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1. Degenerate parabolic p -Laplacian equations: Existence, uniqueness, and asymptotic behavior of solutions

   https://doi.org/10.1515/acv-2024-0060  

   Cruz-Uribe, David; Moen, Kabe; Shao, Yuanzhen (April 2025, Advances in Calculus of Variations)

   Abstract In this paper we study the degenerate parabolicp-Laplacian, ${\partial }_{t}u-v^{-1}\mathrm{div}\left({\left|\sqrt{Q}\nabla u\right|}^{p-2}Q\nabla u\right)=0${\partial_{t}u-v^{-1}\operatorname{div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=0},where the degeneracy is controlled by a matrixQand a weightv.With mild integrability assumptions onQandv, we prove theexistence and uniqueness of solutions on any interval $[0,T]${[0,T]}. If we further assumethe existence of a degenerate Sobolev inequality with gain, thedegeneracy again controlled byvandQ, then we can prove bothfinite time extinction and ultracontractive bounds. Moreover, weshow that there is equivalence between the existence ofultracontractive bounds and the weighted Sobolev inequality. 

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2. Do black holes remember what they are made of?

   https://doi.org/10.1088/1361-6382/ad56ed  

   Bandyopadhyay, Harshraj; Radice, David; Prakash, Aviral; Dhani, Arnab; Logoteta, Domenico; Perego, Albino; Kashyap, Rahul (June 2024, Classical and Quantum Gravity)

   Abstract We study the ringdown signal of black holes formed in prompt-collapse binary neutron star mergers. We analyze data from 47 numerical relativity simulations. We show that the $(\ell =2,m=2)$and $(\ell =2,m=1)$multipoles of the gravitational wave signal are well fitted by decaying damped exponentials, as predicted by black-hole perturbation theory. We show that the ratio of the amplitude in the two modes depends on the progenitor binary mass ratioqand reduced tidal parameter $\tilde{\mathrm{\Lambda }}$. Unfortunately, the numerical uncertainty in our data is too large to fully quantify this dependency. If confirmed, these results will enable novel tests of general relativity in the presence of matter with next-generation gravitational-wave observatories. 

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3. On the Moduli of Lipschitz Homology Classes

   https://doi.org/10.1007/s12220-025-01916-6  

   Kangasniemi, Ilmari; Prywes, Eden (February 2025, The Journal of Geometric Analysis)

   Abstract We define a type of modulus$$\operatorname {dMod}_p$$ ${dMod}_p$for Lipschitz surfaces based on$$L^p$$ $L^p$-integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for Hölder conjugate exponents$$p, q \in (1, \infty )$$ $p,q\in (1,\infty )$, every relative Lipschitzk-homology classchas a unique dual Lipschitz$$(n-k)$$ $(n-k)$-homology class$$c'$$ $c^{\prime }$such that$$\operatorname {dMod}_p^{1/p}(c) \operatorname {dMod}_q^{1/q}(c') = 1$$ ${dMod}_p^{1/p}\left(c\right){dMod}_q^{1/q}\left(c^{\prime }\right)=1$and the Poincaré dual ofcmaps$$c'$$ $c^{\prime }$to 1. As$$\operatorname {dMod}_p$$ ${dMod}_p$is larger than the classical surface modulus$$\operatorname {Mod}_p$$ ${Mod}_p$, we immediately recover a more general version of the estimate$$\operatorname {Mod}_p^{1/p}(c) \operatorname {Mod}_q^{1/q}(c') \le 1$$ ${Mod}_p^{1/p}\left(c\right){Mod}_q^{1/q}\left(c^{\prime }\right)\le 1$, which appears in works by Freedman and He and by Lohvansuu. Our theory is formulated in the general setting of Lipschitz Riemannian manifolds, though our results appear new in the smooth setting as well. We also provide a characterization of closed and exact Sobolev forms on Lipschitz manifolds based on integration over Lipschitzk-chains. 

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4. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter; Zeitlin, Anton M. (September 2023, Communications in Mathematical Physics)

   Abstract We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$ $C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$ $C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$ $X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$ $l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ $ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$ $P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual. 

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5. Waveform uncertainty quantification and interpretation for gravitational-wave astronomy

   https://doi.org/10.1088/1361-6382/acd29b  

   Read, Jocelyn (May 2023, Classical and Quantum Gravity)

   Abstract We demonstrate how to quantify the frequency-domain amplitude and phase accuracy of waveform models, $\delta A$andδφ, in a form that could be marginalized over in gravitational-wave inference using techniques currently applied for quantifying calibration uncertainty. For concreteness, waveform uncertainties affecting neutron-star inspiral measurements are considered, and post-hoc error estimates from a variety of waveform models are made by comparing time-domain and frequency-domain analytic models with multiple-resolution numerical simulations. These waveform uncertainty estimates can be compared to GW170817 calibration envelopes or to Advanced LIGO and Virgo calibration goals. Signal-specific calibration and waveform uncertainties are compared to statistical fluctuations in gravitational-wave observatories, giving frequency-dependent modeling requirements for detectors such as Advanced LIGO Plus, Cosmic Explorer, or Einstein Telescope. Finally, the distribution of waveform error for the GW170817 posterior is computed from tidal models and compared to the constraints onδφor $\delta A$from GWTC-1 by Edelmanet al.In general,δφand $\delta A$can also be interpreted in terms of unmodeled astrophysical energy transfer within or from the source system. 

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