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1. Classical Fisher information for differentiable dynamical systems

   https://doi.org/10.1063/5.0165484  

   Sahbani, Mohamed; Das, Swetamber; Green, Jason R (October 2023, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   Fisher information is a lower bound on the uncertainty in the statistical estimation of classical and quantum mechanical parameters. While some deterministic dynamical systems are not subject to random fluctuations, they do still have a form of uncertainty. Infinitesimal perturbations to the initial conditions can grow exponentially in time, a signature of deterministic chaos. As a measure of this uncertainty, we introduce another classical information, specifically for the deterministic dynamics of isolated, closed, or open classical systems not subject to noise. This classical measure of information is defined with Lyapunov vectors in tangent space, making it less akin to the classical Fisher information and more akin to the quantum Fisher information defined with wavevectors in Hilbert space. Our analysis of the local state space structure and linear stability leads to upper and lower bounds on this information, giving it an interpretation as the net stretching action of the flow. Numerical calculations of this information for illustrative mechanical examples show that it depends directly on the phase space curvature and speed of the flow. 

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2. Simple numerical solutions to the Einstein constraints on various three-manifolds

   https://doi.org/10.1007/s10714-022-03014-2  

   Zhang, Fan; Lindblom, Lee (October 2022, General Relativity and Gravitation)

   Abstract Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature. 

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3. On Stokes' second problem solutions in cylindrical and Cartesian domains

   https://doi.org/10.1063/5.0118838  

   Coxe, Daniel J.; Peet, Yulia T.; Adrian, Ronald J. (October 2022, Physics of fluids)

   It is well known that drag created by turbulent flow over a surface can be reduced by oscillating the surface in the direction transverse to the mean flow. Efforts to understand the mechanism by which this occurs often apply the solution for laminar flow in the infinite half-space over a planar, oscillating wall (Stokes’ second problem) through the viscous and buffer layer of the streamwise turbulent flow. This approach is used for flows having planar surfaces, such as channel flow, and flows over curved surfaces, such as the interior of round pipes. However, surface curvature introduces an additional effect that can be significant, especially when the viscous region is not small compared to the pipe radius. The exact solutions for flow over transversely oscillating walls in a laminar pipe and planar channel flow are compared to the solution of Stokes’ second problem to determine the effects of wall curvature and/or finite domain size. It is shown that a single non-dimensional parameter, the Womersley number, can be used to scale these effects and that both effects become small at a Womersley number of greater than about 6.51, which is the Womersley number based on the thickness of the Stokes’ layer of the classical solution. 

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4. A Semi-Analytic Elastic Rod Model of Pediatric Spinal Deformity

   https://doi.org/10.1115/1.4048400  

   Neelakantan, Sunder; Purohit, Prashant K.; Pasha, Saba (February 2021, Journal of Biomechanical Engineering)

   null (Ed.)

   Abstract The mechanism of the scoliotic curve development in healthy adolescents remains unknown in the field of orthopedic surgery. Variations in the sagittal curvature of the spine are believed to be a leading cause of scoliosis in this patient population. Here, we formulate the mechanics of S-shaped slender elastic rods as a model for pediatric spine under physiological loading. Second, applying inverse mechanics to clinical data of the subtypes of scoliotic spines, with characteristic 3D deformity, we determine the undeformed geometry of the spine before the induction of scoliosis. Our result successfully reproduces the clinical data of the deformed spine under varying loads, confirming that the prescoliotic sagittal curvature of the spine impacts the 3D loading that leads to scoliosis. 

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5. Surface Networks

   Kostrikov, Ilya; Jiang, Zhongshi; Panozzo, Daniele; Zorin, Denis; Bruna, Joan (June 2018, IEEE Conference on Computer Vision and Pattern Recognition)

   We study data-driven representations for three- dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curva- ture directions — this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models Surface Networks (SN). We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs. 

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