1,317 research outputs found

    \v{C}ech-Delaunay gradient flow and homology inference for self-maps

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    We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior. We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for \v{C}ech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive \v{C}ech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.Comment: 22 pages, 8 figure

    Colored fused filament fabrication

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    Fused filament fabrication is the method of choice for printing 3D models at low cost and is the de-facto standard for hobbyists, makers, and schools. Unfortunately, filament printers cannot truly reproduce colored objects. The best current techniques rely on a form of dithering exploiting occlusion, that was only demonstrated for shades of two base colors and that behaves differently depending on surface slope. We explore a novel approach for 3D printing colored objects, capable of creating controlled gradients of varying sharpness. Our technique exploits off-the-shelves nozzles that are designed to mix multiple filaments in a small melting chamber, obtaining intermediate colors once the mix is stabilized. We apply this property to produce color gradients. We divide each input layer into a set of strata, each having a different constant color. By locally changing the thickness of the stratum, we change the perceived color at a given location. By optimizing the choice of colors of each stratum, we further improve quality and allow the use of different numbers of input filaments. We demonstrate our results by building a functional color printer using low cost, off-the-shelves components. Using our tool a user can paint a 3D model and directly produce its physical counterpart, using any material and color available for fused filament fabrication

    Accelerated Parameter Estimation with DALEχ\chi

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    We consider methods for improving the estimation of constraints on a high-dimensional parameter space with a computationally expensive likelihood function. In such cases Markov chain Monte Carlo (MCMC) can take a long time to converge and concentrates on finding the maxima rather than the often-desired confidence contours for accurate error estimation. We employ DALEχ\chi (Direct Analysis of Limits via the Exterior of χ2\chi^2) for determining confidence contours by minimizing a cost function parametrized to incentivize points in parameter space which are both on the confidence limit and far from previously sampled points. We compare DALEχ\chi to the nested sampling algorithm implemented in MultiNest on a toy likelihood function that is highly non-Gaussian and non-linear in the mapping between parameter values and χ2\chi^2. We find that in high-dimensional cases DALEχ\chi finds the same confidence limit as MultiNest using roughly an order of magnitude fewer evaluations of the likelihood function. DALEχ\chi is open-source and available at https://github.com/danielsf/Dalex.git
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