Research & Reviews Discrete Mathematical Structures vol 3 issue 3

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ISSN 2394-1979 (Online)

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Research & Reviews: Discrete Mathematical Structures

Contents

1. Fractional Calculus of the R-Series Mohd. Farman Ali, Manoj Sharma

1

2. R-L F Integral and Triple Dirichlet Average of the R-Series Mohd. Farman Ali

6

3. Dirichlet Average of New Generalized M-series and Fractional Calculus Manoj Sharma

13

4. A Brief Review on Algorithms for Finding Shortest Path of Knapsack Problem Swadha Mishra

17

5. Research and Industrial Insight: Discrete Mathematics 20

Research & Reviews: Discrete Mathematical Structures ISSN: 2394-1979(online) Volume 3, Issue 3 www.stmjournals.com

Fractional Calculus of the R-Series Mohd. Farman Ali1,*, Manoj Sharma2 1

Department of Mathematics, Madhav University, Sirohi, Rajasthan, India Department of Mathematics, Rustamji Institute of Technology, BSF Academy, Tekanpur, Gwalior, Madhya Pradesh, India

2

Abstract The present paper creates a special function called as R-series. This is a special case of Hfunction given by Inayat Hussain. The Hypergeometric function, Mainardi function and Mseries follow R-series and these functions have recently found essential applications in solving problems in physics, biology, bio-science, engineering and applied science etc. Mathematics Subject Classification—26A33, 33C60, 44A15 Keywords: Fractional calculus operators, R−series, Mellin-Barnes integral, special functions

INTRODUCTION TO THE H-FUNCTION The H- function of Inayat Hussain, is a generalization of the familiar H-function of Fox, defined in terms of Mellin-Barnes contour integral [1], as (đ?›ź đ??´ ;đ?›źđ?‘— )1,đ?‘› ,(đ?›źđ?‘—, đ??´đ?‘—; đ?›źđ?‘— )đ?‘›+1,đ?‘? đ?‘—, đ?‘— 1,đ?‘š , (đ?›˝đ?‘—, đ??ľđ?‘— ; đ?‘?đ?‘— )đ?‘š+1,đ?‘ž

đ?‘š,đ?‘› đ??ťđ?‘?,đ?‘ž [đ?‘§(đ?›˝ đ?‘—,đ??ľ đ?‘—)

1

+đ?‘–∞

] = 2đ?œ‹đ?‘– âˆŤâˆ’đ?‘–∞ đ?œƒ(đ?‘ )đ?‘§ đ?‘ đ?‘‘đ?‘

(1)

Where the integrand (or Mellin transform of the H-function) đ?œƒ(đ?‘ ) =

đ?›źđ?‘— đ?‘› âˆ?đ?‘š đ?‘—=1 Γ(đ?›˝đ?‘—− đ??ľđ?‘— đ?‘ ) âˆ?đ?‘—=1[Γ(1=đ?›źđ?‘— +đ??´đ?‘— đ?‘ )]

(2)

đ?‘?đ?‘— đ?‘? âˆ?đ?‘› đ?‘—=đ?‘š+1[Γ(1=đ?›˝đ?‘— +đ??ľđ?‘— đ?‘ )] âˆ?đ?‘—=đ?‘›+1 Γ(đ?›źđ?‘—− đ??´đ?‘— đ?‘ )

Contains fractional powers of some of the involved Γ −functions. Here đ?›źđ?‘— (1đ?‘’đ?‘&#x;đ?‘’đ?‘?) and đ?›˝đ?‘— (1đ?‘›đ?‘‘ đ?‘ž) are complex parameters; đ??´đ?‘— > 0 ( 10đ?‘&#x;đ?‘’đ?‘? ), đ??ľđ?‘— > 0(1 ‌ ‌ ‌ đ?‘ž) ; and exponents; đ?‘Žđ?‘— (đ?‘— = 1 ‌ ‌ ‌ đ?‘›) and đ?‘?đ?‘— (đ?‘— = 1 ‌ ‌ ‌ đ?‘ž) can take noninteger values. Evidently, when all the exponent đ?‘Žđ?‘— and đ?‘?đ?‘— take integer values only, the H-function reduces to the familiar H-function of Fox, [1–3]. The sufficient conditions for the absolute convergence of the contour integral (1), as given by Buschman and Srivastava, are as follows [3]: Ί=

