Rigidity and trace properties of divergence-measure vector fields - Leonardi, G. P.; Saracco, G. (Adv. Calc. Var.)

Existence and blow-up for non-autonomous scalar conservation laws with viscosity - Stefano Bianchini; Giacomo Maria Leccese (Journal of Mathematical Analysis and Applications, 2025)
Keywords: Blow-up; Global existence; Long-term behavior; Scalar conservation laws with viscosity

Paralinearization and extended lifespan for solutions of the $ α$-SQG sharp front equation - Massimiliano Berti; Scipio Cuccagna; Francisco Gancedo; Stefano Scrobogna (Advances in Mathematics, 2025)

Breaking the left-right symmetry in fluttering artificial cilia that perform nonreciprocal oscillations - Ariel S Boiardi; Marchello, Roberto (Meccanica, 2024)
Recent investigations on active materials have introduced a new paradigm for soft robotics by showing that a complex response can be obtained from simple stimuli by harnessing dynamic instabilities. In particular, polyelectrolyte hydrogel filaments actuated by a constant electric field have been shown to exhibit self-sustained oscillations as a consequence of flutter instability. Owing to the nonreciprocal nature of the emerging oscillations, these artificial cilia are able to generate flows along the stimulus. Building upon these findings, in this paper we propose a design strategy to break the left-right symmetry in the generated flows, by endowing the filament with a natural curvature at the fabrication stage. We develop a mathematical model based on morphoelastic rod theory to characterize the stability of the equilibrium configurations of the filament, proving the persistence of flutter instability. We show that the emerging oscillations are nonreciprocal and generate thrust at an angle with the stimulus. The results we find at the level of the single cilium open new perspectives on the possible applications of artificial ciliary arrays in soft robotics and microfluidics.

Comments on the regularity of harmonic maps between singular spaces - Luca Gennaioli; Nicola Gigli; Hui-Chun Zhang; Xi-Ping Zhu (2024)

Computations for Sustainability - Salavatidezfouli, Sajad; Nikishova, Anna; Torlo, Davide; Teruzzi, Martina; Rozza, Gianluigi (Quantitative Sustainability: Interdisciplinary Research for Sustainable Development Goals, 2024)

Data-driven solutions of ill-posed inverse problems arising from doping reconstruction in semiconductors - S. Piani; Luca Heltai; N. Rotundo; W. Lei; P. Farrell (Applied Mathematics in Science and Engineering, 2024)

Dissipative solutions to Hamiltonian systems - Stefano Bianchini; Giacomo Maria Leccese (Kinetic and Related Models, 2024)

Edge transport in Haldane-like models with quasi-periodic disorder - Fabrizio Caragiulo; Vieri Mastropietro; Marcello Porta (2024)
We consider Haldane-like 2d topological insulators on the cylinder, in the presence of weak quasi-periodic disorder. We prove that, at large distances, the boundary correlations agree with the correlations of a renormalized, translation-invariant, massless relativistic model in 1+1 dimensions, multiplied by non-universal oscillatory factors, incommensurate with the lattice spacing. Furthermore, we compute the edge conductance and the edge susceptibility, starting from Kubo formula. We obtain explicit expressions for these response functions, completely determined by the renormalized Fermi velocity of the edge modes. In particular, we prove the quantization of the edge conductance, and the non-universality of the susceptibility. The proof relies on multiscale analysis and rigorous renormalization group methods for quasi-periodic systems, and on lattice Ward identities.

Effect of particle aspect ratio in targeted drug delivery in abdominal aortic aneurysm - Salavatidezfouli, Sajad; Barzegar, Saeid; Sheidani, Armin; Hajisharifi, Arash; Girfoglio, Michele; Stabile, Giovanni; Rozza, Gianluigi (European Journal of Mechanics-B/Fluids, 2024)

First isola of modulational instability of Stokes waves in deep water - Massimiliano Berti; Alberto Maspero; Paolo Ventura (to appear on EMS Surveys in Mathematical Sciences., 2024)

Fourier transform of BV functions, isoperimetry, and discrepancy theory - Thomas Beretti; Luca Gennaioli (2024)

Generative models for the deformation of industrial shapes with linear geometric constraints: Model order and parameter space reductions. - Guglielmo Padula; Francesco Romor; Giovanni Stabile; Gianluigi Rozza (Computer Methods in Applied Mechanics and Engineering, 2024)

A hybrid reduced-order model for segregated fluid-structure interaction solvers in an ALE approach at high Reynolds number - Valentin Eric Brice Nkana Ngan; Giovanni Stabile; Andrea Mola; Gianluigi Rozza (2024)

Infinitely many isolas of modulational instability for Stokes waves - Massimiliano Berti; Livia Corsi; Alberto Maspero; Paolo Ventura (2024)

KAM for Vortex Patches - Massimiliano Berti (Regular and Chaotic Dynamics, 2024)
In the last years substantial mathematical progress has been made in KAM theoryfor quasi-linear/fully nonlinearHamiltonian partial differential equations, notably forwater waves and Euler equations.In this survey we focus on recent advances in quasi-periodic vortex patchsolutions of the $2d$-Euler equation in $\mathbb{R}^{2}$ close to uniformly rotating Kirchhoff elliptical vortices,with aspect ratios belonging to a set of asymptotically full Lebesgue measure.The problem is reformulated into a quasi-linear Hamiltonian equation for a radial displacement from the ellipse. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux – Carathéodory theorem of symplectic rectification, valid in finite dimension.This approach is particularly delicate in an infinite-dimensional phase space: our symplecticchange of variables is a nonlinear modification of the transport flow generated by the angularmomentum itself.

Kinetic data-driven approach to turbulence subgrid modeling - Giulio Ortali; Gabbana, Alessandro; Demo, Nicola; Rozza, Gianluigi; Toschi, Federico (arXiv preprint arXiv:2403.18466, 2024)

Massive Cantor families of periodic solutions of resonant Klein-Gordon equation on $\mathbbS^3$ - Diego Silimbani (2024)
Keywords: 35L05; 37K58; 58E07; 58J45; 83C10; Analysis of PDEs (math.AP); FOS: Mathematics; FOS: Physical sciences; Mathematical Physics (math-ph)

Mesh-Informed Reduced Order Models for Aneurysm Rupture Risk Prediction - D'Inverno, Giuseppe Alessio; Moradizadeh, Saeid; Salavatidezfouli, Sajad; Africa, Pasquale Claudio; Rozza, Gianluigi (arXiv preprint arXiv:2410.03802, 2024)

The Michor-Mumford conjecture in Hilbertian H-type groups - Magnani, Valentino; Tiberio, Daniele (2024)
Keywords: Differential Geometry (math.DG); FOS: Mathematics; Metric Geometry (math.MG)

Minimal actuation and control of a soft hydrogel swimmer from flutter instability - Ariel S Boiardi; Giovanni Noselli (Journal of the Mechanics and Physics of Solids, 2024)
Micro-organisms propel themselves in viscous environments by the periodic, nonreciprocal beating of slender appendages known as flagella. Active materials have been widely exploited to mimic this form of locomotion. However, the realization of such coordinated beating in biomimetic flagella requires complex actuation modulated in space and time. We prove through experiments on polyelectrolyte hydrogel samples that directed undulatory locomotion of a soft robotic swimmer can be achieved by untethered actuation from a uniform and static electric field. A minimal mathematical model is sufficient to reproduce, and thus explain, the observed behavior. The periodic beating of the swimming hydrogel robot emerges from flutter instability thanks to the interplay between its active and passive reconfigurations in the viscous environment. Interestingly, the flutter-driven soft robot exhibits a form of electrotaxis whereby its swimming trajectory can be controlled by simply reorienting the electric field. Our findings trace the route for the embodiment of mechanical intelligence in soft robotic systems by the exploitation of flutter instability to achieve complex functional responses to simple stimuli. While the experimental study is conducted on millimeter-scale hydrogel swimmers, the design principle we introduce requires simple geometry and is hence amenable for miniaturization via micro-fabrication techniques. We believe it may also be transferred to a wider class of soft active materials.
Keywords: Active materials; Flutter instability; Hopf bifurcation; Polyelectrolyte hydrogels; Soft robotics

Modal Analysis of the Wake Shed Behind a Horizontal Axis Wind Turbine with Flexible Blades - Salavatidezfouli, Sajad; Sheidani, Armin; Bakhshaei, Kabir; Safari, Ali; Hajisharifi, Arash; Stabile, Giovanni; Rozza, Gianluigi (Journal of Applied and Computational Mechanics, 2024)

Modeling Droplets Dynamics in Emulsions with Graph Neural Networks - Giulio Ortali; Toschi, Federico; van de Meent, Jan-Willem (ICML 2024 AI for Science Workshop, 2024)

A non local model for cell migration in response to mechanical stimuli - Marchello, Roberto; Colombi, Annachiara; Preziosi, Luigi; Giverso, Chiara (Mathematical Biosciences, 2024)

A note about charts built by Eriksson-Bique and Soultanis on metric measure spaces - Luca Gennaioli; Gigli, Nicola (Canadian Mathematical Bulletin, 2024)

An Online Stabilization Method for Parametrized Viscous Flows - Shafqat Ali; Francesco Ballarin; Gianluigi Rozza (Reduction, Approximation, Machine Learning, Surrogates, Emulators and Simulators., 2024)
The purpose of this work is to investigate the inf-sup stability of reduced basis (RB) method applied to parametric Stokes problem. While performing the Galerkin projection on the reduced space, the inf-sup approximation stability has always been a challenge for the RB community, even if the construction of reduced basis is done using a stable high-fidelity method. In this work we propose a new online stabilization strategy for RB approximation of parametrized Stokes problem. In this strategy, a stable high-fidelity method is used to construct the RB spaces, and then, online solution is improved by a post processing based on rectification method [<a id="ref-link-section-d1765768e608" title=" A two-grid finite-element/reduced basis scheme for the approximation of the solution of parameter dependent PDE. In: 9e Colloque national en calcul des structures (2009)" href="https://link.springer.com/chapter/10.1007/978-3-031-55060-7_1#ref-CR8" data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref">8, <a id="ref-link-section-d1765768e611" title=" RB (Reduced basis) for RB (Rayleigh-Bénard). Comp. Meth. App. Mech. Eng. 261, 132–141 (2013)" href="https://link.springer.com/chapter/10.1007/978-3-031-55060-7_1#ref-CR13" data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref">13, <a id="ref-link-section-d1765768e614" title=" An online intrinsic stabilization strategy for the reduced basis approximation of parametrized advection-dominated problems. Comp. Ren. Math. 354(12), 1188–1194 (2016)" href="https://link.springer.com/chapter/10.1007/978-3-031-55060-7_1#ref-CR16" data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref">16]. This approach involves the computation of less expensive (but less consistent) FE approximation during the online stage and hence the improvement of online solutions using a RB-based rectification method. The consistency of the RB solution is also improved. We compare this approach with existing offline-online stabilization approach presented in our earlier work [<a id="ref-link-section-d1765768e620" title=" Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations. Comp. Math. Appl. 80(11), 2399–2416 (2020)" href="https://link.springer.com/chapter/10.1007/978-3-031-55060-7_1#ref-CR2" data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref">2]. All the numerical simulations are carried out using RBniCS [<a id="ref-link-section-d1765768e624" title="//mathlab.sissa.it/rbnics (2016)" href="https://link.springer.com/chapter/10.1007/978-3-031-55060-7_1#ref-CR4" data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref">4, <a id="ref-link-section-d1765768e627" title=" Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics (2015)" href="https://link.springer.com/chapter/10.1007/978-3-031-55060-7_1#ref-CR14" data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref">14], an open-source reduced order modelling library, built on top of FEniCS [<a id="ref-link-section-d1765768e630" title=" Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)" href="https://link.springer.com/chapter/10.1007/978-3-031-55060-7_1#ref-CR15" data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref">15].

