Under usual assumptions on the Hamiltonian, we prove that any viscosity solution of the corresponding Hamilton-Jacobi equation on the manifold M is locally semiconcave and C loc outside the closure of its singular set (which is nowhere dense in M). Moreover, we prove that, under additional assumptions and in low dimension, any viscosity solution of that Hamilton-Jacobi equation satisfies a gene...

Community detection is one of the most investigated problems in the field of complex networks. Although several methods were proposed, there is still no precise definition of communities. As a step towards a definition, I highlight two necessary properties of communities, separation and internal cohesion, the latter being a new concept. I propose a local method of community detection based on t...

We construct a subset of the Mandelbrot set which is dense on the boundary of the Mandelbrot set and which consists of only infinitely renormalizable points such that the Mandelbrot set is locally connected at every point of this subset. We prove the local connectivity by finding bases of connected neighborhoods directly.

A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n ≥ 3 a complete locally-convex (topological) immersion of a connected (n−1)-manifold into the n-sphere is a surjection onto the boundary of a convex set.

The Intrinsic Shape of Point Clouds Stefan Ohrhallinger, PhD Concordia University, 2012 Given a point cloud, in the form of unorganized points, the problem of automatically connecting the dots to obtain an aesthetically pleasing and piecewise-linear closed interpolating boundary shape has been extensively researched for over three decades. In R, it is even more complicated to find an aesthetic ...