Journal of Prime Research in Mathematics

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.

Abstract: For a connected graph \(G\) of order \(p ≥ 2\), a set \(S ⊆ V (G)\) is an \(x\)-geodominating set of \(G\) if each vertex \(v ∈ V (G)\) lies on an \(x-y\) geodesic for some element y in \(S\). The minimum cardinality of an \(x\)-geodominating set of G is defined as the \(x\)-geodomination number of \(G\), denoted by gx(G). An \(x\)-geodominating set of cardinality \(g_x(G)\) is called a \(g_x\)-set of \(G\). A connected \(x\)-geodominating set of G is an \(x\)-geodominating set S such that the subgraph \(G[S]\) induced by \(S\) is connected. The minimum cardinality of a connected \(x\)-geodominating set of \(G\) is defined as the connected \(x\)-geodomination number of \(G\) and is denoted by \(cg_x(G)\). A connected \(x\)-geodominating set of cardinality \(cg_x(G)\) is called a \(cg_x\)-set of \(G\). We determine bounds for it and find the same for some special classes of graphs. If \(p, a\) and \(b\) are positive integers such that \(2 ≤ a ≤ b ≤ p − 1\), then there exists a connected graph G of order \(p\), \(g_x(G) = a\) and \(cg_x(G) = b\) for some vertex \(x\) in \(G\). Also, if \(p\), \(d\) and \(n\) are integers such that \(2 ≤ d ≤ p − 2\) and \(1 ≤ n ≤ p\), then there exists a connected graph \(G\) of order \(p\), diameter \(d\) and \(cg_x(G) = n\) for some vertex \(x\) in \(G\). For positive integers \(r\), \(d\) and \(n\) with \(r ≤ d ≤ 2r\), there exists a connected graph \(G\) with rad \(G = r\), \(diam G = d\) and \(cg_x(G) = n\) for some vertex \(x\) in \(G\).

Abstract: Richardson’s phenomenological mathematical model of the thrombi growth in microvessels is extended to describe more realistic features of the phenomenon. Main directions of the generalization of Richardson’s model are: 1) the dependence of platelet activation time on the distance from the injured vessel wall; 2) the nonhomogeneity of the platelet distribution in blood flow in the vicinity of the vessel wall. The generalization of the model corresponds to the main experimental results concerning thrombi formation obtained in recent years. The extended model permits to achieve a numerical agreement between model and experimental data.