Optimisation–Based Coupling of Finite Element Model and Reduced Order Model for Computational Fluid Dynamics - Ivan Prusak; Davide Torlo; Monica Nonino; Gianluigi Rozza (2024)

An optimisation–based domain–decomposition reduced order model for parameter–dependent non–stationary fluid dynamics problems - Ivan Prusak; Davide Torlo; Monica Nonino; Gianluigi Rozza (Computers & Mathematics with Applications, 2024)
In this work, we address parametric non–stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the optimisation–based domain decomposition approach, we derive an optimal control problem, for which we present a convergence analysis in the case of non–stationary incompressible Navier–Stokes equations. We discretise the problem with the finite element method and we compare different model order reduction techniques: POD–Galerkin and a non–intrusive neural network procedures. We show that the classical POD–Galerkin is more robust and accurate also in transient areas, while the neural network can obtain simulations very quickly though being less precise in the presence of discontinuities in time or parameter domain. We test the proposed methodologies on two fluid dynamics benchmarks with physical parameters and time dependency: the non–stationary backward–facing step and lid–driven cavity flow.
Keywords: Computational fluid dynamics; Domain decomposition; Optimal control; POD–NN; Proper orthogonal decomposition; Reduced order modelling

A predictive surrogate model for heat transfer of an impinging jet on a concave surface - Salavatidezfouli, Sajad; Rakhsha, Saeid; Sheidani, Armin; Stabile, Giovanni; Rozza, Gianluigi (arXiv preprint arXiv:2402.10641, 2024)

Pure gravity traveling quasi-periodic water waves with constant vorticity - Massimiliano Berti; Luca Franzoi; Alberto Maspero (Comm. Pure Appl. Math., 2024)

Quantitative approximate definable choices - Lerario, Antonio; Rizzi, Luca; Tiberio, Daniele (2024)
Keywords: 14P10; 53C17; Algebraic Geometry (math.AG); Differential Geometry (math.DG); FOS: Mathematics; Metric Geometry (math.MG)

R3MG: R-tree based agglomeration of polytopal grids with applications to multilevel methods - Marco Feder; Andrea Cangiani; Luca Heltai (2024)

Reducibility of Klein-Gordon equations with maximal order perturbations - Massimiliano Berti; Roberto Feola; Michela Procesi; Shulamit Terracina (2024)

Relative heat content asymptotics for sub-Riemannian manifolds - Agrachev, Andrei; Rizzi, Luca; Rossi, Tommaso (Analysis & PDE, 2024)

Relative heat content asymptotics in sub-Riemannian manifolds - Andrei A. Agrachev; Luca Rizzi; Tommaso Rossi (Analysis & PDE, 2024)

Sard properties for polynomial maps in infinite dimension - Lerario, Antonio; Rizzi, Luca; Tiberio, Daniele (2024)
Keywords: 14P10; 53C17; Differential Geometry (math.DG); FOS: Mathematics; Functional Analysis (math.FA); Metric Geometry (math.MG); Optimization and Control (math.OC)

A segregated reduced order model of a pressure-based solver for turbulent compressible flows - Matteo Zancanaro; Valentin Eric Brice Nkana Ngan; Giovanni Stabile; Gianluigi Rozza (2024)

Stochastic Parameter Prediction in Cardiovascular Problems - Bakhshaei, Kabir; Salavatidezfouli, Sajad; Stabile, Giovanni; Rozza, Gianluigi (arXiv preprint arXiv:2411.18089, 2024)

Stokes waves at the critical depth are modulationally unstable - Massimiliano Berti; Alberto Maspero; Paolo Ventura (Comm. Math. Phys., 2024)

Time periodic solutions of completely resonant Klein-Gordon equations on $\mathbbS^3$ - Massimiliano Berti; Beatrice Langella; Diego Silimbani (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2024)

A time-adaptive algorithm for pressure dominated flows: a heuristic estimator - Ivan Prusak; Davide Torlo; Monica Nonino; Gianluigi Rozza (2024)

Two-photon polymerization 3D-printing of micro-scale neuronal cell culture devices - Ali Hosseini; Giovanni Noselli; Michele Giugliano (Journal of Visualized Experiments, 2024)
Neuronal cultures have been a reference experimental model for several decades. However, 3D cell arrangement, spatial constraints on neurite outgrowth, and realistic synaptic connectivity are missing. The latter limits the study of structure and function in the context of compartmentalization and diminishes the significance of cultures in neuroscience. Approximating ex vivo the structured anatomical arrangement of synaptic connectivity is not trivial, despite being key for the emergence of rhythms, synaptic plasticity, and ultimately, brain pathophysiology. Here, two-photon polymerization (2PP) is employed as a 3D printing technique, enabling the rapid fabrication of polymeric cell culture devices using polydimethyl-siloxane (PDMS) at the micrometer scale. Compared to conventional replica molding techniques based on microphotolitography, 2PP micro-scale printing enables rapid and affordable turnaround of prototypes. This protocol illustrates the design and fabrication of PDMS-based microfluidic devices aimed at culturing modular neuronal networks. As a proof-of-principle, a two-chamber device is presented to physically constrain connectivity. Specifically, an asymmetric axonal outgrowth during ex vivo development is favored and allowed to be directed from one chamber to the other. In order to probe the functional consequences of unidirectional synaptic interactions, commercial microelectrode arrays are chosen to monitor the bioelectrical activity of interconnected neuronal modules. Here, methods to 1) fabricate molds with micrometer precision and 2) perform in vitro multisite extracellular recordings in rat cortical neuronal cultures are illustrated. By decreasing costs and future widespread accessibility of 2PP 3D-printing, this method will become more and more relevant across research labs worldwide. Especially in neurotechnology and high-throughput neural data recording, the ease and rapidity of prototyping simplified in vitro models will improve experimental control and theoretical understanding of in vivo large-scale neural systems.

Zoll magnetic systems on the two-torus: A Nash–Moser construction - Luca Asselle; Gabriele Benedetti; Massimiliano Berti (Advances in Mathematics, 2024)
We construct an infinite-dimensional family of smooth integrable magnetic systems on the two-torus which are Zoll, meaning that all the unit-speed magnetic geodesics are periodic. The metric and the magnetic field of such systems are arbitrarily close to the flat metric and to a given constant magnetic field. This extends to the magnetic setting a famous result by Guillemin [19] on the two-sphere. We characterize Zoll magnetic systems as zeros of a suitable action functional S, and then look for its zeros by means of a Nash–Moser implicit function theorem. This requires showing the right-invertibility of the linearized operator dS in a neighborhood of the flat metric and constant magnetic field, and establishing tame estimates for the right inverse. As key step we prove the invertibility of the normal operator dS∘dS⁎ which, unlike in Guillemin's case, is pseudo-differential only at the highest order. We overcome this difficulty noting that, by the asymptotic properties of Bessel functions, the lower order expansion of dS∘dS⁎ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of dS∘dS⁎ restricted to the subspace of functions corresponding to high Fourier modes. The inversion of such a restricted operator is finally achieved by making the crucial observation that lower order Fourier integral operators satisfy asymmetric tame estimates.
Keywords: Bessel functions; Fourier integral operators; Hamiltonian systems; Magnetic geodesics; Nash–Moser implicit function theorem; Zoll flows

Applicable Methodologies for the Mass Transfer Phenomenon in Tumble Dryers: A Review - Sajad Salavatidezfouli; Sajad Hajisharifi; Michele Girfoglio; Giovanni Stabile; Gianluigi Rozza (2023)
Tumble dryers offer a fast and convenient way of drying textiles independent of weather conditions and therefore are frequently used in ordinary households. However, artificial drying of textiles consumes considerable amounts of energy, approximately 8.2 percent of the residential electricity consumption is for drying of textiles in northern European countries (Cranston et al., 2019). Several authors have investigated the aspects of the clothes drying cycle with experimental and numerical methods to understand and improve the process. The first turning point study on understanding the physics of evaporation for tumble dryers was presented by Lambert et al. (1991) in the early 90s. With the aid of Chilton_Colburn analogy, they introduced the concept of area-mass transfer coefficient to address evaporation rate. Afterwards, several experimental or numerical studies were published based on this concept, and furthermore, the model was then developed into 0-dimensional (Deans, 2001) and 1-dimensional (Wei et al., 2017) to gain more accuracy. The evaporation rate is considered to be the main system parameter for dryers with which other performance parameters including drying time, effectiveness, moisture content and efficiency can be estimated. More recent literature focused on utilizing dimensional analysis or image processing techniques to correlate drying indices with system parameters. However, the validity of these regressed models is machine-specific, and hence, cannot be generalized yet. All the previous models for estimating the evaporation rate in tumble dryers are discussed. The review of the related literature showed that all of the previous models for the prediction of the evaporation rate in the clothes dryers have some limitations in terms of accuracy and applicability.

Applicable Methodologies for the Mass Transfer Phenomenon in Tumble Dryers: A Review - Salavatidezfouli, Sajad; Hajisharifi, Arash; Girfoglio, Michele; Stabile, Giovanni; Rozza, Gianluigi (arXiv preprint arXiv:2304.03533, 2023)

Assessment of icing effects on the wake shed behind a vertical axis wind turbine - Sheidani, Armin; Salavatidezfouli, Sajad; Stabile, Giovanni; Barzegar Gerdroodbary, Mostafa; Rozza, Gianluigi (Physics of Fluids, 2023)

Assessment of URANS and LES methods in predicting wake shed behind a vertical axis wind turbine - Sheidani, Armin; Salavatidezfouli, Sajad; Stabile, Giovanni; Rozza, Gianluigi (Journal of Wind Engineering and Industrial Aerodynamics, 2023)

Asymptotics as $s → 0^+$ of the fractional perimeter on Riemannian manifolds - Michele Caselli; Luca Gennaioli (2023)

Benjamin-Feir instability of Stokes waves in finite depth - Massimiliano Berti; Alberto Maspero; Paolo Ventura (Arch. Ration. Mech. Anal., 2023)

Birkhoff normal form and long time existence for periodic gravity water waves - Massimiliano Berti; Feola, Roberto; Pusateri, Fabio (Comm. Pure Appl. Math., 2023)

A comparison of non-matching techniques for the finite element approximation of interface problemsImage 1 - Daniele Boffi; Andrea Cangiani; Marco Feder; Lucia Gastaldi; Luca Heltai (Computers & Mathematics with Applications, 2023)
We perform a systematic comparison of various numerical schemes for the approximation of interface problems. We consider unfitted approaches in view of their application to possibly moving configurations. Particular attention is paid to the implementation aspects and to the analysis of the costs related to the different phases of the simulations.
Keywords: Finite Element Method; Interface problems; Lagrange multiplier; Nitsche method; Non-matching techniques

A Data-Driven Partitioned Approach for the Resolution of Time-Dependent Optimal Control Problems with Dynamic Mode Decomposition - Eleonora Donadini; Maria Strazzullo; Marco Tezzele; Gianluigi Rozza (13th International Conference on Spectral and High Order Methods, ICOSAHOM 2021, 2023)

The deal.II Library, Version 9.5 - Daniel Arndt; Wolfgang Bangerth; Maximilian Bergbauer; Marco Feder; Marc Fehling; Johannes Heinz; Timo Heister; Luca Heltai; Martin Kronbichler; Matthias Maier; Peter Munch; Jean-Paul Pelteret; et al. (Journal of Numerical Mathematics, 2023)

Deep reinforcement learning for the heat transfer control of pulsating impinging jets - Salavatidezfouli, Sajad; Stabile, Giovanni; Rozza, Gianluigi (arXiv preprint arXiv:2309.13955, 2023)

A dimensionality reduction approach for convolutional neural networks. - Laura Meneghetti; Nicola Demo; Gianluigi Rozza (Applied Intelligence, 2023)

On the distribution of the van der Corput sequences - Beretti, Thomas (Archiv der Mathematik, 2023)
For an integer $p\ge 2$, let $\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$ be the p-adic van der Corput sequence. For intervals $[0,\alpha )\subset {\mathbb {T}}$ and for positive integers N, consider the geometrically-shifted discrepancy function $D_{p,N,\alpha }(t)=\sum _{n=0}^{N-1}\mathcal {X}_{[0,\alpha )}(x_n+t)-N\alpha$. In this paper, we give a characterization of the asymptotic behavior of $\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}$ for $N\rightarrow \infty$that depends on the p-adic expansion of $\alpha$.
Keywords: Diaphony; Discrepancy; Uniform distribution; Van der Corput sequence

Explicable hyper-reduced order models on nonlinearly approximated solution manifolds of compressible and incompressible Navier-Stokes equations - Francesco Romor; Giovanni Stabile; Gianluigi Rozza (2023)

Flutter instability in solids and structures, with a view on biomechanics and metamaterials - Davide Bigoni; Francesco Dal Corso; Oleg N. Kirillov; Diego Misseroni; Giovanni Noselli; Andrea Piccolroaz (Proceedings of the Royal Society A, 2023)
The phenomenon of oscillatory instability called ‘flutter’ was observed in aeroelasticity and rotor dynamics about a century ago. Driven by a series of applications involving non-conservative elasticity theory at different physical scales, ranging from nanomechanics to the mechanics of large space structures and including biomechanical problems of motility and growth, research on flutter is experiencing a new renaissance. A review is presented of the most notable applications and recent advances in fundamentals, both theoretical and experimental aspects, of flutter instability and Hopf bifurcation. Open problems, research gaps and new perspectives for investigations are indicated.

