Journal of Prime Research in Mathematics

Abstract: A new class of distributions known as Bivariate Pseudo–Exponential distribution has been defined. The distribution of r–th concomitant and joint distribution of r–th and s–th concomitant of record statistics of the resulting distribution have been derived. Expression for single and product moments has also been obtained for the resulting distributions. A characterization of the k-th concomitant of record statistics for the Pseudoexponential distribution by the conditional expectation is presented.

Abstract: The cardinality of a metric basis of a connected graph \(G\) is called its metric dimension, denoted by \(dim(G)\) and the maximum value of distance between vertices of \(G\) is called its diameter. In this paper, the graphs \(G\) with diameter 2 are characterized when \(dim(G) = 2.\)

Abstract: For two vertices \(u\) and \(v\) in a graph \(G = (V, E)\), the detour distance \(D(u, v)\) is the length of a longest \(u–v\) path in \(G\). A \(u–v\) path of length \(D(u, v)\) is called a \(u–v\) detour. A set \(S ⊆ V\) is called an edge detour set if every edge in \(G\) lies on a detour joining a pair of vertices of \(S\). The edge detour number \(dn_1(G)\) of G is the minimum order of its edge detour sets and any edge detour set of order \(dn_1(G)\) is an edge detour basis of \(G\). A connected graph \(G\) is called an edge detour graph if it has an edge detour set. A subset \(T\) of an edge detour basis \(S\) of an edge detour graph \(G\) is called a forcing subset for \(S\) if \(S\) is the unique edge detour basis containing \(T\). A forcing subset for \(S\) of minimum cardinality is a minimum forcing subset of \(S\). The forcing edge detour number \(fdn_1(S)\) of \(S\), is the minimum cardinality of a forcing subset for \(S\). The forcing edge detour number \(fdn_1(G)\) of \(G\), is \(min{fdn_1(S)}\), where the minimum is taken over all edge detour bases \(S\) in \(G\). The general properties satisfied by these forcing subsets are discussed and the forcing edge detour numbers of certain classes of standard edge detour graphs are determined. The parameters \(dn_1(G)\) and \(fdn_1(G)\) satisfy the relation \(0 ≤ fdn_1(G) ≤ dn_1(G)\) and it is proved that for each pair \(a\), \(b\) of integers with \(0 ≤ a ≤ b\) and \(b ≥ 2\), there is an edge detour graph \(G\) with \(fdn_1(G) = a\) and \(dn_1(G) = b\).

Abstract: A decomposition of a graph G is a collection Ψ of edge-disjoint subgraphs \(H_1,H_2, . . . , H_n\) of \(G\) such that every edge of \(G\) belongs to exactly one \(H_i\). If each \(H_i\) is a path in \(G\), then \(Ψ\) is called a path partition or path cover or path decomposition of \(G\). A divisor path decomposition of a \((p, q)\) graph \(G\) is a path cover \(Ψ\) of \(G\) such that the length of all the paths in \(Ψ\) divides \(q\). The minimum cardinality of a divisor path decomposition of \(G\) is called the divisor path decomposition number of \(G\) and is denoted by \(π_D(G)\). In this paper, we initiate a study of the parameter \(π_D\) and determine the value of \(π_D\) for some standard graphs. Further, we obtain some bounds for \(π_D\) and characterize graphs attaining the bounds.

Abstract: The velocity field and associated shear stress corresponding to the longitudinal oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of Laplace and Hankel transforms. The flow is due to both of the cylinders that at \(t = 0^+\) suddenly begin to oscillate along their common axis with different angular frequencies of their respective velocities. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of Newtonian fluid are also obtained as limiting case. The flows of second grade and Newtonian fluids are compared graphically by plotting their velocity profiles.

Abstract: In this paper we introduce the notions of right F-derivation and left F-derivation of a BCI-algebra and some related properties are explored.