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Issue 3
Volume 8 (2020)

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Authors |
: | M.Arunkumar, E. Sathya and C. Pooja |

Title |
: | Single Variable Generalized Additive - Quadratic and Generalized Cubic- Quartic Functional Equations in Various Banach Spaces |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 1−42 |

**Abstract: ** In this paper, we introduce and examine the generalized Ulam - Hyers stability of single variable generalized additive-quadratic and generalized cubic-quartic functional equations in various Banach spaces with the help of two different methods.

**Keywords: ** Functional equation, Generalized Hyers-Ulam stability, Banach space, (Beta;p) Banach space, Intuitionistic Fuzzy Banach space.

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Authors |
: | R. Savithiri and A. Manonmani |

Title |
: | A Bitopological $(1,2)^*$ - $\mathop{\alpha }\limits^{\bullet } $ - Homeomorphisms |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 43−49 |

**Abstract: ** In this paper, a new concept of homeomorphisms, say $(1,2)^*$ - $\mathop{\alpha }\limits^{\bullet } $ homeomorphisms, $(1,2)^*$ - $\mathop{\alpha }\limits^{\bullet } c$ - homeomorphisms, $(1,2)^*$ - $\alpha gg$-homeomorphisms and $(1,2)^*$ - $\alpha ggc$ - homeomorphisms are introduced and analyzed in bitopological spaces. Also $(1,2)^*$ - $\mathop{\alpha }\limits^{\bullet } $ - $T_{1/2}$ - space and $(1,2)^*$ - $T_{\mathop{\alpha }\limits^{\bullet } }$ - space are introduced and studied.

**Keywords: ** $(1,2)^*$ - $\mathop{\alpha }\limits^{\bullet } $ - homeomorphisms, $(1,2)^*$ - $\mathop{\alpha }\limits^{\bullet } $c - homeomorphisms, $(1,2)^*$ - $\alpha gg$ - homeomorphisms, $(1,2)^*$ - $\alpha ggc$ - homeomorphisms, $(1,2)^*$ - $\mathop{\alpha }\limits^{\bullet } $ - $T_{1/2}$ - space, $(1,2)^*$ - $T_{\mathop{\alpha }\limits^{\bullet } }$ - space.

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Authors |
: | M. P. Aparna and R. Akhila |

Title |
: | Principal Topology on a Rees Matrix Semigroup using Green's Left Quasiorder |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 51−57 |

**Abstract: ** This paper introduces a principal topology on a Rees matrix semigroup using Green's left quasiorder. Since principal topologies are in one-one correspondance with quasiorder relations on a set, the relations are commonly used for constructing such topologies. The basis for the topology is the collection of minimal open neighbourhoods corresponding to each element in a given set. When semigroups are considered with Green's left quasiorder, minimal open neighbourhoods are the principal left ideals. Hence, the collection of principal left ideals will turn out to be a basis for the principal topology on a semigroup. As long as a Rees matrix semigroup is considered, it is observed that these ideals exhibit certain interesting properties. This paper analyses these ideals in the context of a Rees matrix semigroup. The properties thus observed actually determine the number of elements in the so formed principal topology. Further, the topology hence obtained is an example for a finite topology on an infinite set, provided the order of the Rees matrices is finite.

**Keywords: ** Green's left quasiorder, Principal topology, Rees matrix semigroup.

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Authors |
: | S. S. Mahde, S. I. Khalaf, Y. N. Shawawreh, B. Shanmukha, A. M. Nour |

Title |
: | Laplacian Minimum Hub Energy of a Graph |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 59−69 |

**Abstract: ** In this paper, we introduce Laplacian minimum hub energy $LE_H(G)$ of a graph $G$, and compute Laplacian minimum hub energies of some standard graphs, also for a number of well-known families of graphs. Upper and lower bounds for $LE_H(G)$ are established.

**Keywords: ** Minimum hub set, Laplacian minimum hub matrix, Laplacian minimum hub eigenvalue, Laplacian minimum hub energy of a graph.

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Authors |
: | Soham Konar |

Title |
: | Delaunay Triangulations |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 71−76 |

**Abstract: ** We examine a methodology to construct 3D maps. We assume the height above sea level is known only at a finite number of points. Mathematically, this problem is equivalent to reconstructing the graph of a function of two variables when the values of the function are only known at a finite number of points. This leads to a discussion of Delaunay triangulations and the algorithm to compute them. We illustrate the concept discussed with examples.

