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Issue 4
Volume 7 (2019)

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Authors |
: | G. Marimuthu and G. Durga Devi |

Title |
: | Edge Graceful Irregularity Strength of Wheel Related Graphs |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 1−7 |

**Abstract: ** For a simple graph \textit{G}, the edge graceful irregular \textit{s-}labeling is a mapping $f:V\bigcup E\to \left\{1,2,3,\ldots ,s\right\}$ such that if for any two distinct edges \textit{e } and \textit{ g}, $wt\left(e\right)\ne wt\left(g\right)$, $wt\left(uv\right)=\left|f\left(u\right)+f\left(v\right)-f\left(uv\right)\right|$. The edge graceful irregularity strength of \textit{G}, denoted by $egs\left(G\right)$ is the smallest \textit{k} for which \textit{G} has an edge graceful irregular \textit{s-}labeling. In this paper we determine the exact value of an edge graceful irregularity strength of graphs, namely gear, helm, closed helm and flower graph.

**Keywords: ** Graceful irregularity, edge graceful irregularity strength, gear, helm, closed helm, flower graph.

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Authors |
: | H. V. Dedania and H. J. Kanani |

Title |
: | Various Spectral Properties in the Banach Algebra $\mathcal A \times_{c} \mathcal I$ with the Convolution Product |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 9−14 |

**Abstract: ** Let $\mathcal I$ be an ideal of an associative algebra $\mathcal A$ over the field $\mathbb C$. Then the vector space $\mathcal A \times \mathcal I$ with pointwise linear operations becomes an algebra with the product $(a, x) (b, y) = (ab+xy, ay+bx) \; ((a, x), (b, y) \in \mathcal A \times \mathcal I).$ This product is known as the convolution product and this algebra is denoted by $\mathcal A \times_{c} \mathcal I$. In this paper, some well-known spectral properties of Banach algebras are studied for $\mathcal A \times_{c} \mathcal I$.

**Keywords: ** Banach algebras, Convolution product, Topological divisor of zero, Quasi divisor of zero, Topological annihilator condition, Multiplicative Hahn-Banach property, Ditkin's condition, and Tauberian condition.

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Authors |
: | Mengesha Ayene, A. Padma and G. V. Reddy |

Title |
: | Generalized Fractional Integral Operators Pertaining to Galue Type Struve Function |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 15−22 |

**Abstract: ** The Marichev-Saigo-Maeda operators are applied to investigates certain fractional integral image formulas associated with the product of the unified Galu$\acute{e}$ type Struve function and the general class of polynomials. The main formulas are expressed in terms of the generalized Fox-Wright function. Utility or importance are also discussed by giving particular cases to the main results.

**Keywords: ** Galu$\acute{e}$ type Struve function, General class of polynomial, Generalized Marichev-Saigo-Maeda fractional integral operators, Generalized Wright function.

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Authors |
: | Ajay Kumar Chaudhary |

Title |
: | Fixed Point Results for Single Self Maps in Probabilistic Metric Space |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 23−30 |

**Abstract: ** This paper deals with some fixed-point results for single self-maps in probabilistic metric space which is based on two contraction conditions, one is B-contraction and another is C- contraction in Probabilistic metric space.

**Keywords: ** Fixed Point, distribution function, Menger space, contraction condition.

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Authors |
: | Zhaoquan Wang, Shouxu Du and Tao Wang |

Title |
: | Research Schur-Zassenhaus Theorem by Using Anti-homomorphism |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 31−35 |

**Abstract: ** Let $f: G\rightarrow K$ be a function between finite groups. When
the function $f$ is a anti-homomorphism it may preserve group structure.
In this paper, we consider measures of how nearly the group structure is
preserved by an arbitrary function. We first define anti-distributor which is a new way
to build anti-homomorphism from arbitrary function. we demonstrate the applicability
of this theory by constructing anti-homomorphism to prove Schur-Zassenhaus theorem.

**Keywords: ** anti-distributor; anti-homomorphism; Schur-Zassenhaus theorem.

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Authors |
: | S. Jayaprakash and N. Mohamed Thoiyab |

Title |
: | Homomorphism of Neutrosophic Fuzzy L-ideals |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 37−46 |

**Abstract: ** In this paper, we define the concept of homomorphism of Neutrosophic fuzzy L-Ideals and some related properties are discussed. Finally, some results on Neutrosophic fuzzy L-ideals are investigated.

