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1. Exact Mobility Edges for Almost-Periodic CMV Matrices via Gauge Symmetries

   https://doi.org/10.1093/imrn/rnad293  

   Cedzich, Christopher; Fillman, Jake; Li, Long; Ong, Darren C; Zhou, Qi (December 2023, International Mathematics Research Notices)

   Abstract We investigate the symmetries of the so-called generalized extended Cantero–Moral–Velázquez (CMV) matrices. It is well-documented that problems involving reflection symmetries of standard extended CMV matrices can be subtle. We show how to deal with this in an elegant fashion by passing to the class of generalized extended CMV matrices via explicit diagonal unitaries in the spirit of Cantero–Grünbaum–Moral–Velázquez. As an application of these ideas, we construct an explicit family of almost-periodic CMV matrices, which we call the mosaic unitary almost-Mathieu operator, and prove the occurrence of exact mobility edges. That is, we show the existence of energies that separate spectral regions with absolutely continuous and pure point spectrum and exactly calculate them. 

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2. Absolutely continuous spectrum for CMV matrices with small quasi-periodic Verblunsky coefficients

   https://doi.org/10.1090/tran/8696  

   Li, Long; Damanik, David; Zhou, Qi (September 2022, Transactions of the American Mathematical Society)

   We consider standard and extended CMV matrices with small quasi-periodic Verblunsky coefficients and show that on their essential spectrum, all spectral measures are purely absolutely continuous. This answers a question of Barry Simon from 2005. 

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3. Breaking reflection symmetry: evolving long dynamical cycles in Boolean systems

   https://doi.org/10.1088/1367-2630/ad1bdd  

   Ouellet, Mathieu; Kim, Jason Z.; Guillaume, Harmange; Shaffer, Sydney M.; Bassett, Lee C.; Bassett, Dani S. (February 2024, New Journal of Physics)

   Abstract In interacting dynamical systems, specific local interaction rules for system components give rise to diverse and complex global dynamics. Long dynamical cycles are a key feature of many natural interacting systems, especially in biology. Examples of dynamical cycles range from circadian rhythms regulating sleep to cell cycles regulating reproductive behavior. Despite the crucial role of cycles in nature, the properties of network structure that give rise to cycles still need to be better understood. Here, we use a Boolean interaction network model to study the relationships between network structure and cyclic dynamics. We identify particular structural motifs that support cycles, and other motifs that suppress them. More generally, we show that the presence ofdynamical reflection symmetryin the interaction network enhances cyclic behavior. In simulating an artificial evolutionary process, we find that motifs that break reflection symmetry are discarded. We further show that dynamical reflection symmetries are over-represented in Boolean models of natural biological systems. Altogether, our results demonstrate a link between symmetry and functionality for interacting dynamical systems, and they provide evidence for symmetry’s causal role in evolving dynamical functionality. 

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4. An intrinsic approach to relative braid group symmetries on ı$$\imath$$quantum groups

   https://doi.org/10.1112/plms.12562  

   Wang, Weiqiang; Zhang, Weinan (November 2023, Proceedings of the London Mathematical Society)

   Abstract We initiate a general approach to the relative braid group symmetries on (universal) quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi ‐matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on quantum groups are obtained. We establish a number of fundamental properties for these symmetries on quantum groups, strikingly parallel to their well‐known quantum group counterparts. We apply these symmetries to fully establish rank 1 factorizations of quasi ‐matrices, and this factorization property, in turn, helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb–Pellegrini and Dobson–Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time. 

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5. Kaluza-Klein fermion mass matrices from exceptional field theory and $$ \mathcal{N} $$ = 1 spectra

   https://doi.org/10.1007/JHEP03(2021)138  

   Cesàro, Mattia; Varela, Oscar (March 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract Using Exceptional Field Theory, we determine the infinite-dimensional mass matrices for the gravitino and spin-1 / 2 Kaluza-Klein perturbations above a class of anti-de Sitter solutions of M-theory and massive type IIA string theory with topologically-spherical internal spaces. We then use these mass matrices to compute the spectrum of Kaluza-Klein fermions about some solutions in this class with internal symmetry groups containing SU(3). Combining these results with previously known bosonic sectors of the spectra, we give the complete spectrum about some $$ \mathcal{N} $$ N = 1 and some non-supersymmetric solutions in this class. The complete spectra are shown to enjoy certain generic features. 

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