Research & Reviews: Discrete Mathematical Structures
http://computers.stmjournals.com/index.php?journal=RRDMS
<p><strong><span style="text-decoration: underline;">Research & Reviews: Discrete Mathematical Structures </span>(RRDMS) </strong>is a print and e-journal focused towards the rapid publication of fundamental research papers on all areas of Discrete Mathematical Structures.</p><p>Discrete mathematical structures journal deals with discrete objects. Discrete objects are those which are separated from each other like integers, rational numbers, automobiles, houses, peoples etc. are all discrete objects. Some of the major reasons that we adopt Discrete mathematics are. We can handle infinity or large quantity and indefiniteness with them which results from formal approaches are reusable.</p><p>eISSN- 2394-1979</p><p><strong><span style="text-decoration: underline;">Focus & Scope:</span></strong></p><ul><li>Mathematical induction</li><li>logic and Boolean algebra</li><li>set theory</li><li>relations and functions</li><li>sequences and series</li><li>algorithms and theory of computation</li><li>number theory</li><li>matrix theory</li><li>induction and recursion</li><li>counting and discrete probability</li><li>graph theory (trees)</li><li>Calculus of finite differences, discrete calculus or discrete analysis</li><li>Game theory, decision theory, utility theory, social choice theory</li><li>Discrete analogues of continuous mathematics</li><li>Hybrid discrete and continuous mathematics</li></ul>en-USResearch & Reviews: Discrete Mathematical Structures2394-1979<p align="center"><strong>Declaration and Copyright Transfer Form</strong></p><p align="center">(to be completed by authors)</p><p>I/ We, the undersigned author(s) of the submitted manuscript, hereby declare, that the above manuscript which is submitted for publication in the STM Journals(s), is <span>not</span> published already in part or whole (except in the form of abstract) in any journal or magazine for private or public circulation, and, is <strong><span>not</span></strong> under consideration of publication elsewhere.</p><ul><li>I/We will not withdraw the manuscript after 1 week of submission as I have read the Author Guidelines and will adhere to the guidelines.</li><li>I/We Author(s ) have niether given nor will give this manuscript elsewhere for publishing after submitting in STM Journal(s).</li><li>I/ We have read the original version of the manuscript and am/ are responsible for the thought contents embodied in it. The work dealt in the manuscript is my/ our own, and my/ our individual contribution to this work is significant enough to qualify for authorship.</li><li> I/We also agree to the authorship of the article in the following order:</li></ul><p>Author’s name </p><p> </p><p>1. ________________</p><p>2. ________________</p><p>3. ________________</p><p>4. ________________</p><table width="100%" border="0" cellpadding="0"><tbody><tr><td valign="top" width="5%"><p align="center"> </p></td><td valign="top" width="95%"><p>We Author(s) tick this box and would request you to consider it as our signature as we agree to the terms of this Copyright Notice, which will apply to this submission if and when it is published by this journal.</p></td></tr></tbody></table>A Graph Theoretic Approach to quantify the Conflict of Traffic Streams in a Traffic Intersection
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1538
<p>A traffic intersection can be modelled by a graph by considering the streams as vertices and assigning an edge based on the mutually non-conflict streams. In this paper we propose a spectral approach to arrange the steams in order of their conflict and hence determine the most conflicted stream in a traffic intersection.<strong> </strong></p>Ankur Bharali2018-09-212018-09-215FD and DDA of N2 Function
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1695
<p>The object of the present paper is to establish the results of Double Dirichlet average (DDA) of N<sub>2</sub> Function, using Riemann-Liouville Fractional derivative (FD). This function is a modified form of Mittag-Leffler function [15]. The author drives the results between N2-function and fractional operators. The N<sub>2 </sub>Function can be measured as a Dirichlet average and connected with fractional calculus. In this paper the solution comes in compact form of double Dirichlet average of N<sub>2 </sub>Function.</p>Mohd. Farman AliManoj Sharma2018-09-212018-09-215Further Results on Some Degree Based Topological Indices of Sierpi´cski Graphs
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1486
