The Gauss-Green theorem in stratified groups

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.Comment: 69 page

A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up

We introduce the new space $BV^{\alpha}(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha \in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $\alpha$-perimeter and we define its fractional reduced boundary $\mathscr{F}^{\alpha} E$. We are able to show that $W^{\alpha,1}(\mathbb{R}^n)\subset BV^\alpha(\mathbb{R}^n)$ continuously and, similarly, that sets with (locally) finite standard fractional $\alpha$-perimeter have (locally) finite fractional Caccioppoli $\alpha$-perimeter, so that our theory provides a natural extension of the known fractional framework. Our main result partially extends De Giorgi's Blow-up Theorem to sets of locally finite fractional Caccioppoli $\alpha$-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.Comment: 46 page

Simulating the Effects of Shopping Attitudes on Urban Goods Distribution

Studies of urban freight mobility traditionally focused only on restocking flows and usually neglected the linkage with shopping activities even if end consumer's choices in relation to the type of retail undoubtedly impact on freight distribution flows. The paper focuses on the distribution of urban freight facilities, the choices of type of retail and the travel mode used and some models for simulating the choice of retail outlet and the transport mode are presented. The models, jointly with urban freight demand models were used to assess the effects of some land-use scenarios and to define optimal spatial distribution of urban freight facilities able to improve city sustainability and to meet the interests of end consumers, freight operators and society. The results of an application of this method to a test site are also reported and discussed

LUTI models, freight transport and freight facility location

The subject of this book is the new scientific research in the field of modelling the interaction between land use and transport (LUTI modelling). Transport and the location of activities in space have been important themes of study in engineering, social sciences and urban and regional plannin

On sets with finite distributional fractional perimeter

We continue the study of the fine properties of sets having locally finite distributional fractional perimeter. We refine the characterization of their blow-ups and prove a Leibniz rule for the intersection of sets with locally finite distributional fractional perimeter with sets with finite fractional perimeter. As a byproduct, we provide a description of non-local boundaries associated with the distributional fractional perimeter.Comment: 18 page

Fractional divergence-measure fields, Leibniz rule and Gauss–Green formula

Given a E (0, 1] and p E [1, +co], we define the space DMa,p(R-n) of L-p vector fields whose a-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the a-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples

Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula

Given $\alpha\in[0,1]$ and $p\in[1,+\infty]$, we define the space $\mathcal{DM}^{\alpha,p}(\mathbb R^n)$ of $L^p$ vector fields whose $\alpha$-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the $\alpha$-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.Comment: 22 page

Understanding taxi travel demand patterns through Floating Car Data

This paper analyses the current structure of taxi service use in Rome, processing taxi Floating Car Data (FCD). The methodology used to pass from the original data to data useful for the demand analyses is described. Further, the patterns of within-day and day-to-day service demand are reported, considering the origin, the destination and other characteristics of the trips (e.g. travel time). The analyses reported in the paper can help the definition of space-temporal characteristics of future Shared Autonomous Electrical Vehicles (SAEVs) demand in mobility scenarios