Mathematics Interdisciplinary Research
Volume:8 Issue: 3, Summer 2023

‎The minimum edge dominating energy‎, ‎denoted by $EE_{F}(G)$‎, ‎is the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph $G$‎. ‎In this paper‎, ‎we give some bounds and sharp bounds of $EE_{F}(G)$ in terms of matching number‎, ‎the number of positive eigenvalues of the minimum edge dominating matrix‎, ‎and the rank of $G$‎.

‎In this paper‎, ‎reflections of two distinct rays and also of families of orthotomic rays‎, ‎either on connected flat reflectors (as a nonsmooth surface) or on a parametric curved mirror‎, ‎are investigated in 2D Cartesian plane‎. ‎In this way‎, ‎both situations either when the source point is at a finite distance or at infinity are considered‎.  Although we used the usual methods in differential geometry but interestingly‎, ‎in our calculations‎, ‎the differential equations have not been used‎. ‎In fact‎, ‎at first‎, ‎for two distinct rays‎, ‎the intersection point of the reflected rays (which under some conditions is the interference point of simultaneous pulses) is geometrically described‎. ‎Moreover‎, ‎for two joint flat reflectors‎, ‎conditions by which the intersection point (or image) will be in front of the reflector‎, ‎are computed‎, ‎such that they give us an interval for the place of incident point on the second reflector‎. ‎In the continuation‎, ‎considering orthotomic families of rays‎, ‎the locus of interference points of reflected rays on two joint flat walls is obtained‎. ‎Then‎, ‎by the obtained results of two distinct rays (as a movement from discrete to continuous family)‎, ‎it is shown that the caustics of a family of reflected rays on a parametric curved mirror can be obtained‎. Finally‎, ‎finding the caustic for a curved reflector which has self-intersection‎, ‎and a theoretical idea to find the shape of an unknown mirror‎, ‎for a given source and image curve‎, ‎are described‎.

‎In this paper‎, ‎we consider a Sturm-Liouville equation with non-separated boundary conditions on a finite interval‎. ‎We discuss some properties of solutions of the Sturm-Liouville equation‎, ‎where the potential function has a singularity in the finite interval‎. ‎We also calculate eigenvalues and prove the uniqueness of Borg's Theorem of this boundary value problem‎.

‎Let $S$ be a set of $n$ points in the plane‎, ‎$\nabla(S)$ the set of all simple polygons crossing $S$‎, ‎$\gamma_P$ the maximum angle of polygon $P \in \nabla(S)$ and $\theta =min_{P\in\nabla(S)} \gamma_P$‎. ‎In this paper‎, ‎we prove that $\theta\leq 2\pi-\frac{2\pi}{r.m}$ where $m$ and $r$ are the number of edges and inner points of the convex hull of $S$‎, ‎respectively‎. ‎We also propose an algorithm to construct a polygon with the upper bound on its angles‎. ‎Constructing a simple polygon with the angular constraint on a given set of points in the plane can be used for path planning in robotics‎. ‎Moreover‎, ‎we improve our upper bound on $\theta$ and prove that this is tight for $r=1$‎.

‎In this paper‎, ‎we investigate an improved method for stabilizing a class of uncertain chaotic nonlinear dynamical system‎. ‎Our approach follows techniques of optimal principle for time-delayed feedback control and adaptive tracking control theory for stabilizing unstable periodic orbits in a chaotic bounded attractor‎. ‎The uncertain parameters expressed in the system can be separated‎. ‎Analysis and proof are presented using the Lyapunov stability theorem‎. ‎In particular‎, ‎we use the adaptive control theory to design an adaptive law for the estimation of uncertain time-delayed controlled chaotic nonlinear dynamical systems‎. ‎The predictions are presented by numerical simulation through the Rossler system to demonstrate theoretical results‎.

‎Let $R$ be a commutative ring with nonzero identity‎. ‎Throughout this paper we explore some properties of two certain subgraphs of the maximal graph of $R$‎.