Sierpi´cski graphs S(n, k) were defined originally in 1997 by Sandi Klavˇzar<br />and Uroˇs Milutinovi´c. In this paper, degree based topological indices of<br />Sierpi´cski graphs are considered. In particular M1(G), M2(G), M1(G, x), M2(G, x), R(G), (G), ABC(G), ABC4(G), GA(G), GA5(G), H(G), AZI(G), S(G) and F(G) of Sierpi´cski graphs are determined.Padmapriya P2018-09-212018-09-215On Structure and Robustness of Airport Network of India
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1468
This paper uses a complex network approach to study the structure and robustness of Airport Network of India (ANI). The ANI displays small-world (SW) network properties with an average path length of 2.262 and a clustering coefficient of 0.605, and it also exhibits disassortative mixing on degree of nodes, that means the high-degree nodes in the network tend to have connections with low-degree nodes. Being a disassortative network, ANI doesn’t percolate more easily and less robust to node removal. A study of the robustness of ANI is also carried out for targeted node removal.Ankur BharaliDimpee Baruah2018-09-012018-09-015Vertex Odd Divisor Cordial Labeling of Ring Sum of Different Graphs With star Graph
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1696
<p>A graph <em>tt </em>= (<em>V,</em><em> </em><em>E</em>) is said to have a vertex odd divisor cordial labeling if there is a bijection <em>f </em>: <em>{</em>1<em>, </em>3<em>, </em><em>. . . , </em>2<em>n </em><em>− </em>1<em>} </em>such that each edge <em>e </em>= <em>uv </em>is assigned the label 1 if <em>f </em>(<em>u</em>)<em>|</em><em>f </em>(<em>v</em>) or <em>f </em>(<em>v</em>)<em>|</em><em>f </em>(<em>u</em>) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. If a graph admits a vertex odd divisor cordial labeling, then it is called vertex odd divisor cordial graph. In this paper we have derived vertex odd divisor cordial labeling of ring sum of different graphs.</p>D. G. AdaljaG. V. Ghodasara2018-08-312018-08-315Difference Cordial Labeling in context of Joint sum of Graphs
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1570
<p>Suppose G be a (p, q) graph. Suppose f be a map from f(G) to {1,2,...,p}. For each edge xy assign, the label |f(x) – f(y)|. f is difference cordial if f is 1-1 and |e<sub>f</sub>(0) – e<sub>f</sub>(1)| <!--?mso-application progid="Word.Document"?--> 14â‰¤"> 1, where e<sub>f</sub>(1) and e<sub>f</sub>(0) denote the number of edges with labeled 1 except labeled with 1 respectively. A graph which admit difference cordial labeling is called a difference cordial graph.</p><p> In this paper I prove the following results.</p><ol><li>The joint sum of two copies of wheel graph is difference cordial.</li><li>The joint sum of two copies of shell graph is difference cordial.</li><li>The joint sum of two copies of double wheel graph is difference cordial.</li><li>The joint sum of two copies of Petersen graph is difference cordial.</li><li>The joint sum of two copies of coconut tree is difference cordial.</li></ol>amit himmatbhai rokad2018-06-262018-06-265E-Cordial Labeling for Theta Graph
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1463
<strong><em> </em></strong>A binary vertex labeling <em>f </em>:<em> E</em>(<em>G</em>)<em> → </em>{0,1} with induced labeling <em>f <sup>* </sup></em>:<em> V</em>(<em>G</em>)<em> → </em>{0,1} defined by <em>f <sup>*</sup></em>(<em>v</em>) = ∑ { <em>f </em>(<em>uv</em>) : <em>uv </em>ϵ <em>E</em>(<em>G</em>)}(<em>mod 2</em>) is called an E-cordial labeling of graph <em>G</em> if the number of vertices labeled 0 and number of vertices labeled 1 differ by at most 1and the number of edges labeled 0 and number of edges labeled 1 differ by at most 1. A graph which admits E-cordial labeling is called an E-cordial graph. We prove that the graphs obtained by duplication, switching, fusion and open star of theta graph <em>T<sub>α</sub></em> are E-cordial graphs.vishnu prakash2018-06-242018-06-245Analysis of Numerical and Categorical Prediction Algorithms : A Case Study of Hypertension
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1558
<strong>An extremely wide variety and volume of data are generated by the healthcare industries which can be effectively utilized by applying data mining techniques so as to attain significant knowledge that can aid the process of decision making. In this paper an analysis of Prediction Algorithms is performed using clinical data of hypertension patients. Data mining techniques are effectively employed to estimate systolic and diastolic blood pressure of patients and categorize the risk level for each patient.</strong>Ashwini Menon2018-06-192018-06-195Chromatic Curling Number of Certain Derived Graphs
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1229