Friedrichs' systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions - Francesco Romor; Davide Torlo; Gianluigi Rozza (2023)

A general splitting principle on RCD spaces and applications to spaces with positive spectrum - Nicola Gigli; Fabio Marconi (2023)
In this paper we develop a general `analytic' splitting principle for RCD spaces: we show that if there is a function with suitable Laplacian and Hessian, then the space is (isomorphic to) a warped product. Our result covers most of the splitting-like results currently available in the literature about RCD spaces. We then apply it to extend to the non-smooth category some structural property of Riemannian manifolds obtained by Li and Wang.

Hamiltonian paradifferential Birkhoff normal form for water waves - Massimiliano Berti; Alberto Maspero; Federico Murgante (Regul. Chaotic Dyn., 2023)

On the Hausdorff Measure of $\mathbbR^n$ with the Euclidean Topology - Bagnara, Marco; Luca Gennaioli; Giacomo Maria Leccese; Luongo, Eliseo (Real Analysis Exchange, 2023)

On the Minimal Number of Solutions of the Equation φ(n+k)=Mφ(n), M=1,2 - Matteo Ferrari; Lorenzo Sillari (Journal of Integer Sequences, 2023)
We fix a positive integer $k$ and look for solutions $n \in \mathbb{N}$ of the equations $φ(n + k) = φ(n)$ and $φ(n + k) = 2 φ(n)$. For $k \le 12 · 10^{100}$, we prove that Fermat primes can be used to build five solutions for the first equation when $k$ is even, and five for the second one when $k$ is odd. Furthermore, for $k \le 4 · 10^{58}$, we show that for the second equation there are at least three solutions when $k$ is even. Our work increases the previously known minimal number of solutions for both equations.
Keywords: Euler’s phi function

Non-linear manifold reduced-order models with convolutional autoencoders and reduced over-collocation method - Francesco Romor; Giovanni Stabile; Gianluigi Rozza (Journal of Scientific Computing, 2023)

Nonreciprocal oscillations of polyelectrolyte gel filaments subject to a steady and uniform electric field - Giancarlo Cicconofri; Valentina Damioli; Giovanni Noselli (Journal of the Mechanics and Physics of Solids, 2023)
Soft actuators typically require time-varying or spatially modulated control to be operationally effective. The scope of the present paper is to show, theoretically and experimentally, that a natural way to overcome this limitation is to exploit mechanical instabilities. We report experiments on active filaments of polyelectrolyte (PE) gels subject to a steady and uniform electric field. A large enough intensity of the field initiates the motion of the active filaments, leading to periodic oscillations. We develop a mathematical model based on morphoelasticity theory for PE gel filaments beating in a viscous fluid, and carry out the stability analysis of the governing equations to show the emergence of flutter and divergence instabilities for suitable values of the system’s parameters. We confirm the results of the stability analysis with numerical simulations for the nonlinear equations of motion to show that such instabilities may lead to periodic self-sustained oscillations, in agreement with experiments. The key mechanism that underlies such behaviour is the capability of the filament to undergo active shape changes depending on its local orientation relative to the external electric field, in striking similarity with gravitropism, the mechanism that drives shape changes in plants via differential growth induced by gravity. Interestingly, the resulting oscillations are nonreciprocal in nature, and hence able to generate thrust and directed flow at low Reynolds number. The exploitation of mechanical instabilities in soft actuators represents a new avenue for the advancement in engineering design in fields such as micro-robotics and micro-fluidics.

A note on the homogenization of incommensurate thin films - Irene Anello; Andrea Braides; Fabrizio Caragiulo (Mathematical Methods in the Applied Sciences, 2023)
Dimension-reduction homogenization results for thin films have been obtained under hypotheses of periodicity or almost periodicity of the energies in the directions of the mid-plane of the film. In this note, we consider thin films, obtained as sections of a periodic medium with a mid-plane that may be incommensurate, that is, not containing periods other than 0. A geometric almost periodicity argument similar to the cut-and-project argument used for quasicrystals allows to prove a general homogenization result.

An optimisation–based domain–decomposition reduced order model for the incompressible Navier-Stokes equations - Ivan Prusak; Monica Nonino; Davide Torlo; Francesco Ballarin; Gianluigi Rozza (Computers & Mathematics with Applications, 2023)
The aim of this work is to present a model reduction technique in the framework of optimal control problems for partial differential equations. We combine two approaches used for reducing the computational cost of the mathematical numerical models: domain–decomposition (DD) methods and reduced–order modelling (ROM). In particular, we consider an optimisation–based domain–decomposition algorithm for the parameter–dependent stationary incompressible Navier–Stokes equations. Firstly, the problem is described on the subdomains coupled at the interface and solved through an optimal control problem, which leads to the complete separation of the subdomain problems in the DD method. On top of that, a reduced model for the obtained optimal–control problem is built; the procedure is based on the Proper Orthogonal Decomposition technique and a further Galerkin projection. The presented methodology is tested on two fluid dynamics benchmarks: the stationary backward–facing step and lid-driven cavity flow. The numerical tests show a significant reduction of the computational costs in terms of both the problem dimensions and the number of optimisation iterations in the domain–decomposition algorithm.
Keywords: Computational fluid dynamics; Domain decomposition; Optimal control; Proper orthogonal decomposition; Reduced order modelling

Properties of Mixing BV Vector Fields - Stefano Bianchini; Martina Zizza (Communications in Mathematical Physics, 2023)

A reduced basis stabilization for the unsteady Stokes and Navier-Stokes equations - Shafqat Ali; Francesco Ballarin; Gianluigi Rozza (ADVANCES IN COMPUTATIONAL SCIENCE AND ENGINEERING, 2023)
In the Reduced Basis approximation of Stokes and Navier-Stokes problems, the Galerkin projection on the reduced spaces does not necessarily preserved the inf-sup stability even if the snapshots were generated through a stable full order method. Therefore, in this work we aim at building a stabilized Reduced Basis (RB) method for the approximation of unsteady Stokes and Navier-Stokes problems in parametric reduced order settings. This work extends the results presented for parametrized steady Stokes and Navier-Stokes problems in a work of ours. We apply classical residual-based stabilization techniques for finite element methods in full order, and then the RB method is introduced as Galerkin projection onto RB space. We compare this approach with supremizer enrichment options through several numerical experiments. We are interested to (numerically) guarantee the parametrized reduced inf-sup condition and to reduce the online computational costs.

Reducibility for a linear wave equation with Sobolev smooth fast driven potential. - Luca Franzoi (Discr. Cont. Dyn. Syst., 2023)

Relaxed area of $0$-homogeneous maps in the strict $BV$-convergence - Carano, Simone (2023)

Relaxed area of graphs of piecewise Lipschitz maps in the strict $BV$-convergence - Bellettini, Giovanni; Carano, Simone; Scala, Riccardo (2023)

Time quasi-periodic vortex patches of Euler equation in the plane - Massimiliano Berti; Hassainia, Zineb; Masmoudi, Nader (Invent. Math., 2023)

On the analyticity of the Dirichlet-Neumann operator and Stokes waves - Massimiliano Berti; Alberto Maspero; Paolo Ventura (Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 2022)

A behavioral change model to assess vaccination-induced relaxation of social distancing during an epidemic - Buonomo, Bruno; Della Marca, Rossella; Sintayehu Sharbayta, Sileshi (Journal of Biological Systems, 2022)
The success of mass vaccination campaigns may be jeopardized by human risky behaviors. For example, high level of vaccination coverage may induce early relaxation of social distancing. In this paper, we focus on the mutual influence between the decline in prevalence, due to the rise in the overall immunization coverage, and the consequent decrease in the compliance to social distancing measures. We consider an epidemic model where both the vaccination rate and the disease transmission rate are influenced by human behavior, which in turn depends on the current and past information about the spread of the disease. We highlight the impact of the information-related parameters on the transient and asymptotic behavior of the system that is on the early stage of the epidemic and its final outcome. Among the main results, we evidence that sustained oscillations may be triggered by the behavioral memory in the prevalence-dependent vaccination rate. However, the relaxation of social distancing may induce a switch from a cyclic regime to damped oscillations.

Benjamin-Feir instability of Stokes waves - Massimiliano Berti; Alberto Maspero; Paolo Ventura (Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 2022)

A comparison of reduced-order modeling approaches using artificial neural networks for PDEs with bifurcating solutions - Martin W. Hess; Annalisa Quaini; Gianluigi Rozza (ETNA - Electronic Transactions on Numerical Analysis, 2022)

Data-Driven Enhanced Model Reduction for Bifurcating Models in Computational Fluid Dynamics - Martin W. Hess; Annalisa Quaini; Gianluigi Rozza (2022)

A Data-Driven Surrogate Modeling Approach for Time-Dependent Incompressible Navier-Stokes Equations with Dynamic Mode Decomposition and Manifold Interpolation - Martin W. Hess; Annalisa Quaini; Gianluigi Rozza (2022)

The deal.II library, Version 9.4 - Daniel Arndt; Wolfgang Bangerth; Marco Feder; Marc Fehling; Rene Gassmöller; Timo Heister; Luca Heltai; Martin Kronbichler; Matthias Maier; Peter Munch; Jean-Paul Pelteret; Simon Sticko; et al. (Journal of Numerical Mathematics, 2022)

Doubly Intermittent Full Branch Maps with Critical Points and Singularities - Douglas Coates; Stefano Luzzatto; Muhammad Mubarak (2022)
Keywords: 37E05; Dynamical Systems (math.DS); FOS: Mathematics

Doubly Intermittent Maps with Critical Points, Unbounded Derivatives and Regularly Varying Tail - Muhammad Mubarak; Tanja I. Schindler (2022)
Keywords: 37A05; 37A25; 37A50; Dynamical Systems (math.DS); FOS: Mathematics

Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction - Federico Pichi; Maria Strazzullo; F. Ballarin; Gianluigi Rozza (ESAIM: M2AN, 2022)

An example of a weakly mixing BV vector field which is not strongly mixing - Martina Zizza (2022)

Finite element based Model Order Reduction for parametrized one-way coupled steady state linear thermo-mechanical problems - Nirav Shah; Michele Girfoglio; Peregrina Quintela; Gianluigi Rozza; Alejandro Lengomin; Francesco Ballarin; Patricia Barral (Finite Elements in Analysis and Design, 2022)
This contribution focuses on the development of Model Order Reduction (MOR) for one-way coupled steady state linear thermo-mechanical problems in a finite element setting. We apply <a class="topic-link" title="Learn more about Proper Orthogonal Decomposition from ScienceDirect's AI-generated Topic Pages" href="https://www.sciencedirect.com/topics/materials-science/proper-orthogonal-decomposition">Proper Orthogonal Decomposition (POD) for the computation of reduced basis space. On the other hand, for the evaluation of the modal coefficients, we use two different methodologies: the one based on the Galerkin projection (G) and the other one based on Artificial Neural Network (ANN). We aim to compare POD-G and POD-ANN in terms of relevant features including errors and computational efficiency. In this context, both physical and geometrical parametrization are considered. We also carry out a validation of the Full Order Model (FOM) based on customized benchmarks in order to provide a complete computational pipeline. The framework proposed is applied to a relevant industrial problem related to the investigation of thermo-mechanical phenomena arising in blast furnace hearth walls.
Keywords: Artificial Neural Network; Blast furnace; Finite Element Method; Galerkin projection; Geometric and physical parametrization; Proper orthogonal decomposition; Thermo-mechanical problems

Full description of Benjamin-Feir instability of Stokes waves in deep water - Massimiliano Berti; Alberto Maspero; Paolo Ventura (Invent. Math., 2022)

A Gradient Flow Equation for Optimal Control Problems With End-point Cost - Alessandro Scagliotti (Journal of Dynamical and Control Systems, 2022)
In this paper, we consider a control system of the form $\dot x = F(x)u$, linear in the control variable u. Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an end-point cost and the squared 2-norm of the control. This functional induces a gradient flow on the Hilbert space of admissible controls, and we prove a convergence result by means of the Lojasiewicz-Simon inequality. Finally, we show that, if we let the weight of the end-point cost tend to infinity, the resulting family of functionals is Γ-convergent, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.

