arXiv subjects
Publications and source records attributed to Paola Loreti.
We establish some results on the structure of the spectrum of $m$-bonacci numbers. More precisely, we study explicit lower bounds for the distance between elements separated by $N$ positions in the ordered spectrum. The result is further detailed in the Fibonacci and Tribonacci case. The methodology combines the combinatorial structure of $m$-bonacci words and the canonical $m$-bonacci number system.
We study the topology and the Hausdorff dimension of a random Cantor set with overlaps, generated by an iterated function system with scaling ratio equal to the Golden Mean. The results extend known formulas to a case where the Open Set Condition fails. Our methodology is based on the theory of expansions in non-integer bases.
In this paper, we address the question of estimating the energy decay of integro-differential evolution equations with glassy memory. This class of memory kernel was not analyzed in previous studies. Moreover, a detailed analysis provides an explicit estimate of the connection between the kernel function's decay constant and the energy's decay constant.
We begin with a brief overview of the most commonly used fractional derivatives, namely the Caputo and Riemann-Liouville derivatives. We then focus on the study of the fractional time wave equation with the Riemann-Liouville derivative, addressing key questions such as well-posedness, regularity, and a trace result in appropriate interpolation spaces. Additionally, we explore the duality relationship with the Caputo fractional time derivative. The analysis is based on expanding the solution in terms of Mittag-Leffler functions.
We consider initial boundary value problems for time fractional diffusion-wave equations: $$ d_t^α u = -Au + μ(t)f(x) $$ in a bounded domain where $μ(t)f(x)$ describes a source and $α\in (0,1) \cup (1,2)$, and $-A$ is a symmetric ellitpic operator with repect to the spatial variable $x$. We assume that $μ(t) = 0$ for $t > T$:some time and choose $T_2>T_1>T$. We prove the uniqueness in simultaneously determining $f$ in $Ω$, $μ$ in $(0,T)$, and initial values of $u$ by data $u\vert_{ω\times (T_1,T_2)}$, provided that the order $α$ does not belong to a countably infinite set in $(0,1) \cup (1,2)$ which is characterized by $μ$. The proof is based on the asymptotic behavior of the Mittag-Leffler functions.
We consider an initial boundary value problem in a bounded domain $Ω$ over a time interval $(0, T)$ for a time-fractional wave equation where the order of the fractional time derivative is between $1$ and $2$ and the spatial elliptic operator has time-independent coefficients and is not necessarily symmetric. We prove that if for arbitrarily chosen subdomain $ω\subset Ω$ and $T>0$, a solution to the problem vanishes in $ω\times (0,T)$, then $u=0$ in $Ω\times (0, T)$. The uniqueness does not require any geometric condition on $ω$.
We investigate optimal expansions of Kakeya sequences for the representation of real numbers. Expansions of Kakeya sequences generalize the expansions in non-integer bases and they display analogous redundancy phenomena. In this paper, we characterize optimal expansions of Kakeya sequences, and we provide conditions for the existence of unique expansions with respect to Kakeya sequences.
Our goal is to establish a hidden regularity result for solutions of time fractional Petrovsky system where the Caputo fractional derivative is of order $α\in(1,2)$. We achieve such result for a suitable class of weak solutions.
Our purpose is to adapt the Hilbert Uniqueness Method by J.-L. Lions in the case of fractional diffusion-wave equations. The main difficulty is to determine the right shape for the adjoint system, suitable for the procedure of HUM.
Given a positive integer $M$ and a real number $q \in (1,M+1]$, an expansion of a real number $x \in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\ldots,M\}$ is a sequence $(c_i) \in A^{\mathbb N}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. Generalizing many earlier results, we investigate in this paper the topological properties of the set $U_q$ consisting of numbers $x$ having a unique expansion of this form, and the combinatorial properties of the set $U_q'$ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [B] by adapting the method given in [EJK] for the case $M=1$.
Viscoelastic materials have the properties both of elasticity and viscosity. In a previous work we investigate glass relaxation in the framework of viscoelasticity. Here we consider the Burgers model, a first but meaningful step in the general analysis, showing a reachability theorem thanks the analysis of the gap between eigenvalues and the representation of the solution in Fourier series.
We will give some regularity results about fractional diffusion-wave equations.
We investigate a susceptible-infected-susceptible (SIS) epidemic model based on the Caputo-Fabrizio operator. After performing an asymptotic analysis of the system, we study a related finite horizon optimal control problem with state constraints. We prove that the corresponding value function is a viscosity solution of a dynamic programming equation. We then turn to the asymptotic behavior of the value function, proving its convergence to the solution of a stationary problem, as the planning horizon tends to infinity. Finally, we present some numerical simulations providing a qualitative description of the optimal dynamics and the value functions involved.
We study epidemic Susceptible-Infected-Susceptible models in the fractional setting. The novelty is to consider models in which the susceptible and infected populations evolve according to different fractional orders. We study a model based on Caputo derivative, for which we establish existence results of the solutions. Also, we investigate a model based on Caputo-Fabrizio operator, for which we provide existence of solutions and a study of the equilibria. Numerical simulations for both models and a direct numerical comparison are also provided.
We introduce a notion of weak solution for abstract fractional differential equations, motivated by the definition of Caputo derivative. We prove existence results for weak and strong solutions. We also give two examples as application of our results: time-fractional wave equations and time-fractional Petrovsky systems.
We present a general series representation formula for the local solution of Bernoulli equation with Caputo fractional derivatives. We then focus on a generalization of the fractional logistic equation and we present some related numerical simulations.
Expansions in the Golden ratio base have been studied since a pioneering paper of Rényi more than sixty years ago. We introduce closely related expansions of a new type, based on the Fibonacci sequence, and we show that in some sense they behave better.
We address an optimal reachability problem for a planar manipulator in a constrained environment. After introducing the optmization problem in full generality, we practically embed the geometry of the workspace in the problem, by considering some classes of obstacles. To this end, we present an analytical approximation of the distance function from the ellipse. We then apply our method to particular models of hyper-redundant and soft manipulators, by also presenting some numerical experiments.