Dario Corona

• Phd in Mathematics
• PostDoc Position at Università di Camerino

We extend previous works on the multiplicity of solutions to the Allen-Cahn system on closed Riemannian manifolds by considering an arbitrary number of phases. Specifically, we show that on parallelizable manifolds, the number of solutions is bounded from below by topological invariants of the underlying manifold, provided the temperature parameter...

In this paper, we present a comprehensive proof concerning the regularity of critical points for the spline energy functional on Riemannian manifolds, even for the general higher-order case. Although this result is widely acknowledged in the literature, a detailed proof was previously absent. Our proof relies on a generalization of the DuBois-Reymo...

We consider an autonomous, indefinite Lagrangian L admitting an infinitesimal symmetry K whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point p to a flow line $$\gamma =\gamma (t)$$ γ = γ ( t ) of K that does not...

We study the measurable dynamical properties of the interval map generated by the model-case erasing substitution $\rho $ , defined by $$ \begin{align*} \rho(00)=\text{empty word},\quad \rho(01)=1,\quad \rho(10)=0,\quad \rho(11)=01. \end{align*} $$ We prove that, although the map is singular, its square preserves the Lebesgue measure and is strongl...

An error in the statement and proof of Theorem 7 in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Math\'ematique, 360(G4), 315-332, 2024] has been identified. The theorem was used to prove analytical and dynamical properties of a class of interval maps there introduced, called the critical exponent functions. We show he...

The study of dynamic singularity formation in spacetime, focusing on scalar field collapse models, is analyzed. We revisit key findings regarding open spatial topologies, concentrating on minimal conditions necessary for singularity and apparent horizon formation. Moreover, we examine the stability of initial data in the dynamical system governed b...

In this paper, we explore the characteristics of two novel regular spacetimes that exhibit a nonzero vacuum energy term, under the form of a (quasi) anti-de Sitter phase. Specifically, the first metric is spherical, while the second, derived by applying the generalized Newman–Janis algorithm to the first, is axisymmetric. We show that the equations...

We consider an autonomous, indefinite Lagrangian $L$ admitting an infinitesimal symmetry $K$ whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point $p$ to a flow line $\gamma=\gamma(t)$ of $K$ that does not cross $p...

In this paper %dedicated to the memory of Edward Fadell and Sufian Husseini we show how the notion of the Lusternik-Schnirelmann relative category can be used to study a multiplicity problem for brake orbits in a potential well which is homeomorphic to the $N$-dimensional unit disk. The estimate of the relative category of the set of chords with en...

We introduce a variational setting for the action functional of an autonomous and indefinite Lagrangian on a finite dimensional manifold M . Our basic assumption is the existence of an infinitesimal symmetry whose Noether charge is the sum of a one-form and a function on M . Our setting includes different types of Lorentz–Finsler Lagrangians admitt...

A data-driven controller is presented in this paper, which stems from the well known model-free adaptive control approach based on an equivalent linearized dynamical model of the plant. Inspired by the recent paper (Liu and Yang, 2019), the output tracking problem is here solved by a data-driven adaptive sliding-mode controller simultaneously ensur...

We study the measurable dynamical properties of the interval map generated by the model-case erasing substitution ρ, defined by: ρ(00) = empty word, ρ(01) = 1, ρ(10) = 0, ρ(11) = 01. We prove that, although the map is singular, its square preserves the Lebesgue measure and is strongly mixing, thus ergodic, with respect to it. We discuss the extensi...

We introduce a variational setting for the action functional of an autonomous and indefinite Lagrangian on a finite dimensional manifold. Our basic assumption is the existence of an infinitesimal symmetry whose Noether charge is the sum of a one-form and a function. Our setting includes different types of Lorentz-Finsler Lagrangians admitting a tim...