Hamiltonian Birkhoff normal form for gravity-capillary water waves with constant vorticity: almost global existence - Massimiliano Berti; Alberto Maspero; Federico Murgante (Annals of PDEs, 2022)

IGA-Energetic BEM: An Effective Tool for the Numerical Solution of Wave Propagation Problems in Space-Time Domain - Aimi, Alessandra; Ariel S Boiardi (Mathematics, 2022)
The Energetic Boundary Element Method (BEM) is a recent discretization technique for the numerical solution of wave propagation problems, inside bounded domains or outside bounded obstacles. The differential model problem is converted into a Boundary Integral Equation (BIE) in the time domain, which is then written into an energy-dependent weak form successively discretized by a Galerkin-type approach. Taking into account the space-time model problem of 2D soft-scattering of acoustic waves by obstacles described by open arcs by B-spline (or NURBS) parametrizations, the aim of this paper is to introduce the powerful Isogeometric Analysis (IGA) approach into Energetic BEM for what concerns discretization in space variables. The same computational benefits already observed for IGA-BEM in the case of elliptic (i.e., static) problems, is emphasized here because it is gained at every step of the time-marching procedure. Numerical issues for an efficient integration of weakly singular kernels, related to the fundamental solution of the wave operator and dependent on the propagation wavefront, will be described. Effective numerical results will be given and discussed, showing, from a numerical point of view, convergence and accuracy of the proposed method, as well as the superiority of IGA-Energetic BEM compared to the standard version of the method, which employs classical Lagrangian basis functions.
Keywords: energetic BEM; Isogeometric Analysis; soft-scattering; wave propagation

Indeterminacy estimates, eigenfunctions and lower bounds on Wasserstein distances - Nicolò De Ponti; Sara Farinelli (Calculus of Variations and Partial Differential Equations, 2022)
In the paper we prove two inequalities in the setting of $$\mathsf {RCD}(K,∞ )$$spaces using similar techniques. The first one is an indeterminacy estimate involving the p-Wasserstein distance between the positive part and the negative part of an $$L^{∞ }$$function and the measure of the interface between the positive part and the negative part. The second one is a conjectured lower bound on the p-Wasserstein distance between the positive and negative parts of a Laplace eigenfunction.

Isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature - Manini, Davide (2022)
We prove a sharp isoperimetric inequality for Finsler manifolds having non-negative Ricci curvature and Euclidean volume growth. We also prove a rigidity result for this inequality, under the additional hypotheses of boundedness of the isoperimetric set and the finite reversibility of the space. An application to the weighed anisotropic isoperimetric problem in Euclidean cones is presented.

Isoperimetric inequality in noncompact MCP spaces - Cavalletti, Fabio; Manini, Davide (Proc. Am. Math. Soc., 2022)
We prove a sharp isoperimetric inequality for the class of metric measure spaces verifying the synthetic Ricci curvature lower bounds Measure Contraction property (MCP(0, N)) and having Euclidean volume growth at infinity. We avoid the classical use of the Brunn-Minkowski inequality, not available for MCP(0, N), and of the PDE approach, not available in the singular setting. Our approach will be carried over by using a scaling limit of localization.

Kernel-based active subspaces with application to computational fluid dynamics parametric problems using discontinuous Galerkin method - Francesco Romor; Marco Tezzele; Andrea Lario; Gianluigi Rozza (International Journal for Numerical Methods in Engineering, 2022)

Long-time stability of the quantum hydrodynamic system on irrational tori - Roberto Feola; Felice Iandoli; Federico Murgante (Mathematics in Engineering, 2022)
Keywords: Euler-Korteweg; irrational tori; long time stability; QHD system; Small divisors

Mathematical modelling of axonal cortex contractility - Dario Andrini; Balbi, Valentina; Bevilacqua, Giulia; Lucci, Giulio; Pozzi, Giulia; Riccobelli, Davide (Brain Multiphysics, 2022)
The axonal cortex is composed of a regular structure of F-actin and spectrin able to contract thanks to myosin II motors. Such an active tension is of fundamental importance in controlling the physiological shape of axons. Recent experiments show that axons modulate the contraction of the cortex when subject to mechanical deformations, exhibiting a non-trivial coupling between the hoop and the axial active tension. However, the underlying mechanisms are still poorly understood. In this paper, we propose a continuum model of the axon based on the active strain theory. By using the Coleman–Noll procedure, we shed light on the coupling between the hoop and the axial active strain through the Mandel stress tensor. We propose a qualitative analysis of the system under the simplifying assumption of incompressibility, showing the existence of a stable equilibrium solution. In particular, our results show that the axon regulates the active contraction to maintain a homeostatic stress state. Finally, we propose numerical simulations of the model, using a more suitable compressible constitutive law. The results are compared with experimental data, showing an excellent quantitative agreement. Statement of Significance The mechanics of cortical contractility in axons is still poorly understood. Unravelling the mechanisms underlying axial and hoop stress generation in the cortex will give insight on the active regulation of axon diameter. The understanding of this phenomenon may shed new light on the physical causes of axonal morphological degeneration as a consequence of neurodegenerative diseases, viral infections, and traumatic brain injuries.

Mathematical modelling of oscillating patterns for chronic autoimmune diseases - Rossella Della Marca; Maria da Piedade Machado Ramos; Carolina Ribeiro; Ana Jacinta Soares (Mathematical Methods in the Applied SciencesMathematical Methods in the Applied SciencesMath Meth Appl Sci, 2022)
Many autoimmune diseases are chronic in nature, so that in general, patients experience periods of recurrence and remission of the symptoms characterizing their specific autoimmune ailment. In order to describe this very important feature of autoimmunity, we construct a mathematical model of kinetic type describing the immune system cellular interactions in the context of autoimmunity exhibiting recurrent dynamics. The model equations constitute a nonlinear system of integro-differential equations with quadratic terms that describe the interactions between self-antigen presenting cells, self-reactive T cells, and immunosuppressive cells. We consider a constant input of self-antigen presenting cells, due to external environmental factors that are believed to trigger autoimmunity in people with predisposition for this condition. We also consider the natural death of all cell populations involved in our model, caused by their interaction with cells of the host environment. We derive the macroscopic analogue and show positivity and well-posedness of the solution and then we study the equilibria of the corresponding dynamical system and their stability properties. By applying dynamical system theory, we prove that steady oscillations may arise due to the occurrence of a Hopf bifurcation. We perform some numerical simulations for our model, and we observe a recurrent pattern in the solutions of both the kinetic description and its macroscopic analogue, which leads us to conclude that this model is able to capture the chronic behaviour of many autoimmune diseases.
Keywords: autoimmune diseases; cellular interactions; Dynamical systems; Hopf bifurcation; kinetic theory; mathematical biology

On the Michor–Mumford phenomenon in the infinite dimensional Heisenberg group - Magnani, Valentino; Tiberio, Daniele (Revista Matemática Complutense, 2022)

Model hierarchies and higher-order discretisation of time-dependent thin-film free boundary problems with dynamic contact angle - Dirk Peschka; Luca Heltai (Journal of Computational Physics, 2022)

Model order reduction for bifurcating phenomena in fluid-structure interaction problems - Moaad Khamlich; Federico Pichi; Gianluigi Rozza (International Journal for Numerical Methods in FluidsInternational Journal for Numerical Methods in FluidsInt J Numer Meth Fluids, 2022)
Abstract This work explores the development and the analysis of an efficient reduced order model for the study of a bifurcating phenomenon, known as the Coand? effect, in a multi-physics setting involving fluid and solid media. Taking into consideration a fluid-structure interaction problem, we aim at generalizing previous works towards a more reliable description of the physics involved. In particular, we provide several insights on how the introduction of an elastic structure influences the bifurcating behavior. We have addressed the computational burden by developing a reduced order branch-wise algorithm based on a monolithic proper orthogonal decomposition. We compared different constitutive relations for the solid, and we observed that a nonlinear hyper-elastic law delays the bifurcation w.r.t. the standard model, while the same effect is even magnified when considering linear elastic solid.
Keywords: Bifurcation theory; Coandă effect; continuum mechanics; fluid dynamics; monolithic method; parametrized fluid-structure interaction problem; Proper orthogonal decomposition; reduced order modeling

Model Reduction Using Sparse Polynomial Interpolation for the Incompressible Navier-Stokes Equations - Martin W. Hess; Gianluigi Rozza (2022)

The Neural Network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations - Davide Papapicco; Nicola Demo; Michele Girfoglio; Giovanni Stabile; Gianluigi Rozza (Computer Methods in Applied Mechanics and Engineering, 2022)
Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate the Kolmogorov N−width decay thereby obtaining smaller linear subspaces with improved accuracy. These methods however must rely on the knowledge of the characteristic speeds in phase space of the solution, limiting their range of applicability to problems with explicit functional form for the advection field. In this work we approach the problem of automatically detecting the correct pre-processing transformation in a statistical learning framework by implementing a deep-learning architecture. The purely data-driven method allowed us to generalise the existing approaches of linear subspace manipulation to non-linear hyperbolic problems with unknown advection fields. The proposed algorithm has been validated against simple test cases to benchmark its performances and later successfully applied to a multiphase simulation. © 2022 Elsevier B.V.
Keywords: Advection; Computational complexity; Deep neural network; Deep neural networks; Linear subspace; Multiphase simulations; Non linear; Nonlinear hyperbolic equation; Partial differential equations; Phase space methods; Pre-processing; Principal component analysis; reduced order modeling; Reduced order modelling; Reduced-order model; Shifted-POD

The $N$-Link Swimmer in Three Dimensions: Controllability and Optimality Results - Marchello, Roberto; Morandotti, Marco; Shum, Henry; Zoppello, Marta (Acta Applicandae Mathematicae, 2022)
The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal 2-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the 2-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all eight linearly independent directions in the combined configuration and shape space, leading to controllability; the swimmer can move from any starting configuration and shape to any target configuration and shape by operating on the two shape variables. The result is subsequently extended to the $N$-link swimmer. Finally, the minimal time optimal control problem and the minimization of the power expended are addressed and a qualitative description of the optimal strategies is provided.

