Strong approximation is a property that some schemes have, which relates the geometry of their rational points to the geometry of their p-adic points. For S a subset of the places of the rational...Show moreStrong approximation is a property that some schemes have, which relates the geometry of their rational points to the geometry of their p-adic points. For S a subset of the places of the rational numbers, a scheme satisfies strong approximation away from S, if the rational points are dense in some product over all p not in S of the sets of p-adic points. In the last century, a couple of great, general results have been proven which give sufficient or necessary conditions for a scheme or a group scheme to satisfy strong approximation away from some set S. In 2019, Kok and Bright showed, using the Brauer-Manin obstruction, that the scheme representing primitive solutions to the equation X1^2 + 47 X2^2 - 103 X3^2 - 103 * 47 * 17 X4^2 = 0 does not satisfy strong approximation away from infinity. On the other hand, in 2020, Pagano and Bright proved a general result from which it follows that this scheme satisfies strong approximation away from infinity, 17, 47 and 103. This thesis shows that this scheme satisfies strong approximation away from infinity and 17, and it obtains a more general result for some equations of the form a X1^2 + b X2^2 + c X3^2 + d X4^2 = 0.Show less
This bachelor thesis will be concerned with the old English art of ringing church bells called change ringing. The development of change ringing in the early 17th century was mainly due to the...Show moreThis bachelor thesis will be concerned with the old English art of ringing church bells called change ringing. The development of change ringing in the early 17th century was mainly due to the invention of the full-circle wheel on which the bells were mounted. By pulling a rope, a bell would make a rotation of almost 360 degrees with a period of approximately two seconds. The time between two strikes of the same bell could be controlled rather accurately, which made it possible to ring a certain number of bells all after each other and keep repeating this in the same order. A practised bell ringer could also adjust the rotation period of his bell slightly. Two neigboring bells might be swapped by increasing the rotation period of the first bell and decreasing the rotation period of the second. In this fashion one could change the order in which the bells were rung. In such a transition from one ordering of the bells to the order, each bell could not move more than one place, since the rotation period of a bell could only be adjusted a little bit. Bell ringers became interested in ringing the bells in every possible order after each other, whereby the bells were not allowed to be rung more than once in the same order; that is, they wanted to ring an extent. Thus the art of change ringing was born. Along with the development of change ringing came the (mathematical) study of ringing patterns, which is sometimes called campanology (from the Latin word campana, which means ’bell’). Change ringers tried to create methods to ring all possible orderings of the bells in a smart way. An important 17th century study on change ringing was done by Fabian Stedman in his book Campanalogia (1677, see [3]). Initially, most of the campanologists were change ringers, but in the 20th century also mathematicians became interested in this field of study. For an extensive mathematical description of change ringing, the reader may consult [7] and [1]; [2] elaborates on Fabian Stedman as someone using group theoretical tools before the actual development of group theory. This thesis gives a mathematical analysis of change ringing. In Section 2 we will introduce the important notion of a Cayley graph and prove the existence of an extent for an arbitrary number of bells. A historically interesting question about a method called Grandsire Triples will be answered in Section 3, and Section 4 covers Rankin’s campanological theorem, which is closely related to Section 3. Section 5 will be concerned with the existence and construction of extents using specific methods.Show less
