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1. Asymptotic Approximation of a Modified Compressible Navier-Stokes System

   Goh, Ryan; Wayne, C Eugene; Welter, Roland (July 2023, Indiana University Mathematics Journal)

   WestudytheeectsoflocalizationonthelongtimeasymptoticsofamodiedcompressibleNavier-Stokessystem (mcNS) inspired by the previous work of Ho and Zumbrun [4]. We introduce a new decomposition of the momentum eld into its irrotational and incompressible parts, and a new method for approximating solutions of jointly hyperbolic-parabolic equations in terms of Hermite functions in which nth order approximations can be computed for solutions with nth order moments. We then obtain existence of solutions to the mcNS system in weighted spaces and, based on the decay rates obtained for the various pieces of the solutions, determine the optimal choice of asymptotic approximation with respect to the various localization assumptions, which in certain cases can be evaluated explicitly in terms of Hermite functions. 

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2. Asymptotic Approximation of a Modified Compressible Navier-Stokes System

   Goh, Ryan; Wayne, C. Eugene; Welter, Roland (July 2023, Indiana University Mathematics Journal)

   We study the effects of localization on the long-time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun [4]. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of jointly hyperbolic-parabolic equations in terms of Hermite functions in which nth order approximations can be computed for solutions with nth-order moments. We then obtain existence of solutions to the mcNS system in weighted spaces and, based on the decay rates obtained for the various pieces of the solutions, determine the optimal choice of asymptotic approximation with respect to the various localization assumptions, which in certain cases can be evaluated explicitly in terms of Hermite functions. 

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3. Hermite B-Splines: n-Refinability and Mask Factorization

   https://doi.org/10.3390/math9192458  

   Cotronei, Mariantonia; Moosmüller, Caroline (October 2021, Mathematics)

   This paper deals with polynomial Hermite splines. In the first part, we provide a simple and fast procedure to compute the refinement mask of the Hermite B-splines of any order and in the case of a general scaling factor. Our procedure is solely derived from the polynomial reproduction properties satisfied by Hermite splines and it does not require the explicit construction or evaluation of the basis functions. The second part of the paper discusses the factorization properties of the Hermite B-spline masks in terms of the augmented Taylor operator, which is shown to be the minimal annihilator for the space of discrete monomial Hermite sequences of a fixed degree. All our results can be of use, in particular, in the context of Hermite subdivision schemes and multi-wavelets. 

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4. Real operator spaces and operator algebras

   https://doi.org/10.4064/sm230329-23-12  

   Blecher, David P (January 2024, Studia Mathematica)

   We verify that a large portion of the theory of complex operator spaces and operator algebras (as represented by the 2004 book by the author and Le Merdy for specificity) transfers to the real case. We point out some of the results that do not work in the real case. We also discuss how the theory and standard constructions interact with the complexification, which is often as important, but sometimes much less obvious. For example, we develop the real case of the theory of operator space multipliers and the operator space centralizer algebra, and discuss how these topics connect with complexifi- cation. This turns out to differ in some important details from the complex case. We also characterize real structure in complex operator spaces and give ‘real’ characterizations of some of the most important objects in the subject. 

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5. Lattice-Based Proof-of-Work for Post-Quantum Blockchains

   Rouzbeh Behnia, Eamonn W. (October 2021, 5th International Workshop on Cryptocurrencies and Blockchain Technology - CBT 2021 (with ESORICS 2021))

   Proof of Work (PoW) protocols, originally proposed to circumvent DoS and email spam attacks, are now at the heart of the majority of recent cryptocurrencies. Current popular PoW protocols are based on hash puzzles. These puzzles are solved via a brute force search for a hash output with particular properties, such as a certain number of leading zeros. By considering the hash as a random function, and fixing a priori a sufficiently large search space, Grover’s search algorithm gives an asymptotic quadratic advantage to quantum machines over classical machines. In this paper, as a step towards a fuller understanding of post-quantum blockchains, we propose a PoW protocol for which quantum machines have a smaller asymptotic advantage. Specifically, for a lattice of rank n sampled from a particular class, our protocol provides as the PoW an instance of the Hermite Shortest Vector Problem (Hermite-SVP) in the Euclidean norm, to a small approximation factor. Asymptotically, the best known classical and quantum algorithms that directly solve SVP type problems are heuristic lattice sieves, which run in time 20.292n+o(n) and 2 0.265n+o(n) respectively. We discuss recent advances in SVP type problem solvers and give examples of where the impetus provided by a lattice-based PoW would help explore often complex optimization spaces. 

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