Rational Points on Elliptic Curves y^2=x^3+a^3 in f_{p} where p{\equiv}1(mod6) is Prime

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some results are given on the sum of abscissae of these points. We give the number of solutions to y^2{\equiv}x^3+a^3(modp), also given in ([1], p.174), this time by means of the quadratic residue character, in a different way, by using the cubic residue character. Using the Weil conjecture, one can generalize the results concerning the number of points in F_{p} to F_{p^{r}}.Comment: 9 pages; Keywords: Elliptic curves over finite fields, rational point

A p-adic look at the Diophantine equation x^{2}+11^{2k}=y^{n}

We find all solutions of Diophantine equation x^{2}+11^{2k} = y^{n} where x>=1, y>=1, n>=3 and k is natural number. We give p-adic interpretation of this equation.Comment: 4 page

Preface

On behalf of the International Advisory Board and the Local Organizing Committee of The International Congress in Honour of Professor Hari M. Srivastava, which was held in the Auditorium at the Campus of Uludag University, Bursa, Turkey on August 23-26, 2012, and on our own behalves, we would like to express our happiness and gratitude to all participants who attended and actively participated in our Congress. This article is being published in each of the four Special Issues of the SpringerOpen journals, Advances in Difference Equations, Boundary Value Problems, Fixed Point Theory and Applications and Journal of Inequalities and Applications, which are entitled Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

Some Inequalities for the first General Zagreb Index of Graphs and Line Graphs

The first general Zagreb index Mα1(G) of a graph G is equal to the sum of the αth powers of the vertex degrees of G. For α≥0 and k≥1, we obtain the lower and upper bounds for Mα1(G) and Mα1(L(G)) in terms of order, size, minimum/maximum vertex degrees and minimal non-pendant vertex degree using some classical inequalities and majorization technique, where L(G) is the line graph of G. Also, we obtain some bounds and exact values of Mα1(J(G)) and Mα1(Lk(G)), where J(G) is a jump graph (complement of a line graph) and Lk(G) is an iterated line graph of a graph G

On Sombor Index

The concept of Sombor index (SO) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index SO: SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2√, where dG(vi) is the degree of vertex vi in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work

Decision support algorithm under SV-neutrosophic hesitant fuzzy rough information with confidence level aggregation operators

To deal with the uncertainty and ensure the sustainability of the manufacturing industry, we designed a multi criteria decision-making technique based on a list of unique operators for single-valued neutrosophic hesitant fuzzy rough (SV-NHFR) environments with a high confidence level. We show that, in contrast to the neutrosophic rough average and geometric aggregation operators, which are unable to take into account the level of experts' familiarity with examined objects for a preliminary evaluation, the neutrosophic average and geometric aggregation operators have a higher level of confidence in the fundamental idea of a more networked composition. A few of the essential qualities of new operators have also been covered. To illustrate the practical application of these operators, we have given an algorithm and a practical example. We have also created a manufacturing business model that takes sustainability into consideration and is based on the neutrosophic rough model. A symmetric comparative analysis is another tool we use to show the feasibility of our proposed enhancements