In this article we discuss moduli and constants of quasi-Banach space and give some important properties of these moduli and constants. Moreover, we establish relationships of these moduli and constants with each other.
Let \(G=(V,E)\) be a~finite simple graph with \(|V(G)|\) vertices and \(|E(G)|\) edges. An edge-covering of \(G\) is a family of subgraphs \(H_1, H_2, \dots, H_t\) such that each edge of \(E(G)\) belongs to at least one of the subgraphs \(H_i\), \(i=1, 2, \dots, t\). If every subgraph \(H_i\) is isomorphic to a given graph \(H\), then the graph \(G\) admits an \(H\)-covering. A graph \(G\) admitting \(H\) covering is called an \((a,d)\)-\(H\)-antimagic if there is a bijection \(f:V\cup E \to \{1,2,\dots, |V(G)|+|E(G)| \}\) such that for each subgraph \(H’\) of \(G \) isomorphic to \(H\), the sum of labels of all the edges and vertices belonged to \(H’\) constitutes an arithmetic progression with the initial term \(a\) and the common difference \(d\). For \(f(V)= \{ 1,2,3,\dots,|V(G)|\}\), the graph \(G\) is said to be super \((a,d)\)-\(H\)-antimagic and for \(d=0\) it is called \(H\)-supermagic. In this paper, we investigate the existence of super \((a,d)\)-\(C_4\)-antimagic labeling of book graphs, for difference \(d=0,1\) and \(n\geq2\).
In this article, the authors obtain the boundedness of the fractional Marcinkiewicz integral with variable kernel on Morrey-Herz spaces with variable exponents \(\alpha\) and \(p\). The corresponding boundedness for commutators generalized by the Lipschitz function is also considered.
In this article we continue the investigations presented in our previous papers [1,2,3,4], presenting some, for the best of our knowledge, new transformations of the Gauss hypergeometric function (3) and (13). They have been obtained using only elementary methods and stem from a couple of integrals evaluated in terms of complete elliptic integral of first kind by Legendre in [5] Chapter XXVII, at sections II and III.
As an important branch of theoretical chemistry, chemical index calculation has received wide attention in recent years. Its theoretical results have been widely used in many fields such as chemistry, pharmacy, physics, biology, materials, etc. and play a key role in reverse engineering. Its basic idea is to obtain compound characteristics indirectly through the calculation of topological index. As a basic structure, quasi-tree structures are widely found in compounds. In this paper, we obtain the maximal value and the second smallest value of quasi-tree graphs of order \(n.\)
Since the introduction of complex fractals by Mandelbrot they gained much attention by the researchers. One of the most studied complex fractals are Mandelbrot and Julia sets. In the literature one can find many generalizations of those sets. One of such generalizations is the use of the results from fixed point theory. In this paper we introduce in the generation process of Mandelbrot and Julia sets a combination of the S-iteration, known from the fixed point theory, and the s-convex combination. We derive the escape criteria needed in the generation process of those fractals and present some graphical examples.
In this paper, we establish viscosity rule for common fixed points of two nonexpansive mappings in the framework of CAT(0) spaces. The strong convergence theorems of the proposed technique is proved under certain assumptions imposed on the sequence of parameters. The results presented in this work extend and improve some recent announced in the literature.
In this work, we developed homotopy perturbation double Sumudu transform method (HPDSTM) which is obtained by combining homotopy perturbation method, double Sumudu transform and He’s polynomials. The method is applied to find the solution of linear fractional one and two dimensional dispersive KdV and nonlinear fractional KdV equations to illustrate the reliability of the method. It is observed that the solutions obtained by the method converge rapidly to the exact solutions. This method is very powerful, and professional techniques for solving different kinds of linear and nonlinear fractional order differential equations.
In this article, we compute closed forms of M-polynomial for three general classes of convex polytopes. From the M-polynomial, we derive degree-based topological indices such as first and second Zagreb indices, modified second Zagreb index, Symmetric division index, etc.
Let \(G\) be a simple connected molecular graph with vertex set \(V(G)\) and edge set \(E(G)\). One important modification of classical Zagreb index, called hyper Zagreb index \(HM(G)\) is defined as the sum of squares of the degree sum of the adjacent vertices, that is, sum of the terms \({[{{d}_{G}}(u)+{{d}_{G}}(v)]^2}\) over all the edges of \(G\), where \(d_G(v)\) denote the degree of the vertex \(u\) of \(G\). In this paper, the hyper Zagreb index of certain bridge and chain graphs are computed and hence using the derived results we compute the hyper Zagreb index of several classes of chemical graphs and nanostructures.
