and Subhash Chandra Shrivastava Title : Fixed Point of Pseudo Contractive Mapping in a Banach Space Volume, Issue, Year : 8(2)(2020) Pages : 1−5

Abstract: Let X be a Banach space, B a closed ball centred at origin in X, $f : B\to X$ a pseudo contractive mapping i.e. $(\alpha-1)\| x-y\| \le\|(\alpha I-f)(x) - (\alpha I-f)(y)\|$ for all x and y in B and $\alpha> 1$. Here we shown that Mapping f satisfies the property that $f(x) = -f(-x)\ \ \forall \ \ x$ in $\partial B$ called antipodal boundary condition assures existence of fixed point of f in B provided that ball B has a fixed point property with respect to non expansive self mapping. Also included some fixed point theorems which involve the Leray-Schauder condition.

Keywords: Fixed point, Banach space, Non expansive mapping, Pseudo Contractive Mapping, Cauchy Sequence, Lipschitzian Mapping.

 Authors : P. Ramulu, G. L. Reddy and C. Gangadhar Title : On the Zeros of Polynomials with Complex Coefficients Volume, Issue, Year : 8(2)(2020) Pages : 7−13

Abstract: In this paper we prove some extension of the Enestr\"{o}m-Kakeya theorem (Let $P(z)=\sum\limits_{i=0}^{n} a_iz^i$ be a polynomial of degree $n$ such that $0\textless {a_0}\leq {a_1}\leq...\leq{a_n}$ then all the zeros of P(z) lie in $|z|\leq{1}$) by relaxing the hypothesis in different ways we get various other results which in term generalizes.

Keywords: Zeros of polynomial, Enestr\"{o}m-Kakeya theorem.

 Authors : A. K. Shukla and Aneesh Jayswal Title : An Unsteady MHD Flow Past a Vertical Porous Plate Under a Variable Suction Velocity with Soret-Dufour and Second-order Chemical Reaction Volume, Issue, Year : 8(2)(2020) Pages : 15−25

Abstract: The objective of this work to study the combined effect of Soret-Dufour, radiation and second order chemical reaction on unsteady MHD(magnetohydrodynamics) flow past a vertical porous plate immersed in a porous medium under variable suction velocity. In this model, the nonlinear partial differential equation of flow problem has been solved numerically using the Crank-Nicolson implicit finite difference method. Characteristics of velocity, temperature and concentration are shown through graphically and characteristics of skin friction, Nusselt number and Sherwood number are discussed with the help of the table.

Keywords: Magnetohydrodynamics, Heat Transfer, Mass Transfer, Order of chemical reaction, Soret and Dufour effects, Crank-Nicolson finite difference method.

 Authors : Gurupadavva Ingalahalli, S. C. Anil and C. S. Bagewadi Title : A Study on $W_8$-curvature Tensor in Kenmotsu Manifolds Volume, Issue, Year : 8(2)(2020) Pages : 27−34

Abstract: In this paper we study the curvature properties of Kenmotsu manifolds satisfying the conditions $\xi$-$W_8$-flat, $\phi$-$W_8$-semisymmetric, $R(\xi, X)\cdot W_8=0,$ $W_8\cdot R=0,$ $W_8\cdot W_8=0,$ $W_8$-Ricci pseudosymmetric, $W_8\cdot Q=0.$

Keywords: Kenmotsu Manifolds, Curvature tensor, Ricci tensor.

 Authors : Rohit Singh Tomar and R. K. Shrivastav Title : Server Breakdown and Delayed Repair in Three Phases of Service for an M/G/1 Retrial Queueing System Volume, Issue, Year : 8(2)(2020) Pages : 35−47

Abstract: This paper deals with an unreliable server having three phases of heterogeneous service on the basis of M/G/1 queueing system. We suppose that customers arrive and join the system according to a Poisson's process with arrival rate $\lambda$. When the server is working with any phase of service, it may breakdown at any instant. After breakdown, when the server is sent for repair then server stops its service and arrival customers are waiting for repair, which we may called as waiting period of the server. This waiting time stands for delay time/delay repair. In this model, first we derive the joint probability distribution for the server. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalisation of Pollaczek - Khinchin formula. Third, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability analysis of this model. By using a supplementary variable method, we obtain the transient and the steady state solutions for both queueing and reliability measures.

