We study the “combinatorial anabelian geometry” that governs the relationship between the dual semi-graph of a pointed stable curve and various associated profinite fundamental groups of the pointed stable curve. Although many results of this type have been obtained previously in various particular situations of interest under unnecessarily strong hypotheses, the goal of the present paper is to step back from such “typical situations of interest” and instead to consider this topic in the abstract—a point of view which allows one to prove results of this type in much greater generality under very weak hypotheses.
We define a class of wavelet transforms as a continuous and microlocal version of the Littlewood-Paley decompositions. Hörmander's wave front sets as well as the Besov and Triebel-Lizorkin spaces may be characterized in terms of our wavelet transforms.
Silverman's estimate for the number of integral points of the so-called Thue equation is improved in a certain special case. A sufficient condition for the non-existence of rational solutions is also given.
Given two positive continuous functions α and β, necessary and sufficient conditions are given for the system u''(t) = f(t) + A(t)u(t) to have an α -bounded solution u for each β -bounded forcing function f. Applications are given to a nonlinear perturbation problem: u''(t) = A(t)u(t) + F(t, u(t)). Indications are given on how to extend these ideas to {n<SUP>th</SUP>} order equations.