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A new counting polynomial, called Omega, was recently proposed by Diudea. It is defined on the ground of opposite edge strips "ops". The Sadhana polynomial has been defined to evaluate the Sadhana index of a molecular graph. A third one, called Theta polynomial is also derived from ops. In this paper we compute Omega, Sadhana and Theta polynomials of the Cartesian product graphs  

A new counting polynomial, called Omega Ω(G,x), was recently proposed by Diudea. It is defined on the ground of "opposite edge strips" ops. Two related polynomials: Sadhana Sd(G,x) and Theta Θ(G,x) polynomials can also be calculated by ops counting. Close formulas for calculating these three polynomials in infinite nano-structures resulted by embedding the titanium dioxide pattern in plane, cylinder and torus are derived. For the design of titanium dioxide pattern, a procedure based on a sequence of map operations is proposed  

Two edges e and f of a plane graph G are in relation opposite, e op f, if they are opposite edges of an inner face of G. Relation op enables the partition the edge set of G into opposite edge strips ops. On this ground, Diudea defined Omega and Theta polynomial while Ashrafi et al. defined the Sadhana polynomial. In this paper a weighted version of these polynomials was introduced and several relations between them are demonstrated. Some molecular weights are suggested in view of using the derived topological indices in correlational studies  

Let G be a simple graph of order n. We mean by dominating set, a set S C V(G) such that every vertex of G is either in S or adjacent to a vertex in S. The domination polynomial of G is the polynomial Σni=1 (G,i)xi where d((g,i) is the number of dominating sets of G of size i. Two graphs G and H are said to be V-equivalent, written G H, if D((G, x) = D(H, x). The D-equivalence class of G is [G] = { H H G }. Recently, the determination of D-equivalence class of a given graph, has been of interest. In this paper, it is shown that for n ≥ 3, [Kn, n] has size two. We conjecture that the complete bipartite graph Km,n for n - m ≥ 2, is uniquely determined by its domination polynomial  

In this paper, we complement some recent results of L. Márki, J. Meyer, J. Szigeti and L. van Wyk, by investigating the constant-trace representations of a Clifford algebra C(V) of an arbitrary quadratic form q: V→ F (possibly degenerate) and we present some relevant applications. In particular, the existence of the polynomial identities of C(V) of particular form when the characteristic of the base field is zero is looked at. Furthermore, a lower bound is found on the minimal number t, such that C(V) can be embedded in a matrix ring of degree t, over some commutative F-algebra. Also, some results on the dimension of commutative subalgebras of C(V) are obtained. © 2021, The Author(s), under... 

A new methodology has been proposed to enhance inverse dynamics applications in the process of trajectory planning and optimization in Terrain Following Flights (TFF). The new approach uses least square scheme to solve a general two-dimensional (2D) Terrain Following Flight in a vertical plane. In the mathematical process, Chebyshev polynomials are used to model the geographical data of the terrain in a given route in a manner suitable for the aircraft at hand. The aircraft then follows the modeled terrain with sufficient clearance. In this approach the TF problem is effectively converted to an optimal tracking problem. Results show that. this method provides a flexible approach to solve the... 

Let G be a simple graph of order n. A dominating set of G is a set S of vertices of G so that every vertex of G is either in S or adjacent to a vertex in 5. The domination polynomial of G is the polynomial D(G, x) = Σn i=1 d(G, i) xi, where d(G, i) is the number of dominating sets of G of size i. In this paper we show that cycles are determined by their domination polynomials  

Omega polynomial was introduced by Diudea. Following Omega polynomial, the Sadhana polynomial was defined by Ashrafi et al. In this paper we compute Omega and Sadhana polynomials of three classes of dendrimers  

We use the polynomial method to study the existence of partial transversals in the Cayley addition table of Abelian groups  

In this paper, we introduce the concept of universal Gröbner basis for a parametric polynomial ideal. In this direction, we present a new algorithm, called UGS, which takes as input a finite set of parametric polynomials and outputs a universal Gröbner system for the ideal generated by input polynomials, by decomposing the space of parameters into a finite set of parametric cells and for each cell associating a finite set of parametric polynomials which is a universal Gröbner basis for the ideal corresponding to that cell. Indeed, for each values of parameters satisfying a condition set, the corresponding polynomial set forms a universal Gröbner basis for the ideal. Our method relies on the... 

