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Piezoelectric acoustic energy harvesting within the human body has traditionally faced challenges due to insufficient energy levels for biomedical applications. Existing acoustic resonators are often much larger in size, making them impractical for microscale applications. This study investigates the use of acoustically oscillated microbubbles as energy-harvesting resonators. A comparative study was conducted to determine the energy harvested by a freestanding diaphragm and a diaphragm coupled with an oscillating microbubble. The experimental results demonstrated that incorporating a microbubble enabled the flexible piezoelectric diaphragm to harvest seven times more energy than the freestanding diaphragm. These findings were further validated using Laser Doppler Vibrometer (LDV) measurements and stress calculations. Additional experiments with a phantom tissue tank confirmed the feasibility of this technology for biomedical applications. The results indicate that acoustically resonating microbubbles are a promising design for microscale acoustic energy-harvesting resonators in implantable biomedical devices. 

Abstract We propose a monotone and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized infinity Laplacian, which could be related to the family of the so-called two-scale methods. We show that this method is convergent and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right-hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms, are also considered. 

We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, lower semicontinuous energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce the notion of energy solutions, for which we provide existence, uniqueness and certain regularizing effects. We also consider Lipschitz perturbations of this energy. For these problems we provide an a posteriori error estimate and show its reliability. This estimate depends only on the problem data, and imposes no constraints between consecutive time-steps. On the basis of this estimate we provide an a priori error analysis that makes no assumptions on the smoothness of the solution. 

Grating magneto-optical traps are an enabling quantum technology for portable metrological devices with ultracold atoms. However, beam diffraction efficiency and angle are affected by wavelength, creating a single-optic design challenge for laser cooling in two stages at two distinct wavelengths – as commonly used for loading, e.g., Sr or Yb atoms into optical lattice or tweezer clocks. Here, we optically characterize a wide variety of binary gratings at different wavelengths to find a simple empirical fit to experimental grating diffraction efficiency data in terms of dimensionless etch depth and period for various duty cycles. The model avoids complex 3D light-grating surface calculations, yet still yields results accurate to a few percent across a broad range of parameters. Gratings optimized for two (or more) wavelengths can now be designed in an informed manner suitable for a wide class of atomic species enabling advanced quantum technologies. 

We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy minimization, and we discretize the latter with piecewise linear finite elements. For the computation of the discrete solutions, we propose and study a gradient flow and a Newton scheme, and we quantify the effect of Dirichlet data truncation. We also present a wide variety of numerical experiments that illustrate qualitative and quantitative features of fractional minimal graphs and the associated discrete problems. 

Abstract Optimal transport maps and plans between two absolutely continuous measures $$\mu$$ and $$\nu$$ can be approximated by solving semidiscrete or fully discrete optimal transport problems. These two problems ensue from approximating $$\mu$$ or both $$\mu$$ and $$\nu$$ by Dirac measures. Extending an idea from Gigli (2011, On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proc. Edinb. Math. Soc. (2), 54, 401–409), we characterize how transport plans change under the perturbation of both $$\mu$$ and $$\nu$$. We apply this insight to prove error estimates for semidiscrete and fully discrete algorithms in terms of errors solely arising from approximating measures. We obtain weighted $L^2$ error estimates for both types of algorithms with a convergence rate $$O(h^{1/2})$$. This coincides with the rate in Theorem 5.4 in Berman (2018, Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Preprint available at arXiv:1803.00785) for semidiscrete methods, but the error notion is different.