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PubblicazioneDeterministic homogenization under optimal moment assumptions for fast–slow systems. Part 2( 2022)We consider deterministic homogenization for discrete-time fast-slow systems of the form Xk+1 = Xk + n-1an(Xk,Yk) + n-1/2bn(Xk,Yk), Yk+1 = TnYk and give conditions under which the dynamics of the slow equations converge weakly to an Itô diffusion X as n → ∞. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by X are given explicitly. This extends the results of Kelly-Melbourne (J. Funct. Anal. 272 (2017) 4063-4102) from the continuous-time case to the discrete-time case. Moreover, our methods (p-variation rough paths) work under optimal moment assumptions. Combined with parallel developments on martingale approximations for families of nonuniformly expanding maps in Part 1 by Korepanov, Kosloff and Melbourne, we obtain optimal homogenization results when Tn is such a family of maps.
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PubblicazioneA rough path perspective on renormalization( 2019)We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned–Hairer–Zambotti (2016), the links with which are discussed in detail.
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PubblicazioneLarge-scale graph-machine-learning surrogate models for 3D-flowfield prediction in external aerodynamics( 2024)The article presents the application of inductive graph machine learning surrogate models for accurate and efficient prediction of 3D flow for industrial geometries, explicitly focusing here on external aerodynamics for a motorsport case. The final aim is to build a surrogate model that can provide quick predictions, bypassing in this way the unfeasible computational burden of traditional computational fluid dynamics (CFD) simulations. We investigate in this contribution the usage of graph neural networks, given their ability to smoothly deal with unstructured data, which is the typical context for industrial simulations. We integrate an efficient subgraph-sampling approach with our model, specifically tailored for large dataset training. REV-GNN is the chosen graph machine learning model, that stands out for its capacity to extract deeper insights from neighboring graph regions. Additionally, its unique feature lies in its reversible architecture, which allows keeping the memory usage constant while increasing the number of network layers. We tested the methodology by applying it to a parametric Navier–Stokes problem, where the parameters control the surface shape of the industrial artifact at hand, here a motorbike.
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PubblicazioneAcute intermittent hypoxia in neonatal rodent central nervous system facilitates respiratory frequency through the recruitment of hypothalamic areas( 2025)Moderate and acute intermittent hypoxia (IH) facilitates respiration in adults, mostly by recruiting peripheral chemo-/baroreceptors. As central chemoreceptors are widely expressed in immature brains, we hypothesized that IH modulates respiration at birth through a purely neurogenic mechanism involving the hypothalamus. The central nervous system (CNS) isolated from 0- to 3-day-old rats was perfused with four to eight brief (5 min) bouts of mild-hypoxic/normocapnic modified Krebs solution, intermingled with 5-min normoxic episodes, during continuous electrophysiological recordings from upper cervical ventral roots. An IH protocol did not modify bath pH, but superficial ventrolateral medulla and hypothalamic areas experienced lowered oxygen tension, more severe after the second postnatal day, with a partial recovery after each bout. Single exposures to mild hypoxia were well tolerated, and at birth often triggered a spontaneous epoch of irregular baseline activity (< 1 min) superimposed on respiratory events in both whole CNS preparations and spinal cords. Conversely, IH largely halted breathing activity after the second postnatal day, while at birth IH transiently increased the amplitude of respiratory bursts and stably sped up rhythm only when intact suprapontine structures were present. Rhythm acceleration was not directly correlated to instantaneous changes in tissue oxygen tension. After IH, respiratory frequency remained 260% higher than pre-IH control for up to 60 min. Identical modulatory effects were observed with IH supplied through a HEPES buffer solution. Interestingly, IH increased electrical activity and cFos expression in hypothalamic areas without altering total cell number. These observations cast some light on the mechanisms of IH during development, with important insights about pediatric effects of repeated hypoxic episodes.
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PubblicazioneAlgebraic groups in non-commutative probability theory revisited( 2024)The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Schürmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and results in another construction of groups of characters encoding the behavior of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Schürmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of non-crossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Schürmann’s aforementioned work, is nevertheless original on several points and fills a gap in the non-commutative probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively, Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature.