An index theory for uniformly locally finite (ULF) graphs is developed based on the adjacency operator A acting on the space of bounded sequences defined on the vertices. It turns out that the characterization by upper and lower nonnegative eigenvectors is an appropriate tool to overcome the difficulties imposed by the `∞–setting. A distinctive property of the spectral radius r∞(A) in `∞ is the...

I would like to comment on the article by Jino Bak, Jin Hwa Choi, Jae-Sung Kim, and Suk Won Park, “Modified dose difference method for comparing dose distributions,” J. Appl. Clin. Med. Phys. 2012; 13(2): 73-80. Something almost identical to their method was in fact first described by Bakai et al.(1) who called their dimensionless index χ. While it is true that Bak et al. refer to this work in ...

Margulis showed that “most” arithmetic groups are superrigid. Platonov conjectured, conversely, that finitely generated linear groups which are superrigid must be of “arithmetic type.” We construct counterexamples to Platonov’s Conjecture. 1. Platonov’s conjecture that rigid linear groups are aritmetic (1.1) Representation rigid groups. Let Γ be a finitely generated group. By a representation o...

We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension n is two. As pointed out in [7] this case is more subtle than n = 3 or 4 due to the fact that the arguments of the first two authors [11], Burq [1] and Metcalfe [9] showing that local Strichartz estimates for obstacles imply global ones require that the Sobolev index, γ, equal 1/2 when n = 2. We overcome th...

Pareto-type distributions are extreme value distributions for which the extreme value index γ > 0. Classical estimators for γ > 0, like the Hill estimator, tend to overestimate this parameter in the presence of outliers. The empirical influence function plot, which displays the influence that each data point has on the Hill estimator, is introduced. To avoid a masking effect, the empirical infl...

The purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behaviour at infinity. Then, we study the limit as the length scale tends to zero through a Γ-convergence analysis, obta...

We study compactness and Γ-convergence for Ginzburg-Landau type functionals. We only assume that the potential is continuous and positive definite close to one circular well, but allow large zero sets inside the well. We show that the relaxation of the assumptions does not change the results to leading order unless the energy is very large.

A homogenization result is given for a material having soft inclusions arranged in a periodic structure, under the requirement that the interpenetration of matter is forbidden. According to the relation between the softness parameter and the size of the microstucture, three different limit models are deduced via Γ-convergence.

This paper is an extended version of the lecture delivered at the Summer School on Differential Equations and Calculus of Variations (Pisa, September 16-28, 1996). That lecture was conceived as an introduction to the theory of Γ-convergence and in particular to the Modica-Mortola theorem; I have tried to reply the style and the structure of the lecture also in the written version. Thus first co...

The eeective energy of a mixture of two elastic materials in a thin lm is characterized using Gamma-limit techniques. For cylindrical shaped inclusions it is shown that 3D-2D asymptotics and optimal design commute from a variational viewpoint. Regularity of local minimizers for the resulting design is addressed.