**Keywords: ** Neutrosophic set, Neutrosophic Lattices, Neutrosophic fuzzy L-Ideals, Homomorphism of Neutrosophic fuzzy L-ideals.

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Authors |
: | Sarika M Nair and R. Akhila |

Title |
: | Ideal Extension and Idempotents of a Rees Matrix Semigroup |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 47−52 |

**Abstract: ** This paper deals with the method of resolving two relvent problems of Rees matrix semigroups, the ideal extension problem and the structure of sandwich set of idempotents. To resolve the first problem let us consider the ideal extension of Rees matrix semigroup over the multiplicative group $ U(n). $ The study is done using the Rees matrix semigroup containing only the zero element and a semigroup which has more than one element. The idempotents of Rees matrix semigroup is studied using Biordered set and finally we arrive at a conclusion about the structure of sandwich set of idempotents.

**Keywords: ** Rees matrix semigroups, Ideal Extension, Biordered Set.

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Authors |
: | G. G. Muthuri, D. M. Malonza and F. Nyabadza |

Title |
: | Modeling the Effects of Targeted Mass Media Campaigns on Alcohol Abuse in Kenya |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 53−61 |

**Abstract: ** Alcoholism causes serious harmful effects to the addicts and the whole community in general. The mass media campaign against alcohol acts as a source of information to halt alcohol abuse and its potentially harmful effects. We developed a deterministic model for alcohol abuse considering the influence of pre-exposure of mass media campaigns against alcohol abuse. We analyzed the local and global stability of the AFE and the endemic equilibrium point of the model. The nature of the bifurcation of the model was analyzed using Center Manifold Theorem. Numerical simulations were carried out to determine where the campaigns should be targeted for effective control of the abuse. The results showed that mass media campaigns against alcohol consumption reduce alcohol abuse in the community. The model was validated using data from rehabilitation centers in Kenya. The results to policymakers imply that the mass media campaign should be regulated to reduce alcohol addiction.

**Keywords: ** Mass mass campaign, alcohol abuse, reproduction number, alcohol-free equilibrium, endemic equilibrium, sensitivity analysis, numerical simulation.

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Authors |
: | R. Gokilam and R. Thanamani |

Title |
: | Estimating the Approximate Solutions of the Fornberg-Whitham and Oskolkov-Benjamin-Bona-Mahony Equations |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 63−70 |

**Abstract: ** In this paper we study the initial value problems of Fornberg-Whitham(FW) and Oskolkov-Benjamin-Bona-Mahony(OBBM) equations which are locally wellposed in the Sobolev space $H^s$ for $s>\frac{3}{2}$. we define the approximate solutions of FW and OBBM equations and compute the errors. Then we estimate the $H^{\sigma}$ -norm of this errors.

**Keywords: ** Sobolev space, Approximate solutions, Well-posedness, Non-local form.

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Authors |
: | Mirza Farhan Ahmed Beg and Swati Saxena |

Title |
: | Common Fixed Point Theorems in Intuitionistic Fuzzy Metric Space Using Occasionally Weakly Compatible Maps with Integral Type Inequality |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 71−76 |

**Abstract: ** In this paper, we obtain common fixed point theorems in intuitionistic fuzzy metric spaces using occasionally weakly compatible maps with Integral Type Inequality.

**Keywords: ** Intuitionistic fuzzy metric space, Occasionally weakly compatible mappings, Common fixed point, Integral type.

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Authors |
: | Jared Ongaro |

Title |
: | On a Zeuthen-type Problem |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 77−86 |

**Abstract: ** In this paper, we show that every degree $d$ meromorphic function on a smooth connected projective curve $C\subset \PP^2$ of degree $d>4$ is isomorphic to a linear projection from a point $p\in \mathbb {P}^2 \setminus C$ to $\PP^1$. We then pose a Zeuthen-type problem for calculating the plane Hurwitz numbers.