Keywords: First phase of service, second phase of service, third phase of service, random breakdowns, delayed repair, M/G/1 queue, stationary queue size distribution and reliability index.

 Authors : Beletech Alemu Reta and Jula Kabeto Bunkure Title : Production Planning and Controlling by Using Mathematical Programming to Maximize Profit: The Case of Ethiopian Textile Industry Volume, Issue, Year : 8(2)(2020) Pages : 49−65

Abstract: Industrial improvement strategy is expressed by the effective product planning and control use resources at every production stage. The whole process should be carried out in a best possible way and at the lowest cost. Production Manager will have to see that the things proceed as per the plans. This is a control function and has to be carried as meticulously as planning. Both planning and control of production are necessary to produce better quality goods at reasonable prices and in a most systematic manner. The analysis and effective utilization of resources are made sustainable by effective management decision making techniques employed in the industry. A quantitative decision making tool called linear programming , Queue model, Critical path and PERT methods can be used for the optimization problem of product planning/mix. Understanding the concept behind the optimization problem of product mix is essential to the success of the industry for meeting customer needs, service quality/rate, determining its image, focusing on its core business, and inventory management. Apparel manufacturing firms profit mainly depends on the proper allocation and usage of available production time, material, and labor resources. This paper considers 49 Textiles and apparel industrial unit in Ethiopia as a case study. The monthly held resources, product volume, and amount of resources used to produce each unit of product and profit per unit for each product have been collected from the company. The data gathered was used to estimate the parameters of the linear programming model and Queue model. The model was solved using LINGO 16.0/Matlab software. The findings of the study show that the profit of the company can be improved by 49.3 percent, that is, the total profit of Birr 4,445,013.33 per month can be increased to Birr 9,334,528 per month by applying linear programming and Queue models if customer orders have to be satisfied. The profit of the company can be improved by 12.35 percent if the linear programming formulation does not need to consider customer orders. Textle and Apparel industries must produce large quantities in shorter lead times in order to stay alive and compete in the current fashion market. Apparel production needs high level of productivity and production lines should be balanced to get shorter lead time in effective way.

Keywords: Linear Programming, Queue Model, profit.

 Authors : Sushil Chandra Karna Title : Satellites: Equation of Motion in Nechvill's Co-ordinate System Volume, Issue, Year : 8(2)(2020) Pages : 67−70

Abstract: This paper deals with the system of non-linear, non-autonomous and non-homogeneous equations of motion in terms of the Nechvill's Co-ordinate system have been derived. The general solution is beyond our reach.

Keywords: System, Nonlinear, Homogeneous, Frame of reference, constraints.

 Authors : T. Brindha and R. Subiksha Title : Split Total Domination Number of Some Special Graphs Volume, Issue, Year : 8(2)(2020) Pages : 71−79

Abstract: A dominating set for a graph $G = (V,E)$ is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number $\gamma(G)$ is the number of vertices in a smallest dominating set for G. In this paper a new parameter, Split Total Dominating Set $S$ and the Split Total Domination Number $\gamma_{st} (G)$ has been introduced. A dominating set is called split total dominating set if $\langle V-D\rangle$ is disconnected and every vertex $v \in V$ is adjacent to an element of D. The split total domination number is given by $\gamma_{st} (G)$. In this paper the split total domination number for some standard graphs like star, path, cycle, complete, ladder, wheel, bistar, tadpole, comb, barbell, butterfly and fan graphs are found. Also the complement of graphs are obtained.

Keywords: Split Dominating Set, Total Dominating Set, Split Total Dominating Set.

