Boundary Multifractal Behaviour for Harmonic Functions in the Ball

It is well known that if h is a nonnegative harmonic function in the ball of $\mathbb R^{d+1}$ or if h is harmonic in the ball with integrable boundary values, then the radial limit of h exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any β?∈?[0,d], the Hausdorff dimension of the set of points ξ on the sphere such that h(rξ) looks like (1???r)^???β is equal to d???β.     

An investigation of the formation of 1,3,5-heterosubstituted benzene rings by cyclo-condensation of acetyl-substituted organometallic complexes

The cyclocondensation of acetylferrocene and acetylcymantrene catalyzed by SiCl₄ in ethanol yields a mixture of 1,3,5-trisubstituted benzenes and the intermediate 3,1-disubstituted (E)-2-buten-1-ones, including all the homo- and heterometallic species, which were separated and quantified by HPLC. The relative yields of these species are determined by the different ability of the organometallic groups to stabilize the cationic intermediates that participate in the reaction mechanism, which is measurable as the basicity of the starting materials. The X-ray crystal structures of 1-cymantrenyl-3,5-diferrocenylbenzene and (E)-3-cymantrenyl-1-ferrocenyl-2-buten-1-one are described within this report.     

Weierstrass functions in Zygmund's class

Consider the function

where 1$"> and is an almost periodic function. It is well known that the function lives in the so-called Zygmund class. We prove that is generically nowhere differentiable. This is the case in particular if the elementary condition is satisfied. We also give a sufficient condition on the Fourier coefficients of which ensures that is nowhere differentiable.

     

On Measures Driven by Markov Chains

We study measures on \([0,1]\) which are driven by a finite Markov chain and which generalize the famous Bernoulli products.We propose a hands-on approach to determine the structure function \(\tau \) and to prove that the multifractal formalism is satisfied. Formulas for the dimension of the measures and for the Hausdorff dimension of their supports are also provided. Finally, we identify the measures with maximal dimension.