**Keywords: ** Computational Geometry, Delaunay Triangulations, Algorithms.

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Authors |
: | S. A. Bhanotar |

Title |
: | Fibonacci and Lucas Numbers-Continued Fraction Expansion |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 77−84 |

**Abstract: ** In modern science, there are lots of applications of Fibonacci and Lucas numbers have been used in number theory, computer science, applied mathematics and biology. In this paper, I have discussed Fibonacci number, Lucas number, the golden ratio, and their relations to them. I study numerous new properties of these sequences and focused to the generalized the formula for the Fibonacci number, Lucas number and some important identity; some observation of sunflower the head is carried out with the angular momentum velocity and represented continued fraction expansion of such number based on the golden ratio.

**Keywords: ** Fibonacci number, Lucas number, Golden Ratio, Continued fraction expansion.

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Authors |
: | Arun Kumar Rao and Himanshu Pandey |

Title |
: | Estimation of Shape Parameter of Exponentiated Pareto Distribution Via Bayesian Approach |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 85−91 |

**Abstract: ** In this paper, the exponentiated Pareto distribution is considered for Bayesian analysis. The expressions for Bayes estimators of the parameter have been derived under squared error, precautionary, entropy, K-loss, and Al-Bayyati's loss functions by using quasi and gamma priors.

**Keywords: ** Bayesian method, exponentiated Pareto distribution, quasi and gamma priors, squared error, precautionary, entropy, K-loss, and Al-Bayyati's loss functions.

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Authors |
: | Nikky Kumari |

Title |
: | Intuitive Approach for Nonlinear Transportation Problem |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 93−98 |

**Abstract: ** An intuitive approach has been developed for finding an optimal solution of nonlinear transportation problem. Method is presented in the form of an algorithm and illustrated through a numerical example. Approach is simple, easy to understand and apply.

**Keywords: ** Nonlinear transportation problem, intuitive approach, active cell, allocation, optimal solution.

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Authors |
: | Jeffrey Wang |

Title |
: | Oscillation of Bridge: Mathematical Modeling of the Amplitude Under Different Situations |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 99−104 |

**Abstract: ** We consider a function that shows the position of a bridge. This function includes the effect of the force applied by the soldier marching on it. We analyze the function and obtain how the impulse effect the position of the bridge in different cases, such as the resonance case-when the soldier applies a force with the same frequency as the frequency of the bridge oscillations-and the non-resonance case-when the soldier applies a force with a different frequency from the frequency of the bridge oscillations.

**Keywords: ** Bridge, Oscillation, Modeling.

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Authors |
: | S. Arul Ravi |

Title |
: | Best Proximity Point Theorem for $\Phi$-Weak Contractions |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 105−110 |

**Abstract: ** Fixed point theory for generalized $\phi$-weak contractions have been extended for Best proximity point using $p$-property.

**Keywords: ** Best Proximity point, $p$-property, generalized $\phi$-weak contraction, complete metric spaces, lower semi-continuous.

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Authors |
: | P. Jaish and A. Joseph Kennedy |

Title |
: | The Rook Class Partition Algebras |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 111−123 |

**Abstract: ** The Class partition algebras $P_{k}(n,m)$ have been studied independently in \cite{JK} by Kennedy and also studied in \cite{ME} by Martin and Elgamal. We introduce the rook version of class partition algebras $P_{k}(n,m)$, which is subalgebra of the algebra $P_{k}(n)\otimes P_{k}(m)$ tensor product partition algebra. We also find corresponding schur--weyl dualities.

**Keywords: ** Wreath product , Symmetric group, Partition algebra, Centralizer algebra.

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Authors |
: | Chiraag Kambalimath |

Title |
: | Algorithmic Modeling of a Genome Sequence |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 125−139 |

**Abstract: ** We describe how genomes are assembled and read. This involves the implementation of graph theory and the development of algorithms. The algorithms are written in Python.

**Keywords: ** Genome Reconstruction, Algorithms, Bioinformatics.

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Authors |
: | V. Durai Murugan, R. Seethalakshmi and P. Namasivayam |

Title |
: | The Lattice Structure of the Subgroups of Order 42 and 48 in the Subgroup Lattices of $2 \times 2$ Matrices Over $Z_{7}$ |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 141−146 |

**Abstract: ** Let $\mathcal{G}= \left\{\left( \begin{array}{cc}
a & b \\
c & d \end{array}
\right): a,b,c,d \in Z_{p}, ad-bc \neq 0\right\}$.
Then $\mathcal{G}$ is a group under matrix multiplication modulo p. Let $G = \left\{\left( \begin{array}{cc}
a & b \\
c & d \end{array}
\right)\in \mathcal{G} : ad-bc = 1\right\}$.
Then G is a subgroup of $\mathcal{G}$. We have, $o(\mathcal{G}) = p(p^{2}-1)(p-1)$ and $o(G) = p(p^{2}-1)$. Let $L(G)$ denotes the lattice of subgroups of G, where G is the group of $2\times 2$ matrices over $Z_p$ having determinant value 1 under matrix multiplication modulo p, where p is a prime number. In this paper, we give the structure of the subgroups of order 42 and 48 of $L (G)$ in the case when $p=7$.

