We survey the adversarial robustness of neural networks from the perspective of Lipschitz
calculus in a unifying fashion by expressing models, attacks and safety guarantees—that is …

In the first half of the 20th century, analysis went from studying smooth functions to
nonsmooth ones, introducing such notions as weak solutions, Sobolev spaces and …

This is a brief discussion of the properties of solutions to several non-linear elliptic equations
involving diffusive processes of non-local nature. These equation arise in several contexts …

The most important partial differential equation of the second order is the celebrated Laplace
equation. This is the prototype for linear elliptic equations. It is less well known that it also …

These notes are written up after my lectures at the Summer School in Jyväskylä in August
2005. I am grateful to **ao Zhong for his valuable assistance with the practical …

Fix a bounded domain Ω⊂ R d, a continuous function F:∂ Ω→ R, and constants ε> 0 and 1<
p, q<∞ with p− 1+ q− 1= 1. For each x∈ Ω, let u ε (x) be the value for player I of the following …

In this paper we introduce a new family of partial difference operators on graphs and study
equations involving these operators. This family covers local variational p-Laplacian, ∞ …

Conformal dimension measures the extent to which the Hausdorff dimension of a metric
space can be lowered by quasisymmetric deformations. Introduced by Pansu in 1989, this …

Given a weighted graph with N vertices, consider a real-valued regression problem in a
semi-supervised setting, where one observes n labeled vertices, and the task is to label the …

We study the game theoretic p-Laplacian for semi-supervised learning on graphs, and show
that it is well-posed in the limit of finite labeled data and infinite unlabeled data. In particular …