The notes are based on lectures given in St. Flour in 1995, and cover, in greater detail, most
of the course given there. The word" fractal" was coined by Mandelbrot [Man] in the 1970s …

There are several generalizations of the classical theory of Sobolev spaces as they are
necessary for the applications to Carnot-Caratheodory spaces, subelliptic equations …

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in
bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We …

We give an overview over the application of functional equations, namely the classical
Poincaré and renewal equations, to the study of the spectrum of Laplace operators on self …

A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem
is to describe the relationship between the shape (geo metry) of the drum and its sound (its …

The present research monograph is a testimony to the fact that Fractal Analysis is deeply
connected to numerous areas of contemporary Mathematics. Here, we have in mind, in …

We introduce fractional flat space, described by a continuous geometry with constant non-
integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but …

A bstract We construct a theory of fields living on continuous geometries with fractional
Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski …

We study the spectral properties of the Laplacian on infinite Sierpiński gaskets. We prove
that the Laplacian with the Neumann boundary condition has pure point spectrum …

We consider first-and second-order consensus algorithms in networks with stochastic
disturbances. We quantify the deviation from consensus using the notion of network …