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1. Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems

   https://doi.org/10.1002/cpa.22125  

   Colbrook, Matthew J. ; Townsend, Alex ( July 2023 , Communications on Pure and Applied Mathematics)

   Koopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite‐dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure‐preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure‐preserving), which can achieve high‐order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high‐dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046‐dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >10⁵that has a 295,122‐dimensional state space.

    

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2. WR 63: a multiple system (O+O) + WR?

   https://doi.org/10.1093/mnras/stac1762  

   Chené, André-Nicolas ; Mahy, Laurent ; Gosset, Eric ; St-Louis, Nicole ; Dsilva, Karan ; Manick, Rajeev ( September 2022 , Monthly Notices of the Royal Astronomical Society)

   The spectrum of the Wolf–Rayet (WR) star WR 63 contains spectral lines of two different O stars that show regular radial velocity (RV) variations with amplitudes of ∼160 and ∼225 km s−1 on a ∼4.03 d period. The light curve shows two narrow eclipses that are 0.2 mag deep on the same period as the RV changes. On the other hand, our data show no significant RV variations for the WR spectral lines. Those findings are compatible with WR 63 being a triple system composed of two non-interacting late-O stars orbiting a WR star on a period longer than 1000 d. The amplitude of the WR spectral line-profile variability reaches 7–8 per cent of the line intensity and seems related to a 0.04 mag periodic photometric variation. Large wind density structures are a possible origin for this variability, but our data are not sufficient to verify this. Our analysis shows that, should the three stars be bound, they would be coeval with an age of about 5.9 ± 1.4 Myr. The distance to the O stars is estimated to be $3.4\, \pm \, 0.5$ kpc. Their dynamical masses are 14.3 ± 0.1 and 10.3 ± 0.1 M⊙. Using rotating single-star evolutionary tracks, we estimate their initial masses to be 18 ± 2 and 16 ± 2 M⊙ for the primary and the secondary, respectively. Regular spectral monitoring is required in the future to detect RV variations of the WR star that would prove that it is gravitationally bound to the close O+OB system and to determine its mass.

    

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3. Spatial mixing and nonlocal Markov chains

   https://doi.org/10.1002/rsa.20844  

   Blanca, Antonio ; Caputo, Pietro ; Sinclair, Alistair ; Vigoda, Eric ( March 2019 , Random Structures & Algorithms)

   We consider spin systems with nearest‐neighbor interactions on ann‐vertexd‐dimensional cube of the integer lattice graph. We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibrium ofnonlocalMarkov chains. We prove that when SSM holds, the relaxation time (i.e., the inverse spectral gap) of generalblock dynamicsisO(r), whereris the number of blocks. As a second application of our technology, it is established that SSM implies anO(1) bound for the relaxation time of the Swendsen‐Wang dynamics for the ferromagnetic Ising and Potts models. We also prove that formonotonespin systems SSM implies that the mixing time of systematic scan dynamics is. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.

    

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4. Inducing Super-Approximation

   https://doi.org/10.1093/imrn/rnz115  

   Salehi Golsefidy, Alireza ; Zhang, Xin ( July 2019 , International Mathematics Research Notices)

   Let $\Gamma _2\subseteq \Gamma _1$ be finitely generated subgroups of ${\operatorname{GL}}_{n_0}({\mathbb{Z}}[1/q_0])$ where $q_0$ is a positive integer. For $i=1$ or $2$, let ${\mathbb{G}}_i$ be the Zariski-closure of $\Gamma _i$ in $({\operatorname{GL}}_{n_0})_{{\mathbb{Q}}}$, ${\mathbb{G}}_i^{\circ }$ be the Zariski-connected component of ${\mathbb{G}}_i$, and let $G_i$ be the closure of $\Gamma _i$ in $\prod _{p\nmid q_0}{\operatorname{GL}}_{n_0}({\mathbb{Z}}_p)$. In this article we prove that if ${\mathbb{G}}_1^{\circ }$ is the smallest closed normal subgroup of ${\mathbb{G}}_1^{\circ }$ that contains ${\mathbb{G}}_2^{\circ }$ and $\Gamma _2\curvearrowright G_2$ has spectral gap, then $\Gamma _1\curvearrowright G_1$ has spectral gap.

    

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5. Swendsen‐Wang dynamics for general graphs in the tree uniqueness region

   https://doi.org/10.1002/rsa.20858  

   Blanca, Antonio ; Chen, Zongchen ; Vigoda, Eric ( April 2019 , Random Structures & Algorithms)

   The Swendsen‐Wang (SW) dynamics is a popular Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graphG = (V,E). The dynamics is conjectured to converge to equilibrium inO(|V|^1/4) steps at any (inverse) temperatureβ, yet there are few results providingo(|V|) upper bounds. We prove fast convergence of the SW dynamics on general graphs in the tree uniqueness region. In particular, whenβ < β_c(d) whereβ_c(d) denotes the uniqueness/nonuniqueness threshold on infinited‐regular trees, we prove that therelaxation time(i.e., the inverse spectral gap) of the SW dynamics is Θ(1) on any graph of maximum degreed ≥ 3. Our proof utilizes amonotoneversion of the SW dynamics which only updates isolated vertices. We establish that this variant of the SW dynamics has mixing timeand relaxation time Θ(1) on any graph of maximum degreedfor allβ < β_c(d). Our proof technology can be applied to general monotone Markov chains, including for example theheat‐bath block dynamics, for which we obtain new tight mixing time bounds.

    

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