 Authors : K. Iqbal, F. Alam, F. Riaz, A. Hashmi, M. N. Khalid and A. Gohar Title : Application of Numerical Analysis in Real Life Volume, Issue, Year : 8(2)(2020) Pages : 81−85

Abstract: The advancement of numerical analysis and its application over the past decades have provided an incredibly powerful method for researchers to study. The use of such approaches is still not common nonetheless, and documentation of bad practice is all too often present when applied. Probably the reason for this is a lack of knowledge and direction as to the reasonable use of such approaches of study, in particular from codes of practice. It is clear that some sort of initiative is required to improve good exercise and to allow the full potential of this analytical tool to be realized from both a protective and an economic perspective. The paper commences to analyze the major benefits and application of numerical analysis over typical methodologies, and then addressing whether or not it can replace the conventional analytical tools in the design process. Existing literature is used widely to demonstrate the considerations for and against the use of numerical analysis.

Keywords: Numerical analysis, application, real life, validation.

 Authors : A. K. Shukla, R. Yadav and S. K. Chauhan Title : Amazing Effects of Second Order Chemical Reaction, Radiation, and Soret-Dufour on MHD Flow of Walter's B Viscoelastic Fluid Past a Vertical Porous Plate Volume, Issue, Year : 8(2)(2020) Pages : 87−97

Abstract: The focus of this paper is to study magnetohydrodynamics Walter's B viscoelastic fluid flow over a vertical porous plate embedded in a porous medium under the influence of the Soret-Dufour effect, radiation effect, and second order chemical reaction. The contemplated fact is modeled by a system of nonlinear partial differential equations influenced by the boundary conditions. The governing coupled nondimensional partial differential equations solved numerically using Crank-Nicolson implicit finite difference method. Under the influence of governing flow parameters, results for velocity, temperature, and concentration are computed and portrayed through graphs while the numerical values for skin friction, Nusselt number, and Sherwood number are presented with the help of tables.

Keywords: Chemical reaction, Magnetohydrodynamics, Walter's B model, Heat and Mass transfer, Soret and Dufour effects, radiation effect, Crank-Nicolson finite difference method.

 Authors : Shreyan Das Title : Applications of Number Theory in RSA Encryption Systems Volume, Issue, Year : 8(2)(2020) Pages : 99−106

Abstract: Number theory is the branch of mathematics that studies the set of integers. Cryptography is the field of study that has as goal the development of algorithms to send secret messages over public channels like the internet. In this article, we describe applications of number theory to cryptography, specifically within the RSA Encryptions system. This usage allows for this type of Encryption to be significantly advantageous.

Keywords: Number Theory, RSA Encryption, Euclidean Algorithm, Euler's Formula, Cryptography.

 Authors : Wei-Kai Lai and John Risher Title : Generalizing Inequalities Using Power Series Approach Volume, Issue, Year : 8(2)(2020) Pages : 107−112

Abstract: In 1903 Nesbitt introduced a famous inequality: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{3}{2}$ for any positive real numbers $a$, $b$ and $c$. Among all its proofs, Mortici provided a unique approach applying the convergence of power series together with the power means inequality. Adopting this technique, we first generalize several Nesbitt type inequalities to $n$ variable versions. We then combine the knowledge of power series, Young's inequality, and the rearrangement inequality, and deduce some new inequalities.

Keywords: Power Series, Nesbitt's Inequality, Power Means Inequality, Young's Inequality, Rearrangement Inequality.

 Authors : Sanjivappa K Dembare Title : New Function in Fuzzy Topological Spaces Volume, Issue, Year : 8(2)(2020) Pages : 113−119

Abstract: In this paper, we introduce fuzzy generalized semi-open and fuzzy generalized semi-closed maps in fts and obtain certain characterization of the generalized semi-closed and generalized semi- open maps.

Keywords: Fuzzy gs-open maps,fuzzy gs-closed maps,fuzzy gs-closed sets, fuzzy gs-open sets.

