Milnor Fibrations and Regularity Conditions for Real Analytic Mappings
JPRM-Vol. 19 (2023), Issue 1, pp. 82 – 91 Open Access Full-Text PDF
Khurram Shabbir
Abstract: When ƒ: \((\mathbb{R^n},0)\)→\((\mathbb{R^p},0)\)is a surjective real analytic map with isolated critical value, we prove that the(m)-regularity condition (in a sense we define) ensures that \(\frac{f}{||f||}\) is a fibration on small spheres, ƒ induces afibration on the tubes and these fibrations are equivalent.In particular, we make the statement of [12] more precise in the case of an isolated critical point and weextend it to the case of an isolated critical value.
Group Distance Magic Labeling of Product of Graphs
JPRM-Vol. 19 (2023), Issue 1, pp. 73 – 81 Open Access Full-Text PDF
Wasim Ashraf, Hani Shaker
Abstract: A graph is a tool used to build the interconnection network that a system requires. Such networks inter-operability is ensured by specific labeling. There are several labelings in the literature, however the Group Distance Magic Labeling is better for graphs. A graph G is described as ℋ-distance magic graph if for an abelian group ℋ, there exist one-one mapping 𝓁 between group elements and vertex set of graph G such that ⅀ 𝓍 ∈ N(u) 𝓁(x) = µ for all u ∈ V, where µ is the magic constant belongs to abelian group ℋ and N(u) is u′s free neighborhood. In this article, we prove direct product of anti-prism graphs with nth order cycles are ℤ 2st, ℤ2×ℤst, ℤ3 × ℤ2t and ℤ3 × ℤ⅔st distance magic graphs.
Optimal System, Group Invariant Solutions and Conservation Laws of the Non-linear Elastic Wave Equation and Damped Elastic Wave Equation
JPRM-Vol. 19 (2023), Issue 1, pp. 55 – 72 Open Access Full-Text PDF
M. Usman, A. Razzaq, Ali Raza, F.D. Zaman
Abstract: Non-linear wave equations are created by the elastic wave propagation through inelastic material. We obtain the Lie point symmetries for the non-linear elastic wave equation and the optimal system of the symmetry algebra using Lie symmetry approach. Numerous solutions that are group invariant are obtained under the optimal system of subalgebras of Lie algebra. Additionally, the variational symmetries are obtained via Noether approach and the corresponsing conservation laws are presented. The non-linear elastic wave equation with a damping term is also studied. The local conservation laws using the direct approach are also discussed in this study.
Zagreb Connection Numbers on Different Networks
JPRM-Vol. 19 (2023), Issue 1, pp. 44 – 54 Open Access Full-Text PDF
Muhammad Mubashar, Asad Zubair, Muhammad Hussain
Abstract: The first Zagreb index formulated in its approximate form for -electron energy in 1972 and second Zagreb index formulated in 1975 for branching of molecules. Some modification of these indices was proposed in three different ways naming as novel modification, connection indices and leap Zagreb indices. In this paper we proposed and calculated connection indices for Honey Comb Network and triangular benezoid structures. Furthermore as an extension of our work we also formulated connection indices for line graph of subdivision of Honey Comb and triangular benezoid networks.
On the representations of the braid group and the welded braid group
JPRM-Vol. 19 (2023), Issue 1, pp. 34 – 43 Open Access Full-Text PDF
Rana S. Kahil, Mohammad N. Abdulrahim
Abstract: We study the composition of F. R. Cohen’s map Bn → Bnk with the standard representation of Bnk, where Bn is the braid group on n strings. We prove that the obtained representation of Bn is isomorphic to the direct sum of k copies of the standard representation of Bn. A similar work is done for the welded braid group 𝓌Bn.
On the solutions of 2x + 2y = z2 in the Fibonacci and Lucas numbers
JPRM-Vol. 19 (2023), Issue 1, pp. 27 – 33 Open Access Full-Text PDF
Hayder R. Hashim
Abstract: Consider the Diophantine equation 2x + 2y = z2, where x, y and z are nonnegative integers. As thisequation has infinitely many solutions, in this paper we study its solutions in case where the unknowns represent Fibonacci and/or Lucas numbers. In other words, we completely resolve the equation in case of (x, y, z) ∈ {(Fi, Fj , Fk),(Fi, Fj ,Lk),(Li,Lj ,Lk),(Li,Lj , Fk),(Fi,Lj ,Lk),(Fi,Lj , Fk)} with i, j, k ≥ 1 and Fn and Ln denote the general terms of Fibonacci and Lucas numbers, respectively.
