In his paper [14], A. Granville proved several strong results about the distribution of square-free values of polynomials, under the assumption of the abc-conjecture. In our thesis, we generalize...Show moreIn his paper [14], A. Granville proved several strong results about the distribution of square-free values of polynomials, under the assumption of the abc-conjecture. In our thesis, we generalize some of Granville’s results to k-free values of polynomials (i.e., values of polynomials not divisible by the k-th power of a prime) . Further, we generalize a result of Granville on the gaps between consecutive square-free numbers to gaps between integers, such that the values of a given polynomial f evaluated at them are k-free. All our results are under assumption of the abc-conjecture.Show less
We introduce the mixed model of sandpile+anti-sandpile, which is called SA model. In the SA model, we are free to add or remove a particle from a chosen site. Because of the non-abelian property of...Show moreWe introduce the mixed model of sandpile+anti-sandpile, which is called SA model. In the SA model, we are free to add or remove a particle from a chosen site. Because of the non-abelian property of the toppling operators and anti-toppling operators, the SA model becomes subtle, and the group property existing in the pure sandpile model and anti-sandpile model does not hold in the SA model. Because of the non-local property of addition operators and anti-addition operators, the processes related to the sandpile, and anti-sandpile are not Feller. The traditional way of constructing the interacting particle processes in infinite volume, e.g., via Hille-Yoshida, does not work in these cases, other ways of construction are necessary. In the construction of sandpile+anti-sandpile process (SA process), we obtain the semigroup of the process for some special configurations and some special functions by series expansion and then using monotonicity of the process, we can extend it to the general case. The SA process shows a new transition phenomenon: it seems that the stationary measure is the result of a “competition” between the generators. The sandpile model, anti-sandpile model are “self-organized” critical systems. In recent years, this is challenged because the special nature of the dynamics can be considered as an implicit fine tuning that makes sure the system can reach criticality, therefore the “selforganized” critical behavior of these systems can be thought of as a more conventional phase transition between “stabilizable” and “non-stabilizable”. We discuss the conditions for a mixed system to reach a stable state both in finite volume and infinite volume.Show less
What you are reading is my master thesis. I talked to my supervisor Prof. Dr. R. van der Hout about the subject in 2004, but I did not start really until summer 2005. After a few months of...Show moreWhat you are reading is my master thesis. I talked to my supervisor Prof. Dr. R. van der Hout about the subject in 2004, but I did not start really until summer 2005. After a few months of wrestling through all the theory it finally became clear to me. The next thing to do was numerically constructing the behaviour predicted in the theory, and after that writing this report. Most of all I want to thank my supervisor Prof. Dr. R van der Hout for all of his support. We almost met weekly and discussed the progress. I want to thank Dr. J.B. van den Berg for letting me use figures 2.5 and 2.6. I want to thank Peter Bruin for helping me including a flipbook in this thesis of the differential equation I studied. I also want to thank my parents for their support during my study. Finally I want to thank all the other people who supported me during my studies.Show less
