arXiv subjects
Publications and source records attributed to Ferdinand Jacobé de Naurois.
We prove that for every $\varepsilon >0$, there is $δ>0$ such that for every probability measure $μ$ on a free group $\mathbb{F}$, if the operator norm of the action of $μ$ on $L^2\mathbb{F}$ by convolution exceeds $\varepsilon$, then there is a coset of an abelian subgroup of $\mathbb{F}$ with $μ$-mass at least $δ$.
We give an example of a sequence of positive harmonic functions on $\mathbb{Z}^d$, $d\geq 2$, that converges pointwise to a non-harmonic function.
Methods for measuring convexity defects of compacts in R^n abound. However, none of the those measures seems to take into account continuity. Continuity in convexity measure is essential for optimization, stability analysis, global optimality, convergence analysis, and accurate modelling as it ensures robustness and facilitates the development of efficient algorithms for solving convex optimization problems. This paper revisits the axioms underlying convexity measures by enriching them with a continuity hypothesis in Hausdorff's sense. Having provided the concept's theoretical grounds we state a theorem underlining the necessity of restricting ourselves to non-point compacts. We then construct a continuous convexity measure and compare it to existing measures. Importante note : This work is not a research article. It is an undergraduate project undertaken as part of a computer science course at École normale supérieure. It should therefore not be considered as a peer reviewed research paper.