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1. Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

   https://doi.org/10.3934/cpaa.2021174  

   Scott, James M.; Mengesha, Tadele (January 2021, Communications on Pure & Applied Analysis)

   We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript. 

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2. Notes on solutions of KZ equations modulo $p^s$ and $$p$$-adic limit $$s\to\infty$$

   Alexander Varchenko (January 2022, Contemporary mathematics)

   We consider the KZ equations over C in the case, when the hypergeometric solutions are hyperelliptic integrals of genus g. Then the space of solutions is a 2g-dimensional complex vector space. We also consider the same equations modulo ps, where p is an odd prime and s is a positive integer, and over the field Q_p of p-adic numbers. We construct polynomial solutions of the KZ equations modulo ps and study the space Mps of all constructed solutions. We show that the p-adic limit of Mps as s→∞ gives us a g-dimensional vector space of solutions of the KZ equations over Qp. The solutions over Qp are power series at a certain asymptotic zone of the KZ equations. In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the case g=1 of elliptic integrals. The p-adic limit of Mps as s→∞ gives us a one-dimensional space of solutions over Qp at every asymptotic zone. We apply Dwork's theory and show that our germs of solutions over Qp defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over C does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Qp. We describe the Frobenius transformations of solutions of the KZ equations for g=1 and then recover the unit roots of the zeta functions of the elliptic curves defined by the equations y2=βx(x−1)(x−α) over the finite field Fp. Here α,β∈F×p,α≠1 

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3. Data-driven lemma synthesis for interactive proofs

   https://doi.org/10.1145/3563306  

   Sivaraman, Aishwarya; Sanchez-Stern, Alex; Chen, Bretton; Lerner, Sorin; Millstein, Todd (October 2022, Proceedings of the ACM on Programming Languages)

   Interactive proofs of theorems often require auxiliary helper lemmas to prove the desired theorem. Existing approaches for automatically synthesizing helper lemmas fall into two broad categories. Some approaches are goal-directed, producing lemmas specifically to help a user make progress from a given proof state, but they have limited expressiveness in terms of the lemmas that can be produced. Other approaches are highly expressive, able to generate arbitrary lemmas from a given grammar, but they are completely undirected and hence not amenable to interactive usage. In this paper, we develop an approach to lemma synthesis that is both goal-directed and expressive. The key novelty is a technique for reducing lemma synthesis to a data-driven program synthesis problem, whereby examples for synthesis are generated from the current proof state. We also describe a technique to systematically introduce new variables for lemma synthesis, as well as techniques for filtering and ranking candidate lemmas for presentation to the user. We implement these ideas in a tool called lfind, which can be run as a Coq tactic. In an evaluation on four benchmark suites, lfind produces useful lemmas in 68% of the cases where a human prover used a lemma to make progress. In these cases lfind synthesizes a lemma that either enables a fully automated proof of the original goal or that matches the human-provided lemma. 

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4. Understanding the radio relic emission in the galaxy cluster MACS J0717.5+3745: Spectral analysis

   https://doi.org/10.1051/0004-6361/202039428  

   Rajpurohit, K.; Wittor, D.; van Weeren, R. J.; Vazza, F.; Hoeft, M.; Rudnick, L.; Locatelli, N.; Eilek, J.; Forman, W. R.; Bonafede, A.; et al (February 2021, Astronomy & Astrophysics)

   null (Ed.)

   Radio relics are diffuse, extended synchrotron sources that originate from shock fronts generated during cluster mergers. The massive merging galaxy cluster MACS J0717.5+3745 hosts one of the more complex relics known to date. We present upgraded Giant Metrewave Radio Telescope band 3 (300−500 MHz) and band 4 (550−850 MHz) observations. These new observations, combined with published VLA and the new LOFAR HBA data, allow us to carry out a detailed, high spatial resolution spectral analysis of the relic over a broad range of frequencies. The integrated spectrum of the relic closely follows a power law between 144 MHz and 5.5 GHz with a mean spectral slope α  = −1.16 ± 0.03. Despite the complex morphology of this relic, its subregions and the other isolated filaments also follow power-law behaviors, and show similar spectral slopes. Assuming diffusive shock acceleration, we estimated a dominant Mach number of ∼3.7 for the shocks that make up the relic. A comparison with recent numerical simulations suggests that in the case of radio relics, the slopes of the integrated radio spectra are determined by the Mach number of the accelerating shock, with α nearly constant, namely between −1.13 and −1.17, for Mach numbers 3.5 − 4.0. The spectral shapes inferred from spatially resolved regions show curvature, we speculate that the relic is inclined along the line of sight. The locus of points in the simulated color-color plots changes significantly with the relic viewing angle. We conclude that projection effects and inhomogeneities in the shock Mach number dominate the observed spectral properties of the relic in this complex system. Based on the new observations we raise the possibility that the relic and a narrow-angle-tailed radio galaxy are two different structures projected along the same line of sight. 

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5. Black-Hole Models in Loop Quantum Gravity

   https://doi.org/10.3390/universe6080125  

   Bojowald, Martin (August 2020, Universe)

   null (Ed.)

   Dynamical black-hole scenarios have been developed in loop quantum gravity in various ways, combining results from mini and midisuperspace models. In the past, the underlying geometry of space-time has often been expressed in terms of line elements with metric components that differ from the classical solutions of general relativity, motivated by modified equations of motion and constraints. However, recent results have shown by explicit calculations that most of these constructions violate general covariance and slicing independence. The proposed line elements and black-hole models are therefore ruled out. The only known possibility to escape this sentence is to derive not only modified metric components but also a new space-time structure which is covariant in a generalized sense. Formally, such a derivation is made available by an analysis of the constraints of canonical gravity, which generate deformations of hypersurfaces in space-time, or generalized versions if the constraints are consistently modified. A generic consequence of consistent modifications in effective theories suggested by loop quantum gravity is signature change at high density. Signature change is an important ingredient in long-term models of black holes that aim to determine what might happen after a black hole has evaporated. Because this effect changes the causal structure of space-time, it has crucial implications for black-hole models that have been missed in several older constructions, for instance in models based on bouncing black-hole interiors. Such models are ruled out by signature change even if their underlying space-times are made consistent using generalized covariance. The causal nature of signature change brings in a new internal consistency condition, given by the requirement of deterministic behavior at low curvature. Even a causally disconnected interior transition, opening back up into the former exterior as some kind of astrophysical white hole, is then ruled out. New versions consistent with both generalized covariance and low-curvature determinism are introduced here, showing a remarkable similarity with models developed in other approaches, such as the final-state proposal or the no-transition principle obtained from the gauge-gravity correspondence. 

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