đ?‘š đ?‘—=1

đ??ľđ?‘— +

đ?‘› đ?‘—=1

đ?‘Žđ?‘— đ??´đ?‘— −

đ?‘? đ?‘—=đ?‘š+1

đ?‘?đ?‘— đ??ľđ?‘— −

đ?‘ž đ?‘—=đ?‘›+1

1 2

đ??´đ?‘— > 0 and arg(đ?‘§) < đ?œ‹ Ί

THE R-SERIES

The R- series is 0 đ?›ź,đ?›˝ 0 0 đ?›ź,đ?›˝ đ?‘?đ?‘…đ?‘ž (đ?‘Ž1 . . . . đ?‘Žđ?‘? ; , đ?‘?1 . . . . đ?‘?đ?‘ž ; đ?‘§ ) = đ?‘?đ?‘…đ?‘ž (đ?‘§) 0 đ?›ź,đ?›˝ đ?‘?đ?‘…đ?‘ž (đ?‘§)

=

(đ?‘Ž1 )đ?‘˜ . . . . .(đ?‘Žđ?‘? ) đ?‘§đ?‘˜ ∞ đ?‘˜ đ?‘˜=đ?‘œ (đ?‘? ) . . . . .(đ?‘? ) Γ(đ?›źđ?‘˜+đ?›˝)đ?‘˜! 1 đ?‘˜ đ?‘ž đ?‘˜

Here, đ?‘? upper parameters đ?‘Ž1, đ?‘Ž2, 0, đ?‘š > 0 and (đ?‘Žđ?‘— )đ?‘˜ (đ?‘?đ?‘— )đ?‘˜

. . . .

đ?‘Žđ?‘? and đ?‘ž lower parameters

RRDMS (2016) 1-5 Š STM Journals 2016. All Rights Reserved

(3) đ?‘?1, đ?‘?2,

. . . . đ?‘?đ?‘ž ,

đ?›źđ?œ–đ??ś , đ?‘…(đ?›ź) >

Page 1

Research & Reviews: Discrete Mathematical Structures ISSN: 2394-1979(online) Volume 3, Issue 3 www.stmjournals.com

R-L F Integral and Triple Dirichlet Average of the R-Series Mohd. Farman Ali Department of Mathematics, Madhav University, Sirohi, Rajasthan, India

Abstract In this article, we establish the relation between some results of triple Dirichlet average of the R-series and fractional operators. We use a new special function called as R-series, which is a special case of H-function given by Inayat Hussain. In this article, the solution is obtained in compact form of triple Dirichlet average of R-series as well as conversion into single Dirichlet average of R-series, using fractional integral. Keywords: Dirichlet averages, special functions, R-series and Riemann-Liouville fractional integral Mathematics Subject Classification: 2000: Primary: 33E12, 26A33; Secondary: 33C20, 33C65.