<p>The curling number of a graph G is dened as the number of times an element in the degree sequence of G appears the maximum number of times. Graph colouring is an assignment of colours, labels or weights to the vertices or edges of a graph. A colouring C of colours c1, c2,..., cl is said to be a minimum parameter colouring if C consists of minimum number of colours with smallest subscripts. In this paper, we study the chromatic colouring version of curling number of certain derived graphs, with respect to their minimum parameter colourings.</p>Susanth ChandoorSudev NaduvathSunny Joseph Kalayathankal2018-05-282018-05-285Fibonacci Cordial Labeling of Some Graphs
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1416
<p><em>An injective function f: V (G) → {F0, F1, F2, . . . , Fn+1}, where Fj is the j<sup>th</sup> Fibonacci number (j = 0, 1, . . . , n+1), is said to be Fibonacci cordial labeling if the induced function f </em><em>∗</em><em> </em><em>: E(G) → {0, 1} defined by f </em><em>∗</em><em>(uv)</em><em> </em><em>= (f (u) + f (v))(mod2) satisfies the condition |e<sub>f </sub>(0) − e<sub>f</sub> (1)| ≤ 1. A graph which admits Fibonacci cordial labeling is called Fibonacci cordial graph. In this paper, the author investigated the existence of Fibonacci Cordial Labeling of some Graphs.<strong></strong></em></p>amit himmatbhai rokad2018-04-092018-04-095INTEGRAL REPRESENTATION OF K-FUNCTION
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1266
<p><strong>Abstract: </strong>In this article, an attempt to be made to introduce the four-different integral representation of Extended K-function and special cases have also been discussed [1, 2]. The Extended K Function which is introduced by the authors first time in this work is extended version of earlier K-Function of Sharma [3]. We will use Riemann-Liouville fractional calculus operator in finding the integral representation of this extended K- Function. Fractional Calculus operators means calculus operators of arbitrary order (real, complex), So, these operators are useful in finding Integral representations of special functions of fractional calculus.</p><p><strong>Mathematics Subject Classification</strong>: 33C60, 33E12, 82C31, 26A33 </p><p><strong>Keywords</strong>: Riemann-Liouville fractional derivative operator, K-function, Gamma and Beta Function, M-Function.</p><p><strong>Cite this Article</strong></p><p><span style="font-size: medium;">Manoj Sharma, Laxmi Morya, Rajshree Mishra. Integral Representation of K-Function</span><span style="font-size: medium;">. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3):<br /> 81–88p.</span></p>Manoj SharmaLaxmi MoryaRajshree Mishra2018-01-222018-01-225Advanced Computable Extension of New Fractional Kinetic Equation
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1265
<p class="Standard" align="center"><strong><em>Abstract</em></strong><strong><em></em></strong></p><p class="Standard"><em>The aim of this work is to find</em><em> advanced computable extension of kinetic equation with fractional calculus approach. Fractional calculus originated in 1695, shortly after the inversion of classical calculus. The pioneer researchers who studied it were Liouville, Riemann, Leibniz, etc. For a long time, fractional calculus has been regarded as a pure mathematical area without applications. But, in recent decades, such type of synergism has been changed. It has been found that fractional calculus can be useful and even a powerful tool and a précis of the simple information about fractional calculus and its applications can be found in literature. The first use of a derivative of non-integer order is credited to the French mathematician S. F. Lacroix in 1819 who expressed the derivative of non-integer order </em> <em>in terms of Legendre’s factorial symbol </em><em>G</em><em>.</em></p><p><strong><em>Mathematics Subject Classification: </em></strong><em>33C60, 33E12, 82C31, 26A33.</em></p><p><strong><em>Keywords: </em></strong><em>Fractional kinetic equation, </em><em>new modified generalized </em> <em>function</em><em>, Riemann-Liouville operator, Laplace transform</em><em>, modified Riemann-Liouville fractional derivative operator, differential equation</em></p><p><strong>Cite this Article</strong></p><p><span style="font-size: medium;">Sharma M, Tyagi A. Advanced Computable Extension of New Fractional Kinetic Equation. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3): 74–80p.</span></p><p><em><br /></em></p>Manoj SharmaA. Tyagi2018-01-222018-01-225Triple Dirichlet Average of New Generalized M-Function and Fractional Derivative
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1267
