On characteristic polynomial and energy of Sombor matrix
ODAM-Vol. 4 (2021), Issue 3, pp. 29 – 35 Open Access Full-Text PDF
Gowtham Kalkere Jayanna, Ivan Gutman
Abstract:Let \(G\) be a simple graph with vertex set \(V=\{v_1,v_2,\ldots,v_n \}\), and let \(d_i\) be the degree of the vertex \(v_i\). The Sombor matrix of \(G\) is the square matrix \(\mathbf A_{SO}\) of order \(n\), whose \((i,j)\)-element is \(\sqrt{d_i^2+d_j^2}\) if \(v_i\) and \(v_j\) are adjacent, and zero otherwise. We study the characteristic polynomial, spectrum, and energy of \(\mathbf A_{SO}\). A few results for the coefficients of the characteristic polynomial, and bounds for the energy of \(\mathbf A_{SO}\) are established.
A note: Characterization of star, helm, flower and complete graphs by total vertex stress
ODAM-Vol. 4 (2021), Issue 3, pp. 24 – 28 Open Access Full-Text PDF
Johan Kok
Abstract:This note presents the characterization of the families of star, helm, flower and complete graphs by total vertex stress. The note does not present results for many families of graphs but, it highlights important philosophical (math. phil.) aspects for further research. In particular the novelty concepts of forgiven contradictions denoted by, iff\(_f\) as well as iffness and \(f\)-statements are introduced. The author suggests that the characterization of other families of graphs by total vertex stress is possible.
A new version of Ostrowski type integral inequalities for different differentiable mapping
OMS-Vol. 5 (2021), Issue 1, pp. 353 – 359 Open Access Full-Text PDF
M. Iftikhar, A. Qayyum, S. Fahad, M. Arslan
Abstract:In this paper, improved and generalized version of Ostrowski’s type inequalities is established. The parameters used in the peano kernels help us to obtain previous results. The obtained bounds are then applied to numerical integration.
Integral representations for local dilogarithm and trilogarithm functions
OMS-Vol. 5 (2021), Issue 1, pp. 337 – 352 Open Access Full-Text PDF
Masato Kobayashi
Abstract:We show new integral representations for dilogarithm and trilogarithm functions on the unit interval. As a consequence, we also prove (1) new integral representations for Apéry, Catalan constants and Legendre \(\chi\) functions of order 2, 3, (2) a lower bound for the dilogarithm function on the unit interval, (3) new Euler sums.
Some arguments for the wave equation in Quantum theory
OMS-Vol. 5 (2021), Issue 1, pp. 314 – 336 Open Access Full-Text PDF
Tristram de Piro
Abstract:We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.
Moments of generalized order statistics for Pareto-Rayleigh distribution
OMS-Vol. 5 (2021), Issue 1, pp. 306 – 313 Open Access Full-Text PDF
M. Alam, R. U. Khan, Z. Vidović
Abstract:In this paper, we derive the explicit expressions for single and product moments of generalized order statistics from Pareto-Rayleigh distribution using hypergeometric functions. Also, some interesting remarks are presented.