Numerical proof of shell model turbulence closure - Giulio Ortali; Corbetta, Alessandro; Rozza, Gianluigi; Toschi, Federico (Physical Review Fluids, 2022)

Optimal design of planar shapes with active materials - Dario Andrini; Giovanni Noselli; Alessandro Lucantonio (Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2022)
Active materials have emerged as valuable candidates for shape morphing applications, where a body reconfiguration is achieved upon triggering its active response. Given a desired shape change, a natural question is to compare different morphing mechanisms to select the most effective one with respect to an optimality criterion. We introduce an optimal control problem to determine the active strains suitable to attain a target equilibrium shape while minimizing the complexity of the activation. Specifically, we discuss the planar morphing of active, hyperelastic bodies in the absence of external forces and exploit the notion of target metric to encompass a broad set of active materials in a unifying approach. For the case of affine shape changes, we derive explicit conditions on the body reference configuration for the optimality of homogeneous target metrics. More complex shape changes are analysed via numerical simulations to explore the impact on optimal solutions of different objective functionals inspired by features of existing materials. We show how stresses arising from incompatibilities contribute to reduce the complexity of the controls. We believe that our approach may be exploited for the optimal design of active systems and may contribute to gather insight into the morphing strategies of biological systems.

A piecewise conservative method for unconstrained convex optimization - Alessandro Scagliotti; Colli Franzone, P. (Computational Optimization and Applications, 2022)
We consider a continuous-time optimization method based on a dynamical system, where a massive particle starting at rest moves in the conservative force field generated by the objective function, without any kind of friction. We formulate a restart criterion based on the mean dissipation of the kinetic energy, and we prove a global convergence result for strongly-convex functions. Using the Symplectic Euler discretization scheme, we obtain an iterative optimization algorithm. We have considered a discrete mean dissipation restart scheme, but we have also introduced a new restart procedure based on ensuring at each iteration a decrease of the objective function greater than the one achieved by a step of the classical gradient method. For the discrete conservative algorithm, this last restart criterion is capable of guaranteeing a qualitative convergence result. We apply the same restart scheme to the Nesterov Accelerated Gradient (NAG-C), and we use this restarted NAG-C as benchmark in the numerical experiments. In the smooth convex problems considered, our method shows a faster convergence rate than the restarted NAG-C. We propose an extension of our discrete conservative algorithm to composite optimization: in the numerical tests involving non-strongly convex functions with $$\ell ^1$$-regularization, it has better performances than the well known efficient Fast Iterative Shrinkage-Thresholding Algorithm, accelerated with an adaptive restart scheme.

A POD-Galerkin reduced order model for the Navier–Stokes equations in stream function-vorticity formulation - Michele Girfoglio; Annalisa Quaini; Gianluigi Rozza (Computers & Fluids, 2022)
We develop a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for the efficient numerical simulation of the parametric Navier–Stokes equations in the stream function-vorticity formulation. Unlike previous works, we choose different reduced coefficients for the vorticity and stream function fields. In addition, for parametric studies we use a global POD basis space obtained from a database of time dependent full order snapshots related to sample points in the parameter space. We test the performance of our ROM strategy with the well-known vortex merger benchmark and a more complex case study featuring the geometry of the North Atlantic Ocean. Accuracy and efficiency are assessed for both time reconstruction and physical parametrization.
Keywords: Galerkin projection; Navier–Stokes equations; Proper orthogonal decomposition; Reduced order model; Stream function-vorticity formulation

Projection based semi–implicit partitioned Reduced Basis Method for non parametrized and parametrized Fluid–Structure Interaction problems - Monica Nonino; Francesco Ballarin; Gianluigi Rozza; Yvon Maday (2022)
The goal of this manuscript is to present a partitioned Model Order Reduction method that is based on a semi-implicit projection scheme to solve multiphysics problems. We implement a Reduced Order Method based on a Proper Orthogonal Decomposition, with the aim of addressing both time-dependent and time-dependent, parametrized Fluid-Structure Interaction problems, where the fluid is incompressible and the structure is thick and two dimensional.

A Proper Orthogonal Decomposition Approach for Parameters Reduction of Single Shot Detector Networks - Laura Meneghetti; Nicola Demo; Gianluigi Rozza (2022 IEEE International Conference on Image Processing (ICIP), 2022)

A reduced-order model for segregated fluid-structure interaction solvers based on an ALE approach - Valentin Eric Brice Nkana Ngan; Giovanni Stabile; Andrea Mola; Gianluigi Rozza (2022)

The relaxed area of $S^1$-valued singular maps in the strict $BV$-convergence - Bellettini, Giovanni; Carano, Simone; Scala, Riccardo (ESAIM: Control, Optimization and Calculus of Variations, 2022)

Rigidities of Isoperimetric inequality under nonnegative Ricci curvature - Cavalletti, Fabio; Manini, Davide (2022)
The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions [arXiv:1812.05022, arXiv:2012.09490, arXiv:2009.13717, arXiv:2103.08496] culminated by Balogh and Kristaly [arXiv:2012.11862] covering also m.m.s.'s verifying the non-negative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterisation of the equality cases is present in the literature. The scope of this note is to settle this problem by proving, in the same generality of [arXiv:2012.11862], that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extend to weighted Riemannian manifold the rigidity results of [arXiv:2009.13717], it extend to general RCD spaces the rigidity results of [arXiv:2201.04916] and finally applies also to the Euclidean setting by proving that that optimisers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

An SIR–like kinetic model tracking individuals' viral load - Rossella Della Marca; Nadia Loy; Andrea Tosin (Networks and Heterogeneous Media, 2022)

Space-time POD-Galerkin approach for parametric flow control - Francesco Ballarin; Gianluigi Rozza; Maria Strazzullo (Handbook of Numerical Analysis, 2022)

The \textttdeal.II Library, Version 9.4 - Daniel Arndt; Wolfgang Bangerth Marco Feder; Marc Fehling; Rene Gassmöller; Timo Heister; Luca Heltai; Martin Kronbichler; Matthias Maier; Peter Munch; Jean-Paul Pelteret; Simon Sticko; Bruno Turcksin; et al. (Journal of Numerical Mathematics, 2022)

Thermomechanical Modelling for Industrial Applications - Nirav Shah; Michele Girfoglio; Gianluigi Rozza (Progress in Industrial Mathematics at ECMI 2021, 2022)
In this work we briefly present a thermomechanical model that could serve as starting point for industrial applications. We address the non-linearity due to temperature dependence of material properties and heterogeneity due to presence of different materials. Finally a numerical example related to the simplified geometry of blast furnace hearth walls is shown with the aim of assessing the feasibility of the modelling framework.

Variational Approach to Fluid–Structure Interaction via GENERIC - Dirk Peschka; Andrea Zafferi; Luca Heltai; Marita Thomas (Journal of Non-Equilibrium Thermodynamics, 2022)

Vibration Analysis of Piezoelectric Kirchhoff-Love shells based on Catmull-Clark Subdivision Surfaces - Zhaowei Liu; Andrew McBride; Prashant Saxena; Luca Heltai; Yilin Qu; Paul Steinmann (International Journal for Numerical Methods in Engineering, 2022)

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods - Andrea Cangiani; E.H. Georgoulis; Sutton, Oliver J. (Mathematical Models and Methods in Applied Sciences, 2021)
We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the L2(H1) and L∞(L2) norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates.

Approximation of the spectral fractional powers of the Laplace-Beltrami Operator - Bonito, Andrea; Wenyu Lei (arXiv preprint arXiv:2101.05141, 2021)

An artificial neural network approach to bifurcating phenomena in computational fluid dynamics - Federico Pichi; Francesco Ballarin; Gianluigi Rozza; Jan S Hesthaven (2021)

Asymptotic approach to a rotational Taylor swimming sheet - Giovanni Corsi (Comptes Rendus. Mécanique, 2021)

ATHENA: Advanced Techniques for High dimensional parameter spaces to Enhance Numerical Analysis - Francesco Romor; Marco Tezzele; Gianluigi Rozza (Software Impacts, 2021)

Birkhoff normal form for gravity water waves - Massimiliano Berti; Feola, Roberto; Pusateri, Fabio (Water Waves, 2021)

BisPy: Bisimulation in Python - Francesco Andreuzzi (Journal of Open Source Software, 2021)

A CERTIFIED REDUCED BASIS Method FOR LINEAR PARAMETRIZED PARABOLIC OPTIMAL CONTROL PROBLEMS IN SPACE-TIME FORMULATION - Maria Strazzullo; Francesco Ballarin; Gianluigi Rozza (2021)

On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis - Mahmoud Gadalla; Marta Cianferra; Marco Tezzele; Giovanni Stabile; Andrea Mola; Gianluigi Rozza (Computers & Fluids, 2021)
In this work, Dynamic Mode Decomposition (DMD) and Proper Orthogonal Decomposition (POD) methodologies are applied to hydroacoustic dataset computed using Large Eddy Simulation (LES) coupled with Ffowcs Williams and Hawkings (FWH) analogy. First, a low-dimensional description of the flow fields is presented with modal decomposition analysis. Sensitivity towards the DMD and POD bases truncation rank is discussed, and extensive dataset is provided to demonstrate the ability of both algorithms to reconstruct the flow fields with all the spatial and temporal frequencies necessary to support accurate noise evaluation. Results show that while DMD is capable to capture finer coherent structures in the wake region for the same amount of employed modes, reconstructed flow fields using POD exhibit smaller magnitudes of global spatiotemporal errors compared with DMD counterparts. Second, a separate set of DMD and POD modes generated using half the snapshots is employed into two data-driven reduced models respectively, based on DMD mid cast and POD with Interpolation (PODI). In that regard, results confirm that the predictive character of both reduced approaches on the flow fields is sufficiently accurate, with a relative superiority of PODI results over DMD ones. This infers that, discrepancies induced due to interpolation errors in PODI is relatively low compared with errors induced by integration and linear regression operations in DMD, for the present setup. Finally, a post processing analysis on the evaluation of FWH acoustic signals utilizing reduced fluid dynamic fields as input demonstrates that both DMD and PODI data-driven reduced models are efficient and sufficiently accurate in predicting acoustic noises.
Keywords: Dynamic mode decomposition; Ffowcs Williams and Hawkings; Hydroacoustics; Large eddy simulation; Model reduction; Proper orthogonal decomposition

Consistency of the full and reduced order models for Evolve-Filter-Relax Regularization of Convection-Dominated, Marginally-Resolved Flows - Maria Strazzullo; Michele Girfoglio; Francesco Ballarin; T. Iliescu; Gianluigi Rozza (2021)

Data-driven parameter and model order reduction for industrial optimisation problems with applications in naval engineering - Marco Tezzele (SISSA - International School for Advanced Studies, 2021)

A data-driven partitioned approach for the resolution of time-dependent optimal control problems with dynamic mode decomposition - Eleonora Donadini; Maria Strazzullo; Marco Tezzele; Gianluigi Rozza (2021)

The deal.II Library, Version 9.3 - Daniel Arndt; Wolfgang Bangerth; Bruno Blais; Marc Fehling; Rene Gassmöller; Timo Heister; Luca Heltai; Uwe Köcher; Martin Kronbichler; Matthias Maier; Peter Munch; Jean-Paul Pelteret; et al. (Journal of Numerical Mathematics, 2021)

Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems - Alessandro Scagliotti (2021)
Keywords: FOS: Computer and information sciences; FOS: Mathematics; Machine Learning (cs.LG); Optimization and Control (math.OC)

A Differential Perspective on Gradient Flows on CAT(K)-Spaces and Applications - Nicola Gigli; Francesco Nobili (The Journal of Geometric Analysis, 2021)
We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$CAT (κ )$$-spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-E (y_t)$$one uses in the smooth setting and more precisely that $$y_t'$$selects the element of minimal norm in $$-\partial ^-E (y_t)$$. This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of $$L^2$$and CAT(0) valued maps: we define the Laplacian of such $$L^2$$map as the element of minimal norm in $$-\partial ^-E (u)$$, provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$-dense. Basic properties of this Laplacian are then studied.