INTRODUCTION The Dirichlet average of a function is a certain kind integral average with respect to Dirichlet measure. The concept of Dirichlet average was introduced by Carlson in 1977. Carlson has defined Dirichlet averages of functions, which represent certain types of integral average with respect to Dirichlet measure [1–4]. He showed that various important special functions could be derived as Dirichlet averages for the ordinary simple functions like � � ,� � etc. He has also pointed out that the hidden symmetry of all special functions, which provided their various transformations can be obtained by averaging � � ,� � etc. [5, 6]. Thus, he established a unique process towards the unification of special functions by averaging a limited number of ordinary functions [7]. Gupta and Agarwal found that averaging process is not altogether new but directly connected with the old theory of fractional derivative [8, 9]. Carlson overlooked this connection whereas he has applied fractional derivative in so many cases during his entire work. Deora and Banerji have found the double Dirichlet average of ex by using fractional derivatives and they have also found the triple Dirichlet average of xt by using fractional derivatives [10, 11]. Sharma and Jain obtained double Dirichlet average of trigonometry function cos � using fractional derivative and they have also found the triple Dirichlet average of ex by using fractional calculus [12–15]. Recently, Kilbas and Kattuveetti established a correlation among Dirichlet averages of the generalized Mittag-Leffler function with Riemann-Liouville fractional integrals and of the hyper-geometric functions of many variables [16].

DEFINITIONS AND PRELIMINARIES Some definitions are necessary in the preparation of this paper. Standard Simplex in đ?‘šđ?’Œ , đ?’Œ ≼ đ?&#x;?: The standard simplex in đ?‘… đ?‘˜ , đ?‘˜ ≼ 1 by [1]. đ??¸ = đ??¸đ?‘˜ = {đ?‘†(đ?‘˘1, đ?‘˘2 , ‌ đ?‘˘đ?‘˜ ) âˆś đ?‘˘1 ≼ 0, ‌ đ?‘˘đ?‘˜ ≼ 0, đ?‘˘1 + đ?‘˘2 + â‹Ż + đ?‘˘đ?‘˜ ≤ 1}

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Research & Reviews: Discrete Mathematical Structures ISSN: 2394-1979(online) Volume 3, Issue 3 www.stmjournals.com

Dirichlet Average of New Generalized M-series and Fractional Calculus Manoj Sharma Department of Mathematics, Rustamji Institute of Technology, BSF Academy, Tekanpur, Gwalior, Madhya Pradesh, India

Abstract We know that every analytic function can be measured as a Dirichlet average and connected with fractional calculus. In this note, we set up a relation between Dirichlet average of new generalized M-series, and fractional derivative. Fractional derivative is a derivative of arbitrary order i.e. may be real, complex, integer or fractional order. Mathematics Subject Classification: 26A33, 33A30, 33A25 and 83C99. Keywords: Dirichlet average new generalized M-series, fractional derivative, fractional calculus operators

INTRODUCTION Carlson has defined Dirichlet average of functions which represents certain types of integral average with respect to Dirichlet measure [1–5]. He showed that various important special functions can be derived as Dirichlet averages for the ordinary simple functions like� � ,� � etc. He has also pointed out that the hidden symmetry of all special functions which provided their various transformations can be obtained by averaging � � ,� � etc. [6, 7]. Thus he established a unique process towards the unification of special functions by averaging a limited number of ordinary functions. Almost all known special functions and their well-known properties have been derived by this process. In this paper, the Dirichlet average of new generalized M-series has been obtained.

DEFINITIONS We give below some of the definitions which are necessary in the preparation of this paper: Standard Simplex in đ?‘šđ?’? , đ?’? ≼ đ?&#x;?: We denote the standard simplex in đ?‘… đ?‘› , đ?‘› ≼ 1 by Carlson [1]. đ??¸ = đ??¸đ?‘› = {đ?‘†(đ?‘˘1, đ?‘˘2 , ‌ ‌ . . đ?‘˘đ?‘› ) âˆś đ?‘˘1 ≼ 0, ‌ ‌ ‌ . đ?‘˘đ?‘› ≼ 0, đ?‘˘1 + đ?‘˘2 + â‹Ż ‌ ‌ + đ?‘˘đ?‘› ≤ 1}

(1)

Dirichlet Measure Let đ?‘? ∈ đ??ś đ?‘˜ , đ?‘˜ ≼ 2 and let đ??¸ = đ??¸đ?‘˜âˆ’1 be the standard simplex in đ?‘… đ?‘˜âˆ’1 . The complex measure đ?œ‡đ?‘? is defined by đ??¸[1]. 1 đ?‘? −1 đ?‘?đ?‘˜âˆ’1 −1 (1 − đ?‘˘1 − â‹Ż ‌ ‌ ‌ − đ?‘˘đ?‘˜âˆ’1 )đ?‘?đ?‘˜âˆ’1 đ?‘‘đ?‘˘1 ‌ ‌ ‌ ‌ . đ?‘‘đ?‘˘đ?‘˜âˆ’1 (2) đ?‘‘đ?œ‡đ?‘? (đ?‘˘) = đ??ľ(đ?‘?) đ?‘˘11 ‌ ‌ ‌ ‌ ‌ . đ?‘˘đ?‘˜âˆ’1 It will be called a Dirichlet measure. Here,

Γ(đ?‘?1 ) ‌ ‌ ‌ ‌ ‌ . . Γ(đ?‘?đ?‘˜ ) , Γ(đ?‘?1 + â‹Ż ‌ ‌ . . +đ?‘?đ?‘˜ ) đ??ś> = {đ?‘§ ∈ đ?‘§: đ?‘§ ≠0, |đ?‘?â„Ž đ?‘§| < đ?œ‹â „2}, Open right half plane and đ??ś>k is the đ?‘˜ đ?‘Ąâ„Ž Cartesian power of đ??ś>. đ??ľ(đ?‘?) = đ??ľ(đ?‘?1, ‌ ‌ ‌ . đ?‘?đ?‘˜) =

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Page 13

Research & Reviews: Discrete Mathematical Structures ISSN: 2394-1979(online) Volume 3, Issue 3 www.stmjournals.com

A Brief Review on Algorithms for Finding Shortest Path of Knapsack Problem Swadha Mishra* Department of Computer Applications, Invertis University, Bareilly, Uttar Pradesh, India Abstract The gathering knapsack and knapsack problems are summed up to briefest way issue in a class of graphs. An effective calculation is used for finding briefest ways that bend lengths are non-negative. A more effective calculation is portrayed for the non-cyclic which incorporates the knapsack issue. Keywords: knapsack problem, rucksack problem

INTRODUCTION The knapsack problem or rucksack problem is an issue in combinatorial enhancement: Given an arrangement of things, each with a weight and an esteem, decide the number of everything to incorporate into an accumulation so that the aggregate weight is not exactly or equivalent to a given point of confinement and the aggregate esteem is as expansive as could reasonably be expected. Group knapsack problem has been given to: Minimise ∑đ?‘›đ?‘—=1 đ?‘?jjxj (1) Subject to ∑đ?‘›đ?‘—=1 đ?‘?jgj=g0 (2) Where, x1,‌, xn non-negative integers. g0,‌, gn are the subset of the elements of a finite additive abelian group H and c1,‌,cn are non-negative reals. This algorithm for solving this problem has been described by Gomory [1], Shapiro [2, 3], Hu [4] and others. It can be formulated as a shortest path problem in the following way. Let G1 be the graph with node H and arc of the form (h, h+gj) h an arbitrary element of H and j=1,‌, n. The length of such an arc is cj. Let P be the from 0 to g0 in G1 then if xj is the number of arcs of the form (h, h+gj) in P then (x1,‌, xn) is a solution to Eq. (2) and the length of P is Eq. (1). Conversely, if (xn,‌, xn) satisfied Eq. (2), then one may construct paths from 0 to g0. Now a new algorithm is given in this paper to solve this problem [5]. The name knapsack problem applies to:

Maximize ∑đ?‘›đ?‘—=0 đ?‘?jjxj Subject to ∑đ?‘›đ?‘—=0 đ?‘¤jxj=W,

(3) (4)

Where, x0, xi,‌, xn non-negative integers. Where, c0=0, c1,‌, cn are positive reals, w0=1 and w1,‌, wn, W are positive integers. One can formulate a knapsack problem as a longest path problem defining the graph G2 with nodes 0, 1,‌, W and arcs of the form (w, w+wj) of length cj. The knapsack problem is then equivalent to that of finding a longest path from 0 to W [6].

