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Issue 1
Volume 6 (2018)

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Authors |
: | M. R. Rajesh Kanna and R. Pradeep Kumar |

Title |
: | Minimum Dominating Partition Energies of Friendship Graph |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 407−410 |

**Abstract: ** In this paper we compute minimum dominating partition energies of friendship graphs.

**Keywords: ** Minimum dominating set, Minimum dominating partition matrix, Minimum dominating partition eigenvalues, Minimum dominating partition energy of a graph.

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Authors |
: | G. Uthra, K. Thangavelu and R. M. Umamageswari |

Title |
: | Ranking L-R Type Generalized Trapezoidal Fuzzy Numbers |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 411−416 |

**Abstract: ** L-R type fuzzy numbers are quite common in practical life situations involving the triangular fuzzy number, the Gaussian fuzzy number, Cauchy fuzzy number, trapezoidal fuzzy numbers, generalized trapezoidal fuzzy numbers etc. Ranking of these fuzzy numbers plays a vital role in dealing with real life situations with imprecise information. This study presents an average approach for ranking L-R type generalized trapezoidal type fuzzy numbers. Compared with other fuzzy ranking techniques, the ranking proposed in this paper is easily interpreted, flexible and invariant with translating real numbers.

**Keywords: ** Trapezoidal fuzzy numbers, L-R type Generalized Trapezoidal fuzzy numbers, L-R type Generalized Fuzzy Assignment Problem, Ranking of L-R type generalized fuzzy numbers.

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Authors |
: | Thangaraj Beaula and J. Partheeban |

Title |
: | $\alpha $-Triangular Fuzzy Matrix and its Application |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 417−428 |

**Abstract: ** In this paper, a-triangular matrix is introduced with the help of $\alpha$-cut of each its elements and the arithmetic operations on $\alpha$-triangular matrices are defined. The determinant, adjoint, trace of it are also defined and with the help of it some properties are established. The correlation coefficient is defined between two triangular fuzzy numbers in terms of its a-cut and using this idea in $\alpha$-triangular matrix it became feasible to solve a real life problem using analytic hierarchy process (AHP) for $\alpha\in [0,1]$.

**Keywords: ** $\alpha$-triangular matrix, determinant, adjoint, trace, correlation coefficient.

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Authors |
: | Jaison Jose and V. Sangeetha |

Title |
: | On Proper 2-rainbow Domination in Graphs |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 429−434 |

**Abstract: ** For a graph $G$, let $f : V(G) \rightarrow \mathcal{P}(\{1, 2,. . ., k\}$) be a function. If for each vertex $v\in V(G)$ such that $f(v) = \phi$ we have $\cup_{u \in N(v)} f(u) = \{1, 2, . . . , k\}$, then $f$ is called a $k$-rainbow dominating function (or simply $k$RDF) of $G$. The weight $w(f)$, of a $k$RDF $f$ is defined as $w(f) = \sum\limits_{v \in V(G)}^{}|f(v)|$. The minimum weight of a $k$RDF of $G$ is called the $k$-rainbow domination number of $G$, and is denoted by $\gamma_{rk}(G)$. In this paper we define and study a new domination called proper $k$-rainbow domination. A $k$-rainbow dominating function is called a proper $k$-rainbow dominating function if for every pair of adjacent vertices $u$ and $v$, $f(u)\not\subseteq f(v)$ and $f(v)\not\subseteq f(u)$. The weight, $w(f)$, of a proper $k$RDF $f$ is defined as $w(f) = \sum\limits_{v \in V(G)}^{}|f(v)|$. The minimum weight of a proper $k$RDF of $G$ is called the proper $k$-rainbow domination number of $G$, and is denoted by $\gamma_{prk}(G)$. The bounds for 2-rainbow domination and proper 2-rainbow domination for different classes of graphs namely cycles, complete multipartite graph, $P_{n}\times P_{m}$ and Harary graph are found.

**Keywords: ** Rainbow domination, proper rainbow domination, Harary graph.

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Authors |
: | G. R. Ibrahim and S. O. Makanjuola |

Title |
: | Combinatorics Properties of Order-preserving Full Contraction Transformation Semigroup by Their Equivalence Classes |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 435−439 |

**Abstract: ** In this paper the cardinalities of equivalence classes of starred Green's relations in order-preserving full contraction transformation semigroup of a finite set$(OCT_{n})$ and the elements in each equivalence classes were investigated. For each class tables were formed, elements were arrange based on their kernel and image sets, patterns of arrangement observed and formulae were deduced in each case through the combinatorial principles.