**Keywords: ** Hurwitz numbers, Zeuthen numbers.

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Authors |
: | Animesh Gupta and Prahalad Singh Kaurav |

Title |
: | Some Rational Contractions for Coupled Coincidence and Common Coupled Fixed Point Theorems in Complex-Valued Metric Spaces |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 87−93 |

**Abstract: ** The aim of this paper is to obtain a coupled coincidence point theorem and a
common coupled fixed point theorem of contractive type mappings involving rational
expressions in the framework of a complex-valued metric spaces. We also improve the result obtain by \cite{jhade}. The results of this paper generalize and extend the results of Kang \cite{kang}, in complex-valued metric spaces.

**Keywords: ** Coupled fixed point theorem, contractive type mapping, complex
valued metric space.

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Authors |
: | Animesh Gupta and Prahalad Singh Kaurav |

Title |
: | Fixed Points of a New Type of Contractive mappings in $G-$Metric Spaces |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 95−100 |

**Abstract: ** In this paper, using a mapping $F: \mathbb{R}^+ \to \mathbb{R}$, a new notion of contraction in $G$-metric space is introduced and related fixed point theorem which generalizes various known results in $G-$metric spaces. The paper includes an example which shows the validity of our results.

**Keywords: ** $F-$contraction, contractive mapping, fixed point, $G-$metric space.

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Authors |
: | K. Madhu Sudhan and B. Sreenivasula Reddy |

Title |
: | Numerical Study of Magneto-Convective Casson Fluid Flow Past an Exponentially Accelerated Vertical Porous Plate in the Presence of Radiation and Dufour Effects |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 101−110 |

**Abstract: ** A numerical analysis is carried out for an unsteady free convective, chemically reactive, and electrically conducting fluid past an exponentially accelerated vertical porous plate in the presence of heat source, radiation and Dufour effects. The set of non-dimensional governing equations along with boundary conditions are solved numerically. The effect of various physical parameters on flow quantities are studied with the help of graphs. For the physical interest, the variations in skin friction, Nusselt number and Sherwood number are also studied through tables.

**Keywords: ** MHD, Casson fluid, Radiation, Chemical reaction, Heat source, Diffusion thermo effect.

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Authors |
: | M. Reddappa, C. Jaya Subba Reddy and B. Maheswari |

Title |
: | Total Roman Domination in Interval Graph With Adjacent Cliques of Size 3 |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 111−121 |

**Abstract: ** Interval graphs have drawn the attention of many researchers for over 40 years. They form a special class of graphs with many interesting properties and revealed their practical relevance for modeling problems arising in the real world. The theory of domination in graphs introduced by O. Ore [10] and C. Berge [1] has been ever green of graph theory today. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by T.W. Haynes [12, 13]. In this paper a study of total domination and total Roman domination number of an Interval graph with adjacent cliques of size 3 is carried out.

**Keywords: ** Total domination number, Total Roman dominating function, Total Roman domination number, Interval family, Interval graph.

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Authors |
: | T. Srinivasarao and K. Geetha Lakshmi |

Title |
: | Not Distributive Lattice Over a Galois Field |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 123−125 |

**Abstract: ** A prime element in a Euclidean ring and to an irreducible polynomial in a polynomial ring defined over a field are identical. The irreducible polynomial allows us to construct a prime ideal which in turn leading to a maximal ideal. So, the maximal ideal and the Euclidean ring together form a quotient field in which the zero element is the maximal ideal itself. The quotient field is seen as the extended field over the field referred in the beginning. It is easily seen that the actual irreducible polynomial $f(x)$ is now reducible over the extended field. In the present case, we take a finite field and a polynomial from the polynomial ring over this field and verify the members of the field obey the distributive law or not. The purpose of producing a not distributive lattice is to see that enciphering can be done using the members of such a lattice in which it will be difficult to judge the correct deciphered text. Because, there will be multiple results in the deciphering approach. So, which is the correct decipher among the available cipher texts will be a matter of confusion. The present Galois field is over the field of residue classes modulo 3.

