Planetary nebulae (PN) often have weird shapes, due to an inhomogeneous interstellar medium. We investigated the propagation of the shock wave that forms a PN. The form of the shock wave depends on...Show morePlanetary nebulae (PN) often have weird shapes, due to an inhomogeneous interstellar medium. We investigated the propagation of the shock wave that forms a PN. The form of the shock wave depends on the initial density distribution. The equation that describes the shock propagation is a first order non-linear partial differential equation. We found a analytic solution for the equation after a certain assumptions for some basic functions and made estimations for more complex density functions. We also made a model that used toroidal coordinates and one in three dimensions. The toroidal model resembles the Red Rectangle nebula. We also inverted the two-dimensional equation with some assumptions to derive the initial density function from a known shock wave. We used a numerical model to compute the density profile for eleven known planetary nebula. This leads to a qualitative classification into the ellipsoidal, disk and and irregular nebula. Inserting some test shock waves into this equation shows the existence of a extraordinary clover like shape in the density function.Show less
The study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra...Show moreThe study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard example was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infinity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications. This particular paper focuses on Gelfand theory — the relation between multiplicative linear functionals on a commutative Banach algebra and its maximal ideals, as well as with the spectra of its elements. Most of the content of chapters 1 thorough 3 is meant, in one way or another, to lead towards this theory. The central ingredient of Gelfand theory is the well-known Gelfand-Mazur theorem which says that if a Banach algebra is a division algebra then it is isomorphic to C. The first chapter is a purely algebraic one and provides us with all the necessary algebraic techniques, particularly concerning algebras without identity. The second and third chapters introduce normed algebras and Banach algebra and other concepts like the spectrum, and prove several important results among which the Gelfand-Mazur theorem. The fourth chapter is the pivotal one — where Gelfand theory is developed. In the fifth chapter several examples of Banach algebras are discussed in detail, together with their Gelfand representations. Some practical applications of the theory are also mentioned, among which Wiener’s famous theorem about zeroes of functions with absolutely Fourier series, proven entirely from the context of Banach algebras.Show less
Nederland kent aan het begin van de 20e tegengestelde politieke bewegingen. Aan de ene kant de polariserende verzuiling en aan de andere kant het samenbrengende nationalisme. Welke van de ze twee...Show moreNederland kent aan het begin van de 20e tegengestelde politieke bewegingen. Aan de ene kant de polariserende verzuiling en aan de andere kant het samenbrengende nationalisme. Welke van de ze twee bewegingen speelde een belangrijke rol bij de viering van 100 jaar koninkrijk in de gemeente Wateringen.Show less