**Keywords: ** Order-preserving, Contraction mapping, Starred Green's Relation.

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Authors |
: | K. P. R. Rao and A. Sombabu |

Title |
: | Common Coupled Fixed Point Theorems in Dislocated Quasi Fuzzy $B$-Metric Spaces |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 441−455 |

**Abstract: ** In this paper we obtain some unique common coupled fixed point theorems in dislocated quasi fuzzy b-metric spaces. Also we give some examples which support our main results.

**Keywords: ** Fuzzy metric space , Dislocated Quasi fuzzy $b$- metric space, Coupled fixed point, $w$-compatible maps.

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Authors |
: | S. Lekshmi and K. S. Dersanambika |

Title |
: | Some Fixed Point Theorems in Complex Valued b-metric Space Using Weak Commuting Maps |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 457−462 |

**Abstract: ** The aim of the present paper is to establish fixed point theorems for self mappings under weak $\phi$- commuting maps in Complex Valued b-metric space.

**Keywords: ** Complex Valued b-metric space, Weak commuting maps, fixed point.

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Authors |
: | M. A. Patel and N. B. Desai |

Title |
: | An Approximate Analytical Solution of Boussinesq's Equation for Infiltration Phenomenon in Unsaturated Porous Medium |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 463−470 |

**Abstract: ** The present paper discusses the infiltration phenomenon for unsaturated porous medium. The mathematical formulation leads to a one dimensional nonlinear partial differential equation in the form of Boussinesq's equation. An approximate analytical solution of Boussinesq's equation for infiltration phenomenon have been obtained. The homotopy analysis method applied to solve Boussinesq's equation with appropriate boundary conditions. The Mathematica BVPh package for homotopy analysis method has been used to interpret numerically and graphically of solution.

**Keywords: ** Infiltration, Porous medium, Boussinesq's equation, Homotopy analysis method.

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Authors |
: | M. Menaka and M. Marudai |

Title |
: | Fixed Point Theorems for Weakly Generalized $w$-Contraction Mappings and Applications |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 471−481 |

**Abstract: ** In this paper, we prove fixed point theorems for weakly generalized
$w$-contraction mappings in the setting of $w$-distances which generalize results of Wongyat
and Sintunavarat \cite{16}. As an application, we obtain the existence and uniqueness solutions for nonlinear Fredholm
and Volterra integral equations.

**Keywords: ** Fixed point, $w$-distances, Contraction mappings.

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Authors |
: | S. Issar, K. K. Mishra and A. Soni |

Title |
: | Coefficient Estimates for Certain Subclasses of Sakaguchi Type Bi-Univalent Functions |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 483−492 |

**Abstract: ** In this paper, we introduced certain Sakaguchi type subclasses of the function class $\Sigma$ of analytic
and bi-univalent functions associated with quasi-subordination in the open unit disk $\mathbb{U}$. Estimates on the first two Taylor-MacLaurin coefficients $|a_{2}|$ and $|a_{3}|$ for functions belonging to these classes are determined. Certain special cases are also indicated.

**Keywords: ** Analytic functions, Univalent functions, Bi-univalent functions, Quasi-subordination.

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Authors |
: | P. Mariappan and J. Antony Rex Rodrigo |

Title |
: | $M_I^{*}$-closed Sets in Ideal Topological Spaces |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 493−503 |

**Abstract: ** The concept of generalized closed sets was considered by Levine in 1970 \cite{JJ}. In this way we introduce a new generalized closed set via $I_\omega$-open set and study its some basic properties. Also, we investigate the relationship with other types of closed sets.

**Keywords: ** $M_I^{*}$-closed set, $M_I^{*}$-derived set, $M_I^{*}$-frontier.

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Authors |
: | D.Senthilkumar |

Title |
: | Fuglede Putnam Theorem on Class $p-w A(s,t)$ Operator |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 505−508 |

**Abstract: ** In this paper we charcterize Fuglede putnam theorem for class $p-w A(s,t)$ operator.

**Keywords: ** Class $A$ operator, class $p-w A(s,t)$ operator, polar decomposition, Fuglede-Putnam Theorem.

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Authors |
: | S. Muthammai and P. Vidhya |

Title |
: | Inverse Complementary Tree Domination Number of Total Graphs of $P_n$ and $C_n$ |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 509−513 |

**Abstract: ** A non-empty set $D \subseteq V$ of a graph is a dominating
set if every vertex in $V-D$ is adjacent to some vertex in $D$. The
domination number $\gamma(G)$ of $G$ is the minimum cardinality
taken over all the minimal dominating sets of $G$. A dominating set
$D$ is called a complementary tree dominating set if the induced
subgraph $\langle V-D\rangle$ is a tree. The complementary tree domination
number $\gamma_{ctd}(G)$ of $G$ is the minimum cardinality taken
over all minimal complementary tree dominating sets of $G$. Let $D$
be a minimum dominating set of $G$. If $V-D$ contains a dominating
set $D^\prime$, then $D^\prime$ is called the inverse dominating set
of $G$ w.r.t to $D$. The inverse domination number
$\gamma^\prime(G)$ is the minimum cardinality taken over all the
minimal inverse dominating sets of $G$. In this paper, inverse
complementary tree domination in total graphs of $P_n$ and $C_n$ are
obtained.

