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1. Thermal State Preparation via Rounding Promises

   https://doi.org/10.22331/q-2023-10-10-1132  

   Rall, Patrick; Wang, Chunhao; Wocjan, Pawel (October 2023, Quantum)

   A promising avenue for the preparation of Gibbs states on a quantum computer is to simulate the physical thermalization process. The Davies generator describes the dynamics of an open quantum system that is in contact with a heat bath. Crucially, it does not require simulation of the heat bath itself, only the system we hope to thermalize. Using the state-of-the-art techniques for quantum simulation of the Lindblad equation, we devise a technique for the preparation of Gibbs states via thermalization as specified by the Davies generator.In doing so, we encounter a severe technical challenge: implementation of the Davies generator demands the ability to estimate the energy of the system unambiguously. That is, each energy of the system must be deterministically mapped to a unique estimate. Previous work showed that this is only possible if the system satisfies an unphysical 'rounding promise' assumption. We solve this problem by engineering a random ensemble of rounding promises that simultaneously solves three problems: First, each rounding promise admits preparation of a 'promised' thermal state via a Davies generator. Second, these Davies generators have a similar mixing time as the ideal Davies generator. Third, the average of these promised thermal states approximates the ideal thermal state. 

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2. Residual-based a posteriori error estimates of mixed methods for a three-field Biot’s consolidation model

   https://doi.org/10.1093/imanum/draa074  

   Li, Yuwen; Zikatanov, Ludmil T (October 2020, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We present residual-based a posteriori error estimates of mixed finite element methods for the three-field formulation of Biot’s consolidation model. The error estimator are upper and lower bounds of the space-time discretization error up to data oscillation. As a by-product, we also obtain a new a posteriori error estimate of mixed finite element methods for the heat equation. 

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3. The Subelliptic Heat Kernel of the Octonionic Hopf Fibration

   https://doi.org/10.1007/s11118-020-09854-4  

   Baudoin, Fabrice; Cho, Gunhee (July 2020, Potential Analysis)

   null (Ed.)

   We study the sub-Laplacian of the 15-dimensional unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the octonionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of a related sub-Laplacian. As a byproduct we also obtain the spectrum of the sub-Laplacian, the small-time asymptotics of the heat kernel and explicitly compute the sub-Riemannian distance. 

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4. Poincaré constant on manifolds with ends

   https://doi.org/10.1112/plms.12522  

   Grigor'yan, Alexander; Ishiwata, Satoshi; Saloff‐Coste, Laurent (April 2023, Proceedings of the London Mathematical Society)

   Abstract We obtain optimal estimates of the Poincaré constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincaré constant is determined by thesecondlargest end. The proof is based on the argument by Kusuoka–Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends. 

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5. Negative discrete moments of the derivative of the Riemann zeta‐function

   https://doi.org/10.1112/blms.13092  

   Bui, Hung_M; Florea, Alexandra; Milinovich, Micah_B (May 2024, Bulletin of the London Mathematical Society)

   Abstract We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros. 

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