**Keywords: ** Matrix group, subgroups, Lagrange's theorem, Lattice, Atom.

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Authors |
: | Ammar Alsinai, Anwar Alwardi and N. D. Soner |

Title |
: | An Atlas of Different Distances Sets Polynomials of Graphs of Order at most Six |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 147−161 |

**Abstract: ** The different distances sets polynomial of a graph $G$ of order $p$ is
defined as $D_{d}(G,x)=\sum\limits_{i=1}^pd_{d}(G,i)x^i$, where $d_{d}(G,i)$ is the number of different distances sets polynomials of
$G$ of size $i$, \cite{Alsinai}. We call the roots of different distances sets polynomial of a graph the different distances roots of
that graph. In this article, we compute different distances sets polynomial of all graphs of order less than or equal six and their
roots and present them in tables.

**Keywords: ** Different distances sets, different distances sets polynomials.

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Authors |
: | S. R. Jog, S. L. Patil and J. R. Gurjar |

Title |
: | Degree Sum Exponent Distance Index of a Graph |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 163−175 |

**Abstract: ** In this paper, we determine degree sum exponent distance $\chi_{dist}(G)$ for some standard graphs and some graphs arising from complete graph, by taking it as a particular case of general sum-connectivity index.

**Keywords: ** Degree, Distance, Degree sum distance exponent index.

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Authors |
: | Mohand M. Abdelrahim and Abdelilah K. Hassan |

Title |
: | The Use of Adomian Decomposition Method for Solving Nonlinear Wave-Like Equation With Variable Coefficients |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 177−186 |

**Abstract: ** The Adomian Decomposition Method (ADM) has been widely applied in solving partial differential equations which represent various phenomena in engineering and physics. In this paper, nonlinear wave-like equations with variable coefficients are solved using Sawi Adomian De-composition Method (SADM). Sawi decomposition method is a combined form of Sawi transform method and the Adomian decomposition method. The nonlinear term can easily be handled with the help of Adomian polynomials which is considered to be a significant advantage of this technique over the other methods. We shall show that SADM is able to solve this type of equations effectively and accurately.

**Keywords: ** Sawi transform, Sawi A domian decomposition, Wave-like equation.

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Authors |
: | P. R. K. Kishore and M. H. Melesse |

Title |
: | Fuzzy Ideals and Fuzzy Congruences of a Generalised Lattice |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 187−191 |

**Abstract: ** This paper deals with the correspondence between fuzzy ideals, fuzzy filters and fuzzy congruences in a generalised lattice. Proved that lattice of fuzzy ideals (fuzzy filters) of a generalised lattice is isomorphic to the lattice of fuzzy ideals of a corresponding lattice. Finally observed that the lattice of fuzzy ideals of a type of generalised lattice is isomorphic to the lattice of fuzzy congruences of that.

**Keywords: ** Poset, Lattice, Fuzzy lattice, Fuzzy ideal, Fuzzy congruence.

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Authors |
: | S. K. Tiwari and M. Gauratra |

Title |
: | Study of Problem of Some Fixed Point Result For Kannan and Chhatterjea Contraction on Generalized Complex Valued Metric Spaces |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 193−200 |

**Abstract: ** In this paper, we study and established some fixed point theorem for general Kannan and Chhattrjea type contraction in generalized complex valued metric spaces. The results extend and improve the common fixed point result of Elkouch \& Marhrani \cite{elkmar}, which is introduced by Issara et al. \cite{issdee}.

**Keywords: ** Fixed point, Common fixed point, General Kannan contraction, General Chhatterjea contraction, Generalized complex valued metric spaces.

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Authors |
: | Xinran Zheng |

Title |
: | Linear Regression With Only One Feature and Applications |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 201−206 |

**Abstract: ** We review how to obtain, by elementary means, the equations that determine the slope and $y$-intercept of the line that best fits a certain given data. We illustrate the use of these equations with an example.