**Keywords: ** Dominating set, complementary tree dominating set, inverse
complementary dominating set.

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Authors |
: | P. Anuradha Kameswari and L. Jyotsna |

Title |
: | Cryptanalysis of RSA with Small Multiplicative Inverse of $\varphi(N)$ Modulo $e$ and with a Composed Prime Sum $p+q$ |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 515−526 |

**Abstract: ** In this paper, we mount an attack on RSA when $\varphi(N)$ has small multiplicative inverse $k$ modulo $e$, the public encryption exponent. For $k\leq N^\delta,$ the attack bounds for $\delta$ are described by using lattice based techniques. The bound for $\delta$ depends on the prime difference $p-q=N^\beta$ and the maximum bound for $\delta$ is $\alpha-\sqrt{\frac{\alpha}{2}}$ for $e=N^\alpha$ and for $\beta\approx0.5$. If the prime sum $p+q$ is of the form $p+q=2^nk_0+k_1$ where $n$ is a given positive integer and $k_0$ and $k_1$ are two suitably small unknown integers then the maximum bound for $\delta$ can be improved for $\beta\approx0.5$.

**Keywords: ** RSA, Cryptanalysis, Lattices, LLL algorithm, Coppersmith's method.

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Authors |
: | V. Govindan and K. Tamilvanan |

Title |
: | Stability of Functional Equation in Banach Space: Using Two Different Methods |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 527−536 |

**Abstract: ** Using Direct and fixed point method, we prove the Ulam-Hyers stability of a generalized n-dimensional additive functional equation of the form
\[f\left(\sum^n_{i=1}{kx_i}\right)+\sum^n_{j=1}{f\left(-kx_j+\sum^n_{\substack{i=1 \\ i\neq j}}
{kx_i}\right)}=(n-1)\left[\sum^n_{i=1}{\left(2i-1\right)f\left(x_i\right)}\right]\]
Where n is the positive integer with $\mathbb{N}-\{0,1,2\}$ and k is the only odd positive integers in Banach Space is discussed.

**Keywords: ** Additive Functional Equation, Banach Sapce, Fixed Point, Hyers-Ulam Stability.

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Authors |
: | Suzanne A. Allam and Ahmed M. Hashish |

Title |
: | Parameter Estimation for the Exponentiated Weibull Distribution Using SDPWM |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 537−542 |

**Abstract: ** In this paper, the parameters of the Exponentiated Weibull distribution were estimated by the methods of probability weighted moments (PWM) and self-determined probability weighted moments (SDPWM). In addition, simulation experiments were conducted to compare the performance of the two methods. The simulation results showed that the SDPWM method was better than PWM in almost all the cases. In addition, the two methods provided consistent estimators for the unknown parameters.

**Keywords: ** Self-Determined probability weighted moments, probability weighted moments, Exponentiated Weibull distribution.

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Authors |
: | V. Anusuya and B. Nisha |

Title |
: | Decision Making On Triangular Type-2 Fuzzy Soft Set |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 543−548 |

**Abstract: ** Molodtsov.D initiated the concept of soft set theory, which has become an effective mathematical tool for dealing with uncertainty. Especially,fuzzy soft set theory is used to solve decision making problems. In this work, the inter relationship between similarity measure, inclusion measure and entropy measure is obtained using triangular type-2 fuzzy soft sets and its complement. A decision making problem is solved by this relationship.

**Keywords: ** Triangular type-2 fuzzy soft sets, complement of triangular type-2 fuzzy soft sets, similarity measure, inclusion measure, entropy measure.

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Authors |
: | D. K. Thakkar and Neha P. Jamvecha |

Title |
: | Edge-Vertex Domination in Graphs |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 549−555 |

**Abstract: ** In this paper we continue the study of ev-domination (edge-vertex domination) in graphs. We give a characterization of minimal ev-dominating sets in graphs. In particular we prove that in a graph with minimum vertex degree greater than or equal to 2, the complement of a minimal ev-dominating set is an edge dominating set. We also state and prove necessary and sufficient condition under which the ev-domination number increases or decreases when a vertex is removed from the graph. We also consider the operation of removing an edge from the graph and prove that the ev-domination number does not decrease when an edge is removed from the graph.

**Keywords: ** ev-dominating set, minimal ev-dominating set, minimum ev-dominating set, ev-domination number, edge dominating set, private vertex neighbourhood of edge.