**Keywords: ** Distributive Lattice, Galois Field, Euclidean ring.

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Authors |
: | Ramya and B. Sooryanarayana |

Title |
: | Radio Multiplicative Number of Certain Classes of Transformation Graphs |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 127−148 |

**Abstract: ** The concept of radio labeling motivated by the channel assignment
problem is generalised herein to include various other types of
radio labelings. Let $\mathbb{M}$ be a subset of non-negative
integers and $(\mathbb{M},\star)$ be a monoid with the identity $e$.
We define a radio $\star$-labeling of graph $G(V,E)$ as a mapping
$f:V\rightarrow \mathbb{M}$ such that $| f(u) - f(v)|\star\ d(u,
v)\geq diam(G)+1-e $, for all $u,v\in V$.
The radio $\star$-number $rn^\star(f)$ of a radio $\star$-labeling $f$ of $G$ is the maximum label assigned to a vertex of $G$. The \textit{radio $\star$-number} of $G$ denoted by $rn^\star(G)$ is $min\{ rn^\star(f)\} $ taken over all radio $\star$-labeling $f$ of $G$. In this paper we completely determine $rn^\times(G)$ of some transformation graphs of path and cycle.

**Keywords: ** Radio labeling, radio multiplicative labeling, radio multiplicative number.

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Authors |
: | Jaydeep Parejiya |

Title |
: | On the Connectedness of the Complement of a Unit Graph of a Commutative Ring |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 149−152 |

**Abstract: ** The rings considered in this article are commutative with identity. We denote the set of all maximal ideals of a ring $R$ by $Max(R)$ and we denote the Jacobson radical of $R$ by $J(R)$. Let $R$ be a ring. Recall from [2] that the {\it unit graph} of $R$, denoted by $G(R)$, is an undirected graph whose vertex set of all elements of $R$ and distinct vertices $x, y$ are joined by an edge in this graph if and only if $x + y\in U(R)$. In this article, we studied Complement of unit graph and we denoted it $(UG(R))^{c}$. Hence, in this graph two elements $x, y$ are joined by an edge in $(UG(R))^{c}$ if and only if $ x + y \in NU(R)$. In this article we proved some results on connectedness of $(UG(R))^{c}$.

**Keywords: ** Connected graph, Max(R).

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Authors |
: | Nirbhay Narang |

Title |
: | Deriving the Formula for Linear Regression |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 153−159 |

**Abstract: ** The goal of this research paper is to determine a generalised formula for the line and plane of best fit given a set of points in 2 and 3 dimensions respectively, as well as generalising a formula for a function of best fit in $n$ dimensions. The applications are wide and varied, including statistical analysis and data science.

**Keywords: ** Linear Regression, Matrices, Machine Learning, Planes of best fit, Linear Algebra.

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Authors |
: | Nistala V.E.S. Murthy and Emandi Gouthami |

Title |
: | About Generalized Soft Quotient Groups |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 161−173 |

**Abstract: ** The aim of this paper is to introduce the notions of generalized soft group, generalized soft (normal) subgroup, generalized soft quotient group etc., generalizing the corresponding notions of a soft group over a group and show that several of the crisp group theoretic results naturally extended to these new objects too.

**Keywords: ** (Generalized) Soft set, (Generalized) Soft group, (Generalized) Soft quotient group, (Generalized) Soft (normal) subgroup.

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Authors |
: | Jaise Jose and D. Pradeep |

Title |
: | Finite Dimensional Approximation of Simplified Gauss-Newton Scheme for Nonlinear Ill-Posed Problems |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 175−184 |

**Abstract: ** We consider the finite dimensional approximation of simplified Gauss-Newton iterative scheme presented in \cite{rajan1} for solving nonlinear ill-posed problems. The convergence and convergent analysis of this scheme is carried out with both an a priori and an a posteriori parameter choice strategies. The error estimates are derived accordingly. We propose an order optimal parameter choice strategy for the regularization parameter, which gives the optimal convergence rate. Finally, we present numerical examples to verify the theoretical results.

