Uniform approximation of the integrated density of states for long-range percolation Hamiltonians

In this paper we study the spectrum of long-range percolation graphs. The underlying geometry is given in terms of a finitely generated amenable group. We prove that the integrated density of states (IDS) or spectral distribution function can be approximated uniformly in the energy variable. Using this, we are able to characterise the set of discontinuities of the IDS

Conflicts in the learning of real numbers and limits

question: "Is 0.999... (nought point nine recurring) equal to one, or just less than one?". Many answers contained infinitesimal concepts: "The same, because the difference between them is infinitely small." " The same, for at infinity it comes so close to one it can be considered the same." "Just less than one, but it is the nearest you can get to one without actually saying it is one." "Just less than one, but the difference between it and one is infinitely small." The majority of students thought that 0.999... was less than one. It may be that a few students had been taught using infinitesimal concepts, or that the phrase “just less than one ” had connotations for the students different from those intended by the questioner; but it seems more likely that the answers represent the students ’ own rationalisations made in an attempt to resolve conflicts inherent in the students ’ previous experience of limiting processes. Some conscious and subconscious conflicts Most of the mathematics met in secondary school consists of sophisticated idea

Affine Processes and Pseudo-Differential Operators with Unbounded Coefficients

The concept of pseudo-differential operators allows one to study stochastic processes through their symbol. This approach has generated many new insights in recent years. However, most results are based on the assumption of bounded coefficients. In this thesis, we study Levy-type processes with unbounded coefficients and, especially, affine processes. In particular, we establish a connection between pseudo-differential operators and affine processes which are well-known from mathematical finance. Affine processes are an interesting example in this field since they have linearly growing and hence unbounded coefficients. New techniques and tools are developed to handle the affine case and then expanded to general Levy-type processes. In this way, the convergence of a simulation scheme based on a Markov chain approximation, results on path properties, and necessary conditions for the symmetry of operators were proven