In this contribution we define a new class of non-linear integrals based on decomposition integrals. These integrals are motivated by greediness of many real-life situations. Another view on this new class of integrals is that it is a generalization of both the Shilkret and PAN integrals. Moreover, it can be seen as an iterated Shilkret integral. Also, an example in time-series analysis is prov...

This thesis is focused on the theoretical characterization of topological order in non-Abelian fractional quantum Hall (FQH) states. The first part of the thesis is concerned with the ideal wave function approach to FQH states, where the idea is to try to obtain model wave functions and model Hamiltonians for all possible FQH states and to have a physical way of characterizing their topological...

We prove that the zeros of the polynomials Pm(a) of degree m, defined by Borosh and Moll via Pm(a) = 2m+3/2 π (a + 1) Z ∞ 0 dx (x4 + 2ax2 + 1)m+1 , approach the lemniscate {ζ ∈ C : |ζ − 1| = 1, Rζ < 0}, as m diverges.

The cyclicity of period annuli of some classes of reversible and non-Hamiltonian quadratic systems under quadratic perturbations are studied. The argument principle method and the centroid curve method are combined to prove that the related Abelian integral has at most two zeros.

We show that for each genus there are only finitely many algebraically primitive Teichmüller curves C, such that i) C lies in the hyperelliptic locus and ii) C is generated by an abelian differential with two zeros of order g − 1. We prove moreover that for these Teichmüller curves the trace field of the affine group is not only totally real but cyclotomic.