Cubic Pell’s Equation

This is an initial study of Pell’s equations of higher degree, which is an open problem in Number Theory. The rst step is to investigate the Pell’s equation of the form x³-dy³ = 1. Later, we consider the form N(θ) = x³ + cy³ + c²z³ – 3cxyz = 1, where...

Strukturen hos amöbor och koamöbor i två dimensioner

We study complex functions of two variables through topological studies of their amoebas and coamoebas. The number of holes in the coamoeba to certain polynomials are carefully investigated. Using winding numbers it is proven that a maximum number of holes in the...

Uniqueness of Solutions to the Poisson Equation

We show that the Poisson equation has at most one solution where M is open and bounded. As intermediate steps we also show two properties for harmonic functions. The first that for a harmonic function v, the value at any given point will equal the mean value of v in a...

Konvexa funktioner

Denna uppsats behandlar begreppet konvexa funktioner utifrån ett matematiskt perspektiv. Tyngdpunkten ligger på envariabla funktioner, men i det fjärde och sista kapitlet definieras även en flervariabel konvex funktion. Förutom bevis på att vissa funktioner är konvexa...