**Keywords: ** Linear Regression, Elementary Means, Data Science.

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Authors |
: | Matthew Song |

Title |
: | Melanoma Detection From Images of Moles Using Neural Networks |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 207−210 |

**Abstract: ** Moles can be analyzed for certain characteristics to determine whether the mole is malignant or benign and if a patient is at risk for melanoma, a type of skin cancer. A standard multilayer perceptron and convolutional neural network are used to investigate whether machine learning can serve as a substitution for a human examination of such moles. Both neural networks were developed using the Tensor Flow python module, and images of moles were fed as training points and test points to be classified. Both types of networks were able to outperform random guessing, with accuracies of 0.82 and 0.84 for the multilayer perceptron and convolutional neural network respectively.

**Keywords: ** Melanoma Detection, Neural Networks, Skin cancer.

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Authors |
: | Dereje Tigabu Muluneh |

Title |
: | A New Approach to Solve Simultaneous Linear Diophantine Equations in Two Variables |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 211−215 |

**Abstract: ** In this paper, we describe a new approach for solving simultaneous linear Diophantine equations in two variables. We also introduce a new theorem that computes the solution of simultaneous linear Diophantine equations in two variables and proved theorem with an elaborated example. The theorem also helps to decide the existence of integer solutions to the simultaneous linear Diophantine equations in two variables.

**Keywords: ** Diophantine equation, a system of linear Diophantine equations, greatest common divisor(gcd), and integer solution.

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Authors |
: | RD Beulah and T. Saradhadevi |

Title |
: | Regularity of Intuitionistic Fuzzy Soft Hypergraphs |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 217−223 |

**Abstract: ** An Intuitionistic Fuzzy Soft Set(IFSS) is an extension of fuzzy soft set to deal with vague information corresponding to their different parameters. The IFSS is the most efficient tool to deal with uncertain information than fuzzy soft set. Hypergraphs are used to represent almost any complex situation involving objects and a relationship among them. The concept of IFSS is applied to hypergraphs and presented the notion of Intuitionistic Fuzzy Soft Hypergraphs(IFSHGs). Further, we defined regular, totally regular and perfectly regular IFSHGs with their properties and also illustrated with its examples.

**Keywords: ** Intuitionistic Fuzzy Soft Hypergraphs, Regular, Totally Regular, Perfectly Regular.

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Authors |
: | Seungwan Son |

Title |
: | System Representation of the Lorentz Gas |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 225−231 |

**Abstract: ** We introduce a random environment system modeling a Lorentz Gas with semi-randomly positioned, periodic scatterers. Each system is determined through a probabilistic method producing a deterministic walk. We prove that the path within the generalized system must be periodic and calculate the expected number of loops within the path of a 1-door system. For n-door systems, where $n>1$, we describe an algorithm to calculate the expected number of loops of its path by making use of its bijection to the Lorentz Gas.

**Keywords: ** Lorentz Gas, Random Environment, Random walk, Algorithm.

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Authors |
: | Rana A. Bakoban, Maha A. Aldahlan and Leena S. Alzahrani |

Title |
: | New Statistical Properties for Beta Inverted Exponential Distribution and Application on Covid-19 Cases in Saudi Arabia |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 233−254 |

**Abstract: ** In this article new statistical properties of the beta inverted exponential distribution are investigated. The harmonic mean, mode, quantile function, median, skewness as well as kurtosis and interquartile range are studied. Important measures, which have a wide range of application in many fields, are considered. Among these measures stress\textendash strength reliability and R{\'{e}}nyi and Shannon entropies as well as order statistics. Also, the maximum likelihood estimation for the unknown parameters, survival and hazard functions are derived based on complete samples. The performance of the estimators is considered in terms of their biases and mean square error via Monte Carlo simulation. The potentiality of this distribution is illustrated through the daily COVID-19 cases observed in Jeddah, Saudi Arabia from $2^{nd}$ May to $6^{th}$ July 2020 as well as two other real data sets.

**Keywords: ** Beta inverted exponential distribution, COVID-19, Hypergeometric function, Maximum likelihood estimation, Monte Carlo simulation, Newton-Raphson method, R{\'{e}}nyi and Shannon entropies, Statistical characterizations, Stress\textendash strength reliability.

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Authors |
: | Michael Lu |

Title |
: | Applications of Number Theory to Cryptography |

Volume, Issue, Year |
: | 8(3)(2020) |

Pages |
: | 255−261 |

**Abstract: ** Number theory is the branch of mathematics that studies the set of integers. Cryptography is the field of study that develops algorithms to send secret messages over public channels like the internet. In this article, we describe applications of number theory to cryptography.

**Keywords: ** Cryptography, Number theory, Algorithms.