Journal of Prime Research in Mathematics

Abstract: Let \(K\) be a commutative field. In this paper we study the action of the automorphism group of the rational function field \(K(X)\) on the set of all valuations of \(K(X)\) which are trivial on \(K\). We apply this study in finding a classification of some simple algebraic extension of \(K\).

Abstract:  Let K be a commutative field and let \(K((X))\) be the field of Laurent series in one variable \(X\), consider with its natural \(X-\)adic topology. In this paper we prove that any closed subfield \(K ⊂ L ⊂ K((X))\) is of the form \(L = K((f))\) and \(K((X))\) is algebraic over L of degree \(ord_{X}(f)\). Some other properties of \(L\) are studied.

Abstract: A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some result concerning both Ramsey saturated and unsaturated graph. In particular, we show that a cycle \(C_{n}\) and a Jahangir \(J_{m}\) Ramsey unsaturated or saturated graphs of \(R(C_n, W_m)\) and \(R(P_n, J_m)\), respectively. We also suggest an open problems.

Abstract: Computer graphics environment requires realistic visual models of data generated. These data can be either 2D or 3D and corresponding visual models are called curves and surfaces. In this paper, a piecewise rational cubic function [8] has been extended to rational bicubic function. Simple constraints are derived on the free parameters in the description of rational bicubic function to preserve the shape of monotonic data.

Abstract: The aim of the present paper is to investigate a new subclass of starlike functions of complex order \(b\neq 0\). Let \(f(z)=z+a_{2}z^{2}+…\) be an analytic function in the unit disc \(D=\{z| |z|<1\}\) which satisfies \(1+\frac{1}{b}(z\frac{f'(z)}{f(z)}-1)=\frac{1+A\omega z}{1+B\omega z}\), for some \(\omega \in \Omega\) and for all \(z \in D\). Then f is called a Janowski starlike function of complex order b, where A and B are complex numbers such that \(Re(1-A\overline{B})\geq |A-B|, im(1-A\overline{B}<|A-B|, |B|<1\) and \(\omega(z)\) ) is a Schwarz function in the unit disc D [1], [10], [12]. The class of these functions is denoted by \(S^{∗}(A, B, b)\). In this paper we will give the representation theorem, distortion theorem, two point distortion theorem, Koebe domain under the montel normalization, and coefficient inequality for this class.

Abstract: Let G be a graph with vertex set \(V = V (G)\) and edge set \(E = E(G)\) and let \(e = |E(G)|\) and \(v = |V (G)|\). A one-to-one map \(λ\) from \(V ∪ E\) onto the integers \(\{1, 2, …, v + e\}\) is called vertex magic total labeling if there is a constant \(k\) so that for every vertex \(x\), \(λ+\sum λ(xy)=k\). where the sum is over all vertices \(y\) adjacent to \(x\). Let us call the sum of labels at vertex x the weight \(w_{λ}(x)\) of the vertex under labeling \(λ\); we require \(w_{λ}(x) = k\) for all \(x\). The constant \(k\) is called the magic constant for \(λ\). In this paper, we present the vertex magic total labelings of disconnected graph, in particular, two copies of isomorphic generalized Petersen graphs \(2P(n, m)\), disjoint union of two non-isomorphic suns \(S_{m} ∪ S_{n}\) and t copies of isomorphic suns \(tS_{n}\).