Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds

The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities

Separation functions and mild topologies

Given M and N Hausdorff topological spaces, we study topologies on the space C-0 (M; N) of continuous maps f : M? N. We review two classical topologies, the "strong" and the "weak" topology. We propose a definition of "mild topology" that is coarser than the "strong" and finer than the "weak" topology. We compare properties of these three topologies, in particular with respect to proper continuous maps f : M? N, and affine actions when N = R-n. To define the "mild topology" we use "separation functions;" these "separation functions" are somewhat similar to the usual "distance function d(x, y)" in metric spaces (M, d), but have weaker requirements. Separation functions are used to define pseudo balls that are a global base for a T2 topology. Under some additional hypotheses, we can define "set separation functions" that prove that the topology is T6. Moreover, under further hypotheses, we will prove that the topology is metrizable. We provide some examples of uses of separation functions: one is the aforementioned case of the mild topology on C-0(M; N). Other examples are the Sorgenfrey line and the topology of topological manifolds

Broadband and Tunable Light Harvesting in Nanorippled MoS2 Ultrathin Films

Nanofabrication of flat optic silica gratings conformally layered with two-dimensional (2D) MoS2 is demonstrated over large area (cm2), achieving a strong amplification of the photon absorption in the active 2D layer. The anisotropic subwavelength silica gratings induce a highly ordered periodic modulation of the MoS2 layer, promoting the excitation of Guided Mode Anomalies (GMA) at the interfaces of the 2D layer. We show the capability to achieve a broadband tuning of these lattice modes from the visible (VIS) to the near-infrared (NIR) by simply tailoring the illumination conditions and/or the period of the lattice. Remarkably, we demonstrate the possibility to strongly confine resonant and nonresonant light into the 2D MoS2 layers via GMA excitation, leading to a strong absorption enhancement as high as 240% relative to a flat continuous MoS2 film. Due to their broadband and tunable photon harvesting capabilities, these large area 2D MoS2 metastructures represent an ideal scalable platform for new generation devices in nanophotonics, photo- detection and -conversion, and quantum technologies

ColDoc Project

There are two main frameworks currently used to present information (in particular, related to scientific fields). A document redacted with the standard LaTeX / PDF toolbox. Pros: these documents are state-of-the-art quality. Cons: the final user has no way of interacting with a PDF document (other than sending an e-mail to the original authors) The Web 2.0 way (think of: Wikipedia or Stack Exchange ). Pros: in those frameworks, content is continuously developed and augmented by users. Cons: the end result, though, is fragmented, and cannot (in general) be presented as an unified document. A ColDoc tries to get the best of two publishing frameworks

Regularity of Solutions to Hamilton-Jacobi Equations

We formulate an Hamilton\u2013Jacobi partial differential equation $H(x, Du(x)) = 0$ on a n dimensional manifold M , with assumptions of uni- form convexity of H(x, \ub7) and regularity of H in a neighborhood of {H = 0} in T 17 M ; we define the \u201cmin solution\u201d u, a generalized solution, which often co- incides with the viscosity solution; the definition is suited to proving regularity results about u; in particular, we prove that the closure of the set where u is not regular is a H^(n 121) \u2013rectifiable set

Database degli esercizi (EDB)

This web site contains a mathematics project, called \u201cEDB\u201d. This presents mathematical exercises for first year students, that I collected in a decade of teaching. It is published at the address https://coldoc.sns.it/CD/EDB EDB contains elements of theory, roughly 1000 exercises and 600 solutions. EDB implements a simple game to engage students. Theory and exercises are public, whereas solutions are reserved. But solutions may be bought: each student has an allowance of eulercoins to buy solutions; a student may also earn more eulercoins by writing the solution to an exercise and sending it to me

On Asymmetric Distances

In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space

Probability measures on infinite-dimensional Stiefel manifolds

An interest in infinite-dimensional manifolds has recently appeared in Shape Theory. An example is the Stiefel manifold, that has been proposed as a model for the space of immersed curves in the plane. It may be useful to define probabilities on such manifolds. Suppose that H is an infinite-dimensional separable Hilbert space. Let S 82H be the sphere, p 08S. Let \u3bc be the push forward of a Gaussian measure \u3b3 from TpS onto S using the exponential map. Let v 08TpS be a Cameron--Martin vector for \u3b3; let R be a rotation of S in the direction v, and \u3bd=R#\u3bc be the rotated measure. Then \u3bc,\u3bd are mutually singular. This is counterintuitive, since the translation of a Gaussian measure in a Cameron--Martin direction produces equivalent measures. Let \u3b3 be a Gaussian measure on H; then there exists a smooth closed manifold M 82H such that the projection of H to the nearest point on M is not well defined for points in a set of positive \u3b3 measure. Instead it is possible to project a Gaussian measure to a Stiefel manifold to define a probability