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We present a new class of preconditioned iterative methods for solving linear systems of the form Ax=b. Our methods are based on constructing a low-rank Nyström approximation to A using sparse random matrix sketching. This approximation is used to construct a preconditioner, which itself is inverted quickly using additional levels of random sketching and preconditioning. We prove that the convergence of our methods depends on a natural average condition number of A, which improves as the rank of the Nyström approximation increases. Concretely, this allows us to obtain faster runtimes for a number of fundamental linear algebraic problems: 1. We show how to solve any n×n linear system that is well-conditioned except for k outlying large singular values in O~(n^2.065+k^ω) time, improving on a recent result of [Dereziński, Yang, STOC 2024] for all k≳n^0.78. 2. We give the first O~(n^2+d_λ^ω) time algorithm for solving a regularized linear system (A+λI)x=b, where A is positive semidefinite with effective dimension d_λ=tr(A(A+λI)^{−1}). This problem arises in applications like Gaussian process regression. 3. We give faster algorithms for approximating Schatten p-norms and other matrix norms. For example, for the Schatten 1-norm (nuclear norm), we give an algorithm that runs in O~(n ^{2.11}) time, improving on an O~(n ^{2.18}) method of [Musco et al., ITCS 2018]. All results are proven in the real RAM model of computation. Interestingly, previous state-of-the-art algorithms for most of the problems above relied on stochastic iterative methods, like stochastic coordinate and gradient descent. Our work takes a completely different approach, instead leveraging tools from matrix sketching. 

We consider magnetic Weyl metals as a platform to achieve current control of magnetization textures with transport currents utilizing their underlying band geometry. We show that the transport current in a Weyl semimetal produces an axial magnetization due to orbital magnetic moments of the Weyl electrons. The associated axial magnetization can generate a torque acting on the localized magnetic moments. For the case of a magnetic vortex in a nanodisk of Weyl materials, this current-induced torque can be used to reverse its circulation and polarity. We discuss the axial magnetization torques in Weyl metals on general symmetry grounds and compare their strength to current-induced torques in more conventional materials. 

Abstract BackgroundHydrogels are one of the most ubiquitous polymeric materials. Among them gelatin, agarose and polyacrylamide-based formulations have been effectively utilized in a variety of biomedical and defense-related applications including ultrasound-based therapies and soft tissue injury investigations stemming from ballistic and blast exposures. Interestingly, while in most cases accurate prediction of the mechanical response of these surrogate gels requires knowledge of the underlying finite deformation, high-strain rate material properties, it is these properties that have remained scarce in the literature. ObjectiveBuilding on our prior works using Inertial Microcavitation Rheometry (IMR), here we present a comprehensive list of the high-strain rate (> 10$$^3$$ ${}^3$1/s) mechanical properties of these three popular classes of hydrogel materials characterized via laser-based IMR, further showing that the choice in finite-deformation, rate-dependent constitutive model can be informed directly by the type of crosslinking mechanism and resultant network structure of the hydrogel, thus providing a chemophysical basis of the the choice of phenomenological constitutive model. MethodsWe analyze existing experimental gelatin IMR datasets and compare the results with prior data on polyacrylamide. ResultsWe show that a Neo-Hookean Kelvin-Voigt (NHKV) model can suitably simulate the high-rate material response of dynamic, physically crosslinked hydrogels like gelatin, while the introduction of a strain-stiffening parameter through the use of the quadratic Kelvin-Voigt (qKV) model was necessary to appropriately model chemically crosslinked hydrogels such as polyacrylamide due to the nature of the static,covalent bonds that comprise their structure. ConclusionsIn this brief we show that knowledge of the type of underlying polymer structure, including its bond mobility, can directly inform the appropriate finite deformation, time-dependent viscoelastic material model for commonly employed tissue surrogate hydrogels undergoing high strain rate loading within the ballistic and blast regimes.