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1. Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases

   https://doi.org/10.1103/PhysRevB.110.195149  

   Pace, Salvatore D; Zhu, Chenchang; Beaudry, Agnès; Wen, Xiao-Gang (November 2024, Physical Review B)

   We study the nonlinear $$\sigma$$-model in $${(d+1)}$$-dimensional spacetime with connected target space $$K$$ and show that, at energy scales below singular field comfigurations (such as vortices), it has an emergent non-invertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $$d$$-representations of a discrete $$d$$-group $$\mathbb{G}^{(d)}$$ (i.e. the emergent symmetry is the dual of the invertible $$d$$-group $$\mathbb{G}^{(d)}$$ symmetry). The $$d$$-group $$\mathbb{G}^{(d)}$$ is determined such that its classifying space $$B\mathbb{G}^{(d)}$$ is given by the $$d$$-th Postnikov stage of $$K$$. In $(2+1)$D and for finite $$\mathbb{G}^{(2)}$$, this symmetry is always holo-equivalent to an invertible $${0}$$-form---ordinary---symmetry with potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $$\sigma$$-model spontaneously breaks this symmetry, and when $$\mathbb{G}^{(d)}$$ is finite, it is described by the deconfined phase of $$\mathbb{G}^{(d)}$$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free $S^2$ nonlinear $$\sigma$$-model in $${(3+1)}$$D and show that it has an emergent non-invertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this non-perturbative result is different from the results obtained using the $S^N$ and $$\mathbb{C}P^{N-1}$$ nonlinear $$\si$$-models in the large-$$N$$ limit. 

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2. Applications of the Clifford torus to material textures

   https://doi.org/10.1107/S160057672400219X  

   De_Graef, Marc (June 2024, Journal of Applied Crystallography)

   This paper introduces a new 2D representation of the orientation distribution function for an arbitrary material texture. The approach is based on the isometric square torus mapping of the Clifford torus, which allows for points on the unit quaternion hypersphere (each corresponding to a 3D orientation) to be represented in a periodic 2D square map. The combination of three such orthogonal mappings into a single RGB (red–green–blue) image provides a compact periodic representation of any set of orientations. Square torus representations of five different orientation sampling methods are compared and analyzed in terms of the Rieszsenergies that quantify the uniformity of the samplings. The effect of crystallographic symmetry on the square torus map is analyzed in terms of the Rodrigues fundamental zones for the rotational symmetry groups. The paper concludes with example representations of important texture components in cubic and hexagonal materials. The new RGB representation provides a convenient and compact way of generating training data for the automated analysis of material textures by means of neural networks. 

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3. Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures

   https://doi.org/DOI 10.1515/crelle-2020-0005  

   Bridgeman, M; Wu, Yunhui (January 2021, Journal für die reine und angewandte Mathematik)

   De Gruyter (Ed.)

   In this article we show that for every finite area hyperbolic surface X of type (g; n) and any harmonic Beltrami differential 􏰚 on X , then the magnitude of 􏰚 at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of 􏰚 over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity. 

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4. 2-chains and square roots of Thompson’s group

   https://doi.org/10.1017/etds.2019.14  

   KOBERDA, THOMAS; LODHA, YASH (September 2020, Ergodic Theory and Dynamical Systems)

   We study 2-generated subgroups $$\langle f,g\rangle <\operatorname{Homeo}^{+}(I)$$ such that $$\langle f^{2},g^{2}\rangle$$ is isomorphic to Thompson’s group $$F$$ , and such that the supports of $$f$$ and $$g$$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $$C^{2}$$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $$F$$ as part of a general phenomenon among subgroups of $$\operatorname{Homeo}^{+}(I)$$ . 

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5. Hybrid Perturbations in Stacked Patch–Ring Circularly Polarized Microstrip Antennas for CubeSat Applications

   https://doi.org/10.1109/MAES.2022.3141436.  

   M. M. Hossain, S. I. (February 2022, IEEE aerospace and electronic systems magazine)

   A hybrid perturbation scheme is used in this article to achieve wide axial ratio (AR) bandwidth and beamwidth from circularly polarized (CP) microstrip patch–ring antennas using a single probe feed. Perturbations in the diagonal corners of a square ring and a square patch arranged in a stacked configuration are introduced to achieve the circular polarization. First, an enhanced AR bandwidth is obtained when a combination of a square ring and a square patch with negative perturbations is used as parasitic and driven elements, respectively. Next, circular polarization with wider AR bandwidth, wider beamwidth, and lower cross-polarization is obtained when a combination of a driven square patch with positive perturbation and a parasitic square ring with negative perturbations, termed as hybrid perturbations, is used. This antenna has a footprint suitable for small satellite applications (e.g., CubeSats) and its operating frequencies cover the allocated S-band downlink frequencies of NASA Deep Space Network and NASA Near Earth Network. 

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