**Keywords: ** Nonlinear Ill-posed Problems, Regularization, Inverse Problems, Iterative Method, Parameter Choice Rule.

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Authors |
: | Motulo, Trisilowati and Tamsih |

Title |
: | Dynamical Analysis of Tumor Growth Model with Immunotherapy |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 185−192 |

**Abstract: ** In this paper, the analysis of tumor growth model with immunotherapy involving dendritic cells is discussed. The model consists of four compartments namely the tumor cells, the active CTLs, the T-helper cells, and the dendritic cells. The growth rate of the tumor cells in this model follows the logistic model. The dendritic cell therapy functions as an inhibitor of tumor growth without causing side effects on the other cells so that the spread of tumor cells can be minimized. Next, dynamical analysis is performed by determining the stability analysis of the equilibrium point. It shows that the model has six equilibria consisting of three tumor-free equilibria namely $E_0, E_1, E_2$ and three tumor equilibria namely $E_3,E_4,E_5$. The equilibria points $E_0$ and $E_3$ are not stable since there are positive eigenvalues while other equilibria will be stable if those meet certain conditions. Furthermore, the simulation results support the analysis result.

**Keywords: ** Dynamical system, Tumor, Immunotherapy.

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Authors |
: | H. V. Dedania and H. J. Kanani |

Title |
: | Gel'fand Theory of the Commutative Banach Algebra $\mathcal A \times_{c} \mathcal I$ with the Convolution Product |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 193−199 |

**Abstract: ** Let $\mathcal A$ be an algebra and $\mathcal I$ be an ideal in $\mathcal A$. Then $\mathcal A \times \mathcal I$ is an algebra with pointwise linear operations and the convolution product $(a, x) (b, y) = (ab+xy, ay+xb) \; ((a, x), (b, y) \in \mathcal A \times \mathcal I)$; it will be denoted by $\mathcal A \times_{c} \mathcal I$. If $\mathcal A$ is a commutative Banach algebra and $\mathcal I$ is a closed ideal in $\mathcal A$, then $\mathcal A \times_{c} \mathcal I$ is also a commutative Banach algebra with some suitable norm. In this paper, we shall study the Gel'fand theory, uniqueness properties, and regularity of $\mathcal A \times_{c} \mathcal I$.

**Keywords: ** Convolution product, UUNP, U$C^{\ast}$NP, regular algebra, uniform algebra

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Authors |
: | Sanjay K. Srivastava, Neha Wadhwa and Neeti Bhandari |

Title |
: | A New Technique in Stability of Infinite Delay Differential Equations With Impulsive Effects |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 201−207 |

**Abstract: ** In this work, we consider the stability of impulsive infinite delay differential equations. A new technique is derived to establish the stability criteria for impulsive infinite delay differential equations. By using Lyapunov functions and Razumikhin technique, some results are obtained which are more general than ones existing in literature. Lyapunov functionals are adopted and components of x are divided into several groups, correspondingly, several functions $V_j\left(t,x^{\left(j\right)}\right)$, ($j=1,2,\dots,m$) are employed. It is shown that impulses do contribute to yield stability properties even when the underlying system does not enjoy any stability behaviour. An example is also presented to illustrate the efficiency of the result obtained.

**Keywords: ** Impulsive infinite delay differential equations, Uniform stability, Lyapunov functions, Razumikhin technique.

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Authors |
: | V. R. Kulli |

Title |
: | Some Multiplicative Neighborhood Dakhayani Indices of Certain Nanostructures |

Volume, Issue, Year |
: | 7(4)(2019) |

Pages |
: | 209−217 |

**Abstract: ** A topological index is a numerical parameter mathematically derived from the graph structure. Connectivity indices are applied to measure the chemical characteristics of chemical compounds in Chemistry. In this paper, we introduce the multiplicative neighborhood Dakshayani indices, multiplicative hyper neighborhood Dakshayani indices, multiplicative sum connectivity neighborhood Dakshayani index, multiplicative product connectivity neighborhood Dakshayani index, general first and second multiplicative neighborhood Dakshayani indices of a molecular graph and determine exact formulas for these indices of line graphs of subdivision graphs of 2-\textit{D} lattice, nanotube and nanotorus of $TUC_{4}C_{8}[p, q]$.

**Keywords: ** Multiplicative neighborhood Dakshayani indices, multiplicative sum connectivity neighborhood Dakshayani index, multiplicative product connectivity neighborhood Dakshayani index, nanostructure.