A Note on Powers in Finite Fields

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation which is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.Comment: 4 page

The circle equation over finite fields

Interesting patterns in the geometry of a plane algebraic curve C can be observed when the defining polynomial equation is solved over the family of finite fields. In this paper, we examine the case of C the classical unit circle defined by the circle equation x2 + y2 = 1. As a main result, we establish a concise formula for the number of solutions to the circle equation over an arbitrary finite field. We also provide criteria for the existence of diagonal solutions to the circle equation. Finally, we give a precise description of how the number of solutions to the circle equation over a prime field grows as a function of the prime.Mathematics Subject Classification (2010): 11G20, 11D45, 11A07, 14G15.Keywords: Diophantine geometry, prime numbers, siamese twin prime

Generating Geospatially Realistic Driving Patterns Derived From Clustering Analysis Of Real EV Driving Data

In order to provide a vehicle fleet that realistically represents the predicted Electric Vehicle (EV) penetration for the future, a model is required that mimics people driving behaviour rather than simply playing back collected data. When the focus is broadened from on a traditional user-centric smart charging approach to be more grid-centric, it suddenly becomes important to know not just when- and how much the vehicles charge, but also where in the grid they plug in. Since one of the main goals of EV-grid studies is to find the saturation point, it is equally important that the simulation scales, which calls for a statistically correct, yet flexible model. This paper describes a method for modelling EV, based on non-categorized data, which takes into account the plug in locations of the vehicles. By using clustering analysis to extrapolate and classify the primary locations where the vehicles park, the model can be transferred geographically using known locations of the same classification