<p><strong>Abstract</strong></p><p>The aim of present work to establish some results of Triple Dirichlet average of New Generalized M-Function, using fractional derivative.</p><p><strong>Keywords and Phrases</strong>: Dirichlet average, New Generalized M-Function fractional derivative and Fractional calculus operators.</p><p class="IEEEAbtract"><strong>Mathematics Subject Classification:</strong> 26A33, 33A30, 33A25 and 83C99.</p><p> </p><p><strong><span style="font-size: medium;">Cite this Article</span></strong></p><p> </p><p>Manoj Sharma. Triple Dirichlet Average of New Generalized M-Function and Fractional Derivative. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3): 89–93p.</p><p><span style="font-size: medium;">.</span></p>Manoj Sharma2018-01-222018-01-225Reliability Analysis of an Infinite Range Failure Model under Different Prior Distributions
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1219
<p align="center"><strong><em>Abstract</em></strong><strong><em></em></strong></p><p><em>A failure rate model has been selected for the present study and Bayesian estimates of reliability and hazard rate functions of that model have been obtained. The prior distributions considered for this study are, uniform and inverted gamma distributions. Since, the same mathematics is required for the study of survival analysis that is why the estimate of survival function has also been obtained in this paper.</em></p><p><strong><em>Keywords:</em></strong><em> Reliability function, hazard function, maximum likelihood estimate, inverted gamma distribution, survival function, conception rate function</em></p><p><strong>Cite this Article</strong><strong></strong></p><p>Dev Singh. Reliability Analysis of an Infinite Range Failure Model under Different Prior Distributions. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3): 51–73p.<strong></strong></p><p><em><br /></em></p>Dev Singh2018-01-112018-01-115Vector of Auto-Regressive Modeling for Agricultural Crop Production
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1210
<p align="center"><strong><em>Abstract</em></strong><strong><em></em></strong></p><p><em>The present investigation was carried out to study the trends in area and production of paddy crop, grown in Tamil Nadu during the period 1950–51 to 2009–10, using the multivariate time series modeling. The multivariate time-series model, VAR (p) model of order 1 was found suitable to study the trends in area as well as production of these crops. Decreasing trends in area as well production of these crops have been observed.</em></p><p><strong><em>Keywords: </em></strong><em>Cross correlation, partial canonical correlations<strong>,</strong> vector of auto-regression,</em><em> Portmanteau test, Jarque-Bera normality test</em></p><p><strong>Cite this Article</strong></p><p>Vetriselvi R, Rajarathinam A. Vector of Auto-Regressive Modeling for Agricultural Crop Production. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3): 42–50p.<strong></strong></p><p><em><br /></em></p>R. VetriselviRajarathinam Arunachalam2018-01-112018-01-115Cointegration and Error Correaction Modeling for BSE and NSE Stock Prices Time Series Data
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1198
<p class="Sous-auteur1" align="center"><strong><span lang="EN-GB">Abstract</span></strong></p><p class="Sous-auteur1"><span lang="FR">Econometrician always have been observed that most of the economics time series are non-stationary. The ordinary least square technique could be applied to estimate the model parameters only if the variables have been found to be stationary, i.e., they do not have unit roots. Otherwise, an alternative approach has to be followed, which is ‘co-integration’. Co-integration is a method of finding out the long-term relationship between economic variables under consideration. In the present study, it is necessary to know whether National Stock Exchange (NSE) and Bombay Stock Exchange (BSE) daily closing stock prices time séries data will have a long-term movements or relationship between them for which the co-integration analysis is being proposed. The error correction model (ECM) captures the impact of long-run equilibrium on the short run dynamics, provided that the variables are co-integrated. The ECM developed by Engle and Granger reconciles the short run behavior of an economic variable with its log run behavior. The ECM results showed that the coefficient of the error correction term ONELAGU, of about -0.020 suggests that only about 2% of the discrepancy between long-term and short-term LOGNSE is corrected, suggesting a slow rate of adjustment to equilibrium. Further, the validity of ECM showed that all three tests conducted through spurious of the model and serial correlation of residual and normality of residual are in favour of the model. Since, all tests in favour of ECM, the results of the model can be used for policy decisions.</span> </p><p class="Sous-auteur1"><strong><span lang="EN-GB">Keywords:</span></strong><span lang="EN-GB">Stationarity, co-integration, error correction model, </span><span lang="FR">breusch-godfrey test, granger representation theorem</span></p><p><strong>Cite this Article</strong><strong></strong></p><p>Rajarathinam A, Balamurugan D. Cointegration and Error Correaction Modeling for BSE and NSE Stock Prices Time Series Data. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3): 30–41p.<strong></strong></p><p class="Sous-auteur1"><span lang="FR"><br /></span></p>Rajarathinam ArunachalamBalamurugan D.2018-01-112018-01-115Self-Similarity Solution of Gravitating Cylindrical Shock Wave in the Theory of Flare-ups in Novae I
