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1. Decomposing multitwists

   https://doi.org/10.1007/s11854-023-0301-4  

   Fletcher, Alastair N; Vellis, Vyron (April 2024, Journal d'Analyse Mathématique)

   The Decomposition Problem in the class $$LIP(\S^2)$$ is to decompose any bi-Lipschitz map $$f:\S^2 \to \S^2$$ as a composition of finitely many maps of arbitrarily small isometric distortion. In this paper, we construct a decomposition for certain bi-Lipschitz maps which spiral around every point of a Cantor set $$X$$ of Assouad dimension strictly smaller than one. These maps are constructed by considering a collection of Dehn twists on the Riemann surface $$\S^2 \setminus X$$. The decomposition is then obtained via a bi-Lipschitz path which simultaneously unwinds these Dehn twists. As part of our construction, we also show that $$X \subset \S^2$$ is uniformly disconnected if and only if the Riemann surface $$\S^2 \setminus X$$ has a pants decomposition whose cuffs have hyperbolic length uniformly bounded above, which may be of independent interest. 

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2. Weyl remainders: an application of geodesic beams

   https://doi.org/10.1007/s00222-023-01178-5  

   Canzani, Yaiza; Galkowski, Jeffrey (June 2023, Inventiones mathematicae)

   Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M ,  g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams. 

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3. Yang-Mills as a Liouville theory

   https://doi.org/10.1016/j.physletb.2023.138229  

   Stieberger, Stephan; Taylor, Tomasz R; Zhu, Bin (November 2023, Physics Letters B)

   We propose a description of the gluon scattering amplitudes as the inverse Mellin transforms of the conformal correlators of light operators in two-dimensional Liouville theory tensored with WZW-like chiral currents on the celestial sphere. The dimensions of operators are Mellin dual to gluon light cone energies while their positions are determined by the gluon momentum directions. Tree-level approximation in Yang-Mills theory corresponds to the semiclassical limit of Liouville theory. By comparing subleading corrections, we find b^2=(8π^2)^{−1}b_0g^2(M), where b is the Liouville coupling constant, g(M) is the Yang Mills coupling at the renormalization scale M and b_0 is the one-loop coefficient of the Yang-Mills beta function. 

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4. Joint Low-Rank Factorizations with Shared and Unshared Components: Identifiability and Algorithms

   https://doi.org/10.23919/EUSIPCO.2019.8903050  

   Sorensen, Mikael; Sidiropoulos, Nikolaos D. (September 2019, 27th European Signal Processing Conference (EUSIPCO), A Coruna, Spain, 2019)

   We study the joint low-rank factorization of the matrices X=[A B]G and Y=[A C]H, in which the columns of the shared factor matrix A correspond to vectorized rank-one matrices, the unshared factors B and C have full column rank, and the matrices G and H have full row rank. The objective is to find the shared factor A, given only X and Y. We first explain that if the matrix [A B C] has full column rank, then a basis for the column space of the shared factor matrix A can be obtained from the null space of the matrix [X Y]. This in turn implies that the problem of finding the shared factor matrix A boils down to a basic Canonical Polyadic Decomposition (CPD) problem that in many cases can directly be solved by means of an eigenvalue decomposition. Next, we explain that by taking the rank-one constraint of the columns of the shared factor matrix A into account when computing the null space of the matrix [X Y], more relaxed identifiability conditions can be obtained that do not require that [A B C] has full column rank. The benefit of the unconstrained null space approach is that it leads to simple algorithms while the benefit of the rank-one constrained null space approach is that it leads to relaxed identifiability conditions. Finally, a joint unbalanced orthogonal Procrustes and CPD fitting approach for computing the shared factor matrix A from noisy observation matrices X and Y will briefly be discussed. 

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5. Microphysically modified magnetosonic modes in collisionless, high-β plasmas

   https://doi.org/10.1017/S0022377823000429  

   Majeski, S.; Kunz, M.W.; Squire, J. (June 2023, Journal of Plasma Physics)

   With the support of hybrid-kinetic simulations and analytic theory, we describe the nonlinear behaviour of long-wavelength non-propagating (NP) modes and fast magnetosonic waves in high- $$\beta$$ collisionless plasmas, with particular attention to their excitation of and reaction to kinetic micro-instabilities. The perpendicularly pressure balanced polarization of NP modes produces an excess of perpendicular pressure over parallel pressure in regions where the plasma $$\beta$$ is increased. For mode amplitudes $$|\delta B/B_0| \gtrsim 0.3$$ , this excess excites the mirror instability. Particle scattering off these micro-scale mirrors frustrates the nonlinear saturation of transit-time damping, ensuring that large-amplitude NP modes continue their decay to small amplitudes. At asymptotically large wavelengths, we predict that the mirror-induced scattering will be large enough to interrupt transit-time damping entirely, isotropizing the pressure perturbations and morphing the collisionless NP mode into the magnetohydrodynamic (MHD) entropy mode. In fast waves, a fluctuating pressure anisotropy drives both mirror and firehose instabilities when the wave amplitude satisfies $$|\delta B/B_0| \gtrsim 2\beta ^{-1}$$ . The induced particle scattering leads to delayed shock formation and MHD-like wave dynamics. Taken alongside prior work on self-interrupting Alfvén waves and self-sustaining ion-acoustic waves, our results establish a foundation for new theories of electromagnetic turbulence in low-collisionality, high- $$\beta$$ plasmas such as the intracluster medium, radiatively inefficient accretion flows and the near-Earth solar wind. 

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