Papers

LNCT Bhopal & Technocrats Institute of Technology Bhopal India

 In this paper we generalize the concept of weakly projective and weakly injective module. We define, Let R is a ring and let M be a semi simple right module. Then there exists an infinite cardinal R such that M(+) E(M^R) is Quasi-weakly injective. Consequently this result holds also for any right R-module with essential socle. Moreover, if R is a right q.f. ring then the result holds for any right module M. A right module M is quasi-weakly projective if and only if M is weakly Rⁿ-Projective for all positive integers n. Let M be a quasi-weakly projective right module over a semi-perfect ring. If the projective cover P(M) may be expressed as S(+) Kwith S finitely generated then M has a direct summand isomorphic to S. in particular a finitely generated an indecomposable quasi-weakly projective right module is projective. Let R be a left prefect ring such that every projective right R-module is quasi weakly injective [i.e. R is weakly R-injective). Then R is a quasi-Frobenius ring [i.e. QF ring].

Let R is a QF-ring and let M_R is a R-module and M=E(+)K, where E=(+)sum_i=1^k (e_iR)^ais a projective module and K is a singular module. If we write k/rad k= sum_i=1^k (eR/ej)^B(+)any soc(k)= sum_i=1^k (eR/ej)ⁿ then.

(i) M is quasi-weakly projective iff for all i = 1, 2 …….k if B_i =/ 0then ais infinite.

(ii) M is quasi-injective iff for all i = 1, 2 ……k if y=/0 thena: is infinite.

¹Researcher Scholar, Bharathiar University, Coimbatore & Assistant Professor, Deptt. of Mathematics Ethiraj College for Women Chennai India

²Associate Professor & Head Deptt. of Mathematics, D.G. Vaishnav College, Arunbakkam Chennai India

 In most of the fuzzy multicriteria decision analysis problem, the final step results in choosing the best among the alternatives, represented in terms of Fuzzy numbers. In this paper, we propose a modified ranking method to rank fuzzy numbers particularly for non convex and nonnormal fuzzy numbers) of all types. This modified method is based on the geometrical interpretation, and is applicable in most of the situations. Comparison with other methods is also done here. 
 

¹Researcher Scholar, Bharathiar University, Coimbatore India

²Associate Professor & Head Deptt. of Mathematics, D.G. Vaishnav College, Arunbakkam Chennai India

In this paper, we discuss the acyclic vertex coloring and acyclic chromatic number of middle graph, central graph and total graph of wheel graph. 
General terms : Middle graph, central graph, total graph of wheel graph Wn are denoted by M(Wn), C(Wn) and T (Wn) respectively.

School of Studies in Statistics, Pt Ravishankar Shukla UniversityRaipur India

Email:- profsksingh@hotmail.com

 The purpose of the present paper is to stochastically analysis of a system having two parallel operating units subject to inspection and replacement. Whenever any operating unit fails it is inspected to diagnose the possibility of its either repair or replacement. There is a single repair facility to repair the failed units on first come first serve basis. All the failure time distributions are assumed to be negative exponential while as inspection, repair and replacement time distributions are taken to be arbitrarily. Various reliability characteristics are obtained in order to analyze the expected profit earned by the system. Also, a consistent asymptotically normal (CAN) estimator, a 100(1-alpha)% asymptotic confidence interval for the steady state availability of the system and the maximum likelihood estimator (MLE) of the system reliability are obtained. Lastly a numerical example is discussed to highlight the important results. 

¹Department of Mathematics, S.M.L (P.G.)  College Jhunjhunu Rajasthan India

²Department of Mathematics, Shri Jagdish Prasad Jhaermal Tibrewal University Chundela Jhunjhunu Rajasthan India

Email:- ysingh23@yahoo.co.in

We derive three compositions of the fractional integral operators associated with a product of I-function and a general class of polynomials due to Srivastava. The result obtained are of general character and apply for the general composition of the various fractional integration operators associated with special functions of mathematical Physics and Chemistry appeared in various recent publications. 
 

¹Department of Mathematics Bharath Institute of Science and Technology Bharath University Chennai Tamil Nadu India

²Associate Professor & Head Deptt. of Mathematics, D.G. Vaishnav College, Arunbakkam Chennai India

Email:*kr2210@hotmail.com

 In this paper, we present a unique way of coloring of Bi-magic 4-regular graphs with girth 3 and 4. It is interesting to note that the relation between bi-magic numbers and chromatic numbers are of importance in the coloring of the vertices