In this paper, an accurate series solution in conjunction with an energy formulation for the treatment of piezocomposite plates with arbitrary geometry and aspect ratio, under both electrical and mechanical loadings are proposed. A remedy for dealing with nonhomogeneous boundary conditions is also presented. Through introduction of amending polynomials of order pk for the kth layer, the accuracy and convergence rate are dramatically improved. These polynomials ensure continuity of the generalized displacement fields across the interfaces, while their derivatives can have the required discontinuities up to a desired order. Moreover, depending on the nature of the physical problem under... 

In this paper, we study graphs whose matching polynomials have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs. We show that apart from K7(set minus)(E(C3)∪E(C4)) there is no connected k-regular matching integral graph if k≥2. It is also shown that if G is a graph with a perfect matching, then its matching polynomial has a zero in the interval (0,1]. Finally, we describe all claw-free matching integral graphs  

In this paper, we study graphs whose matching polynomials have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs. We show that apart from K7∖(E(C3)∪E(C4)) there is no connected k-regular matching integral graph if k≥2. It is also shown that if G is a graph with a perfect matching, then its matching polynomial has a zero in the interval (0,1]. Finally, we describe all claw-free matching integral graphs. © 2017 Elsevier B.V  

In this paper we try to unify the frameworks of definitions of semantic security, indistinguishability and non-malleability by defining semantic security in comparison based framework. This facilitates the study of relations among these goals against different attack models and makes the proof of the equivalence of semantic security and indistinguishability easier and more understandable. Besides, our proof of the equivalence of semantic security and indistinguishability does not need any intermediate goals such as non devidability to change the definition framework  

This paper presents comparison among methods of determination of the characteristic polynomial coefficients. First, the resultant systems from the methods are compared based on frequency criteria such as the closed loop bandwidth, gain and phase margins. Then the step responses of the resultant systems are compared on the basis of the transient behavior criteria including overshoot, rise time, settling time and error (via IAE, ITAE, ISE and ITSE integral indices). Also relative stability of the systems is compared together. Finally the best choices in regards to the above diverse criteria are presented. COPYRIGHT © ENFORMATIKA  

Here we are concerned with the best approximation by polynomials to rational functions in the uniform norm. We give some new theorems about the best approximation of 1/(1+x) and 1/(x-a) where a>1. Finally we extend this problem to that of computing the best approximation of the Chebyshev expansion in uniform norm and give some results and conjectures about this. © 2005 IMACS. Published by Elsevier B.V. All rights reserved  

A proper orientation of a graph G=(V,E) is an orientation D of E(G) such that for every two adjacent vertices v and u, dD -(v) ≠ dD -(u) where dD -(v) is the number of edges with head v in D. The proper orientation number of G is defined as χ→(G)=minD∈Γmaxv∈V(G)d D -(v) where Γ is the set of proper orientations of G. We have χ(G)-1≤χ→(G)≤Δ(G), where χ(G) and Δ(G) denote the chromatic number and the maximum degree of G, respectively. We show that, it is NP-complete to decide whether χ→(G)=2, for a given planar graph G. Also, we prove that there is a polynomial time algorithm for determining the proper orientation number of 3-regular graphs. In sharp contrast, we will prove that this problem... 

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a0 + a1x + ⋯ + anxn (resp. right polynomial a0 + xa1 + ⋯ + xnan) over K such that a0 + a1a + ⋯ + anan = 0 (resp. a0 + aa1 + ⋯ + anan). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not... 

In this paper we present a sampling result about continuous-domain black and white images that form a convex shape. In particular, we will study shapes whose boundaries belong to the zero-level sets (roots) of bivariate polynomials. In [1] it was shown that generalized 2D moments of the image can lead to annihilation equations for the coefficients of the bivariate polynomial that determine the boundary of the shape. More precisely, when the bivariate polynomial is of degree n (which has n(n+3)/2 non-trivial coefficients), the results in [1] indicate that for invertibility of the linear annihilation equations, 2D moments of up to degree 3n-1 (overall equation moments) are required. In this... 

In the touring polygons problem (TPP), for a given sequence (s= P0, P1, ⋯, Pk, t = Pk+1) of polygons in the plane, where s and t are two points, the goal is to find a shortest path that starts from s, visits each of the polygons in order and ends at t. In the constrained version of TPP, there is another sequence (F0, ⋯, Fk) of polygons called fences, and the portion of the path from Pi to Pi+1 must lie inside the fence Fi. TPP is NP-hard for disjoint non-convex polygons, while TPP and constrained TPP are polynomially solvable when the polygons are convex and the fences are simple polygons. In this work, we present the first polynomial time algorithm for solving constrained TPP when the...