# Journal of Prime Research in Mathematics

Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing two issues in June and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year.

Latest Published Articles

### Weakened condition for the stability to solutions of parabolic equations with “maxima”

JPRM-Vol. 1 (2013), Issue 1, pp. 01 – 10 Open Access Full-Text PDF
D. Kolev, T. Donchev, K. Nakagawa
Abstract: A class of reaction-diffusion equations with nonlinear reaction terms perturbed with a term containing ”maxima” under initial and boundary conditions is studied. The similar problems that have no ”maxima” have been studied during the last decade by many authors. It would be of interest the standard conditions for the reaction function to be weakened in the sense that the partial derivative of the reaction function, w.r.t. the unknown, to be bounded from above by a rational function containing $$(1 + t) ^{−1}$$ where $$t$$ is the time. When we slightly weaken the standard condition imposed on the reaction function then the solution still decays to zero not necessarily in exponential order. Then we have no exponential stability for the solution of the considered problem. We establish a criterion for the nonexponential stability. The asymptotic behavior of the solutions when $$t → +∞$$ is discussed as well. The parabolic problems with ”maxima” arise in many areas as the theory of automation control, mechanics, nuclear physics, biology and ecology.

### Weighted homogeneous polynomials with isomorphic milnor algebras

JPRM-Vol. 1 (2012), Issue 1, pp. 106 – 114 Open Access Full-Text PDF
Imran Ahmed
Abstract: We recall first some basic facts on weighted homogeneous functions and filtrations in the ring A of formal power series. We introduce next their analogues for weighted homogeneous diffeomorphisms and vector fields. We show that the Milnor algebra is a complete invariant for the classification of weighted homogeneous polynomials with respect to right-equivalence, i.e. change of coordinates in the source and target by diffeomorphism.

### Covering cover pebbling number for square of a cycle

JPRM-Vol. 1 (2012), Issue 1, pp. 102 – 105 Open Access Full-Text PDF
A. Lourdusamy, T. Mathivanan
Abstract: Let $$G$$ be a connected graph. Let p be the number of pebbles distributed on the vertices of $$G$$. A pebbling move is defined by removing two pebbles from one vertex and put a pebble on an adjacent vertex. The covering cover pebbling number, $$σ(G)$$, is the least p such that after a sequence of pebbling moves, the set of vertices should form a covering for $$G$$ from every configuration of p pebbles on the vertices of $$G$$. In this paper, we determine the covering cover pebbling number for square of a cycle.

### On two families of graphs with constant metric dimension

JPRM-Vol. 1 (2012), Issue 1, pp. 95 – 101 Open Access Full-Text PDF
M. Ali, M. T. Rahim, G. Ali
Abstract: If $$G$$ is a connected graph, the distance d(u, v) between two vertices $$u, v ∈ V (G)$$ is the length of a shortest path between them. Let $$W = {w_1, w_2, …., w_k}$$ be an ordered set of vertices of $$G$$ and let $$v$$ be a vertex of $$G$$. The representation r(v|W) of $$v$$ with respect to $$W$$ is the k-tuple $$(d(v, w_1), d(v, w_2), ….., d(v, w_k))$$. If distinct vertices of $$G$$ have distinct representations with respect to W, then W is called a resolving set or locating set for $$G$$. A resolving set of minimum cardinality is called a basis for $$G$$ and this cardinality is the metric dimension of $$G$$, denoted by $$dim(G)$$. A family G of connected graphs is a family with constant metric dimension if $$dim(G)$$ does not depend upon the choice of $$G$$ in $$G$$. In this paper, we show that the graphs (D^{∗}_{p}\) and $$D^{n}_{p}$$, obtained from prism graph have constant metric dimension.

### Continuity estimate of the optimal exercise boundary with respect to volatility for the american foreign exchange put option

JPRM-Vol. 1 (2012), Issue 1, pp. 85 – 94 Open Access Full-Text PDF
Nasir Rehman, Sultan Hussain, Malkhaz Shashiashvili
Abstract: In this paper we consider the Garman-Kohlhagen model for the American foreign exchange put option in one-dimensional diffusion model where the volatility and the domestic and foreign currency risk-free interest rates are constants. First we make preliminary estimate regarding the optimal exercise boundary of the American foreign exchange put option and then the continuity estimate with respect to volatility for the value functions of the corresponding options. Finally we establish the continuity estimate for the optimal exercise boundary of the American foreign exchange put option with respect to the volatility parameter.

### Some results of accretive operators and convex sets in 2-probabilistic normed space

JPRM-Vol. 1 (2012), Issue 1, pp. 76 – 84 Open Access Full-Text PDF
P. K. Harikrishnan, K. T. Ravindran
Abstract: In this paper we introduce the concept of accretive operators, discuss some properties of resolvents of an accretive operator in 2- probabilistic normed spaces and focusing on the results of convex sets in 2-probabilistic normed spaces.