Mass concentration for Ergodic Choquard Mean-Field Games

We study concentration phenomena in the vanishing viscosity limit for second-order stationary Mean-Field Games systems defined in the whole space $\mathbb{R}^N$ with Riesz-type aggregating nonlocal coupling and external confining potential. In this setting, every player of the game is attracted toward congested areas and the external potential discourages agents to be far away from the origin. Focusing on the mass-subcritical regime $N-\gamma'<\alpha<N$, we study the behavior of solutions in the vanishing viscosity limit, namely when the diffusion becomes negligible. First, we investigate the asymptotic behavior of rescaled solutions as $\varepsilon\to0$, obtaining existence of classical solutions to potential free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around minima of the potential.Comment: accepted in ESAIM: COC

Existence and compactness of conformal metrics on the plane with unbounded and sign-changing Gaussian curvature

We show that the prescribed Gaussian curvature equation in $\mathbb{R}^2$ $$-\Delta u= (1-|x|^p) e^{2u},$$ has solutions with prescribed total curvature equal to $\Lambda:=\int_{\mathbb{R}^2}(1-|x|^p)e^{2u}dx\in \mathbb{R}$, if and only if $$p\in(0,2) \qquad \text{and} \qquad (2+p)\pi\le\Lambda<4\pi$$ and prove that such solutions remain compact as $\Lambda\to\bar{\Lambda}\in[(2+p)\pi,4\pi)$, while they produce a spherical blow-up as $\Lambda\uparrow4\pi$

Un risultato di h-principle per curve a curvatura costante in R^n

Oggetto di questa tesi è mostrare che ogni curva liscia immersa nello spazio euclideo n-dimensionale, può essere deformata in una curva con curvatura costante, mediante una perturbazione arbitrariamente piccola della curva iniziale e delle sue rette tangenti. Tale valore costante della curvatura deve essere maggiore o uguale del massimo della curvatura della curva iniziale, e si dimostra che tale limite inferiore è ottimale. Questo significa che le curve lisce di curvatura costante, soddisfano, secondo la terminologia di Gromov, il `relative C^1-dense h-principle' nello spazio delle curve immerse in R^n

La formula di Faà di Bruno

La formula di Faà di Bruno fornisce una regola per calcolare la derivata n-esima di una funzione composta. In questa tesi studieremo la forma fattoriale della formula di Faà di Bruno e poi tratteremo la formula dal punto di vista combinatorio, andando ad associare alle partizioni di insiemi le derivate di funzioni composte. Infine ci occuperemo della generalizzazione della formula nel caso di funzioni in più variabili

Leveraging probabilistic reasoning in deterministic planning for large-scale autonomous Search-and-Tracking

Search-And-Tracking (SaT) is the problem of searching for a mobile target and tracking it once it is found. Since SaT platforms face many sources of uncertainty and operational constraints, progress in the field has been restricted to simple and unrealistic scenarios. In this paper, we propose a new hybrid approach to SaT that allows us to successfully address large-scale and complex SaT missions. The probabilistic structure of SaT is compiled into a deterministic planning model and Bayesian inference is directly incorporated in the planning mechanism. Thanks to this tight integration between automated planning and probabilistic reasoning, we are able to exploit the power of both approaches. Planning provides the tools to efficiently explore big search spaces, while Bayesian inference, by readily combining prior knowledge with observable data, allows the planner to make more informed and effective decisions. We offer experimental evidence of the potential of our approach

Deterministic versus Probabilistic Methods for Searching for an Evasive Target

Several advanced applications of autonomous aerial vehicles in civilian and military contexts involve a searching agent with imperfect sensors that seeks to locate a mobile target in a given region. Effectively managing uncertainty is key to solving the related search problem, which is why all methods devised so far hinge on a probabilistic formulation of the problem and solve it through branch-and-bound algorithms, Bayesian filtering or POMDP solvers. In this paper, we consider a class of hard search tasks involving a target that exhibits an intentional evasive behaviour and moves over a large geographical area, i.e., a target that is particularly difficult to track down and uncertain to locate. We show that, even for such a complex problem, it is advantageous to compile its probabilistic structure into a deterministic model and use standard deterministic solvers to find solutions. In particular, we formulate the search problem for our uncooperative target both as a deterministic automated planning task and as a constraint programming task and show that in both cases our solution outperforms POMDPs methods