Journal of Prime Research in Mathematics

Abstract: Here first we will prove the existence of fixed points for a weakly continuous, strictly quasi bounded operator on a reflexive Banach space and a completely continuous, strictly quasi bounded operator on any normed linear space. Using these results we can deduce the existence of eigen values and surjectivity of quasi bounded operator in similar situations.

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 a detour set of \(G\) if every vertex in \(G\) lies on a detour joining a pair of vertices of \(S\). The detour number \(dn(G)\) of G is the minimum order of its detour sets and any detour set of order \(dn(G)\) is a detour basis of \(G\). A set \(S ⊆ V\) is called a connected detour set of \(G\) if S is detour set of \(G\) and the subgraph \(G[S]\) induced by S is connected. The connected detour number \(cdn(G)\) of \(G\) is the minimum order of its connected detour sets and any connected detour set of order \(cdn(G)\) is called a connected detour basis of \(G\). Graphs G with detour diameter \(D ≤ 4\) are characterized when \(cdn(G) = p\), \(cdn(G) = p−1\), \(cdn(G) = p−2\) or \(cdn(G) = 2\). A subset \(T\) of a connected detour basis \(S\) of \(G\) is a forcing subset for \(S\) if \(S\) is the unique connected detour basis containing \(T\). The forcing connected detour number \(fcdn(S)\) of \(S\) is the minimum cardinality of a forcing subset for \(S\). The forcing connected detour number \(fcdn(G)\) of \(G\) is \(min{fcdn(S)}\), where the minimum is taken over all connected detour bases \(S\) in \(G\). The forcing connected detour numbers of certain classes of graphs are determined. It is also shown that for each pair \(a\), \(b\) of integers with \(0 ≤ a < b\) and \(b ≥ 3\), there is a connected graph \(G\) with \(fcdn(G) = a\) and \(cdn(G) = b\).

Abstract: The velocity field and the shear stress corresponding to the unsteady flow of a generalized Oldroyd-B fluid in an infinite circular cylinder subject to a longitudinal time-dependent shear stress are determined by means of Hankel and Laplace transforms. The exact solutions, written in terms of the generalized \(G\)-functions, satisfy all imposed initial and boundary conditions. The similar solutions for ordinary Oldroyd-B, ordinary and generalized Maxwell, ordinary and generalized second grade as well as for Newtonian fluids are obtained as limiting cases of our general solutions.

Abstract: For given graphs \(G\) and \(H\), the Ramsey number \(R(G, H)\) is the least natural number n such that for every graph \(F\) of order \(n\) the following condition holds: either \(F\) contains \(G\) or the complement of \(F\) contains \(H\). Beaded wheel \(BW_{2,m}\) is a graph of order \(2m + 1\) which is obtained by inserting a new vertex in each spoke of the wheel \(W_m\). In this paper, we determine the Ramsey number of paths versus Beaded wheels: \(R(P_n, BW_{2,m}) = 2n − 1\) or \(2n\) if \(m ≥ 3\) is even or odd, respectively, provided \(n ≥ 2m^2 − 5m + 4\).

Abstract: This paper presents a computational technique for Fredholm and Volterra integral equations of the second kind. The method based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and legendre polynomials are presented. The operational matrices of integration and product are utilized to reduce the computation of integral equation into some algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Abstract: A two-parameter family of piecewise \(C^2\) rational quintic functions is presented along with an error investigation for the approximation of an arbitrary \(C^3\) function. The two parameters have a direct geometric interpretation making their use straightforward. Illustrations of their effect on the shape of the rational function are given. The relaxed continuity constraints and the increased flexibility via the two parameters make the
proposed function a suitable candidate for interactive CAD.