A Dimensionality Reduction Approach for Convolutional Neural Networks - Laura Meneghetti; Nicola Demo; Gianluigi Rozza (2021)

On Dini derivatives of real functions - Giuliano Klun; Alessandro Fonda; Andrea Sfecci (2021)

Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation - Nirav Shah; Martin W. Hess; Gianluigi Rozza (Numerical Mathematics and Advanced Applications ENUMATH 2019, 2021)
The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time.

Displacement convexity of Entropy and the distance cost Optimal Transportation - Fabio Cavalletti; Nicola Gigli; Flavia Santarcangelo (Annales de la Faculté des sciences de Toulouse : Mathématiques, 2021)

Doulbeault and J-invariant Cohomologies on Almost Complex Manifolds - Lorenzo Sillari; Adriano Tomassini (Complex Analysis and Operator Theory, 2021)
In this paper we relate the cohomology of $J$-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.
Keywords: Almost Complex Manifolds; Cohomology of Lie Algebra; Compact four-manifold; Dolbeault Cohomology; Frölicher Spectral Sequence; Solvmanifold

A dynamic mode decomposition extension for the forecasting of parametric dynamical systems - Francesco Andreuzzi; Nicola Demo; Gianluigi Rozza (arXiv preprint arXiv:2110.09155, 2021)

A dynamic model for viscoelasticity in domains with time-dependent cracks - Francesco Sapio (Nonlinear Differential Equations and Applications NoDEA, 2021)
In this paper, we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracking domains.

Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method - Moreno Pintore; Federico Pichi; Martin W. Hess; Gianluigi Rozza; Claudio Canuto (Advances in Computational Mathematics, 2021)
The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.

An efficient computational framework for naval shape design and optimization problems by means of data-driven reduced order modeling techniques - Nicola Demo; Giulio Ortali; Gianluca Gustin; Gianluigi Rozza; Gianpiero Lavini (Bolletino dell Unione Matematica Italiana, 2021)
This contribution describes the implementation of a data-driven shape optimization pipeline in a naval architecture application. We adopt reduced order models in order to improve the efficiency of the overall optimization, keeping a modular and equation-free nature to target the industrial demand. We applied the above mentioned pipeline to a realistic cruise ship in order to reduce the total drag. We begin by defining the design space, generated by deforming an initial shape in a parametric way using free form deformation. The evaluation of the performance of each new hull is determined by simulating the flux via finite volume discretization of a two-phase (water and air) fluid. Since the fluid dynamics model can result very expensive—especially dealing with complex industrial geometries—we propose also a dynamic mode decomposition enhancement to reduce the computational cost of a single numerical simulation. The real-time computation is finally achieved by means of proper orthogonal decomposition with Gaussian process regression technique. Thanks to the quick approximation, a genetic optimization algorithm becomes feasible to converge towards the optimal shape.

Equilibrium measure for a nonlocal dislocation energy with physical confinement - Mora, Maria Giovanna; Alessandro Scagliotti (2021)

Exactness of Linear Response in the Quantum Hall Effect - Sven Bachmann; De Roeck, Wojciech; Fraas, Martin; Markus Lange (Annales Henri Poincaré, 2021)

An existence result for the fractional Kelvin–Voigt’s model on time-dependent cracked domains - Maicol Caponi; Francesco Sapio (Journal of Evolution Equations, 2021)
We prove an existence result for the fractional Kelvin–Voigt’s model involving Caputo’s derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin–Voigt’s model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique.

AN EXTENDED PHYSICS INFORMED NEURAL NETWORK FOR PRELIMINARY ANALYSIS OF PARAMETRIC OPTIMAL CONTROL PROBLEMS - Nicola Demo; Maria Strazzullo; Gianluigi Rozza (2021)

A first-order condition for the independence on p of weak gradients - Nicola Gigli; Francesco Nobili (2021)

Heat content asymptotics in sub-Riemannian manifolds - Luca Rizzi; Tommaso Rossi (Journal de Mathématiques Pures et Appliquées, 2021)

Hierarchical model reduction techniques for flow modeling in a parametrized setting - Matteo Zancanaro; F. Ballarin; Simona Perotto; Gianluigi Rozza (Multiscale Modeling and Simulation, 2021)
In this work we focus on two different methods to deal with parametrized partial differential equations in an efficient and accurate way. Starting from high fidelity approximations built via the hierarchical model reduction discretization, we consider two approaches, both based on a projection model reduction technique. The two methods differ for the algorithm employed during the construction of the reduced basis. In particular, the former employs the proper orthogonal decomposition, while the latter relies on a greedy algorithm according to the certified reduced basis technique. The two approaches are preliminarily compared on two-dimensional scalar and vector test cases.

Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing - Nicola Demo; Marco Tezzele; Andrea Mola; Gianluigi Rozza (Journal of Marine Science and Engineering, 2021)
In the field of parametric partial differential equations, shape optimization represents a challenging problem due to the required computational resources. In this contribution, a data-driven framework involving multiple reduction techniques is proposed to reduce such computational burden. Proper orthogonal decomposition (POD) and active subspace genetic algorithm (ASGA) are applied for a dimensional reduction of the original (high fidelity) model and for an efficient genetic optimization based on active subspace property. The parameterization of the shape is applied directly to the computational mesh, propagating the generic deformation map applied to the surface (of the object to optimize) to the mesh nodes using a radial basis function (RBF) interpolation. Thus, topology and quality of the original mesh are preserved, enabling application of POD-based reduced order modeling techniques, and avoiding the necessity of additional meshing steps. Model order reduction is performed coupling POD and Gaussian process regression (GPR) in a data-driven fashion. The framework is validated on a benchmark ship.

Hybrid Neural Network Reduced Order Modelling for Turbulent Flows with Geometric Parameters - Matteo Zancanaro; Markus Mrosek; Giovanni Stabile; Carsten Othmer; Gianluigi Rozza (Fluids, 2021)

Independence of synthetic curvature dimension conditions on transport distance exponent - Afiny Akdemir; Andrew Colinet; Robert McCann; Fabio Cavalletti; Flavia Santarcangelo (Trans. Amer. Math. Soc., 2021)

Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group - Tommaso Rossi (Advances in Calculus of Variations, 2021)

Large KAM tori for quasi-linear perturbations of KdV - Massimiliano Berti; Kappeler, Thomas; Montalto, Riccardo (Arch. Ration. Mech. Anal., 2021)

A local approach to parameter space reduction for regression and classification tasks - Francesco Romor; Marco Tezzele; Gianluigi Rozza (arXiv preprint arXiv:2107.10867, 2021)
Submitted

Local well posedness of the Euler-Korteweg equations on {$\Bbb T^d$} - Massimiliano Berti; Alberto Maspero; Federico Murgante (Journal of Dynamics and Differential Equations, 2021)
We consider the Euler–Korteweg system with space periodic boundary conditions $x \in {\mathbb {T}}^d$. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.

On master test plans for the space of BV functions - Francesco Nobili; Enrico Pasqualetto; Timo Schultz (2021)

Model Order Reduction for Nonlinear and Time-Dependent Parametric Optimal Flow Control Problems - Maria Strazzullo (SISSA, 2021)

A Monolithic and a Partitioned, Reduced Basis Method for Fluid–Structure Interaction Problems - Monica Nonino; F. Ballarin; Gianluigi Rozza (Fluids, 2021)
The aim of this work is to present an overview about the combination of the Reduced Basis Method (RBM) with two different approaches for Fluid–Structure Interaction (FSI) problems, namely a monolithic and a partitioned approach. We provide the details of implementation of two reduction procedures, and we then apply them to the same test case of interest. We first implement a reduction technique that is based on a monolithic procedure where we solve the fluid and the solid problems all at once. We then present another reduction technique that is based on a partitioned (or segregated) procedure: the fluid and the solid problems are solved separately and then coupled using a fixed point strategy. The toy problem that we consider is based on the Turek–Hron benchmark test case, with a fluid Reynolds number Re=100.

Monotonicity formulas for harmonic functions in RCD(0,N) spaces - Nicola Gigli; Ivan Yuri Violo (2021)
We generalize to the <span id="MathJax-Element-2-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-13" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 5.909em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 4.825em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1004.7em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-17" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">R<span id="MathJax-Span-18" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">C<span id="MathJax-Span-19" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">D<span id="MathJax-Span-20" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-21" class="mn" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">0<span id="MathJax-Span-22" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-23" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">N<span id="MathJax-Span-24" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with non-negative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in [AFM] we also introduce the notion of electrostatic potential in <span id="MathJax-Element-3-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-25" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.715em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.233em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.17em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-29" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">R<span id="MathJax-Span-30" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">C<span id="MathJax-Span-31" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">D spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in <span id="MathJax-Element-4-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-32" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 6.392em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 5.247em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1005.13em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-36" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">R<span id="MathJax-Span-37" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">C<span id="MathJax-Span-38" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">D<span id="MathJax-Span-39" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-40" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">K<span id="MathJax-Span-41" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-42" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">N<span id="MathJax-Span-43" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">) spaces and on a new functional version of the `(almost) outer volume cone implies (almost) outer metric cone' theorem.

Multi-fidelity data fusion for the approximation of scalar functions with low intrinsic dimensionality through active subspaces - Francesco Romor; Marco Tezzele; Gianluigi Rozza (Proceedings in Applied Mathematics & Mechanics, 2021)

Multi-fidelity data fusion through parameter space reduction with applications to automotive engineering - Francesco Romor; Marco Tezzele; Markus Mrosek; Carsten Othmer; Gianluigi Rozza (arXiv preprint arXiv:2110.14396, 2021)

Multiscale coupling of one-dimensional vascular models and elastic tissues - Luca Heltai; Alfonso Caiazzo; Lucas Müeller (Annals of Biomedical Engineering, 2021)

The Neural Network shifted-Proper Orthogonal Decomposition: a Machine Learning Approach for Non-linear Reduction of Hyperbolic Equations - Davide Papapicco; Nicola Demo; Michele Girfoglio; Giovanni Stabile; Gianluigi Rozza (2021)

Non-intrusive data-driven ROM framework for hemodynamics problems - Michele Girfoglio; Leonardo Scandurra; Francesco Ballarin; Giuseppe Infantino; Francesca Nicolò; Andrea Montalto; Gianluigi Rozza; Roberto Scrofani; Marina Comisso; Francesco Musumeci (Acta Mechanica Sinica, 2021)

Non-well-ordered lower and upper solutions for semilinear systems of PDEs - Alessandro Fonda; Giuliano Klun; Andrea Sfecci (Communications in Contemporary MathematicsCommunications in Contemporary Mathematics, 2021)
We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

A novel iterative penalty method to enforce boundary conditions in Finite Volume POD-Galerkin reduced order models for fluid dynamics problems - Kelbij Star; Giovanni Stabile; Francesco Belloni; Gianluigi Rozza; Joris Degroote (Communications in Computational Physics, 2021)
A Finite-Volume based POD-Galerkin reduced order model is developed for fluid dynamic problems where the (time-dependent) boundary conditions are controlled using two different boundary control strategies: the control function method, whose aim is to obtain homogeneous basis functions for the reduced basis space and the penalty method where the boundary conditions are enforced in the reduced order model using a penalty factor. The penalty method is improved by using an iterative solver for the determination of the penalty factor rather than tuning the factor with a sensitivity analysis or numerical experimentation. The boundary control methods are compared and tested for two cases: the classical lid driven cavity benchmark problem and a Y-junction flow case with two inlet channels and one outlet channel. The results show that the boundaries of the reduced order model can be controlled with the boundary control methods and the same order of accuracy is achieved for the velocity and pressure fields. Finally, the speedup ratio between the reduced order models and the full order model is of the order 1000 for the lid driven cavity case and of the order 100 for the Y-junction test case.

