Magic and Modulo Magicness of Paley Digraph

A special digraph arises in round robin tournaments. More exactly, a tournament Tq with q players 1, 2, ... , q in which there are no draws. This gives rise to a digraph in which either (u, v) or (v, u) is an arc for each pair u, v. Graham and Spencer defined the tournament as, The nodes of digraph Dp are {0, 1, ... , p -1} and Dp contains the arc (u, v) if and only if u - v is a quadratic residue modulo p where p  3(mod 4) be a prime. This digraph is referred as the Paley tournament. Raymond Paley was a person raised Hadamard matrices by using this quadratic residues. So to honor him this tournament was named as Paley tournament. These results were extended by Bollobas for prime powers. Modular super edge trimagic labeling and modular super vertex magic total labeling has been investigated in this paper.

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Quadratic residue modulo a power of 2

10.31219/osf.io/a6sv4 ◽

2018 ◽

Author(s):

Thinh D. Nguyen

Keyword(s):

Prime Power ◽

Quadratic Residue ◽

Residue Modulo ◽

Quadratic Residues ◽

Powers Of 2 ◽

The Subject ◽

Quadratic Residue Modulo

Quadratic residue modulo an odd prime power has been studied for centuries. Many mathematical tools have been devised to deal with those odd prime power. The left moduli of powers of 2 are thus naturally the subject of this article. We set the objectives of showing some intriguing properties of quadratic residues modulo an even prime power. Though humble in its signiﬁcance, our results are achieved after years of reading Prasolov’s and I. F. Sharygin’s maths books.

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On the Hybrid Fourth Power Mean Involving Legendre’s Symbol and One Kind Two-Term Exponential Sums

Journal of Mathematics ◽

10.1155/2021/5560902 ◽

2021 ◽

Vol 2021 ◽

pp. 1-9

Author(s):

Jingzhe Wang

Keyword(s):

Exponential Sums ◽

Quadratic Residue ◽

Fourth Power ◽

Power Mean ◽

Residue Modulo ◽

Fourth Power Mean ◽

Elementary Methods ◽

Quadratic Residue Modulo

The main purpose of this article is using the elementary methods and the properties of the quadratic residue modulo an odd prime p to study the calculating problem of the fourth power mean of one kind two-term exponential sums and give an interesting calculating formula for it.

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Extraordinary circumstances: Covid-19 – Italian Serie A scenario

Proceedings of the Institution of Mechanical Engineers Part P Journal of Sports Engineering and Technology ◽

10.1177/17543371211019289 ◽

2021 ◽

pp. 175433712110192

Author(s):

Milan Radojicic ◽

Aleksandar Djokovic ◽

Nikola Cvetkovic

Keyword(s):

Cluster Analysis ◽

Hierarchical Cluster Analysis ◽

Hierarchical Cluster ◽

Round Robin ◽

The Past ◽

Novel Approach ◽

Round Robin Tournaments

Unpredictable and uncontrollable situations have happened throughout history. Inevitably, such situations have an impact on various spheres of life. The coronavirus disease 2019 has affected many of them, including sports. The ban on social gatherings has caused the cancellation of many sports competitions. This paper proposes a methodology based on hierarchical cluster analysis (HCA) that can be applied when a need occurs to end an interrupted tournament and the conditions for playing the remaining matches are far from ideal. The proposed methodology is based on how to conclude the season for Serie A, a top-division football league in Italy. The analysis showed that it is reasonable to play 14 instead of the 124 remaining matches of the 2019–2020 season to conclude the championship. The proposed methodology was tested on the past 10 seasons of the Serie A, and its effectiveness was confirmed. This novel approach can be used in any other sport where round-robin tournaments exist.

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