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1120
<p class="Normal13pt" align="center"><strong><span>Abstract</span></strong><strong></strong></p><p class="Normal13pt"><span>In this paper, the characteristics of the flow of a cylindrical shock wave are investigated where finite amount of energy in an infinitely concentrate form and total energy remain constant. The present paper deals with the problem in which a finite amount of energy in an infinitely concentrated form is suddenly released along a straight line in a conducting gas subjected to gravitational forces into account.</span></p><p class="Normal13pt"><strong>Keywords:</strong> Cylindrical shock; finite amount of energy, infinitely concentrate, Energy constant</p><p><strong>Cite this Article</strong></p><p>Jitendra Kumar Soni. Self-Similarity Solution of Gravitating Cylindrical Shock Wave in the Theory of Flare-ups in Novae I. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3): <br /> 18–21p.<strong></strong></p>Jitendra Kumar Soni2017-12-052017-12-055Intelligent Ranking Techniques for Software Reliability Growth Models
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1177
<p align="center"><strong><em>Abstract</em></strong><em></em></p><p><em>Reliability is gaining a lot of importance in the modern society, be it gadgets, automation, hardware, software or any other application. We are getting dependent more on machines and so is the need for relives ability gaining more weightage. This paper focuses on software reliability and the complexity involved in the prediction of reliability of the developed software. The paper introduces various assumptions and pre-requisites for software reliability modeling. Different researchers have proposed various software reliability growth models to predict the reliability of software, but none of these models may fit for all software development scenarios. The paper explores the existing ranking and selection techniques for choosing the best model out of the available options.</em></p><p class="IndexTerms"><strong><em>Keywords: </em></strong><em>Software reliability, software quality, modeling, SRGM, failure data </em></p><p><strong>Cite this Article</strong><strong></strong></p><p>Archana Kumar, Abhinav Juneja. Intelligent Ranking Techniques for Software Reliability Growth Models. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3): 22–29p.<strong></strong></p><p class="IndexTerms"><em><br /></em></p><p class="IndexTerms"> </p>Dr. Archana KumarAbhinav Juneja2017-12-052017-12-055Topological Integer Additive Set-Valued Graphs: A Review
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1054
<p align="center"><strong><em><span style="font-family: Times New Roman; font-size: medium;">Abstract</span></em></strong></p><p><em><span style="font-family: Times New Roman;">Let </span></em> <em><span style="font-family: Times New Roman;"> denote a set of non-negative integers and </span></em> <em><span style="font-family: Times New Roman;"> be the collection of all non-empty subsets of </span></em> <em><span style="font-family: Times New Roman;">. An integer additive set-labeling (IASL) of a graph </span></em> <em><span style="font-family: Times New Roman;"> is an injective set-valued function </span></em> <em><span style="font-family: Times New Roman;"> where induced function </span></em> <em><span style="font-family: Times New Roman;"> is defined by</span></em> <em><span style="font-family: Times New Roman;">, where </span></em> <em><span style="font-family: Times New Roman;"> is the sumset of </span></em> <em><span style="font-family: Times New Roman;"> and </span></em> <em><span style="font-family: Times New Roman;">. A set-labeling </span></em> <em><span style="font-family: Times New Roman;"> is said to be a topological set-labeling if </span></em> <em><span style="font-family: Times New Roman;"> is a topology on the ground set </span></em> <em><span style="font-family: Times New Roman;"> and a set-labeling </span></em> <em><span style="font-family: Times New Roman;"> is said to be a topogenic set-labeling if </span></em> <em><span style="font-family: Times New Roman;"> is a topology on </span></em> <em><span style="font-family: Times New Roman;">. In this article, we critically review some interesting studies on the properties and characteristics of different topological and topogenic integer additive set-labeling of certain graphs.