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Authors |
: | N.U. Khan, M. Kamarujjama and T. Kashmin |

Title |
: | Generating Function Associated with the Product of Gegenbauer Polynomials |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 557−561 |

**Abstract: ** This paper deals with general expansions which gives as special cases involving Legendre, Ultraspherical, and Tchebcheff polynomials, Appell, Bessel's and generalized hypergeometric functions of mathematical physics. The results extended Pathan and Kamarujjama expansion \cite{7}, Exton's generating function \cite{5} and Feldheim's expansion \cite{3}. A number of known and new generating functions are shown to follow as an application of the main result.

**Keywords: ** Legendre, Gegenbauer, Tchebcheff, and Ultraspherical polynomials, Appell's, Bessel's and Generalized hypergeometric functions, Pochhammer symbol.

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Authors |
: | S. K. Pandey |

Title |
: | A Remark on Derivations and Bounded Nilpotence Index |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 563−565 |

**Abstract: ** Pace P. Nielsen and Michal Ziembowski has constructed a nil ring $R$ which has bounded nilpotence index 2 and proved that this ring has a derivation $\delta $ for which the differential polynomial ring $R[x;\delta ]$ is not even prime radical and thus obtained some interesting algebraically important results. The purpose of this note is to establish that the ring $R$ constructed by these authors is nothing but an even square ring whose every element is a nil element.

**Keywords: ** Even square rings, nil elements, nilpotent elements, nil ideals, rings of bounded nilpotence index.

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Authors |
: | T. Iswarya, L. Rajendran and KE. Sathappan |

Title |
: | Modelling of the Non-linear Biotransformation Process |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 567−577 |

**Abstract: ** Mathematical modelling of production of l-malic acid from fumaric acid which is performed by the action of the fumarase isolated from porcine heart and the fumarase in permeabilized baker's yeast cells are discussed. This model is based on nonlinear system of equations containing a non-linear term related to rate of the reactions. The effects of various parameters on the concentrations are also discussed. Our analytical results are also compared with numerical results (Sci lab program) and satisfactory agreement is noted. Simple analytical expressions for the concentration of substrate and product have been derived for all values of reaction parameters using the new homotopy perturbation method (NHPM). Kinetic parameters are also obtained from the enzyme reaction rate.

**Keywords: ** l-Malic acid; Fumarase; Baker's yeast; Permeabilization; Mathematical Modelling; non-linear differential equation.

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Authors |
: | Manju Somanath, J. Kannan and K. Raja |

Title |
: | The Extendibility of Diophantine Pairs |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 579−582 |

**Abstract: ** Let $n$ be a non-zero integer. A set $\{a_1, a_2,\cdots, a_m\}$ of $m$ distinct
positive integers is called a Diophantine $m-tuples$ with the property $D(n),$
if $a_ia_j+n$ is a perfect square for all $1\leq i

**Keywords: ** Diophantine triple, Perfect square, Quadratic Polynomial.

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Authors |
: | Ganesh Kumar |

Title |
: | To Determine the Value of Game in Fuzzy Environment |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 583−588 |

**Abstract: ** In this research paper an optimization technique under uncertainty with payoffs as octagonal fuzzy numbers is developed. In present paper octagonal numbers in fuzzy environment are converted in crisp values to apply the classical methods to determine the game value and best strategies of players. Uncertainty and imperfect information arise only when players are unaware the observations of the actions of choices of each other.

**Keywords: ** Octagonal fuzzy numbers, crispness, fuzzy payoffs.

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Authors |
: | B. Sridevi and K. Suvarna |

Title |
: | Commutativity of Periodic Rings with Some Identities in the Center |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 589−596 |

**Abstract: ** Let R be an periodic ring. In this paper, we prove that an $(n+1)n$-tortion free periodic ring satisfying the properties $(ab)^{n}-ba \in Z(R) $, $(ab)^{n+1}- ba\in Z (R)$, $a^{n}(ab)-(ba)a^{n}\in Z(R)$ for all $a,b \in R$ is commutative, then R is commutative.

**Keywords: ** Commutativity of Periodic rings, Periodic rings, Center.

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Authors |
: | Mukund V. Bapat |

Title |
: | Double Crown Related E Cordial Graphs |

Volume, Issue, Year |
: | 6(1−C)(2018) |

Pages |
: | 597−602 |

**Abstract: ** WE obtain different families of graphs by attaching two pendent edges at each vertex of G. We call these graphs as double crown graphs and denote them by $G^{++}$. We show that $C_{n}^{++}$ and $S_{4}^{++}$ and $W_{n+1}^{++}$ are E-cordial families.

**Keywords: ** Double crown, E-cordial, double, pendent edge, wheel, shel labeling.