A numerical approach for heat flux estimation in thin slabs continuous casting molds using data assimilation - Umberto Emil Morelli; Patricia Barral; Peregrina Quintela; Gianluigi Rozza; Giovanni Stabile (International Journal for Numerical Methods in Engineering, 2021)

Nutations in growing plant shoots as a morphoelastic flutter instability - Daniele Agostinelli; Giovanni Noselli; Antonio DeSimone (Phil. Trans. R. Soc. A, 2021)
Growing plant shoots exhibit spontaneous oscillations that Darwin observed, and termed "circumnutations". Recently, they have received renewed attention for the design and optimal actuation of bioinspired robotic devices. We discuss a possible interpretation of these spontaneous oscillations as a Hopf-type bifurcation in a growing morphoelastic rod. Using a three-dimensional model and numerical simulations, we analyse the salient features of this flutter-like phenomenon (e.g. the characteristic period of the oscillations) and their dependence on the model details (in particular, the impact of choosing different growth models) finding that, overall, these features are robust with respect to changes in the details of the growth model adopted.

Nutations in plant shoots: Endogenous and exogenous factors in the presence of mechanical deformations - Daniele Agostinelli; Antonio DeSimone; Giovanni Noselli (Frontiers in Plant Science, 2021)
We present a three-dimensional morphoelastic rod model capable to describe the morphogenesis of growing plant shoots driven by differential growth. We discuss the evolution laws for endogenous oscillators, straightening mechanisms, and reorientations to directional cues, such as gravitropic reactions governed by the avalanche dynamics of statoliths. We use this model to investigate the role of elastic deflections due to gravity loading in circumnutating plant shoots. We show that, in the absence of endogenous cues, pendular and circular oscillations arise as a critical length is attained, thus suggesting the occurrence of an instability triggered by exogenous factors. When also oscillations due to endogenous cues are present, their weight relative to those associated with the instability varies in time as the shoot length and other biomechanical properties change. Thanks to the simultaneous occurrence of these two oscillatory mechanisms, we are able to reproduce a variety of complex behaviors, including trochoid-like patterns, which evolve into circular orbits as the shoot length increases, and the amplitude of the exogenous oscillations becomes dominant.

Parallel transport on non-collapsed $\mathsfRCD(K,N)$ spaces - Emanuele Caputo; Nicola Gigli; Enrico Pasqualetto (2021)
We provide a general theory for parallel transport on non-collapsed <span id="MathJax-Element-2-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-13" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.474em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.992em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1001.93em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-17" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">R<span id="MathJax-Span-18" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">C<span id="MathJax-Span-19" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">D spaces obtaining both existence and uniqueness results. Our theory covers the case of geodesics and, more generally, of curves obtained via the flow of sufficiently regular time dependent vector fields: the price that we pay for this generality is that we cannot study parallel transport along a single such curve, but only along almost all of these (in a sense related to the notions of Sobolev vector calculus and Regular Lagrangian Flow in the nonsmooth setting). The class of <span id="MathJax-Element-3-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-20" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.619em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.956em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1002.9em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-24" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">n<span id="MathJax-Span-25" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">c<span id="MathJax-Span-26" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">R<span id="MathJax-Span-27" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">C<span id="MathJax-Span-28" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">D spaces contains finite dimensional Alexandrov spaces with curvature bounded from below, thus our construction provides a way of speaking about parallel transport in this latter setting alternative to the one proposed by Petrunin (1998). The precise relation between the two approaches is yet to be understood.

Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces - Alessandro Fonda; Giuliano Klun; Andrea Sfecci (Mediterranean Journal of Mathematics, 2021)
We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.

A POD-Galerkin reduced order model for a LES filtering approach - Michele Girfoglio; Annalisa Quaini; Gianluigi Rozza (Journal of Computational Physics, 2021)
We propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for an implementation of the Leray model that combines a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficients. For the reconstruction of the pressures fields, we use a pressure Poisson equation approach. We test our ROM on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0≤Re≤100. The accuracy of the reduced order model is assessed against results obtained with the full order model. For the 2D case, a parametric study with respect to the filtering radius is also presented.

A POD-Galerkin reduced order model of a turbulent convective buoyant flow of sodium over a backward-facing step - Kelbij Star; Giovanni Stabile; Gianluigi Rozza; Joris Degroote (Applied Mathematical Modelling, 2021)
A Finite-Volume based POD-Galerkin reduced order modeling strategy for steady-state Reynolds averaged Navier–Stokes (RANS) simulation is extended for low-Prandtl number flow. The reduced order model is based on a full order model for which the effects of buoyancy on the flow and heat transfer are characterized by varying the Richardson number. The Reynolds stresses are computed with a linear eddy viscosity model. A single gradient diffusion hypothesis, together with a local correlation for the evaluation of the turbulent Prandtl number, is used to model the turbulent heat fluxes. The contribution of the eddy viscosity and turbulent thermal diffusivity fields are considered in the reduced order model with an interpolation based data-driven method. The reduced order model is tested for buoyancy-aided turbulent liquid sodium flow over a vertical backward-facing step with a uniform heat flux applied on the wall downstream of the step. The wall heat flux is incorporated with a Neumann boundary condition in both the full order model and the reduced order model. The velocity and temperature profiles predicted with the reduced order model for the same and new Richardson numbers inside the range of parameter values are in good agreement with the RANS simulations. Also, the local Stanton number and skin friction distribution at the heated wall are qualitatively well captured. Finally, the reduced order simulations, performed on a single core, are about 105 times faster than the RANS simulations that are performed on eight cores.

Propagating geometry information to finite element computations - Luca Heltai; Wolfgang Bangerth; Martin Kronbichler; Andrea Mola (Transactions on Mathematical Software, 2021)

PyGeM: Python Geometrical Morphing - Marco Tezzele; Nicola Demo; Andrea Mola; Gianluigi Rozza (Software Impacts, 2021)
PyGeM is an open source Python package which allows to easily parametrize and deform 3D object described by CAD files or 3D meshes. It implements several morphing techniques such as free form deformation, radial basis function interpolation, and inverse distance weighting. Due to its versatility in dealing with different file formats it is particularly suited for researchers and practitioners both in academia and in industry interested in computational engineering simulations and optimization studies.
Keywords: Free form deformation; Geometrical morphing; Inverse distance weighting; Python; Radial basis functions interpolation

Quadratic life span of periodic gravity-capillary water waves - Massimiliano Berti; Feola, R.; Luca Franzoi (Water Waves, 2021)

Quantitative lower bounds to the Euclidean and the Gaussian Cheeger constants - Julin, V.; Saracco, G. (Ann. Fenn. Math., 2021)

Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Critical Potentials and dimensionality - Dirk Hundertmark; Michal Jex; Markus Lange (arXiv:2107.14128, 2021)

Quasi-optimal mesh sequence construction through Smoothed Adaptive Finite Element Methods - Ornela Mulita; Stefano Giani; Luca Heltai (SIAM Journal on Scientific Computing, 2021)

Quasistatic Limit of a Dynamic Viscoelastic Model with Memory - Gianni Dal Maso; Francesco Sapio (Milan Journal of Mathematics, 2021)
We study the behaviour of the solutions to a dynamic evolution problem for a viscoelastic model with long memory, when the rate of change of the data tends to zero. We prove that a suitably rescaled version of the solutions converges to the solution of the corresponding stationary problem.

Rectifiability of the free boundary for varifolds - Luigi De Masi (Indiana Univ. Math. J., 2021)

A Reduced Order Cut Finite Element method for geometrically parametrized steady and unsteady Navier–Stokes problems - Efthymios N Karatzas; Monica Nonino; F. Ballarin; Gianluigi Rozza (Computer & Mathematics With Applications, 2021)
We focus on steady and unsteady Navier–Stokes flow systems in a reduced-order modeling framework based on Proper Orthogonal Decomposition within a levelset geometry description and discretized by an unfitted mesh Finite Element Method. This work extends the approaches of [1], [2], [3] to nonlinear CutFEM discretization. We construct and investigate a unified and geometry independent reduced basis which overcomes many barriers and complications of the past, that may occur whenever geometrical morphings are taking place. By employing a geometry independent reduced basis, we are able to avoid remeshing and transformation to reference configurations, and we are able to handle complex geometries. This combination of a fixed background mesh in a fixed extended background geometry with reduced order techniques appears beneficial and advantageous in many industrial and engineering applications, which could not be resolved efficiently in the past.
Keywords: Cut Finite Element Method; Navier–Stokes equations; Parameter–dependent shape geometry; Reduced Order Models; Unfitted mesh

Reduced Order Methods for Parametrized Non-linear and Time Dependent Optimal Flow Control Problems, Towards Applications in Biomedical and Environmental Sciences - Maria Strazzullo; Zakia Zainib; F. Ballarin; Gianluigi Rozza (Numerical Mathematics and Advanced Applications ENUMATH 2019, 2021)
We introduce reduced order methods as an efficient strategy to solve parametrized non-linear and time dependent optimal flow control problems governed by partial differential equations. Indeed, the optimal control problems require a huge computational effort in order to be solved, most of all in physical and/or geometrical parametrized settings. Reduced order methods are a reliable and suitable approach, increasingly gaining popularity, to achieve rapid and accurate optimal solutions in several fields, such as in biomedical and environmental sciences. In this work, we employ a POD-Galerkin reduction approach over a parametrized optimality system, derived from the Karush-Kuhn-Tucker conditions. The methodology presented is tested on two boundary control problems, governed respectively by (1) time dependent Stokes equations and (2) steady non-linear Navier-Stokes equations.

Reduced order models for the incompressible Navier-Stokes equations on collocated grids using a `discretize-then-project' approach - Kelbij Star; Benjamin Sanderse; Giovanni Stabile; Gianluigi Rozza; Joris Degroote (International Journal for Numerical Methods in Fluids, 2021)

The relative heat content for submanifolds in sub-Riemannian geometry - Tommaso Rossi (Actes du séminaire Théorie Spectrale et Géométrie, 2021)

A remark on two notions of flatness for sets in the Euclidean space - Ivan Yuri Violo (2021)
In this note we compare two ways of measuring the <span id="MathJax-Element-1-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-1" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-3" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">n-dimensional "flatness" of a set <span id="MathJax-Element-2-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-4" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.921em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.197em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.148em, 1003.2em, 2.474em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-6" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">S<span id="MathJax-Span-7" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">⊂<span id="MathJax-Span-8" class="msubsup" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.137em, 1000.73em, 4.162em, -999.997em); top: -3.975em; left: 0em;"><span id="MathJax-Span-11" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_AMS;">R<span id="MathJax-Span-12" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.7374px; font-family: MathJax_Math-italic;">d, where <span id="MathJax-Element-3-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-13" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.197em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.595em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.691em, 1002.6em, 2.775em, -999.997em); top: -2.528em; left: 0em;"><span id="MathJax-Span-15" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">n<span id="MathJax-Span-16" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">∈<span id="MathJax-Span-17" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal;"><span id="MathJax-Span-19" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_AMS;">N and <span id="MathJax-Element-4-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-20" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.137em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.534em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.53em, 2.474em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-22" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">d<span id="MathJax-Span-23" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">><span id="MathJax-Span-24" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">n. 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display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">) between <span id="MathJax-Element-9-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-51" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.847em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1000.67em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-53" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">S and <span id="MathJax-Element-10-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-54" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-56" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">n-dimensional affine subspaces of <span id="MathJax-Element-11-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-57" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.389em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 1.148em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(-0.057em, 1001.15em, 1.208em, -999.997em); top: -1.022em; left: 0em;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(3.137em, 1000.73em, 4.162em, -999.997em); top: -3.975em; left: 0em;"><span id="MathJax-Span-62" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_AMS;">R<span id="MathJax-Span-63" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-size: 11.7374px; font-family: MathJax_Math-italic;">d. The second is an `intrinsic' approach in which we view the same set <span id="MathJax-Element-12-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-64" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.847em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1000.67em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-66" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">S as a metric space (endowed with the induced Euclidean distance). Then we consider numbers <span id="MathJax-Element-13-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-67" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.378em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.775em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.65em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-71" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">a<span id="MathJax-Span-72" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-73" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x<span id="MathJax-Span-74" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-75" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r<span id="MathJax-Span-76" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">)'s, that are the scaling-invariant Gromov-Hausdorff distances between balls centered at <span id="MathJax-Element-14-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-77" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.54em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-79" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x of radius <span id="MathJax-Element-15-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-80" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.545em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.425em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.42em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-82" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r in <span id="MathJax-Element-16-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-83" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.847em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1000.67em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-85" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">S and the <span id="MathJax-Element-17-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-86" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-88" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">n-dimensional Euclidean ball of the same radius. As main result of our analysis we make rigorous a phenomenon, first noted by David and Toro, for which the numbers <span id="MathJax-Element-18-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-89" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.378em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.775em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.65em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-93" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_SansSerif;">a<span id="MathJax-Span-94" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-95" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x<span id="MathJax-Span-96" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-97" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r<span id="MathJax-Span-98" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">)'s behaves as the square of the numbers <span id="MathJax-Element-19-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-99" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 3.559em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 2.896em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.329em, 1002.77em, 2.655em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-101" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">α<span id="MathJax-Span-102" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">(<span id="MathJax-Span-103" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">x<span id="MathJax-Span-104" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">,<span id="MathJax-Span-105" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.184em; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">r<span id="MathJax-Span-106" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Main;">)'s. Moreover we show how this result finds application in extending the Cheeger-Colding intrinsic-Reifenberg theorem to the biLipschitz case. As a by-product of our arguments, we deduce analogous results also for the Jones' numbers <span id="MathJax-Element-20-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-107" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.545em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.088em, 1000.54em, 2.354em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-109" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">β's (i.e. the one-sided version of the numbers <span id="MathJax-Element-21-Frame" class="MathJax" style="display: inline; line-height: normal; font-size: 13.608px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif;"><span id="MathJax-Span-110" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.847em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; width: 0.666em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-112" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; font-family: MathJax_Math-italic;">α's).

