The presented techniques can be utilized to explore novel aspects of both classical and quantum dynamics in negatively curved spaces, and to realise the emerging models of topological hyperbolic matter. Our experiments showcase both a versatile platform to emulate hyperbolic lattices in tabletop experiments, and a set of methods to verify the effective hyperbolic metric in this and other platforms. We will then show how to write these quantities in cylindrical and spherical coordinates. We use a lattice regularization of hyperbolic space in an electric-circuit network to measure the eigenstates of a `hyperbolic drum', and in a time-resolved experiment we verify signal propagation along the curved geodesics. In this paper we provide an integral representation of thefractional Laplace-Beltrami operator for general riemannian manifoldswhich has several interesting applications. In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. Here we discuss and experimentally demonstrate that the spectral ordering of Laplacian eigenstates for hyperbolic (negatively curved) and flat two-dimensional spaces has a universally different structure. Like the Laplacian, the LaplaceBeltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. A key input for the Laplace equation is the curvature of space. The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. spacetime metric, 90 stellar mass, 90 black hole binaries, 89.
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