ALGORITHM The graph G1 and G2 of the previous section are examples of a class of graphs which for the purposes of this paper, we call knapsack graphs. Definition A graph G with nodes N and arcs A is a knapsack graph if, (1) The arcs A can be partitioned into n disjoint sets A1,‌,AN; (2) The length of each arc belonging to Aj is lj; (3) Let P=(i0, i1,...,ip) be a path between an arbitrary pair of nodes i0, ip. Suppose that (it-1, it)∈Amt, for t=1,‌., p. Then for any re-ordering, n1,‌., np of the indices m1,‌, mp, there exist a path Q=(j0,j1,‌, jp), where j0=i0, jp=ip and (jt-1=jt) ∈Ant for t=1,‌., p. For shortest path problems with non-negative

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Research and Industrial Insight: Discrete Mathematics Math’s Maze Runner Mazes are in vogue right now, from NBO's West world, to the arrival of the British faction TV arrangement, The Crystal Maze. Be that as it may, labyrinths have been around for centuries and a standout amongst the most acclaimed labyrinths, the Labyrinth home of the Minotaur, assumes a featuring part in Greek mythology. The most part acknowledged that a maze contains just a single way, regularly spiraling around and collapsing back on it, in constantly diminishing circles, though a labyrinth contains expanding ways, giving the voyager decisions and the potential for getting, exceptionally lost. Design a maze is very tough task and for that human should be rewarded. Many algorithms for maze have been created by computer scientists and mathematicians. These algorithms based on two principles: one which begin with a solitary, limited space and afterward sub-separate it with dividers (and entryways) to deliver ever littler sub-spaces; and others which begin with a world brimming with detached rooms and after that devastate dividers to make ways/courses between them. Escape Plan There are many techniques by which you can run away from maze but firstly you need to the details about the maze from which you want to escape. Most strategies work for "basic" labyrinths, that is, ones with no tricky alternate routes by means of scaffolds or "entry circles" – round ways that lead back to where they began. Along these lines, accepting it is a straightforward labyrinth, the strategy that many individuals know is "follow the wall”. Basically, you put one hand on a mass of the labyrinth (it doesn't make a difference which hand the length of you are reliable) and after that continue strolling, keeping up contact between your hand and the divider. In the end, you will get out. This is on account of on the off chance that you envision getting the mass

of a labyrinth and extending its edge to evacuate any corners, you will in the long run frame something circle-like, a portion of which must shape part of the labyrinth's external limit. This strategy for escape may not work, be that as it may, if the begin or complete areas are in the labyrinth's middle. Be that as it may, a few labyrinths are purposely intended to disappoint, for example, the Escot Gardens' beech fence labyrinth in Devon, which contains no less than five extensions, thus a long way from "straightforward". There is another method by which maze escape will be easy i.e., Tremaux’s Algorithm, it works in all the cases. Envision that, as Hansel and Gretel in the pixie story, you can leave a trail of "breadcrumbs" behind you as you explore your way through the labyrinth and afterward recall these guidelines: in the event that you touch base at an intersection you have not beforehand experienced (there will be no scraps as of now on the trail ahead), then haphazardly select an approach. On the off chance that that leads you to an intersection where one way is different to you yet the other is not, then select the unexplored way. What's more, if picking between an on more than one occasion utilized way, pick the way utilized once, then leave another, second trail behind you. The cardinal administer is never, ever select a way as of now containing two trails. This technique is ensured, in the end, to get you out of any labyrinth. Mazes in everyday life Now you think how is this maze thing useful for us? Maze is interesting and adventurous but not for our daily day-to-day life when we are on work or we are doing something important. Bill Hillier, a theorist in 1980s, find that most of the housing estates that have a layout like maze. This reise q question that we can measure the maze-iness of a house?

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