</span></em></p><p><span style="font-family: Times New Roman;"><strong><em>Keywords</em></strong><strong><em>:</em></strong><em> Integer additive set-labeled graphs, topological additive set-labeled graphs, topogenic integer additive set-labeled graphs, integer additive set-filter graphs</em></span></p><p> </p><p><strong><span style="font-family: Times New Roman; font-size: medium;">Cite this Article</span></strong></p><p><strong></strong><span style="font-family: Times New Roman;">Sudev NK, Chithra KP, Germina KA. Topological Integer Additive Set-Valued Graphs: A Review. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(3): 1–17p.</span></p><p><span style="font-family: Times New Roman;"><em><br /></em></span></p>N. K. SudevK. P. ChithraK. A. Germina2017-11-142017-11-145Triple Dirichlet Average of New M-Function and Fractional Derivative
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1103
<p><strong>Abstract</strong>:<strong></strong></p><p>The aim of present paper to establish some results of Triple Dirichlet average of <strong>M-Function</strong>, using fractional derivative. Fractional calculus is powerful branch of mathematics that has also sporadically been applied to structural dynamics problems. Indeed, it appears to us to be a solution in search of a good problem. In that sense, it has some of the character of the laser shortly after its development. Fractional calculus is also inherent in the physics of dynamic structural mechanics or whether it is a convenient mathematical tool for manipulating equations derived using more conventional arguments. It is generally known that integer-order derivatives and integrals have clear physical and geometric interpretations.</p><p><strong>Keywords and Phrases</strong>: Dirichlet average, <em>New </em><strong>M-Function </strong>fractional derivative and Fractional calculus operators.</p><p><strong>Cite this Article</strong></p><p><span style="font-size: medium;">Mohd. Farman Ali, Manoj Sharma, Renu Jain. Triple Dirichlet Average of New M-Function and Fractional Derivative. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(2): 37–41p.</span></p>Mohd. Farman AliManoj SharmaRenu Jain2017-10-122017-10-125Dirichlet Average of Extended Mainardi Function and Fractional Derivative
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1102
<p align="center"><strong><em>Abstract</em></strong></p><p><em>In this paper, we establish a relation of Dirichlet average of Extended Mainardi function, using fractional derivative. The fractional calculus deals with integrals and derivatives of arbitrary order. Special roles in the applications of fractional calculus operators are played by the transcendental function of the Mittag-Leffler, generalized M-series function, M-function Generalized Miller-Ross function, Wright’s functions and more generally by the Meijer’s G-functions, Fox’s H-functions and Saxena’s I-function. </em><strong><em> </em></strong></p><p><strong><em>Mathematics Subject Classification:</em></strong><em> 26A33, 33A30, 33A25 and 83C99</em></p><p><strong><em>Keywords: </em></strong><em>Dirichlet average, Mainardi function, Extended Mainardi function fractional derivative and Fractional calculus operators</em></p><p><em><br /></em></p>Santosh VermaManoj Sharma2017-10-122017-10-125Integral of Generalized M-Function Involving Product of Special Functions
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1100
<p align="center"><strong><em>Abstract</em></strong></p><p><em>In this paper, an attempt to be made for study about </em><em>integrals of generalized M-Function multiplied with Jacobi polynomial.</em></p><p><strong><em>Keywords: </em></strong><em>Generalized Mittag-Leffler function, Generalized M-Function.</em></p><p><strong>Cite this Article</strong></p><p><span style="font-size: medium;">Laxmi Morya, Manoj Sharma, Rajshree Mishra. Integral of Generalized M-Function Involving Product of Special Functions. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(2): 33–36p.</span></p><p><em><br /></em></p>Laxmi MoryaManoj SharmaRajshree Mishra2017-10-122017-10-125Computable Extension of New Advanced Fractional Kinetic Equation
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1099
