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1. Canonical Stratifications Along Bisheaves

   https://doi.org/10.1007/978-3-030-43408-3_15  

   Nanda, V; Patel, A (June 2020, Topological Data Analysis)

   null (Ed.)

   A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible. 

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2. Estimating Loads Along Elastic Rods

   https://doi.org/10.1109/ICRA.2019.8794301  

   Aloi, Vincent A.; Rucker, D. Caleb (May 2019, 2019 International Conference on Robotics and Automation (ICRA))
3. Estimating Forces Along Continuum Robots

   https://doi.org/10.1109/LRA.2022.3188905  

   Aloi, Vincent; Dang, Khoa T.; Barth, Eric J.; Rucker, Caleb (October 2022, IEEE Robotics and Automation Letters)
4. Diffusing orbits along chains of cylinders

   https://doi.org/10.3934/dcds.2022121  

   Gidea, Marian; Marco, Jean-Pierre (January 2022, Discrete and Continuous Dynamical Systems)

   We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on A^3. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are –dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections. Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the tilt map and the homoclinic maps associated to its essential invariant circles. We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems. 

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5. Matrix Displacement Convexity Along Density Flows

   https://doi.org/10.1007/s00205-024-02021-8  

   Shenfeld, Yair (October 2024, Archive for Rational Mechanics and Analysis)