Measure transfer and SS-adic developments for subshifts

Based on previous work of the authors, to any $S$-adic development of a subshift $X$ a "directive sequence" of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given $S$-adic development. The issuing rich picture enables one to deduce results about $X$ with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d \geq 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$

SS-adic expansions related to continued fractions (Natural extension of arithmetic algorithms and S-adic system)

"Natural extension of arithmetic algorithms and S-adic system". July 20～24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider S-adic expansions associated with continued fraction algorithms, where an S-adic expansion corresponds to an infinite composition of substitutions. Recall that a substitution is a morphism of the free monoid. We focus in particular on the substitutions associated with regular continued fractions (Sturmian substitutions), and with Arnoux-Rauzy, Brun, and Jacobi{Perron (multidimensional) continued fraction algorithms. We also discuss the spectral properties of the associated symbolic dynamical systems under a Pisot type assumption

Control-Oriented Reduced Order Modeling of Dipteran Flapping Flight

Flying insects achieve flight stabilization and control in a manner that requires only small, specialized neural structures to perform the essential components of sensing and feedback, achieving unparalleled levels of robust aerobatic flight on limited computational resources. An engineering mechanism to replicate these control strategies could provide a dramatic increase in the mobility of small scale aerial robotics, but a formal investigation has not yet yielded tools that both quantitatively and intuitively explain flapping wing flight as an "input-output" relationship. This work uses experimental and simulated measurements of insect flight to create reduced order flight dynamics models. The framework presented here creates models that are relevant for the study of control properties. The work begins with automated measurement of insect wing motions in free flight, which are then used to calculate flight forces via an empirically-derived aerodynamics model. When paired with rigid body dynamics and experimentally measured state feedback, both the bare airframe and closed loop systems may be analyzed using frequency domain system identification. Flight dynamics models describing maneuvering about hover and cruise conditions are presented for example fruit flies (Drosophila melanogaster) and blowflies (Calliphorids). The results show that biologically measured feedback paths are appropriate for flight stabilization and sexual dimorphism is only a minor factor in flight dynamics. A method of ranking kinematic control inputs to maximize maneuverability is also presented, showing that the volume of reachable configurations in state space can be dramatically increased due to appropriate choice of kinematic inputs

Quantum meets optimization and machine learning

With the advent of the quantum era, what role the quantum computer will play in optimization and machine learning becomes a natural and salient question. The development of novel quantum computing techniques is essential to showcase the quantum advantage in these fields. At the same time, new findings in classical optimization and machine learning algorithms also have the potential to stimulate quantum computing research. In the dissertation, we explore the fascinating connections between quantum computing, optimization, and machine learning, paving the way for transformative advances in all three fields. First, on the quantum side, we present efficient quantum algorithms for fundamental problems in sampling, optimization, and quantum physics. Our results highlight the practical advantages of quantum computing in these fields. In addition, we introduce new approaches to quantum complexity theory for characterizing the quantum hardness of optimization and machine learning problems. Second, on the optimization side, we improve the efficiency of the state-of-the-art classical algorithms for solving semi-definite programming (SDP), Fourier sensing, dynamic distance estimation, etc. Our classical results are closely intertwined with quantum computing. Some of them serve as stepping stones to new quantum algorithms, while others are motivated by quantum applications or inspired by quantum techniques. Third, on the machine learning side, we develop fast classical and quantum algorithms for training over-parameterized neural networks with provable guarantees of convergence and generalization. Furthermore, we contribute to the security aspect of machine learning by theoretically investigating some potential approaches to (classically) protect private data in collaborative machine learning and to (quantumly) protect the copyright of machine learning models. Fourth, we investigate the concentration and discrepancy properties of hyperbolic polynomials and higher-order random walks, which could potentially be applied to quantum computing, optimization, and machine learning.Computer Science

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Aeroelastic instabilities of an airfoil in transitional flow regimes