<p align="center"><strong><em>Abstract</em></strong><strong><em></em></strong></p><p><em>First, we obtain </em><em>computable extensions of advanced fractional kinetic equation. </em><em>The aim of this survey is to explore the behavior of physical and biological systems from the point of view of fractional calculus. Fractional calculus, integration and differentiation of an arbitrary or fractional order, provide new tools that expand the descriptive power of calculus beyond the familiar integer-order concepts of rates of change and area under a curve. Fractional calculus adds new functional relationships and new functions to the familiar family of exponentials and sinusoids that arise in the area of ordinary linear differential equations</em><em>. Among such functions that play an important role, we have the Euler Gamma function, the Euler Beta function, the Mittag-Leffler functions, the Wright and Fox functions, M-Function, K-Function. </em><em>In late years, fractional kinetic equations are considered because of their helpfulness and significance in mathematical physics, particularly in astrophysical issues. In astrophysics, kinetic equations assign an arrangement of differential equations, portraying the rate of progress of chemical composition of a star for every species as far as the response rates for pulverization and creation of that species are concerned. Techniques for demonstrating procedures of pulverization and generation of stars have been created for bio-chemical reactions and their insecure harmony states and for concoction response systems with unstable states, oscillations and hysteresis. The point of present paper is to discover the arrangement of summed up fragmentary request motor condition, utilizing another exceptional capacity. The outcome got here is decently widespread in nature. Exceptional cases, identifying with the Mittag-Leffler work are additionally considered</em><em>.</em><em> </em></p><p><strong><em>Mathematics Subject Classification: </em></strong><em>33C60, 33E12, 82C31, 26A33</em></p><p><strong><em>Keywords: </em></strong><em>Fractional kinetic equation, </em><em>Generalized </em> <em>function</em><em>, Riemann-Liouville operator, Laplace transform</em><em>, modified Riemann-Liouville fractional derivative operator, differential equation</em></p><p><strong>Cite this Article</strong></p><p><span style="font-size: medium;">Manoj Sharma, Laxmi Morya, Rajshree Mishra. Computable Extension of New Advanced Fractional Kinetic Equation. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(2): 26–32p.</span></p><p><em><br /></em></p>Manoj SharmaLaxmi MoryaRajshree Mishra2017-10-112017-10-115Triple Dirichlet Average of M- Function and Fractional Derivative
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1060
<p align="center"><strong><em>Abstract</em></strong><strong><em></em></strong></p><p><em>The aim of present paper is to establish some results of triple Dirichlet average of </em> <strong><em> </em></strong><em>function, using fractional derivative.</em></p><p><strong><em>Keywords:</em></strong><em> Dirichlet average, </em> <em> function<strong> </strong>fractional derivative and Fractional calculus operators</em></p><p><strong>Cite this Article</strong><strong></strong></p><p>Manoj Sharma,Kiran Sharma, Susheel Dhanelia. Triple Dirichlet Average of M- Function and Fractional Derivative. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(2): 21–25p.</p>Manoj SharmaKiran SharmaSusheel Dhanelia2017-09-012017-09-015Fractional Free Electron Laser Equation and K-Function
http://computers.stmjournals.com/index.php?journal=RRDMS&page=article&op=view&path%5B%5D=1058
<p align="center"><strong><em>Abstract</em></strong><strong><em></em></strong></p><p><em>In this era, </em><em>fractional </em><em>free electron laser (FEL) </em><em>equations are studied due to their utility and importance in mathematical physics and engineering. The aim of present paper is to find the solution of generalized fractional order </em><em>free electron laser (FEL) </em><em>equation, using K-function. The results obtained here are moderately universal in nature. Special cases, relating to the exponential function are also considered. The special functions were introduced in 18th century to define solutions of differential equations emerging as mathematical models of certain problems in astronomy, physics and biological science. The adjective ‘special’ of this nomenclature can be attributed to the simple fact that these functions owed their origin to special circumstances. Here we present a brief survey of the hypergeometric functions and their generalized forms due to the prime importance of hypergeometric functions in the study of special functions.</em></p><p><strong><em>Keywords: </em></strong><em>Fractional </em><em>free electron laser (FEL) </em><em>equation, K-function, generalized </em><em>M-series</em><em>, Riemann-Liouville operator</em></p><p><strong>Cite this Article</strong><strong></strong></p><p>Manoj Sharma, Kishan Sharma. Fractional Free Electron Laser Equation and K-Function. <em>Research & Reviews: Discrete Mathematical Structures</em>. 2017; 4(2): 17–20p.</p>Manoj SharmaKishan Sharma2017-09-012017-09-015