 Authors : Suraj Oruganti Title : Applications of Number Theory to Cryptography: A Look Into the Diffie-Hellman Key Exchange and the RSA Cipher Volume, Issue, Year : 8(2)(2020) Pages : 121−131

Abstract: Number theory is the study of integer values, and as Carl Friedrich Gauss once coined it, "the queen of mathematics." More specifically, a field of study in number theory is the behavior of prime numbers. Cryptography is the development of algorithms in order to transmit information securely. Number Theory is intertwined with, and is an integral part of cryptography. Therefore, in this paper we will be discussing the applications of number theory to cryptography.

Keywords: Cryptography, Diffie-Hellman Key Exchange, RSA Cipher.

 Authors : Preeti Nagar and Rajendra Pandey Title : Some Generalized and Extension Results for $q$-Bessel Type Wavelets Volume, Issue, Year : 8(2)(2020) Pages : 133−141

Abstract: In this paper, we use the concept of wavelet functions in the context of $q-$theory. We precisely extend and generalized the results of $q-$Bessel wavelets and then developed for the new context. Reconstruction and Placherel type formulas are proved.

Keywords: Wavelets, Besel function, $q-$Bessel function, modified Bessel functions, generalized $q-$Bessel functions, $q-$Bessel wavelets.

 Authors : Oyombe Aluala, Oduor Okoya Michael, Obogi Robert and Kerongo Joash Title : Lie Symmetry Solution of Third Order Nonlinear Ordinary Differential Equation Volume, Issue, Year : 8(2)(2020) Pages : 143−154

Abstract: In this paper we used the method of Lie symmetry to solve and get a mathematical solution to a third order first degree nonlinear ordinary differential equation (ODE) of fourth degree in second derivative, which is common in waves of systems like water in shallow oceans because it yields exact solutions without depending on initial boundary values.

Keywords: Lie groups of transformations, Lie algebras, infinitesimal transformations, extended transformations, invariance under transformations, variation symmetries, Lie theory of differential equations, reduction of order and integrating factors.

 Authors : Andanje Mulambula, D. B. Oduor and B. O. Kwach Title : Volatility Estimation Using European-Logistic Brownian Motion with Jump Diffusion Process Volume, Issue, Year : 8(2)(2020) Pages : 155−163

Abstract: Volatility is the measure of how we are uncertain about the future of stock or asset prices. Black-Scholes model formed the foundation of stock or asset pricing. However, some of its assumptions like constant volatility and interest among others are practically impossible to implement hence other option pricing models have been explored to help come up with a much reliable way of predicting the price trends of options. The measure of volatility and good forecasts of future volatility are crucial for implementation, evaluation of asset and derivative pricing of asset. In particular, volatility has been used in financial markets in assessment of risk associated with short-term fluctuations in financial time-series. Constant volatility is not true in practical sense especially in short term intervals because stock prices are able to reproduce the leptokurtic feature and to some extent the volatility smile". To address the above problem the Jump-Diffusion Model and the Kou Double-Exponential Jump-Diffusion Model were presented. But still they have not fully addressed the issue of reliable prediction because the observed implied volatility surface is skewed and tends to flatten out for longer maturities; the two models abilities to produce accurate results are reduced. This study ventures into a research that will involve volatility estimation using European logistic-type option pricing with jump diffusion. The knowledge of logistic Brownian motion will be used to develop a logistic Brownian motion with jump diffusion model for price process.

Keywords: Black-Scholes formula, Brownian motion, Logistic Brownian motion, Jump diffusion, Volatility.

 Authors : Vivek Ily Title : Using the SIR Model as a Guideline for Vaccination during an Infectious Disease Volume, Issue, Year : 8(2)(2020) Pages : 165−170

Abstract: In this paper, we discuss and analyze the well-known Susceptible-Infected-Recovered (SIR) Model to address the following questions: (Q1) What total proportion of the population will get infected? (Q2) What proportion of the population should be vaccinated in order to suppress the epidemic? To answer these questions, we discuss and analyze and the SIR model and its variations to predict the outcome of an epidemic and any measures needed to suppress the outbreak of the disease.

Keywords: SIR Model, Mathematical Modeling, Infectious Diseases, Vaccination.