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PubblicazionePolynomial‐order oscillations in geometric discrepancy( 2026)Let $C\subset\mathbb{R}^2$ be a convex body, and for a positive integer $N$, let $\mathcal{P}$ be a configuration of $N$ points in $[0,1)^2$. The discrepancy of $\mathcal{P}$ with respect to $C$ is defined by \begin{equation*} \mathcal{D}(\mathcal{P},\, C)=\sum_{\mathbf{p}\in\mathcal{P}}\sum_{\mathbf{n}\in\mathbb{Z}^2}\mathds{1}_C(\mathbf{p}+\mathbf{n})-N|C|, \end{equation*} and one may estimate how $\mathcal{P}$ deviates from uniformity by averaging the latter quantity over a family of sets. When considering quadratic averages over translated and dilated copies of $C$, one gets the \textit{homothetic quadratic discrepancy} \begin{equation*} \mathcal{D}_2(\mathcal{P},\, C)=\int_{0}^{1}\int_{[0,1)^2}\left|\mathcal{D}( \mathcal{P},\,\boldsymbol{\tau}+\delta C)\right|^2\,{\rm d}\boldsymbol{\tau}\,{\rm d} \delta. \end{equation*} We investigate the behaviour of the optimal \textit{homothetic quadratic discrepancy}, that is \begin{equation*} \inf_{\# \mathcal{P}=N} \mathcal{D}_2(\mathcal{P},\, C)\quad\text{as}\quad N\to+\infty. \end{equation*} Beck~\cite{MR915529} and Beck and Chen~\cite{MR1489133} showed that the optimal \textit{h.q.d.} of convex polygons has an order of growth of $\log N$, and more recently, Brandolini and Travaglini~\cite{MR4358540} proved that the optimal \textit{h.q.d.} of planar convex bodies with a $\mathcal{C}^2$ boundary has an order of growth of $N^{1/2}$. We show that, in general, a single order of growth for the optimal \textit{h.q.d.} need not exist. First, by an implicit geometric construction of $C$, we obtain prescribed oscillations between $\log N$ and $N^{1/2}$. Second, by a subtler design of $\partial C$ and via Fourier-analytic methods, we obtain prescribed polynomial-order oscillations in the range $N^\alpha$ with $\alpha\in(2/5,1/2)$. Moreover, we show that the set of planar convex bodies whose optimal \textit{h.q.d.} does not admit a single order of growth is residual in the (Hausdorff) metric space of planar convex bodies.
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PubblicazioneLyapunov Exponents of Linear Switched Systems( 2025)We explicitly compute the maximal Lyapunov exponent for a switched system on and the corresponding switching function which realizes the maximal exponent. This computation is reduced to the characterization of optimal trajectories for an optimal control problem on the Lie group.
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PubblicazioneNeuropeptide Y—Graphene Oxide Complexes Inhibit Amygdala NPY-Receptor Expressing Glutamatergic Pathways and Selectively Remove Aversive Memory In Vivo( 2026)Therapeutic needs to modulate brain circuits highlight graphene-based materials (GBMs) as an emerging tool to engineer specific interventions to treat neuro-diseases. In this context, graphene oxide (GO) nanosheets offer new drug delivery strategies to reach neural cells and signaling networks selectively. To complex and transport neuropeptide Y (NPY), GO was engineered as GO:NPY, and its activity was investigated in regulating excitatory neurotransmission in NPY-positive synaptic pathways, when delivered to the amygdala, a structure mediating fear memory responses. First, it was shown that in vitro GO:NPY specifically and selectively inhibited glutamate release and suppressed synaptic enhancement via NPY receptors. In a rat model of anxiety disorder, when injected into the amygdala, GO:NPY suppressed contextual fear memory responses via activation of NPY receptors in specific synaptic pathways. This easy-to-tune GBM-based nanoplatforms promise advances in co-delivery vectors preserving the synaptic specificity needed for treating specific pathological conditions.
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PubblicazioneOn the distribution of the van der Corput sequences( 2023)For an integer p≥ 2 , let {xn}n∈N⊂T be the p-adic van der Corput sequence. For intervals [0 , α) ⊂ T and for positive integers N, consider the geometrically-shifted discrepancy function Dp,N,α(t)=∑n=0N-1X[0,α)(xn+t)-Nα. In this paper, we give a characterization of the asymptotic behavior of ‖Dp,N,α(·)‖L2(T) for N→ ∞ that depends on the p-adic expansion of α.
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PubblicazioneOn Dold-Whitney's parallelizability of 4-manifolds( 2025)We present a proof of a theorem by Dold and Whitney, according to which a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. Our proof is based on an argument by R. Kirby on the classification of SO(4)-principal bundles over the 4-sphere by means of their Euler and first Pontryagin classes.