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds - Francesco Nobili; Ivan Yuri Violo (2021)
<blockquote class="abstract mathjax" style="line-height: 1.55; font-size: 1.05em; margin-bottom: 21.6px; margin-block: 14.4px 21.6px; margin-inline: 40px; background-color: white; border-left: 0px; padding: 0px; color: #000000; font-family: 'Lucida Grande', Helvetica, Arial, sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial;">We prove that if <span id="MathJax-Element-1-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-1" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.269em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.028em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.148em, 1001.03em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-2" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-3" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">M is a closed <span id="MathJax-Element-2-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-4" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-5" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-6" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">n-dimensional Riemannian manifold, <span id="MathJax-Element-3-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-7" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 3.017em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.474em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.389em, 1002.41em, 2.534em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-8" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-9" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">n<span id="MathJax-Span-10" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">≥<span id="MathJax-Span-11" class="mn" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">3, with <span id="MathJax-Element-4-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-12" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 6.392em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 5.247em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.389em, 1005.19em, 2.534em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-13" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-14" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-15" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-16" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">R<span id="MathJax-Span-17" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">i<span id="MathJax-Span-18" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">c<span id="MathJax-Span-19" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">≥<span id="MathJax-Span-20" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.304em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">n<span id="MathJax-Span-21" class="mo" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.244em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">−<span id="MathJax-Span-22" class="mn" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px 0px 0px 0.244em; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">1 and for which the optimal constant in the critical Sobolev inequality equals the one of the <span id="MathJax-Element-5-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-23" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 0.787em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 0.606em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.389em, 1000.61em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-24" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-25" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">n-dimensional sphere <span id="MathJax-Element-6-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-26" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.329em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.088em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(0.124em, 1001.09em, 1.208em, -999.997em); top: -1.022em; left: 0em;"><span id="MathJax-Span-27" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-28" class="msubsup" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.088em; height: 0px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(3.077em, 1000.54em, 4.162em, -999.997em); top: -3.975em; left: 0em;"><span id="MathJax-Span-29" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-30" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-31" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_AMS;">S<span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; top: -4.397em; left: 0.545em;"><span id="MathJax-Span-32" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-size: 11.7374px; font-family: MathJax_Math-italic;">n, then <span id="MathJax-Element-7-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-33" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.269em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.028em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.148em, 1001.03em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-34" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-35" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">M is isometric to <span id="MathJax-Element-8-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-36" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.329em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.088em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(0.124em, 1001.09em, 1.208em, -999.997em); top: -1.022em; left: 0em;"><span id="MathJax-Span-37" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-38" class="msubsup" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.088em; height: 0px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(3.077em, 1000.54em, 4.162em, -999.997em); top: -3.975em; left: 0em;"><span id="MathJax-Span-39" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-40" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-41" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_AMS;">S<span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; top: -4.397em; left: 0.545em;"><span id="MathJax-Span-42" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-size: 11.7374px; font-family: MathJax_Math-italic;">n. An almost-rigidity result is also established, saying that if equality is almost achieved, then <span id="MathJax-Element-9-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-43" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.269em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.028em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.148em, 1001.03em, 2.173em, -999.997em); top: -1.986em; left: 0em;"><span id="MathJax-Span-44" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-45" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Math-italic;">M is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the <span id="MathJax-Element-10-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-46" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.715em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.233em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1002.17em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-47" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-48" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-49" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-50" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">R<span id="MathJax-Span-51" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-52" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds. An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compact <span id="MathJax-Element-11-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-53" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.871em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.51em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1001.45em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-54" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-55" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-56" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-57" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-58" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D space, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences of <span id="MathJax-Element-12-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-59" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.715em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.233em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1002.17em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-60" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-61" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-62" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-63" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">R<span id="MathJax-Span-64" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-65" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D spaces and on a Polya-Szego inequality of Euclidean-type in <span id="MathJax-Element-13-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-66" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.871em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 1.51em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1001.45em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-67" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-68" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-69" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-70" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-71" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D spaces. As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in the <span id="MathJax-Element-14-Frame" class="MathJax" style="display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 13.608px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;"><span id="MathJax-Span-72" class="math" style="transition: none 0s ease 0s; display: inline-block; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.715em;"><span style="transition: none 0s ease 0s; display: inline-block; position: relative; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; width: 2.233em; height: 0px; font-size: 16.6018px;"><span style="transition: none 0s ease 0s; display: inline; position: absolute; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; clip: rect(1.329em, 1002.17em, 2.414em, -999.997em); top: -2.227em; left: 0em;"><span id="MathJax-Span-73" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-74" class="texatom" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-75" class="mrow" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none;"><span id="MathJax-Span-76" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">R<span id="MathJax-Span-77" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">C<span id="MathJax-Span-78" class="mi" style="transition: none 0s ease 0s; display: inline; position: static; border: 0px; padding: 0px; margin: 0px; vertical-align: 0px; line-height: normal; text-decoration: none; font-family: MathJax_Main;">D-setting.

The sharp quantitative isocapacitary inequality - Guido De Philippis; Michele Marini; Ekaterina Mukoseeva (Revista Matematica Iberoamericana, 2021)
We prove a sharp quantitative form of the classical isocapacitary inequality. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. This provides a positive answer to a conjecture of Hall, Hayman, and Weitsman (J. Analyse Math.'91). © 2021 Real Sociedad Matemática Española
Keywords: Fraenkel asymmetry; isocapacitary inequality; Stability estimates

The sharp quantitative isocapacitary inequality (the case of p-capacity) - Ekaterina Mukoseeva (Advances in Calculus of Variations, 2021)
We prove a sharp quantitative form of isocapacitary inequality in the case of a general p. This work is a generalization of the author's paper with Guido De Philippis and Michele Marini, where we treated the case of 2-capacity. © 2021 Walter de Gruyter GmbH, Berlin/Boston 2021.
Keywords: isocapacitary inequality; Stability estimates

Smoothed-adaptive perturbed inverse iteration for elliptic eigenvalue problems - Stefano Giani; Luka Grubisic; Luca Heltai; Ornela Mulita (Computational Methods in Applied Mathematics, 2021)

A sufficient criterion to determine planar self-Cheeger sets - Saracco, G. (J. Convex Anal., 2021)

A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems - Nicola Demo; Marco Tezzele; Gianluigi Rozza (SIAM Journal on Scientific Computing, 2021)
In this work, we present an extension of the genetic algorithm (GA) which exploits the active subspaces (AS) property to evolve the individuals on a lower dimensional space. In many cases, GA requires in fact more function evaluations than others optimization method to converge to the optimum. Thus, complex and high-dimensional functions may result intractable with the standard algorithm. To address this issue, we propose to linearly map the input parameter space of the original function onto its AS before the evolution, performing the mutation and mate processes in a lower dimensional space. In this contribution, we describe the novel method called ASGA, presenting differences and similarities with the standard GA method. We test the proposed method over n-dimensional benchmark functions – Rosenbrock, Ackley, Bohachevsky, Rastrigin, Schaffer N. 7, and Zakharov – and finally we apply it to an aeronautical shape optimization problem.

Traveling quasi-periodic water waves with constant vorticity - Massimiliano Berti; Luca Franzoi; Alberto Maspero (Arch. Ration. Mech. Anal., 2021)

Trotter product formulae for $*$-automorphisms of quantum lattice systems - Sven Bachmann; Markus Lange (2021)

Uniqueness and continuous dependence for a viscoelastic problem with memory in domains with time dependent cracks - Federico Cianci; Gianni Dal Maso (2021)

A vanishing-inertia analysis for finite-dimensional rate-independent systems with nonautonomous dissipation and an application to soft crawlers - Paolo Gidoni; Filippo Riva (Calculus of Variations and Partial Differential Equations, 2021)
We study the approximation of finite-dimensional rate-independent quasistatic systems, via a vanishing-inertia asymptotic analysis of dynamic evolutions. We prove the uniform convergence of dynamic solutions to a rate-independent one, employing the variational concept of energetic solution. Motivated by applications in soft locomotion, we allow time-dependence of the dissipation potential, and translation invariance of the potential energy.

A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences - G. Carere; Maria Strazzullo; Francesco Ballarin; Gianluigi Rozza; R. Stevenson (Computers and Mathematics with Applications, 2021)

Well-Ordered and Non-Well-Ordered Lower and Upper Solutions for Periodic Planar Systems - Alessandro Fonda; Giuliano Klun; Andrea Sfecci (Advanced Nonlinear Studies, 2021)

Advances in reduced order methods for parametric industrial problems in computational fluid dynamics - Gianluigi Rozza; M.H. Malik; Nicola Demo; Marco Tezzele; Michele Girfoglio; Giovanni Stabile; Andrea Mola (Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018, 2020)
Reduced order modeling has gained considerable attention in recent decades owing to the advantages offered in reduced computational times and multiple solutions for parametric problems. The focus of this manuscript is the application of model order reduction techniques in various engineering and scientific applications including but not limited to mechanical, naval and aeronautical engineering. The focus here is kept limited to computational fluid mechanics and related applications. The advances in the reduced order modeling with proper orthogonal decomposition and reduced basis method are presented as well as a brief discussion of dynamic mode decomposition and also some present advances in the parameter space reduction. Here, an overview of the challenges faced and possible solutions are presented with examples from various problems.

On the Approximation of Quasistatic Evolutions for the Debonding of a Thin Film via Vanishing Inertia and Viscosity - Filippo Riva (Journal of Nonlinear Science, 2020)
In this paper, we contribute to studying the issue of quasistatic limit in the context of Griffith’s theory by investigating a one-dimensional debonding model. It describes the evolution of a thin film partially glued to a rigid substrate and subjected to a vertical loading. Taking viscosity into account and under suitable assumptions on the toughness of the glue, we prove that, in contrast to what happens in the undamped case, dynamic solutions converge to the quasistatic one when inertia and viscosity go to zero, except for a possible discontinuity at the initial time. We then characterise the size of the jump by means of an asymptotic analysis of the debonding front.