Cette thèse porte sur l'étude de l'instabilité aéroélastique provenant de l'interaction fluide–structure, dans le cas d'une aile rigide montée sur un ressort en torsion. L'étude est centrée sur le phénomène de flottement dû à un décollement laminaire, et plus précisément sur les oscillations (en torsion) auto-entretenues détectées expérimentalement pour un profil NACA0012 à faible incidence, dans la gamme de nombre de Reynolds dits transitionnels (Re in [10^4 – 10^5]), caractérisé par un décollement de la couche limite initialement laminaire, suivi d'une transition et d'un rattachement. L'objectif principal de la thèse est d'expliquer ce phénomène en se basant sur des concepts d'instabilité. Pour ce faire, différentes approches numériques ont été conduites: des simulations numériques bidimensionnelles et des simulations numériques tridimensionnelles (DNS). Ces approches ont en suite servi de base à des analyses de stabilité linéaire (LSA) autour d'un champ moyen ou d'un champ périodique (analyse de Floquet). Le deuxième objectif vise à explorer les différents scénarios non linéaires qui apparaissent dans cette gamme de Reynolds. La première partie de la thèse est consacrée à la caractérisation de l'écoulement autour de l'aile pour des angles d'incidence fixes. Des simulations temporelles bidimensionnelles montrent l'apparition d'oscillations à haute fréquence associées au détachement tourbillonnaire en aval du profil à partir de Re = 8000. Une analyse de stabilité hydrodynamique (Floquet) est réalisée pour caractériser la transition vers un écoulement tridimensionnel. Des simulations tridimensionnelles sont ensuite réalisées pour Re = 50000 afin de caractériser l'écoulement instantané et moyenné. L'analyse des forces moyennes exercées sur l'aile à incidence fixe permettent de détecter une rigidité aérodynamique négative (rapport moment-incidence) pour la gamme |alpha| 0°), où des solutions chaotiques et quasi-périodiques coexistent pour les mêmes paramètres structuraux, et évolue vers un scénario où les oscillations se font autour de alpha = 0°. La dernière partie de la thèse essaie d'expliquer la déstabilisation des positions d'équilibre non nulles conduisant à un comportement quasi-périodique à l'aide d'analyses LSA autour des champs moyens et périodiques à incidence fixe. Même si ces analyses sont incapables de prédire un mode propre instable, nous concluons que l'inclusion du terme des contraintes de Reynolds dans la dynamique de perturbation de l'écoulement moyen a un effet important.This thesis investigates aeroelastic instability phenomena arising in coupled fluid–structure interactions, considering the flow around a rigid airfoil mounted on a torsion spring. The focus is on the laminar separation flutter phenomenon, namely a self-sustained pitch oscillation detected experimentally on a NACA0012 airfoil in the transitional Reynolds number regime (Re in [10^4 – 10^5]) at low incidences, characterised by a detachment of an initially laminar boundary layer followed by its transition and subsequent reattachment. The main objective of the thesis is to explain this phenomenon in terms of instability concepts. For this, a combination of numerical approaches involving two- and three-dimensional Navier–Stokes simulations—the latter refereed to as Direct Numerical Simulations (DNS)—along with linear stability analyses (LSA) around a mean flow or a periodic flow (Floquet analysis) is employed. A second objective is to numerically explore the different nonlinear scenarios appearing in the low-to-moderate Reynolds number regime. The first part of the thesis is devoted to the characterisation of the fluid flow around the airfoil considering fixed incidences. Two-dimensional time-marching simulations are first employed, showing the emergence of high-frequency vortex shedding oscillations for Re = 8000. A hydrodynamic stability analysis (Floquet) is then employed to characterise the transition to a three-dimensional flow and DNS is eventually used to characterise both instantaneous and averaged flow quantities at Re = 50000. An analysis of the mean forces exerted on a fixed-incidence wing allows to detect a negative aerodynamic stiffness (torque-to-incidence ratio) in the range |alpha| < 2°, indicating a static instability. The second part of the thesis is devoted to the characterisation of the primary instability of the coupled fluid–structure system using LSA around the mean and periodic flow fields. Considering the symmetrical equilibrium position alpha = 0°, the analysis shows the presence of an unstable static mode, in accordance with the existence of a negative aerodynamic stiffness. In the third part of the thesis, the emergence of self-sustained flutter oscillations is investigated via two-dimensional aeroelastic simulations. The investigation shows that the system first transitions towards a pitch oscillation around the nonsymmetrical equilibrium position (alpha > 0°), with coexistence of chaotic and quasi-periodic solutions for the same structural parameters, and subsequently transitions towards a pitch oscillation around the symmetrical position (alpha = 0°) as the Reynolds number increases. In the last part of the thesis, an attempt is made to explain the destabilisation of the nonsymmetrical equilibrium positions leading to a quasi-periodic behaviour using LSA around the mean and periodic flow fields at fixed incidences. Even if these analyses are unable to predict an unstable eigenmode, we conclude that the inclusion of the Reynolds stress term in the mean flow perturbation dynamics has an important effect

Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems

Manibo CNC. Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems. Bielefeld: Universität Bielefeld; 2019.In this work, we consider primitive inflation rules as generators of aperiodic tilings, and subsequently, of aperiodic point sets (which are toy models for quasicrystals) deemed adequate for diffraction analysis. We harvest the combinatorial-geometric properties of these systems to obtain renormalisation relations for the pair correlation functions, which carry over to measures that generate the diffraction measure. This yields a measure-valued renormalisation satisfied by each of the components of the diffraction. Using tools from the theory of Lyapunov exponents, we provide a sufficient criterion to rule out the presence of absolutely continuous components in the diffraction and a necessary condition to have a non-trivial absolutely continuous part. Moreover, we provide a computable bound which one can use to use invoke this criterion. We show that this holds for large classes of systems, and, as a sanity check, show that the necessary criterion for existence is satisfied by systems which are a priori known to have absolutely continuous diffraction. Furthermore, we present the recovery of known singularity results and point out connections to number-theoretic quantities which naturally arise from these objects, such as logarithmic Mahler measures