We study the non-autonomous variational problem: \begin{equation*} \inf_{(\phi,\theta)} \bigg\{\int_0^1 \bigg(\frac{k}{2}\phi'^2 + \frac{(\phi-\theta)^2}{2}-V(x,\theta)\bigg)\text{d}x\bigg\} \end{equation*} where $k>0$, $V$ is a bounded continuous function, $(\phi,\theta)\in H^1([0,1])\times L^2([0,1])$ and $\phi(0)=0$ in the sense of traces. Th...

We consider Hamiltonian functions of the classical type, namely, even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton’s equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamil...

We consider Hamiltonian functions of classical type, namely even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton's equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonia...

The critical exponent of a word w over a given alphabet is the supremum of the reals for which w contains an -power. We study the maps associating to every real in [0; 1] the inverse of the critical exponent of its base-n expansion. We strengthen a combinatorial result by J.D. Currie and N. Rampersad to show that these maps have infinitely many non...

This paper introduces a complete work-flow for the translation of dynamic isolated signs based on data acquired from a data-glove. A sign language translation system based on a wearable device represents indeed a more efficient solution with respect to cameras or position trackers for helping speech-impaired people on a daily basis. The paper descr...

In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a \begin{document}$ N $\end{document}-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least \...

Objectives Many people diagnosed with autism spectrum disorder (ASD) have difficulty in developing communication skills, and they suffer from speech disorders. They must use an augmentative and alternative communication (AAC) system that helps with ordinary speech, such as picture exchange communication systems, text systems, and voice output devic...

Consider a compact manifold with boundary, homeomorphic to the N-dimensional disk, and a Tonelli Lagrangian function defined on the tangent bundle. In this paper, we study the multiplicity problem for Euler-Lagrange orbits that satisfy the conormal boundary conditions and that lay on the boundary only in their extreme points. In particular, for sui...

In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a N-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least N orthogonal Finsler geodesic chord...

In this paper, we study the multiplicity problem for Euler–Lagrange orbits that satisfy the conormal boundary conditions for a suitable class of reversible Lagrangian functions on compact manifolds. Such a class contains, e.g. the energy function of reversible Finsler metrics that satisfy a convexity condition on the boundary.

Nowadays a commercial product for sign language translation is still not available. This paper presents our latest results towards this goal, presenting a functional prototype called Talking Hands. Talking Hands uses a data-glove to detect the hand movements of the user, and a smartphone application to gather all the data and translates them into v...

The aim of this work is characterizing the class of LPV systems that admit steady-state trajectories depending exclusively on the scheduling parameter. In particular, it will be shown that only certain parameter dependent steady-state profiles are admissible and can be reached by means of a suitable control input. Furthermore, the asymptotic stabil...

In this paper, the problem of optimizing the output regulation of a weakly dual redundant plant is addressed. When the system is underactuated, only a subset of the outputs can be arbitrarily controlled and the remaining ones are constrained. With a specific focus on the asymptotic output tracking problem for single-input systems, we investigate th...

A wearable device for sign language translation, called Talk- ing Hands, is presented. It is composed by a custom data glove, which is designed to optimize the data acquisition, and a smartphone applica- tion, which others user personalizations. Although Talking Hands can not translate a whole sign language, it others an effective communication to...

In this paper the problem of optimizing the output regulation of an underactuated LPV system is considered. When the system is underactuated, only a subset of the outputs can be arbitrarily controlled, and the remaining ones are constrained. Having identified a special class of LPV systems admitting steady-states, the problem of finding the input t...

In this paper the problem of optimizing the output regulation of a weakly dual redundant plant is addressed. When the system is underactuated, only a subset of the outputs can be arbitrarily controlled, and the remaining ones are constrained. With a specific focus on quasiperiodic references, i.e. signals that can be written as trigonometric series...

In this paper the problem of optimizing the output regulation of a weakly dual redundant plant is addressed. When the system is under-actuated, only a subset of the outputs can be arbitrarily controlled, and the remaining ones are constrained. We investigate the problem of finding the input that minimize a cost function of the overall output tracki...